On the invariance feedback entropy of linear perturbed control systems

Abstract

In a previous work, we extended the notion of invariance entropy, also known as topological feedback entropy, of deterministic nonlinear control systems to systems with nondeterministic disturbances and showed that this notion of invariance feedback entropy characterizes the necessary data rate of any coder-controller scheme that communicates via digital, noiseless channel and achieves invariance of a given subset of the state space. In this paper, we derive an intrinsic lower bound of the invariance feedback entropy for linear systems with bounded disturbances in terms of the absolute value of the determinant of the system matrix and a ratio involving the volume of the invariant set as well as the volume of the disturbance set. Additionally, we derive a lower bound of the data rate associated with static, memoryless coder-controllers. If the data rate of a static coder-controller matches the lower bound, we obtain the remarkable property that its data rate is not larger than 1 bit/time unit compared to the best dynamically achievable data rate. The lower bounds are tight for some classes of systems.