Deficient Excitation in Parameter Learning

2025-03-04

AI Summary

This paper addresses system identification under deficient excitation (DE)—a scenario where the input excitation matrix is rank-deficient and fails to satisfy conventional persistent excitation conditions. Methodologically, it establishes: (1) the first systematic DE theory, enabling exact online subspace decomposition to distinguish identifiable from unidentifiable subspaces; (2) a model-free recursive least-squares estimator that achieves exponential convergence of estimation error within the identifiable subspace under noise-free conditions; and (3) a distributed cooperative consensus protocol, wherein multiple local estimators collaboratively fuse information under complementary DE conditions—requiring only collective excitation sufficiency to attain globally consistent, optimal estimates. Experimental results demonstrate the framework’s effectiveness and robustness in system identification tasks, even under severe excitation limitations.

Technology Category

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Application Category

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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.

Problem

• Investigates parameter learning under deficient excitation conditions.
• Proposes an online algorithm for optimal parameter estimation.
• Develops a cooperative protocol for distributed parameter learning.

Innovation

• Online algorithm calculates identifiable subspaces.
• Cooperative protocol enhances distributed parameter learning.
• Exponential convergence in identifiable subspace without persistent excitation.