Hybrid Systems, Iterative Learning Control, and Non-minimum Phase
HHybrid Systems, Iterative Learning Control, andNon-minimum Phase byIsaac A. SpiegelA dissertation submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy(Mechanical Engineering)in the University of Michigan2021Doctoral Committee:Associate Professor Kira Barton, ChairAssociate Research Scientist Tulga ErsalAssociate Professor Chinedum OkwudireAssociate Professor Necmiye Ozay a r X i v : . [ c s . S Y ] M a r losed-formrepresentation ofpiecewise defined systemsStable inversionof piecewiseaffine systemsFundamental contribution Enabled class of control systemsCh. 4 Ch. 5Ch. 3ILC ofsmoothNMPsystemsInvert-LinearizeILC R e l a x a t i o n o f s u ffi c i e n t c o nd i t i o n s f o r s m oo t hn o n li n e a r s t a b l e i n v e r s i o n ILC ofhybridinverse-stable sys.Incorporation ofautomatic differentiationinto Newton ILCNewton ILC formalizationfor time-varying systemsof relative degree ≥ saac A. [email protected] iD: 0000-0002-4415-9190© Isaac A. Spiegel 2021o those who value this, and to those who benefit from it.ii CKNOWLEDGMENTS
It is difficult to find the right words for this acknowledgement. There are so many people I want toaddress in so many different ways.This dissertation would not have been possible without funding from the University of Michi-gan Department of Mechanical Engineering, the National Science Foundation, and the NationalInstitute of Standards and Technology. Equally essential are the members of my doctoral com-mittee: Kira Barton, Tulga Ersal, Chinedum Okwudire, and Necmiye Ozay. And I am especiallygrateful for all the collaborators that have contributed to this research: Kira Barton, Tom Oomen,Nard Strijbosch, Robin de Rozario, Patrick Sammons, and Tom van de Laar.As I conclude this doctoral program, I have also been reflecting on a bigger picture that sprawlsbeyond the walls of graduate schools and my time within them. To me, this dissertation has notbeen 5 years, but rather 13-20 years in in the making. Because I see this as the culmination of mylife to date, I keep thinking back over my life’s full span. There’s a lot to be humble about.I bring this up because over time I’ve accrued a great list people to whom I’m deeply indebted,but have not repaid and probably cannot repay, and that’s a humbling feeling. To thank in words isnot enough. These are debts of friendship, time, effort, grace, pain, forgiveness, and love. I wish Icould give you all what you are worth or at least give you the feeling deep inside that you are worthmore than can be given, and that you are loved in kind. I realize this is a bit dramatic, but whatcan I do? So much of me has been shaped by you. I’m carried by the gifts you’ve given me. And Iknow that you are carried by gifts given to you by others. There are all these gifts begetting gifts,rippling outward. So if it is not within my power to repay you, I hope to at least pay it forward asbest I can.You’ve made my life special. I will do my best by you, and by those I meet as time goes on.So here’s to those that shaped me, from the beginning:iiiisa SpiegelDavid SpiegelDaniel SpiegelBreanna SpiegelLauren RuggeJessica RuggeSean KennedyDick RuggeMarilyn StamperHannah RosenfeldColleen RosenfeldMarc RosenfeldIan RosenfeldMaster Mark W. Pattison,
Sir
Master Danielle Page-PattisonKevin ShawDani GradisherLauren ShawAnita HayworthDave ShawHenry SweatPauline DischAlex SweatJola ProscenoMark SweatRyan GizaJohn McBlairMaster Matt GoodsellMerl GoodsellMaster Adam RosenbergAmanda RosenbergMaster David LamDan GehlhaarHeidi HillMichael Eckstein Dave LambillotteNick LococoErika Shaw,
Honorary High Admiral V
Matthew EliceiriMargie WilhelmJudy HollandLinda RohmundJon RasmussenSteven WalkerEric SmithTyler WorkingerChristian WorkingerLynn LightSally PtakSydney SpiegelEzra SpiegelCharlene SpiegelCharles SpiegelLily-Jean CiccotelliTony CiccotelliSteve CiccotelliFred CiccotelliEleonore StumpDonald StumpNathan StumpAaron StumpMonica StumpCathy NoordThomas NoordMick NoordFrancesca NoordJake BertJason MarshalMcKenna TaylorAndrew Rappolt Ben LoveJen WilsonMatt Golman,
Founder
Asa Puckette,
Co-Founder
Alec Asperslag,
Honorary High Admiral I
Grant Lawrence Thompson,
Honorary Vice Admiral I
Elena Cl´emenc¸on-CharlesMegan GaffneyAndrea ReyesAna ReyesMicael Maya-PeinlJulian NobleNatalie Van ValkenbergGeoff Van ValkenbergAlex Van ValkenbergMaria Van ValkenbergWilliam Van ValkenbergMax Van ValkenbergGeorge StimsonRie TsuboiDarlene BlanchardJustin ConnMichael SantosTatiana RoySuki BerryAnnie TarabiniLonnie SafarikAmy KuoEmma LindleyAriel Jones,
Honorary High Admiral III
Stan Austin,
Honorary High Admiral IV ivammy NessSavonnah TurnerJaden PrattMike ShookBlaze NewmanRyan CardenesCaroline PollockScott HuntleyJason BerendKaren des JardinsStephen des JardinDarryl WaltonBobbie WaltonNick FooteKevin BriceWillie SaakeKaren SaakeAndrew DalagerGabe CemajAndy CollettaMark LeonDana PedeBetty HuangKyla WilsonVictoria LyOlivia PerryLinda LamLogan MercerZack MayedaCassidy MayedaLaura O’HaganDustin AtlasBen AtlasTory BaderEvan WongZyad Hammad Megan BradleySean HolcombGreg JacksonYukina TanakaHaruka NomuraSatoru TakagiAli CandlinZo¨e WinkworthMichael Reza FarziSydney LefabvreEmily DohertyMichael McCutchenLara HardingAlyx BarbeauDelani DavisDrew SpillerHaley KovacsClaire LiJessica ResnickAdam ResnickAndy PackardEmily JensenOliver O’ReillyJames CaseyAmando MillerChris BulpittCynthia TanDanny WilsonLouis MalitoBardia GanjiRoshena MacphersonJohn WallaceBen ChenBrian GrafNicole SchauserDerek Chou Pol LladoAlex CuevasMandy HuoMarc RusselStephen RhodesTaito NabeshimaHiroo YugamiFumitada IguchiMakoto ShimizuAsaka KohiyamaKyotaka KonnoHiroaki KobayashiMari SuzukiKunihiko YanagisawaHiroki SatoKasemchai ChaiprasobpholSyo OnoderaTaro FukushigeJun SakaiYuta FujiwaraYoshikazu ShibataRyusuke MiharaTakahiro KumagaiShoya MurayamaShoma OnukiDaniel Kenji PedersonLea RossanderFrej RossanderEivind RossanderJenny RossanderTatsuki MomoseSayaka OnoShogo OnishiNick GustafsonCharlie HeckrothLei Leeverry HobbsYuri HobbsNick OngJuri MizukiHaruka OchisaiMeng HanRobert BaumannGracie BurdeosOskar S¨odergrenKaori OhruiMarian LumongsodRyan ThompsonTakuya YashimaMizuki TemmaLauri PokkaSamuli KoivuPhilip GrajetzkiMegumi SuganoMidori HiroseHeraldo StefanonChristopher TacubAmrita SrinivasanAmmamaTathappaGanesh SrinivasanTerri SrinivasanGita SrinivasanMichael SrinivasanRaghu SrinivasanLisa TranMelis S¸ ahin¨ozNarayanan KidambiKatie McLaughlin Xiao-Yu FuCJ LintonMeg BrennanKira BartonAlex ShorterEfe BaltaIlya KovalenkoShreyas KousikMiguel SaezZheng WangDeema TotahMike QuannDing ZhangDelaramm AfkhamiNazanin FarjamAnne GuJoaquin GabaldonMeghna MenonMax WuDory YangYaqing XuLoubna BaroudiMax ToothmanMingjie BiTyler TonerYassine QamsaneLai-Yu Leo TseChristopher PannierBerk AltınMax ToothmanBo FuAllison WhiteC. David Remy Robin Rodr´ıguezAhmet MazaciogluNils Smit-AnseeuwAudrey SedalMario MedinaSuhak LeeMatthew PorterRachel VitaliPatrick HolmesDan BruderKim IngrahamMisaki NozawaHasnaa RabbatRemy PelzerNico MbolamenaSherry LinShuyu LongTom OomenNard StrijboschNoud MoorenNic DirkxRobin de RozarioLennart BlankenEnzo EversFahim SakibCamiel BeckersRobert van der WeijstJoey ReindersChyannie AmarillioRishi MohanDaniel VeldmanFrans VerbruggenTom van de Laarvi
REFACE
Technology advances. As it does so, both the complexity of our machines and the performancedemands upon them grow. To meet these performance demands, ever more sophisticated auto-matic control schemes are developed for these machines. Most controllers are designed based onknowledge—i.e. a mathematical model—of a machine’s dynamics, and the performance achiev-able by such controllers is usually a function of how well the dynamical model represents the truesystem behavior. The simultaneous increases in system complexity and performance requirementsthus clash when the system complexity exceeds the ability of a mathematical modeling frameworkto capture.The development of new, more flexible dynamical modeling frameworks (hybrid systems) isa clear response to this problem, one to which the controls engineering community has takenheartily [1, 2, 3]. However, improved model fidelity is only half the battle. There must also be compatibility between a controller synthesis method and a model framework. In other words, asecond clash occurs when model complexity exceeds the ability of a control framework to utilize.The main purpose of this dissertation is to bridge the gap between high fidelity modeling andcontroller synthesis for systems required to perform a task repetitively, such as manufacturingrobots producing many copies of the same product. This gap-bridging is done by contributingnew mathematical tools to both the theory of modeling and the theory of controller synthesis forrepetitive systems (Iterative Learning Control (ILC)).There are three primary contributions. First is a new mathematical representation of hybridsystems that admits several mathematical operations crucial to ILC synthesis, but that had beenundefined for prior hybrid system representations. The second and third contributions deal withthe particularly challenging case of dynamical models that have unstable inverses (often calledNon-minimum Phase (NMP) models). Because many ILC syntheses use model inversion [4], suchNMP models can cause instability of the overall controlled system. To combat this problem forboth hybrid and non-hybrid systems, the second contribution is a new ILC framework that enablesthe incorporation of alternative notions of model inversion with ILC, circumventing the instabilityproblem. Finally, the third contribution is a method for deriving such an alternative, stable inversespecifically for a class of hybrid NMP models. Thus, the three primary contributions culminate inviihe ability to perform ILC with hybrid models, even when the models have unstable inverses.These results address challenges faced by a broad array of systems, from pick-and-place robotsto piezoactuators [5, ch. 6], [6, 7]. However, the original impetus for this research is Electrohydro-dynamic Jet (e-jet) Printing, which epitomizes the challenges arising from the tandem growth ofperformance demands and system complexity. E-jet printing, like commonplace inkjet printing,revolves around the ejection of liquid from a nozzle onto a target surface. Unlike inkjet printing,e-jet printing is driven by the delicate interaction of applied electric fields with particles in theliquid meniscus at the nozzle tip. See Figure P.1 for illustration. This gives e-jet the capacityfor submicron resolution patterning for printed optics and electronics [8, 9]. But e-jet printers aresensitive machines, and require automatic control to perform reliably. Unfortunately, the ejectionprocess is too small and too fast for online control of the fluid flow during ejection to be practical.Thus, the clearest path to control e-jet printing is by analyzing performance after each ejection andseeking to improve performance from ejection to ejection, i.e. ILC.However, the system electrohydrodynamics are complicated enough that no traditional control-oriented modeling effort has been able to capture the complete ejection process’s dynamics. Thusbegan research on the hybrid modeling of e-jet printing and the ILC of hybrid systems, resulting inthe first primary contribution. The hybrid e-jet modeling research is provided in this dissertationas an application-focused contribution motivating the theoretical work.Because the complexity and sensitivity of e-jet printing produces substantial confounding fac-tors, preliminary physical implementation of this new theory was attempted on a comparativelysimple inkjet printhead positioning system. The ILC scheme in focus failed to produce stable con-trol. This spurred the research into the ILC failure mechanisms, leading to the discovery of theILC scheme’s fundamental inability to leverage models with unstable inverses, which the print-head positioning system happened to have. The second and third major contributions address thisproblem.In other words, this dissertation contributes new broadly applicable control-theoretic tools bornfrom challenges encountered in practice on physical systems.viii - M i c r o ca p ill a r y N o zz l e Fluid
Meniscus(i.e. fluid-airinterface) S ub s t r a t e N a nopo s iti on i ng S t a g e NozzleWall
Figure P.1:
Top:
Photograph of the e-jet printer at the University of Michigan. The conductive nozzleand substrate are connected by a high voltage amplifier to apply electric fields to the fluid at the nozzletip.
Bottom:
Schematic and time lapse photography illustrating the ejection process (with set at thebeginning of a voltage pulse). The meniscus base radius 𝑟 𝑀 , determined by the nozzle outer diameter, is µ m here, and in general can be as small as µ m . The speed of the ejection is determined largely by thefluid viscosity, and can be as fast as µ s for inviscid fluids such as water. The . process shown hereis for a fluid 300 times the viscosity of water, i.e. about the viscosity of castor oil. Further details are givenin Chapter 2. ix ABLE OF CONTENTS
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxChapter1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enabling ILC of Hybrid Systems:Closed-Form Hybrid System Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˜ 𝜂 ( ) Selection and Implementation . . . . . . . . . . 824.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.4.1 Benchmark Technique: Gradient ILC . . . . . . . . . . . . . . . . . . . 844.4.2 Example System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.4.3 Simulation and Analysis Methods . . . . . . . . . . . . . . . . . . . . . 874.4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 . . . . . . . . . . . . . . . . . . . . . . . 𝜇 𝑔 = . . . . . . . . . . . . . . . . . . . . 965.2.2 Non-unique Exact Inversion For 𝜇 𝑔 ≥ . . . . . . . . . . . . . . . . . . 985.2.3 Unique Exact Inverses For 𝜇 𝑔 ∈ { , } . . . . . . . . . . . . . . . . . . 1035.3 Stable Inversion of PWA Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.3.1 Exact Stable Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.3.2 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.4 Validation: Application to ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.4.1 Example System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.4.2 ILC Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.4.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124xi Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
IST OF FIGURES
P.1
Top:
Photograph of the e-jet printer at the University of Michigan. The conductivenozzle and substrate are connected by a high voltage amplifier to apply electric fieldsto the fluid at the nozzle tip.
Bottom:
Schematic and time lapse photography illus-trating the ejection process (with set at the beginning of a voltage pulse). Themeniscus base radius 𝑟 𝑀 , determined by the nozzle outer diameter, is µ m here, andin general can be as small as µ m . The speed of the ejection is determined largely bythe fluid viscosity, and can be as fast as µ s for inviscid fluids such as water. The . process shown here is for a fluid 300 times the viscosity of water, i.e. about theviscosity of castor oil. Further details are given in Chapter 2. . . . . . . . . . . . . . ix1.1 Castle of Control Contributions. Rectangular “building blocks” represent the funda-mental control-theoretic contributions. Conical “spires” represent a class of nonlinear,potentially time-varying systems supported by the underlying theoretical contribu-tions. Note that more elevated rectangular building blocks also depend on the build-ing blocks beneath them. For example, theory for the stable inversion of PiecewiseAffine (PWA) systems requires theory for the conventional inversion of PWA systems.The validation of the theory in each chapter is executed via control of an example sys-tem from the corresponding spire. Practical target applications for each class include(i) piecewise mass-spring-dampers (e.g. ankle-foot orthosis emulation [47, ch. 3-4]), (ii) cart-and-pendulum systems (e.g. bridge and tower cranes [48, 49]), and (iii)micro-positioning systems (See Chapter 5.4). . . . . . . . . . . . . . . . . . . . . . . 82.1 ( Top ) Schematic of e-jet printing setup. Dimensions represent signed displacementsin the direction of the single-sided arrows with respect to the inertial ( 𝑥, 𝑧 ) coordinatesystem. The volumetric flow rate 𝑄 , labeled via block arrow, is also a signed quantity,but its sign is with respect to the control volume denoted by the dashed box. ( Bottom )Time lapse photography of an ejection with time stamps from the rising edge of avoltage pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Ejection process in terms of the physical phenomena at the liquid-air interface. Dis-tinction is drawn between the process under a constant DC applied voltage, for whichejections begin and end repeatedly under the natural high voltage electrohydrodynam-ics, and under a single subcritical pulse, which yields a single ejection whose cessationis induced by the falling of the pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Schematic of the system as a hybrid automaton. The transition inequalities are labeledexplicitly on the automaton edges for completeness, and are derived in section 2.2.1.3. 17xiii.4 To-scale schematic of a critically deformed meniscus and the prolate spheroidal co-ordinate system (with horizontal axis at 𝑧 (cid:48) = ) used to describe the geometry of themeniscus and its local electric field (ellipses) and electric potential contours (hyperbo-lae). The focus is marked by the white dot at 𝑧 (cid:48) = 𝑎 . For 𝑧 (cid:48) < the field and potentialcontours are assumed “Cartesian.” Note that by the 𝑧 -axis sign convention, ℎ -valuesare negative, and 𝑎 is positive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 Training error, 𝐸𝑟𝑟 , and standard deviation of 𝑄 in the build-up state with respectto 𝑐 𝛿𝐸 and 𝑐 𝑟 𝐼 . The color bar across the top of the figure applies to both plots, andgives the logarithmic relationship between pixel brightness and the magnitude of themean and standard deviation in picoLiters per millisecond ( pLms ). The hatched regionsrepresent ( 𝑐 𝛿𝐸 , 𝑐 𝑟 𝐼 ) combinations corresponding to Inf error values. The circledpoint in both images represents the global minimum
𝐸𝑟𝑟 found in this analysis. . . . 312.6 Time constants ( 𝜏 ) and DC Gains ( 𝐺 ) of 𝑄 and ℎ jetting models versus 𝑉 ℎ . Forcomparability between 𝑄 and ℎ , 𝜏 and 𝐺 are normalized by their mean value. Nocorrelation is apparent between 𝜏 and 𝑉 ℎ . However, DC Gain and 𝑉 ℎ are clearlycorrelated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.7 Complete model error given as the RMSE normalized by the range of measured datawithin each location. Each bar is computed from 100 measured timeseries averaging23 points each for build-up, 34 points for Jetting, and 237 points for relaxation. Errorbars represent plus/minus one standard deviation. . . . . . . . . . . . . . . . . . . . . 342.8 Plot of the simulation and measured data for ℎ , 𝑄 , and the individual pressure com-ponents contributing to 𝑄 for 𝑇 𝑝 = and 𝑉 ℎ = . 𝑃 𝐸 and 𝑃 𝑔 are combinedbecause 𝑃 𝑔 is constant and they are both always positive, meaning that their sumshould be balanced against 𝑃 𝛾 with the assistance of 𝑃 𝜇 . . . . . . . . . . . . . . . . . 352.9 Paraboloidal and hyperboloidal electrode geometry approximations overlaid on noz-zle/meniscus photographs at a low voltage equilibrium (left), and at the critical Taylorcone preceding jetting (right). The “underdetermined” hyperboloid is given only atthe critical meniscus because the necessary prior information (in this case, the valueof 𝜉 ) is only known for the critical meniscus (see section 2.2.1.3). . . . . . . . . . . . 362.10 Schematic of the two CVs used in this work superimposed over an illustration of a jetimmediately after breaking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.11 System Architecture. Top:
Automaton illustrating the timed switching behavior of thesystem model and the reset determining the initial contiguity droplet volume. Eachautomaton location is accompanied by photographs of the first and final samples ofthe corresponding partial process from a video with 𝑉 ℎ = and 𝑇 𝑝 = . . Bottom:
Block diagram illustrating breakdown of a complete input-to-droplet-volumemodel into a nozzle flow rate model 𝑓 𝑄 and a droplet volume model 𝑓 V 𝑑 , both of whichare piecewise defined to capture the switching and reset behavior of the automaton. . 402.12 Measured data from a particular ejection video illustrating the motivation for a propor-tional 𝑄 𝑑 model. Left:
Half-outline of fluid body (the jet is roughly symmetric about 𝑧 -axis) at the onset of contiguity and at the end of the voltage pulse, representing a increase in volume outside the droplet CV. Right:
Flow rate time series data duringcontiguity illustrating roughly proportional signals between 𝑄 and 𝑄 𝑑 . . . . . . . . . 41xiv.13 Depiction of the two CVs described in Section 2.3.1.1, the distinction between the di-rect and reflection-augmented measurement regions, and the estimated jet break posi-tion superposed on the final contiguity frame and first retraction frame for a particulartrial ( 𝑉 ℎ = , 𝑇 𝑝 = . ). Δ 𝑧 is the distance from either fluid body tip to thejet break position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.14 Retraction-stage droplet volume measurement taken by direct and reflection-augmentedmeasurement techniques. Light and dark points represent raw and filtered data, re-spectively. As V 𝑑 is expected to be constant during retraction, this plot illustrates thereflection-augmented technique’s superiority in that it maintains a roughly constantvalue of . after the transient (i.e. for 𝑡 ≥ . ), while the direct measurementsteadily decreases until about 𝑡 = . . . . . . . . . . . . . . . . . . . . . . . . . . 482.15 Percent error in final deposited droplet volume, V 𝑑 ( 𝑡 𝑟 ) , using measured and simulated 𝑄 . The lowest voltage experiment exceeds 100% error in both cases. Each bar rep-resents the mean value of 𝑁 = samples. Measured 𝑄 results illustrate the highquality of equation (2.54) for all but the lowest voltage case. Simulated 𝑄 results il-lustrate increased error associated with increased uncertainty in the cascaded model,motivating future flow rate modeling work. . . . . . . . . . . . . . . . . . . . . . . . 492.16 Time series plots of droplet volume V 𝑑 for a representative experiment and the exper-iment of lowest high voltage 𝑉 ℎ . Plotted measured data is the mean of the validationdata ( 𝑁 = samples for each time series) with an envelope of plus or minus thestandard deviation. The data suggests that the reset is the main source of error in low 𝑉 ℎ experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.17 Jet break positions of each experiment’s validation data against high voltage. Thenozzle outlet is located at µ m and the substrate at − µ m . 𝑁 = samples forall high voltages except 𝑉 ℎ = , for which 𝑁 = samples because four pulsewidths are tested at 𝑉 ℎ = . The modest spread of data points at 𝑉 ℎ = suggests that high voltage (equivalent to the difference between high and low voltagein this data set) has a greater influence on jet break position than pulse width in thesubcritical jetting regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.18 Mean unsigned percent error and standard deviation (given by error bars) of finaldroplet volume over all validation trials except those of lowest high voltage ( 𝑁 = samples). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1 Block diagram of the closed-form Piecewise Defined (PWD) system representation . . 573.2 An R topology representable by (3.6) with 14 hyperplanes (dashed lines) having 1convex ( 𝔞 ) and 2 nonconvex ( 𝔟 and 𝔠 ) dynamic regimes. . . . . . . . . . . . . . . . . 593.3 Mass-spring-damper system used for validation. The spring stiffness is a piecewisedefined function of the spring extension and the applied force is a nonlinear functionof actuator voltage 𝑢 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.4 Above:
Histogram of final-trial NRMSE for all experiments. All but the highest errorbin are defined by a 0.005 range of NRMSE.
Middle:
Average trajectory of NRMSEvs. Trial number for all completely convergent simulations.
Bottom:
Same on alogarithmic scale, for perspective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.5 Timeseries evolution over multiple trials for a simulation with (cid:107) 𝑒 𝜃 (cid:107) = . . Thissimulation took 8 trials to completely converge. . . . . . . . . . . . . . . . . . . . . . 68xv.6 Percentage of experiments that converge for each 0.05 range of relative model errorfrom 0 to 1. The dashed line is a least squares model of the decay in the probabilityof convergence as model error increases. . . . . . . . . . . . . . . . . . . . . . . . . 684.1 Cart and pendulum system. Dimension, position, and mass annotations are in grey.Force and torque annotations are in black. . . . . . . . . . . . . . . . . . . . . . . . . 854.2 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.3 System Block Diagram. The control law outputting 𝑢 is synthesized from the controlmodels defined by the behatted parameters of Table 4.1 and by 𝜔 𝑐 ( 𝑘 ) = 𝜔 𝑦 ( 𝑘 ) = . The plant and controller gain blocks are defined with the truth model parametersgenerated according to Section 4.4.3. Inter-trial signals from trial ℓ are stored andused to compute the input for trial ℓ + . . . . . . . . . . . . . . . . . . . . . . . . . 884.4 Representative input solution trajectories from low- and high-model-error ILILC sim-ulations compared with the solution to the zero-model-error problem. The zero-model-error solution is the input trajectory that would be chosen for feedforwardcontrol in the absence of learning, and differs notably from both minimum-error tra-jectories found by ILILC with stable inversion. . . . . . . . . . . . . . . . . . . . . . 904.5 Top:
Histogram giving the percentage of simulations converged in each bin of themodel error metric (cid:107) 𝑒 𝜃 (cid:107) . Bottom:
Mean value of NRMSE for each ILC trial overall simulations that are convergent for both gradient ILC and ILILC with stable inver-sion. This illustrates that for comparable robustness to model error, ILILC convergessubstantially faster than gradient ILC. . . . . . . . . . . . . . . . . . . . . . . . . . . 915.1 Photo of desktop inkjet printer with the case removed. The motor actuates the print-head motion along a guide rail via a timing belt, and the motion is measured by alinear optical encoder with resolution of µ m (about 600 dots per inch). . . . . . . . 1135.2 System block diagram. The plant block uses the truth model of the printer systemobtained by experimental system identification while the ILC law is synthesized usingthe control model. The downsample and upsample blocks account for the differencein sample period between the ILC law and the truth model. . . . . . . . . . . . . . . . 1135.3 Bode plot of the experimental plant data, truth model of the plant, and control modelof the plant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.4 Reference. The reference is 1999 samples long for the truth model, and is downsam-pled to 1000 samples for the control model. . . . . . . . . . . . . . . . . . . . . . . . 1175.5 NRMSE of each ILC trial, illustrating convergence of ILILC to a plateau determinedby the noise injected to the system, and dramatically surpassing the convergence speedof both benchmark ILC techniques. The NRMSE of the learning-free stable inversionsimulation is also pictured. It is 4% smaller than the feedback-only simulation (ILILCtrial 0), but is much larger than the performance achievable with learning. Becausethe noise injected in this chapter is of the same distribution and injection location asthe noise in Chapter 4, the same convergence threshold can be used to approximatethe minimum NRMSE achievable by ILILC. . . . . . . . . . . . . . . . . . . . . . . 121xvi.6 Error (top) and Input (bottom) time series data for the feedback-only simulation,learning-free stable inversion simulation, and the final trial of the ILILC simulation.Both stable inversion and ILILC perform as expected, but due to model error learningis required to reap the full benefit of feedforward control. . . . . . . . . . . . . . . . . 1225.7 Error (top) and Input (bottom) time series data for the three ILC simulations. Even theworst-performing ILC technique—P-type ILC—yields a reduction in maximum errormagnitude when compared to the learning-free techniques of Figure 5.6, but Invert-Linearize Iterative Learning Control (ILILC) clearly yields the lowest-error perfor-mance. This superiority is in spite of ILILC acquiring more high frequency contentvia learning than the other ILC schemes, which appear less noisy but appear to containhigher amplitude, lower frequency oscillations that degrade performance. . . . . . . . 1236.1 Castle of Control Contributions, revisited for visualization of this dissertation’s poten-tial broader impacts. These may take the form of other new classes of control systemsleveraging the theoretical contributions presented here. . . . . . . . . . . . . . . . . . 127A.1 NRMSE versus trial number of past works’ ILC schemes (A.2) applied with learn-ing gain (A.4) to the system (A.3). These NRMSEs monotonically increase, con-firming the inability of the past work on ILC with discrete-time nonlinear systems toaccount for unstable inverses. The NRMSE trajectory yielded by the stable-inversion-supported ILILC scheme proposed by this article is also displayed. The convergenceof this ILC scheme when applied to (A.3) reiterates its ability to control such non-minimum phase systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131xvii IST OF TABLES Ω of 𝜔 = ( 𝑇 𝑝 , 𝑉 ℎ ) pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3 𝔭 -values and correlation coefficients for LTI Metrics vs. 𝑉 ℎ . . . . . . . . . . . . . . 332.4 Transition Timing Error Mean ( 𝑒 𝜇 ) & Standard Deviation ( 𝑒 𝜎 ) . . . . . . . . . . . . 332.5 Experimental High Voltage and Pulse Width Pairs . . . . . . . . . . . . . . . . . . . 444.1 Cart-Pendulum Control Model Parameters . . . . . . . . . . . . . . . . . . . . . . . 884.2 Transient Convergence Rates for ILILC and Gradient ILC . . . . . . . . . . . . . . . 915.1 Simulation Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.2 NRMSE and Peak Error Magnitude of Truth Model Simulations . . . . . . . . . . . . 121xviii IST OF ABBREVIATIONS AM Additive Manufacturing µ-AM
Microscale Additive Manufacturing CV Control Volume
DoD
Drop on Demand e-jet
Electrohydrodynamic Jet
FDM
Fused Deposition Modeling
ILC
Iterative Learning Control
ILILC
Invert-Linearize Iterative Learning Control
LTI
Linear Time Invariant
MLD
Mixed Logical Dynamical
MPC
Model Predictive Control
NILC
Newton Iterative Learning Control
NMP
Non-minimum Phase
ODE
Ordinary Difference/Differential Equation
PWA
Piecewise Affine
PWD
Piecewise Defined
ZPETC
Zero Phase Error Tracking Control xix
BSTRACT
Hybrid systems have steadily grown in popularity over the last few decades because they easethe task of modeling complicated nonlinear systems. Legged locomotion, robotic manipulation,and additive manufacturing are representative examples of systems benefiting from hybrid model-ing. They are also prime examples of repetitive processes; gait cycles in walking, product assemblytasks in robotic manipulation, and material deposition in additive manufacturing. Thus, they wouldalso benefit substantially from Iterative Learning Control (ILC), a class of feedforward controllersfor repetitive systems that achieve high performance in output reference tracking by learning fromthe errors of past process cycles. However, the literature is bereft of ILC syntheses from hybridmodels. The main thrust of this dissertation is to provide a broadly applicable theory of ILC fordeterministic, discrete-time hybrid systems, i.e. piecewise defined (PWD) systems.A type of ILC called Newton ILC (NILC) serves as the foundation for this mission due to itsadmittance of an unusually broad range of nonlinearities. Preventing the synthesis of NILC fromhybrid models is the fact that contemporary hybrid modeling frameworks do not admit closed-formfunction composition of a single state transition formula capturing the complete hybrid systemdynamics. This dissertation offers a new, closed-form PWD modeling framework to solve thisproblem.However, NILC itself is not without flaw. This dissertation’s research reveals that it generallyfails to converge when synthesized from models with unstable inverses (i.e. non-minimum phase(NMP) models), a class that includes flexible-link robotic manipulators. Thus, to fulfill the goalof providing the most broadly applicable control theory possible, improvement to NILC must bexxade to avoid the operation that causes divergence when applied to NMP systems (a particularmatrix inversion).Stable inversion—a technique for generating stable state trajectories from unstable systems bydecoupling their stable and unstable modes—is identified as a valuable tool in this endeavor. Thisconcept is well-explored for linear time invariant systems, but stable inversion for hybrid systemshas not been explored by the prior art. Thus, to focus the research, this dissertation specificallyexamines piecewise affine (PWA) systems (a subset of PWD systems) for the study of NMP hybridsystem control. For PWA systems (and their PWD superset), in addition to a lack of stable inver-sion, a general, closed-form solution to the conventional inversion problem is also absent from theliterature. Having a closed-form conventional inverse model is a prerequisite for stable inversion,but inversion of PWA models is nontrivial because the uniqueness of PWA system inverses is notguaranteed as it is for ordinary affine systems. Therefore, to achieve the first ILC of a hybrid sys-tem with an unstable inverse, theory for both conventional inversion and stable inversion must bedelivered for PWA systems.In summary, the three main gaps addressed by this dissertation are (1) the lack of compatibilitybetween existing hybrid modeling frameworks and ILC synthesis techniques, (2) the failure ofNILC for NMP systems, and (3) the lack of inversion and stable inversion theory for PWA systems.These issues are addressed by (1) developing a closed-form representation for PWD systems, (2)developing a new ILC framework informed by NILC but free of matrix inversion, and (3) derivingconventional and stable model inversion theories for PWA systems.xxi
HAPTER 1
Introduction
The main contributions to hybrid systems and control theory are presented in Chapters 3-5. Thesechapters validate their theoretical contributions with simulations on a diverse set of example sys-tems derived from mechanics principles or data collected from physical systems. Additionally, thesimulations feature numerous types of model errors in order to stress test the proposed controllers.However, it is desirable to further ground theoretical controls research in the needs of application-based research. To this end Chapter 2 presents Electrohydrodynamic Jet (e-jet) printing in detail,specifically the first efforts to create end-to-end Ordinary Difference/Differential Equation (ODE)-based models of the fluid flow from the nozzle to the substrate over the course of an ejection.Two models are derived. The first is focused on extending physics-based nonlinear modelingas much as possible to achieve an end-to-end dynamical model of the flow rate of fluid throughthe nozzle outlet. The second is focused on capturing the volume of fluid actually deposited onthe substrate. Additionally, the second model seeks to accomplish this while limiting the systemnonlinearities to a Piecewise Affine (PWA) definition of the dynamics.As explained thoroughly in the chapter, this modeling research is a direct response to the fu-ture desire to achieve high performance control of deposited droplet volume. In this endeavor,automatic control is required due to the high uncertainty in system behavior, and Iterative Learn-ing Control (ILC) is the clearest choice because physical sensing limitations preclude real-timefeedback control. The desire for ILC motivates the desire for an ODE-based model for controllersynthesis. As demonstrated in Chapter 2.2, hybridness is ultimately necessary to capture the end-to-end process dynamics. This is in spite of contributing substantial improvements to the physics-based smooth nonlinear electrohydrodynamics model, resulting in a five-fold increase to the rangeof well-modeled meniscus deformation. Thus, in addition to raw scientific contributions to e-jetmodeling, Chapter 2 concretely establishes the need for a theory merging ILC and hybrid systems.Note however, that this dissertation does not contain a physical implementation of the de-1eloped ILC theory on an e-jet printer. This is because the main objective here is to deliver afoundational theory of ILC for hybrid systems, and e-jet printing presents additional challenges toILC beyond model hybridness. These challenges are explained in the description of future work,Section 6.3. For a foundational ILC theory such issues were deemed less critical than the treatmentof the Non-minimum Phase (NMP) behavior described later in this chapter, which fundamentallyimpacts the stability of ILC for many systems. In other words, while e-jet printing is represen-tative of systems requiring iterative learning for control and requiring hybridness for end-to-endODE-based modeling, more e-jet-printing-specific research is necessary before safe and effectiveautomatic control is achievable.
Concretely defined, ILC is the process of learning an optimal feedforward control input over mul-tiple trials of a repetitive process based on feedback measurements from previous trials. ILC isused when typical real-time-feedback and/or feedforward control techniques yield too much out-put tracking error (as determined by case-specific criteria) because of their reactive nature or modelerror, respectively. Notable past applications include robot-assisted stroke rehabilitation [10], highspeed train control [11], and laser additive manufacturing [12], all of which use nonlinear models.In fact, while the majority of ILC literature focuses on linear systems, the prevalence of nonlineardynamics in real-world systems has motivated the development of numerous ILC syntheses fromdiscrete-time nonlinear models [13, 14, 15, 16].Nonlinear modeling can be extremely challenging. Chapter 2 illustrates that there are repetitivesystems for which even extensive traditional (i.e. continuous) nonlinear modeling is insufficientfor capturing a process’s full dynamics. Hybrid system modeling offers a more flexible formalmodeling framework.The term “hybrid systems” encompasses a wide variety of modeling frameworks that are com-posed of a set of traditional dynamical models—e.g. systems of ODEs—and a set of rules regardingswitching between which of these “component” models is governing the system state evolution ata particular point in time [2]. The set of switching rules can usually be interpreted as a discreteevent system [3, 1]. Each discrete state, or “location,” is associated with one of the componentdynamical models. The transitioning between locations is conditioned on the dynamical statesand control inputs. This dissertation specifically considers deterministic hybrid systems, in whichsatisfaction of any transition condition enforces a location switch at the moment in time the con-dition is satisfied. This is in opposition to frameworks based on transition guard conditions andlocation invariant conditions, which may allow a transition if satisfied, but do not typically enforce switching. Deterministic switching behavior (and state resets) can in general be captured by piece-2ise definition of a state space system via augmentation of the state dimension [17]. Thus, mostdeterministic hybrid system frameworks are a subset of Piecewise Defined (PWD) systems [2].In the past couple decades, hybrid systems have become broadly popular because their abilityto stitch simple dynamical systems together to produce complicated state trajectories has greatlyeased the task of modeling many physical systems. Examples include cyber-physical systems ingeneral [1], automobile driver behavior [18], power systems [19], legged locomotion [20], con-veyor systems [21], and additive manufacturing [22].Clearly, there is overlap between repetitive systems and systems well-modeled as hybrid sys-tems: gait cycles in legged locomotion are repetitive, as are many manufacturing processes. Infact, of myriad robotics-related uses, manufacturing is one of the primary fields in which ILC isapplied [4]. However, the combination of ILC and hybrid modeling is absent from the literature.This is the main gap of Chapter 3.The first step to achieving ILC of PWD systems is to choose a particular ILC scheme to buildoff of. However, literature on the ILC of nonlinear systems can trend towards hyper-specializationwith respect to the system model, making it more restrictive than desired. Even amongst the moregeneral literature such as [13, 14, 15, 16, 23] mentioned above, nearly all published ILC theory fordiscrete-time nonlinear systems feature at least one of the following model restrictions(R1) relative degree of either 0 or 1 [14, 15],(R2) affineness in the input [13, 14, 15, 23],(R3) time-invariance [13, 16], and(R4) smoothness of the state transition formula and output functions (Lipschitz continuity at themost relaxed) [13, 14, 15, 16, 23].This is problematic because many practical systems violate these constraints. (R1) may be vio-lated in the position control of myriad systems including piezoactuators [24], motors [25], roboticmanipulators [26], and vehicles [27]. (R2) may be violated by piezoactuators [24], electric powerconverters [28], wind energy systems [29], magnetic levitation systems [30], e-jet printing (Chapter2.2) and flexible-link manipulators [26]. (R3) may be violated by any feedforward-input-to-outputmodel of systems using both feedforward and feedback control, as is often done for robotic ma-nipulation [31]. Finally, and of primary concern here, (R4) may be violated by gain switchingfeedback control systems (e.g. for motor control [32]), power converters [28], legged locomotion[33], e-jet printing (Chapter 2), and robotic manipulation [6]. The fact that many of these examplesystems violate multiple restrictions illustrates that it can be challenging to find a model-based ILCsynthesis scheme appropriate for many real-world applications. Indeed, flexible-link manipulatorsviolate all four, and they are relevant to the fast and cost-effective automation of pick-and-place3nd assembly tasks as well as to the control of large structures such as cranes [5, ch. 6]. Suchapplication spaces would benefit from having a versatile ILC scheme free from (R1)-(R4).Additionally, while ILC seeks to converge to a satisfactorily low error, this learning is notimmediate, and trials executed before the satisfactory error threshold is passed may be seen ascostly failures from the perspective of the process specification. It is thus desirable to develop ILCschemes that converge as quickly as possible.There is one published ILC scheme that meets the need for versatility and speed: the applicationof Newton’s root finding algorithm to a complete finite error time series (as opposed to individualpoints in time). This technique was first proposed in [34], and is called Newton Iterative LearningControl (NILC) here. NILC’s synthesis procedure and convergence analysis are unusually broadin that they are free of (R1)-(R4) [34]. Additionally, Newton’s method has been shown to deliverfaster convergence in ILC than more basic schemes such as P-type ILC [35, ch. 5].However, while the convergence conditions of NILC do not preclude their application to hy-brid systems, the synthesis of NILC requires a closed-form lifted system model. This lifted modelis a vector-input-vector-output function taking in the control input time series and outputting thesystem output time series. To construct this model from state space systems requires function com-position and differentiation of a single closed-form state transition formula capturing the model’sentire dynamics. Existing hybrid system formalisms rarely possess such monolithic state transitionformulas, and the ones that do cannot explicitly nest calls to them via function composition (furtherdetails on this point are given in the introduction to Chapter 3). Thus the approach to addressingthe first main gap, i.e. the first main contribution, is the development of a closed-form state spacerepresentation of PWD systems. This enables ILC of a large swath of hybrid systems.To support this main contribution, Chapter 3 also delivers two ancillary contributions relatedto the implementation of NILC. The original NILC literature [34] assumes an appropriate liftedmodel is given. Synthesis of this model from state-space systems requires careful handling of thesystem relative degree, which has been neglected in subsequent works leveraging NILC, such as[36]. Neither have time-varying nonlinear system dynamics been considered in the lifted systemderivation. Thus, the first ancillary contribution is an explicit formalization of NILC for time-varying nonlinear systems of any relative degree ≥ . Additionally, the differentiation in Newton’smethod has been challenging for past authors due to computational cost. The second ancillary con-tribution is the incorporation of automatic differentiation (see [37, 38]) into NILC implementation,dramatically reducing this cost. 4 .3 Aim 2: Versatile, Fast ILC of NMP Nonlinear Systems NILC has one of the least restrictive sets of sufficient conditions for convergence published in theprior art. However, Chapter 4 reveals that when synthesized from models with unstable inverses ,NILC typically generates control signals that diverge to enormous magnitudes. In other wordsNILC is not compatible with these models, which are often called NMP models. This is prob-lematic because a number of important physical systems are well represented by NMP models.Examples include piezoactuators [7], electric power converters [28], wind energy systems [29],DC motor and tachometer assemblies [39], and flexible-link manipulators [26]. Thus, incompati-bility of NILC and NMP models is the main gap of Chapter 4.Note that the full original definition of NMP refers to the property of a frequency responsefunction having the minimum possible phase change from 𝜔 = to 𝜔 → ∞ for a given magnitudetrajectory. For Linear Time Invariant (LTI) systems this is achieved if and only if the system andits inverse are causal and stable. In other words, strictly proper systems cannot be minimum phaseregardless of the stability of their inverses. However, because the lack of causality is rarely anobstacle in feedforward control when the entire reference is known in advance, NMP is often usedas jargon for inverse instability (equivalently, as an abbreviation for “non-minimum phase zerodynamics ”) in the feedforward control community, and thus in this dissertation as well.For linear models with unstable inverses, a common way to obtain feedforward control signalsis to systematically synthesize approximate dynamical models with stable inverses by individuallychanging the model zeros and poles, e.g. Zero Phase Error Tracking Control (ZPETC) [40]. How-ever, it is difficult to prescribe analogous systematic approximation methods for nonlinear modelsbecause the poles and zeros do not necessarily manifest as distinct binomial factors that can beindividually inverted or modified in the system transfer function.An alternative is to harness the fact that a scalar difference equation that is unstable whenevolved forward in time from an initial condition is stable if evolved backwards in time froma terminal condition. If the stable and unstable modes of a system are decoupled and evolved inopposite directions, a stable total trajectory can be obtained. This process is called stable inversion.For linear systems on a bi-infinite timeline, with boundary conditions at time ±∞ , stable inversiongives an exact solution to the output tracking problem posed by the unstable inverse model. Inpractice on a finite timeline, a high-fidelity approximation is obtained by ensuring the referenceis designed with sufficient room for pre- and post-actuation, i.e. with a “flat” beginning and end.Additionally, unlike ILC, stable inversion alone cannot account for model error. To address this,[41] details stable inversion and presents an ILC scheme for linear systems that incorporates a More specifically: models for which the inverse’s linearization about the input trajectory is unstable. See Section4.2.1 for details.
