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Transient Response Shaping via Pole Assignment in Specified Regions

The Proportional-Integral (PI) and Proportional Integral Derivative (PID) controller provide perfect steady state tracking and disturbance rejection of steps if the closed loop can be stabilized. The problem of achieving a "good" transient response over the stabilizing set remains a difficult open problem. In this paper we develop a technique to address this problem by assigning closed loop characteristic roots to prescribed regions of the complex plane. The specific regions chosen for study are a rectangle R, circle C and a triangle T. These regions reflect fast settling time (poles to the left of a line), small overshoot (poles to the right of a line) and limited frequency of oscillation (imaginary parts of poles within prescribed bounds). By using D-decomposition techniques and the Edge Theorem we attempt to determine useful subsets of the stabilizing set over which "good" transient response may be obtained. The results are illustrated by examples.