- Email: [email protected]
Robust H∞ state-feedback design for discrete-time descriptor systems⁎
9th IFAC Symposium on Robust Control Design 9th IFAC Symposium on Robust Control Design Florianopolis, Brazil, September 3-5, 2018Design 9th IFAC Symposium on Robust Control Available online at www.sciencedirect.com Florianopolis, Brazil, September 3-5, 2018 9th IFAC Symposium on Robust Control Florianopolis, Brazil, September 3-5, 2018Design Florianopolis, Brazil, September 3-5, 2018
ScienceDirect
IFAC PapersOnLine 51-25 (2018) 78–83
Robust H state-feedback design for ∞ Robust H design for ∞ state-feedback Robust H state-feedback design for ∞ discrete-time descriptor systems Robust H state-feedback design for ∞ discrete-time descriptor systems discrete-time descriptor systems discrete-time descriptor systems Carlos Rodr´ıguez ∗∗ , Karina A. Barbosa ∗∗ , Daniel Coutinho ∗∗ ∗∗
Carlos Rodr´ıguez ∗∗∗ , Karina A. Barbosa ∗∗∗ , Daniel Coutinho ∗∗ ∗∗ Carlos Rodr´ıguez ∗ , Karina A. Barbosa ∗ , Daniel Coutinho ∗∗ ∗∗ Carlos Rodr´ ıguez , Karina A. Barbosa , Daniel Coutinho ∗ of Electrical Engineering, Universidad de Santiago de ∗ Department Department of Electrical Engineering, Universidad de Santiago de ∗ ∗ DepartmentChile, of Electrical Engineering, Universidad de Santiago de Av. 3519, Chile ∗ Chile, Av. Ecuador Ecuador 3519, Santiago, Santiago, Chile Department of Electrical Engineering, Universidad de Santiago de Chile, Av. Ecuador 3519, Santiago, Chile E-mail: [email protected], [email protected] E-mail: [email protected], [email protected] ∗∗ Chile, Av. Ecuador 3519, Santiago, Chile Federal de of Automation and Systems, Universidade E-mail: [email protected], [email protected] ∗∗ Department of Automation and Systems, Universidade Federal de ∗∗ Department E-mail: [email protected], [email protected] ∗∗ Santa Catarina (UFSC), Florian´ o polis, SC 88040-900, Brazil of Automation and Systems, Universidade ∗∗ Department Santa Catarina (UFSC), Florian´ o polis, SC 88040-900,Federal Brazil de Department ofE-mail: Automation and Systems, Universidade Santa Catarina (UFSC), Florian´ opolis, SC 88040-900,Federal Brazil de [email protected] E-mail: [email protected] Santa Catarina (UFSC), Florian´ opolis, SC 88040-900, Brazil E-mail: [email protected] E-mail: [email protected] Abstract: This This paper paper deals deals with with the the design of of robust H H∞ controllers for discrete-time descriptor Abstract: ∞ controllers for discrete-time descriptor Abstract: This paper deals with the design design of robust robustinHall for discrete-time descriptor systems with uncertain time-varying parameters state space It assumed ∞ controllers ∞ systems with uncertain time-varying parameters inHall state spaceformatrices. matrices. It is isdescriptor assumed Abstract: This paper deals with the functions design of robust discrete-time ∞ controllers systems with uncertain time-varying parameters in all state spaceparameters matrices. It is bounded assumed that the system matrices are affine of uncertain bounded with that the system matrices are affine functions of uncertain bounded parameters with bounded systems with uncertain time-varying parameters in all state space matrices. It is assumed variations. Firstly, a novel version of the Bound Real Lemma for linear discrete time varying that the system matrices are affine functions of uncertain bounded parameters with bounded variations. Firstly, a novelare version offunctions the Bound Real Lemma for linear discretewith timebounded varying that the system matrices affine of uncertain bounded parameters descriptor systems is presented. Then, the latter result is specialized for state-feedback design variations. Firstly, a novel version of the Bound Real Lemma for linear discrete time varying descriptor systems is presented. Then, the latter result is specialized for state-feedback design variations. Firstly, ais novel version ofconstraints the Lemma for the linear discrete time varying descriptor systems presented. Then, theBound latterinReal result is specialized for state-feedback design in terms of linear matrix inequality order to ensure closed-loop admissibility in terms of linear matrix inequality constraints in order to ensure the closed-loop admissibility descriptor systems isanpresented. Then, the latter result is specialized for state-feedback design -norm for all admissible uncertainty. while guaranteeing upper-bound on the system H in terms of linear matrix inequality constraints in order to ensure the closed-loop admissibility -norm forthe allclosed-loop admissible admissibility uncertainty. while guaranteeing an upper-bound on the system H∞ ∞to in terms of examples linear matrix inequalitytoconstraints in order ensure Numerical are presented illustrate the proposed control design technique. -norm for all admissible uncertainty. while guaranteeing an upper-bound on the system H ∞ ∞ Numerical examplesan are presented to on illustrate the proposed control design technique. for all admissible uncertainty. while guaranteeing the system H∞ -norm Numerical examples areupper-bound presented to illustrate the proposed control design technique. © 2018, IFAC (International Federationtoofillustrate Automaticthe Control) Hosting by Elsevier Ltd. All rights reserved. Numerical examples are presented proposed control design technique. Keywords: control, parameter-dependent parameter-dependent system system matrices, matrices, Keywords: descriptor descriptor systems, systems, H H∞ control, Keywords: descriptor systems, H∞ time-varying discrete-time systems, convex optimization. control, parameter-dependent system matrices, ∞ ∞ convex optimization. time-varying discrete-time systems, parameter-dependent system matrices, Keywords: descriptor systems, H∞ control, time-varying discrete-time systems, convex optimization. time-varying discrete-time systems, convex optimization. 1. in 1. INTRODUCTION INTRODUCTION in terms terms of of LMIs LMIs to to ensure ensure the the system system admissibility admissibility while while guaranteeing a bound on its H norm. The latter result 1. INTRODUCTION in terms of LMIs to ensure the system admissibility while ∞ guaranteeing a bound on its H norm. The latter result ∞ 1. (or INTRODUCTION in terms of LMIs to ensure the system admissibility while was extended in (Chadli and Darouach, 2012) and (Feng guaranteeing a bound on its H latter result ∞ norm. The Descriptor systems singular systems) can naturally ∞ extended ainbound (Chadli and Darouach, 2012)latter and (Feng Descriptor systems (or singular systems) can naturally was guaranteeing on its H norm. The result ∞ was extended in (Chadli and Darouach, 2012) and (Feng and 2013) to more suitable conditions for describe many practical systems with advantage of Descriptor systems (or singular naturally and Yagoubi, Yagoubi, 2013) to derive derive suitable conditions for describe many practical systems systems) with the the can advantage of was extended in (Chadli and more Darouach, 2012) and (Feng Descriptor systems (or singular systems) can naturally the H design problem. It is worth mentioning describe many practical systemsstructure with theby advantage of and Yagoubi, 2013) to derive more suitable conditions for preserving the original model considering ∞ control the H control design problem. It is worth mentioning preserving the original model structure by considering ∞ and Yagoubi, 2013) to derive more suitable conditions for describe many practical systems withstate thebyand advantage of that the aforementioned results deal with uncertainty-free the H control design problem. It is worth mentioning ∞ dynamic and static relations among algebraic preserving the original model structure considering ∞ aforementioned results deal with uncertainty-free the dynamic and static relations among statebyand algebraic that the H control design problem. It is worth mentioning preserving the original model structure considering ∞ discrete-time descriptor that the aforementioned results deal with uncertainty-free variables. Nowadays, many developed for standard dynamic and static relations among state and descriptor systems. systems. variables. Nowadays, many results results developed for algebraic standard discrete-time that the aforementioned results deal with uncertainty-free dynamic and static relations among state and algebraic discrete-time descriptor systems. state-space systems have been extended to deal with variables. Nowadays, many results developed for standard state-space systems many have results been extended tofordeal with discrete-time problem for The robust H descriptor systems. ∞ control variables. Nowadays, developed standard control problem for discrete-time discrete-time descripdescripThe robust H state-space systems have been extended to deal with ∞ descriptor models; see, for the descriptor dynamical dynamical models; see,extended for instance, instance, the texttexttor systems subject to parametric uncertainties been problem for discrete-timehas descripThe robust H ∞ control ∞ state-space systems have been to deal with tor systems subject to parametric uncertainties has been books of Xu and Lam (2006) and Duan (2010). descriptor dynamical models; see, for instance, the textproblem for discrete-time descriprobust H ∞ control tor systems subject to Darouach, parametric uncertainties has been books of Xu and Lam models; (2006) and Duan (2010). the text- The studied in (Chadli and 2014), (Coutinho et al., descriptor dynamical see, for instance, studied in (Chadli and Darouach, 2014), (Coutinho etbeen al., books of Xu and Lam (2006) and Duan (2010). tor systems subject to parametric uncertainties has 2014), (Ji et al., 2007), (Xu and Lam, 2006) and (Belov studied in (Chadli and Darouach, 2014), (Coutinho et al., Since many control systems are affected by external disbooks of Xu and Lam (2006) and Duan (2010). 2014), (Ji et al., 2007), (Xu and Lam, 2006) and (Belov Since many control systems are affected by external dis- studied in (Chadli and Darouach, 2014), (Coutinho et al., and Andrianova, 2016). In particular, Coutinho et 2014), (Ji et al., 2007), (Xu and Lam, 2006) and (Belov turbances, it is often desired to make the controlled sysSince many control systems are affected by external disAndrianova, 2016).