PID Control of Planar Nonlinear Uncertain Systems in the Presence of Actuator Saturation

This paper investigates PID control design for a class of planar nonlinear uncertain systems in the presence of actuator saturation. Based on the bounds on the growth rates of the nonlinear uncertain function in the system model, the system is placed in a linear differential inclusion. Each vertex system of the linear differential inclusion is a linear system subject to actuator saturation. By placing the saturated PID control into a convex hull formed by the PID controller and an auxiliary linear feedback law, we establish conditions under which an ellipsoid is contractively invariant and hence is an estimate of the domain of attraction of the equilibrium point of the closed-loop system. The equilibrium point corresponds to the desired set point for the system output. Thus, the location of the equilibrium point and the size of the domain of attraction determine, respectively, the set point that the output can achieve and the range of initial conditions from which this set point can be reached. Based on these conditions, the feasible set points can be determined and the design of the PID control law that stabilizes the nonlinear uncertain system at a feasible set point with a large domain of attraction can then be formulated and solved as a constrained optimization problem with constraints in the form of linear matrix inequalities (LMIs). Application of the proposed design to a magnetic suspension system illustrates the design process and the performance of the resulting PID control law.