Given a finite or countable family of continuous vector fields {fi(x)}i∈ I , we show that there exists a feedback k : Rn → I such that the switched system x = fk(x)(x) is globally asymptotically stable whenever there exists a smooth control Liapunov function V such that for all x ≠ 0, ∇V(x)fi(x)<0, for some i ∈ I.
Ancona, F. and Bressan, A.1999: Patchy vector fields and asymptotic stabilization. ESAIM: Control, Optimisation and Calculus of Variations4, 445-72.
2.
Artstein, Z.1983: Stabilization with relaxed controls. Nonlinear Analysis , Theory, Methods and Applications7, 1163-73.
3.
Bacciotti, A.2004: Stabilization by means of state space depending switching rules. Systems and Control Letters53, 195-201.
4.
Bacciotti, A. and Rosier, L.2001: Liapunov functions and stability in control theory. Lecture Notes in Control and Information Sciences 267, Springer Verlag.
Clarke, F.H., Ledyaev, Yu.S. and Stern, R.J.1998: Asymptotic stability and smooth Lyapunov functions. Journal of Differential Equations149, 69-114.
8.
Filippov, A.F.1988: Differential equations with discontinuous righthand sides . Kluwer.
9.
Geromel, J.C. and Colaneri, P.2006: Stability and stabilization of continuous-time switched linear systems. SIAM Journal Control Optimization45, 1915-30.
10.
Hájek, O.1979: Discontinuous differential equations, I. Journal of Differential Equations32, 149-70.
11.
Hu, B., Zhai, G. and Michel, A.2002: Common quadratic Lyapunov-like functions with associated switching regions for two unstable second-order LTI systems. International Journal of Control75, 1127-35.
12.
Liberzon, D.2003: Switching in systems and control. Birkhäuser.
13.
Paden, B.E. and Sastry, S.S.1987: A calculus for computing Filippov’s differential inclusion with application to the variable structure control of robot manipulators . IEEE Transactions on Circuits and Systems34, 73-81.
14.
Petterson, S.2003: Synthesis of switched linear systems. Proceedings of the 42nd IEEE-CDC, 5283-88.
15.
Petterson, S. and Lennarston, B.2001: Stabilization of hybrid systems using a Min-Projection Strategy. Proceedings of the American Control Conference , Arlington, VA, 223-228.
16.
Pogromsky, A.Yu., Heemels, W.P.H. and Nijmijer, H.2003: On solution concepts and well-posedness of linear relay systems. Automatica39, 2139-47.
17.
Shevitz, D. and Paden, B.1994: Lyapunov stability theory of nonsmooth systems. IEEE Transactions on Automatic Control39, 1910-14.
18.
Skafidas, E., Evans, R.J., Savkin, A.V. and Petersen, I.R.1999: Stability results for switched controller systems. Automatica35, 553-64.
19.
Sontag, E.D.1989: A ‘‘universal’’ construction of Artstein’s Theorem on nonlinear stabilization. Systems and Control Letters13, 117-23.
20.
Sontag, E.D.1990: Mathematical control theory. Springer-Verlag .
21.
Sun, Z. and Ge, S.S.2005: Switched linear systems. Springer-Verlag .
22.
Xu, X., Zhai, G. and He, S.2007: Stabilizability and practical stabilizability of continuous-time switched systems: a unified view. Proceedings of American Control Conference. New York.
23.
Wicks, M., Peleties, P. and DeCarlo, R.A.1998: Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems. European Journal of Control4, 140-47.