Asymptotic Model Matching for LPV Systems

2nd IFAC Workshop on Linear Parameter Varying Systems 2nd IFAC Workshop on Linear Parameter Varying Systems Florianopolis, Brazil,on September 3-5, 2018Varying 2nd IFAC Linear Systems 2nd IFAC Workshop Workshop Linear Parameter Parameter Varying Systems Florianopolis, Brazil,on September 3-5, 2018 Available online at www.sciencedirect.com Florianopolis, Brazil, September 3-5, 2018 2nd IFAC Workshop on Linear Parameter Varying Systems Florianopolis, Brazil, September 3-5, 2018 Florianopolis, Brazil, September 3-5, 2018

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IFAC PapersOnLine 51-26 (2018) 173–178

Asymptotic Asymptotic Asymptotic Asymptotic

Model Model Matching Matching for for LPV LPV Model Matching for LPV Systems Model Matching for LPV Systems Systems G. Conte ∗∗∗ A. Systems M. Perdon ∗∗∗ E. Zattoni ∗∗ ∗∗ ∗∗

G. Conte ∗ A. M. Perdon ∗ E. Zattoni ∗∗ G. G. Conte Conte ∗∗ A. A. M. M. Perdon Perdon ∗∗ E. E. Zattoni Zattoni ∗∗ ∗∗ G. di Conte A. M. Perdon E. Zattoni Dipartimento Ingegneria dell’Informazione, a Dipartimento di Ingegneria dell’Informazione, Universit` Universit` a Politecnica Politecnica Dipartimento di Ingegneria dell’Informazione, Universit` a delle Marche, 60131 Ancona, Italy. Dipartimento didelle Ingegneria dell’Informazione, Universit` a Politecnica Politecnica Marche, 60131 Ancona, Italy. Dipartimento di Ingegneria dell’Informazione, Universit` a Politecnica delle Marche, 60131 Ancona, Italy. Italy. (e-mail: {gconte, perdon}@univpm.it) delle Marche, 60131 Ancona, (e-mail: {gconte, perdon}@univpm.it) ∗∗ delle Marche, 60131 Ancona, Italy. (e-mail: {gconte, perdon}@univpm.it) di dell’Energia Elettrica ∗∗ ∗∗ Dipartimento (e-mail: {gconte, perdon}@univpm.it) Dipartimento di Ingegneria Ingegneria dell’Energia Elettrica ee ∗∗ ∗∗ Dipartimento (e-mail: {gconte, perdon}@univpm.it) di Ingegneria dell’Energia Elettrica ee dell’Informazione ”G. Marconi” Alma Mater Studiorum, Universit` a Dipartimento di Ingegneria dell’Energia Elettrica dell’Informazione ”G. Marconi” Alma Mater Studiorum, Universit` a di di ∗∗ Dipartimento di Ingegneria dell’Energia Elettrica e dell’Informazione ”G. Marconi” Alma Mater Studiorum, Universit` a Bologna, 40136 Bologna, Italy. (e-mail: [email protected]) dell’Informazione Marconi” Alma [email protected]) Studiorum, Universit` a di di Bologna, 40136”G. Bologna, Italy. (e-mail: dell’Informazione Marconi” Alma [email protected]) Studiorum, Universit` a di Bologna, 40136 40136”G. Bologna, Italy. (e-mail: [email protected]) Bologna, Bologna, Italy. (e-mail: Bologna, 40136 Bologna, Italy. (e-mail: [email protected]) Abstract: Abstract: We We consider consider the the problem problem of of forcing forcing the the output output of of aa given given Linear Linear Parameter Parameter Varying Varying Abstract: We consider the problem of forcing the output of a given Linear Parameter Varying (LPV) plant to match asymptotically that of a given LPV model using a static or Abstract: We consider the problem of forcing the output of a given Linear Parameter Varying (LPV) plant to match asymptotically that of a given LPV model using a static or aa dynamic dynamic Abstract: We consider the problem of forcing the output of a given Linear Parameter Varying (LPV) plant to match asymptotically that of a given LPV model using a static or a dynamic feedback control law that also quadratically stabilizes the plant. The problem is investigated (LPV) plant to match asymptotically that of a given LPV model using a static or a dynamic feedback control law that also quadratically stabilizes the plant. The problem is investigated (LPV) plant to match asymptotically that of a given LPV model using a static or a dynamic feedback control law that also quadratically stabilizes the plant. The problem is investigated from a structural point of view and solvability conditions are expressed in geometric terms. feedback control law thatof also stabilizes the plant. The problem is investigated from a structural point viewquadratically and solvability conditions are expressed in geometric terms. feedback control law that also quadratically stabilizes the plant. The problem is investigated from a structural point of view and solvability conditions are expressed in geometric terms. Under suitable hypotheses, checkable sufficient conditions and viable procedures for constructing from asuitable structural point of checkable view andsufficient solvability conditions expressed in geometric terms. Under hypotheses, conditions andare viable procedures for constructing from a structural point of view and solvability conditions are expressed in geometric terms. Under suitable hypotheses, checkable sufficient conditions and viable procedures for constructing solutions are provided. Under suitable hypotheses, checkable sufficient conditions and viable procedures for constructing solutions are provided. Under suitable hypotheses, checkable sufficient conditions and viable procedures for constructing solutions are provided. solutions are provided. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. solutions are provided. Keywords: Keywords: Linear Linear Parameter Parameter Varying Varying Systems, Systems, Model Model Matching Matching Problem, Problem, Quadratic Quadratic Stability. Stability. Keywords: Keywords: Linear Linear Parameter Parameter Varying Varying Systems, Systems, Model Model Matching Matching Problem, Problem, Quadratic Quadratic Stability. Stability. Keywords: Linear Parameter Varying Systems, Model Matching Problem,ofQuadratic Stability. 1. INTRODUCTION requires differentiabity the time varying 1. INTRODUCTION requires differentiabity of the time varying parameter parameter of of 1. INTRODUCTION requires differentiabity of the time varying parameter of the plant. Stability of the closed-loop was not discussed, 1. INTRODUCTION requires differentiabity of the time varying parameter of the plant. Stability of the closed-loop was not discussed, 1. INTRODUCTION requires differentiabity of the time parameter of the Stability the closed-loop was not discussed, but the use of LMIs of methods for its varying study was suggested. the plant. plant. Stability of the closed-loop was was not discussed, In this paper we consider the problem of compensating a but the use of LMIs methods for its study suggested. the plant. Stability of the closed-loop was was not suggested. discussed, In this paper we consider the problem of compensating a but the use of LMIs methods for its study Here, we consider LPV systems whose coefficient matrices, but the use of LMIs methods for its study was suggested. Here, we consider LPV systems whose coefficient matrices, In paper consider compensating given plant inwe such a waythe to problem force itsof output to matchaa In this thisplant paperin wesuch consider problem compensating but the of LMIs methods for its study was suggested. given a waythe to force itsof output to match Here, we consider LPV systems whose coefficient matrices, in state space form, are affine combinations with time Here, weuse consider LPV systems whose coefficient matrices, In thisplant paperin consider the compensating in state space form, are affine combinations with time given such aa a to force output asymptotically of given model in case in which given plant inwethat such way to problem force its its output to match asymptotically that of away given model inofthe the caseto in match whicha in Here, we consider LPV systems whose coefficient matrices, state space form, are affine combinations with time dependent coefficients of a finite set of constant matrices. in state space form, are affine combinations with time given plant in such a way to force its output to match asymptotically that of a given model in the case in which both the plant and the model are linear parameter varying dependent coefficients of a finite set of constant matrices. asymptotically thatthe of model a givenare model inparameter the case invarying which dependent both the plant and linear in state space combinations with time coefficients of of matrices. For a given LPVform, plant are and afinite givenset LPV model we consider dependent coefficients of aa affine finite set of constant constant matrices. asymptotically that of model a given model inparameter the case invarying which both the plant the are linear systems. Linearand parameter varying systems, or LPV sys- For a given LPV plant and a given LPV model we consider both the plant and the model are linear parameter varying dependent coefficients