Revised CPA method to compute Lyapunov functions for nonlinear systems. (English) Zbl 1317.34134
Summary: The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov functions for nonlinear systems with asymptotically stable equilibria. In [the second author, Dyn. Syst. 20, No. 3, 281–299 (2005; Zbl 1102.34038)] 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. 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 \(C^2\) 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. In this paper, we cover general \(n\)-dimensional systems.
MSC:
| 34D20 | Stability of solutions to ordinary differential equations |
Keywords:
Lyapunov function; nonlinear system; exponential stability; basin of attraction; CPA function; piecewise linear function; linear programmingCitations:
Zbl 1102.34038References:
| [1] | Baier, R.; Grüne, L.; Hafstein, S., Linear programming based Lyapunov function computation for differential inclusions, Discrete Contin. Dyn. Syst. Ser. B, 17, 33-56 (2012) · Zbl 1235.93222 |
| [2] | Ban, H.; Kalies, W., A computational approach to Conleyʼs decomposition theorem, J. Comput. Nonlinear Dynam., 1, 4, 312-319 (2006) |
| [3] | Bartels, R.; Stewart, G., Algorithm 432: Solution of the matrix equation \(A X + X B = C\), Commun. ACM, 15, 820-826 (1972) · Zbl 1372.65121 |
| [4] | Bertsimas, D.; Tsitsiklis, J., Introduction to Linear Optimization (1997), Athena Scientific |
| [5] | Clarke, F.; Ledyaev, Y.; Stern, R., Asymptotic stability and smooth Lyapunov functions, J. Differential Equations, 149, 69-114 (1998) · Zbl 0907.34013 |
| [6] | Conley, C., Isolated Invariant Sets and the Morse Index, CBMS, vol. 38 (1978), American Math. Soc. · Zbl 0397.34056 |
| [7] | Eghbal, N.; Pariz, N.; Karimpour, A., Discontinuous piecewise quadratic Lyapunov functions for planar piecewise affine systems, J. Math. Anal. Appl., 399, 586-593 (2012) · Zbl 1254.93126 |
| [8] | Giesl, P., Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math., vol. 1904 (2007), Springer · Zbl 1121.34001 |
| [9] | Giesl, P.; Hafstein, S., Existence of piecewise affine Lyapunov functions in two dimensions, J. Math. Anal. Appl., 371, 233-248 (2010) · Zbl 1205.34064 |
| [10] | Giesl, P.; Hafstein, S., Construction of Lyapunov functions for nonlinear planar systems by linear programming, J. Math. Anal. Appl., 388, 463-479 (2012) · Zbl 1248.34083 |
| [11] | Giesl, P.; Hafstein, S., Existence of piecewise linear Lyapunov functions in arbitrary dimensions, Discrete Contin. Dyn. Syst., 32, 3539-3565 (2012) · Zbl 1266.37010 |
| [12] | Grüne, L., Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization, Lecture Notes in Math., vol. 1783 (2002), Springer · Zbl 0991.37001 |
| [13] | Hafstein, S., A constructive converse Lyapunov theorem on exponential stability, Discrete Contin. Dyn. Syst., 10, 657-678 (2004) · Zbl 1070.93044 |
| [14] | Hafstein, S., A constructive converse Lyapunov theorem on asymptotic stability for nonlinear autonomous ordinary differential equations, Dyn. Syst., 20, 281-299 (2005) · Zbl 1102.34038 |
| [15] | Hafstein, S., An Algorithm for Constructing Lyapunov Functions, Electron. J. Differ. Equ. Monogr., vol. 8 (2007) · Zbl 1131.37021 |
| [16] | Hahn, W., Stability of Motion (1967), Springer · Zbl 0189.38503 |
| [17] | Johansen, T., Computation of Lyapunov functions for smooth nonlinear systems using convex optimization, Automatica, 36, 1617-1626 (2000) · Zbl 0971.93069 |
| [18] | Johansson, M.; Rantzer, A., On the computation of piecewise quadratic Lyapunov functions, (Proceedings of the 36th IEEE Conference on Decision and Control (1997)) |
| [19] | Julian, P., A high level canonical piecewise linear representation: Theory and applications (1999), Universidad Nacional del Sur: Universidad Nacional del Sur Bahia Blanca, Argentina, Ph.D. Thesis |
| [20] | Julian, P.; Guivant, J.; Desages, A., A parametrization of piecewise linear Lyapunov function via linear programming, Internat. J. Control, 72, 702-715 (1999) · Zbl 0963.93036 |
| [21] | Kalies, W.; Mischaikow, K.; VanderVorst, R., An algorithmic approach to chain recurrence, Found. Comput. Math., 5, 4, 409-449 (2005) · Zbl 1099.37010 |
| [22] | Khalil, H., Nonlinear Systems (2002), Prentice Hall · Zbl 1003.34002 |
| [23] | Lyapunov, A., The General Problem of the Stability of Motion (1992), Taylor & Francis, translated by A.T. Fuller · Zbl 0786.70001 |
| [24] | Marinosson, S., Stability analysis of nonlinear systems with linear programming: A Lyapunov functions based approach (2002), Gerhard-Mercator-University: Gerhard-Mercator-University Duisburg, Germany, Ph.D. Thesis · Zbl 1344.37001 |
| [25] | Marinosson, S., Lyapunov function construction for ordinary differential equations with linear programming, Dyn. Syst., 17, 137-150 (2002) · Zbl 1013.34051 |
| [26] | Norton, D., The fundamental theorem of dynamical systems, Comment. Math. Univ. Carolin., 36, 3, 585-597 (1995) · Zbl 0847.58049 |
| [27] | Oishi, Y., Probabilistic design of a robust state-feedback controller based on parameter-dependent Lyapunov functions, (Proceedings of the 42nd IEEE Conference on Decision and Control (2003)), 1920-1925 |
| [28] | Papachristodoulou, A.; Prajna, S., The construction of Lyapunov functions using the sum of squares decomposition, (Proceedings of the 41st IEEE Conference on Decision and Control (2002)), 3482-3487 |
| [29] | Parrilo, P., Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization (2000), Caltech: Caltech Pasadena, USA, Ph.D. Thesis |
| [30] | Peet, M., Exponentially stable nonlinear systems have polynomial Lyapunov functions on bounded regions, IEEE Trans. Automat. Control, 54, 979-987 (2009) · Zbl 1367.93442 |
| [31] | Peet, M.; Papachristodoulou, A., A converse sum-of-squares Lyapunov result: An existence proof based on the Picard iteration, (49th IEEE Conference on Decision and Control (2010)), 5949-5954 |
| [32] | Ratschan, S.; She, Z., Providing a basin of attraction to a target region of polynomial systems by computation of Lyapunov-like functions, SIAM J. Control Optim., 48, 7, 4377-4394 (2010) · Zbl 1215.65188 |
| [33] | Rezaiee-Pajand, M.; Moghaddasie, B., Estimating the region of attraction via collocation for autonomous nonlinear systems, Struct. Engrg. Mech., 41, 2, 263-284 (2012) |
| [34] | Sanderson, C., Armadillo: An open source C++ linear algebra library for fast prototyping and computationally intensive experiments (2010), NICTA, Technical Report |
| [35] | Sastry, S., Nonlinear Systems: Analysis, Stability, and Control (1999), Springer · Zbl 0924.93001 |
| [36] | Zubov, V., Methods of A.M. Lyapunov and Their Application (1964), P. Noordhoff: P. Noordhoff Groningen, Netherlands · Zbl 0115.30204 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