The aforementioned lack of stable inversion theory for Hybrid systems provides an obvious finalgap that must be filled to achieve the first ILC of a hybrid system with unstable inverse dynamics.However, such a gap statement belies the fact that there is no published general solution to theconventional inversion of PWD systems, let alone stable inversion. To focus the research on solvingboth the inversion and stable inversion problems for a class of hybrid system, Chapter 5 considersa subset of PWD systems called PWA systems.PWA systems are simply PWD systems with the component dynamics restricted to affine mod-els. Examples include current transformers [42], one-sided spring supports [43], gain switching[32], and e-jet printing (Chapter 2.3). As with hybrid systems in general, the mathematical rigorprovided by the PWA framework facilitates analysis and control theory development for thesesystems. Examples include stabilizing state feedback control [44] and model reference adaptivecontrol [45].To date, such research has focused primarily on feedback control. Feedforward control has notbeen thoroughly addressed in the PWA literature. In addition to preventing the control of systemswith specific needs for feedforward control, this gap inhibits the implementation of existing feed-back control theory that requires feedforward control components. Indeed, [46] presents a solutionto the output reference tracking problem for a class of PWA systems using both feedback and feed-forward control elements, but does not present a method to compute the feedforward signal. Thevalidation is instead limited to a master-slave synchronization example in which the feedforwardinput to the master system is known in advance.In other words, while Chapter 5’s contributions of rigorous theory for the inversion and stableinversion of PWA systems do yield the first ILC of an NMP hybrid system, they also have broader6amifications for the control of PWA systems.
One can see that the control-theoretic contributions of Aims 1-3 are not wholly independent ofone another. Instead, they build off one another and share responsibility for enabling the ILC ofdifferent classes of systems. Figure 1.1 gives a graphical representation of this synergy by treatingthe fundamental contributions as building blocks that work together to hold up both each other andthe specific classes of systems for which they enable control.
Chapter 2 prioritizes the modeling of e-jet printing, while Chapters 3-5 prioritize hybrid systemsand ILC theory. Thus, other than the most basic conventions, effort to unify the notation betweenthese two sections has been limited. Overloaded symbol definitions occur, but are highly distin-guished by context. For example 𝑄 is used to indicate volumetric flow rate in Chapter 2 and a setof subspaces of the real vector space R 𝑛 in Chapters 3-5.Within Chapters 3-5 notation is unified with one exception. Chapters 3-4 represent the timeargument in functions of discrete time as 𝑥 ( 𝑘 ) , with 𝑘 as the time step index. Chapter 5 uses thesubscript notation 𝑥 𝑘 . This is done to compact a number of otherwise very long expressions.7 losed-formrepresentationof PWD systemsStable inversionofPWA systemsFundamental contribution Enabled class of control systemsCh. 4 Ch. 5Ch. 3ILC ofsmoothNMPsystemsInvert-LinearizeILC(ii) (iii) R e l a x a t i o n o f s u ffi c i e n t c o nd i t i o n s f o r s m oo t hn o n li n e a r s t a b l e i n v e r s i o n ILC ofhybridinverse-stable sys.(i)Incorporation ofautomatic differentiationinto NILCNILC formalization fortime-varying systemsof relative degree ≥ Figure 1.1: Castle of Control Contributions. Rectangular “building blocks” represent the fundamentalcontrol-theoretic contributions. Conical “spires” represent a class of nonlinear, potentially time-varyingsystems supported by the underlying theoretical contributions. Note that more elevated rectangular build-ing blocks also depend on the building blocks beneath them. For example, theory for the stable inversion ofPWA systems requires theory for the conventional inversion of PWA systems. The validation of the theory ineach chapter is executed via control of an example system from the corresponding spire. Practical target ap-plications for each class include (i) piecewise mass-spring-dampers (e.g. ankle-foot orthosis emulation [47,ch. 3-4]), (ii) cart-and-pendulum systems (e.g. bridge and tower cranes [48, 49]), and (iii) micro-positioningsystems (See Chapter 5.4). HAPTER 2
Repetitive Processes Needing Hybrid Modeling:Electrohydrodynamic Jet Printing Studies
Additive Manufacturing (AM) is a growing class of processes that fabricate components in alayer-by-layer fashion. However, several obstacles inhibit the widespread adoption of AM. Chiefamong these is the lack of appropriate process descriptions for both implementing satisfactoryprocess planners and integrating process feedback control to enable repeatable, accurate part fab-rication. A major reason for this is that many AM processes are governed by complex physicalphenomena, e.g. the melting, heat transfer and solidification processes in Laser Engineered NetShaping (LENS) and Selective Laser Melting (SLM) [50], and the jetting and binder-burnout pro-cesses in Binder Jetting [51], resulting in strongly nonlinear mappings between process inputs andprocess outputs. Because of these nonlinearities, models that may be useful for control design areonly applicable in small regions of the operating space, accounting for only short periods within thetotal process. Therefore, in order to enable control that addresses AM processes in a holistic man-ner, there is a need to develop models that are capable of describing the complex, interconnected,physical phenomena of AM processes.One important subdivision of AM that has attracted significant attention is Microscale Ad-ditive Manufacturing (µ-AM). µ-AM comprises a number of processes characterized by theirability to produce feature sizes on the order of
100 nm to µ m . Processes typically classedas µ-AM include ink jet [52] and other direct-write technologies such as dip-pen nanolithogra-phy [53], and stereolithography [54]. Applications for these processes include patterning, printed Content of this chapter also published as:I. A. Spiegel, P. Sammons and K. Barton, “Hybrid Modeling of Electrohydrodynamic Jet Printing,”in
IEEE Transactions on Control Systems Technology , vol. 28, no. 6, pp. 2322-2335, Nov. 2020,https://doi.org/10.1109/TCST.2019.2939963 ©IEEE 2020. Reprinted with permission.I. A. Spiegel, T. van de Laar, T. Oomen and K. Barton, “A Control-Oriented Dynamical Model of DepositedDroplet Volume in Electrohydrodynamic Jet Printing.” in
Proceedings of the ASME 2020 Dynamic Systems and Con-trol Conference . Virtual: ASME, 2020. https://doi.org/10.1115/DSCC2020-3238 ©ASME 2020. Reprinted withpermission. ∼ µ m minimum [61].Reliably fulfilling the potential for submicron resolution requires closed-loop control for therejection of disturbances introduced by variations in nozzle shape, fluid properties, and environ-mental factors like temperature and humidity [62]. However, the computer-vision-based measure-ment process, [22], is too slow compared to the jetting process for traditional real-time feedbackcontrol to be practical. Thus, recent research has focused on closing the loop in the iteration do-main rather than the time domain. Specifically, this research uses measurements from previoustrials of an e-jet printing task in conjunction with dynamical models of nominal system behaviorto inform the feedforward control input signal for the subsequent trial [63, 64, 65], a techniqueknown as iterative learning control (ILC).Thus far, this research has been limited to using models of droplet spreading over a substrate todetermine the droplet volume necessary for achieving a desired final print topography. It has notused knowledge of the dynamics between the applied voltage signal and the volume of fluid ulti-10ately ejected from the nozzle, instead assuming perfectly known static relationships. Dynamicalmodels relating applied voltage to fluid flow could be used to decouple the problems of learning thecorrect volume to deposit and learning the applied voltage signal necessary to achieve that volume,potentially decreasing the number of trials required to achieve satisfactory performance.For a model to be compatible with ILC it must be founded on ordinary differential or dif-ference equations (ODEs), but the majority of current modeling efforts for e-jet printing can bebroadly classified as either partial-differential-equation-driven finite element/volume methods orstatic scaling laws. In the former category, a significant amount of work has been aimed at study-ing the fluid-air surface profile as a function of electric field and, to a lesser extent, specific materialproperties such as fluid conductivity [66, 67, 68]. Additionally, there has been work towards de-veloping models of the spreading and coalescence of e-jet-printed droplets on substrates [69, 70],as well as developing relationships between input parameters, (e.g. applied voltage), and depositeddroplet properties (e.g. contact angle and volume). In the latter category, several authors proposedstatic scaling law relationships between process inputs such as applied voltage and certain processoutputs such as the frequency at which jets issued from the fluid [71, 72, 73]. While both the highfidelity simulation models and the static scaling models provide benefits for some applications,they are not suitable for use in model-based feedback control algorithms. In particular, there isa lack of compact models that are capable of holistically describing the distinct dynamic regimesin e-jet printing. An attractive framework for accomplishing this task is that of hybrid dynamicalsystems, as they enable the capture of complicated, varying dynamics while still maintaining afoundation in ODEs rather than in partial differential equations.After some necessary further technical details on e-jet printing in Section 2.1, this chap-ter presents two hybrid modeling frameworks for e-jet printing. First, Section 2.2 presents acontinuous-time model with nonlinear components that highlights the necessity of hybridness forODE-based modeling of e-jet printing by using as much physics-based modeling as possible andthen capturing the remaining unmodeled parts of the ejection process with data driven techniques.These physics- and data-driven models are linked by a hybrid framework. This constitutes the first-ever complete model of the ejection process based on ODEs. However, this model only capturesfluid flow at the nozzle outlet, and does not explicitly output the final droplet volume depositedon the substrate. Thus, Section 2.3 presents a discrete-time, piecewise affine (PWA) model featur-ing simplified flow dynamics and a new framework for capturing the final droplet volume. Bothsections feature empirical validation of their models.11 .1 Electrohydrodynamic Jet Printing The conventional e-jet printing setup requires an ink-filled emitter, typically a conductive (or con-ductively coated) microcapillary nozzle, and an attractor, typically a flat, conducting, groundedsubstrate. The emitter is positioned vertically above the attractor and an electric potential is ap-plied between the two components. Figure 2.1 gives a schematic of this configuration, labelingboth the important geometric constants of the printer setup and two important dynamical processvariables: volumetric flow rate of ink out of the nozzle, 𝑄 , and the position of the meniscus tip inspace, ℎ . Volumetric flow rate 𝑄 is important because of its obvious relevance to the fluid volumeultimately deposited on the substrate. Meniscus position ℎ is important both because it capturesimportant “milestones” in the ejection process (e.g. the transition between dynamic regimes, im-pingement of the jet on the substrate, and the breaking of the jet) and because its dynamics arecoupled to those of 𝑄 by capturing the change in the system’s capacitor geometry.The electric field induced between the liquid meniscus and the attractor causes the meniscusto deform. This deformation can be stable at low voltages, and as applied voltage increases theequilibrium shape of the meniscus changes from a spherical cap to a sharp point known as a Taylorcone. If the applied voltage is high enough, the meniscus becomes unstable and a jet of ink issuesfrom the tip of the Taylor cone towards the attractor. While the voltage difference between theemitter and attractor is maintained and there exists material contiguity between the emitter andattractor, a redistribution of charge occurs in the fluid until the electrically induced surface stressbecomes weak enough that the natural liquid surface tension causes the jet to retract, leaving adroplet of ink on the attractor.If voltage is held high after this ejection, charge will again accumulate at the meniscus andanother ejection will occur. The indefinite repetition of this cycle at constant high voltage is termedDC printing. Drop on Demand (DoD) printing is an alternative method in which distinct pulsesof length 𝑇 𝑝 and high voltage 𝑉 ℎ are used against a constant low voltage bias 𝑉 𝑙 to control thetiming and size of droplet deposition [22]. DoD printing can be further subdivided into subcriticalprinting and the complementary supercritical printing. Subcritical printing, upon which this workfocuses, is defined as DoD printing in which 𝑇 𝑝 is short enough that the high voltage pulse fallsbefore the natural cessation of the jet can begin [22]. In other words, subcritical printing is whereeach pulse corresponds to a single ejection which is stopped artificially by the falling edge of thepulse. (Supercritical printing is any DoD printing that is not subcritical, and is not considered inthis work).The remaining sections in this chapter subdivide the ejection process itself into a set of distinctdynamic regimes. These regimes are the locations of the hybrid model. For both hybrid models inthis chapter, the division of the total ejection process into partial processes is based on physically12 O ZZ L E W A LL INK S U BS T R A T E +- TOTALCONTROLVOLUME M I C R O P I P E TT E N O ZZ L E FILLLINE
Figure 2.1: (
Top ) Schematic of e-jet printing setup. Dimensions represent signed displacements in thedirection of the single-sided arrows with respect to the inertial ( 𝑥, 𝑧 ) coordinate system. The volumetricflow rate 𝑄 , labeled via block arrow, is also a signed quantity, but its sign is with respect to the controlvolume denoted by the dashed box. ( Bottom ) Time lapse photography of an ejection with time stamps fromthe rising edge of a voltage pulse.
To leverage as much as possible existing knowledge of e-jet printing physics, this section dividesthe ejection process into the following three partial processes. The “build-up” regime describesthe initial deformation of the meniscus into a Taylor Cone when voltage is stepped high. The“jetting” regime describes the development of the jet at the Taylor cone tip, its approach towardsthe attractor, the fluid flow while the jet is fully developed, and the retraction of the jet back to theTaylor cone. Finally, the “relaxation” regime describes the settling of the meniscus from a Taylorcone back to a stable equilibrium position while the voltage is low. A complete ejection consists ofswitching from build-up to jetting to relaxation. This process, and the distinction between dynamicregimes and actuation methods, is illustrated in Figure 2.2.The rest of the section is structured as follows. Section 2.2.1 defines the mathematical rep-resentation of e-jet as a hybrid dynamical system, and derives the individual model components.Section 2.2.2 describes the experimental system, the procedure for collecting and processing data,and the system identification techniques for defining the data-driven portions of the model. Sec-tion 2.2.3 presents a validation of the identified model. Finally, Section 2.2.4 provides a conclusivesummary of the section.
A hybrid dynamical system is a synthesis of a discrete event system and a dynamical systemgoverned by differential or difference equations. Because this section emphasizes the physics-driven modeling of e-jet printing, continuous-time differential equations are used . Like discreteevent systems, hybrid systems are often formalized as a set, called a hybrid automaton, in whicheach element describes a different feature of the system behavior. Here, the hybrid automatondefinition of Cassandras and Lafortune [3] serves as a basis for this work’s formalism, in whicha hybrid system is given as a 10-tuple G = ( 𝑃, 𝑋, 𝑈, 𝑌 , 𝑓 , 𝑔, 𝜙, 𝜓, 𝑝 , x ) . 𝑃 is a set of discretestates or “locations,” 𝑋 = R 𝑛 is the dynamical state space, 𝑈 = R 𝑚 is the dynamical control inputspace, 𝑌 = R 𝑞 is the dynamical output space, 𝑓 : 𝑃 × 𝑋 × 𝑈 → 𝑋 is a vector field denoting the This section is the exception to the rule of using discrete-time systems in this dissertation. This section thus usesa somewhat different hybrid system formalization than Section 2.3 onwards. - +- Ink Taylorconeforms +- Electrical stressoverwhelmssurface tension
Nozzle
SubstrateCharges +- CriticalchargedepositionChargeaccumulationrenews VoltagefallingElectricalstress fallsdue to:
Subcritical DoDActuation: DCRegime: Build-up Jetting Relaxation
Figure 2.2: Ejection process in terms of the physical phenomena at the liquid-air interface. Distinctionis drawn between the process under a constant DC applied voltage, for which ejections begin and endrepeatedly under the natural high voltage electrohydrodynamics, and under a single subcritical pulse, whichyields a single ejection whose cessation is induced by the falling of the pulse. 𝑔 : 𝑃 × 𝑋 → 𝑌 is the arithmetic map from the dynamical states to theoutput space in a given location, 𝜙 : 𝑃 × 𝑌 → 𝑃 is the transition function determining the discretestate based on the dynamical output (usually an inequality condition), 𝜓 : 𝑃 × 𝑃 × 𝑋 → 𝑋 is thereset function which can instantaneously change the dynamical state when a transition occurs, and 𝑝 , x are initial conditions of the discrete and dynamical states. This definition is reminiscent ofa PWD nonlinear system. However, here the automaton structure helps guide the modeling effort,and the use of continuous time rather than discrete time complicates the translation of resets into astrict PWD framework, which usually relies on a notion of sample period [17].Furthermore, this definition features one significant structural simplification when comparedto most general contemporary hybrid systems, which imposes a limitation on system behavior. Itdoes not allow for uncertain discrete transitions because it does not have invariants on the locationsor guards on the transitions. (Invariants and guards allow for separate, potentially overlapping,restrictions on the viability of each location and transition). Instead, the transition function 𝜙 enforces a one-to-one mapping from the dynamic outputs and the current location to the nextlocation. This limitation is imposed to keep the scope of the mathematical framework comparableto the physical modeling objectives of this work—in which system stochasticity is not considered.In this work, the e-jet system is modeled with three locations, 𝑃 = { 𝑝 , 𝑝 , 𝑝 } correspondingto build-up, jetting, and relaxation, respectively. Thus, at any point in time, the state of the systemis described by 𝑝 ( 𝑡 ) ∈ 𝑃 and the dynamical state vector x ( 𝑡 ) ∈ 𝑋 , and is driven by the input vector u ( 𝑡 ) ∈ 𝑈 via the differential equations (cid:164) x ( 𝑡 ) = 𝑓 ( 𝑝 ( 𝑡 ) , x ( 𝑡 ) , u ( 𝑡 )) = 𝑓 ( x ( 𝑡 ) , u ( 𝑡 )) 𝑝 ( 𝑡 ) = 𝑝 𝑓 ( x ( 𝑡 ) , u ( 𝑡 )) 𝑝 ( 𝑡 ) = 𝑝 𝑓 ( x ( 𝑡 ) , u ( 𝑡 )) 𝑝 ( 𝑡 ) = 𝑝 (2.1) y ( 𝑡 ) = 𝑔 ( 𝑝 ( 𝑡 ) , x ( 𝑡 )) = 𝑔 ( x ( 𝑡 )) 𝑝 ( 𝑡 ) = 𝑝 𝑔 ( x ( 𝑡 )) 𝑝 ( 𝑡 ) = 𝑝 𝑔 ( x ( 𝑡 )) 𝑝 ( 𝑡 ) = 𝑝 (2.2) 𝑝 ( 𝑡 ) = 𝜙 ( 𝑝 − ( 𝑡 ) , y ( 𝑡 )) (2.3)Note that (2.3) captures the instantaneousness of discrete transitions, with 𝑝 − ( 𝑡 ) being the value of 𝑝 ( 𝑡 ) prior to 𝜙 evaluation.In this work the outputs are the physical system values, y ( 𝑡 ) = [ ℎ ( 𝑡 ) , 𝑄 ( 𝑡 )] 𝑇 . The exactdefinitions of x and u depend on the details of the differential equations to be derived, but for the16 uild-up Jetting Relaxation Figure 2.3: Schematic of the system as a hybrid automaton. The transition inequalities are labeled explicitlyon the automaton edges for completeness, and are derived in section 2.2.1.3. sake of clarity, they are preemptively defined as x ( 𝑡 ) = 𝛿ℎ ( 𝑡 ) 𝛿𝑄 ( 𝑡 ) 𝛿 (cid:164) 𝑄 ( 𝑡 ) = ℎ ( 𝑡 ) − 𝑄 ( 𝑡 ) − (cid:164) 𝑄 ( 𝑡 ) 𝑝 ( 𝑡 ) ∈ { 𝑝 , 𝑝 } ℎ ( 𝑡 ) − ℎ 𝑁 𝑄 ( 𝑡 ) − 𝑄 𝑓 (cid:164) 𝑄 ( 𝑡 ) 𝑝 ( 𝑡 ) = 𝑝 (2.4) u ( 𝑡 ) = 𝑢 ( 𝑡 ) = 𝑉 ( 𝑡 ) − 𝑝 ( 𝑡 ) ∈ { 𝑝 , 𝑝 } 𝑉 ( 𝑡 ) − 𝑉 ℎ 𝑝 ( 𝑡 ) = 𝑝 (2.5)where ℎ 𝑁 is the displacement of the substrate from the nozzle outlet and 𝑄 𝑓 is the flow rate at theend of the voltage pulse. This model structure allows for the dynamical states 𝛿ℎ and 𝛿𝑄 to beequal to the outputs ℎ and 𝑄 during build-up and relaxation (i.e. when the models are physics-based), and equal to deviations from output equilibria ℎ 𝑁 and 𝑄 𝑓 during jetting (i.e. when themodel is a black box).A graphical representation of the hybrid automaton for an e-jet ejection is given in Figure 2.3.The remainder of this section focuses on deriving 𝑓 , 𝑓 , 𝑓 , 𝜙 , and 𝜓 . For all these derivations,it is assumed that nozzle and the substrate have zero velocity with respect to one another, and thatthe substrate is flat and clean (i.e. free of pre-deposited substances). Physically, the build-up location represents the accumulation of charge and mass in the menis-cus under high voltage leading up to jetting. The relaxation state represents the convergence ofthe meniscus to an equilibrium shape at low voltage. Despite involving nontrivial changes to themeniscus volume and shape, throughout both of these states the liquid’s form factor is approx-imately that of a typical pendant droplet, whose deformations under applied electric fields have17een the subject of much study [74, 75, 76]. This prior art enables these locations to be modeledfrom a perspective of physical first-principles. Such a model was first proposed by Wright, Krein,and Chato [77] in 1993, further developed by Yang, Kim, Cho and Chung [78] in 2014, and servesas a foundation for the physics-driven modeling in this work.The premise of Wright’s dynamic model is a 1-dimensional (along the nozzle axis) Newtonianforce balance across the plane of the nozzle outlet (i.e. across the boundary of the control volumevia which 𝑄 is defined). Algebraic manipulation of this force balance yields the derivative of flowrate as a function of pressures at the plane. This is represented as (cid:164) 𝑄 = 𝜋𝑟 𝐼 𝜌𝐿 𝑃 𝑛𝑒𝑡 (2.6)where 𝑟 𝐼 is the inner radius of the nozzle outlet, 𝜌 is the mass density of the ink, 𝐿 is the length ofthe fluid column, and 𝑃 𝑛𝑒𝑡 is the net pressure difference across the nozzle outlet, equal to the sumof pressure changes arising from the dominant physical phenomena in the system. These physicalphenomena are the pressure due to gravity 𝑃 𝑔 (taken to be hydrostatic pressure), the pressure dueto viscous flow 𝑃 𝜇 (assumed equal to Hagen-Poiseuille flow in the nozzle), the pressure due tosurface tension 𝑃 𝛾 (from the Young-Laplace equation applied to the meniscus tip), and the electricpressure due to applied field 𝑃 𝐸 (equal to the electrical energy density of the system at the meniscustip [79, 80]). Mathematically, these pressures are given by 𝑃 𝑔 = 𝜌𝑔𝐿 (2.7) 𝑃 𝜇 ( 𝑡 ) = − 𝜇𝐿𝑄 ( 𝑡 ) 𝜋𝑟 𝐼 (2.8) 𝑃 𝛾 ( 𝑡 ) = − 𝛾𝑅 ( 𝑡 ) (2.9) 𝑃 𝐸 ( 𝑡 ) = 𝜀𝐸 ( 𝑡 ) (2.10)where 𝑔 is the gravitational constant, 𝜇 is the dynamic viscosity of the ink, 𝑟 𝐼 is a measure of thenozzle shaft inner radius, 𝛾 is the liquid-air surface tension coefficient of the ink, 𝑅 is the radiusof curvature of the meniscus tip, 𝜀 is the permittivity of the medium between the meniscus and theground plate (usually air), and 𝐸 is the electric field at the liquid-air interface at the meniscus tip.This model provides a concrete ODE-based representation of flow rate dynamics in e-jet print-ing. However, Wright does not perform any empirical validations of this model. The model variantpresented by Yang in [78] sees some validation, but is limited by the fact that it relies on a time-varying variable that is not explicitly modeled, and whose value must be gleaned from measureddata and updated at each time-step or approximated as constant. Indeed, while the current work18ses equations (2.6-2.10) virtually unchanged as a model framework, several of their constitutiveelements are substantially modified to facilitate the model’s practical implementation. Specifically, 𝑃 𝜇 ( 𝑡 ) , 𝑅 ( 𝑡 ) , and 𝐸 ( 𝑡 ) are addressed in this work.In past works 𝑃 𝜇 is given by the Hagen-Poisuille equation, which describes the pressure dropthrough a long tube of constant cross-section due to shear forces, and uses 𝑟 𝐼 = 𝑟 𝐼 . However,the nozzles used in e-jet printing do not have constant cross-sections. To achieve the small outletdiameters required, the nozzles are made from pulled-glass micropipettes, which do not have easilydefined axial cross-sections. Thus, this work treats 𝑟 𝐼 in (2.8) as an effective inner tube radius, 𝑟 𝐼 = 𝑐 𝑟 𝐼 𝑟 𝐼 (2.11)where 𝑐 𝑟 𝐼 is a correction coefficient to be found via a system identification process described insection 2.2.2.2. This modeling strategy keeps the shear term physically meaningful while enablinga degree of flexibility required for dealing with uncertain nozzle geometry.Both 𝑅 ( 𝑡 ) and 𝐸 ( 𝑡 ) depend on the shape of the meniscus. For the purpose of modeling 𝑅 ( 𝑡 ) inthe surface tension term, equation (2.9), Wright and Yang both assume the meniscus is a sphericalcap. This assumption is clearly only valid for small deformations of the meniscus under an electricfield. As the meniscus sharpens into a Taylor cone, it achieves a much smaller 𝑅 ( 𝑡 ) than can becaptured with a spherical model.Wright does not offer a physics-based model for 𝐸 ( 𝑡 ) , but Yang models the meniscus (assumedto be conducting) and nozzle together as the one half of a two-sheeted hyperboloid of revolution.To fully determine this hyperboloid the following constraints are imposed:(C2.1) the surface vertex position is fixed to the meniscus tip,(C2.2) the surface center of symmetry is fixed to the substrate,(C2.3) 𝑅 ( 𝑡 ) is equal to the meniscus tip radius of curvature measured from photographs.These constraints enable the use of an analytical solution for electric field, but (C2.3) makes themodel dependent on an enormous quantity of measured data, as each time step of every voltagestep (characterized by the combination of 𝑉 𝑙 and 𝑉 ℎ ) requires a different value of 𝑅 ( 𝑡 ) . Yang cir-cumvents this need by considering only small meniscus deformations, up to roughly 40% increasesin ℎ ( 𝑡 ) from an initial condition where the spherical cap assumption holds, and assuming 𝑅 ( 𝑡 ) isconstant. Additionally, to make up for differences between the theoretical electrode configurationdescribed above and the physical system, the theoretical electric field is multiplied by a time-varying model adjustment coefficient, which itself is a function requiring the system identificationof two parameters. 19he current work presents a model seeking to capture large meniscus deformations (greaterthan 200% increases in ℎ ( 𝑡 ) ) while reducing the need for measured data to define the model. This isdone by modeling the meniscus as a paraboloid of revolution for all dynamical equation derivations(rather than the mixed spherical and hyperboloidal paradigm), and by leveraging knowledge of theforce balance at low voltage equilibrium.To fully determine the paraboloid at any point in time, only two constraints are necessary:(C2.1) and(C2.4) The surface intersects the nozzle-ink interface (i.e. the outer edge of the nozzle outlet).Under these constraints, the radius of curvature and theoretical electric field are given by [81] 𝑅 ( 𝑡 ) = 𝑟 𝑀 − ℎ ( 𝑡 ) (2.12) 𝐸 𝑡 ( 𝑡 ) = 𝑉 ( 𝑡 ) 𝑅 ( 𝑡 ) ln (cid:16) ( ℎ ( 𝑡 )− ℎ 𝑁 ) 𝑅 ( 𝑡 ) (cid:17) (2.13)The adjusted electrical field to be plugged into equation (2.10) is chosen to be 𝐸 ( 𝑡 ) = 𝑐 𝐸 𝑒𝑞 𝐸 𝑡,𝑒𝑞 + 𝑐 𝛿𝐸 𝛿𝐸 𝑡 ( 𝑡 ) (2.14) 𝛿𝐸 𝑡 ( 𝑡 ) = 𝐸 𝑡 ( 𝑡 ) − 𝐸 𝑡,𝑒𝑞 (2.15)where 𝐸 𝑡,𝑒𝑞 is the unadjusted field at low voltage equilibrium, 𝛿𝐸 𝑡 is the change in unadjusted fieldfrom 𝐸 𝑡,𝑒𝑞 , and the correction coefficients 𝑐 𝐸 𝑒𝑞 and 𝑐 𝛿𝐸 are both constants. Only 𝑐 𝛿𝐸 need be foundvia system identification involving measured timeseries data. 𝑐 𝐸 𝑒𝑞 can be found analytically fromthe fact that at equilibrium the sum of all forces (equivalent to 𝑃 𝑔 + 𝑃 𝜇 + 𝑃 𝛾 + 𝑃 𝐸 in this case) mustbe zero. This relation yields 𝑐 𝐸 𝑒𝑞 = − 𝑟 𝑀 ln (cid:16) ℎ 𝑒𝑞 ( ℎ 𝑁 − ℎ 𝑒𝑞 ) 𝑟 𝑀 (cid:17) √︃ − 𝑔𝐿 𝜌𝑟 𝑀 − 𝛾ℎ 𝑒𝑞 √ 𝜀ℎ 𝑒𝑞 𝑉 𝑙 (2.16)where the only empirical information required is the meniscus position at low voltage equilibrium, ℎ 𝑒𝑞 , which is a single datum as opposed to a model parameter that must be regressed on a data set.Thus the paraboloidal meniscus approximation enables modeling of larger meniscus deformationsby capturing the sharpening of the meniscus as it grows in size via equation (2.12). Equation(2.12) also eliminates the need for 𝑅 ( 𝑡 ) to be defined by measured timeseries data, and basing theelectric field adjustment scheme on perturbation from equilibrium via (2.14) and (2.16) eliminatesa parameter in 𝐸 ( 𝑡 ) requiring system identification, thereby reducing the model’s dependence onempirical data to be fully defined. 