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

Yacine Chitour

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 6, page 291-331
  • ISSN: 1292-8119

Abstract

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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 ( x u ) ∥ 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 14.5cm G ( H , p , q ) ( X ) = def sup ∥ u ∥ p = X ∥ H ( x u ) ∥ q , 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 k L ( x ) = - ( 1 1 ) T x , we determine the profiles corresponding to the main output maps. In particular, if σ 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 ¨ = σ 0 ( - 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 ¨ = σ 0 ( - x - x ˙ - ( x ˙ ) 3 + u ) with x ( 0 ) = x ˙ ( 0 ) = 0 , are finite.

How to cite

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Chitour, Yacine. "On the Lp-stabilization of the double integrator subject to input saturation." ESAIM: Control, Optimisation and Calculus of Variations 6 (2010): 291-331. <http://eudml.org/doc/197300>.

@article{Chitour2010,
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 α such that $\|H(x_u)\|_q\leq \alpha(\|u\|_p)$, where xu 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 $$ G\_\{(H,p,q)\}(X)\stackrel\{\rm def\}\{=\}\sup\_\{\|u\|\_p=X\}\|H(x\_u)\|\_q, $$ 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 kF 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. },
author = {Chitour, Yacine},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Nonlinear control systems; Lp-stabilization; input-to-state stability; finite-gain stability; input saturation; Lyapunov function.; nonlinear control systems; -stabilization; input-to-state stability; Lyapunov function},
language = {eng},
month = {3},
pages = {291-331},
publisher = {EDP Sciences},
title = {On the Lp-stabilization of the double integrator subject to input saturation},
url = {http://eudml.org/doc/197300},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Chitour, Yacine
TI - On the Lp-stabilization of the double integrator subject to input saturation
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
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 $1\leq p\leq q\leq \infty$, we assume that there exists a ${\cal {K}}_{\infty}$-function α such that $\|H(x_u)\|_q\leq \alpha(\|u\|_p)$, where xu 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 $$ G_{(H,p,q)}(X)\stackrel{\rm def}{=}\sup_{\|u\|_p=X}\|H(x_u)\|_q, $$ 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 kF 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.
LA - eng
KW - Nonlinear control systems; Lp-stabilization; input-to-state stability; finite-gain stability; input saturation; Lyapunov function.; nonlinear control systems; -stabilization; input-to-state stability; Lyapunov function
UR - http://eudml.org/doc/197300
ER -

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