Eigenvalue Characterization of Persistence of Excitation for Convex Adaptive Predictive Control

Abstract

Adaptive model predictive control (MPC) has emerged as a viable solution to the dual-control problem, which refers to the combined challenges of control and system identification. To guarantee estimates convergence, several architectures employ some form of persistence of excitation (PoE), an input design requirement that translates into a nonlinear time-varying constraint with generally nonconvex solution set. However, this is seldom addressed structurally in the context of adaptive MPC. In this article, we provide new insights over its requirements and time-varying nature, within the context of noise-free linear time-invariant systems in state feedback. We first recast the PoE constraint as a bilinear matrix inequality and implement two distinct convexification approaches that approximate its nonconvex solution. We then employ an eigenvalue analysis to characterize an additional approximation and an explicit geometric index that relates to the achievable size of the solution set within a constrained input space. Numerical simulations show a (saturated) linear relationship between the geometric index and the size of solution sets, both true and approximated. More importantly, the time-varying characteristic of the PoE constraint dominates the size of the convex approximations, independent of the principle behind convexification.