Controllability of second-order switched linear systems

PHYSCON 2011, León, Spain, September, 5–September, 8 2011 CONTROLLABILITY OF SECOND-ORDER SWITCHED LINEAR SYSTEMS J. Clotet M.I. Garcı́a-Planas M.D. Magret UPC Spain josep.clotet@upc.edu UPC Spain maria.isabel.garcia@upc.edu UPC Spain m.dolors.magret@upc.edu Abstract We consider second-order switched linear systems and obtain sufficient conditions for such systems to be controllable. Key words Switched linear system, controllability. 1 Introduction Many second order linear systems appear in mechanical and electrical engineering: forced mass-spring damper systems and LRC circuits, for example, exhibit second order behaviour. Controllability of these systems was studied by the authors in [Clotet and Garcı́aPlanas, 2006], [Garcı́a-Planas, 2007] and [Garcı́aPlanas, 2008]. Switched systems constitute a particular kind of hybrid systems which have been studied with growing interest in the last years. In particular, different authors have studied second-order switched systems. In [Sun and Ge, 2005], the controllability of switched linear systems is characterized. The conjecture which states that Hurwitz stability of the convex hull of a set of Metzler matrices is a necessary and sufficient condition for the asymptotic stability of the associated switched linear system under arbitrary switching is proved to be true for systems constructed from a pair of secondorder Metzler matrices, systems constructed from an arbitrary finite number of second-order Metzler matrices and that the conjecture is in general false for higher order systems. In [Xuping and Antsaklis, 1999], necessary and sufficient conditions for stabilizability of second-order LTI systems are found. In [ Zhang, Yangzhou and Pingyuan, 2005], a necessary and sufficient condition for the origin to be asymptotically stable under the predesigned switching law is obtained. In this paper, we obtain sufficient conditions for a second-order switched linear system to be controllable. 2 Second-order Switched Linear Systems Roughly speaking, a switched system is a set of several of continuous-time (or discrete-time) dynamical subsystems and a rule (switching law) orchestrating the switching between them, specifying at any time instant which subsystem is active. We deal with switched second-order linear systems, that is to say, switching systems consisting of several second-order linear subsystems. Let us consider a well-defined switching path θ : [t0 , T ) −→ M , t0 < T ≤ ∞, for some index set M , initial time t0 , and a switching sequence of θ over + + [t0 , T ), {(t0 , θ(t+ 0 )), (t1 , θ(t1 )), . . . , (tℓ , θ(tℓ ))}. Definition 2.1. A singular second-order switched linear system is a system which consists of several linear second-order subsystems and a switching well-defined path σ taking values into the index set M = {1, . . . , ℓ} which indexes the different subsystems.  Eσ δ 2 (x) = Aσ δx + Bσ x + Cσ u y = Dσ x (1) where Eσ , Aσ , Bσ ∈ Mn (R), Cσ ∈ Mn×m (R), Dσ ∈ Mp×n (R). In the continuous case, such a system can be mathematically described by  Eσ ẍ(t) = Aσ ẋ(t) + Bσ x(t) + Cσ u(t) y(t) = Dσ x(t) (2) where Eσ , Aσ , Bσ ∈ Mn (R), Cσ ∈ Mn×m (R), Dσ ∈ Mp×n (R). In the sequential case, by  Eσ x(k + 2) = Aσ x(k + 1) + Bσ x(k) + Cσ u(k) y(k) = Dσ x(k) (3) where Eσ , Aσ , Bσ ∈ Mn (R), Cσ ∈ Mn×m (R), Dσ ∈ Mp×n (R). Calling z = δx, we can linearize a second-order switched linear system thus obtaining:  In Proposition 3.2. ([Garcı́a-Planas, 2007]) A second order singular linear system         δx 0 In x 0 = + u Eσ Bσ Aσ δz z Cσ is controllable if, and only if, the following 2n2 ×((2n− 2)n + 2nm)-matrix and will write EδX = AX + Bu (4)  −E   C=  x where X = ( δx ). In the case where Ei = In , for all i ∈ M the switched system will be called non-singular. Obviously systems where rk Ei = n for all i ∈ M can be reduced to the non-singular case. 3 Controllable States. Controllability We will denote, as usual, by Φ(t, t0 , x0 , u, σ) the state trajectory at time t of system (1) starting from t0 with initial value x0 , input u and switching well-defined path σ. Definition 3.1. A switched singular second-order linear system is said to be controllable when for any t0 ∈ R, xf and w ∈ Rn , there exists a real number tf > t0 , a switching well-defined path σ : [t0 , tf ] −→ M and an input u : [t0 , tf ] −→ Rm such that | .. 0 0 0 0 0 0 {z Proposition 3.1. ([Clotet and Garcı́a-Planas, 2006]) A second order singular linear system E ẍ(t) = Aẋ(t) + Bx(t) + Cu(t) y(t) = Dx(t) have full rank for all s ∈ C. ... −E 0 0 0 ... C 0 0 ... −A 0 0 0 ... 0 C 0 ... B |0 0 0 ...{z0 0 C } } 2nm 1. There exists i ∈ {1, . . . , ℓ} such that
i) rk Ei Ci = n, and
ii) rk s2 Ei − sAi − Bi Ci = n, ∀s ∈ C.  −Ei 0 ... −Ai −Ei ...  Bi −Ai ...  Ci =    0 0 0 .. 0 0 0 Ci 0 0 ... 0 Ci 0 ... 0 0 Ci ... 0 0 0 .. . ... −Ei ... −Ai ... Bi {z (2n−2)n 0 0 0 }| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . ... Ci 0 0 ... 0 Ci 0 ... 0 0 Ci {z       } 2nm We are interested in the non-trivial case where all subsystems are not controllable. From now on, we will consider all subsystems are non-controllable. Proposition 3.4. A switched second-order linear system  Eσ δ 2 (x) = Aσ δx + Bσ x + Cσ u y = Dσ x (1) is controllable if, and only if, the linearized switched linear system Eσ δX = Aσ X + Bσ u: is controllable if, and only if, both matrices EC
s2 E − sA − B C    . Proposition 3.3. Sufficient conditions for controllability of system (1) are the following ones. | We remember that we can determine whether any subsystem is controllable using the following characterizations. i) ii) .. From here we deduce the first sufficient conditions for controllability of system (1). (1) is controllable, the system is also controllable.
