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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

Robust Static and Fixed-order Dynamic Output Feedback Control of Discrete-time Lur’e Systems Kwang Ki Kevin Kim ∗ Richard D. Braatz ∗∗ ∗

University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (email: [email protected]) and Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA (email: [email protected]) ∗∗ Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA (email: [email protected])

Abstract: This paper considers the design of static and fixed-order dynamic output feedback controllers for discrete-time Lur’e systems with sector-bounded nonlinearities and polytopic parametric uncertainty. Controller design equations are derived for systems with multiple states, outputs, and nonlinearities in terms of linear matrix inequalities (LMIs). The design methods are based on parameter-dependent Lyapunov functions (PDLFs) combined with the latest in theoretical and iterative numerical methods for solving certain classes of nonconvex inequalities. The design methods are illustrated in several numerical examples. Keywords: Lur’e system, static output feedback control, fixed-order dynamic output feedback control, parameter-dependent Lyapunov function, robust control. 1. INTRODUCTION Lyapunov methods provide simple but powerful ways to analyze and design stabilizing controllers for nonlinear dynamical systems (e.g., see Boyd et al. [1994], VanAntwerp and Braatz [2000] and citations therein). Many well-known stability results were developed for a benchmark problem known as the Lur’e problem (Meyer [1965], Narendra and Taylor [1973], Sharma and Singh [1981]). The discrete-time Lur’e system consists of the interconnection of a linear time-invariant (LTI) system in feedback with a nonlinear operator: x[k+1] = Ax[k] + Bp p[k] , q[k] = Cq x[k] + Dqp p[k] ,

p[k] = −φ(q[k] , k),

(1)

where A ∈ Rn×n , Bp ∈ Rn×np , Cq ∈ Rnq ×n , Dqp ∈ Rnq ×np , and the nonlinear operator φ ∈ Φ, where Φ is a set of static functions that satisfy φ(0, k) ≡ 0 for all k ∈ Z+ and have some specified input-output characteristics, such as satisfying a sector bound or having a slope within some specified range (the detailed mathematical descriptions for the nonlinear operators are given later in this paper). Model uncertainties are typically represented as parametric variations or as unmodeled dynamics. A matrix polytope is a standard representation for real parametric uncertainties (for example, see Barmish [1994], Bhattacharyya et al. [1995]). A commonly used method for deriving a robust stability test for an uncertain system with state matrices described by polytopes is based on a single quadratic Lyapunov function for the entire uncertainty set. The conservatism of this approach motivated some ? This work was supported by the National Science Foundation under Grant #0426328.

Copyright by the International Federation of Automatic Control (IFAC)

227

researchers to reduce conservatism by using parameterdependent Lyapunov functions (PDLFs). These Lyapunov functions, which are quadratic in the state and have an affine dependence on the uncertain parameters, have been applied to derive linear matrix inequality (LMI)-based robust stability conditions for continuous-time linear systems (Feron et al. [1996], Ramos and Peres [2002]) and discrete-time linear systems (de Oliveira et al. [1999]). The robust stability tests involve the solution of parameterized LMIs. The design of stabilizing controllers for systems with nonlinearities and uncertainties is important in control theory and practice. Static output feedback (SOF) is the simplest control to implement in practice (e.g., see the survey paper by Syrmos et al. [1997] and citations therein). The direct application of Lyapunov analysis to SOF design, even for linear time-invariant systems, results in optimization over bilinear matrix inequalities (BMIs), which are not convex Goh et al. [1994], VanAntwerp et al. [1997]. These optimization problems can be solved very slowly using global optimization methods or a local solution can be determined by iterative linear matrix inequality (ILMI) approaches. Several ILMI-based algorithms have been developed for SOF controller design of LTI systems (for example, Iwasaki and Skelton [1994a], Geromel et al. [1994], Beran and Grigoriadis [1997], Ghaoui et al. [1997]). This paper considers the design of robust stabilizing SOF controllers for Lur’e systems with polytopic parametric uncertainty. Depending on the specific control objectives, the design methods are written in terms of LMIs or ILMIs. For the same control objectives, the results are extended to

