H∞ Output-Feedback Gain-Scheduled Control for Discrete-Time Linear Systems Affected by Time-Varying Parameters*

Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Proceedings of the 20th World Proceedings of the 20th9-14, World Toulouse, France, July 2017 The International Federation of Congress Automatic Control Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017

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H∞ Output-Feedback Gain-Scheduled Control for H Output-Feedback Gain-Scheduled Control for H∞ Output-Feedback Gain-Scheduled Control Linear Systems Affected by for ∞ Discrete-Time Discrete-Time Linear Systems by ⋆ Discrete-Time LinearParameters Systems Affected Affected by Time-Varying ⋆ Time-Varying Parameters ⋆ Time-Varying Parameters ∗ ∗ ∗

Tábitha E. Rosa Cecília F. Morais Ricardo C. L. F. Oliveira Tábitha E. Rosa ∗∗ Cecília F. Morais ∗∗ Ricardo C. L. F. Oliveira ∗∗ Tábitha ∗E. Rosa Cecília F. Morais Ricardo C. L. F. Oliveira School of Electrical and Computer Engineering, ∗ of Electrical and Computer Engineering, University of Campinas – UNICAMP, Campinas, SP, Brazil. ∗ School School of Electrical and Computer Engineering, University of Campinas – UNICAMP, Campinas, SP, Brazil. (e-mail: {tabitha, cfmorais, ricfow}@dt.fee.unicamp.br). University of Campinas – UNICAMP, Campinas, SP, Brazil. (e-mail: {tabitha, cfmorais, ricfow}@dt.fee.unicamp.br). (e-mail: {tabitha, cfmorais, ricfow}@dt.fee.unicamp.br). Abstract: This paper deals with the problem of H∞ reduced order dynamic output feedback control Abstract: This paper withaffected the problem of H∞ reduced order dynamic output feedback control for discrete-time lineardeals systems by time-varying parameters with polynomial dependency and Abstract: This paper deals with the problem of H∞ reduced order dynamic output feedback control for discrete-timeterms. linearThe systems affected by time-varying parameters with polynomial dependency and norm-bounded main motivation comes from recent discretization methods for uncertain for discrete-time linear systems affected by time-varying parameters with polynomial dependency and norm-bounded terms. polynomially The main motivation comes from discretized recent discretization methods for uncertain systems, that produce parameter-dependent systems of arbitrary degree with norm-bounded terms. polynomially The main motivation comes from discretized recent discretization for uncertain systems, that produce parameter-dependent ofmethods arbitrary degree with norm-bounded terms. The design conditions are provideddiscretized in terms ofsystems sufficient parameter-dependent systems, that produce polynomially parameter-dependent systems of arbitrary degree with norm-bounded The with design conditions arebeing provided in terms of sufficient parameter-dependent LMI conditionsterms. combined scalar searches, capable to synthesize robust or gain-scheduled norm-bounded terms. The with design conditions arebeing provided in terms of sufficient parameter-dependent LMI conditions searches, capable to synthesize or gain-scheduled controllers. The combined approach iswith alsoscalar particularized handle the popular class ofrobust time-varying polytopic LMI conditions combined scalar searches, to being capable to synthesize robust or gain-scheduled controllers. The approach is also particularized to handle the popular class of time-varying polytopic systems, having as novelty no requirement of special treatment for the output measured matrix. controllers. The approach is also particularized to handle the popular class of time-varying polytopic systems, having as are novelty no requirement of potentialities special treatment for the output measured matrix. Numerical examples provided to illustrate the of the approach to cope with discretized systems, having as novelty no requirement of special treatment for the output measured matrix. Numerical examples are provided torelaxations illustrate thewhen potentialities of with the approach to cope with discretized systems and the efficiency of the compared the existing methods for gainNumerical examples are provided torelaxations illustrate thewhen potentialities of with the approach to cope with discretized systems and the efficiency of the compared the existing methods for gainscheduled or robust stabilization of polytopic time-invariant and time-varying systems. systems and the efficiency of the relaxations when compared with the existing methods for gainscheduled or robust stabilization of polytopic time-invariant and time-varying systems. scheduled or robust stabilization of polytopic time-invariant time-varying systems. © 2017, IFAC (International Federation of Automatic Control)and Hosting by Elsevier Ltd. All rights reserved. Keywords: LPV discrete-time systems, H∞ output-feedback control, gain-scheduled control, LMI Keywords: LPV discrete-time systems, H∞ output-feedback control, gain-scheduled control, LMI relaxations. Keywords: LPV discrete-time systems, H∞ output-feedback control, gain-scheduled control, LMI relaxations. relaxations. are concerned with static output-feedback controller design for 1. INTRODUCTION 1. INTRODUCTION are concerned with static output-feedback for discrete-time (Du and Yang, 2008; controller Braga et design al., 2015; are concernedLPV with static output-feedback for 1. INTRODUCTION LPV (Du or and Yang, 2008; controller Braga(LTI) et design al.,systems 2015; De Caigny et al., 2010) linear time-invariant Linear parameter-varying (LPV) models comprise an important discrete-time discrete-time LPV (Du and Yang, 2008; Braga et al., 2015; Caigny et al.,2010; 2010) or linear time-invariant systems Linear models important (Agulhari et al., 2015). When(LTI) it is possible class ofparameter-varying systems that can (LPV) describe linearcomprise systemsan by De De Caignyetetal., al., 2010)Chang or linear time-invariant (LTI) systems Linear parameter-varying (LPV) models comprise anaffected important (Agulhari et al., 2010; Chang et al., 2015). When it is possible class of systems that can describe linear systems affected by to read or estimate the uncertain time-varying parameters, the time-varying parameters and also nonlinear systemsaffected in terms by of (Agulhari et al., 2010; Chang et al., 2015). When it is possible class of systems that can describe linear systems to read or estimate the uncertain time-varying parameters, the parameters andasalso nonlinear systems termsand of controller can be scheduled by the values of the parameters, atime-varying family of linear models, it has been shown in inRugh read or estimate the uncertain time-varying parameters, the parameters andasalso nonlinear systems inRugh termsand of to controller can be scheduled by the values of the parameters, atime-varying family of linear models, it has been shown in potentiallycan improving the results respectofto the the robust case. Shamma class of can shown model, in forRugh example, be scheduled by with the values parameters, aShamma family (2000). of linearThis models, as systems it has been and controller potentially improving theisresults respect to theand robust (2000). This classas of systems can model,mobile for example, This class of controllers calledwith gain-scheduled has case. been mechanical systems such the nonholomonic robot potentially improving the results with respect to the robust case. Shamma (2000). Thissuch classas of the systems can model,mobile for example, class ofapplied controllers is called gain-scheduled and has2013; been mechanical systems nonholomonic robot This studied and for many years (Sato and Peaucelle, studied in Huang et al. (2014) and missile autopilots designed class of controllers is called gain-scheduled and has been mechanical systems such as the nonholomonic mobile robot This studied and applied for many years (Sato and Peaucelle, 2013; studied in Huang et al. (2014) and missile autopilots designed and Shamma, Mohammadpour andPeaucelle, Scherer, 2012). in WhiteinetHuang al. (2007). analysis and design Rugh studied and applied 2000; for many years (Sato and 2013; studied et al.Concerning (2014) andstability missile autopilots designed in