Non-Asymptotic State-Space Identification of Closed-Loop Stochastic Linear Systems using Instrumental Variables

Abstract

The paper suggests a generalization of the Sign-Perturbed Sums (SPS) finite sample system identification method for the identification of closed-loop observable stochastic linear systems in state-space form. The solution builds on the theory of matrix-variate regression and instrumental variable methods to construct distribution-free confidence regions for the state-space matrices. Both direct and indirect identification are studied, and the exactness as well as the strong consistency of the construction are proved. Furthermore, a new, computationally efficient ellipsoidal outer-approximation algorithm for the confidence regions is proposed. The new construction results in a semidefinite optimization problem which has an order-of-magnitude smaller number of constraints, as if one applied the ellipsoidal outer-approximation after vectorization. The effectiveness of the approach is also demonstrated empirically via a series of numerical experiments.

Figures (3)

• Figure 1: Empirical coverage probability of MIV-SPS EOA as a function of the exploitation parameter for different feedback setups, using direct identification, for a 2-dimensional LSS, $n=500$, $s=500$.
• Figure 2: Empirical coverage probability of MIV-SPS EOA as a function of the exploitation parameter for different feedback setups, using indirect identification, for a 2-dimensional LSS, $n=500$, $s=500$.
• Figure 3: Empirical coverage probability of MIV-SPS EOA as a function of the sample size $n$ in an open-loop ($\varepsilon = 0$) setting, using direct identification, for a 2-dimensional LSS, $s=500$.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Definition 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• proof : Proof of Lemma \ref{['lemma4']}
• proof : Proof of Theorem \ref{['theorem1']}
• proof : Proof of Theorem \ref{['theorem2']}