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

the computation of fixed–order dynamic output feedback controllers. This paper is organized as follows. Section 2 introduces the discrete-time Lur’e systems and summarizes theoretical results used in the rest of the paper. Section 3 derives stabilizing static and fixed-order dynamic output feedback control designs for nominal Lur’e systems and Section 4 derives corresponding results for Lur’e systems with polytopic parametric uncertainty. The SOF control design methods proposed in this paper are demonstrated and compared in numerical examples in Section 5. Section 6 concludes the paper. 2. MATHEMATICAL PRELIMINARIES

φ(0, k) ≡ 0 for all k ∈ Z+ and have some specified inputoutput characteristics described in Definition 1. Beyond Lur’e systems with fixed–values of system matrices, we also consider Lur’e systems in which the system matrices and control function χ that are parameter–dependent. More specifically, these maps are defined by sets that affinely depend on the uncertain parameter θ ∈ Θ ⊂ Rnθ . 2.3 Stability Analysis and State Feedback Control Consider the Lur’e system (2) with a control affine term χ(x[k] , u[k] , k) = Bu u[k] with controllable pair (A, Bu ) and the design objective of determining a linear state feedback controller u[k] = Ks x[k] ,

(3)

where Ks is the control gain matrix of compatible dimension. Applying the feedback controller (3) to the system (2) results in the closed-loop system:

2.1 Notation and Definitions The notation is quite standard. Z+ and R+ denote the set of all nonnegative integers and the set of all nonnegative real numbers, respectively. 0 and I denote the matrix whose components are all zeros and the identity matrix of compatible dimension, respectively. X  0 denotes that the matrix X is positive definite, X  0 denotes that X is positive semidefinite, and X ≺ 0 and X  0 denote negative definite and semidefinite matrices, respectively. Sym(X) := X + X T and X ⊥ denotes a full-rank matrix orthogonal to X. Through this paper, the nonlinearity φ is taken to be a member of some specific classes of nonlinear operators. Definition 1. (Definitions of classes of nonlinear operators) A nonlinearity φ : Rnq × Z+ → Rnq is of family    |α| Φsb if αi−1 φi (σ, k) + σ αi−1 φi (σ, k) − σ ≤ 0, and is of |µ| σ ,k) i (ˆ family Φsr if −µi ≤ φi (σ,k)−φ ≤ µi for all σ, σ ˆ ∈ Rnq , σ−ˆ σ k ∈ Z+ , and i = 1, . . . , nq , where the subscript i denotes the ith element of the vector. A nonlinearity φ : Rnq × ¯ α if kφ(σ, k)k ≤ αkσk holds Z+ → Rnp is of family Φ sb nq for all σ ∈ R , k ∈ Z+ . A nonlinear mapping φ : Rnq × ¯ µ if kφ(σ, k)−φ(ˆ Z+ → Rnp is of family Φ σ , k)k ≤ µkσ − σ ˆk sr for all σ 6= σ ˆ ∈ Rnq , k ∈ Z+ .

x[k+1] = (A + Bu Ks )x[k] − Bp φ(q[k] , k).

(4)

The following lemma provides a sufficient LMI condition for the linear state feedback controller (3) to stabilize the closed-loop system (4). |α|

Lemma 2. The closed-loop system (4) with φ ∈ Φsb is globally asymptotically stabilized by the state feedback controller u[k] = Ks x[k] with Ks = W Y −1 if the LMI   −Y ∗ ∗ ∗ 0 −T ∗ ∗  ≺ 0, AY + Bu W −Bp T −Y ∗ Cq Y 0 0 −Sα T



(5)

is feasible for Y = Y T  0, a diagonal matrix T > 0, and W , where Sα = diag{1/α12 , · · · , 1/αn2 p }. If the LMI (5) with Sα = 1/α2 I and T = I is feasible, then the closed¯ α is globally asymptotically loop system (4) with φ ∈ Φ sb stabilized by the state feedback control law (3) with Ks = W Y −1 . The LMI (5) follows from the S-procedure (Yakubovich [1977]) and the Shur complement lemma (Boyd et al. [1994]).