White et al. (2007). Concerning stability analysisundoubtedly and design Rugh and Shamma, 2000; Mohammadpour and Scherer, 2012). methods, possibly including performance criteria, Rugh and Shamma, 2000; Mohammadpour and Scherer, 2012). The purpose of this paper is to provide a synthesis procein White et al. (2007). Concerning stability analysis and design methods, possibly including criteria, undoubtedly the Lyapunov stability theoryperformance is the one that provided the most The of this paper is to provide a synthesis procedure purpose for gain-scheduled stabilizing dynamic output-feedback methods, possibly including criteria, undoubtedly the Lyapunov stability theoryperformance is the one that the most purpose of this paper is to provide a synthesis procegeneral and relevant techniques developed soprovided far for LPV sys- The dure for gain-scheduled stabilizing dynamic output-feedback controllers for linear discrete-time systems affected by timethe Lyapunov stability theory is the one that provided the most general and relevant techniques developed so far for employed LPV sys- dure for gain-scheduled stabilizing dynamic output-feedback tems. In this context, the first results in the literature controllers for linear discrete-time systems affected by timevarying parameters with polynomial dependency of arbitrary general and relevant techniques developed so far for LPV systems. In thisofcontext, the stability first results in can the literature employed controllers for linear discrete-time systems affected by timethe notion quadratic (that be conservative in varying parameters with polynomial dependency of arbitrary degree and norm-bounded uncertainty. This particular structure tems. In thisofcontext, the first results in can the literature employed the notion quadratic (that be conservative in degree varyingand parameters with polynomial dependency of structure arbitrary the general case) and, asstability an attempt improve accuracy and norm-bounded uncertainty. the uncertainty is adopted aiming toThis treatparticular discretized uncerthe notion of quadratic stability (thatto can be conservative in of the general case) and, as an attempt to improve accuracy and degree and norm-bounded uncertainty. This particular structure precision, parameter-dependent Lyapunov functions with affine of the uncertainty is adopted aiming to treat discretized uncertain continuous-time systems, that is a recent and promising the general case) and, as an attempt to improve accuracy and precision, parameter-dependent Lyapunov functions with affine of the uncertainty is adopted aiming to treat discretized uncerand polytopic structure were proposed in, for instance, Scherer tain continuous-time systems, that is a recent and promising field of research. In this context, the proposed conditions are precision, parameter-dependent Lyapunov for functions with affine tain continuous-time systems, that is a recent and promising and polytopic structure were proposed instance, (1996); Apkarian and Adams (1998); in, Apkarian et al. Scherer (2000); field of research. In this context, the proposedand conditions are the first to cope with this kind of uncertainty they are the and polytopic structure were proposed in, for instance, Scherer (1996); Apkarian and Adams (1998);Daafouz Apkarianand et al. (2000); field of research. In this context, the proposedand conditions are Lee (2006); de Souza et al. (2006); Bernussou the first to cope with this kind of uncertainty they are the contribution of the paper. Moreover, the possibility of (1996); Apkarian and Adams (1998);Daafouz Apkarianand et al. (2000); main Lee (2006); de Yang Souza et al. (2006); Bernussou the first to cope with this kind of uncertainty and they are the (2001); Du and (2008), and homogeneous polynomially contribution of the paper. Moreover, the possibility of designing output-feedback controllers (gain-scheduled or roLee (2006); de Yang Souza(2008), et al. (2006); Daafouz and Bernussou main (2001); Du and and homogeneous main contribution of the paper. Moreover, the possibility of parameter-dependent matrices in, for example, Depolynomially Caigny et al. designing output-feedback controllers (gain-scheduled or robust) of arbitrary order (including static controllers), without (2001); Du and Yang (2008), and homogeneous polynomially parameter-dependent matrices in, for example, De Caigny et al. designing output-feedback controllers (gain-scheduled or ro(2010); Agulhari et al. (2010). These methods are less conserbust) of arbitrary order (including static controllers), without usual constraints imposed on the matrix associatedwithout to the parameter-dependent for example, De et al. the (2010);and, Agulhari et al.matrices (2010).in, These methods areCaigny less conserof arbitrary order (including static controllers), vative in general, quadratic stability results as bust) the usual constraints imposed on the as matrix associated to the measured output, can also be viewed a novelty of the paper. (2010);and, Agulhari et al. contain (2010). the These methods are less conservative in general, contain the quadratic stability results as the usual constraints imposed on the as matrix associated to the avative particular case. measured output, can alsoclear be viewed a novelty of method, the paper. To strengthen and make the potential of the and, in general, contain the quadratic stability results as a particular case. measured output, can also be viewed as a novelty of the paper.a To strengthen and make clear the potential of the method, condition to cope with time-varying polytopic systems,a aInparticular case. situations, it may be difficult to measure design many practical To strengthen andto make cleartime-varying the potentialpolytopic of the method, a design cope with systems, In many practical situations, it may be difficult to measure that have extensively investigated in the literature, is also proor to estimate the states of real-time control systems, making design condition condition to cope with time-varying polytopic systems, In many practical situations, it may be difficult to measure that have extensively investigated in the literature, is also proor to estimate states real-time based controlon systems, making vided. Numerical examples for this class of systems show that worthless the the design of of controllers state-feedback have extensively investigated in the literature, is also proor to estimate the states of real-time based controlon systems, making that Numerical examples for thisless class of systemsresults show that worthless the cases, design of employment controllers state-feedback the proposed methods can provide conservative laws. In these the of output-feedback tech- vided. vided. Numerical examples for this class of systems show than that worthless the design of controllers based on state-feedback the proposed methods can provide less conservative results than laws. In these cases, the employment of output-feedback techmore recent methods available in the literature for both niques becomes a more viable alternative due to the easetechand the proposed methods can provide less conservative results than laws. In these cases, the employment of output-feedback the more recent methods available in the literature for both niques becomes a more viableSeveral alternative due into the the literature ease and time-invariant and time-varying systems. The design conditions low cost of implementation. studies more recent methods available in the literature for both niques becomes a more viableSeveral alternative due into the the literature ease and the time-invariant andterms time-varying The nowadays design conditions low cost of implementation. studies are provided in of robustsystems. LMIs that can be time-invariant and time-varying systems. The design conditions low cost of implementation. Several studies in the literature ⋆ Supported by the Brazilian agencies CAPES, CNPq (Proc. 132220/2015-6), are provided in terms of robust LMIs that nowadays can be are provided in terms of robust LMIs that nowadays can be ⋆ Supported by the Brazilian agencies CAPES, CNPq (Proc. 132220/2015-6),

