Chapter 7 Robust Stabilization and Disturbance Attenuation of Switched Linear Parameter-Varying Systems in Discrete Time

Transcription

1 Chapter 7 Robust Stabilization and Disturbance Attenuation of Switched Linear Parameter-Varying Systems in Discrete Time Ji-Woong Lee and Geir E. Dullerud Abstract Nonconservative analysis of discrete-time switched linear parametervarying systems is achieved via switching path-dependent Lyapunov and Kalman Yakubovich Popov inequalities. Exact convex conditions for the synthesis of a class of state-feedback controllers are then expressed in terms of nested unions of linear matrix inequalities. The resulting controllers are robust in the sense that their coefficients depend solely on a finite number of the most recent past modes and parameters, but not on the current mode or parameter. 7.1 Introduction A linear parameter-varying (LPV system is defined by a parameterized collection of linear state-space models and a set of admissible parameter trajectories [1,2,13]. An LPV system typically arises from the abstraction of a nonlinear model, where the precise nonlinear dependence on trajectories is replaced by a covering abstraction given in terms of varying parameters. The attraction with such abstracted models is that they can be significantly simpler to analyze, while at the same time because they admit more behaviors than the original nonlinear system can be used to infer guaranteed properties about the original system. Counterbalancing this potential ease of analysis is that, if the abstraction is too coarse or conservative, it may not be J.-W. Lee ( Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802, USA jiwoong@psu.edu G.E. Dullerud Department of Mechanical Science and Engineering, The University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA dullerud@illinois.edu J. Mohammadpour and C.W. Scherer (eds., Control of Linear Parameter Varying Systems with Applications, DOI / , Springer Science+Business Media, LLC

2 158 J.-W. Lee and G.E. Dullerud possible to prove anything about the abstracted model even though the nonlinear system on which it is based has all the properties sought. In particular, such conservatism can occur in the situation where the underlying nonlinear dynamics are operating about several different equilibrium points. Although frequently possible to use a pure LPV model to abstract the nonlinear dynamics in this scenario, there is potentially much to gain in terms of reducing conservatism of the abstraction by explicitly modeling the logical switches between equilibrium points. This motivates the class of switched LPV systems considered in this chapter. Further motivating the switched LPV class, even when simply operating around a single equilibrium point, is the situation where a nonlinear system to be analyzed exhibits multiple modes of operation due to jumps in the dynamics; in this case, it is natural to abstract the system by multiple LPV models and a switching logic between them, so that a switch from one LPV model to another corresponds to a jump from one nonlinear dynamical equation to another. Whenever such a switched LPV abstraction of a nonlinear system satisfies certain stability and performance specifications over all admissible parameter trajectories and mode switching sequences, the original nonlinear system is expected to satisfy the same stability and performance requirements. For these reasons, switched LPV approaches have found applications to a variety of nonlinear analysis and control problems such as gain-scheduled missile autopilot [10], active magnetic bearing system design [11], high-performance aircraft control [12], and multi-objective control of a wind turbine [9]. In this chapter, we focus on discrete-time switched LPV systems, where each LPV model is associated with a parameter polytope. Such systems have already been considered in the literature in the context of stabilization, H -type disturbance attenuation, and model reduction [14, 15]. However, these results are based on a conservative analysis of stability and disturbance attenuation properties. Thus, our first objective is to present a nonconservative, convex analysis of these properties by extending the existing nonconservative analysis results for LPV systems [6] and switched linear systems [7, 8] to switched LPV systems. The resulting analysis conditions are expressed in terms of an increasing union of Lyapunov inequalities (for stability and Kalman Yakubovich Popov inequalities (for disturbance attenuation performance indexed by the number of most recent past system modes and parameters that the associated quadratic Lyapunov function depends on. Our next objective is to use these analysis conditions to obtain nonconservative, convex synthesis conditions for a certain type of robust state-feedback controllers that guarantee stability and disturbance attenuation bounds. These controllers are robust in the sense that they do not depend on the current system mode or parameter value. However, we assume that the system mode and parameter become available to the controller with a unit delay, and that the state-feedback gain matrix is parameterized by a finite number of past modes and parameters. Our synthesis conditions thus complement existing results, which are limited to controllers that do not recall past modes or parameters. The organization of the chapter is as follows. In Sects. 7.2 and 7.3, wepresent stability analysis and robust stabilization results. Then these results are generalized

3 7 Switched LPV Systems in Discrete Time 159 to disturbance attenuation problems in Sects. 7.4 and 7.5. The analysis and synthesis results are illustrated and compared with existing results via numerical examples in Sect Then a concluding remark is made in Sect Notation Denoted by R, N, andn 0 are the spaces of real numbers, positive integers, and nonnegative integers, respectively. For x R n, denoted by x is the Euclidean norm of x defined by x = ( x T x 1/2.ForX R n n, we write X > 0 (resp. X < 0 to indicate that X is symmetric and positive definite (resp. negative definite. The identity matrix (resp. zero matrix is denoted by I (resp. 0 with its dimension understood. 7.2 Stability Let N, M 1,...,M N N be given. Let Θ {1,...,N} be a nonempty set of infinite sequences in {1,...,N}. For each i {1,...,N} let Λ i be the set of all probability distributions on {1,...,M i }; namely, Λ i = { ( λ = λ (1,...,λ (M i R M i : λ (1,...,λ (Mi 0and Then, for each θ =(θ(0,θ(1,... Θ,define M i j=1 } λ ( j = 1. Λ θ = Λ θ(0 Λ θ(1 = { (σ(0,σ(1,...: σ(t Λ θ(t for all t N 0 }, so that Λ θ is the space of all infinite sequences (σ(0,σ(1,..., whereσ(t is a probability distribution on {1,...,M θ(t } for each t N 0. Let n N and A ij R n n be given for i = 1,..., N and j = 1,...,M i. Write A iλ = M i j=1 ( for i {1,...,N} and λ = λ (1,...,λ (M i polyhedron defined by λ ( j A ij A i = {A iλ : λ Λ i } R n n Λ i. Then the polytope (i.e., bounded

