Chapter_1_Motivating_Problems_and_Systems.ipynb

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    "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Chapter-1---Motivating-Problems-and-Systems\" data-toc-modified-id=\"Chapter-1---Motivating-Problems-and-Systems-1\">Chapter 1 - Motivating Problems and Systems</a></span><ul class=\"toc-item\"><li><span><a href=\"#1.2-Averaging-algorithms-in-wireless-sensor-networks\" data-toc-modified-id=\"1.2-Averaging-algorithms-in-wireless-sensor-networks-1.1\">1.2 Averaging algorithms in wireless sensor networks</a></span><ul class=\"toc-item\"><li><span><a href=\"#Create-the-Graph-in-networkx-and-define-$A$\" data-toc-modified-id=\"Create-the-Graph-in-networkx-and-define-$A$-1.1.1\">Create the Graph in networkx and define $A$</a></span></li><li><span><a href=\"#Simulate-and-visualize-Process\" data-toc-modified-id=\"Simulate-and-visualize-Process-1.1.2\">Simulate and visualize Process</a></span></li><li><span><a href=\"#Interactive-network-presentation\" data-toc-modified-id=\"Interactive-network-presentation-1.1.3\">Interactive network presentation</a></span></li></ul></li><li><span><a href=\"#Exercise-1.4-Simulating-the-averaging-dynamics\" data-toc-modified-id=\"Exercise-1.4-Simulating-the-averaging-dynamics-1.2\">Exercise 1.4 Simulating the averaging dynamics</a></span></li><li><span><a href=\"#Exercise-1.5-(Section-1.6-ff.)-Bugs-Dynamic\" data-toc-modified-id=\"Exercise-1.5-(Section-1.6-ff.)-Bugs-Dynamic-1.3\">Exercise 1.5 (Section 1.6 ff.) Bugs Dynamic</a></span><ul class=\"toc-item\"><li><span><a href=\"#Cyclic-Pursuit\" data-toc-modified-id=\"Cyclic-Pursuit-1.3.1\">Cyclic Pursuit</a></span></li><li><span><a href=\"#Cyclic-Balancing\" data-toc-modified-id=\"Cyclic-Balancing-1.3.2\">Cyclic Balancing</a></span></li></ul></li><li><span><a href=\"#Exercise-1.6-Robotic-coordination\" data-toc-modified-id=\"Exercise-1.6-Robotic-coordination-1.4\">Exercise 1.6 Robotic coordination</a></span></li></ul></li></ul></div>"
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    "%matplotlib widget\n",
    "\n",
    "# Import packages\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib as mpl\n",
    "import networkx as nx\n",
    "\n",
    "# Import self defined functions\n",
    "import ch1_lib  # Chapter 1 specific library\n",
    "import lib  # General library\n",
    "\n",
    "# For interactive graphs\n",
    "import ipywidgets as widgets\n",
    "\n",
    "# Settings\n",
    "custom_figsize= (6, 4) # Might need to change this value to fit the figures to your screen\n",
    "custom_figsize_square = (5, 5)  "
   ]
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   "source": [
    "# Chapter 1 - Motivating Problems and Systems\n",
    "These Jupyter Notebook scripts contain some examples, visualization and supplements accompanying the book \"Lectures on Network Systems\" by Francesco Bullo http://motion.me.ucsb.edu/book-lns/. These scripts are published with the MIT license. **Make sure to run the first cell above to import all necessary packages and functions and adapt settings in case.** After that, you can jump to each example and follow the description there to execute cell by cell (Tip: Use the shortcut Shift+Enter to execute each cell). Most of the functions are kept in separate files to keep this script neat.\n",
    "\n",
    "## 1.2 Averaging algorithms in wireless sensor networks\n",
    "\n",
    "This example is based on first work of Sean Jaffe. It contains the example from Chapter 1.2 for discrete averaging network dynamics with wireless sensor network, enhanced examples and discussion will occur in Chapter 5, this is an introductory visulization. \n",
    "\n",
    "\n",
    "### Create the Graph in networkx and define $A$\n",
    "\n",
    "Execute next cell to create the example graph and define the matrix $A_{wsn}$.\n",
    "\n",
    "Note: In Chapter 5 we will introduce an elegant algorithm do define the weights as in this example, the equal-neighbour-model."