(Xu In and particular, Coutinho et al. al. turbances, itcontrol is oftensystems desiredare to make thebycontrolled sys2014), (Ji et considered al., 2007), Lam,descriptor 2006) andsystems (Belov Since many affected external dis- and and Andrianova, 2016). discrete-time In particular, Coutinho et al. (2014) have tem less sensitive to exogenous disturbances by applying turbances, it is often desired to make the controlled sys(2014) have considered discrete-time descriptor systems tem less sensitive to exogenous disturbances by applying Andrianova, 2016). In particular, Coutinho et al. turbances, it is often desired to disturbances make the controlled sys- and (2014) have considered discrete-time descriptor systems with parametric uncertainties and affine theory. In of the H tem less sensitive to exogenous by applying ∞ control with polytopic-type polytopic-type parametric uncertainties and an an affine theory. In the the context context of continuous-time continuous-time the H ∞ control (2014) have considered discrete-time descriptor systems tem less sensitive to exogenous disturbances by applying parameter-dependent Lyapunov function was considered control theory. In the context of continuous-time the H with polytopic-type parametric uncertainties and an affine descriptor systems, the control problem of ∞ ∞ Lyapunovuncertainties function wasand considered descriptor systems, the H HIn∞ control problem of linear linear timetime- parameter-dependent ∞the with polytopic-type parametric an affine theory. context of continuous-time the H∞ control to derive aa solution in terms of bilinear matrix Lyapunov function was inequalities considered invariant systems has been by Wang al. (1998) descriptor systems, the H∞ control problem of et linear time- parameter-dependent ∞addressed to derive solution in terms of bilinear matrix inequalities invariant systems has been addressed by Wang et al. (1998) parameter-dependent Lyapunov function was considered descriptor systems, the H control problem of linear timeto derive a solution in terms of bilinear matrix inequalities (BMIs). Norm-bounded uncertainties were investigated in which proposed and conditions based invariant systemsnecessary has been∞addressed by Wang et al. (1998) Norm-bounded uncertainties were investigated in which proposed and sufficient sufficient conditions based (BMIs). to deriveand a solution in terms ofbased bilinear matrix inequalities invariant systemsnecessary has been addressed by Wang etThen, al. (1998) (BMIs). Norm-bounded uncertainties were investigated in (Chadli Darouach, 2014), on the results derived on two generalized algebraic Riccati equations. Mawhich proposed necessary and sufficient conditions based (Chadli and Darouach, 2014), based on the results derived on two generalized algebraic Riccati equations. Then, Ma(BMIs). Norm-bounded uncertainties were investigated in which proposed necessary and sufficient conditions based in (Chadli and Darouach, 2012), also by means of BMI on two generalized algebraic Riccati equations. Then, Ma(Chadli and Darouach, 2014), based on the results derived subuchi et Xia al. and et al. (Chadli and Darouach, 2012), also by means of BMI subuchi et al. al. (1997), (1997), Xia et etRiccati al. (2005) (2005) and Zhang Zhang etMaal. in (Chadli and Darouach, 2014), based on the results derived on two generalized algebraic equations. Then, control design Belov Andrianova (Chadli andconditions. Darouach,Recently, 2012), also byand means of BMI (2003) have proposed in terms subuchi et al. (1997), equivalent Xia et al. solutions (2005) and etlinal. in design Recently, Belov Andrianova (2003) have proposed equivalent in Zhang terms of of in (Chadli andconditions. Darouach, 2012), also byand means of BMI subuchi et inequality al. (1997), (LMI) Xia etconstraints. al. solutions (2005) and Zhang etlinal. control control design conditions. Recently, Belov and Andrianova (2016) proposed a novel result to the robust H control ear matrix ∞ (2003) have proposed equivalent solutions in terms of lin(2016) proposed a novel result to the robust ear matrix constraints. ∞ control control design conditions. Recently, Belov and H Andrianova (2003) haveinequality proposed (LMI) equivalent solutions in terms of lin- problem for discrete-time descriptor systems in terms of ear matrix inequality (LMI) constraints. (2016) proposed a novel result to the robust H control ∞ problem for discrete-time descriptor in∞ terms of On the hand, several (2016)constraints proposed abased novel on result toBRL thesystems robust H control ear inequality (LMI)exist constraints. ∞ LMI the provided in (Feng for discrete-time descriptor systems in terms of On matrix the other other hand, there there exist several different different versions versions problem LMI constraints based on the BRL provided in (Feng of the Real Lemma (BRL) for the discrete-time for discrete-time descriptor systems in extended terms of On theBounded other hand, there exist several different versions problem and Yagoubi, 2013). The latter result has been LMI constraints based on the BRL provided in (Feng of the Bounded Real Lemma (BRL) for the discrete-time On theBounded otherHsiung hand, exist several different versions and Yagoubi, 2013). Theonlatter result has been extended of the Real there Lemma (BRL) for theXu discrete-time counterpart. and Lee (1999) and and Yang LMI constraints based the BRL provided in (Feng and Yagoubi, 2013). The latter result has been extended and Andrianova (2017) to counterpart. Hsiung and Lee(BRL) (1999)for and and Yang in ∞ state-feedback of the Bounded Real Lemma theXu discrete-time in Belov Belov and 2013). Andrianova (2017) to H H state-feedback ∞ been counterpart. Hsiung and Leethe (1999) and Xu and Yang (2000) versions of BRL by of aa set and Yagoubi, The latter result has extended for LTI discrete-time descriptor with normin Belov and Andrianova (2017) to systems H ∞ (2000) proposed proposed versions of the BRL and by means means of Yang set design ∞ state-feedback counterpart. Hsiung and Lee (1999) Xu and design for LTI discrete-time descriptor systems with normof nonlinear matrix inequalities. Although it plays a key in Belov and Andrianova (2017) to H state-feedback (2000) proposed versions of the BRL by means of a set ∞ bounded parametric uncertainties in all system matrices. design for LTI discrete-time descriptor systems with normof nonlinear matrix inequalities. Although it plays a key (2000) proposed versions of the BRL by means of a set bounded parametric uncertainties in all system matrices. role on theoretically solving the H the latter forparametric LTI discrete-time descriptor systems with normof nonlinear matrix inequalities. Although it plays a key design ∞ problem, bounded uncertainties in all system matrices. role on theoretically solving the H problem, the latter ∞ of matrix inequalities. itZhang plays a key However, there are few results in the literature addressing rolenonlinear on are theoretically solving the Although H∞ problem, the latter results hard to be numerically et al. bounded parametric uncertainties in all system matrices. ∞solved. However, there are few results in the literature addressing results are hard to be numerically Zhang et al. role on have theoretically solving the H∞solved. problem, theof latter control problem for linear varying the H there are few results in the literature addressing (2008) introduced a version BRL results are hard to be numerically solved. Zhang al. However, the H∞ problem for discrete discrete linear time time varying (2008) have introduced a more more tractable tractable version of et BRL ∞ control However, there are few results in the literature addressing results are hard to be numerically solved. Zhang al. (DLTV) descriptor systems considering robust feedback the H problem for discrete linear time varying ∞ control (2008) have introduced a more tractable version of et BRL ∞ (DLTV) descriptor systems considering robust feedback This work the H∞ control problem for considering discrete linear time varying (2008) havehas introduced a more tractable version of BRL (DLTV) been supported by “Fondo Nacional de Desarrollo descriptor systems robust feedback controllers. Motivated by this fact, the present work deals This work has been supported by “Fondo Nacional de Desarrollo controllers. Motivated by thisconsidering fact, the present work deals (DLTV) descriptor systems robust feedback Cient´ ıfico y Tecnol´ ogico” - Fondecyt, Chile,Nacional under grant 1151199, This work has been supported by “Fondo de Desarrollo with the H control problem for a class of DLTV uncontrollers. Motivated by this fact, the present work ∞ control problem for a class of DLTVdeals Cient´ıfico y Tecnol´ ogico” - Fondecyt, Chile, under grant 1151199, with the H un∞ This work has been supported by “Fondo de Desarrollo controllers. Motivated by this fact, the present work deals and D. Coutinho has been partially supported by CNPq - Brazil, Cient´ ıfico y Tecnol´ ogico” - Fondecyt, Chile,Nacional under grant 1151199, certain descriptor systems by considering a robust statewith the H problem for a class of DLTV un∞ control and D. Coutinho has been partially supported by CNPq - Brazil, ∞ certain descriptor systems by considering a robust stateCient´ ıfico y Tecnol´ o gico” Fondecyt, Chile, under grant 1151199, underD.grant 304351/2014-8. with thedescriptor H∞ control problem for a classaofrobust DLTV unand Coutinho has been partially supported by CNPq - Brazil, certain systems by considering stateunder grant 304351/2014-8. and D.grant Coutinho has been partially supported by CNPq - Brazil, under 304351/2014-8. certain descriptor systems by considering a robust state-
under grant© 304351/2014-8. 2405-8963 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2018 IFAC 114 Copyright 2018 IFAC 114 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 114 10.1016/j.ifacol.2018.11.085 Copyright © 2018 IFAC 114
IFAC ROCOND 2018 Florianopolis, Brazil, September 3-5, 2018 Carlos Rodríguez et al. / IFAC PapersOnLine 51-25 (2018) 78–83
feedback control law. The proposed result is inspired on a dual version of the BRL proposed in (Barbosa et al., 2017) which has introduced necessary and sufficient LMIbased conditions to the admissibility analysis of DLTV descriptor systems based on time-varying Lyapunov functions. In particular, a necessary and sufficient condition is firstly introduced for the existence of a static statefeedback controller such that the resulting closed-loop system is admissible with a guaranteed upper-bound on the H∞ performance index. Then, the proposed result is tailored for H∞ control design in terms of a set of LMI constraints. Numerical examples are provided to illustrate the approach showing the effectiveness of the proposed H∞ control design method for DLTV uncertain descriptor systems. Notation. Zi is the set of integers equal to or larger than i, C is the set of complex numbers, R+ is the set of positive real numbers, Rn is the n-dimensional Euclidean space, Rm×n is the set of m × n real matrices, In is the n×n identity matrix, 0n and 0m×n are the n×n and m×n matrices of zeros, respectively, and diag{· · · } stands for a block-diagonal matrix. For a real matrix S, S T denotes its transpose, He{S} stands for S+S T and S > 0 means that S is symmetric and positive-definite. For a symmetric block matrix, the symbol denotes the transpose of the blocks outside the main diagonal block. 2 denotes the space of square summable vector sequences over [0, ∞) with norm ·2 . For a given convex bounded polyhedral domain B, V(B) denotes the set of all the vertices of B. The one-step delayed and the one-step ahead shift of the time argument k will be denoted by k− := k−1 and k+ := k+1, respectively. 2. DESCRIPTOR SYSTEMS Descriptor systems may exhibit a more complex behavior than ordinary dynamical system such as the existence of solution and impulsive modes (or causality in the discrete-time case). In this section, some key concepts and two canonical representations related to DLTV descriptor systems, such as admissibility and stability, are recalled as proposed in (Barbosa et al., 2018). To this end, consider the following DLTV descriptor systems: E(k)x(k+1) = A(k)x(k)+B(k)w(k), x(0)=x0 ,
(1)
where x(k) ∈ Rn is the state, w(k) ∈ Rnw is the disturbance input vector assumed to belong to 2 , E(k), A(k) and B(k) are bounded matrix functions with appropriate dimensions, and x0 is a consistent initial condition. Furthermore, E(k) might be a singular matrix but it is assumed that rank{E(k)} = r ≤ n, ∀ k ∈ Z0 . Definition 1. Consider the DLTV descriptor system (1) with x0 being a consistent initial condition. Then: (a) The system is said to be regular if for any w(k) ∈ Rnw and x0 there exists a solution x(k) for all k ∈ Z0 and it is unique. (b) The system is said to be causal if it is regular and the solution x(k) for any x0 and w(k) ∈ Rnw is a function of x0 and w(0), . . . , w(k−1), for all k ∈ Z0 . (c) The system is said to be exponentially stable if it is regular for any x0 and w(k) ≡ 0, and there exist real scalars α > 0 and β ∈ (0, 1) such that 115
79
x(k) ≤ α β k x0 , ∀ k ∈ Z1 .
(d) The system is said to be admissible if it is causal and exponentially stable. Let z(k) = C(k)x(k) + D(k)w(k),
(2)
be the output performance vector of the DLTV descriptor systems in (1), where z(k) ∈ Rnz , and C(k) and D(k) are bounded matrix functions with appropriate dimensions. Hence, the H∞ -norm of system (1)-(2) can be characterized by the following definition. Definition 2. Consider the system in (1)-(2). Then the system induced 2 -gain from w(k) to z(k), or simply the system H∞ -norm, is given by: G wz ∞ = sup
w∈2
z(k)2 : w(k) ≡ 0, E(0)x(0) = 0 w(k)2
where G wz stands for the operator from w(k) to z(k).
