On the structure of H∞ control systems and related extensions

An existence condition of a H^∞ controller which achieves a prescribed norm bound of the closed-loop transfer function is derived in the frequency domain based on a generalization of the notion of J-lossless systems. This condition is regarded as a frequency-domain representation of the well-known existence condition in the state space represented in terms of the two algebraic Riccati equations. A notion of J-orthogonal complement, which is introduced as a generalization of the usual orthogonal complement, play an important role in clarifying the fundamental frequency domain structure of the model-matching problem and simplifying the computation of controllers. The results are extended to the nonstandard case where the direct feedthrough from the input to the error or from the exogenous signal to the output is no longer of full rank. It is shown that the proper controller achieving the prescribed norm bound exists even in this case. In nonstandard cases, the controller order can be smaller than the plant order