Rapid determination of approximate closed-loop poles of feedback control systems

In a control system, the transient response can be determined accurately only when the closed-loop poles of the system are known. The determination of the closed-loop poles involves in general a solution of a high-degree equation, which is difficult when the degree of the equation exceeds two. While mathematicians have attempted to solve such equations analytically, the control engineers have preferred approximate graphical solutions. The present paper suggests an extension of an already existing graphical method. The technique followed requires an asymptotic gain-frequency plot of the given open-loop transfer function. The plot is divided into three regions, viz., the high-gain region, the low-gain region, and the region near the crossover frequency. The closed-loop real poles are determined for high-gain and low-gain regions from known relationships of closed-loop and open-loop functions. From the knowledge derived therefrom, the complex-conjugate roots (along with real roots, if any) are determined for the crossover frequency region.