4 160 J.-W. Lee and G.E. Dullerud is the convex hull of matrices A i1,..., A imi for each i = 1,..., N. With A = {A 1,...,A N }, the pair (A,Θ defines the discrete-time switched polytopic linear parametervarying (or switched LPV system, whose state-space description takes the form x(t + 1=A θ(tσ(t x(t, t N 0, (7.1 for mode sequences θ =(θ(0,θ(1,... Θ and parameter sequences σ = (σ(0,σ(1,... Λ θ. In the special case of N = 1 (i.e., single mode, the switched LPV system reduces to an LPV system A 1, where the time-varying parameter σ(t determines the state matrix A 1σ(t for (7.1 from a polytope of matrices (i.e., the convex hull of A 11,...,A 1M1. On the other hand, in the spacial case of M 1 = = M N = 1, the switched LPV system (A,Θ reverts to the switched linear system ({A 11,...,A N1 },Θ, where the time-varying mode θ(t determines the state matrix A θ(t1 for (7.1 from a finite set of matrices. Our stability requirement for the switched LPV system is that the state x(t of the state-space model (7.1 converges to the origin with a single exponential decay rate uniformly in time and also uniformly over mode sequences and parameter sequences. Definition 7.1. The switched LPV system (A,Θ is said to be uniformly exponentially stable if there exist c 1andλ (0, 1 such that the state-space model (7.1 satisfies x(t cλ t t 0 x(t 0 (7.2 for all t, t 0 N 0 with t t 0,forallx(t 0 R n,forallθ Θ, andforallσ Λ θ. The stability of the switched LPV system (A,Θ is closely related to that of an associated switched linear system. Let  = {A 11,...,A 1M1,...,A N1,...,A NMN }. An infinite sequence of (pairs of indices (i 0 j 0,i 1 j 1,... is a switching sequence for A if it {1,...,N} and j t {1,...,M it } for all t N 0.Let Θ be the set of all switching sequences for A restricted to the mode sequences in Θ;thatis, Θ = { } (i 0 j 0,i 1 j 1,...: (i 0,i 1,... Θ, j t = 1,...,M it, t N 0. Then the pair (Â, Θ defines the discrete-time switched linear system whose state-space description is given by (7.1 for switching sequences (θ,σ = (θ(0σ(0,θ(1σ(1,... Θ. The stability requirement for this switched linear system is consistent with that for the linear LPV system.

5 7 Switched LPV Systems in Discrete Time 161 Definition 7.2. The switched linear system A, Θ is said to be uniformly exponentially stable if there exist c 1andλ (0,1 such that the state-space model (7.1 satisfies (7.2forallt, t 0 N 0 with t t 0,forallx(t 0 R n,andforall (θ,σ Θ. To simplify notation, set θ(t =0andσ(t ( =1(i.e.,Λ ( 0 = {1} fort < 0 whenever θ Θ and σ Λ θ.definel M (Θ resp. L M Θ as the set of all switching ( paths of length M N 0 that appear in at least one of the mode sequences in Θ resp. switching sequences in Θ : L M (Θ= { } (θ(t M,...,θ(t: θ Θ, t N 0, (7.3a { ( L M Θ = ˆθ(t M,..., ˆθ(t : ˆθ Θ, } t N 0. (7.3b Then let N M (Θ be the largest subset of L M (Θ satisfying the following: For each (i 0,...,i M N M (Θ, thereexistk N with K > M and (i M+1,...,i K such that (i K M,...,i K =(i ( 0,...,i M and (i k,...,i k+m N( M (Θ for 0 k K M. Similarly, let N M Θ be the largest subset of L M Θ satisfying the following: For each (i 0 j 0,...,i M j M N M Θ,thereexistK N with K > M and (i M+1 j M+1,...,i K j K such that (i K M j K M,...,i K j K =(i 0 j 0,...,i M j M and (i k j k,...,i k+m j k+m N M Θ for 0 k K M. We will use the convention that (i k,...,i l =0(resp. (i k j k,...,i l j l =01 if k > l; otherwise, (i k,...,i l (resp. (i k j k,...,i l j l is a switching path of length l k. The following theorem gives an exact convex condition for the stability of switched LPV systems in terms of linear matrix inequalities. Theorem 7.1. The following are equivalent: (a The switched LPV system (A(,Θ is uniformly exponentially stable. (b The switched linear system A, Θ is uniformly exponentially stable. (c There exist a path length M N 0 and an indexed (finite family of matrices Y (i1 j 1,...,i M j M R n n such that Y (i0 j 0,...,i M 1 j M 1 > 0, (7.4a A im j M Y (i0 j 0,...,i M 1 j M 1 A T i M j M Y (i1 j 1,...,i M j M < 0 (7.4b for all (i 0 j 0,...,i M j M N M Θ. (d There exist a path length M N 0, real numbers α, β > 0, and an indexed (uncountably infinite family of matrices Y (i1 λ 1,...,i M λ M R n n such that αi Y (i0 λ 0,...,i M 1 λ M 1 β I, A im λ M Y (i0 λ 0,...,i M 1 λ M 1 A T i M λ M Y (i1 λ 1,...,i M λ M αi for all (i 0,...,i M N M (Θ and for all (λ 0,...,λ M Λ i0 Λ im. (7.5a (7.5b

6 162 J.-W. Lee and G.E. Dullerud Moreover, if (c holds with M N, then (d is satisfied with Y (i0 λ 0,...,i M 1 λ M 1 = Y (i1 λ 1,...,i M λ M = M i0 j 0 =1 M i1 j 1 =1 M im 1 j M 1 =1 M im j M =1 λ ( j 0 0 λ ( j M 1 M 1 Y (i0 j 0,...,i M 1 j M 1, (7.6a λ ( j 1 1 λ ( j M M Y (i 1 j 1,...,i M j M (7.6b ( for (λ 0,...,λ M Λ i0 Λ im,whereλ k = λ (1 k,...,λ (M i k k,k= 0,...,M. If (c holds with M = 0, then (d is satisfied with Y (i0 λ 0,...,i M 1 λ M 1 = Y (i1 λ 1,...,i M λ M = Y 01. Proof. The proof extends that of [6, Theorem 1]. We will show that (a (b (c (a; the equivalence (c (d will follow as a by-product. It is clear that (a implies (b. Due to [7, Corollary 3.4], condition (b implies the existence of an M N 0 and matrices X (i1 j 1,...,i M j M > 0 such that A T i M j M X (i1 j 1,...,i M j M A im j M X (i0 j 0,...,i M 1 j M 1 < 0 for all (i 0 j 0,...,i M j M N M Θ. The Schur complement formula, along with Y (i1 j 1,...,i M j M = X 1 (i 1 j 1,...,i M j M, then yields (c. Suppose (c holds true, so that (7.4 is satisfied for all (i 0 j 0,...,i M j M N M Θ. Similarly to the proof of [7, Corollary 3.4], run the following algorithm to enlarge the set N M Θ to L M Θ : Step 0. Set L = N M Θ. Step 1. If L = L M Θ, then stop; otherwise, choose a switching path (i 0 j 0,..., i M j M L M Θ L such that (i 1 j 1,...,i M+1 j M+1 L for some i M+1 {1,...,N} and j M+1 {1,...,M im+1 }. Step 2. If i 0 j 0,...,i } M 1 j M 1, ˆk M ˆl M / L for any ˆk M {1,..., M} and ˆl M {1,...,, then choose a Y > 0 such that MˆkM A im j M YA T i M j M Y (i1 j 1,...,i M j M < 0, put Y (i0 j 0,...,i M 1 j M 1 = Y, and go to Step 4.

