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src/year2/distributed-autonomous-systems/sections/_averaging_systems.tex
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\chapter{Averaging systems}
\section{Graphs}
\subsection{Definitions}
\begin{description}
\item[Directed graph (digraph)] \marginnote{Directed graph}
Pair $G = (I, E)$ where $I=\{1, \dots, N\}$ is the set of nodes and $E \subseteq I \times I$ is the set of edges.
\item[Undirected graph] \marginnote{Undirected graph}
Digraph where $\forall i,j: (i, j) \in E \Rightarrow (j, i) \in E$.
\item[Subgraph] \marginnote{Subgraph}
Given a graph $(I, E)$, $(I', E')$ is a subgraph of it if $I' \subseteq I$ and $E' \subset E$.
\begin{description}
\item[Spanning subgraph] Subgraph where $I' = I$.
\end{description}
\item[In-neighbor] \marginnote{In-neighbor}
A node $j \in I$ is an in-neighbor of $i \in I$ if $(j, i) \in E$.
\begin{description}
\item[Set of in-neighbors] \marginnote{Set of in-neighbors}
The set of in-neighbors of $i \in I$ is the set:
\[ \mathcal{N}_i^\text{IN} = \{ j \in I \mid (j, i) \in E \} \]
\item[In-degree] \marginnote{In-degree}
Number of in-neighbors of a node $i \in I$:
\[ \indeg[i] = | \mathcal{N}_i^\text{IN} | \]
\end{description}
\item[Out-neighbor] \marginnote{Out-neighbor}
A node $j \in I$ is an out-neighbor of $i \in I$ if $(i, j) \in E$.
\begin{description}
\item[Set of out-neighbors] \marginnote{Set of in-neighbors}
The set of out-neighbors of $i \in I$ is the set:
\[ \mathcal{N}_i^\text{OUT} = \{ j \in I \mid (i, j) \in E \} \]
\item[Out-degree] \marginnote{Out-degree}
Number of out-neighbors of a node $i \in I$:
\[ \outdeg[i] = | \mathcal{N}_i^\text{OUT} | \]
\end{description}
\item[Balanced digraph] \marginnote{Balanced digraph}
A digraph is balanced if $\forall i \in I: \indeg[i] = \outdeg[i]$.
\item[Periodic graph] \marginnote{Periodic graph}
Graph where there exists a period $k > 1$ that divides the length of any cycle.
\begin{remark}
A graph with self-loops is aperiodic.
\end{remark}
\item[Strongly connected digraph] \marginnote{Strongly connected digraph}
Digraph where each node is reachable from any node.
\item[Connected undirected graph] \marginnote{Connected undirected graph}
Undirected graph where each node is reachable from any node.
\item[Weakly connected digraph] \marginnote{Weakly connected digraph}
Digraph where its undirected version is connected.
\end{description}
\subsection{Weighted digraphs}
\begin{description}
\item[Weighted digraph] \marginnote{Weighted digraph}
Triplet $G=(I, E, \{a_{i, j}\}_{(i,j) \in E})$ where $(I, E)$ is a digraph and $a_{i,j} > 0$ is a weight for the edge $(i,j)$.
\begin{description}
\item[Weighted in-degree] \marginnote{Weighted in-degree}
Sum of the weights of the inward edges:
\[ \indeg[i] = \sum_{j=1}^N a_{j, i} \]
\item[Weighted out-degree] \marginnote{Weighted out-degree}
Sum of the weights of the outward edges:
\[ \outdeg[i] = \sum_{j=1}^N a_{i, j} \]
\end{description}
\item[Weighted adjacency matrix] \marginnote{Weighted adjacency matrix}
Non-negative matrix $\matr{A}$ such that $\matr{A}_{i,j} = a_{i,j}$:
\[
\begin{cases}
\matr{A}_{i,j} > 0 & \text{if $(i, j) \in E$} \\
\matr{A}_{i, j} = 0 & \text{otherwise}
\end{cases}
\]
\item[In/out-degree matrix] \marginnote{In/out-degree matrix}
Matrix where the diagonal contains the in/out-degrees:
\[
\matr{D}^\text{IN} = \begin{bmatrix}
\indeg[1] & 0 & \cdots & 0 \\
0 & \indeg[2] \\
\vdots & & \ddots \\
0 & \cdots & 0 & \indeg[N] \\
\end{bmatrix}
\qquad
\matr{D}^\text{OUT} = \begin{bmatrix}
\outdeg[1] & 0 & \cdots & 0 \\
0 & \outdeg[2] \\
\vdots & & \ddots \\
0 & \cdots & 0 & \outdeg[N] \\
\end{bmatrix}
\]
\begin{remark}
Given a digraph with adjacency matrix $\matr{A}$, its reverse digraph has adjacency matrix $\matr{A}^T$.
