Stabilization for Switched Linear Systems with Constant Input via Switched Observer

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1 Stabilization for Switched Linear Systems with Constant Input via Switched Observer Takuya Soga and Naohisa Otsuka Graduate School of Advanced Science and Technology, Tokyo Denki University, Hatayama-Machi, Hiki-Gun, Saitama , Japan Division of Science, School of Science and Engineering, Tokyo Denki University, Hatayama-Machi, Hiki-Gun, Saitama , Japan Abstract. In this paper, sufficient conditions for equilibrium points related to switched linear systems with constant input to be globally asymptotically stable via switched observer are presented. The obtained result is an extension of the result of [7] to switched linear systems with constant input which does not contain uncertain parameters. 1 Introduction Switched system is one of the so-called hybrid systems which consists of a family of subsystems and a switching rule among them. The aspect of the switched system is found in various fields such as aircraft industry, mobile robot and animal world etc[1]. Further, the idea of switching has also been used to design intelligent control which are based on the switching between different controllers. As important problems in such switched systems there are stability problem with arbitrary switching and stabilization problem via appropriate switching rule. Until now many results on stability and stabilization problems for various types of switched linear systems without input have been studied (e.g., [3],[4],[5]-[9]). In addition, it is also important to consider the case which contains the control input for practical applications. In particular, Deaecto et al. [2] gave sufficient conditions for equilibrium point related to continuous-time switched linear system with constant input to be globally asymptotically stable via state feedback switched rule, and then the results were applied to DC-DC converters control design. However, the same problem via switched observer have not been investigated. The objective of this paper is to study sufficient conditions for equilibrium point related to continuous-time switched linear systems with constant input to be globally asymptotically stable via switched observer. In Section 2 the main result for the switched linear systems with constant input is given. Section 3 gives an illustrative numerical example. Finally, concluding remarks are given in Section 4. 91

2 2 Takuya Soga and Naohisa Otsuka 2 Stabilization via switched observer At first, the following notations which will be needed throughout this study. Notations λ max (M) is the maximum eigenvalue of a symmetric matrix M R n n. M is the maximum singular value of a matrix M R n m. ( i.e., M 2 = λ max (M T M) ) For two matrices M 1, M 2 R n n, M 1 > M 2 implies that M 1 M 2 is positive definite. ( i.e., M 1 M 2 > 0 ) A λ is the convex combination of A 1, A 2,, A N. ( i.e., A λ = N i=1 λ ia i, N i=1 λ i = 1, λ i 0 ) Λ := {λ = (λ 1, λ 2,, λ N ) N i=1 λ i = 1, λ i 0}. N := {1, 2,, N}. arg mins :=the minimum index number of minimum element of an ordered i N set S = {s 1, s 2,, s N }. Next, consider the following continuous-time switched linear system Σ σ : { ẋ(t) = Aσ(ˆx,t) x(t) + B σ(ˆx,t) u(t) x(0) = x 0, y(t) = C σ(ˆx,t) x(t), where x(t) R n is the state, u(t) = u R m is the constant input, y(t) R p is the output, ˆx(t) R n is the state of switched linear observer described as d dt ˆx(t) = A σ(ˆx,t)ˆx(t) + L σ(ˆx,t) {y(t) C σ(ˆx,t)ˆx(t)} + B σ(ˆx,t) u, (1) σ(ˆx, t) : R n R + N is the switching strategy which depends on the observer state ˆx and time t and L σ(ˆx,t) is the observer gain. Now, consider the following closed-loop system Σ σ combined by the switched system Σ σ and the switched observer (1). Σ σ : [ ] d Aσ(ˆx,t) dt x(t) = L σ(ˆx,t) C σ(ˆx,t) x(t) + 0 A σ(ˆx,t) L σ(ˆx,t) C σ(ˆx,t) [ Bσ(ˆx,t) 0 ] u, where x(t) := [ ˆx T (t) (x(t) ˆx(t)) T ] T ( R 2n ) is the extended state vector. The following lemma will be used to prove our main result. 92

