Two extensions of topological feedback entropy

Abstract

Topological feedback entropy (TFE) measures the intrinsic rate at which a continuous, fully observed, deterministic control system generates information for controlled set-invariance. In this paper we generalise this notion in two directions; one is to continuous, partially observed systems, and the other is to discontinuous, fully observed systems. In each case, we show that the corresponding generalised TFE coincides with the smallest feedback bit rate that allows a form of controlled invariance to be achieved.

Appendices

Appendix A: Proof for Proposition 2.1

Assume (Ob) and (WCI). Suppose that $s$ and $v_0^{s-1}$ satisfy (Ob). We construct $\alpha , \tau , G$ that satisfy (C) as follows: Set $\tau$ equal to $t$ in (WCI). For $\alpha$, we first observe the following. By (WCI), the continuity of $f_u$, and openness of $\mathrm{int }K$, for any $x_s \in X$ there is an open set $O(x_s)$ in $X$ that contains $x_s$ and is such that $f_{H_{\tau -1}(x_s)} \cdots f_{H_0(x_s)}(O(x_s)) \subset \mathrm{int }K$. By ranging $x_s$ in $f_{v_0^{s-1}}(X)$ we obtain an open cover $\{ O(x_s): x_s \in f_{v_0^{s-1}}(X) \}$ of $f_{v_0^{s-1}}(X)$ in $X$.

We then use the continuous map $h_{v_0^{s-1}}:=f_{v_0^{s-1}}g_{v_0^{s-1}}^{-1}$ to form the collection of open sets in $g_{v_0^{s-1}}(X)$, $\{ h_{v_0^{s-1}}^{-1}(O(x_s)): x_s \in f_{v_0^{s-1}}(X) \}$. Note that the union of sets $h_{v_0^{s-1}}^{-1}(O(x_s))$ is $g_{v_0^{s-1}}(X)$. Since $g_{v_0^{s-1}}(X)$ is equipped with the subspace topology of $Y^{s+1}$, each open set $h_{v_0^{s-1}}^{-1}(O(x_s))$ in $g_{v_0^{s-1}}(X)$ can be expanded to an open set $S^{\prime }(x_s)$ in $Y^{s+1}$ so that $h_{v_0^{s-1}}$ maps points in $S^{\prime }(x_s) \cap g_{v_0^{s-1}}(X)$ into $O(x_s)$. Now $\{S^{\prime }(x_s): x_s \in f_{v_0^{s-1}}(X) \}$ is an open cover of $g_{v_0^{s-1}}(X)$ in $Y^{s+1}$.

The open cover $\alpha$ of $g_{v_0^{s-1}}(X)$ in $Y^{s+1}$ is then chosen to be a finite subcover $\{ L^1, \ldots , L^r \}$ of $\{S^{\prime }(x_s)\}$. The existence of a finite subcover is guaranteed by the compactness of $g_{v_0^{s-1}}(X)$ as the image of the compact set $X$ under the continuous map $g_{v_0^{s-1}}$. Let $x_s^q$ be the point in $f_{v_0^{s-1}}(X)$ such that $L^q=S^{\prime }(x_s^q)$, and let

$$\begin{aligned} G_k(L^q)=H_k \left( x_s^q \right) , \quad 1 \le q \le r, \, 0 \le k \le \tau -1. \end{aligned}$$

By construction we have that if $x_0 \in g_{v_0^{s-1}}^{-1}(L^q)$ then

$$\begin{aligned} f_{v_0^{s-1}G(L^q)} (x_0) = f_{v_0^{s-1}{\{ H_k(x_s^q)\}}_{k=0}^{\tau -1}}(x_0) \in \mathrm{int }K. \end{aligned}$$

This confirms the feasibility of (C) under (Ob) and (WCI).

Appendix B: Proof for Proposition 2.2

The openness of $B_j$ is confirmed by writing it as

$$\begin{aligned} B_j&= g_{v_0^{s-1}}^{-1}(A_0) \cap \Phi _{G(A_0)}^{-1} g_{v_0^{s-1}}^{-1}(A_1) \cap \Phi _{G(A_0)}^{-1} \Phi _{G(A_1)}^{-1} g_{v_0^{s-1}}^{-1}(A_2) \\&\cap \cdots \cap \Phi _{G(A_0)}^{-1} \cdots \Phi _{G(A_{j-2})}^{-1} g_{v_0^{s-1}}^{-1}(A_{j-1}), \end{aligned}$$

where $\Phi _{G(A)}:= f_{v_0^{s-1}G(A)}$ is a continuous map.

