Deficient Excitation in Parameter Learning

TL;DR

The paper extends parameter learning beyond persistent excitation by introducing Deficient Excitation (DE), which yields identifiable and non-identifiable subspaces of parameters. It develops an online algorithm that identifies these subspaces and achieves least-squares optimal learning with exponential convergence in the identifiable subspace, even under DE, and provides a distributed scheme where multiple agents cooperatively estimate parameters over directed graphs under complementary DE. The approach is demonstrated through system-identification applications for both single linear systems and interconnected networks, showing that global knowledge can be recovered from locally DE measurements. The work offers a robust, scalable framework for distributed parameter learning in large-scale or sensor-limited settings, with provable convergence guarantees under deterministic conditions and bounded-noise scenarios.

Abstract

This paper investigates parameter learning problems under deficient excitation (DE). The DE condition is a rank-deficient, and therefore, a more general evolution of the well-known persistent excitation condition. Under the DE condition, a proposed online algorithm is able to calculate the identifiable and non-identifiable subspaces, and finally give an optimal parameter estimate in the sense of least squares. In particular, the learning error within the identifiable subspace exponentially converges to zero in the noise-free case, even without persistent excitation. The DE condition also provides a new perspective for solving distributed parameter learning problems, where the challenge is posed by local regressors that are often insufficiently excited. To improve knowledge of the unknown parameters, a cooperative learning protocol is proposed for a group of estimators that collect measured information under complementary DE conditions. This protocol allows each local estimator to operate locally in its identifiable subspace, and reach a consensus with neighbours in its non-identifiable subspace. As a result, the task of estimating unknown parameters can be achieved in a distributed way using cooperative local estimators. Application examples in system identification are given to demonstrate the effectiveness of the theoretical results developed in this paper.

Figures (5)

• Figure 1: Subspace parameter learning error when $u=10\sin(t+2)$.
• Figure 3: Subspace parameter learning error when $u=10\sin(t+2)$ in the presence of white noise.
• Figure 5: Parameter learning error when $u=10\sin(t+2)$ (where $\tilde{\theta}_{[i]}(t)$ is the $i$th element of $(\hat{\theta}(t)-\theta$).
• Figure 7: Communication graph.
• Figure 8: Distributed parameter learning error at each estimator.

Theorems & Definitions (6)

• Definition 1: Persistent Excitation
• Definition 2: Deficient Excitation
• proof
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• proof
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