and FAPESP by (Proc. the 2014/22881-1). Brazilian agencies CAPES, CNPq (Proc. 132220/2015-6), ⋆ Supported the 2014/22881-1). Brazilian agencies CAPES, CNPq (Proc. 132220/2015-6), andSupported FAPESP by (Proc. and FAPESP (Proc. 2014/22881-1). Copyright © 2017, 2017 IFAC 8952Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright ©under 2017 responsibility IFAC 8952Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 8952 10.1016/j.ifacol.2017.08.2465

Proceedings of the 20th IFAC World Congress Tábitha E. Rosa et al. / IFAC PapersOnLine 50-1 (2017) 8618–8623 Toulouse, France, July 9-14, 2017

solved by polynomial approximations (relaxations) and robust LMI parsers. Notation: For matrices or vectors, the symbol ′ denotes the transpose. ⋆ represents transposed blocks in a symmetric matrix, He(X) = X + X ′ is used to shorten formulas. The set of natural numbers is denoted by N and the set of vectors (matrices) of order n (n × m) with real numbers is given by Rn (Rn×m ). To state that a symmetric matrix P is positive (negative) definite, it is used P > 0 (P < 0). The symbol ! stands for factorial. 2. PROBLEM STATEMENT Consider the following linear discrete-time system affected by time-varying parameters x(k + 1) =A∆ (α (k))x(k) + B∆ (α (k))u(k) + E∆ (α (k))w(k) z(k) =Cz (α (k))x(k) + Dz (α (k))u(k) + Ez (α (k))w(k) y(k) =Cy (α (k))x(k) + Ey (α (k))w(k) (1) where x(k) ∈ Rnx is the state vector, u(k) ∈ Rm is the control input, w(k) ∈ Rr is the exogenous input, z(k) ∈ R p is the controlled output variable, y(k) ∈ Rq is the measurement output and α (k) = [α1 (k), . . . , αN (k)] is a vector of bounded time-varying parameters. Matrices A∆ (α (k)), B∆ (α (k)) and E∆ (α (k)) are given by A∆ (α (k)) = A(α (k)) + ∆A(α (k)) B∆ (α (k)) = B(α (k)) + ∆B(α (k)) (2) E∆ (α (k)) = E(α (k)) + ∆E(α (k)) where the terms ∆A(α (k)), ∆B(α (k)) and ∆E(α (k)) represent unstructured uncertainties that are limited as |∆A(α (k))| ≤ δA2 I, |∆B(α (k))| ≤ δB2 I, (3) |∆E(α (k))| ≤ δE2 I, where (δA , δB , δe ) are known values. Matrices A(α (k)), B(α (k)), E(α (k)), Cz (α (k)), Dz (α (k)), Ez (α (k)), Cy (α (k)) and Ey (α (k)) are polynomial matrices of a fixed degree on α (k) whose coefficients of the monomials are known. For example, if all matrices are affine (degree one) on α (k), the system fits into the so called LPV representation. Moreover, if the parameters lie in the unit simplex given by � �

