Semiglobal stability of PID for uncertain nonaffine systems

Abstract

Proportional–integral–derivative (PID) controller is the most widely used form of feedback control by far. However, a satisfactory theory is still lacking to explain the rationale of the widespread PID control, and the gap between theory and practice of PID needs to be filled by control scientists. It is worth noting that a recent work (Zhao and Guo, 2022) proved that the classical PID controller has the ability to globally stabilize a basic class of uncertain nonaffine nonlinear systems, while the system nonlinear function is required to have a linear growth rate with respect to the state variables. In this paper, we will consider the more general case where the system nonlinear function may have super-linear growth rates, and show that the classical PID control can always be designed to stabilize such systems in the semiglobal sense, provided that the smallest eigenvalue of the control gain matrix has a positive lower bound. Moreover, a three dimensional parameter set for the stabilizing PID parameters will be constructed. Furthermore, under some additional knowledge about the system uncertain functions, we will propose a simpler PD controller with desired compensation, and show that the proposed controllers can still be designed to achieve semiglobal stability and desired tracking targets.