20he final point to be addressed in the physics-driven modeling of flow rate is the differencebetween the build-up and relaxation states, which revolves around 𝑐 𝛿𝐸 . The high voltage duringbuild-up, which pushes the system well beyond the region of attraction of any stable equilibria,makes 𝛿𝐸 𝑡 a significant overapproximation, necessitating 𝑐 𝛿𝐸 < . However, in relaxation, whenthe system is under a low voltage yielding a stable equilibrium, the changes in unadjusted field aresmaller. Reducing them is both unnecessary for convergence to equilibrium, and can cause surfacetension to unrealistically overwhelm electric stress. Thus during relaxation, 𝑐 𝛿𝐸 is simply set to 1.This fully defines the flow rate dynamics for build-up and relaxation, leaving the dynamics ofmeniscus position to be derived. To do this, an equation for the volume of the fluid outside thenozzle may be differentiated. This yields 𝑄 as a function of (cid:164) ℎ , which can then be rearranged.The volume equation is that of a paraboloid truncated at the nozzle outlet: V ( 𝑡 ) = − 𝜋 𝑟 𝑀 ℎ ( 𝑡 ) (2.17)The time derivative of this equation, rearranged, yields (cid:164) ℎ ( 𝑡 ) = − 𝜋𝑟 𝑀 𝑄 ( 𝑡 ) (2.18)Equations (2.6) and (2.18) can be used to describe the evolution of flow rate and meniscusposition during build-up, but two small formalities must be addressed before it can be incorporatedwith the hybrid model. First, it is a model in 𝑄 and ℎ whereas the hybrid system definition hasdynamical states of 𝛿𝑄 and 𝛿ℎ . Second, this model is not second-order in 𝑄 , while the hybridsystem definition is.Both of these issues are remedied by choosing 𝑔 ( x ( 𝑡 )) = 𝑔 ( x ( 𝑡 )) = (cid:34) (cid:35) x ( 𝑡 ) (2.19)This implies that during build-up, 𝛿𝑄 = 𝑄 , 𝛿ℎ = ℎ . Thus, the final form of 𝑓 and 𝑓 is 𝑓 , ( x , u ) = − 𝜋𝑟 𝑀 𝛿𝑄 𝜋𝑟 𝐼 𝜌𝐿 (cid:16) 𝜌𝑔𝐿 − 𝜇𝐿𝜋𝑟 𝐼 𝛿𝑄 + 𝛾𝑟 𝑀 𝛿ℎ + 𝜀 𝐸 (cid:17) (2.20) 𝐸 = − (cid:118)(cid:116) − ℎ 𝑒𝑞 𝛾 − 𝑔𝐿𝑟 𝑀 𝜌𝜀𝑟 𝑀 − 𝑐 𝛿𝐸 𝛿ℎ √ 𝑢𝑟 𝑀 ln (cid:16) 𝛿ℎ ( ℎ 𝑁 − 𝛿ℎ ) 𝑟 𝑀 (cid:17) (2.21)21here time argument ( 𝑡 ) has been dropped for compactness, 𝑐 𝛿𝐸 < for 𝑓 , and 𝑐 𝛿𝐸 = for 𝑓 .Finally, because the actuating voltage pulses in e-jet typically rise while the system is at somelow-voltage equilibrium, the initial conditions of the system are within the build-up state. Here,we assume simulations to start from a stationary meniscus. Thus x = (cid:104) ℎ 𝑒𝑞 (cid:105) 𝑇 , 𝑝 = 𝑝 (2.22) The physics-driven models 𝑓 and 𝑓 describe meniscus deformations up to the critical Taylorcone. However, as the jet forms at the cone tip, the capacitor geometry, charge flow, and fluid flowbecome more complicated, and these models cease to represent the system’s dynamics. At thispoint, the system transitions to the data-driven model of jetting dynamics. This model is developedby finding differential equations yielding signal shapes similar to those of empirical data and fittingthe parameters of those differential equations to measurements.Classification of the empirical signal profiles is dependent on the actuation method. DC print-ing and frequently supercritical DoD printing can yield highly nonlinear dynamics in that flowrate will fall and the jet will cease automatically due to critical charge ejection despite a constantinput. In subcritical printing, however, the falling edge of the voltage pulse is directly responsiblefor the fall in flow rate and jet cessation. This causality enables the input/output dynamics to bewell-captured by a linear time invariant (LTI) model defined in terms of change from the locallysteady high-voltage flow, i.e. from the flow with a fully developed contiguous jet. Thus, this worklimits the scope of the jetting model to subcritical printing.Based on qualitative observations of measured jetting signals, a second order LTI system ischosen for flow rate, with model coefficients to be found by least squares regression. For mathe-matical consistency with the physics-driven models, the continuous-time model 𝛿 (cid:165) 𝑄 ( 𝑡 ) = 𝑎 𝑄 𝛿 (cid:164) 𝑄 ( 𝑡 ) + 𝑎 𝑄 𝛿𝑄 ( 𝑡 ) + 𝑏 𝑄 𝑢 ( 𝑡 ) (2.23)is desired. However, because jetting is a high-speed, small-scale phenomenon, sampling can berelatively coarse and noisy. Regression performance is thus significantly improved when using thediscrete time model given by 𝛿𝑄 ( 𝑘 + ) = ˜ 𝑎 𝑄 𝛿𝑄 ( 𝑘 + ) + ˜ 𝑎 𝑄 𝛿𝑄 ( 𝑘 ) + ˜ 𝑏 𝑄 𝑢 ( 𝑘 ) (2.24)were 𝑘 is the discrete time index.To get the continuous-time model coefficients 𝑎 𝑄 , 𝑎 𝑄 , and 𝑏 𝑄 from the regressed discrete22ime coefficients ˜ 𝑎 𝑄 , ˜ 𝑎 𝑄 , and ˜ 𝑏 𝑄 , the poles and final values of the two systems are matched.First, the two poles of the discrete time system are converted to their continuous-time equivalentsvia 𝑝 𝑠𝑖 = ln ( 𝑝 𝑧𝑖 ) 𝑇 𝑠 𝑖 ∈ { , } (2.25)where 𝑝 𝑠𝑖 is a continuous-time pole, 𝑝 𝑧𝑖 is a discrete time pole, and 𝑇 𝑠 is the sampling period.Solving the characteristic equation of the continuous-time system then yields 𝑎 𝑄 = 𝑝 𝑠 + 𝑝 𝑠 𝑎 𝑄 = − 𝑝 𝑠 𝑝 𝑠 (2.26)For the input coefficient, the final value theorem expressions at low voltage for the continuous anddiscrete time systems can be equated to yield lim 𝑠 → 𝑠 𝛿𝑄 ( 𝑠 ) 𝑈 ( 𝑠 ) ( 𝑉 𝑙 − 𝑉 ℎ ) 𝑠 = lim 𝑧 → ( 𝑧 − ) 𝛿𝑄 ( 𝑧 ) 𝑈 ( 𝑧 ) 𝑧 ( 𝑉 𝑙 − 𝑉 ℎ ) 𝑧 − 𝑏 𝑄 = ˜ 𝑏 𝑄 𝑎 𝑄 ˜ 𝑎 𝑄 + ˜ 𝑎 𝑄 − (2.27)where 𝛿𝑄 () 𝑈 () is the transfer function from input to 𝛿𝑄 .The above analysis fully defines the flow rate model for jetting, and one may note that it isindependent of meniscus position. However, the meniscus position is still important in the jettingmodel, particularly during jet retraction, because it will trigger the transition to the relaxationlocation.Inspection of measured meniscus position signals during jet retraction suggest it may be ap-proximated by exponential decay, i.e. a first order LTI system. However, the retraction does notbegin until some time after the falling edge of the voltage pulse. This is accounted for by addinga delay on the input. This system will be identified in much the same way as flow-rate, withregression and delay identification being performed on the discrete time system 𝛿ℎ ( 𝑘 + ) = ˜ 𝑎 ℎ 𝛿ℎ ( 𝑘 ) + ˜ 𝑏 ℎ 𝑢 ( 𝑘 − 𝑑 ) (2.28)where 𝑑 is the delay in time steps between the step down in voltage and the beginning of retrac-tion. The continuous-time model parameters arising from pole matching and final value theoremmatching, respectively, are 𝑎 ℎ = ln ( ˜ 𝑎 ℎ ) 𝑇 𝑠 𝑏 ℎ = ˜ 𝑏 ℎ 𝑎 ℎ ˜ 𝑎 ℎ − (2.29)23hich are then plugged into the continuous-time form of the system, given by 𝛿 (cid:164) ℎ ( 𝑡 ) = 𝑎 ℎ 𝛿ℎ ( 𝑡 ) + 𝑏 ℎ T − 𝑑𝑇 𝑠 𝑢 ( 𝑡 ) (2.30)where T is the shift operator, making T − 𝑑𝑇 𝑠 𝑢 ( 𝑡 ) = 𝑢 ( 𝑡 − 𝑑𝑇 𝑠 ) .With this, 𝑓 is fully defined as (cid:164) x ( 𝑡 ) = 𝑓 ( x ( 𝑡 ) , 𝑢 ( 𝑡 )) = 𝑎 ℎ 𝑎 𝑄 𝑎 𝑄 x ( 𝑡 ) + 𝑏 ℎ T − 𝑑𝑇 𝑠 𝑏 𝑄 𝑢 ( 𝑡 ) (2.31) 𝑔 will account for the fact that unlike the physics-driven models, the jetting model is notderived in terms of absolute 𝑄 and ℎ . Instead, it is assumed that in subcritical DoD printing, theflow at the falling edge of the voltage pulse is fully developed and locally steady. Thus y ( 𝑡 ) = 𝑔 ( x ( 𝑡 )) = (cid:34) (cid:35) x + (cid:34) ℎ 𝑁 𝑄 𝑓 (cid:35) (2.32)where 𝑄 𝑓 is the empirically determined flow rate at 𝑡 = 𝑇 𝑝 . This work uses analysis of system stability to concretely define the points in the ejection processat which transitions between the physics- and data-driven models should occur. As described insection 2.1, up to a critical magnitude of deformation each meniscus shape is a stable equilibriumof the system for a particular input. Deformation in excess of that critical shape marks the transitionfrom build-up to jetting. Likewise, the complementary condition—the point during jet retractionat which the meniscus shape becomes a stable equilibrium for some applied voltage—is used tomark the transition from jetting to relaxation.This notion of the transition condition was first proposed by the authors in [22], which leveragesthe physics-based work of Yarin, Koombhongse, and Reneker on deriving the range of stable equi-libria in terms of meniscus shape [82]. In this work, Yarin models the meniscus as a hyperboloidin a prolate spheroidal coordinate system ( 𝜂, 𝜉 ) with the horizontal axis free to move between thenozzle and substrate, equivalent to replacing (C2.2) and (C2.3) with (C2.4) in the surface definition.This results in hyperboloids that are underdetermined given the information available for dynamicsimulation, but can fit the true meniscus shape more closely if extra information is available at agiven point.In the prolate spheroidal coordinate system, 𝜉 ∈ [ , ] denotes a hyperbola, and 𝜂 ∈ [ , ∞) igure 2.4: To-scale schematic of a critically deformed meniscus and the prolate spheroidal coordinatesystem (with horizontal axis at 𝑧 (cid:48) = ) used to describe the geometry of the meniscus and its local electricfield (ellipses) and electric potential contours (hyperbolae). The focus is marked by the white dot at 𝑧 (cid:48) = 𝑎 .For 𝑧 (cid:48) < the field and potential contours are assumed “Cartesian.” Note that by the 𝑧 -axis sign convention, ℎ -values are negative, and 𝑎 is positive. denotes an ellipse (see Figure 2.4). Yarin shows that regardless of surface tension coefficient, themaximum stable deformation of the meniscus corresponds to the shape of the critical hyperbola 𝜉 ∗ ≈ . . This extra datum fully determines the surface of the critical Taylor cone.In [22], 𝜉 is estimated from measured data, and the jetting state is isolated from the build-upstate empirically. For simulations independent of measured data, the transition condition must bedefined in terms of the model’s dynamical states. In other words, for this work a critical meniscusposition ℎ ∗ must be found in terms of 𝜉 ∗ .The derivation of this relationship starts with the transformation equations between the 2Dcross section of the prolate spheroidal coordinate system and a Cartesian coordinate system ( 𝑥 (cid:48) , 𝑧 (cid:48) ) with the same horizontal and vertical axes: 𝜂 = 𝑎 (cid:18)√︃ 𝑥 (cid:48) + ( 𝑧 (cid:48) + 𝑎 ) + √︃ 𝑥 (cid:48) + ( 𝑧 (cid:48) − 𝑎 ) (cid:19) (2.33) 𝜉 = 𝑎 (cid:18)√︃ 𝑥 (cid:48) + ( 𝑧 (cid:48) + 𝑎 ) − √︃ 𝑥 (cid:48) + ( 𝑧 (cid:48) − 𝑎 ) (cid:19) (2.34) 𝑧 (cid:48) = 𝑎𝜂𝜉 (2.35)While both coordinate systems are two-dimensional, three equations are needed to relate them.This is because in addition to 𝜉 and 𝜂 , the prolate spheroidal coordinate system is also parame-25erized by the distance 𝑎 from the horizontal 𝑧 (cid:48) -axis to the focus of the coordinate system. Theapplication of (C2.4) to equation (2.34) and (C2.1) to equation (2.35) yields 𝜉 ∗ = √︃ 𝑟 𝑀 + ( 𝑎 − ℎ 𝑐 ∗ ) − √︃ 𝑟 𝑀 + (− 𝑎 − ℎ 𝑐 ∗ ) 𝑎 (2.36) ℎ ∗ = 𝑎𝜉 ∗ + ℎ 𝑐 ∗ (2.37)where ℎ 𝑐 ∗ is the displacement of the prolate spheroidal coordinate system from the inertial coordi-nate system ( 𝑥, 𝑧 ) .As is, the system is underdetermined, having unknowns 𝑎 , ℎ ∗ , and ℎ 𝑐 ∗ . This is remedied byreformulating the theory in [82] to yield ℎ 𝑐 ∗ as a function of ℎ ∗ and constant system properties via ℎ 𝑐 ∗ = √︁ 𝛼 + 𝛼ℎ ∗ − 𝛼 (2.38)where 𝛼 = 𝜋𝛾ℎ 𝑁 (cid:0) − 𝜉 ∗ (cid:1) (cid:16) ln + 𝜉 ∗ − 𝜉 ∗ (cid:17) ( 𝐶𝑉 ∗ ) , (2.39) 𝐶 ≈ . E − kg m s V is the conversion from volts to statvolts in SI units, and 𝑉 ∗ is the maximumnon-jetting applied voltage. 𝑉 ∗ is empirically determined, providing the extra datum required tofully determine the critical hyperboloid.Now solving the system (2.36), (2.37) will yield four sets of solutions for ℎ ∗ , and 𝑎 . Three ofthese solutions are spurious: two for returning meniscus positions inside the nozzle and another foryielding 𝜂 outside of its domain when plugged into equation (2.33). This leaves a single solutionfor the maximum stable meniscus length, thereby defining the transitions into and out of jetting viathe transition function 𝑝 + = 𝜙 ( 𝑝 − , y ) = 𝑝 ℎ ≥ ℎ ∗ ∧ 𝑝 − = 𝑝 𝑝 ℎ < ℎ ∗ 𝑝 ℎ ≥ ℎ ∗ ∧ 𝑝 − ≠ 𝑝 (2.40)where ℎ ∗ = 𝛼 − 𝛽 − 𝜉 ∗ − √︄ 𝛼 (cid:0) 𝛼 (cid:0) − 𝜉 ∗ (cid:1) − 𝛽 (cid:1) − 𝜉 ∗ (2.41)and 𝛽 = √︃(cid:0) − 𝜉 ∗ (cid:1) (cid:0) 𝛼 (cid:0) − 𝜉 ∗ (cid:1) − 𝑟 𝑀 𝜉 ∗ (cid:1) (2.42)The ( − ) subscript indicates the state value preceding transition and the ( + ) subscript indicates thestate value after transition. The transition is instantaneous.While the outputs y representing the absolute flow rate and meniscus position should be con-26inuous over transitions between locations, the dynamical states x must undergo a reset at eachtransition because the build-up and jetting dynamics use different set points from which 𝛿ℎ and 𝛿𝑄 are defined.The reset’s objective is thus to ensure y is continuous despite discontinuities in x . For 𝛿ℎ and 𝛿𝑄 , this amounts to subtracting and adding the output offsets of the jetting model when enteringand exiting the jetting state, respectively. For 𝛿 (cid:164) 𝑄 , the flow rate dynamics in equations (2.20-2.21)can be used to set the initial 𝛿 (cid:164) 𝑄 when entering jetting, and the state can be set to zero when exitingjetting, as it is unused by the build-up dynamics.Thus x + = 𝜓 ( 𝑝 − , 𝑝 + , x − ) = x − + − ℎ 𝑁 − 𝑄 𝑓 (cid:104) (cid:105) 𝑓 ( x − , u − ) 𝑝 − 𝑝 + = 𝑝 𝑝 x − + ℎ 𝑁 𝑄 𝑓 (cid:104) − (cid:105) 𝑓 ( x − , u − ) 𝑝 − 𝑝 + = 𝑝 𝑝 (2.43)Now, with the dynamics of all locations and the process for transitioning between them de-fined, a complete end-to-end model of the e-jet cycle under subcritical actuation is achieved. Thefollowing sections discuss the experimental procedures and data processing involved in modelverification. There are two main components of the process used for collecting data for system identificationand validation of the hybrid e-jet model proposed above: the physical system used to generate highspeed videos and image processing code that is used to extract signals of interest from the raw highspeed videos.The experimental setup consists of an e-jet printer (custom built at the University of Michigan),and a high speed camera (Vision Research, Phantom V9.0) which are automated and synchronizedvia the drivers of an X-Y-Z nanopositioning stage (Aerotech, “Planar DL ”) and software writtenin the Aerotech A3200 Motion Composer Integrated Development Environment. The high speedcamera is fixed with a 20x microscope lens assembly yielding . µ mpixel resolution. Norland OpticalAdhesive 81, a UV curable monomer-photocatalyst mixture, is used as the ink for all experimentsin this work. A silicon wafer serves as the substrate. The microcapillary nozzle is supplied by WordPrecision Instruments (product TIP30TW1), and sputter coated in house with gold-palladium alloy27able 2.1: Experimental Setup and Process ParametersParameter Symbol ValueNozzle outlet inner radius 𝑟 𝐼 µ m Nozzle outlet outer radius(meniscus radius) 𝑟 𝑀 . µ m Fluid column height 𝐿 Substrate position ℎ 𝑁 µ m Ink Density 𝜌 kgm [83]Ink Dynamic Viscosity 𝜇 . [84]Surface Tension Coefficient 𝛾 . Nm [69]Low Voltage 𝑉 𝑙
525 V
Sample Period 𝑇 𝑠 µ s Table 2.2: The set Ω of 𝜔 = ( 𝑇 𝑝 , 𝑉 ℎ ) pairs 𝑉 ℎ [V] 1100 1150 1200 1250 1300 1300 1300 1300 1350 1370 𝑇 𝑝 [ ms ] 2.0 2.0 2.0 2.0 1.5 1.8 2.0 2.3 2.0 2.0for conductivity.Important physical parameters of the printer setup and video capture process are tabulated intable 2.1. Twenty single-droplet ejections are recorded for each element of a set Ω of ten ( 𝑇 𝑝 , 𝑉 ℎ ) pairs, listed in table 2.2. All trials begin from the low voltage equilibrium meniscus position overa clean region of substrate with no prior fluid depositions. For each pair, 10 of the 20 recordingsare designated as training data, and the remaining as validation data. These pairs are chosen toapproximately cover the range of voltages within the subcritical regime for the given printer con-figuration and ink, as well as to offer some variation in 𝑇 𝑝 where possible (the subcritical regimemay be “narrow” at particularly low or high values of 𝑉 ℎ ).The experimental setup and procedure (other than the specific ( 𝑇 𝑝 , 𝑉 ℎ ) pairs tested) is identicalto those of [22], which may be referenced for greater detail. Additionally, Figure P.1 contains aphotograph of the experimental system.Once the videos are captured, ℎ and 𝑄 signals are extracted for each trial via an image process-ing protocol consisting of substrate position identification, ink-nozzle interface identification, edgefinding, region area/volume computation, and numerical differentiation. These operations are alsodescribed in detail in [22]. The only procedure change from [22] is that where the previous workdiscarded trials for which the algorithm failed to identify the ink-nozzle interface (a task made dif-ficult by the dullness of the feature corners), this work retains all recorded video and uses manual28nterface identification for trials where the algorithm failed. For the physics-driven locations, there are two major parameters that must be found via systemidentification: 𝑐 𝑟 𝐼 and 𝑐 𝛿𝐸 . The system identification is done via the minimization of flow rateerror in simulations performed over a mesh of parameter test values.Each parameter is given a set of 100 linearly spaced test values on the ranges 𝑐 𝑟 𝐼 ∈ [ , ] and 𝑐 𝛿𝐸 ∈ [ . , . ] , yielding 10,000 models in total. For 𝑐 𝑟 𝐼 , this range allows 𝑟 𝐼 to vary from thenozzle outlet radius to approximately the radius of the main body of the micropipette. For 𝑐 𝛿𝐸 ,this range is found by trial and error with boundaries chosen such that no 𝑟 𝐼 in the aforementionedrange yields simulation timeseries sufficiently approximating the empirical signals under holisticqualitative assessment.Each of the 10,000 models is given a scalar error value, 𝐸𝑟𝑟 , by the the metric
𝐸𝑟𝑟 ( 𝑐 𝛿𝐸 , 𝑐 𝑟 𝐼 ) = mean 𝜔 ∈ Ω ,𝑖 ∈ 𝐼 (cid:32) RMS 𝑘 ∈ 𝐾 𝑝 ( 𝜔,𝑖 ) (cid:16) 𝑄 𝜔,𝑖 ( 𝑘𝑇 𝑠 ) − ˆ 𝑄 𝜔,𝑐 𝛿𝐸 ,𝑐 𝑟𝐼 ( 𝑘𝑇 𝑠 ) (cid:17) (cid:33) (2.44)where 𝑖 ∈ 𝐼 = { , , · · · , } is the index of the training data recording for a given 𝜔 , 𝐾 𝑝 ( 𝜔, 𝑖 ) isthe set of time steps in the build-up state for the ( 𝜔, 𝑖 ) recording, 𝑄 𝜔,𝑖 is the measured flow ratetrajectory of the ( 𝜔, 𝑖 ) recording, and ˆ 𝑄 𝜔,𝑐 𝛿𝐸 ,𝑐 𝑟𝐼 is the simulated flow rate trajectory generated withthe given test values of 𝑐 𝛿𝐸 and 𝑐 𝑟 𝐼 , and the input dictated by 𝜔 . These simulations are performedusing a fourth order Runge-Kutta method with step-size 𝑇 𝑠 =
100 ns .In addition to 𝑐 𝑟 𝐼 and 𝑐 𝛿𝐸 , the low-voltage equilibrium position ℎ 𝑒𝑞 , the maximum non-jettingvoltage 𝑉 ∗ , and the input delay 𝑑 for the jetting dynamics of meniscus position must be determinedempirically. ℎ 𝑒𝑞 is found by averaging the empirical ℎ ( ) over all training trials. 𝑉 ∗ is set to based on the results in [69], which uses the same ink and printer setup as the current work. For each 𝜔 ∈ Ω , 𝑑 is the median number of time steps during jetting for which 𝑉 ( 𝑡 ) = 𝑉 𝑙 and | 𝛿ℎ ( 𝑡 ) | ≤ tol ,where tol is a tolerance close to 0.For the jetting location, this work assumes that(A2.1) each ( 𝑇 𝑝 , 𝑉 ℎ ) pair requires a different set of LTI model parameters, and(A2.2) these parameter sets must be derived independently from one another.The LTI models for each ( 𝑇 𝑝 , 𝑉 ℎ ) pair are thus derived from only the 10 training recordings corre-sponding to that pair. The discrete-time LTI model parameters for 𝑄 , ˜ 𝑎 𝑄 , ˜ 𝑎 𝑄 , and ˜ 𝑏 𝑄 , are to befit by basic least squares regression. For ℎ , however, a standard least squares regression runs therisk of returning an LTI model whose steady state value under low voltage is near ℎ ∗ but does not29ross ℎ ∗ . Such a model would prevent a transition from jetting to relaxation from ever happening.Thus the system identification for meniscus position will constrain the least squares optimizationsuch that the final value theorem applied to the model under low voltage yields ℎ ( 𝑡 → ∞) ≥ . ℎ ∗ (2.45)Because the optimization is carried out over the decision variables ˜ 𝑎 ℎ and ˜ 𝑏 ℎ , this inequalityconstraint is realized as (cid:104) ℎ 𝑁 − . ℎ ∗ 𝑉 ℎ − 𝑉 𝑙 (cid:105) (cid:34) ˜ 𝑎 ℎ ˜ 𝑏 ℎ (cid:35) ≤ ℎ 𝑁 − . ℎ ∗ (2.46)which can be implemented via MATLAB’s lsqlin function. This section first presents the results of the system identification processes described above, anduses these results to assess this work’s approach to integrating data-driven components into e-jetmodeling. Then, an error breakdown of the fully defined model against the validation data ispresented, and analysis is given.
Figure 2.5 gives the error and standard deviation of the build-up 𝑄 models over the 100-by-100array of 𝑐 𝑟 𝐼 and 𝑐 𝛿𝐸 choices. The parameter values yielding minimum error are 𝑐 𝑟 𝐼 = . and 𝑐 𝛿𝐸 = . , which are applied to all ( 𝑇 𝑝 , 𝑉 ℎ ) pairs for the given ink and printer configuration.This value of 𝑐 𝑟 𝐼 yields an effective inner shaft radius of 𝑟 𝐼 = µ m . This is nearly an order ofmagnitude larger than the nozzle outlet inner radius, 𝑟 𝐼 = µ m . However, it is still significantlycloser to 𝑟 𝐼 than it is to the capillary radius preceding the taper ( ∼ - µ m ), which makes up ∼ of the fluid column. This verifies the necessity of using 𝑐 𝑟 𝐼 to find an effective radius forthe fluid column in the physics-based modeling, as the true nozzle outlet radius and mean columnradius would both fall outside of the low error valley seen in Figure 2.5, regardless of choice of 𝑐 𝛿𝐸 .For the choice of 𝑐 𝛿𝐸 itself, the low error valley apparent in Figure 2.5 may tempt one tobelieve that the global minimum within this valley is not substantially different from other pointsalong some “minimum trajectory” across the mesh, and that choosing a point in the valley yieldingthe least aggressive model adjustment (i.e. 𝑐 𝛿𝐸 closest to ) may be a better modeling choice.However, in fact the model behavior changes as 𝑐 𝛿𝐸 increases and 𝑐 𝑟 𝐼 correspondingly decreases.30 igure 2.5: Training error, 𝐸𝑟𝑟 , and standard deviation of 𝑄 in the build-up state with respect to 𝑐 𝛿𝐸 and 𝑐 𝑟 𝐼 . The color bar across the top of the figure applies to both plots, and gives the logarithmic relationshipbetween pixel brightness and the magnitude of the mean and standard deviation in picoLiters per millisecond( pLms ). The hatched regions represent ( 𝑐 𝛿𝐸 , 𝑐 𝑟 𝐼 ) combinations corresponding to Inf error values. The circledpoint in both images represents the global minimum
𝐸𝑟𝑟 found in this analysis. igure 2.6: Time constants ( 𝜏 ) and DC Gains ( 𝐺 ) of 𝑄 and ℎ jetting models versus 𝑉 ℎ . For comparabilitybetween 𝑄 and ℎ , 𝜏 and 𝐺 are normalized by their mean value. No correlation is apparent between 𝜏 and 𝑉 ℎ . However, DC Gain and 𝑉 ℎ are clearly correlated. Specifically, the pressure due to surface tension ( 𝑃 𝛾 ) loses influence over the flow rate dynamicscompared to the pressures due to electric field ( 𝑃 𝐸 ) and shear forces ( 𝑃 𝜇 ), as can be seen byinspection of equation (2.20). This degrades the model’s ability to capture the system’s transientresponse to the step increase in applied voltage, which is governed largely by the balance of 𝑃 𝐸 and 𝑃 𝛾 due to 𝑃 𝜇 being zero at 𝑡 = . Thus, while the error increases incurred by choosing 𝑐 𝛿𝐸 closerto 1 may be relatively small, this choice has nontrivial ramifications for the physical meaning ofthe model, which justifies the selection of 𝑐 𝛿𝐸 based on global minimum RMSE.To assess the system identification approach taken for the jetting location, specifically the ne-cessity of assumptions (A2.1) and (A2.2), Figure 2.6 illustrates how the transient and steady statebehavior of the LTI models vary with applied voltage. This is done by plotting the time constants, 𝜏 , and DC gains, 𝐺 , for each 𝜔 against 𝑉 ℎ . 𝜏 and 𝐺 come from the roots of a system’s characteris-tic polynomial (which determine the rate of exponential decay of transient responses in stable LTIsystems) and the final value theorem applied to a system with a unit step input, respectively, andare given by 𝜏 ℎ = − 𝑎 ℎ 𝜏 𝑄 = − 𝑎 𝑄 (2.47) 𝐺 ℎ = − 𝑏 ℎ 𝑎 ℎ 𝐺 𝑄 = − 𝑏 𝑄 𝑎 𝑄 (2.48)Pearson correlation coefficients, 𝑟 , and the corresponding 𝔭 -values (stylized with a fraktur fontto distinguish 𝔭 from the hybrid system’s discrete state, 𝑝 ( 𝑡 ) ) of the 𝜏 and 𝐺 magnitudes versus 𝑉 ℎ are given in table 2.3. 𝑟 ∈ [− , ] quantifies the degree to which two variables are linearly32able 2.3: 𝔭 -values and correlation coefficients for LTI Metrics vs. 𝑉 ℎ LTI Metric 𝔭 𝑟 | 𝜏 ℎ | . Irrelevant | 𝜏 𝑄 | . Irrelevant | 𝐺 ℎ | E − − . | 𝐺 𝑄 | . − . Table 2.4: Transition Timing Error Mean ( 𝑒 𝜇 ) & Standard Deviation ( 𝑒 𝜎 )Transition 𝑒 𝜇 [ ms ] 𝑒 𝜎 [ ms ] Build-up to Jetting − .
26 0 . Jetting to Relaxation − .
04 0 . correlated (and the direction in which they are correlated). 𝔭 -values indicate the probability thatthe measured 𝑟 could arise from randomness given a true relationship of zero correlation. Thus,correlation coefficients associated with high 𝔭 -values are marked irrelevant. 𝑟 and 𝔭 are computedwith respect to 𝑉 ℎ because the input to the dynamic system is based on squared voltage.The fact that the 𝜏 and 𝐺 values are far from constant across all 𝑉 ℎ clearly demonstrates thevalidity of (A2.1). (A2.2) is more interesting. Time constant shows no correlation with 𝑉 ℎ . Thismeans that there is unlikely to be a meaningful linear relationship between 𝜏 and 𝜔 , and that someindependent system identification may be necessary for each 𝜔 . However, DC Gain exhibits strongcorrelation. This is a desirable result because it suggests the possibility of reducing the strictnessof (A2.2) by incorporating the dependence of DC Gain on 𝜔 into the framework of jetting-statedynamics, which may reduce the burden of system identification in future work. To represent the results of the complete model, Figure 2.7 gives a normalized root mean squarederror (NRMSE) breakdown of the model for each location with respect to the measured data acrossall ( 𝑇 𝑝 , 𝑉 ℎ ) pairs from the rising edge of the voltage pulse, 𝑡 = , to 𝑡 = .
65 ms . For each location 𝑝 ∈ { 𝑝 , 𝑝 , 𝑝 } and each output 𝑦 ∈ { ℎ, 𝑄 } the NRMSE is calculated by 𝑁 𝑅𝑀 𝑆𝐸 ( 𝑝, 𝑦 ) = RMS 𝜔 ∈ Ω ,𝑖 ∈ 𝐼,𝑘 ∈ 𝐾 𝑝 ( 𝜔,𝑖 ) 𝑦 𝜔,𝑖 ( 𝑘𝑇 𝑠 ) − ˆ 𝑦 𝜔 ( 𝑘𝑇 𝑠 ) range 𝜔 ∈ Ω ,𝑖 ∈ 𝐼,𝑘 ∈ 𝐾 𝑝 ( 𝜔,𝑖,𝑝 ) 𝑦 𝜔,𝑖 ( 𝑘𝑇 𝑠 ) where ˆ 𝑦 𝜔 is the simulated output trajectory generated with the input dictated by 𝜔 , and 𝐾 𝑝 ( 𝜔, 𝑖 ) isthe set of time-steps for which the physical system trial ( 𝜔, 𝑖 ) is in state 𝑝 .33 mpirical data range overlocation's timespan Figure 2.7: Complete model error given as the RMSE normalized by the range of measured data withineach location. Each bar is computed from 100 measured timeseries averaging 23 points each for build-up,34 points for Jetting, and 237 points for relaxation. Error bars represent plus/minus one standard deviation.
Error of state transition times is reported in table 2.4. These errors are given as a simple meanand standard deviation, rather than an RMSE-style metric. This is done in order to preserve thesign of the error, and thus indicate whether the simulation transitions early (positive error) or late(negative error).Finally, a representative timeseries plot showing the simulated output trajectories against themeasured trajectories is given in Figure 2.8.A qualitative assessment of Figure 2.8 shows that despite some notably erroneous features,the overall model reproduces the empirical timeseries satisfactorily. This holistic satisfactionis supported quantitatively by the NRMSE values presented in Figure 2.7, which illustrates thatthe range-normalized timeseries errors average to only 11% for both outputs across all locations(where the average is evenly weighted with respect to the locations, not individual points in time).However, the standard deviations on these NRMSEs are relatively large, such that the maximumsum of standard deviation and NRMSE reaches 33%. While the average sum of the NRMSE andstandard deviation is still just 18%, indicating that overall the error is acceptable despite somenontrivial spread, the standard deviations still warrant discussion.A key reason the standard deviation is large is that the error within a single timeseries is notevenly distributed. Instead, there are areas of small and large errors. Take, for example, themeniscus position during jetting, which has the largest NRMSE and standard deviation. Figure2.8 shows that towards the end of the jetting state, there is very little error between the measuredand simulated ℎ trajectories. However at the beginning of the measured trajectory’s jetting state(upon which the NRMSE computation is based), there is a brief period of large error while thesimulated system is still in the build-up state. As can be seen from table 2.4, the simulations’delayed transition into jetting (and thus the large error in ℎ during the period where the locationof the simulation and experiment are mismatched) is consistent, and is likely the cause of the34 igure 2.8: Plot of the simulation and measured data for ℎ , 𝑄 , and the individual pressure componentscontributing to 𝑄 for 𝑇 𝑝 = and 𝑉 ℎ = . 𝑃 𝐸 and 𝑃 𝑔 are combined because 𝑃 𝑔 is constant and theyare both always positive, meaning that their sum should be balanced against 𝑃 𝛾 with the assistance of 𝑃 𝜇 . yperboloid (underdetermined without prior information) Low Voltage Equilibrium Critical Meniscus
Figure 2.9: Paraboloidal and hyperboloidal electrode geometry approximations overlaid on nozzle/meniscusphotographs at a low voltage equilibrium (left), and at the critical Taylor cone preceding jetting (right). The“underdetermined” hyperboloid is given only at the critical meniscus because the necessary prior informa-tion (in this case, the value of 𝜉 ) is only known for the critical meniscus (see section 2.2.1.3). relatively large standard deviations. In fact, if the error analysis is restricted to the time span inwhich the measured and simulated trajectories are in the same location, the sum of the standarddeviation and NRMSE for ℎ in jetting decreases by 23%.While location mismatch is not the only factor contributing to the observed errors, it is worthfocusing on in particular not only because of its numerical impact illustrated above, but also be-cause it can be ascribed to a specific modeling assumption: that of the meniscus’s shape. While theparaboloidal shape assumption is clearly more accurate than the spherical cap assumption, it stillincreasingly overestimates the volume of the meniscus as the meniscus elongates, with the criticalmeniscus volume being overestimated by 16% on average (compare to 75% for the spherical cap).This means the model requires a greater liquid volume change per unit change in ℎ than the truesystem. Consequently, for the same 𝑄 the simulated ℎ in this work grows more slowly than thephysical system the nearer ℎ draws to ℎ ∗ . On top of this, any retardation of ℎ due to geometricmodel mismatch is amplified through the positive coupling of ℎ and 𝑄 . In other words, the geo-metric mismatch also causes a reduction in (cid:164) 𝑄 , which further slows meniscus elongation beyond thedirect effect of geometric mismatch on (cid:164) ℎ . This combination of factors ultimately results in delayedtransitions to jetting.Note that a hyperboloidal shape assumption is not intrinsically better than the paraboloidalmodel in this regard. Indeed, while combining constraints (C2.1), (C2.2), and (C2.4) can be usedto produce fully determined hyperboloids for use in the dynamic equations, these hyperboloids arevirtually identical to the mathematically simpler paraboloids used in this work. This is illustratedin Figure 2.9. In other words, the ability of hyperboloids to capture the sharpness of the criticalmeniscus is contingent on the relaxation of (C2.2), and the alternative constraint used to derive thetransition condition ( 𝜉 = 𝜉 ∗ ) is only valid at the critical meniscus.36hus, while there is room for future studies to polish the model minutiae, the above resultsdemonstrate the efficacy of the given hybrid system framework, along with this work’s contribu-tions to each of the component models, in capturing the end-to-end dynamics of an e-jet printingprocess. In summary, this section delivers the first end-to-end ODE-driven model of an e-jet printing pro-cess. This is achieved by the combination of three major types of contribution. First, a hybridsystems framework is presented for combining multiple partial process models into an end-to-endmodel. Second, the scope of prior partial process models is expanded by means such as increasingthe sophistication of geometric modeling, leveraging equilibrium information to structure dynam-ical equations, and analysis of the necessary areas in which to incorporate data-driven modelingelements. Lastly, transitions between the hybrid system’s component models are determined viaconsideration of the partial process’s stability.
The physics-focused model of Section 2.2 does not explicitly model the deposited fluid volume,instead only modeling the volumetric flow rate of fluid out of the nozzle. This is in large partbecause past works have considered droplet volume ill-defined until the jet breaks, at which pointthe droplet volume was considered constant. Thus, there remains a gap in the satisfaction ofthe requirements that a model be both control-oriented (i.e. ODE-based) and explicitly outputdeposited droplet volume.The main contribution of the present section is a hybrid system model framework for e-jetprinting that bridges this gap. Specifically, this section• defines the droplet volume as a dynamical state variable that may evolve over time,• presents a new division of the ejection process into partial processes to facilitate droplet volumemodeling,• proposes and experimentally validates a mapping between nozzle flow rate and deposited dropletvolume enabled by the preceding bullets, and• presents a new computer vision technique for taking consistent droplet volume measurementsfrom high speed microscope video. 37 ubstrate Nozzle WallFluid Nozzle-ConnectedFluid BodySubstrate-ConnectedFluid Body
Total Control VolumeDroplet Control Volume
Figure 2.10: Schematic of the two CVs used in this work superimposed over an illustration of a jet immedi-ately after breaking.
The remainder of the section is organized as follows. Section 2.3.1 presents the model (i.e. thefirst three bullets). Section 2.3.2 presents the experimental methods for measurement, system iden-tification, and model verification, including the final bullet. Section 2.3.3 presents and discussesthe model verification results. Finally, concluding remarks are given in section 2.3.4.
The plane of the nozzle outlet and the solid substrate surface provide obvious boundaries for aControl Volume (CV) through which the total volume of fluid outside the nozzle,
V ( 𝑡 ) , and thetotal flow rate through the nozzle outlet, 𝑄 ( 𝑡 ) , may be analyzed and modeled. Similarly, to analyzedeposited droplet volume, V 𝑑 ( 𝑡 ) , as a dynamically evolving variable a droplet CV must be defined.This CV cannot be the same as the total fluid CV because only a fraction of the cumulative flowout of the nozzle up until the jet breaks is deposited on the substrate. The remainder of the fluidis retracted back into the nozzle under the power of surface tension after the jet breaks and thenozzle-connected fluid body and substrate-connected fluid body become disjoint.This work introduces a CV with an upper boundary at the 𝑧 -coordinate ℎ 𝑏 , where the jet ulti-mately pinches closed and breaks into two disjoint fluid bodies, as shown in Figure 2.10. This CVallows measurements of droplet volume to be made as time series data while remaining consistentwith prior notions of droplet volume in that after the jet breaks the volume of fluid in the dropletCV remains constant (assuming negligible evaporation).To avoid dramatic increases in complexity, this work’s model does not explicitly use ℎ 𝑏 . The jetbreak point is only used to facilitate defining droplet volume as a time series signal and for extract-ing time series measurements of droplet volume from microscope videos. Theoretical derivation38f the jet break point’s position is thus beyond the scope of this work, and it is estimated indepen-dently for each material ejection as described in Section 2.3.2.3. Ultimately, this work’s model is given as the cascading of two discrete-time state-space systemswith state transition formulas of the form 𝑓 : 𝑋 × 𝑈 × 𝑇 → 𝑋 where 𝑋 is the state vector space, 𝑈 is the input vector space, and 𝑇 is the time vector, all over the field R . These two systems are theinput-to-nozzle-flow-rate model (cid:34) 𝑄 ( 𝑡 + 𝑇 𝑠 ) 𝑄 ( 𝑡 + 𝑇 𝑠 ) (cid:35) = 𝑓 𝑄 (cid:32) (cid:34) 𝑄 ( 𝑡 ) 𝑄 ( 𝑡 + 𝑇 𝑠 ) (cid:35) , 𝑉 ( 𝑡 ) − 𝑉 𝑙 , 𝑡 (cid:33) , (2.49)and the nozzle-flow-rate-to-droplet-volume model (cid:34) V ( 𝑡 + 𝑇 𝑠 )V 𝑑 ( 𝑡 + 𝑇 𝑠 ) (cid:35) = 𝑓 V 𝑑 (cid:32) (cid:34) V ( 𝑡 )V 𝑑 ( 𝑡 ) (cid:35) , (cid:34) 𝑄 ( 𝑡 ) 𝑄 ( 𝑡 + 𝑇 𝑠 ) (cid:35) , 𝑡 (cid:33) . (2.50) 𝑇 𝑠 is the sample period in seconds, 𝑡 is the time from the rising edge of the voltage pulse in seconds,and 𝑉 ( 𝑡 ) is the applied voltage signal.Both 𝑓 𝑄 and 𝑓 V 𝑑 are piecewise defined to capture switching between partial process dynamicsand to capture state resets—functions that execute upon certain switches and alter the dynamicalstates before the first evaluation of the newly active partial process dynamics.In Section 2.2 the division of the material ejection process into partial processes was doneto maximize the use of physics-driven first principles model components, and was based on thestretching of the meniscus beyond its maximum stable non-jetting equilibrium extension.This work presents an alternate breakdown of the material ejection process designed to facil-itate modeling of the deposited droplet volume. This breakdown revolves around whether or notthere exists a contiguous fluid stream between the nozzle and substrate, as the cessation of thiscontiguity is synonymous with the cessation of flow into or out of the droplet CV. Specifically,the complete process is broken into an “approach” stage, a “contiguity” stage, and a “retraction”stage. To better focus on the mapping between 𝑄 and V 𝑑 , instead of modeling the meniscus tipposition dynamics this work assumes that the timing of jet impingement and breaking are deter-mined solely by the pulse parameters 𝑉 𝑙 , 𝑉 ℎ , and 𝑇 𝑝 . Switching is thus governed by time: withthe rising edge of the voltage pulse set to 𝑡 = , transition from approach to contiguity occurswhen 𝑡 exceeds 𝑡 𝑐 ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) and transition from contiguity to retraction occurs when 𝑡 exceeds 𝑡 𝑟 ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) , where 𝑡 𝑐 and 𝑡 𝑟 are identified from data for each set of pulse parameters as describedin Section 2.3.2.3. 39 pproach Contiguity Retraction Figure 2.11: System Architecture.