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mulated as the search for a static output-feedback gain, denoted by � � Ac (α (k)) Bc (α (k)) ∈ R(m+nc )×(q+nc ) , (5) Θ(α (k)) = Cc (α (k)) Dc (α (k))

for the augmented system ˜ + B˜ ∆ (α (k))u(k) ˜ + E˜ ∆(α (k))w(k) x(k ˜ + 1) = A˜ ∆ (α (k))x(k) ˜ + D˜ z (α (k))u(k) ˜ + E˜ z (α (k))w(k) z˜(k) = C˜z (α (k))x(k) ˜ y(k) ˜ = Cy (α (k))x(k) ˜ + E˜ y (α (k))w(k) (6) ′ ′ with x(k) ˜ = [x(k)′ xc (k)′ ] , u(k) ˜ = [xc (k + 1)′ u(k)′ ] , y(k) ˜ = ′ [xc (k)′ y(k)′ ] , z˜(k) = z(k) and   A˜ ∆ (α (k)) E˜ ∆ (α (k)) B˜ ∆ (α (k))  C˜z (α (k)) E˜z (α (k)) D˜ z (α (k))  = 0 C˜y (α (k)) E˜y (α (k))   A∆ (α (k)) 0 E∆ (α (k)) 0 B∆ (α (k)) 0 0 0 I 0      Cz (α (k)) 0 Ez (α (k)) 0 Dz (α (k))  .   0 I 0 0 0 Cy (α (k)) 0 Ey (α (k)) 0 0 Therefore, the closed-loop matrices are given by � � � � A˜ cl (α (k)) B˜ cl (α (k)) A˜ ∆ (α (k)) E˜∆ (α (k)) = ˜ C˜cl (α (k)) D˜ cl (α (k)) Cz (α (k)) E˜ z (α (k)) � � (7) � � B˜ ∆ (α (k)) ˜ ˜ + ˜ Θ(α (k)) Cy (α (k)) Ey (α (k)) , Dz (α (k))

As a performance criterion, the controller Θ(α (k)) must minimize an upper bound µ for the H∞ norm of the system (7) following the well known definition (see for instance De Caigny et al. (2010)), that establishes that for any input w(k) ∈ ℓ2 , the system output z(k) satisfies ||z(k)||2 < µ ||w(k)||2 , µ > 0, ∀α (k) ∈ Λ, k ≥ 0.

Finally, the following lemma (Zhou and Khargonekar, 1988) is necessary to prove the conditions presented in the next section. Lemma 1. Given a scalar λ > 0 and matrices M and N of compatible dimensions, then MN + N ′ M ′ ≤ λ MM ′ + λ −1 N ′ N. 3. MAIN RESULTS

N

Λ≡

ζ ∈ RN : ∑ ζi = 1, ζi ≥ 0, i = 1, . . . ,N i=1

for all k ≥ 0, then the system is also known as a time-varying polytopic system. The motivation to consider matrices A∆ (α (k)), B∆ (α (k)) and E∆ (α (k)) in the form (2), i.e. that combine polynomial matrices of arbitrary degree plus norm-bounded terms, comes from the discretization of polytopic continuous-time uncertain systems. Using a Taylor series expansion of arbitrary fixed degree, as proposed in Braga et al. (2014), the resulting discretized system can be represented with this particular structure. Although the discretization procedure is not investigated in this paper, this more general representation is adopted. The purpose of this paper is the design of a stabilizing dynamic output-feedback controller with order nc ≤ nx given by � xc (k + 1) = Ac (α (k))xc (k) + Bc (α (k))y(k) C= (4) u(k) = Cc (α (k))xc (k) + Dc (α (k))y(k) According to Mårtensson (1985), the problem of designing a dynamic output-feedback controller of order nc can be refor-

Sufficient parameter-dependent LMI conditions for the synthesis of H∞ dynamic output-feedback gain-scheduled controllers for system (1) are proposed in this section. The main technical contribution of this paper is introduced in the sequence. Theorem 1. If there exist parameter-dependent symmetric matrices P(α (k)) > 0 ∈ R(nx +nc )×(nx +nc ) , parameter-dependent matrices 1 F(α¯ (k)) and G(α¯ (k)) ∈ R(nx +nc )×(nx +nc ) , L(α (k)) ∈ R(m+nc )×(q+nc) and S(α (k)) ∈ R(q+nc )×(q+nc ) , scalar variables µ , λA , λB and λE , and given scalar parameters γ1 , γ2 , γ3 , ε and ξ such that Q + C ′ S ′ + S C < 0, (8) ∀α (k) ∈ Λ, where Q is given in (9) with � ˜ α (k))L(α (k))Cy (α (k)) Γ11 = He ξ B( � ˜ α (k))F (α¯ (k)) − P(α (k + 1)) + ξ A( + λAδA2 I + λBδB2 I + λE δE2 I ˜ α (k))G(α¯ (k)))′ Γ21 = − ξ F(α¯ (k)) + ε (A( ˜ α (k))L(α (k))Cy (α (k)))′ + ε (B(

1

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� � The parameter vector α¯ (k) represents α¯ (k) = α (k), α (k + 1) .

