具有吸引域先验知识的非线性系统辨识
Nonlinear System Identification with Prior Knowledge of the Region of Attraction
摘要：
研究了当平衡点的吸引域（ROA）上存在先验知识时的非线性系统辨识问题。我们提出了一种识别方法的形式的最优化问题，最大限度地减少拟合误差，并保证所需的稳定性。通过联合辨识系统的动态特性和验证系统稳定性的李雅普诺夫函数来解决该问题。在这种情况下，假设集是一个再生核希尔伯特空间，并且相对于ROA的给定子集的每个点，李雅普诺夫函数的李导数不等式施加了约束。该问题是一个具有无穷多个约束的非凸无穷维优化问题。为了获得一个易于处理的配方，只有一个适当设计的有限子集的约束被认为是。由此产生的问题承认一个解决方案的形式的线性组合的部分内核及其衍生物。推导了一个等价的具有二次费用函数的线性和双线性约束的最优化问题。适当的变量变量变换给出了该问题的凸变换形式。为了减少超参数的数量，最优化问题适用于对角核的情况。通过实例说明了该方法。
英文摘要：
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 hyperparameters, the optimization problem is adapted to the case of diagonal kernels. The method is demonstrate by means of an example.
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