Robust Output Regulation of Linear Passive Systems with Multivalued Upper Semicontinuous Controls
aa r X i v : . [ c s . S Y ] D ec Robust Output Regulation ofLinear Passive Systems withMultivalued Upper Semicontinuous Controls
F´elix A. Miranda and Fernando Casta˜nosJune 23, 2018
Abstract
The use of multivalued controls derived from a special maximal mono-tone operator are studied in this note. Starting with a strictly passivelinear system (with possible parametric uncertainty and external distur-bances) a multivalued control law is derived, ensuring regulation of theoutput to a desired value. The methodology used falls in a passivity-basedcontrol context, where we study how the multivalued control affects thedissipation equation of the closed-loop system, from which we derive itsrobustness properties. Finally, some numerical examples together withimplementation issues are presented to support the main result.
Sometimes it is useful to have an interpretation of the action of the controllerin energetic terms. Among the most important methodologies of passivity-based control (PBC) that achieve this interpretation are the so-called energyshaping techniques. The purpose of energy shaping, as its name suggests, is tochange the energy function (by means of the control action) in such a way thatstabilization and performance objectives are satisfied. Although energy-shapingstrategies have proved to be very useful yielding an easy interpretation of thecontroller in energetic terms [16], robustness against external perturbations andmodel uncertainty is still a topic of research.On the other hand, the study of differential inclusions for modelling andanalysis of processes in control theory is extensive (e.g. [1, 8, 13]), whereasthe problem of designing a multivalued control in order to achieve a desiredresponse is less explored, except for the case of sliding mode control, whichtakes advantage of the multivalued nature of the signum multifunction in orderto ensure robustness of the closed-loop system.An important family of differential inclusions (more general than those ob-tained by using sliding modes techniques) are those for which its right-hand sideis represented by maximal monotone operators. In the case of linear plants, the1losed-loop system is sometimes called a multivalued Lur’e dynamical system,for which results about existence and uniqueness of solutions have been provedin [2, 4, 5, 22]. This kind of systems are related to complementarity and pro-jected dynamical systems [3], which makes its study important for a broad rangeof applications coming from different fields such as automatic control, economics,mechanics, etc.The main contribution of this note consists in a design procedure for amultivalued-control — where the multivalued part is represented by the sub-differential of some proper, convex and lower semicontinuous function — whichachieves finite-time regulation of the desired output together with insensitivityagainst a family of bounded and unmatched perturbations.The proposed multivalued control strategy differs remarkably from thosewhich are common in Sliding-Mode Control in the sense that we obtain finite-time regulation and disturbance rejection without a discontinuous right-handside and therefore without the necessity of solutions of the associated system inthe the sense of Filippov.This note is organized as follows. In Section 2 the class of systems that weconsider is established in conjunction with the class of perturbations that it willbe treated. The multivalued structure of the controller is presented and well-posedness of the closed-loop system is established. In Section 3, we introducethe main result of this note. Namely, robustness and finite time convergenceof the closed-loop system are demonstrated. Section 4 touches the point aboutimplementation of the multivalued control law by introducing a regularizationof the multivalued map. Some examples are presented showing the closed loopproperties. The note ends with some conclusions and future research lines inSection 5.
Throughout this note, all vectors are column vectors, even the gradient of ascalar function that we denote by ∇ H ( x ) = ∂H ( x ) ∂x . A matrix A ∈ R n × n iscalled positive definite (denoted as A > w ⊤ Aw > w ∈ R n \ { } (note that we are not assuming A symmetric).A set-valued function or multifunction F : R n → R n is a map that associateswith any w ∈ R n a subset F ( w ) ⊂ R n . The domain of F is given byDom F = { w ∈ R n : F ( w ) = ∅} , related with the definition of a multifunction is the concept of its graph,Graph F = { ( w, z ) ∈ R n × R n : z ∈ F ( w ) } . The graph is used to define the concept of monotonicity of a multifunction inthe following way: A set-valued function F is said to be monotone if for all( w, z ) ∈ Graph F and all ( w ′ , z ′ ) ∈ Graph F the relation h z − z ′ , w − w ′ i ≥
02s preserved, where h· , ·i denotes the usual scalar product on R n . A monotonemap F is called maximal monotone if, for every pair ( ˆ w, ˆ z ) ∈ R n × R n \ Graph F ,there exits ( w, z ) ∈ Graph F with h z − ˆ z, w − ˆ w i <
0, or in other words, if noenlargement of its graph is possible in R n × R n without destroying monotonicity.Let f : R n → R ∪ { + ∞} be a proper, convex and lower semi-continuousfunction. The effective domain of f is given byDom f = { w ∈ R n : f ( w ) < ∞} . We say that f is proper if its effective domain is non empty. The subdifferential ∂f ( w ) of f ( · ) at w ∈ R n is defined by ∂f ( w ) = { ζ ∈ R n : f ( σ ) − f ( w ) ≥ h ζ, σ − w i for all σ ∈ R n } . An important convex function is the indicator function of a convex set S , definedby ψ S ( w ) = ( w ∈ S + ∞ if w / ∈ S .
