Weighted \(H_\infty \) model reduction for linear switched systems with time-varying delay.

*(English)*Zbl 1154.93326Summary: This paper is concerned with \(\mathcal H_\infty \) model reduction for continuous-time linear switched systems with time-varying delay. For a given stable switched system, our attention is focused on construction of a reduced-order model such that the error system is exponentially stable with a prescribed weighted \(\mathcal H_\infty \) performance. By applying the average dwell time approach and the piecewise Lyapunov function technique, delay-dependent/delay-independent sufficient conditions are proposed in terms of Linear Matrix Inequality (LMI) to guarantee the exponential stability and the weighted \(\mathcal H_\infty \) performance for the error system. The model reduction problem is solved by using the projection approach, which casts the model reduction problem into a sequential minimization problem subject to LMI constraints by employing the cone complementary linearization algorithm. A numerical example is provided to illustrate the effectiveness of the proposed theory.

##### MSC:

93B11 | System structure simplification |

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

93B18 | Linearizations |

##### Keywords:

exponential stability; \(\mathcal H_\infty \) performance; model reduction; switched systems; time-varying delay
PDF
BibTeX
XML
Cite

\textit{L. Wu} and \textit{W. X. Zheng}, Automatica 45, No. 1, 186--193 (2009; Zbl 1154.93326)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Boukas, E.K.; Al-Muthairi, N.F., Delay-dependent stabilization of singular linear systems with delays, International journal of innovative computing, information and control, 2, 283-291, (2006) |

[2] | Boukas, E.K.; Shi, P.; Nguang, S.K., Robust \(\mathcal{H}_\infty\) control for linear Markovian jump systems with unknown nonlinearities, Journal of mathematical analysis and applications, 282, 241-255, (2003) · Zbl 1029.93064 |

[3] | Chen, B.; Lam, J.; Xu, S., Memory state feedback guaranteed cost control for neutral delay systems, International journal of innovative computing, information and control, 2, 293-303, (2006) |

[4] | Daafouz, J.; Riedinger, P.; Iung, C., Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach, IEEE transactions on automatic control, 47, 11, 1883-1887, (2002) · Zbl 1364.93559 |

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

[6] | El Ghaoui, L.; Oustry, F.; Ait Rami, M., A cone complementarity linearization algorithm for static output-feedback and related problems, IEEE transactions on automatic control, 42, 8, 1171-1176, (1997) · Zbl 0887.93017 |

[7] | Gao, H.; Lam, J.; Wang, C., Model simplification for switched hybrid systems, Systems & control letters, 55, 12, 1015-1021, (2006) · Zbl 1120.93311 |

[8] | Gao, H.; Lam, J.; Wang, C.; Xu, S., \(\mathcal{H}_\infty\) model reduction for discrete time-delay systems: delay independent and dependent approaches, International journal of control, 77, 321-335, (2004) · Zbl 1066.93009 |

[9] | Glover, K., All optimal Hankel-norm approximations of linear multivariable systems and their \(\mathcal{L}_\infty\)-error bounds, International journal of control, 39, 6, 1115-1193, (1984) · Zbl 0543.93036 |

[10] | Grigoriadis, K.M., Optimal \(\mathcal{H}_\infty\) model reduction via linear matrix inequalities: continuous- and discrete time cases, Systems & control letters, 26, 5, 321-333, (1995) · Zbl 0877.93017 |

[11] | Hale, J.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer Berlin |

[12] | Hespanha, J.P.; Morse, A.S., Switching between stabilizing controllers, Automatica, 38, 11, 1905-1917, (2002) · Zbl 1011.93533 |

[13] | Hespanha, J P, & Morse, A S (1999). Stability of switched systems with average dwell time. In: Proc. 38th conf. decision control (pp. 2655-2660) |

[14] | Hu., B.; Michel, A.N., Stability analysis of digital feedback control systems with time-varying sampling periods, Automatica, 36, 897-905, (2000) · Zbl 0941.93034 |

[15] | Ishii, H.; Francis, B.A., Stabilizing a linear system by switching control with Dwell time, IEEE transactions on automatic control, 47, 12, 1962-1973, (2002) · Zbl 1364.93641 |

[16] | Liberzon, D., Switching in systems and control, (2003), Birkhauser Boston · Zbl 1036.93001 |

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

[18] | Moon, Y.S.; Park, P.; Kwon, W.H.; Lee, Y.S., Delay-dependent robust stabilization of uncertain state-delayed systems, International journal of control, 74, 1447-1455, (2001) · Zbl 1023.93055 |

[19] | Moore, B., Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE transactions on automatic control, 26, 17-31, (1981) · Zbl 0464.93022 |

[20] | Morse, A.S., Supervisory control of families of linear set-point controllers part I: exact matching, IEEE transactions on automatic control, 41, 10, 1413-1431, (1996) · Zbl 0872.93009 |

[21] | Shi, P.; Boukas, E.K.; Agarwal, R.K., Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay, IEEE transactions on automatic control, 44, 11, 2139-2144, (1999) · Zbl 1078.93575 |

[22] | Sun, X.M.; Zhao, J.; Hill, D.J., Stability and \(\mathcal{L}_2\)-gain analysis for switched delay systems: a delay-dependent method, Automatica, 42, 1769-1774, (2006) · Zbl 1114.93086 |

[23] | Wu, L.; Shi, P.; Gao, H.; Wang, C., \(\mathcal{H}_\infty\) mode reduction for two-dimensional discrete state-delayed systems, IEE Proceedings-vision, image and signal processing, 153, 6, 769-784, (2006) |

[24] | Xu, S.; Lam, J.; Huang, S.; Yang, C., \(\mathcal{H}_\infty\) model reduction for linear time-delay systems: continuous time case, International journal of control, 74, 11, 1062-1074, (2001) · Zbl 1022.93008 |

[25] | Zhai, G.; Hu, B.; Yasuda, K.; Michel, A.N., Disturbance attenuation properties of time-controlled switched systems, Journal of franklin institute, 338, 7, 765-779, (2001) · Zbl 1022.93017 |

[26] | Zhai, G.; Lin, H.; Kim, Y.; Imae, J.; Kobayashi, T., \(\mathcal{L}_2\) gain analysis for switched systems with continuous-time and discrete-time subsystems, International journal of control, 78, 15, 1198-1205, (2005) · Zbl 1088.93010 |

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.