Sufficient Conditions for Minimum-Phase Second-Order Linear Systems

Abstract

Several sufficient conditions are derived for a second-order linear system to be minimum phase. It is commonly known that a second-order linear system with collocated sensors and actuators is minimum phase. In general, this common knowledge is correct mathematically when the input influence matrix is the transpose of the output influence matrix. In this paper, we extend this knowledge to the case with noncollocated actuators and sensors. First, a conventional approach is used to prove some sufficient conditions of sensor and actuator locations for a system to be minimum phase. Second, a geometrical approach is introduced to discuss the sufficient conditions, and then used to derive other more useful sufficient conditions for a minimum-phase second-order linear system. Many illustrative examples are provided for the readers to better understand the theories developed in this paper.

Keywords

Transmission zeros minimum-phase linear systems robust control structural control

Get full access to this article

View all access options for this article.

References

Goodwin, G.C. and Sin, K.S. , 1984, Adaptive Filtering, Prediction, and Control, Prentice-Hall, Englewood Cliffs, NJ.

Juang, J.N. and Phan, M. , 1992, "Robust controller designs for second-order dynamic systems: A virtual passive approach," Journal of Guidance, Control, and Dynamics 15(5), 1192-1198.

Kailath, T. , 1980, Linear Systems, Prentice-Hall , Englewood Cliffs, NJ.

Kida, T. , Ohkami, Y. , and Sambongi, S. , 1985, "Poles and transmission zeros of flexible spacecraft control systems," Journal of Cuidance, Cuntrol, and Dynamics 8(2), 208-213.

Kwakernaak, H. and Sivan, R. , 1972, Linear Optimal Control Systems, Wiley-Interscience , New York.

Morris, K.A. and Juang, J.N. , 1994, "Dissipative controller designs for second-order dynamic systems," IEEE Transactions on Automatic Control 39(5), 1056-1063.

Williams, T. , 1989, "Transmission zero bounds for large structures, with applications ," Journal of Guidance, Control, and Dynamics 12(1), 33-38.

Williams, T. , 1992, "Model order effects on the transmission zeros of flexible space structures," Journal of Guidance, Control, and Dynamics 15(2), 540-543.

Williams, T. , 1992, "Transmission zeros of non-collocated flexible structures: Finite-dimensional effects," AIAA Dynamics Specialists Conference, Dallas, TX, April, 1992.

10.

Williams, T. and Juang, J.N. , 1990, "Pole/zero cancellations in flexible space structures ," Journal of Guidance, Control, and Dynamics 13(4), 684-690.

11.

Williams, T. and Juang, J.N. , 1992, "Sensitivity of the transmission zeros of flexible space structures," Journal of Guidance, Control, and Dynamics , 15(2) 368-375.