It is easy to see that when f ( · ) is equal to the indicator function of a closedconvex set S , then the subdifferential coincides with the normal cone of the set S at the point w ∈ S , i.e., ∂ψ S ( w ) = N S ( w ) = { ξ ∈ R n : 0 ≥ h ξ, σ − w i for all σ ∈ S } . Note that if w is in the interior of S then N S ( w ) = { } . If w / ∈ S then N S ( w ) = ∅ . Consider the following affine system:Σ : ( ˙ x ( t ) = Ax ( t ) + B u u ( t ) + B v v ( t ) y ( t ) = Cx ( t ) + Du ( t ) , (2.1)where x ∈ R n denotes the system state, u , y ∈ R m are the port variables avail-able for interconnection, which are conjugated in the sense that their producthas units of power, and matrices A, B u , B v , C, D are constant and of suitabledimensions. The term v ∈ R m accounts for an uncertain exogenous input whichis considered bounded. Moreover, without loss of generality, the external signal v ( t ) can be decomposed as the sum of a constant term v + and a bounded signal ν ( t ).The robust output regulation problem consists in regulating the output y to a desired value y d , even in the presence of the external perturbation v ( t ) andparametric uncertainties. Remark . Notice that, for D = 0 and B u = B v , the problem reduces to astandard sliding-mode control problem with matched disturbances. We departfrom these standard assumptions and make the following instead.3 ssumption 1. There exists a (possibly unknown) matrix P = P ⊤ > suchthat (cid:20) P A + A ⊤ P P B u − C ⊤ B ⊤ u P − C − ( D + D ⊤ ) (cid:21) < . (2.2)Assumption 1 is a rewrite of the strict passivity property of plant (2.1)with respect to the input u and output y [12]. Moreover, is easy to seethat the passivity assumption implies that D is positive definite. Equivalently,Assumption 1 can be rewritten in terms of the energy-balance equation. Moreprecisely, there exists a continuously differentiable function H : R n → R , calledthe storage function, such that for all t ≥ H ( x ( t )) − H ( x (0)) = Z t u ⊤ ( τ ) y ( τ ) dτ − d( t ) , where the function d( t ) ≥ H ( x ) is bounded from below (see, e.g., [16]). Remark . The strict passivity assumption allows us to admit the quadraticfunction H ( x ) = x ⊤ P x as a storage function with P satisfying (2.2).It is worth noting that in the linear case, the class of passive systems isequivalent to the class of Port-Hamiltonian (PH) systems described in [20, Ch.4], i.e., Σ can be written as˙ x = F ∇ H ( x ) + g u u + g v vy = h ( x ) + ju with F = AP − = J − R , where J = − J ⊤ and R = R ⊤ ≥ g u = B u , g v = B v , h ( x ) = Cx and j = D .Along this note we will use both representations of Σ with the purpose ofexpressing the related computations in the context of basic interconnection anddamping assignment (IDA) [17, 20]. In this subsection a multivalued control law is introduced by using maximalmonotone operators. It will be shown later on that these are robust in the faceof parametric and additive uncertainties.Let u ∈ R m and y ∈ R m be the port variables available for interconnectionassociated to the controller. The multivalued control input is defined in termsof the graph of a multifunction U : R m → R m by( u , y ) ∈ Graph U . Remark . It is worthy to mention that, in the case when the multifunction U is monotone, the relation ( u , y ) ∈ Graph U defines a static, incrementally4 - +- Figure 1: Interconnection of a controller to a plant.passive map . Furthermore, if 0 ∈ U (0), then the relation between u and y defines a static passive map inasmuch as h u , y i ≥ u , y ) ∈ Graph U . Previous lines motivate the following assumption.
Assumption 2.
The multifunction U is maximal monotone, and defines astatic passive relation between the input u and the output y . The multivalued nature of the proposed control motivates us to depart fromthe classical intelligent control paradigm and to make use of the behaviouralframework proposed by Willems [21] instead. In this context, the plant andthe controller are interconnected using a power preserving pattern as shown inFigure 1 satisfying: y = y =: y , − u = u =: u and therefore u y + u y = 0 . The interconnected system (plant and controller) results in˙ x = Ax − B u u + B v v (2.3a) y = Cx − Du (2.3b) u ∈ U ( y ) , (2.3c)where our task is to determine U ( y ) such that y is regulated to some fixedvalue y d , even in the presence of uncertainties in the system parameters and theexternal perturbation v . Note that the previous argument rules out the trivialcontrol u = D − ( Cx − y d ). In fact, even if all the system parameters and thestate x were known, that control would not be admissible, since it is not passive(see Assumption 2).It is well known that when U ( y ) is given as the subdifferential of a proper,convex and lower semicontinuous function Φ( · ) (i.e. U ( y ) = ∂ Φ( y )), it is amaximal monotone operator [18, Cor. 31.5.2]. Therefore, we will focus oncontrols of the form u ∈ ∂ Φ( y ) , (2.4) A multivalued map F is called incrementally passive if h y − y ′ , u − u ′ i ≥ u, y ) ∈ Graph F and for all ( u ′ , y ′ ) ∈ Graph F . R m → R m . Morespecifically, in Section 3 we will prove that, for some closed convex set S , robustregulation of the output y is obtained for the case when Φ( y ) = ( ϕ + ψ S )( y ),where ϕ ( · ) is proper, convex and lower semicontinuous with effective domaincontaining S and ψ S ( · ) is the indicator function of the set S . In other words, Φis the restriction of ϕ to S . Before presenting the main result of this note about robustness of the closed-loop system (2.3), is important first to establish its well-posedness. Specifically,well-posedness of the closed-loop system comprises two issues. The first ques-tion is: Is there always a control input u ∈ ∂ Φ( y )? and the second one isabout uniqueness and existence of solutions of the associated differential inclu-sion (2.3).For the second issue about a solution of the differential inclusion (2.3), well-posedness was proved previously in [4, 5], where the subdifferential of the con-jugate function of Φ( y ) together with passivity of the associated system plays acrucial role.The first issue deserves more explanation. At first we need y ∈ S for alltime t , this comes from the definition of the subdifferential, i.e., u ∈ ∂ Φ( y ) isequivalent to Φ( σ ) − Φ( y ) ≥ h u, σ − y i for all σ ∈ R m , where, in the case of Φ( y ) = ( ϕ + ψ S )( y ) we have ϕ ( σ ) − ϕ ( y ) + ψ S ( σ ) − ψ S ( y ) ≥ h u, σ − y i for all σ ∈ R m , (2.5)and it is clear that if y / ∈ S , then we will have ∂ Φ( y ) = ∅ . Then, we mustguarantee that, no matter what the initial conditions are, it is possible to findan output y ∈ S , such that u ∈ ∂ Φ( y ) is well defined.In the case where ϕ ≡ D is symmetric, well-posedness iseasy to show. Since u ∈ ∂ψ S ( y ) = N S ( y ), from the definition of normal conewe have 0 ≥ h u, σ − y i for all σ ∈ S , which in view of (2.3b) translates to0 ≥ h D − ( Cx − y ) , σ − y i = ( σ − y ) ⊤ D − ( Cx − y ) = h Cx − y, σ − y i D − ¡++¿ for all σ ∈ S , with the inner product weighted by D − . From [10, p. 117]we have that the above inequality is the characterization of the projection of Cx onto the set S with the induced norm k · k D − , i.e. y = Proj D − S ( Cx ) = arg min σ ∈ S k Cx − σ k D − , (2.6)and the control input u transforms into u = D − (cid:16) Cx − Proj D − S ( Cx ) (cid:17) ∈ N S ( y ) . (2.7)6herefore, in the case of D symmetric, we can find an expression for the output y in terms of the projection operator Proj D − S ( · ) (note that this implies y ∈ S independently of the state x ). Moreover, due to the Lipschitzian property of theprojection operator [10, p. 118], substitution of u in (2.3) leads to a well-posedordinary differential equation (not a differential inclusion!) with a Lipchitzianright-hand side (see [14] for a detailed development in the scalar case).For the general case where ϕ is not the zero function, and removing theassumption about the symmetry of D , from (2.5) we have that the problemconsists in finding y ∈ S such that0 ≤ h D − y − D − Cx, σ − y i + ϕ ( σ ) − ϕ ( y ) for all σ ∈ S , (2.8)where we made use of (2.3b). The inequality (2.8) is an hemivariational in-equality , for which existence and uniqueness of solutions can be deduced from D − as the following Lemma extracted from [9] shows. Lemma 1 (Lemma 5.2.1, [9]) . Suppose that F ( y ) is continuous and stronglymonotone, i.e. h F ( y ) − F ( y ′ ) , y − y ′ i ≥ η k y − y ′ k for all y, y ′ ∈ R m and some η > . Then, for each g ∈ R n there exists an uniquesolution y ∗ ∈ S to h F ( y ) − g, σ − y i + Φ( σ ) − Φ( y ) ≥ for all σ ∈ S .
Since D is positive definite, it is straightforward to see that D − is positivedefinite too. Furthermore, the linear map y D − y is strongly monotone (as aconsequence of applying Rayleigh’s inequality). Then using Lemma 1 we havethat the hemivariational inequality problem (2.8) has a unique solution for eachstate x . In other words: For all x ∈ R n , there always exists a unique y ∈ S suchthat the control u ∈ ∂ Φ( y ) is well defined. Remark . The computation of the control input which forces y ∈ S obviouslydepends on the solution of the hemivariational inequality (2.8) and therefore itdepends implicitly on the actual state of the plant x . This dependency of thestate, induces a partition in the phase space. Remark . For the case when ϕ ≡ D is symmetric, we might be temptedto use (2.7) as control input (because it is passive), but unfortunately it isnot implementable in our setting because it depends explicitly on the systemparameters and state. The role of (2.7) is analogous to the role of the equivalentcontrol in sliding modes [19], in the sense that it is not implementable buthelps to determine the dynamics associated to the closed-loop system. See [14]for an example of the use of the control (2.7) in the scalar case and someimplementation issues. The interested reader is referred to [7, 9, 15], and references therein for more informationand properties about variational and hemivariational inequalities.
