DAS small changes

src/year2/distributed-autonomous-systems/sections/_averaging_systems.tex

@@ -284,7 +284,7 @@
         \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.
+            By \Cref{th:lti_continuous}, row/column stochasticity is not required for consensus. Instead, the requirement is for the matrix to be the Laplacian.
         \end{remark}
 \end{description}
 
@@ -314,7 +314,7 @@
 \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.
+    If a weighted digraph $G$ is strongly connected, then $\lambda = 0$ is a simple eigenvalue of $\matr{L}$.
 \end{lemma}
 
 \begin{theorem}[Continuous-time consensus] \marginnote{Continuous-time consensus}

src/year2/distributed-autonomous-systems/sections/_containment.tex

@@ -3,7 +3,7 @@
 
 \begin{description}
     \item[Leader-follower network] \marginnote{Leader-follower network}
-        Consider agents partitioned into $N_f$ followers and $N-N_f$ leaders.
+        Consider $N$ agents partitioned into $N_f$ followers and $N-N_f$ leaders.
         
         The state vector can be partitioned as:
         \[ \x = \begin{bmatrix} \x_f \\ \x_l \end{bmatrix} \]
@@ -15,7 +15,7 @@
         \]
         where $\lap_f$ is the followers' Laplacian, $\lap_l$ the leaders', and $\lap_{fl}$ is the part in common.
 
-        Assume that leaders and followers run the same Laplacian-based distributed control law (i.e., an normal averaging system), the system can be formulated as:
+        Assume that leaders and followers run the same Laplacian-based distributed control law (i.e., a normal averaging system), the system can be formulated as:
         \[
             \begin{bmatrix} \dot{\x}_f(t) \\ \dot{\x}_l(t) \end{bmatrix} =
             - \begin{bmatrix} \lap_f & \lap_{fl} \\ \lap_{fl}^T & \lap_l \end{bmatrix}
@@ -30,7 +30,7 @@
                 \begin{bmatrix}
                     \dot{x}_1(t) \\ \dot{x}_2(t) \\ \dot{x}_3(t) \\ \dot{x}_4(t)
                 \end{bmatrix} = 
-                \begin{bmatrix}
+                - \begin{bmatrix}
                     \begin{tabular}{ccc|c}
                         1   & -1    & 0     & 0     \\
                         -1  & 2     & -1    & 0     \\
@@ -122,7 +122,7 @@
                         \x_f^T \lap_f \x_f &\geq 0 & & \forall \x_f
                     \end{aligned}
                 \]
-            \item The only case when $\x^T \lap \x = 0$ for $\x \neq 0$ is with $\x = \alpha\vec{1}$ for $\alpha \neq 0$. As $\forall \x_f: \bar{\x} \neq \alpha\vec{1}$, it holds that $\forall \x_f: \x_f^T \lap_f \x_f \neq 0$.
+            \item The only case when $\x^T \lap \x = 0$ for $\x \neq 0$ is with $\x = \alpha\vec{1}$ for $\alpha \neq 0$. As $\forall \x_f: \bar{\x} \neq \alpha\vec{1}$, it holds that $\forall \x_f \neq 0: \x_f^T \lap_f \x_f \neq 0$.
         \end{enumerate}
         Therefore, $\lap_f$ is positive definite as $\forall \x_f \neq 0: \x_f^T \lap_f \x_f > 0$.
     \end{proof}
@@ -182,7 +182,7 @@
                 Therefore, we have that:
                 \[
                     \begin{aligned}
-                        \left( \sum_{j=1}^N a_{ij} \right) x_{E,i} &= \sum_{j=1}^N a_{ij} x_{E,j} & & \forall i \in \{ 1, \dots, N_f \} \\
+                        \left( \sum_{k=1}^N a_{ik} \right) x_{E,i} &= \sum_{j=1}^N a_{ij} x_{E,j} & & \forall i \in \{ 1, \dots, N_f \} \\
                         x_{E,i} &= \sum_{j=1}^N \frac{a_{ij}}{\sum_{k=1}^N a_{ik}} x_{E,j} & & \forall i \in \{ 1, \dots, N_f \} \\
                     \end{aligned}
                 \]
@@ -211,7 +211,7 @@
 \end{description}
 
 \begin{theorem}[Containment with non-static leaders non-equilibrium]
-    Naive containment with non-static leaders do not have an equilibrium.
+    Naive containment with non-static leaders does not have an equilibrium.
 
