A Controller Degree Bound for $\mathcal{H}^\infty $-Optimal Control Problems of the Second Kind
David J. N. Limebeer, George D. Halikias · SIAM Journal on Control and Optimization · 1988
This paper is a continuation of our work on $\mathcal{H}^\infty $-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 $P_{12} (s)$ and $P_{21} (s)$ are square (problems of the first kind). If the McMillan degree of $P(s)$ is n, it was shown that there exist $\mathcal{H}^\infty $-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 $P_{12} (s)$ to have more rows than columns (with $P_{21} (s)$ square), or alternatively, we allow $P_{21} (s)$ to have more columns than rows (with $P_{12} (s)$ 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.