   ]
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  {
   "cell_type": "code",
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   "source": [
    "# Creating the Network System example as in Chapter 1.2\n",
    "G = nx.DiGraph()\n",
    "G.add_edges_from([(0,1), (1,0), (1,2), (2,1), (2,3), (3,2), (1,3), (3,1)])\n",
    "\n",
    "# Define position of nodes in graph plot\n",
    "pos = {0:[0.2,0.2],1:[.4,.2],2:[.4,.6],3:[.7,.6]}\n",
    "\n",
    "# Define the adjacency matrix A\n",
    "A = np.array([[0.5,0.5, 0., 0.],\n",
    "              [1/4, 1/4, 1/4, 1/4],\n",
    "              [0., 1/3, 1/3, 1/3],\n",
    "              [0., 1/3, 1/3, 1/3]\n",
    "])\n",
    "\n",
    "fig, ax1 = plt.subplots(figsize = custom_figsize)  # Init figure\n",
    "nx.draw_networkx(G, pos, node_size=200, ax = ax1)  # Draw network\n",
    "ax1.margins(0.05) # Zooming out for better visualization\n",
    "\n",
    "print(\"A matrix: \\n\", A)"
   ]
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   "cell_type": "markdown",
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   "source": [
    "### Simulate and visualize Process\n",
    "Choose the initial state vector x (randomized or chosen) and the number to time steps to simulate.\n",
    "\n",
    "We plot our simulation with the initial average as dashed-line to show, that we do not reach average-consensus (more on that in Chapter 5)\n",
    "\n",
    "Note: Discrete-time linear system $x(k+1) = A x(k)$, $x(0)=x_0$ or, equivalently $x(k) = A^k x_0$"
   ]
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   "cell_type": "code",
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    "# Initialize random state vector or define your own one\n",
    "x_0 = np.random.rand(4)\n",
    "#x_0 = np.array([0.2, 0.4, 0.6, 0.8])\n",
    "# Choosing the range of time step\n",
    "t = 15\n",
    "\n",
    "# Initialize figure\n",
    "fig, ax2 = plt.subplots(figsize=custom_figsize)\n",
    "\n",
    "states = lib.simulate_network(A,x_0, t)  # Simulate network and save states for each time step in a t*n np.array\n",
    "lib.plot_node_val_2D(states, x_0, t, ax2)  # Visualize states in a 2D Graph"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Interactive network presentation\n",
    "Below you can find an interactive network simulation and different visualization, where you can use the slide bar to visualize all values at each node for each time step."
   ]
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    "fig, ax3 = plt.subplots(figsize=custom_figsize)\n",
    "fig, v_bound, pos = lib.init_network_sim_plot(G, states, fig, pos=pos)\n",
    "\n",
    "def inter(timestep):\n",
    "    lib.update_network(timestep['new'], G=G, states_m=states, ax=ax3, vbound=v_bound, pos=pos)\n",
    "    #ax3.margins(0.20) # Zooming out for better visualization\n",
    "    return None\n",
    "\n",
    "# Plot initial configuration\n",
    "lib.update_network(0, G=G, states_m=states, ax=ax3, vbound=v_bound, pos=pos)\n",
    "\n",
    "\n",
    "# Widget\n",
    "# If this cell is executed twice we are making sure in the following, that the previous widget instances are all closed\n",
    "try:\n",
    "    [c.close() for c in widget12.children]  # Note: close_all() does also affect plot, thus list compr.\n",
    "except NameError:  # Only want to except not defined variable error\n",
    "    pass\n",
    "\n",
    "widget12 = lib.create_widgets_play_slider(fnc=inter, minv=0, maxv=t-1, step=1, play_speed=1000)\n",
    "\n",
    "display(widget12)"
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    "## Exercise 1.4 Simulating the averaging dynamics\n",
    "\n",
    "In this chapter the results for exercise 1.4 are presented for comparison and discussing the averaging dynamics of these different systems."
