Numerical Construction of Lyapunov Functions Using Homotopy Continuation Method

Abstract

Lyapunov functions are frequently used for investigating the stability of linear and nonlinear dynamical systems. Though there is no general method of constructing these functions, many authors use polynomials in $ p-forms $ as candidates in constructing Lyapunov functions while others restrict the construction to quadratic forms. By focussing on the positive and negative definiteness of the Lyapunov candidate and the Hessian of its derivative, and using the sum of square decomposition, we developed a method for constructing polynomial Lyapunov functions that are not necessarily in a form. The idea of Newton polytope was used to transform the problem into a system of algebraic equations that were solved using the polynomial homotopy continuation method. Our method can produce several possibilities of Lyapunov functions for a given candidate. The sample test conducted demonstrates that the method developed is promising

Keywords

• p-form
• Sum of Square
• Homotopy
• PHClab
• Polytope.

Kaynakça

1. [1] A. A. Ahmadi and A. P Parrilo. Sum of squares and polynomial convexity, (2009).
2. [2] A. A. Ahmadi and A. P Parrilo. A convex polynomial that is not sos-convex, Mathematical Programming, 135 (2012):275-292.
3. [3] E. Alexander, and A. Khovanskii. Elimination theory and Newton polytopes, Functional Analysis and Other Mathematics, 1 (2006):45-71.
4. [4] E. Eivind. Principal minors and the Hessian, https://www.dr-eriksen.no/teaching/GRA6035/2010/lecture5.pdf, 1 (2010):1- 25.
5. [5] E. Mohlmann. Automatic stability verification via Lyapunov functions, representations, transformations, and practical issues, (2018).
6. [6] G. E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decompostion, Automata Theory and Formal Languages, 33 (1975): 134-183.
7. [7] G. Shuhong. Absolute Irreducibility of Polynomials via Newton Polytopes, Journal of Algebra, 237 (2001):501-520.
8. [8] G. V. Yun. PHClab A MATLAB/Octave Interface to PHCpack, Software for Algebraic Geometry, 148 (2008):15-32.

9. [9] J. Zhenyi et al. Constructing the Lyapunov Function through Solving Positive Dimensional Polynomial System, Journal of Applied Mathematic, 5 (2013).
10. [10] K. Miroslav et al. Nonlinear and Adaptive Control Design, John Wiley and Sons, New York, 1 (1995).
11. [11] K. Ramazan and P. Arcady. Lyapunov stability of the generalized stochastic pantograph equation. Journal of Mathematics, 9 (2018). [12] L. Gasinski, et al. Nonlinear Analysis, Series In Mathematical Analysis And Applications,Chapman and Hall/CRC Press, Taylor and Francis Group (2005).
12. [13] M. Ansari et al. Stability analysis of the homogeneous nonlinear dynamical systems using Lyapunov function generation based on the basic functions, SN Appl. Sci.,2 (2020):219.
13. [14] M. Anthony and L. Hou. Stability theory of dynamical systems involving multiple non-monotonic Lyapunov functions, Mathematics in Engineering, Science and Aerospace, 7 (2016): 3482-3487.
14. [15] M. Jankovic et al. Constructive Lyapunov stabilization of nonlinear cascade systems, IEEE transactions on automatic control, 41 (1996): 1723-1735.
15. [16] P. Antonis, and P. Stephen. On the construction of Lyapunov functions using the sum of squares decomposition, Proceedings of the 41st IEEE Conference on Decision and Control, 3 (2002):3482-3487.
16. [17] P. Victoria and W. Thorsten. An Algorithm for Sums of Squares of Real Polynomials, Journal of Pure and Applied Algebra, 127 (1998):99-104.
17. [18] S. Bernd. Polynomial Equations and Convex Polytopes, The American Mathematical Monthly, 105 (2018):907-922.
18. [19] S. Peter, et al. Simplification Methods for Sum-of-Squares Programs, Optimization and Control, 2 (2013).
19. [20] S. Zhikun et al. Nonlinear Analysis: Hybrid Systems, A semi-algebraic approach for asymptotic stability analysis, 4 (2009) : 588-596.
20. [21] T. V. Nguyen, et al. Existence Conditions of a common quadratic Lyapunov function for a set of second-order systems, Trans. of the Society of Instrument and Control Engineers, 42 (2006):241-246.
21. [22] , V. Lakshmikantham. Vector Lyapunov functions and stability analysis of nonlinear systems, Mathematics and its application, 1 (1991).
22. [23] Z. Guopeng, et al. Suficient and necessary conditions for Lyapunov stability of genetic networks with SUM regulatory logic, Cognitive neurodynamics, 9.4 (2015):447-458.