On Algebraic Proofs of Stability for Homogeneous Vector Fields
On Algebraic Proofs of Stability for Homogeneous Vector Fields
复制标题
关于齐次向量场稳定性的代数证明
DOI：
10.1109/tac.2019.2914968
复制
发表时间：
2018
影响因子：
6.8
通讯作者：
Bachir El Khadir
中科院分区：
文献类型：
--
作者：
Amir Ali Ahmadi;Bachir El Khadir
我们证明，如果齐次、连续可微的向量场是渐近稳定的，那么它承认李亚普诺夫函数，它是两个多项式（即有理函数）的比率。我们进一步证明，当向量场是多项式时，有理函数及其导数上的李雅普诺夫不等式都有平方和证明，因此，这样的李雅普诺夫函数总是可以通过半定规划找到。这概括了渐近稳定线性系统承认二次李亚普诺夫函数的经典事实，该函数满足一定的线性矩阵不等式。除了齐次矢量场之外，该结果还可以通过证明非齐次系统最低阶齐次分量的渐近稳定性来显示非齐次系统的局部渐近稳定性。本文还包括一些负面结果：我们表明，在不存在同质性的情况下，全局渐近稳定的多项式向量场可能无法承认全局有理李雅普诺夫函数，并且在存在同质性的情况下，有理李雅普诺夫函数的分子次数可能需要任意高（即使对于固定次数和维数的向量场）。另一方面，我们还给出了一系列齐次多项式向量场，它们允许低次有理李雅普诺夫函数，但需要任意高次的多项式李雅普诺夫函数。这显示了使用有理李雅普诺夫函数的潜在好处，特别是当我们保证其存在具有结构化分母并且搜索成本并不比多项式函数更昂贵时。
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.