Now, it is introduced two canonical representations of DLTV descriptor systems tailored for control design which will be instrumental to derive the main result of this paper. Then, consider firstly the following descriptor system: E(k)x(k+1) = A(k)x(k) + B(k)w(k) + Bu (k)u(k) (3) z(k) = C(k)x(k) + D(k)w(k) + Du (k)u(k) (4) where u(k) ∈ Rnu is the control input vector, and Bu (k) and Du (k) are bounded matrix functions. The singular value decomposition (SVD) state-space representation of system (3)-(4) is defined as follows: ¯ ¯ ¯ ¯u (k)u(k) Eξ(k+1) = A(k)ξ(k)+ B(k)w(k) +B (5) ¯ z(k) = C(k)ξ(k)+D(k)w(k) +Du (k)u(k) where ξ(k) = N −1 (k)x(k) and
0 Ir ¯ , E = M (k)E(k)N (k + 1) = 0 0n−r A1 (k) A2 (k) ¯ A(k) = M (k)A(k)N (k) = , A3 (k) A4 (k) B1 (k) ¯ , B(k) = M T(k)B(k) = B2 (k) Bu1 (k) T ¯ Bu (k) = M (k)Bu (k) = , Bu2 (k)
(6)
¯ C(k) = C(k)N (k) = [ C1 (k) C2 (k) ] ,
with M (k), N (k) being nonsingular n × n real matrices for ¯ ¯ ¯u (k) and all k ∈ Z. In addition, the matrices A(k), B(k), B ¯ ¯ C(k) are partitioned accordingly to E. Next, assuming that system (3)-(4) is admissible, it is possible to obtain the following Weierstrass form: u (k)u(k) B(k)w(k) +B Eζ(k+1) = A(k)ζ(k)+ z(k) = C(k)ζ(k)+D(k)w(k) +Du (k)u(k)
−1 (k)x(k) and where ζ(k) = N
(7)
IFAC ROCOND 2018 80 Florianopolis, Brazil, September 3-5, 2018 Carlos Rodríguez et al. / IFAC PapersOnLine 51-25 (2018) 78–83
0 =M (k)E(k)N (k + 1) = Ir E 0 0n−r , 0 Af (k) , A(k) = M (k)A(k)T (k) = 0 In−r Bf (k) T B(k) = M (k)B(k) = A−1 4 (k)B2 (k) Buf (k) u (k) = M T(k)Bu (k) = B A−1 4 (k)Bu2 (k) with
(8)
Lemma 1. (Xie et al. (1993)) The following LTV system x ˜(k + 1) = Af (k)˜ x(k) + Bf (k)w(k) z(k) = Cf (k)˜ x(k) + Df (k)w(k)
C(k) = C(k)T (k) = [ Cf (k) C2 (k) ] ,
Af (k) = A1 (k) − A2 (k)A−1 4 (k)A3 (k),
Bf (k) = B1 (k) − A2 (k)A−1 4 (k)B2 (k),
Buf (k) = Bu1 (k) − A2 (k)A−1 4 (k)Bu2 (k), Cf (k) = C1 (k) −
In this paper, the robust H∞ control problem is addressed for the class of uncertain system as defined in (10). More precisely, an LMI-based result will be proposed in the following section for designing a static state feedback control law u(k) = Kx(k) in order to ensure the robust admissibility of the closed-loop system with a guaranteed upperbound on the system H∞ -norm for all (θ(k), ∆θ(k)) ∈ B. Before ending this section, the following version of BRL for ordinary DTLTV systems is recalled.
(9)
C2 (k)A−1 4 (k)A3 (k).
(k) and N (k) are nonsingular n × n real Moreover, M u (k) and C(k) matrices for all k ∈ Z, and A(k), B(k), B are partitioned accordingly to E. 3. PROBLEM STATEMENT
Consider the following uncertain discrete-time descriptor systems: E(θ(k))x(k+1) = A(θ(k))x(k)+B(θ(k))w(k) +Bu (θ(k))u(k) (10) z(k) = C(θ(k))x(k)+D(θ(k))w(k) +Du (θ(k))u(k) where x(k) ∈ Rn is the state, w(k) ∈ Rnw is the disturbance input belonging to 2 , u(k) ∈ Rnu is the control input, z(k) ∈ Rnz is the performance output, and θ(k) = [ θ1 (k), . . . , θp (k) ] T ∈ Rp is a vector of bounded uncertain time-varying parameters. The matrices E(θ(k)), A(θ(k)), B(θ(k)), Bu (θ(k)), C(θ(k)), D(θ(k)) and Du (θ(k)) are affine functions of θ(k) with appropriate dimension with E(θ(k) allowed to be a singular matrix but satisfying rank{E(θ(k))} = r ≤ n for all k ∈ Z0 .
Further, considering ∆θ(k) := θ(k)−θ(k − 1), it is assumed that the parameters θi (k) and their variations ∆θi (k), for i = 1, . . . , p and k ∈ Z0 , are such that θ i ≤ θi (k) ≤ θi and δ i ≤ ∆θi (k) ≤ δ i , where θ i , θi , δ i and δ i are known extremum values of θi (k) and ∆θi (k), respectively. Moreover, let B be the polytope representing the set of admissible values of (θ, ∆θ) and assume that the admissible values of θ(k) and ∆θ(k) are such that if (θ(k), ∆θ(k)) ∈ B then θ(k −1) also belongs to B. Similarly as in de Souza et al. (2006), a polytope B having this property will be referred to as a consistent polytope. Note that a consistent polytope must include ∆θ = 0 as an admissible value of ∆θ(k). In the sequel, θ ∈ B means that (θ(k), 0) ∈ B and the dependence of θ(k) and ∆θ(k) on k will be often omitted to ease the notation. Next, we introduce the notion of robust admissibility to be considered throughout this paper. Definition 3. System (10) is said to be robustly admissible if it is admissible for all (θ, ∆θ) ∈ B . 116
(11)
is exponentially stable and Gwz ∞ ≤ γ with γ being a given positive scalar, for all k ∈ Z0 , if and only if there exists a bounded matrix function Q(k) > 0 such that: −Q(k + 1) Q(k)AT (k) −Q(k) f < 0, ∀k ∈ Z0 (12) BfT (k) 0 −γI 0 Cf (k)Q(k) Df (k) −γI 4. MAIN RESULTS This section proposes an LMI based approach to the H∞ control problem. Firstly, it is introduced a version BRL for DLTV descriptor system for robust admissibility analysis while guaranteeing a bound on the system H∞ -norm. Then, the analysis result is extended to cope with robust state-feedback synthesis. 4.1 H∞ Performance Analysis The following result provides two equivalent necessary and sufficient conditions to ensure the robust admissibility of the unforced system in (10) while providing a guaranteed upper-bound on the system H∞ -norm. In the next theorem, the admissible values of θ(k) and ∆θ(k) are assumed to belong to a compact set D which is not necessarily a polytope as the set B. Theorem 2. Consider the uncertain descriptor system in (10) with u(k) ≡ 0. Let D be a compact set of admissible (θ, ∆θ), γ be a given positive scalar, and E0 (θ) ∈ Rn×(n−r) be any full column-rank matrix function of θ such that E(θ)E0 (θ) = 0, ∀ θ ∈ D. Assume that Ker{E(θ)} ⊆ Ker{C(θ)}, for all θ ∈ D. Then, system (10) is admissible and its H∞ -norm is smaller than γ, if and only if one of the following conditions holds: (a) There exist bounded matrix functions X(θ) > 0 and R(θ) such that:
Σ1 (θ) X(θ−∆θ)AT (θ) −X(θ−∆θ) < 0, T 0 −γI Bw(θ) C(θ)E0 (θ−∆θ)R(θ) C(θ)X(θ−∆θ) D(θ) −γI ∀ (θ, ∆θ) ∈ D, (13) where Σ1 (θ) = −E(θ)X(θ)E T (θ) + He{A(θ)E0 (θ−∆θ)R(θ)}.
IFAC ROCOND 2018 Florianopolis, Brazil, September 3-5, 2018 Carlos Rodríguez et al. / IFAC PapersOnLine 51-25 (2018) 78–83
(b) There exist bounded matrix functions F (θ), G(θ), H(θ), S(θ) and P (θ) > 0 such that Γ1 (θ) C(θ)F(θ) −γI T T < 0, Γ2 (θ, ∆θ) G (θ)C (θ) Γ3 (θ, ∆θ) −H(θ) 0 0 −P (θ) DT (θ) 0 0 −γI B T (θ) ∀ (θ, ∆θ) ∈ D, (14) where Γ1 (θ) = He{A(θ)F (θ) + H(θ)E T (θ)}, Γ2 (θ, ∆θ) = E0 (θ−∆θ)S(θ) + GT (θ)AT (θ) − F (θ),
Γ3 (θ, ∆θ) = P (θ−∆θ) − He{G(θ)}.
Proof. Firstly, for convenience of notation, the dependence of matrix functions on θ and (θ −∆θ) will be represented as functions of k and k − 1, respectively. That is, let Ω(·) be a matrix function of either θ or ∆θ. Then, Ω(k) = Ω(θ) and Ω(k− ) = Ω(θ − ∆θ). As a result, the uncertain descriptor system in (10), with u(k) ≡ 0, will read as in (1)-(2). Sufficiency of (a): Consider that the descriptor system is in SVD form given by (5) and assumed that X1 (k) X2 (k) ¯ , X(k) := N −1(k + 1)X(k)N −T (k + 1) = X2T (k) X4 (k) (15) where the dimensions of the partitions are compatible with ¯ as defined in (6). those of E Thus, pre- and post-multiplying (13) by T
−1
T = diag{M (k), N (k), I, I} and T , respectively, and considering (6) and (15), we obtain that: ¯ 1 (k) Σ X(k T ¯ −) ¯ − )A¯ (k) −X(k (16) < 0, ¯ T(k) B 0 −γI w ¯ 2 (k) ¯ X(k ¯ − ) D(k) −γI Σ C(k) where −1 ¯ ¯ 1(k) =He{A(k)N (k)E0 (k− )R(k)M T (k) Σ ¯ X(k) ¯ E ¯ T (k)}, −E(k)
−1 ¯ ¯ 2 (k) = C(k)N (k)E0 (k− )R(k)M T (k). Σ
Now, it will be shown that if (13) is satisfied then the unforced system (10) is causal. Note that if (13) holds ¯ 1 (k) < 0, then (16) is also satisfied, which implies that Σ for all k ∈ Z0 .
In the light of (6) and similarly as in (Ishihara and Terra, 2003), E0 (k) can be factorized in the following form: (17) E0 (k− ) = N (k) 0r×(n−r) In−r W (k),
where W (k) ∈ R(n−r)×(n−r) is any nonsingular matrix for all k ∈ Z0 . Next, let (18) W (k)R(k)M T (k) = R1 (k) R2 (k) , where R1 (k) ∈ R(n−r)×r and R2 (k) ∈ R(n−r)×(n−r) . ¯ 1 (k) accordingly to X(k), ¯ Partitioning Σ i.e.,
117
81
¯ 12 (k) ¯ (k) Σ ¯ 1 (k) = Σ11 Σ T ¯ (k) Σ ¯ 22 (k) , Σ 12
and considering (6) and (18), it can be easily established that ¯ 22 (k) = A4 R2 + RT AT < 0, ∀ k ∈ Z0 , Σ 2 4 which implies that the matrix A4 (k) is nonsingular for all k ∈ Z0 . Then, from Lemma 2 in (Barbosa et al., 2018), the causality of the unforced system (10) is ensured. Next, it will be shown that (13) also guarantees that the unforced system (10) is exponentially stable and G wz ∞ < γ. To this end, notice since the matrix A4 (k) is nonsingular for all k ∈ Z0 that the unforced system (10) can be represented in the Weierstrass form as given in (7)(8). Hence, similarly as before, the matrix inequality in (16) also holds with M (k), N (k), W (k), R1 (k) and R2 (k) (k), N (k), W (k), R 1 (k) and respectively replaced by M (k) and N (k) are Weierstrass form trans2 (k), where M R formation matrices and W (k) is any nonsingular matrix for all k ∈ Z0 . Now, define the matrix
2 (k) 1 (k) X X −1 −T X(k) := N (k+1)X(k)N (k+1) = T 4 (k) , (19) X2 (k) X where the partitions of X(k) are accordingly to those of as given in (8). Moreover, the assumption Ker{E(k)} ⊆ E Ker{C(k)} implies that C2 (k) = 0 for all k ∈ Z0 in (8). In addition, let the following matrices (k), N −1 (k), I, I}, T T = diag{M T Ia Ia IbT Ia 0 0 I(nz +nw ), I = 0 0 IcT Id IdT Id
Ia = [ Ir 0 ] , Ib = [ 0 Ir ] , Ic = [In−r 0 ] , Id = [ 0 In−r ] . Thus, considering (8) and (17)-(19), and pre- and postmultiplying (16) by TT (k) = IT T and T(k), respectively, the matrix inequality in (13) can be cast as follows: T −T Ψ1 (k) Ψ2 (k) −1 Ψ(k) = T 2 (k) Ψ 3 (k) T < 0, ∀ k ∈ Z0 , (20) Ψ where Ψ(k) denotes the left-hand side of (13), and 1 (k) −X T 1 (k) = X1 (k− )Af −X1 (k− ) , Ψ BT 0 −γI f 1 (k− ) D(k) −γI 0 Cf X T (k− ) B2 (k) Cf (k)X 2 (k) 1 (k) X R 2 2 (k) = Ψ , T (k− ) 0 T (k− )AT −X 0 X 2 f 2 2 (k) + R T (k) X 3 (k− ) R 2 Ψ3 (k) = 3 (k− ) . 3 (k− ) −X X
1 (k) < 0, which from Note that (20) ensures that Ψ Lemma 1 implies that system (11), and also the unforced system (10), is exponentially stable and its H∞ -norm is smaller than γ, for all k ∈ Z0 , with Q(k) = X(k− ).
