Fundamental limitations of performance in the presence of finite capacity feedback

This paper addresses a fundamental limitation of performance for feedback systems, in the presence of a communication channel. The feedback loop comprises a discrete-time, linear and time-invariant plant, a channel, an encoder and a decoder which may also embody a controller. Measurements of the plant's output must be encoded for transmission over the channel. Information, at the other end of the channel, is decoded and used to generate a control signal, which is additively disturbed by a Gaussian and stationary stochastic process. We derive an inequality of the form L/spl I.bar/ /spl ges/ /spl Sigma/ max{0, log(|/spl lambda//sub i/(A)|)} - C/sub channel/, where L/spl I.bar/ is a measure of disturbance rejection, A is the open loop dynamic matrix and C/sub channel/ is the Shannon capacity of the channel. Our measure L/spl I.bar/ is non-positive and smaller L/spl I.bar/ indicates better rejection (attenuation), while L/spl I.bar/ = 0 signifies no rejection. Previous results show that C/sub channel/ > /spl Sigma/max{0, log(|/spl lambda//sub i/(A)|)} is a necessary condition for stability and now we show that the extra rate C/sub channel/ - /spl Sigma/max{0, log(|/spl lambda//sub i/(A)|)} determines a fundamental limitation for disturbance rejection. Additionally, we prove that, under stationarity assumptions, L/spl I.bar/ admits a log-sensitivity integral representation. We contrast our condition with Rode's integral formula and the water-bed effect. The new inequality shows explicitly how the capacity of the channel limits closed loop performance.