Uniform stabilization for linear systems with persistency of excitation. The neutrally stable and the double integrator cases

Abstract : Consider the controlled system $dx/dt = Ax + \alpha(t)Bu$ where the pair $(A,B)$ is stabilizable and $\alpha(t)$ takes values in $[0,1]$ and is persistently exciting, i.e., there exist two positive constants $\mu,T$ such that, for every $t\geq 0$, $\int_t^{t+T}\alpha(s)ds\geq \mu$. In particular, when $\alpha(t)$ becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback $u=Kx$, with $K$ only depending on $(A,B)$ and possibly on $\mu,T$, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when $A$ is neutrally stable and when the system is the double integrator.
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Submitted on : Friday, February 23, 2007 - 10:55:58 AM
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Antoine Chaillet, Yacine Chitour, Antonio Loría, Mario Sigalotti. Uniform stabilization for linear systems with persistency of excitation. The neutrally stable and the double integrator cases. Mathematics of Control, Signals, and Systems, Springer Verlag, 2008, 20 (2), pp.135-156. ⟨10.1007/s00498-008-0024-1⟩. ⟨hal-00132968⟩

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