Synthesis of control systems based on approximation to a third-order system

IN RECENT YEARS the high-speed performance and accuracy demanded of military applications and industrial processes have placed increasing emphasis on the dynamic characteristics of control systems. However, system design is most effective when both steady-state and transient performance specifications can be realized. Ordinarily the designer works from the open-loop to the closed-loop specifications, i.e., trial-and-error modifications are made to the fixed portion of the system until a compensation network is found that eventually leads to an acceptable closed-loop performance. In contrast, another approach, synthesis, proceeds from the closed-loop specifications to an appropriate open-loop system, which upon closure of the feedback path, satisfies the specifications. The methods of Guillemin1 and Aaron2 accomplish this transition, but not without difficulties and complications which arise from the fact that a closed-loop transfer function must be selected that simultaneously: 1. exhibits a pole-zero excess equal to or greater than the excess of the fixed portion of the system, and 2. satisfies a set of closed-loop specifications. Recently, Aseltine3 introduced an inverse-root locus method that graphically determines the open-loop pole locations from a given closed-loop pole-zero configuration. However, none of these methods provides a simple procedure indicating how a closed-loop transfer function can be formulated directly from a detailed set of transient and steady-state specifications. This paper presents a quick and accurate method of synthesizing a linear, continuous, unity-feedback system from a set of specifications that eliminates the foregoing disadvantages. Attention is confined to a type I system that is compensated by an R-C realizable transfer function. To be R-C realizable, the poles of this function must be simple on the negative real axis, the origin and infinity excluded, i.e., its pole-zero excess must be equal to or greater than zero.