\end{remark}
\begin{remark}
It holds that:
\[
\matr{D}^\text{IN} = \text{diag}(\matr{A}^T \vec{1})
\quad
\matr{D}^\text{OUT} = \text{diag}(\matr{A} \vec{1})
\]
where $\vec{1}$ is a vector of ones.
\end{remark}
\begin{remark}
A digraph is balanced iff $\matr{A}^T \vec{1} = \matr{A} \vec{1}$.
\end{remark}
\end{description}
\subsection{Laplacian matrix}
\begin{description}
\item[(Out-degree) Laplacian matrix] \marginnote{Laplacian matrix}
Matrix $\matr{L}$ defined as:
\[ \matr{L} = \matr{D}^\text{OUT} - \matr{A} \]
\begin{remark}
The vector $\vec{1}$ is always an eigenvector of $\matr{L}$ with eigenvalue $0$:
\[ \matr{L}\vec{1} = (\matr{D}^\text{OUT} - \matr{A})\vec{1} = \matr{D}^\text{OUT}\vec{1} - \matr{D}^\text{OUT}\vec{1} = 0 \]
\end{remark}
\item[In-degree Laplacian matrix] \marginnote{In-degree Laplacian matrix}
Matrix $\matr{L}^\text{IN}$ defined as:
\[ \matr{L}^\text{IN} = \matr{D}^\text{IN} - \matr{A}^T \]
\begin{remark}
$\matr{L}^\text{IN}$ is the out-degree Laplacian of the reverse graph.
\end{remark}
\end{description}
\section{Distributed algorithm}
\begin{description}
\item[Distributed algorithm] \marginnote{Distributed algorithm}
Given a network of $N$ agents that communicate according to a (fixed) digraph $G$ (each agent receives messages from its in-neighbors), a distributed algorithm computes:
\[ x_i^{k+1} = \stf_i(x_i^k, \{ x_j^k \}_{j \in \mathcal{N}_i^\text{IN}}) \quad i \in \{ 1, \dots, N \} \]
\[ x_i^{k+1} = \stf_i(x_i^k, \{ x_j^k \}_{j \in \mathcal{N}_i^\text{IN}}) \quad \forall i \in \{ 1, \dots, N \} \]
where $x_i^k$ is the state of agent $i$ at time $k$ and $\stf_i$ is a local state transition function that depends on the current input states.
\begin{remark}
@ -178,7 +24,8 @@
\end{description}
\subsection{Discrete-time averaging algorithm}
\section{Discrete-time averaging algorithm}
\begin{description}
\item[Linear averaging distributed algorithm (in-neighbors)] \marginnote{Linear averaging distributed algorithm (in-neighbors)}
@ -192,7 +39,7 @@
\[ x_i^{k+1} = \sum_{j=1}^N a_{ij} x_j^k \quad i \in \{ 1, \dots, N \} \]
In matrix form, it becomes:
\[ x^{k+1} = \matr{A}^T x^k \]
\[ \vec{x}^{k+1} = \matr{A}^T \vec{x}^k \]
where $\matr{A}$ is the adjacency matrix of $G^\text{comm}$.
\begin{remark}
@ -204,7 +51,7 @@
Given a fixed sensing digraph with self-loops $G^\text{sens} = (I, E)$ (i.e., $(i, j) \in E$ indicates that $j$ sends messages to $i$), the algorithm is defined as:
\[ x_i^{k+1} = \sum_{j \in \mathcal{N}_i^\text{OUT}} a_{ij} x_j^k = \sum_{j=1}^{N} a_{ij} x_j^k \]
In matrix form, it becomes:
\[ x^{k+1} = \matr{A} x^k \]
\[ \vec{x}^{k+1} = \matr{A} \vec{x}^k \]
where $\matr{A}$ is the weighted adjacency matrix of $G^\text{sens}$.