Necessity of (a): Assume that the unforced system (10) is admissible and Gwz ∞ < γ, ∀ k ∈ Z0 . Consider that the system is in the SVD representation as given in (6)
IFAC ROCOND 2018 82 Florianopolis, Brazil, September 3-5, 2018 Carlos Rodríguez et al. / IFAC PapersOnLine 51-25 (2018) 78–83
and notice that A4 (k) is nonsingular due to the system admissibility. In addition, similarly as before, it is possible to obtain an equivalent representation as in (11) with (9). Hence, Lemma 1 holds since the system is exponentially stable and Gwz ∞ < γ. Also, there exits a Weierstrass representation of system (1)-(2) as given in (7)-(8).
where
(k), N (k), W (k), R 1 (k) and Now, consider the matrices M 2 (k) as defined in the sufficiency of item (a). It can be R readily verified that if (12) is satisfied then Ψ(k) < 0, ∀ k ∈ Z0 in (20), with 2 (k) = 0, X 4 (k) > 0, R 1 (k) = 0, 1 (k) = Q(k+ ), X X ∀ k ∈ Z0 , (21) and R2 (k) being such that
Υ4 (θ)=P (θ−∆θ) − He{G}, with p + θ i Pi P (θ) = P 0
4 (k− )+He{R 2 (k)} < 0, ∀ k ∈ Z1 . X Then, it turns out that (13) holds with the matrices T (k) diag X 1 (k), X 4 (k) N (k), X(k) = N −1(k) 0 R −1 (k), 2 (k) N R(k) = W 1 (k), X 4 (k), and R 2 (k) are as defined in (21). where X
Υ2 (θ)=E0 (θ−∆θ)S + GT AT (θ) + Y T BuT (θ) − µG,
Υ3 (θ)=(C(θ)G+Du (θ)Y )T ,
i=1
p (θi −∆θi )Pi P (θ − ∆θ) = P + 0
(23)
i=1
Then, the closed-loop system of (10) with u(k) = Y G−1 x(k) is robustly admissible and Gwz ∞ < γ, for all (θ, ∆θ) ∈ B. The proof of the above theorem is straightforward from the proof of Theorem 2 by considering G(θ) = G, F (θ) = µG, H(θ) = H and (23), and it will be omitted for brevity. 5. NUMERICAL EXAMPLES
(a) ⇔ (b): This part of the proof follows similar steps as the proof of (Barbosa et al., 2018, Theorem 1) and it will be omitted for brevity. ∇∇∇ Remark 1. Assumption Ker{E(θ)} ⊆ Ker{C(θ)} implies that the performance output variable z(k) will only depend on the dynamic part of the state vector and it has been also considered by Takaba (1998). 4.2 H∞ Control Design
Based on the H∞ analysis result established in Theorem 2(b), a robust state feedback control design approach will be derived in the sequel for ensuring the admissibility of the closed-loop system (10) while guaranteeing an upper-bound γ on the system H∞ -norm. To obtain computationally tractable conditions, it is further assumed that there exists an affine matrix function E0 (θ) ∈ Rn×(n−r) , having full-column rank for all θ in B, such that E(θ)E0 (θ) = 0, ∀ θ ∈ B. In addition, the matrices F, G and H are constrained to be constant and P (θ) to be an affine function of θ. The next theorem presents sufficient conditions in terms of a finite set of linear matrix inequalities (LMIs) to the state feedback design of discrete time descriptor systems subject to time-varying uncertainties in all system matrices. Theorem 3. Consider the descriptor system in (10). Let B be a given polytope of admissible (θ, ∆θ), γ be a given positive scalar, E0 (θ) ∈ Rn×(n−r) be any full columnrank affine matrix function of θ such that E(θ)E0 (θ) = 0, ∀ θ ∈ B, and µ be a given positive scalar. Assume that Ker{E(θ)} ⊆ Ker{C(θ)}, ∀ θ ∈ B. Suppose there exist matrices G, S, H, Y and Pi , i = 0, 1, . . . , p, with appropriate dimensions, satisfying the following LMI: Υ1 (θ) µC(θ)G+µDu (θ)Y −γI Υ3 (θ) Υ4 (θ) Υ2 (θ) < 0, −H 0 0 −P (θ) DT (θ) 0 0 −γI B T (θ) ∀ (θ, ∆θ) ∈ V(B),
Υ1 (θ)=He{µA(θ)G + µBu (θ)Y + HE T (θ)},
(22)
118
Example 1. Consider the uncertainty descriptor system in (10) with the following state-space matrices:
1+0.2θ2 0 0 0 1 0 E(θ) = 0 0 0
0.8(1+αθ1 ) 0 −0.32(1+0.2θ2 ) 1 0 0 A(θ) = 0 1+0.2θ2 −1
0.4 θ1 Bu (θ) = , Bw (θ) = 0.25 ∗ Bu (θ) −1 − θ2
0.1θ1 0.8 0 0.1θ2 C(θ) = , Dw (θ) = Du (θ) = 0 0.01θ2 0 0.1(1+θ1 ) where θ(k) is a time-varying parameters such that |θ(k)| ≤ 1, for all k ∈ Z0 and rank{E(θ(k))} = 2 for all admissible θ(k). In (Barbosa et al., 2018), it has been shown that the system is admissible for α ≤ 0.256. In this example, a robust state feedback controller is designed to ensure the closed-loop admissibility for α = 0.4, while minimizing the system H∞ -norm considering a maximum admissible parameter variation, i.e., |∆θ(k)| ≤ 2, ∀ k ∈ Z0 . Hence, considering E0 (θ) = [ 0 0 1 ]T and applying the optimization problem: min γ subject to (22). (24) µ,G,S,H,Y,P0 ,P1
leads to the following control law u(k) = [ −1.1316 0.0779 0.3674 ] x(k), for µ = 0.2123. The resulting closed-loop system is robustly admissible and Gwz ∞ < 0.6152, for all admissible parameter uncertainty. It is worth mentioning that a line search on µ has been performed in (24) in order to obtain a minimized upper-bound on the system H∞ -norm. Example 2. Consider the descriptor system as in (10) with the following state-space matrices: 1 −1 −6 −1 −1 1 −6 1 0 3+2θ 1 0 0 5+2θ −1 0 , A(θ) = , E(θ) = 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0
IFAC ROCOND 2018 Florianopolis, Brazil, September 3-5, 2018 Carlos Rodríguez et al. / IFAC PapersOnLine 51-25 (2018) 78–83
2 0 1 00 0 0 Bu (θ) = Bw (θ) = , C(θ) = , D(θ) = 1 1 0 00 1 1 where θ(k) is a time-varying parameter such that |θ(k)| ≤ 1.0, for all k ∈ Z0 . Notice that the open-loop system is non-causal for all k ∈ Z0 , and rank{E(θ(k))} = 2 for all admissible θ(k). Thus, in this example, a robust feedback control law is determined to ensure the closed-loop system admissibility while minimizing an upper-bound on the system H∞ -norm considering ∆θ(k) ≤ 1, for all k ∈ Z0 . Note that, since the parameter variation is different from the maximum admissible variation and the matrix E(θ) is uncertain, it is not available in the literature can approach suitable to solve this problem. However, applying the optimization problem in (24) considering µ = 0.50 and (31 + 12θ) 1 1 0 . E0 (θ) = (5 + 2θ) 0 0 −1 we obtain u(k) = [ 0.0716 0.8830 −4.0003 −2.8498 ]x(k), which guarantees the closed-loop admissibility and Gwz ∞ < 4.1074, for all admissible uncertainty. Similarly to Example 1, the latter result has been derived by performing a line search on µ. 6. CONCLUSION This paper has addressed the robust H∞ state-feedback design problem for discrete-time descriptor systems subject to time-varying parameters in all state-space matrices. The paper contribution is two fold: (i) a new version of the BRL for general discrete time-varying descriptor systems is proposed; and (ii) the latter result is specialized for robust state-feedback design guaranteeing the closed-loop admissibility with a prescribed upper-bound on the system H∞ -norm. Two numerical examples have demonstrated the potentials of the proposed approach. REFERENCES Barbosa, K., Coutinho, D., de Souza, C., and Rodrguez, C. (2017). Bounded real lemma for discrete linear timevarying descriptor systems. In 2017 11th Asian Control Conference (ASCC), 1835–1840. Barbosa, K., de Souza, C., and Coutinho, D. (2018). Admissibility analysis of discrete linear time-varying descriptor systems. Automatica, 91, 136–143. Belov, A. and Andrianova, O. (2016). Robust statefeedback H∞ control for discrete-time descriptor systems with norm-bounded parametric uncertainties. Preprint submitted to Automatica, arXiv:1611.01825. Belov, A. and Andrianova, O. (2017). On LMI approach to robust state-feedback H∞ control for discrete-time descriptor systems with uncertainties in all matrices. In 20th World Congress The International Federation of Automatic Control, IFAC-PapersOnLine, volume 50, 15483–15487. Elsevier. Chadli, M. and Darouach, M. (2012). Novel bounded real lemma for discrete-time descriptor systems: application to H∞ control design. Automatica, 48(2), 449–453. 119
83
Chadli, M. and Darouach, M. (2014). Further enhancement on robust H∞ control design for discrete-time singular systems. IEEE Transactions on Automatic Control, 59(2), 494–499. Coutinho, D., de Souza, C., and Barbosa, K. (2014). Robust H∞ control of discrete-time descriptor systems. In Control Conference (ECC), 2014 European, 1915– 1920. IEEE. 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. on Signal Processing, 54(6), 2110–2118. Duan, G.R. (2010). Analysis and Design of Descriptor Linear Systems. Springer. Feng, Y. and Yagoubi, M. (2013). On state feedback H∞ control for discrete-time singular systems. IEEE Transactions on Automatic Control, 58(10), 2674–2679. Hsiung, K. and Lee, L. (1999). Lyapunov inequality and bounded real lemma for discrete-time descriptor systems. IEE Proceedings-Control Theory and Applications, 146(4), 327–331. Ishihara, J. and Terra, M. (2003). A new Lyapunov equation for discrete-time descriptor systems. In Proc. of the 2003 American Control Conference, 2003, 5078– 5082. Denver, USA. Ji, X., Su, H., and Chu, J. (2007). Robust state feedback H∞ control for uncertain linear discrete singular systems. IET Control Theory & Applications, 1(1), 195– 200. Masubuchi, I., Kamitane, Y., Ohara, A., and Suda, N. (1997). H∞ control for descriptor systems: a matrix inequalities approach. Automatica, 33(4), 669–673. Takaba, K. (1998). Robust H2 control of descriptor system with time-varying uncertainty. International Journal of Control, 71(4), 559–579. Wang, H., Yung, C., and Chang, F. (1998). Bounded real lemma and H∞ control for descriptor systems. IEE Proceedings-Control Theory and Applications, 145(3), 316–322. Xia, Y., Shi, P., Liu, G., and Rees, D. (2005). Robust mixed H2 /H∞ state-feedback control for continuoustime descriptor systems with parameter uncertainties. Circuits, Systems and Signal Processing, 24(4), 431–443. Xie, L., de Souza, C.E., and Wang, Y. (1993). Robust control of discrete time dynamical systems. Automatica, 29, 1133–1137. Xu, S. and Lam, J. (2006). Robust control and filtering of singular systems. Lecture Notes in Control and Information Sciences. Springer. Xu, S. and Yang, C. (2000). H∞ state feedback control for discrete singular systems. IEEE Transactions on Automatic Control, 45(7), 1405–1409. Zhang, G., Xia, Y., and Shi, P. (2008). New bounded real lemma for discrete-time singular systems. Automatica, 44(3), 886–890. Zhang, L., Huang, B., and Lam, J. (2003). LMI synthesis of H2 and mixed H2 /H∞ controllers for singular systems. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 50(9), 615–626.