Proceedings of the 20th IFAC World Congress 8620 Tábitha E. Rosa et al. / IFAC PapersOnLine 50-1 (2017) 8618–8623 Toulouse, France, July 9-14, 2017

 Γ11 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ Γ22 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆  Γ21    2I Γ Γ − µ ⋆ ⋆ ⋆ ⋆ ⋆ 31 32    0 Γ43 −I ⋆ ⋆ ⋆ ⋆  Γ41  Q=  L(α (k))′ B( ˜ α (k))′ 0 L(α (k))′ Dz (α (k))′ 0 0 ⋆ ⋆ ⋆     ⋆  ξ F(α¯ (k)) ε G(α¯ (k)) 0 0 0 −λA I ⋆   ξ L(α (k))Cy (α (k)) ε L(α (k))Cy (α (k)) 0 L(α (k))Ey (α (k)) L(α (k)) 0 −λB I ⋆ 0 0 0 I 0 0 0 − λE I 

    ¯ ′ ξ (SCy − Cy F) γ1Cy ′ ′ ′ ¯ ′  ε (SCy − Cy G) γ2Cy      Cˆ =  0  , Sˆ =  0   (SE − E )′   0  y y γ3 I S′ The inequality in (10) can be rewritten as Qˆ + Cˆ′ X B + B ′ X ′ Cˆ < 0, with X = S and � � ¯ 0 (Ey − S−1Ey ) I ¯ ε (Cy − S−1Cy G) B = ξ (Cy − S−1Cy F)

Γ31 = ξ Cz (α (k))F (α¯ (k)) + ξ Dz (α (k))L(α (k))Cy (α (k)) ˜ α (k))′ + (B( ˜ α (k))L(α (k))Ey (α (k)))′ Γ41 = E( Γ22 = − ε (G(α¯ (k)) + G(α¯ (k))′ ) + P(α (k)) Γ32 = ε Cz (α (k))G(α¯ (k)) + ε Dz (α (k))L(α (k))Cy (α (k)) Γ43 = Ez (α (k))′ + (Dz (α (k))L(α (k))Ey (α (k)))′ and C and S are given, respectively, by C = [γ1Cy (α (k)) γ2Cy (α (k)) 0 0 γ3 I 0 0 0] ,   ξ (S(α (k))Cy (α (k)) − Cy (α (k))F (α¯ (k)))′ ′  ε (S(α (k))Cy (α (k)) − Cy (α (k))G(α¯ (k)))    0   ′   (S( α (k))E ( α (k)) − E ( α (k))) y y  , S = ′  S( α (k))     0   0

′

Adopting B ⊥ as follows ′  I 0 0 0 ξ (S−1Cy F¯ − Cy )′ −1 ′ 0 I 0 0 ε (S Cy G¯ − Cy )   0 0 I 0 0 −1 ′ 0 0 0 I (S Ey − Ey )

0

then Θ(α (k)) = L(α (k))S(α (k))−1 is a dynamic output feedback stabilizing gain-scheduled controller and µ is an H∞ guaranteed cost for system (7). Proof. To save space, the dependency on α is be omitted, P(α (k + 1)) is denoted by P+ and matrices G(α¯ (k)) and ¯ The first step relies F(α¯ (k)) are represented as G¯ and F. on recovering the LMIs that treat the original matrices of the system (A˜ ∆ , B˜ ∆ , E˜ ∆ ) which contain the norm bounded uncertainties. Knowing the relation given in (3) and using Lemma 1 sequentially with the following choices � � M ′ = ∆E˜ ′ 0 0 0 0 0 0 , N = [0 0 0 I 0 0 0] , and λ = λE , then � � M ′ = ∆B˜ ′ 0 0 0 0 0 , λ = λB , N = [ξ LCy ε LCy 0 LEy L 0] , and finally � � � � M ′ = ∆A˜ ′ 0 0 0 0 , N = ξ F¯ ε G¯ 0 0 0 ,

(9)

λ = λA , and, next, by applying the Schur’s complement, one obtains that if (8) holds, then the following statement is also verified (10) Qˆ + Cˆ′ Sˆ′ + SˆCˆ < 0 for   ⋆ ⋆ ⋆ ⋆ ϒ11 ′ ¯ ¯ ϒ21 −ε (G + G ) + P ⋆ ⋆ ⋆    Qˆ = ξ Cz F¯ + ξ Dz LCy ε Cz G¯ + ε Dz LCy − µ 2 I ⋆ ⋆  E˜ ′ + (B˜ LE )′ 0 ϒ43 −I ⋆ y ∆ ∆ ′ ′ ˜ L B∆ 0 L′ D′z 0 0 with � � ϒ11 = He ξ B˜ ∆ LCy + ξ A˜ ∆F¯ − P+ ¯ ′ + ε (B˜ ∆ LCy )′ ϒ21 = − ξ F¯ + ε (A˜ ∆ G) ′ ′ ϒ43 = Ez + (Dz LEy ) and

