Sequential Gradient-Descent Optimization of Data-Dependent, Rank-Deficient Cost Functions with a Unique Common Global Minimizer

Online system identification entails optimization of a sequence of cost functions that are updated by data as it becomes available. The goal is to obtain a sequence of minimizers that converge to the true parameters. If the true parameters are the unique common minimizer of all of the costs in the sequence, then convergence of a sequence of minimizers to the true parameters can be viewed as convergence of the state of a dynamical system to an equilibrium. This paper investigates global asymptotic stability of the equilibrium of a dynamical system defined by online gradient-based optimization of a sequence of quadratic cost functions that have a unique common minimizer, but may individually have multiple global minimizers due to rank deficiency. Under a weak persistency-type condition, it is shown that global asymptotic stability can be guaranteed for this class of costs. These results are specialized to the case of least squares costs and illustrated by examples.