On Asymptotic Stability of Linear Control Systems
Affiliation(s)
Department of Mathematics, Faculty of Science, Anadolu University, Eskisehir, Turkey.
Department of Mathematics, Faculty of Science, Anadolu University, Eskisehir, Turkey.
ABSTRACT
Asymptotic stability of linear systems is closely related to Hurwitz stability of the system matrices. For uncertain linear systems we consider stability problem through common quadratic Lyapunov functions (CQLF) and problem of stabilization by linear feedback.
Asymptotic stability of linear systems is closely related to Hurwitz stability of the system matrices. For uncertain linear systems we consider stability problem through common quadratic Lyapunov functions (CQLF) and problem of stabilization by linear feedback.
Cite this paper
Yılmaz, Ş. , Büyükköroğlu, T. and Dzhafarov, V. (2015) On Asymptotic Stability of Linear Control Systems. Applied Mathematics, 6, 71-77. doi: 10.4236/am.2015.61008.
Yılmaz, Ş. , Büyükköroğlu, T. and Dzhafarov, V. (2015) On Asymptotic Stability of Linear Control Systems. Applied Mathematics, 6, 71-77. doi: 10.4236/am.2015.61008.
References
[1] Liberzon, D. (2003) Switching in System and Control. Birkhäuser, Boston.
http://dx.doi.org/10.1007/978-1-4612-0017-8 [2] Boyd, S. and Yang, Q. (1989) Structured and Simultaneous Lyapunov Functions for System Stability Problems. International Journal of Control, 49, 2215-2240.
http://dx.doi.org/10.1080/00207178908559769 [3] Büyükköroglu, T., Esen, Ö. and Dzhafarov, V. (2011) Common Lyapunov Functions for Some Special Classes of Stable Systems. IEEE Transactions on Automatic Control, 56, 1963-1967.
http://dx.doi.org/10.1109/TAC.2011.2137510 [4] Cheng, D., Guo, L. and Huang, J. (2003) On Quadratic Lyapunov Functions. IEEE Transactions on Automatic Control, 48, 885-890.
http://dx.doi.org/10.1109/TAC.2003.811274 [5] Mason, O. and Shorten, R. (2006) On the Simultaneous Diagonal Stability of a Pair of Positive Linear Systems. Linear Algebra and Its Applications, 413, 13-23.
http://dx.doi.org/10.1016/j.laa.2005.07.019 [6] Narendra, K.S. and Balakrishnan, J. (1994) A Common Lyapunov Function for Stable LTI Systems with Commuting A-Matrices. IEEE Transactions on Automatic Control, 39, 2469-2471.
http://dx.doi.org/10.1109/9.362846 [7] Shorten, R.N. and Narendra, K.S. (2002) Necessary and Sufficient Conditions for the Existence of a Common Quadratic Lyapunov Function for a Finite Number of Stable Second Order Linear Time-Invariant Systems. International Journal of Adaptive Control and Signal Processing, 16, 709-728.
http://dx.doi.org/10.1002/acs.719 [8] Liberzon, D. and Tempo, R. (2004) Common Lyapunov Functions and Gradient Algorithms. IEEE Transactions on Automatic Control, 49, 990-994.
http://dx.doi.org/10.1109/TAC.2004.829632 [9] Marcus, M. and Minc, H.A. (1992) Survey of Matrix Theory and Matrix Inequalities. Dover, New York. [10] Polyak, B.T. and Shcherbakov, P.S. (1999) Numerical Search of Stable or Unstable Element in Matrix or Polynomial Families: A Unified Approach to Robustness Analysis and Stabilization. Robustness in Identification and Control Lecture Notes in Control and Information Sciences, 254, 344-358.
http://dx.doi.org/10.1007/BFb0109879
[1] Liberzon, D. (2003) Switching in System and Control. Birkhäuser, Boston.
http://dx.doi.org/10.1007/978-1-4612-0017-8 [2] Boyd, S. and Yang, Q. (1989) Structured and Simultaneous Lyapunov Functions for System Stability Problems. International Journal of Control, 49, 2215-2240.
http://dx.doi.org/10.1080/00207178908559769 [3] Büyükköroglu, T., Esen, Ö. and Dzhafarov, V. (2011) Common Lyapunov Functions for Some Special Classes of Stable Systems. IEEE Transactions on Automatic Control, 56, 1963-1967.
http://dx.doi.org/10.1109/TAC.2011.2137510 [4] Cheng, D., Guo, L. and Huang, J. (2003) On Quadratic Lyapunov Functions. IEEE Transactions on Automatic Control, 48, 885-890.
http://dx.doi.org/10.1109/TAC.2003.811274 [5] Mason, O. and Shorten, R. (2006) On the Simultaneous Diagonal Stability of a Pair of Positive Linear Systems. Linear Algebra and Its Applications, 413, 13-23.
http://dx.doi.org/10.1016/j.laa.2005.07.019 [6] Narendra, K.S. and Balakrishnan, J. (1994) A Common Lyapunov Function for Stable LTI Systems with Commuting A-Matrices. IEEE Transactions on Automatic Control, 39, 2469-2471.
http://dx.doi.org/10.1109/9.362846 [7] Shorten, R.N. and Narendra, K.S. (2002) Necessary and Sufficient Conditions for the Existence of a Common Quadratic Lyapunov Function for a Finite Number of Stable Second Order Linear Time-Invariant Systems. International Journal of Adaptive Control and Signal Processing, 16, 709-728.
http://dx.doi.org/10.1002/acs.719 [8] Liberzon, D. and Tempo, R. (2004) Common Lyapunov Functions and Gradient Algorithms. IEEE Transactions on Automatic Control, 49, 990-994.
http://dx.doi.org/10.1109/TAC.2004.829632 [9] Marcus, M. and Minc, H.A. (1992) Survey of Matrix Theory and Matrix Inequalities. Dover, New York. [10] Polyak, B.T. and Shcherbakov, P.S. (1999) Numerical Search of Stable or Unstable Element in Matrix or Polynomial Families: A Unified Approach to Robustness Analysis and Stabilization. Robustness in Identification and Control Lecture Notes in Control and Information Sciences, 254, 344-358.
http://dx.doi.org/10.1007/BFb0109879