of a finite set of constant matrices. systems. Linear parameter varying systems, or LPV sysFor a given LPV plant and a given LPV model we consider the problem of finding a static or a dynamic feedback For aproblem given LPV plant and given or LPV model we feedback consider both the and thesystems model are linear parameter varying of finding a astatic a dynamic systems. Linear parameter varying systems, or LPV systems, areplant dynamical whose defining equations in the systems. Linear parameter varying systems, or LPV sysFor aproblem given LPV plant and given LPV model weto consider tems, are dynamical systems whose defining equations in the of finding aa astatic or aa the dynamic feedback compensator that forces the output of plant match the problem of finding static or dynamic feedback systems. Linear parameter varying systems, or LPV syscompensator that forces the output of the plant to match tems, are dynamical systems whose defining equations in state space form depend on a parameter θ, which takes tems, space are dynamical systems defining in compensator state form depend on awhose parameter θ, equations which takes the problem of finding a static or a dynamic feedback that forces the output of the plant to match asymptotically that of the model for all inputs and all inicompensator that forces the output of the plant to match tems, are dynamical systems whose defining equations in state space form depend on a parameter θ, which takes values in a given parameter set Θ, and are linear for asymptotically that of the model for all inputs and all inistate space on a set parameter values in a form givendepend parameter Θ, and θ,arewhich lineartakes for asymptotically compensator that forces the output of the plant to match that of the model for all inputs and all initial conditions, while quadratically stabilizing the closedasymptotically that of the model for all inputs and all inistate space on scheduling a set parameter θ,are values in given parameter Θ, linear for each value ofform θ ∈depend Θ. Using parameters, LPV conditions, while the closedvaluesvalue in aaof given parameter set Θ, and and arewhich lineartakes for tial asymptotically that ofquadratically the modelon forstabilizing all inputs stability and all inieach θθ ∈ Θ. Using scheduling parameters, LPV tial conditions, while quadratically stabilizing the closedloop. The reason for focusing quadratic is tial conditions, while quadratically stabilizing the closedvalues in a given parameter set Θ, and are linear for loop. The reason for focusing on quadratic stability is each value of ∈ Θ. Using scheduling parameters, LPV systems canofbeθ employed to scheduling describe nonlinear behaviors each value ∈ Θ. Using parameters, LPV loop. tial conditions, while quadratically stabilizing the closedsystems can be employed to describe nonlinear behaviors The reason for focusing on quadratic stability is that it can be efficiently characterized, for the considered loop. The reason for focusing on quadratic stability is each value of θ ∈ Θ. Using scheduling parameters, LPV that it can be efficiently characterized, for the considered systems can be employed to describe nonlinear behaviors and, to some extent, this makes it possible to solve specific systems can be employed to describe nonlinear behaviors and, to some extent, this makes it possible to solve specific that loop. it The forLMIs. focusing on aquadratic stability is can be efficiently characterized, for the considered systems, in reason terms of Using structural geometric that it can be efficiently characterized, for the considered systems can be employed to describe nonlinear behaviors and, to some extent, this makes it possible to solve specific nonlinear problems by means of linear control methods. systems, in terms of LMIs. Using a structural geometric and, to some extent,by this makesofitlinear possible to solve specific systems, that it can be efficiently characterized, for the considered nonlinear problems means control methods. in terms of LMIs. Using a structural geometric approach, sufficient conditions for solving the problem, systems, in terms of LMIs. Using a structural geometric and, to some extent, this makes possible solve specific nonlinear problems by means of control methods. In general, the problem of compensating atoplant in such approach, sufficient conditions for solving the problem, nonlinear problems by means ofitlinear linear control methods. systems, insufficient terms of conditions LMIs. Using structural In general, the problem of compensating a in such approach, for solving the problem, with aa static compensator in the state approach, sufficient conditions foraor, solving the geometric problem, by means of desired linear control methods. either with static compensator or, in case case the state In general, the problem of compensating aa plant plant in such anonlinear way thatproblems its output achieves performances can either In general, the problem of compensating plant in such approach, sufficient conditions for solving the problem, a way that its output achieves desired performances can either with a static compensator or, in case the state of the model is not accessible for measurement, with a either with a static compensator or, in case the state In general, the problem of compensating a plant in such abe way that its output achieves desired performances can tackled by considering a model whose output exhibits of the model is not accessible for measurement, with a a way that by itsconsidering output achieves desired performances can of be tackled a model whose output exhibits either with a static compensator or, in case the state the model is not accessible for measurement, with a dynamic one, are given. A viable algorithmic procedure to of the model is not accessible for measurement, with a a that by its output achieves performances cana dynamic one, are given. A viable algorithmic procedure to be tackled considering aa model whose output exhibits the desired behaviour and by desired designing, if possible, beway tackled by considering model whose output exhibits of the model is not accessible for measurement, with a the desired behaviour and by designing, if possible, a dynamic one, are given. A viable algorithmic procedure to check solvability conditions and to synthesize a suitable dynamic one, are given. A viable algorithmic procedure to be bybehaviour considering a model whose that output exhibits the desired by to designing, if possible, controller that forces theand plant match model (seeaa check solvability conditions and to synthesize a suitable thetackled desired behaviour and by designing, if possible, dynamic one, are given. A viable algorithmic procedure to controller that forces the plant to match that model (see check solvability conditions and to synthesize a suitable compensator is presented. check solvability conditions and to synthesize a suitable the desired behaviour by to designing, if possible, compensator is presented. controller that forces the plant match that model (see Ichikawa (1985)). This motivates the interest in finding controller that forces theand plant to match that model (seea compensator check solvability conditions and to synthesize a suitable Ichikawa (1985)). This motivates the interest in finding is presented. The paper is organized as follows. The asymptotic matchcompensator is presented. controller that forces the plant to match that model (see Ichikawa (1985)). This motivates the interest in finding solvability conditions for the model matching problem and The paper is organized as follows. The asymptotic matchIchikawa (1985)). This interest in finding solvability conditions formotivates the model the matching problem and The compensator is presented. paper is organized as follows. The asymptotic matching problem is formally stated in Section 2, after describing The paper is organized as follows. The asymptotic matchIchikawa This interest finding solvability conditions for the matching problem and in defining(1985)). viable procedures for thethe synthesis ofinsolutions problem is formally stated in Section 2, after describing solvability conditions formotivates the model model matching problem and ing The paper is organized as follows. The asymptotic matchin defining viable procedures for the synthesis of solutions ing problem is formally stated in Section 2, after describing the class of the considered LPV systems, making dising problem is formally stated in Section 2, after describing solvability conditions for the model problem in