′

one has, by the Projection Lemma, that (8) implies B ⊥ QB ⊥ < 0, that is,   ˜   ¯ ′ ′   + −P ⋆ ⋆ ⋆ ξF Acl ⋆  −I   ε G¯ ′    0 P ⋆ (11)  0 0 −µ 2 I ⋆  + He  ˜   0   < 0 Ccl 0 0 B˜ ′cl 0 D˜ ′cl −I and A˜ cl , B˜ cl , C˜cl , D˜ cl are given in (7). Multiplying (11) on the left by R ′ and on the right by   I 0 0 ′ ′ A˜ C˜ 0 R =  cl cl  < 0 0 I 0 0 0 I one has   ⋆ ⋆ A˜ cl PA˜ ′cl − P+  C˜ ′ PA˜ ′ −µ 2 I + C˜cl PC˜cl′ ⋆  < 0 cl cl ′ −I D˜ ′cl B˜ cl which can be recognized as the Bounded Real Lemma (de Souza et al., 2006, Lemma 3) applied to the system (7). The parameter-dependent inequalities of Theorem 1 are only linear with respect to the optimization variables if the scalars γi , ε and ξ are given, and a discussion about this subject is given in Section 4. The main feature associated to Theorem 1 is the treatment of the output matrix Cy (α (k)), that does not require any special structure or constraint. Note that the first LMI based output feedback methods required that this matrix was constant, parameter-independent and constrained to the form Cy (α (k)) = [I 0] (Peres et al., 1994; Geromel et al., 1996). More recent methods relieved these constrains, but in general still requiring similarity transformations (Dong and Yang, 2008, 2013). Another interesting feature presented in the design conditions is the fact that the slack variables F(α (k)) and G(α (k)) are parameter-dependent, what is unusual in synthesis conditions for the design of robust gains. Aiming to treat the popular class of time-varying polytopic systems (affine dependency on the parameters) that are free of norm-bounded terms, i.e., ∆A(α (k)) = ∆B(α (k)) = ∆E(α (k)) =

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0, the following corollary presents an adaptation of Theorem 1 to cope with this particular scenario that has been extensively investigated in the literature. Corollary 1. If there exist parameter-dependent symmetric matrices P(α (k)) > 0 ∈ R(nx +nc )×(nx +nc ) , parameter-dependent matrices F(α¯ (k)) and G(α¯ (k))∈R(nx +nc )×(nx +nc ) , L(α (k)) ∈ R(m+nc )×(q+nc ) and S(α (k)) ∈ R(q+nc )×(q+nc ) , a scalar variable µ , and given scalar parameters γ1 , γ2 , γ3 , ε and ξ , such that (10) holds, then Θ(α (k)) = L(α (k))S(α (k))−1 is a dynamic outputfeedback stabilizing gain-scheduled controller and µ is an H∞ guaranteed cost for system (7), with ∆A(α (k)) = ∆B(α (k)) = ∆E(α (k)) = 0. As a by-product, the condition of Theorem 1 (as well Corollary 1) can be straightforwardly adapted to provide a static state-feedback gain-scheduled controller (nc = 0), as shown in the following corollary. Corollary 2. If there exist parameter-dependent symmetric matrices P(α (k)) > 0 ∈ Rnx ×nx , parameter-dependent matrices F(α¯ (k)) and G(α¯ (k)) ∈ Rnx ×nx , L(α (k)) ∈ Rm×nx and S(α (k)) ∈ Rnx ×nx , scalar variables µ , λA , λB and λE , and given scalar parameters γ1 , γ2 , γ3 , ε and ξ , Cy (α (k)) = I and Ey (α (k)) = 0, such that (8) holds, then Θ(α (k)) = L(α (k))S(α (k))−1 is a state-feedback stabilizing gain-scheduled controller and µ is an H∞ guaranteed cost for system (7). Theorem 1 can be also adapted to synthesize robust dynamic output-feedback controllers for uncertain LTI systems, as shown in the next remark. Remark 1. If system (1) is LTI, then the design of a stabilizing robust (parameter-independent) controller Θ is obtained by solving (8), omitting the time-dependence of α (k). The resulting conditions are simpler to be solved, and a stabilizing controller guarantees that the closed-loop eigenvalues are inside the unit circle. 4. FINITE DIMENSIONAL TESTS The time-dependence associated to all matrices in Theorem 1 can be dropped since it is assumed that α (k) ∈ Λ for all k ≥ 0. At this point, an important observation concerns matrix P(α (k + 1)) since α (k + 1) and α (k) may depend on each other (bounded rate of variation) or not (arbitrary fast variation) (Oliveira and Peres, 2009). The latter case is considered in the experiments presented next (considering α (k + 1) = β ∈ Λ), but the bounded case could be handled as well using the methodology proposed in De Caigny et al. (2010). After theses considerations, the design conditions proposed in Section 3 are not programmable yet, i.e. as parameter-dependent (or robust) LMIs. This class of infinite dimensional optimization problem is hard to solve. However, polynomial approximations became an effective tool to address the problem in terms of the so called relaxations. By fixing the optimization variables as polynomials (more precisely, homogeneous polynomials) of fixed degree, the positivity (or negativity) of the resulting polynomial inequalities can be checked in terms of a finite set of LMIs using, for instance, Pólya’s Theorem based methods. Recently, such set of LMIs could be automatically obtained by using the ROLMIP (Robust LMI Parser) toolbox (Agulhari et al., 2012), that works jointly with Yalmip (Löfberg, 2004). Regarding the choice of the polynomial degrees for the optimization variables, some remarks are important. The variables L(α (k)) and S(α (k)) define the structure of the controller, and if a robust controller (parameter-independent) is desired, then

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the associated degrees must be zero. If at least one of the degrees is not zero, then the synthesized controller is gainscheduled and the vector of parameters α (k) must be available on-line (measured or estimated). The degrees associated with the other variables only influence the conservativeness of the solutions and, as a general rule, higher degrees produce better solutions at the price of a larger computational effort. To perform a fair comparison with the other methods from the literature, these variables are kept with degree one. As seen in Oliveira and Peres (2007), the number of monomials N (g) of homogeneous polynomials with N parameters and degree g is given by 2 N (g) = (N + g − 1)!/(g!(N − 1)!). Therefore, the number of scalar variables (V (g)) in the Theorem 1 can be calculated by � � � nx + nc + 1 V (g) = N (g) (nx + nc ) 2N (g)(nx + nc ) + 2 � +(q + nc)(m + q + 2nc) + 1.

