Ill-Conditioned Plants and Integral Relations

We consider a standard robust performance problem for two input, two output diagonal systems. It has been conjectured that this problem is inherently difficult for plants that are ill-conditioned. We describe this difficulty qualitatively using the integral relationship between gain and phase. We then use the classical sensitivity integral to derive an upper bound upon the average value of the plant condition number at low frequencies. This bound must be satisfied if robust performance is to be achieved.