On the L -stabilization of the double integrator subject to input saturation

Abstract : We consider a finite-dimensional control system (Σ) x(t) = f(x(t), u(t)), such that there exists a feedback stabilizer k that renders x = f(x; k(x)) globally asymptotically stable. Moreover, for (H, p, q) with H an output map and 1 ≤ p ≤ q ∞, we assume that there exists a K∞-function α such that ‖H(xu)‖q ≤α(‖u‖p), where xu is the maximal solution of (Σ)k x(t) = f(x(t), k(x(t))+u(t)), corresponding to u and to the initial condition x(0) = 0. Then, the gain function G(H, p, q) of (H, p, q) given by is well-defined. We call profile of k for (H, p, q) any K∞-function which is of the same order of magnitude as G(H, p, q). For the double integrator subject to input saturation and stabilized by kL(x) = −(1 1)T x, we determine the profiles corresponding to the main output maps. In particular, if 0 is used to denote the standard saturation function, we show that the L2-gain from the output of the saturation nonlinearity to u of the system x = σ0(−x−x+u) with x(0) = x(0) = 0, is finite. We also provide a class of feedback stabilizers kF that have a linear profile for (x, p, p), 1 ≤ p ≤ ∞. For instance, we show that the L2-gains from x and x_ to u of the system x =σ0(−x − x − (x)3 + u) with x(0) = x (0) = 0, are finite. © 2001 EDP Sciences, SMAI.
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Yacine Chitour. On the L -stabilization of the double integrator subject to input saturation. ESAIM - Control, Optimisation and Calculus of Variations, 2001, 6, pp.291-331. ⟨10.1051/cocv2001112⟩. ⟨hal-02320856⟩



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