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

## Partial eigenvalue assignment with time delay robustness

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

Received  December 2011 Revised  January 2013 Published  April 2013

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.
Citation: Xiaobin Mao, Hua Dai. Partial eigenvalue assignment with time delay robustness. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 207-221. doi: 10.3934/naco.2013.3.207
##### References:
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Datta, "Numerical Methods for Linear Control Systems Design and Analysis,", Elsevier Academic Press, (2003).   Google Scholar  B. N. Datta, An algorithm to assign eigenvalues in a Hessenberg matrix,, IEEE Transactions on Automatic Control, AC-32 (1987), 414.  doi: 10.1109/TAC.1987.1104622. Google Scholar  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. Google Scholar  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), (2002).   Google Scholar  T. Hu and J. Lam, On optimizing performance indices with pole assignment constraints,, Journal of System Control Engineering, 212 (1998), 327.   Google Scholar  J. Kautsky, N. K. Nichols and P. Van Dooren, Robust pole assignment in linear state feedback,, International Journal of Control, 41 (1985), 1129.  doi: 10.1080/0020718508961188. Google Scholar  J. Lam and W. Yan, A gradient flow approach to robust pole-placement problem,, International Journal of Robust Nonlinear Control, 5 (1995), 175.  doi: 10.1002/rnc.4590050303. Google Scholar  J. Lam and W. Yan, Pole assingnment with optimal spectral conditioning,, Systems Control Letters, 29 (1997), 241.  doi: 10.1016/S0167-6911(97)90007-4. Google Scholar  J. Lam, W. Yan and T. Hu, Pole assignment with eigenvalue and stability robustness,, International Journal of Control, 72 (1999), 1165.  doi: 10.1080/002071799220326. Google Scholar  V. Mehrmann and H. Xu, An analysis of the pole placement problem I. The single-input case,, Electron. Trans. Numer. Anal., 4 (1996), 89. Google Scholar  V. Mehrmann and H. Xu, An analysis of the pole placement problem II. The multi-input case,, Electron. Trans. Numer. Anal., 5 (1997), 77. Google Scholar  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.  doi: 10.1137/S0895479896302382. Google Scholar  G. Miminis and C. C. Paige, An algorithm for pole assignment of time invariant linear systems,, International Journal of Control, 35 (1982), 341.  doi: 10.1080/00207178208922623. Google Scholar  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.  doi: 10.1145/210089.210094. Google Scholar  N. K. Nichols, Robustness in partial pole placement,, IEEE Transactions on automatic control, AC-32 (1987), 728.  doi: 10.1109/TAC.1987.1104703. Google Scholar  T. J. Owens and J. O'Reilly, Parametric state-feedback control for arbitrary eigenvalue assignment with minimum sensitivity,, IEE Proceedings D, 136 (1989), 307.   Google Scholar  B. Porter and R. Crossley, "Model Control: Theory and Applications,", Barnes Noble, (1972).   Google Scholar  Y. M. Ram, A. Singh and J. E. Mottershead, State feedback control with time delay,, Mechanical Systems and Signal Processing, 23 (2009), 1940.  doi: 10.1016/j.ymssp.2008.04.012. Google Scholar  Y. Saad, Projection and deflation methods for partial pole assignment in linear state feedback,, IEEE Transactions on Automatic Control, 33 (1988), 290.  doi: 10.1109/9.406. Google Scholar  D. R. Sarkissian, "Theory and Computations of Partial Eigenvalue and Eigenstructure Assignment Problems in Matrix Second-order and Distributed-parameter Systems",, Ph.D thesis, (2001).   Google Scholar  X. Shi and Y. Wei, A sharp version of Bauer-Fike's theorem,, J. Comput. Appl. Math., 236 (2012), 3218.  doi: 10.1016/j.cam.2012.02.021. Google Scholar  E. De Souza and S. P. Bhattacharyya, Controllability, observability and the solution of AX-XB=C ,, Linear Algebra Appl., 39 (1981), 167.  doi: 10.1016/0024-3795(81)90301-3. Google Scholar  J. G. Sun, On numerical methods for robust pole assignment in control system design,, J. Comput. Math., 5 (1987), 119. Google Scholar  J. G. Sun, On numerical methods for robust pole assignment in control system design II,, J. Comput. Math., 5 (1987), 352. Google Scholar  J. G. Sun, On measures of robustness of a control system,, Math. Numer. Sinica(In Chinese), 9 (1987), 319.   Google Scholar  A. Varga, A Schur method for pole assignment,, IEEE Transactions on Automatic Control, AC-26b (1981), 517.  doi: 10.1109/TAC.1981.1102605. Google Scholar  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, (2000).   Google Scholar