   ]
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    "# Define x_0\n",
    "xinitial = np.array([-2., -1., 0., +1., +2.])\n",
    "\n",
    "# Defining the 3 different systems.\n",
    "# Complete graph\n",
    "Acomplete = np.ones((5,5)) / 5\n",
    "\n",
    "# Cycle graph\n",
    "Acycle = np.array([\n",
    "    [1/3, 1/3, 0, 0, 1/3],\n",
    "    [1/3, 1/3, 1/3, 0, 0],\n",
    "    [0, 1/3, 1/3, 1/3, 0],\n",
    "    [0, 0, 1/3, 1/3, 1/3],\n",
    "    [1/3, 0, 0, 1/3,  1/3] ]  )\n",
    "\n",
    "\n",
    "# Star topology. center = node 1\n",
    "Astar = np.array([\n",
    "    [1/5, 1/5, 1/5, 1/5, 1/5],\n",
    "    [1/2, 1/2, 0, 0, 0], \n",
    "    [1/2, 0, 1/2, 0, 0],\n",
    "    [1/2, 0, 0, 1/2, 0],\n",
    "    [1/2, 0, 0, 0, 1/2]   ])\n",
    "\n",
    "# Defining simulation time\n",
    "ts = 10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Complete Graph Simulation**"
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "# Initialize figure\n",
    "fig, ax141 = plt.subplots(figsize=custom_figsize)\n",
    "\n",
    "states_complete = lib.simulate_network(Acomplete,xinitial, ts)  # Simulate network and save states for each time step in a t*n np.array\n",
    "lib.plot_node_val_2D(states_complete, xinitial, ts, ax141)  # Visualize states in a 2D Graph"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Cycle graph Simulation**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "# Initialize figure\n",
    "fig, ax142 = plt.subplots(figsize=custom_figsize)\n",
    "\n",
    "states_cycle = lib.simulate_network(Acycle,xinitial, ts)  # Simulate network and save states for each time step in a t*n np.array\n",
    "lib.plot_node_val_2D(states_cycle, xinitial, ts, ax142)  # Visualize states in a 2D Graph"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Star Graph Simulation**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Initialize figure\n",
    "fig, ax143 = plt.subplots(figsize=custom_figsize)\n",
    "\n",
    "states_star = lib.simulate_network(Astar,xinitial, ts)  # Simulate network and save states for each time step in a t*n np.array\n",
    "lib.plot_node_val_2D(states_star, xinitial, ts, ax143)  # Visualize states in a 2D Graph"
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "# Verfying the results (this will be discussed in more detail in Chapter 5)\n",
    "eigval, eigvec = np.linalg.eig( Astar.transpose() )\n",
    "dom_eigvec = eigvec[0:5,1]/eigvec[0:5,1].sum()\n",
    "print(\"Dominant eigenvector: \", dom_eigvec)\n",
    "print(\"Final values :\", xinitial@dom_eigvec*np.ones(5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1.5 (Section 1.6 ff.) Bugs Dynamic\n",
    "In this example the bugs dynamics as in exercise 1.5 are visualized for the cyclic pursuit or the cyclic balancing case.\n",
    "### Cyclic Pursuit\n",
    "Initialize and simulate the bugs results. If wanted, change some of the parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "radius = 1.0  # Circle radius\n",
    "n_bugs = 6 # Number of bugs\n",
    "n_steps = 50  # Number of simulation steps (states)\n",
    "gain = 0.4  # Gain kappa\n",
    "\n",
    "# Initialize the bugs\n",
    "bugs_init, states_bugs_init = ch1_lib.init_bugs(radius, n_bugs, n_steps)\n",
    "# Simulate the bugs\n",
    "states_bugs = ch1_lib.simulate_bugs_cyclic_pursuit(radius, n_bugs, n_steps, gain, bugs_init, states_bugs_init)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below you find an interactive visulization of the bugs dynamics for cyclic pursuit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": false
