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. (English) Zbl 1027.93037
Let \(\{A_1,\dots, A_M\}\) be a finite set of \(2\times 2\) Hurwitz matrices. Various conditions are given that are necessary and sufficient for the existence of \(P> 0\) such that
\[
A^*_i P+ PA_i\leq 0,\qquad i= 1,\dots, M.
\]
These conditions are easily checkable. Several example are given.
Reviewer: Vladimir Răsvan (Craiova)
MSC:
| 93D30 | Lyapunov and storage functions |
| 15A39 | Linear inequalities of matrices |
| 34D20 | Stability of solutions to ordinary differential equations |
References:
| [1] | Boyd, Linear Algebra and its Applications 49 pp 2215– (1989) |
| [2] | Narendra, IEEE Transactions on Automatic Control 39 pp 2469– (1994) |
| [3] | A solution to the common Lyapunov function problem for continuous time systems. In Proceedings of 36th Conference on Decision and Control, 1997. |
| [4] | Necessary and Sufficient Conditions for the Existence of a Common Lyapunov Function for Two Second Order Systems. Technical Report, Center for Systems Science, Report 9806, 1998. |
| [5] | Liberzon, IEEE Control Systems Magazine 19 pp 59– |
| [6] | Barabonov, Siberian Mathematical Journal 29 pp 12– (1986) |
| [7] | A result on common quadratic Lyapunov functions. IEEE Transactions on Automatic Control, (Technical Report, Signals and Systems Group, NUI, Maynooth, NUIM-SS-2002-01. |
| [8] | Loewy, Linear Algebra and its Applications 13 pp 79– (1976) |
| [9] | On the existence of a common quadratic Lyapunov function for linear stable switching systems. In Proceedings of the 10th Yale Workshop on Adaptive and Learning Systems, 1998. |
| [10] | Convex Sets. MacGraw Hill: New York, 1965.. |
| [11] | Cohen, IEEE Transactions on Automatic Control 38 pp 611– (1993) |
| [12] | Stable Adaptive Systems. Prentice-Hall, Englewood Cliffs, NT, 1989.. · Zbl 0758.93039 |
| [13] | On the stability and existence of common Lyapunov functions for linear stable switching systems. Technical Report, Report 9805, Center for Systems Science, April 1998. |
| [14] | A sufficient condition for the existence of a common Lyapunov function for two second order systems: Part 1. Technical Report, Report 9701, Center for Systems Science, Yale University, 1997. |
| [15] | Calculus and Analytic Geometry. Addison-Wesley: Reading, MA, 1984. |
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