Robust Implementable Regulator Design of General Linear Systems
RRobust Implementable Regulator Design of General Linear Systems
Lei Wang a , Lorenzo Marconi b , Christopher M. Kellett c a Australia Centre for Field Robotics, The University of Sydney, Australia. b C.A.S.Y - DEI, University of Bologna, Italy. c Research School of Electrical, Energy, and Materials Engineering, Australian National University, Australia.
Abstract
Robust implementable output regulator design approaches are studied for general linear continuous-time systems with periodically sampledmeasurements, consisting of both the regulation errors and extra measurements that are generally non-vanishing in steady state. A digitalregulator is first developed via the conventional emulation-based approach, rendering the regulation errors asymptotically bounded witha small sampling period. We then develop a hybrid design framework by incorporating a generalized hold device, which transforms theoriginal problem into the problem of designing an output feedback controller fulfilling two conditions for a discrete-time system. Weshow that such a controller can always be obtained by designing a discrete-time internal model, a discrete-time washout filter, and adiscrete-time output feedback stabilizer. As a result, the regulation errors are shown to be globally exponentially convergent to zero, whilethe sampling period is fixed but can be arbitrarily large. This design framework is further developed for a multi-rate digital regulator witha large sampling period of the measurements and a small control execution period.
Key words:
Robust output regulation; Sampling; Washout filter; Internal model; Generalized hold device
The problem of output regulation is to design a controller soas to achieve asymptotic trajectory tracking and/or disturbancecompensation in the presence of reference/disturbance signalsthat are trajectories of an autonomous system (the so-calledexosystem). Taking robustness into consideration, the internalmodel principle has been shown as the most effective designand analysis tool in the seminal work [1] for linear continuous-time systems. Internal model-based methods have been welldeveloped for continuous-time nonlinear systems with continu-ous measurements (e.g., [2,3,4]), hybrid systems (e.g., [5,6,7])and networked systems (e.g., [8,9]).In general, the internal model-based regulator consists oftwo main components: the internal model and the stabilizer.There are two kinds of control architectures, referred to as post-processing and pre-processing schemes (see e.g. [10]), depend-ing on the topology used to connect the internal model and sta-bilizing units. As for the pre-processing scheme, the stabilizer isdirectly cascaded with the controlled plant, processing the reg-ulation errors. In post-processing schemes, on the other hand,the internal model is cascaded with the plant by processing theregulation errors. For single-input single-error (SISE) systems,both schemes are fundamentally equivalent with pre-processingschemes that have been shown to be more constructive in somecases, such as for nonlinear systems with non-vanishing extrameasurements (e.g. [4,11]), and multi-rate systems (e.g. [12]).The measurements for feedback design are generally ob-tained from periodically sampling sensors, whose measure isnot accessible continuously, and the regulator is practicallyimplemented by digital devices or in combination with somesimple analog devices, for example generalized zero-order
Email addresses: [email protected] (Lei Wang), [email protected] (Lorenzo Marconi), [email protected] (Christopher M. Kellett). hold devices. In [13] for linear continuous-time systems, astate-feedback solution is studied locally by proposing a fullydiscrete-time regulator, which fulfills the internal model prin-ciple solely at the sampling time by thus guaranteeing onlypractical regulation. To achieve asymptotic regulation, in [14]a hybrid internal model is proposed. It is shown that thecontinuous-time steady state input can be generated by meansof continuous-time internal models acting as “generalized sig-nal re-constructors” and there always exists a discrete-timestabilizer achieving the desired regulation purpose, though inthe absence of robustness analysis. Motivated by this work, [5]further develops a robust solution for SISE linear systems. Inaddition, [15,16,17] adopt the emulation-based method by sam-pling the measurements and the control inputs in the absenceof samples, which requires the sampling period to be small andrenders the regulation errors bounded in a general sense. Inall the aforementioned results, the controlled continuous-timesystems are required to be detectable by the regulation errors,which might not be fulfilled in practice, for example, as withthe inverted pendulum on a cart considered in Section 5 below.Motivated by the previous analysis, this paper studies therobust sampled-data regulation problem of general linearcontinuous-time systems, for which the detectability propertyis fulfilled by the whole set of measurements, consisting ofboth the regulation errors and possible extra non-vanishingmeasurements. Motivated by [15,16], we first present anemulation-based solution, which under a small sampling pe-riod, renders the regulation errors asymptotically bounded by aconstant, depending on the time derivative of the steady statesof both extra measurements and control inputs, and adjustableby the sampling period. In view of this, we then develop ahybrid design framework by incorporating a generalized holddevice. By cascading this device with the controlled plant, it isshown that the desired regulation objective can be achieved bydesigning a discrete-time output feedback stabilizer fulfillingtwo conditions, i.e., stabilizing the closed loop at the originand compensating for the steady state input. Inspired by [4],to fulfill both conditions, we further propose a discrete-time
