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**On practical asymptotic stabilizability of switched affine systems.**
*(English)*
Zbl 1157.93502

Summary: We report some new results on practical asymptotic stabilizability of switched systems consisting of affine subsystems. We first briefly review some practical asymptotic stabilizability notions and some results from our previous papers. Then we propose a new approach to estimate the region of attraction for switched affine systems. Based on this new approach, we present several new sufficient conditions for the practical asymptotic stabilizability and global practical asymptotic stabilizability of such systems. Finally, a computational approach to check the new sufficient conditions is proposed and it is applied to several numerical examples.

### MSC:

93D20 | Asymptotic stability in control theory |

93C15 | Control/observation systems governed by ordinary differential equations |

93C05 | Linear systems in control theory |

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\textit{X. Xu} et al., Nonlinear Anal., Hybrid Syst. 2, No. 1, 196--208 (2008; Zbl 1157.93502)

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### References:

[1] | Blanchini, F., Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions, IEEE transactions on automatic control, 39, 2, 428-433, (1994) · Zbl 0800.93754 |

[2] | P. Bolzern, W. Spinelli, Quadratic stabilization of a switched affine system about a nonequilibrium point, in: Proceedings of the 2004 American Control Conference, Boston, MA, 2004, pp. 3890-3895 |

[3] | P. Colaneri, J.C. Geromel, A. Astolfi, Stabilization of continuous-time nonlinear switched systems, in: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, Seville, Spain, December 2005, pp. 3309-3314 · Zbl 1129.93042 |

[4] | DeCarlo, R.; Branicky, M.S.; Pettersson, S.; Lennartson, B., Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE, 88, 7, 1069-1082, (2000) |

[5] | E. Feron, Quadratic stabilizability of switched systems via state and output feedback, Technical Report CICS-P-468, MIT Center for Intelligent Control Systems, 1996 |

[6] | Khalil, H.K., Nonlinear systems, (2002), Prentice-Hall Upper Saddle River, NJ |

[7] | Koutsoukos, X.D.; Antsaklis, P.J., Design of stabilizing switching control laws for discrete- and continuous-time linear systems using piecewise-linear Lyapunov functions, International journal of control, 75, 12, 932-945, (2002) · Zbl 1016.93062 |

[8] | Lakshmikantham, V.; Leela, S.; Martynyuk, A.A., Practical stability of nonlinear systems, (1990), World Scientific · Zbl 0753.34037 |

[9] | LaSalle, J.P.; Lefschetz, S., Stability by lyapunov’s direct method with applications, (1961), Academic Press |

[10] | Liberzon, D.; Morse, A.S., Basic problems in stability and design of switched systems, IEEE control systems magazine, 19, 5, 59-70, (1999) · Zbl 1384.93064 |

[11] | Liberzon, D., Switching in systems and control, (2003), Birkhäuser · Zbl 1036.93001 |

[12] | H. Lin, P.J. Antsaklis, Uniformly ultimate boundedness control for uncertain switched linear systems, ISIS Technical Report, ISIS-2003-004, University of Notre Dame, August 2003. Available at: http://www.nd.edu/ isis/tech.html |

[13] | Lin, H.; Antsaklis, P.J., A necessary and sufficient condition for robust asymptotic stabilizability of continuous-time uncertain switched linear systems, (), 3690-3695 |

[14] | H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: A short survey of recent results, in: Proceedings of the 2005 IEEE International Symposium on Intelligent Control, Limassol, Cyprus, June 2005, pp. 24-29 |

[15] | H. Lin, Robust analysis and synthesis of uncertain linear hybrid systems with networked control applications, Ph.D. Thesis, Department of Electrical Engineering, University of Notre Dame, July 2005 |

[16] | Michel, A.N., Quantitative analysis of simple and interconnected systems: stability, boundedness and trajectory behavior, IEEE transactions on circuit theory, CT-17, 3, 292-301, (1970) |

[17] | Michel, A.N.; Porter, D.W., Analysis of discontinuous large-scale systems: stability, transient behavior and trajectory bounds, International journal of systems science, 2, 77-95, (1971) · Zbl 0217.58203 |

[18] | S. Pettersson, B. Lennartson, Stabilization of hybrid systems using a min-projection strategy, in: Proceedings of the 2001 American Control Conference, Arlington, VA, June 2001, pp. 223-228 |

[19] | Picasso, B.; Colaneri, P., Practical stability analysis for quantized control systems via small-gain theorem, () |

[20] | R.N. Shorten, F. Wirth, O. Mason, K. Wulff, C. King, Stability criteria for switched and hybrid systems, Draft, June 2005. Available at: http://www.hamilton.ie/bob/SwitchedStability.pdf · Zbl 1127.93005 |

[21] | Su, R.; Abdelwahed, S.; Neema, S., Computing finitely reachable containable region for switching systems, IEE Proceedings — control theory and applications, 152, 477-486, (2005) |

[22] | Weiss, L.; Infante, E.F., Finite time stability under perturbing forces and on product spaces, IEEE transactions on automatic control, 12, 1, 54-59, (1967) · Zbl 0168.33903 |

[23] | M.A. Wicks, R.A. DeCarlo, Solution of coupled Lyapunov equations for the stabilization of multimodal linear systems, in: Proceedings of the 1997 American Control Conference, Albuquerque, NM, 1997, pp. 1709-1713 |

[24] | Wicks, M.; Peleties, P.; DeCarlo, R., Switched controller synthesis for the quadratic stabilisation of a pair of unstable linear systems, European journal of control, 4, 140-147, (1998) · Zbl 0910.93062 |

[25] | Xu, X.; Antsaklis, P.J., Stabilization of second-order LTI switched systems, International journal of control, 73, 14, 1261-1279, (2000) · Zbl 0992.93078 |

[26] | X. Xu, Practical stabilizability of a class of switched systems, in: Proceedings of 2004 American Control Conference, Boston, MA, June 2004, pp. 4537-4542 |

[27] | Xu, X.; Zhai, G., On practical stability and stabilization of hybrid and switched systems, (), 615-630 · Zbl 1135.93370 |

[28] | Xu, X.; Zhai, G., Practical stability and stabilization of hybrid and switched systems, IEEE transactions on automatic control, 50, 11, 1897-1903, (2005) · Zbl 1365.93359 |

[29] | X. Xu, G. Zhai, New results on practical stabilization and practical reachability of switched systems, in: Proceedings of 2005 American Control Conference, Portland, OR, June 2005, pp. 3784-3789 |

[30] | X. Xu, G. Zhai, Some results on practical asymptotic stabilizability of switched systems, in: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, Seville, Spain, December 2005, pp. 3998-4003 |

[31] | X. Xu, G. Zhai, S. He, Stabilizability and practical stabilizability of continuous-time switched systems: A unified view, in: Proceedings of the 2007 American Control Conference, New York City, NY, July 2007, pp. 663-668. Longer version available at: http://ecse.bd.psu.edu/ xxx12/publications/ACC07a.pdf |

[32] | G. Zhai, A.N. Michel, On practical stability of switched systems, in: Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, December 2002, pp. 3488-3493 |

[33] | G. Zhai, A.N. Michel, Generalized practical stability analysis of discontinuous dynamical systems, in: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, HI, December 2003, pp. 1663-1668 |

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