     \begin{proof}
         Ideally, the equilibria for followers' and leader's dynamics are:
@@ -235,7 +235,7 @@
             \end{split}
         \]
         
-        By inspecting the value of the containment error $\vec{e}(t)$ when it reaches equilibrium we have that:
+        By inspecting the value of the containment error $\vec{e}(t)$ when it reaches equilibrium, we have that:
         \[
             \begin{split}
                 0 &= \dot{\vec{e}}(t) \\
@@ -331,7 +331,7 @@
 
 \begin{description}
     \item[Containment with discrete-time] \marginnote{Containment with discrete-time}
-        Containment can be discretized using the forward-Eurler discretization. Its dynamics is defined as:
+        Containment can be discretized using the forward-Euler discretization. Its dynamics is defined as:
         \[
             \begin{aligned}
                 \dot{\x}_i(t) &= - \sum_{j \in \mathcal{N}_i} a_{ij} (x_i(t) - x_j(t)) & & \forall i \in \{1, \dots, N_f\} \\
@@ -373,5 +373,5 @@
         \[
             \dot{\x}(t) = - \lap \otimes \matr{I}_d \x(t)
         \]
-        where $\otimes$ is the Kronecker product.
+        where $\otimes$ is the Kronecker product (i.e., apply the same matrix across each dimension).
 \end{description}

src/year2/distributed-autonomous-systems/sections/_cooperative_robotics.tex

@@ -6,7 +6,7 @@
         Problem where $N$ agents want to optimize their positions $\z_i \in \mathbb{R}^2$ to perform multi-robot surveillance in an environment with:
         \begin{itemize}
             \item A static target to protect $\r_0 \in \mathbb{R}^2$.
-            \item Static intruders/opponents $\r_i \in \mathbb{R}^2$, each assigned to an agent $i$.
+            \item Static intruders/opponents $\r_i \in \mathbb{R}^2$, each assigned to the respective agent $i$.
         \end{itemize}
 
         The average position of the agents define the barycenter:
@@ -16,7 +16,7 @@
         \[
             l_i(\z_i, \sigma(\z)) = 
             \gamma_i \underbrace{\Vert \z_i - \r_i \Vert^2}_{\text{close to opponent}} + 
-            \underbrace{\Vert \sigma(\z) - \r_0 \Vert^2}_{\text{barycenter close to protectee}}
+            \underbrace{\Vert \sigma(\z) - \r_0 \Vert^2}_{\text{barycenter close to target}}
         \]
         Note that the opponent component only depends on local variables while the target component needs global information.
 
@@ -69,8 +69,9 @@
                 &\frac{\partial}{\partial z_i} \left.\left( \sum_{j=1}^{N} l_j(z_j, \sigma(z_1, \dots, z_N)) \right) \right|_{z_j=z_j^k} \\
                 &= 
                     \left.\frac{\partial}{\partial z_i} l_i(z_i, \sigma) \right|_{\substack{z_i = z_i^k,\\\sigma = \sigma(\z^k)}} + 
-                    \left.\left(\sum_{j=1}^{N} \frac{\partial}{\partial \sigma} l_j(z_j, \sigma) \right)\right|_{\substack{z_j = z_j^k,\\\sigma = \sigma(\z^k)}} \cdot
+                    \sum_{j=1}^{N} \left( \left. \left( \frac{\partial}{\partial \sigma} l_j(z_j, \sigma) \right)\right|_{\substack{z_j = z_j^k,\\\sigma = \sigma(\z^k)}} 
-                    \left.\frac{\partial}{\partial z_i} \sigma(z_1, \dots, z_N)\right|_{\substack{z_j=z_j^k}}
+                    \cdot
+                    \left.\frac{\partial}{\partial z_i} \sigma(z_1, \dots, z_N)\right|_{\substack{z_j=z_j^k}} \right)
             \end{split}
         \]
 

