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

Free access article

Issue		ESAIM: COCV Volume 6, 2001


Page(s)		291 - 331
DOI		10.1051/cocv:2001112

DOI: 10.1051/cocv:2001112

ESAIM: COCV, March 2001, Vol. 6, pp. 291-331

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

Yacine Chitour

Université Paris XI, Département de Mathématiques, 91405 Orsay, France; (Yacine.Chitour@math.u-psud.fr)

(Received November 22, 1999. Revised January, 2001.)

Abstract
We consider a finite-dimensional control system $(\Sigma) \dot x(t)=f(x(t),u(t))$ , such that there exists a feedback stabilizer k that renders $\dot x=f(x,k(x))$ globally asymptotically stable. Moreover, for (H,p,q) with H an output map and $1\leq p\leq q\leq \infty$ , we assume that there exists a ${\cal {K}}_{\infty}$ -function $\alpha$ such that $\Vert H(x_u)\Vert _q\leq \alpha(\Vert u\Vert _p)$ , where x_u is the maximal solution of $(\Sigma)_k \dot x(t)=f(x(t),k(x(t))+u(t))$ , 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 14.5cm

$\begin{displaymath}G_{(H,p,q)}(X)\stackrel{\rm def}{=}\sup_{\Vert u\Vert _p=X}\Vert H(x_u)\Vert _q, \end{displaymath}$

is well-defined. We call profile of k for (H,p,q) any ${\cal {K}}_{\infty}$ -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 k_L(x)=-(1 1)^Tx, we determine the profiles corresponding to the main output maps. In particular, if $\sigma_0$ 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 $\ddot x=\sigma_0(-x-\dot x+u)$ with $x(0)= \dot x(0)=0$ , is finite. We also provide a class of feedback stabilizers k_F that have a linear profile for (x,p,p), $1\leq p\leq \infty$ . For instance, we show that the L₂-gains from x and $\dot x$ to u of the system $\ddot x=\sigma_0(-x-\dot x-(\dot x)^3+u)$ with $x(0)= \dot x(0)=0$ , are finite.

AMS Subject: 93D15, 93D21, 93D30.

Key words: Nonlinear control systems, L^p-stabilization, input-to-state stability, finite-gain stability, input saturation, Lyapunov function.

© EDP Sciences, SMAI 2001

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