Deficient Excitation in Parameter Learning

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.