ScienceDirect
IFAC PapersOnLine 51-25 (2018) 78–83
Robust H state-feedback design for ∞ Robust H design for ∞ state-feedback Robust H state-feedback design for ∞ discrete-time descriptor systems Robust H state-feedback design for ∞ discrete-time descriptor systems discrete-time descriptor systems discrete-time descriptor systems Carlos Rodr´ıguez ∗∗ , Karina A. Barbosa ∗∗ , Daniel Coutinho ∗∗ ∗∗
Carlos Rodr´ıguez ∗∗∗ , Karina A. Barbosa ∗∗∗ , Daniel Coutinho ∗∗ ∗∗ Carlos Rodr´ıguez ∗ , Karina A. Barbosa ∗ , Daniel Coutinho ∗∗ ∗∗ Carlos Rodr´ ıguez , Karina A. Barbosa , Daniel Coutinho ∗ of Electrical Engineering, Universidad de Santiago de ∗ Department Department of Electrical Engineering, Universidad de Santiago de ∗ ∗ DepartmentChile, of Electrical Engineering, Universidad de Santiago de Av. 3519, Chile ∗ Chile, Av. Ecuador Ecuador 3519, Santiago, Santiago, Chile Department of Electrical Engineering, Universidad de Santiago de Chile, Av. Ecuador 3519, Santiago, Chile E-mail: [email protected], [email protected] E-mail: [email protected], [email protected] ∗∗ Chile, Av. Ecuador 3519, Santiago, Chile Federal de of Automation and Systems, Universidade E-mail: [email protected], [email protected] ∗∗ Department of Automation and Systems, Universidade Federal de ∗∗ Department E-mail: [email protected], [email protected] ∗∗ Santa Catarina (UFSC), Florian´ o polis, SC 88040-900, Brazil of Automation and Systems, Universidade ∗∗ Department Santa Catarina (UFSC), Florian´ o polis, SC 88040-900,Federal Brazil de Department ofE-mail: Automation and Systems, Universidade Santa Catarina (UFSC), Florian´ opolis, SC 88040-900,Federal Brazil de [email protected] E-mail: [email protected] Santa Catarina (UFSC), Florian´ opolis, SC 88040-900, Brazil E-mail: [email protected] E-mail: [email protected] Abstract: This This paper paper deals deals with with the the design of of robust H H∞ controllers for discrete-time descriptor Abstract: ∞ controllers for discrete-time descriptor Abstract: This paper deals with the design design of robust robustinHall for discrete-time descriptor systems with uncertain time-varying parameters state space It assumed ∞ controllers ∞ systems with uncertain time-varying parameters inHall state spaceformatrices. matrices. It is isdescriptor assumed Abstract: This paper deals with the functions design of robust discrete-time ∞ controllers systems with uncertain time-varying parameters in all state spaceparameters matrices. It is bounded assumed that the system matrices are affine of uncertain bounded with that the system matrices are affine functions of uncertain bounded parameters with bounded systems with uncertain time-varying parameters in all state space matrices. It is assumed variations. Firstly, a novel version of the Bound Real Lemma for linear discrete time varying that the system matrices are affine functions of uncertain bounded parameters with bounded variations. Firstly, a novelare version offunctions the Bound Real Lemma for linear discretewith timebounded varying that the system matrices affine of uncertain bounded parameters descriptor systems is presented. Then, the latter result is specialized for state-feedback design variations. Firstly, a novel version of the Bound Real Lemma for linear discrete time varying descriptor systems is presented. Then, the latter result is specialized for state-feedback design variations. Firstly, ais novel version ofconstraints the Lemma for the linear discrete time varying descriptor systems presented. Then, theBound latterinReal result is specialized for state-feedback design in terms of linear matrix inequality order to ensure closed-loop admissibility in terms of linear matrix inequality constraints in order to ensure the closed-loop admissibility descriptor systems isanpresented. Then, the latter result is specialized for state-feedback design -norm for all admissible uncertainty. while guaranteeing upper-bound on the system H in terms of linear matrix inequality constraints in order to ensure the closed-loop admissibility -norm forthe allclosed-loop admissible admissibility uncertainty. while guaranteeing an upper-bound on the system H∞ ∞to in terms of examples linear matrix inequalitytoconstraints in order ensure Numerical are presented illustrate the proposed control design technique. -norm for all admissible uncertainty. while guaranteeing an upper-bound on the system H ∞ ∞ Numerical examplesan are presented to on illustrate the proposed control design technique. for all admissible uncertainty. while guaranteeing the system H∞ -norm Numerical examples areupper-bound presented to illustrate the proposed control design technique. © 2018, IFAC (International Federationtoofillustrate Automaticthe Control) Hosting by Elsevier Ltd. All rights reserved. Numerical examples are presented proposed control design technique. Keywords: control, parameter-dependent parameter-dependent system system matrices, matrices, Keywords: descriptor descriptor systems, systems, H H∞ control, Keywords: descriptor systems, H∞ time-varying discrete-time systems, convex optimization. control, parameter-dependent system matrices, ∞ ∞ convex optimization. time-varying discrete-time systems, parameter-dependent system matrices, Keywords: descriptor systems, H∞ control, time-varying discrete-time systems, convex optimization. time-varying discrete-time systems, convex optimization. 1. in 1. INTRODUCTION INTRODUCTION in terms terms of of LMIs LMIs to to ensure ensure the the system system admissibility admissibility while while guaranteeing a bound on its H norm. The latter result 1. INTRODUCTION in terms of LMIs to ensure the system admissibility while ∞ guaranteeing a bound on its H norm. The latter result ∞ 1. (or INTRODUCTION in terms of LMIs to ensure the system admissibility while was extended in (Chadli and Darouach, 2012) and (Feng guaranteeing a bound on its H latter result ∞ norm. The Descriptor systems singular systems) can naturally ∞ extended ainbound (Chadli and Darouach, 2012)latter and (Feng Descriptor systems (or singular systems) can naturally was guaranteeing on its H norm. The result ∞ was extended in (Chadli and Darouach, 2012) and (Feng and 2013) to more suitable conditions for describe many practical systems with advantage of Descriptor systems (or singular naturally and Yagoubi, Yagoubi, 2013) to derive derive suitable conditions for describe many practical systems systems) with the the can advantage of was extended in (Chadli and more Darouach, 2012) and (Feng Descriptor systems (or singular systems) can naturally the H design problem. It is worth mentioning describe many practical systemsstructure with theby advantage of and Yagoubi, 2013) to derive more suitable conditions for preserving the original model considering ∞ control the H control design problem. It is worth mentioning preserving the original model structure by considering ∞ and Yagoubi, 2013) to derive more suitable conditions for describe many practical systems withstate thebyand advantage of that the aforementioned results deal with uncertainty-free the H control design problem. It is worth mentioning ∞ dynamic and static relations among algebraic preserving the original model structure considering ∞ aforementioned results deal with uncertainty-free the dynamic and static relations among statebyand algebraic that the H control design problem. It is worth mentioning preserving the original model structure considering ∞ discrete-time descriptor that the aforementioned results deal with uncertainty-free variables. Nowadays, many developed for standard dynamic and static relations among state and descriptor systems. systems. variables. Nowadays, many results results developed for algebraic standard discrete-time that the aforementioned results deal with uncertainty-free dynamic and static relations among state and algebraic discrete-time descriptor systems. state-space systems have been extended to deal with variables. Nowadays, many results developed for standard state-space systems many have results been extended tofordeal with discrete-time problem for The robust H descriptor systems. ∞ control variables. Nowadays, developed standard control problem for discrete-time discrete-time descripdescripThe robust H state-space systems have been extended to deal with ∞ descriptor models; see, for the descriptor dynamical dynamical models; see,extended for instance, instance, the texttexttor systems subject to parametric uncertainties been problem for discrete-timehas descripThe robust H ∞ control ∞ state-space systems have been to deal with tor systems subject to parametric uncertainties has been books of Xu and Lam (2006) and Duan (2010). descriptor dynamical models; see, for instance, the textproblem for discrete-time descriprobust H ∞ control tor systems subject to Darouach, parametric uncertainties has been books of Xu and Lam models; (2006) and Duan (2010). the text- The studied in (Chadli and 2014), (Coutinho et al., descriptor dynamical see, for instance, studied in (Chadli and Darouach, 2014), (Coutinho etbeen al., books of Xu and Lam (2006) and Duan (2010). tor systems subject to parametric uncertainties has 2014), (Ji et al., 2007), (Xu and Lam, 2006) and (Belov studied in (Chadli and Darouach, 2014), (Coutinho et al., Since many control systems are affected by external disbooks of Xu and Lam (2006) and Duan (2010). 2014), (Ji et al., 2007), (Xu and Lam, 2006) and (Belov Since many control systems are affected by external dis- studied in (Chadli and Darouach, 2014), (Coutinho et al., and Andrianova, 2016). In particular, Coutinho et 2014), (Ji et al., 2007), (Xu and Lam, 2006) and (Belov turbances, it is often desired to make the controlled sysSince many control systems are affected by external disAndrianova, 2016).(Xu In and particular, Coutinho et al. al. turbances, itcontrol is oftensystems desiredare to make thebycontrolled sys2014), (Ji et considered al., 2007), Lam,descriptor 2006) andsystems (Belov Since many affected external dis- and and Andrianova, 2016). discrete-time In particular, Coutinho et al. (2014) have tem less sensitive to exogenous disturbances by applying turbances, it is often desired to make the controlled sys(2014) have considered discrete-time descriptor systems tem less sensitive to exogenous disturbances by applying Andrianova, 2016). In particular, Coutinho et al. turbances, it is often desired to disturbances make the controlled sys- and (2014) have considered discrete-time descriptor systems with parametric uncertainties and affine theory. In of the H tem less sensitive to exogenous by applying ∞ control with polytopic-type polytopic-type parametric uncertainties and an an affine theory. In the the context context of continuous-time continuous-time the H ∞ control (2014) have considered discrete-time descriptor systems tem less sensitive to exogenous disturbances by applying parameter-dependent Lyapunov function was considered control theory. In the context of continuous-time the H with polytopic-type parametric uncertainties and an affine descriptor systems, the control problem of ∞ ∞ Lyapunovuncertainties function wasand considered descriptor systems, the H HIn∞ control problem of linear linear timetime- parameter-dependent ∞the with polytopic-type parametric an affine theory. context of continuous-time the H∞ control to derive aa solution in terms of bilinear matrix Lyapunov function was inequalities considered invariant systems has been by Wang al. (1998) descriptor systems, the H∞ control problem of et linear time- parameter-dependent ∞addressed to derive solution in terms of bilinear matrix inequalities invariant systems has been addressed by Wang et al. (1998) parameter-dependent Lyapunov function was considered descriptor systems, the H control problem of linear timeto derive a solution in terms of bilinear matrix inequalities (BMIs). Norm-bounded uncertainties were investigated in which proposed and conditions based invariant systemsnecessary has been∞addressed by Wang et al. (1998) Norm-bounded uncertainties were investigated in which proposed and sufficient sufficient conditions based (BMIs). to deriveand a solution in terms ofbased bilinear matrix inequalities invariant systemsnecessary has been addressed by Wang etThen, al. (1998) (BMIs). Norm-bounded uncertainties were investigated in (Chadli Darouach, 2014), on the results derived on two generalized algebraic Riccati equations. Mawhich proposed necessary and sufficient conditions based (Chadli and Darouach, 2014), based on the results derived on two generalized algebraic Riccati equations. Then, Ma(BMIs). Norm-bounded uncertainties were investigated in which proposed necessary and sufficient conditions based in (Chadli and Darouach, 2012), also by means of BMI on two generalized algebraic Riccati equations. Then, Ma(Chadli and Darouach, 2014), based on the results derived subuchi et Xia al. and et al. (Chadli and Darouach, 2012), also by means of BMI subuchi