A Graphical Interpretation of SISO H-infinity Controller Approximation
P.J. Goddard and K. Glover
Abstract
Given a stabilising controller which satisfies a $\hinf$ performance bound on the closed loop transfer fuction, sufficient conditions for any other controller to be stabilising and satisfy the same $\hinf$ performance bound are presented in [Goddard and Glover 1993,1994].
Such controllers are said to be $\pga$, where $P$ is the model of the plant under consideration and $\gamma$ is the required level of prespecified $\hinf$ 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 of[Goddard and Glover 1993,1994]. 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 $\pga$ controller maximising the product of the determinants of the weights always leads to a {\em 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.
Keywords Controller Approximation, Frequency weighted model reduction, Graphical Methods, H-infinity.