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:
[1]

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.

[2]

R. Byers and S. G. Nash, Approaches to robust pole assignment,, International Journal of Control, 49 (1989), 97.

[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, (1998), 585.

[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. 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, 28 (2001), 241.

[6]

B. N. Datta, "Numerical Methods for Linear Control Systems Design and Analysis,", Elsevier Academic Press, (2003).

[7]

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.

[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.

[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), (2002).

[10]

T. Hu and J. Lam, On optimizing performance indices with pole assignment constraints,, Journal of System Control Engineering, 212 (1998), 327.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[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.

[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.

[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. 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. 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. doi: 10.1145/210089.210094.

[20]

N. K. Nichols, Robustness in partial pole placement,, IEEE Transactions on automatic control, AC-32 (1987), 728. 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.

[22]

B. Porter and R. Crossley, "Model Control: Theory and Applications,", Barnes Noble, (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. 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. 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, (2001).

[26]

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.

[27]

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.

[28]

J. G. Sun, On numerical methods for robust pole assignment in control system design,, J. Comput. Math., 5 (1987), 119.

[29]

J. G. Sun, On numerical methods for robust pole assignment in control system design II,, J. Comput. Math., 5 (1987), 352.

[30]

J. G. Sun, On measures of robustness of a control system,, Math. Numer. Sinica(In Chinese), 9 (1987), 319.

[31]

A. Varga, A Schur method for pole assignment,, IEEE Transactions on Automatic Control, AC-26b (1981), 517. 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, (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. doi: 10.1137/S0895479895294885.

[2]

R. Byers and S. G. Nash, Approaches to robust pole assignment,, International Journal of Control, 49 (1989), 97.

[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, (1998), 585.

[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. 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, 28 (2001), 241.

[6]

B. N. Datta, "Numerical Methods for Linear Control Systems Design and Analysis,", Elsevier Academic Press, (2003).

[7]

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.

[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.

[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), (2002).

[10]

T. Hu and J. Lam, On optimizing performance indices with pole assignment constraints,, Journal of System Control Engineering, 212 (1998), 327.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[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.

[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.

[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. 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. 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. doi: 10.1145/210089.210094.

[20]

N. K. Nichols, Robustness in partial pole placement,, IEEE Transactions on automatic control, AC-32 (1987), 728. 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.

[22]

B. Porter and R. Crossley, "Model Control: Theory and Applications,", Barnes Noble, (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. 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. 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, (2001).

[26]

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.

[27]

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.

[28]

J. G. Sun, On numerical methods for robust pole assignment in control system design,, J. Comput. Math., 5 (1987), 119.

[29]

J. G. Sun, On numerical methods for robust pole assignment in control system design II,, J. Comput. Math., 5 (1987), 352.

[30]

J. G. Sun, On measures of robustness of a control system,, Math. Numer. Sinica(In Chinese), 9 (1987), 319.

[31]

A. Varga, A Schur method for pole assignment,, IEEE Transactions on Automatic Control, AC-26b (1981), 517. 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, (2000).

[1]

Nguyen H. Sau, Vu N. Phat. LP approach to exponential stabilization of singular linear positive time-delay systems via memory state feedback. Journal of Industrial & Management Optimization, 2018, 14 (2) : 583-596. doi: 10.3934/jimo.2017061

[2]

B. Cantó, C. Coll, A. Herrero, E. Sánchez, N. Thome. Pole-assignment of discrete time-delay systems with symmetries. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 641-649. doi: 10.3934/dcdsb.2006.6.641

[3]

Wing-Cheong Lo. Morphogen gradient with expansion-repression mechanism: Steady-state and robustness studies. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 775-787. doi: 10.3934/dcdsb.2014.19.775

[4]

Xing Li, Chungen Shen, Lei-Hong Zhang. A projected preconditioned conjugate gradient method for the linear response eigenvalue problem. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 389-412. doi: 10.3934/naco.2018025

[5]

K.H. Wong, C. Myburgh, L. Omari. A gradient flow approach for computing jump linear quadratic optimal feedback gains. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 803-808. doi: 10.3934/dcds.2000.6.803

[6]

Qingwen Hu, Huan Zhang. Stabilization of turning processes using spindle feedback with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4329-4360. doi: 10.3934/dcdsb.2018167

[7]

Huawen Ye, Honglei Xu. Global stabilization for ball-and-beam systems via state and partial state feedback. Journal of Industrial & Management Optimization, 2016, 12 (1) : 17-29. doi: 10.3934/jimo.2016.12.17

[8]

Lorena Bociu, Steven Derochers, Daniel Toundykov. Feedback stabilization of a linear hydro-elastic system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1107-1132. doi: 10.3934/dcdsb.2018144

[9]

István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773

[10]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369

[11]

Jinzhi Lei, Frederic Y. M. Wan, Arthur D. Lander, Qing Nie. Robustness of signaling gradient in drosophila wing imaginal disc. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 835-866. doi: 10.3934/dcdsb.2011.16.835

[12]

Rubén Caballero, Alexandre N. Carvalho, Pedro Marín-Rubio, José Valero. Robustness of dynamically gradient multivalued dynamical systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1049-1077. doi: 10.3934/dcdsb.2019006

[13]

Benjamin B. Kennedy. A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1633-1650. doi: 10.3934/dcdsb.2013.18.1633

[14]

Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 47-66. doi: 10.3934/dcdss.2020003

[15]

Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185

[16]

Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23

[17]

Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014

[18]

Dinh Cong Huong, Mai Viet Thuan. State transformations of time-varying delay systems and their applications to state observer design. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 413-444. doi: 10.3934/dcdss.2017020

[19]

Yacine Chitour, Jean-Michel Coron, Mauro Garavello. On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 643-672. doi: 10.3934/dcds.2006.14.643

[20]

Peng Cui, Hongguo Zhao, Jun-e Feng. State estimation for discrete linear systems with observation time-delayed noise. Journal of Industrial & Management Optimization, 2011, 7 (1) : 79-85. doi: 10.3934/jimo.2011.7.79

 Impact Factor: 

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]