Discrete Time Systems Part 13 ppt

Stability and L2 Gain Analysis of Switched Linear Discrete-Time Descriptor Systems 349 [3] K Takaba, N Morihira, and T Katayama, “A generalized Lyapunov theorem for descriptor systems,” Systems & Control Letters, vol 24, no 1, pp 49–51, 1995 [4] I Masubuchi, Y Kamitane, A Ohara, and N Suda, “H ∞ control for descriptor systems: A matrix inequalities approach,” Automatica, vol 33, no 4, pp 669–673, 1997 [5] E Uezato and M Ikeda, “Strict LMI conditions for stability, robust stabilization, and H ∞ control of descriptor systems,” in Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, USA, pp 4092–4097, 1999 [6] M Ikeda, T W Lee, and E Uezato, “A strict LMI condition for H2 control of descriptor systems," in Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, pp 601–604, 2000 [7] D Liberzon and A S Morse, “Basic problems in stability and design of switched systems,” IEEE Control Systems Magazine, vol 19, no 5, pp 59–70, 1999 [8] R DeCarlo, M S Branicky, S Pettersson, and B Lennartson, “Perspectives and results on the stability and stabilizability of hybrid systems,” Proceedings of the IEEE, vol 88, no 7, pp 1069–1082, 2000 [9] D Liberzon, Switching in Systems and Control, Birkhäuser, Boston, 2003 [10] Z Sun and S S Ge, Switched Linear Systems: Control and Design, Springer, London, 2005 [11] M S Branicky, “Multiple Lyapunov functions and other analysis tools for switched and hybrid Systems," IEEE Transactions on Automatic Control, vol 43, no 4, pp 475-482, 1998 [12] K S Narendra and J Balakrishnan, “A common Lyapunov function for stable LTI systems with commuting A-matrices,” IEEE Transactions on Automatic Control, vol 39, no 12, pp 2469–2471, 1994 [13] D Liberzon, J P Hespanha, and A S Morse, “Stability of switched systems: A Lie-algebraic condition,” Systems & Control Letters, vol 37, no 3, pp 117–122, 1999 [14] G Zhai, B Hu, K Yasuda, and A N Michel, “Disturbance attenuation properties of time-controlled switched systems," Journal of The Franklin Institute, vol 338, no 7, pp 765–779, 2001 [15] G Zhai, B Hu, K Yasuda, and A N Michel, “Stability and L2 gain analysis of discrete-time switched systems," Transactions of the Institute of Systems, Control and Information Engineers, vol 15, no 3, pp 117–125, 2002 [16] G Zhai, D Liu, J Imae, and T Kobayashi, “Lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems," IEEE Transactions on Circuits and Systems II, vol 53, no 2, pp 152–156, 2006 [17] G Zhai, R Kou, J Imae, and T Kobayashi, “Stability analysis and design for switched descriptor systems," International Journal of Control, Automation, and Systems, vol 7, no 3, pp 349–355, 2009 [18] G Zhai, X Xu, J Imae, and T Kobayashi, “Qualitative analysis of switched discrete-time descriptor systems," International Journal of Control, Automation, and Systems, vol 7, no 4, pp 512–519, 2009 [19] D Liberzon and S Trenn, “On stability of linear switched differential algebraic equations," in Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, China, pp 2156–2161, 2009 [20] S Xu and C Yang, “Stabilization of discrete-time singular systems: A matrix inequalities approach," Automatica, vol 35, no 9, pp 1613–1617, 1999 350 Discrete Time Systems [21] G Zhai and X Xu, “A unified approach to analysis of switched linear descriptor systems under arbitrary switching," in Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, China, pp 3897-3902, 2009 20 Robust Stabilization for a Class of Uncertain Discrete-time Switched Linear Systems Songlin Chen, Yu Yao and Xiaoguan Di Harbin Institute of Technology P R China Introduction Switched systems are a class of hybrid systems consisting of several subsystems (modes of