\end{description}
@ -225,38 +72,46 @@
\end{description}
\begin{lemma}
An adjacency matrix $\matr{A}$ is doubly stochastic if it is row stochastic and the graph $G$ associated to it is weight balanced and has positive weights.
\end{lemma}
\begin{lemma} \phantomsection\label{th:strongly_connected_eigenvalues}
Given a digraph $G$ with adjacency matrix $\matr{A}$, if $G$ is strongly connected and aperiodic, and $\matr{A}$ is row stochastic, its eigenvalues are such that:
\begin{itemize}
\item $\lambda = 1$ is a simple eigenvalue (i.e., algebraic multiplicity of 1),
\item All others $\mu$ are $|\mu| < 1$.
\end{itemize}
\indenttbox
\begin{remark}
For the lemma to hold, it is necessary and sufficient that $G$ contains a globally reachable node and the subgraph of globally reachable nodes is aperiodic.
\end{remark}
\end{lemma}
\begin{theorem}[Consensus] \marginnote{Consensus}
\subsection{Consensus}
\begin{theorem}[Discrete-time consensus] \marginnote{Discrete-time consensus}
Consider a discrete-time averaging system with digraph $G$ and weighted adjacency matrix $\matr{A}$. Assume $G$ strongly connected and aperiodic, and $\matr{A}$ row stochastic.
It holds that there exists a left eigenvector $\vec{w} \in \mathbb{R}^N$, $\vec{w} > 0$ such that the consensus converges to:
\[
\lim_{k \rightarrow \infty} x^k
= \vec{1}\frac{\vec{w}^T x^0}{\vec{w}^T\vec{1}}
= \begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix} \frac{\sum_{i=1}^N w_i x_i^0}{\sum_{i=1}^N w_i}
\lim_{k \rightarrow \infty} \vec{x}^k
= \vec{1}\frac{\vec{w}^T \vec{x}^0}{\vec{w}^T\vec{1}}
= \begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix} \frac{\sum_{i=1}^N w_i x_i^0}{\sum_{j=1}^N w_j}
= \begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix} \sum_{i=1}^N \frac{w_i}{\sum_{j=1}^N w_j} x_i^0
\]
where $\tilde{w}_i = \frac{w_i}{\sum_{i=j}^N w_j}$ are all normalized and sum to 1 (i.e., they produce a convex combination).
Moreover, if $\matr{A}$ is doubly stochastic (e.g., $G$ weight balanced with positive weights), then it holds that the consensus is the average:
Moreover, if $\matr{A}$ is doubly stochastic, then it holds that the consensus is the average:
\[
\lim_{k \rightarrow \infty} x^k = \vec{1} \frac{1}{N} \sum_{i=1}^N x_i^0
\lim_{k \rightarrow \infty} \vec{x}^k = \vec{1} \frac{1}{N} \sum_{i=1}^N x_i^0
\]
\begin{proof}[Sketch of proof]
Let $\matr{T} = \begin{bmatrix} \vec{1} & \vec{v}^2 & \cdots & \vec{v}^N \end{bmatrix}$ be a change in coordinates that transforms an adjacency matrix into its Jordan form $\matr{J}$:
\[ \matr{J} = \matr{T}^{-1} \matr{A} \matr{T} \]
As $\lambda=1$ is a simple eigenvalue, it holds that:
As $\lambda=1$ is a simple eigenvalue (\Cref{th:strongly_connected_eigenvalues}), it holds that:
\[
\matr{J} = \begin{bmatrix}
1 & 0 & \cdots & 0 \\
@ -267,18 +122,18 @@
\]
where the eigenvalues of $\matr{J}_2 \in \mathbb{R}^{(N-1) \times (N-1)}$ lie inside the open unit disk.
