### 6.14. ROOT-LOCUS METHOD FOR NEGATIVE-FEEDBACK SYSTEMS

The root-locus method is a technique for determining the roots of the closed-loop characteristic equation of a system as a function of the static gain. This method is based on the relationship that exists between the roots of the closed-loop transfer function and the poles and zeros of the open-loop transfer function. The root-locus method, which was conceived by Evans [23, 25], has several distinct advantages.

Knowledge of the location of the closed-loop roots permits the very accurate determination of a control system’s relative stability and transient performance. Alternatively, approximate solutions may be obtained, with a considerable reduction of labor, if very accurate solutons are not required. This and the following section present the graphical method of constructing the root locus and of interpreting the results for negative- and postive-feedback systems, respectively. Working digital computer programs for obtaining the root locus are presented in Section 6.17 (using MATLAB) and 6.18. The technique for synthesizing a system utilizing the root-locus method is discussed in Section 7.9. The method is a useful one and should be part of the designer’s bag of tricks.

Let us consider the general feedback control system illustrated in Figure 6.55. In order to find the poles of the closed-loop transfer function, we require that

**Figure 6.55** A nonunity-feedback control system.

where *n* = 0, ±1, ±2, ... Equation (6.131) ...