Improving the Plant’s Phase Response by Asymmetrical Lead/Lag Compensation of a given Phase and Slope Deficit

A proposal for enhancement of classical lead/lag loop shaping is presented in this work. Although being a rather old method, the Bode-diagram design of phase shifting compensators is still very popular with many users because of its simplicity. In the traditional way, lead or lag filters or PID controllers are used to optimize the phase response of the plant. First the phase deficit is determined at the desired gain crossover frequency $\omega_{c}$ of the open loop, then a lead or lag compensator is designed, doing the necessary phase correction at $\omega_{c}$ with its phase maximum or minimum. In this new approach, the missing phase amount isn’t generated at the maximum or minimum of the compensator’s symmetrical phase curve, but in its rising or falling area. The positive or negative slope of the controller’s phase response at $\omega_{c}$ is hereby used to adjust additionally the slope of the open loop phase to an optimal value. This measure improves the step response of the closed control loop in most cases and often results in an increase of robustness. Interestingly, the design procedure generates in some cases a generalized lead/lag filter, which can be of non-minimum phase type or even instable. The positive effect of the modified design procedure is shown through Monte Carlo Simulations. The design procedure based on this new approach needs strong numerical support, but it can be easily implemented with MATLAB or a similar math package.