7 7 Switched LPV Systems in Discrete Time 163 Step 3. Choose an ε > 0suchthat εa im j M Y (i0 j 0,...,i M 1 j M 1 A T i M j M Y (i1 j 1,...,i M j M < 0 and substitute Y (i0 j 0,...,i M 1 j M 1 with εy (i0 j 0,...,i M 1 j M 1. Whenever there exist K N and (i K j K,...,i 1 j 1 such that (i k K j k K,..., i k K+M j k K+M L for all k = 0,..., K, then scale Y (ik K j k K,..., i k K+M j k K+M with the same scaling factor ε for all k = 0,..., K as well. Step 4. Substitute L with L {(i 0 j 0,...,i M j M } andgotostep 1. By the definition of N M Θ, each step of this algorithm (including Step 3 is well defined. Moreover, the algorithm produces an extended set of matrices Y (i1 j 1,...,i M j M such that (7.4 holds for all (i 0 j 0,...,i M j M L M Θ. Assume M N without loss of generality. Since L M Θ is a finite set, there exist α, β > 0 such that αi Y (i0 j 0,...,i M 1 j M 1 β I (7.7a and A im j M Y (i0 j 0,...,i M 1 j M 1 A T i M j M Y (i1 j 1,...,i M j M < αi for all (i 0 j 0,...,i M j M L M Θ. Applying the Schur compliment formula to the last inequality yields [ ] αi Y (i1 j 1,...,i M j M A im j M Y (i0 j 0,...,i M 1 j M 1 < 0. (7.7b Y (i0 j 0,...,i M 1 j M 1 Given ( (i 0,...,i M L M (Θ, choose (λ 0,...,λ M Λ i0 Λ im with λ k = λ (1 k,...,λ (M it k for k = 0,...,M, whereλ 0 = {1}. DefineY (i0 λ 0,...,i M 1 λ M 1 and Y (i1 λ 1,...,i M λ M as in (7.6. Taking the weighted sum of (7.7 with weights given by (λ 0,...,λ M then yields (7.5a and [ ] αi Y (i1 λ 1,...,i M λ M A im λ M Y (i0 λ 0,...,i M 1 λ M 1 < 0. Y (i0 λ 0,...,i M 1 λ M 1 Taking the Schur complement of Y (i0 λ 0,...,i M 1 λ M 1 from this inequality, we obtain (7.5b. This shows that, if (c holds, then (7.5 is satisfied for all (i 0,...,i M L M (Θ and (λ 0,...,λ M Λ i0 Λ im. Because this implies (d and because (c is a special case of (d, it is immediate that (c and (d are equivalent. To complete the proof, we will show (a holds true given that (7.5 is satisfied for all (i 0,...,i M L M (Θ and (λ 0,...,λ M Λ i0 Λ im. Choose a mode sequence θ Θ and a parameter sequence σ Λ θ.put A(t=A θ(tσ(t, Y(t=Y (θ(t Mσ(t M,...,θ(t 1σ(t 1

8 164 J.-W. Lee and G.E. Dullerud for all t N 0,sothat αi Y(t β I, A(tY(tA(t T Y(t + 1 αi for all t N 0. If we put X(t=Y(t 1, then there exists an η > 0, independent of θ and σ, such that β 1 I X(t α 1 I, A(t T X(t + 1A(t X(t ηi for all t N 0. Thus, by specializing [5, Corollary 12] to pure stability analysis, we deduce that there exist c 1andλ (0,1 such that the linear time-varying system (7.1 satisfies (7.2forallt, t 0 N 0 with t t 0.Sinceθ Θ and σ Λ θ are arbitrary, and since the constants c and λ can be determined solely from α 1, β 1, and η (see, e.g., [8, Lemma 4], we conclude that (a holds true. According to Theorem 7.1, only the mode switching paths in N M (Θ are relevant to stability. This is because the switching paths outside N M (Θ cannot appear more than once in any mode sequence in Θ, and because the number of such switching paths is finite for each path length M. Although there is no upper bound on the path length M that is required for our stability test, it is usually the case in practice that one only needs to try the first few path lengths M. This agrees with the fact that the common Lyapunov function approach (i.e., the case of M = 0 and the multiple Lyapunov function approaches (i.e., versions of the case of M = 1 have been very useful in practice. What Theorem 7.1 gives us is the option to go beyond M = 0andM = 1ifweare willing and able to pay additional computational cost in return for potentially better stability analysis. 7.3 Stabilization Let m, n N, A ij R n n,andb ij R n m be given for i = 1,..., N and j = 1,..., M i. Write A iλ = M i j=1 whenever i {1,...,N} and λ = λ ( j A ij and B iλ = M i j=1 λ ( j B ij ( λ (1,...,λ (M i Λ i. The polytopes defined by A i = {A iλ : λ Λ i } R n n, B i = {B iλ : λ Λ i } R n m

9 7 Switched LPV Systems in Discrete Time 165 are the convex hulls of A i1,..., A imi and B i1,..., B imi, respectively, for each i = 1,..., N. As in the previous section, let Θ {1,...,N} be a nonempty set of mode sequences. Then, with G = {(A 1,B 1,...,(A N,B N }, the pair (G,Θ defines the controlled version of the discrete-time switched LPV system described by x(t + 1=A θ(tσ(t x(t+b θ(tσ(t u(t, t N 0, (7.8 for mode sequences θ Θ, parameter sequences σ Λ θ, and control sequences u =(u(0,u(1,... We will consider all linear state-feedback controllers that generate the control input u(t at each time t N 0 based on a finite number L N 0 of past mode sequences θ(t L,..., θ(t 1 and parameter sequences σ(t L,..., σ(t 1 as well as the perfectly observed current state x(t. As in the previous section, let θ(t=0andσ(t=1fort < 0 whenever θ Θ and σ Λ θ. Also, write (θσ L (t=(θ(t Lσ(t L,...,θ(tσ(t, (θσ L (t =(θ(t Lσ(t L,...,θ(t 1σ(t 1, (θσ L (t + =(θ(t L + 1σ(t L + 1,...,θ(tσ(t for L, t N 0, θ Θ, andσ Λ θ. For a fixed path length L N 0,letΛ 0 = {1} and K = {K (i1 λ 1,...,i L λ L : λ k Λ ik,i k = 0,1,...,N, k = 1,...,L} R m n if L > 0, and let K = {K 01 } R m n be a singleton if L = 0. Then K defines a robust L-path-dependent state-feedback controller described by u(t=k (θσl (t x(t, t N 0, (7.9 if L > 0, and u(t=k 01 x(t, t N 0,ifL = 0. For example, if L = 2, then and u(0=k (01,01 x(0, u(1=k (01,θ(0σ(0 x(1, u(t=k (θ(t 2σ(t 2,θ(t 1σ(t 1 x(t for t 2. Clearly, the case of L = 0 corresponds to the robust state-feedback controller in the usual sense; if L > 0, on the other hand, the controller perfectly observes the mode and parameter sequences with a unit delay (or less and performs gain scheduling based on the most recent past L modes and parameters.

