The servo problem in discontinuous control systems

Discontinuous control systems, and relay feedback systems in particular, are one of the most important types of nonlinear systems. The term “relay” comes from electrical applications where on-off control has long been used. To describe the nonlinear phenomenon typical of electrical relays, the nonlinear function also received the name “relay” comprising a number of discontinuous nonlinearities. Thus, when applied to any type of control system, the term “relay” is now associated not with applications but with the kind of nonlin-earities that are found in the system models. However, what we traditionally call the “relay system” cannot always be described by the relay system model. For example, the vibrational voltage regulator is not a relay system in the full sense, as the charge and discharge time constants are different and the regulator is better described as a variable structure system (which is, however, a discontinuous control system).

Applications of the discontinuous feedback principle have evolved from vi-brational voltage regulators and missile thruster servomechanisms of the 1940s to numerous on-off process parameter closed-loop control systems, sigma-delta modulators, process identification and automatic tuning of proportional-integral-derivative (PID) controller techniques, and DC motor, hydraulic and pneumatic servo systems, to name a few. The enormous number of residential temperature control systems available throughout the world illustrates how popular discontinuous control systems are. A number of industrial examples of relay systems were given in the classic book on relay systems [94]. Furthermore, many existing sliding mode algorithms can be considered and analyzed via the relay control principle. Perhaps, some aspects of hybrid systems can also be analyzed via application of discontinuous control system theory. In fact, discontinuous control systems are probably the most conventional type of control system in history.