Necessary and sufficient conditions for non-overshooting step responses for LTI systems

In this paper we consider a class of systems for which a necessary and sufficient condition is given for the determination of whether or not its impulse response starting from zero initial conditions is nonnegative. The class of plants considered is SISO, LTI, stable and strictly proper. We show that the system step response is nonovershooting by converting it to a root finding problem of a polynomial. The derived necessary and sufficient condition can then be used to develop other sufficiency conditions for more general plants where the poles need not satisfy the special form given above. Further we derive a necessary and sufficient condition guaranteeing a nonovershooting step response for another class of SISO, LTI stable and strictly proper systems. The poles of this class of systems are assumed real and of the form λ_i = iλ, λ < 0, i = 1, 2,..., n. The conditions should be useful in practical control system design where a nonovershooting step response is desirable. In addition necessary and sufficient conditions are derived for a SISO plant with a complex conjugate pole pair. The latter result is then used to state conditions for a third order plant with no zeros.