10 166 J.-W. Lee and G.E. Dullerud The feedback interconnection of the controlled system (G,Θ, described by (7.8, and a robust path-dependent controller K, described by (7.9, gives rise to a closed-loop system whose state evolves according to x(t + 1= ( A θ(tσ(t + B θ(tσ(t K (θσl (t x(t, t N0. (7.10 We now present an exact convex condition for the existence of a stabilizing robust path-dependent controller, and a synthesis procedure guaranteed to yield such a controller, if it exists. Definition 7.3. The switched LPV system (G,Θ is said to be uniformly exponentially stabilizable if there exist c 1, λ (0,1, L N 0, and a robust L-path-dependent state-feedback controller such that the closed-loop state-space model (7.10 satisfies (7.2 forallt, t 0 N 0 with t t 0,forallx(t 0 R n,forall θ Θ,andforallσ Λ θ. Theorem 7.2. The switched LPV system (G,Θ is uniformlyexponentiallystabilizable if and only if there exist a path length M N 0 and indexed (finite families of matrices W (i1 j 1,...,i M j M R m n and Y (i1 j 1,...,i M j M R n n such that [ ] Y (i1 j 1,...,i M j M A im j M Y (i0 j 0,...,i M 1 j M 1 + B im j M W (i0 j 0,...,i M 1 j M 1 < 0 Y (i0 j 0,...,i M 1 j M 1 (7.11 for all (i 0 j 0,...,i M j M N M Θ. Moreover, if (7.11 holds with M N, thena robust M-path-dependent state-feedback controller K that uniformly exponentially stabilizes the system (G,Θ is given by K (i0 λ 0,...,i M 1 λ M 1 = W (i0 λ 0,...,i M 1 λ M 1 Y 1 (i 0 λ 0,...,i M 1 λ M 1 (7.12a for all (i 0,...,i M N M (Θ and for all (λ 0,...,λ M Λ i0 Λ im,where Y (i0 λ 0,...,i M 1 λ M 1 = W (i0 λ 0,...,i M 1 λ M 1 = M i0 j 0 =1 M i0 j 0 =1 M im 1 j M 1 =1 M im 1 j M 1 =1 λ ( j 0 0 λ ( j M 1 M 1 Y (i0 j 0,...,i M 1 j M 1, (7.12b λ ( j 0 0 λ ( j M 1 M 1 W (i0 j 0,...,i M 1 j M 1 (7.12c ( for (λ 0,...,λ M 1 Λ i0 Λ im 1 with λ k = λ (1 k,...,λ (M i k k,k= 0,...,M 1. If (7.11 holds with M = 0, then a robust uniformly exponentially stabilizing statefeedback controller K is given by K 01 = W 01 Y 1 01.

11 7 Switched LPV Systems in Discrete Time 167 Proof. The proof is an extension of [6, Theorem 2]. Suppose that the closed-loop system with a robust L-path-dependent controller K is uniformly exponentially stable. Then there exist c 1andλ (0,1 such that (7.2 holds for all t, t 0 N 0 with t t 0,forallx(t 0 R n,forallθ Θ,andforallσ Λ θ. Following the proof of [8, Lemma 4(a], it is readily seen that there exists an M L, constants α, β > 0 (which depend solely on c and λ, and matrices Y (i1 j 1,...,i M j M > 0 such that αi Y (θσm (t β I,  (θσl (ty (θσm (t  T (θσ L (t Y (θσ M (t + < αi for all t N 0, θ Θ, andσ Λ θ,where  (θσl (t = A θ(tσ(t + B θ(tσ(t K (θσl (t are the closed-loop state matrices. In particular, this holds whenever θ M (t N L (Θ and σ M (t Λ θ(t M Λ θ(t,andso αi Y (i0 λ 0,...,i M 1 λ M 1 β I (7.13a and  (im L λ M L,...,i M λ M Y (i0 λ 0,...,i M 1 λ M 1 ÂT (i M L λ M L,...,i M λ M Y (i1 λ 1,...,i M λ M < αi for all (i 0,...,i M N M (Θ and (λ 0,...,λ M Λ i0 Λ im.asm L, the L-path-dependent controller K can be taken to be M-path-dependent, so we can assume L = M > 0 without loss of generality. Now, applying the Schur complement formula to the last inequality gives [ ] αi Y (i1 λ 1,...,i M λ M AiM λ M + B im λ M K (i0 λ 0,...,i M 1 λ M 1 Y(i0 λ 0,...,i M 1 λ M 1 < 0 Y (i0 λ 0,...,i M 1 λ M 1 (7.13b for all (i 0,...,i M N M (Θ and (λ 0,...,λ M Λ i0 Λ im. Specializing (7.13 to the associated switched linear system over all switching( sequences in Θ, and using (7.12a, we obtain (7.11forall(i 0 j 0,...,i M j M N M Θ. This establishes necessity. To show sufficiency, suppose (7.11 holds for all (i 0 j 0,...,i M j M N M Θ. Since (7.11 defines a finite number of inequalities over a finite number of matrix variables, there exist α, β > 0 such that, along with (7.12, we have (7.13 for

