Lead and lag compensators with complex poles and zeros design formulas for modeling and loop shaping

Complex lead and lag compensators are new additions to the repertoire of compensator structures for loop shaping. This article facilitates the use of these compensators by providing explicit formulas that relate the parameters of the compensators to features of their frequency responses. Two examples illustrate the utility of these compensators for system modeling and controller design. While the examples involve low-order plants, the principles of employing the complex lead and lag compensators remain the same for higher-order systems. We plan to use these compensators as weighting functions with automated robust design tools. A weighting function is a transfer function whose frequency response magnitude is used to bound closed-loop response or modeling uncertainty. The complex lead and lag compensators provide new degree of freedom for selecting weighting functions. In particular, the steep magnitude slope in the transition region of these compensators more closely approximates an ideal step function than weighting functions appearing in the literature (Packard et al.)