Partial Excitation in Parameter Learning

This paper investigates parameter learning problems under Partial Persistent Excitation (PPE). The PPE condition is a rank-deficient, and therefore, a more general evolution of the well-known Persistent Excitation (PE) condition. Under the PPE condition, a proposed online algorithm is able to calculate the PE and non-PE subspaces, and finally gives an optimal parameter estimate in the sense of least squares. In particular, the learning error within the PE subspace exponentially converges to zero in the noise-free case. The PPE 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 PPE condition. This protocol allows each local estimator to operate locally in its PE subspace, and reach a consensus with neighbors in its non-PE 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.