Merge pull request #4 from liuktc/patch-4

src/year2/distributed-autonomous-systems/das.tex

@@ -60,6 +60,7 @@
     \include{./sections/_formation_control.tex}
     \include{./sections/_cooperative_robotics.tex}
     \include{./sections/_safety_controllers.tex}
+    \include{./sections/_feedback_optimization.tex}
     \include{./sections/_neural_networks.tex}
 
 \end{document}

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

@@ -42,7 +42,7 @@
         where:
         \begin{itemize}
             \item $\z = (\z_1, \dots, \z_N)$ with $\z_i \in \mathbb{R}^{n_i}$,
-            \item $l_i: \mathbb{R}^{n_i} \rightarrow \mathbb{R}^d$ is the loss function of the agent $i$,
+            \item $l_i: \mathbb{R}^{n_i} \times \mathbb{R}^d \rightarrow \mathbb{R}$ is the loss function of the agent $i$,
             \item $\sigma(\z)$ is an aggregation function generically defined as $\sigma(\z) = \frac{1}{N} \sum_{i=1}^{N} \phi_i(\z_i)$, for some $\phi_i: \mathbb{R}^{n_i} \rightarrow \mathbb{R}^d$
         \end{itemize}
 

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

@@ -0,0 +1,210 @@
+\chapter{Feedback optimization}
+
+
+% Single system:
+
+% \begin{enumerate}
+%     \item Optimization algorithm: generate a ``reference'' for the control system (algorithmic level).
+%     \item Control system: move the robot and use the output as an input of the optimization algorithm (physical level).
+% \end{enumerate}
+
+\begin{description}
+    \item[Feedback optimization] \marginnote{Feedback optimization}
+        Consider a continuous-time non-linear dynamical system:
+        \[
+            \dot{\x}(t) = f(\x(t), \u(t)) \quad \x(0) = \x_0
+        \]
+        
+        Assume that there exists a steady-state map $h: \mathbb{R}^m \rightarrow \mathbb{R}^n$ (i.e., sort of oracle) that given the input $\bar{\u}$ returns the equilibrium state $\bar{\x}$ associated to it:
+        \[
+            % \begin{gathered}
+                % \forall \bar{\u} \in \mathbb{R}^m: \bar{\u} \mapsto \bar{\x} = h(\bar{\u}) \\
+                f(h(\bar{\u}), \bar{\u}) = 0
+            % \end{gathered}
+        \]
+        % Assume that the system is pre-stabilized around the equilibra. i.e., 
+        % \[
+        %     \begin{split}
+        %         \bar{\u} \in \R^m \\
+        %         \bar{\u} \mapsto \bar{\x} = h(\bar{\u}) \\
+        %         f(\bar{\x}, \bar{\u}) = 0 \\
+        %     \end{split}
+        % \]
+        Moreover, it is assumed that, for any $\bar{\u} \in \mathbb{R}^m$, $\bar{\x} = h(\bar{\u})$ is a globally exponentially stable equilibrium for the system $f(\x(t), \bar{\u})$.
+
+        The goal of feedback optimization is to design a dynamic feedback law $\u(t) = \kappa(\x(t), t)$ such that $\u(t)$ and $\x(t)$ converge to the solution of the problem:
+        \[
+            \begin{gathered}
+                \min_{\z \in \R^n, \u \in \R^m} l(\z) \\
+                \text{subject to } \z = h(\u)
+            \end{gathered}
+        \]
+        where $h$ is not explicitly available and the control law can only access the current state $\x(t)$ ($\nabla h$ can however be used).
+
+    \item[Reduced feedback optimization problem] \marginnote{Reduced feedback optimization problem}
+        By injecting the constraint of the feedback optimization problem into the cost function, the problem becomes unconstrained:
+        \[
+            \min_{\u \in \R^m} l(h(\u))
+        \]
+
+        % Can apply the gradient method:
+        % \[
+        %     \u^{k+1} = \u^k - \alpha \nabla h(\u^k) \nabla l(h(\u^k))
+        % \]
+        and can be solved using the gradient flow:
+        \[
+            \dot{\u}(t) = -\delta \nabla h(\u(t)) \nabla l(h(\u(t)))
+        \]
+        where $\delta$ is a hyperparameter (note that it is not the step size as it is not needed in continuous time).
+
+        \begin{figure}[H]
+            \centering
+            \includegraphics[width=0.35\linewidth]{./img/feedback_reduced_naive.png}
+        \end{figure}
+
+        \begin{remark}
+            By assumption, $h$ is not available.
+        \end{remark}
