Optimum switching criteria for higher order contactor servo with interrupted circuits

IN A contactor servo, the controlling elements determine only the instants of time at which the contactor operates. In between these moments the controlled system is left alone to follow its own equations of motion and is not in any way affected by the error and the time derivatives of the error. This is also true for continuous control systems with saturable elements in response to a large reference input. For optimum operation of a second-order system of this nature, Hopkin1 and McDonald2 showed that the switching criterion can be expressed as a nonlinear relationship between the error and its first time derivative. Their analyses were in terms of the phaseplane technique and the optimum switching boundary was shown to be a trajectory on the phase plane. Bogner and Kazda3 generalized the phase-plane criteria for second-order contactor servos1,2 to a phase-space criteria for higher order contactor servos. An implicit but basic assumption of their analysis is that in an n-order contactor servo the error and its derivatives up to the n-1 order are continuous at the moment of switching. What does this assumption imply physically?