On the ${L}^{p}$-stabilization of the double integrator subject to input saturation
ESAIM: Control, Optimisation and Calculus of Variations, Volume 6 (2001), pp. 291-331.

We consider a finite-dimensional control system $\left(\Sigma \right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\stackrel{˙}{x}\left(t\right)=f\left(x\left(t\right),u\left(t\right)\right)$, such that there exists a feedback stabilizer $k$ that renders $\stackrel{˙}{x}=f\left(x,k\left(x\right)\right)$ globally asymptotically stable. Moreover, for $\left(H,p,q\right)$ 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 $\parallel H\left({x}_{u}\right){\parallel }_{q}\le {\alpha \left(\parallel u\parallel }_{p}\right)$, where ${x}_{u}$ is the maximal solution of ${\left(\Sigma \right)}_{k}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\stackrel{˙}{x}\left(t\right)=f\left(x\left(t\right),k\left(x\left(t\right)\right)+u\left(t\right)\right)$, corresponding to $u$ and to the initial condition $x\left(0\right)=0$. Then, the gain function ${G}_{\left(H,p,q\right)}$ of $\left(H,p,q\right)$ given by

 ${G}_{\left(H,p,q\right)}\left(X\right)\stackrel{\mathrm{def}}{=}\underset{{\parallel u\parallel }_{p}=X}{sup}{\parallel H\left({x}_{u}\right)\parallel }_{q},$
is well-defined. We call profile of $k$ for $\left(H,p,q\right)$ any ${𝒦}_{\infty }$-function which is of the same order of magnitude as ${G}_{\left(H,p,q\right)}$. For the double integrator subject to input saturation and stabilized by ${k}_{L}\left(x\right)=-{\left(1\phantom{\rule{4pt}{0ex}}1\right)}^{T}x$, 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 $\stackrel{¨}{x}={\sigma }_{0}\left(-x-\stackrel{˙}{x}+u\right)$ with $x\left(0\right)=\stackrel{˙}{x}\left(0\right)=0$, is finite. We also provide a class of feedback stabilizers ${k}_{F}$ that have a linear profile for $\left(x,p,p\right)$, $1\le p\le \infty$. For instance, we show that the ${L}_{2}$-gains from $x$ and $\stackrel{˙}{x}$ to $u$ of the system $\stackrel{¨}{x}={\sigma }_{0}\left(-x-\stackrel{˙}{x}-{\left(\stackrel{˙}{x}\right)}^{3}+u\right)$ with $x\left(0\right)=\stackrel{˙}{x}\left(0\right)=0$, are finite.

Classification: 93D15,  93D21,  93D30
Keywords: nonlinear control systems, ${L}^{p}$-stabilization, input-to-state stability, finite-gain stability, input saturation, Lyapunov function
@article{COCV_2001__6__291_0,
author = {Chitour, Yacine},
title = {On the $L^p$-stabilization of the double integrator subject to input saturation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {291--331},
publisher = {EDP-Sciences},
volume = {6},
year = {2001},
zbl = {0996.93082},
mrnumber = {1824105},
language = {en},
url = {http://archive.numdam.org/item/COCV_2001__6__291_0/}
}
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
DA  - 2001///
SP  - 291
EP  - 331
VL  - 6
PB  - EDP-Sciences
UR  - http://archive.numdam.org/item/COCV_2001__6__291_0/
UR  - https://zbmath.org/?q=an%3A0996.93082
UR  - https://www.ams.org/mathscinet-getitem?mr=1824105
LA  - en
ID  - COCV_2001__6__291_0
ER  - 
%0 Journal Article
%A Chitour, Yacine
%T On the $L^p$-stabilization of the double integrator subject to input saturation
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2001
%P 291-331
%V 6
%I EDP-Sciences
%G en
%F COCV_2001__6__291_0
Chitour, Yacine. On the $L^p$-stabilization of the double integrator subject to input saturation. ESAIM: Control, Optimisation and Calculus of Variations, Volume 6 (2001), pp. 291-331. http://archive.numdam.org/item/COCV_2001__6__291_0/

[1] V. Blondel, E. Sontag, M. Vidyasagar and J. Willems, Open Problems in Mathematical Systems and Control Theory. Springer-Verlag, London (1999). | MR | Zbl

[2] J.C. Doyle, T.T. Georgiou and M.C. Smith, The parallel projection operators of a nonlinear feedback system1992) 1050-1054. | MR

[3] A.T. Fuller, In the large stability of relay and saturated control with linear controllers. Internat. J. Control 10 (1969) 457-480. | MR | Zbl

[4] D.J. Hill, Dissipative nonlinear systems: Basic properties and stability analysis1992) 3259-3264.

[5] 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. | Zbl

[6] 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 Control 5 (1995) 381-398. | MR | Zbl

[7] Z. Lin, A. Saberi and A.R. Teel, Simultaneous ${L}^{p}$-stabilization and internal stabilization of linear systems subject to input saturation - state feedback case. Systems Control Lett. 25 (1995) 219-226. | Zbl

[8] A. Megretsky, A gain scheduled for systems with saturation which makes the closed loop system ${L}^{2}$-bounded. Preprint (1996).

[9] E.D. Sontag, Mathematical theory of control. Springer-Verlag, New York (1990). | MR | Zbl

[10] 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). | MR | Zbl

[11] 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.

[12] A.R. Teel, Global Stabilization and restricted tracking for multiple integrators with bounded controls. Systems Control Lett. 24 (1992) 165-171. | MR | Zbl

[13] Y. Yang, H.J. Sussmann and E.D. Sontag, Stabilization of linear systems with bounded controls. IEEE Trans. Automat. Control 39 (1994) 2411-2425. | MR | Zbl

[14] Y. Yang, Global stabilization of linear systems with bounded feedbacks. Ph.D. Thesis, Rutgers University (1993).

[15] A.J. Van Der Schaft, ${L}^{2}$-gain and passivity techniques in nonlinear control. Springer, London, Lecture Notes in Control and Inform. Sci. (1996). | Zbl