Top:
Automaton illustrating the timed switching behavior of the systemmodel and the reset determining the initial contiguity droplet volume. Each automaton location is accom-panied by photographs of the first and final samples of the corresponding partial process from a video with 𝑉 ℎ = and 𝑇 𝑝 = . . Bottom:
Block diagram illustrating breakdown of a complete input-to-droplet-volume model into a nozzle flow rate model 𝑓 𝑄 and a droplet volume model 𝑓 V 𝑑 , both of which arepiecewise defined to capture the switching and reset behavior of the automaton. The lone reset in the system is applied to V 𝑑 upon the switch from approach to contiguity. Thecomplete model architecture is thus visualized by Figure 2.11. If the shape of the fluid-air interface were constant over time and the fluid incompressible, thevolumetric flow rate out of the nozzle, 𝑄 , would be equal to the volumetric flow rate into the dropletcontrol volume, 𝑄 𝑑 . However, observation of video data indicates that the interface broadensslowly but steadily while voltage is high during contiguity. This implies that 𝑄 𝑑 is only somefraction of 𝑄 . The video observation is corroborated by observation of the extracted time seriesdata, such as that shown in Figure 2.12. This motivates a simple proportional model between 𝑄 and 𝑄 𝑑 during contiguity: 𝑄 𝑑 ( 𝑡 ) = 𝑏 𝑄 𝑑 𝑄 ( 𝑡 ) (2.51)where 𝑏 𝑄 𝑑 is a constant.Trapezoidal integration of 𝑄 𝑑 ( 𝑡 ) in equation (2.51) yields the first-order discrete-time dropletvolume model for the contiguity stage V 𝑑 ( 𝑡 + 𝑇 𝑠 ) = V 𝑑 ( 𝑡 ) + 𝑇 𝑠 𝑏 𝑄 𝑑 𝑄 ( 𝑡 + 𝑇 𝑠 ) + 𝑇 𝑠 𝑏 𝑄 𝑑 𝑄 ( 𝑡 ) (2.52)where V 𝑑 is the droplet volume and 𝑇 𝑠 is the sample period.40 ozzle OutletFluidRadius Fluid-AirInterfaceDroplet CV BoundAir Figure 2.12: Measured data from a particular ejection video illustrating the motivation for a proportional 𝑄 𝑑 model. Left:
Half-outline of fluid body (the jet is roughly symmetric about 𝑧 -axis) at the onset of contiguityand at the end of the voltage pulse, representing a increase in volume outside the droplet CV. Right:
Flow rate time series data during contiguity illustrating roughly proportional signals between 𝑄 and 𝑄 𝑑 . Volume is also added to the droplet CV during the approach stage when the tip of the meniscuscrosses the upper boundary of the droplet CV but has not yet struck the substrate. However, eachejection video only provides a few samples of this situation, as the jet traverses the distance fromthe jet break position to the substrate relatively quickly. It may thus be impractical to identify adynamical model of volume increase during the approach stage. Instead, this work sets V 𝑑 ( 𝑡 + 𝑇 𝑠 ) to V 𝑑 ( 𝑡 ) during approach, and uses a reset to give V 𝑑 an initial condition in the contiguity stage,which accounts for the fluid added to the droplet CV during approach.The total volume of the fluid outside the nozzle, V , at the moment of jet collision with thesubstrate, 𝑡 𝑐 , may be roughly modeled as the volume of the Boolean union of a cylinder and acone arranged to approximate the fluid body shape. V 𝑑 is then some fraction of the cylindervolume. This is equivalent to some fraction, 𝜓 , of the total volume minus the Boolean differenceof the cone and the cylinder, 𝜓 . Physics-driven modeling of the jet diameter and break positionnecessary to explicitly calculate 𝜓 and 𝜓 are beyond the scope of this work, but the structure ofthe mapping between V and V 𝑑 at 𝑡 = 𝑡 𝑐 arising from this geometric analysis may still be used: V + 𝑑 = V − 𝑑 + 𝜓 V − + 𝜓 (2.53)where 𝜓 and 𝜓 require data-driven identification and the subscripts + and − indicate a state’svalue before and after reset. V − 𝑑 will be 0 unless there was already fluid in the droplet controlvolume (e.g. if a second pulse is fired over an existing droplet).Equation (2.53) requires total volume V be captured by the state dynamics, which can be41one with a trapezoidal integration of the input 𝑄 similar to that of equation (2.51). This additioncompletes the hybrid model of droplet volume evolution in terms of total flow rate input, whichcan be given in totality as (cid:34) V ( 𝑡 + 𝑇 𝑠 )V 𝑑 ( 𝑡 + 𝑇 𝑠 ) (cid:35) = 𝑓 V 𝑑 (cid:32) (cid:34) V ( 𝑡 )V 𝑑 ( 𝑡 ) (cid:35) , (cid:34) 𝑄 ( 𝑡 ) 𝑄 ( 𝑡 + 𝑇 𝑠 ) (cid:35) , 𝑡 (cid:33) = V ( 𝑡 ) + 𝒬 V 𝑑 ( 𝑡 ) 𝑡 < 𝑡 𝑐 ∨ 𝑡 ≥ 𝑡 𝑟 V ( 𝑡 ) + 𝒬 𝜓 V ( 𝑡 ) + V 𝑑 ( 𝑡 ) + 𝑏 𝑄 𝑑 𝒬 + 𝜓 𝑡 𝑐 ≤ 𝑡 < 𝑡 𝑐 + 𝑇 𝑠 V ( 𝑡 ) + 𝒬 V 𝑑 ( 𝑡 ) + 𝑏 𝑄 𝑑 𝒬 𝑡 𝑐 + 𝑇 𝑠 ≤ 𝑡 < 𝑡 𝑟 (2.54)where 𝒬 = 𝑇 𝑠 ( 𝑄 ( 𝑡 + 𝑇 𝑠 ) + 𝑄 ( 𝑡 )) (2.55)and 𝑡 < 𝑡 𝑐 corresponds to the approach stage, 𝑡 ≥ 𝑡 𝑟 corresponds to the retraction stage, 𝑡 𝑐 + 𝑇 𝑠 ≤ 𝑡 <𝑡 𝑟 corresponds to all but the first time step of the contiguity stage, and 𝑡 𝑐 ≤ 𝑡 < 𝑡 𝑐 + 𝑇 𝑠 correspondsto the first time step of contiguity, in which the reset is applied. The main focus of this chapter is the development and validation of the mapping between nozzleflow rate 𝑄 and deposited droplet volume V 𝑑 . This could be done by simply injecting measured 𝑄 data into equation (2.54) and assessing the generated V 𝑑 signals against measured droplet volumes.However, for control there must ultimately be a model with input based on applied voltage 𝑉 ( 𝑡 ) rather than 𝑄 ( 𝑡 ) . To demonstrate the viability of equation (2.54) for this purpose, this sectionpresents a simple 𝑉 -to- 𝑄 model based on Section 2.2, which is cascaded with the V 𝑑 model.The jetting model in Section 2.2 is a second-order LTI system, which may be represented indiscrete time as 𝑄 ( 𝑡 + 𝑇 𝑠 ) = 𝑎 𝑄, ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) 𝑄 ( 𝑡 + 𝑇 𝑠 ) + 𝑎 𝑄, ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) 𝑄 ( 𝑡 ) + 𝑏 𝑄, ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) 𝑢 ( 𝑡 ) (2.56)where the input 𝑢 ( 𝑡 ) is given as 𝑢 ( 𝑡 ) = 𝑉 ( 𝑡 ) − 𝑉 𝑙 (2.57)This choice of input is made because the physics-based first principles models of flow rate are42riven by the applied voltage squared, and because at the low voltage stable equilibrium, 𝑄 shouldbe zero. The model parameters 𝑎 𝑄, , 𝑎 𝑄, , and 𝑏 𝑄, are identified independently for each pulsedefinition in Section 2.2.Section 2.2 uses equation (2.56) when the nonlinear physics first principles models cease tocapture the observed dynamics. This happens during contiguity and in approach and retractionwhen the meniscus is sufficiently elongated. Because the nonlinear models cannot capture theentirety of approach or contiguity, incorporating them into this work’s deposited-volume-focusedswitching framework would substantially complicate the model. Thus to preserve the model’sfocus and manage complexity while still accounting for changes in dynamical behavior over thecourse of ejection, the structure of equation (2.56) is applied to the entire model with separateparameters identified for contiguity and non-contiguity partial processes. This results in the model (cid:34) 𝑄 ( 𝑡 + 𝑇 𝑠 ) 𝑄 ( 𝑡 + 𝑇 𝑠 ) (cid:35) = 𝑓 𝑄 (cid:32) (cid:34) 𝑄 ( 𝑡 ) 𝑄 ( 𝑡 + 𝑇 𝑠 ) (cid:35) , 𝑢 ( 𝑡 ) , 𝑡 (cid:33) = 𝑎 𝑄, ˜ 𝑎 𝑄, 𝑄 ( 𝑡 ) 𝑄 ( 𝑡 + 𝑇 𝑠 ) + 𝑏 𝑄, 𝑢 ( 𝑡 ) 𝑡 < 𝑡 𝑐 ∨ 𝑡 ≥ 𝑡 𝑟 𝑎 𝑄, 𝑎 𝑄, 𝑄 ( 𝑡 ) 𝑄 ( 𝑡 + 𝑇 𝑠 ) + 𝑏 𝑄, 𝑢 ( 𝑡 ) 𝑡 𝑐 ≤ 𝑡 < 𝑡 𝑟 , (2.58)where the tilde-topped and overlined parameters are separately identified (and have the input argu-ments ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) dropped for compactness) and correspond to the approach and retraction stagesand the contiguity stage, respectively. The e-jet printing setup is identical to that of Section 2.2.2.1.Twelve sets of ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) parameters, referred to as “experiments,” are tested, with 20 trialsof each experiment being recorded. All trials begin from the low voltage equilibrium meniscusposition over a clean region of substrate with no prior fluid depositions: 𝑉 ( ) = 𝑉 𝑙 , 𝑄 ( ) = 𝑄 ( 𝑇 𝑠 ) = , V 𝑑 ( ) = , and V ( ) is the small total fluid volume outside the nozzle at low voltageequilibrium (see Figure 2.1, ). Time 𝑡 = is defined at the rising edge of the voltage pulse. 𝑉 𝑙 =
525 V for all experiments. 𝑉 ℎ and 𝑇 𝑝 values are tabulated in Table 2.5.43able 2.5: Experimental High Voltage and Pulse Width Pairs 𝑉 ℎ [V] 𝑇 𝑝 [ ms ]1100 2.01150 2.01200 2.01250 2.0 𝑉 ℎ [V] 𝑇 𝑝 [ ms ]1300 1.51300 1.81300 2.01300 2.3 𝑉 ℎ [V] 𝑇 𝑝 [ ms ]1350 2.01370 2.01420 1.51470 1.5 Each frame of video is a grayscale image containing the nozzle tip, the fluid outside the nozzle,and—if there is fluid near enough to the substrate—a reflection of the fluid off of the substrate.The image processing protocol used to extract time series measurements of V and V 𝑑 from theseimages is nearly the same as that of [22]. Edge finding identifies the ( 𝑥, 𝑧 ) coordinates for thesilhouette of the nozzle tip, fluid, and reflection. Corner finding and extremum finding identify the 𝑧 -coordinates of the nozzle-fluid interface and the substrate-fluid interface. Finally, the width ofthe silhouette at each 𝑧 -coordinate is treated as the diameter of a disk of height equal to the imageresolution (i.e. the height of one pixel) for volume determination. 𝑄 and 𝑄 𝑑 measurements arenumerical derivatives of V and V 𝑑 measurements. On top of this established procedure, this workintroduces a method to harness the reflection for improving volume measurement consistency nearthe substrate and a method for estimating jet break position, both of which are illustrated in Figure2.13 and explained in detail below.In previous works, the reflection was eliminated from volume calculations entirely. However,due to the quantization error associated with fixing the substrate position measurement to a pixeledge (and possibly other optical or image processing imperfections), this direct calculation leads tononzero flow of fluid through the bottom of the droplet CV as the droplet spreads. This change indroplet volume measurement during the retraction stage makes identifying a final droplet volumevalue difficult. Thus this work introduces reflection-augmented volume measurement to maintainconservation of volume near the substrate. Given images that extend Δ 𝑝 𝑟 pixels below the esti-mated substrate position, a region Δ 𝑝 𝑟 pixels tall centered on the estimated substrate position isdefined as the “reflection-augmented measurement region.” The volume of fluid contained in theupper half of this region (i.e. the portion of the region containing direct fluid silhouette rather thanreflection) is taken as half the volume computed in the total reflection-augmented measurementregion. In other words, the true fluid volume is taken as the average of the silhouette volume andreflection volume.The jet break position for each trial is measured from the first video frame in which there aretwo disjoint fluid bodies (i.e. the first frame of retraction). The jet break position is estimated to be44 otal Control VolumeDroplet Control VolumeDirect Measurement RegionReflection-AugmentedJet Break Position Measurement RegionIdentified Fluid Silhouette Figure 2.13: Depiction of the two CVs described in Section 2.3.1.1, the distinction between the direct andreflection-augmented measurement regions, and the estimated jet break position superposed on the finalcontiguity frame and first retraction frame for a particular trial ( 𝑉 ℎ = , 𝑇 𝑝 = . ). Δ 𝑧 is thedistance from either fluid body tip to the jet break position. the midpoint between the tips of these bodies based on the assumption that the initial droplet andmeniscus tip velocities and accelerations are equal and opposite at the moment the jet breaks. Thisassumption is driven by the fact that the initial jet-breaking and retraction behavior is dominatedby surface tension. While this assumption neglects much of the complexity of the true retractionphysics, Figure 2.13 suggests it does an acceptable job of identifying the thinnest portion of the jetimmediately before breaking, and it circumvents image resolution and noise issues associated withdirectly computing the thinnest jet point in the final frame of contiguity. The parameters to be identified can be grouped into two categories. First are the primary modelparameters making up the dynamics of the 𝑄 -to- V 𝑑 model (2.54): 𝑏 𝑄 𝑑 , 𝜓 , and 𝜓 , which areconstant over all experiments. Second are the supporting model parameters determining the tim-ing of switching in the volume model (2.54) and the simulated nozzle flow rate model (2.58): 𝑡 𝑐 ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) , 𝑡 𝑟 ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) , ˜ 𝑎 𝑄, ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) , ˜ 𝑎 𝑄, ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) , ˜ 𝑏 𝑄, ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) , 𝑎 𝑄, ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) , 𝑎 𝑄, ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) , and 𝑏 𝑄, ( 𝑉 𝑙 , 𝑉 ℎ , 𝑇 𝑝 ) , whose values vary with the the experiment parameters. Themeasured data is divided evenly into training and validation data. For each experiment, 10 trialsare reserved for training and 10 for validation.In order to keep the timing of switching fixed to a particular sample, for each experiment 𝑡 𝑐 and45 𝑟 are taken as the median time of the first frame of contiguity and retraction over the 10 trainingtrials of that experiment (1 parameter from 10 samples). All other parameters are identified vialeast squares regression. The model coefficients for equation (2.58) are trained independently foreach experiment on all the available data in the corresponding process stage (3 parameters from353 samples for approach and 194 samples for contiguity, on average). The reset parameters 𝜓 and 𝜓 are trained on the set of first frames of contiguity (i.e. the 𝑡 𝑐 ≤ 𝑡 < 𝑡 𝑐 + 𝑇 𝑠 sample) fromall trials (2 parameters from 120 samples). Finally, 𝑏 𝑄 𝑑 is trained on all contiguity training data (1parameter from 2337 samples). This section presents metrics for validating the reflection-augmented volume measurement tech-nique, and the ability of the total model to predict deposited droplet volume.The improvement yielded by reflection-augmented volume measurement over direct measure-ment is quantified as the mean percent decrease in total variation of the filtered retraction-stage V 𝑑 timeseries between the two techniques. The total variation 𝐿 of a time-varying parameter is its totalchange (as opposed to net change) over a given period of time 𝑇 𝐿 . In theory, the total variation of V 𝑑 in the retraction stage is zero, making its reduction a practical improvement metric. However,high frequency measurement noise also contributes to 𝐿 . To moderate noise’s influence, the V 𝑑 signal is filtered before its total variation is computed. Thus, the total variation of a trial 𝑗 is givenby 𝐿 𝑗 = 𝑡 𝑟 + 𝑇 𝐿 ∑︁ 𝑡 = 𝑡 𝑟 |V 𝑓𝑑 ( 𝑡 + 𝑇 𝑠 ) − V 𝑓𝑑 ( 𝑡 ) | (2.59)where V 𝑓𝑑 is the filtered volume signal. The final metric for reflection-augmented volume mea-surement performance is Δ 𝐿 % =
100 mean 𝑗 ∈ All Validation Trials 𝐿 direct 𝑗 − 𝐿 augmented 𝑗 𝐿 direct 𝑗 (2.60). Here, a Savitzky-Golay filter with a window size of 15 samples is used. 𝑇 𝐿 =
10 ms (200samples), roughly the time it takes for the droplet to spread and settle to its final shape on thesubstrate.The efficacy of the overall model in predicting deposited droplet volume is measured by themean unsigned error between the modeled V 𝑑 ( 𝑡 𝑟 ) (equal to V 𝑑 ( 𝑡 > 𝑡 𝑟 ) ) and the measured final46roplet volume. Measured final droplet volume is taken as V final 𝑑, 𝑗 = mean 𝑡 ∈[ 𝑡 𝑟 ,𝑡 𝑟 +
10 ms ] V meas 𝑑, 𝑗 ( 𝑡 ) (2.61)where V meas 𝑑, 𝑗 ( 𝑡 ) is the measured droplet volume time series for a particular trial 𝑗 , making themean unsigned error 𝑒 V 𝑑 = mean 𝑗 ∈ 𝐽 |V final 𝑑, 𝑗 − V 𝑑, 𝑗 ( 𝑡 𝑟 ) | (2.62)This metric is evaluated over multiple sets of trials 𝐽 . In addition to an aggregate 𝑒 V 𝑑 in which 𝐽 contains the validation trials of all but one experiment (that of lowest 𝑉 ℎ , see Section 2.3.3.2 fordiscussion of this exclusion), individual 𝑒 V 𝑑 values are computed for each experiment. This is doneto examine how model performance changes with the pulse parameters. Additionally, for each ofthese sets 𝐽 , both an 𝑒 V 𝑑 using V 𝑑 ( 𝑡 𝑟 ) generated from injecting measured nozzle flow rate intoequation (2.54) and an 𝑒 V 𝑑 using V 𝑑 ( 𝑡 𝑟 ) generated from nozzle flow rate simulated by equation(2.58) are computed. This is done to enable both focus on the quality of equation (2.54) and broaderconsideration of the ultimate needs for a control-oriented e-jet printing model, respectively. Finally,along with each 𝑒 V 𝑑 , a corresponding standard deviation of the signed error is presented. Equations (2.59) and (2.60) show that the reflection-augmented image processing yields a 42%decrease in total variation of measured droplet volume time series data compared to direct mea-surement, with an associated standard deviation of 10%. This substantial performance improve-ment can be visualized through the example retraction-stage droplet volume time series in Figure2.14, in which the direct measurement yields a steady decrease while the reflection-augmentedmeasurement yields a relatively constant droplet volume.However, in both measurement schemes there is a steep transient at the start of retraction. Thisarises from an inability of these measurement techniques to conserve volume over the collapse ofthe relatively tall and thin droplet tail (observable in Figure 2.13, right) into the larger and widermain droplet body.Thus, these results demonstrate that the reflection-augmented volume measurement scheme isan effective tool that may be useful for future e-jet printing research, but does not address ev-ery artifact associated with video-based measurement, which may serve as the subject of futureinvestigations. 47 igure 2.14: Retraction-stage droplet volume measurement taken by direct and reflection-augmented mea-surement techniques. Light and dark points represent raw and filtered data, respectively. As V 𝑑 is expectedto be constant during retraction, this plot illustrates the reflection-augmented technique’s superiority in thatit maintains a roughly constant value of . after the transient (i.e. for 𝑡 ≥ . ), while the directmeasurement steadily decreases until about 𝑡 = . Figure 2.15 presents the mean percent error in the final droplet volume as computed by equation(2.62), for each experiment. The experiment of lowest high voltage ( 𝑉 ℎ = ) clearly repre-sents an outlier in this data, having a percent error of 130% for the predictions driven by measurednozzle flow rate and 170% for the predictions using simulated nozzle flow rate, more than triplethe next highest percent error.To better discuss this outlying experiment, time series plots of V 𝑑 ( 𝑡 ) and a plot of each trial’sestimated jet break position, ℎ 𝑏 , versus 𝑉 ℎ are presented in Figures 2.16 and 2.17, respectively.From the time series plot, one observes that the reset—the initial step change from zero to non-zero volume—is the most clearly erroneous feature of the low 𝑉 ℎ time series. The reset causes alarge overestimation of the initial volume in the contiguity stage that cannot be compensated forby the contiguity dynamics models, which only capture the change in droplet volume from thebeginning to the end of contiguity.The plot of ℎ 𝑏 in Figure 2.17 lends insight into why this reset error may arise. While the exper-iments well within the subcritical jetting regime (those from to ) show comparable ℎ 𝑏 values, the experiment of lowest 𝑉 ℎ shows a jet break position markedly closer to the sub-strate. Because ℎ 𝑏 marks the upper boundary of the droplet control volume (a condition necessaryfor droplet flow rate to be zero after the jet breaks), this lowered jet break position substantiallyreduces the fraction of total volume that is in the droplet control volume at the first moment of con-tiguity. This change in the fraction of total volume is not accounted for by the reset model (2.53),which assumes only the total volume itself is changing (e.g. because of jet diameter variations over48 roplet Volume Model Driven by Measured Nozzle Flow Rate0%20%40%60%80%100% High Voltage [V] Pulse Width [ms] M ea n U n s i gn e d P e r ce n t E rr o r [ % ] Standard DeviationN=10 per Experiment High Voltage [V] Pulse Width [ms] M ea n U n s i gn e d P e r ce n t E rr o r [ % ] Droplet Volume Model Driven by Simulated Nozzle Flow Rate
Figure 2.15: Percent error in final deposited droplet volume, V 𝑑 ( 𝑡 𝑟 ) , using measured and simulated 𝑄 . Thelowest voltage experiment exceeds 100% error in both cases. Each bar represents the mean value of 𝑁 = samples. Measured 𝑄 results illustrate the high quality of equation (2.54) for all but the lowest voltage case.Simulated 𝑄 results illustrate increased error associated with increased uncertainty in the cascaded model,motivating future flow rate modeling work. igure 2.16: Time series plots of droplet volume V 𝑑 for a representative experiment and the experiment oflowest high voltage 𝑉 ℎ . Plotted measured data is the mean of the validation data ( 𝑁 = samples for eachtime series) with an envelope of plus or minus the standard deviation. The data suggests that the reset is themain source of error in low 𝑉 ℎ experiments. applied voltage).Because the given model structure does not account for the changing jet break position nearthe boundaries of the subcritical jetting regime, the low 𝑉 ℎ experiment is deemed to be outside theapplicable domain of the model, and is thus removed from the aggregate model error data, given inFigure 2.18. The lower error yielded when the droplet volume model is driven by measured nozzleflow rate illustrates the validity of the 𝑄 -to- V 𝑑 model (2.54). When equation (2.54) is driven bythe nozzle flow rate simulated by equation (2.58), making a complete model from 𝑉 to V 𝑑 , theerror increases. This is due to increased model uncertainty associated with equation (2.58) andits cascading with equation (2.54). While reducing this model uncertainty will be an importantfuture endeavor, these results demonstrate the foundation of a dynamical 𝑉 -to- V 𝑑 model that maybe integrated with iterative learning control for the sake of e-jet printing control. This section presents a hybrid system model framework for dynamical droplet volume modeling ine-jet printing based on contiguity of the fluid jet between the nozzle and the substrate. This over-arching modeling framework involves the contributions of several novel model elements whose50
100 1150 1200 1250 1300 1350 1400 1450 1500
High Voltage [V] -120-100-80-60 J e t B r ea k P o s iti on [ µ m ] Validation Trial Data PointsMean & Standard Deviation
Figure 2.17: Jet break positions of each experiment’s validation data against high voltage. The nozzle outletis located at µ m and the substrate at − µ m . 𝑁 = samples for all high voltages except 𝑉 ℎ = ,for which 𝑁 = samples because four pulse widths are tested at 𝑉 ℎ = . The modest spread of datapoints at 𝑉 ℎ = suggests that high voltage (equivalent to the difference between high and low voltagein this data set) has a greater influence on jet break position than pulse width in the subcritical jetting regime. Driven by Measured Q Driven by Simulated Q M ea n U n s i gn e d P e r ce n t E rr o r [ % ] Aggregate Percent Error in Final Deposited Droplet VolumeN = 110
Figure 2.18: Mean unsigned percent error and standard deviation (given by error bars) of final dropletvolume over all validation trials except those of lowest high voltage ( 𝑁 = samples). This chapter has contributed two hybrid models for e-jet printing: one focused on fidelity to thefirst principles of the physical system and the other on capturing the final volume of the droplet onthe substrate.Specifically, the former contributes new geometrical and equilibrium analysis, extending theamount of the ejection process that can be modeled by first principles and reducing model relianceon measured data. Additionally, Section 2.2 introduces and validates hybrid modeling as a meansto merge physics-driven modeling with data-driven modeling to produce an ODE-based model ofthe end-to-end electrohydrodynamic ejection process.The latter hybrid model of Section 2.3 introduces an alternative division of the total ejectionprocess into partial process and defines a new CV for droplet volume. These contributions en-able dynamical modeling of the droplet volume ultimately deposited on the substrate via PWAframework.The bigger picture delivered by this chapter is that e-jet printing exemplifies a class of physicalsystems for which hybrid modeling is thus far the only path to ODE-based modeling, and thus tocontrol-oriented modeling. E-jet printing also represents a class of systems for which ILC is nec-essary to improve performance because of the impracticality of real-time feedback. Thus, while it52s certainly not the only motivation, e-jet printing provides a concrete motivation for the integra-tion of hybrid systems theory and ILC theory, upon which the subsequent chapters focus. Finally,because e-jet’s modeling and control challenges are shared by numerous other AM technologies,the validation of this modeling philosophy may lower the boundary for the development of similarmodels for these AM processes. Fused Deposition Modeling (FDM), perhaps the most ubiquitousAM technology, serves as a prime example. While formal hybrid modeling has not been attempted,[85] identifies several distinct regimes for the filament deposition dynamics. As these regimes arisedue to the physical state of the printhead and filament, these dynamics may be well-unified by ahybrid framework. 53
HAPTER 3
Enabling ILC of Hybrid Systems:Closed-Form Hybrid System Representation
In most cases, even when hybrid system control takes inspiration from preexisting controltheories, significant work must be done to redevelop the theory specifically for hybrid systems[45, 86, 87]. One reason for this difficulty is that the hybrid systems are, in general, mathemati-cally represented as a type of automaton [1, 3] rather than as a closed-form system of ODEs as istypical for control systems. This lack of a closed-form system representation prevents the direct ap-plication of many analytical mathematical operations that would ordinarily accompany dynamicalsystem analysis and controller design (e.g. function composition, Jacobian).The importance of performing such operations on piecewise functions, and thus the importanceof having closed form representations of piecewise functions, was originally identified by the non-linear circuit theory community well before the modern notion of hybrid systems were developed.In fact, in 1977 Chua and Kang published a canonical, closed-form representation of piece-wiselinear functions to close this gap [88]. However, their representation was not designed with dy-namical systems in mind, and thus has several features making it incompatible with hybrid sys-tems analysis and control. Most importantly, it is based on interpolation between the breakpointsof the piecewise functions, precluding it from representing nonlinear dynamics within a particulardiscrete state. Secondly, rather than representing each “piece” of a piecewise function indepen-dently, [88] treats each “piece” as a superposition of components from all “pieces” correspondingto lesser values of the independent variable. Finally, higher dimensional functions are representedby nesting functions of 1 or 2 dimensions, which results in structures far more convoluted than thestate-space models employed by today’s control engineers.
Content of this chapter also published as:I. A. Spiegel and K. Barton, “A Closed-Form Representation of Piecewise Defined Systems and their Integrationwith Iterative Learning Control,” in , Philadelphia, PA, USA, 2019, pp.2327-2333, https://doi.org/10.23919/ACC.2019.8814823 ©IEEE 2019. Reprinted with permission.
54n the more contemporary literature, Bemporad and Morari’s seminal Mixed Logical Dynam-ical (MLD) Systems seek to provide a more analytical hybrid system representation, which iscomposed entirely of systems of algebraic equalities and inequalities [89]. However, like the sys-tems of [88], MLD systems cannot contain nonlinear dynamics within a discrete state. Moreimportantly, while MLD systems integrate readily with control frameworks that involve online op-timization, such as Model Predictive Control (MPC), they can be difficult to integrate with otherclasses of controllers because they require the solution of a mixed integer program at each timestep to determine the system’s discrete state.This restriction on controller options is particularly problematic when real time feedback isunavailable, or when model errors are substantial enough to prevent the fulfillment of performancegoals. In such cases, ILC is attractive. However, to date no implementation of ILC has beenmade with hybrid system models. This is in part because the mathematical operations required tosynthesize the controller cannot be performed on contemporary hybrid system representations.The primary contribution of the present chapter is to deliver a closed-form representation of aparticular class of hybrid systems: PWD systems (i.e. a generalization of the popular PWA systemclass). This closed-form representation is shown to bridge the gap between system representationand control via the application of ILC to an example hybrid system. Minor contributions are madeto the selected ILC method (applicable to both hybrid and non-hybrid systems): it is reformalizedto enable application to systems of any relative degree, and a novel recommendation is made vis-`a-vis implementation in order to facilitate direct application of the control theory and to improvecontroller scalability. No special modifications are made for the hybrid nature of the examplesystem, thereby illustrating the utility of the closed-form representation.The rest of the chapter is organized as follows. Section 3.1 presents the closed-form piecewisedefined system representation, i.e. this work’s main contribution, via a proposition and constructiveproof. Section 3.2 details the iterative learning controller to be used. Section 3.3 presents theexample hybrid system, the methods of controller performance analysis, and presents and discussesthe results of the simulation experiments. Finally, Section 3.4 provides concluding remarks andsuggests future work.
Definition 3.1 (PWD System) . A discrete-time PWD system is a system defined by 𝑥 ( 𝑘 + ) = 𝑓 𝑞 ( 𝑥 ( 𝑘 ) , 𝑢 ( 𝑘 ) , 𝑘 ) 𝑦 ( 𝑘 ) = ℎ 𝑞 ( 𝑥 ( 𝑘 ) , 𝑢 ( 𝑘 ) , 𝑘 ) for (cid:34) 𝑥 ( 𝑘 ) 𝑢 ( 𝑘 ) (cid:35) ∈ 𝑄 𝑞 (3.1)55here 𝑘 is the discrete time index, 𝑥 ∈ R 𝑛 𝑥 is the state vector, 𝑢 ∈ R 𝑛 𝑢 is the control input, 𝑦 ∈ R 𝑛 𝑦 is the output vector, 𝑞 ∈ { , , · · · , | 𝑄 |} , 𝑄 𝑞 ∈ 𝑄 = { 𝑄 , 𝑄 , · · · , 𝑄 | 𝑄 | } is a convex polytope (i.e.an intersection of half-spaces) in R 𝑛 𝑥 + 𝑛 𝑢 , 𝑓 𝑞 : R 𝑛 𝑥 × R 𝑛 𝑢 × Z → R 𝑛 𝑥 is a closed-form functionrepresenting the potentially nonlinear state dynamics in 𝑄 𝑞 , and ℎ 𝑞 : R 𝑛 𝑥 × R 𝑛 𝑢 × Z → R 𝑛 𝑦 is thepotentially nonlinear closed-form output function in 𝑄 𝑞 . All polytopes in 𝑄 are disjoint, and theirunion is equal to R 𝑛 𝑥 + 𝑛 𝑢 .The following theorem and constructive proof constitute the primary contribution of this chap-ter. Theorem 3.1 (Closed-Form PWD System Representation) . Any PWD system (3.1) can be repre-sented in closed form as 𝑥 ( 𝑘 + ) = 𝑓 ( 𝑥 ( 𝑘 ) , 𝑢 ( 𝑘 ) , 𝑘 ) 𝑦 ( 𝑘 ) = ℎ ( 𝑥 ( 𝑘 ) , 𝑢 ( 𝑘 ) , 𝑘 ) (3.2) where 𝑓 and ℎ explicitly encapsulate both the component dynamics and switching behavior of thePWD system.Proof. Because 𝑄 𝑞 are convex polytopes, their boundaries are hyperplanes, which can be repre-sented via 𝑝 𝑇 (cid:34) 𝑥 ( 𝑘 ) 𝑢 ( 𝑘 ) (cid:35) = 𝑏 (3.3)where 𝑝 ∈ R 𝑛 𝑥 + 𝑛 𝑢 describes the orientation of the hyperplane and 𝑏 ∈ R is an offset. To capturethe partitioning of R 𝑛 𝑥 + 𝑛 𝑢 into | 𝑄 | regions by 𝑛 𝑃 hyperplanes in the system dynamics, this workintroduces the auxiliary logical state vector 𝛿 ( 𝑘 ) = 𝑓 𝛿 ( 𝑥 ( 𝑘 ) , 𝑢 ( 𝑘 )) = 𝐻 (cid:32) 𝑃 (cid:34) 𝑥 ( 𝑘 ) 𝑢 ( 𝑘 ) (cid:35) − 𝛽 (cid:33) (3.4) 𝑃 = (cid:104) 𝑝 𝑝 · · · 𝑝 𝑛 𝑃 (cid:105) 𝑇 𝛽 = (cid:104) 𝑏 𝑏 · · · 𝑏 𝑛 𝑃 (cid:105) 𝑇 where 𝐻 is the element-wise Heaviside step function with convention 𝐻 ( ) = , and the relation-ship between | 𝑄 | and 𝑛 𝑃 depends on the exact configuration of the hyperplanes. The 𝑖 th elementof 𝛿 indicates on which side of the 𝑖 th hyperplane the system vector (cid:104) 𝑥 ( 𝑘 ) 𝑇 , 𝑢 ( 𝑘 ) 𝑇 (cid:105) 𝑇 falls (withpoints on hyperplanes being included in the space corresponding to 𝑝 𝑇𝑖 (cid:104) 𝑥 ( 𝑘 ) 𝑇 , 𝑢 ( 𝑘 ) 𝑇 (cid:105) 𝑇 − 𝑏 𝑖 > , Some communities may not consider the Heaviside function to be closed-form. Many others do [90, 91, 92,93, 94, 95, 96]. This classification as closed-form is supported by its seamless integration into most symbolic mathsoftware, in which its derivative is well-defined. ( k + ) (cid:34) x ( k ) u ( k ) (cid:35) ... ... ... K | Q | f | Q | K × (product) × (product) f δ Σ PolytopeLocalization SelectorFunctionsBitBitVector f Figure 3.1: Block diagram of the closed-form PWD system representation see Remark 3.2). In this manner, each 𝑄 𝑞 uniquely corresponds to a particular value of 𝛿 , whichwill be denoted 𝛿 ∗ 𝑞 .As the system evolves 𝛿 will be compared to each 𝛿 ∗ 𝑞 via a set of selector functions given by 𝐾 𝑞 ( 𝛿 ( 𝑘 )) = (cid:107) 𝛿 ∗ 𝑞 − 𝛿 ( 𝑘 ) (cid:107) = 𝛿 ∗ 𝑞 = 𝛿 ( 𝑘 ) 𝛿 ∗ 𝑞 ≠ 𝛿 ( 𝑘 ) (3.5)Note that this equation is equivalent to the Kronecker delta of and (cid:13)(cid:13) 𝛿 ∗ 𝑞 − 𝛿 ( 𝑘 ) (cid:13)(cid:13) for any norm, butis left in the given zero exponential form for explicitness.With these selector functions, the original state dynamics and outputs can be represented bythe closed-form equations 𝑥 ( 𝑘 + ) = 𝑓 ( 𝑥 ( 𝑘 ) , 𝑢 ( 𝑘 ) , 𝑘 ) = | 𝑄 | ∑︁ 𝑞 = 𝑓 𝑞 ( 𝑥 ( 𝑘 ) , 𝑢 ( 𝑘 ) , 𝑘 ) 𝐾 𝑞 ( 𝑓 𝛿 ( 𝑥 ( 𝑘 ) , 𝑢 ( 𝑘 ))) 𝑦 ( 𝑘 ) = ℎ ( 𝑥 ( 𝑘 ) , 𝑢 ( 𝑘 )) = | 𝑄 | ∑︁ 𝑞 = ℎ 𝑞 ( 𝑥 ( 𝑘 ) , 𝑢 ( 𝑘 ) , 𝑘 ) 𝐾 𝑞 ( 𝑓 𝛿 ( 𝑥 ( 𝑘 ) , 𝑢 ( 𝑘 ))) (3.6)This completes the closed-form representation of (3.1). A block diagram of the representation’sstructure is given in Figure 3.1 (cid:3) Remark . This representation makes no assumptions regarding the continuity ofeither the state transition formula 𝑓 or the output function ℎ over the switching hyperplanes. Therepresentation may model both continuous and discontinuous hybrid systems. Remark . The use of the Heaviside function in equation(3.4) systematically includes each hyperplane in the polytopes lying to a particular side. In other57ords, a hyperplane serving as a boundary/face for two adjacent polytopes is included in onlyone of them, such that one polytope is closed on that face and the other polytope is open. Towhich side the hyperplane belongs can be chosen by manipulation of 𝑃 and 𝛽 . For example,( 𝑃 = [ , ] , 𝛽 = ) and ( 𝑃 = [ , − ] , 𝛽 = ) both define the horizontal axis in R . Under(3.4), the former definition yields the same value of 𝛿 ( 𝛿 = ) for points on the hyper plane and in top half of R , while the bottom half of R excluding the horizontal axis yields 𝛿 = . Thelatter definition yields 𝛿 = for the hyperplane and the bottom half of R , while the top half of R excluding the hyperplane yields 𝛿 = . In this manner, the hyperplane can be included in eitherthe top or bottom polytope, but not both. The inability to include the hyperplane in both polytopesis desirable because if the polytopes have a non-null intersection there is uncertainty regarding thestate dynamics in the intersection.In many cases this method of determining which polytope contains a face (i.e. which polytopeis closed) and which polytope is open at the same face is amply flexible, and ensures that there arean equal number of polytopes and 𝛿 ∗ values. However, if a system designer desires greater flexi-bility, the Heaviside function can be replaced with the signum function. In this case, each polytopeinterior and each face uniquely corresponds to a different 𝛿 , and the designer must assemble thecomplete polytope from an interior and the desired faces. For example, if 𝑄 𝑞 corresponds to theinterior and face associated with 𝛿 ∗ 𝑞, and 𝛿 ∗ 𝑞, , then 𝐾 𝑞 ( 𝛿 ) = (cid:13)(cid:13)(cid:13) 𝛿 ∗ 𝑞, − 𝛿 ( 𝑘 ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) 𝛿 ∗ 𝑞, − 𝛿 ( 𝑘 ) (cid:13)(cid:13)(cid:13) . Remark . Despite the con-vexity requirements of the R 𝑛 𝑥 + 𝑛 𝑢 partitioning, and consequential affine-in- 𝑥 requirement of theswitching behavior governed by equations (3.4) and (3.5), the representation (3.6) can encom-pass a wide range of complex system structures. There need not be a one-to-one correspondencebetween the quantity of discrete dynamic regimes and the quantity of polytopes in system (3.6).Indeed, while each polytope 𝑄 𝑞 must be convex, one may construct many interesting nonconvextopologies by stitching together adjacent polytopes 𝑄 𝑒 and 𝑄 𝑑 by setting 𝑓 𝑒 = 𝑓 𝑑 , ℎ 𝑒 = ℎ 𝑑 . This isillustrated in Figure 3.2. Additionally, nonlinear switching conditions such as sin ( 𝑥 𝑖 ) > (where 𝑥 𝑖 is the 𝑖 th element of 𝑥 ) can obviously be accommodated by simply making the nonlinear expres-sion of 𝑥 into a new state. This of course increases the dimension of the system, but it notablydoes not increase the dimension of the NILC problem, which depends solely on the outputs andquantity of samples in a trial timeseries. NILC, first introduced in terms of abstract Banach space operators by [34], is the use of theNewton-Raphson root-finding algorithm to derive a trial-varying learning matrix 𝐿 ℓ for the classi-58 igure 3.2: An R topology representable by (3.6) with 14 hyperplanes (dashed lines) having 1 convex ( 𝔞 )and 2 nonconvex ( 𝔟 and 𝔠 ) dynamic regimes. cal ILC law u ℓ + = u ℓ + 𝐿 ℓ e ℓ (3.7)where u is a control input timeseries vector, or “lifted” vector, e is a lifted error vector, and ℓ ∈{ , , · · · } is the trial index .NILC was first formalized for discrete-time state space systems by [36]. However, the treat-ment of NILC in [36] is limited in that it only considers time-invariant systems and is only appli-cable to systems with a relative degree, 𝜇 , of 1. For discrete-time systems, the relative degree of anoutput is the number of time steps that must transpire before the explicit representation of the out-put, in terms of inputs and initial conditions, contains any input [97]. In many cases this notion ofrelative degree is adequate for the synthesis of NILC from PWD systems. However, there are alsoPWD models for which alternative notions of relative degree are useful for managing issues relatedto the switching behavior of hybrid systems. A rigorous treatment of such issues and the definitionof an alternative relative degree is given in Chapter 5. For now, the present section presents anNILC framework generalized for time-varying discrete-time SISO systems of any relative degree ≥ . For all systems, the purpose of incorporating knowledge of the relative degree into NILC syn-thesis is to guarantee that 𝐿 ℓ is well-defined regardless of model relative degree. Specifically, 𝐿 ℓ being well-defined is contingent on 𝐿 ℓ being invertible, and any notion of relative degree yielding In the literature, 𝑗 or 𝑘 is usually used for the trial index. Here, ℓ is used for the trial index because 𝑖 and 𝑗 will be used for matrix element indexing, 𝑘 is used for the discrete time index, 𝑡 is avoided to prevent confusion withcontinuous time, and ℓ is the next letter in the alphabet and thus commonly used for indexing. ˆ 𝑥 ℓ ( 𝑘 + ) = ˆ 𝑓 ( ˆ 𝑥 ℓ ( 𝑘 ) , 𝑢 ℓ ( 𝑘 ) , 𝑘 ) (3.8a) ˆ 𝑦 ℓ ( 𝑘 ) = ˆ ℎ ( ˆ 𝑥 ℓ ( 𝑘 )) (3.8b)where ˆ 𝑥 ∈ R 𝑛 𝑥 , 𝑢 ∈ R , ˆ 𝑦 ∈ R , 𝑓 : R 𝑛 𝑥 × R × Z → R 𝑛 𝑥 , and ℎ : R 𝑛 𝑥 → R . Hats, ˆ , are usedto emphasize that (3.8) is an imperfect model of some true system, though it is assumed that thecontrol input and initial condition are perfectly known.Let the system describe a repetitive process with finite duration. This translates to the assump-tions:(A3.1) A trial must have a duration of 𝑁 time steps, 𝑘 ∈ { , , · · · , 𝑁 } .(A3.2) The initial conditions are trial invariant, ˆ 𝑥 ℓ ( ) = 𝑥 ∀ ℓ .(A3.3) The desired output values, 𝑟 ( 𝑘 ) ∈ R , are trial invariant.By (A3.1), the input and output trajectories can be represented as timeseries vectors of the samelength for every trial: ˆ y ℓ = (cid:104) ˆ 𝑦 ℓ ( 𝜇 ) ˆ 𝑦 ℓ ( 𝜇 + ) · · · ˆ 𝑦 ℓ ( 𝑁 ) (cid:105) 𝑇 ∈ R 𝑁 − 𝜇 + (3.9) u ℓ = (cid:104) 𝑢 ℓ ( ) 𝑢 ℓ ( ) · · · 𝑢 ℓ ( 𝑁 − 𝜇 ) (cid:105) 𝑇 ∈ R 𝑁 − 𝜇 + (3.10)Note that ˆ y ℓ is time-shifted forward by the relative degree 𝜇 , which is a function of the specificdefinitions of 𝑓 and ℎ .By (A3.2) and (3.8), if 𝑥 is known then ˆ y ℓ is entirely a function u ℓ given by ˆ g : R 𝑁 − 𝜇 + → R 𝑁 − 𝜇 + ˆ y 𝑖ℓ = ˆ g 𝑖 ( u ℓ ) = ˆ 𝑦 ℓ ( 𝜇 + 𝑖 − ) (3.11a) ˆ 𝑦 ℓ ( 𝑘 ) = ˆ ℎ (cid:16) ˆ 𝑓 ( 𝑘 − ) ( u ℓ ) (cid:17) 𝑘 ∈ { 𝜇, 𝜇 + , · · · , 𝑁 } (3.11b)where the non-parenthetical superscript 𝑖 denotes the 𝑖 th element of a vector, indexing from 1, andthe parenthetical superscript ( 𝑘 ) denotes function composition of the form ˆ 𝑓 ( 𝑘 ) ( u ℓ ) = ˆ 𝑓 ( ˆ 𝑥 ℓ ( 𝑘 ) , 𝑢 ℓ ( 𝑘 ) , 𝑘 ) = ˆ 𝑓 (cid:16) ˆ 𝑓 (cid:16) · · · , 𝑢 ℓ ( 𝑘 − ) , 𝑘 − (cid:17) , 𝑢 ℓ ( 𝑘 ) , 𝑘 (cid:17) (3.12)60he recursion of the state dynamics 𝑓 expressed by (3.12) has a terminal condition of ˆ 𝑓 ( ˆ 𝑥 ℓ ( ) , 𝑢 ℓ ( ) , ) . Because ˆ 𝑥 ℓ ( ) = 𝑥 is known in advance and the time argument is determinedby the element index of the lifted representation, ˆ y ℓ is a function of only u ℓ . Note that because thefirst element of ˆ y ℓ is ˆ 𝑦 ℓ ( 𝜇 ) it explicitly depends on u = 𝑢 ℓ ( ) .Similarly, by (A3.3), the reference r = (cid:104) 𝑟 ( 𝜇 ) 𝑟 ( 𝜇 + ) · · · 𝑟 ( 𝑁 ) (cid:105) 𝑇 ∈ R 𝑁 − 𝜇 + (3.13)is fixed, causing the output error timeseries, e ℓ to be approximable by a function of only u ℓ . Thisrelationship is given by e ℓ ( u ℓ ) = r − y ℓ = r − g ( u ℓ ) ≈ r − ˆ g ( u ℓ ) (3.14)where g : R 𝑁 − 𝜇 + → R 𝑁 − 𝜇 + represents the unknowable dynamics of the true system, and y ℓ is themeasured value of this true system output. Newton’s method can thus be applied to iteratively findan argument u ℓ bringing e ℓ ( u ℓ ) toward 0. This naturally yields a control law of the form (3.7) with 𝐿 ℓ = (cid:18) 𝜕 ˆ g 𝜕 u ( u ℓ ) (cid:19) − (3.15)where 𝜕 ˆ g 𝜕 u is the Jacobian (in numerator layout) of ˆ g with respect to u as a function of u .However, (3.7), (3.15) is not strictly Newton’s method applied to e ℓ because ˆ g ( u ℓ ) is, like allmodels, only an approximation of the true physical dynamics g ( u ℓ ) . Because of this, convergenceof y ℓ to r is linear (rather than quadratic) and guaranteed if the following sufficient conditions aresatisfied [34].(A3.4) The true dynamics g ( u ℓ ) are continuously differentiable and their Jacobian 𝜕 g 𝜕 u is Lipschitzcontinuous with respect to u within some ball around the solution trajectory u soln .(A3.5) The inverse of the model Jacobian (i.e. the learning matrix) always has a bounded norm: (cid:107) 𝐿 ℓ (cid:107) < 𝜀 ∈ R > ∀ ℓ (A3.6) The learning matrix is sufficiently similar to the inverse of the true lifted system Jacobian: (cid:13)(cid:13)(cid:13) 𝐼 − 𝐿 ℓ 𝜕 g 𝜕 u ( u ℓ ) (cid:13)(cid:13)(cid:13) < ∀ ℓ (A3.7) The initial guess u is in some basin of attraction around the solution u soln (guaranteed toexist by (A3.4)-(A3.6)), (cid:107) u − u soln (cid:107) < 𝜀 ∈ R > Note that the continuous differentiability and Lipschitz continuity conditions are on the sufficientcondition for the true system, not the model. 61n the implementation of this controller, the Jacobian as a function of u ℓ only needs to be de-rived once (in advance of trial 0). Still, large sample quantities (i.e. long trial durations and/or hightemporal resolution) have historically made deriving this matrix a significant computational bur-den. This has forced past authors to approximate 𝜕 ˆ g 𝜕 u using techniques ranging from the coarse andsimple chord method [34], to more accurate but computationally expensive methods like gradient-based optimization [98]. However, advances in automatic differentiation techniques, such as thosein the software tool CasADi [37], now enable rapid derivation of Jacobian functions that are exactto nearly machine precision [38]. This largely eliminates the need for Jacobian approximationmethods in many scenarios where a system model is available.Finally, note that the class of systems (3.1) treated by the closed-form representation in thiswork is broader than the class of systems (3.8) to which the controller may be applied, and that(3.8) itself is broader than the class of systems guaranteed to satisfied (A3.4)-(A3.6). However,this work chooses to present a broad closed-form representation both because there may be othercontrollers for which the representation is useful, and because there exists systems of class (3.1)that do converge under the given controller and are not contained by narrower popular subclassessuch as piecewise affine and affine-in-the-input systems. An example of such a system is given inSection 3.3. This section demonstrates both the closed form system representation and NILC through the sim-ulated control of a nonlinear mass-spring-damper system over an increasing degree of mismatchbetween the truth model and control model.
The truth model is given by the continuous-time mass-spring-damper system pictured in Figure3.3. The system’s equation of motion is (cid:165) 𝑦 ( 𝑡 ) = − 𝜅𝑚 𝑦 ( 𝑡 ) − 𝜈𝑚 (cid:164) 𝑦 ( 𝑡 ) + 𝜌 𝑚 𝑢 ( 𝑡 ) + 𝜌 𝑚 tan − ( 𝑢 ( 𝑡 )) (3.16) 𝜅 = 𝜅 𝑦 ( 𝑡 ) > 𝜅 𝑦 ( 𝑡 ) ≤ (3.17)where 𝑦 ( 𝑡 ) is the displacement of the mass from the neutral position (positive in direction of springextension), 𝑢 ( 𝑡 ) is an applied actuator voltage, 𝑚 is the mass, 𝜅 is the stiffness coefficient, 𝜈 is62 y ( t ) ρ m u ( t ) + ρ m tan − ( u ( t )) κ = κ y ( t ) > κ y ( t ) ≤ κ m y ( t ) ν m ˙ y ( t ) Figure 3.3: Mass-spring-damper system used for validation. The spring stiffness is a piecewise definedfunction of the spring extension and the applied force is a nonlinear function of actuator voltage 𝑢 . a damping coefficient, and 𝜌 , 𝜌 are constant coefficients mapping from applied voltage to ap-plied force. The stiffness coefficient depends on whether the system is in compression (softer) orextension (stiffer), and the spring hardens with displacement in either direction due to the cubic ex-ponent on 𝑦 ( 𝑡 ) in (3.16). The “measurements” given to the controller after each trial are generatedfrom this model using Runge Kutta 4-step integration and a step size of 𝑇 𝑠 = .
001 s .The controller itself requires a discrete time model, which is derived from (3.16) using theforward Euler method. This yields output and switching equations identical to those of thecontinuous-time system, but the state dynamics become ˆ 𝑥 ( 𝑘 + ) = (cid:104) ˆ 𝑦 ( 𝑘 + ) ˆ 𝑦 ( 𝑘 + ) (cid:105) 𝑇 (3.18)where ˆ 𝑦 ( 𝑘 + ) = 𝑦 ( 𝑘 + ) − ˆ 𝑦 ( 𝑘 ) + ˆ 𝑇 𝑠 ˆ 𝑚 (cid:18) − ˆ 𝜅 ˆ 𝑦 ( 𝑘 ) − ˆ 𝜈 ˆ 𝑇 𝑠 ˆ 𝑦 ( 𝑘 + ) + ˆ 𝜌 𝑢 ( 𝑘 ) + ˆ 𝜌 tan − ( 𝑢 ( 𝑘 )) (cid:19) (3.19)where ˆ 𝑇 𝑠 = .
01 s , an order of magnitude coarser than the truth model. This model is nonlinear inthe states and the input, has relative degree 𝜇 = , 2 dynamic regimes and can be represented inclosed form with one auxiliary variable. The hyperplane corresponding to this auxiliary variable isgiven by 𝑃 = (cid:104) − (cid:105) 𝛽 = (cid:104) (cid:105) (3.20)which gives rise to 2 polytopes.Note that because neither the switching function nor the output function in equation (3.17)depends on the input, and because the system is SISO, this system’s closed from representation633.6) is equivalent to (3.8), the system form accepted by the ILC.Assumptions (A3.1)-(A3.3) are enforced as follows. Each trial lasts 9.5 seconds (approxi-mately 𝜋 seconds), making the length of ˆ g ( u ℓ ) equal to 𝑁 − 𝜇 + = . The trial invariant initialconditions are 0 for both ˆ 𝑦 ( 𝑘 ) and ˆ 𝑦 ( 𝑘 + ) . The trial invariant reference is chosen to be 𝑟 ( 𝑘 ) = sin ( ( 𝑘 − ) ˆ 𝑇 𝑠 ) 𝑘 > 𝑘 = (3.21)where the delay in the sinusoid and the leading 0 ensure that the reference does not demand thatthe controller alter the initial conditions.For the derivation of the control law (3.7), (3.15) from the closed form system representation,this work uses the open source software tool CasADi. CasADi is chosen because its combinationof symbolic framework, sparse matrix storage, and automatic differentiation make for the rapidcomputation of exact ˆ g ( u ℓ ) and 𝜕 ˆ g 𝜕 u functions. However, the Heaviside function in CasADi usesthe trinary “0.5-at-origin” convention rather than the binary “1-at-origin” convention, making itequivalent to signum in the context of determining on which side of a hyperplane a point lies.Because of this, selector functions in CasADi must be written by assembling the polytope interiorsand faces as in Remark 3.2. Here this assembly is done to be equivalent to a “1-at-origin” Heavisideconvention. There are two simulation design objectives. The first is to assess the utility of the closed formsystem representation in enabling the control of a hybrid system via a controller developed fornon-hybrid systems. The second is to assess the sensitivity of the combination of NILC with thissystem representation. Here, “sensitivity” is characterized by the likelihood that the controllerwill diverge given a particular degree of modeling error. Both of these goals are accomplishedwith simulations wherein the controller is synthesized via increasingly erroneous models and itsconvergence behavior is analyzed.In this study, a “simulation” is an attempt to produce r within 20 process trials under NILC fora particular set of mismatched truth and control model parameters. The control model is identicalfor all simulations, with parameters given by the vector ˆ 𝜃 = (cid:104) ˆ 𝑚 ˆ 𝜌 ˆ 𝜌 ˆ 𝜈 ˆ 𝜅 ˆ 𝜅 (cid:105) 𝑇 (3.22) = (cid:104) (cid:105) 𝑇 (3.23)64he truth model parameters are perturbations of the control model parameters by a random relativeerror. Mathematically this is given by the random vector 𝜃 = (cid:16) × 𝑒 𝑇𝜃 (cid:12) 𝐼 + 𝐼 (cid:17) ˆ 𝜃 (3.24)where × ∈ R is a vector with every element equal to 1, 𝑒 𝜃 ∈ R is a random vector, (cid:12) isthe matrix Hadamard product, and 𝐼 is the identity matrix. Each element of 𝑒 𝜃 is the (positive ornegative) relative error 𝑒 𝑖𝜃 = 𝜃 𝑖 − ˆ 𝜃 𝑖 ˆ 𝜃 𝑖 (3.25)and is regenerated for each simulation. A scalar parameter describing the degree of modeling errorcan thus be given by (cid:107) 𝑒 𝜃 (cid:107) .Simulations are organized with respect to increasing (cid:107) 𝑒 𝜃 (cid:107) . There are 20 sets of 50 simulationseach. Sets are defined by bounds on (cid:107) 𝑒 𝜃 (cid:107) , and each simulation in a set is characterized by arandom 𝑒 𝜃 such that (cid:107) 𝑒 𝜃 (cid:107) falls within the specified bounds. These bounds are given by 0.05increments from 0 to 1. For the sake of comparison, one additional simulation is run with zeromodeling error ( (cid:107) 𝑒 𝜃 (cid:107) = ). To ensure the model mismatch is truly 0, the truth model in thissimulation is taken to be the forward Euler discrete model rather than a Runge Kutta integration ofthe continuous model.The efficacy of the controller in a simulation is quantified by the normalized root mean squareerror (NRMSE) between r and y . The RMSE is normalized by the amplitude of r , which in thiswork is 1. Thus the normalization does not alter the numerics here, but it formally nondimen-sionalizes the results. y ℓ is said to have “completely” converged to r if this NRMSE is less than0.005.Holistic analysis of the integration of the closed-form system representation and ILC is basedon the spread of NRMSE values over all sets and the average trajectory of NRMSE versus trialnumber ℓ for all completely convergent simulations. Comparisons between simulation sets aremade by the percentage of simulations within each set that completely converge. For the controller derivation, on a desktop computer with
16 GB of RAM and a
CPU,CasADi performs the repeated function composition to acquire ˆ g and the differentiation to acquirethe 949-by-949 Jacobian 𝜕 ˆ g 𝜕 u in 29 seconds. Once these functions are derived for the first time,before the first learning operation, they need not be derived again, and can be called ordinarily togenerate the trial-varying learning matrix for each subsequent trial. MATLAB symbolic toolbox isused as a traditional symbolic math tool for comparison on the same task. MATLAB fails to com-65lete the task, running out of memory after 1.3 hours. This illustrates the substantial computationaladvantage of automatic differentiation in this context, and its ability to enable direct application ofNewton’s method in ILC.To confirm the efficacy of the combination of NILC and the closed-form hybrid system rep-resentation, Figure 3.4 presents the trajectory of output convergence over trial iteration. The con-troller reaches the complete convergence threshold in 8 trials under 0 model error, in 12 trials forthe mean trajectory over all convergent simulations with erroneous models, and in 14 trials for asimulation with one standard deviation greater NRMSE than the mean trajectory. This illustratesthat while increases in model error can slow convergence, this retardation is modest. Additionally,it can be noted that nearly all experiments (96.5%) either completely converge or diverge entirely,yielding final trial NRMSE values in excess of the zero-input NRMSE. This is illustrated by thelarge gap in the Figure 3.4 histogram between the final-trial RMSEs. The remaining experimentsare either almost convergent (likely requiring several more trials to completely converge), or aredivergent but with the “blowing up” limited to very few points at the very end of the timeseries,thereby having only a small affect on NRMSE. Finally, to supplement above statistical analysisand to show that the converged simulations indeed yield qualitatively reasonable input trajectories,the timeseries data for a representative simulation is shown in Figure 3.5.The results in the preceding paragraph follow the convergence behavior predicted by [34]for a generic input-output model under NILC with static modeling error. The adherence of thiswork’s results to prior predictions illustrates that, as expected, the hybrid nature of the examplesystem does not intrinsically necessitate special accommodation in the controller. Instead, all thatis needed to apply the controller is the closed-form input-output model, the construction of whichis enabled by the closed-form piecewise defined system representation presented.Finally, to practically evaluate (A3.6), Figure 3.6 gives the percentage of experiments that con-verge within each experiment set. Up to (cid:107) 𝑒 𝜃 (cid:107) = . , 100% of the experiments converge. Beyonda relative model error of 0.15, the convergent experiment percentage decays in a reasonably linearfashion. Taking the conservative assumption that the probability of experiment convergence beginsto decay for model error greater than 0, a linear least squares model yields a decay rate of − convergence probability per unit increase in relative model error (or − . percent convergenceprobability per unit increase in percent model error), with an RMSE of between the decaymodel and data.Naturally, the convergence probability of other systems may behave differently under increas-ing (cid:107) 𝑒 𝜃 (cid:107) , and the above analysis clearly does not constitute a definitive theory. However, becausethe set of values of 𝑒 𝜃 for which (A3.6) is satisfied is usually impossible to compute in prac-tice (because g is unknown), it is important to have practical references for understanding systemconvergence. Because mass-spring-damper-like oscillators are ubiquitous across most fields of66 NRMSEof 1 st Trial35 Experiments Over Bracketed Range
Figure 3.4:
Above:
Histogram of final-trial NRMSE for all experiments. All but the highest error bin aredefined by a 0.005 range of NRMSE.
Middle:
Average trajectory of NRMSE vs. Trial number for allcompletely convergent simulations.
Bottom:
Same on a logarithmic scale, for perspective. I npu t [ V ] Time [s] -1.5-1-0.500.511.5 O u t pu t [ m ] ReferenceTrial 0Trial 2Trial 4Trial 6Trial 8
Figure 3.5: Timeseries evolution over multiple trials for a simulation with (cid:107) 𝑒 𝜃 (cid:107) = . . This simulationtook 8 trials to completely converge.Figure 3.6: Percentage of experiments that converge for each 0.05 range of relative model error from 0 to1. The dashed line is a least squares model of the decay in the probability of convergence as model errorincreases. This chapter contributes a closed-form representation of piecewise defined dynamical systems.The utility of having this representation is demonstrated by the application to a hybrid system of aniterative learning controller based on Newton’s method, which was hitherto impossible because ofthe controller’s need for a dynamical system model supporting function composition and Jacobianoperations.Additionally, the controller derivation itself is formally generalized for systems of relative de-gree greater than 1, and for time-varying systems. Additionally, automatic differentiation is usedto enable faster and more accurate implementation of Newton’s method than prior works, whichrequired coarser and/or more computationally expensive approximation of the model Jacobian.Future work for the closed-form system representation revolves around expanding the repre-sentation to facilitate specification of more complicated switching logic and dynamical state resetsupon discrete transitions. While all discrete-time hybrid automaton topologies may be captured bypiecewise definition [17], doing so often requires augmentation of the state vector and may makemodel synthesis more challenging. 69
HAPTER 4
Nonlinear Systems with Unstable Inverses:Invert-Linearize ILC and Stable Inversion
Recall from Chapter 1 that NILC is chosen as the foundational ILC technique for this disser-tation because of its fast convergence rate and because it is synthesizable from a broader range ofsystems than is considered by other publications on ILC synthesis from nonlinear models. Specif-ically, other published ILC laws are subject to at least one of the following restrictions on thecontrol model:(R1) having relative degree of either 0 or 1 [14, 15],(R2) being affine in the input [13, 14, 15, 23],(R3) being time-invariant [13, 16], and(R4) being smooth (Lipschitz continuous at the most relaxed) [13, 14, 15, 16, 23]from all of which NILC is free. Models used to synthesize NILC are, however, subject to therestriction of(R5) having a stable inverse.Note that the non-NILC prior art [13, 14, 15, 16] also suffers from (R5). While the prior art doesnot explicitly reveal this shortcoming, evidence of it is given in the appendix. The ultimate goalof this chapter is a new ILC framework inheriting the benefits of NILC while surmounting thisshortcoming.
Content of this chapter also to be submitted as:I. A. Spiegel, N. Strijbosch, T. Oomen and K. Barton, “Iterative Learning Control with Discrete-Time NonlinearNon-minimum Phase Models via Stable Inversion,” in
IEEE Transactions on Control Systems Technology.
Copyrightmay be transferred without notice, after which this version may no longer be accessible.
The first step of stable inversion is deriving the conventional inverse. To synthesize a minimalinverse system representation, first assume (3.8) is in the normal form ˆ 𝑥 𝑖 ( 𝑘 + ) = ˆ 𝑥 𝑖 + ( 𝑘 ) 𝑖 < 𝜇 (4.1a) ˆ 𝑥 𝑖 ( 𝑘 + ) = ˆ 𝑓 𝑖 ( ˆ 𝑥 ( 𝑘 ) , 𝑢 ( 𝑘 ) , 𝑘 ) 𝑖 ≥ 𝜇 (4.1b) ˆ 𝑦 ( 𝑘 ) = ˆ 𝑥 (4.1c)where ˆ 𝑥 ( ) = , and the superscripts 𝑖 indicate the vector element index, starting from 1. Notethe ILC trial index subscript ℓ is omitted in this section, as stable inversion on its own does notinvolve incrementing ℓ . Equation (4.1a) captures the time delay arising from the system relative72egree, while equation (4.1b) captures the remaining system dynamics. One method of derivingthis normal form from a system not in normal form is given in [102].Given this normal form, use (4.1c) to replace the first 𝜇 state variables with output variablesvia ˆ 𝑥 𝑖 ( 𝑘 ) = ˆ 𝑦 ( 𝑘 + 𝑖 − ) 𝑖 ≤ 𝜇 (4.2)Similarly, replace the 𝜇 th state variable incremented by one time step (i.e. the left side of (4.1b) for 𝑖 = 𝜇 ) with an output variable via ˆ 𝑥 𝜇 ( 𝑘 + ) = ˆ 𝑦 ( 𝑘 + 𝜇 ) (4.3)These substitutions are made to facilitate the inversion of system (4.1), as the inverse of a systemwith relative degree 𝜇 ≥ is necessarily acausal with dependence on some subset of { ˆ 𝑦 ( 𝑘 ) , ˆ 𝑦 ( 𝑘 + ) , · · · , ˆ 𝑦 ( 𝑘 + 𝜇 )} at each time step 𝑘 . For notational compactness, define the ˆ 𝑦 -preview vector ˆ 𝓎 ( 𝑘 ) (cid:66) [ ˆ 𝑦 ( 𝑘 ) , · · · , ˆ 𝑦 ( 𝑘 + 𝜇 )] 𝑇 . Then inverting (4.1b) with 𝑖 = 𝜇 yields the conventional inverseoutput function 𝑢 ( 𝑘 ) = ˆ 𝑓 𝜇 − (cid:18) (cid:104) ˆ 𝑥 𝜇 + , · · · , ˆ 𝑥 𝑛 𝑥 (cid:105) 𝑇 , ˆ 𝓎 ( 𝑘 ) , 𝑘 (cid:19) (4.4)where ˆ 𝑓 𝜇 − is the inverse of ˆ 𝑓 𝜇 , i.e. (4.1b, 𝑖 = 𝜇 ) solved for 𝑢 ( 𝑘 ) . This output equation issubstituted into (4.1b) with 𝑖 > 𝜇 along with (4.2)-(4.3) to yield the entire inverse state dynamics ˆ 𝜂 ( 𝑘 + ) = ˆ 𝑓 𝜂 ( ˆ 𝜂 ( 𝑘 ) , ˆ 𝓎 ( 𝑘 ) , 𝑘 ) (4.5a) 𝑢 ( 𝑘 ) = ˆ 𝑓 𝜇 − ( ˆ 𝜂 ( 𝑘 ) , ˆ 𝓎 ( 𝑘 ) , 𝑘 ) (4.5b)where ˆ 𝜂 ∈ R 𝑛 𝜂 ( 𝑛 𝜂 = 𝑛 𝑥 − 𝜇 ) is the inverse state vector defined ˆ 𝜂 𝑖 ( 𝑘 ) (cid:66) ˆ 𝑥 𝜇 + 𝑖 ( 𝑘 ) (4.6)and ˆ 𝑓 𝜂 : R 𝑛 𝜂 × R 𝜇 + × Z → R 𝑛 𝜂 is the inverse state dynamics ˆ 𝑓 𝑖𝜂 ( ˆ 𝜂 ( 𝑘 ) , ˆ 𝓎 ( 𝑘 ) , 𝑘 ) (cid:66) ˆ 𝑓 𝑖 + 𝜇 ( ˆ 𝑥 ( 𝑘 ) , 𝑢 ( 𝑘 ) , 𝑘 ) (4.7)Next, this inverse system is to be similarity transformed to decouple the stable and unstablemodes of its linearization about the initial condition. Consider the Jacobian 𝐴 = 𝜕 ˆ 𝑓 𝜂 𝜕 ˆ 𝜂 (cid:16) ˆ 𝜂 = , ˆ 𝓎 = ˆ 𝓎 † , 𝑘 = (cid:17) (4.8)73here ˆ 𝓎 † is the solution to ˆ 𝑓 𝜂 ( , ˆ 𝓎 † , ) = . Then let 𝑉 be the similarity transform matrix such that ˜ 𝐴 = 𝑉 − 𝐴𝑉 = (cid:34) ˜ 𝐴 𝓈
00 ˜ 𝐴 𝓊 (cid:35) (4.9)where ˜ 𝐴 𝓈 ∈ R 𝑛 𝓈 × 𝑛 𝓈 has all eigenvalues inside the unit circle, and ˜ 𝐴 𝓊 ∈ R 𝑛 𝜂 − 𝑛 𝓈 × 𝑛 𝜂 − 𝑛 𝓈 has alleigenvalues outside the unit circle. This can be satisfied by deriving the real block Jordan form of 𝐴 . The corresponding inverse system state dynamics are ˜ 𝜂 ( 𝑘 + ) = ˜ 𝑓 𝜂 ( ˜ 𝜂 ( 𝑘 ) , ˆ 𝓎 ( 𝑘 ) , 𝑘 ) ≔ 𝑉 − ˆ 𝑓 𝜂 ( 𝑉 ˜ 𝜂 ( 𝑘 ) , ˆ 𝓎 ( 𝑘 ) , 𝑘 ) (4.10)where the tilde on ˜ 𝑓 𝜂 indicates application to ˜ 𝜂 rather than ˆ 𝜂 . Note that despite using a linearization-derived linear similarity transform, (4.10) describes the same nonlinear time-varying dynamics as(4.5a), but with the linear parts of the stable and unstable modes decoupled.If (3.8) has an unstable inverse, then (4.10) is unstable and ˜ 𝜂 ( 𝑘 ) will be unbounded as 𝑘 in-creases. However, given an infinite timeline in the positive and negative direction, the equation ˜ 𝜂 ( 𝑘 ) = ∞ ∑︁ 𝜅 = −∞ 𝜙 ( 𝑘 − 𝜅 ) (cid:0) ˜ 𝑓 𝜂 ( ˜ 𝜂 ( 𝜅 − ) , ˆ 𝓎 ( 𝜅 − ) , 𝜅 − ) − ˜ 𝐴 ˜ 𝜂 ( 𝜅 − ) (cid:1) (4.11)where 𝜙 ( 𝑘 ) = ˜ 𝐴 𝑘 𝓈 𝑛 𝓈 × 𝑛 𝜂 − 𝑛 𝓈 𝑛 𝜂 − 𝑛 𝓈 × 𝑛 𝓈 𝑛 𝜂 − 𝑛 𝓈 × 𝑛 𝜂 − 𝑛 𝓈 𝑘 > 𝐼 𝑛 𝓈 × 𝑛 𝓈 𝑛 𝓈 × 𝑛 𝜂 − 𝑛 𝓈 𝑛 𝜂 − 𝑛 𝓈 × 𝑛 𝓈 𝑛 𝜂 − 𝑛 𝓈 × 𝑛 𝜂 − 𝑛 𝓈 𝑘 = 𝑛 𝓈 × 𝑛 𝓈 𝑛 𝓈 × 𝑛 𝜂 − 𝑛 𝓈 𝑛 𝜂 − 𝑛 𝓈 × 𝑛 𝓈 − ˜ 𝐴 𝑘 𝓊 𝑘 < (4.12)is an exact, bounded solution to (4.10) provided the right hand side of (4.11) exists for all 𝑘 ∈ Z .However, (4.11) is implicit, and thus cannot be directly evaluated. A fixed-point problem solver—past work uses Picard iteration—must be used to find ˜ 𝜂 , and sufficient conditions for the solverconvergence and solution uniqueness must be determined.74 .2 Novel ILC Analysis and Development The NILC scheme (3.7), (3.15) provides convergence of e ℓ to 0 in theory. However, this as-sumes perfect computation of the matrix inversion in (3.15). In practice, the precision to which (cid:16) 𝜕 ˆ g 𝜕 u ( u ℓ ) (cid:17) − can be accurately computed is directly dependent on the condition number of 𝜕 ˆ g 𝜕 u ( u ℓ ) .If the condition number of a matrix is large enough, the values computed for its inverse may be-come arbitrary, and their order of magnitude may grow directly with the order of magnitude ofthe condition number [103, ch. 3.2], [104]. This “blowing up” of the matrix inverse can causedivergence of (3.7), (3.15).Large 𝜕 ˆ g 𝜕 u condition numbers have been previously observed for NMP linear systems, bothtime-invariant [105], [106, ch. 5.3-5.4] and time-varying [107], [108, ch. 4.1.1]. The fact thatthe minimum singular value of 𝜕 ˆ g 𝜕 u ( u ℓ ) decreases with increases in the system frequency responsefunction magnitude at the Nyquist frequency [109] contributes to this ill-conditioning. For linearsystems, this magnitude is directly dependent on the zero magnitudes, and thus on the inverse sys-tems’ stability. These phenomena generalize to nonlinear systems because the Jacobian evaluatedat a particular input trajectory, 𝜕 ˆ g 𝜕 u ( u ∗ ) , is equal to the constant matrix 𝜕 ¯ g 𝜕 u where ¯ g is the liftedinput-output model of the linearization of (3.8) about the trajectory u ∗ .To illustrate this equality, first consider that the elements of 𝜕 ˆ g 𝜕 u ( u ∗ ) are given by (3.11b) andthe chain rule as 𝜕 ˆ 𝑦 ( 𝑘 ) 𝜕𝑢 ( 𝑗 ) ( u ∗ ) = 𝜕 ˆ ℎ𝜕 ˆ 𝑥 (cid:16) ˆ 𝑓 ( 𝑘 − ) ( u ∗ ) (cid:17) 𝜕 ˆ 𝑓 ( 𝑘 − ) 𝜕 u ( u ∗ ) 𝜕 u 𝜕𝑢 ( 𝑗 ) (4.13)where 𝜕 u 𝜕𝑢 ( 𝑗 ) = (cid:104) × 𝑗 × 𝑁 − 𝜇 + − 𝑗 (cid:105) 𝑇 (4.14)and 𝜕 ˆ ℎ𝜕 ˆ 𝑥 is a row vector.Then consider the linearization of (3.8) about u ∗ : 𝛿 ˆ 𝑥 ( 𝑘 + ) = ¯ 𝑓 ( 𝛿 ˆ 𝑥 ( 𝑘 ) , 𝛿𝑢 ( 𝑘 ) , 𝑘 ) (4.15a) = 𝜕 ˆ 𝑓𝜕 ˆ 𝑥 ( ˆ 𝑥 ∗ ( 𝑘 ) , 𝑢 ∗ ( 𝑘 ) , 𝑘 ) 𝛿 ˆ 𝑥 ( 𝑘 ) + 𝜕 ˆ 𝑓𝜕𝑢 ( ˆ 𝑥 ∗ ( 𝑘 ) , 𝑢 ∗ ( 𝑘 ) , 𝑘 ) 𝛿𝑢 ( 𝑘 ) 𝛿 ˆ 𝑦 ( 𝑘 ) = ¯ ℎ ( 𝛿 ˆ 𝑥 ( 𝑘 )) = 𝜕 ˆ ℎ𝜕 ˆ 𝑥 ( ˆ 𝑥 ∗ ( 𝑘 )) 𝛿 ˆ 𝑥 ( 𝑘 ) (4.15b)where ˆ 𝑥 ∗ ( 𝑘 ) = ˆ 𝑓 ( 𝑘 − ) ( u ∗ ) and the 𝛿 notation denotes 𝛿 ˆ 𝑥 ( 𝑘 ) = ˆ 𝑥 ( 𝑘 ) − ˆ 𝑥 ∗ ( 𝑘 ) for ˆ 𝑥 and similar for 𝑢 .75ifting (4.15) in the same manner as (3.8) yields the output perturbation as a function of theinput perturbation time series 𝛿 u via 𝛿 ˆ 𝑦 ( 𝑘 ) = 𝜕 ˆ ℎ𝜕 ˆ 𝑥 (cid:16) ˆ 𝑓 ( 𝑘 − ) ( u ∗ ) (cid:17) ¯ 𝑓 ( 𝑘 − ) ( 𝛿 u ) (4.16)Because of (4.15)’s linearity, ¯ 𝑓 ( 𝑘 − ) ( 𝛿 u ) can be explicitly expanded as ¯ 𝑓 ( 𝑘 − ) ( 𝛿 u ) = (cid:32) 𝑘 − (cid:214) 𝜅 = 𝜕 ˆ 𝑓 ( 𝜅 ) 𝜕 ˆ 𝑓 ( 𝜅 − ) ( u ∗ ) (cid:33) 𝛿 ˆ 𝑥 ( ) + 𝜕 ˆ 𝑓 ( 𝑘 − ) 𝜕 u ( u ∗ ) 𝛿 u (4.17)where (cid:206) is ordered with the factor of least ℓ on the right and the factor of greatest ℓ on the left.The terminal condition of the recursive function composition is ˆ 𝑓 (− ) = ˆ 𝑥 ( ) . From (4.16) and(4.17) it is clear that the elements of 𝜕 ¯ g 𝜕𝛿 u are given by 𝜕𝛿 ˆ 𝑦 ( 𝑘 ) 𝜕𝛿𝑢 ( 𝑗 ) = 𝜕 ˆ ℎ𝜕 ˆ 𝑥 (cid:16) ˆ 𝑓 ( 𝑘 − ) ( u ∗ ) (cid:17) 𝜕 ˆ 𝑓 ( 𝑘 − ) 𝜕 u ( u ∗ ) 𝜕𝛿 u 𝜕𝛿𝑢 ( 𝑗 ) (4.18)which is equal to (4.13) because 𝜕𝛿 u 𝜕𝛿𝑢 ( 𝑗 ) = 𝜕 u 𝜕𝑢 ( 𝑗 ) due to the identical structures (3.10) of u and 𝛿 u with respect to 𝑢 and 𝛿𝑢 time indexing. Thus, if (3.8) is such that its linearization (4.15) is unstableit will suffer ill-conditioning and (3.15) may be so difficult to compute in practice that attempts todo so yield a matrix with large erroneous elements. Such a learning gain matrix may in turn cause u ℓ + to contain large erroneous elements, causing the learning law to diverge.Therefore, for the learning law (3.7) to converge for a system with an unstable inverse in prac-tice, a learning matrix synthesis that does not require matrix inversion of 𝜕 ˆ g 𝜕 u ( u ℓ ) is desired. To circumvent issues associated with inverting 𝜕 ˆ g 𝜕 u ( u ℓ ) this work introduces a new learning matrixdefinition seeking to satisfy the requirements (A3.4)-(A3.6) in the spirit of Newton’s method, butwithout the matrix inversion requirement of (3.15). The new learning matrix is given by 𝐿 ℓ = 𝜕 ˆ g − 𝜕 ˆ y ( y ℓ ) (4.19)where ˆ g − : R 𝑁 − 𝜇 + → R 𝑁 − 𝜇 + is a lifted model of the inverse of (3.8). This makes 𝜕 ˆ g − 𝜕 ˆ y afunction of the output of (3.8), namely ˆ y ℓ . As stated in Section 3.2, ˆ y ℓ is merely a prediction of theaccessible, measured output y ℓ . Hence y ℓ is used as the input to 𝜕 ˆ g − 𝜕 ˆ y . In short, this work proposesusing the linearization of the inverse of (3.8) rather than the inverse of the linearization, and thus76he new framework (3.7), (4.19) will be referred to as “Invert-Linearize ILC” (ILILC).A direct method of inverting (3.8) is to solve ˆ 𝑦 ℓ ( 𝑘 + 𝜇 ) = ˆ ℎ ( ˆ 𝑓 ( 𝑘 + 𝜇 − ) ( u ℓ )) (4.20)for 𝑢 ℓ ( 𝑘 ) , and substitute the resulting function of { ˆ 𝑦 ℓ ( 𝑘 ) , ˆ 𝑦 ℓ ( 𝑘 + ) , · · · , ˆ 𝑦 ℓ ( 𝑘 + 𝜇 )} into (3.8a).However, if (3.8) has an unstable inverse, this method of inversion will yield unbounded states ˆ 𝑥 ℓ ( 𝑘 ) as 𝑘 increases. Thus, ˆ g − is derived via stable inversion rather than direct inversion. Note,though, that (4.19) also admits the use of other stable approximate inverse models for ˆ g − shouldthey be available. This section proves a relaxed set of sufficient conditions for the convergence of Picard iteration tothe unique solution to the stable inversion problem, i.e. the unique solution to (4.11) from Section4.1. This enables stable inversion—and thus ILC—for a new class of system representations cap-turing simultaneous feedback and feedforward control. Additionally, a new initial Picard iterateprescription is given to suit the broadened scope of stable inversion, and a procedure for practicalimplementation is described. This procedure enables the derivation of ˆ g − . The standard Picard iterative solver [110, ch. 9] for (4.11) is ˜ 𝜂 ( 𝑚 + ) ( 𝑘 ) = ∞ ∑︁ 𝜅 = −∞ 𝜙 ( 𝑘 − 𝜅 ) (cid:0) ˜ 𝑓 𝜂 (cid:0) ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) , ˆ 𝓎 ( 𝜅 − ) , 𝜅 − (cid:1) − ˜ 𝐴 ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) (cid:1) (4.21)where the parenthetical subscript ( 𝑚 ) ∈ Z ≥ is the Picard iteration index.To prove that (4.21) converges to a unique solution, [101] makes the assumptions that(Z1) ˆ 𝑓 ( , , 𝑘 ) = ∀ 𝑘 , and(Z2) ˜ 𝜂 ( ) ( 𝑘 ) = ∀ 𝑘 .The first assumption is violated for many representations of systems incorporating both feedbackand feedforward control. An example of such a system is given in Section 4.4, where 𝑢 is thefeedforward control input and the feedback control is part of the time-varying dynamics of ˆ 𝑓 . Thisfeedback control influences ˆ 𝑥 regardless of whether or not 𝑢 ( 𝑘 ) = . While there may often be The continuous-time literature also makes these assumptions [99, 100]
77 change of variables that enables satisfaction of (Z1), (4.1a-4.1c) already imposes constraints onthe states and outputs, and for many systems it is unlikely for there to exist a change of variablessatisfying both assumptions.Furthermore, while for systems satisfying (Z1), (Z2) may be the zero-input state trajectory, thisis untrue for systems violating (Z1). For these systems, the zero state trajectory (Z2) is essentiallyarbitrary, and may degrade the quality of low- 𝑚 Picard iterates if far from the solution trajectory.This seriously jeopardizes convergence because the computational complexity of the Picard iter-ation solution grows exponentially with the number of iterations. It is thus desirable to reach asatisfactory solution in as few iterations as possible, i.e. it is desirable to have high-quality low- 𝑚 iterates.Thus, this work presents a new set of sufficient conditions for the unique convergence of (4.21)that relaxes (Z1), (Z2). Before proof of this, several definitions are presented. Definition 4.1 (Lifted Matrices and Third-Order Tensors) . Given the vector and matrix functionsof time 𝑎 ( 𝑘 ) ∈ R 𝑛 and 𝐵 ( 𝑘 ) ∈ R 𝑛 × 𝑛 , the corresponding lifted matrix and third order tensor aregiven by upright bold notation: a ∈ R 𝑛 ×K and B ∈ R 𝑛 × 𝑛 ×K . K is the time dimension, and may be ∞ . Elements of the lifted objects are a 𝑖,𝑘 (cid:66) 𝑎 𝑖 ( 𝑘 ) and B 𝑖, 𝑗,𝑘 (cid:66) 𝐵 𝑖, 𝑗 ( 𝑘 ) . Definition 4.2 (Matrix and Third-Order Tensor Norms) . (cid:107)·(cid:107) ∞ refers to the ordinary ∞ -norm whenapplied to vectors, and is the matrix norm induced by the vector norm when applied to matrices(i.e. the maximum absolute row sum). Additionally, the entry-wise (∞ , ) -norm is defined for thematrices and third-order tensors a and B from Definition 4.1 as (cid:107) a (cid:107) ∞ , (cid:66) ∑︁ 𝑘 ∈K (cid:107) 𝑎 ( 𝑘 ) (cid:107) ∞ (cid:107) B (cid:107) ∞ , (cid:66) ∑︁ 𝑘 ∈K (cid:107) 𝐵 ( 𝑘 ) (cid:107) ∞ (4.22) Definition 4.3 (Local Approximate Linearity [99, 101]) . ˜ 𝑓 𝜂 is locally approximately linear in ˜ 𝜂 ( 𝑘 ) and its ˜ 𝜂 ( 𝑘 ) = dynamics, in a closed 𝑠 -neighborhood around ( ˜ 𝜂 ( 𝑘 ) = , ˜ 𝑓 𝜂 ( , ˆ 𝓎 ( 𝑘 ) , 𝑘 ) = ) ,with Lipschitz constants 𝐾 , 𝐾 > if ∃ 𝑠 > such that for any vectors• 𝑎 ( 𝑘 ) , 𝑏 ( 𝑘 ) ∈ R 𝑛 𝜂 with (cid:107)·(cid:107) ∞ ≤ 𝑠 ∀ 𝑘 , and• 𝒶 ( 𝑘 ) , 𝒷 ( 𝑘 ) ∈ R 𝜇 + such that (cid:13)(cid:13) ˜ 𝑓 𝜂 ( , 𝒶 ( 𝑘 ) , 𝑘 ) (cid:13)(cid:13) ∞ , (cid:13)(cid:13) ˜ 𝑓 𝜂 ( , 𝒷 ( 𝑘 ) , 𝑘 ) (cid:13)(cid:13) ∞ ≤ 𝑠 ∀ 𝑘 the following is true ∀ 𝑘 (cid:13)(cid:13)(cid:0) ˜ 𝑓 𝜂 ( 𝑎 ( 𝑘 ) , 𝒶 ( 𝑘 ) , 𝑘 ) − 𝐴𝑎 ( 𝑘 ) (cid:1) − (cid:0) ˜ 𝑓 𝜂 ( 𝑏 ( 𝑘 ) , 𝒷 ( 𝑘 ) , 𝑘 ) − 𝐴𝑏 ( 𝑘 ) (cid:1)(cid:13)(cid:13) ∞ ≤ 𝐾 (cid:107) 𝑎 ( 𝑘 ) − 𝑏 ( 𝑘 ) (cid:107) ∞ + 𝐾 (cid:13)(cid:13) ˜ 𝑓 𝜂 ( , 𝒶 ( 𝑘 ) , 𝑘 ) − ˜ 𝑓 𝜂 ( , 𝒷 ( 𝑘 ) , 𝑘 ) (cid:13)(cid:13) ∞ (4.23)78ith these definitions a new set of sufficient conditions for Picard iteration convergence maybe established. Theorem 4.1.