operation) and a switching rule indicating the active subsystem at each instant of time In recent years, considerable efforts have been devoted to the study of switched system The motivation of study comes from theoretical interest as well as practical applications Switched systems have numerous applications in control of mechanical systems, the automotive industry, aircraft and air traffic control, switching power converters, and many other fields The basic problems in stability and design of switched systems were given by (Liberzon & Morse, 1999) For recent progress and perspectives in the field of switched systems, see the survey papers (DeCarlo et al., 2000; Sun & Ge, 2005) The stability analysis and stabilization of switching systems have been studied by a number of researchers (Branicky, 1998; Zhai et al., 1998; Margaliot & Liberzon, 2006; Akar et al., 2006) Feedback stabilization strategies for switched systems may be broadly classified into two categories in (DeCarlo et al., 2000) One problem is to design appropriate feedback control laws to make the closed-loop systems stable for any switching signal given in an admissible set If the switching signal is a design variable, then the problem of stabilization is to design both switching rules and feedback control laws to stabilize the switched systems For the first problem, there exist many results In (Daafouz et al., 2002), the switched Lyapunov function method and LMI based conditions were developed for stability analysis and feedback control design of switched linear systems under arbitrary switching signal There are some extensions of (Daafouz et al., 2002) for different control problem (Xie et al., 2003; Ji et al., 2003) The pole assignment method was used to develop an observer-based controller to stabilizing the switched system with infinite switching times (Li et al., 2003) It is should be noted that there are relatively little study on the second problem, especially for uncertain switched systems Ji had considered the robust H∞ control and quadratic stabilization of uncertain discrete-time switched linear systems via designing feedback control law and constructing switching rule based on common Lyapunov function approach (Ji et al., 2005) The similar results were given for the robust guaranteed cost control problem of uncertain discrete-time switched linear systems (Zhang & Duan, 2007) Based on multiple Lyapunov functions approach, the robust H∞ control problem of uncertain continuous-time switched linear systems via designing switching rule and state feedback was studied (Ji et al., 2004) Compared with the switching rule based on common Lyapunov function approach (Ji et al., 2005; Zhang & Duan, 2007), the one based on multiple Lyapunov 352 Discrete Time Systems functions approach (Ji et al., 2004) is much simpler and more practical, but discrete-time case was not considered Motivated by the study in (Ji et al., 2005; Zhang & Duan, 2007; Ji et al., 2004), based on the multiple Lyapunov functions approach, the robust control for a class of discrete-time switched systems with norm-bounded time-varying uncertainties in both the state matrices and input matrices is investigated It is shown that a state-depended switching rule and switched state feedback controller can be designed to stabilize the uncertain switched linear systems if a matrix inequality based condition is feasible and this condition can be dealt with as linear matrix inequalities (LMIs) if the associated scalar parameters are selected in advance Furthermore, the parameterized representation of state feedback controller and constructing method of switching rule are presented All the results can be