Compositional Quantification of Invariance Feedback Entropy for Networks of Uncertain Control Systems

Abstract

In the context of uncertain control systems, the notion of invariance feedback entropy (IFE) quantifies the state information required by any controller to render a subset $Q$ of the state space invariant. IFE equivalently also quantifies the smallest bit rate, from the coder to the controller in the feedback loop, above which $Q$ can be made invariant over a digital noiseless channel. In this letter, we consider discrete-time uncertain control systems described by difference inclusions and establish three results for IFE. First, we show that the IFE of a discrete-time uncertain control system $\Sigma $ and a nonempty set $Q$ is upper bounded by the largest possible IFE of $\Sigma $ and any member of any finite partition of $Q$ . Second, we consider two uncertain control systems, $\Sigma _{1}$ and $\Sigma _{2}$ , which are identical except for the transition function, such that the behavior of $\Sigma _{1}$ is included within that of $\Sigma _{2}$ . For a given nonempty subset of the state space, we show that the IFE of $\Sigma _{2}$ is larger or equal to the IFE of $\Sigma _{1}$ . Third, we establish an upper bound for the IFE of a network of uncertain control subsystems in terms of the IFEs of smaller subsystems. Further, via an example, we show that the upper bound is tight for some systems. Finally, to illustrate the effectiveness of the results, we compute an upper bound and a lower bound of the IFE of a network of uncertain, linear, discrete-time subsystems describing the evolution of temperature of 100 rooms in a circular building.