et al. al. (1997), (1997), Xia et etRiccati al. (2005) (2005) and Zhang Zhang etMaal. in (Chadli and Darouach, 2014), based on the results derived on two generalized algebraic equations. Then, control design Belov Andrianova (Chadli andconditions. Darouach,Recently, 2012), also byand means of BMI (2003) have proposed in terms subuchi et al. (1997), equivalent Xia et al. solutions (2005) and etlinal. in design Recently, Belov Andrianova (2003) have proposed equivalent in Zhang terms of of in (Chadli andconditions. Darouach, 2012), also byand means of BMI subuchi et inequality al. (1997), (LMI) Xia etconstraints. al. solutions (2005) and Zhang etlinal. control control design conditions. Recently, Belov and Andrianova (2016) proposed a novel result to the robust H control ear matrix ∞ (2003) have proposed equivalent solutions in terms of lin(2016) proposed a novel result to the robust ear matrix constraints. ∞ control control design conditions. Recently, Belov and H Andrianova (2003) haveinequality proposed (LMI) equivalent solutions in terms of lin- problem for discrete-time descriptor systems in terms of ear matrix inequality (LMI) constraints. (2016) proposed a novel result to the robust H control ∞ problem for discrete-time descriptor in∞ terms of On the hand, several (2016)constraints proposed abased novel on result toBRL thesystems robust H control ear inequality (LMI)exist constraints. ∞ LMI the provided in (Feng for discrete-time descriptor systems in terms of On matrix the other other hand, there there exist several different different versions versions problem LMI constraints based on the BRL provided in (Feng of the Real Lemma (BRL) for the discrete-time for discrete-time descriptor systems in extended terms of On theBounded other hand, there exist several different versions problem and Yagoubi, 2013). The latter result has been LMI constraints based on the BRL provided in (Feng of the Bounded Real Lemma (BRL) for the discrete-time On theBounded otherHsiung hand, exist several different versions and Yagoubi, 2013). Theonlatter result has been extended of the Real there Lemma (BRL) for theXu discrete-time counterpart. and Lee (1999) and and Yang LMI constraints based the BRL provided in (Feng and Yagoubi, 2013). The latter result has been extended and Andrianova (2017) to counterpart. Hsiung and Lee(BRL) (1999)for and and Yang in ∞ state-feedback of the Bounded Real Lemma theXu discrete-time in Belov Belov and 2013). Andrianova (2017) to H H state-feedback ∞ been counterpart. Hsiung and Leethe (1999) and Xu and Yang (2000) versions of BRL by of aa set and Yagoubi, The latter result has extended for LTI discrete-time descriptor with normin Belov and Andrianova (2017) to systems H ∞ (2000) proposed proposed versions of the BRL and by means means of Yang set design ∞ state-feedback counterpart. Hsiung and Lee (1999) Xu and design for LTI discrete-time descriptor systems with normof nonlinear matrix inequalities. Although it plays a key in Belov and Andrianova (2017) to H state-feedback (2000) proposed versions of the BRL by means of a set ∞ bounded parametric uncertainties in all system matrices. design for LTI discrete-time descriptor systems with normof nonlinear matrix inequalities. Although it plays a key (2000) proposed versions of the BRL by means of a set bounded parametric uncertainties in all system matrices. role on theoretically solving the H the latter forparametric LTI discrete-time descriptor systems with normof nonlinear matrix inequalities. Although it plays a key design ∞ problem, bounded uncertainties in all system matrices. role on theoretically solving the H problem, the latter ∞ of matrix inequalities. itZhang plays a key However, there are few results in the literature addressing rolenonlinear on are theoretically solving the Although H∞ problem, the latter results hard to be numerically et al. bounded parametric uncertainties in all system matrices. ∞solved. However, there are few results in the literature addressing results are hard to be numerically Zhang et al. role on have theoretically solving the H∞solved. problem, theof latter control problem for linear varying the H there are few results in the literature addressing (2008) introduced a version BRL results are hard to be numerically solved. Zhang al. However, the H∞ problem for discrete discrete linear time time varying (2008) have introduced a more more tractable tractable version of et BRL ∞ control However, there are few results in the literature addressing results are hard to be numerically solved. Zhang al. (DLTV) descriptor systems considering robust feedback the H problem for discrete linear time varying ∞ control (2008) have introduced a more tractable version of et BRL ∞ (DLTV) descriptor systems considering robust feedback This work the H∞ control problem for considering discrete linear time varying (2008) havehas introduced a more tractable version of BRL (DLTV) been supported by “Fondo Nacional de Desarrollo descriptor systems robust feedback controllers. Motivated by this fact, the present work deals This work has been supported by “Fondo Nacional de Desarrollo controllers. Motivated by thisconsidering fact, the present work deals (DLTV) descriptor systems robust feedback Cient´ ıfico y Tecnol´ ogico” - Fondecyt, Chile,Nacional under grant 1151199, This work has been supported by “Fondo de Desarrollo with the H control problem for a class of DLTV uncontrollers. Motivated by this fact, the present work ∞ control problem for a class of DLTVdeals Cient´ıfico y Tecnol´ ogico” - Fondecyt, Chile, under grant 1151199, with the H un∞ This work has been supported by “Fondo de Desarrollo controllers. Motivated by this fact, the present work deals and D. Coutinho has been partially supported by CNPq - Brazil, Cient´ ıfico y Tecnol´ ogico” - Fondecyt, Chile,Nacional under grant 1151199, certain descriptor systems by considering a robust statewith the H problem for a class of DLTV un∞ control and D. Coutinho has been partially supported by CNPq - Brazil, ∞ certain descriptor systems by considering a robust stateCient´ ıfico y Tecnol´ o gico” Fondecyt, Chile, under grant 1151199, underD.grant 304351/2014-8. with thedescriptor H∞ control problem for a classaofrobust DLTV unand Coutinho has been partially supported by CNPq - Brazil, certain systems by considering stateunder grant 304351/2014-8. and D.grant Coutinho has been partially supported by CNPq - Brazil, under 304351/2014-8. certain descriptor systems by considering a robust state-
under grant© 304351/2014-8. 2405-8963 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2018 IFAC 114 Copyright 2018 IFAC 114 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 114 10.1016/j.ifacol.2018.11.085 Copyright © 2018 IFAC 114
IFAC ROCOND 2018 Florianopolis, Brazil, September 3-5, 2018 Carlos Rodríguez et al. / IFAC PapersOnLine 51-25 (2018) 78–83
feedback control law. The proposed result is inspired on a dual version of the BRL proposed in (Barbosa et al., 2017) which has introduced necessary and sufficient LMIbased conditions to the admissibility analysis of DLTV descriptor systems based on time-varying Lyapunov functions. In particular, a necessary and sufficient condition is firstly introduced for the existence of a static statefeedback controller such that the resulting closed-loop system is admissible with a guaranteed upper-bound on the H∞ performance index. Then, the proposed result is tailored for H∞ control design in terms of a set of LMI constraints. Numerical examples are provided to illustrate the approach showing the effectiveness of the proposed H∞ control design method for DLTV uncertain descriptor systems. Notation. Zi is the set of integers equal to or larger than i, C is the set of complex numbers, R+ is the set of positive real numbers, Rn is the n-dimensional Euclidean space, Rm×n is the set of m × n real matrices, In is the n×n identity matrix, 0n and 0m×n are the n×n and m×n matrices of zeros, respectively, and diag{· · · } stands for a block-diagonal matrix. For a real matrix S, S T denotes its transpose, He{S} stands for S+S T and S > 0 means that S is symmetric and positive-definite. For a symmetric block matrix, the symbol denotes the transpose of the blocks outside the main diagonal block. 2 denotes the space of square summable vector sequences over [0, ∞) with norm ·2 . For a given convex bounded polyhedral domain B, V(B) denotes the set of all the vertices of B. The one-step delayed and the one-step ahead shift of the time argument k will be denoted by k− := k−1 and k+ := k+1, respectively. 2. DESCRIPTOR SYSTEMS Descriptor systems may exhibit a more complex behavior than ordinary dynamical system such as the existence of solution and impulsive modes (or causality in the discrete-time case). In this section, some key concepts and two canonical representations related to DLTV descriptor systems, such as admissibility and stability, are recalled as proposed in (Barbosa et al., 2018). To this end, consider the following DLTV descriptor systems: E(k)x(k+1) = A(k)x(k)+B(k)w(k), x(0)=x0 ,
(1)
where x(k) ∈ Rn is the state, w(k) ∈ Rnw is the disturbance input vector assumed to belong to 2 , E(k), A(k) and B(k) are bounded matrix functions with appropriate dimensions, and x0 is a consistent initial condition. Furthermore, E(k) might be a singular matrix but it is assumed that rank{E(k)} = r ≤ n, ∀ k ∈ Z0 . Definition 1. Consider the DLTV descriptor system (1) with x0 being a consistent initial condition. Then: (a) The system is said to be regular if for any w(k) ∈ Rnw and x0 there exists a solution x(k) for all k ∈ Z0 and it is unique. (b) The system is said to be causal if it is regular and the solution x(k) for any x0 and w(k) ∈ Rnw is a function of x0 and w(0), . . . , w(k−1), for all k ∈ Z0 . (c) The system is said to be exponentially stable if it is regular for any x0 and w(k) ≡ 0, and there exist real scalars α > 0 and β ∈ (0, 1) such that 115
79
x(k) ≤ α β k x0 , ∀ k ∈ Z1 .
(d) The system is said to be admissible if it is causal and exponentially stable. Let z(k) = C(k)x(k) + D(k)w(k),
(2)
be the output performance vector of the DLTV descriptor systems in (1), where z(k) ∈ Rnz , and C(k) and D(k) are bounded matrix functions with appropriate dimensions. Hence, the H∞ -norm of system (1)-(2) can be characterized by the following definition. Definition 2. Consider the system in (1)-(2). Then the system induced 2 -gain from w(k) to z(k), or simply the system H∞ -norm, is given by: G wz ∞ = sup
w∈2
z(k)2 : w(k) ≡ 0, E(0)x(0) = 0 w(k)2
where G wz stands for the operator from w(k) to z(k).
Now, it is introduced two canonical representations of DLTV descriptor systems tailored for control design which will be instrumental to derive the main result of this paper. Then, consider firstly the following descriptor system: E(k)x(k+1) = A(k)x(k) + B(k)w(k) + Bu (k)u(k) (3) z(k) = C(k)x(k) + D(k)w(k) + Du (k)u(k) (4) where u(k) ∈ Rnu is the control input vector, and Bu (k) and Du (k) are bounded matrix functions. The singular value decomposition (SVD) state-space representation of system (3)-(4) is defined as follows: ¯ ¯ ¯ ¯u (k)u(k) Eξ(k+1) = A(k)ξ(k)+ B(k)w(k) +B (5) ¯ z(k) = C(k)ξ(k)+D(k)w(k) +Du (k)u(k) where ξ(k) = N −1 (k)x(k) and
0 Ir ¯ , E = M (k)E(k)N (k + 1) = 0 0n−r A1 (k) A2 (k) ¯ A(k) = M (k)A(k)N (k) = , A3 (k) A4 (k) B1 (k) ¯ , B(k) = M T(k)B(k) = B2 (k) Bu1 (k) T ¯ Bu (k) = M (k)Bu (k) = , Bu2 (k)
(6)
¯ C(k) = C(k)N (k) = [ C1 (k) C2 (k) ] ,
with M (k), N (k) being nonsingular n × n real matrices for ¯ ¯ ¯u (k) and all k ∈ Z. In addition, the matrices A(k), B(k), B ¯ ¯ C(k) are partitioned accordingly to E. Next, assuming that system (3)-(4) is admissible, it is possible to obtain the following Weierstrass form: u (k)u(k) B(k)w(k) +B Eζ(k+1) = A(k)ζ(k)+ z(k) = C(k)ζ(k)+D(k)w(k) +Du (k)u(k)
−1 (k)x(k) and where ζ(k) = N
(7)
IFAC ROCOND 2018 80 Florianopolis, Brazil, September 3-5, 2018 Carlos Rodríguez et al. / IFAC PapersOnLine 51-25 (2018) 78–83
0 =M (k)E(k)N (k + 1) = Ir E 0 0n−r , 0 Af (k) , A(k) = M (k)A(k)T (k) = 0 In−r Bf (k) T B(k) = M (k)B(k) = A−1 4 (k)B2 (k) Buf (k) u (k) = M T(k)Bu (k) = B A−1 4 (k)Bu2 (k) with
(8)
Lemma 1. (Xie et al. (1993)) The following LTV system x ˜(k + 1) = Af (k)˜ x(k) + Bf (k)w(k) z(k) = Cf (k)˜ x(k) + Df (k)w(k)
C(k) = C(k)T (k) = [ Cf (k) C2 (k) ] ,
Af (k) = A1 (k) − A2 (k)A−1 4 (k)A3 (k),
Bf (k) = B1 (k) − A2 (k)A−1 4 (k)B2 (k),
Buf (k) = Bu1 (k) − A2 (k)A−1 4 (k)Bu2 (k), Cf (k) = C1 (k) −
In this paper, the robust H∞ control problem is addressed for the class of uncertain system as defined in (10). More precisely, an LMI-based result will be proposed in the following section for designing a static state feedback control law u(k) = Kx(k) in order to ensure the robust admissibility of the closed-loop system with a guaranteed upperbound on the system H∞ -norm for all (θ(k), ∆θ(k)) ∈ B. Before ending this section, the following version of BRL for ordinary DTLTV systems is recalled.