+\end{description}
+
+
+
+\section{Centralized feedback optimization}
+
+\begin{description}
+    \item[Feedback optimization based on time-scale separation] \marginnote{Feedback optimization based on time-scale separation}
+        Gradient flow of the reduced feedback optimization problem where the steady-state $h(\u(t))$ is approximated with the current state $\x(t)$.
+
+        \begin{figure}[H]
+            \centering
+            \includegraphics[width=0.35\linewidth]{./img/feedback_timescale.png}
+        \end{figure}
+
+        Intuitively, with $\delta > 0$ sufficiently small, we are creating a time-scale separation (i.e., dynamics working on different time-scales) between plant and optimization dynamics:
+        \begin{itemize}
+            \item The plant $\x(t)$ is a fast dynamics that tracks $h(\u(t))$ (i.e., the equilibrium for the current $\u$) so that $\nabla l(\x(t)) \approx \nabla l(h(\u(t)))$ for the optimization dynamics.
+            \item The optimization dynamics is a slow dynamics that changes the input, thus refining the equilibrium that minimizes the loss.
+        \end{itemize}
+        % (i.e., $\x(t)$ is a fast dynamic tracking  (an equilibrium) and the optimization dynamics is slow that changes the input and thus the equilibrium).
+
+        % If $\delta = 0$, $\dot{\u} = 0$ and $\bar{\x} = h(\bar{\u})$ is globally exponentially stable so that:
+        % \[
+        %     \x(t) \rightarrow \bar{\x} = h(\bar{\u})
+        % \]
+
+    \begin{remark}[Time-scale separation theory]
+        Method to analyze a dynamical system by identifying a slow and fast dynamics, and studying them separately. Auxiliary systems are introduced:
+        \begin{descriptionlist}
+            \item[Boundary-layer system] with $\u$ constant to model the fast system. It is exponentially stable by assumption and can be associated with the Lyapunov function $U(\x)$.
+            \item[Reduced system] with the gradient flow without the state $\x$ to model the slow system. It can be shown that it is asymptotically stable with the Lyapunov function $W(\u)$.
+        \end{descriptionlist}
+        The stability of the whole dynamics is studied by using $U+W$ as Lyapunov function and a sufficiently small $\delta$.
+
+        \begin{figure}[H]
+            \centering
+            \includegraphics[width=0.95\linewidth]{./img/feedback_time_scale_theory.png}
+        \end{figure}
+        % \begin{itemize}
+        %     \item Slow dynamics: optimization dynamics,
+        %     \item Fast dynamics: plant dynamics
+        % \end{itemize}
+    
+        % Treat them separately with two systems:
+        % \begin{itemize}
+        %     \item Boundary-layer system. It has $\u$ constant (thus exponentially stable by assumption).
+        %     \item Reduced system. It is the gradient flow without state $\x$. Can be shown to be asymptotically stable.
+        % \end{itemize}
+    
+        % Use Lyapunov functions $U(\x)$ and $W(\u)$, one for each system, to prove stability. Then, combine then $U+W$ with sufficiently small $\delta$ to show overall stability.
+    \end{remark}
+\end{description}
+
+\begin{theorem}[Feedback optimization based on time-scale separation convergence]
+    Consider a closed-loop system and assume that:
+    \begin{itemize}
+        \item $\bar{\x} = h(\bar{\u})$ is a globally exponentially stable equilibrium,
+        \item $l(h(\cdot))$ is radially unbounded and possibly non-convex,
+        \item $\nabla l(h(\cdot))$, $\nabla l$, $h$, and $f$ are Lipschitz continuous.
+    \end{itemize}
+    It holds that, for a sufficiently small $\delta > 0$, the trajectory $(\x(t), \u(t))$ of the system satisfies:
+    \[
+        \lim_{t \rightarrow \infty} \texttt{dist}\left( \begin{bmatrix} \x(t) - h(\u(t)) \\ \u(t) \end{bmatrix}, \begin{bmatrix} 0 \\ \u^* \in \mathcal{U}^* \end{bmatrix} \right) = 0
+    \]
+    where $\texttt{dist}$ is the set distance and $\mathcal{U}^*$ is the set of stationary points of $l(h(\cdot))$.
+\end{theorem}
+
+
+
+\section{Distributed feedback optimization}
+
+\begin{description}
