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

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Abstract

Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A, B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, \({\int_t^{t+T}\alpha(s){\rm d}s \geq \mu}\). In particular, when α(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 μ, 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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Authors and Affiliations

  1. Laboratoire des Signaux et Systèmes, Supélec, 3, Rue Joliot Curie, 91192, Gif s/Yvette, France

    Antoine Chaillet, Yacine Chitour & Antonio Loría

  2. Université Paris Sud, Orsay, France

    Antoine Chaillet & Yacine Chitour

  3. C.N.R.S., 54506, Vandœuvre-lès-Nancy, France

    Antonio Loría

  4. INRIA, Institut Élie Cartan, UMR 7502 Nancy-Université/CNRS/INRIA, POB 239, 54506, Vandœuvre-lès-Nancy, France

    Mario Sigalotti

Authors
  1. Antoine Chaillet
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  2. Yacine Chitour
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  3. Antonio Loría
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  4. Mario Sigalotti
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Corresponding author

Correspondence to Yacine Chitour.

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Notation

A continuous function \({\phi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}}\) is of class \({\mathcal{K}\, (\phi \in \mathcal{K})}\), if it is strictly increasing and \({\phi(0) = 0.\, \psi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}}\) is of class \({\mathcal{L}\, (\psi \in \mathcal{L})}\) if it is continuous, non-increasing and tends to zero as its argument tends to infinity. A function \({\beta : \mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}}\) is said to be a class \({\mathcal{KL}}\)-function if, \({\beta(\cdot,t) \in \mathcal{K}}\) for any t ≥ 0, and \({\beta(s, \cdot) \in \mathcal{L}}\) for any s ≥ 0. We use |·| for the Euclidean norm of vectors and the induced L 2-norm of matrices.

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Chaillet, A., Chitour, Y., Loría, A. et al. Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases. Math. Control Signals Syst. 20, 135–156 (2008). https://doi.org/10.1007/s00498-008-0024-1

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  • DOI: https://doi.org/10.1007/s00498-008-0024-1

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