The main result of this note is presented in this section. Namely, from anenergy-shaping point of view, we show that the multivalued control (2.4) canbe expressed as a basic IDA controller plus a robustifying term denoted by η ,affecting directly the dissipation of the closed-loop system and yielding to theoutput regulation despite the presence of external and parametric disturbances.From the closed-loop equation of the system, we have˙ x = Ax − B u u + B v v (3.1a) y = Cx − Du (3.1b) u ∈ ∂ Φ( y ) (3.1c)with Φ( · ) = ( ϕ + ψ S )( · ) for some convex set S . The perturbation input v ( t ),decomposed as a constant term v + and a bounded unknown signal ν ( t ), affectsthe dissipation equation in the following way.Let ¯ x be the equilibrium point of (3.1) associated to a constant perturbation( ν ( t ) ≡
0) and input u = 0, i.e.,0 = A ¯ x + B v v + , (3.2)and let H be the storage function of system (3.1) (i.e. H ( x ) = x ⊤ P x with P satisfying (2.2)). We obtain0 = F ∇ H (¯ x ) + B v v + = F ∇ H (¯ x ) ± F ∇ H ( x ) + B v v + = − F ( ∇ H ( x ) − ∇ H ( x )) + B v v + with H ( x ) = ( x − ¯ x ) ⊤ P ( x − ¯ x ). Now, defining H a ( x ) = H ( x ) − H ( x ) we havethe basic IDA controller equation [6] for v + as F ∇ H a ( x ) = B v v + . Then, we have that the term v + acts as an energy-shaping control changing thestorage function of the uncontrolled system H to H and therefore changingthe equilibrium of the system. The closed-loop system results in˙ x = F ∇ H ( x ) − B u u + B v ν (3.3a) y = Cx − Du (3.3b) u ∈ ∂ Φ( y ) (3.3c)8or the case ν = 0, a control input can be designed in order to obtain theasymptotic regulation of the output y to y d using an energy-shaping interpre-tation as follows. Lemma 2.
For system (3.3) , let x ∗ be an admissible equilibrium associated tothe constant control ¯ u = D − ( Cx ∗ − y d ) , i.e. x ∗ satisfies Ax ∗ − B u D − ( Cx ∗ − y d ) + B v v + . (3.4) Then, ¯ u achieves regulation of the output to y d when ν = 0 . Furthermore, ¯ u isa basic IDA controller and satisfies F ∇ H b ( x ) = − B u ¯ u , with H b ( x ) = H ( x ) − H ( x ) and H ( x ) = ( x − x ∗ ) ⊤ P ( x − x ∗ ) .Proof. Let x ∗ be an equilibrium of system (3.3) satisfying (3.4). Then, from (3.2)we have that 0 = A ( x ∗ − ¯ x ) − B u D − ( Cx ∗ − y d ) , or, in terms of the storage functions H and H ,0 = − F ( ∇ H ( x ) − ∇ H ( x )) − B u D − ( Cx ∗ − y d ) . Therefore, we obtain a change in the storage function from H ( x ) with minimumat ¯ x to H ( x ) with minimum at x ∗ which implies convergence of the state x to x ∗ . Also, for u = ¯ u in (3.3b) we have y = Cx − DD − ( Cx ∗ − y d ) = C ( x − x ∗ ) + y d and y → y d as x → x ∗ .The control ¯ u described in Lemma 2 shapes the energy by changing thestorage function. For the new storage function H we have that the controlinput u = ¯ u + η establishes a new dissipation equation as˙ H ( x ) = ∇ H ( x ) ⊤ F ∇ H ( x ) − ∇ H ( x ) ⊤ B u η + ∇ H ( x ) ⊤ B v ν = 12 ( x − x ∗ ) ⊤ ( A ⊤ P + P A )( x − x ∗ ) − ( x − x ∗ ) ⊤ P B u η − ( y − Cx + D (¯ u + η )) ⊤ η + ( x − x ∗ ) ⊤ P B v ν = 12 (cid:2) ( x − x ∗ ) ⊤ η ⊤ (cid:3) (cid:20) A ⊤ P + P A − P B u + C ⊤ − B ⊤ u P + C − ( D + D ⊤ ) (cid:21) (cid:20) x − x ∗ η (cid:21) − ( y − y d ) ⊤ η + ( x − x ∗ ) ⊤ P B v ν , (3.5)where in the case of ν = 0 we obtain the energy-balancing equation changingthe output to − ( y − y d ). 9 emark . Note that the control ¯ u achieves the asymptotic regulation of theoutput y via a change in the storage function H ( x ) but, once again, ¯ u is not im-plementable, as it requires perfect knowledge of the state and system parametersand would lead to a closed-loop system which is not robust.In Section 2.2 it was established that, when Φ( y ) = ψ S ( y ) and D is sym-metric, we have y = Proj D − S ( Cx ) . This equation evidently shows the robustness property of the multivalued controllaw u ∈ N S ( y ), since it is not necessary to maintain the state x at a precisepoint. Instead, it is sufficient to maintain x in the set of points for which itsprojection over S is equal to y d . This result obviously depends of the shape ofthe set S and in order to achieve robust regulation it is necessary that y d ∈ ∂S and int N S ( y d ) = ∅ (see Figures 2, 3 below).The previous argument can be extended for the more general case where u ∈ ∂ Φ( y ) with Φ( y ) = ( ϕ + ψ S )( y ) and ϕ an arbitrary proper, convex andlower semicontinuous function, i.e. we can achieve robust output regulation fora family of controls parametrized by ϕ . Theorem 1 (Main result) . Consider system (3.3) , and suppose that Assump-tion 1 holds. Then, the