We firstly consider the approximation of linear time invariant (LTI) controllers which have been designed for LTI models and satisfy a prespecified level of closed loop H-infinity performance. We derive sufficient conditions for any other controller to be stabilising and satisfy the same level of H-infinity performance. Such controllers are said to be (P,gamma)-admissible, where P is the model of the plant under consideration and gamma is the required level of prespecified H-infinity 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. The design of reduced order (P,gamma)-admissible controllers is then formulated as a frequency weighted model reduction problem. We demonstrate that when used in conjunction with a combined model reduction/convex optimisation scheme, the proposed design procedures are effective in substantially reducing controller complexity.
It is advantageous for the required weights to be large in some sense. Hence we characterise solutions which maximise either the trace or the determinant of the weights. For scalar controllers a graphical interpretation of the derived sufficient conditions is presented. We show that for scalar weights the procedure for maximising the trace of the weights, and the procedure for maximising the determinant of the weights, both result in the best possible weights. Hence in the scalar case the two procedures are equivalent.
We then consider the approximation of linear parameter varying (LPV) controllers which have been designed for LPV models. Two well known LTI approximation methods -- balanced truncation and optimal Hankel norm approximation -- are extended to the LPV framework. Given a high order quadratically stabilising controller we then derive a bound on the degradation in quadratic performance when the high order controller is replaced with a low order approximant.
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