2.2 Discrete-Time Lur’e Systems

2.4 Parameter-Dependent Lyapunov Functions

This paper considers the design of static and fixed-order dynamic output feedback controllers for some classes of Lur’e systems with multi-valued nonlinear mappings in a negative feedback interconnection. The global (or local) asymptotic (or exponential) stability of the closed-loop system is guaranteed in the presence of the internal and/or external perturbations. The discrete-time Lur’e systems are considered:

Robustness analysis and synthesis have been widely studied for linear systems with polytopic uncertainty. A widely used approach to these problems is to search for a common quadratic Lyapunov function that is reformulated into a sufficient condition written in terms of matrix inequalities. The use of a single quadratic Lyapunov function can result in highly conservative results, which motivated subsequent efforts that reduce conservative by using parameterdependent Lyapunov functions (PDLFs). To illustrate the use of PDLFs while presenting some theoretical results used later in the paper, consider the uncertain system

x[k+1] = Ax[k] + Bp p[k] + χ(x[k] , u[k] , k), q[k] = Cq x[k] , p[k] = −φ(q[k] , k),

(2)

y[k] = Cy x[k] ,

where x ∈ Rn and y ∈ Rny denote the state and the measurement vector, respectively, q ∈ Rnq and p ∈ Rnp are the input and output of the nonlinearity, respectively, and u ∈ Rnu is the control input. In addition, the nonlinear function χ : Rn × Rnu × Z+ → Rn is assumed to be Lipschitz in the first argument. The nonlinear operator φ ∈ Φ, where Φ is a set of static functions that satisfy 228

x[k+1] = A(θ[k] )x[k] , nθ

(6)

where A is affine in θ[k] ∈ Θ ⊂ R , k ∈ Z+ . Consider a Lyapunov matrix that is also affine inPthe parametric nv uncertainty vector θ, i.e., X(θ[k] ) = j=1 ρj (θ[k] )Xj , Pnv where ρ (θ ) = 1, ρ (θ ) ∈ [0, 1] for all θ[k] ∈ j j [k] [k] j=1 Θ ⊂ Rnθ , and Xj = XjT is real for each j = 1, . . . , nv . In addition, suppose that Θ is a convex hull with a finite

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

set of vertices Θv , i.e., Θ = Co(Θv ). It is straightforwrd to apply Lyapunov analysis to show that, if the matrix inequality AT (θ

[k] )X(θ[k+1] )A(θ[k] )

− X(θ[k] ) ≺ 0

(7)

holds for all θ[k] , θ[k+1] ∈ Θ ⊂ Rnθ at each k ∈ Z+ , then the origin of the uncertain system (6) is globally asymptotically stable (g.a.s.) Since the parameter-dependent matrix (7) is not jointly convex in θ[k] and θ[k+1] , it is desirable to find an equivalent LMI condition to (7). Lemma 3. (Polytopic parameter-dependent systems) The origin of the uncertain system (6) is g.a.s. for any timevarying uncertain vector θ[k] ∈ Θ ⊂ Rnθ if any of the following inequalities hold for the specified variables: (i) there exists a Lyapunov matrix X(θ[k] ) = X T (θ[k] ) = Pnv j=1 ρj (θ[k] )Xj  0 such that AT (θ[k] )X(θ[k+1] )A(θ[k] ) − X(θ[k] ) ≺ 0, ∀θ ∈ Θ;

(8)

T

(ii) P there exists a Lyapunov matrix Y (θ(k)) = Y (θ(k)) = nv j=1 ρj (θ(k))Yj  0 such that   Y (θ[k] ) Y (θ[k+1] )AT (θ[k] ) 0 A(θ[k] )Y (θ[k+1] ) Y (θ[k+1] )

(9)

for every θ ∈ Θ; (iii) there exists a Lyapunov matrix X(θ[k] ) = X T (θ[k] ) = Pnv j=1 ρj (θ[k] )Xj  0 and G of compatible dimensions such that   T