show all references

##### References:
  M. Arnold and B. N. Datta, Single-input eigenvalue assignment algorithms: A close look,, SIAM J. Matrix Anal. Appl., 19 (1998), 444.  doi: 10.1137/S0895479895294885. Google Scholar  R. Byers and S. G. Nash, Approaches to robust pole assignment,, International Journal of Control, 49 (1989), 97. Google Scholar  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, (1998), 585.   Google Scholar  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.  doi: 10.1016/S0024-3795(99)00123-8. Google Scholar  D. Calvetti, B. Lewis and L. Reichel, Partial eigenvalue assignment for large linear control systems,, Contemporary Mathematics, 28 (2001), 241. Google Scholar  B. N. Datta, "Numerical Methods for Linear Control Systems Design and Analysis,", Elsevier Academic Press, (2003).   Google Scholar  B. N. Datta, An algorithm to assign eigenvalues in a Hessenberg matrix,, IEEE Transactions on Automatic Control, AC-32 (1987), 414.  doi: 10.1109/TAC.1987.1104622. Google Scholar  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. Google Scholar  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), (2002).   Google Scholar  T. Hu and J. Lam, On optimizing performance indices with pole assignment constraints,, Journal of System Control Engineering, 212 (1998), 327.   Google Scholar  J. Kautsky, N. K. Nichols and P. Van Dooren, Robust pole assignment in linear state feedback,, International Journal of Control, 41 (1985), 1129.  doi: 10.1080/0020718508961188. Google Scholar  J. Lam and W. Yan, A gradient flow approach to robust pole-placement problem,, International Journal of Robust Nonlinear Control, 5 (1995), 175.  doi: 10.1002/rnc.4590050303. Google Scholar  J. Lam and W. Yan, Pole assingnment with optimal spectral conditioning,, Systems Control Letters, 29 (1997), 241.  doi: 10.1016/S0167-6911(97)90007-4. Google Scholar  J. Lam, W. Yan and T. Hu, Pole assignment with eigenvalue and stability robustness,, International Journal of Control, 72 (1999), 1165.  doi: 10.1080/002071799220326. Google Scholar  V. Mehrmann and H. Xu, An analysis of the pole placement problem I. The single-input case,, Electron. Trans. Numer. Anal., 4 (1996), 89. Google Scholar  V. Mehrmann and H. Xu, An analysis of the pole placement problem II. The multi-input case,, Electron. Trans. Numer. Anal., 5 (1997), 77. Google Scholar  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.  doi: 10.1137/S0895479896302382. Google Scholar  G. Miminis and C. C. Paige, An algorithm for pole assignment of time invariant linear systems,, International Journal of Control, 35 (1982), 341.  doi: 10.1080/00207178208922623. Google Scholar  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.  doi: 10.1145/210089.210094. Google Scholar  N. K. Nichols, Robustness in partial pole placement,, IEEE Transactions on automatic control, AC-32 (1987), 728.  doi: 10.1109/TAC.1987.1104703. Google Scholar  T. J. Owens and J. O'Reilly, Parametric state-feedback control for arbitrary eigenvalue assignment with minimum sensitivity,, IEE Proceedings D, 136 (1989), 307.   Google Scholar  B. Porter and R. Crossley, "Model Control: Theory and Applications,", Barnes Noble, (1972).   Google Scholar  Y. M. Ram, A. Singh and J. E. Mottershead, State feedback control with time delay,, Mechanical Systems and Signal Processing, 23 (2009), 1940.  doi: 10.1016/j.ymssp.2008.04.012. Google Scholar  Y. Saad, Projection and deflation methods for partial pole assignment in linear state feedback,, IEEE Transactions on Automatic Control, 33 (1988), 290.  doi: 10.1109/9.406. Google Scholar  D. R. Sarkissian, "Theory and Computations of Partial Eigenvalue and Eigenstructure Assignment Problems in Matrix Second-order and Distributed-parameter Systems",, Ph.D thesis, (2001).   Google Scholar  X. Shi and Y. Wei, A sharp version of Bauer-Fike's theorem,, J. Comput. Appl. Math., 236 (2012), 3218.  doi: 10.1016/j.cam.2012.02.021. Google Scholar  E. De Souza and S. P. Bhattacharyya, Controllability, observability and the solution of AX-XB=C ,, Linear Algebra Appl., 39 (1981), 167.  doi: 10.1016/0024-3795(81)90301-3. Google Scholar  J. G. Sun, On numerical methods for robust pole assignment in control system design,, J. Comput. Math., 5 (1987), 119. Google Scholar  J. G. Sun, On numerical methods for robust pole assignment in control system design II,, J. Comput. Math., 5 (1987), 352. Google Scholar  J. G. Sun, On measures of robustness of a control system,, Math. Numer. Sinica(In Chinese), 9 (1987), 319.   Google Scholar  A. Varga, A Schur method for pole assignment,, IEEE Transactions on Automatic Control, AC-26b (1981), 517.  doi: 10.1109/TAC.1981.1102605. Google Scholar  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, (2000).   Google Scholar
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