PID controller design with guaranteed gain and phase margins

This paper considers the problem of designing PID controllers that achieve the guaranteed gain and phase margin specifications for a given but arbitrary single-input single-output plant. Using definitions of gain and phase margins, the given specifications are translated into simultaneous stabilization of two families of polynomials. Thereafter, an extension of the results of PID stabilization to the case of complex polynomials is shown to be useful in providing a solution to this simultaneous stabilization problem. A computational characterization of all admissible PID controllers is given. This characterization for all admissible PID controllers involves the solution of a linear programming problem. An example is presented to illustrate use of the design procedure in detail.