Preprint submitted to Automatica 1 March 2021 a r X i v : . [ m a t h . O C ] F e b nternal model and a discrete-time washout filter, which inturn simplifies the problem to the design of a discrete-timeoutput feedback stabilizer for an augmented discrete-time lin-ear system that is stabilizable and detectable. As a result, theregulation errors are shown to be globally exponentially con-vergent to zero, while the sampling period is fixed but can bearbitrarily large. By regarding both the generalized hold deviceand the discrete-time internal model as a hybrid internal modelunit, we note that the proposed robust implementable regula-tor is consistent with the pre-processing internal model-basedstructure proposed in [4]. Furthermore, this design frameworkis developed for a multi-rate digital regulator with a large sam-pling period of the measurements and a small control executionperiod. We show that given almost any large sampling periodof the measurements, the regulation errors can be rendered tobe bounded by a constant, depending on the time derivative ofthe desired steady states of control inputs, and adjustable bythe control execution period.This paper is organized as follows. In Section 2 the con-sidered problem is explicitly formulated and some standingassumptions are presented. Section 3 presents the emulation-based approach, which then motivates us to propose a new im-plementable regulator design framework via generalized holddevices in Section 4. In Section 5, this framework is furtherexplored for a multi-rate digital regulator. To show the effec-tiveness of the proposed approach, the linearly approximatedmodel of the inverted pendulum on the cart is studied in Sec-tion 6. Conclusions are presented in Section 7. This paper isdifferent from the conference version [18] by additionally pre-senting the motivating emulation-based approach in Section 3and developing the multi-rate digital regulator in Section 5. Consider the output feedback regulation problem for linearsystems ˙ w = Sw ˙ x = A x + B u + P wy = C x + Q w (1)with exogenous states w ∈ R d , states x ∈ R n , inputs u ∈ R m and measurements y ∈ R q . We deal with a general classof linear systems in which the measurements y consist ofregulation errors e ( t ) := C e x ( t ) + Q e w ( t ) ∈ R q e , to besteered to zero asymptotically, and also extra measurements y m ( t ) := C m x ( t ) + Q m w ( t ) ∈ R q m on which no specificregulation requirements are fixed, with q = q e + q m . Con-sidering the practical situation in which the measurements aregenerally obtained in a discrete-time manner, i.e., periodicallysampled with sample time T > , the measurements avail-able for feedback are given by the sampled regulation errors ˆ e ( t ) := C e x ( t k ) + Q e w ( t k ) and the sampled extra measure-ments ˆ y m ( t ) := C m x ( t k ) + Q m w ( t k ) for t ∈ [ t k , t k +1 ) , t k = kT and k ∈ N . As customary in the field of output reg-ulation, we assume that S is neutrally stable and there existsan invariant compact set W ∈ R d such that w ( t ) ∈ W for all t ≥ . For convenience, we set |W| := max w ∈W (cid:107) w (cid:107) .In this setting, the control objective is to design a robustimplementable regulator driven by the sampled measurements (ˆ e ( t ) , ˆ y m ( t )) such that the resulting closed-loop trajectories are bounded, and the continuous-time regulation errors e ( t ) asymp-totically converge to zero. As in [1,4], we are interested in a robust solution, i.e., the above control objective is guaranteedeven if all system matrices of (1) except S vary in a (small)neighborhood of their nominal forms . Additionally, this paperpresents an implementable solution that can be directly imple-mented by purely digital devices, or together with some simpleanalog devices, such as generalized hold devices (see [14]).Due to the presence of the sampled measurements, the re-sulting system is fundamentally hybrid. In this paper, we willfollow terminologies from [22] to denote the time t ∈ [ t k , t k +1 ) by a hybrid time domain ( t, k ) , and represent a hybrid systemas a combination of flow and jump dynamics, which are re-spectively described by differential and difference equations.The action of sampling the measurements e and y m leads to ameasurement model of the kind ˙ τ = 1˙ˆ e = 0˙ˆ y m = 0 , for ( τ, ˆ e, ˆ y m ) ∈ [0 , T ) × R q τ + = 0ˆ e + = C e x + Q e w ˆ y + m = C m x + Q m w , for ( τ, ˆ e, ˆ y m ) ∈ { T } × R q (2)in which τ is a clock state, and ˆ e, ˆ y m are the sampled mea-surements available for feedback. In the subsequent Sections 3and 4, the flow and jump conditions are governed by the clock τ only as in (2), i.e., the flow occurs for τ ∈ [0 , T ) and thejump occurs for τ = T , which will be occasionally omitted forsimplicity.Throughout this paper, for any square matrix M , we denoteby σ ( M ) its spectrum. We make some standard assumptionspreviously used for a robust continuous-time solution (e.g. [1]). Assumption 1 (i) The matrix triplet ( A, B, C ) is stabilizable and detectable;(ii) There holds the non-resonance conditionrank (cid:34) A − λI n BC e (cid:35) = n + q e , ∀ λ ∈ σ ( S ) . (3)Assumption 1 immediately implies that for any pair of ma-trices ( P, Q e ) , there exist Π x ∈ R n × d and Ψ ∈ R m × d suchthat the regulator equations Π x S = A Π x + B Ψ + P C e Π x + Q e (4)are satisfied.As in [13,14], in order to preserve the stabilizability anddetectability of system (1) after discretization, the followingassumption is made. If there exist uncertainties on the matrix S , then the idea of anadaptive internal model (e.g. [20,21]) can be employed. ssumption 2 The sampling period T is not pathological fromthe pair ( A, S ) . That is, for any distinct λ i , λ j ∈ σ ( A ) (cid:83) σ ( S ) , λ i − λ j (cid:54) = 2 kπ/T , for any k ∈ N . (5)Note that condition (5) is generically satisfied for all sam-pling time T ∈ R + , given any matrices A, S . Remark 1