family of controls that satisfy u ∈ ∂ Φ( y ) , with Φ( y ) =( ϕ + ψ S )( y ) for some proper, convex and lower semicontinuous function ϕ andsome closed convex set S specified in the proof, yields the robust output regulation y = y d in finite time whenever h D − ( y d − Cx ∗ ) , y d i < D ϕ ( y d , − y d ) , (3.6) and k ν k ≤ B , for some B > specified along the proof. Here, D ϕ ( y , d ) is the directionalderivative of the function ϕ at the point y in the direction d and x ∗ is theequilibrium associated to the basic IDA design (Lemma 2). Furthermore, in thefamily of all controls, there exist at least one that is passive.Proof. Applying the control input u ∈ ∂ Φ( y ) automatically implies that y ∈ S (see Subsection 2.2). Then, if we want the regulation of y to y d a necessarycondition is y d ∈ S . Consider the following convex set S = conv { , y d } (3.7)where the operator conv { a, b } refers to the convex hull of two points a ∈ R m and b ∈ R m , i.e.conv { a, b } = { c ∈ R m : c = λa + (1 − λ ) b, λ ∈ [0 , } . and consider the following half-spaceΩ d = { x ∈ R n : h D − ( y d − Cx ) , y d i ≤ D ( y d , − y d ) } . y is equal to y d whenever x ∈ Ω d . Assuming x ∈ Ω d implies h D − ( y d − Cx ) , y d i ≤ D ( y d , − y d ) = inf ρ> ϕ ( y d − ρy d ) − ϕ ( y d ) ρ ≤ ϕ ( µy d ) − ϕ ( y d )1 − µ for all µ ∈ [0 , , where we did the change of variables ρ = 1 − µ . Because the term 1 − µ ispositive, we have h D − ( y d − Cx ) , (1 − µ ) y d i ≤ ϕ ( µy d ) − ϕ ( y d ) for all µ ∈ [0 , S can be represented as σ = µy d ∈ S for some µ ∈ [0 , h D − ( y d − Cx ) , σ − y d i + ϕ ( σ ) − ϕ ( y d ) ≥ σ ∈ S .
That is, y d is a solution of the hemivariational inequality (2.8) when x ∈ Ω d ,and considering the uniqueness of solutions, the output y must be equal to y d .It remains to show that (even in the presence of the external perturbation ν ), the system state x enters the interior of the set Ω d in finite time and remainstherein for all future time. In terms of the equilibrium x ∗ , we have from (3.6)that x ∗ ∈ Ω d . We will prove that for some δ > E = { x ∈ R n : ( x − x ∗ ) ⊤ P ( x − x ∗ ) ≤ δ } ⊂ int Ω d around x ∗ that isattractive and invariant.Considering the dissipation equation (3.5), it is clear that η = u − ¯ u is welldefined, where ¯ u is the basic IDA control from Lemma 2, and u ∈ ∂ Φ( y ). Then,equation (3.5) transforms into˙ H ( x ) = 12 (cid:2) ( x − x ∗ ) ⊤ − ( u − ¯ u ) ⊤ (cid:3) (cid:20) A ⊤ P + P A P B u − C ⊤ B ⊤ u P − C − ( D + D ⊤ ) (cid:21) (cid:20) x − x ∗ − ( u − ¯ u ) (cid:21) − ( y − y d ) ⊤ ( u − ¯ u ) + ( x − x ∗ ) ⊤ P B v ν, where the term − ( y − y d ) ⊤ ( u − ¯ u ) is negative for all y = y d (i.e., for all x / ∈ Ω d ).Indeed, we have from (3.6) and Lemma 2 that −h ¯ u, y d i < D ( y d , − y d ) , and from the definition of subdifferential we have u ∈ ∂ Φ( y ) ⇔ ϕ ( σ ) − ϕ ( y ) ≥ h u, σ − y i for all σ ∈ S. Specifically, for σ = y d we obtain −h u, y − y d i ≤ ϕ ( y d ) − ϕ ( y ). Moreover, forall y ∈ S \ { y d } we can write y = λy d with λ ∈ [0 , − ( y − y d ) ⊤ ( u − ¯ u ) ≤ ϕ ( y d ) − ϕ ( λy d ) − (1 − λ ) y ⊤ d ¯ u< ϕ ( y d ) − ϕ ( λy d ) + (1 − λ ) inf µ> ϕ ( y d − µy d ) − ϕ ( y d ) µ ≤ ϕ ( y d ) − ϕ ( λy d ) + (1 − λ ) ϕ ( y d − µy d ) − ϕ ( y d ) µ µ >
0. Setting µ = 1 − λ > − ( y − y d ) ⊤ ( u − ¯ u ) < y = y d .From (3.3b) we have that u must satisfy u = D − ( Cx − y ). Substituting u and ¯ u in (3.5) and applying the Lambda inequality to the term ( x − x ∗ ) ⊤ P B v ν ,we have˙ H ≤ − w ⊤ Rw − ( y − y d ) ⊤ ( u − ¯ u ) + ( x − x ∗ ) ⊤ Λ( x − x ∗ ) + ν ⊤ B v P Λ − P B v ν where Λ = Λ ⊤ > w ⊤ = (cid:2) ( x − x ∗ ) ⊤ ( y − y d ) ⊤ D −⊤ (cid:3) R = − (cid:20) A ⊤ P + P A − C ⊤ D −⊤ B ⊤ u P − P B u D − C P B u − C ⊤ D −⊤ DB ⊤ u P − D ⊤ D − C − ( D + D ⊤ ) (cid:21) . It follows that − R <
0, since it is obtained applying the following non singularcongruence transformation (cid:20) I − D − C I (cid:21) to (2.2). Now, setting Λ in a way that R Λ = R − (cid:20) Λ 00 0 (cid:21) > , we have ˙ H ( x ) ≤ − w ⊤ R Λ w − ( y − y d ) ⊤ ( u − ¯ u ) + ν ⊤ B v P Λ − P B v ν and therefore˙ H ( x ) ≤ − λ min ( R Λ ) k w k + λ max ( B v P Λ − P B v ) k ν k ≤ − λ min ( R Λ ) k x − x ∗ k + λ max ( B v P Λ − P B v ) k ν k Considering that we are looking for stability of the ellipsoid E defined above,we have that, for all x / ∈ E , k x − x ∗ k > δλ max ( P ) , and therefore, if ν satisfies k ν k ≤ δλ min ( R Λ )2 λ max ( P ) λ max ( B ⊤ v P Λ − P B v ) = B , we conclude that ˙ H < x / ∈ E , i.e. the set E is attractive and invari-ant [11]. Now, for passivity of the controller, we have from (2.8) that h u, y