Necessary and sufficient regressor conditions for the global asymptotic stability of recursive least squares

Abstract

In recursive least squares (RLS), a persistently exciting (PE) regressor guarantees global asymptotic stability (GAS) of the estimation error relative to the zero equilibrium, and hence convergence of the parameter estimates to their true values. It is known, however, that PE is sufficient but not necessary for GAS, and thus GAS might be achieved even if the regressor is not PE. Since analyses based on PE are ubiquitous in the recursive estimation literature, the existence of non-PE regressors that ensure GAS raises the question of how PE can be directly generalized into a necessary and sufficient condition for the GAS of RLS, and whether or not such a generalization would provide a simple characterization of non-PE regressors for which RLS is GAS. In this paper, we introduce WPE, a direct generalization of PE that is necessary and sufficient for GAS, and explain how it can be understood as a “non-uniform” extension of PE to specific classes of summation windows and lower bound sequences. Next, we show that WPE is equivalent to a condition that emerges from extending certain proofs of non-negative series divergence, such as Oresme’s divergence proof for the harmonic series, to sequences of real symmetric positive-semidefinite matrices.