We note that this paper follows the internal model-based regulator design framework, where solutions of the reg-ulator equations (4) are not used for feedback design, and theinternal model is designed to compensate for the steady stateinput Ψ w ( t ) . Thus, as in other relevant works [1,3,4], the pre-sented designs yield a robust regulation with respect to smallvariations of the system matrices of (1) except S . Before presenting the proposed framework, this section aimsto present how the conventional emulation-based approach[15,16] can be applied to solve the considered problem.To apply the emulation-based approach, we first assume thatthe continuous-time measurements y ( t ) are accessible and in-vestigate a robust regulator driven by y ( t ) = ( e ( t ) , y m ( t )) .Typically, there are two internal model-based control schemes:post-processing [1] and pre-processing [4] methods, both lead-ing to a robust dynamical regulator of the form ˙ x c = A c x c + B c y , x c ∈ R n c u = C c x c (6)where matrices A c ∈ R n c × n c , B c ∈ R n c × q , C c ∈ R m × n c aredesigned such that the following two requirements hold (alwayssatisfiable under Assumption 1):(i) the matrix A := (cid:34) A BC c B c C A c (cid:35) is Hurwitz;(ii) for all (Π x , Ψ) satisfying (4), the matrix equations Π x c S = A c Π x c + B c (cid:34) Y m (cid:35) , Ψ = C c Π x c with Y m = C m Π x + Q m , admit a solution Π x c .With these matrices A c , B c , C c in hand, following theemulation-based method, it is always possible to design aregulator driven by the sampled measurements ˆ y = (ˆ e, ˆ y m ) fulfilling the dynamics (2), of the form ˙ τ = 1 , ˙ x c = A c x c + B c ˆ y, ˙ u = 0 for ( τ, x c , u ) ∈ [0 , T ) × R n c + m ,τ + = 0 , x + c = x c , u + = C c x c for ( τ, x c , u ) ∈ { T } × R n c + m . (7)By setting ˆ x c := e − A c τ x c − (cid:90) τ e − A c r d r B c ˆ y , (7) can berewritten into a discrete-time equivalent form ˙ˆ x c = 0 , ˙ u = 0ˆ x + c = e A c T ˆ x c − (cid:82) T e − A c ( r − T ) d rB c ˆ yu + = C c [ e A c T ˆ x c − (cid:82) T e − A c ( r − T ) d rB c ˆ y ] , (8) which clearly can be implemented by digital devices.Fundamental to show the properties of the closed-loop sys-tem (1)-(8) is the following lemma, which is adapted from [15,Lemma 2]. Lemma 1
There exist a symmetric positive definite matrix P ∈ R ( n + n c ) × ( n + n c ) , κ > and γ ∗ > such that (cid:34) PA + AP + 2 κ P + γ ∗ C (cid:62) C PBB (cid:62) P − γ ∗ I q (cid:35) ≤ (9) with B = (cid:34) BB c (cid:35) and C = (cid:34) CA CBC c C c B c C C c A c (cid:35) . Let τ max ( κ, γ ) := κr arctan( r ) , if γ > κ κ , if γ = κ κr arctanh ( r ) , if γ < κ (10)where r = (cid:113) | γ κ − | . Proposition 1
Suppose Assumption 1 holds. Choose κ, γ ∗ according to Lemma 1. Then for all γ ≥ γ ∗ , and T ∈ (0 , τ max ( κ, γ )] ,(i) the trajectories of the resulting closed-loop system (1) withregulator (8) are bounded, and(ii) there exists a k em > , independent of γ, κ such that lim t + j →∞ (cid:107) e ( t, j ) (cid:107)≤ k em √ γκ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:34) Y m S Ψ S (cid:35)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (11)We observe that the emulation-based regulator (8) renders theclosed-loop trajectories bounded, provided that the samplingperiod T is smaller than τ max ( κ, γ ) . That is, this method maynot be effective when a large T is desired. On the other hand,from (11) it is seen that the regulation error e ( t ) is asymptoti-cally bounded by a constant, depending on the time-derivativeof the desired steady states of y m and u , and on the samplinginterval (i.e., γ, κ ). This bound can be arbitrarily decreased byincreasing γ , which in turn implies a smaller upper bound forallowable T . Practical (and not asymptotic) regulation is essen-tially due to the fact that the control input is also sampled andthus the continuous-time steady state control input, which is Ψ w ( t ) , is not exactly generated by the controller (7). Motivatedby these restrictions, we develop a design technique, which cannot only solve the problem under a very large T , but also renderthe regulation errors e ( t ) exponentially convergent to zero. It is well-known (see [1]) that the steady state input forcingthe desired regulation objective of system (1) is a continuous-time signal of the form u ss ( t ) = Ψ w ( t ) with Ψ provided by theregulator equations (4). Note that, in general, this continuous-time steady state input cannot be perfectly re-constructed by a3iscrete-time compensator. This motivates us to embed into theregulator a continuous-time signal reconstructor, which gener-ates the steady-state control input during flows. Motivated by[14], we deal with a cascade of the controlled plant (1) and a generalized hold device , the latter described by ˙ ζ = (Φ ⊗ I q e ) ζ , ζ + = v ζ (12)with state ζ ∈ R dq e and input v ζ ∈ R dq e used to reset ζ at thesampling time, and Φ ∈ R d × d being a matrix whose minimalpolynomial is coincident with that of S , the latter denoted by P S ( λ ) = (cid:80) d − i =0 s i λ i + λ d . To ease subsequent analysis, thereis no loss of generality to let
Φ = (cid:34) d − I d − − s ( − s · · · − s d − ) (cid:35) , with d − being a zero column vector of dimension d − .The feedback control law is designed as u = Lζ + v u (13)where v u ∈ R m denotes the residual control input that will bedesigned as digital, i.e., ˙ v u = 0 , and L ∈ R m × dq e is such thatthe pair (Φ ⊗ I q e , L ) is observable.By augmenting (12) and (13) with (1), we obtain ˙ w = S w ˙ x = A x + B L ζ + B v u + P w ˙ ζ = (Φ ⊗ I q e ) ζ (14)during the flow, and during the jump, w + = w , x + = x , ζ + = v ζ (15)with input v := col ( v u , v ζ ) satisfying ˙ v = 0 during flow.The problem at hand is thus to design a digital controllerof the hybrid system (14)-(15) with control inputs v andmeasurements (ˆ e, ˆ y m ) fulfilling the dynamics (2) such thatthe resulting closed-loop system trajectories are bounded and lim t + j →∞ e ( t, j ) = 0 . By letting w D ( τ ) := e − Sτ wx D ( τ ) := e − Aτ x − (cid:82) τ e − A r
BLe (Φ ⊗ I qe )( r − τ ) d r ζ − (cid:82) τ e − A r d r B v u − (cid:82) τ e − A r
P e S ( r − τ ) d r wζ D ( τ ) := e − (Φ ⊗ I qe ) τ ζ , (16)it immediately follows that the “sample-data” discrete-time sys-tem linked to (14)-(15) is given by ˙ w D = 0 , ˙ x D = 0 , ˙ ζ D = 0 w + D = S D w D x + D = A D x D + L D ζ D + B D v u + P D w D ζ + D = v ζ ˆ y = (cid:34) ˆ e ˆ y m (cid:35) = (cid:34) C e x D + Q e w D C m x D + Q m w D (cid:35) (17) with S D := e S T , A D := e A T , B D := (cid:82) T e A r d r B , L D := (cid:82) T e A ( T − r ) BLe (Φ ⊗ I qe ) r d r , and P D := (cid:82) T e A ( T − r ) ·· P e
S r d r . For system (17) the following holds. Lemma 2
Suppose Assumptions 1 and 2 hold.(i) The system (17) is stabilizable and detectable w.r.t. inputs v and outputs ˆ y when w D = 0 .(ii) Let Φ D := e Φ T . For any Π x in (4), there exists Π ζ ∈ R dq e × d such that Π x S D = A D Π x + L D Π ζ + P D Π ζ S D = (Φ D ⊗ I q e )Π ζ C e Π x + Q e . (18) Proof. Proof of (i).