12 168 J.-W. Lee and G.E. Dullerud all (i 0,...,i M N M (Θ and (λ 0,...,λ M Λ i0 Λ im. Taking the Schur complement of Y (i0 λ 0,...,i M 1 λ M 1 in (7.13b gives αi Y (i0 λ 0,...,i M 1 λ M 1 β I,  (i0 λ 0,...,i M λ M Y (i0 λ 0,...,i M 1 λ M 1 ÂT (i 0 λ 0,...,i M λ M Y (i 1 λ 1,...,i M λ M αi for all (i 0,...,i M N M (Θ and (λ 0,...,λ M Λ i0 Λ im,where  (i0 λ 0,...,i M λ M = A im λ M + B im λ M K (i0 λ 0,...,i M 1 λ M 1 are the closed-loop state matrices. Now, the equivalence of (a and (d in Theorem 7.1 implies that the closed-loop system is uniformly exponentially stable. According to Theorem 7.2, only the feedback gain matrices K (θσm (t over θ M (t N M (Θ, t N 0,andθ Θ are relevant to the stability of the closed-loop system. The remaining feedback gain matrices can be chosen arbitrarily. Note that Theorem 7.2 gives an exact synthesis condition, but that it is limited to the cases where the mode and parameter are observed with a unit delay. If either the current mode or the current parameter is available for measurement, then one can use the results in [15]. 7.4 Performance Analysis In this section, we will address the problem of evaluating the worst-case l 2 -induced norm (i.e., the disturbance attenuation property of a switched LPV system. Given l, m, n N, and given A ij R n n, B ij R n m, C ij R l n,andd ij R l m for i = 1,...,N and for j = 1,...,M i, consider the state-space model x(t + 1 =A θ(tσ(t x(t+b θ(tσ(t w(t, t N 0 ; z(t =C θ(tσ(t x(t+d θ(tσ(t w(t, t N 0, (7.14 over mode sequences θ Θ, parameter sequences σ Λ θ, and disturbance sequences w =(w(0,w(1,...; the error output sequence is given by z = (z(0,z(1,... Writing A i = {A iλ : λ Λ i } R n n, B i = {B iλ : λ Λ i } R n m, C i = {C iλ : λ Λ i } R l n, D i = {D iλ : λ Λ i } R l m

13 7 Switched LPV Systems in Discrete Time 169 for i {1,...,N},let S = {(A 1,B 1,C 1,D 1,...,(A N,B N,C N,D N }. If Θ is a nonempty subset of {1,...,N}, then the pair (S,Θ defines the discretetime switched LPV system whose l 2 -induced norm under given θ Θ and σ Λ θ is defined by the supremum of the square root of t=0 z(t 2 / t=0 w(t 2 over all w with t=0 w(t 2 <. We are concerned with evaluating the worst-case l 2 - induced norm over all θ and σ. The system (S,Θ shall be said to be uniformly exponentially stable if (A,Θ is uniformly exponentially stable. Definition 7.4. A uniformly exponentially stable switched LPV system (S,Θ is said to satisfy uniform disturbance attenuation level γ > 0 if there exists γ (0,γ such that t=0 z(t 2 γ 2 w(t 2 (7.15 t=0 for all θ Θ,forallσ Λ θ,andforallw with t=0 w(t 2 <. As in the case of pure stability, the disturbance attenuation property of the switched LPV system (S,Θ is closely related to that of the switched linear system (Ŝ, Θ,where Ŝ = {(A ij,b ij,c ij,d ij : i = 1,...,N, j = 1,...,M i }, Θ = { } (i 0 j 0,i 1 j 1,...: (i 0,i 1,... Θ, j t = 1,...,M it, t N 0. The state-space description of the switched linear system (Ŝ, Θ is given by (7.14, and hence the same as that of the switched LPV system (S,Θ, except that it is restricted to switching sequences (θ,σ=(θ(0σ(0,θ(1σ(1,... Θ. The system (Ŝ Θ (, is said to be uniformly exponentially stable if A, Θ is uniformly exponentially stable. The performance requirement for (Ŝ, Θ is consistent with that for (S,Θ. Definition 7.5. A uniformly exponentially stable switched linear system (Ŝ, Θ is said to satisfy uniform disturbance attenuation level γ > 0 if there exists γ (0,γ such that (7.15 holds for all (θ,σ Θ and for all w with t=0 w(t 2 <. We will continue to use the convention that θ(t=0andσ(t=1(i.e.,λ 0 = {1} for t < 0 whenever θ Θ and σ Λ θ.letl M (Θ and L M Θ be as in (7.3for path lengths M N 0. The following theorem gives an exact convex condition for the stability and performance of switched LPV systems in terms of linear matrix inequalities.

14 170 J.-W. Lee and G.E. Dullerud Theorem 7.3. Let γ > 0. The following are equivalent: (a The switched LPV system (S,Θ is uniformly exponentially stable and satisfies uniform disturbance attenuation level γ. (b The switched linear system (Ŝ, Θ is uniformly exponentially stable and satisfies uniform disturbance attenuation level γ. (c There exist a path length M N 0 and an indexed (finite family of matrices Y (i1 j 1,...,i M j M R n n such that Y (i0 j 0,...,i M 1 j M 1 > 0, (7.16a [ ][ ][ ] T AiM j M B im j Y(i0 M j 0,...,i M 1 j M 1 0 AiM j M B im j M C im j M D im j M 0 I C im j M D im j M [ ] Y(i1 j 1,...,i M j M 0 0 γ 2 < 0 (7.16b I for all (i 0 j 0,...,i M j M L M Θ. (d There exist a path length M N 0, real numbers α, β > 0, and an indexed (uncountably infinite family of matrices Y (i1 λ 1,...,i M λ M R n n such that αi Y (i0 λ 0,...,i M 1 λ M 1 β I, (7.17a [ AiM λ M B im λ M ][ ][ Y(i0 λ 0,...,i M 1 λ M 1 0 AiM λ M B im λ M ] T C im λ M D im λ M 0 I [ ] Y(i1 λ 1,...,i M λ M 0 0 γ 2 αi I C im λ M D im λ M (7.17b for all (i 0,...,i M L M (Θ and for all (λ 0,...,λ M Λ i0 Λ im. Moreover, if (c holds with M ( N, then (d is satisfied with (7.6 for (λ 0,...,λ M Λ i0 Λ im,whereλ k = λ (1 k,...,λ (M i k k,k= 0,...,M. If (c holds with M = 0, then (d is satisfied with Y (i0 λ 0,...,i M 1 λ M 1 = Y (i1 λ 1,...,i M λ M = Y 01. Proof. We will show (a (b (c (d (a. Clearly (a implies (b. Suppose (b holds. Then, due to the proof of the necessity part of [7, Theorem 3.3] and a simple scaling argument to take into account γ 1, there exist X (i0 j 0,...,i M 1 j M 1 > 0 satisfying [ ] T [ ][ ] AiM j M B im j M X(i1 j 1,...,i M j M 0 AiM j M B im j M C im j M D im j M 0 I [ ] X(i0 j 0,...,i M 1 j M γ 2 < 0 I C im j M D im j M

