On Algebraic Proofs of Stability for Homogeneous Vector Fields

We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function, which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the ra...

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Bibliographic Details
Published in:IEEE transactions on automatic control 2020-01, Vol.65 (1), p.325-332
Main Authors: Ahmadi, Amir Ali, El Khadir, Bachir
Format: Article
Language:English
Subjects:
Algebraic methods in control
Asymptotic properties
Asymptotic stability
converse lyapunov theorems
Fields (mathematics)
Heart rate variability
Homogeneity
Liapunov functions
Linear matrix inequalities
Linear systems
Lyapunov methods
Mathematical analysis
Mathematical model
Matrix algebra
nonlinear dynamics
Polynomials
Programming
Rational functions
rational Lyapunov functions
Semidefinite programming
Stability
Testing
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Summary:We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function, which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sum of squares certificates and, hence, such a Lyapunov function can always be found by semidefinite programming. This generalizes the classical fact that an asymptotically stable linear system admits a quadratic Lyapunov function, which satisfies a certain linear matrix inequality. In addition to homogeneous vector fields, the result can be useful for showing local asymptotic stability of nonhomogeneous systems by proving asymptotic stability of their lowest order homogeneous component. This paper also includes some negative results: We show that in absence of homogeneity, globally asymptotically stable polynomial vector fields may fail to admit a global rational Lyapunov function, and in presence of homogeneity, the degree of the numerator of a rational Lyapunov function may need to be arbitrarily high (even for vector fields of fixed degree and dimension). On the other hand, we also give a family of homogeneous polynomial vector fields that admit a low-degree rational Lyapunov function but necessitate polynomial Lyapunov functions of arbitrarily high degree. This shows the potential benefits of working with rational Lyapunov functions, particularly as the ones whose existence we guarantee have structured denominators and are not more expensive to search for than polynomial ones.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2019.2914968