Directionality, coupling, and multivariable loop-shaping

The author presents results useful in applying multivariable loop-shaping ideas to ill-conditioned plants. First, he examines the relation between open- and closed-loop properties at one loop-breaking point. He also investigates how feedback properties at one point are related, both nominally as well as robustly, to those at the other point, and describes the role played by the directional properties of an ill-conditioned plant. By combining these two sets of results, he shows how the open-loop transfer function at one point should be shaped to obtain satisfactory feedback properties at the other point. Finally, he studies design limitations imposed when the plant is ill-conditioned and investigates the sensitivity of feedback properties to violations of the alignment conditions