A Controller Degree Bound for H^∞-Optimal Control Problems of the Second Kind

This paper is a continuation of our work on H^∞-optimal control problems which may be embedded in the linear fractional configuration of Fig. 1. In two previous articles [19], [20], a controller degree bound was established for problems in which both P₁₂ (s) and P₂₁ (s) are square (problems of the first kind). If the McMillan degree of $P(s)$ is n, it was shown that there exist H^∞-optimal controllers with McMillan degree no greater than $n - 1$. Here we switch our attention to problems of the second kind. That is, we allow P₁₂ (s) to have more rows than columns (with P₂₁ (s) square), or alternatively, we allow P₂₁ (s) to have more columns than rows (with P₁₂ (s) square). Our main result shows that the degree bound derived previously for problems of the first kind carries over to problems of the second kind without change. In addition to the controller degree bound, our analysis suggests a number of modifications which are easily made to currently available computer programs [7], [26]. Test calculations (for problems of the second kind) show that these improvements result in a marked reduction in computation time and also enhance the numerical robustness of the software.