Partial eigenvalue assignment with time delay robustness

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2013, 3(2): 207-221. doi: 10.3934/naco.2013.3.207

Partial eigenvalue assignment with time delay robustness

Xiaobin Mao ^1, and Hua Dai ^1,

1.	Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China, China

Received December 2011 Revised January 2013 Published April 2013

Abstract
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The partial eigenvalue assignment problem concerns reassigning a few of undesired eigenvalues of a linear system to suitably chosen locations and keeping the other large number of eigenvalues and eigenvectors unchanged (no spill-over). This paper considers the partial eigenvalue assignment problem with time delay robustness. A time delay robustness measure is presented by analyzing the sensitivity of the assigned eigenvalues with respect to time delay. The problem is formulated as an unconstrained minimization problem with the cost function involving the time delay robustness measure. A numerical algorithm with analytical formulation of the gradient for the cost function is provided. A numerical example is included to show the effectiveness of the proposed method.

Keywords: time delay robustness, Linear system, gradient., sensitivity, state feedback, partial eigenvalue assignment.

Mathematics Subject Classification: Primary: 93B55, 93B52; Secondary: 65F18, 15A2.

Citation: Xiaobin Mao, Hua Dai. Partial eigenvalue assignment with time delay robustness. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 207-221. doi: 10.3934/naco.2013.3.207

References:

[1]	M. Arnold and B. N. Datta, Single-input eigenvalue assignment algorithms: A close look, SIAM J. Matrix Anal. Appl., 19 (1998), 444-467. doi: 10.1137/S0895479895294885.
[2]	R. Byers and S. G. Nash, Approaches to robust pole assignment, International Journal of Control, 49 (1989), 97-117.
[3]	D. Calvetti, B. Lewis and L. Reichel, On the solution of the single input pole placement problem, Mathematical Theory of Networks and Systems (MTNS 98)(Eds: A. Beghi, L. Finesso and G. Picci), Il Poliografo, Padova, (1998), 585-588.
[4]	D. Calvetti, B. Lewis and L. Reichel, On the selection of poles in the single input pole placement problem, Linear Algebra Appl., 302/303 (1999), 331-345. doi: 10.1016/S0024-3795(99)00123-8.
[5]	D. Calvetti, B. Lewis and L. Reichel, Partial eigenvalue assignment for large linear control systems, Contemporary Mathematics, American Mathematical Society, Providence, RI, 28 (2001), 241-254.
[6]	B. N. Datta, "Numerical Methods for Linear Control Systems Design and Analysis," Elsevier Academic Press, New York, 2003.
[7]	B. N. Datta, An algorithm to assign eigenvalues in a Hessenberg matrix, IEEE Transactions on Automatic Control, AC-32 (1987), 414-417. doi: 10.1109/TAC.1987.1104622.
[8]	B. N. Datta and Y. Saad, Arnoldi methods for large Sylvester-like observer matrix equations and an associated algorithm for partial spectrum assignment, Linear Algebra Appl., 154/156 (1991), 225-244.
[9]	B. N. Datta and D. R. Sarkissian, Partial eigenvalue assignment in linear systems: existence, uniqueness and numerical solution, Proceedings of the Mathematical Theory of Networks and Systems (MTNS), Notre Dame, August 2002.
[10]	T. Hu and J. Lam, On optimizing performance indices with pole assignment constraints, Journal of System Control Engineering, 212 (1998), 327-337.
[11]	J. Kautsky, N. K. Nichols and P. Van Dooren, Robust pole assignment in linear state feedback, International Journal of Control, 41 (1985), 1129-1155. doi: 10.1080/0020718508961188.
