Loop shaping to improve the multivariable phase margin

H_∞ controller design is based upon minimizing the infinity-norm of a transfer function related to the system. Different design problems such as sensitivity to input disturbances or sensitivity to model uncertainty can be solved by using the appropriate transfer function(s). As this design procedure is norm based it does not directly account for the influence of phase and, consequently, can lead to a system that has a poor multivariable phase margin. Since pure phase perturbations can arise in real physical systems due to sensor misalignment, it is important that a designer account for such uncertainty. This paper presents an example where H _∞ techniques are used to successfully design a compensator that will improve a plant´s poor performance characteristics while maintaining its ability to tolerate model uncertainty. Despite being robust to norm based uncertainty, the resulting H_∞ compensated system has a poor multivariable phase margin. We then demonstrate a follow-on technique which can be used to design a secondary controller which improves the phase margin without adversely affecting the improvements made by the H_∞ compensator. This secondary controller is designed to reduce the slope of the singular values of the open loop H_∞ compensated plant in the gain crossover region while minimizing the effect on the singular values within the regions defined by the performance specifications. Our technique is based upon observations of the magnitude characteristics that affect the phase margin of single-input single-output systems