Explicit Determination of State Feedback Matrices

Abstract

Methods which calculate state feedback matrices explicitly for uncontrollable systems are considered in this paper. They are based on the well-known method of the entire eigenstructure assignment. The use of a particular similarity transformation exposes certain intrinsic properties of the closed loop w-eigenvectors together with their companion z-vectors. The methods are extended further to deal with multi-input control systems. Existence of eigenvectors solution is established. A differentiation property of the z-vectors is proved for the repeated eigenvalues assignment case. Two examples are worked out in detail.

Share and Cite:

El-Ghezawi, O. (2015) Explicit Determination of State Feedback Matrices. Advances in Pure Mathematics, 5, 403-412. doi: 10.4236/apm.2015.57040.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] El-Ghezawi, O.M.E. (2003) Explicit Formulae for Eigenstructure Assignment. Proceedings of the 5th Jordanian International Electrical and Electronic Engineering Conference, Amman, 13-16 October 2003, 183-187.
[2] Porter, B. and D’Azzo, J.J. (1977) Algorithm for the Synthesis of State-Feedback Regulators by Entire Eigenstructure Assignment. Electronic Letters, 13, 230-231.
http://dx.doi.org/10.1049/el:19770167
[3] D’azzo, J.J. and Houpis, C.H. (1995) Linear Control Systems: Analysis and Design. 4th Edition, McGraw-Hill, New York.
[4] 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
[5] White, B.A. (1995) Eigenstructure Assignment: A Survey. Proceedings of the Institution of Mechanical Engineers, 209, 1-11.
http://dx.doi.org/10.1080/00207179408923068
[6] Liu, G.P. and Patton, R.J. (1998) Eigenstructure Assignment for Control System Design. John Wiley & Sons, New York.
[7] Mimins, G. and Paige, C.C. (1982) An Algorithm for Pole Assignment of Time Invariant Linear Systems. International Journal of Control, 35, 341-354.
http://dx.doi.org/10.1080/00207178208922623
[8] Ramadan, M.A. and El-Sayed, E.A. (2006) Partial Eigenvalue Assignment Problem of Linear Control Systems Using Orthogonality Relations. Acta Montanistica Slovaca Roník, 11, 16-25.
[9] Datta, B.N. and Sarkissian, D.R. (2002) Partial Eigenvalue Assignment in Linear Systems: Existence, Uniqueness and Numerical Solution.
http://www3.nd.edu/~mtns/papers/70_3.pdf
http://www.math.niu.edu/~dattab/psfiles/paper.mtns.2002.pdf
[10] Lancaster, P. and Tismentasky, M. (1985) The Theory of Matrices with Applications. 2nd Edition, Academic Press, Waltham, Massachusetts.
[11] Graybill, F.A. (1983) Matrices with Applications in Statistics. Wadsworth Publishing Company, Belmont.
[12] Green, P.E. and Carroll, J.D. (1976) Mathematical Tools for Applied Multivariate Analysis. Academic Press, Waltham, Massachusetts.
[13] Schott, J.R. (1997) Matrix Analysis for Statistics. John Wiley, New York.

Copyright © 2021 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.