With Assumption 1.(i) and 2, according to[19], it can be deduced that ( A D , B D , C ) is stabilizable anddetectable. Then using the PBH test, it can be verified thatsystem (17) is stabilizable and detectable w.r.t. inputs v andoutputs ˆ y when w D = 0 . Proof of (ii).
As for the solution of (18), we observe that, forany Ψ ∈ R m × d , since (Φ ⊗ I q e , L ) is observable, there alwaysexists a unique solution Π ζ such that Π ζ S = (Φ ⊗ I q e )Π ζ Ψ = L Π ζ . (19)In view of the fact that (18) is the discretized form of (4)(derived by Assumption 1.(ii)) and (19), this indicates that such (Π x , Π ζ ) is also the solution of (18), completing the proof. (cid:4) The desired robust regulator can be completed by designingan output feedback controller for the discrete-time system (17),having the form ˙ z = 0 , z + = A z z + B z ˆ y , z k ∈ R n z ,v = K z z + L z ˆ y . (20) Theorem 1
Suppose Assumptions 1 and 2 hold. The robustsampled output regulation problem of system (1) is solved bythe regulator (12), (13), and (20) if (a) the origin of the closed-loop discrete-time system (17), (20)with w D = 0 is globally exponentially stable;(b) for any Π x , Π ζ satisfying (18), letting Y m = C m Π x + Q m ,there exists a solution Π z ∈ R n z × d of Π z S D = A z Π z + B z (cid:104) d × q e Y (cid:62) m (cid:105) (cid:62) (cid:34) m × d Π ζ S D (cid:35) = K z Π z + L z (cid:104) d × q d Y (cid:62) m (cid:105) (cid:62) . (21)In the following lemma, we claim that if the number ofcontrol inputs matches the one of regulation errors then theabove sufficient conditions are also necessary. Lemma 3
Suppose Assumptions 1 and 2 hold. The sampledrobust output regulation problem of system (1) with m = q e issolved by the regulator (12), (13), and (20) if and only if therequirements (a) and (b) in Theorem 1 are fulfilled. In the previous subsection, we have shown that the originalrobust output regulation problem with sampled measurementscan be transformed into the design of a discrete-time outputfeedback controller (20) such that the requirements (a) and (b)in Theorem 1 are fulfilled.Observe that by partitioning K z and L z as (cid:34) K z,u K z,ζ (cid:35) and (cid:34) L z,u L z,ζ (cid:35) consistently with the partition of v in v u and v ζ , thesecond equation of (21) can be partitioned into two parts m × d = K z,u Π z + L z,u (cid:104) d × q e Y (cid:62) m (cid:105) (cid:62) (22) Π ζ S D = K z,ζ Π z + L z,ζ (cid:104) d × q e Y (cid:62) m (cid:105) (cid:62) . (23)From (22), one can see that the controller (20) is required toblock the steady state of the extra measurements ˆ y m , denoted by Y m w D . In addition, (23) expresses the ability of the controller(20) to reproduce the ideal steady state of input v ζ , which is Π ζ S D w D .In the following, we show how to systematically design thecontroller (20) so that the conditions of Theorem 1 are fulfilled.Motivated by [4], the fulfillment of (22) suggests the design ofa washout digital filter to ˆ y m so as to block its steady state.The digital filter takes the form ˙ ξ = 0 , ξ + = F f ξ + G f ˆ y m y f = ˆ y m − Γ f ξ (24)where y f is the filter output, ( F f , Γ f ) ∈ R dq m × dq m × R q m × dq m is an observable pair with all eigenvalues of F f lying strictlywithin the unit circle, and G f ∈ R dq m × q m is such that Φ D ⊗ I q m = F f + G f Γ f .Similarly, the fulfillment of (23) suggests to consider adiscrete-time internal model of the form ˙ η = 0 , η + = (Φ D ⊗ I q e ) η + v η v ζ = η + ¯ v ζ (25)with η ∈ R dq e and ¯ v ζ , v η ∈ R dq e to be determined later. Lemma 4
Suppose Assumptions 1 and 2 hold. The augmenteddiscrete-time system (17), (24), and (25) is stabilizable anddetectable w.r.t. inputs ¯ v := ( v u , ¯ v ζ , v η ) and outputs y := col (ˆ e, y f ) , when w D = 0 . The proof of Lemma 4 is given in Appendix D. This lemmanaturally suggests to design a discrete-time output feedback stabilizer for the augmented discrete-time system (17), (24),and (25) when w D = 0 . Let ˙ ϑ = 0 , ϑ + = A ϑ ϑ + B ϑ y , ϑ ∈ R n z − dq ¯ v = C ϑ ϑ + D ϑ y . (26)be such a controller. The following theorem can be then proved. Theorem 2
Let (26) be a stabilizer for the augmented system(17), (24), and (25). Then, with system (20) defined by thecascade of (24), (25), and (26), the requirements (a) and (b) inTheorem 1 are fulfilled.Proof.
It is clear that by designing the controller (20) as thecascade of (24), (25) and (26), the requirement (a) in Theorem1 is fulfilled. Now we proceed to show that the requirement (b)is also fulfilled.We first note that since ( F f , Γ f ) is observable and Φ D ⊗ I q m = F f + G f Γ f , given any Y m ∈ R q m × d , there exists a uniquesolution Π f ∈ R q m d × d for the linear matrix equation Π f S D = F f Π f + G f Y m Y m − Γ f Π f . (27)Then, for any Π ζ satisfying (18), we have Π η S D = (Φ D ⊗ I q e ) Π η (28)with Π η = Π ζ S D . Thus, it can be easily verified thatwith the controller (20) as the cascade of (24), (25) and(26), the resulting matrix equations (21) permit a solution Π z = (cid:104) Π (cid:62) f Π (cid:62) η d × ( n z − dq ) (cid:105) (cid:62) , i.e., the resulting requirement(b) in Theorem 1 is fulfilled. (cid:4) We conclude the section by observing that the proposed con-troller fits in the “pre-processing” structure delineated in [4] andthat, consistently with the prescriptions of [14,5], the internalmodel unit contains q e copies of the continuous-time exosys-tem and q e copies of the discretized exosystem (see Figure 1). Fig. 1. Structure of the proposed regulator.
In the previous section, we have proposed a robust hybridregulator solving the considered problem globally and expo-nentially, even if the measurements are sampled with a verylarge interval. The regulator is implemented in a hybrid man-ner, i.e., a combination of a digital controller and a general-ized hold device. In this section, this hybrid regulator is further5eveloped for a pure digital regulator. Recalling that the sam-pling interval of measurements can be almost arbitrarily largein Theorem 2, we thus adopt multi-rate samplings for system(1), i.e., the control execution period is TN where T , that canbe very large, is the sampling period of measurements and pa-rameter N ∈ N + is used to determine the control executioninterval. In this respect, the control signal u can be modelledby the following hybrid form (cid:40) ˙ τ c = 1˙ u = 0 , for ( τ c , u ) ∈ [0 , TN ) × R m (cid:40) τ + c = 0 u + = ˆ u , for ( τ c , u ) ∈ { TN } × R m (29)where τ c is a clock state and ˆ u is an input to be determined.Following the design in Section 4, we design ˆ u as ˆ u = Lζ + K z,u z + L z,u ˆ y ˙ τ = 1˙ ζ = (Φ ⊗ I q e ) ζ ˙ z = 0 , for ( τ, ζ, z ) ∈ [0 , T ) × R n c τ + = 0 ζ + = K z,ζ z + L z,ζ ˆ yz + = A z z + B z ˆ y , for ( τ, ζ, z ) ∈ { T } × R n c (30)with (Φ ⊗ I q e , L ) observable. Let K z = (cid:34) K z,u K z,ζ (cid:35) and L z = (cid:34) L z,u L z,ζ (cid:35) . The resulting closed-loop stability is formulated below,with proof given in Appendix E. Theorem 3
Suppose that Assumption 1 holds. Let
T > be any number satisfying Assumption 2, and choose A z , B z , K z , L z according to Theorem 2. Then there exist a N ∗ ∈ N + and a γ † > such that for all N ≥ N ∗ , the tra-jectories of the resulting closed-loop system (1) with a digitalregulator (29)-(30) are bounded, and lim t + j →∞ (cid:107) e ( t, j ) (cid:107)≤ γ † √ N (cid:107) Ψ S (cid:107) . (31) Remark 2
In contrast with (11) in Proposition 1, (31) demon-strates that the regulation error e eventually converges to a setin relation to the time derivative of the desired steady states of u , independent of that of y m . In this respect, when the desiredsteady state of u is constant and that of y m is time-varying, thedigital regulator (29)-(30) can guarantee that the regulationerrors exponentially converge to zero, while there is no suchguarantee for the emulation-based approach by Proposition 1. Consider the output regulation problem of an inverted pen-dulum on a cart [24,25], whose linearly approximated model is described by m ¨ q = − mgθ − µ f ˙ q + u + P wm (cid:96) ¨ θ = ( m + m ) gθ + µ f ˙ q − u + P w (32)where q is the distance of the cart from the zero reference, θ isthe angle of the pendulum w.r.t. the vertical axis, input u is thehorizontal force applied to the cart, and P w and P w denoteperturbations to q -dynamics and θ -dynamics, respectively, withexogenous variable w being simply generated by an oscillatorof the form ˙ w = Sw , S = (cid:34) − Ω (cid:35) . All other parameters are as in [8]. Suppose both q and θ are mea-sured periodically by sensors, with the sample period T = 0 . .In this setting, the problem in question is to design an imple-mentable regulator taking advantage of the sampled measure-ments such that all closed-loop signals are bounded and the reg-ulation output e ( t ) := θ ( t ) asymptotically converges to zero.Denote y m ( t ) := q ( t )+ (cid:96)θ ( t ) , which is periodically availableas q ( t ) and θ ( t ) are measured periodically. Thus, by setting x := col ( q + (cid:96)θ, ˙ q + (cid:96) ˙ θ, θ, ˙ θ ) , we can rewrite (32) in the form(1) with A = g
00 0 0 10 µ f m (cid:96) ( m + m ) gm (cid:96) − µ f m , B = − m o (cid:96) ,C e = (cid:104) (cid:105) , C m = (cid:104) (cid:105) ,P (cid:62) = (cid:104) P (cid:62) + P (cid:62) m P (cid:62) m (cid:96) (cid:105) , Q e = Q m = 0 . It is clear that the detectability property is fulfilled by thewhole vector ( e, y m ) , i.e., the pair ( A, C ) , instead of ( A, C e ) is detectable, with C = [ C (cid:62) e C (cid:62) m ] (cid:62) . Letting m = 0 . , m =2 , µ f = 0 . , g = 9 . , (cid:96) = 0 . and Ω = 5 , straightforward cal-culations show that Assumptions 1 and 2 are fulfilled.Following the design paradigm proposed in Section 4, wedesign the generalized zero-order hold device (12) and thefeedback law (13) with L = [1 0] . Let Φ D = exp (Φ T ) and Γ f = [1 0] . We then design the compensator (25), and thewashout filter (24) via the H ∞ control [26] with F f = (cid:34) . . − .