(9)
C2 (k)A−1 4 (k)A3 (k).
(k) and N (k) are nonsingular n × n real Moreover, M u (k) and C(k) matrices for all k ∈ Z, and A(k), B(k), B are partitioned accordingly to E. 3. PROBLEM STATEMENT
Consider the following uncertain discrete-time descriptor systems: E(θ(k))x(k+1) = A(θ(k))x(k)+B(θ(k))w(k) +Bu (θ(k))u(k) (10) z(k) = C(θ(k))x(k)+D(θ(k))w(k) +Du (θ(k))u(k) where x(k) ∈ Rn is the state, w(k) ∈ Rnw is the disturbance input belonging to 2 , u(k) ∈ Rnu is the control input, z(k) ∈ Rnz is the performance output, and θ(k) = [ θ1 (k), . . . , θp (k) ] T ∈ Rp is a vector of bounded uncertain time-varying parameters. The matrices E(θ(k)), A(θ(k)), B(θ(k)), Bu (θ(k)), C(θ(k)), D(θ(k)) and Du (θ(k)) are affine functions of θ(k) with appropriate dimension with E(θ(k) allowed to be a singular matrix but satisfying rank{E(θ(k))} = r ≤ n for all k ∈ Z0 .
Further, considering ∆θ(k) := θ(k)−θ(k − 1), it is assumed that the parameters θi (k) and their variations ∆θi (k), for i = 1, . . . , p and k ∈ Z0 , are such that θ i ≤ θi (k) ≤ θi and δ i ≤ ∆θi (k) ≤ δ i , where θ i , θi , δ i and δ i are known extremum values of θi (k) and ∆θi (k), respectively. Moreover, let B be the polytope representing the set of admissible values of (θ, ∆θ) and assume that the admissible values of θ(k) and ∆θ(k) are such that if (θ(k), ∆θ(k)) ∈ B then θ(k −1) also belongs to B. Similarly as in de Souza et al. (2006), a polytope B having this property will be referred to as a consistent polytope. Note that a consistent polytope must include ∆θ = 0 as an admissible value of ∆θ(k). In the sequel, θ ∈ B means that (θ(k), 0) ∈ B and the dependence of θ(k) and ∆θ(k) on k will be often omitted to ease the notation. Next, we introduce the notion of robust admissibility to be considered throughout this paper. Definition 3. System (10) is said to be robustly admissible if it is admissible for all (θ, ∆θ) ∈ B . 116
(11)
is exponentially stable and Gwz ∞ ≤ γ with γ being a given positive scalar, for all k ∈ Z0 , if and only if there exists a bounded matrix function Q(k) > 0 such that: −Q(k + 1) Q(k)AT (k) −Q(k) f < 0, ∀k ∈ Z0 (12) BfT (k) 0 −γI 0 Cf (k)Q(k) Df (k) −γI 4. MAIN RESULTS This section proposes an LMI based approach to the H∞ control problem. Firstly, it is introduced a version BRL for DLTV descriptor system for robust admissibility analysis while guaranteeing a bound on the system H∞ -norm. Then, the analysis result is extended to cope with robust state-feedback synthesis. 4.1 H∞ Performance Analysis The following result provides two equivalent necessary and sufficient conditions to ensure the robust admissibility of the unforced system in (10) while providing a guaranteed upper-bound on the system H∞ -norm. In the next theorem, the admissible values of θ(k) and ∆θ(k) are assumed to belong to a compact set D which is not necessarily a polytope as the set B. Theorem 2. Consider the uncertain descriptor system in (10) with u(k) ≡ 0. Let D be a compact set of admissible (θ, ∆θ), γ be a given positive scalar, and E0 (θ) ∈ Rn×(n−r) be any full column-rank matrix function of θ such that E(θ)E0 (θ) = 0, ∀ θ ∈ D. Assume that Ker{E(θ)} ⊆ Ker{C(θ)}, for all θ ∈ D. Then, system (10) is admissible and its H∞ -norm is smaller than γ, if and only if one of the following conditions holds: (a) There exist bounded matrix functions X(θ) > 0 and R(θ) such that:
Σ1 (θ) X(θ−∆θ)AT (θ) −X(θ−∆θ) < 0, T 0 −γI Bw(θ) C(θ)E0 (θ−∆θ)R(θ) C(θ)X(θ−∆θ) D(θ) −γI ∀ (θ, ∆θ) ∈ D, (13) where Σ1 (θ) = −E(θ)X(θ)E T (θ) + He{A(θ)E0 (θ−∆θ)R(θ)}.
IFAC ROCOND 2018 Florianopolis, Brazil, September 3-5, 2018 Carlos Rodríguez et al. / IFAC PapersOnLine 51-25 (2018) 78–83
(b) There exist bounded matrix functions F (θ), G(θ), H(θ), S(θ) and P (θ) > 0 such that Γ1 (θ) C(θ)F(θ) −γI T T < 0, Γ2 (θ, ∆θ) G (θ)C (θ) Γ3 (θ, ∆θ) −H(θ) 0 0 −P (θ) DT (θ) 0 0 −γI B T (θ) ∀ (θ, ∆θ) ∈ D, (14) where Γ1 (θ) = He{A(θ)F (θ) + H(θ)E T (θ)}, Γ2 (θ, ∆θ) = E0 (θ−∆θ)S(θ) + GT (θ)AT (θ) − F (θ),
Γ3 (θ, ∆θ) = P (θ−∆θ) − He{G(θ)}.
Proof. Firstly, for convenience of notation, the dependence of matrix functions on θ and (θ −∆θ) will be represented as functions of k and k − 1, respectively. That is, let Ω(·) be a matrix function of either θ or ∆θ. Then, Ω(k) = Ω(θ) and Ω(k− ) = Ω(θ − ∆θ). As a result, the uncertain descriptor system in (10), with u(k) ≡ 0, will read as in (1)-(2). Sufficiency of (a): Consider that the descriptor system is in SVD form given by (5) and assumed that X1 (k) X2 (k) ¯ , X(k) := N −1(k + 1)X(k)N −T (k + 1) = X2T (k) X4 (k) (15) where the dimensions of the partitions are compatible with ¯ as defined in (6). those of E Thus, pre- and post-multiplying (13) by T
−1
T = diag{M (k), N (k), I, I} and T , respectively, and considering (6) and (15), we obtain that: ¯ 1 (k) Σ X(k T ¯ −) ¯ − )A¯ (k) −X(k (16) < 0, ¯ T(k) B 0 −γI w ¯ 2 (k) ¯ X(k ¯ − ) D(k) −γI Σ C(k) where −1 ¯ ¯ 1(k) =He{A(k)N (k)E0 (k− )R(k)M T (k) Σ ¯ X(k) ¯ E ¯ T (k)}, −E(k)
−1 ¯ ¯ 2 (k) = C(k)N (k)E0 (k− )R(k)M T (k). Σ
Now, it will be shown that if (13) is satisfied then the unforced system (10) is causal. Note that if (13) holds ¯ 1 (k) < 0, then (16) is also satisfied, which implies that Σ for all k ∈ Z0 .
In the light of (6) and similarly as in (Ishihara and Terra, 2003), E0 (k) can be factorized in the following form: (17) E0 (k− ) = N (k) 0r×(n−r) In−r W (k),
where W (k) ∈ R(n−r)×(n−r) is any nonsingular matrix for all k ∈ Z0 . Next, let (18) W (k)R(k)M T (k) = R1 (k) R2 (k) , where R1 (k) ∈ R(n−r)×r and R2 (k) ∈ R(n−r)×(n−r) . ¯ 1 (k) accordingly to X(k), ¯ Partitioning Σ i.e.,
117
81
¯ 12 (k) ¯ (k) Σ ¯ 1 (k) = Σ11 Σ T ¯ (k) Σ ¯ 22 (k) , Σ 12
and considering (6) and (18), it can be easily established that ¯ 22 (k) = A4 R2 + RT AT < 0, ∀ k ∈ Z0 , Σ 2 4 which implies that the matrix A4 (k) is nonsingular for all k ∈ Z0 . Then, from Lemma 2 in (Barbosa et al., 2018), the causality of the unforced system (10) is ensured. Next, it will be shown that (13) also guarantees that the unforced system (10) is exponentially stable and G wz ∞ < γ. To this end, notice since the matrix A4 (k) is nonsingular for all k ∈ Z0 that the unforced system (10) can be represented in the Weierstrass form as given in (7)(8). Hence, similarly as before, the matrix inequality in (16) also holds with M (k), N (k), W (k), R1 (k) and R2 (k) (k), N (k), W (k), R 1 (k) and respectively replaced by M (k) and N (k) are Weierstrass form trans2 (k), where M R formation matrices and W (k) is any nonsingular matrix for all k ∈ Z0 . Now, define the matrix
2 (k) 1 (k) X X −1 −T X(k) := N (k+1)X(k)N (k+1) = T 4 (k) , (19) X2 (k) X where the partitions of X(k) are accordingly to those of as given in (8). Moreover, the assumption Ker{E(k)} ⊆ E Ker{C(k)} implies that C2 (k) = 0 for all k ∈ Z0 in (8). In addition, let the following matrices (k), N −1 (k), I, I}, T T = diag{M T Ia Ia IbT Ia 0 0 I(nz +nw ), I = 0 0 IcT Id IdT Id
Ia = [ Ir 0 ] , Ib = [ 0 Ir ] , Ic = [In−r 0 ] , Id = [ 0 In−r ] . Thus, considering (8) and (17)-(19), and pre- and postmultiplying (16) by TT (k) = IT T and T(k), respectively, the matrix inequality in (13) can be cast as follows: T −T Ψ1 (k) Ψ2 (k) −1 Ψ(k) = T 2 (k) Ψ 3 (k) T < 0, ∀ k ∈ Z0 , (20) Ψ where Ψ(k) denotes the left-hand side of (13), and 1 (k) −X T 1 (k) = X1 (k− )Af −X1 (k− ) , Ψ BT 0 −γI f 1 (k− ) D(k) −γI 0 Cf X T (k− ) B2 (k) Cf (k)X 2 (k) 1 (k) X R 2 2 (k) = Ψ , T (k− ) 0 T (k− )AT −X 0 X 2 f 2 2 (k) + R T (k) X 3 (k− ) R 2 Ψ3 (k) = 3 (k− ) . 3 (k− ) −X X
1 (k) < 0, which from Note that (20) ensures that Ψ Lemma 1 implies that system (11), and also the unforced system (10), is exponentially stable and its H∞ -norm is smaller than γ, for all k ∈ Z0 , with Q(k) = X(k− ).