i ≥ ϕ ( y ) − ϕ (0) . ϕ such that ϕ ( y ) ≥ ϕ (0) for all y ∈ S , then the control u will be passive.Finally, finite-time convergence of the output is obtained automatically fromthe proof. Namely, E ⊂ int Ω d together with attractivity and invariance of E implies that there exist a time t ∗ < ∞ such that the state will cross the boundaryof Ω d and will remain inside of Ω d for all t > t ∗ .Figures 2 and 3 show a picture of the convergence of the term Cx to theinterior of the set { y d } + N S ( y d ) in the output space for the sets S = conv { , y d } and S = conv { [0 , , [ y d , , [0 , y d ] , [ y d , y d ] } respectively, for the case ϕ = 0, D = I n and m = 2. Note that, Cx − y d ∈ N S ( y d ) is equivalent to y d =Proj S ( Cx ) and from (2.6) we obtain y = y d .Figure 2: Trajectory of Cx converging to interior of { y d } + N S ( y d ) with the mul-tivalued control u ∈ N S ( y ) and S = conv { , y d } . This implies that y convergesto y d . Remark . From Figure 2 it is possible to see that, if x ∗ satisfies the condi-tion (3.6) for y d ∈ S , then we can achieve robust output regulation for anyother desired value ¯ y d in the relative interior of S by redefining the set S to¯ S = conv { , ¯ y d } . Moreover, in a more general setting, condition (3.6) allowsus to attack the problem of robust tracking in the following way. Let y d ( t ) bethe desired reference signal. If, for all values of the function y d : R → R m ,condition (3.6) is satisfied together with the bound in ν ( t ), then robust outputtracking is possible as shown in Example 1 below. Remark . It is worth to note that a similar result can be obtained (with possiblydifferent bounds in the external perturbation and different condition in x ∗ ), ifwe change the form of the set S . For example, for Φ( y ) = ψ S ( y ) a possible set S could be as the one given in Figure 3, where the point y d is still in the boundaryof S and the normal cone to S at y d has no empty interior, the details are leftto the reader. 13igure 3: Trajectory Cx converging to interior of { y d } + N S ( y d ) with the mul-tivalued control u ∈ N S ( y ) and S = conv { [0 , , [ y d , , [0 , y d ] , [ y d , y d ] } . Up to this point, we have shown that whenever u ∈ ∂ Φ( y ) the membership of y to the set S , together with robust output regulation are assured. Our nextstep is to develop a way to recover an explicit expression for the values of thecontrol input u in terms of the measured output y and independent of the systemparameters and state.Note that exact values of input u can be computed by solving the hemivari-ational inequality (2.8) at each time instant t , and making use of (3.3b), butthis approach requires knowledge of the system parameters and state x .As another alternative, it is worth noting that the approach of continuousselections does not yield the desired features. For example, in the case of ϕ = 0,a continuous selection of the multifunction ∂ Φ( · ) = N S ( · ) is u = 0, (in fact u = 0 is the unique continuous selection). However, with that control thestorage function of (3.3) is given by H with minimum at ¯ x and consequently,neither robust output regulation nor y ∈ S properties are (in general) obtained.Similar results can be obtained when ϕ ∈ C , since u = ∇ ϕ ( · ) is always acontinuous selection of ∂ Φ( · ).Instead of looking for continuous selections of ∂ Φ( · ), we are going to focuson a regularization of Graph ∂ Φ in the sense used in [14]. More precisely,˜ u − ∇ ϕ ( y ) ∈ N S ( y − ε [˜ u − ∇ ϕ ( y )]) (4.1)is a regularization of the inclusion u ∈ ∂ Φ( y ). Namely, note that for ε = 0we recover ˜ u ∈ ∂ Φ( y ) (because ∂ Φ( y ) = ∇ ϕ ( y ) + N S ( y )). Moreover, with theprevious definition we are allowing outputs y not necessarily in S . Instead wenow require y ∈ { ε [˜ u − ∇ ϕ ( y )] } + S .The well-posedness of inclusion (4.1) together with a single valued expressionfor ˜ u are established below in Theorem 2. The following Lemma will be usefulwhen proving it. 14 emma 3. The map f : R m → R m given by f ( z ) := ( I + εD − ) − z , where D − + D − T > , is a contraction for all ε > .Proof. Defining ζ = f ( z ) , we have ( I + εD − ) ζ = z and direct computation gives k z − z k = (cid:2) ( I + εD − )( ζ − ζ ) (cid:3) ⊤ (cid:2) ( I + εD − )( ζ − ζ ) (cid:3) = k ζ − ζ k + ε ( ζ − ζ ) ⊤ (cid:2) D − + D −⊤ (cid:3) ( ζ − ζ )+ ε ( ζ − ζ ) ⊤ D −⊤ D − ( ζ − ζ ) ≥ k ζ − ζ k + ελ min (cid:0) D − + D −⊤ (cid:1) k ζ − ζ k + ε λ min (cid:0) D −⊤ D − (cid:1) k ζ − ζ k . Therefore, k f ( z ) − f ( z ) k ≤ p ελ min ( D − + D −⊤ ) + ε λ min ( D −⊤ D − ) k z − z k . Theorem 2.
