On the transient control of Linear Time Invariant systems

Practical problems require the synthesis of a set of stabilizing controllers that guarantee transient performance specifications such as a bound on the overshoot of its closed loop step response. A majority of these specifications for linear time invariant (LTI) systems can be converted to the requirement of synthesizing a set of stabilizing controllers guaranteeing the non-negative impulse response of an appropriate transfer function whose coefficients are functions of the controller parameters. 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 (z, K)/D (z, K) has a decaying, non-negative impulse response. For single input single output (SISO) LTI systems, one may assume that the coefficients of the polynomials N (z, K) and D (z, K) are affine in K. An earlier result by the authors provides an approximation of the set of stabilizing controller parameters in terms of unions of polyhedral sets. In this paper, we provide necessary and sufficient conditions for a rational proper stable transfer function to have a non-negative impulse response. For the synthesis problem, we show that these conditions translate into a sequence of polynomial matrix inequalities in K using Markov-Lucaks´ theorem. We propose an outer approximation of the feasible set of matrix inequalities using Lasserre´s moment method.