55 0 . (cid:35) , G f = (cid:34) . − . (cid:35) . With the above design, the remaining problem is to design adiscrete-time output feedback stabilizer for the correspondingdiscrete-time system (17), (24), and (25), which can be easilysolved via the H ∞ control [26] again.As seen from simulation results in Fig. 2, it can be seenthat the regulation error e ( t ) converges to zero and y m ( t ) isbounded. Regarding a multi-rate digital regulator, the controlaction is executed with a period T . The simulation results arepresented in Fig. 3, where both e ( t ) and y m ( t ) are bounded.6n contrast, we apply the emulation-based approach to design adigital regulator (8). When the sampling period is T , it is foundthat the trajectories of e ( t ) and y m ( t ) are unbounded. Whenthe sampling period is T , the simulation results are given inFig. 4, where both e ( t ) and y m ( t ) are bounded. Fig. 2. Trajectories of e ( t ) , y m ( t ) with a generalized hold device.Fig. 3. Trajectories of e ( t ) , y m ( t ) with a multi-rate regulator.Fig. 4. Trajectories of e ( t ) , y m ( t ) using the emulation-based approachwith sampling period T . In this paper, the robust implementable output regula-tor design problem has been investigated for general linearcontinuous-time systems with periodically sampled measure-ments, consisting of both the regulation errors and extra mea-surements that are generally non-vanishing in steady state. Weshowed that the conventional emulation-based solution cannotbe used to handle the problems when the sampling period islarge and the asymptotic regulation is desired. Motivated bythis, we proposed a design framework by incorporating a gen-eralized zero-order hold device, which transforms the originalproblem into the problem of designing an output feedbackcontroller fulfilling two conditions for a discrete-time system. With the design of a discrete-time compensator and a discrete-time washout filter, it has been shown that there always existsa discrete-time output feedback stabilizer for the resulting aug-mented system, which, together with the previously designedcompensator and filter, completes the design of the controller.The resulting regulator structure aligns with the pre-processingscheme proposed in [4], by regarding the generalized zero-order hold device and the discrete-time compensator as aninternal model. Furthermore, the framework was generalizedby proposing a multi-rate digital regulator, guaranteeing theregulation errors bounded by a constant, depending on thetime derivative of the desired steady states of control inputs,and adjustable by the control execution period.
A Proof of Proposition 1
The proof mainly follows the idea of [15]. Let ˜ x = x − Π x w , ˜ x c = x c − Π x c w , ˜ y = y − ˆ y , ˜ u = C c x c − u , which rewrites the resulting closed-loop system into the formconsisting of (i) the flow dynamics ˙ τ = 1˙˜ x = A ˜ x + B C c ˜ x c − B ˜ u ˙˜ x c = A c ˜ x c + B c C ˜ x − B c ˜ y ˙˜ y = CA ˜ x + CBC c ˜ x c − CB ˜ u + (cid:34) Y m Sw (cid:35) ˙˜ u = C c A c ˜ x c + C c B c C ˜ x − C c B c ˜ y + C c Π x c Sw for τ ∈ [0 , T ) , and (ii) the jump dynamics τ + = 0 , ˜ x + = ˜ x , ˜ x + c = ˜ x c , ˜ y + = 0 , ˜ u + = 0 , for τ = T . With τ max defined in (10), we let λ ∈ (0 , and φ : [0 , τ max ] → R be the solution of ˙ φ = − κφ − γ ( φ + 1) , φ (0) = λ − . According to [23], we have φ ∈ [ λ, λ − ] . Then we can choosea Lyapunov function as V ( τ, ˜ x, ˜ x c , ˜ y ) = (cid:34) ˜ x ˜ x c (cid:35) (cid:62) P (cid:34) ˜ x ˜ x c (cid:35) + φ ( τ ) (cid:34) ˜ y ˜ u (cid:35) (cid:62) (cid:34) ˜ y ˜ u (cid:35) which implies ˙ V ≤ − κV + 2 γ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:34) Y m SwC c Π x c Sw (cid:35)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) during flow and V + ≤ V during jump. Therefore, it can beeasily verified that the statement (i) is true, and lim t + j →∞ (cid:107) ˜ x ( t, j ) (cid:107) ≤ γκ (cid:107) P (cid:107) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:34) Y m SwC c Π x c Sw (cid:35)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , k em = (cid:107) C e (cid:107)|W| / (cid:112) (cid:107) P (cid:107) . This completesthe proof. (cid:4) B Proof of Theorem 1