Let $x^k = \matr{T}\bar{x}^k$, then we have that:
Let $\vec{x}^k = \matr{T}\bar{\vec{x}}^k$, then we have that:
\[
\begin{split}
x^{k+1} &= \matr{A} x^{k} \iff \\
\matr{T} \bar{x}^{k+1} &= \matr{A} (\matr{T} \bar{x}^k) \iff \\
\bar{x}^{k+1} &= \matr{T}^{-1} \matr{A} (\matr{T} \bar{x}^k) = \matr{J}\bar{x}^k
&\vec{x}^{k+1} = \matr{A} \vec{x}^{k} \\
&\iff \matr{T} \bar{\vec{x}}^{k+1} = \matr{A} (\matr{T} \bar{\vec{x}}^k) \\
&\iff \bar{\vec{x}}^{k+1} = \matr{T}^{-1} \matr{A} (\matr{T} \bar{\vec{x}}^k) = \matr{J}\bar{\vec{x}}^k
\end{split}
\]
Therefore:
\[
\begin{gathered}
\lim_{k \rightarrow \infty} \bar{x}^k = \bar{x}_1^0 \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \\
\lim_{k \rightarrow \infty} \bar{\vec{x}}^k = \bar{x}_1^0 \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \\
\bar{x}_1^{k+1} = \bar{x}_1^k \quad \forall k \geq 0 \\
\lim_{k \rightarrow \infty} \bar{x}_i^{k} = 0 \quad \forall i = 2, \dots, N \\
\end{gathered}
@ -299,6 +154,10 @@
\end{example}
\section{Discrete-time averaging algorithm over time-varying graphs}
\subsection{Time-varying digraphs}
\begin{description}
@ -330,10 +189,13 @@
\begin{description}
\item[Linear time-varying (LTV) discrete-time system] \marginnote{Linear time-varying (LTV) discrete-time system}
In matrix form, it can be formulated as:
\[ x^{k+1} = \matr{A}(k) x^k \]
\[ \vec{x}^{k+1} = \matr{A}(k) \vec{x}^k \]
\end{description}
\end{description}
\subsection{Consensus}
\begin{theorem}[Discrete-time consensus over time-varying graphs] \marginnote{Discrete-time consensus over time-varying graphs}
Consider a time-varying discrete-time average system with digraphs $\{G(k)\}_{k \geq 0}$ (all with self-loops) and weighted adjacency matrices $\{\matr{A}(k)\}_{k \geq 0}$. Assume:
\begin{itemize}
@ -343,12 +205,136 @@
It holds that there exists a vector $\vec{w} \in \mathbb{R}^N$, $\vec{w} > 0$ such that the consensus converges to:
\[
\lim_{k \rightarrow \infty} x^k
= \vec{1}\frac{\vec{w}^T x^0}{\vec{w}^T\vec{1}}
\lim_{k \rightarrow \infty} \vec{x}^k
= \vec{1}\frac{\vec{w}^T \vec{x}^0}{\vec{w}^T\vec{1}}
\]
Moreover, if each $\matr{A}(k)$ is doubly stochastic, it holds that the consensus is the average:
\[
\lim_{k \rightarrow \infty} x^k = \vec{1} \frac{1}{N} \sum_{i=1}^N x_i^0
\lim_{k \rightarrow \infty} \vec{x}^k = \vec{1} \frac{1}{N} \sum_{i=1}^N x_i^0
\]
\end{theorem}
\end{theorem}
\section{Continuous-time averaging algorithm}
\subsection{Laplacian dynamics}
\begin{description}
\item[Network of dynamic systems] \marginnote{Network of dynamic systems}
Network described by the ODEs:
\[ \dot{x}_i(t) = u_i(t) \quad \forall i \in \{ 1, \dots, N \} \]
with states $x_i \in \mathbb{R}$, inputs $u_i \in \mathbb{R}$, and communication following a digraph $G$.
\item[Laplacian dynamics system] \marginnote{Laplacian dynamics system}
Consider a network of dynamic systems where $u_i$ is defined as a proportional controller (i.e., only communicating $(i, j)$ have a non-zero weight $a_{ij}$):
\[
\begin{split}
u_i(t)
&= - \sum_{j \in \mathcal{N}_i^\text{OUT}} a_{ij} \Big( x_i(t) - x_j(t) \Big) \\
&= - \sum_{j=1}^{N} a_{ij} \Big( x_i(t) - x_j(t) \Big)
\end{split}
\]
\begin{remark}
With this formulation, consensus can be seen as the problem of minimizing the error defined as the difference between the states of two nodes.
\end{remark}
\begin{remark}
A definition with in-neighbors also exists.
