0
Research Papers

# On the Existence of Controlled Maximal Contractive Sets for Multi Input Linear Systems

[+] Author and Article Information
Andrea Cristofaro

School of Science and Technology,
University of Camerino,
62032 Camerino (MC), Italy
e-mail: andrea.cristofaro@unicam.it

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement, and Control. Manuscript received March 8, 2012; final manuscript received December 6, 2012; published online March 28, 2013. Assoc. Editor: Sergey Nersesov.

J. Dyn. Sys., Meas., Control 135(3), 031017 (Mar 28, 2013) (8 pages) Paper No: DS-12-1078; doi: 10.1115/1.4023402 History: Received March 08, 2012; Revised December 06, 2012

## Abstract

In this work, the existence of maximal contractive sets for unstable multi input linear systems is discussed. It is shown that, under suitable algebraic conditions, linear feedback laws can be designed such that the set of values satisfying the saturation constraints is an invariant set for the closed-loop system, which is asymptotically stable.

<>

## References

Blanchini, F., 1999, “Set Invariance in Control,” Automatica, 35, pp. 1747–1767.
Castelan, E. B., and Hennet, J. C., 1993, “On Invariant Polyhedra of Continuous-Time Linear Systems,” IEEE Trans. Autom. Control, 38, pp. 1680–1685.
Dorea, C., 2009, “Output-Feedback Controlled-Invariant Polyhedra for Constrained Linear Systems,” Proceedings IEEE Conference on Decision and Control 2009, Shanghai, China, pp. 5317–5322.
Dorea, C., and Hennet, J. C., 1999, “(A, B)-Invariant Polyhedral Sets of Linear Discrete Time Systems,” J. Optim. Theory Appl., 103, pp. 521–542.
Hu, T., and Lin, Z., 2001, “Exact Characterization of Invariant Ellipsoids for Linear Systems With Saturating Actuators,” Proceedings IEEE Conference on Decision and Control 2001, Orlando, FL, pp. 4669–4674.
O'Dell, B. D., and Misawa, E. A., 2000, “Semi-Ellipsoidal Controlled Invariant Sets for Constrained Linear Systems,” Proceedings of the American Control Conference 2000, Chigago, IL, ACC, pp. 1779–1783.
Zhou, B., and Duan, G., 2009, “On Analytical Approximation of the Maximal Invariant Ellipsoids for Linear Systems With Bounded Controls,” IEEE Trans. Autom. Control, 54, pp. 346–353.
Sontag, E. D., and Sussman, H. J., 1990, “Nonlinear Output Feedback Design for Linear Systems With Saturating Controls,” Proceedings of the 29thIEEE Conference on Decision and Control, pp. 3414–3416.
Sussman, H. J., and Yang, Y., 1991, “On the Stabilizability of Multiple Integrators by Means of Bounded Feedback Controls,” Proceedings of the 30th IEEE Conference on Decision and Control, pp. 70–72.
Lin, Z., 1998, “Global Control of Linear Systems With Saturating Actuators,” Automatica, 34, pp. 897–905.
Lin, Z., and Saberi, A., 1993, “Semi-Global Exponential Stabilization of Linear Systems Subject to ‘Input Saturation’ via Linear Feedbacks,” Syst. Control Lett., 21, pp. 225–239.
Lin, Z., Saberi, A., and Teel, A. R., 1995, “Control of Linear Systems With Saturating Actuators,” Proceedings of the 34th IEEE Conference on Decision and Control, pp. 285–289.
Hu, T., Lin, Z., and Qiu, L., 2001, “Stabilization of Exponentially Unstable Linear Systems With Saturating Actuators,” IEEE Trans. Autom. Control, 46, pp. 973–979.
Hu, T., Lin, Z., and Qiu, L., 2002, “An Explicit Description of Null Controllable Region of Linear Systems With Saturating Actuators,” Syst. Control Lett., 30, pp. 65–78.
Hu, T., Teel, A. R., and Zaccarian, L., 2006, “Stability and Performance for Saturated Systems via Quadratic and Nonquadratic Lyapunov Functions,” IEEE Trans. Autom. Control, 51, pp. 1770–1786.
Saberi, A., Han, J., and Stoorvogel, A. A., 2002, “Constrained Stabilization Problems for Linear Plants,” Automatica, 38, pp. 639–654.
Stoorvogel, A. A., Saberi, A., and Shi, G., 2004, “Properties of Recoverable Region and Semi-Global Stabilization in Recoverable Region for Linear Systems Subject to Constraints,” Automatica, 40, pp. 1481–1494.
Teel, A. R., 1999, “Anti-Windup for Exponentially Unstable Linear Systems,” Int. J. Robust Nonlinear Control, 9, pp. 701–716.
Lasserre, J. B., 1993, “Reachable, Controllable Sets and Stabilizing Control of Constrained Linear Systems,” Automatica, 29, pp. 531–536.
Corradini, M. L., Cristofaro, A., and Giannoni, F., 2009, “Asymptotic Stabilization of Planar Unstable Linear Systems by a Finite Number of Saturating Actuators,” European Control Conference 2009, Budapest, Hungary.
Corradini, M. L., Cristofaro, A., and Giannoni, F., 2009, “On the Asymptotic Stabilization of Unstable Linear Systems With Bounded Controls,” Far East J. Math. Sci., 35, pp. 233–247.
Benzaouia, A., 1994, “The Resolution of Equation XA + XBX = HX and the Pole Assignment Problem,” IEEE Trans. Autom. Control, 39, pp. 2091–2095.
Castelan, E. B., Gomes da Silva, J. M., Jr., and Cury, J. E. R., 1996, “A Reduced-Order Framework Applied to Linear Systems With Constrained Controls,” IEEE Trans. Autom. Control, 41, pp. 249–255.
Cristea, M., 2007, “A Note on Global Implicit Function Theorem,” Pure Appl. Math., 3, pp. 128–143.
Antsaklis, P. J., and Michel, A. N., 2006, Linear Systems, Birkhäuser, Boston.

## Figures

Fig. 1

Evolution of the state norm ||x(t)||

Fig. 2

Evolution of the controls u1(t)=K1*x(t) (dashed line), u2(t)=K2*x(t) (dotted line), and u3(t)=K3*x(t) (continuous line)

## Errata

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections