Linear Control of Non-Linear Systems

For some simple relay control systems it is shown that if the control signal input to the relay is made an appropriate linear combination of the state variables, nearly optimum performance can be achieved. The performance index used is integral -square-error; its value when the switching function is linear is calculated and compared with the value obtained when the switching function is the optimal (usually non-linear) one. E.g. for a second-order plant and a command signal which is a step of random magnitude having a rectangular probability distribution, linear switching can give a mean integral-square-error which exceeds the optimal value by less than 1%.