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.

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.

Cite this paper

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

El-Ghezawi, O. (2015) Explicit Determination of State Feedback Matrices.

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.

[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.