[12]	J. Lam and W. Yan, A gradient flow approach to robust pole-placement problem, International Journal of Robust Nonlinear Control, 5 (1995), 175-185. doi: 10.1002/rnc.4590050303.
[13]	J. Lam and W. Yan, Pole assingnment with optimal spectral conditioning, Systems Control Letters, 29 (1997), 241-253. doi: 10.1016/S0167-6911(97)90007-4.
[14]	J. Lam, W. Yan and T. Hu, Pole assignment with eigenvalue and stability robustness, International Journal of Control, 72 (1999), 1165-1174. doi: 10.1080/002071799220326.
[15]	V. Mehrmann and H. Xu, An analysis of the pole placement problem I. The single-input case, Electron. Trans. Numer. Anal., 4 (1996), 89-105.
[16]	V. Mehrmann and H. Xu, An analysis of the pole placement problem II. The multi-input case, Electron. Trans. Numer. Anal., 5 (1997), 77-97.
[17]	V. Mehrmann and H. Xu, Choosing poles so that the single-input pole placement is well conditioned, SIAM J. Matrix Anal. Appl., 19 (1998), 664-681. doi: 10.1137/S0895479896302382.
[18]	G. Miminis and C. C. Paige, An algorithm for pole assignment of time invariant linear systems, International Journal of Control, 35 (1982), 341-354. doi: 10.1080/00207178208922623.
[19]	G. Miminis and H. Roth, Algorithm 747: A Fortran subroutine to solve the eigenvalue assignment problem for multiinput systems using state feedback, ACM Trans. Math. Software, 21 (1995), 299-326. doi: 10.1145/210089.210094.
[20]	N. K. Nichols, Robustness in partial pole placement, IEEE Transactions on automatic control, AC-32 (1987), 728-732. doi: 10.1109/TAC.1987.1104703.
[21]	T. J. Owens and J. O'Reilly, Parametric state-feedback control for arbitrary eigenvalue assignment with minimum sensitivity, IEE Proceedings D, 136 (1989), 307-313.
[22]	B. Porter and R. Crossley, "Model Control: Theory and Applications," Barnes Noble, New York, 1972.
[23]	Y. M. Ram, A. Singh and J. E. Mottershead, State feedback control with time delay, Mechanical Systems and Signal Processing, 23 (2009), 1940-1945. doi: 10.1016/j.ymssp.2008.04.012.
[24]	Y. Saad, Projection and deflation methods for partial pole assignment in linear state feedback, IEEE Transactions on Automatic Control, 33 (1988), 290-297. doi: 10.1109/9.406.
[25]	D. R. Sarkissian, "Theory and Computations of Partial Eigenvalue and Eigenstructure Assignment Problems in Matrix Second-order and Distributed-parameter Systems", Ph.D thesis, Northerm Illinnois University, 2001.
[26]	X. Shi and Y. Wei, A sharp version of Bauer-Fike's theorem, J. Comput. Appl. Math., 236 (2012), 3218-3227. doi: 10.1016/j.cam.2012.02.021.
[27]	E. De Souza and S. P. Bhattacharyya, Controllability, observability and the solution of AX-XB=C , Linear Algebra Appl., 39 (1981), 167-188. doi: 10.1016/0024-3795(81)90301-3.
[28]	J. G. Sun, On numerical methods for robust pole assignment in control system design, J. Comput. Math., 5 (1987), 119-134.
[29]	J. G. Sun, On numerical methods for robust pole assignment in control system design II, J. Comput. Math., 5 (1987), 352-363.
[30]	J. G. Sun, On measures of robustness of a control system, Math. Numer. Sinica(In Chinese), 9 (1987), 319-326.
[31]	A. Varga, A Schur method for pole assignment, IEEE Transactions on Automatic Control, AC-26b (1981), 517-519. doi: 10.1109/TAC.1981.1102605.
[32]	A. Varga, Robust pole assignment via Sylvester equation based state feedback parameterization, Proceedings of the 2000 IEEE International Symposium on Computer-Aided Control System Design, Anchorage, Alaska, USA, 2000.

