Stabilization of 2-D Linear Parameter-Varying Systems using Parameter-Dependent Lyapunov Function: An LMI Approach

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1 Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul Korea July Stabilization of 2-D Linear Parameter-Varying Systems using Parameter-Dependent Lyapunov Function: An LMI Approach Sung Wook Yun Yun Jong Choi PooGyeon Park Electrical and Computer Engineering Division POSTECH Pohang Korea (Tel: wook2 scope75 Abstract: This paper proposes a parameter-dependent state-feedback controller for the 2-D discrete linear parameter-varying (LPV) system with the Fornasini-Machesini (FM) firstmodel To find the stabilizing conditions of the system we first transform the closed-loop system to a Roesser-type model and then derive the conditions to linear matrix inequalities (LMIs) using a parameter-dependent Lyapunov function (PDLF) and a relaxation technique The simulation results show that the designed controller is valid and the system asymptotically converges to the origin Keywords: 2-D system FM first model linear matrix inequalities (LMIs) linear parameter-varying (LPV) parameter-dependent Lyapunov function (PDLF) 1 INTRODUCTION For several decades there has been much interest in the analysis of two-dimensional (2-D) systems which has introduced some 2-D dynamic models: for examples Fornasini-Machesini (FM) first (Fornasini and Marchesini 1976 Kar and Singh 23 Tong Zhou 26) and second model (Fornasini and Marchesini 1978 Hinamoto 1997 Ooba 2) Roesser model (Roesser 75) and so on These 2-D system theories can be applied not only to the theoretical areas such as iterative learning control (ILC) and signal processing but also to the practical systems such as thermal processes water steam heating etc (Bose 1982 Kaczorek 1985 Du and Xie 22) Many of practical systems have included non-linearities such as sector-bounded conditions input saturations or other non-linear functions In this case these systems can be often modeled in the linear parameter-varying (LPV) systems Thus in the paper we consider that the 2-D system has a certain non-linearity which is described as the LPV system (Becker et al 1993 Apkarian et al 1995 Apkarian and Tuan 2 Park and Choi 21) Here our goal is to propose a method for stabilizing the 2-D LPV system using a parameter-dependent Lyapunov function (PDLF) the system is regarded as the FM This research was supported by the MIC (Ministry of Information and Communication) Korea under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Advancement) (IITA-28- C ) This research was supported by the MIC (Ministry of Information and Communication) Korea under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Advancement) (IITA-28- C ) first model and the controller is designed as the statefeedbackto the best of our knowledge not much academic research has been studied on this area The resulting conditions for stabilization can be expressed withlinearmatrixinequalities (LMIs) usingtherelaxation technique in Park and Choi 21 which is solved by the LMI toolbox in MATLAB In addition the numerical example shows that the proposed controller is valid The paper is organized as follows Section 2 describes the system handling in the paper and explains the problem Section 3 explains a new approach and designs the controller for stabilization of 2-D LPV system in detail Section 4 shows the simulation results Finally section 5 concludes the paper with summarization 2 PROBLEM FORMULATION Consider the 2-D discrete LPV system with the FM first model: x(t + 1k + 1) = A 1 (θ t )x(tk + 1) + A 2 (θ t )x(t + 1k) +A 3 (θ t )x(tk) + B(θ t )u(tk) (1) x(tk) R n is the state vector u(tk) R m is the control input and θ t R r denotes a time-varying parameter vector of time-varying parameters θ ti with respect to the index t such that for i = 1 r θ t = θ t1 θ tr T (2) which satisfies the following condition θ ti = 1 θ ti 1 θ ti θ t 1i δ i (3) /8/$2 28 IFAC / KR-11467

2 17th IFAC World Congress (IFAC'8) Seoul Korea July Then the system matrices can be described such as A 1 (θ t ) = A 3 (θ t ) = θ ti A 1i A 2 (θ t ) = θ ti A 3i B(θ t ) = θ ti A 2i θ ti B i (4) The objective of the paper is to design a parameterdependentstate-feedback controller forstabilizingthesystem (1) which is chosen as u(tk) = K 1 (θ t θ t 1 )x(tk + 1) + K 2 (θ t θ t 1 )x(t + 1k) +K 3 (θ t θ t 1 )x(tk) (5) θ t 1 denotes the one-step-past vector of timevarying parameters θ t Substituting (5) into (1) we have the closed-loop system as follows: Let x(t +1k + 1) = Ā1(θ t θ t 1 )x(tk + 1) +Ā2(θ t θ t 1 )x(t + 1k) + Ā3(θ t θ t 1 )x(tk) (6) Ā 1 (θ t θ t 1 ) = A 1 (θ t ) + B(θ t )K 1 (θ t θ t 1 ) Ā 2 (θ t θ t 1 ) = A 2 (θ t ) + B(θ t )K 2 (θ t θ t 1 ) Ā 3 (θ t θ t 1 ) = A 3 (θ t ) + B(θ t )K 3 (θ t θ t 1 ) z(tk + 1) = x(t + 1k + 1) Ā1(θ t θ t 1 )x(tk + 1) then (6) can be described as a Roesser-type model: x(t + 1k) x(tk) E(θ t θ t 1 ) = F(θ z(tk + 1) t θ t 1 ) (7) z(tk) E(θ t θ t 1 ) = F(θ t θ t 1 ) = I Ā2(θ t θ t 1 ) I Ā1 (θ t θ t 1 ) I Ā 3 (θ t θ t 1 ) 3 STABILIZING THE 2-D LPV SYSTEM USING THE PARAMETER-DEPENDENT LAYPUNONV FUNCTION Let us consider a candidate PDLF for the 2-D discrete system such as x(tk) V ( z(tk) ) = whose difference is given by x(tk) V ( z(tk) ) = T x(tk) P(θ z(tk) t 1) x(tk) (8) z(tk) T x(t + 1k) x(t + 1k) P(θ z(tk + 1) t ) z(tk + 1) T x(tk) x(tk) P(θ z(tk) t 1) < z(tk) (9) P(θ t 1 ) = P 1 (θ t 1 ) P 2 (θ t 1 ) P(θ t ) = P 1 (θ t ) P 2 (θ t ) and means the direct sum Using (9) in the following theorem we provide a PDLFbased stabilizer in terms of parameterized linear matrix inequalities (PLMIs) Theorem 1 The closed-loop system (7) is globally asymptotically stable if there exist matrices P1 (θ t ) P2 (θ t ) P 1 (θ t 1 ) P2 (θ t 1 ) W 1 (θ t ) W 2 (θ t ) W 3 (θ t ) K1 (θ t θ t 1 ) K 2 (θ t θ t 1 ) and K 3 (θ t θ t 1 ) such that P(θ t 1 ) ( ) ( ) F(θ t θ t 1 ) P(θ t 1 ) E(θ tθ t 1 )W T (θ t ) ( ) +( ) > W T (θ t ) P(θt ) P(θ t 1 ) > P(θt ) > P(θ t 1 ) = P 1 (θ t 1 ) P 2 (θ t 1 ) P(θ t ) = P 1 (θ t ) P 2 (θ t ) W T W T (θ t ) = 1 (θ t ) W2 T (θ t ) W3 T (θ t ) In this case the state-feedback controller (5) is given by K 1 (θ t θ t 1 ) = K 1 1 (θ t θ t 1 ) P 1 (θ t 1 ) K 2 (θ t θ t 1 ) = K 2 (θ t θ t 1 )W1 T (θ t ) K 3 (θ t θ t 1 ) = K 1 3 (θ t θ t 1 ) P 1 (θ t 1 ) (1) Proof From the Lyapunov difference (9) we can derive the following PLMI: for P(θ t 1 ) = P 1 (θ t 1 ) P(θt ) = P 1 (θ t ) P(θ t 1 ) ( ) F(θ t θ t 1 ) P(θ > t 1 ) (22) (22) = E(θ t θ t 1 ) P(θ t )E T (θ t θ t 1 ) Using the following relation for (22) of (11) (E W P 1 ) P(E T P 1 W T ) (11) E PE T EW T + WE T W P 1 W T (12) and using the Schur-complement with respect to W P 1 W T then we have the resultant PLMIs (1) Unfortunately it is extremely difficult to directly solve the parameter-dependent conditions of the Theorem 1 because it is required to solve an infinite number of LMIs Therefore it is important to reduce the solvability of PLMI to the solvability of a finite number of LMIs To do this we use the convex relaxation technique in Park and Choi 21 with the following assumptions: 1154

3 17th IFAC World Congress (IFAC'8) Seoul Korea July P 1 (θ t ) = P 2 (θ t ) = P 2 (θ t 1 ) = K 1 (θ t θ t 1 ) = K 2 (θ t θ t 1 ) = K 3 (θ t θ t 1 ) = W 1 (θ t ) = W 3 (θ t ) = θ ti P1i P1 (θ t 1 ) = θ ti P2i + θ t 1i P2i + θ ti K1ti + θ ti K2ti + θ ti K3ti + j=1 θ t 1i P1i θ ti θ tj P2ij j=1 θ t 1i θ t 1j P2ij θ t 1i K1t 1i θ t 1i K2t 1i θ t 1i K3t 1i θ ti W 1i W 2 (θ t ) = θ ti W 3i θ ti W 2i Corollary 2 (Simplified stabilizing condition via LMI) The closed-loop system (7) is globally asymptotically stable if there exist matrices P1i P2i P2ij K1ti K 1t 1i K2ti K2t 1i K3ti K3t 1i W 1i W 2i W 3i Λ Λ i Σ Σ i and Ξ i such that for all i = 1 r j = 1 r Υ ( ) ( ) Γ ij r 1 ij r r ( ) > (13) Ω ij r 1 Π ij r r Ψ ij r r P 1i > P2i > P2ij > (14) Υ = N Γ i1 = M 1 i + N 1 i Λ + Λ T Λ i + Λ T i (15) Σ + Σ T Σ i + Σ T i (16) Ω i1 = M 2 i + N 2 i M3 ii + N ij = 3 ii i = j M 3 ij + N 3 ij i j M4 ii + N Π ij = 4 ii i = j M 4 ij i j Ξ i + Ξ T i (17) Ψ ij = M 5 ij + N 5 ij W1i W Θ 2i W1i W 2i M 1 i = W 3i W 3i 1 P 2 1i 1 P 2 2i M 2 i = 1 P 2 1i 1 P 2 2i P 2i Θ Θ ( ) ( ) ( ) Bi K1ti A M 3 ii = B i K3ti 2i W T 1i ( ) B i K2ti P 2ii Bi K1tj M T A 3 ij = B i K3tj 2i W T 1j Θ B i K2tj 1 2 P 2ij A 1i P1j + M T B 4 ij = i K1t 1j A 3i P1j + Θ B i K2t 1j B i K3t 1j P ( ) ( ) 2ij M 5 ij = P2ij Θ ( ) Θ = 2n 2n N = (Λ + Λ T ) + (Σ + Σ T ) N 1 i = ( Λ i ) + ( 2Λ) δi 2 (Ξ i + Ξ T i ) N 2 i = ( Σ i ) + ( 2Σ) (Λ + Λ N 3 ij = T ) + (Λ i + Λ T i ) + (Ξ i + Ξ T i ) i = j Λ i j N 4 ii = ( 2Ξ i ) (Σ + Σ N 5 ij = T ) + (Σ i + Σ T i ) + (Ξ i + Ξ T i ) i = j (Σ + Σ T ) i j Proof With the following vector: I θ t1 I θ tr I θ t 11 I θ t 1r I T PLMIs (1) can be relaxed into the LMIs conditions (13) (17) which are thediscrete-time version ofparkandchoi 21 For more details of relaxation refer to Park and Choi 21 4 SIMULATION RESULTS We consider the 2-D LPVdiscrete-time FMfirstmodel (1) under the conditions: 1155

4 17th IFAC World Congress (IFAC'8) Seoul Korea July r = 2 δ 1 = δ 2 = 1 θ t1 = sin 2 t θ t2 = cos 2 t A 11 = A = A 21 = A = A 31 = A = 2 5 B 1 = B 1 2 = 16 Assume that the boundary conditions x(1k)x(t1) t 1k 1} of the state are set to uniformly distributed random numbers between and 1 except the initial conditions: x(11) = 5 25 x(21) = x(12) = (18) By the LMI toolbox in the Matlab solutions of (13) can be obtained as follows: K 1t1 = K1t 11 = technique and constraint elimination the conditions for stabilization were reduced to an LMI term Numerical examples verified that the designed controller was valid and the state asymptotically converged to the origin As further works we shall handle the 2-D discrete LPV system with thefmsecondmodelandthe outputfeedback case State x 1 (db) k 6 Fig 1 The state trajectory of x 1 with respect to t and k 6 4 t 2 K 1t2 = K1t 12 = K 2t1 = K2t 11 = K 2t2 = K2t 12 = K 3t1 = K3t 11 = K 3t2 = K3t 12 = P 11 = P12 = P 21 = P22 = P 211 = P221 = P 212 = P222 = W 11 = W = W 21 = W = W 31 = W = Fig 1 and Fig 2 show the state trajectory according to t and k As shown in the figures the 2-D LPV system is asymptotically stablized by the designed parameterdependent controller 5 CONCLUSION In this paper we proposed the parameter-dependent state feedback controller for the 2-D discrete LPV system with the FM first model To find the stabilizing conditions of the system we firsttransformed the FMfirstmodel tothe Roesser model and then using the PDLF the relaxation State x 2 (db) k 6 Fig 2 The state trajectory of x 2 with respect to t and k 6 REFERENCES P Apkarian PGahinet and GBecker Self-scheduled H control of linear parameter-varying systems: a design example Automatica 31: P Apkarian and HD Tuan Parameterized LMIs in control theory SIAM Journal on Control and Optimization 38: NK Bose Applied Multidimensional Systems Theory Van Nostrand Reinhold New York 1982 G Becker A Packard D Philbrick and G Balas Control of parameterically-dependent linear systems: a single quadratic Lyapunov approach Proceedings ACC Ch Du and L Xie H Control and Filtering of Two-dimensional Systems Springer Berlin Heidelberg 22 E Fornasini and G Marchesini State-space realization theory of two-dimensional filters IEEE Transactionson Automatic Control 21: t

5 17th IFAC World Congress (IFAC'8) Seoul Korea July E Fornasini and G Marchesini Doubly-indexed dynamical systems: State-space models and structural properties Mathematical Systems Theory 12: T Hinamoto Stability of 2-D discrete systems described by the Fornasini-Marchesini second model IEEE Transactions on Circuits and Systems 44: T Kaczorek Two-Dimensional Linear Systems Springer Berlin Germany 1985 H Kar and V Singh Stability of 2-D Systems Described by the Fornasini-Marchesini First Model IEEE Transactions on Signal Processing 51: T Ooba On stability analysis of 2-D systems based on 2-D Lyapunov matrix inequalities IEEE Transactions on Circuits and Systems 47: P Park and DJ Choi LPV controller design for the nonlinear RTAC system International Journal of Robust and Nonlinear Control 11: RP Roesser Adiscrete state-space model for linear image processing IEEE Transactions of Automatic Control 2: Tong Zhou Stability and Stability Margin for a Two- Dimensional System IEEE Transactions on Signal Processing 54: