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

Chitour, Yacine. "On the $L^p$-stabilization of the double integrator subject to input saturation." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 291-331. <http://eudml.org/doc/90596>.

@article{Chitour2001,
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\le p\le q\le \infty $, we assume that there exists a $\{\mathcal \{K\}\}_\{\infty \}$-function $\alpha $ such that $\Vert H(x_u)\Vert _q\le \alpha (\Vert u\Vert _p)$, 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\[ G\_\{(H,p,q)\}(X)\stackrel\{\rm def\}\{=\}\sup \_\{\Vert u\Vert \_p=X\}\Vert H(x\_u)\Vert \_q, \]is well-defined. We call profile of $k$ for $(H,p,q)$ any $\{\mathcal \{K\}\}_\{\infty \}$-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 $L_2$-gain from the output of the saturation nonlinearity to $u$ of the system $\ddot\{x\}=\sigma _0(-x-\dot\{x\}+u)$ with $x(0)= \dot\{x\}(0)=0$, is finite. We also provide a class of feedback stabilizers $k_F$ that have a linear profile for $(x,p,p)$, $1\le p\le \infty $. For instance, we show that the $L_2$-gains from $x$ and $\dot\{x\}$ to $u$ of the system $\ddot\{x\}=\sigma _0(-x-\dot\{x\}-(\dot\{x\})^3+u)$ with $x(0)= \dot\{x\}(0)=0$, are finite.},
author = {Chitour, Yacine},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {nonlinear control systems; $L^p$-stabilization; input-to-state stability; finite-gain stability; input saturation; Lyapunov function; -stabilization},
language = {eng},
pages = {291-331},
publisher = {EDP-Sciences},
title = {On the $L^p$-stabilization of the double integrator subject to input saturation},
url = {http://eudml.org/doc/90596},
volume = {6},
year = {2001},
}

TY - JOUR
AU - Chitour, Yacine
TI - On the $L^p$-stabilization of the double integrator subject to input saturation
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2001
PB - EDP-Sciences
VL - 6
SP - 291
EP - 331
AB - 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\le p\le q\le \infty $, we assume that there exists a ${\mathcal {K}}_{\infty }$-function $\alpha $ such that $\Vert H(x_u)\Vert _q\le \alpha (\Vert u\Vert _p)$, 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\[ G_{(H,p,q)}(X)\stackrel{\rm def}{=}\sup _{\Vert u\Vert _p=X}\Vert H(x_u)\Vert _q, \]is well-defined. We call profile of $k$ for $(H,p,q)$ any ${\mathcal {K}}_{\infty }$-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 $L_2$-gain from the output of the saturation nonlinearity to $u$ of the system $\ddot{x}=\sigma _0(-x-\dot{x}+u)$ with $x(0)= \dot{x}(0)=0$, is finite. We also provide a class of feedback stabilizers $k_F$ that have a linear profile for $(x,p,p)$, $1\le p\le \infty $. For instance, we show that the $L_2$-gains from $x$ and $\dot{x}$ to $u$ of the system $\ddot{x}=\sigma _0(-x-\dot{x}-(\dot{x})^3+u)$ with $x(0)= \dot{x}(0)=0$, are finite.
LA - eng
KW - nonlinear control systems; $L^p$-stabilization; input-to-state stability; finite-gain stability; input saturation; Lyapunov function; -stabilization
UR - http://eudml.org/doc/90596
ER -