Synthesis of PID controllers with guaranteed non-overshooting transient response

This paper presents a new method for approximating the set of PID controllers satisfying a class of transient specifications. The problem of designing a controller to satisfy transient specifications such as the maximum allowable overshoot to a given input or the response being required to be within an envelope can be cast as a problem of guaranteeing the impulse response of an appropriate closed loop error transfer function to be non-negative. Stabilizing PID controllers for Linear Time Invariant (LTI) systems can be synthesized as a union of convex polygons in k_i - k_d space for k_p´s lying in a specific range. In this paper, we provide a method to restrict the stabilizing set for LTI systems further by using Widder´s theorem and Markov-Lucaks representation for polynomials that are non-negative on the positive real axis. Widder´s theorem provides necessary and sufficient conditions for the error response to be non-negative and upon an application of Widder´s theorem, we obtain a sequence of polynomials, whose coefficients are polynomial functions of k_p, k_i and k_d to be non-negative. For every polynomial in the sequence and for a specified k_p, using Markov-Lucaks theorem and Minkowski´s projection, we obtain a polynomial inequality in k_i and k_d that must be satisfied by every controller satisfying the desired transient specification. We also provide a method to arbitrarily tighten this set of desired controllers.