## Abstract and Applied Analysis

### Optimal Pole Assignment of Linear Systems by the Sylvester Matrix Equations

#### Abstract

The problem of state feedback optimal pole assignment is to design a feedback gain such that the closed-loop system has desired eigenvalues and such that certain quadratic performance index is minimized. Optimal pole assignment controller can guarantee both good dynamic response and well robustness properties of the closed-loop system. With the help of a class of linear matrix equations, necessary and sufficient conditions for the existence of a solution to the optimal pole assignment problem are proposed in this paper. By properly choosing the free parameters in the parametric solutions to this class of linear matrix equations, complete solutions to the optimal pole assignment problem can be obtained. A numerical example is used to illustrate the effectiveness of the proposed approach.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 301375, 7 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412278769

Digital Object Identifier
doi:10.1155/2014/301375

Mathematical Reviews number (MathSciNet)
MR3253576

Zentralblatt MATH identifier
07022114

#### Citation

He, Hua-Feng; Cai, Guang-Bin; Han, Xiao-Jun. Optimal Pole Assignment of Linear Systems by the Sylvester Matrix Equations. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 301375, 7 pages. doi:10.1155/2014/301375. https://projecteuclid.org/euclid.aaa/1412278769

#### References

• D. Z. Zheng, Linear System Theory, Tsinghua University Press, 2nd edition, 2002.
• M. G. Safonov and M. Athans, “Gain and phase margin for multiloop $LQG$ regulators,” IEEE Transactions on Automatic Control, vol. AC-22, no. 2, pp. 173–179, 1977.
• M. H. Amin, “Optimal pole shifting for continuous multivariable linear systems,” International Journal of Control, vol. 41, no. 3, pp. 701–707, 1985.
• D. P. Iracleous and A. T. Alexandridis, “A simple solution to the optimal eigenvalue assignment problem,” IEEE Transactions on Automatic Control, vol. 44, no. 9, pp. 1746–1749, 1999.
• K. Sugimoto, “Partial pole placement by LQ regulators: an inverse problem approach,” IEEE Transactions on Automatic Control, vol. 43, no. 5, pp. 706–708, 1998.
• R. Lu, Y. Xu, and A. Xue, “${H}_{\infty }$ filtering for singular systems with communication delays,” Signal Processing, vol. 90, no. 4, pp. 1240–1248, 2010.
• R. Lu, H. Li, and Y. Zhu, “Quantized ${H}_{\infty }$ filtering for singular time-varying delay systems with unreliable communication channel,” Circuits, Systems, and Signal Processing, vol. 31, no. 2, pp. 521–538, 2012.
• R. Q. Lu, H. Y. Wu, and J. J. Bai, “New delay-dependent robust stability criteria for uncertain neutral systems with mixed delays,” Journal of the Franklin Institute: Engineering and Applied Mathematics, vol. 351, no. 3, pp. 1386–1399, 2014.
• Y. J. Liu, C. L. P. Chen, G. X. Wen, and S. C. Tong, “Adaptive neural output feedback tracking control for a class of uncertain discrete-time nonlinear systems,” IEEE Transactions on Neural Networks, vol. 22, no. 7, pp. 1162–1167, 2011.
• Y. J. Liu and S. C. Tong, “Adaptive neural network tracking control of uncertain nonlinear discrete-time systems with nonaffine dead-zone input,” IEEE Transactions on Cybernetics, 2014.
• Z. Wu, P. Shi, H. Su, and J. Chu, “Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled-data,” IEEE Transactions on Cybernetics, vol. 43, no. 6, pp. 1796–1806, 2013.
• Z. Wu, P. Shi, H. Su, and J. Chu, “Passivity analysis for discrete-time stochastic Markovian jump neural networks with mixed time delays,” IEEE Transactions on Neural Networks, vol. 22, no. 10, pp. 1566–1575, 2011.
• B. Zhou, Truncated Predictor Feedback for Time-Delay Systems, Springer, Heidelberg, Germany, 2014.
• M. M. Fahmy and J. O\textquotesingle Reilly, “Use of the design freedom of time-optimal control,” Systems and Control Letters, vol. 3, no. 1, pp. 23–30, 1983.
• K. Sugimoto and Y. Yamamoto, “Solution to the inverse regulator problem for discrete-time systems,” International Journal of Control, vol. 48, no. 3, pp. 1285–1300, 1988.
• R. L. Kosut, “Suboptimal control of linear time-invariant systems subject to control structure constraints,” IEEE Transactions on Automatic Control, vol. 15, no. 5, pp. 557–563, 1970.
• J. Wu and T. Lee, “A new approach to optimal regional pole placement,” Automatica, vol. 33, no. 10, pp. 1917–1921, 1997.
• T. C. Yang, T. J. Li, and Y. H. Xu, “Design of optimal regulator with preassigned closed-loop poles by inverse problem approach,” Acta Automatica Sinica, vol. 10, no. 4, pp. 317–323, 1984.
• T. J. Yu and G. Z. Dai, “Necessary conditions for closed-loop poles of linear quadratic optimal control systems,” Control Theory & Applications, vol. 8, no. 2, pp. 230–233, 1991.
• S. W. Mei, T. L. Shen, and K. Z. Liu, Modern Robust Control Theory and Application, Tsinghua University Press, 2003.
• Z. Li and Y. Wang, “Weighted steepest descent method for solving matrix equations,” International Journal of Computer Mathematics, vol. 89, no. 8, pp. 1017–1038, 2012.
• B. Zhou, Z. Li, and Z. Lin, “Discrete-time ${l}_{\infty }$ and ${l}_{2}$ norm vanishment and low gain feedback with their applications in constrained control,” Automatica, vol. 49, no. 1, pp. 111–123, 2013.
• B. Zhou, Z. L. Lin, and G. R. Duan, “${L}_{\infty }$ and ${L}_{2}$ low-gain feedback: their properties, characterizations and applications in constrained control,” IEEE Transactions on Automatic Control, vol. 56, no. 5, pp. 1030–1045, 2011.
• B. Zhou, Z. Li, and Z. Lin, “Observer based output feedback control of linear systems with input and output delays,” Automatica, vol. 49, no. 7, pp. 2039–2052, 2013.
• B. Zhou, Z. Lin, and G. Duan, “Truncated predictor feedback for linear systems with long time-varying input delays,” Automatica, vol. 48, no. 10, pp. 2387–2399, 2012.
• B. Zhou and G. Duan, “A new solution to the generalized Sylvester matrix equation $AV-EVF=BW$,” Systems and Control Letters, vol. 55, no. 3, pp. 193–198, 2006.
• S. G. Wang and Z. H. Yang, Generalized Inverse Matrix and Its Application, Beijing Industrial University Press, 1996. \endinput