Since $g_{v_0^{s-1}}$ is continuous and $\alpha$ is an open cover of  $g_{v_0^{s-1}}(X)$ in $Y^{s+1}$, the collection $\{ g_{v_0^{s-1}}^{-1}(A) : A \in \alpha \}$ is an open cover of $X$. Hence any $x_0 \in X$ must be in some $g_{v_0^{s-1}}^{-1}(A_0)$, $A_0 \in \alpha$. Then constraint (C) implies that the $s+\tau$ inputs $v_0^{s-1}G(A_0)$ forces $x_{s+\tau } \in \mathrm{int }K \subset X$. Repeating this process indefinitely, we see that for any $x_0 \in X$ there is a sequence $A_0, A_1, \ldots$ of sets in $\alpha$ such that $x_{i(s+\tau )} \in g_{v_0^{s-1}}^{-1}(A_i)$ when the input sequences $u_{i(s+\tau )}^{(i+1)(s+\tau )-1}$ are used.

Appendix C: Proof for Lemma 2.1

For each $j, k \in \mathbb N$, the collection $\beta _{j+k}$ consists of all sets of the form

$$\begin{aligned} B_{j+k}\!&= \! \left[ \! g_{v_0^{s-1}}^{-1}(A_0) \!\cap \! \Phi _{G(A_0)}^{-1} g_{v_0^{s-1}}^{-1}(A_1) \cap \cdots \cap \Phi _{G(A_0)}^{-1} \cdots \Phi _{G(A_{j-2})}^{-1} g_{v_0^{s-1}}^{-1}(A_{j-1}) \!\right] \!\cap \! \Phi _{G(A_0)}^{-1}\\&\cdots \Phi _{G(A_{j-1})}^{-1} \Big ( g_{v_0^{s-1}}^{-1}(A_j) \cap \Phi _{G(A_j)}^{-1} g_{v_0^{s-1}}^{-1}(A_{j+1}) \cap \cdots \cap \Phi _{G(A_j)}^{-1}\\&\cdots \Phi _{G(A_{j+k-2})}^{-1} g_{v_0^{s-1}}^{-1}(A_{j+k-1}) \Big ), \end{aligned}$$

where $\Phi _{G(A)}= f_{v_0^{s-1}G(A)}$ with $A_0, \ldots , A_{j+k-1}$ ranging over $\alpha$. Note that the expression inside the square brackets runs over all sets in $\beta _j$, while the expression inside the large parentheses runs over all sets in $\beta _k$. Constrain ${\{ A_i \}}_{i=0}^{j-1}$ to index sets in a minimal subcover $\beta _j^{\prime }$ of $\beta _j$, and ${\{ A_i \}}_{i=j}^{j+k-1}$ to those in a minimal subcover $\beta _k^{\prime }$ of $\beta _k$. Denote the constrained family of sets $B_{j+k}$ thus formed by $\beta _{j+k}^{*}$.

We claim that $\beta _{j+k}^{*}$ is still an open cover for $X$. To see this, observe that any $x_0 \in X$ must lie in a set $B_j^{\prime } \in \beta _{j}^{\prime }$ indexed by some sequence ${\{ A_i \}}_{i=0}^{j-1}$ in $\alpha$. Furthermore, $\Phi _{G(A_{j-1})} \cdots \Phi _{G(A_0)}(x_0) \in \mathrm{int }K \subset X$ and thus lies in some set $B_k^{\prime }$ in the minimal subcover $\beta _k^{\prime }$. Hence $x_0 \in \Phi _{G(A_0)}^{-1} \cdots \Phi _{G(A_{j-1})}^{-1} (B_k^{\prime })$ and so $\beta _{j+k}^{*}$ is still a cover for $X$. As there are $N(\beta _j)$ sets in $\beta _j^{\prime }$, and to each there correspond $N(\beta _k)$ possible sets in $\beta _k^{\prime }$, the number of distinct elements in $\beta _{j+k}^{*}$ is less than or equal to $N(\beta _j) N(\beta _k)$. By the definition of minimal subcovers we have that for any $j, k \in \mathbb N$,

$$\begin{aligned} N(\beta _{j+k}) \le N(\beta _j) N(\beta _k). \end{aligned}$$

Take logarithm with base $2$ on each side of the inequality.