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

Chitour, Yacine

ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 291-331.

We consider a finite-dimensional control system $\left(\Sigma \right)\dot{x}\left(t\right)=f(x\left(t\right),u\left(t\right))$, such that there exists a feedback stabilizer $k$ that renders $\dot{x}=f(x,k\left(x\right))$ 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 ${𝒦}_{\infty }$-function $\alpha$ such that $\parallel H\left(x_u\right){\parallel }_q\le {\alpha (\parallel u\parallel }_p)$, where $x_u$ is the maximal solution of ${\left(\Sigma \right)}_k\dot{x}\left(t\right)=f(x\left(t\right),k\left(x\left(t\right)\right)+u\left(t\right))$, corresponding to $u$ and to the initial condition $x\left(0\right)=0$. Then, the gain function $G_{(H,p,q)}$ of $(H,p,q)$ given by

$$G_{(H,p,q)}\left(X\right)\stackrel{\mathrm{def}}{=}\underset{{\parallel u\parallel }_p=X}{sup}{\parallel H\left(x_u\right)\parallel }_q,$$

is well-defined. We call profile of $k$ for $(H,p,q)$ any ${𝒦}_{\infty }$-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 $k_L\left(x\right)=-{\left(11\right)}^{T}x$, 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 $\ddot{x}={\sigma }_0(-x-\dot{x}+u)$ with $x\left(0\right)=\dot{x}\left(0\right)=0$, 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 $\ddot{x}={\sigma }_0(-x-\dot{x}-{\left(\dot{x}\right)}^3+u)$ with $x\left(0\right)=\dot{x}\left(0\right)=0$, are finite.

[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