defining viable forincluding, thematching synthesis of solutions solutions for many classes of systems, besides thatand of the class of the considered LPV systems, making disin defining viable procedures procedures for the synthesis of ing problem is formally stated in Section 2, after describing for many classes of systems, including, besides that of the class of the considered LPV systems, making distinction between the situation in which the state of the the classbetween of the the considered systems, in viablethose procedures forincluding, thesystems, synthesis of solutions tinction situationLPV in which the making state of disthe for many classes of besides that of linear systems, of nonlinear periodic sysfordefining many classes of systems, systems, including, besides thatsysof tinction the class of the considered LPV systems, making dislinear systems, those of nonlinear systems, periodic between the situation in which the state of the model is accessible for measurement and a static feedback tinction between the situation in which the state of the for many classes of systems, including, besides that of linear systems, those of nonlinear systems, periodic systems, time-delay systems, systems over rings, 2D systems, model is accessible for measurement and a static feedback linear time-delay systems, those of nonlinear systems, sys- model tems, systems, systems over rings,periodic 2D systems, tinction between situation in which state of the is accessible for measurement and aathe static feedback compensator may the solve the problem and that in which model is accessible for measurement and static feedback linear systems, those of nonlinear systems, periodic systems, time-delay systems, systems over rings, 2D systems, time-varying systems, swithching systems (see e.g. (Moore compensator may solve the problem and that in which the tems, time-delay systems, systemssystems over rings, 2D systems, model accessible for measurement andmeasurement a static feedback time-varying systems, swithching (see e.g. (Moore compensator may the problem that in the state ofis the model is not accessible for and compensator may solve solve theaccessible problem and and that in which which the tems, time-delay systems, systems over rings, systems, time-varying systems, swithching systems (see e.g. (Moore and Silverman, 1972; Moog et al., 1991; Di2D Benedetto state of the model is not for measurement and time-varying systems, swithching systems (see e.g. (Moore compensator may solve the problem and that in which the and Silverman, 1972; Moog et al., 1991; Di Benedetto state of the model is not accessible for measurement and a dynamic feedback compensator is required. In Section 3, state of the model is not accessible for measurement and time-varying systems, swithching systems (see e.g. (Moore a dynamic feedback compensator is required. In Section 3, and Silverman, 1972; Moog et al., 1991; Di Benedetto Grizzle, 1994; Conte and Perdon, 1995; Colaneri and Silverman, 1972; Moog al., 1991; Benedetto and Grizzle, 1994; Conte and et Perdon, 1995;Di Colaneri and athe state of the model is not accessible for measurement and dynamic feedback compensator is required. In Section 3, geometric tools that are used to solve the problem are a dynamic feedback compensator is required. In Section 3, Silverman, 1972; Moog et al., 1991; Di Benedetto and Grizzle, 1994; Conte and Perdon, 1995; Colaneri and Kuˇ c era, 1997; Picard et al., 1998; Marinescu and Bourles, the geometric tools that are used to solve the problem are andcera, Grizzle, Conte and Perdon, 1995; Colaneri and the Kuˇ 1997;1994; Picard et al., 1998; Marinescu and Bourles, athe dynamic feedback compensator is required. In Sectionare 3, geometric that are to solve the problem introduced andtools described andused general sufficient solvability geometric tools that are used to solve the problem are and Grizzle, 1994; Conte and Perdon, 1995; Colaneri and Kuˇ c era, 1997; Picard et al., 1998; Marinescu and Bourles, 2003; Marinescu, 2009; Perdon et al., 2016)). introduced and described and general sufficient solvability Kuˇcera, 1997; Picard etPerdon al., 1998; Marinescu and Bourles, introduced the geometric tools that are used to solve the problem are 2003; Marinescu, 2009; et al., 2016)). and described and general sufficient solvability conditions for the case in which the state of the model introduced for andthe described general solvability Kuˇ cthe era, 1997; etPerdon al., 1998; Bourles, case inand which the sufficient state of the model 2003; Marinescu, 2009; et al., 2016)). In area of Picard LPV systems, the problem ofand forcing the conditions 2003; Marinescu, 2009; Perdon et Marinescu al., 2016)). introduced and described and general sufficient solvability In the area of LPV systems, the problem of forcing the conditions for the case in which the state of the model is accessible for measurement are stated in Theorem 1. conditions for the case in which the state of the model 2003; Marinescu, 2009; Perdon et al., 2016)). is accessible for measurement are stated in Theorem 1. In the area of LPV systems, the problem of forcing the output of a given plant to track asymptotically a reference In the area of LPV systems, problem of forcing the is output of a given plant to trackthe asymptotically a reference conditions for the case in which the state of the model accessible for measurement are stated in Theorem 1. Proposition 2 provides slightly more conservative sufficent is accessible for measurement are stated in Theorem 1. In the area of LPV systems, the problem of forcing the output of a given plant to track asymptotically a reference signal, which can be viewed as the output of a suitable Proposition 2 provides slightly more conservative sufficent output of a given to track signal, which canplant be viewed asasymptotically the output of aareference suitable Proposition is accessiblewhich measurement are stated in checkable Theorem 1. 22for provides slightly more conservative sufficent conditions, have the advantage of being by Proposition provides slightly more conservative sufficent output of a given plant tointrack asymptotically aaareference signal, which can be viewed as the of suitable model, was considered Balas et output al. (2004). In that conditions, which have the advantage of being checkable by signal, which can be viewed as the output of suitable Proposition 2 provides slightly more conservative sufficent model, was considered in Balas et al. (2004). In that conditions, which have the advantage of being checkable by an algorithmic procedure. If they are verified, a compenconditions, which have the advantage being checkable by signal, which can be viewed the of constructa suitable an algorithmic procedure. If they areofverified, a compenmodel, was considered in et al. In paper, the authors proposed aas methodology for model, the wasauthors considered in Balas Balas et output al. (2004). (2004). In that that an conditions, which have the advantage of being checkable by paper, proposed a methodology for constructalgorithmic procedure. If they are verified, aa algorithcompensator can be practically constructed by aa viable an algorithmic procedure. If they are verified, compenmodel, was considered in Balas et al. (2004). In that sator can be practically constructed by viable algorithpaper, the authors proposed a methodology for constructing a compensator that achieves the tracking by means paper, the authors proposed a methodology for constructing a compensator that achieves the tracking by means sator an algorithmic procedure. If they are verified, a compencan be practically constructed by a viable algorithmic procedure. Proposition 3 consider the case in which sator can be practically constructed by a viable algorithpaper, the authors proposed a methodology for constructing aa compensator that achieves the by of dynamic inversiont. Implementation of such procedure procedure. Proposition 33 consider case in which ingdynamic compensator thatImplementation achieves the tracking tracking by means means mic sator can be practically constructed by the a viable algorithof inversiont. of such procedure Proposition the case which mic procedure. procedure. Proposition 3 consider consider the case in in which ing a compensator thatImplementation achieves the tracking by means mic of inversiont. of of dynamic dynamic inversiont. Implementation of such such procedure procedure mic procedure. Proposition 3 consider the case in which of dynamic inversiont. Implementation of such procedure ∗ ∗ ∗ ∗ ∗ ∗