\end{remark}
% \[
% \dot{x}_i(t) =
% -\sum_{j \in \mathcal{N}_i^\text{OUT}} a_{ij} (x_i(t) - x_j(t))
% -\sum_{j=1}^N a_{ij} (x_i(t) - x_j(t))
% \]
% $a_{ij} = 0$ if $(i, j) \notin E$.
\begin{theorem}[Linear time invariant (LTI) continuous-time system] \phantomsection\label{th:lti_continuous} \marginnote{Linear time invariant (LTI) continuous-time system}
With $\vec{x} = \begin{bmatrix} x_1 & \dots & x_N \end{bmatrix}^T$, the system can be written in matrix form as:
\[ \dot{\vec{x}}(t) = - \matr{L} \vec{x}(t) \]
where $\matr{L}$ is the Laplacian associated with the communication digraph $G$.
\begin{proof}
The system is defined as:
\[
\dot{x}_i(t) = - \sum_{j=1}^{N} a_{ij} \Big( x_i(t) - x_j(t) \Big)
\]
By rearranging, we have that:
\[
\begin{split}
\dot{x}_i(t)
&= - \left( \sum_{j=1}^{N} a_{ij} \right) x_i(t) + \sum_{j=1}^{N} a_{ij} x_j(t) \\
&= -\outdeg[i] x_i(t) + (\matr{A}\vec{x}(t))_i
\end{split}
\]
Which in matrix form is:
\[
\begin{split}
\dot{\vec{x}}(t)
&= - \matr{D}^\text{OUT} \vec{x}(t) + \matr{A} \vec{x}(t) \\
&= - (\matr{D}^\text{OUT} - \matr{A}) \vec{x}(t)
\end{split}
\]
By definition, $\matr{L} = \matr{D}^\text{OUT} - \matr{A}$. Therefore, we have that:
\[ \dot{\vec{x}}(t) = - \matr{L} \vec{x}(t) \]
\end{proof}
\end{theorem}
\begin{remark}
By \Cref{th:lti_continuous}, row/column stochasticity is not required for consensus. Instead, the requirement is for the matrix to be Laplacian.
\end{remark}
\end{description}
\subsection{Consensus}
\begin{lemma}
It holds that:
\[
\matr{L}\vec{1}
= \matr{D}^\text{OUT} \vec{1} - \matr{A}\vec{1}
= \begin{bmatrix} \outdeg[1] \\ \vdots \\ \outdeg[i] \end{bmatrix} - \begin{bmatrix} \outdeg[1] \\ \vdots \\ \outdeg[i] \end{bmatrix}
= 0
\]
\end{lemma}
\begin{lemma} \phantomsection\label{th:weighted_laplacian_eigenvalues}
The Laplacian $\matr{L}$ of a weighted digraph has an eigenvalue $\lambda=0$ and all the others have strictly positive real part.
\end{lemma}
\begin{lemma}
Given a weighted digraph $G$ with Laplacian $\matr{L}$, the following are equivalent:
\begin{itemize}
\item $G$ is weight balanced.
\item $\vec{1}$ is a left eigenvector of $\matr{L}$: $\vec{1}^T\matr{L} = 0$ with eigenvalue $0$.
\end{itemize}
\end{lemma}
\begin{lemma} \phantomsection\label{th:connected_simple_eigenvalue}
If a weighted digraph $G$ is strongly connected, then $\lambda = 0$ is a simple eigenvalue.
\end{lemma}
\begin{theorem}[Continuous-time consensus] \marginnote{Continuous-time consensus}
Consider a continuous-time average system with a strongly connected weighted digraph $G$ and Laplacian $\matr{L}$. Assume that the system follows the Laplacian dynamics $\dot{\vec{x}}(t) = - \matr{L}\vec{x}(t)$ for $t \geq 0$.
It holds that there exists a left eigenvector $\vec{w}$ of $\matr{L}$ with eigenvalue $\lambda=0$ such that the consensus converges to:
\[
\lim_{t \rightarrow \infty} \vec{x}(t) = \vec{1} \left( \frac{\vec{w}^T \vec{x}(0)}{\vec{w}^T \vec{1}} \right)
\]
Moreover, if $G$ is weight balanced, then it holds that the consensus is the average:
\[
\lim_{t \rightarrow \infty} \vec{x}(t) = \vec{1} \frac{\sum_{i=1}^N x_i(0)}{N}
\]
% \begin{proof}
% \end{proof}
\end{theorem}
\begin{remark}
The result also holds for unweighted digraphs as $\vec{1}$ is both a left and right eigenvector of $\matr{L}$.
\end{remark}