System Identification with Variance Minimization via Input Design

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.