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

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 <inline-formula> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$\Sigma $ </tex-math></inline-formula> and a nonempty set <inline-formula> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> is upper bounded by the largest possible IFE of <inline-formula> <tex-math notation="LaTeX">$\Sigma $ </tex-math></inline-formula> and any member of any finite partition of <inline-formula> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula>. Second, we consider two uncertain control systems, <inline-formula> <tex-math notation="LaTeX">$\Sigma _{1}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\Sigma _{2}$ </tex-math></inline-formula>, which are identical except for the transition function, such that the behavior of <inline-formula> <tex-math notation="LaTeX">$\Sigma _{1}$ </tex-math></inline-formula> is included within that of <inline-formula> <tex-math notation="LaTeX">$\Sigma _{2}$ </tex-math></inline-formula>. For a given nonempty subset of the state space, we show that the IFE of <inline-formula> <tex-math notation="LaTeX">$\Sigma _{2}$ </tex-math></inline-formula> is larger or equal to the IFE of <inline-formula> <tex-math notation="LaTeX">$\Sigma _{1}$ </tex-math></inline-formula>. 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.