Gradient Projection Anti-windup Scheme on Constrained Planar LTI Systems

The gradient projection anti-windup (GPAW) scheme was recently proposed as an anti-windup method for nonlinear multi-input-multi-output systems/controllers, the solution of which was recognized as a largely open problem in a recent survey paper. This report analyzes the properties of the GPAW scheme applied to an input constrained first order linear time invariant (LTI) system driven by a first order LTI controller, where the objective is to regulate the system state about the origin. We show that the GPAW compensated system is in fact a projected dynamical system (PDS), and use results in the PDS literature to assert existence and uniqueness of its solutions. The main result is that the GPAW scheme can only maintain/enlarge the exact region of attraction of the uncompensated system. We illustrate the qualitative weaknesses of some results in establishing true advantages of anti-windup methods, and propose a new paradigm to address the anti-windup problem, where results relative to the uncompensated system are sought.