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the state of the model is not accessible for measurement and it provides sufficient conditions for the solution of the problem by exploiting the result of Proposition 2. The last section contains conclusions and description of future work. 2. PRELIMINARIES AND PROBLEM STATEMENT A Linear Parameter Varying, or LPV, system Σθ is a dynamical structure defined by a compact subset Θ ⊂ Rk and functions A(.) : Θ → Rn×n , B(.) : Θ → Rn×m , C(.) : Θ → Rp×n . For any function θ(.) : R+ → Θ in a given class F, that is called the class of admissible functions, the LPV system Σθ evolves according to the equations  x(t) ˙ = A(θ(t))x(t) + B(θ(t))u(t) (1) Σθ = y(t) = C(θ(t))x(t)

where t ∈ R+ is the time variable, x ∈ X = Rn is the state, u ∈ U = Rm is the input, y ∈ Y = Rp is the output. The class F of all admissible functions can be specified, if needed, in different ways. Here, we will only assume that any element of F is a piecewise continuous function with only a finite number of discontinuities in any finite time interval, so to avoid chattering phenomena in the evolution of Σθ . Note that the output of the system Σθ depends on the initial condition x(0), on the input signal u(.) : R+ → U and also on the choice of the function θ(.) in the class of admissible ones. By applying to Σθ a state feedback u(t) = F (θ(t))x(t), where F (.) : Θ → Rm×n , we obtain a compensated system that will be denoted by ΣF θ , whose closed-loop dynamics is defined by the parameter dependent matrix (A(θ(t)) + B(θ(t))F (θ(t))). Writing θ¯ = (θ¯1 ...θ¯k ) ∈ Θ ⊂ Rk , we assume that the matrices in (1) have the following parametric structure  ¯ = A0 + θ¯i Ai A(θ) i=1,...,k ¯ = B0 + B(θ) θ¯i Bi (2)  i=1,...,k ¯ = C0 + C(θ) θ¯i Ci i=1,...,k

where Ai , Bi , Ci for i = 0, ..., k are constant real matrices of suitable dimensions. Denoting respectively by θi− and by θi+ the minimum and the maximum of θ¯i in Θ, we have for the second member in the first equation (2) the equality  θ¯i Ai = A0 + i=1,...,k    θ¯i − θi− + − − θi + + (θ − θi ) Ai = A0 + i=1,...,k θi − θi− i   θ¯i − θ−   + − i (A0 + θi− Ai ) + + − (θi − θi Ai ) = θ − θi i=1,...,k i=1,...,k i  ˜ A˜0 + θi A˜i i=1,...,k