The Picard iteration (4.21) converges to a unique solution to (4.10) if the followingsufficient conditions are met.(A4.1) (cid:13)(cid:13) ˜ η ( ) (cid:13)(cid:13) ∞ , ≤ 𝑠 (A4.2) ∀ 𝑘 ∃ ˆ 𝓎 ( 𝑘 ) = ˆ 𝓎 † ( 𝑘 ) such that ˜ 𝑓 𝜂 ( , ˆ 𝓎 † ( 𝑘 ) , 𝑘 ) = (A4.3) ˜ 𝑓 𝜂 is locally approximately linear in the sense of (4.23)(A4.4) 𝐾 (cid:107) ϕ (cid:107) ∞ , < (A4.5) (cid:107) ϕ (cid:107) ∞ , 𝐾 (cid:107) ˜ f 𝜂 ( , ˆ 𝓎 ) (cid:107) ∞ , −(cid:107) ϕ (cid:107) ∞ , 𝐾 ≤ 𝑠 where (cid:13)(cid:13) ˜ f 𝜂 ( , ˆ 𝓎 ) (cid:13)(cid:13) ∞ , = (cid:205) ∞ 𝑘 = −∞ (cid:13)(cid:13) ˜ 𝑓 𝜂 ( , ˆ 𝓎 ( 𝑘 ) , 𝑘 ) (cid:13)(cid:13) ∞ and ϕ are defined by Definition 4.1; i.e. ϕ is thelifted tensor version of (4.12).Proof. This proof shares the approach of [101] in establishing the Cauchy nature of the Picardsequence. It is also influenced by the proofs of Picard iterate local approximate linearity forcontinuous-time systems in [99].Proof that (4.21) converges to a unique fixed point begins with an induction showing that ˜ 𝜂 ( 𝑚 ) ( 𝑘 ) remains in the locally approximately linear neighborhood ∀ 𝑘 , 𝑚 . The base case of thisinduction is given by (A4.1). Then under the premise (cid:13)(cid:13) ˜ η ( 𝑚 ) (cid:13)(cid:13) ∞ , ≤ 𝑠 (4.24)the induction proceeds as follows. Here, ellipses indicate the continuation of a line of mathematics.By the Picard iterative solver (4.21): (cid:13)(cid:13) ˜ η ( 𝑚 + ) (cid:13)(cid:13) ∞ , = ∞ ∑︁ 𝑘 = −∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ∑︁ 𝜅 = −∞ 𝜙 ( 𝑘 − 𝜅 ) (cid:0) ˜ 𝑓 𝜂 (cid:0) ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) , ˆ 𝓎 ( 𝜅 − ) , 𝜅 − (cid:1) − 𝐴 ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ · · · (4.25)By the triangle inequality: · · · ≤ ∞ ∑︁ 𝑘 = −∞ ∞ ∑︁ 𝜅 = −∞ (cid:13)(cid:13) 𝜙 ( 𝑘 − 𝜅 ) (cid:0) ˜ 𝑓 𝜂 (cid:0) ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) , ˆ 𝓎 ( 𝜅 − ) , 𝜅 − (cid:1) − 𝐴 ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) (cid:1)(cid:13)(cid:13) ∞ · · · (4.26)79y the fact that for matrix norms induced by vector norms (cid:107) 𝐵𝑎 (cid:107) ≤ (cid:107) 𝐵 (cid:107) (cid:107) 𝑎 (cid:107) for matrix 𝐵 andvector 𝑎 : · · · ≤ ∞ ∑︁ 𝑘 = −∞ ∞ ∑︁ 𝜅 = −∞ (cid:107) 𝜙 ( 𝑘 − 𝜅 ) (cid:107) ∞ (cid:13)(cid:13) ˜ 𝑓 𝜂 (cid:0) ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) , ˆ 𝓎 ( 𝜅 − ) , 𝜅 − (cid:1) − 𝐴 ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) (cid:13)(cid:13) ∞ · · · (4.27) · · · = ∞ ∑︁ 𝜅 = −∞ (cid:13)(cid:13) ˜ 𝑓 𝜂 (cid:0) ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) , ˆ 𝓎 ( 𝜅 − ) , 𝜅 − (cid:1) − 𝐴 ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) (cid:13)(cid:13) ∞ ∞ ∑︁ 𝑘 = −∞ (cid:107) 𝜙 ( 𝑘 − 𝜅 ) (cid:107) ∞ · · · (4.28)By the fact that (cid:205) ∞ 𝑘 = −∞ (cid:107) 𝜙 ( 𝑘 − 𝜅 ) (cid:107) ∞ has the same value ∀ 𝜅 · · · = (cid:107) ϕ (cid:107) ∞ , ∞ ∑︁ 𝜅 = −∞ (cid:13)(cid:13) ˜ 𝑓 𝜂 (cid:0) ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) , ˆ 𝓎 ( 𝜅 − ) , 𝜅 − (cid:1) − 𝐴 ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) (cid:13)(cid:13) ∞ · · · (4.29)By (A4.2): · · · = (cid:107) ϕ (cid:107) ∞ , ∞ ∑︁ 𝜅 = −∞ (cid:13)(cid:13)(cid:0) ˜ 𝑓 𝜂 (cid:0) ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) , ˆ 𝓎 ( 𝜅 − ) , 𝜅 − (cid:1) − 𝐴 ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) (cid:1) − (cid:16) ˜ 𝑓 𝜂 (cid:16) , ˆ 𝓎 † ( 𝜅 − ) , 𝜅 − (cid:17) − 𝐴 ( ) (cid:17)(cid:13)(cid:13)(cid:13) ∞ · · · (4.30)By (A4.3): · · · ≤ (cid:107) ϕ (cid:107) ∞ , ∞ ∑︁ 𝜅 = −∞ 𝐾 (cid:13)(cid:13) ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) (cid:13)(cid:13) ∞ + 𝐾 (cid:13)(cid:13) ˜ 𝑓 𝜂 ( , ˆ 𝓎 ( 𝜅 − ) , 𝜅 − ) (cid:13)(cid:13) ∞ · · · (4.31) · · · = (cid:107) ϕ (cid:107) ∞ , (cid:16) 𝐾 (cid:13)(cid:13) ˜ η ( 𝑚 ) (cid:13)(cid:13) ∞ , + 𝐾 (cid:13)(cid:13) ˜ f 𝜂 ( , ˆ 𝓎 ) (cid:13)(cid:13) ∞ , (cid:17) · · · (4.32)By (A4.4), both sides of (A4.5) can be multiplied by the denominator in (A4.5) without changingthe inequality direction. Thus by (4.24) and algebraic rearranging of (A4.5) · · · ≤ (cid:107) ϕ (cid:107) ∞ , (cid:16) 𝐾 𝑠 + 𝐾 (cid:13)(cid:13) ˜ f 𝜂 ( , ˆ 𝓎 ) (cid:13)(cid:13) ∞ , (cid:17) ≤ 𝑠 (4.33) ∴ (cid:13)(cid:13) ˜ η ( 𝑚 ) (cid:13)(cid:13) ∞ , ≤ 𝑠 ∀ 𝑚 . Because (cid:13)(cid:13) ˜ η ( 𝑚 ) (cid:13)(cid:13) ∞ , ≥ (cid:13)(cid:13) ˜ 𝜂 ( 𝑚 ) ( 𝑘 ) (cid:13)(cid:13) ∞ ∀ 𝑘 , this implies that ˜ 𝜂 ( 𝑚 ) ( 𝑘 ) is within thelocally approximately linear neighborhood ∀ 𝑚 , 𝑘 .To show that (4.21) converges to a unique fixed point, define Δ ˜ 𝜂 ( 𝑚 ) ( 𝑘 ) (cid:66) ˜ 𝜂 ( 𝑚 + ) ( 𝑘 ) − ˜ 𝜂 ( 𝑚 ) ( 𝑘 ) (4.34)80hen, by a nearly identical induction (cid:13)(cid:13) Δ ˜ η ( 𝑚 + ) (cid:13)(cid:13) ∞ , ≤ (cid:107) ϕ (cid:107) ∞ , 𝐾 (cid:13)(cid:13) Δ ˜ η ( 𝑚 ) (cid:13)(cid:13) ∞ , (4.35)By (A4.4) lim 𝑚 →∞ (cid:13)(cid:13) Δ ˜ η ( 𝑚 ) (cid:13)(cid:13) ∞ , = (4.36)which implies lim 𝑚 →∞ (cid:13)(cid:13) Δ ˜ 𝜂 ( 𝑚 ) ( 𝑘 ) (cid:13)(cid:13) ∞ = ∀ 𝑘 (4.37) ∴ ∀ 𝑘 the sequence { ˜ 𝜂 𝑚 ( 𝑘 )} is a Cauchy sequence, and thus the fixed point ˜ 𝜂 ( 𝑘 ) = lim 𝑚 →∞ ˜ 𝜂 ( 𝑚 ) ( 𝑘 ) is unique. (cid:3) Neither the preceding presentation nor the nonlinear stable inversion prior art [101] explicitlydiscusses the intuitive foundation of stable inversion: evolving the stable modes of an inversesystem forwards in time from an initial condition and evolving the unstable modes backwards intime from a terminal condition. Unlike for linear time invariant (LTI) systems, this intuition is notput into practice directly for nonlinear systems because the similarity transforms that completelydecouple the stable and unstable modes of linear systems do not necessarily decouple the stableand unstable modes of nonlinear systems. However, the same principle underpins this work. Thisis evidenced by the fact that the intuitive LTI stable inversion is recovered from (4.11) when ˆ 𝑓 isLTI, as illustrated briefly below.For LTI ˆ 𝑓 , ˜ 𝑓 takes the form ˜ 𝜂 ( 𝑘 + ) = ˜ 𝐴 ˜ 𝜂 ( 𝑘 ) + ˜ 𝐵 ˆ 𝓎 ( 𝑘 ) (4.38) (cid:34) ˜ 𝜂 𝓈 ( 𝑘 + ) ˜ 𝜂 𝓊 ( 𝑘 + ) (cid:35) = (cid:34) ˜ 𝐴 𝓈
00 ˜ 𝐴 𝓊 (cid:35) (cid:34) ˜ 𝜂 𝓈 ( 𝑘 ) ˜ 𝜂 𝓊 ( 𝑘 ) (cid:35) + (cid:34) ˜ 𝐵 𝓈 ˜ 𝐵 𝓊 (cid:35) ˆ 𝓎 ( 𝑘 ) (4.39)Then the implicit solution (4.11) becomes the explicit solution ˜ 𝜂 ( 𝑘 ) = ∞ ∑︁ 𝜅 = −∞ 𝜙 ( 𝑘 − 𝜅 ) ˜ 𝐵 ˆ 𝓎 ( 𝜅 − ) (4.40) (cid:34) ˜ 𝜂 𝓈 ( 𝑘 ) ˜ 𝜂 𝓊 ( 𝑘 ) (cid:35) = (cid:34) (cid:205) 𝑘𝜅 = −∞ ˜ 𝐴 𝑘 − 𝜅 𝓈 ˜ 𝐵 𝓈 ˆ 𝓎 ( 𝜅 − )− (cid:205) ∞ 𝜅 = 𝑘 + ˜ 𝐴 𝑘 − 𝜅 𝓊 ˜ 𝐵 𝓊 ˆ 𝓎 ( 𝜅 − ) (cid:35) (4.41) = (cid:34) ˜ 𝐴 𝓈 ˜ 𝜂 𝓈 ( 𝑘 − ) + ˜ 𝐵 𝓈 ˆ 𝓎 ( 𝑘 − ) ˜ 𝐴 − 𝓊 ˜ 𝜂 𝓊 ( 𝑘 + ) − ˜ 𝐴 − 𝓊 ˜ 𝐵 𝓊 ˆ 𝓎 ( 𝑘 ) (cid:35) (4.42)which is the forward evolution of the stable modes and backward evolution of the unstable modes81here the initial and terminal conditions at 𝑘 = ±∞ are zero. ˜ 𝜂 ( ) Selection and Implementation
This subsection addresses the need to select a new initial Picard iterate ˜ 𝜂 ( ) ( 𝑘 ) in the absenceof (Z2). Also addressed is the fact that (4.21) is a purely theoretical, rather than implementable,solution because it contains infinite sums along an infinite timeline.In the context of ILC, the learned feedforward control action is often intended to be a relativelyminor adjustment to the primary action of the feedback controller. Thus, choosing ˜ 𝜂 ( ) ( 𝑘 ) to bethe feedback-only trajectory, i.e. the zero-feedforward-input trajectory, is akin to warm-startingthe fixed-point solving process. This trajectory is given by ˆ 𝑥 ( 𝑘 + ) = ˆ 𝑓 ( ˆ 𝑥 ( 𝑘 ) , , 𝑘 ) ˆ 𝑥 ( ) = 𝑛 𝑥 ˜ 𝜂 ( ) ( 𝑘 ) = 𝑉 − (cid:34) 𝜇 × 𝜇 𝜇 × 𝑛 𝜂 𝑛 𝜂 × 𝜇 𝐼 𝑛 𝜂 × 𝑛 𝜂 (cid:35) ˆ 𝑥 ( 𝑘 ) (4.43)for 𝑘 ∈ { , · · · , 𝑁 − 𝜇 } .An implementable version of (4.21) is given by ˜ 𝜂 ( 𝑚 + ) ( 𝑘 ) = 𝑁 − 𝜇 + ∑︁ 𝜅 = 𝜙 ( 𝑘 − 𝜅 ) (cid:0) ˜ 𝑓 𝜂 (cid:0) ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) , ˆ 𝓎 ( 𝜅 − ) , 𝜅 − (cid:1) − ˜ 𝐴 ˜ 𝜂 ( 𝑚 ) ( 𝜅 − ) (cid:1) (4.44)for 𝑘 ∈ { , ..., 𝑁 − 𝜇 } , fixing the initial condition ˜ 𝜂 ( 𝑚 ) ( ) = 𝑛 𝜂 ∀ 𝑚 .Note that (4.44) is equivalent to assuming ˜ 𝜂 ( 𝑚 ) ( 𝑘 ) = , ˆ 𝓎 ( 𝑘 ) = , and ˜ 𝑓 𝜂 ( , , 𝑘 ) = for 𝑘 ∈ (−∞ , − ] ∪ [ 𝑁 − 𝜇 + , ∞) and extracting the 𝑘 ∈ [ , 𝑁 − 𝜇 ] elements of ˜ 𝜂 ( 𝑚 + ) ( 𝑘 ) generatedby (4.21). These assumptions correspond to a lack of control action prior to 𝑘 = and a referencetrajectory that brings the system back to its zero initial condition with enough trailing zeros for thesystem to settle by 𝑘 = 𝑁 − 𝜇 . This is typical of repetitive motion processes, but admittedly maypreclude some other ILC applications.Furthermore, for the first Picard iteration ( 𝑚 + = ) these assumptions yield identical (4.44)-and (4.21)-generated ˜ 𝜂 ( ) ( 𝑘 ) on 𝑘 ∈ [ , 𝑁 − 𝜇 ] . Because output tracking of NMP systems ingeneral requires preactuation, for this range of 𝑘 to contain a practical control input trajectorythere must be sufficient leading zeros in the reference starting at 𝑘 = . For the following Picarditerates the theoretical and implementable trajectories are unlikely to be equal, but can be madecloser the more leading zeros are included in the reference.Ultimately, applying (4.44) for any number of iterations 𝑚 final ≥ yields an expression foreach time step of ˜ 𝜂 ( 𝑚 final ) ( 𝑘 ) whose only variable parameters are the elements of ˆ y . This is because82he recursion calling ˜ 𝜂 ( 𝑚 final ) ( 𝑘 ) terminates at the known trajectory ˜ 𝜂 ( ) ( 𝑘 ) , and because ˆ 𝑦 ( 𝑘 ) = for 𝑘 ∈ { , ..., 𝜇 − } due to the known initial condition ˆ 𝑥 ( ) = . The concatenation of theseexpressions plugged into the inverse output function (4.5b) yields the lifted inverse system model ˆ g − ( ˆ y ) = ˆ 𝑓 𝜇 − (cid:0) ˜ 𝜂 ( 𝑚 final ) ( ) , ˆ 𝓎 ( ) , (cid:1) ˆ 𝑓 𝜇 − (cid:0) ˜ 𝜂 ( 𝑚 final ) ( ) , ˆ 𝓎 ( ) , (cid:1) ... ˆ 𝑓 𝜇 − (cid:0) ˜ 𝜂 ( 𝑚 final ) ( 𝑁 − 𝜇 ) , ˆ 𝓎 ( 𝑁 − 𝜇 ) , 𝑁 − 𝜇 (cid:1) (4.45)which enables the synthesis of the ILILC learning matrix (4.19). With this, the complete syn-thesis of ILILC with stable inversion—starting from a model in the normal form (4.1)—can besummarized by Procedure 4.1. Procedure 4.1
ILILC Synthesis with Stable Inversion Derive the minimal state space representation ˆ 𝑓 𝜂 and ˆ 𝑓 𝜇 − (from (4.5)) of the conventionalinverse of (4.1). Apply similarity transform 𝑉 (from (4.9)) to derive the inverse state dynamics representation ˜ 𝑓 𝜂 (from (4.10)) with decoupled stable and unstable linear parts. Use (4.43)-(4.44) to derive the inverse system state ˜ 𝜂 ( 𝑚 final ) as a function of ˆ y at each point intime 𝑘 ∈ { , · · · , 𝑁 − 𝜇 } . Derive the lifted inverse model ˆ g − via (4.45). Use an automatic differentiation tool to derive 𝜕 ˆ g − 𝜕 ˆ y as a function of y , i.e. the learning matrix 𝐿 ℓ from (4.19). Compute 𝐿 ℓ = 𝜕 ˆ g − 𝜕 ˆ y ( y ℓ ) at each trial for the ILC law (3.7).// Steps 3-4 are greatly facilitated by using a computer algebra system. CasADi can providethis functionality in addition to automatic differentiation. This section presents validation of the fundamental claim that the original NILC fails for modelswith unstable inverses and that the newly proposed ILILC framework—when used with stableinversion—succeeds. Additionally, while the intent of ILC is to account for model error, overlyerroneous modeling can cause violation of (C2.3), which may cause divergence of the ILC law.Thus this section also probes the performance and robustness of ILILC with stable inversion overincreasing model error in physically motivated simulations.The ILILC law (3.7), (4.19) is applied as a reference shaping tool to a feedback control system(sometimes called “series ILC”). This represents the common scenario of applying a higher level83ontroller to “closed source” equipment. The resultant system (3.8) is a nonlinear time-varyingsystem with relative degree 𝜇 = .Modeling error is simulated by synthesizing the ILC laws from a nominal “control model” ofthe example system, and applying the resultant control inputs to a set of “truth models” featuringrandom parameter errors and the injection of process and measurement noise. Finally, to give con-text to the results for ILILC with stable inversion, identical simulations are run with a benchmarktechnique that does not require modification for NMP systems: gradient ILC. Aside from those using NILC, the authors know of no prior art explicitly addressing ILC synthesisfrom discrete-time nonlinear (particularly nonlinear in the input) time-varying models with relativedegree greater than 1. However, with automatic differentiation, “gradient ILC” is nearly as easilysynthesized from this class of models as it is from the LTI models it was proposed for in [111].The most straightforward form of Gradient ILC is gradient descent applied to the optimizationproblem arg min u e 𝑇 e (4.46)which yields the ILC law u ℓ + = u ℓ + 𝛾 𝜕 ˆ g 𝜕 u ( u ℓ ) 𝑇 e 𝑗 (4.47)where 𝛾 > is the gradient descent step size. Note that (4.47) is free of the matrix inversion thathistorically inhibited the application of NILC to systems with unstable inverses. 𝛾 is a tuning parameter that influences the performance-robustness trade off of (4.47). Reducing 𝛾 improves the probability that (4.47) will converge for some unknown model error, but may alsoreduce the rate of convergence. For the sake of comparing the convergence rates between gradientILC and ILILC, here we choose 𝛾 such that the two methods have comparable probabilities ofconvergence over the battery of random model errors tested: 𝛾 = . . Consider the system pictured in Figure 4.1, consisting of a pendulum fixed to the mass center of acart on a rail. This subsection presents the first-principles continuous-time equations of motion forthis plant, the method for converting these dynamics to the discrete-time normal form (4.1), andthe control architecture of the system.The cart is subjected to an applied force 𝑐 , and viscous damping occurs both between the cartand the rail and between the pendulum and the cart. Equations of motion for this plant are given84 igure 4.1: Cart and pendulum system. Dimension, position, and mass annotations are in grey. Force andtorque annotations are in black. by (cid:165) 𝜓 = − (cid:16) 𝐻 𝑀 𝑝 ( 𝑐 + 𝜔 𝑐 ) cos ( 𝜓 ) + 𝑑 𝑝 ( 𝑀 𝑐 + 𝑀 𝑝 ) (cid:164) 𝜓 + 𝐻 𝑀 𝑝 sin ( 𝜓 ) cos ( 𝜓 ) (cid:164) 𝜓 + ℊ 𝐻 (cid:16) 𝑀 𝑐 𝑀 𝑝 + 𝑀 𝑝 (cid:17) sin ( 𝜓 )− 𝑑 𝑐 𝐻 𝑀 𝑝 cos ( 𝜓 ) (cid:164) 𝑧 (cid:17) 𝐻 𝑀 𝑝 (cid:0) ( 𝑀 𝑐 + 𝑀 𝑝 ) − 𝑀 𝑝 cos ( 𝜓 ) (cid:1) (4.48) (cid:165) 𝑧 = (cid:0) 𝐻 ( 𝑐 + 𝜔 𝑐 ) + 𝑑 𝑝 cos ( 𝜓 ) (cid:164) 𝜓 + 𝐻 𝑀 𝑝 sin ( 𝜓 ) (cid:164) 𝜓 + ℊ 𝐻 𝑀 𝑝 sin ( 𝜓 ) cos ( 𝜓 )− 𝑑 𝑐 𝐻 (cid:164) 𝑧 (cid:1) 𝐻 (cid:0) ( 𝑀 𝑐 + 𝑀 𝑝 ) − 𝑀 𝑝 cos ( 𝜓 ) (cid:1) (4.49)where 𝜓 ( 𝑘 ) is the pendulum angle, 𝑧 ( 𝑘 ) is the cart’s horizontal position, ℊ = . ms is gravita-tional acceleration, and the process noise 𝜔 𝑐 ( 𝑘 ) is a random sample from a normal distributionwith mean and standard deviation . E − . 𝐻 is the pendulum half-length, 𝑀 𝑐 and 𝑀 𝑝 arethe cart and pendulum masses, and 𝑑 𝑐 and 𝑑 𝑝 are the cart-rail and pendulum-cart damping coef-ficients, respectively. The time argument of 𝜔 𝑐 , 𝜓 , 𝑧 and their derivatives has been dropped forcompactness.The output to be tracked is the pendulum tip’s horizontal position, 𝑦 . Obtaining a discrete-timestate space model of this system in the normal form (4.1) requires first a change of coordinates85uch that the desired output is a state, and then discretization. The change of coordinates is 𝜓 = arcsin (cid:16) 𝑦 − 𝑧 𝐻 (cid:17) (4.50)with associated derivative substitutions (cid:164) 𝜓 = (cid:164) 𝑦 − (cid:164) 𝑧 𝐻 √︃ − ( 𝑦 − 𝑧 ) 𝐻 (4.51) (cid:165) 𝜓 = sec ( 𝜓 ) (cid:0) (cid:165) 𝑦 − (cid:165) 𝑧 + 𝐻 sin ( 𝜓 ) (cid:164) 𝜓 (cid:1) 𝐻 (4.52)Then the equations of motion are solved for in terms of the new coordinates. In the present case(4.48)-(4.52) can be solved for (cid:165) 𝑦 ( 𝑘 ) and (cid:165) 𝑧 ( 𝑘 ) as functions of 𝑦 ( 𝑘 ) , 𝑧 ( 𝑘 ) , (cid:164) 𝑦 ( 𝑘 ) , and (cid:164) 𝑧 ( 𝑘 ) . Next,forward Euler discretization is applied recursively to the equations of motion to reformulate thestate dynamics in terms of discrete time increments rather than derivatives, as is required by thenormal form. The innermost layer of the recursion is the first derivatives (cid:164) 𝑦 ( 𝑘 ) = 𝑦 ( 𝑘 + ) − 𝑦 ( 𝑘 ) 𝑇 𝑠 (cid:164) 𝑧 ( 𝑘 ) = 𝑧 ( 𝑘 + ) − 𝑧 ( 𝑘 ) 𝑇 𝑠 (4.53)where the sample period 𝑇 𝑠 = .
016 s in this case. These can be plugged into (cid:165) 𝑦 ( 𝑘 ) and (cid:165) 𝑧 ( 𝑘 ) toeliminate their dependence on derivatives. The next—and in this case final—layer is the forwardEuler discretization of the second derivatives. The outermost layer can be rearranged to yield thediscrete-time equations of motion 𝑦 ( 𝑘 + ) = (cid:165) 𝑦 ( 𝑘 ) 𝑇 𝑠 + 𝑦 ( 𝑘 + ) − 𝑦 ( 𝑘 ) 𝑧 ( 𝑘 + ) = (cid:165) 𝑧 ( 𝑘 ) 𝑇 𝑠 + 𝑧 ( 𝑘 + ) − 𝑧 ( 𝑘 ) , (4.54)which are directly used to define the state dynamics 𝑓 in terms of the state vector 𝑥 ( 𝑘 ) = [ 𝑦 ( 𝑘 ) , 𝑦 ( 𝑘 + ) , 𝑧 ( 𝑘 ) , 𝑧 ( 𝑘 + )] 𝑇 . The explicit expressions of (4.54) are too long to print here,but can be easily obtained in Mathematica, MATLAB symbolic toolbox, etc. via the algebra de-scribed in (4.50)-(4.54).The output must track the reference 𝑟 ( 𝑘 ) given in Figure 4.2. To accomplish this the plant isequipped with a full-state feedback controller modeled as 𝑐 ( 𝑘 ) = 𝜅 𝑟 ∗ ( 𝑘 ) − (cid:104) 𝜅 𝜅 𝜅 𝜅 (cid:105) 𝑥 ( 𝑘 ) (4.55) 𝑟 ∗ ( 𝑘 ) = 𝑟 ( 𝑘 ) + 𝑢 ( 𝑘 ) (4.56)Here, 𝑟 ∗ ( 𝑘 ) is the effective reference and 𝑢 ( 𝑘 ) is the control input generated by the ILC law. In86 Time [s] R e f e r e n ce [ m ] Figure 4.2: Reference other words, the ILC law adjusts the reference delivered to the feedback controller to eliminatethe error transients inherent to feedback control. Finally, the error signal input to the ILC law issubject to measurement noise 𝜔 𝑦 ( 𝑘 ) 𝑒 ( 𝑘 ) = 𝑟 ( 𝑘 ) − 𝑦 ( 𝑘 ) − 𝜔 𝑦 ( 𝑘 ) (4.57)where the noise’s distribution has 0 mean and standard deviation E − .The ILC law itself is synthesized from a “control model” that is identical in structure to the“truth model” presented above, but has ˆ 𝜔 𝑐 = ˆ 𝜔 𝑦 = and uses the model parameters tabulatedin Table 4.1. Stable inversion for the synthesis of learning matrix (4.19) is performed with asingle Picard iteration, i.e. 𝑚 final = in (4.45). To simulate model error, the hatless truth modelparameters differ from the behatted control model parameters in a manner detailed in Section 4.4.3.This ultimately results in the system block diagram given in Figure 4.3. Let ˆ 𝑒 𝜃 ∈ R be a vector of the control model parameters in Table 4.1. Then a truth model can bespecified by the vector 𝜃 , generated via 𝑒 𝜃 = (cid:16) × 𝑒 𝑇𝑒 𝜃 (cid:12) 𝐼 + 𝐼 (cid:17) ˆ 𝑒 𝜃 (4.58)where (cid:12) is the Hadamard product and 𝑒 𝑒 𝜃 ∈ R is a random sample of a uniform distribution.Under (4.58), each element of 𝑒 𝑒 𝜃 is the relative error between the corresponding elements of 𝑒 𝜃 and ˆ 𝑒 𝜃 . Thus, (cid:13)(cid:13) 𝑒 𝑒 𝜃 (cid:13)(cid:13) provides a scalar metric for the model error between the control model anda given truth model. The range (cid:13)(cid:13) 𝑒 𝑒 𝜃 (cid:13)(cid:13) ∈ [ , . ] is divided into 20 bins of equal width, and 5087able 4.1: Cart-Pendulum Control Model ParametersParameter Symbol ValueCart Mass ˆ 𝑀 𝑐 . Pendulum Mass ˆ 𝑀 𝑝 .
25 kg
Pendulum Half-Length ˆ 𝐻 .
225 m
Cart-Rail Damping Coefficient ˆ 𝑑 𝑐 kgs Pendulum-Cart Damping Coefficient ˆ 𝑑 𝑝 . kg m s Full State Feedback Gain 0 ˆ 𝜅 Full State Feedback Gain 1 ˆ 𝜅 − Full State Feedback Gain 2 ˆ 𝜅 Full State Feedback Gain 3 ˆ 𝜅 − Full State Feedback Gain 4 ˆ 𝜅 ω c r r ∗ κ c Plant f x (cid:102) κ κ κ κ (cid:103) (cid:102) × n x − (cid:103) ILC Law y − u e ω y Intra-trial signalInter-trial signal − Figure 4.3: System Block Diagram. The control law outputting 𝑢 is synthesized from the control modelsdefined by the behatted parameters of Table 4.1 and by 𝜔 𝑐 ( 𝑘 ) = 𝜔 𝑦 ( 𝑘 ) = . The plant and controller gainblocks are defined with the truth model parameters generated according to Section 4.4.3. Inter-trial signalsfrom trial ℓ are stored and used to compute the input for trial ℓ + . 𝑢 ( 𝑘 ) = ∀ 𝑘 . A full set of 50 trials of one of the ILC laws applied to a single truthmodel is referred to as a “simulation.” The results of these simulations are used to characterize theprobability of convergence and rate of convergence of each ILC law.For each iteration of a simulation, the normalized root mean square error (NRMSE) is given byNRMSE ℓ (cid:66) RMS ( e ℓ )(cid:107) r (cid:107) ∞ (4.59)A simulation is deemed convergent if there exists ℓ ∗ such that NRMSE ℓ is less than some tolerancefor all ℓ ≥ ℓ ∗ . This work uses a tolerance of E − , which is close to the NRMSE floor created bynoise.Let ℓ 𝛽,𝜏,𝜆 be the minimum ℓ ∗ for truth model 𝜏 ∈ [ , ] in bin 𝛽 ∈ [ , ] under ILC law 𝜆 ∈ { ILILC , gradient ILC } , and let C be the set of all ( 𝛽, 𝜏 ) for which both ILILC and gradientILC converge. Then the mean transient convergence rate R ℓ = mean C ,ℓ ∈[ ,ℓ 𝛽,𝜏,𝜆 ] (cid:32) NRMSE 𝛽,𝜏,𝜆ℓ
NRMSE 𝛽,𝜏,𝜆ℓ − (cid:33) (4.60)offers a numerical performance metric. Note that [34] gives a theoretical convergence analysis forthe ILC structure (3.7) in general (covering NILC, ILILC, and gradient ILC). This analysis canbe used to lower bound performance (i.e. upper bound convergence rate) via multiple parameterscomputed from the learning matrix 𝐿 ℓ and the true dynamics g . The mean transient convergencerate (4.60) may thus serve as a specific, measurable counterpart to any theoretical worst-case-scenario analyses performed via the formulas in [34].Finally, to verify the fundamental necessity and efficacy of ILILC for systems with unstableinverses, 2 trials of traditional stable-inversion-free NILC (3.7), (3.15) are applied to each truthmodel. The condition number of 𝜕 ˆ g 𝜕 u ( u ) is E . Attempted inversion of this matrix in MATLAB yieldsan inverse matrix with average nonzero element magnitude of E and max element magnitude of E . Consequently, u generated by (3.7), (3.15) has an average element magnitude of E
10 m anda max element magnitude of E
11 m , which is so large that y and 𝜕 ˆ g 𝜕 u ( u ) contain NaN elements forall simulations. Conversely, while some simulations using ILILC, i.e. (3.7), (4.19), diverge due toexcessive model error, the majority converge. This validates the fundamental claim that the directapplication of Newton’s method in NILC is insufficient for systems with unstable inverses, and89 igure 4.4: Representative input solution trajectories from low- and high-model-error ILILC simulationscompared with the solution to the zero-model-error problem. The zero-model-error solution is the inputtrajectory that would be chosen for feedforward control in the absence of learning, and differs notably fromboth minimum-error trajectories found by ILILC with stable inversion. that slight modification of the learning matrix and the incorporation of stable inversion addressesthis gap.To accompany the quantitative metric (cid:107) 𝑒 𝜃 (cid:107) , Figure 4.4 offers a qualitative sense of the degreeof model error in this study by comparing two representative ILILC solution trajectories 𝑢 ( 𝑘 ) with the solution to the (cid:107) 𝑒 𝜃 (cid:107) = , 𝜔 𝑐 ( 𝑘 ) = 𝜔 𝑦 ( 𝑘 ) = scenario. The lower-model-error rep-resentative solution is from within the range of (cid:107) 𝑒 𝜃 (cid:107) for which all simulations converged, whilethe higher-model-error solution comes from a bin in which some simulations diverged. A moredetailed analysis of the boundaries in 𝜃 -space determining convergence or divergence of a simu-lation is beyond the scope of this work. However, the given trajectories illustrate that even in theconservative subspace defined by the 100% convergent bins learning bridges a visible performancegap, and that beyond this subspace there are far greater performance gains to be had.Finally, a statistical comparison of the performance and robustness of ILILC with stable inver-sion and gradient ILC is given in Figure 4.5. The tuning of gradient ILC indeed yields comparablerobustness to ILILC, with ILILC 97% as likely to converge as gradient ILC over all simulations.The convergence rates of the two ILC schemes, however, differ substantially, with gradient ILCtaking over 3 times as many trials as ILILC to converge on average. The mean transient conver-gence rate values tabulated in Table 4.2 give a more portable quantification of ILILC’s advantage,having a convergence rate nearly half that of gradient ILC’s.This analysis confirms that ILILC with stable inversion is an important addition to the engi-neer’s toolbox because it enables ILC synthesis from nonlinear non-minimum phase models and90 LILC + Stab. Inv.Gradient ILC
ILILC + Stab. Inv. Mean (714 samples)Gradient ILC Mean (714 samples)Convergence ThresholdStandard DeviationStandard Deviation
Figure 4.5:
Top:
Histogram giving the percentage of simulations converged in each bin of the model errormetric (cid:107) 𝑒 𝜃 (cid:107) . Bottom:
Mean value of NRMSE for each ILC trial over all simulations that are convergent forboth gradient ILC and ILILC with stable inversion. This illustrates that for comparable robustness to modelerror, ILILC converges substantially faster than gradient ILC.
Table 4.2: Transient Convergence Rates for ILILC and Gradient ILCILC Law Mean Standard DeviationGradient ILC 0.76 0.17ILILC + Stab. Inv. 0.41 0.27delivers the fast convergence characteristic of algorithms based on Newton’s method.
This chapter introduces and validates a new ILC synthesis scheme applicable to nonlinear time-varying systems with unstable inverses and relative degree greater than 1. This is done with thesupport of nonlinear stable inversion, which is advanced from the prior art via proof of convergencefor an expanded class of systems and methods for improved practical implementation. In all, thisresults in a new, broadly implementable ILC scheme displaying a competitive convergence speedunder benchmark testing. 91uture work may focus on further broadening the applicability of ILILC by relaxing referenceand initial condition repetitiveness assumptions, and on the extension of ILILC with a potentiallyadaptive tuning parameter or other means to enable the exchange of some speed for robustnesswhen called for. Levenberg-Marquardt-Fletcher algorithms may offer one source of inspiration forsuch work. 92
HAPTER 5
Hybrid Systems with Unstable Inverses:Stable Inversion of Piecewise Affine Systems
As discussed in Chapter 1, despite the call for it [46], little work has been done on the feedfor-ward control of PWA systems. Naturally, a lack of feedforward control research in general impliesa lack of research on the more specific problem of feedforward control for NMP PWA systems.Thus, while the combination of ILILC and stable inversion is promising for the ILC of PWAsystems with NMP dynamics, there are two major gaps in the literature that must be filled beforesuch control can be realized. First, there does not exist any published theory on stable inversionfor PWA systems (or hybrid systems of any kind). Second, there does not exist sufficient literatureon the conventional closed-form PWA system inversion prerequisite for stable inversion.The most relevant prior art on PWA system inversion is Sontag’s foundational work on piece-wise linear systems [112]. Sontag proposes that these systems are potentially invertible-with-delay,and explains the signal time shifting necessary to accommodate this delay. This serves as a begin-ning for the concept of the relative degree of a PWA system. However, these concepts requirefurther development to account for the fact that a PWA system’s apparent relative degree maychange during switching between component models and the fact that PWA systems may havemultiple inverses. Additionally, Sontag leaves the derivation of the inverse dynamics other thanthe delay open as a “nontrivial part” of the inversion process. Finally, for reference tracking it isdesirable to invert a system without delay. This frequently results in anticausal inverses—those inwhich future reference values are required to compute a current input—but this is no problem fortypical feedforward control scenarios where the entire reference is known in advance.Further work of Sontag [113] delves into the abstract algebra of piecewise linear functions andincludes the useful facts that it is decidable whether two sets are isomorphic under piecewise linear
Content of this chapter also to be submitted as:I. A. Spiegel, N. Strijbosch, R. de Rozario, T. Oomen and K. Barton, “Stable Inversion of Piecewise AffineSystems with Application to Feedforward and Iterative Learning Control,” in
IEEE Transactions on Automatic Control .Copyright may be transferred without notice, after which this version may no longer be accessible.
A variety of similar discrete-time PWA system definitions appear in the literature. For the sakeof familiarity and simplicity, the following time-varying system definition uses a representationsimilar to [2] and [114].
Definition 5.1 (PWA System) . A PWA system is given by 𝑥 𝑘 + = 𝐴 𝑞,𝑘 𝑥 𝑘 + 𝐵 𝑞,𝑘 𝑢 𝑘 + 𝐹 𝑞,𝑘 𝑦 𝑘 = 𝐶 𝑞,𝑘 𝑥 𝑘 + 𝐷 𝑞,𝑘 𝑢 𝑘 + 𝐺 𝑞,𝑘 for 𝑥 𝑘 ∈ 𝑄 𝑞 (5.1)94here 𝑘 ∈ Z is the time-step index, 𝑥 ∈ R 𝑛 𝑥 , 𝑢 ∈ R 𝑛 𝑢 , and 𝑦 ∈ R 𝑛 𝑦 are the state, input, and outputvectors, and 𝑄 𝑞 ∈ 𝑄 where 𝑄 is the set of “locations,” i.e. a set of disjoint regions with unionequal to R 𝑛 𝑥 . Each location 𝑄 𝑞 is the union of a set of disjoint convex polytopes. Here, a convexpolytope is defined simply as an intersection of half spaces. Additionally, let the relative degree ofthe 𝑞 th component model be denoted 𝜇 𝑞 for 𝑞 ∈ (cid:74) , | 𝑄 | (cid:75) ( (cid:74) (cid:75) indicates a closed set of integers).To facilitate both mathematical analysis and controller synthesis, the remainder of the chapteruses the equivalent closed-form representation 𝑥 𝑘 + = A 𝑘 𝑥 𝑘 + B 𝑘 𝑢 𝑘 + F 𝑘 (5.2a) 𝑦 𝑘 = C 𝑘 𝑥 𝑘 + D 𝑘 𝑢 𝑘 + G 𝑘 (5.2b)with the upright, bold, capital letter notation defined as M 𝑘 ≔ | 𝑄 | ∑︁ 𝑞 = 𝑀 𝑞,𝑘 𝐾 𝑞 ( 𝛿 𝑘 ) (5.3) 𝐾 𝑞 ( 𝛿 𝑘 ) ≔ (cid:206) | Δ ∗ 𝑞 | 𝑖 = (cid:13)(cid:13)(cid:13) 𝛿 ∗ 𝑞,𝑖 − 𝛿 𝑘 (cid:13)(cid:13)(cid:13) = 𝛿 𝑘 ∈ Δ ∗ 𝑞 otherwise (5.4) 𝛿 𝑘 = 𝛿 ( 𝑥 𝑘 ) (cid:66) 𝐻 ( 𝑃𝑥 𝑘 − 𝛽 ) (5.5)where M and 𝑀 𝑞,𝑘 stand in for any of { A , B , F , C , D , G } , and { 𝐴 𝑞,𝑘 , 𝐵 𝑞,𝑘 , 𝐹 𝑞,𝑘 , 𝐶 𝑞,𝑘 , 𝐷 𝑞,𝑘 , 𝐺 𝑞,𝑘 } ,respectively. 𝐻 is the Heaviside step function evaluated element-wise on its vector argument, 𝐾 𝑞 is the binary-output selector function for the 𝑞 th location, Δ ∗ 𝑞 = { 𝛿 ∗ 𝑞, , 𝛿 ∗ 𝑞, , · · · } is the set of binaryvector signatures of the 𝑞 th location, 𝑃 ∈ R 𝑛 𝑃 × 𝑛 𝑥 is a matrix consisting of concatenated hyperplaneorientation vectors, and 𝛽 ∈ R 𝑛 𝑃 is a vector of hyperplane offsets. For more information aboutclosed form representations of piecewise defined systems, see Chapter 3.Furthermore, the following assumptions are made for all systems(A5.1) 𝑥 ∈ 𝑋 , where 𝑋 is the set of initial conditions from which all locations 𝑄 𝑞 ∈ 𝑄 arereachable in finite time(A5.2) the system is single-input-single-output (SISO), 𝑛 𝑢 = 𝑛 𝑦 = (A5.3) switching depends only on the states, not the input(A5.4) all component models have the same relative degree, 𝜇 𝑐 , for all time, 𝜇 𝑞 = 𝜇 𝑐 ∀ 𝑞 ∈ (cid:74) , | 𝑄 | (cid:75) and ∀ 𝑘 Definition 5.2 (Global Dynamical Relative Degree) . The global dynamical relative degree of aSISO PWA system is the smallest number 𝜇 𝑔 ≥ such that the explicit expression of 𝑦 𝑘 + 𝜇 𝑔 interms of component state space matrices, selector functions, 𝑥 𝑘 , and 𝑢 𝑖 , 𝑖 ≥ 𝑘 contains 𝑢 𝑘 outsideof a selector function for all switching sequences on the interval (cid:74) 𝑘, 𝑘 + 𝜇 𝑔 (cid:75) .This 𝜇 𝑔 is essentially the traditional relative degree, but neglecting inputs appearing in selectorfunctions. This neglect is introduced to avoid situations in which the only explicit appearance ofthe input in the output function is within the Heaviside function, leading to a potentially infinitenumber of input values yielding the same output. Such non-injectiveness would make inversionunusually challenging. The assumption (A5.3) is made for similar reasons. This section presents the conventional exact inverses of PWA systems under assumptions (A5.1)-(A5.4). Conventional inversion is the process of(Step 1) obtaining an expression for the previewed output 𝑦 𝑘 + 𝜇 𝑔 in terms of 𝑥 𝑘 , M 𝑘 + 𝜅 , and 𝑢 𝑘 + 𝜅 where 𝜅 ≥ ,(Step 2) solving the previewed output equation for 𝑢 𝑘 in terms of 𝑥 𝑘 , M 𝑘 + 𝜅 , and 𝑢 𝑘 + 𝜅 + , and finally(Step 3) taking this expression of 𝑢 𝑘 as the output function of the inverse system, and plugging itin to (5.2a) to obtain the state transition formula of the inverse system.Note that 𝑦 𝑘 + 𝜇 𝑔 is necessarily an explicit function of 𝑢 𝑘 by Definition 5.2.For systems with 𝜇 𝑔 = , this inverse system is unique, and can be expressed explicitly. Forsystems with 𝜇 𝑔 ≥ , there may be multiple solutions to the problem of solving 𝑦 𝑘 + 𝜇 𝑔 for 𝑢 𝑘 (Step2), and thus the inverse system cannot be expressed explicitly without additional assumptions. Thissection gives both the general, implicit inverse system for 𝜇 𝑔 ≥ systems and sufficient conditionsfor the uniqueness of system inversion for 𝜇 𝑔 ∈ { , } systems along with the correspondingexplicit inverse systems. 𝜇 𝑔 = Lemma 1 (Relative Degree of 0) . The global dynamical relative degree of a reachable SISO PWAsystem, i.e. a PWA system satisfying (A5.1) and (A5.2), is 0 if and only if the relative degree of all omponent models are 0 for all time: 𝜇 𝑞 = ∀ 𝑞 ∈ (cid:74) , | 𝑄 | (cid:75) , ∀ 𝑘 ⇐⇒ 𝜇 𝑔 = (5.6) Proof.
