Scaling and redundancy for ill-conditioned two input, two output plants

It is well known in the multivariable control literature that ill-conditioned plants may pose significant difficulties in control design. In this paper, we explore two causes of ill-conditioning. The first arises when one input has much smaller gain than does the other. The second occurs when the inputs are almost redundant, in the sense that they affect the plant outputs in approximately the same manner. To quantify these two sources of ill-conditioning, we define input and output redundancy angles, and relate these angles to the size of the plant inverse and condition number. We also investigate the dependence of the redundancy angles upon scaling, and show that certain plants have nearly redundant inputs regardless of the choice of units.