On Two Problems for Matrix Polytopes

DOI: 10.4236/am.2014.517253   PDF   HTML   XML   2,140 Downloads   2,569 Views   Citations


We consider two problems from stability theory of matrix polytopes: the existence of common quadratic Lyapunov functions and the existence of a stable member. We show the applicability of the gradient algorithm and give a new sufficient condition for the second problem. A number of examples are considered.

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Yılmaz, Ş. and Büyükköroğlu, T. (2014) On Two Problems for Matrix Polytopes. Applied Mathematics, 5, 2650-2656. doi: 10.4236/am.2014.517253.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Boyd, S. and Yang, Q. (1989) Structured and Simultaneous Lyapunov Functions for System Stability Problems. International Journal of Control, 49, 2215-2240.
[2] 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.
[3] Cheng, D., Guo, L. and Huang, J. (2003) On Quadratic Lyapunov Functions. IEEE Transactions on Automatic Control, 48, 885-890.
[4] Dayawansa, W.P. and Martin, C.F. (1999) A Converse Lyapunov Theorem for a Class of Dynamical Systems Which Undergo Switching. IEEE Transactions on Automatic Control, 44, 751-760.
[5] King, C. and Shorten, R. (2004) A Singularity Test for the Existence of Common Quadratic Lyapunov Functions for Pairs of Stable LTI Systems. Proceedings of the American Control Conference, Boston, 30 June-2 July 2004, 3881-3884.
[6] 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.
[7] 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.
[8] 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.
[9] Shorten, R.N., Mason, O., Cairbre, F.O. and Curran, P. (2004) A Unifying Framework for the SISO Circle Criterion and Other Quadratic Stability Criteria. International Journal of Control, 77, 1-8.
[10] Liberzon, D. and Tempo, R. (2004) Common Lyapunov Functions and Gradient Algorithms. IEEE Transactions on Automatic Control, 49, 990-994.
[11] Polyak, B.T. and Shcherbakov, P.S. (2005) Hard Problems in Linear Control Theory: Possible Approaches to Solution. Automation and Remote Control, 66, 681-718.
[12] 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, 245, 344-358.
[13] Horn, R.A. and Johnson, C.R. (1985) Matrix Analysis. Cambridge University Press, Cambridge.

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