  As is well-known, the classical PID control plays a dominating role in
various control loops of industrial processes. However, a theory that can
explain the rationale why the linear PID can successfully deal with the
ubiquitous uncertain nonlinear dynamical systems and a method that can provide
explicit design formulae for the PID parameters are still lacking. This paper
is a continuation of the authors recent endeavor towards establishing a
theoretical foundation of PID. We will investigate the rationale of PID control
for a general class of high dimensional second order non-affine uncertain
systems. We will show that a three dimensional parameter set can be constructed
explicitly, such that whenever the PID parameters are chosen from this set, the
closed-loop systems will be globally stable and the regulation error will
converge to zero exponentially fast, under some suitable conditions on the
system uncertainties. Moreover, we will show that the PD(PI) control can
globally stabilize several special classes of high dimensional uncertain
nonlinear systems. Furthermore, we will apply the Markus-Yamabe theorem in
differential equations to provide a necessary and sufficient condition for the
choice of the PI parameters for a class of one dimensional non-affine uncertain
systems. These theoretical results show explicitly that the controller
parameters are not necessary to be of high gain, and that the ubiquitous PID
control does indeed have strong robustness with respect to both the system
structure uncertainties and the selection of the controller parameters.

Optimization and Control