3 Stabilization for Switched Systems with Constant Input 3 Lemma 1 [7] Suppose that ϵ > 0, η > 0, a positive-definite matrix P 1 R n n (> 0) and L σ R n p C σ R p n in (1). If α > max H σ 2 /ϵη ( > 0 ), σ N where H σ := P 1 L σ C σ, then the following matrix [ ] ϵi P 1 L σ C σ P σ := (P 1 L σ C σ ) T αηi is positive-definite. Then, the following theorem is our main result. Theorem 2 Suppose that the switched system Σ σ with constant input u(t) = u and let x e R n be given. If the following two conditions (i) and (ii) are satisfied, then lim t x(t) = x e for an arbitrary initial state x 0 R n via the switching rule σ(ˆx, t) = arg min i N ˆξ T P 1 (A iˆx + B i u), ˆξ := ˆx xe (2) which depends on the state ˆx of switched observer (1) with L σ := P 1 2 Y σ. (i) There exist λ Λ, a positive-definite matrix P 1 ( > 0 ) and ϵ > 0 such that A T λ P 1 + P 1 A λ < ϵi, (3) A λ x e + B λ u = 0. (4) (ii) There exist a positive-definite matrix P 2 ( > 0 ) and Y i R n p such that A T 1 P 2 + P 2 A 1 C1 T Y1 T Y 1 C 1 < ηi. A T N P 2 + P 2 A N CN TY N T Y N C N < ηi (5) for some η > 0. Proof. Define ξ := [ˆξ T (ξ ˆξ) T ] T (ξ := x x e ) and a functional V ( ξ) as [ ] V ( ξ) = ξ T P1 0 P ξ, P :=, 0 αp 2 where P is a positive definite matrix and α > max H σ 2 /ϵη ( > 0 ), where σ N := P 1 L σ C σ. Then, it follows from (2), (3), (4) and (5) that the time H σ 93

4 4 Takuya Soga and Naohisa Otsuka derivative of V ( ξ) satisfies the equations: [ ] [ ] V ( ξ) = x T P1 0 ξ + 0 αp ξ T P1 0 x 2 0 αp 2 = 2ˆξ T P 1 (A σ ˆx+B σ u)+2ˆξ T P 1 L σ C σ (x ˆx) +2α(ξ ˆξ) T P 2 (A σ L σ C σ )(x ˆx) = 2 min i N {ˆξ T P 1 (A iˆx+b i u)}+2ˆξ T P 1 L σ C σ {(ξ+x e ) (ˆξ+x e )} +2α(ξ ˆξ) T P 2 (A σ L σ C σ ){(ξ + x e ) (ˆξ + x e )} = min i N {ˆξ T (A T i P 1 +P 1 A i )ˆξ+2ˆξ T P 1 (A i x e +B i u)}+ 2ˆξ T P 1 L σ C σ (ξ ˆξ) +α(ξ ˆξ) T (A T σ P 2 +P 2 A σ C T σ Y T σ < min λ Λ {ˆξ T (A T λ P 1 +P 1 A λ )ˆξ+2ˆξ T P 1 (A λ x e + B λ u)} Y σ C σ )(ξ ˆξ) +2ˆξ T P 1 L σ C σ (ξ ˆξ) αη(ξ ˆξ) T (ξ ˆξ) < ϵ ˆξ T ˆξ + 2ˆξ T P 1 L σ C σ (ξ ˆξ) αη(ξ ˆξ) T (ξ ˆξ) = [ ˆξT (ξ ˆξ) ] [ ] [ ] ϵi P 1 L σ C σ ˆξ T (P 1 L σ C σ ) T αηi ξ ˆξ. Since α > max σ N H σ 2 /ϵη ( > 0 ), it follows from Lemma 1 that [ ] ϵi P 1 L σ C σ P σ :=, (P 1 L σ C σ ) T αηi is positive definite. Hence d dt V ( ξ) < 0 which implies that V ( ξ) is a Lyapunov function for the extended switched system Σ σ, that is, lim ˆξ(t) = lim (ˆx(t) t t x e ) = 0 and lim (ξ(t) ˆξ(t)) = lim (x(t) ˆx(t)) = 0. Thus, lim x(t) = x e for t t t an arbitrary initial state x 0 R via the switching rule (2). 3 An illustrative numerical example In this section, an illustrative numerical example for the Theorem 2 will be shown. Consider the two-dimensional switched linear system Σ σ with constant input which consists of two subsystems and switched observer (1). Here, each subsys- 94

