Non-monotonic Lyapunov functions for stability of nonlinear and switched systems : theory and computation
Non-monotonic Lyapunov functions for stability of nonlinear and switched systems : theory and computation
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用于非线性和切换系统稳定性的非单调李亚普诺夫函数：理论与计算
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发表时间：
2008
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通讯作者：
Amir Ali Ahmadi
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作者：
Amir Ali Ahmadi
李雅普诺夫直接法基于状态的标量函数的存在，该函数沿轨道单调递减，它仍然是建立非线性系统稳定性的主要工具。由于基于Lyapunov理论的稳定性分析的主要挑战总是找到合适的Lyapunov函数，因此弱化对Lyapunov函数的要求是非常有意义的。在本文中，我们放宽了李雅普诺夫定理的单调性要求，扩大了能够提供稳定性证明的函数类。既包括离散时间情况，也包括连续时间情况。在整篇论文中，特别关注了凸优化中的技术，这些技术允许以计算上容易处理的方式搜索Lyapunov函数。因此，我们的理论贡献服从于凸规划公式。在离散时间情形下，我们给出了全局渐近稳定的两个新的充分条件，它们允许Lyapunov函数局部增加，但保证每隔几步平均减少一次。我们的第一个条件是非凸的，但允许直观地解释。第二个条件是凸的，它包含了第一个条件作为特例，它可以作为一个半定规划。我们证明了当存在非单调Lyapunov函数时，可以构造一个更复杂的单调递减函数。我们通过三类不同的动力系统的例子证明了我们的方法相对于标准的Lyapunov理论的优越性。首先，我们考虑多项式动力学，其中我们利用平方和编程的技巧。其次，对分段仿射系统进行了分析。在这里，建立了与分段二次Lyapunov函数方法的联系。最后，我们研究了具有任意切换的系统
Lyapunov’s direct method, which is based on the existence of a scalar function of the state that decreases monotonically along trajectories, still serves as the primary tool for establishing stability of nonlinear systems. Since the main challenge in stability analysis based on Lyapunov theory is always to find a suitable Lyapunov function, weakening the requirements of the Lyapunov function is of great interest. In this thesis, we relax the monotonicity requirement of Lyapunov’s theorem to enlarge the class of functions that can provide certificates of stability. Both the discrete time case and the continuous time case are covered. Throughout the thesis, special attention is given to techniques from convex optimization that allow for computationally tractable ways of searching for Lyapunov functions. Our theoretical contributions are therefore amenable to convex programming formulations. In the discrete time case, we propose two new sufficient conditions for global asymptotic stability that allow the Lyapunov functions to increase locally, but guarantee an average decrease every few steps. Our first condition is nonconvex, but allows an intuitive interpretation. The second condition, which includes the first one as a special case, is convex and can be cast as a semidefinite program. We show that when non-monotonic Lyapunov functions exist, one can construct a more complicated function that decreases monotonically. We demonstrate the strength of our methodology over standard Lyapunov theory through examples from three different classes of dynamical systems. First, we consider polynomial dynamics where we utilize techniques from sum-of-squares programming. Second, analysis of piecewise affine systems is performed. Here, connections to the method of piecewise quadratic Lyapunov functions are made. Finally, we examine systems with arbitrary switching