15 7 Switched LPV Systems in Discrete Time 171 for all (i 0 j 0,...,i M j M L M Θ. Then the Schur complement formula, along with Y (i0 j 0,...,i M 1 j M 1 = γ 2 X 1 (i 0 j 0,...,i M 1 j M 1, gives ((c. Suppose (c holds, and assume M N without loss of generality. Since L M Θ is a finite set, there exist α, β > 0suchthat αi Y (i0 j 0,...,i M 1 j M 1 β I (7.18a and [ ][ ][ AiM j M B im j Y(i0 M j 0,...,i M 1 j M 1 0 AiM j M B im j M C im j M D im j M 0 I C im j M D im j M [ ] [ ] Y(i1 j 1,...,i M j M 0 αi 0 0 γ 2 < I 0 αi for all (i 0 j 0,...,i M j M L M Θ. Applying the Schur compliment formula to the last inequality yields αi Y (i1 j 1,...,i M j M 0 A im j M Y (i0 j 0,...,i M 1 j M 1 B im j M αi γ 2 IC im j M Y (i0 j 0,...,i M 1 j M 1 D im j M Y (i0 j 0,...,i M 1 j M 1 0 < 0. (7.18b I Given ( (i 0,...,i M L M (Θ, choose (λ 0,...,λ M Λ i0 Λ im with λ k = λ (1 k,...,λ (M it k for k = 0,...,M, whereλ 0 = {1}. DefineY (i0 λ 0,...,i M 1 λ M 1 and Y (i1 λ 1,...,i M λ M as in (7.6. Taking the weighted sum of (7.18 with weights given by (λ 0,...,λ M then yields (7.17a and αi Y (i1 λ 1,...,i M λ M 0 A im λ M Y (i0 λ 0,...,i M 1 λ M 1 B im λ M αi γ 2 IC im λ M Y (i0 λ 0,...,i M 1 λ M 1 D im λ M Y (i0 λ 0,...,i M 1 λ M 1 0 < 0. I Using the Schur complement formula once more, we obtain (7.17b. Thus, (d holds true. It remains to show (d implies (a. Suppose (d holds, so that (7.17 is satisfied for all (i 0,...,i M L M (Θ and (λ 0,...,λ M Λ i0 Λ im. Assume M N without loss of generality, and fix a θ Θ and a σ Λ θ.put A(t=A θ(tσ(t, B(t=B θ(tσ(t, C(t=C θ(tσ(t, D(t=D θ(tσ(t, ] T

16 172 J.-W. Lee and G.E. Dullerud and for t N 0,sothat Y(t=Y (θ(t Mσ(t M,...,θ(t 1σ(t 1 [ A(t B(t C(t D(t ][ Y(t 0 0 I αi Y(t β I, ][ ] T A(t B(t C(t D(t [ ] Y(t γ 2 αi I for all t N 0. If we put X(t=γ 2 Y(t 1, then there exists an η > 0, independent of θ and σ, such that [ A(t B(t C(t D(t γ 2 β 1 I X(t γ 2 α 1 I, ] T [ ][ ] X(t A(t B(t 0 I C(t D(t [ ] X(t 0 0 γ 2 ηi I for all t N 0. Thus, by [5, Corollary 12] with an appropriate scaling argument, the linear time-varying system (7.14 is uniformly exponentially stable and satisfies uniform disturbance attenuation level γ.sinceθ Θ and σ Λ θ are arbitrary, and since α 1, β 1,andη can be chosen independently of (θ,σ, we conclude that (a holds true. Note that, in Theorem 7.3, the Kalman Yakubovich Popov (KYP inequality (7.16 is required to be satisfied over all switching paths in L M Θ, including the transient paths that contain the dummy mode-parameter pair 01. Compare this with Theorem 7.1, where the ( Lyapunov inequality (7.4 isrequiredoverasmaller set of switching paths N M Θ. If the mode sequence and parameter sequence are fixed, then this agrees with the intuition that, while only the switching paths that occur infinitely many times in the mode sequence is relevant to uniform exponential stability, every switching path including those that never occur more than once in the mode sequence counts as far as disturbance attenuation performance is concerned. Theorems 7.1 and 7.3 make this intuition precise for the case where the mode and parameter sequences are nondeterministic. 7.5 Performance Optimization Given l, m 1, m 2, n N, and given A ij R n n, B 1,ij R n m 1, B 2,ij R n m 2, C ij R l n, D 1,ij R l m 1,andD 2,ij R l m for i = 1,...,N and for j = 1,...,M i, consider the controlled state-space model x(t + 1 =A θ(tσ(t x(t+b 1,θ(tσ(t w(t+b 2,θ(tσ(t u(t, t N 0 ; z(t =C θ(tσ(t x(t+d 1,θ(tσ(t w(t+d 2,θ(tσ(t u(t, t N 0. (7.19

17 7 Switched LPV Systems in Discrete Time 173 Our objective in this section is to extend the result of the previous section to the problem of designing a state-feedback controller that optimizes the disturbance attenuation performance of the closed-loop system. Defining matrix polytopes A i = {A iλ : λ Λ i } R n n, B 1,i = {B 1,iλ : λ Λ i } R n m 1, B 2,i = {B 2,iλ : λ Λ i } R n m 2, C i = {C iλ : λ Λ i } R l n, D 1,i = {D 1,iλ : λ Λ i } R l m 1, D 2,i = {D 2,iλ : λ Λ i } R l m 2 for i = 1,...,N, let T = {(A 1,B 1,1,B 2,1,C 1,D 1,1,D 2,1,...,(A N,B 1,N,B 2,N,C N,D 1,N,D 2,N }. If Θ is a nonempty subset of {1,...,N}, then the pair (T,Θ defines the controlled version of the discrete-time switched LPV system (S,Θ considered in the previous section. The system (T,Θ is said to be uniformly exponentially stable if (A,Θ is uniformly exponentially stable. We will consider all linear state-feedback controllers that recall L most recent past modes and parameters for some L N 0.Let K = {K (i1 λ 1,...,i L λ L : λ k Λ ik,i k = 0,1,...,N, k = 1,...,L} R m 2 n if L > 0, and let K = {K 01 } R m 2 n be a singleton if L = 0. Then K defines such a controller, which we call a robust L-path-dependent state-feedback controller. The feedback interconnection of the controlled system (T,Θ and a robust pathdependent controller K gives rise to a closed-loop system of the form x(t + 1 = ( A θ(tσ(t + B 2,θ(tσ(t K (θσl (t x(t+b1,θ(tσ(t w(t, t N 0 ; z(t = C θ(tσ(t + D 2,θ(tσ(t K (θσl (t x(t+d1,θ(tσ(t w(t, t N 0. (7.20 Definition 7.6. Let γ > 0andL N 0.ArobustL-path-dependent state-feedback controller K is said to achieve uniform disturbance attenuation level γ for the switched LPV system (T,Θ if there exists γ (0,1 such that the closed-loop state-space model (7.20 is uniformly exponentially stable and satisfies (7.15 for all θ Θ,forallσ Λ θ,andforallw with t=0 w(t 2 <. Theorem 7.4. Let γ > 0. There existsarobust finite-path-dependentstate-feedback controller that stabilizes and achieves uniform disturbance attenuation level γ for the switched LPV system (T,Θ if and only if there exist a path length M N 0 and indexed (finite families of matrices W (i1 j 1,...,i M j M R m 2 n and Y (i1 j 1,...,i M j M