As a final comment, note that the proposed conditions require the scalar parameters γi , ε and ξ to be given, otherwise the conditions are bilinear matrix inequalities (BMIs). This paper does not investigate how to perform a search in these parameters. Instead, a set of values already tested in Vieira et al. (2015), that employs a similar approach to investigate state-feedback control for LTI systems, is used. The set is:

γ1 = γ2 = 0,γ3 = −105, ε = 1, ξ ∈ {−0.9, − 0.8, . . .,0.8,0.9}. (12) 5. NUMERICAL EXAMPLES The conditions proposed in this paper were implemented in M ATLAB (R2013b) using the parser ROLMIP (Agulhari et al., 2012), the parser Yalmip (Löfberg, 2004) and the solvers SeDuMi 1.3 (Sturm, 1999) and Mosek (ApS, 2015), the latter being used only in Example 5.1. Computer specification: Ubuntu Linux, Intel Core i7-4770 (3.40 GHz), 8.0 GB RAM. 5.1 Example 1 - Polynomial LPV System Consider the LPV continuous-time system that represents a linearized dynamic equation of a VTOL helicopter presented in Keel et al. (1988) (where more details about the physical description of the dynamic equations can be found) with the following matrices:   −0.0366 0.0271 0.0188 −0.4555  0.0482 −1.0100 0.0024 −4.0208 Ac (α (t)) =  , 0.1002 p(t) −0.7070 1.4200  0 0 1 0     (13) 0.4422 0.1761 0.05  3.5446 −7.5922 0.01 Bc (α (t)) = −5.5200 4.4900 , Ec (α (t)) =  0 , 0 0 0

where p(t) is a parameter that varies arbitrarily inside the interval 0.3681 ± 0.05. The discrete-time polynomial matrices A∆ (α (k)), B∆ (α (k)), and E∆ (α (k)), can be computed from (13) through the discretization method proposed in Braga et al. (2014), considering, for instance, a Taylor series expansion of degree ℓ = 2 for a sampling period of T = 0.01s. For outputfeedback control purposes, consider also 2 The value can be slightly different in Corollaries 1 and 2 or when the gain is robust (L(α ) and S(α ) of degree zero).

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   1 00 0 1 0 | 0.1 Cz = , [Dz |Ez ] = , Ey = [0.1 0.05 0 0]′ , 0 10 0 0 1 | 0.2

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Cy (α (k)) = 0.5I α1 (k) + I α2 (k), where the uncertainties in the output matrix can represent, for example, a failure in the measuring sensor.

60

50

The aim of this example is to apply Theorem 1 in order to obtain stabilizing static output-feedback controllers assuring an upper bound to the H∞ norm of the closed-loop system. For this project, the robust stabilization case (L(α (k)) = L and S(α (k)) = S) and the affine scheduled control (L(α (k)) and S(α (k)) with degree 1 of parameter dependence) are considered. Additionally, the flexibility of the method associated with the scalar parameter search is evaluated by employing Theorem 1 with the set of scalars presented in (12) or fixing the value of ξ = 0 (no search performed). Table 1 summarizes the results obtained for the H∞ guaranteed costs associated with the designed static output-feedback controllers. The best robust controller obtained through the scalar searches from the set (12) was acquired with ξ = −0.9 and is given by (truncated with four decimal digits)   −1.4574 −0.6769 0.4679 0.7212 Θ= . −0.5063 0.8386 −0.2711 −1.1103 Note that, as expected, the use of gain-scheduled instead of robust controllers combined with the search in the scalar variables, allows achieving less conservative results for this example. Table 1. H∞ guaranteed costs associated to the controllers designed by Theorem 1 for Example 1. Scalars

(12) with ξ = 0

(12)

Robust Gain-scheduled

8.1504 6.6586

1.8187 0.9765

5.2 Example 2 - Polytopic LPV System The system to be investigated can be found in De Caigny et al. (2009), consisting in a polytopic time-varying system whose vertices of the system matrices are given by     1 0 −2 | 0 0 −1 0 |0 [A1 |A2 ] = η 2 −1 1 | 1 −1 0 , [E1 |E2 ] = 1 | 0 , −1 1 0 | 0 −2 −1 0 |1     1 1 0 0 Ezi = Dzi = 0, i = 1, 2 [Bi ] = 0 ,Cyi = 0 1 0 , Czi = [1 1 1] , 0 The aim of this example is to compare the conditions of Corollary 1 (C1), Theorem 8 from De Caigny et al. (2010) (dCCOPS), and Theorem 4 from Du and Yang (2008) (DY). The search in the scalar variables is performed using (12). The example is reproduced for the synthesis of H∞ robust (C1rob , dCCOPSrob , DY) and parameter-dependent (C1 and dCCOPS) static output-feedback controllers with degree 1 of parameter dependence. Figure 1 shows the H∞ guaranteed costs obtained by the techniques for each η belonging to the interval [0.43, 0.445]. Note that, as expected, less conservative results are obtained employing gain-scheduled controllers. Besides, observe that in both cases (robust and gain-scheduled), Corollary 1 yielded the lowest guaranteed costs.

