Nonlinear System Identification with Prior Knowledge of the Region of Attraction
Mohammad Khosravi and Roy S. Smith Corresponding authorThis research project is part of the Swiss Competence Center for Energy Research SCCER FEEB、D of the Swiss Innovation Agency Innosuisse.The authors are with Automatic Control Lab, ETH Zurich, Switzerland
Abstract
We consider the problem of nonlinear system identification when prior knowledge is available on the region of attraction (ROA) of an equilibrium point. We propose an identification method in the form of an optimization problem, minimizing the fitting error and guaranteeing the desired stability property. The problem is approached by joint identification the dynamics and a Lyapunov function verifying the stability property. In this setting, the hypothesis set is a reproducing kernel Hilbert space, and with respect to each point of the given subset of the ROA, the Lie derivative inequality of the Lyapunov function imposes a constraint. The problem is a non-convex infinite-dimensional optimization with infinite number of constraints. To obtain a tractable formulation, only a suitably designed finite subset of the constraints are considered. The resulting problem admits a solution in form of a linear combination of the sections of the kernel and its derivatives. An equivalent optimization problem with a quadratic cost function subject to linear and bilinear constraints is derived. A suitable change of variable gives a convex reformulation of the problem. To reduce the number of hyperpa
中文速览
用已知的吸引域（region of attraction, ROA）先验知识来约束非线性系统辨识，是一个兼顾数据拟合与稳定性保证的难题。作者将其转化为一个联合优化问题：同时学习未知向量场（用光滑向量值再生核希尔伯特空间 smooth vector-valued reproducing kernel Hilbert space 建模）和一个二次李雅普诺夫函数（Lyapunov function），要求在给定的 ROA 子集上李雅普诺夫导数不等式成立。原问题有无穷维变量和无穷多约束，作者引入"(α,β)-网格"这一有限点集，证明只需在这些离散点上满足约束便可蕴含整个区域上的稳定性条件，从而将问题化为有限维的二次目标加线性与双线性约束的优化；再通过一个非显然的变量替换，进一步得到凸重构，使问题可高效求解。方法还针对对角核做了简化以减少超参数，并通过数值示例验证了有效性。这项工作的意义在于，它提供了一套严格且可计算的框架，把"系统在某区域内稳定"这类物理先验知识无缝融入数据驱动的非线性系统辨识，避免了事后再做稳定性校正的繁琐。
原文 arXiv:2003.12330;中英对照 + 大白话阅读 https://aha.fim.ai/paper/2003.12330v1