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_ ≥ Σ max{0, log(|λ_i(A)|)} - C_channel, where L_ is a measure of disturbance rejection, A is the open loop dynamic matrix and C_channel is the Shannon capacity of the channel. Our measure L_ is non-positive and smaller L_ indicates better rejection (attenuation), while L_ = 0 signifies no rejection. Previous results show that C_channel > Σmax{0, log(|λ_i(A)|)} is a necessary condition for stability and now we show that the extra rate C_channel - Σmax{0, log(|λ_i(A)|)} determines a fundamental limitation for disturbance rejection. Additionally, we prove that, under stationarity assumptions, L_ 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.