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On existence of stabilizing switching laws within a class of unstable linear systems. (English) Zbl 1421.93108
Summary: The equivalence of two conditions, condition (3) and condition (4) stated in problem statement section, regarding the existence of stabilizing switching laws between two unstable linear systems first appeared in [E. Feron, Quadratic stabilization of switched system via state and output feedback. Tech. Rep., CICS-F-468, MIT, Cambridge, MA, USA (1996)]. Although Feron never published this result, it has been referenced in almost every survey on switched systems; see, for example, [D. Liberzon and A. S. Morse, IEEE Control Sys. 19, No. 5, 59–70 (1999; Zbl 1384.93064)]. This paper proposes another way to prove the equivalence of two conditions regarding the existence of stabilizing switching laws between two unstable linear systems. One is effective for theoretical derivation, while the other is implementable, and a class of stabilizing switching laws have been explicitly constructed by M. A. Wicks, P. Peleties and R. A. DeCarlo [“Construction of piecewise Lyapunov functions for stabilizing switched systems”, in: Proceedings of the 33rd IEEE conference on decision and control. Los Alamitos, CA: IEEE Computer Society. 3492–3497 (1994; doi:10.1109/CDC.1994.411687)]. With the help of the equivalent relation, a condition for the existence of controllers and stabilizing switching laws between two unstabilizable linear control systems is then proposed. Then, the study is further extended to the issue concerning the construction of quadratically stabilizing switching laws among \(N\) unstable linear systems and \(N\) unstabilizable linear control systems. The obtained results are employed to study the existence of control laws and quadratically stabilizing switching laws within a class of unstabilizable linear control systems. The numerical examples are illustrated and simulated to show the feasibility and effectiveness of the proposed methods.

MSC:
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
93C05 Linear systems in control theory
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