Let ϕ be a strictly convex, lower semicontinuous function that is C and satisfies ϕ ( y ) ≥ ϕ (0) for all y ∈ S . ∇ ϕ : R m → R m is Lipschitz continuous with constant L such that L < λ min (cid:18) D − + D −⊤ (cid:19) . Then, for ε > sufficiently small, the regularized control ˜ u can be expressed as: ˜ u = y − Proj S ( y ) ε + ∇ ϕ ( y ) (4.2) Furthermore, ˜ u is passive respect to y .Proof. From (4.1) we have that for all σ ∈ S the following holds:0 ≥ h ˜ u − ∇ ϕ ( y ) , σ − y + ε [˜ u − ∇ ϕ ( y )] i . (4.3)Multiplying by ε > y on the left-hand side of theinner product we obtain0 ≥ h y − y + ε [˜ u − ∇ ϕ ( y )] , σ − y + ε [˜ u − ∇ ϕ ( y )] . y − ε [˜ u − ∇ ϕ ( y )] = Proj S ( y ) , from which we obtain (4.2). Now we show that the interconnection of theplant (3.3a)–(3.3b) with the regularized control (4.2) is well-posed. It is easy tosee that well-posedness of the closed-loop system is equivalent to proving that,for any state x ∈ R n , the equations˜ u = y − Proj S ( y ) ε + ∇ ϕ ( y ) , ˜ u = D − ( Cx − y ) , have a unique solution. Proceeding with the substitution of the second equationand after some manipulations we have y = (cid:0) I + εD − (cid:1) − (cid:2) Proj S ( y ) − ε ∇ ϕ ( y ) + εD − Cx (cid:3) = ( f ◦ g ) ( y ) , with f as in Lemma 3 and g : R m → R m given by g ( z ) = Proj S ( z ) − ε ∇ ϕ ( z ) + εD − Cx .
We argue that the composition mapping f ◦ g is a contraction for ε sufficientlysmall. Indeed, making use of Lemma 3 we have that (cid:13)(cid:13) ( f ◦ g ) ( y ) − ( f ◦ g ) ( y ) (cid:13)(cid:13) ≤ β ( ε ) k g ( y ) − g ( y ) k ≤ εLβ ( ε ) k y − y k where β ( ε ) = (cid:2) ελ min (cid:0) D − + D −⊤ (cid:1) + ε λ min (cid:0) D −⊤ D − (cid:1)(cid:3) / . Note that theterm εLβ ( ε ) is equal to 1 for ε = 0 and ddε (cid:18) εLβ ( ε ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = L − λ min (cid:18) D − + D −⊤ (cid:19) < , i.e. the term εLβ ( ε ) is strictly decreasing in a neighbourhood of ε = 0 and thusit is less than 1 for ε sufficiently small. Therefore, f ◦ g is a contraction and theinterconnection is well-posed. It only rests to prove the passivity property of ˜ u .From (4.3) we have for σ = 0 ∈ S h ˜ u, y i ≥ h∇ ϕ ( y ) , y i + ε k ˜ u − ∇ ϕ ( y ) k . Note that for ε = 0 we have y ∈ S and from the strictly convexity assumption(see e.g. [10, p. 183]), h∇ ϕ ( y ) , y i > ϕ ( y ) − ϕ (0) ≥ y ∈ S .
In other words we havelim ε ↓ h∇ ϕ ( y ) , y i + ε k ˜ u − ∇ ϕ ( y ) k > . Consequently, h ˜ u, y i ≥ ε > Remark . Note that Theorem 2 is still true if we change the first assumptionby ϕ ( y ) ≥ ϕ (0) for all y ∈ Dom ϕ with ϕ a convex function.16 .2 Example 1 Consider the circuit described by the diagram of Figure 4. We wish to regulatethe outputs y and y to a desired value y d . Taking as state variables the +-+- Figure 4: Circuit diagram of Example 1, where the goal is to regulate the voltageat the outputs y and y .fluxes in inductors and charges in capacitors, we have the following state-spacerepresentation:˙ x = − R C − L − R C C − R + R + R L L R L − R C − R C − L R L C − R + R L L x + R L
00 00 R L u + R R v (4.4a) y = " R L L R L L x + (cid:20) R L R L (cid:21) u (4.4b)where x = (cid:2) x x x x (cid:3) ⊤ are the charge in capacitor C , flux in inductor L , charge in capacitor C and flux in inductor L , respectively, u = (cid:2) u u (cid:3) ⊤ are the control inputs (currents) and y = (cid:2) y y (cid:3) ⊤ are the voltages in resis-tances R L and R L , respectively. Assume that we want to control the outputsto y d = (cid:2) f ( t ) (cid:3) ⊤ , where f ( t ) is a sawtooth wave function with amplitude 0 . R = R = R = 1Ω, R L = 2Ω, R L = 3Ω, L = 1H, L = 2H, C = 1F, C = 3F, v =10 + 50 sin( t ) sign(sin( πt )). Taking the convex function ϕ = 0, simple algebrashows that condition (3.6) is equal to h D − ( y d − Cx ∗ ) , y d i = − − f ( t ) + 56 f ( t ) , f ( t ) ∈ ( − . , . ε = 1 × − and S the convex, time-varying set S ( t ) = conv (cid:26)(cid:20) (cid:21) , (cid:20) f ( t ) (cid:21)(cid:27) . Figure 5 shows the convergence of the output to the desired reference, evenin the presence of the external perturbation v . Moreover, is easy to see that thecondition y ∈ S is satisfied. The computed control input is shown in Figure 6. Figure 5: Output response of plant (4.4) with regularized control (4.2). Thepicture shows convergence to the desired output y d (with ε = 1 × − and ϕ = 0) subject to the perturbation v ( t ) = 10 + 50 sin( t ) sign(sin( πt )). Consider the following affine system˙ x = Ax + B u u + B v v (4.5a) y = Cx + Du (4.5b)18 Figure 6: Time trajectory of regularized control (4.2) with ε = 1 × − and ϕ = 0.with A = − . . − . . − . − . − . . − . − . − . − . . . − . − . , B u = − . − . . . . − . − . . ,C = (cid:20) . . . . . . . . (cid:21) , B v = . . . . − . . − . ,D = (cid:20) . . − . . (cid:21) . where the external perturbation signal v ( t ) is decomposed as v ( t ) = (cid:20) (cid:21) + (cid:20) f ( t ) f ( t ) (cid:21) (4.6)with f ( t ) a sinusoidal function with amplitude 2 and frequency of 10 Hz and f ( t ) corresponds to a sawtooth wave with amplitude 3 and frequency of π Hz.Suppose that we want to regulate the output to the set-point y d = (cid:2) − (cid:3) .Let us verify the assumptions of Theorem 1. The equilibrium point x ∗ is x ∗ = (cid:2) . . − . . (cid:3) ⊤ . and it satisfies h D − ( y d − Cx ∗ ) , y d i = − . . ϕ ( y ) = log ( e y + e y ), which is properand C , we have that D ϕ ( y d , − y d ) = −h∇ ϕ ( y d ) , y d i = − . . Condition (3.6) is satisfied. Using the SDPT3 software to solve (2.2) we obtain P = . . − . . . . − . . − . − . . . . . . . , which is positive definite with eigenvalues in { . , . , . , . } .Figure 7 shows the output response for a regularized control ˜ u with ε = 1 × − , where finite time convergence toward the desired set-point can be verifieddespite the external parametric disturbances of the system. Control and statetrajectories are shown in Figures 8 and 9, respectively. Figure 7: Output’s time trajectory for plant (4.5) showing the convergence tothe desired value y d = [ − ⊤ . This note presents an extension (for the m -dimensional case) of the multival-ued control presented in [14]. Moreover, more general multivalued functions ofthe form u ∈ ∂ Φ( y ) are considered, assuring finite time convergence together20 Figure 8: Time trajectory of regularized control (4.2) applied to the plant (4.5)with ε = 1 × − and ϕ = log( e y + e y ). Figure 9: State’s time trajectories of regulated plant (4.5) with the regularizedcontrol (4.2) (with ε = 1 × − and ϕ = log( e y + e y )) and perturbation (4.6).with, robust output regulation in the face of parametric and external (bounded)21isturbances.The effect of the multivalued control relies directly on the dissipation termmodifying the rate of convergence of the storage function H to x ∗ and leavingwithout change the interconnection matrix J .Between the main assumptions considered, the fact that D is invertible playsan essential role. A research line is the case of no D (i.e. y = Cx ).The implemented control (4.2) acts in fact as a high gain controller when y / ∈ S and coincides with the continuous selection of ∂ Φ( y ) when y ∈ S . However,since the output contains a feedthrough component of the input, the high gaindoes not result in arbitrary large controls. That is, the control converges toa bounded, well-defined value as ε →
0. It is worth noting that the resultingcontroller is passive and independent of the system parameters and of the systemstate.The well-suited structure of Port-Hamiltonian systems together with passiv-ity opens the opportunity to investigate the robust output regulation problemin the nonlinear setting.
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