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

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

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

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topChitour, 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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