Analysis and synthesis of feedback compensated third-order control systems via the coefficient plane

Third-order unity numerator control systems arise from the use of feedback compensation in order to meet the system specification. The normalization technique described in the paper permits the complete analysis and synthesis of third-order unity numerator linear or quasi-linear control systems by reference to the coefficient plane. Attention is concentrated on systems with a pair of complex roots plus a real root so orientated in the s plane that all three poles significantly affect the system response. Thus compared with certain existing design techniques, no approximations arise from the neglect of the third system pole when such criteria as bandwidth, maximum percentage overshoot, and velocity constant are determined. Conventional synthesis involves the determination of the system transfer function given the system performance specification. Provided that three independent adjustable parameters are available, three independent performance criteria can be met at one and the same time. It is shown that the coefficient plane yields the appropriate transfer function coefficients directly from a simply constructed coefficient plane contour. For systems with only two adjustable parameters, a performance compromise is necessary. The coefficient plane is ideal for the rapid determination of the best possible compromise between, say, peak amplitude ratio and bandwidth which can be achieved with such a restricted system. An alternative, and even simpler synthesis proposition based on standard forms of transfer function is also illustrated.