By setting χ := col ( x D , ζ D , z ) , the resulting closed-loop(17), (20) can be compactly described by ˙ w D = 0 , ˙ χ = 0 ,w + D = S D w D , χ + = A cl χ + P cl w D (B.1)for some appropriately defined matrices A cl , P cl . With the re-quirement (a), it immediately follows that | σ ( A cl ) | < , i.e., alleigenvalues of A cl lie within the unit circle. Thus there existsa unique Π cl ∈ R ( n + dq e + n z ) × d such that Π cl S D = A cl Π cl + P cl . (B.2)With condition (b), there exists a solution Π z for the equations(21). With Π x , Π ζ being a solution of (18), it can be easilyconcluded that Π cl = col (Π x , Π ζ , Π z ) is a solution of (B.2),and thus the unique one.Let ρ := col ( x, ζ, e, y m , z ) , which compactly expresses thehybrid system (14), (15), and (20) as the form ˙ w = Sw , w + = w ˙ ρ = F cl ρ + P F wρ + = J cl ρ + P J w (B.3)with some appropriately defined matrices F cl , J cl , P F , P J , with | σ ( J cl e F cl T ) | < by the requirement (a). Thus system (B.3) isexponentially stable at the invariant set M = { ( τ, w, ρ ) : ρ = (cid:98) Π cl ( τ ) w } with (cid:98) Π cl ( τ ) : [0 , T ) → R ( n + dq e + n z ) × d being the unique solution of the equations d (cid:98) Π cl ( τ ) dτ + (cid:98) Π cl ( τ ) S = F cl (cid:98) Π cl ( τ ) + P F (cid:98) Π cl (0) = J cl (cid:98) Π cl ( T ) + P J . (B.4)With (4), (19), and (21), simple calculations show that (cid:98) Π cl ( τ ) := col (Π x , Π ζ , , Y m e − Sτ , Π z e − Sτ ) is a solution of (B.4), and thus is the unique one. Since C e Π x + Q e = 0 in (4), it indicates that e ( t ) := C e x ( t ) + Q e w ( t ) vanishes in M . The proof is thus completed. (cid:4) C Proof of Lemma 3
The “if” part has been proved in Theorem 1. As for theproof of “only if” part, we can see that the requirement (a) isclear. Thus, we now focus on the proof of the requirement (b).Using the notations in the proof of Theorem 1, we use (B.3) todenote the resulting hybrid closed-loop system (1), (12), (13),and (20). Simple calculations show that (B.4) has the uniquesolution ˆΠ cl ( τ ) , which can be partitioned as (cid:98) Π cl ( τ ) := col (Π x ( τ ) , Π ζ ( τ ) , , Y m e − Sτ , Π z e − Sτ ) and C e Π x ( τ ) + Q e . (C.1)To be explicit, we can equivalently rewrite (B.4) as (cid:40) ˙Π x ( τ ) = − Π x ( τ ) S + A Π x ( τ ) + B Ψ( τ ) + P Π x (0) = Π x ( T )˙Π ζ ( τ ) = − Π ζ ( τ ) S + (Φ ⊗ I q e )Π ζ ( τ )Π z S D = A z Π z + B z (cid:104) d × q e Y (cid:62) m (cid:105) (cid:62) (C.2)where Y m = C m Π x (0) + Q m , and Ψ( τ ) = Ψ ζ ( τ ) + Ψ v u e − Sτ with Ψ ζ ( τ ) = L Π ζ ( τ ) and (cid:34) Ψ v u Π ζ (0) S D (cid:35) = K z Π z + L z (cid:104) d × q d Y (cid:62) m (cid:105) (cid:62) . Putting (C.1) and the first part of (C.2) together, by As-sumption 1 and m = q e , we observe that they reduce to thecontinuous-time regulator equations (4) and have the uniqueconstant solution Π x ( τ ) , Ψ( τ ) , i.e., independent of τ . On theother hand, taking the second of (C.2) into consideration, wecan deduce that Π ζ is also independent of τ since it allows fora solution Π ζ ( τ ) if and only if ˙Π ζ ( τ ) = 0 . Furthermore, weobserve that Ψ v u is constant, and satisfies Ψ v u = [Ψ − Ψ ζ ] e Sτ .To make this equality holds for constant Ψ and Ψ ζ , there nec-essarily holds Ψ = Ψ ζ , leading to Ψ v u = 0 . In view of theprevious observations, the requirement (b) can be easily con-cluded by using the fact that P, Q are arbitrary matrices. (cid:4)
D Proof of Lemma 4
Instrumental to the subsequent analysis is the followinglemma.
Lemma 5
Suppose Assumptions 1 and 2 hold. Let T ( λ ) = col (1 , λ, . . . , λ d − ) ⊗ I q e . Thenrank (cid:34) A D − λI n L D T ( λ ) C e (cid:35) = n + q e (D.1) holds for all λ ∈ σ ( S D ) .Proof. Consider the auxiliary system ˙ ξ = (Φ ⊗ I q m ) ξ , ˙ x = Ax + BLζ , ˙ ζ = (Φ ⊗ I q e ) ζe = C e x , y m = C m x − Γ f ξ which, by Assumption 1 and with the construction that (Φ ⊗ I q m , Γ f ) and (Φ ⊗ I q e , L ) are observable, respectively, can beeasily inferred to be detectable. Thus, with Assumption 2 andaccording to [19] again, it can be seen that its discretized form,as ξ k +1 = (Φ D ⊗ I q m ) ξ k , x k +1 = A D x + L D ζ k ζ k +1 = (Φ D ⊗ I q e ) ζ k e k = C e x k , y m,k = C m x k − Γ f ξ k (Φ D − λI d ) ⊗ I q m A D − λI n L D D − λI d ) ⊗ I q e C e − Γ f C m is full-column-rank for all λ ∈ { λ ∈ C | | λ |≥ } . Then con-sidering λ ∈ σ (Φ D ) , (D.1) can be concluded by some simplecolumn transformation. (cid:4) We now proceed to use the PBH test and Lemma 5 to ver-ify the stabilizability and detectability of (17). Regarding thestabilizability, it holds if and only if all rows of the followingmatrix F f − λI G f C m A D − λI L D
00 0 − λI I D − λI ) ⊗ I q e (cid:124) (cid:123)(cid:122) (cid:125) ˆ A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B D I
00 0 I (cid:124) (cid:123)(cid:122) (cid:125) ˆ B are independent for all λ ∈ { λ ∈ C | | λ |≥ } . We note that F f − λI is nonsingular for all λ ∈ { λ ∈ C | | λ |≥ } by constructionand ( A D , B D ) is stabilizable by [19] and Assumptions 1.(i).Thus, it can be easily verified that the above matrix is full-row-rank for all λ ∈ { λ ∈ C | | λ |≥ } .To further explore the detectability, it is true if and only iffor all λ ∈ { λ ∈ C | | λ |≥ } , the matrix ˆ A ˆ C , with ˆ C = (cid:34) C e − Γ f C m (cid:35) is full-column-rank. Since F f + G f Γ f = Φ D ⊗ I q m by con-struction, the above verification reduces to show (Φ D − λI ) ⊗ I q m A D − λI L D