T

X(θ[k] ) A (θ[k] )G 0 GA(θ[k] ) Sym(G) − X(θ[k+1] )

(10)

for every θ ∈ Θ, or equivalently,   T Xj AT j G GAj Sym(G) − Xi

 0, ∀i, j = 1, . . . , nv ;

(11)

(iv) there exists a Lyapunov matrix Y (θ[k] ) = Y T (θ[k] ) = Pnv j=1 ρj (θ[k] )Yj  0 and H of compatible dimensions such that   Sym(H) − Y (θ[k] ) H T AT (θ[k] ) 0 A(θ[k] )H Y (θ[k+1] )

 0, ∀i, j = 1, . . . , nv .

(Ad + Bu,d Ko Cy,d )Xd + Xd (Ad + Bu,d Ko Cy,d )T ≺ 0,

(16)

where Ad ,

h

−0.5I 0 A −0.5I

i

h i 0 , Bu,d , B , Cy,d , [Cy 0] , u

and Xd = diag{X, X} with X = Y −1 . The LMI (16) contains bilinear terms in the unknown (decision) matrices X and Ko , separated by constant system matrices. Checking the feasibility of the inequality (16) is a nonconvex problem known to be NP-hard (Blondel and Tsitsiklis [2000]). This nonconvex inequality (16) can be written in terms of a simpler set of coupled linear matrix inequalities (Iwasaki and Skelton [1995], Griagoriadis and Skelton [1996]). Lemma 5. The matrix inequality (16) (or (15)) holds for some Ko and X (or Y ) if and only if X (or Y ) satisfies the two matrix inequalities: ⊥ ⊥ T Bu (AX + XAT )(Bu ) T ⊥ T T ⊥ T (Cy ) (A Y + Y A)((Cy ) )

≺ 0, ≺ 0,

(17)

where XY = Y X = I. Finding X = X T  0 and Y = Y T  0 that jointly satisfy the two matrix inequalities (17) with Y X = XY = I is still a nonconvex problem, but search methods have been developed based on iterative sequential solutions of the two LMI problems with respect to X and Y . By substituting a solution X (or Y ) fo (17) into (16) (or (15)), a stabilizing static output feedback control gain matrix Ko can be computed for the system (14). 3. OUTPUT FEEDBACK CONTROL OF DISCRETE-TIME LUR’E SYSTEMS

(13)

3.1 Static Output Feedback Controller Design Consider the SOF control problem for the Lur’e system x[k+1] = Ax[k] + Bu u[k] − Bp φ(q[k] , k) y[k] = Cy x[k] ,

q[k] = Cq x[k] ,

(18)

¯ α or Φ|α| . The triplet where φ(·, ·) is in a specific class Φ sb sb (A, Bu , Cy ) is assumed to be stabilizable and detectable. For SOF controller synthesis for the nominal Lur’e system (18), the resulting optimization problem is min γ

2.5 Static Output Feedback for LTI Systems

Y,Ko

−Y ∗ ∗ ∗ 0 −I ∗ ∗  s.t. Y  0,  ≺ 0. AY + Bu Ko Cy Y −Bp −Y ∗ Cq Y 0 0 −γI



The closed-loop LTI system y[k] = Cy x[k] ,

(15)

some Y = Y T  0, which can be written in terms of an equivalent inequality

This section derives static and fixed-order dynamic output feedback controller design equations for discrete-time ¯ α or φ ∈ Φ|α| . Lur’e systems with φ ∈ Φ sb sb

Proof: See de Oliveira et al. [1999]. Remark 4. The inequalities (10) and (12) are jointly affine in the uncertain parameter vectors θ[k] and θ[k+1] so that no product terms of Xj , Yj , and Aj appear in (10)(13), which is indispensable in reducing the corresponding controller synthesis problems in the next sections to LMIs or ILMIs.

x[k+1] = Ax[k] + Bu u[k] ,

(A + Bu Ko Cy )T Y (A + Bu Ko Cy ) − Y ≺ 0.

(12)

for every θ ∈ Θ, or equivalently,   Sym(H) − Yj H T AT j Aj H Yi

the existence of a gain matrix Ko that satisfies the matrix inequality

(14)

is g.a.s. with an output feedback controller u[k] = Ko y[k] if and only if the matrix A + Bu Ko Cy is Schur, i.e., the eigenvalues of A + Bu Ko Cy are inside the open unit circle in the domain of complex variables. This is equivalent to 229



(19)

The LMI constraint (19) can be rewritten in the same form as (16): ¯γ + B ¯u Ko C ¯y )Y¯ + Y¯ (A ¯γ + B ¯ u Ko C ¯y )T ≺ 0, (A

(20)

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

system, i.e., nc ≤ n, the design problem can be reformulated as an equivalent static output feedback design problem, in the same manner as for LTI systems (Iwasaki and Skelton [1994b]). Consider a state-space realization of the dynamical output feedback controller

where −0.5I

¯γ A ¯y C

0 0 0 0 0 −0.5I 0 0  ¯ 0    , , Bu , , A −Bp −0.5I 0 Bu Cq 0 0 −0.5γI 0   , Cy 0 0 0 , Y¯ , diag{Y, I, Y, I}.







ζ[k+1] = Ac ζ[k] + Bc y[k] ,

A sufficient LMI condition for the nonconvex problem (16) derived by Crusius and Trofino [1999] is used below to develop a suboptimal static output feedback controller design method. Theorem 1. Consider the system (18) with Cy of full row rank and the convex optimization min γ Y,N

(21)

¯ T ≺ 0. ¯u N C ¯y + C ¯TN TB ¯γ Y¯ + Y¯ A ¯T + B s.t. Y  0, A u y γ ∗ Φα sb ,

∗

The where α = √ closed-loop system (18) with φ ∈ 1/ γ ∗ and γ ∗ is the optimal value of (21), is globally asymptotically stabilized by the static output feedback controller Ko = N M −1 where the full rank matrix M satisfies M Cy = Cy Y . Theorem 2. Consider the system (18) with Bu of full column rank and the convex optimization min γ X,N

(22)

¯A ¯γ + A ¯T X ¯ +B ¯u N C ¯y + C ¯TN TB ¯ T ≺ 0, s.t. X  0, X γ y u

¯ = diag{X, I, X, I}. The closed-loop system (18) where X √ ∗ ∗ ∗ ∗ for φ ∈ Φα sb where α = 1/ γ and γ is the optimal value of (21) is globally asymptotically stabilized by the static output feedback controller Ko = M −1 N where the full rank matrix M satisfies Bu M = XBu . The proofs of Thms. 1 and 2, which are similar to the proofs in Crusius and Trofino [1999], are in Kim [2009]. The next theorem follows from (20) and Lemma 5. Theorem 3. There exists a stabilizing SOF controller gain ¯ α and upper matrix Ko for the system (18) with φ ∈ Φ sb √ sector bound α , 1/ γ if there exists Y = Y T  0 such that
¯⊥ A ¯γ Y¯ + Y¯ A ¯T (B ¯ ⊥ )T ≺ 0, B u γ u




¯ T )⊥ X ¯A ¯γ + A ¯T X ¯ ((C ¯ T )⊥ )T ≺ 0, (C y γ y

(23)

¯ = Y¯ −1 . where Y¯ = diag{Y, I, Y, I} and X ¯ α can be reformulated for the Lur’e Theorem 3 for φ ∈ Φ sb system whose nonlinear operator is described by a more |α| general class of sector conditions, φ ∈ Φsb , which is defined as being componentwise. To do this, replace the matrices A¯γ and Y¯ in the LMIs of Thm. 3 by −0.5I 0  0 0 ¯S ,  A α

0 A Cq

u[k] = Cc ζ[k] + Dc y[k] ,

(24) −1

which has the transfer function C(z) = Cc (zI − Ac ) Dc , where uz = C(z)yz ,

Bc + (25)

and uz and yz are the z-transformations of u[k] and y[k] , respectively. The closed-loop Lur’e system obtained with (24) can be written as ¯x[k] + B ¯u u[k] − B ¯p φ(¯ x ¯[k+1] = A¯ q[k] , k), ¯ ¯ y¯[k] = Cy x ¯[k] , q¯[k] = Cq x ¯[k] ,

(26)

  ¯u , diag{Bu , I}, B ¯pT , BpT 0 , where A¯ , diag{A, 0}, B C¯y , diag{Cy , I}, C¯q , [Cq 0], x ¯ , (xT , ζ T )T is the concatenated state, and u[k] is the output of a static output feedback controller h i u[k] = Kdof y¯[k] ;

Kdof ,

Dc Cc . Bc Ac

(27)

This indicates that all results for static output feedback control problems in this paper can apply to dynamic fixedorder feedback control problems when nc ≤ n by using the transformed state-space realization " ¯ ¯ ¯ # ¯ G(z) ,

A Bu Bp ¯y 0 0 C ¯q 0 0 C

.

(28)

4. OUTPUT FEEDBACK CONTROL FOR POLYTOPIC DISCRETE-TIME LUR’E SYSTEMS This section derives static output feedback controller design equations for polytopic uncertain discrete-time Lur’e ¯ α or φ ∈ Φ|α| in Fig. 1. systems with φ ∈ Φ sb sb 4.1 With Parametric Uncertainties in the Output Channel Consider the system x[k+1] = A(θ[k] )x[k] + Bu u[k] − Bp φ(q[k] , k), y[k] = Cy (θ[k] )x[k] ,

q[k] = Cq x[k] ,

(29)

where x[k] ∈ Rn is the state and u[k] ∈ Rnu is the control input at time k ∈ Z+ , and θ[k] ∈ Θ specifies the parametric uncertainty where Θ ⊂ Rnθ is a polytope that is closed and compact. Assume that the mappings A : Θ → Rn×n and Cy : Θ → Rny ×n are continuous in θ(k) ∈ Θ which is Lebesgue measurable for all k ∈ Z+ . The parametric uncertainty is described in terms of a polytopic linear differential inclusion (PLDI) (Boyd et al.

−0.5I 0 0  , −Bp −0.5I 0 0 0 −0.5Sα

Y¯T , diag{Y, T, Y, T },

respectively, where Sα = diag{1/α12 , · · · , 1/αn2 p } and T  0 is a diagonal matrix. 3.2 Fixed-Order Dynamic Output Feedback Control When the desired order of a dynamic output feedback controller is less than or equal to the order of the nominal 230

Fig. 1. Polytopic uncertain Lur’e system controlled by output feedback.

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

[1994]) in which the state and output matrices in (29) are affinely dependent on the time-varying parameter vector θ : Z+ → Θ,   v A(θ) Cy (θ) ∈ ΩAC , Co (ΩAC ) ,

∀θ ∈ Θ,

(30)

Theorem 8. (Robust Stabilizing SOF) Consider the system (34) where (A(θ[k] ), Bu (θ[k] )) are within a PLDI (35) and Cy is assumed P to be full row rank. If there exist the nv ρj (θ[k] )Yj , G, Mg , and Ng such matrices Y (θ[k] ) = j=1 that

where ΩvAC , {[A1 Cy,1 ] , · · · , [Anv Cy,nv ]}, and nv = 2nθ . Two different SOF controller design schemes are proposed for the system (29) with PDLFs given in Section 2.4. Theorem 6. (Robust Stabilizing SOF) Consider the system (29) with (A(θ[k] ), Cy (θ[k] )) represented as a PLDI (30) and Bu assumed to be P of full column rank. If there nv ρj (θ[k] )Xj , G, Mg , and exist the matrices X(θ[k] ) = j=1 Ng such that Xj ∗ ∗ 0 I ∗  GAj + Bu Ng Cy,j −GBp Sym(G) − Xi Cq 0 0

∗ ∗ 0 ∗ γI



(31)

are feasible for all i, j = 1, . . . , nv , then Ko = Mg−1 Ng is a stabilizing SOF control gain, i.e., the feedback control signal u[k] = Mg−1 Ng y[k]

