Simplified Methods for Eigenvalue Assignment

Omar Moh’d El-Basheer El-Ghezawi^*

Electrical Engineering Department, The University of Jordan, Amman, Jordan.

DOI: 10.4236/apm.2015.57037                

Abstract

A state feedback method of reduced order for eigenvalue assignment is developed in this paper. It offers immediate assignment of m eigenvalues, with freedom to assign the remaining n-m eigenvalues. The method also enjoys a systematic one-step application in the case where the system has a square submatrix. Further simplification is also possible in certain cases. The method is shown to be applicable to uncontrollable systems, offering the simplest control law when having maximum uncontrollable eigenvalues. 

Keywords

Eigenvalue Assignment, Pole Placement, Recursive Methods

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Hassan, M.M. and Amin, M.H. (1987) Recursive Eigenstructure Assignment in Linear Systems. International Journal of Control, 45, 291-310. http://dx.doi.org/10.1080/00207178708933729
[2]	Graybill, F.A. (1983) Matrices with Applications in Statistics. Wadsworth Publishing Company, Belmont.
[3]	Green, P.E. and Carroll, J.D. (1976) Mathematical Tools for Applied Multivariate Analysis. Academic Press, New York.
[4]	El-Ghezawi, O.M.E. (1991) A Two-Stage Method for Eigenvalueigen Vector Assignment. Dirasat, 17, 65-77.
[5]	D’azzo, J.J. and Houpis, C.H. (1995) Linear Control Systems: Analysis and Design. 4th Edition, McGraw-Hill, New York.
[6]	El-Ghezawi, O.M.E. (2010) Unification and Improvement of Certain Methods for Eigenvalue Assignment. Dirasat, 37, 206-213.
[7]	Ackermann, J. and Utkin, V.I. (1998) Sliding Mode Control Design Based on Ackermann’s Formula. IEEE Transactions on Automatic Control, 43, 234-237. http://dx.doi.org/10.1080/00207178708933729
[8]	Petkov, P., Christov, N. and Konstantinov, M. (1991) Computational Methods for Linear Control Systems. Prentice Hall, Upper Saddle River.
[9]	Lancaster, P. and Tismentasky, M. (1985) The Theory of matrices with Applications. 2nd Edition, Academic Press, New York.
[10]	Schott, J.R. (1997) Matrix Analysis for Statistics. John Wiley, Hoboken.
[11]	El-Ghezawi, O.M.E. (1997) Recursive and Modified Recursive Eigenstructure Assignment of Uncontrollable Systems. Dirasat, 24, 620-628.
[12]	Liu, G.P. and Patton, R.J. (1998) Eigenstructure Assignment for Control System Design. John Wiley & Sons, New York.
[13]	White, B.A. (1995) Eigenstructure Assignment: A Survey. Proceedings of the Institution of Mechanical Engineers, 209, 1-11. http://dx.doi.org/10.1243/pime_proc_1995_209_357_02
[14]	Sobel, K.M., Shapiro, E.Y. and Andry, A.N. (1994) Eigenstructure Assignment. International Journal of Control, 59, 13-37. http://dx.doi.org/10.1080/00207179408923068
[15]	Friedland, B. (2005) Control System Design: Introduction to State Space Methods. Dover Publications, New York.