18 174 J.-W. Lee and G.E. Dullerud R n n such that Y (i1 j 1,...,i M j M 0 F (i0 j 0,...,i M j M B 1,iM j M γ 2 I G (i0 j 0,...,i M j M D 1,iM j M Y (i0 j 0,...,i M 1 j M 1 0 < 0, I (7.21a with F (i0 j 0,...,i M j M = A im j M Y (i0 j 0,...,i M 1 j M 1 + B 2,iM j M W (i0 j 0,...,i M 1 j M 1, (7.21b G (i0 j 0,...,i M j M = C im j M Y (i0 j 0,...,i M 1 j M 1 + D 2,iM j M W (i0 j 0,...,i M 1 j M 1 (7.21c for all (i 0 j 0,...,i M j M L M Θ. Moreover, if (7.21 holds with M N, then a robust M-path-dependent state-feedback controller K that achieves uniform disturbance attenuation level γ for the system (T,Θ is given by (7.12 for all (i 0,...,i M L M (Θ and for all (λ 0,...,λ M Λ i0 Λ im.if (7.21 holds with M = 0, then a robust uniformly exponentially stabilizing state-feedback controller K is given by K 01 = W 01 Y Proof. The proof is based on Theorem 7.3 but otherwise parallels that of Theorem 7.2, so it is omitted. Theorem 7.4 gives an exact, convex condition for the existence of suboptimal robust finite-path-dependent state-feedback controllers. If an optimal (stabilizing controller exists, then one can run the sequence of semidefinite programs that minimize γ 2 subject to linear matrix inequalities (7.21 for path lengths M = 0,1,... As in the case of pure stabilization, the minimal value of γ 2 saturates fast as one goes down this sequence of semidefinite programs, and thus trying only the first few path lengths M often suffices in practice. Theorem 7.4 is a direct extension of Theorem 7.2 to performance optimization, and, hence, it is limited to the synthesis of robust state-feedback controllers that observe the mode and parameter with a unit delay. Again, if the current mode or parameter is available for measurement, then the results in [15] can be used instead. 7.6 Illustrative Examples Example 1 In this example, we will apply Theorem 7.1 to analyze the stability of a simple switched LPV system (A,Θ. LetN = 2, M 1 = M 2 = 2, and Θ = {1,2} (i.e., the mode sequence is unconstrained. Let A have

19 7 Switched LPV Systems in Discrete Time 175 A 11 = [ ] , A 12 = [ ] α 0, A 21 = A 11, and A 22 = A 12, 1 α where α > 0. Since A 11 = A 21 and A 12 = A 22, it is easily seen that the switched LPV system (A,Θ is equivalent to the LPV system {A 1λ : λ Λ 1 } considered in [6, Example 1], which in turn is equivalent to the switched linear system ({A 11,A 12 },Θ. For each path length M N 0,letα M denote the largest value of α such that the system of Lyapunov inequalities (7.4 is feasible for all (i 0 j 0,...,i M j M N M Θ,where N 0 Θ = {11,12,21,22}, N 1 Θ = {(11,11,(11,12,(11,21,(11,22,(12,11,..., (21,22,(22,11,(22,12,(22,21,(22,22}, and so on. Then, we obtain α 0 = 0.301, α 1 = 0.478, and α 2 = α 3 = = Thus, we conclude that α = is the largest value of α for which the switched LPV system is uniformly exponentially stable. This result agrees with that of [6, Example 1]. Restricting our attention to parameter-dependent Lyapunov functions as in [4] and[3] would yield suboptimal stability bounds α = and α = 0.478, respectively. Example 2 We will borrowan examplefrom [14] anduse Theorem7.4to illustrate how optimal disturbance attenuation is achieved for switched LPV systems. Let N = 6, M 1 = = M 6 = 2, and Θ = {θ} be a singleton, where is of period 10. Let T have θ =(1,2,3,4,5,6,5,4,3,2, 1,2,3,4,5,6,5,4,3,2,... [ ] [ ] A i1 = + ρ i, B 1,i1 = C i1 = [ ] + ρ i [ ], B 2,i1 = B 1,i1, D 1,i1 = D 2,i1 = 0, [ ] [ ] ρ i,

20 176 J.-W. Lee and G.E. Dullerud and A i2 = [ ] [ ] ρ i, B 1,i2 = C i2 = [ ] + ρ i [ ], B 2,i2 = B 1,i2, D 1,i2 = D 2,i2 = 0, [ ] [ ] ρ i, where ρ i = cos((i 1π/5 for i = 1,...,6. Our objective is to achieve optimal disturbance attenuation performance for the switched LPV system (T,Θ via robust finite-path-dependent state feedback. We have L 0 (Θ ={1,2,3,4,5,6}, L 1 (Θ ={(0,1,(1,2,(2,3,(3,4,(4,5,(5,6,(6,5,...,(2,1}, L 2 (Θ ={(0,0,1,(0,1,2,(1,2,3,(2,3,4,(3,4,5,(4,5,6, (5,6,5,(6,5,4,(5,4,3,(4,3,2,(3,2,1,(2,1,2}, and so on. It is readily seen that, for this particular Θ, no path length M > 2 needs to be considered because all path lengths M 2 result in the same system of linear matrix inequalities. It is straightforward to obtain L 2 ( Θ ={(01,01,11,(01,01,12,(01,11,21,(01,11,22,(01,12,21,..., (21,12,22,(22,11,21,(22,11,22,(22,12,21,(22,12,22}. This set contains 86 switching paths. If B 2,i1 and B 2,i2 were zero for i = 1,...,6, then minimizing γ 2 subject to the system of KYP inequalities (7.16, with B ij = B 1,ij and D ij = D 1,ij, over all (i 0 j 0,i 1 j 1,i 2 j 2 L 2 Θ would yield γ = This is the minimum disturbance attenuation level of the uncontrolled system. However, minimizing γ 2 subject( to the system of linear matrix inequalities (7.21 over all (i 0 j 0,i 1 j 1,i 2 j 2 L 2 Θ gives γ = 1.55, which is the minimum performance bound achievable by a robust finite-path-dependent state-feedback controller. An optimal solution to (7.21 is given by 43 pairs of W (i0 j 0,i 1 j 1 and Y (i0 j 0,i 1 j 1.The resulting optimal controller takes the form K (01,01 x(0 if t = 0; u(t= K (01,θ(0σ(0 x(1 if t = 1; K (θ(t 2σ(t 2,θ(t 1σ(t 1 x(t, if t 2, where, whenever ( θ(t 2=i 0, σ(t 2=λ 0 = ( θ(t 1=i 1, σ(t 1=λ 1 = λ (1 0 λ (1 1 (2,λ 0,,,λ (2 1