First, the forward implication is proven directly. 𝜇 𝑞 = ∀ 𝑞, 𝑘 = ⇒ 𝐷 𝑞,𝑘 ≠ ∀ 𝑞, 𝑘 (5.7) 𝐷 𝑞,𝑘 ≠ ∀ 𝑞, 𝑘 = ⇒ D 𝑘 ≠ ∀ 𝑘 (5.8) D 𝑘 ≠ ∀ 𝑘 = ⇒ (5.2b) always explicitly contains 𝑢 𝑘 (5.9) ∴ 𝜇 𝑞 = ∀ 𝑞 ∈ 𝑄 = ⇒ 𝜇 𝑔 = (5.10)Now the backwards implication is proven by proving the contrapositive. ∃ 𝑞, 𝑘 s.t. 𝜇 𝑞 ≠ = ⇒ ∃ 𝑞, 𝑘 s.t. 𝐷 𝑞,𝑘 = (5.11)By (A5.1), there exists some finite sequence of inputs to bring the system to this location 𝑄 𝑞 atsome time step 𝑘 such that D 𝑘 = 𝐷 𝑞,𝑘 = , and (5.2b) becomes 𝑦 𝑘 = C 𝑘 𝑥 𝑘 + G 𝑘 (5.12)which is not an explicit function of 𝑢 𝑘 , meaning 𝜇 𝑔 ≠ . ∴ ¬ (cid:0) 𝜇 𝑞 = ∀ 𝑞 ∈ (cid:74) , | 𝑄 | (cid:75) , ∀ 𝑘 (cid:1) = ⇒ ¬ (cid:0) 𝜇 𝑔 = (cid:1) (5.13)By (5.10) and (5.13), (5.6) must be true. (cid:3) Theorem 5.1 ( 𝜇 𝑔 = PWA System Inverse) . The inverse of a PWA system satisfying (A5.1)-(A5.4)with 𝜇 𝑐 = is itself a PWA system satisfying (A5.1)-(A5.4) and is given by 𝑥 𝑘 + = A 𝑘 𝑥 𝑘 + B 𝑘 𝑦 𝑘 + F 𝑘 (5.14) 𝑢 𝑘 = C 𝑘 𝑥 𝑘 + D 𝑘 𝑦 𝑘 + G 𝑘 (5.15) where A 𝑘 = A 𝑘 − B 𝑘 D − 𝑘 C 𝑘 B 𝑘 = B 𝑘 D − 𝑘 F 𝑘 = F 𝑘 − B 𝑘 D − 𝑘 G 𝑘 C 𝑘 = − D − 𝑘 C 𝑘 D 𝑘 = D − 𝑘 G 𝑘 = − D − 𝑘 G 𝑘 such that 𝑦 𝑘 is the input of the inverse system and 𝑢 𝑘 is the output. roof. (Step 1) is satisfied by (5.2b) and (Step 2) by 𝑢 𝑘 = D − 𝑘 ( 𝑦 𝑘 − C 𝑘 𝑥 𝑘 − G 𝑘 ) (5.16)Equation (5.16) is always well-defined because by Lemma 1 𝜇 𝑐 = = ⇒ 𝜇 𝑔 = , which in turnimplies D 𝑘 ≠ ∀ 𝑘 , and because D is always scalar because the system is SISO by (A5.2). Plugging(5.16) into (5.2a) yields the inverse system state transition formula, satisfying (Step 3). (cid:3) 𝜇 𝑔 ≥ For systems with 𝜇 𝑔 ≥ , (5.2b) does not explicitly contain 𝑢 𝑘 because D 𝑘 = ∀ 𝑘 ; this is corollaryto Lemma 1, Definition 5.2, and (A5.4). Consequently, (Step 1) necessitates the derivation of anexplicit formula for the output preview 𝑦 𝑘 + 𝜇 𝑔 , i.e. the output at a time after the current time step 𝑘 .This preview of future output is necessary for deriving an output equation that explicitly dependson the current input 𝑢 𝑘 . Lemma 2 ( 𝜇 𝑔 ≥ PWA System Output Preview) . Given a PWA system satisfying (A5.1)-(A5.4)with known global dynamical relative degree 𝜇 𝑔 , the output function with minimum preview suchthat the function is explicitly dependent on an input term outside of the selector functions for anyswitching sequence is given by 𝑦 𝑘 + 𝜇 𝑔 = C 𝑘 𝑥 𝑘 + D 𝑘 𝑢 𝑘 + G 𝑘 + Ψ 𝑘 ( 𝑢 𝑘 + , · · · , 𝑢 𝑘 + 𝜇 𝑔 − ) (5.17) with C 𝑘 ≔ C 𝑘 + 𝜇 𝑔 (cid:169)(cid:173)(cid:171) 𝜇 𝑔 − (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:170)(cid:174)(cid:172) D 𝑘 ≔ C 𝑘 + 𝜇 𝑔 (cid:169)(cid:173)(cid:171) 𝜇 𝑔 − (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:170)(cid:174)(cid:172) B 𝑘 G 𝑘 ≔ C 𝑘 + 𝜇 𝑔 𝜇 𝑔 − ∑︁ 𝑠 = (cid:169)(cid:173)(cid:171)(cid:169)(cid:173)(cid:171) 𝜇 𝑔 − (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:170)(cid:174)(cid:172) F 𝑘 + 𝑠 (cid:170)(cid:174)(cid:172) + G 𝑘 + 𝜇 𝑔 Ψ 𝑘 ( 𝑢 𝑘 + , · · · ) ≔ C 𝑘 + 𝜇 𝑔 𝜇 𝑔 − ∑︁ 𝑠 = (cid:169)(cid:173)(cid:171)(cid:169)(cid:173)(cid:171) 𝜇 𝑔 − (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:170)(cid:174)(cid:172) B 𝑘 + 𝑠 𝑢 𝑘 + 𝑠 (cid:170)(cid:174)(cid:172) where the factors in the products generated by (cid:206) are ordered sequentially by the index 𝑚 . Thefactor corresponding to the greatest value of the index must be on the left, and the factor corre- ponding to the smallest value of the index must be on the right. For example, (cid:214) 𝑚 = A 𝑘 + 𝑚 ≡ A 𝑘 + A 𝑘 + A 𝑘 + (cid:46) A 𝑘 + A 𝑘 + A 𝑘 + (5.18) because matrices do not necessarily commute. Additionally, if the lower bound on the product in-dex exceeds the upper bound on the product index (an “empty product”), then the product resolvesto the identity matrix. Similarly, empty sums resolve to . For example (cid:214) 𝑚 = anything ≡ 𝐼 ∑︁ 𝑠 = anything ≡ (5.19) Proof.
Let the base case of the proof by induction be 𝜇 𝑔 = such that 𝑦 𝑘 + = C 𝑘 + ( A 𝑘 𝑥 𝑘 + F 𝑘 + B 𝑘 𝑢 𝑘 ) + G 𝑘 + (5.20)which is achieved equivalently from (5.17) and from the system definition by plugging (5.2a) into(5.2b) incremented by one time step (i.e. plugging the equation for 𝑥 𝑘 + into the equation for 𝑦 𝑘 + ). The preview is minimal because, by Definition 5.2, C 𝑘 + B 𝑘 ≠ for all switching sequenceson (cid:74) 𝑘, 𝑘 + (cid:75) , and by Lemma 1 𝜇 𝑔 ≥ = ⇒ D 𝑘 = ∀ 𝑘 . In other words, for 𝜇 𝑔 = , (5.20) isalways an explicit function of 𝑢 𝑘 . Thus, regardless of switching sequence, outputs further in thefuture, such as 𝑦 𝑘 + , never need to be considered in order to explicitly relate the current input to anoutput.Then consider (5.17) with 𝜇 𝑔 = 𝜈 as the foundation of the induction step. To prove (5.17) holdsfor 𝜇 𝑔 = 𝜈 + , first increment (5.17) with 𝜇 𝑔 = 𝜈 by one time step, yielding 𝑦 𝑘 + 𝜈 + = C 𝑘 + 𝜈 + (cid:32) 𝜈 − (cid:214) 𝑚 = A 𝑘 + + 𝑚 (cid:33) 𝑥 𝑘 + + C 𝑘 + 𝜈 + (cid:32) 𝜈 − (cid:214) 𝑚 = A 𝑘 + + 𝑚 (cid:33) B 𝑘 + 𝑢 𝑘 + + C 𝑘 + 𝜈 + 𝜈 − ∑︁ 𝑠 = (cid:32) (cid:32) 𝜈 − (cid:214) 𝑚 = 𝑠 + A 𝑘 + + 𝑚 (cid:33) F 𝑘 + + 𝑠 (cid:33) + G 𝑘 + + 𝜈 + C 𝑘 + 𝜈 + 𝜈 − ∑︁ 𝑠 = (cid:32) (cid:32) 𝜈 − (cid:214) 𝑚 = 𝑠 + A 𝑘 + + 𝑚 (cid:33) B 𝑘 + + 𝑠 𝑢 𝑘 + + 𝑠 (cid:33) (5.21)This can be simplified by first factoring out C 𝑘 + 𝜈 + and adjusting the product Π indices to subsume99he constant + in A 𝑘 + + 𝑚 : 𝑦 𝑘 + 𝜈 + = C 𝑘 + 𝜈 + (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:33) 𝑥 𝑘 + + (cid:32) 𝜈 (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:33) B 𝑘 + 𝑢 𝑘 + + 𝜈 − ∑︁ 𝑠 = (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:33) F 𝑘 + + 𝑠 (cid:33) + 𝜈 − ∑︁ 𝑠 = (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:33) B 𝑘 + + 𝑠 𝑢 𝑘 + + 𝑠 (cid:33) (cid:33) + G 𝑘 + 𝜈 + (5.22)Similarly, the sum (cid:205) indices may be adjusted to subsume the constant + in F 𝑘 + + 𝑠 , B 𝑘 + + 𝑠 , and 𝑢 𝑘 + + 𝑠 . Because the sum index 𝑠 also appears in the lower bound of the products, 𝑚 = 𝑠 + , theproduct index lower bound must also be adjusted. 𝑦 𝑘 + 𝜈 + = C 𝑘 + 𝜈 + (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:33) 𝑥 𝑘 + + (cid:32) 𝜈 (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:33) B 𝑘 + 𝑢 𝑘 + + 𝜈 ∑︁ 𝑠 = (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:33) F 𝑘 + 𝑠 (cid:33) + 𝜈 ∑︁ 𝑠 = (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:33) B 𝑘 + 𝑠 𝑢 𝑘 + 𝑠 (cid:33) (cid:33) + G 𝑘 + 𝜈 + (5.23)Finally, the two input terms (those containing 𝑢 , arising from D 𝑘 + and Ψ 𝑘 + ) can be combined toachieve 𝑦 𝑘 + 𝜈 + = C 𝑘 + 𝜈 + (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:33) 𝑥 𝑘 + + 𝜈 ∑︁ 𝑠 = (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:33) F 𝑘 + 𝑠 (cid:33) + 𝜈 ∑︁ 𝑠 = (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:33) B 𝑘 + 𝑠 𝑢 𝑘 + 𝑠 (cid:33) (cid:33) + G 𝑘 + 𝜈 + (5.24)which is a function of 𝑥 𝑘 + and potentially 𝑢 𝜅 for 𝜅 ∈ (cid:74) 𝑘 + , 𝑘 + 𝜈 + (cid:75) . The dependence of 𝑦 𝑘 + 𝜈 + on the input terms is conditioned on the switching sequence. Definition 5.2 implies that if 𝜇 𝑔 = 𝜈 + there exists some switching sequence on (cid:74) 𝑘, 𝑘 + 𝜈 + (cid:75) such that the input coefficientsin (5.24) are zero, i.e. ∃ { 𝑥 𝜅 | 𝜅 ∈ (cid:74) 𝑘, 𝑘 + 𝜈 + (cid:75) } s.t. ∀ 𝑠 ∈ (cid:74) , 𝜈 (cid:75) C 𝑘 + 𝜈 + (cid:32) 𝜈 (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:33) B 𝑘 + 𝑠 = (5.25)Thus, to guarantee the expression for 𝑦 𝑘 + 𝜈 + explicitly contains the input for all switching se-quences, 𝑥 𝑘 + in (5.24) must be expanded (via (5.2a)) to be in terms of 𝑥 𝑘 and 𝑢 𝑘 explicitly. The100esulting expression can be rearranged as follows: 𝑦 𝑘 + 𝜈 + = C 𝑘 + 𝜈 + (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:33) ( A 𝑘 𝑥 𝑘 + B 𝑘 𝑢 𝑘 + F 𝑘 )+ 𝜈 ∑︁ 𝑠 = (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:33) F 𝑘 + 𝑠 (cid:33) (cid:33) + Ψ 𝑘 ( 𝑢 𝑘 + , · · · , 𝑢 𝑘 + 𝜈 ) + G 𝑘 + 𝜈 + (5.26) 𝑦 𝑘 + 𝜈 + = C 𝑘 + 𝜈 + (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:33) A 𝑘 𝑥 𝑘 + (cid:32) 𝜈 (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:33) B 𝑘 𝑢 𝑘 + 𝜈 ∑︁ 𝑠 = (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:33) F 𝑘 + 𝑠 (cid:33) + (cid:32) 𝜈 (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:33) F 𝑘 + G 𝑘 + 𝜈 + (cid:33) + Ψ 𝑘 ( 𝑢 𝑘 + , · · · , 𝑢 𝑘 + 𝜈 ) (5.27) 𝑦 𝑘 + 𝜈 + = C 𝑘 + 𝜈 + (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:33) 𝑥 𝑘 + (cid:32) 𝜈 (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:33) B 𝑘 𝑢 𝑘 + 𝜈 ∑︁ 𝑠 = (cid:32) (cid:32) 𝜈 (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:33) F 𝑘 + 𝑠 (cid:33) (cid:33) + G 𝑘 + 𝜈 + + Ψ 𝑘 ( 𝑢 𝑘 + , · · · , 𝑢 𝑘 + 𝜈 ) (5.28)Equation (5.28) is equation (5.17) for 𝜇 𝑔 = 𝜈 + , thereby proving the lemma. (cid:3) Using the output preview equation (5.17), a general PWA system inverse for 𝜇 𝑔 ≥ can befound in the same manner as for 𝜇 𝑔 = . Theorem 5.2 (General 𝜇 𝑔 ≥ PWA System Inverse) . Given any PWA system satisfying (A5.1)-(A5.4) with known global dynamical relative degree 𝜇 𝑔 ≥ , the inverse system with 𝑢 𝑘 as theoutput is given by the implicit, anticausal system 𝑥 𝑘 + = A 𝑘 𝑥 𝑘 + B 𝑘 𝑦 𝑘 + 𝜇 𝑔 + F 𝑘 − B 𝑘 D − 𝑘 Ψ 𝑘 (cid:16) 𝑢 𝑘 + , · · · , 𝑢 𝑘 + 𝜇 𝑔 − (cid:17) (5.29) 𝑢 𝑘 = C 𝑘 𝑥 𝑘 + D 𝑦 𝑘 + 𝜇 𝑔 + G 𝑘 − D − 𝑘 Ψ 𝑘 (cid:16) 𝑢 𝑘 + , · · · , 𝑢 𝑘 + 𝜇 𝑔 − (cid:17) (5.30) where A 𝑘 = A 𝑘 + B 𝑘 C 𝑘 B 𝑘 = B 𝑘 D 𝑘 F 𝑘 = F 𝑘 + B 𝑘 G 𝑘 C 𝑘 = − D 𝑘 C 𝑘 D 𝑘 = D − 𝑘 G 𝑘 = − D 𝑘 G 𝑘 Proof.
The sole term in Lemma 2’s (5.17) containing 𝑢 𝑘 outside of a selector function is D 𝑘 𝑢 𝑘 . The101oefficient D 𝑘 = C 𝑘 + 𝜇 𝑔 (cid:16)(cid:206) 𝜇 𝑔 − 𝑚 = A 𝑘 + 𝑚 (cid:17) B 𝑘 is always scalar because the system is SISO, (A5.2), andalways nonzero by Definition 5.2. Thus (5.17) can be divided by D 𝑘 and 𝑢 𝑘 can be arithmeticallymaneuvered onto one side of the equation by itself, yielding (5.30).Equation (5.30) is implicit in general because (5.17) cannot be uniquely solved for 𝑢 𝑘 in gen-eral. This is proven by presenting an example in which multiple input trajectories have the sameoutput trajectory. Consider the two-location system 𝐴 ,𝑘 = (cid:34) (cid:35) 𝐴 ,𝑘 = (cid:34) (cid:35) 𝐵 ,𝑘 = 𝐵 ,𝑘 = (cid:34) (cid:35) 𝐶 ,𝑘 = 𝐶 ,𝑘 = (cid:104) (cid:105) 𝑃 = (cid:104) (cid:105) 𝛽 = . Δ ∗ = { } Δ ∗ = { } (5.31)with 𝐹 , 𝐷 , and 𝐺 matrices all equal to zero. Given 𝑥 𝑘 = [ , ] 𝑇 , both 𝑢 𝑘 = and 𝑢 𝑘 = yield 𝑦 𝑘 + 𝜇 𝑔 = . Because there is not a unique solution to (5.17) for 𝑢 𝑘 , there does not exist an explicitformula for the solution (5.30).The system is necessarily anticausal because 𝑢 𝑘 is necessarily a function of 𝑦 𝑘 + 𝜇 𝑔 and 𝜇 𝑔 > . (cid:3) Remark . Analytically, the implicitness of (5.30) arises in C 𝑘 through C 𝑘 ,which is a function of C 𝑘 + 𝜇 𝑔 by definition, and C 𝑘 + 𝜇 𝑔 is a function of 𝑥 𝑘 + 𝜇 𝑔 by (5.3)-(5.5). Finally, 𝑥 𝑘 + 𝜇 𝑔 is a function of 𝑢 𝑘 via 𝑥 𝑘 + 𝜇 𝑔 = (cid:169)(cid:173)(cid:171) 𝜇 𝑔 − (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:170)(cid:174)(cid:172) 𝑥 𝑘 + 𝜇 𝑔 − ∑︁ 𝑠 = (cid:169)(cid:173)(cid:171)(cid:169)(cid:173)(cid:171) 𝜇 𝑔 − (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:170)(cid:174)(cid:172) F 𝑘 + 𝑠 (cid:170)(cid:174)(cid:172) + 𝜇 𝑔 − ∑︁ 𝑠 = (cid:169)(cid:173)(cid:171)(cid:169)(cid:173)(cid:171) 𝜇 𝑔 − (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:170)(cid:174)(cid:172) B 𝑘 + 𝑠 𝑢 𝑘 + 𝑠 (cid:170)(cid:174)(cid:172) + (cid:169)(cid:173)(cid:171) 𝜇 𝑔 − (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:170)(cid:174)(cid:172) B 𝑘 𝑢 𝑘 (5.32)following from 𝑦 𝑘 + 𝜇 𝑔 = C 𝑘 + 𝜇 𝑔 𝑥 𝑘 + 𝜇 𝑔 + G 𝑘 + 𝜇 𝑔 and Lemma 2. Remark . Note Ψ 𝑘 is written as a functionof a set of previewed 𝑢 -values The written set of 𝑢 -values is the maximum quantity of 𝑢 -valuesthat may be required by Ψ 𝑘 . Depending on the switching sequence, fewer previewed 𝑢 -values maybe required. In fact, by the definition of global dynamical relative degree 𝜇 𝑔 , there must exist aswitching sequence for which no previewed 𝑢 -values are required, because otherwise 𝜇 𝑔 would be102maller. In other words, Definition 5.2 and Lemma 2 imply ∃{ 𝑥 𝜅 | 𝜅 ∈ (cid:74) 𝑘, 𝑘 + 𝜇 𝑔 (cid:75) } s.t. ∀ 𝑠 ∈ (cid:74) , 𝜇 𝑔 − (cid:75) C 𝑘 + 𝜇 𝑔 (cid:169)(cid:173)(cid:171) 𝜇 𝑔 − (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:170)(cid:174)(cid:172) B 𝑘 + 𝑠 = ∧ C 𝑘 + 𝜇 𝑔 (cid:169)(cid:173)(cid:171) 𝜇 𝑔 − (cid:214) 𝑚 = A 𝑘 + 𝑚 (cid:170)(cid:174)(cid:172) B 𝑘 ≠ (5.33)For affine time-invariant systems without piecewise definition, there is never required inputpreview because the expressions claimed equal to zero in (5.33) reduce as C 𝑘 + 𝜇 𝑔 (cid:169)(cid:173)(cid:171) 𝜇 𝑔 − (cid:214) 𝑚 = 𝑠 + A 𝑘 + 𝑚 (cid:170)(cid:174)(cid:172) B 𝑘 + 𝑠 = 𝐶 𝐴 𝜇 𝑔 − − 𝑠 𝐵 𝑠 ∈ (cid:74) , 𝜇 𝑔 − (cid:75) (5.34)which are always zero regardless of state sequence. However, it is important to emphasize thatthis is not the case for PWA systems. Even when the relative degrees of all component models areequal to 𝜇 𝑐 , (A5.4), the inter-location relative degree may not be equal to 𝜇 𝑐 . More formally 𝐶 ,𝑘 + 𝐵 ,𝑘 = ∧ 𝐶 ,𝑘 + 𝐵 ,𝑘 = (cid:54) = ⇒ 𝐶 ,𝑘 + 𝐵 ,𝑘 = (5.35)and 𝐶 ,𝑘 + 𝐵 ,𝑘 ≠ ∧ 𝐶 ,𝑘 + 𝐵 ,𝑘 ≠ (cid:54) = ⇒ 𝐶 ,𝑘 + 𝐵 ,𝑘 ≠ (5.36)and this lack of conclusiveness regarding the inter-state relative degree generalizes to larger 𝜇 𝑐 . Inshort, the inter-state relative degree may be either lower or higher than 𝜇 𝑐 , and may be different fordifferent switching sequences (with a maximum value of 𝜇 𝑔 as given by Definition 5.2).Because of the implicitness and potential requirement for input preview in the general PWAsystem inverse, it is nontrivial to use it for computing input trajectories from output trajectories.However, there are conditions under which inversion of a PWA system with 𝜇 𝑔 ≥ is unique, andthe inverse itself becomes an explicit PWA system as is the case for 𝜇 𝑔 = . The remainder of thissection provides such sufficient conditions for the cases of 𝜇 𝑔 = and 𝜇 𝑔 = . 𝜇 𝑔 ∈ { , } First the “location-independent output function” assumption is introduced:103A5.5) C 𝑘 = 𝐶 𝑘 , D 𝑘 = 𝐷 𝑘 , G 𝑘 = 𝐺 𝑘 , with 𝐶 𝑘 , 𝐷 𝑘 , 𝐺 𝑘 indicating parameters that potentially varywith time but that are identical ∀ 𝑞 ∈ (cid:74) , | 𝑄 | (cid:75) Lemma 3 (Relative Degree of 1) . A PWA system satisfying (A5.1)-(A5.5) has a global dynamicalrelative degree of 1 if and only if the relative degree of all component models are 1 for all time: 𝜇 𝑐 = ⇐⇒ 𝜇 𝑔 = (5.37) Proof.
The foreward implication follows directly from (A5.4), (A5.5), and 𝜇 𝑐 = : 𝜇 𝑐 = = ⇒ 𝐷 𝑘 + = ∧ 𝐶 𝑘 + 𝐵 𝑞,𝑘 ≠ ∀ 𝑞 ∈ (cid:74) , | 𝑄 | (cid:75) (5.38) = ⇒ D 𝑘 + = ∧ C 𝑘 + B 𝑘 ≠ ∀ 𝑘 (5.39)This is equivalent to implying that the inter-location relative degree can be neither higher nor lowerthan 𝜇 𝑐 = under (A5.4) and (A5.5) ∴ 𝜇 𝑐 = = ⇒ 𝜇 𝑔 = (5.40)The backward implication follows from (5.33) (i.e. the combination of Definition 5.2 and Lemma2), which implies 𝜇 𝑔 = = ⇒ ∃{ 𝑥 𝑘 , 𝑥 𝑘 + } s.t. C 𝑘 + B 𝑘 ≠ (5.41)Then, by (A5.4), (A5.5), and the fact that 𝜇 𝑔 = = ⇒ D 𝑘 = by Lemma 1, one finds that C 𝑘 + B 𝑘 = 𝐶 𝑘 + B 𝑘 ≠ = ⇒ 𝜇 𝑐 = . Therefore (5.37) is true. (cid:3) Corollary 5.2.1 (Unique Inverse of 𝜇 𝑔 = PWA Systems) . The inverse of a PWA system satisfying(A5.1)-(A5.5) with 𝜇 𝑐 = is given by the following explicit, anticausal PWA system. 𝑥 𝑘 + = A 𝑘 𝑥 𝑘 + B 𝑘 𝑦 𝑘 + + F 𝑘 (5.42) 𝑢 𝑘 = C 𝑘 𝑥 𝑘 + D 𝑘 𝑦 𝑘 + + G 𝑘 (5.43) where A 𝑘 = A 𝑘 + B 𝑘 C 𝑘 B 𝑘 = B 𝑘 D 𝑘 F 𝑘 = F 𝑘 + B 𝑘 G 𝑘 C 𝑘 = − D 𝑘 𝐶 𝑘 + A 𝑘 D 𝑘 = ( 𝐶 𝑘 + B 𝑘 ) − G 𝑘 = − D 𝑘 ( 𝐶 𝑘 + F 𝑘 + 𝐺 𝑘 + ) roof. 𝜇 𝑔 = by Lemma 3. Plugging (A5.5) and 𝜇 𝑔 = into (5.30) yields 𝑢 𝑘 = ( 𝐶 B 𝑘 ) − ( 𝑦 𝑘 + − 𝐶 ( A 𝑘 𝑥 𝑘 + F 𝑘 ) − 𝐺 ) (5.44)which is explicit. Note that ( 𝐶 B 𝑘 ) − is always well defined because it is scalar by (A5.2) and isnonzero by 𝜇 𝑔 = and Definition 5.2. (cid:3) To derive explicit inverses for PWA systems with 𝜇 𝑔 = , (A5.5) is used along with the new“output-based switching” assumption. Assuming 𝜇 𝑔 > , this assumption is expressed as(A5.6) 𝑃 = 𝑃 𝑜 C 𝑘 and 𝛽 = 𝛽 𝑜 − 𝑃 𝑜 G 𝑘 where 𝑃 𝑜 and 𝛽 𝑜 contain the orientation vectors and offsets of hyperplanes in the output space R 𝑛 𝑦 . Corollary 5.2.2 (Unique Inverse of 𝜇 𝑔 = PWA Systems) . The inverse of a PWA system satisfy-ing (A5.1)-(A5.6) with known global dynamical relative degree 𝜇 𝑔 = is given by the followingexplicit, anticausal PWA system. 𝑥 𝑘 + = A 𝑘 𝑥 𝑘 + B 𝑘 𝑦 𝑘 + + F 𝑘 (5.45) 𝑢 𝑘 = C 𝑘 𝑥 𝑘 + D 𝑘 𝑦 𝑘 + + G 𝑘 (5.46) where A 𝑘 = A 𝑘 + B 𝑘 C 𝑘 B 𝑘 = B 𝑘 D 𝑘 F 𝑘 = F 𝑘 + B 𝑘 G 𝑘 C 𝑘 = − D 𝑘 𝐶 𝑘 + A 𝑘 + A 𝑘 D 𝑘 = ( 𝐶 𝑘 + A 𝑘 + B 𝑘 ) − G 𝑘 = − D 𝑘 ( 𝐶 𝑘 + A 𝑘 + F 𝑘 + 𝐶 𝑘 + F 𝑘 + + 𝐺 𝑘 + ) Proof.
Plugging (A5.5) and 𝜇 𝑔 = into (5.30) yields 𝑢 𝑘 = ( 𝐶 𝑘 + A 𝑘 + B 𝑘 ) − ( 𝑦 𝑘 + − 𝐶 𝑘 + A 𝑘 + A 𝑘 𝑥 𝑘 − 𝐶 𝑘 + A 𝑘 + F 𝑘 − 𝐶 𝑘 + F 𝑘 + − 𝐺 𝑘 + ) (5.47)In general, (5.47) would be implicit because of A 𝑘 + and F 𝑘 + ’s dependence on 𝑥 𝑘 + , and thus 𝑢 𝑘 ,via the selector functions 𝐾 𝑞 ( 𝛿 ( 𝑥 𝑘 + )) = (cid:206) Δ ∗ 𝑞𝑖 = (cid:13)(cid:13)(cid:13) 𝛿 ∗ 𝑞,𝑖 − 𝐻 ( 𝑃 ( A 𝑘 𝑥 𝑘 + B 𝑘 𝑢 𝑘 + F 𝑘 )− 𝛽 ) (cid:13)(cid:13)(cid:13) (5.48)However, under (A5.6) (in combination with (A5.5)) this becomes 𝐾 𝑞 ( 𝛿 ( 𝑥 𝑘 + )) = (cid:206) Δ ∗ 𝑞𝑖 = (cid:13)(cid:13)(cid:13) 𝛿 ∗ 𝑞,𝑖 − 𝐻 ( 𝑃 𝑜 𝐶 𝑘 + ( A 𝑘 𝑥 𝑘 + F 𝑘 )− 𝛽 𝑜 + 𝑃 𝑜 𝐺 𝑘 + ) (cid:13)(cid:13)(cid:13) (5.49)105hich is not a function of 𝑢 𝑘 and is thus explicit.The reduction of (5.48) to (5.49) relies on the fact that 𝐶 𝑘 + B 𝑘 = ∀ 𝑘 . This is true for 𝜇 𝑔 = systems under (A5.5) because 𝜇 𝑔 = = ⇒ 𝜇 𝑐 > by Lemma 3. (cid:3) Remark 𝜇 𝑐 , 𝜇 𝑔 Relationship) . Note that unlike for relative degrees of 0 and 1, 𝜇 𝑐 = (cid:54) = ⇒ 𝜇 𝑔 = under assumptions (A5.1)-(A5.6). If 𝑛 𝑥 > , there exists systems for which 𝜇 𝑐 = but 𝜇 𝑔 > due to inter-location dynamics. For many PWA systems, evolving the inverse systems derived in Section 5.2 forward in time froman initial state at time 𝑘 = 𝑘 and with a bounded reference 𝑦 𝑘 + 𝜇 𝑔 = 𝑟 𝑘 + 𝜇 𝑔 will yield an inversesystem trajectory 𝑢 𝑘 that is bounded for all 𝑘 ≥ 𝑘 and suitable for feedforward control. However,inverse PWA system instabilities may arise from NMP component dynamics, causing 𝑢 𝑘 to becomeunbounded under this conventional system evolution scheme, despite the bounded reference. Insuch cases, a bounded 𝑢 𝑘 may still be achievable on a bi-infinite timeline via stable inversion.Formally, the stable inversion problem may be given as follows. Definition 5.3 (PWA Stable Inversion Problem Statement) . Given an explicit inverse PWA systemrepresentation 𝑥 𝑘 + = A 𝑘 𝑥 𝑘 + B 𝑘 𝑦 𝑘 + 𝜇 𝑔 + F 𝑘 (5.50a) 𝑢 𝑘 = C 𝑘 𝑥 𝑘 + D 𝑘 𝑦 𝑘 + 𝜇 𝑔 + G 𝑘 (5.50b)and a reference trajectory 𝑦 𝑘 + 𝜇 𝑔 = 𝑟 𝑘 + 𝜇 𝑔 known for all 𝑘 ∈ Z , a two point boundary value problemis formed by (5.50a) and the boundary conditions 𝑥 −∞ = 𝑥 ∞ = . The solution to the stableinversion problem is the bounded bi-infinite time series 𝑢 𝑘 ∈ R ∀ 𝑘 , which is generated by (5.50b)and the bounded bi-infinite solution 𝑥 𝑘 to the boundary value problem.The following assumptions on system parameter boundedness and boundary conditions arecommon in some form across much stable inversion literature.(A5.7) There exists a supremum to the norms of the inverse system matrices: sup 𝑘 (cid:13)(cid:13)(cid:13) A 𝑘 (cid:13)(cid:13)(cid:13) , sup 𝑘 (cid:13)(cid:13)(cid:13) B 𝑘 (cid:13)(cid:13)(cid:13) , sup 𝑘 (cid:13)(cid:13)(cid:13) F 𝑘 (cid:13)(cid:13)(cid:13) , sup 𝑘 (cid:13)(cid:13)(cid:13) C 𝑘 (cid:13)(cid:13)(cid:13) , sup 𝑘 (cid:13)(cid:13)(cid:13) D 𝑘 (cid:13)(cid:13)(cid:13) , sup 𝑘 (cid:13)(cid:13)(cid:13) G 𝑘 (cid:13)(cid:13)(cid:13) ∈ R (5.51)Any vector norm may be used, and the matrix norm is that induced by the vector norm.106A5.8) The reference 𝑦 𝑘 + 𝜇 𝑔 and bias terms F 𝑘 , G 𝑘 decay to zero at the extremities of the bi-infinitetime series: ∀ 𝜀 ∈ R > ∃ 𝜂 , 𝜂 ∈ Z s.t. (cid:13)(cid:13) 𝑦 𝑘 + 𝜇 𝑔 (cid:13)(cid:13) , (cid:13)(cid:13)(cid:13) F 𝑘 (cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13) G 𝑘 (cid:13)(cid:13)(cid:13) < 𝜀 ∀ 𝑘 ∈ (−∞ , 𝜂 (cid:75) ∪ (cid:74) 𝜂 , ∞) (5.52)Additionally, stable inversion of PWA systems involves two challenges not faced in the stableinversion of linear systems. First, the dynamics of all locations and the inter-location dynamicsmust be simultaneously accounted for when decoupling the stable and unstable system modes.Second, there must be a way to manage switching in the two partial system evolutions. Thesechallenges are manifested in the following assumptions.(A5.9) There exists a similarity transform matrix 𝑉 ∈ R 𝑛 𝑥 × 𝑛 𝑥 that decouples the stable and unstablemodes of (5.50a). Formally this decoupling can be expressed as 𝑉 A 𝑘 𝑉 − = ˜ A 𝓈 𝑘 𝑛 𝓈 × 𝑛 𝓊 𝑛 𝓊 × 𝑛 𝓈 ˜ A 𝓊 𝑘 ∀ 𝑘 (5.53)where 𝑛 𝓈 is the number of stable modes, 𝑛 𝓊 is the number of unstable modes, 𝑛 𝓈 + 𝑛 𝓊 = 𝑛 𝑥 , ˜ A 𝓊 𝑘 has all eigenvalue magnitudes > ∀ 𝑘 , and the free systems 𝑧 𝓈 𝑘 + = ˜ A 𝓈 𝑘 𝑧 𝓈 𝑘 𝑧 𝓊 𝑘 + = (cid:0) ˜ A 𝓊 𝑘 (cid:1) − 𝑧 𝓊 𝑘 (5.54)with appropriately sized state vectors 𝑧 𝓈 , 𝑧 𝓊 are globally uniformly asymptotically stableabout the origin.(A5.10a) Switching is exclusively dependent on the stable modes: 𝑃𝑉 − = (cid:104) ˜ 𝑃 𝓈 𝑛 𝑃 × 𝑛 𝓊 (cid:105) (5.55)(A5.10b) Switching is exclusively dependent on the unstable modes and all unstable states arisingfrom (5.50) are reachable in one time step from some predecessor state for all 𝑘 : (cid:16) 𝑃𝑉 − = (cid:104) 𝑛 𝑃 × 𝑛 𝓈 ˜ 𝑃 𝓊 (cid:105) (cid:17) ∧ (cid:0) ∀ 𝑘, ∀ ˜ 𝑥 𝓊 𝑘 + ∈ X 𝓊 ⊆ R 𝑛 𝓊 Pre ({ ˜ 𝑥 𝓊 𝑘 + }) ≠ ∅ (cid:1) (5.56)where ˜ 𝑥 𝓊 𝑘 = (cid:104) 𝑛 𝓊 × 𝑛 𝓈 𝐼 𝑛 𝓊 × 𝑛 𝓊 (cid:105) 𝑉 𝑥 𝑘 . X 𝓊 is a set containing at least all solution values of ˜ 𝑥 𝓊 𝑘 + (see Section 5.3.2 for elaboration). Pre (X) is the set of predecessor states whose107ne-step successors belong to the set X .For detailed theorems on the sufficient conditions for uniform asymptotic stability of systemsof the form (5.54), see [115].As implied by the separation of (A5.10) into two opposing assumptions, the challenges asso-ciated with switching management precipitate different stable inversion procedures for the stable-mode-dependent switching and unstable-mode-dependent switching cases. In general, the trajec-tory of the modes upon which switching is dependent are computed first. This allows the switchingsignal for the overall system to be computed and given as an exogenous input to the evolution ofthe remaining modes.The theorems for these cases are supported by the following notation for the decoupled systemin addition to the above-defined ˜ A 𝓈 𝑘 , ˜ A 𝓊 𝑘 , ˜ 𝑃 𝓈 , ˜ 𝑃 𝓊 . ˜ 𝑥 𝓈 𝑘 (cid:66) ℐ 𝓈 𝑉 𝑥 𝑘 ˜ B 𝓈 𝑘 (cid:66) ℐ 𝓈 𝑉 B 𝑘 ˜ F 𝓈 𝑘 (cid:66) ℐ 𝓈 𝑉 ˜ F 𝑘 ˜ 𝑥 𝓊 𝑘 (cid:66) ℐ 𝓊 𝑉 𝑥 𝑘 ˜ B 𝓊 𝑘 (cid:66) ℐ 𝓊 𝑉 B 𝑘 ˜ F 𝓊 𝑘 (cid:66) ℐ 𝓊 𝑉 ˜ F 𝑘 (5.57)where ℐ 𝓈 (cid:66) (cid:104) 𝐼 𝑛 𝓈 × 𝑛 𝓈 𝑛 𝓈 × 𝑛 𝓊 (cid:105) ℐ 𝓊 (cid:66) (cid:104) 𝑛 𝓊 × 𝑛 𝓈 𝐼 𝑛 𝓊 × 𝑛 𝓊 (cid:105) (5.58) Theorem 5.3 (PWA Stable Inversion with Stable-Mode-Dependent Switching) . Given an explicitinverse PWA system (5.50) satisfying (A5.7)-(A5.9) and (A5.10a), the solution to the stable inver-sion problem exists and can be found by first computing the stable mode time series ˜ 𝑥 𝓈 𝑘 and locationtime series 𝛿 𝑘 ∀ 𝑘 forwards in time via 𝛿 𝑘 = 𝐻 (cid:0) ˜ 𝑃 𝓈 ˜ 𝑥 𝓈 𝑘 − 𝛽 (cid:1) (5.59) ˜ 𝑥 𝓈 𝑘 + = ˜ A 𝓈 𝑘 ˜ 𝑥 𝓈 𝑘 + ˜ B 𝓈 𝑘 𝑦 𝑘 + 𝜇 𝑔 + ˜ F 𝓈 𝑘 (5.60) The location time series being now known, the unstable mode time series ˜ 𝑥 𝓊 𝑘 can be computedbackwards in time via ˜ 𝑥 𝓊 𝑘 = (cid:0) ˜ A 𝓊 𝑘 (cid:1) − (cid:16) ˜ 𝑥 𝓊 𝑘 + − ˜ B 𝓊 𝑘 𝑦 𝑘 + 𝜇 𝑔 − ˜ F 𝓊 𝑘 (cid:17) (5.61) with 𝛿 𝑘 input directly to the selector functions in (5.3). Finally the solution 𝑢 𝑘 is computed via(5.50b) with 𝑥 𝑘 = 𝑉 − (cid:20) (cid:16) ˜ 𝑥 𝓈 𝑘 (cid:17) 𝑇 , (cid:16) ˜ 𝑥 𝓊 𝑘 (cid:17) 𝑇 (cid:21) 𝑇 .Proof. The prescribed formula represents a solution to the stable inversion problem because• by (A5.10a), (5.59) is equivalent to (5.5), 108 by (A5.9), the concatenated evolutions of (5.60) and (5.61) are equivalent to (5.50a), and• by (A5.7) and (A5.8), (5.60) and (5.61) decay to the form of (5.54) in the limits as 𝑘 approaches ∞ or −∞ , and thus by (A5.9) the boundary conditions at these limits are satisfied.The solution is guaranteed to exist because• (5.59)-(5.61) and (5.50b) are all explicit functions with all variables in the right-hand side knowndue to the order of time series computation, and• the outputs of (5.59)-(5.61) exist because the system parameters and input signals are boundedby (A5.7) and (A5.8), and ˜ A 𝓊 𝑘 is guaranteed invertible by the eigenvalue condition of (A5.9). (cid:3) Theorem 5.4 (PWA Stable Inversion with Unstable-Mode-Dependent Switching) . Given an ex-plicit inverse PWA system (5.50) satisfying (A5.7)-(A5.9) and (A5.10b), the solution to the sta-ble inversion problem exists and can be found in the following manner. First solve the implicitbackward-in-time evolution of the unstable modes, (5.61), for ˜ 𝑥 𝓊 𝑘 at each time step using any of theapplicable algorithms (e.g. brute force search over all locations or computational geometry meth-ods [116], see Section 5.3.2 for elaboration). For each time step at which one of the potentiallymultiple solution values of ˜ 𝑥 𝓊 𝑘 is chosen (any selection method is valid), the location vector 𝛿 𝑘 maybe computed by 𝛿 𝑘 = 𝐻 (cid:0) ˜ 𝑃 𝓊 ˜ 𝑥 𝓊 𝑘 − 𝛽 (cid:1) (5.62) The location time series being computed, 𝛿 𝑘 may be directly plugged in to the selector functionsin (5.3) to make the forward-in-time evolution of the stable modes, (5.60), explicit such that itcan be evaluated at each time step for ˜ 𝑥 𝓈 𝑘 . The solution 𝑢 𝑘 is then computed via (5.50b) with 𝑥 𝑘 = 𝑉 − (cid:20) (cid:16) ˜ 𝑥 𝓈 𝑘 (cid:17) 𝑇 , (cid:16) ˜ 𝑥 𝓊 𝑘 (cid:17) 𝑇 (cid:21) 𝑇 , as in Theorem 5.3.Proof. The prescribed formula represents a solution to the stable inversion problem for the samereasons as Theorem 5.3, but with the first proposition of (A5.10b)—i.e. 𝑃𝑉 − = (cid:2) 𝑛 𝑃 × 𝑛 𝓈 , ˜ 𝑃 𝓊 (cid:3) —used in place of (A5.10a). The solution is guaranteed to exist because• the second proposition of (A5.10b) guarantees the solution set of (5.61) is non-empty,• the PWA nature of the original inverse system (5.50) enables application of existing algorithmsguaranteed to find Pre ({ ˜ 𝑥 𝓊 𝑘 + }) and thus solve (5.61) [116], and• with solutions to (5.61) chosen ∀ 𝑘 , the remaining equations are explicit with bounded outputsfor the same reasons as in Theorem 5.3. 109 Note that while [116] focuses on PWA systems with time-invariant components, only the one-step predecessor set need be computed at each time step. Thus, because a time-varying system isindistinguishable from a time-invariant system over a single time step, the algorithms of [116] arestill applicable.
