##
**Optimal piecewise state feedback control for impulsive switched systems.**
*(English)*
Zbl 1145.93352

Summary: We consider a class of the optimal state feedback control problems involving switched systems with state jumps. The planning time horizon is subdivided into \(N\) subintervals, where the breakpoints are called the re-setting times. One system is used during each subinterval with a piecewise state feedback control form. The re-setting time, the feedback gain matrices and the heights of the state jumps at the re-setting times are considered as decision variables. It is shown, through a time scaling transform, that this class of optimal control problems is equivalent to an optimal parameter selection problem, where the varying re-setting time points are mapped into fixed time points. The gradient formula of the transformed cost function is derived. With this gradient formula, the optimal parameter selection problem can be solved as a nonlinear optimization problem, and the optimal control software MISER 3.3 can be used. Two illustrative examples are solved using the proposed method.

### MSC:

93B52 | Feedback control |

49K25 | Optimal control problems with equations with ret.arguments (nec.) (MSC2000) |

PDF
BibTeX
XML
Cite

\textit{R. Li} et al., Math. Comput. Modelling 48, No. 3--4, 468--479 (2008; Zbl 1145.93352)

Full Text:
DOI

### References:

[1] | Chen, D.; Sun, J.; Wu, Q., Impulsive control and its application to lu’s chaotic system, Chaos, solitons and fractals, 21, 1135-1142, (2004) · Zbl 1060.93517 |

[2] | Hu, S.; Lakshmikantham, V.; Leela, S., Impulsive differential systems and the pulse phenomena, Journal of mathematical, analysis and applications, 137, 605-612, (1989) · Zbl 0684.34003 |

[3] | Khadra, A.; Liu, X.; Shen, X., Application of impulsive synchronization to communication security, IEEE transactions on circuits and systems, 50, 341-350, (2003) · Zbl 1368.94105 |

[4] | Rogovchenko, Y.V., Impulsive evaluation systems: main results and new trends, Dynamics of continuous, discrete, and impulsive systems, 3, 57-88, (1997) · Zbl 0879.34014 |

[5] | Yang, T., Impulsive control theory, (2001), Springer-Verlag Berlin, Germany |

[6] | Bensoussan, A.; Menaldi, J.L., Hybrid control and dynamic programming, Dynamics of continuous, discrete and impulse systems, 3, 395-442, (1997) · Zbl 0897.49022 |

[7] | Barles, G., Deterministic impulse control problems, SIAM journal on control and optimization, 23, 419-432, (1985) · Zbl 0571.49020 |

[8] | Branicky, M.S.; Borkar, V.S.; Mitter, S.K., A unified framework for hybrid control: model and optimal control theory, Institute of electrical and electronics engineers. transactions on automatic control, 43, 31-45, (1998) · Zbl 0951.93002 |

[9] | Wu, C.Z.; Teo, K.L., Global impulsive optimal control computation, Journal of industrial and management optimization, 2, 435-450, (2006) · Zbl 1112.49031 |

[10] | Aubin, J.P.; Lygeros, J.; Quincampoix, M.; Sastry, S.; Seube, N., Impulse differential inclusions: A viability approach to hybrid systems, Institute of electrical and electronics engineers. transactions on automatic control, 47, 1-20, (2001) |

[11] | X. Xu, P.J. Antsaklis, An approach to general switched linear quadratic optimal control problems with state jumps, in: Proceedings of the Fifteenth International Symposium on Mathematical Theory of Networks and Systems, Notre Dame, Indiana, August 2002 |

[12] | Liu, Y.; Teo, K.L.; Jennings, L.S.; Wang, S., On a class of optimal control problems with state jump, Journal of optimization theory and applications, 98, 65-82, (1998) · Zbl 0908.49023 |

[13] | Haddad, W.M.; Chellaboina, V.; Kablar, N.A., Non-linear impulsive dynamical systems, part II: stability of feedback interconnections and optimality, International journal of control, 74, 1659-1677, (2001) · Zbl 1101.93317 |

[14] | Savkin, A.V.; Petersen, I.R.; Skafidas, E.; Evans, R.J., Hybrid dynamical systems: robust control synthesis problems, System and control letters, 9, 81-90, (1996) · Zbl 0866.93038 |

[15] | A.V. Savkin, I.R. Petersen, E. Skafidas, R.J. Evans, Robust control via controlled switching, in: Proceedings of CESA’ 96, Lille, France, 1996 pp. 1117-1122 |

[16] | Skafidas, E.; Evans, R.J.; Mareels, I.M.Y.; Nerode, A., (), 341-355 |

[17] | S. Hedlund, A. Rantzer, Optimal control for hybrid systems, in: Proceedings of the IEEE Conference on Decision and Control, Phoenix, AR, USA, 1999, pp. 3972-3977 |

[18] | Bemporad, A.; Morari, M., Control of systems integrating logic, dynamics and constraints, Automatica, 35, 407-427, (1999) · Zbl 1049.93514 |

[19] | Lee, H.W.J.; Teo, K.L.; Rehbock, V.; Jennings, L.S., Control parametrization enhancing technique for time optimal control problems, Dynamical systems and applications, 6, 243-261, (1997) · Zbl 0894.49018 |

[20] | L.S. Jennings, M.E. Fisher, K.L. Teo, C.J. Goh, MISER3.3 Optimal control software: Theory and user manual, Department of Mathematics and Statistics, The University of Western Australia, Australia, 2004 |

[21] | C.Y.F. Ho, B.W.K. Ling, Y.Q. Liu, P.K.S. Tam, K.L. Teo, Fuzzy switching systems: Minimizing discontinuities and ripple magnitude and energy, Complex Systems, Intelligence and Modern Technology Applications, CSIMTA, Cherbourg, September 2004, pp. 139-144 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.