Lp-stabilization of integrator chains subject to input saturation using Lyapunov-based homogeneous design.

Abstract : Consider the $n$th integrator $\dot x=J_nx+\sigma(u)e_n$, where $x\in\mathbb{R}^n$, $u\in\mathbb{R}$, $J_n$ is the $n$th Jordan block and $e_n=(0\ \cdots 0\ 1)^T\in\mathbb{R}^n$. We provide easily implementable state feedback laws $u=k(x)$ which not only render the closed-loop system globally asymptotically stable but also are finite-gain $L_p$-stabilizing with arbitrarily small gain, as in [A. Saberi, P. Hou, and A. Stoorvogel, IEEE Trans. Automat. Control, 45 (2000), pp. 1042--1052]. These $L_p$-stabilizing state feedbacks are built from homogeneous feedbacks appearing in finite-time stabilization of linear systems. We also provide additional $L_\infty$-stabilization results for the case of both internal and external disturbances of the $n$th integrator, namely, for the perturbed system $\dot x=J_nx+e_n\sigma (k(x)+d)+D$, where $d\in\mathbb{R}$ and $D\in\mathbb{R}^n$.
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Yacine Chitour, M. Harmouche, Salah Laghrouche. Lp-stabilization of integrator chains subject to input saturation using Lyapunov-based homogeneous design.. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2015, 53 (4), pp.2406-2423. ⟨10.1137/140997725⟩. ⟨hal-01271283⟩

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