Revised CPA method to compute Lyapunov functions for nonlinear systems |
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Authors: | Peter A. Giesl Sigurdur F. Hafstein |
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Affiliation: | 1. Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom;2. School of Science and Engineering, Reykjavik University, Menntavegur 1, IS-101 Reykjavik, Iceland |
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Abstract: | The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov functions for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium?s basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C2 and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems. |
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Keywords: | Lyapunov function Nonlinear system Exponential stability Basin of attraction CPA function Piecewise linear function Linear programming |
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