21 7 Switched LPV Systems in Discrete Time 177 we have K (01,θ(0σ(0 = W (01,i0 λ 0 Y 1 (01,i 0 λ 0, K (θ(t 2σ(t 2,θ(t 1σ(t 1 = W (i0 λ 0,i 1 λ 1 Y 1 (i 0 λ 0,i 1 λ 1 with Y (01,i0 λ 0 = λ (1 0 Y (01,i0 1 + λ (2 0 Y (01,i 0 2, W (01,i0 λ 0 = λ (1 0 W (01,i λ (2 0 W (01,i 0 2, Y (i0 λ 0,i 1 λ 1 = λ (1 0 λ (1 1 Y (i 0 1,i λ (2 0 λ (2 1 Y (i 0 2,i 1 2, W (i0 λ 0,i 1 λ 1 = λ (1 0 λ (1 1 W (i 0 1,i λ (2 0 λ (2 1 W (i0 2,i 1 2. Example 3 We will now consider the example studied in [15,Sect.5].LetN = 2, M 1 = M 2 = 2, and Θ = {1,2}.LetT have A 11 = 0 0 1, B 1,11 = 0.1, B 2,11 = 0.1, A 12 = A 21 = A 22 = C 11 = [ ], D 1,11 = D 2,11 = 0, 0.3, B 1,12 = 0.1, B 2,12 = C 12 = [ ], D 1,12 = D 2,12 = 0, 0.3, B 1,21 = 0.1, B 2,21 = C 21 = [ ], D 1,21 = D 2,21 = 0, 0.3, B 1,22 = 0.1, B 2,22 = 0.8 C 22 = [ ], D 1,22 = D 2,22 = 0., ,,

22 178 J.-W. Lee and G.E. Dullerud The objective is to obtain a robust finite-path-dependent state-feedback controller that achieves a desired disturbance attenuation performance for the switched LPV system (T,Θ. We have L 0 (Θ={1,2}, L 1 (Θ={(0,1,(0,2,(1,1,(1,2,(2,1,(2,2}, L 2 (Θ={(0,0,1,(0,0,2,(0,1,1,(0,1,2,(0,2,1,(0,2,2,(1,1,1, (1,1,2,(1,2,1,(1,2,2,(2,1,1,(2,1,2,(2,2,1,(2,2,2}, and so on. With path length M = 0, minimizing γ 2 subject to (7.21 for all modeparameter pairs in L 0 Θ gives γ = 4.38, and a robust state-feedback controller u(t=k 01 x(t, t N 0, that achieves this performance level is given by K 01 = W 01 Y 1 01 = [ ] = [ ]. This result coincides with that in [15, Sect. 5]. However, if past modes and parameters are available to the controller, then a better-performing controller can be obtained by considering a path length M > 0. Indeed, if γ M denotes the minimum achievable disturbance attenuation level by a robust M-path-dependent statefeedback controller, then we have γ 0 = 4.38,γ 1 = 4.28,γ 2 = 4.14,γ 3 = 4.14,... Moreover, if either the current mode or the current parameter is available to the controller, then one can lower the performance level further by using modedependent and parameter-dependent controllers as in [15] Conclusion We extended existing nonconservative analysis and synthesis results for switched linear systems and polytopic LPV systems to switched LPV systems. These extensions are again nonconservative, and provide convex analysis and synthesis conditions in terms of linear matrix inequalities. In particular, the stability and performance analysis conditions cover existing but conservative results in the literature, and allow us to pay additional computational cost in return for a better analysis. On the other hand, the controller synthesis conditions are useful for the case where neither the system mode nor the system parameter is observed without delay, and thus complements existing results. We envision that the results of this work could play an important role in automated analysis and synthesis for control of nonlinear systems.

23 7 Switched LPV Systems in Discrete Time 179 References 1. Apkarian P, Gahinet P (1995 A convex characterization of gain-scheduled H controllers. IEEE Trans Automat Contr 40(5: Blanchini F, Miani S (2003 Stabilization of LPV systems: state feedback, state estimation, and duality. SIAM J Contr Optim 42(1: Daafouz J, Bernussou J (2001 Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties. Syst Contr Lett 43(5: de Oliveira MC, Bernussou J, Geromel JC (1999 A new discrete-time robust stability condition. Syst Contr Lett 37(4: Dullerud GE, Lall S (1999 A new approach for analysis and synthesis of time-varying systems. IEEE Trans Automat Contr 44(8: Lee J-W (2006 On uniform stabilization of discrete-time linear parameter-varying control systems. IEEE Trans Automat Contr 51(10: Lee J-W, Dullerud GE (2006 Optimal disturbance attenuation for discrete-time switched and Markovian jump linear systems. SIAM J Contr Optim 45(4: Lee J-W, Dullerud GE (2006 Uniform stabilization of discrete-time switched and Markovian jump linear systems. Automatica 42(2: Lescher F, Zhao JY, Borne P (2006 Switching LPV controllers for a variable speed pitch regulated wind turbine. Int J Comp Commun Contr 1(4: Lim S, How JP (2003 Modeling and H control for switched linear parameter-varying missile autopilot. IEEE Trans Contr Syst Technol 11(6: Lu B, Wu F (2004 Switching LPV control designs using multiple parameter-dependent Lyapunov functions. Automatica 40(11: Lu B, Wu F, Kim SW (2006 Switching LPV control of an F-16 aircraft via controller state reset. IEEE Trans Contr Syst Technol 14(2: Shamma JS, Athans M (1991 Guaranteed properties of gain scheduled control for linear parameter-varying plants. Automatica 27(3: Zhang L, Shi P (2008 l 2 l model reduction for switched LPV systems with average-dwell time. IEEE Trans Automat Contr 53(10: Zhang L, Shi P, Boukas EK, Wang C (2006 H control of switched linear discrete-time systems with polytopic uncertainties. Optim Contr Applicat Meth 27(5: