Robust H Control for Uncertain Two-Dimensional Discrete Systems Described by the General Model via Output Feedback Controllers

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1 International Robust Journal H Control of Control for Uncertain Automation wo-dimensional and Systems Discrete vol. Systems 6 no. 5 Described pp by the October General 008 Model via Output 785 Robust H Control for Uncertain wo-dimensional Discrete Systems Described by the General Model via Output Feedback Controllers Abstract: his paper considers the problem of robust H control for uncertain -D discrete systems in the General Model via output feedback controllers. he parameter uncertainty is assumed to be norm-bounded. he purpose is the design of output feedback controllers such that the closed-loop system is stable while satisfying a prescribed H performance level. In terms of a linear matrix inequality a sufficient condition for the solvability of the problem is obtained and an explicit expression of desired output feedback controllers is given. An example is provided to demonstrate the application of the proposed method. Keywords: Discrete systems general model linear matrix inequality robust H control twodimensional systems uncertain systems. 1. INRODUCION wo-dimensional (-D systems have received much attention during the past decades [10] since -D systems have extensive applications in image processing seismographic data processing thermal processes water stream heating modeling of partial differential equations and other areas [38]. Different kinds of -D models such as -D Roesser models and -D Fornasini-Marchesini models [] etc. have been proposed. hese models have also been extended to multidimensional systems; see e.g. [1113]. A great number of fundamental notions and results of onedimensional (1-D discrete systems were generalized to -D discrete systems [17]. Since the introduction of the general state space model of -D systems (-D Manuscript received May ; revised December 1 007; accepted May 008. Recommended by Editorial Board member Guang-Hong Yang under the direction of Editor Jae Weon Choi. his work was supported by NSF of P. R. China under Grants the Jiangsu Planned Projects for Postdoctoral Research Funds Bthe National Science Foundation for Distinguished Young Scholars of P. R. China under Grant and the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Huiling Xu is with the Department of Applied Mathematics Nanjing University of Science and echnology Nanjing and the Research Institute of Automation Southeast University P. R. China ( xuhuiling_005@yahoo. com.cn. Yun Zou and Shengyuan Xu are with the Department of Automation Nanjing University of Science and echnology P. R. China ( s: syxu0@yahoo.com.cn zouyun@vip. 163.com. Lei Guo is with the School of instrument Science and Opto- Electronics Engineering Beihang University Beijing China ( l.guo@seu.edu.cn. GM in [9] a lot of research topics such as controllability [5] minimum energy control [4] internal stability [1] computation of -D eigenvalues and the transfer function matrix [0] related to -D GM have been studied in the literature. Since the late 1980s the problem of H control and filtering for linear systems has drawn considerable attention; many relevant results have been reported in the literature; see e.g. [14] and the references therein. he robust H control problem for discrete-time systems was addressed in [19]. Very recently robust H control and filtering for -D systems described by Roesser models and Fornasini- Marchesini model have been studied; see e.g. [16 18] and the references therein. However the problem of robust H control for -D discrete systems described by General models has not been investigated up to date. In this paper we are concerned with the problem of robust H control for -D discrete systems in the General model with parameter uncertainties. he parameter uncertainty is assumed to be unknown but norm bounded. he problem to be addressed is to design an output feedback controller such that the resulting closed-loop system is asymptotically stable and satisfies a prescribed H performance for all admissible uncertainties. A sufficient condition for the solvability of this problem is proposed in terms of a linear matrix inequality (LMI. When this LMI is feasible an explicit expression of desired output feedback controllers is given. An example is provided to demonstrate the applicability of the proposed methods. Notation. hroughout this paper for Hermitian matrices X and Y the notation X Y (respectively X

2 786 >Y means that the matrix X Y is positive semidefinite (respectively positive definite. I is the identity matrix with appropriate dimension. he superscript represents the transpose and the complex conjugate transpose. he notation x stands for the Euclidean norm of the vector x. Matrices if not explicitly stated are assumed to have compatible dimensions.. PROBLEM FORMULAION AND PRELIMINARIES Consider the following class of -D discrete-time systems described by the general model [9]: xi ( j ( A1 A1 xi ( j ( A A x( i j 1 ( A A x( i j ( Σ : = +Δ Δ Δ 0 + Bu 1 ( i+ + Bu( i j+ 1 + B0u( i j + Lω( i+ + L ω( i j+ 1 + L ω( i j (1 z( i j = Hx( i j + Gω( i j + Bu( i j ( yi ( j = ( C+Δ Cxi ( j + Dω( i j (3 with the boundary conditions: x(0 j x x(0 i = x i i j = 01 L (4 = 0 j 0 n x( i j R ui ( j R and yi ( j R are respectively the local state vector the control input and measurement output of the plant. q z( i j R is the controlled output ω( i j R is the disturbance input which is assumed to belong to l { [ 0 [ 0 } A n Bn L n C B D H and G are known real constant matrices with appropriate dimensions. ΔAn and ΔC are unknown matrices representing norm-bounded parameter uncertainties and are assumed to be of the form: ΔA M 1 1 A M Δ = A0 M0 Fi ( jn Δ ΔC M c m p l (5 q l Fi ( j R is an unknown real matrix satisfying F( i j 1 (6 and M n ( n = 01 M c and N are known real constant matrices with appropriate dimensions. An unforced -D GM extracted from (1-(3 is given by ( ( ( + A xij ( + Lω( i+ ( Σ 1: xi+ 1 j+ 1 = Axi j + Axij (7 + Lω( i j+ 1 + L0ω( i j z( i j = Hx( i j + Gω( i j. (8 hen the transfer function of system is ( Σ 1 as follows: ( = ( z z H z z I z A z A A ( zl + z L + L + G Definition 1 [3]: he -D GM ( Σ 1 is asymptotically stable if sup xi ( j < and i j i j lim xi ( j = 0 and zero input ω( i j 0 and any boundary condition such that sup xi ( 0 < i and sup x(0 j <. j Definition : Consider the -D GM ( Σ 1 with zero boundary condition. Given a scalar γ > 0 -D GM ( Σ 1 is said to be asymptotically stable with and H noise attenuation γ if it is asymptotically stable and satisfies zi ( j < γ ω( i j for any nonzero { [ [ } ω( i j l 0 0 zi ( j = zi ( + zi ( j+ 1 zi ( j ω( i j = ω( i+ ω( i j+ 1 ω( i j. Remark 1: Note that the definition is a natural extension of -D Fornasini-Marchesini local statespace (FM LSS model to General Model (GM case []. Remark : By the -D Parseval s theorem [1] it is easy to see that the condition zi ( j < γ ω( i j under zero-initial conditions for any nonzero ω ( i j { [ 0 [ 0 } l is equivalent to j θ j θ j θ j θ e ( 1 e = sup σ[ e ( 1 e ] < γ. θ1 θ [ 0π] σ ( represents the maximum singular value of matrix ( Now we consider the following full-order dynamic output feedback controller:

3 Robust H Control for Uncertain wo-dimensional Discrete Systems Described by the General Model via Output 787 ( 1k ˆ( A xij ˆ( 1 A xij ˆ( ( Σ k: xˆ i+ 1 j+ 1 = A x i+ 1 j + k + + 0k + B1ky( i+ + B1ky( i+ + B y( i j 0k ˆ ( ( (9 ui ( j = C xi j + D y i j (10 k ˆ ( k x i j is the controller state Ank B nk ( n = 01 C k and D k are matrices to be determined later. Applying this controller to the uncertain D GM ( results in the following closed-loop system: ( 1c 1 ( Σ : ξ i+ 1 j+ 1 = ( A +Δ A ξ( i+ + ( Ac +Δ Ac ξ ( i j+ 1 + ( A0c +Δ A0c ξ( i j + L1cω( i+ + L ω( i j+ 1 + L ω( i j c 0c c (11 z( i j = ( C +Δ C ξ( i j + D ω( i j (1 x( i j ξ ( i j = xˆ( i j M c BDkM c c c c M = N = [ N ] n Mn + BnDkMc = Bnk M c 0 Δ Anc = MnFN Δ Cc = McFN. and An + BnDkC BnCk Anc = BnkC A Dc = G+ BDkD nk Ln + BnDkD Cc = [ H + BDkC BCk] Lnc = Bnk D n = 01. he robust H control problem to be addressed in this paper can be formulated as follows: given a scalar γ >0 find an output feedback controller in the form of ( Σ k such that the resulting closed-loop system ( Σ is asymptotically stable with an H noise attenuation γ. We end this section by presenting Lemmas that will be essential in the proof of our main results in the next section. Lemma 1 [15]: ϒ be a nonsingular symmetric matrix and F satisfy (6. If M and N are constant matrices of appropriate dimensions and there exists a constant ε >0 such that ε I NN > 0 then ( A+ MFN ( A+ MFN A ( εmm A+ ε N N. Lemma [7]: he -D GM ( 1 is asymptotically stable if there exist matrices P 1 > 0 P > 0 P 0 > 0 such that the following LMI holds: A PA P < 0 P = diag{ P P P } A = [ A A A ] P = P 1 + P + P BOUNDED REAL LEMMA he following bounded real lemma will play an important role in solving the robust H control problem formulated in next section. heorem 1: Given a scalar γ > 0 the -D GM (Σ 1 is asymptotically stable with an H noise attenuation γ if there exist matrices P > 0 P 1 > 0 P > 0 such that the following LMI holds: A PA P+ H H A PL+ H G < 0 LPA+ G H γ I+ LPL+ GG (13 P = diag P P P P P { 1 1} = { } = { } = [ ] A = [ A A A ]. H diag H H H G diag G G G L L L L Proof : By (13 it is easy to see that A PA P < 0. Noting this and Lemma we have that system ( Σ 1 ω i j = 0 is asymptotically stable. herefore with ( Gz ( 1 z is analytic in z 1 1 z 1. Next we shall show jθ1 jθ * jθ1 jθ jθ1 jθ Ue ( e = γ I ( e e e ( e > 0 for all θ θ [ 0π] 1. o this end we note that (13 implies that there exist a matrix Φ > 0 such that A PA P+ H H +Φ A PL+ H G < 0 LPA+ G H γ I+ LPL+ GG (14 Φ= diag{ Φ 00}. Pre-multiplying and post-multiplying (14 by jθ1 jθ e I e I I jθ jθ e I e I I and jθ1 jθ e I e I I jθ1 jθ e I e I I

4 788 respectively we obtain that for all θ θ [ 0π] ( θ θ 1 A j 1 j PA( jθ1 jθ L( jθ1 jθ L( jθ1 jθ P+Φ+ 3H H 3H G + < 0 3G H 3γ I + 3G G (15 1 θ = jθ1 1+ jθ θ = jθ1 1+ jθ + 0 A( jθ j e A e A A L( jθ j e L e L L. herefore from (15 we have that for all θ θ 0π 1 [ ] 1 1 Φ P+ 3 H H + A( jθ jθ P A( jθ jθ < 0 and Φ P + 3 H H + A( j j P A( j j + [ A( j j PL( j j + 3 H G] (16 Λ ( jθ jθ [ L( jθ jθ PA( jθ jθ + 3 G H] < 0 Λ( jθ jθ θ1 θ θ1 θ θ1 θ θ1 θ = 3γ I 3 G G L( jθ jθ PL( jθ jθ > 0. Ω( jθ jθ 1 θ1 θ θ1 θ Λ ( jθ1 jθ [ L( jθ1 jθ PA( jθ1 jθ + 3 G H ]. =Φ+ [ A( j j PL( j j + 3 H G] It easy to see that Ω ( jθ1 jθ > 0. hen (16 can be rewritten 1 1 H H jθ1 jθ Ω( jθ jθ PA( jθ jθ P + 3 +Ω ( < 0. ( jθ1 jθ 1 1 R( jθ jθ = e I A jθ jθ. (17 hen it is east to see that the asymptotic stability of the system ( Σ 1 implies that R( jθ jθ is invertible for all θ θ [ 0π] 1. By (16 we have that 1 jθ1 jθ * jθ1 jθ jθ1 jθ 3 Ue ( e = 3γ I 3 ( e e e ( e G H] Ω( jθ1 jθ θ1 θ θ1 θ + Λ( jθ jθ [ L( jθ jθ PA( jθ jθ [ A( j j PL( j j 3 H G]. (18 Note that ( jθ1 jθ [ A( jθ1 jθ PL( jθ1 jθ H G ( jθ1 jθ L( jθ1 jθ ( θ θ + 3 ] =Φ > 0. Ω + 3 ] Λ [ PA j j G H 1 (19 hen by the Schur complement formula (19 can be written Λ( jθ1 jθ ( θ1 θ ( θ1 θ ( θ1 θ ( θ1 θ Ω( jθ jθ A j j PL j j + 3H G L j j PA j j + 3G H > 0. 1 j From (18 and (0 we have that 1 j ( all θ θ [ 0π] (0 U e θ θ > 0 for 1.hus the -D GM ( Σ 1 has an H noise attenuation γ. his completes the proof. Remark 3: In heorem 1 degenerate to the existing bounded real lemma for discrete -D Fornasini- Marchesini local state-space (FM LSS model []. hus heorem 1 can be regarded as a natural generalization of bounded real lemma of -D Fornasini-Marchesini local state-space (LSS model to General Model (GM case. 4. ROBUS H CONROL In this section an LMI approach will be developed to solve the robust H control problem. heorem : Consider the uncertain -D GM ( Σ. Given a scalar γ > 0 then there full-order dynamic output feedback controller in the form of ( Σ k such that the resulting closed-loop system is asymptotically stable with an H noise attenuation γ if there exist matrices X > 0 Y > 0 Π > 0 Π > 0 D k Z n n that P1 Φ Ψ and scalar 0 ( n 01 Θ1 0 J E 0 U 0 Θ V F 0 0 J V P% 0 Q 0 < 0 E F 0 Θ3 K Q K Θ4 0 U Θ5 n P ε > = such { % P P P} Θ 1 = diag Π +ΠP Π P +Π 1 1 (1

5 Robust H Control for Uncertain wo-dimensional Discrete Systems Described by the General Model via Output 789 { γ γ γ } Θ = diag I I I { } { ε ε ε } Θ 3 = diag I I I Θ = diag I I I 4 { ε ε ε } Θ = diag I I I 5 E = diag{ E1 E1 E1} [ K ] diag{ F1 F1 F1} F = G+ BDk D { 1 1 1} 1 k c { } [ ] E HY B H BD C 1 = + Ψ + F = 1 K = diag K K K K = BD M U = diag U U U U = NY N J = [ J 1 J J 3 ] AnY + BnΨ An + BnDkC Jn = Zn X An +ΦnC V [ 1 V V 3 ] Ln + BnDkD Vn = X L n +Φ n D Q [ 1 Q Q 3 ] Y I P% = I X Mn + BnDkMc Qn = X Mn +ΦnMc n = 01. In this case a desired dynamic output feedback controller is given in the form of ( Σ with parameters as follows: nk = ( n n nk n k n k nk = ( Φn n k k ( k 01. A S Z X A Y SB CY X B C W X B D CY W B S X B D ( C = Ψ D CY W n= (3 S and W are any nonsingular matrices satisfying SW = I XY. (4 Proof : First from (1 it is easy to see Y I < 0 I X which I XY is nonsingular. his ensures that there always exist nonsingular matrices S and W such that (4 is satisfied. Now we introduce the following nonsingular matrices: Y Ω = W I 1 1. P =Ω Ω 0 I X Ω = 0 S. (5 hen X S P = S Ξ Ξ = W Y X Y YW > 0. ( (6 We have P > 0. By calculations it can be verified that the LMI in (1 can be re-written as Θ1 0 J E 0 U 0 Θ V F 0 0 J V P 0 Q 0 < 0 E F 0 Θ3 K Q K Θ4 0 U Θ5 (7 Ω1 ( P1 P Ω1 Ω1 PΩ 1 Θ 1 = diag Ω1 PΩ 1+ Ω1 P1Ω 1 J = ΩA 1cΩ1 ΩAcΩ1 ΩA0cΩ 1 V = ΩL 1c ΩLc ΩL 0c Q = ΩM1 ΩM Ω M 0 P =ΩP Ω E = diag C Ω C Ω C Ω F = diag D D D { } { } c 1 c 1 c 1 c c c = { Ω1 Ω1 Ω1} = diag{ Mc Mc Mc}. U diag N N N K Now pre- and post-multiplying (7 by diag{ Ω1 Ω1 Ω1 I I I Ω I I I I I I I I I} and its transpose respectively and by the Schur complement formula and Lemma 1 the desired result follows immediately. Remark 4: heorem provides us with a sufficient condition for the solvability of the robust H control of uncertain -D discrete-time systems described by General Model. herefore using the similar approach in [3] it is easy to solve (1 and (3. 5. NUMERICAL EXAMPLE In this example we consider the thermal processes in chemical reactors heat exchangers and pipe furnaces which can be described by the partial differential equation [3]: ( ( x t x t x ( U( t = x t + t (8

6 790 ( x t is usually the temperature at x( space 0 x f and t ( time [ 0 ] and U( t is a given force function. Define x( i j = ( i j ( i j ( i x j t = Δ Δ. It is easy to see that the equation (8 can be converted into the following -D GM: x( i+ 1 j+ 1 = a1x( i+ + a0x( i j (9 + bu( i+ + l w i j 1 0 ( Δt Δt a1 = 1 Δt. a0 = b1 =Δt. Δx Δx Assume that the controlled output and the measurement output are given by ( = 0.4 ( ( ( = 10 ( + 10 (. z i j x i j u i j y i j x i j w i j a = 0.4 a = 0.5 Δ x= 0. Δ t = 0.1 b = 0.1 l = By heorem a desired output feedback controller can be constructed as ( + j+ = xˆ( i xˆ( i j y( i+ y( i j ( = ˆ ( (. xi ˆ 1 1 u i j x i j y i j By the Matlab LMI Control oolbox the minimum γ is obtained as γ * = CONCLUSIONS his paper has studied the problem of robust H control for uncertain discrete -D GM systems. A sufficient solvability condition has been proposed. his desired output feedback controllers can be designed by solving a certain LMI. Examples have shown that the proposed approach is effective. REFERENCES [1] M. Bisiacco E. Fornasini and G. Marchesini On some connections between BIBO and internal stability of two-dimensional filters IEEE rans. on Circuits and Systems vol. 3 pp [] C. Du and L. Xie H Control and Filtering of wo-dimensional Systems Springer-Verlag Heidelberg 00. [3]. Kaczorek wo-dimensional Linear Systems Springer-Verlag Berlin [4]. Kaczorek Minimum energy control for general model of -D linear systems Int. J. Control vol. 47 pp [5]. Kaczorek Local controllability and minimum energy control of continuous -D linear systems with variable coefficients Multidimensional Systems and Signal Processing vol. 6 pp [6] H. Kar and V. Singh Stability analysis of -D state-space digital filters with overflow nonlinearities IEEE rans. Circuits Syst. I vol. 47 pp [7] H. Kar and V. Singh Stability of -D systems described by the Fornasini-Marchesini first model IEEE rans. on Signal Processing vol. 6 pp [8] J. Kevorkian Partial Differential Equations: Analytical Solution echniques Chapman & Hall London [9] J. E. Kurek he general state-space model for a two-dimensional linear digital system IEEE rans. on Automatic Control vol. 30 pp [10] Z. Lin and L. Bruton BIBO stability of inverse -D digital filters in the presence of nonessential singularities of the second kind IEEE rans. Circuits Syst. vol. 36 pp [11] Z. Lin J. Ying and L. Xu An algebraic approach to strong stabilizability of linear n-d MIMO systems IEEE rans. on Automatic Control vol. 47 pp [1] W.-S. Lu and A. Antoniou wo-dimensional Digital Filters Marcel Dekker New York 199. [13] Y. Minory L. Xu and S. Osami D modelfollowing servo system Multidimensional Systems and Signal Processing vol. 10 pp [14] M. Sampei. Mita and M. Nakamichi An algebraic approach to H output feedback control problems Systems & Control t. vol. 14 pp [15] Y. Wang L. Xie and C. E. de Souza Robust control of a class of uncertain nonlinear systems Systems & Control t. vol. 19 pp [16] H. Xu Y. Zou and S. Xu Robust H control for a class of uncertain nonlinear twodimensional systems Int. J. Innovative Computing Information and Control vol. 1 pp [17] S. Xu J. Lam Z. Lin and K. Galkowski Positive real control for uncertain twodimensional systems IEEE rans. Circuits Syst. I vol. 49 pp [18] R. Yang L. Xie and C. Zhang H and mixed

7 Robust H Control for Uncertain wo-dimensional Discrete Systems Described by the General Model via Output 791 H /H control of two-dimensional systems in Roesser model Automatica vol. 4 pp [19] L. Yuan L. E. K. Achenie and W. Jiang Robust H control for linear discrete-time systems with norm-bounded time varying uncertainty Systems & Control t. vol. 7 pp [0] Y. Zou and C. Yang An algorithm for computation of -D eigenvalues IEEE rans. on Automatic Control vol. 39 pp Huiling Xu received the Ph.D. degree in Control Science and Control Engineering from Nanjing University of Science and echnology in 005. Her areas of interest include robust filtering and control singular systems two-dimensional systems. Yun Zou received the Ph.D. degree in Automation from Nanjing University of Science and echnology in His areas of interest include singular systems multidimensional systems and transient stability of power systems. Shengyuan Xu received the Ph.D. degree in Control Science and Control Engineering from Nanjing University of Science and echnology in His areas of interest include robust filtering and control singular systems time-delay systems and nonlinear systems. Lei Guo received the Ph.D. degree in Control Engineering from Southeast University (SEU P. R. China in His research interests include robust control stochastic systems fault detection and nonlinear control and applications.