show all references

References:

[1]	M. Arnold and B. N. Datta, Single-input eigenvalue assignment algorithms: A close look, SIAM J. Matrix Anal. Appl., 19 (1998), 444-467. doi: 10.1137/S0895479895294885.
[2]	R. Byers and S. G. Nash, Approaches to robust pole assignment, International Journal of Control, 49 (1989), 97-117.
[3]	D. Calvetti, B. Lewis and L. Reichel, On the solution of the single input pole placement problem, Mathematical Theory of Networks and Systems (MTNS 98)(Eds: A. Beghi, L. Finesso and G. Picci), Il Poliografo, Padova, (1998), 585-588.
[4]	D. Calvetti, B. Lewis and L. Reichel, On the selection of poles in the single input pole placement problem, Linear Algebra Appl., 302/303 (1999), 331-345. doi: 10.1016/S0024-3795(99)00123-8.
[5]	D. Calvetti, B. Lewis and L. Reichel, Partial eigenvalue assignment for large linear control systems, Contemporary Mathematics, American Mathematical Society, Providence, RI, 28 (2001), 241-254.
[6]	B. N. Datta, "Numerical Methods for Linear Control Systems Design and Analysis," Elsevier Academic Press, New York, 2003.
[7]	B. N. Datta, An algorithm to assign eigenvalues in a Hessenberg matrix, IEEE Transactions on Automatic Control, AC-32 (1987), 414-417. doi: 10.1109/TAC.1987.1104622.
[8]	B. N. Datta and Y. Saad, Arnoldi methods for large Sylvester-like observer matrix equations and an associated algorithm for partial spectrum assignment, Linear Algebra Appl., 154/156 (1991), 225-244.
[9]	B. N. Datta and D. R. Sarkissian, Partial eigenvalue assignment in linear systems: existence, uniqueness and numerical solution, Proceedings of the Mathematical Theory of Networks and Systems (MTNS), Notre Dame, August 2002.
[10]	T. Hu and J. Lam, On optimizing performance indices with pole assignment constraints, Journal of System Control Engineering, 212 (1998), 327-337.
[11]	J. Kautsky, N. K. Nichols and P. Van Dooren, Robust pole assignment in linear state feedback, International Journal of Control, 41 (1985), 1129-1155. doi: 10.1080/0020718508961188.
[12]	J. Lam and W. Yan, A gradient flow approach to robust pole-placement problem, International Journal of Robust Nonlinear Control, 5 (1995), 175-185. doi: 10.1002/rnc.4590050303.
[13]	J. Lam and W. Yan, Pole assingnment with optimal spectral conditioning, Systems Control Letters, 29 (1997), 241-253. doi: 10.1016/S0167-6911(97)90007-4.
[14]	J. Lam, W. Yan and T. Hu, Pole assignment with eigenvalue and stability robustness, International Journal of Control, 72 (1999), 1165-1174. doi: 10.1080/002071799220326.
[15]	V. Mehrmann and H. Xu, An analysis of the pole placement problem I. The single-input case, Electron. Trans. Numer. Anal., 4 (1996), 89-105.
[16]	V. Mehrmann and H. Xu, An analysis of the pole placement problem II. The multi-input case, Electron. Trans. Numer. Anal., 5 (1997), 77-97.
[17]	V. Mehrmann and H. Xu, Choosing poles so that the single-input pole placement is well conditioned, SIAM J. Matrix Anal. Appl., 19 (1998), 664-681. doi: 10.1137/S0895479896302382.
[18]	G. Miminis and C. C. Paige, An algorithm for pole assignment of time invariant linear systems, International Journal of Control, 35 (1982), 341-354. doi: 10.1080/00207178208922623.
[19]	G. Miminis and H. Roth, Algorithm 747: A Fortran subroutine to solve the eigenvalue assignment problem for multiinput systems using state feedback, ACM Trans. Math. Software, 21 (1995), 299-326. doi: 10.1145/210089.210094.
[20]	N. K. Nichols, Robustness in partial pole placement, IEEE Transactions on automatic control, AC-32 (1987), 728-732. doi: 10.1109/TAC.1987.1104703.
[21]	T. J. Owens and J. O'Reilly, Parametric state-feedback control for arbitrary eigenvalue assignment with minimum sensitivity, IEE Proceedings D, 136 (1989), 307-313.
[22]	B. Porter and R. Crossley, "Model Control: Theory and Applications," Barnes Noble, New York, 1972.
[23]	Y. M. Ram, A. Singh and J. E. Mottershead, State feedback control with time delay, Mechanical Systems and Signal Processing, 23 (2009), 1940-1945. doi: 10.1016/j.ymssp.2008.04.012.
[24]	Y. Saad, Projection and deflation methods for partial pole assignment in linear state feedback, IEEE Transactions on Automatic Control, 33 (1988), 290-297. doi: 10.1109/9.406.
[25]	D. R. Sarkissian, "Theory and Computations of Partial Eigenvalue and Eigenstructure Assignment Problems in Matrix Second-order and Distributed-parameter Systems", Ph.D thesis, Northerm Illinnois University, 2001.
[26]	X. Shi and Y. Wei, A sharp version of Bauer-Fike's theorem, J. Comput. Appl. Math., 236 (2012), 3218-3227. doi: 10.1016/j.cam.2012.02.021.
[27]	E. De Souza and S. P. Bhattacharyya, Controllability, observability and the solution of AX-XB=C , Linear Algebra Appl., 39 (1981), 167-188. doi: 10.1016/0024-3795(81)90301-3.
[28]	J. G. Sun, On numerical methods for robust pole assignment in control system design, J. Comput. Math., 5 (1987), 119-134.
[29]	J. G. Sun, On numerical methods for robust pole assignment in control system design II, J. Comput. Math., 5 (1987), 352-363.
[30]	J. G. Sun, On measures of robustness of a control system, Math. Numer. Sinica(In Chinese), 9 (1987), 319-326.
[31]	A. Varga, A Schur method for pole assignment, IEEE Transactions on Automatic Control, AC-26b (1981), 517-519. doi: 10.1109/TAC.1981.1102605.
[32]	A. Varga, Robust pole assignment via Sylvester equation based state feedback parameterization, Proceedings of the 2000 IEEE International Symposium on Computer-Aided Control System Design, Anchorage, Alaska, USA, 2000.

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