with A˜0 = A0 + and θ˜i =

θ¯i −θi− . θi+ −θi−



θ − Ai , i=1,...,k i

(3) + − ˜ Ai = (θi − θi )Ai

Similar equalities hold for the second

members in the other equations (2) and then, since 0 ≤ θ¯i −θi− θi+ −θi−

≤ 1, we can assume without loss of generality,

possibly renaming the terms at issue, that the parameter θ¯i , for i = 1, ..., k ranges in the closed real interval [0, 1] or, 556

equivalently, that Θ is the unit hypercube with a vertex in the origin contained in the positive cone in Rk . ¯ = ImB0 . Note that for θ¯ = (0...0) ∈ Θ we have ImB(θ) In the rest of the paper, we assume that, for all θ¯ ∈ Θ, the ¯ satisfies the following condition matrix B(θ) ¯ ¯ = B0 H(θ) (4) B(θ) ¯ In particular, for some nonsingular square matrix H(θ). ¯ = ImB0 . Note that condition (4) is this implies ImB(θ) ¯ = B for all θ¯ ∈ Θ and hence verified, in particular, if B(θ) it holds for the LPV systems considered in Bokor and Balas (2005). Assume we are given two LPV systems ΣP θ and ΣM θ of the form (1), called respectively the plant and the model, and defined respectively by the plant equations  x˙ P (t) = AP (θ(t)) xP (t) + BP (θ(t)) w(t) ΣP θ ≡ (5) y(t) = CP (θ(t)) xP (t) 



with state xP ∈ XP = Rn , input w ∈ W = Rm , output y ∈ Y = Rp , and by the model equations  x˙ M (t) = AM (θ(t)) xM (t) + BM (θ(t)) u(t) ΣM θ ≡ (6) yM (t) = CM (θ(t)) xM (t)

with state xM ∈ XM = Rn , input u ∈ U = Rm and output yM ∈ Y = Rp . Without loss of generality, we assume that ¯ and BM (θ) ¯ are full-column rank for the matrices BP (θ) all θ ∈ Θ. Note that, while the input and state spaces of the two systems above differ and may have different dimensions, their output spaces are assumed to coincide. Letting both ΣP θ and ΣM θ evolve according to the same function θ(.) (this hypothesis is not restrictive, as it will be shown later on), we can consider the problem of synthesizing a controller for the plant that stabilizes it and that forces its response to match asymptotically that of the model. The notion of stability we consider here is that of quadratic stability. We recall that quadratic stability of the LPV system Σθ is characterized by the existence of a symmetric positive definite matrix P of suitable dimensions such that ¯ + P A(θ) ¯ < 0 for all θ¯ ∈ Θ. Quadratic stability A (θ)P implies asymptotic stability and it is of interest in practice due to the fact that it can be characterized and studied using computationally efficient tools by means of linear matrix inequalities (LMIs) (Boyd et al. (1994)). Focusing on quadratic stability, although it is a stronger requirement than global asymptotic stability, we have the possibility to exploit previous results (notably those of Conte et al. (2015)) and to use efficient computational methods for checking this property. Since it is natural to consider models whose dynamics is asymptotically stable, in our context we will assume that the model is quadratically stable. Matching problems of the above kind can be formulated as disturbance decoupling problems with (quadratic) stability for a suitable system, that essentially compares the output of the plant and that of the model (Emre and Hautus (1980)). This can be seen by considering the outputdifference LPV system ΣDθ , defined by the equations  x˙ M (t) = AM (θ(t)) xM (t) + BM (θ(t)) u(t) x˙ P (t) = AP (θ(t)) xP (t) + BP (θ(t)) w(t) ΣDθ ≡ y¯(t) = CM (θ(t)) xM (t) − CP (θ(t)) xP (t) (7) Compensating the system ΣDθ by means of a static or dynamic feedback, through the (control) input w(t), in

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such a way to (quadratically) stabilize it and, at the same time, to annihilate its forced response (that is: to annihilate its response for null initial conditions) for every (disturbance) input u(t) and for all admissible θ(.) (namely, to decouple y¯(t) from u(t) with (quadratic) stability) is equivalent to (quadratically) stabilize the plant and, at the same time, to force its output to match asymptotically that of the model for every input u(t) and every initial condition of the plant, of the model and, in case it is dynamic, of the compensator. In the following, we will formally state two different problems, depending on whether the state of the model is accessible (i.e. measurable) or not. In the first case, compensation can be achieved by static feedback of the plant state and of the model state. In the second case, when the state of the model is not accessible, it is necessary to employ a dynamic feedback compensation. Problem 1. Given the plant (5) and the model (6), assume that the latter is quadratically stable and that its state is measurable. Then, the Static Model Matching Problem with Quadratic Stability (SMMPQS) consists in finding a static feedback control law of the form w(t) = FM (θ(t)) xM (t)+FP (θ(t)) xP (t)+G(θ(t)) u(t) (8) for the disturbed output-difference system (7) such that the compensated system  x˙ M (t) = AM (θ(t)) xM (t) + BM (θ(t)) u(t)     x˙ P (t) = BP (θ(t)) FM (θ(t)) xM (t) +(AP (θ(t)) + BP (θ(t)) FP (θ(t))) xP (t) ΣCθ ≡   +B P (θ(t)) G(θ(t)) u(t)   y¯(t) = CM (θ(t)) xM (t) − CP (θ(t)) xP (t) (9) is quadratically stable and its forced response (that is: its response for initial condition (xM (0), xP (0)) = (0, 0)) is null for every input u(t) (equivalently, the output y¯(t) is decoupled from the input u(t)) for all admissible θ(.). Problem 2. Given the plant (5) and the model (6), assume that the latter is quadratically stable. Then, the Dynamic Model Matching Problem with Quadratic Stability (DMMPQS) consists in finding an integer q¯ and a dynamic, LPV feedback compensator of the form  x˙ (t) = AE (θ(t)) xE (t) + AE (θ(t)) xP (t)   E +K(θ(t)) u(t) (10) w(t) = F (θ(t)) xE (t) + FP (θ(t)) xP (t)   +G(θ(t)) u(t) with xE ∈ XE = Rq¯ for the disturbed output-difference system (7) such that the compensated system  x˙ M (t) = AM (θ(t)) xM (t) + BM (θ(t)) u(t)     x˙ P (t) = (AP (θ(t)) + BP (θ(t)) FP (θ(t))) xP (t)     +BP (θ(t)) F (θ(t)) xE (t) +BP (θ(t)) G(θ(t)) u(t) ΣexCθ ≡   (t) = AE (θ(t)) xE (t) + AE (θ(t)) xP (t) x ˙  E    +K(θ(t)) u(t)   y¯(t) = CM (θ(t)) xM (t) − CP (θ(t)) xP (t) (11) is quadratically stable and its forced response (that is: its response for initial condition (xM (0), xP (0), xE (0)) = (0, 0, 0)) is null for every input u(t) (equivalently, the output y¯(t) is decoupled from the input u(t)) for all admissible θ(.). Note that quadratic stability of (9) and (11) implies, in particular, limt→+∞ y¯(t) = 0 if u(t) is identically 557

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zero, for every initial conditions (xM (0), xP (0)) of the plant and of the model in the static feedback case or (xM (0), xP (0), xE (0)) of the plant, of the model and of the compensator in the dynamic feedback case. Together with nullity of the forced responses, this implies that the output yM (t) = CP (θ(t)) xP (t) of the plant matches asymptotically the output yM (t) = CM (θ(t)) xM (t) of the model if the plant is compensated by means of (8) or, respectively, of (10), for every input u(t), for all admissible θ(.) and for all initial conditions. Moreover, quadratic stability of (9) and (11) implies in particular quadratic stability of the compensated plant. In fact, considering (9), from     ¯ P1 P2 0 AM (θ) + ¯ ¯ ¯ ¯ ¯ P2 P 3  BP (θ)FM(θ) AP (θ) + BP (θ) FP (θ) ¯ P1 P2 AM (θ) 0 <0 ¯ M (θ) ¯ AP (θ) ¯ + BP (θ) ¯ FP (θ) ¯ BP (θ)F P2 P3 (12)   0 multiplying both members on the left by and on xP   0 the right by we get xP   ¯ ¯ ¯  P3 + x P  AP (θ) + BP (θ) FP (θ)  (13) ¯ + BP (θ) ¯ FP (θ) ¯ xP < 0 P3 AP (θ) for all xp ∈ XP . Similar computations can be carried on also considering (11). Remark 1. In case the plant and the model depends on two different parameters, namely θ1 (t) and θ2 (t), we can consider a single parameter θ(t) = (θ1 (t) θ2 (t) ) and write all the defining matrices accordingly by adding a suitable number of null  components: for instance, wit¯ = A0 + ing AP (θ¯1 ) = A0 + θ¯1i Ai as AP (θ)   i=1,...,k θ¯1i Ai + θ¯2i Ai with Ai = 0 for i = i=1,...,k

i=1,...,k

1, ..., k2 .

3. SOLUTION OF THE MATCHING PROBLEMS In line with the previous section, we address the model matching problems defined therein as disturbance decoupling problems with quadratic stability for LPV systems. In particular, the case of static feedback has been studied by Conte et al. (2015) and a solution was found by exploiting the geometric concepts and tools that we recall herein for the reader’s convenience. Definition 1. Given a LPV system Σθ of the form (1), a subspace V ⊆ X is said to be a controlled invariant ¯ ⊆ V + ImB(θ) ¯ holds subspace for Σθ if and only if A(θ)V for all θ¯ ∈ Θ. It is known ((Basile and Marro, 1992)) that V is a controlled invariant subspace for Σθ if and only if for ¯ such that (A(θ) ¯ + all θ¯ ∈ Θ there exists a matrix F (θ) ¯ (θ))V ¯ B(θ)F ⊆ V. Then, the feedback u(t) = F (θ(t))x(t) makes V invariant for the closed-loop dynamics of the compensated system ΣF θ for any admissible θ. Any such feedback or, equivalently, the indexed family of matrices ¯ θ¯ ∈ Θ} that defines it, is called a friend of V. {F (θ), Proposition 1. Given a LPV system Σθ described by (1), (2) and satisfying (4), a subspace V ⊆ X is a controlled invariant subspace for Σθ if and only if

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Ai V ⊆ V + ImB0

(14)

Proof. See Conte et al. (2015) Proposition 1. The family of controlled invariant subspaces for Σθ contained in a given subspace W ⊆ X , denoted by ¯ B(θ), ¯ W), is closed with respect to sum of subV(A(θ), spaces (Basile and Marro (1992)) and, therefore, it ¯ B(θ), ¯ W). has a maximum element, denoted by V ∗ (A(θ), By Proposition 1, the controlled invariant subspace ¯ B(θ), ¯ W) can be computed as the last term of V ∗ (A(θ), the sequence of subspaces Vj defined by V0 = W,  Vj+1 = Vj

i=0,...,k

 A−1 i (Vi + ImB0 ) ,

(15)

with j = 0, . . . , , where  ≤ dim W is the least integer such that V+1 = V . Note that the situation considered herein is slightly more general than that considered by Bokor and Balas (2005) and the algorithm slightly differs from that proposed by Conte et al. (1991). In the rest of the paper, we will be interested in the maximum element of the ¯ B(θ), ¯ K), where K =  ¯ KerC(θ) ¯ = family V(A(θ), θ∈Θ  ∗ ¯ ¯ KerC . If no confusion arises, V (A( θ), B( θ), K) i i=0,...,k ∗ will be simply denoted by V . The general result about the solution of the SMMPQS is stated in the following theorem. Theorem 1. Given an LPV plant ΣP θ and a quadratically stable LPV model ΣM θ of the form (1), (2), both satisfying (4), the related SMMPQS is solvable if there exists a controlled invariant subspace V for  the output-difference system ΣDθ contained in K = i=0,...,k Ker (CM i − CP i )   ¯ = FM (θ) ¯ FP (θ) ¯ , θ¯ ∈ θ} such that with a friend {F  (θ) 1) the condition     ¯ 0 BM (θ) ⊆ V + Im (16) Im ¯ 0 BP (θ) holds for all θ¯ ∈ Θ, 2) the system ΣF Dθ , obtained  by compensating ΣDθ by xM (t) u(t) = F (θ(t)) , is quadratically stable. xP (t)