Aj G + Bu,j Ng Cy Cq G

(32)

stabilizes the system (29) whose uncertain model is represented by the PLDI (30). Theorem 7. (Robust Stabilizing SOF–ILMIs) There exists a stabilizing SOF control gain matrix Ko for the system in Thm. 6 if there exist matrices G, H, and X(θ[k] ) = Pnv X T (θ[k] ) = j=1 ρj (θ[k] )Xj  0 that satisfy   −1 ∗ ∗ I ∗ −Bp Sym(G−1 ) 0 0 0 G−T

∗ ∗ j 0 I ∗ ∗  (j) T  (η2 )  0, GAj −GBp Sym(G) − Xi ∗ Cq 0 0 γI

for all i, j = 1, . . . , nv , where η1 , diag{I, I, Bu⊥ , I, I} and (j) T ⊥ η2 , diag{(Cy,j ) , I, I, I}. 4.2 With Parametric Uncertainties in the Input Channel

y[k] = Cy x[k] ,

q[k] = Cq x[k] ,

n

(37)

j

(j) η3

I ∗ ∗  (j) T (η3 )  0, −Bp Yi ∗ 0 0 γI

0 Aj G Cq G



 

η4 

0 HAj Cq G−1

∗ ∗ I ∗ −HBp Sym(H) − Yi−1 0 0 0 0

∗ ∗ ∗ ∗  ∗ ∗  η4T  0,  γI ∗ −1 0 Yj



(38)

(j)

⊥ hold for all i, j = 1, . . . , nv , where η3 , diag{I, I, Bu,j , I} T ⊥ and η4 , diag{(Cy ) , I, I, I, I}.

5. NUMERICAL EXAMPLES The theoretical results in Section 4 are applied to design static output feedback controllers for uncertain Lur’e systems with multiple states, outputs, and nonlinearities. The LMIs were solved using off-the-shelf software (Sturm [1999], L¨ofberg [2008]). Example 1. In this example, uncertainty only occurs in the A-matrix so all of the design methods in Section 4 can be compared. Consider the system (29) or (34) with h i h i 

(34)

nu

∀θ ∈ Θ,

(36)



0.0758 −0.6711 , Bp = , 0.7576 −0.4003





Cq = −0.0071 0.2107 , Cy = 0.3939 0.0303 ,

where x[k] ∈ R is the state and u[k] ∈ R is the control input at time k ∈ Z+ , and the parameter vector θ[k] ∈ Θ where Θ ⊂ Rnθ is a polytope that is closed and compact. The mappings A : Θ → Rn×n and Bu : Θ → Rn×nu are assumed to be affine and continuous in θ ∈ Θ which is Lebesgue measurable:   v A(θ) Bu (θ) ∈ ΩAB , Co (ΩAB ) ,



Theorem 9. (Robust Stabilizing SOF: Coupled LMIs) There exists a stabilizing SOF control gain matrix Ko for the system in Thm. P8nvif there exist the matrices G, Y (θ[k] ) = Y T (θ[k] ) = j=1 ρj (θ[k] )Yj  0, and H such that Sym(G) − Y ∗ ∗ ∗ 

A(θ[k] ) ∈ Co({A1 , A2 }), Bu =

Consider the system x[k+1] = A(θ[k] )x[k] + Bu (θ[k] )u[k] − Bp φ(q[k] , k),

∗ ∗ 0 ∗ γI

u[k] = Ng−1 Mg y[k]

(33)

 ∗

X (j) η2

∗ ∗ ∗ ∗  T ∗ ∗   η1  0, γI ∗ 0 Xi−1

∗ ∗ Yi 0

are feasible for all i, j = 1, . . . , nv , then Ko = Ng−1 Mg is a stabilizing SOF control gain, i.e., the feedback control signal

Sym(G−1 )

Sym(H) − Xj 0  η1  Aj H  Cq H 0

∗ I −Bp 0

stabilizes the system (34) whose uncertain model is represented by the PLDI (35).