+    \item[Distributed feedback optimization] \marginnote{Distributed feedback optimization} 
+        Network of $N$ agents each with dynamics:
+        \[
+            \dot{\x}_i(t) = f_i(\x_i(t), \u_i(t))
+        \]
+
+        The goal is to design a control law $\u_i(t) = \kappa_i(\{ \x_j(t) \}_{j \in \mathcal{N}_i}, t)$ such that $\u_1(t), \dots, \u_N(t)$ and $\x_1(t), \dots, \x_N(t)$ converge to the solution of the problem:
+        \[
+            \begin{gathered}
+                \min_{\substack{(\z_1, \dots, \z_N)\\(\u_1, \dots, \u_N)}} \sum_{i=1}^{N} l_i(\z_i, \sigma(\z)) \\
+                \text{subject to } \z_i = h_i(\u_i)
+            \end{gathered}
+        \]
+        where $\sigma(\z) = \frac{1}{N} \sum_{i=1}^{N} \phi_i(\z_i)$ is an aggregation function.
+
+    \item[Reduced distributed feedback optimization] \marginnote{Reduced distributed feedback optimization}
+        By injecting the constraint into the cost function, the problem becomes:
+        \[
+            \min_{(\u_1, \dots, \u_N)} \sum_{i=1}^{N} l_i(h_i(\u_i), \sigma(h(\u)))
+        \]
+        and can be solved using the parallel gradient flow:
+        \[
+            \small
+            \dot{\u}_i = - \delta_1 \nabla h_i(\u_i) \left( \nabla_{[h_i(\u_i)]} l_i(h_i(\u_i), \sigma(h(\u))) + \left( \sum_{j=1}^{N} \nabla_{[\sigma(h(\u))]} l_j(h_j(\u_j), \sigma(h(\u))) \right) \frac{1}{N}\nabla \phi_i(h_i(\u_i)) \right)
+        \]
+        where $\delta_1 > 0$ is a hyperparameter.
+
+        \begin{remark}
+            This formulation uses $h$ which is not available and requires global information.
+        \end{remark}
+
+    \item[Aggregative tracking feedback] \marginnote{Aggregative tracking feedback}
+        Use $\x_i$ to approximate $h_i(\u_i)$ and dynamic average consensus for the aggregation function. The dynamics is:
+        \[
+            \begin{split}
+                \dot{\u}_i &= -\delta_1 \nabla h_i(\u_i) \left( \nabla_{[\x_i]} l_i(\x_i, \phi_i(\x_i)+\w_i) + \left( \nabla_{[\phi_i(\x_i)+\w_i]} l_i(\x_i, \phi_i(\x_i)+\w_i) + \v_i \right) \nabla \phi_i(\x_i) \right) \\
+                \delta_2 \dot{\w}_i &= - \sum_{j \in \mathcal{N}_i} a_{ij} (\w_i - \w_j) - \sum_{j \in \mathcal{N}_i} a_{ij} (\phi_i(\x_i) - \phi_i(\x_j)) \\
+                \delta_2 \dot{\v}_i &= - \sum_{j \in \mathcal{N}_i} a_{ij} (\v_i - \v_j) - \sum_{j \in \mathcal{N}_i} a_{ij} (\nabla_{[\phi_i(\x_i)+\w_i]} l_i(\x_i, \phi_i(\x_i)+\w_i) - \nabla_{[\phi_j(\x_j)+\w_j]} l_j(\x_j, \phi_j(\x_j)+\w_j)) \\
+            \end{split}
+        \]
+        where $\delta_1 > 0$ is used to slow down $\dot{\u}_i$ and $\delta_2 > 0$ to speed up $\dot{\w}_i$ and $\dot{\v}_i$.
+
+        \begin{remark}
+            The formulation for dynamic average consensus is for the continuous case with the Laplacian.
+        \end{remark}
+
+        \begin{remark}
+            The overall system works on three time-scales.
+            \begin{figure}[H]
+                \centering
+                \includegraphics[width=0.6\linewidth]{./img/feedback_distributed.png}
+            \end{figure}
+        \end{remark}
+\end{description}
+
+\begin{theorem}[Aggregative tracking feedback convergence]
+    If:
+    \begin{itemize}
+        \item The aggregative tracking feedback algorithm is initialized so that $\sum_{i=1}^{N} \texttt{col}(\w_i(0), \v_i(0)) = 0$,
+        \item $G$ is strongly connected and weight balanced,
+        \item $\bar{\x}_i = h_i(\bar{\u}_i)$ is globally exponentially stable,
+        \item $\sum_{i=1}^{N} l_i(h_i(\cdot), \sigma(h(\cdot)))$ is radially unbounded and possibly non-convex,
+        \item $\nabla_{[\cdot]} l_i$, $\phi_i$, and $h_i$ are Lipschitz continuous.
+    \end{itemize}
+    Then, for small $\delta_1 > 0$ and $\delta_2 > 0$, the trajectory of the system satisfies:
+    \[
+        \lim_{t \rightarrow \infty} \texttt{dist}\left( \begin{bmatrix} \x(t) - h(\u(t)) \\ \u(t) \end{bmatrix}, \begin{bmatrix} 0 \\ \u^* \in \mathcal{U}^* \end{bmatrix} \right) = 0
+    \]
+    where $\mathcal{U}^*$ is the set of stationary points of $\sum_{i=1}^{N} l_i(h_i(\cdot), \sigma(h(\cdot)))$.
+\end{theorem}