considered as extensions of the existing results for both switched and non-switched systems Problem formulation Firstly, we consider a class of uncertain discrete-time switched linear systems described by ⎧x( k + 1) = ( Aσ ( k ) + ΔAσ ( k ) ) x( k ) + ( Bσ ( k ) + ΔBσ ( k ) ) u( k ) ⎪ Bσ ( k ) Aσ ( k ) ⎨ ⎪ y( k ) = C x( k ) σ (k) ⎩ (1) where x( k ) ∈ R n is the state, u( k ) ∈ R m is the control input, y( k ) ∈ R q is the measurement output and σ ( k ) ∈ Ξ = {1, 2, Ν } is a discrete switching signal to be designed Moreover, σ ( k ) = i means that the ith subsystem ( Ai , Bi , C i ) is activated at time k (For notational simplicity, we may not explicitly mention the time-dependence of the switching signal below, that is, σ ( k ) will be denoted as σ in some cases) Here Ai , Bi and C i are constant matrices of compatible dimensions which describe the nominal subsystems The uncertain matrices ΔAi and ΔBi are time-varying and are assumed to be of the forms as follows ΔAi = M Fai ( k )N ΔBi = Mbi Fbi ( k )N bi (2) where M , N , Mbi , N bi are given constant matrices of compatible dimensions which characterize the structures of the uncertainties, and the time-varying matrices Fai ( k ) and Fbi ( k ) satisfy T T T Fai ( k )Fai ( k ) ≤ I , Fbi ( k )Fbi ( k ) ≤ I ∀i ∈ Ξ (3) where I is an identity matrix We assume that no subsystem can be stabilized individually (otherwise the switching problem will be trivial by always choosing the stabilized subsystem as the active subsystem) The problem being addressed can be formulated as follows: For the uncertain switched linear systems (1), we aim to design the switched state feedback controller u( k ) = Kσ x( k ) (4) and the state-depended switching rule σ ( x( k )) to guarantee the corresponding closed-loop switched system Robust Stabilization for a Class of Uncertain Discrete-time Switched Linear Systems x( k + 1) = [ Aσ + ΔAσ + ( Bσ + ΔBσ )Kσ ]x( k ) 353 (5) is asymptotically stable for all admissible uncertainties under the constructed switching rule Main results In order to derive the main result, we give the two main lemmas as follows Lemma 1: (Boyd, 1994) Given any constant ε and any matrices M , N with compatible dimensions, then the matrix inequality MFN + N T F T M T < ε MM T + ε −1N T N holds for the arbitrary norm-bounded time-varying uncertainty F satisfying FT F ≤ I Lemma 2: (Boyd, 1994) (Schur complement lemma) Let S , P , Q be given matrices such that Q = QT , P = PT , then ⎡ P ST ⎤ T −1 ⎢ ⎥ < ⇔ Q < 0, P − S Q S < ⎢S Q ⎦ ⎥ ⎣ A sufficient condition for existence of such controller and switching rule is given by the following theorem Theorem 1: The closed-loop system (5) is asymptotically stable when ΔAi = ΔBi = if there exist symmetric positive definite matrices Xi ∈ R n×n , matrices Gi ∈ R n×n , Yi ∈ R m×n , scalars ε i > (i ∈ Ξ ) and scalars λij < (i , j ∈ Ξ , λii = −1) such that ⎡ ∑ λij (GiT + Gi − Xi ) * ⎢ j∈Ξ ⎢ −Xi Ai Gi + BiYi ⎢ ⎢ Gi ⎢ ⎢ Gi ⎢ ⎢ ⎢ Gi ⎢ ⎣ * * * * λi−1X1 * λi−1X2 0 ⎤ ⎥ ⎥ * ⎥ ⎥ * ⎥ < ∀i ∈ Ξ * ⎥ ⎥ * ⎥ ⎥ −1 λiN XN ⎥ ⎦ * (6) is satisfied ( ∗ denotes the corresponding transposed block matrix due to symmetry), then the state feedback gain matrices can be given by (4) with K i = YiGi−1 (7) and the corresponding switching rule is given by σ ( x( k )) = arg min{ x T ( k )Xi−1x( k )} i∈Ξ Proof Assume that there exist Gi , Xi , Yi , ε i and λij such that inequality (6) is satisfied By the symmetric positive definiteness of matrices Xi , we get (Gi − Xi )T Xi−1 (Gi − Xi ) ≥ (8) 354 Discrete Time Systems which is equal to GiT Xi−1Gi ≥ GiT + Gi − Xi It follows from (6) and λij < that ⎡ ∑ λij GiT Xi−1Gi ⎢ j∈Ξ ⎢ ⎢ AiGi + BiYi ⎢ Γi ⎢ ⎣ where Γi = [Gi , Gi , * −Xi * ⎤ ⎥ ⎥ * ⎥