The necessity and sufficiency of anytime capacity for stabilization of a linear system over a noisy communication link, Part II: vector systems

In part I, we reviewed how Shannon’s classical notion of capacity is not sufficient to characterize a noisy communication channel if the channel is intended to be used as part of a feedback loop to stabilize an unstable scalar linear system. While classical capacity is not enough, a sense of capacity (parametrized by reliability) called “anytime capacity” is both necessary and sufficient for channel evaluation in this context. The rate required is the log of the open-loop system gain and the required reliability comes from the desired sense of stability. Sufficiency is maintained even in cases with noisy observations and without any explicit feedback between the observer and the controller. This established the asymptotic equivalence between scalar stabilization problems and delay-universal communication problems with feedback. Here in part II, the vector-state generalizations are established and it is the magnitudes of the unstable eigenvalues that play an essential role. To deal with such systems, the concept of the anytime rate-region is introduced. This is the region of rates that the channel can support while still meeting potentially different anytime reliability targets for parallel message streams. All the scalar results generalize on an eigenvalue by eigenvalue basis. When there is no explicit feedback of the noisy channel outputs, the intrinsic delay of the unstable system tells us what the feedback delay needs to be while evaluating the anytime-rate-region for the channel. An example involving a binary erasure channel is used to illustrate how differentiated service is required in any separation-based control architecture.