论文标题
学习混合系统的Lyapunov功能
Learning Lyapunov Functions for Hybrid Systems
论文作者
论文摘要
我们提出了一种基于抽样的方法来学习lyapunov的功能，用于接受混合代表的一类离散时间自动级混合系统。 Such systems include autonomous piecewise affine systems, closed-loop dynamics of linear systems with model predictive controllers, piecewise affine/linear complementarity/mixed-logical dynamical system in feedback with a ReLU neural network controller, etc. The proposed method comprises an alternation between a learner and a verifier to find a valid Lyapunov function inside a convex set of Lyapunov function candidates.在每次迭代中，学习者使用状态样本的集合通过参数空间中的凸面程序选择Lyapunov函数候选。然后，验证者在状态空间中求解一个混合构成二次程序，以验证提出的lyapunov函数候选者或用反例，即lyapunov条件失败的状态。然后将此反例添加到学习者的样本集中，以完善Lyapunov功能候选者的集合。通过根据凸优化的分析中心切割平面方法设计学习者和验证者，我们表明，当Lyapunov函数集在参数空间中是全约维时，我们的方法在有限的步骤中找到了Lyapunov函数。我们证明了我们在闭环MPC动力系统和Relu神经网络控制的PWA系统上的稳定分析方法。
We propose a sampling-based approach to learn Lyapunov functions for a class of discrete-time autonomous hybrid systems that admit a mixed-integer representation. Such systems include autonomous piecewise affine systems, closed-loop dynamics of linear systems with model predictive controllers, piecewise affine/linear complementarity/mixed-logical dynamical system in feedback with a ReLU neural network controller, etc. The proposed method comprises an alternation between a learner and a verifier to find a valid Lyapunov function inside a convex set of Lyapunov function candidates. In each iteration, the learner uses a collection of state samples to select a Lyapunov function candidate through a convex program in the parameter space. The verifier then solves a mixed-integer quadratic program in the state space to either validate the proposed Lyapunov function candidate or reject it with a counterexample, i.e., a state where the Lyapunov condition fails. This counterexample is then added to the sample set of the learner to refine the set of Lyapunov function candidates. By designing the learner and the verifier according to the analytic center cutting-plane method from convex optimization, we show that when the set of Lyapunov functions is full-dimensional in the parameter space, our method finds a Lyapunov function in a finite number of steps. We demonstrate our stability analysis method on closed-loop MPC dynamical systems and a ReLU neural network controlled PWA system.