5 Stabilization for Switched Systems with Constant Input 5 tem s matrices A i, input matrices B i and output matrices C i are as follows. [ ] [ ] [ ] [ ] A 1 =, A 2 =, B 1 =, B 2 =, C 1 = [ 4 2 ], C 2 = [ 6 1 ]. Now, if we choose an input u = 1, a positive definite matrix P 1 = I 2, ϵ = 1 parameters λ 1 = 0.4, λ 2 = 0.6 ( λ 1 + λ 2 = 1 ), and x e = [2 1] T, then the condition (i) of Theorem 2 is satisfied. Next, if we choose the observer gain where P 2 := I 2, Y 1 := L 1 := P2 1 Y 1, L 2 := P2 1 Y 2, [ ] [ ] 3 5, Y 2 2 := and a parameter η = 1/8, then the 1 condition (ii) of Theorem 2 is satisfied. Thus, x e is globally asymptotically stable via the following switched rule σ(ˆx, t) = arg min i {1,2} ˆξ T P 1 (A iˆx + B i u). (6) In fact, for example, if we choose an initial state x 0 = [3 6] T of the system Σ σ and an initial state ˆx 0 = [3 5.5] T of the switched observer (1), then we have { ˆξT P 1 (A 1 ˆx 0 + B 1 u) = , ˆξ T P 1 (A 2 ˆx 0 + B 2 u) = According to the switching rule σ(ˆx, t) in (6), σ(ˆx 0, 0) = 2 is chosen. Further, if we choose the number of subsystems according to the switching rule in the same way, then we can see the states x(t) and ˆx(t) go to the x e as t tends to (see Fig. 1). 4 Concluding Remarks In this paper, the stabilization problem via switched observer for continuoustime switched linear system with constant input which was studied by Deaecto et al. [2] was investigated. At first, sufficient conditions for equilibrium points related to the switched linear system with constant input to be globally asymptotically stable via switched observer were presented. Further, a numerical example to illustrate the main result was also shown. The obtained result is an extension of the result of [7] to switched linear systems with constant input which does not contain uncertain parameters. 95

6 6 Takuya Soga and Naohisa Otsuka x xhat x2, xhat x e x1, xhat1 Fig. 1. State trajectories of x(t) and ˆx(t) References 1. W.P. Dayawansa and C.F. Martin, A converse Lyapunov theorem for a class of dynamical systems which undergo switching, IEEE Transactions on Automatic Control, vol.44, no.4, pp , G.S. Deaecto, J.C. Geromel, F.S. Garcia and J.A. Pomilio, Switched Affine Systems Control Design With Appliction To DC-DC Converters, IET Control Theory and Applictions vol.4, no.7, pp , E.Feron, Quadratic stabilizibility of switched systems via state and output feedback, MIT Technical report CICSP-468, pp. 1 13, J.C.Geromel, P.Colaneri and P.Bolzern, Dynamic output feedback control of switched linear systems, IEEE Transactions on Automatic Control, vol.53, no.3, pp , D.Liberzon, Switching in systems and control, Systems & Control : Foundation & Applications, Birkhäuser, A. V. Savkin and R. J. Evans, Hybrid dynamical systems : Controller and Sensor Switching Problems, Birkhäuser, T.Soga, N.Otsuka, Quadratic stabilizability for polytopic uncertain switched linear systems via switched observer, submitted for publication. 8. Z.D.Sun, S.S.Ge, Stability theory of switched dynamical systems, Springer-verlag, M.Wicks, P.Peleties and R.DeCarlo, Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems, European Journal of Control, vol.4, no.2, pp ,