Design of PID control for planar uncertain nonlinear systems with input delay

This article presents a PID control design for a class of planar uncertain nonlinear systems with bounded time-varying input delay. The delay is unknown with a known bound. The class of systems is characterized by the bounds on the growth rates of the nonlinear function in the system model. Based on these bounds on the growth rates, the system is placed in a linear differential inclusion, a convex hull with four linear systems as vertices. For a given set of PID control coefficients, a Lyapunov-Krasovskii functional is constructed that leads to sufficient conditions, in the form of linear matrix inequalities in the bound on the time delay, a positive scalar and two positive definite matrices, under which the equilibrium solution corresponding to the desired set point of the outputs of all vertex linear systems and, hence, that of the original uncertain nonlinear system, is globally uniformly asymptotically stable. With the bound on the time delay as a variable, an optimization problem with linear matrix inequalities as constraints can be formulated to determine the largest admissible bound on the time delay for the closed-loop system under the given PID controller. By viewing the PID control coefficients as additional free variables, the optimization problem is then adapted for the design of the PID coefficients for the largest admissible bound on the time delay. The design is then applied to a magnetic suspension system. Simulation results illustrate the design algorithm and the performance of the resulting PID controller.