dCCOPS dCCOPSrob DY C1rob C1

µ 40 30

20

10 0.43

0.435

0.44

η

0.445

Fig. 1. H∞ guaranteed costs associated with parameterdependent or robust (rob) static output-feedback controllers computed by the methods C1, dCCOPS, and DY for Example 2. 5.3 Example 3 - Polytopic LTI System Consider the LTI polytopic discrete-time system given in Example 3 of Chang et al. (2015). Robust static output-feedback controllers are synthesized in order to compare the H∞ performance of Corollary 1 (C1) proposed in this paper applied to LTI systems, Theorem 4 (CPZT 4 ) of Chang et al. (2015), Theorem 1 (Remark 1) of Morais et al. (2013) (MBOP), adapted to obtain an output-feedback gain, quadratic stabilization (QS), and the two-stage procedure proposed in Theorem 1/ Theorem 3 of Agulhari et al. (2010) (AOP). The search in the scalar variables for Corollary 1 is given in (12). For the Theorem 4 of Chang et al. (2015), β = 0.13 and ρ = 0.09 are used (the best result reported in that paper). First, a precisely known output matrix, as given in the original example in Agulhari et al. (2010), is adopted: (14) Cy (α ) = Cy = [0 0 0 0 2] , with the corresponding H∞ guaranteed costs referred as µ1 in Table 2. Secondly, a polytopic output matrix whose vertices are described by Cy1 = Cy , and Cy2 = 0.5Cy , with Cy as given in (14) is considered. The associated H∞ guaranteed costs are referred as µ2 in Table 2, that also shows the computational complexity in terms of the numbers of variables and LMI rows. Table 2. H∞ guaranteed costs (µ1 and µ2 ) obtained for Example 3. Methods

µ1

µ2

LMI rows

Variables

C1 CPZ15T 4 MBOP QS AOP

4.7790 8.4041 8.8669 Infeasible 36.4517

3.8157 13.8880 – – 9.5892

62 49 46 29 52

133 83 73 13 85

Note that for both cases, the best results were obtained employing the conditions of Corollary 1, at the price of an acceptable increase in the computational complexity. 6. CONCLUSION This paper proposed new synthesis conditions to deal with the problem of output-feedback gain-scheduled control of discretetime linear systems subject to polynomial time-varying parameters and norm-bounded uncertainty. Numerical experiments

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showed that the proposed method is more general and can be less conservative than other techniques from the literature in terms of H∞ performance, even in the case of LTI systems. The main reasons for such accomplishment is the employment of parameter-dependent slack variables and an improved treatment for the measured output matrix, that is main issue when dealing with output-feedback using LMI based methods. REFERENCES Agulhari, C.M., Oliveira, R.C.L.F., and Peres, P.L.D. (2010). Robust H∞ static output-feedback design for time-invariant discrete-time polytopic systems from parameter-dependent state-feedback gains. In Proc. 2010 Amer. Control Conf., 4677–4682. Baltimore, MD, USA. Agulhari, C.M., Oliveira, R.C.L.F., and Peres, P.L.D. (2012). Robust LMI parser: A computational package to construct LMI conditions for uncertain systems. In XIX CBA, 2298– 2305. Campina Grande, PB, Brasil. Apkarian, P. and Adams, R.J. (1998). Advanced gainscheduling techniques for uncertain systems. IEEE Trans. Control Syst. Technol., 6(1), 21–32. Apkarian, P., Pellanda, P.C., and Tuan, H.D. (2000). Mixed H2 /H∞ multi-channel linear parameter-varying control in discrete time. Syst. Control Lett., 41(5), 333–346. ApS, M. (2015). The MOSEK optimization software. http: //www.mosek.com. Braga, M.F., Morais, C.F., Tognetti, E.S., Oliveira, R.C.L.F., and Peres, P.L.D. (2014). Discretisation and control of polytopic systems with uncertain sampling rates and networkinduced delays. Int. J. Control, 87(11), 2398–2411. Braga, M.F., Morais, C.F., Tognetti, E.S., Oliveira, R.C., and Peres, P.L. (2015). Discretization and event triggered digital output feedback control of LPV systems. Syst. Control Lett., 86, 54–65. Chang, X.H., Park, J.H., and Zhou, J. (2015). Robust static output feedback H∞ control design for linear systems with polytopic uncertainties. Syst. Control Lett., 85, 23–32. Daafouz, J. and Bernussou, J. (2001). Parameter dependent Lyapunov functions for discrete time systems with time varying parameter uncertainties. Syst. Control Lett., 43(5), 355–359. De Caigny, J., Camino, J.F., Oliveira, R.C.L.F., Peres, P.L.D., and Swevers, J. (2009). Gain-scheduled H∞ -control for discrete-time polytopic LPV systems using homogeneous polynomially parameter-dependent Lyapunov functions. In Proc. 6th IFAC Symp. Robust Control Design, 19–24. Haifa, Israel. De Caigny, J., Camino, J.F., Oliveira, R.C.L.F., Peres, P.L.D., and Swevers, J. (2010). Gain-scheduled H2 and H∞ control of discrete-time polytopic time-varying systems. IET Control Theory & Appl., 4(3), 362–380. de Souza, C.E., Barbosa, K.A., and Trofino, A. (2006). Robust H∞ filtering for discrete-time linear systems with uncertain time-varying parameters. IEEE Trans. Signal Process., 54(6), 2110–2118. Dong, J. and Yang, G.H. (2008). Robust static output feedback control for linear discrete-time systems with time-varying uncertainties. Syst. Control Lett., 57(2), 123–131. Dong, J. and Yang, G.H. (2013). Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties. Automatica, 49(6), 1821–1829. Du, X. and Yang, G.H. (2008). LMI conditions for H∞ static output feedback control of discrete-time systems. In Proc. 47th IEEE Conf. Dec. Control, 5450–5455. Cancun, Mexico.

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