# 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

## Access Full Article

top## Abstract

top## How to cite

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 -

## References

top- V. Blondel, E. Sontag, M. Vidyasagar and J. Willems, Open Problems in Mathematical Systems and Control Theory. Springer-Verlag, London (1999). Zbl0945.93005
- J.C. Doyle, T.T. Georgiou and M.C. Smith, The parallel projection operators of a nonlinear feedback system, in Proc. 31st IEEE Conf. Dec and Control. Tucson, AZ, IEEE Publications, Piscataway, NJ (1992) 1050-1054.
- A.T. Fuller, In the large stability of relay and saturated control with linear controllers. Internat. J. Control10 (1969) 457-480. Zbl0176.39302
- D.J. Hill, Dissipative nonlinear systems: Basic properties and stability analysis, in Proc. 31st IEEE Conf. Dec and Control. Tucson, AZ, IEEE Publications, Piscataway, NJ (1992) 3259-3264.
- W. Liu, Y. Chitour and E.D. Sontag, On finite-gain stabilization of linear systems subject to input saturation. SIAM J. Control Optim.4 (1996) 1190-1219. Zbl0855.93077
- Z. Lin and A. Saberi, A semi-global low and high gain design technique for linear systems with input saturation stabilization and disturbance rejection. Internat. J. Robust Nonlinear Control5 (1995) 381-398. Zbl0833.93046
- Z. Lin, A. Saberi and A.R. Teel, Simultaneous Lp-stabilization and internal stabilization of linear systems subject to input saturation - state feedback case. Systems Control Lett.25 (1995) 219-226. Zbl0877.93101
- A. Megretsky, A gain scheduled for systems with saturation which makes the closed loop system L2-bounded. Preprint (1996).
- E.D. Sontag, Mathematical theory of control. Springer-Verlag, New York (1990). Zbl0703.93001
- E.D. Sontag, Stability and stabilization: Discontinuities and the effect of disturbances, in Nonlinear analysis, differential equation and control, edited by F.H. Clarke and R.J. Stern, Nato Sciences Series C 528 (1999). Zbl0937.93034
- E.D. Sontag and H.J. Sussmann, Remarks on continuous feedbacks, in Proc. IEEE Conf. Dec and Control. Albuquerque, IEEE Publications, Piscataway, NJ (1980) 916-921.
- A.R. Teel, Global Stabilization and restricted tracking for multiple integrators with bounded controls. Systems Control Lett.24 (1992) 165-171. Zbl0752.93053
- Y. Yang, H.J. Sussmann and E.D. Sontag, Stabilization of linear systems with bounded controls. IEEE Trans. Automat. Control39 (1994) 2411-2425. Zbl0811.93046
- Y. Yang, Global stabilization of linear systems with bounded feedbacks. Ph.D. Thesis, Rutgers University (1993).
- A.J. Van Der Schaft, L2-gain and passivity techniques in nonlinear control. Springer, London, Lecture Notes in Control and Inform. Sci. (1996).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.