Classical design in the s -plane

Linear system responses to stimulus have two components: steady-state terms that are directly related to the input and transient terms that are either exponential or oscillatory with an envelope of exponential form. The root locus method is a control system design technique that determines the roots of the characteristic equation (closed-loop poles) when the open-loop gain-constant “K” is increased from zero to infinity. The locus of the roots or closed-loop poles is plotted in the s-plane. The design method requires the closed-loop poles to be plotted in the s-plane because K is varied from zero to infinity, and then a value of K is selected to provide the necessary transient response as required by the performance specification. The loci always start at open-loop poles (denoted by x) and terminate at open-loop zeros (denoted by o) when they exist. The root locus method provides a very powerful tool for control system design. The objective is to shape the loci so that closed-loop poles can be placed in the s-plane at positions that produce a transient response that meets a given performance specification. The Routh-Hurwitz stability criterion is described in the chapter.