PID Control for a Class of Non-Affine Uncertain Systems

Although the classical PID controller is the most widely used ones in industrial processes which are typically nonlinear with uncertainties, almost all of the design methods for PID control are focused on linear systems. Recently, we have presented a mathematical theory for the design of linear PID control of nonlinear uncertain systems in [15], [16], where the input signal is assumed to be additive with no uncertainties in the input channel. The purpose of this paper is to establish a theory for the PID control of a class of second order non-affine uncertain systems, where the input channel may have uncertainties and is not necessarily additive. We will show that a 3-dimensional manifold can be constructed from which the three PID parameters can be arbitrarily chosen to make the closed-loop control system globally stable and the regulation error approaches to zero asymptotically. Moreover, for the case where the setpoint is also an equilibrium of the open-loop nonlinear system, we give a simple necessary and sufficient condition for the choice of the PD controller parameters by using the Markus- Yamabe theorem in differential equations.