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

ESAIM: Control, Optimisation and Calculus of Variations (2001)

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

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topChitour, 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 -

## References

top- [1] V. Blondel, E. Sontag, M. Vidyasagar and J. Willems, Open Problems in Mathematical Systems and Control Theory. Springer-Verlag, London (1999). Zbl0945.93005MR1727924
- [2] J.C. Doyle, T.T. Georgiou and M.C. Smith, The parallel projection operators of a nonlinear feedback system, in Proc. 31${}^{st}$ IEEE Conf. Dec and Control. Tucson, AZ, IEEE Publications, Piscataway, NJ (1992) 1050-1054. MR1205325
- [3] A.T. Fuller, In the large stability of relay and saturated control with linear controllers. Internat. J. Control 10 (1969) 457-480. Zbl0176.39302MR258494
- [4] D.J. Hill, Dissipative nonlinear systems: Basic properties and stability analysis, in Proc. 31${}^{st}$ IEEE Conf. Dec and Control. Tucson, AZ, IEEE Publications, Piscataway, NJ (1992) 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. Zbl0855.93077
- [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. Zbl0833.93046MR1346406
- [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. Zbl0877.93101
- [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). Zbl0703.93001MR1070569
- [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). Zbl0937.93034MR1695014
- [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. Zbl0752.93053MR1153225
- [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. Zbl0811.93046MR1337566
- [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). Zbl0925.93004

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