src/year2/distributed-autonomous-systems/sections/_formation_control.tex

@@ -16,7 +16,7 @@
         \[
             F_{e,i}(x) = -a_{i,i-1}(x_i-x_{i-1}) - a_{i,i+1}(x_i - x_{i+1})
         \]
-        Equivalently, it is possible to express the elastic force as the negative gradient of the elastic energy:
+        Equivalently, it is possible to express the elastic force as the negative gradient of the elastic potential energy:
         \[
             F_{e,i}(x) = -\frac{\partial}{\partial x_i}\left( \frac{1}{2} a_{i,i-1} \Vert x_i - x_{i-1} \Vert^2 + \frac{1}{2} a_{i,i+1} \Vert x_i - x_{i+1} \Vert^2 \right)
         \]
@@ -86,7 +86,7 @@
             \end{figure}
         \end{remark}
         
-        By adding a damping coefficient (i.e., dispersion of velocity) $c=1$, the overall system dynamics can be defined as:
+        By adding a constant damping coefficient (i.e., dispersion of velocity) $c=1$, the overall system dynamics can be defined as:
         \[
             \begin{split}
                 \dot{x}_i &= v_i \\
@@ -148,7 +148,7 @@
 
 \begin{description}
     \item[Formation control] \marginnote{Formation control}
-        Consider $N$ agents with states $\x_i(t) \in \mathbb{R}^d$ and communicating according to a fixed undirected graph $G$, and a set of distances $d_{ij} = d_{ji}$. The goal is to position each agent respecting the desired distances between them:
+        Consider $N$ agents with states $\x_i(t) \in \mathbb{R}^d$ and communicating according to a fixed undirected graph $G$. The goal is to position each agent respecting the desired distances $d_{ij} = d_{ji}$ between them:
         \[
             \forall (i,j) \in E: \Vert \x_i^\text{form} - \x_j^\text{form} \Vert = d_{ij}
         \]

src/year2/distributed-autonomous-systems/sections/_optimization.tex

@@ -307,8 +307,9 @@
                 \[
                     \begin{aligned}
                         V(\tilde{\z}^{k+1}) - V(\tilde{\z}^k) &= \Vert \tilde{\z}^{k+1} \Vert^2 - \Vert \tilde{\z}^k \Vert^2 \\
-                        &= \cancel{\Vert \tilde{\z}^k \Vert^2} - 2\alpha(\vec{u}^k)^T\tilde{\z}^k + \alpha^2 \Vert \vec{u}^k \Vert^2 - \cancel{\Vert \tilde{\z}^k \Vert^2} &&& \text{\Cref{th:strong_convex_lipschitz_gradient}} \\
+                        &= \Vert \tilde{\z}^{k} - \alpha \vec{u}^{k} \Vert^2 - \Vert \tilde{\z}^k \Vert^2 \\
-                        &\leq -2\alpha\gamma_1 \Vert\tilde{\z}^k\Vert^2 + \alpha(\alpha-2\gamma_2) \Vert\vec{u}^k\Vert^2
+                        &= \cancel{\Vert \tilde{\z}^k \Vert^2} - 2\alpha(\vec{u}^k)^T\tilde{\z}^k + \alpha^2 \Vert \vec{u}^k \Vert^2 - \cancel{\Vert \tilde{\z}^k \Vert^2} \\
+                        &\leq -2\alpha\gamma_1 \Vert\tilde{\z}^k\Vert^2 + \alpha(\alpha-2\gamma_2) \Vert\vec{u}^k\Vert^2 &&& \text{\Cref{th:strong_convex_lipschitz_gradient}} 
                     \end{aligned}
                 \]
 
@@ -344,7 +345,7 @@
     \[
         \min_{\z} \frac{1}{2}\z^T \matr{Q} \z + \vec{r}^T \z
         \qquad
-        \nabla l = \matr{Q} \z^k + \vec{r}
+        \nabla l = \matr{Q} \z + \vec{r}
     \]
     The gradient method can be reduced to an affine linear system:
     \[