Bang-bang control of a second-order non-linear unstable plant with third-order non-linearity

Abstract

In this paper the design of a controller for a relay-controlled second-order non-linear unstable plant with third-order non-linearity is considered. The task of the controller is the simultaneous reduction of output and output derivative to zero with the input being at zero. It is established that, provided the initial values of error and error derivative fall in a ‘ controllable region ’, it is possible to reduce error and error derivative to zero, simultaneously and in the shortest possible time with at most one switching reversal of the relay. It is also shown that, through simple transformation of error and error derivative, the equation of the switching curve can be made independent of the constant gain of the plant, as well as of the coefficient of the non-linear term.