On Controlling the Transient Response of Linear Time Invariant Systems

The main topic of investigation of this paper is to find a bound for the set of control parameters, K, so that a rational, proper transfer function, N(s,K) D(s,K) has a decaying, non-negative impulse response. It is assumed that the coefficients of the polynomials N(s, K) and D(s, K) are affine in K. A broad class of transient response control problems can be formulated in this way. An earlier result by the authors enables one to constructively find the set of controllers K that render D(s, K) Hurwitz as a countable union of polyhedral sets. Another result by the authors provides two necessary conditions for the transfer function N(s,K) D(s,K) to have a non-negative impulse response: the dominant root of D(s, K) be real and no real root of N(s, K) be greater than the dominant root of D(s, K). Using these two results, and the Descartes’ rule of signs, a procedure is outlined to construct an outer approximation (as a union of polyhedral sets) of the set of controllers K so that N(s,K) D(s,K) has a non-negative and decaying impulse response.