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

@@ -306,13 +306,16 @@
 
         By using the unicycle model dynamics, it becomes:
         \[
-            \dot{\x}^\text{int} = \begin{bmatrix}
+            \begin{split}
+                \dot{\x}^\text{int} &= \begin{bmatrix}
                     \cos(\theta) & -\rho\sin(\theta) \\
                     \sin(\theta) & \rho\cos(\theta) \\
                 \end{bmatrix}
                 \begin{bmatrix}
                     v \\ \omega
-            \end{bmatrix}
+                \end{bmatrix} \\
+                \dot{\theta} &= \omega
+            \end{split}
         \]
 
         By formulating $v$ and $\omega$ as a state-feedback control with input $\u^\text{int} \in \mathbb{R}^2$ as:
@@ -326,5 +329,10 @@
                 -\frac{1}{\rho} \sin(\theta) & \frac{1}{\rho} \cos(\theta)
             \end{bmatrix} \u^\text{int}
         \]
-        The result is a single-integrator $\dot{\x}^\text{int} = \u^\text{int}$.
+        The result is a single-integrator $\dot{\x}^\text{int} = g(\x)\u^\text{int}$.
+
+        A choice of $\u$ can be:
+        \[
+            \u^\text{int} = k (\x^\text{int} - \x^\text{dest}) + \dot{\x}^\text{dest}
+        \]
 \end{description}