The most immediate issue with Theorems 5.3 and 5.4 is that, while they provide an exact solutionto the stable inversion problem, their procedures cannot be implemented because of the infinitenature of the time series involved. This issue applies to past works on stable inversion as well, andthe same means of addressing the issue is taken here. Namely, an approximate solution is obtainedby prescribing a finite reference 𝑟 𝑘 + 𝜇 𝑔 for 𝑘 ∈ (cid:74) , 𝑁 − 𝜇 𝑔 (cid:75) and strictly enforcing the boundaryconditions on only the initial/terminal states of the stable/unstable mode evolution. In other words, 𝑥 𝓈 = and 𝑥 𝓊 𝑁 − 𝜇 𝑔 = but 𝑥 𝓈 𝑁 − 𝜇 𝑔 and 𝑥 𝓊 may be nonzero.In general the closer 𝑥 𝓈 𝑘 and 𝑥 𝓊 𝑘 come to decaying to zero by 𝑘 = 𝑁 − 𝜇 𝑔 and 𝑘 = , respectively,the higher quality the approximation of the 𝑢 𝑘 time series will be. In other words, the closer 𝑢 𝑘 comes to returning 𝑦 𝑘 + 𝜇 𝑔 = 𝑟 𝑘 + 𝜇 𝑔 when input to the original system from which the inverse (5.50)was derived. To achieve this high quality approximation, one may specify 𝑟 𝑘 + 𝜇 𝑔 to begin andend with a number of zero elements to allow space for the 𝑢 𝑘 time series to contain the pre- andpost-actuation typically necessary for the control of NMP systems. The number of zero elementsrequired to achieve a satisfactorily low error is case dependent. One typical heuristic is to ensurethat the durations of the leading and trailing zeros are approximately equal to the system settlingtime.In addition to the practical need for finite references with leading and trailing zeros, the case ofunstable-mode-dependent switching warrants special attention regarding implementation.First, consider methods to solve the implicit equation (5.61) at each time step. Any method willconsist of two parts: identifying a set of valid solutions and then choosing one of them. Choos-ing a solution can be formalized as the minimization of some cost function, (cid:13)(cid:13) 𝑥 𝓊 𝑘 + − 𝑥 𝓊 𝑘 (cid:13)(cid:13) beinga straightforward and universally applicable option. If the inverse system (5.50) has exclusivelyunstable modes, input-based costs such as (cid:107) 𝑢 𝑘 (cid:107) and (cid:107) 𝑢 𝑘 + − 𝑢 𝑘 (cid:107) may also be used. Note that anysuch optimization is combinatorial, i.e. the decision variable can only take on values from a partic-ular finite set. For the problem considered here, the cardinality of this set is at most the number oflocations | 𝑄 | in the system. This is because each location contains at most one solution to (5.61)due to the eigenvalue condition in (A5.9) making ( ˜ A 𝓊 𝑘 ) − full rank and thus one-to-one.110aving an upper bound of one solution per location leads to a direct method for deriving theset of valid solutions to (5.61). For each location 𝑄 𝑞 ∈ 𝑄 , compute (cid:16) ˜ 𝐴 𝓊 𝑞,𝑘 (cid:17) − (cid:16) ˜ 𝑥 𝓊 𝑘 + − 𝐵 𝓊 𝑞,𝑘 𝑟 𝑘 + 𝜇 𝑔 − 𝐹 𝓊 𝑞,𝑘 (cid:17) (5.63)and check whether the result lies in 𝑄 𝑞 . If so, the result is a solution to (5.61). Naturally, logicrelating to solution selection criteria may be incorporated to reduce computational cost, e.g. check-ing locations in order of proximity to the current location to avoid checking all locations in the casethat the solution selection cost function is something like (cid:13)(cid:13) 𝑥 𝓊 𝑘 + − 𝑥 𝓊 𝑘 (cid:13)(cid:13) . Alternatively, Pre (cid:16)(cid:8) ˜ 𝑥 𝓊 𝑘 + (cid:9)(cid:17) may be derived whole using the computational-geometry-supported algorithms of [116]. As notedin [116], the algorithm of least cost may be case-dependent.Finally, consider verification of the existence of a solution to the stable inversion problemwith unstable-mode-based switching. The verification method recommended here is to first ver-ify (A5.7)-(A5.9) and the first proposition of (A5.10b) directly, then run the procedure given inTheorem 5.4 for finding a solution. If (A5.7)-(A5.9) and the first proposition of (A5.10b) havebeen verified, then the procedure is guaranteed to find a solution to (5.61) if a solution exists.Equivalently, failure to find a solution implies no solution exists.This method is recommended over directly attempting to verify the second proposition of(A5.10b) because this second proposition may be conservative, difficult to verify, and yield verylimited computational savings over the recommended method. The conservativeness arises fromthe choice of set of possible ˜ 𝑥 𝓊 𝑘 + values, X 𝓊 . It is unlikely for one to possess knowledge of thesolution ˜ 𝑥 𝓊 𝑘 + values prior to actually solving the stable inversion problem, so X 𝓊 may need to beset much larger than necessary to ensure it contains the solution trajectory. This containment isnecessary for truth of the proposition to imply verification of solution existence. Conversely, theproposition may evaluate to false despite containing a true solution if X 𝓊 also contains unreachablestates. Selection of X 𝓊 may thus be a delicate, challenging task.The expectation of high computational cost arises from the subtle discrepancy between the ca-pabilities of established PWA system verification methods and the second proposition of (A5.10b).Multiple methods exist for verifying the reachability/controllability of a PWA system to a targetset X 𝓊 [116, 117]. However, these methods typically verify whether at least one element of thetarget set is reachable, whereas (A5.10b) requires verification that every element of the target setis reachable. In other words, the computational savings one might expect from existing reachabil-ity/controllability verification schemes may not be available.In short, the second proposition of (A5.10b) is useful for specifying a condition under which asolution is guaranteed to exist, and thus for the derivation and proof of Theorem 5.4. But it is notrecommended as a tool for existence verification. This is not a significant loss, however, because111he procedure given in Theorem 5.4 for finding a solution is itself a valid verification tool. This section uses the stable inversion theory of Section 5.3, and thus also the conventional in-version theory of Section 5.2, to simulate the application of ILILC to a PWA system with NMPcomponent dynamics: an inkjet printhead positioning system. This system uses feedback andfeedforward control simultaneously. In addition to ILILC, for benchmarking purposes a numberof other controllers are applied to the system:• feedback-only control (i.e. zero feedforward input),• learning-free PWA stable inversion (i.e. stable inversion without ILILC),• gradient ILC, and• P-type ILC.P-type ILC is used as a benchmark in addition to the gradient ILC benchmark introduced in Chapter4 because P-type ILC is among the most common forms of ILC used in industry, is considered bymany to be a form of “model-free” ILC, and like gradient ILC does not necessarily cause instabilitywhen applied to systems with NMP dynamics. Details for the formulation of all ILC schemes aregiven in Section 5.4.2.These simulations are subject to a variety of model errors and other disturbances to validatethe stable-inversion-supported learning controller’s practicality. In other words, the controller issynthesized from a “control model” and applied to a “truth model” representing a physical system.The control model features mismatches in parameter values, sample rate, model order, and relativedegree.Additionally, the truth model is subject to copious process noise and measurement noise. Whilethe physical system has virtually no output-measurable noise, the injection here is done as a pre-liminary test to ensure that noise does not corrupt the learning process beyond the remedial powerof conventional filtering.
The truth model is based on the physical desktop inkjet printhead positioning testbed at the Eind-hoven University of Technology, pictured in Figure 5.1. The input to this system is an appliedmotor voltage, 𝑐 𝑘 , and the output is the printhead position along a . guide rail, 𝑦 𝑃𝑘 , measuredby a linear optical encoder with a resolution of µ m . The applied motor voltage 𝑐 𝑘 is the sum of112 igure 5.1: Photo of desktop inkjet printer with the case removed. The motor actuates the printhead motionalong a guide rail via a timing belt, and the motion is measured by a linear optical encoder with resolutionof µ m (about 600 dots per inch). ω c r k e k c k y Pk u k ω y − Feedback A C , B C Controller C C , D C Plant A P , B P C P , D P y Ck Intra-trial signalInter-trial signalILC Law
Upsample Down-sampleDown-sample
Figure 5.2: System block diagram. The plant block uses the truth model of the printer system obtained byexperimental system identification while the ILC law is synthesized using the control model. The down-sample and upsample blocks account for the difference in sample period between the ILC law and the truthmodel. a feedback component, 𝑦 𝐶𝑘 , and feedforward component, 𝑢 𝑘 . Finally, to add additional disturbanceto the simulation, Gaussian white process noise 𝜔 𝑐 (zero mean, standard deviation .
03 V ) andmeasurement noise 𝜔 𝑦 (zero mean, standard deviation µ m ) are added at the input and output ofthe plant, respectively. This ultimately results in the block diagram of Figure 5.2.System identification of the printer yields a discrete-time LTI model, which is used as thetruth model. Truncation-based model order reduction by 1 (MATLAB function balred ), zero-order-hold-based sample period reduction by a factor of 2 (MATLAB function d2d ), and randomperturbation of model parameters results yields a new LTI model, which is used as the controlmodel and can be represented by the state space system ( ˆ 𝐴 𝑃 , ˆ 𝐵 𝑃 , ˆ 𝐶 𝑃 , ˆ 𝐷 𝑃 ) with state vector ˆ 𝑥 𝑃 ∈ R 𝑛 ˆ 𝑥𝑃 .To account for the change in sample period, truth model output signals are decimated by a113 igure 5.3: Bode plot of the experimental plant data, truth model of the plant, and control model of the plant. factor of 2 before being input to the ILC law and ILC law output signals are upsampled by a factorof 2 with a zero order hold before being applied to the truth model. Parameters for the truth andcontrol models are given in terms of pole, zero, and gain values in Table 5.1. A Bode plot of theexperimental data, truth model, and control model are given in Figure 5.3.The feedback controller is of identical structure and tuning for the truth and control models,but has different parameter values due to the difference in sample period between the two models.In either case, the feedback controller is composed of a second order lowpass filter given by thediscrete-time transfer function 𝐶 𝐿𝑃 ( 𝑧 ) = 𝑏𝑧 ( 𝑧 + ) 𝑧 + 𝑎 𝑧 + 𝑎 (5.64)in series with a hybrid Proportional-Derivative (PD) controller. The proportional gain 𝐾 𝑝 is set toa high value when the reference-output error is greater than some magnitude threshold, and is set114o a lower value otherwise: 𝐶 𝑃𝐷 ( 𝑧 ) = (cid:16) 𝐾 𝑝 + 𝐾 𝑑 𝑇 𝑠 (cid:17) 𝑧 − 𝐾 𝑑 𝑇 𝑠 𝑧 (5.65) 𝐾 𝑝 = 𝐾 𝑝, | 𝑒 𝑘 − | ≤ 𝑒 switch 𝐾 𝑝, | 𝑒 𝑘 − | > 𝑒 switch (5.66)where 𝑇 𝑠 is the sample period in seconds and 𝐾 𝑑 is the derivative gain. Note that switching is theerror of the previous time step rather than the current time step in order for switching to be state-based, and thus satisfy (A5.3). Mathematically this switching is made state-based by augmentingthe minimal state-space representation of 𝐶 𝑃𝐷 ( 𝑧 ) 𝐶 𝐿𝑃 ( 𝑧 ) with an extra state that stores the errorinput to the lowpass filter. In other words, the feedback controller model is given by ˆ 𝐴 𝐶 = − 𝑎 − 𝑎
00 0 0 ˆ 𝐵 𝐶 = (5.67) ˆ 𝐶 𝐶 = − 𝑏 (cid:104) 𝐾 𝑑 ( + 𝑎 ) 𝑇 𝑠 + 𝐾 𝑝 𝑎 𝐾 𝑑 𝑎 𝑇 𝑠 + 𝐾 𝑝 ( 𝑎 − ) (cid:105) (5.68) ˆ 𝐷 𝐶 = (cid:104) 𝑏 (cid:16) 𝐾 𝑝 + 𝐾 𝑑 𝑇 𝑠 (cid:17) (cid:105) (5.69)with state vector ˆ 𝑥 𝐶𝑘 ∈ R 𝑛 ˆ 𝑥𝐶 having its final element equal to 𝑒 𝑘 − . Parameter values for thefeedback controller are given in Table 5.1. For both the truth model and the control model, thelowpass filter has a roll off frequency of
40 Hz and a damping ratio of 0.7.To perform stable inversion of the system dynamics from the feedforward input to the output,a monolithic PWA model of the form (5.2) is needed. This is given by ˆ 𝑥 𝑘 + = ˆ A 𝑘 ˆ 𝑥 𝑘 + ˆ B 𝑘 𝑢 𝑘 + ˆ F 𝑘 (5.70) ˆ 𝑦 𝑘 = ˆ C 𝑘 ˆ 𝑥 𝑘 + ˆ D 𝑘 𝑢 𝑘 + ˆ G 𝑘 (5.71)where ˆ A 𝑘 = ˆ 𝐴 𝑃 − ˆ 𝐵 𝑃 ˆ 𝐷 𝐶 ˆ 𝐶 𝑃 ˆ 𝐵 𝑃 ˆ 𝐶 𝐶 − ˆ 𝐵 𝐶 ˆ 𝐶 𝑃 ˆ 𝐴 𝐶 ˆ B 𝑘 = ˆ 𝐵 𝑃 𝑛 ˆ 𝑥𝐶 × ˆ F 𝑘 = ˆ 𝐵 𝑃 ˆ 𝐷 𝐶 ˆ 𝐵 𝐶 𝑟 𝑘 (5.72) ˆ C 𝑘 = (cid:104) ˆ 𝐶 𝑃 × 𝑛 ˆ 𝑥𝐶 (cid:105) ˆ D 𝑘 = G 𝑘 = 𝑥 𝑘 = ˆ 𝑥 𝑃𝑘 ˆ 𝑥 𝐶𝑘 (5.73)115able 5.1: Simulation Model ParametersTruth Model Control ModelPlant Poles . ± . 𝑖 . ± . 𝑖 .
00 0 . .
00 1 . N/APlant Zeros − .
10 33 . − . − . .
16 0 . Plant Gain . E − . E − 𝑎 − . − . 𝑎 .
70 0 . 𝑏 .
027 0 . 𝐾 𝑑 𝐾 𝑝,
40 40 𝐾 𝑝,
160 160 𝑒 switch 𝑇 𝑠 .
001 s 0 .
002 s
This system is NMP, as it has all the zeros of the plant model given in Table 5.1 (as well asadditional zeros).The monolithic control model has two locations based on the switching of 𝐾 𝑝 in ˆ 𝐶 𝐶 and ˆ 𝐷 𝐶 .Let the location 𝑞 = correspond to low error with 𝐾 𝑝, and 𝑞 = correspond to high error and 𝐾 𝑝, . Then the switching parameters are 𝑃 = (cid:34) × 𝑛 ˆ 𝑥𝑃 + 𝑛 ˆ 𝑥𝐶 − − × 𝑛 ˆ 𝑥𝑃 + 𝑛 ˆ 𝑥𝐶 − (cid:35) 𝛽 = (cid:34) − 𝑒 switch − 𝑒 switch (cid:35) (5.74) Δ ∗ = (cid:40) (cid:34) (cid:35) (cid:41) Δ ∗ = (cid:40) (cid:34) (cid:35) , (cid:34) (cid:35) (cid:41) (5.75)Note that 𝛿 = [ , ] 𝑇 is not reachable, and thus need not be included in Δ ∗ .This monolithic model is of global dynamical relative degree 𝜇 𝑔 = and satisfies (A5.1)-(A5.5), enabling the use of Corollary 5.2.1 for derivation of the conventional inverse. The resultantinverse system satisfies (A5.7)-(A5.10a), enabling the use of stable inversion for the generation ofstable inverse state trajectories. The decoupling similarity transform 𝑉 is derived by the MATLABfunction canon applied to the dynamics of location 1.Finally, the reference 𝑟 𝑘 for the truth model to track is given in Figure 5.4.116 Time [s] P r i n t h ea d P o s iti on [ m ] Figure 5.4: Reference. The reference is 1999 samples long for the truth model, and is downsampled to 1000samples for the control model.
In Chapter 4, ILILC is presented as a means to derive the trial-varying learning matrix 𝐿 ℓ of theclassical ILC law u ℓ + = u ℓ + 𝐿 ℓ ( r − y ℓ ) (3.7)where ℓ ∈ Z ≥ is the iteration index, u , r , y ∈ R 𝑁 − 𝜇 𝑔 + are the lifted vectors (i.e. time seriesvectors) u ℓ = (cid:104) 𝑢 ℓ, 𝑢 ℓ, · · · 𝑢 ℓ,𝑁 − 𝜇 𝑔 (cid:105) 𝑇 (5.76) r ℓ = (cid:104) 𝑟 ℓ,𝜇 𝑔 𝑟 ℓ,𝜇 𝑔 + · · · 𝑟 ℓ,𝑁 (cid:105) 𝑇 (5.77) y ℓ = (cid:104) 𝑦 ℓ,𝜇 𝑔 𝑦 ℓ,𝜇 𝑔 + · · · 𝑦 ℓ,𝑁 (cid:105) 𝑇 (5.78)and 𝑁 ∈ Z >𝜇 𝑔 is the number of time steps in a trial of the output reference tracking task (thenumber of samples is 𝑁 + ).To derive 𝐿 ℓ , ILILC calls for a lifted input-output model inverse ˆ g − : R 𝑁 − 𝜇 𝑔 + → R 𝑁 − 𝜇 𝑔 + taking in the measured output y and outputting the control signal u predicted to yield y when inputto the true, unknown system. Equivalently, ˆ g − takes in the model output ˆ y and outputs the controlsignal u that yields ˆ y when input to the known model approximating the true system.117his ˆ g − must be closed-form, such that the ILILC learning matrix 𝐿 ℓ = 𝜕 ˆ g − 𝜕 ˆ y ( y ℓ ) (4.19)can be derived via an automatic differentiation tool such as CasADi [37]. Here, 𝜕 ˆ g − 𝜕 ˆ y is the Jacobian(in numerator layout) of ˆ g − with respect to ˆ y . Furthermore, as stated in Chapter 4, when theknown system model is NMP, 𝜕 ˆ g − 𝜕 ˆ y is likely to be ill-conditioned unless ˆ g − is synthesized usingstable inverse trajectories. For a PWA system satisfying (A5.7)-(A5.10a), Theorem 5.3 provides amethod for generating these closed-form state trajectories; given trial-invariant conditions ˜ 𝑥 𝓈 ℓ, = and ˜ 𝑥 𝓊 ℓ,𝑁 − 𝜇 𝑔 = , each element of the time series 𝑥 𝑘 and 𝛿 𝑘 is a function only of ˆ y . Then ˆ g − isgiven by ˆ g − = ˆ C ˆ 𝑥 + ˆ D 𝑦 𝜇 𝑔 + ˆ G ˆ C ˆ 𝑥 + ˆ D 𝑦 + 𝜇 𝑔 + ˆ G ... ˆ C 𝑁 − 𝜇 𝑔 ˆ 𝑥 𝑁 − 𝜇 𝑔 + ˆ D 𝑁 − 𝜇 𝑔 𝑦 𝑁 + ˆ G 𝑁 − 𝜇 𝑔 (5.79)Finally, for the particular example system studied here, the common practice (see, e.g. [4])of adding filters to the ILC law is implemented. Two filters are used. First, a zero-phase-shiftversion of the feedback controller’s lowpass filter is applied to the input and output of the ILClaw. In lifted form, the feedback controller’s filter is given by the lower diagonal, square, Toeplitzmatrix ℱ whose first column is the unit magnitude impulse response of the lowpass filter (5.64)on 𝑘 ∈ (cid:74) , 𝑁 − 𝜇 𝑔 (cid:75) . The zero phase shift is achieved by first filtering the signals forwards in time,and then backwards in time. The resultant lifted zero-phase-shift lowpass filter is 𝒬 = (cid:1) 𝐼 ℱ (cid:1) 𝐼 ℱ (5.80)where (cid:1) 𝐼 is a square matrix with ones on the antidiagonal and zeros elsewhere. Second, to eliminatetime series edge effects the first 35 and last 35 samples of the 1000 sample ILC law output areforced to zero. These edge effects may arise because the finite stable inversion trajectories havenearly zero—rather than zero—initial conditions for the unstable modes and similar for the termi-nal conditions of the stable modes. In lifted form this filter is given by the identity matrix with thefirst 35 and last 35 diagonal elements set to zero, notated as ℰ .Thus, the ILILC law used here is ultimately given by u ℓ + = ℰ𝒬 ( u ℓ + 𝐿 ℓ ( r − 𝒬 y ℓ )) (5.81)with u = and 𝑦 ℓ,𝑘 = 𝑦 𝑃ℓ,𝑘 + 𝜔 𝑦 . 118 .4.2.2 Gradient and Lifted P-Type ILC Gradient ILC and lifted P-type ILC both also use the filtered ILC law (5.81), but with differentdefinitions of 𝐿 ℓ .Gradient ILC uses 𝐿 ℓ = 𝛾 𝜕 ˆ g 𝜕 u ( u ℓ ) 𝑇 (5.82)as in Chapter 4, where 𝛾 is the learning gain and ˆ g is the lifted system input-output model ˆ y ℓ = ˆ g ( u ℓ ) (5.83)which can be synthesized via (3.11)-(3.12).P-type ILC is typically expressed without lifting as [118] 𝑢 ℓ + ,𝑘 = 𝑢 ℓ,𝑘 + 𝒫 (cid:0) 𝑟 ℓ,𝑘 + 𝜇 − 𝑦 ℓ,𝑘 + 𝜇 (cid:1) (5.84)where 𝒫 is a constant scalar learning gain and 𝜇 is the system relative degree (here the globaldynamical relative degree 𝜇 𝑔 is used). In lifted form, (5.84) manifests as the trial invariant learningmatrix 𝐿 ℓ = 𝒫 𝐼 𝑁 − 𝜇 𝑔 + × 𝑁 − 𝜇 𝑔 + ∀ ℓ (5.85)which can be plugged into the filtered learning law (5.81). At this time, there is no literatureprescribing a stable synthesis procedure for the P-type ILC of PWA systems. In fact, the literaturelacks application of either gradient ILC or P-type ILC to PWA models. However, because ofits simplicity it is ubiquitous in industry and often synthesized heuristically, much like the PIDfeedback control for which it was named. Thus, P-type ILC makes for an important benchmark.Tuning of the learning gains for both benchmark methods is described in Section 5.4.3. The presented stable inversion theory’s ability to derive the inverse of a non-minimum phase PWAmodel is tested by applying u = ˆ g − ( r ) to the control model. Then, to assess a more practicalutility, five independent simulations with the truth model are performed. First, a simulation isrun with the feedforward input fixed to zero for all time, yielding a feedback-only simulationto serve as a baseline against which the four feedforward controllers can be compared. Next,learning-free stable inversion is applied to the truth model. In other words, a simulation is runwith u = ˆ g − ( r ) . The remaining three simulations each use one of the ILC techniques describedSection 5.4.2 (ILILC, gradient ILC, or P-type ILC) with 9 trials (8 learning operations). For allILC simulations, u is the zero vector. 119he primary metric for assessing control performance in a given trial is the normalized rootmean square error (NRMSE) of the truth model, defined asNRMSE = RMS 𝑘 ∈ (cid:74) ,𝑁 (cid:75) ( 𝑒 𝑘 )(cid:107) r (cid:107) ∞ (5.86)The peak error magnitude max 𝑘 (| 𝑒 𝑘 |) is also considered.For gradient ILC and P-type ILC, tuning of the learning gains is done to achieve the mostaggressive stable controller possible. Specifically 𝛾 and 𝒫 are chosen as the largest whole numbersuch that the NRMSE decreases monotonically over all trials or drops below the convergencetolerance, set to 0.0005 here. These numbers are found via a line search over many ILC simulationsusing the given truth and control model, varying only the learning gains. By this method, 𝛾 = (dimensionless) and 𝒫 = Vm . If the learning gains are increased above these values, thebenchmark ILC schemes begin to exhibit instability. When u = ˆ g − ( r ) is input to the noise-free control model (5.70)-(5.71) from which it was derived,the resulting NRMSE and peak error magnitude are E − and
49 nm . This is nearly zero comparedto the other simulation errors, tabulated in Table 5.2, and what error there is can be attributed to theapproximation error expected of finite-time stable inversion, as discussed in Section 5.3.2. Thisvalidates the fundamental theoretical contributions of this chapter: the stable inversion—and thusalso conventional inversion—of PWA systems.To analyze the more practical application of these techniques to the truth model, Figure 5.5plots the evolution of the ILC schemes’ NRMSEs over the iteration process, and Figures 5.6 and5.7 plot the input and error time series for the five simulations. Specifically, Figure 5.6 comparesthe learning-free simulations to ILILC and Figure 5.7 compares ILILC against the other benchmarklearning techniques. The NRMSE and peak error magnitude for these simulations are tabulated inTable 5.2.Because the model error is relatively small, learning-free stable inversion does yield someimprovement (4%) over the feedback-only NRMSE. (Naturally, one expects that the NRMSE oflearning-free stable inversion would grow if the model error increased). Learning-free stable in-version also reduces the peak error magnitude, which is a critical safety criterion in many appli-cations, by 28%. However, because of the model error that does exist, all of the ILC schemesdefeat learning-free stable inversion in both metrics. P-type ILC shows the least improvement, butstill yields a 40% reduction in NRMSE from learning-free stable inversion. This improvement isdwarfed by that of ILILC, however, which yields a 70% NRMSE improvement from gradient ILC120able 5.2: NRMSE and Peak Error Magnitude of Truth Model SimulationsNRMSE Peak Error MagnitudeFeedback-Only . E − . Learning-Free Stable Inversion . E − . P-type ILC - Final Trial . E − . Gradient ILC - Final Trial . E − . ILILC - Final Trial . E − .
34 mm
Figure 5.5: NRMSE of each ILC trial, illustrating convergence of ILILC to a plateau determined by thenoise injected to the system, and dramatically surpassing the convergence speed of both benchmark ILCtechniques. The NRMSE of the learning-free stable inversion simulation is also pictured. It is 4% smallerthan the feedback-only simulation (ILILC trial 0), but is much larger than the performance achievable withlearning. Because the noise injected in this chapter is of the same distribution and injection location asthe noise in Chapter 4, the same convergence threshold can be used to approximate the minimum NRMSEachievable by ILILC. igure 5.6: Error (top) and Input (bottom) time series data for the feedback-only simulation, learning-freestable inversion simulation, and the final trial of the ILILC simulation. Both stable inversion and ILILCperform as expected, but due to model error learning is required to reap the full benefit of feedforwardcontrol. igure 5.7: Error (top) and Input (bottom) time series data for the three ILC simulations. Even the worst-performing ILC technique—P-type ILC—yields a reduction in maximum error magnitude when comparedto the learning-free techniques of Figure 5.6, but ILILC clearly yields the lowest-error performance. Thissuperiority is in spite of ILILC acquiring more high frequency content via learning than the other ILCschemes, which appear less noisy but appear to contain higher amplitude, lower frequency oscillations thatdegrade performance.
This chapter has derived theory for the inversion of a class of PWA systems. This includes theimplicit inverse formula for systems of any relative degree and the explicit formulas for systemswith global dynamical relative degree (a concept introduced in this work) of 0, 1, or 2, alongwith the proof of sufficient conditions for the inverse of the original PWA system to be explicit.Additionally, for cases in which the inverse system is unstable, a stable inversion procedure iscreated, along with proof of the sufficient conditions for the procedure to be applicable.The ability to analytically produce inverse system models for hybrid systems has multipleapplications in controls. Demonstrated here is the newfound ability to apply ILC to PWA systemswith unstable inverses to achieve low error output reference tracking.There are many avenues for future work. Of particular interest is the relaxation of the con-straints on relative degree. For some hybrid systems, it may be desirable to have locations in whichthe input cannot affect the output, and the state is governed by natural dynamics alone. In suchcases the global dynamical relative degree would be undefined (infinite), which is not consideredhere. There may also be more cases in which different locations feature different component rel-ative degrees, which would violate (A5.4). Relaxing these constraints would dramatically expandthe class of systems addressed. Extension to multi-input-multi-output systems and input-basedswitching would also be significant contributions.124
HAPTER 6
Conclusion
Spurred by the long-term goal of achieving high performance droplet volume control in e-jet print-ing, this dissertation makes substantial contributions to both the modeling of e-jet printing and thecontrol theory necessary to leverage those models. At each stage, practical obstacles give rise tonovel scientific and mathematical research.First, while Chapter 2 presents significant new progress in the traditional physics-based mod-eling of meniscus electrohydrodynamics, the inability of traditional models to completely capturethe ejection process from end to end motivates the development of hybrid e-jet modeling frame-works. Likewise, the inability to process the computer-vision-based jet measurements fast enoughfor real-time feedback control motivates the study of ILC for this system. Finally, the lack ofILC theory for hybrid systems in the preexisting literature serves as the impetus for the controlsresearch making up Chapters 3-5.The first of these chapters lays the groundwork for the remaining hybrid systems and ILCresearch by providing a closed-form PWD system representation and integrating it with NILC.Based on the original convergence analysis for NILC, one might expect the performance of thisintegration to be mostly uniform across all PWD models. However, while the lifted model deriva-tion given in Chapter 3 is guaranteed to have a theoretically invertible Jacobian if the relativedegree is constant over the trial, this does not account for the practical ability to compute the in-verse. As found in Chapter 4, this practical ability is compromised when the system model hasan unstable inverse. Such NMP systems are not negligible edge cases. They arise when modelingmany practical motion control devices—a key application space for ILC—from piezoactuators [7]to DC motor and tachometer assemblies [39]. Thus, to leave the issue of NMP hybrid systemsunaddressed would be to deliver more of a minefield than a control theory.Chapters 4-5 sweep this minefield by introducing the new ILILC framework, integrating itwith stable inversion, and developing the first theory for the stable inversion of hybrid systems.125ltimately this enables ILC synthesis from a model of the physical device that originally soundedthe alarm on NILC of NMP systems, an inkjet printhead positioning system.
Both the hybrid e-jet modeling research and the control theory research in this dissertation haveramifications beyond the validations presented here, and beyond the intended future use for dropletvolume control. In fact, the broader impacts of hybrid e-jet modeling have already begun to mani-fest.Within the world of e-jet printing, the hybrid models provided by this dissertation serve asan end-to-end process model that is more easily interpreted and analyzed by human researchersthan computational multiphysics models. Because of this, the physics-driven hybrid model is cur-rently being used for the benchmarking and development of such sophisticated partial-differential-equation-based simulations.This dissertation’s modeling contributions have also gained attention in the broader AM com-munity, outside of e-jet-specific research. The promotion and validation of hybrid modeling fore-jet printing helped pave the way for the proposal of a more general hybrid AM modeling frame-work, which was used for modeling the multi-level workflow of FDM [119]. There is also greatpromise for more thorough hybrid modeling of the physical dynamics of FDM: research has beenpublished identifying distinct dynamic regimes FDM may occupy depending on the physical stateof the printhead and filament [85]. Additionally, because AM processes are mostly open loop atthis time, they are compelling candidates for ILC. In other words, there is evidence of a rich fieldof systems whose modeling would benefit from following in the footsteps of this dissertation, andwhose control may benefit directly.To see the broader impacts of Chapters 3-5, it may be beneficial to revisit the “castle-of-building-blocks” visualization from Figure 1.1. Such a revisitation is given in Figure 6.1, whichillustrates a number of new classes of control systems that may be enabled by the fundamentalcontributions of this work. Most obviously, where there are contributions to inversion and stableinversion theory, there is opportunity for feedforward control. While this dissertation focuses onILC because of the performance advantages it yields over learning-free methods, one should notdiscount the importance of ordinary feedforward control, for it can be applied in non-repetitivescenarios where the learning mechanisms of most ILC schemes may falter.126
LC ofsmoothNMPsystems ILC ofhybridNMPsystems Stable inversion ofPWA systemsConventional inversion ofPWA systemsILC ofhybridinverse-stable sys.NILC formalization for time-varyingsystems of relative degree ≥ Figure 6.1: Castle of Control Contributions, revisited for visualization of this dissertation’s potential broader impacts. These may take the form ofother new classes of control systems leveraging the theoretical contributions presented here. he ability to generate feedforward control signals can also be useful in scenarios where feed-back control is a primary focus. A concrete example is given by [46], where a combined feed-back/feedforward control scheme for PWA system reference tracking is derived, but is limitedbased on the availability of the feedforward signal. This dissertation alleviates that limitation.Finally, a further impact on feedback control may be possible with the closed-form PWD rep-resentation. The new ability to holistically differentiate a hybrid state transition function or outputfunction may facilitate the synthesis of a diverse range of controllers, e.g. feedback linearization.
Clearly, these potential broader impacts themselves constitute a large arena for future work. Inparticular, investigations into the hybrid modeling of other AM technologies and the use of theclosed-form PWD system representation for feedback control synthesis may be valuable contribu-tions. Of course, further progress towards the original long-term objective of e-jet droplet volumecontrol is of primary interest.The three main challenges to overcome here are the need for point-to-point ILC, managing thetrade-off between time-based location transitions and model stability, and encoding the limits ofthe safe printing region into the model.Clearly, the droplet volume modeling of the PWA e-jet model (the second proposed model,Section 2.3) is necessary to achieve droplet volume control. However, it is undesirable to requirereference specification for the entire droplet volume time series; only the final droplet volumevalue matters. ILC with reference specification for only a subset of the trial time series exists andis called point-to-point ILC [120], but has not been combined with either NILC or hybrid models.This leaves a gap between the prior art and the needs of e-jet printing.Additionally, while the PWA e-jet model introduces droplet volume modeling, it removes state-based transitions and replaces them with time-based transitions. Time-based transitions cannot bealtered by ILC, but the time of transition will certainly change in practice if the input voltagechanges in magnitude from trial to trial. This makes state-based switching desirable. To achievestate-based transitions, dynamical modeling of the meniscus position is necessary. This presentsan issue for ILC because a linear model fit to meniscus position data during the approach locationis likely to be unstable (this may be deduced from Figure 2.8), and asymptotic model stability isprerequisite to ILC in general [4].Constrained system identification or extension of the physics-focused e-jet model’s build-up orjetting locations may enable stable modeling of the meniscus position. However, this is unlikelyto completely solve the e-jet ILC problem. Stable linear modeling during jetting implies that themeniscus position may be completely controllable during jetting (potentially with all control ac-128ions experiencing the same time delay modeled for the physics-focused model’s jetting location).There is no evidence to suggest this degree of controllability in practice. Basing ILC on this as-sumption may have undesirable results. For example, if a droplet volume is desired to be smallerthan it was for trial ℓ , ILC may request a large negative voltage during jetting to arrest the jet orremove material from the substrate. This may be possible according to the LTI component models,but may not be possible in reality, and attempts to do so may take the physical system outside theregime in which the model is applicable. Such risks lead to the final gap identified for the ILC ofe-jet printing: encoding of the safe input range.In other words, a major area for future work in e-jet modeling is the prediction and encodingof the boundaries of the subcritical regime. This is both a performance and safety issue. Attemptsto project subcritical regime behavior beyond the regime limits may result in failure to eject, mis-placed droplets due to tilted ejection angles, or destruction of the nozzle via flooding or arcing.Beyond e-jet printing, there are also exciting control theory developments to be built directlyoff the contributions of this dissertation. Two meaningful areas for future work are identified.First is the investigation of compatibility between the closed-form PWD representation and formsof ILC not using lifted models. While ILILC’s sufficient conditions for convergence are verybroad, in some cases it may be desirable to prioritize computational cost. In such cases the largematrix operations in lifted ILC may be a disadvantage. Thus, the use of the closed-form PWDrepresentation to synthesize filter-based ILC, such as that of [121], may be valuable.Second is the relaxation of the assumptions under which an NMP PWA system may be con-trolled via ILILC. Currently, these assumptions are those of PWA stable inversion, but ILILC canadmit other inverse system approximations as well. Thus, assumption relaxation efforts could fo-cus on improving PWA stable inversion or introducing a new stable inverse approximation method.Finally, it must be remembered that the research and new engineering tools provided here weredeveloped in large part as a response to the unexpected challenges that leapt up from physicalsystems. Surely nature’s surprises are not exhausted.129 PPENDIX A
Neglect of Inverse Instability by Non-NILC Prior Art
This appendix demonstrates that the sufficient conditions for convergence proposed by past works[13, 14, 15, 16] on ILC for discrete-time nonlinear systems are in actuality not sufficient for somecases of systems having unstable inverses. This is done by running model-error-free ILC simula-tions that are guaranteed to converge by the past works, and observing them to diverge instead.Each of [13, 14, 15, 16] proposes sufficient conditions for the convergence lim ℓ →∞ e ℓ = 𝑁 − 𝜇 + of a particular ILC scheme applied to a particular class of nonlinear dynamics. All of these classesof nonlinear dynamics are supersets of the SISO LTI dynamics 𝑥 ℓ ( 𝑘 + ) = 𝐴𝑥 ℓ ( 𝑘 ) + 𝐵𝑢 ℓ ( 𝑘 ) (A.1a) 𝑦 ℓ ( 𝑘 ) = 𝐶𝑥 ℓ ( 𝑘 ) (A.1b)with relative degree 𝜇 = , i.e. 𝐶 𝐵 ≠ . Additionally, assume (A.1) is stable and 𝑥 ℓ ( ) is such that 𝑦 ℓ ( ) = 𝑟 ℓ ( ) ∀ ℓ . Given a system of this structure, the ILC schemes and convergence conditionsof the past work reduce to the following.From [13] the learning law is 𝑢 ℓ + ( 𝑘 ) = 𝑢 ℓ ( 𝑘 ) + 𝐿 ℓ ( 𝑘 ) ( 𝛾 𝑒 ℓ ( 𝑘 + ) + 𝛾 𝑒 ℓ ( 𝑘 )) (A.2)where 𝐿 ∈ R is a potentially time-varying and trial-varying part of the learning gain and 𝛾 , 𝛾 ∈ R are trial-invariant, time-invariant learning gains with 𝛾 ≠ . The learning laws of [14, 15, 16] arespecial cases of (A.2): [14] sets 𝛾 = , 𝛾 = − , [15] sets 𝛾 = , 𝛾 = , and [16] sets 𝛾 = andleaves 𝛾 free.Each work presents a different variation of convergence analysis, but all propose a sufficientcondition of the form(CA.1) | − 𝐿 ℓ ( 𝑘 ) 𝛾 𝐶 𝐵 | < ∀ 𝑘, ℓ .In [13, 15, 16] (CA.1) is used exactly, while the convergence analysis in [14] implies the additionalsufficient condition 130 Trial Number -20 -10 N R M S E Past ILC with = 1, = 0Past ILC with = 1, ILILC with Stable Inversion
Figure A.1: NRMSE versus trial number of past works’ ILC schemes (A.2) applied with learning gain (A.4)to the system (A.3). These NRMSEs monotonically increase, confirming the inability of the past work onILC with discrete-time nonlinear systems to account for unstable inverses. The NRMSE trajectory yieldedby the stable-inversion-supported ILILC scheme proposed by this article is also displayed. The convergenceof this ILC scheme when applied to (A.3) reiterates its ability to control such non-minimum phase systems. (CA.2) (cid:107) 𝐴 (cid:107) > where any consistent norm may be chosen for (cid:107)·(cid:107) .Consider the example system and learning gain 𝐴 = − . − .
79 0 .
530 0 . − .
36 0 . 𝐵 = . 𝐶 = (cid:104) . . − . (cid:105) 𝑥 ℓ ( ) = ∀ ℓ (A.3) 𝐿 ℓ ( 𝑘 ) = . ( 𝐶 𝐵 ) − ∀ 𝑘, ℓ (A.4)with the reference given in Figure 4.2. This system has an unstable inverse.The plant (A.3) satisfies (CA.2), and with (A.4) it satisfies (CA.1) for 𝛾 = . Thus, accordingto [13, 14, 15, 16] the ILC scheme (A.2) is guaranteed to yield tracking error convergence in amodel-error-free simulation. However, Figure A.1 shows that the tracking error diverges under(A.2), meaning that satisfaction of (CA.1) and (CA.2) is not actually sufficient for the convergenceof all systems (A.1) under the learning law (A.2). This illustrates that the failure to account forphenomena arising from inverse instability is not unique to NILC, but rather pervades the literatureon ILC with discrete-time nonlinear systems. 131 IBLIOGRAPHY [1] P. Tabuada,
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