Bu Mg = GBu , Xj  0



Mg Cy = Cy G, Yj  0

 Sym(G) − Y j 0 

Table 1. The maximal upper bound on the sector and optimal SOF control gains for Examples 1, 2 and 3. Ex. Design Methods α∗ Ko∗ Ex. 1

Thm. Thm. Thm. Thm.

6 7 8 9

1.0827 1.4497 1.3412 1.4497

Ex. 2

Thm. 6

0.6277

Thm. 7

0.8972

Thm. 8

0.5808

Thm. 9

0.8833

(35)

where ΩvAB , {[A1 Bu,1 ] , · · · , [Anv Bu,nv ]}, nv = 2nθ , and θ : Z+ → Θ is a time-varying vector. Two SOF controller design schemes are proposed for the system (34) with PDLFs given in Section 2.4. 231

Ex. 3

−0.6507 −0.2701 −0.8133 −0.9091 h i 0.3858 −0.2707 −0.0245 −0.0195 h i 0.1572 −0.0695 h−0.0695 −0.0038i 0.4782 −0.8366 h−0.0852 0.1144 i 0.3556 −0.1069 −0.1069 0.1295

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

where h

A1 =

i

0.9697 0.1515 −0.3030 0.5152

, A2 =

h

i

0.9753 0.1235 −0.2469 0.2346

,

¯α . and nonlinearities are within the set φ ∈ Φ sb Example 2. This example compares the two different design methods in Section 4.1. Consider the system (29) with " # −0.12 1 0 0 0.1 + θ1,[k] 0 , 0 0 0.6 + θ2,[k]

A(θ[k] ) =

"

#

1 0 0 1 , Bp = 1 −1

Bu =

"

#

0.6 0.4 −0.4 −0.6 , −0.35 −0.65





1 + 1.4θ1,[k] 0 −2 1 0 0 Cq = , , Cy (θ[k] ) = 1 1 + θ2,[k] 0 0 1 0

h

i

¯ α , and bounds for nonlinearities within the set φ ∈ Φ sb the uncertain parameters as θ1,[k] ∈ [−0.5, 0] and θ2,[k] ∈ [0, 0.5] for all k ∈ Z+ . Example 3. This example compares the two different design methods in Section 4.2. Consider the system (34) with " # A(θ[k] ) =

−0.12 1 0 0 0.1 + θ1,[k] 0 , 0 0 0.6 + θ2,[k]

" Bu (θ[k] ) =

" Bp =

#

1 0 0 1 + 1.4θ1,[k] , 1 + 1.2θ2,[k] −1

#

h i h i 0.6 0.4 1 0 0 1 0 −2 −0.4 −0.6 , Cq = , Cy = , 0 1 0 1 1 0 −0.35 −0.65

¯ α , and bounds for the nonlinearities within the set φ ∈ Φ sb uncertain parameters as θ1,k ∈ [−0.5, 0] and θ2,k ∈ [0, 0.5] for all k ∈ Z+ . Suppose that the control objective is to maximize the upper bound α on the sector such that the closed-loop system (29) or (34) is stabilized by the output feedback controller u[k] = Ko y[k] . The values for α∗ and Ko∗ computed from Thms. 6-9 are reported in Table 1. According to Table 1, the different design methods can produce controllers with different levels of conservatism for systems that only have uncertainty in the state matrix A. The least conservatism was obtained by Thms. 7 and 9. 6. CONCLUSIONS Static and fixed-order dynamic output feedback control design methods are derived for polytopic uncertain Lur’e systems with sector-bounded nonlinearities. The nonconvex matrix inequality formulations for output feedback controller design are provided in a mathematical form for which iterative numerical algorithms have been developed. Each iteration of the numerical algorithms is formulated in terms of linear matrix inequalities that are solved using off-the-shelf software. The design methods are compared in three numerical examples. REFERENCES B. R. Barmish. New Tools for Robustness of Linear Systems. MacMillan, New York, 1994. E. Beran and K. M. Grigoriadis. Computational issues in alternating projection algorithms for fixed-order control design. In Proc. of the ACC, pages 81–85, 1997.

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