¯ : Proof. Let condition 1) be satisfied and let G(θ) W  → U be, for all θ¯ ∈ Θ, a linear map such that ¯ BM (θ) ⊆ V for all θ¯ ∈ Θ and consider the Im ¯ ¯ BP (θ)G(θ) static, LPV feedback control law w(t) = FM (θ(t))xM (t)+FP (θ(t))xP (t)+G(θ(t))u(t) (17) for the output-difference system ΣDθ described by (7). Letting V be a matrix whose columns are a basis of V, consider a change of basis in XM ⊕ XP of the form z  = T x,  z1  with T = (V T ). In the new basis, writing z = z2  with z1 ∈ V and z2 ∈ ImT , the compensated system ΣCθ evolves according to equations of the form  z˙ (t) = A11 (θ(t))z1 (t) + A12 (θ(t))z2 (t)+   1 D1 (θ(t))u(t) ΣCθ = .  z˙2 (t) = A22 (θ(t))z2 (t)  y(t) = C2 (θ(t))z2 (t) (18) 558

Hence, for all admissible θ(.), the chosen feedback control law achieves the decoupling of y(t) from u(t). At the same time, as the free dynamics of ΣCθ and of ΣF Dθ coincide, it quadratically stabilizes, thank to condition 2), the output-difference LPV system. This implies, as seen at the end of Section 2, that the plant ΣP θ , compensated by w(t) = FM (θ(t))xM (t) + FP (θ(t))xP (t) + G(θ(t))u(t), is quadratically stable and that its output matches asymptotically that of the model ΣM θ for all admissible θ(.) and for all initial condition. Remark 2. Given a controlled invariant subspace V ⊆ X , thank to (4), condition 1) of Theorem 1 can be easily seen to be equivalent to     0 BM 0 . (19) ⊆ V + Im Im 0 BP 0 ¯ = In fact, (19) is implied by (16) and, being ImBM (θ) ¯ = ImBP 0 , the converse is obvious. ImBM 0 and ImBP (θ) As in the case of the disturbance decoupling problem considered in Conte et al. (2015), solvability of the problem depends on a structural condition, namely condition 1), and a qualitative condition, namely condition 2). Their coupling makes the search for a controlled invariant subspace and a friend which satisfy both quite complicated. Actually, in general situations, no algorithmic procedure to perform the search is known. The following proposition gives sufficient conditions for solving the SMMPQS that can be checked in a finite number of steps and, at the same time, it indicates a viable procedure for the construction of solutions, if any exists. Proposition 2. In the same hypotheses as in Theorem 1, let V ∗ denote the maximum controlled invariant subspace for  the output-difference system ΣDθ contained in K = − CP i ). Then, the related SMMPQS i=0,...,k Ker (CM i is solvable if the condition     ¯ 0 BM (θ) ⊆ V ∗ + Im (20) Im ¯ 0 BP (θ) holds for all θ¯ ∈ Θ and there exists a family of matrices F0 , ..., Fk , with Fi = (FM i FP i ) for i = 0, ..., k, that satisfy the following conditions 1)



AM i 0 0 AP i

for i = 0, ..., k 2)



+



0 BP 0



 Fi V ∗ ⊆ V ∗



    AM i 0 0 P+ + F BP 0  i   0 AP i   AM i 0 0 P + Fi < 0 BP 0 0 AP i

(21)

(22)

for some positive definite, symmetric matrix P and i = 0, ..., k.     0 0 ¯ = Proof. By (4), we have HP (θ), ¯ BP 0 BP (θ) ¯ is a nonsingular square matrix, for all θ¯ ∈ Θ. where HP (θ) Consider, for θ¯ = (θ¯1 ...θ¯k ), the matrix  ¯ = H −1 (θ)(F ¯ 0+ F (θ) (23) θ¯i Fi ) P i=1,...,k

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and apply the feedback w(t) = F (θ(t))x(t) to the system ΣDθ . The closed-loop matrix satisfies, for all θ¯ ∈ Θ, the equality     ¯ 0 AM (θ) 0 ¯ ¯ F (θ) = ¯ + BP (θ) A P (θ)  0  AM 0 + 0 θ¯i AM i i=1,...,k  +  0 AP 0 + θ¯i AP i i=1,...,k    0 −1 ¯ ¯ HP (θ)HP (θ)(F0 + θ¯i Fi ) = BP 0   i=1,...,k    0 AM 0 0 F + + BP 0  0  0 AP 0    AM i 0 0 ¯ + Fi . θi 0 AP i BP 0 i=1,...,k (24) By (21), we have      ¯ 0 0 AM (θ) ¯ V∗ = + F ( θ) ¯ ¯ BP (θ) AP(θ)   0  AM 0 0 0 F0 V ∗ + + BP 0   0 AP 0    A 0 0 M i + Fi V ∗ ⊆ V ∗ θ¯i BP 0 0 AP i i=1,...,k (25) ¯ θ¯ ∈ Θ} is a friend of V ∗ . and hence {F (θ), Moreover, recalling that θ¯i ≥ 0 for i = 1, ..., k, by (22) we have      ¯ 0 AM (θ) 0 ¯ + F ( θ) P+ ¯ ¯ AP (θ)  B P (θ)    0 ¯ 0 AM (θ) 0 ¯ P ¯ F (θ) = ¯ + BP (θ) 0 AP (θ)      AM 0 0 0 + F0 P+ B 0 A P0   P 0   AM 0 0 0 P + F0 + B 0 A P0 P0       AM i 0 0 ¯ θi P + + F0 BP 0 0 AP i i=1,...,k       AM i 0 0 P <0 + Fi BP 0 0 AP i

(26) and hence the closed-loop dynamics of ΣF is quadratically Dθ stable. The conclusion, then, follows from Theorem 1. It is useful to remark that the conditions of the above Proposition can be practically checked and that a solution to the SMMPQS can be constructed in a finite number of steps by adapting in an obvious way the procedure given in Section 3 of Conte et al. (2015) for constructing a solution of the disturbance decoupling problem considered there. The same conditions as in Proposition 2 guarantee solvability, by means of a dynamic feedback compensator, also in case the state of the model cannot be directly observed and measured, as shown by the following proposition. Proposition 3. In the same hypotheses as in Proposition 2, let V ∗ denote the maximum controlled invariant subspace for the output-difference system ΣDθ contained in K = i=0,...,k Ker (CM i − CP i ). Then, the related DMMPQS is solvable if condition (20) holds for all θ¯ ∈ Θ and there exists a family of matrices F0 , ..., Fk , with Fi = 559

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(FM i FP i ) for i = 0, ..., k, that satisfy conditions 1) and 2) of Proposition 2.   VM Hint of proof. Let V = be a matrix of suitable VP dimensions whose columns form a basis of V ∗ . Then, by conditions 1) and 2) of Proposition 2, we can write     0 AM i 0 V = V Li + (−FM i VM − FP i VP ) BP 0 0 AP i (27) for suitable matrices Li for all i = 1, ..., k. Using (22) it is possible to show that there exist symmetic, positive definite matrices P1 , P2 and P3 , of suitable dimensions, such that the following inequalities hold for all i = 0, ..., k: L i P 1 + P1 L i < 0 ((AP i + BP 0 FP i ) P2 + P2 (AP i + BP 0 FP i )) < 0 A M i P3 + P 3 A M i < 0

(28)

Now, consider the extended output-difference system  x˙ (t) = AM (θ(t)) xM (t) + BM (θ(t)) u(t)   M x˙ P (t) = AP (θ(t)) xP (t) + BP (θ(t)) w(t) ΣexDθ ≡ x   ˙ E (t) = AE (θ(t)) xE (t) + BE (θ(t)) u(t) y¯(t) = CM (θ(t)) xM (t) − CP (θ(t)) xP (t) (29) ¯ = AE0 +  ¯ where AE (θ) i=1,...k θi AEi , with AEi = Li for ¯ is constructed in i = 0, ..., k, and, exploiting (20), BE (θ) such a way that   ¯ BM (θ) ¯ = V BE (θ) (30) ¯ θ) ¯ BP (θ)G(

¯ for all θ¯ ∈ Θ. The subspace Vex hold, for a suitable G(θ), of the state space  of Σ exDθ spanned by the columns of the VM VP matrix Vex = is a controlled invariant subspace I ¯ ¯ − CP (θ(t)) 0) for ΣexDθ , contained in Ker(CM (θ(t)) ¯ for any θ ∈ Θ, since       AM i 0 VM 0 0 VP 0 AP i 0 + BP 0 (0 FP i FM i VM ) 0 0 AEi I 0   VM = VP L i I (31) for all i = 0, ...k and in virtue of Proposition 1. We see from (31) that Vex can be made invariant by using a friend  ¯  {F (θ) = (0 FP 0 FM 0 VM )+ (32) F V ), θ¯ ∈ Θ} θ¯ (0 F i=1,...,k

i

Pi

Mi M

which does not exploit the knowledge of xM (t). Since the relation     ¯ 0 BM (θ) ¯  (33) Im  0  ⊆ Vex + Im  BP (θ) ¯ 0 BE (θ)

holds for all θ¯ ∈ Θ thank to (30), by applying the feedback   xM  w(t) = F (θ) xP + G(θ)u(t) to the extended outputxE difference system ΣexDθ , the output is decoupled from u(t). Moreover, in the basis defined by the columns of the

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 I 0 VM matrix 0 I VP , the matrix of the dynamics of the 0 0 I compensated system takes the form   ¯ AM (θ) 0 0 ¯ + BP 0 FP (θ) ¯ 0 =  0 AP (θ) ¯ 0 0 L(θ)   AM 0 0 0 0 A P 0 B P 0 FP 0 0 + (34) 0 0 L 0   θ¯i AM i 0 0   0 θ¯i (AP i + BP 0 FP i ) 0  i=1,...,k 0 0 θ¯i Li for all θ¯ ∈ Θ and this shows, by (28), that the same feedback quadratically stabilizes ΣexDθ . Remark 3. The matching problems we have considered could also be addressed by exploiting the notion of timevarying controlled invariant subspace introduced in Ilchmann (1989), which is more general that that given in Definition 1. Using the results of Ilchmann (1989), instead of those of Conte et al. (2015), it is possible to find less conservative conditions for the existence of a feedback compensator that decouples the output of the output-difference system ΣDθ from u(t). Such feedback would achieve an exact matching between the output of the plant and that of the model if both the systems are suitably initialized (that is: if ΣDθ is initialized at a point of the maximum time-varying controlled invariant subspace contained in the Kernel of its output map). For a different initialization, however, asymptotic matching is not guaranteed, since the results of Ilchmann (1989) do not assure (quadratic) stability of the closed-loop. In general, for time-varying controlled invariant subspaces, it is not known how to characterize the existence of a friend that (quadratically) stabilizes the closed-loop. Therefore, the condition of Theorem 1 for solving the asymptotic model matching problem for all initial conditions of the plant and of the model, as required in the statements of Section 2, cannot be easily weakened using the alternative approach suggested by Ilchmann (1989). 4. CONCLUSION Sufficient conditions for solving the model matching problem with quadratic stability for LPV systems have been given using a structural geometric approach, both in the case in which the state of the model is accessible for measurement and in the case in which it is not. Future work will aim at finding less conservative conditions, at weakining the stability requirement and at finding a minimum for the dimension of the state space of the dynamic feedback compensator that possibly solves the DMMPQS. In addition, the case in which the state of the plant is not accessible for measurement will be taken into account. REFERENCES Balas, G., Bokor, J., and Szab´ o, Z. (2004). Tracking of continuous LPV systems using dynamic inversion. In 43rd IEEE Conference on Decision and Control, 2929– 2933. 560

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