00 0 − λI I D − λI ) ⊗ I q e C e − Γ f C m is full-column-rank for all λ ∈ { λ ∈ C | | λ |≥ } . For all λ ∈{ λ ∈ C | | λ |≥ , λ / ∈ σ (Φ D ) } , the above matrix is full-column-rank if and only if ( A D , C ) is detectable, which is clearly trueby [19] and Assumptions 1.(i).With this being the case, we turn to investigate the case that λ ∈ σ (Φ D ) . Since (Φ D ⊗ I q m , Γ f ) is observable by construc- tion, by taking appropriate column transformation, the previousverification reduces to showrank (cid:34) A D − λI L D T ( λ ) C e (cid:35) = n + q e ∀ λ ∈ σ (Φ D ) , which clearly is true by recalling (D.1) and the fact that σ (Φ D ) = σ ( S D ) . The proof is thus completed. (cid:4) E Proof of Theorem 3
Let ˜ x D := x D − Π x w D , ˜ ζ D := ζ D − Π ζ w D , and ˜ z := z − Π z w D . The closed-loop discrete-time system (17)-(20) canbe compactly rewritten as ˙ τ = 0 , ˙˜ X cl = 0 , for τ ∈ [0 , T ) τ + = 0 , ˜ X + cl = A cl ˜ X cl , for τ = T with ˜ X cl := col (˜ x D , ˜ ζ D , ˜ z ) and A cl := A D + B D L z,u L D K z,u L z,ζ C K z,ζ B z C A z . With the controller (20) satisfying Theorem 1, there exists asymmetric positive definite matrix P cl and a positive constant λ cl < such that V cl = ˜ X (cid:62) cl P cl ˜ X cl satisfies ˙ V cl = 0 , V + cl = ˜ X (cid:62) cl A (cid:62) cl P cl A cl ˜ X cl ≤ λ cl V cl . (E.1)To ease the subsequent analysis, we propose a clock τ cl dy-namics as ˙ τ cl = 1 , for τ cl ∈ (cid:83) Ni =1 [0 , iN T ) τ + cl = (1 − (cid:98) τ cl T (cid:99) ) τ , for (cid:98) Nτ cl T (cid:99) ∈ N + . With such a clock, we set ε := u − ˆ u , and can rewrite theclosed-loop system (1)-(29)-(30) under the error coordinates ˜ X cl = col (˜ x D , ˜ ζ D , ˜ z ) as ˙ τ cl = 1˙ w D = 0˙˜ X cl = (cid:34) e − Aτ cl Bε dq e + n z (cid:35) ˙ ε = − L (Φ ⊗ I q e ) e (Φ ⊗ I qe ) τ cl ˜ ζ D − L (Φ ⊗ I q e )Π ζ e τ cl S w D for τ cl ∈ (cid:83) Ni =1 [0 , iN T ) , and τ + cl = (1 − (cid:98) τ cl T (cid:99) ) τ cl w + D = [(1 − (cid:98) τ cl T (cid:99) ) + (cid:98) τ cl T (cid:99) S D ] w D ˜ X + cl = (cid:18) (1 − (cid:98) τ cl T (cid:99) ) I + (cid:98) τ cl T (cid:99) A cl (cid:19) ˜ X cl ε + = 0 (cid:98) Nτ cl T (cid:99) ∈ N + .Let k = T ln λ cl , and α ∗ = argmin α ∈ R + φ e α + 8 φ λ cl αk σ m ( P cl ) λ cl with φ = max τ cl ∈ [0 ,T ] (cid:107) e − Aτ cl B (cid:107)(cid:107) P cl (cid:107) and φ = max τ cl ∈ [0 ,T ] (cid:107) L (cid:107)(cid:107) Φ e Φ τ cl (cid:107) .Then set k := α ∗ k N and N ≥ N ∗ := 32 φ e α ∗ + 8 φ λ cl α ∗ k σ m ( P cl ) λ cl , and choose a Lyapunov function as U = e − k τ cl V cl + e − k s (cid:107) ε (cid:107) with s = τ cl − (cid:98) Nτ cl T (cid:99) TN . It is clear that s ≤ TN , and ˙ s = 1 for τ cl ∈ (cid:83) Ni =1 [0 , iN T ) and s + = 0 for (cid:98) Nτ cl T (cid:99) ∈ N + .During jumps, we consider two cases: (i) τ cl < T and (ii) τ cl = T . For τ cl < T , we have U + = e − k τ cl V cl ≤ U . For τ cl = T , by (E.1), we have U + = λ cl (cid:101) X (cid:62) D P cl (cid:101) X D ≤ U . Hence,during jumps we always have U + ≤ U .During flows, with (19) and (E.1) we compute the timederivative of U as ˙ U ≤ − k e − k τ cl ˜ X (cid:62) cl P cl ˜ X cl + 2 e − k τ cl (cid:107) ˜ X (cid:62) cl P cl (cid:107)(cid:107) e − Aτ cl Bε (cid:107)− k e − k s (cid:107) ε (cid:107) +2 e − k s (cid:107) ε (cid:107) ( φ (cid:107) ˜ ζ D (cid:107) + (cid:107) L (Φ ⊗ I q e )Π ζ w (cid:107) ) ≤ − ( k e − k τ cl σ m ( P cl ) − φ k ) (cid:107) ˜ X cl (cid:107) +2 e − k τ cl φ (cid:107) ˜ X cl (cid:107)(cid:107) ε (cid:107)− k e − k TN (cid:107) ε (cid:107) + k (cid:107) Ψ Sw (cid:107) ≤ − k e − k T σ m ( P cl ) (cid:107) ˜ X cl (cid:107) − k e − k TN (cid:107) ε (cid:107) + k (cid:107) Ψ Sw (cid:107) . Thus, we have lim t + j →∞ (cid:107) ε ( t, j ) (cid:107) ≤ e α ∗ α ∗ k N λ cl (cid:107) Ψ Sw (cid:107) lim t + j →∞ (cid:107) ˜ X cl ( t, j ) (cid:107) ≤ α ∗ k Nλ cl σ m ( P cl ) (cid:107) Ψ Sw (cid:107) . (E.2)This then yields that the trajectories of the resulting closed-loopsystem (1)-(29)-(30) are bounded.Regarding the bound of (cid:107) e (cid:107) , we note that by (16) and (18),we have e = C e ˜ x with ˜ x = e Aτ cl ˜ x D + (cid:82) τ cl e − A ( r − τ cl ) BLe (Φ ⊗ I qe ) r d r ˜ ζ D + (cid:82) τ cl e − A ( r − τ cl ) d r B ( K z,u ˜ z + L z,u ˜ x D ) . Thus, using the latter of (E.2) and τ cl ∈ [0 , T ] , it can be verifiedthat there exists a γ † such that (31) holds. (cid:4) References [1] B. A. Francis and W. M. Wonham. “The internal model principle ofcontrol theory”.
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