Neural Lyapunov Control for Discrete-Time Systems
Neural Lyapunov Control for Discrete-Time Systems
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DOI：
10.48550/arxiv.2305.06547
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发表时间：
2023-05
期刊：
影响因子：
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通讯作者：
Junlin Wu;Andrew Clark;Y. Kantaros;Yevgeniy Vorobeychik
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文献类型：
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作者：
Junlin Wu;Andrew Clark;Y. Kantaros;Yevgeniy Vorobeychik
虽然确保线性系统的稳定性是众所周知的，但它仍然是非线性系统的主要挑战。在这种情况下，一般的方法是计算一个李雅普诺夫函数和相关的控制策略的组合。然而，寻找一般非线性系统的李雅普诺夫函数是一项具有挑战性的任务。为了解决这一挑战，已经提出了几种方法，使用神经网络表示李雅普诺夫函数。然而，这种方法要么集中在连续时间系统，或高度限制类的非线性动力学。我们提出了第一种方法学习神经李亚普诺夫控制在广泛的一类离散时间系统。三个关键因素使我们能够有效地学习可证明稳定的控制策略。首先是一种新的混合整数线性规划方法，用于验证离散时间李雅普诺夫稳定性条件，利用这些条件的特定结构。第二个是一种新的方法来计算验证子水平集。第三种是一种启发式的基于梯度的方法，用于快速找到反例，以显着加快李雅普诺夫函数学习。我们在四个标准基准上的实验表明，我们的方法显着优于最先进的基线。例如，在路径跟踪基准测试中，我们在运行时间和吸引区域的大小方面都优于最近的神经Lyapunov控制基线，并且在四个基准测试中的两个（cartpole和PVTOL）上，我们是第一个自动返回可证明稳定控制器的方法。我们的代码可在https://github.com/jlwu002/nlc_discrete上获取。
While ensuring stability for linear systems is well understood, it remains a major challenge for nonlinear systems. A general approach in such cases is to compute a combination of a Lyapunov function and an associated control policy. However, finding Lyapunov functions for general nonlinear systems is a challenging task. To address this challenge, several methods have been proposed that represent Lyapunov functions using neural networks. However, such approaches either focus on continuous-time systems, or highly restricted classes of nonlinear dynamics. We propose the first approach for learning neural Lyapunov control in a broad class of discrete-time systems. Three key ingredients enable us to effectively learn provably stable control policies. The first is a novel mixed-integer linear programming approach for verifying the discrete-time Lyapunov stability conditions, leveraging the particular structure of these conditions. The second is a novel approach for computing verified sublevel sets. The third is a heuristic gradient-based method for quickly finding counterexamples to significantly speed up Lyapunov function learning. Our experiments on four standard benchmarks demonstrate that our approach significantly outperforms state-of-the-art baselines. For example, on the path tracking benchmark, we outperform recent neural Lyapunov control baselines by an order of magnitude in both running time and the size of the region of attraction, and on two of the four benchmarks (cartpole and PVTOL), ours is the first automated approach to return a provably stable controller. Our code is available at: https://github.com/jlwu002/nlc_discrete.