Stability verification for monotone systems using homotopy algorithms
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A monotone self-mapping of the nonnegative orthant induces a monotone discrete-time dynamical system which evolves on the same orthant. If with respect to this system the origin is attractive then there must exist points whose image under the monotone map is strictly smaller than the original point, in the component-wise partial ordering. Here it is shown how such points can be found numerically, leading to a recipe to compute order intervals that are contained in the region of attraction and where the monotone map acts essentially as a contraction. An important application is the numerical verification of so-called generalized small-gain conditions that appear in the stability theory of large-scale systems.
KeywordsMonotone systems Stability theory Homotopy algorithms
Mathematics Subject Classifications (2010)93C55 47H07 65H20
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- 3.Allgower, E.L., Georg, K.: Numerical continuation methods. In: Springer Series in Computational Mathematics, vol. 13. Springer, Berlin (1990)Google Scholar
- 4.Allgower, E.L., Georg, K.: Numerical path following. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. V, pp. 3–207. North-Holland, Amsterdam (1997)Google Scholar
- 5.Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Bertini: Software for Numerical Algebraic Geometry (2006). Available at http://www.nd.edu/~sommese/bertini
- 8.Gao, K., Lin, Y.: On equivalent notions of input-to-state stability for nonlinear discrete time systems. In: Proc. of the IASTED Int. Conf. on Control and Applications, pp. 81–87 (2000)Google Scholar
- 10.Karafyllis, I., Jiang, Z.-P.: A vector small-gain theorem for general nonlinear control systems. arXiv:0904.0755v1 [math.OC] (2009)