In Section 2.2, we motivate the need of the novel notion of invariance feedback entropy. We define the IFE and establish various elementary properties in Section 2.3. In Section 2.4, we establish the data rate theorem. In Section 2.5, we derive a lower bound on the IFE of uncertain linear control systems. The lower bound is invariant under state space trans-formations and recovers the well-known minimal data rate (sum of the logarithms of the unstable eigenvalues of the system matrix) in the absence of uncertainties. Additionally, we derive a lower bound of the data rate of any static, memoryless coder-controller. We show that the lower bounds are tight for certain classes of systems.
In Section 3.2, we show three additional useful properties of the IFE. In Section 3.3, we show how one can approximate the IFE of a network of uncertain control systems and a set Q using the IFEs of subsystems. In Section 3.4.1, by an example, we demonstrate that this upper bound can be tight. Finally, in Section 3.4.2, we compute an upper bound and a lower bound of the IFE of an uncertain, linear, discrete-time system, that describes the evolution of temperature of 100 rooms in a circular building.
Section 4.2 presents the fundamental definitions for invariance entropy of deterministic systems (IED). In Section 4.3, we describe in detail the implementational steps of our al-gorithm to compute an upper bound of the IED and illustrate them by a two-dimensional linear example. Section 4.4 presents two upper bounds for the IFE. Section 4.5 describes the relationship between the discussed upper bounds for IED and IFE in the case of de-terministic systems. The results of our proposed algorithms for dede-terministic systems are illustrated on a linear and two nonlinear examples in Section 4.6, in which we also present upper bounds of the IFE computed for a two-dimensional uncertain linear sys-tem. Moreover, for the uncertain linear example, we analytically compute a lower bound for comparison. Finally, Section 4.7 contains some comments on the performance of our algorithms.
Chapter 2
Invariance Feedback Entropy
2.1 Introduction
In this chapter we study the notion of invariance feedback entropy (IFE) and establish some of its properties. IFE quantifies the smallest asymptotic average bit rate, from the coder to the controller in the feedback loop, above which a subset Q of the state set can be made invariant over a digital noiseless channel.
2.1.1 Contributions
The contents of this chapter have been published in the journal IEEE Transactions on Automatic Control [75]. It is a joint work with Dr. Matthias Rungger and Prof. Majid Zamani. I established the Theorems 1, 4 and 5, and the Lemma 9. I revised the Example 2, the proof of the Lemmas 4 and 2 and the Remark 2. I also revised the Theorems 7 and 8 to improve the lower bounds through subspace projection and also added the Remark 1.
Rest of the work was done by Dr. Matthias Rungger. Prof. Majid Zamani supervised the work.
We establish a number of elementary properties of the IFE, e.g., we provide conditions that ensure that the IFE is finite and show that we recover the well-known notion of entropy for deterministic control systems. When there is a feedback refinement relation [61] from one system to another one, we show that the entropy of the former is not larger than the latter. This result generalizes the fact that the invariance entropy of deterministic control systems cannot increase under semiconjugation [19, Thm 3.5], [38, Prp. 2.13]. We prove the data rate theorem, which shows that the invariance entropy is a tight lower bound of the data rate of any coder-controller that achieves invariance in the closed loop. To this end, we introduce a history-dependent notion of data rate. We discuss possible alternative data rate definitions and motivate our particular choice by two examples. We analyze uncertain linear control systems and derive a universal lower bound of the IFE. The lower bound depends on the absolute value of the determinant of the system matrix and a ratio involving the volume of the invariant set and the set of uncertainties. The lower bound is invariant under state space transformations and recovers the well-known minimal data
rate [60] in the absence of uncertainties. Furthermore, we derive a lower bound of the data rate of any static, memoryless coder-controller. Both lower bounds are intimately related and for certain cases it is possible to bound the performance loss due to the restriction to static coder-controllers by 1 bit/time unit. We show that the lower bounds are tight for certain classes of systems.
2.1.2 Notations
We denote by N, Z and R the set of natural, integer and real numbers, respectively. We annotate those symbols with subscripts to restrict the sets in the obvious way, e.g. R>0 denotes the positive real numbers. We denote the closed, open and half-open intervals in R with endpoints a and b by [a, b], ]a, b[, [a, b[, and ]a, b], respectively. The corresponding intervals inZare denoted by [a;b], ]a;b[, [a;b[, and ]a;b], i.e., [a;b] = [a, b]∩Zand [a;a[ =∅. For a set A, we use #A∈Z≥0∪ {∞} to denote the number of elements of A, i.e., if A is finite we have #A ∈Z≥0 and #A=∞ otherwise. Given two sets A and B, we say that A is smaller (larger) than B if #A≤#B (#A≥#B) holds. A setJ of subsets of A is said tocover B, whereB ⊆A, ifB is a subset of the union of the elements ofJ. Acover of a setB, is a set of subsets of B that covers B.
We use ∃a∈Ax = a to refer to: there exists a in A such that x = a. In a similar way,
∀a∈Ax = a is used. Given two sets A, B ⊆ Rn, we define the set addition by A+B :=
{x ∈ Rn | ∃a∈A,∃b∈B x = a +b}. For A = {a}, we slightly abuse notation and use a+B ={a}+B. The symbols clA, intA and ℘(A) denote the closure, the interior and the power set of a set A, respectively. We call a set A ⊆ Rn measurable if it is Lebesgue measurable and use µ(A) to denote its measure [72]. We use id to denote an identity map.
For a linear space E, we denote it’s dimension by dim(E).
We follow [63] and use f: A⇒ B to denote a set-valued map from A into B, whereas f:A →B denotes an ordinary map. Iff is set-valued, thenf isstrict if for everya∈Awe havef(a)6=∅. The restriction of f to a subset M ⊆A is denoted byf|M. By convention we set f|∅ :=∅. The composition of f :A ⇒B and g :C ⇒ A, (f ◦g)(x) = f(g(x)) is denoted by f ◦g. We use BA to denote the set of all functions f :A→B. For a relation R⊆A×B and D⊆A, we define R(D) :=∪d∈DR(d).
The concatenation of two functions x : [0;a[ → X and y : [0;b[ → X with a ∈ N and b ∈ N∪ {∞} is denoted by xy which we define by xy(t) := x(t) for t ∈ [0;a[ and xy(t) := y(t−a) for t∈[a, a+b[. We use inf∅=∞, log2∞=∞ and 0· ∞= 0.
For scalars a, b and sets A, B, by a·b and A×B we denote the scalar product and the Cartesian product, respectively. For a set A, a partition is a collection of disjoint nonempty subsets of A that have A as their union. By [a0a1. . . aN−1], ai ∈N, we denote a finite sequence of integers of length N, also called a word. An element of the set NZ is referred to as a bi-infinite word. We use the notation|·|for the absolute value of a complex number. For ann×n matrixB, byλ(B), ρ(B) andBi,j we denote an eigenvalue of B, the spectral radius of B and the entry in the j-th column of the i-th row of B, respectively.