The exact and unique solution for phase-lead and phase-lag compensation

Phase-lead and -lag compensation is one of the most commonly used techniques for designing control systems in the frequency domain, especially when the Bode diagram or root locus is used. In most cases, the graphic-based approximation or trial-and-error approach has been utilized in the design process. This paper presents the exact and unique solution to the design of phase-lead and phase-lag compensation when the desired gains in the magnitude and phase are known at a given frequency. It also gives the concise condition for determining the existence of single-stage lead or lag compensation.