基于输入设计的方差最小化系统辨识
System Identification with Variance Minimization via Input Design
摘要：
子空间方法是线性系统的主流系统辨识方法之一，其基本思想是将系统参数矩阵投影到与输入输出相关的子空间中进行估计。然而，现有的大多数子空间方法由于缺乏估计的封闭形式表达而无法保证统计性能。同时，传统的子空间方法不能处理噪声的不确定性，不能得到稳定的辨识结果。本文从输入设计的角度提出了一种新的改进子空间方法，该方法保证了辨识结果的一致性和稳定性，同时保证了辨识结果的方差最小。具体地说，我们首先得到一个封闭的形式估计的系统矩阵，然后分析的统计性能，通过推导出的最大识别偏差。这个辨识偏差最大化问题是非凸的，并通过将其分成两个子问题来解决，并保证最优解。其次，提出了一种输入设计方法，通过最小化方差来处理不确定性，获得稳定的辨识结果。这个问题被公式化为一个有约束的最小-最大优化问题。在保证安全约束的前提下，通过预测控制的方法将代价函数转化为凸函数，得到最优解。我们证明了所提出的方法的一致性和收敛性。仿真结果表明了该方法的有效性。
英文摘要：
The subspace method is one of the mainstream system identification method of linear systems, and its basic idea is to estimate the system parameter matrices by projecting them into a subspace related to input and output. However, most of the existing subspace methods cannot have the statistic performance guaranteed since the lack of closed-form expression of the estimation. Meanwhile, traditional subspace methods cannot deal with the uncertainty of the noise, and thus stable identification results cannot be obtained. In this paper, we propose a novel improved subspace method from the perspective of input design, which guarantees the consistent and stable identification results with the minimum variance. Specifically, we first obtain a closed-form estimation of the system matrix, then analyze the statistic performance by deriving the maximum identification deviation. This identification deviation maximization problem is non-convex, and is solved by splitting it into two sub-problems with the optimal solution guaranteed. Next, an input design method is proposed to deal with the uncertainty and obtain stable identification results by minimizing the variance. This problem is formulated as a constrained min-max optimization problem. The optimal solution is obtained from transforming the cost function into a convex function while ensuring the safety constraints through the method of predictive control. We prove the consistency and the convergence of the proposed method. Simulation demonstrates the effectiveness of our method.
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