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

Chitour, Yacine

ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 291-331.

We consider a finite-dimensional control system $\left(\Sigma \right)\dot{x}\left(t\right)=f(x\left(t\right),u\left(t\right))$, such that there exists a feedback stabilizer $k$ that renders $\dot{x}=f(x,k\left(x\right))$ globally asymptotically stable. Moreover, for $(H,p,q)$ with $H$ an output map and $1\le p\le q\le \infty$, we assume that there exists a ${𝒦}_{\infty }$-function $\alpha$ such that $\parallel H\left(x_u\right){\parallel }_q\le {\alpha (\parallel u\parallel }_p)$, where $x_u$ is the maximal solution of ${\left(\Sigma \right)}_k\dot{x}\left(t\right)=f(x\left(t\right),k\left(x\left(t\right)\right)+u\left(t\right))$, corresponding to $u$ and to the initial condition $x\left(0\right)=0$. Then, the gain function $G_{(H,p,q)}$ of $(H,p,q)$ given by

$$G_{(H,p,q)}\left(X\right)\stackrel{\mathrm{def}}{=}\underset{{\parallel u\parallel }_p=X}{sup}{\parallel H\left(x_u\right)\parallel }_q,$$

is well-defined. We call profile of $k$ for $(H,p,q)$ any ${𝒦}_{\infty }$-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 $k_L\left(x\right)=-{\left(11\right)}^{T}x$, 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 $L_2$-gain from the output of the saturation nonlinearity to $u$ of the system $\ddot{x}={\sigma }_0(-x-\dot{x}+u)$ with $x\left(0\right)=\dot{x}\left(0\right)=0$, is finite. We also provide a class of feedback stabilizers $k_F$ that have a linear profile for $(x,p,p)$, $1\le p\le \infty$. For instance, we show that the $L_2$-gains from $x$ and $\dot{x}$ to $u$ of the system $\ddot{x}={\sigma }_0(-x-\dot{x}-{\left(\dot{x}\right)}^3+u)$ with $x\left(0\right)=\dot{x}\left(0\right)=0$, are finite.

Classification : 93D15, 93D21, 93D30
Mots clés : nonlinear control systems, $L^p$-stabilization, input-to-state stability, finite-gain stability, input saturation, Lyapunov function

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