Necessity of (a): Assume that the unforced system (10) is admissible and Gwz ∞ < γ, ∀ k ∈ Z0 . Consider that the system is in the SVD representation as given in (6)
IFAC ROCOND 2018 82 Florianopolis, Brazil, September 3-5, 2018 Carlos Rodríguez et al. / IFAC PapersOnLine 51-25 (2018) 78–83
and notice that A4 (k) is nonsingular due to the system admissibility. In addition, similarly as before, it is possible to obtain an equivalent representation as in (11) with (9). Hence, Lemma 1 holds since the system is exponentially stable and Gwz ∞ < γ. Also, there exits a Weierstrass representation of system (1)-(2) as given in (7)-(8).
where
(k), N (k), W (k), R 1 (k) and Now, consider the matrices M 2 (k) as defined in the sufficiency of item (a). It can be R readily verified that if (12) is satisfied then Ψ(k) < 0, ∀ k ∈ Z0 in (20), with 2 (k) = 0, X 4 (k) > 0, R 1 (k) = 0, 1 (k) = Q(k+ ), X X ∀ k ∈ Z0 , (21) and R2 (k) being such that
Υ4 (θ)=P (θ−∆θ) − He{G}, with p + θ i Pi P (θ) = P 0
4 (k− )+He{R 2 (k)} < 0, ∀ k ∈ Z1 . X Then, it turns out that (13) holds with the matrices T (k) diag X 1 (k), X 4 (k) N (k), X(k) = N −1(k) 0 R −1 (k), 2 (k) N R(k) = W 1 (k), X 4 (k), and R 2 (k) are as defined in (21). where X
Υ2 (θ)=E0 (θ−∆θ)S + GT AT (θ) + Y T BuT (θ) − µG,
Υ3 (θ)=(C(θ)G+Du (θ)Y )T ,
i=1
p (θi −∆θi )Pi P (θ − ∆θ) = P + 0
(23)
i=1
Then, the closed-loop system of (10) with u(k) = Y G−1 x(k) is robustly admissible and Gwz ∞ < γ, for all (θ, ∆θ) ∈ B. The proof of the above theorem is straightforward from the proof of Theorem 2 by considering G(θ) = G, F (θ) = µG, H(θ) = H and (23), and it will be omitted for brevity. 5. NUMERICAL EXAMPLES
(a) ⇔ (b): This part of the proof follows similar steps as the proof of (Barbosa et al., 2018, Theorem 1) and it will be omitted for brevity. ∇∇∇ Remark 1. Assumption Ker{E(θ)} ⊆ Ker{C(θ)} implies that the performance output variable z(k) will only depend on the dynamic part of the state vector and it has been also considered by Takaba (1998). 4.2 H∞ Control Design
Based on the H∞ analysis result established in Theorem 2(b), a robust state feedback control design approach will be derived in the sequel for ensuring the admissibility of the closed-loop system (10) while guaranteeing an upper-bound γ on the system H∞ -norm. To obtain computationally tractable conditions, it is further assumed that there exists an affine matrix function E0 (θ) ∈ Rn×(n−r) , having full-column rank for all θ in B, such that E(θ)E0 (θ) = 0, ∀ θ ∈ B. In addition, the matrices F, G and H are constrained to be constant and P (θ) to be an affine function of θ. The next theorem presents sufficient conditions in terms of a finite set of linear matrix inequalities (LMIs) to the state feedback design of discrete time descriptor systems subject to time-varying uncertainties in all system matrices. Theorem 3. Consider the descriptor system in (10). Let B be a given polytope of admissible (θ, ∆θ), γ be a given positive scalar, E0 (θ) ∈ Rn×(n−r) be any full columnrank affine matrix function of θ such that E(θ)E0 (θ) = 0, ∀ θ ∈ B, and µ be a given positive scalar. Assume that Ker{E(θ)} ⊆ Ker{C(θ)}, ∀ θ ∈ B. Suppose there exist matrices G, S, H, Y and Pi , i = 0, 1, . . . , p, with appropriate dimensions, satisfying the following LMI: Υ1 (θ) µC(θ)G+µDu (θ)Y −γI Υ3 (θ) Υ4 (θ) Υ2 (θ) < 0, −H 0 0 −P (θ) DT (θ) 0 0 −γI B T (θ) ∀ (θ, ∆θ) ∈ V(B),
Υ1 (θ)=He{µA(θ)G + µBu (θ)Y + HE T (θ)},
(22)
118
Example 1. Consider the uncertainty descriptor system in (10) with the following state-space matrices:
1+0.2θ2 0 0 0 1 0 E(θ) = 0 0 0
0.8(1+αθ1 ) 0 −0.32(1+0.2θ2 ) 1 0 0 A(θ) = 0 1+0.2θ2 −1
0.4 θ1 Bu (θ) = , Bw (θ) = 0.25 ∗ Bu (θ) −1 − θ2
0.1θ1 0.8 0 0.1θ2 C(θ) = , Dw (θ) = Du (θ) = 0 0.01θ2 0 0.1(1+θ1 ) where θ(k) is a time-varying parameters such that |θ(k)| ≤ 1, for all k ∈ Z0 and rank{E(θ(k))} = 2 for all admissible θ(k). In (Barbosa et al., 2018), it has been shown that the system is admissible for α ≤ 0.256. In this example, a robust state feedback controller is designed to ensure the closed-loop admissibility for α = 0.4, while minimizing the system H∞ -norm considering a maximum admissible parameter variation, i.e., |∆θ(k)| ≤ 2, ∀ k ∈ Z0 . Hence, considering E0 (θ) = [ 0 0 1 ]T and applying the optimization problem: min γ subject to (22). (24) µ,G,S,H,Y,P0 ,P1
leads to the following control law u(k) = [ −1.1316 0.0779 0.3674 ] x(k), for µ = 0.2123. The resulting closed-loop system is robustly admissible and Gwz ∞ < 0.6152, for all admissible parameter uncertainty. It is worth mentioning that a line search on µ has been performed in (24) in order to obtain a minimized upper-bound on the system H∞ -norm. Example 2. Consider the descriptor system as in (10) with the following state-space matrices: 1 −1 −6 −1 −1 1 −6 1 0 3+2θ 1 0 0 5+2θ −1 0 , A(θ) = , E(θ) = 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0
IFAC ROCOND 2018 Florianopolis, Brazil, September 3-5, 2018 Carlos Rodríguez et al. / IFAC PapersOnLine 51-25 (2018) 78–83
2 0 1 00 0 0 Bu (θ) = Bw (θ) = , C(θ) = , D(θ) = 1 1 0 00 1 1 where θ(k) is a time-varying parameter such that |θ(k)| ≤ 1.0, for all k ∈ Z0 . Notice that the open-loop system is non-causal for all k ∈ Z0 , and rank{E(θ(k))} = 2 for all admissible θ(k). Thus, in this example, a robust feedback control law is determined to ensure the closed-loop system admissibility while minimizing an upper-bound on the system H∞ -norm considering ∆θ(k) ≤ 1, for all k ∈ Z0 . Note that, since the parameter variation is different from the maximum admissible variation and the matrix E(θ) is uncertain, it is not available in the literature can approach suitable to solve this problem. However, applying the optimization problem in (24) considering µ = 0.50 and (31 + 12θ) 1 1 0 . E0 (θ) = (5 + 2θ) 0 0 −1 we obtain u(k) = [ 0.0716 0.8830 −4.0003 −2.8498 ]x(k), which guarantees the closed-loop admissibility and Gwz ∞ < 4.1074, for all admissible uncertainty. Similarly to Example 1, the latter result has been derived by performing a line search on µ. 6. CONCLUSION This paper has addressed the robust H∞ state-feedback design problem for discrete-time descriptor systems subject to time-varying parameters in all state-space matrices. The paper contribution is two fold: (i) a new version of the BRL for general discrete time-varying descriptor systems is proposed; and (ii) the latter result is specialized for robust state-feedback design guaranteeing the closed-loop admissibility with a prescribed upper-bound on the system H∞ -norm. Two numerical examples have demonstrated the potentials of the proposed approach. REFERENCES Barbosa, K., Coutinho, D., de Souza, C., and Rodrguez, C. (2017). Bounded real lemma for discrete linear timevarying descriptor systems. In 2017 11th Asian Control Conference (ASCC), 1835–1840. Barbosa, K., de Souza, C., and Coutinho, D. (2018). Admissibility analysis of discrete linear time-varying descriptor systems. Automatica, 91, 136–143. Belov, A. and Andrianova, O. (2016). Robust statefeedback H∞ control for discrete-time descriptor systems with norm-bounded parametric uncertainties. Preprint submitted to Automatica, arXiv:1611.01825. Belov, A. and Andrianova, O. (2017). On LMI approach to robust state-feedback H∞ control for discrete-time descriptor systems with uncertainties in all matrices. In 20th World Congress The International Federation of Automatic Control, IFAC-PapersOnLine, volume 50, 15483–15487. Elsevier. Chadli, M. and Darouach, M. (2012). Novel bounded real lemma for discrete-time descriptor systems: application to H∞ control design. Automatica, 48(2), 449–453. 119
83
Chadli, M. and Darouach, M. (2014). Further enhancement on robust H∞ control design for discrete-time singular systems. IEEE Transactions on Automatic Control, 59(2), 494–499. Coutinho, D., de Souza, C., and Barbosa, K. (2014). Robust H∞ control of discrete-time descriptor systems. In Control Conference (ECC), 2014 European, 1915– 1920. IEEE. 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. on Signal Processing, 54(6), 2110–2118. Duan, G.R. (2010). Analysis and Design of Descriptor Linear Systems. Springer. Feng, Y. and Yagoubi, M. (2013). On state feedback H∞ control for discrete-time singular systems. IEEE Transactions on Automatic Control, 58(10), 2674–2679. Hsiung, K. and Lee, L. (1999). Lyapunov inequality and bounded real lemma for discrete-time descriptor systems. IEE Proceedings-Control Theory and Applications, 146(4), 327–331. Ishihara, J. and Terra, M. (2003). A new Lyapunov equation for discrete-time descriptor systems. In Proc. of the 2003 American Control Conference, 2003, 5078– 5082. Denver, USA. Ji, X., Su, H., and Chu, J. (2007). Robust state feedback H∞ control for uncertain linear discrete singular systems. IET Control Theory & Applications, 1(1), 195– 200. Masubuchi, I., Kamitane, Y., Ohara, A., and Suda, N. (1997). H∞ control for descriptor systems: a matrix inequalities approach. Automatica, 33(4), 669–673. Takaba, K. (1998). Robust H2 control of descriptor system with time-varying uncertainty. International Journal of Control, 71(4), 559–579. Wang, H., Yung, C., and Chang, F. (1998). Bounded real lemma and H∞ control for descriptor systems. IEE Proceedings-Control Theory and Applications, 145(3), 316–322. Xia, Y., Shi, P., Liu, G., and Rees, D. (2005). Robust mixed H2 /H∞ state-feedback control for continuoustime descriptor systems with parameter uncertainties. Circuits, Systems and Signal Processing, 24(4), 431–443. Xie, L., de Souza, C.E., and Wang, Y. (1993). Robust control of discrete time dynamical systems. Automatica, 29, 1133–1137. Xu, S. and Lam, J. (2006). Robust control and filtering of singular systems. Lecture Notes in Control and Information Sciences. Springer. Xu, S. and Yang, C. (2000). H∞ state feedback control for discrete singular systems. IEEE Transactions on Automatic Control, 45(7), 1405–1409. Zhang, G., Xia, Y., and Shi, P. (2008). New bounded real lemma for discrete-time singular systems. Automatica, 44(3), 886–890. Zhang, L., Huang, B., and Lam, J. (2003). LMI synthesis of H2 and mixed H2 /H∞ controllers for singular systems. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 50(9), 615–626.