Bang-bang control of a second-order non-linear stable plant with second-order nonlinearity

In this paper the design of a controller for a relay-controlled second-order non-linear stable plant with second-order nonlinearity is considered. The task of the controller is the simultaneous reduction of plant's output and output derivative to zero with the input to the feedback system being at z;ro. It will be shown that for all initial values ot output and output derivative it would be possible to bring them to zero, simultaneously and in the shortest possible time with at most one switching reversal of the relay. Through simple transformation, it would be shown that the equation of switching curve can be made independent of constant gain of the plant as well as the coefficient of its non-linear term.