Abstract
For a class of linear switched systems in continuous time a controllability condition implies that state feedbacks allow to achieve almost sure stabilization with arbitrary exponential decay rates. This is based on the Multiplicative Ergodic Theorem applied to an associated system in discrete time. This result is related to the stabilizability problem for linear persistently excited systems.
Highlights
IntroductionAlmost sure stabilization, arbitrary rate of convergence, Lyapunov exponents, persistent excitation, Multiplicative Ergodic Theorem
Let N be a positive integer and consider the family of N control systems xi(t) = Aixi(t) + αi(t)Biui(t), i ∈ {1, . . . , N }, where, for i ∈ {1, . . . , N }, xi(t) ∈ Rdi is the state of the subsystem i, ui(t) ∈ Rmi is the control input of the subsystem i, di and mi are non-negative integers, Ai and Bi are matrices with real entries and appropriate dimensions, and αi : R+ → {0, 1} is a switching signal determining the activity of the control input on the i-th subsystem
This paper analyzes the stabilizability of all subsystems in (1.1) by linear feedback laws ui(t) = Kixi(t) under randomly generated switching signals α1, . . . , αN satisfying (1.2), and the maximal almost sure exponential decay rates that can be achieved with such feedbacks
Summary
Almost sure stabilization, arbitrary rate of convergence, Lyapunov exponents, persistent excitation, Multiplicative Ergodic Theorem. FRITZ COLONIUS AND GUILHERME MAZANTI random switching signals, such as in the monographs Costa, Fragoso, and Todorov [13] and Davis [15], and papers such as Benaïm, Le Borgne, Malrieu, and Zitt [3], Cloez and Hairer [12], and Guyon, Iovleff, and Yao [21] Such systems are useful models in several applications, ranging from air traffic control, electronic circuits, and automotive engines to chemical processes and population models in biology. The main result of our paper, Theorem 5.1, implies that, if one requires the feedback to stabilize (1.3) for almost every randomly generated signal α (with respect to the random model described in Section 2), one can retrieve stabilizability with arbitrary decay rates, giving a positive answer to an open problem stated by Chitour and Sigalotti (personal communication). For N ∈ N∗ we let N := {1, ..., N } and R+ := [0, ∞), R∗+ := (0, ∞)
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