 C 0 0 ... 0 0 0 0 C 0 ... 0 0 0 0 0 C ... 0 0 0 . (2n−2)n has full rank.  0 0 0 has full rank. Remark 3.1. Obviously, when one of the subsystems of system Eσ δ 2 (x) = Aσ δx + Bσ x + Cσ u y = Dσ x 0 ... −A −E ... B −A ... 2. There exists i ∈ {1, . . . , ℓ} such that matrix Φ(tf , t0 , x0 , u, σ) = xf , Φ̇(tf , t0 , x0 , u, σ) = w  E ẍ(t) = Aẋ(t) + Bx(t) + Cu(t) y(t) = Dx(t) (5)  In Eσ    δx δz is controllable.  =  0 In Bσ Aσ      0 x   +  u Cσ z Let us consider the
following matrix sequence. N0 = B 1 . . . B ℓ Nk = A · diag(Nk−1 , . . . , Nk−1 ) for k > 0,
where A = I2n A1 . . . A2n−1 . . . . Aℓ . . . A2n−1 1 ℓ Remark 3.2. Note that 4 Controllability and feedback transformations Let us consider a subsystem ẍ = Ai ẋ + Bi x + Ci u (6) for some i ∈ M . We introduce state and derivative feedbacks: rk N0 ≤ rk N1 ≤ · · · ≤ rk N2n = rk N2n+1 = . . . u = Fi ẋ + Gi x + v Proposition 3.5. A necessary and sufficient condition for controllability of singular second-order switched linear system (2) is rk N2n = 2n ẍ = (Ai + Ci Fi )ẋ + (Bi + Ci Gi )x + Ci v Proof. It follows from the algebraic characterization of controllable switched linear systems in [Sun and Ge, 2005]. Corollary 3.1. If there exists j ∈ {1, . . . , 2n} such that Nj has full rank, then the singular second-order switched linear system (2) is controllable. Then the following sufficient condition for controllability may be stated. In particular, if
rk B1 . . . Bℓ = n then system (2) is controllable. For all (i1 , . . . , iℓ ) permutations of {1, . . . , ℓ}, let us denote by   A i1 A i2 . . . A iℓ   0 I2n   Ai =  , ..   . I2n   Bi =   . B i1 B i 2 . . . B i ℓ   Theorem 3.1. A sufficient condition for controllability of the singular second-order switched linear system is that there exists i ∈ {1, . . . , ℓ} such that
rk Bi Ai Bi . . . Ai2n−1 Bi − 2n = = (1 − m)rk Bi1 . . . Biℓ (7) Theorem 4.1. The system (6) is controllable if and only if system (7) is. Corollary 4.1. Suppose that the system (6) verifies Cj = Ci , Aj = Ai + Ci Fi and Bj = Bi + Ci Gi for some couple 1 ≤ i < j ≤ ℓ. Then, the system is controllable if and only if ẍ = Aσ̄ ẋ + Bσ̄ x + Cσ̄ u is, where σ̄ is the path taking values into the index set M̄ = {1, . . . , j − 1, ĵ, j + 1, . . . , ℓ}. Corollary 4.2. Suppose that the system (6) verifies Ci = C, Ai+1 = Ai + Ci Fi and Bi+1 = Bi + Ci Gi for all i = 1, . . . , ℓ. Then, the system is controllable if, and only if, ẍ = A1 ẋ + B1 x + C1 u is.
Remark 4.1. Condition rk Ei Ci = n ensures that there exists a second order derivative feedback Fi such that Ei + Ci Fi is regular and pre-multiplying the system by (Ei + Ci Fi )−1 the new system is standard. We conclude that in this case we can apply sufficient conditions for controllability obtained for standard systems. Acknowledgements This work was partially supported by MTM201019356-C02-02. B i 1 B i 2 . . . B iℓ .. so that the system is transformed into
References Clotet, J., Garcı́a-Planas, M.I. (2006) Second order generalized linear systems. Structural invariants and controllability. International Journal of Pure and Applied Mathematics, 27 - 4, pp. 465-476. Garcı́a-Planas, M.I. (2007) Controllability matrix of second order generalized linear systems. Systems theory and Scientific computation. WSEAS Press, pp. 276-279. Garcı́a-Planas, M.I. (2008) Bounding the distance from a controllable two order system to a noncontrollable one. International Journal of Contemporary Mathematical Sciences 3, 1-4, pp. 17-23. Gurvits, L., Shorten, R., Mason, O. (2007) On the Stability of Switched Positive Linear Systems. IEEE Transactions on Automatic Control, 52-6, pp. 10991103. Sun, Z., Ge, S.S. (2005) Switched Linear Systems. London England. Springer. Xuping, X., Antsaklis, J. (1999) Stabilization of Second-Order LTI Switched Systems. Proceedings of the 38th Conference on Decision and Control. Phoenix, USA. Zhang, L., Yangzhou, C., Pingyuan, C. (2005) Stabilization for a class of second-order switched systems. Nonlinear Analysis, 62, pp. 1527-1535.