Performance Limitations of Nonminimum Phase Affine Nonlinear Systems

Abstract

This paper presents an analytical method to determine the minimum norm of the regulation error as well as that of the trajectory tracking error for nonminimum phase affine nonlinear systems. The proposed method derives the Hamilton–Jacobi–Bellman equation associated with the cheap control problem. Depending on the order of the system's internal dynamics, the proposed technique either constructs an algebraic equation with parametric coefficients (for systems with first order internal dynamics) or gives a differential equation (in the case of systems with second-order or higher internal dynamics). The energy of the solution of the derived equation is then used as the performance limitation index to represent the minimum value of the error norm. The proposed index is well suited for analytical as well as numerical calculations for any arbitrary set of initial values for the internal states. Simulation studies are performed to show how the proposed method can be applied to obtain the minimum norm of the output for a system.