A Unified Approach to Reduced-Order H2 Output Feedback Controller Design

Yung-Shan  Chou; Kun-Chih Hsieh; Yong-Cheng Chu

doi:10.6180/jase.2006.9.4.08

A Unified Approach to Reduced-Order H2 Output Feedback Controller Design

Yung-Shan Chou This email address is being protected from spambots. You need JavaScript enabled to view it.¹, Kun-Chih Hsieh¹ and Yong-Cheng Chu¹

¹Department of Electrical Engineering, TamKang University, Tamsui, Taiwan 251, R.O.C.

Received: September 16, 2005
Accepted: February 13, 2006
Publication Date: December 1, 2006

Download Citation: ||https://doi.org/10.6180/jase.2006.9.4.08

ABSTRACT

This brief note deals with the synthesis of reduced-order H₂ dynamic output feedback controllers for single-input-single-output linear systems. In particular, magnitude constraint on the controller coefficients, which is of practical interest but has been ignored in the literature, is also taken into account during the design. Sufficient linear matrix inequality (LMI) conditions for the existence of such controllers of arbitrary orders are derived in a unified manner and explicit formulas for computing the controllers are given. In comparison with several existing techniques in the literature, the advantages of our new approach are three-folds: (i) practical issue like the magnitude constraint on the controller coefficients is taken into account in the design, (ii) numerically reliable computation for the proposed LMI method is available, and (iii) the technique can be easily extended to the design of reduced-order controllers with other performances which admit LMI representations, e.g., H_∝ performance, passivity property, generalized H₂ performance. A numerical example is provided to demonstrate the potential of the proposed method.

Keywords: H2 Controller, Reduced-Order, Coefficient Constraint, Linear Matrix Inequality (LMI)

REFERENCES

[1] Doyle, J. C., Glover, K., Khargonekar P. and Francis, B. A., “State Space Solutions to Standard H2 and H Control Problems,” IEEE Trans. on Automat. Contr., Vol. 34, pp. 831847 (1989).
[2] Zhou, K. and Doyle, J. C., Essentials of Robust Control, Prentice Hall, Inc. Upper Saddle River, New Jersey, U.S.A. (1998).
[3] Gahinet, P. and Ignat, A., “Low-Order H Synthesis via LMIs,” Proc. Amer. Contr. Conf., Baltimore, Maryland, pp. 14991500 (1994).
[4] Iwasaki, T. and Skelton, R. E., “All Controllers for the General H Control Problem: LMI Existence Conditions and State Space Formulas,” Automatica, Vol. 30, pp. 13071317 (1994).
[5] Tanaka, H. and Sugie, T., “New Characterization of Fixed-Order Controllers Based on LMI,” Int. J. Contr., Vol. 72, pp. 5874 (1999).
[6] Du, H. and Shi, X., “Low-Order H Controller Design Using LMI and Genetic Algorithm,” Proc. Amer. Contr. Conf., Anchorage, AK, pp. 511512 (2002).
[7] Grigoriadis, K. M. and Skelton, R. E., “Low-Order Control Design for LMI Problem Using Alternating Projection Method,” Automatica, Vol. 32, pp. 1117 1125 (1996).
[8] Mattei, M., “Sufficient Conditions for the Synthesis of H Fixed-Order Controllers,” Int. J. Robust and Nonl. Contr.l, pp. 12371248 (2000).
[9] Asai, T. and Hara, S., “A Unified Approach to LMIBased Reduced Order Self-Scheduling Control Synthesis,” Sys. and Contr. Lett., Vol. 36, pp. 7586 (1999).
[10] Wang, S. and Chow, J. H., “Low-Order Controller Design for SISO Systems Using Coprime Factors and LMI,” IEEE Trans. on Automat. Contr., Vol. 45, pp. 11661169 (2000).
[11] Xin, X., “A Unified LMI Approach to Reduced-Order Controllers: A Matrix Pencil Perspective,” Proc. IEEE Conf. Decis. and Contr., Maui, Hawaii, pp. 51375142 (2003).
[12] Henrion, D., “LMI Optimization for Fixed-Order H Controller Design,” Proc. IEEE Conf. Decis. and Contr., Maui, Hawaii, pp. 46464651 (2003).
[13] Henrion, D., Sebek, M. and Kucera, V., “Positive Polynomials and Robust Stabilization with Fixed-Order Controllers,” IEEE Trans. on Automat. Contr., Vol. 48, pp. 11781186 (2003).
[14] Trofino, A., “Robust, Stable and Reduced Order Dynamic Output Feedback Controllers with Guaranteed H2 Performance,” Proc. IEEE Conf. Decis. and Contr., Maui, Hawaii, pp. 34703475 (2002).
[15] Collins, E. G. Jr., Sivaprasad, A. and Selekwa, M. F., “Insights on Reduced Order H2 Optimal Controller Design Methods,” Proc. Amer. Contr. Conf., Denver, Colorado, pp. 53575362 (2003).
[16] Chou, Y. S., Hsieh, K. C. and Chuang, C. M., “On the Design of Reduced-Order H2 Controllers with Coefficient Constraint,” IEEE International Midwest Symposium on Circuits and Systems, Hiroshima, Japan, pp. III-129III-132 (2004).
[17] Nise, N. S., Control Systems Engineering, Wiley International Edition, fourth edition (2004).
[18] Souza, C. E. De and Shaked, U., “An LMI Method for Output-Feedback H Control Design for System with Real Parameter Uncertainty,” Proc. IEEE Conf. Decision and Control, Tampa, Florida USA, pp. 1777 1779 (1998).
[19] Scherer, C. A., Gahinet, P. and Chilali, M., “Multiobjective Output-Feedback Control via LMI Optimization,” IEEE Trans. on Automat. Contr., Vol. 42, pp. 896911 (1997).
[20] Crusius, C. A. R. and Trofino, A., “Sufficient LMI Conditions for Output Feedback Control Problems,” IEEE Trans. on Automat. Contr., Vol. 44, pp. 1053 1057 (1999).
[21] Gahinet, P., Nemirovski, A., Laub, A. J. and Chilali, M., Manual of LMI Control Toolbox, the Math Works, Inc (1995).
[22] Hsieh, K. C., “On the Design of Fixed-Order Output Feedback Controllers,” MS thesis, Department of Electrical Engineering, Tamkang University, Taipei, Taiwan, R.O.C. (2004).
[23] Henrion, D. and Sebek, M., Fixed-order robust controller design with the Polynomial, Toolbox 3.0, Feb 21 (2003).