A graphical interpretation of SISO H∞ controller approximation

Given a stabilising controller which satisfies a H_∞ performance bound on the closed loop transfer function, sufficient conditions for any other controller to be stabilising and satisfy the same H_∞ performance bound have been presented. Such controllers are said to be (P,γ)-admissible, where P is the model of the plant under consideration and γ is the required level of prespecified H_∞ performance. The conditions are expressed as norm bounds on particular frequency weighted errors where the weights are selected to make a specific transfer function a contraction. Subject to this constraint, approximation is made easier if the weights are as large as possible. Here we show that for scalar controllers the sufficient conditions have a natural frequency by frequency interpretation as either the inside or outside of a circle in the complex plane. We show that given a multivariable (P,γ)-admissible controller maximising the product of the determinants of the weights always leads to a direction where the weights are the best possible. Further, we show that for scalar weights maximising the product of the traces of the weights always leads to a direction where the weights are the best possible ones