Properties of single input, two output feedback systems

We describe properties of multivariable feedback control systems for which the plant has a single input and two outputs. Of particular interest are “algebraic” design trade-offs that occur between different feedback properties at the same frequency. We define concepts of plant and controller direction and alignment, and show that the closed loop response depends upon the alignment between plant and controller. In particular, if alignment is poor, then using high gain will cause output sensitivity to be large and the closed loop system will exhibit large interactions. We also discuss alignment in the case where gain is not necessarily large, and show that alignment is not important beyond the closed loop bandwidth. Also, the closed loop properties are not sensitive to modest amounts of misalignment. Finally, we show there exists a class of controllers that are always poorly aligned with the plant independently of the scaling used for the system outputs