Feedback and uncertainty: Some basic problems and results

Abstract This paper will review some fundamental results in the understanding of several basic problems concerning feedback and uncertainty. First, we will consider adaptive control of linear stochastic systems, in particular, the global stability and optimality of the well-known self-tuning regulators, designed by combining the least-squares estimator with the minimum variance controller. This natural and seemingly simple case had actually been a longstanding central problem in the area of adaptive control, and its solution offers valuable insights necessary for understanding more complex problems. Next, we will discuss the theoretical foundation of the classical proportional-integral-derivative (PID) control, to understand the rationale behind its widespread successful applications in control practice where almost all of the systems to be controlled are nonlinear with uncertainties, by presenting some theorems on the global stability and asymptotic optimality of the closed-loop systems, and by providing a concrete design method for the PID parameters. Finally, we will consider more fundamental problems on the maximum capability and limitations of the feedback mechanism in dealing with uncertain nonlinear systems, where the feedback mechanism is defined as the class of all possible feedback laws. Some extensions and perspectives will also be discussed in the paper.