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   "source": [
    "fig, ax151 = plt.subplots(figsize=custom_figsize)\n",
    "\n",
    "def interactive_bugs(timestep):\n",
    "    ax151.clear()\n",
    "    ch1_lib.plot_circle_and_bugs(ax151, radius, states_bugs[timestep['new']])  # Take the new value received from the slider dict\n",
    "    return None\n",
    "\n",
    "# Plot initial configuration\n",
    "ch1_lib.plot_circle_and_bugs(ax151, radius, states_bugs[0])\n",
    "\n",
    "# Widget\n",
    "# If this cell is executed twice we are making sure in the following, that the previous widget instances are all closed\n",
    "try:\n",
    "    [c.close() for c in widget151.children]  # Note: close_all() does also affect plot, thus list compr.\n",
    "except NameError:  # Only want to except not defined variable error\n",
    "    pass\n",
    "\n",
    "widget151 = lib.create_widgets_play_slider(fnc=interactive_bugs, minv=0, maxv=n_steps-1, step=1, play_speed=500)\n",
    "\n",
    "display(widget151)"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cyclic Balancing\n",
    "\n",
    "Using some of the previous defined values, showing cycling balancing dynamics."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Initialize the bugs\n",
    "bugs_init2, states_bugs_init2 = ch1_lib.init_bugs(radius, n_bugs, n_steps)\n",
    "# Simulate the bugs\n",
    "states_bugs2 = ch1_lib.simulate_bugs_cyclic_balancing(radius, n_bugs, n_steps, gain, bugs_init2, states_bugs_init2)\n",
    "\n",
    "fig, ax152 = plt.subplots(figsize=custom_figsize)\n",
    "\n",
    "def interactive_bugs2(timestep):\n",
    "    ax152.clear()\n",
    "    ch1_lib.plot_circle_and_bugs(ax152, radius, states_bugs2[timestep['new']])  # Take the new value received from the slider dict\n",
    "    return None\n",
    "\n",
    "# Plot initial configuration\n",
    "ch1_lib.plot_circle_and_bugs(ax152, radius, states_bugs2[0])\n",
    "\n",
    "# Widget\n",
    "# If this cell is executed twice we are making sure in the following, that the previous widget instances are all closed\n",
    "try:\n",
    "    [c.close() for c in widget152.children]  # Note: close_all() does also affect plot, thus list compr.\n",
    "except NameError:  # Only want to except not defined variable error\n",
    "    pass\n",
    "\n",
    "widget152 = lib.create_widgets_play_slider(fnc=interactive_bugs2, minv=0, maxv=n_steps-1, step=1, play_speed=500)\n",
    "\n",
    "display(widget152)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1.6 Robotic coordination\n",
    "\n",
    "Choose the initial position below and run the cell to visualize the rendezvous coordination. As to be proven in exercise 1.6, if the robots start in a square position, they will stay in a square position.\n",
    "\n",
    "First, setting up and running the simulation for the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Set up initial position matrix\n",
    "initial_positions = 2*np.random.random((4,2))-1\n",
    "\n",
    "# If you want to use square matrix\n",
    "#initial_positions = np.array([[.9, .9], [-.9, .9], [-.9, -.9], [.9, -.9]], dtype=float)\n",
    "\n",
    "# Number of timesteps and dt\n",
    "n_dt = 200\n",
    "dt = 0.02\n",
    "\n",
    "current_positions = initial_positions.copy()\n",
    "\n",
    "all_positions = np.zeros((n_dt, 4, 2))\n",
    "all_positions[0] = initial_positions.copy()\n",
    "\n",
    "# Simulate the robotic coordination dynamics and save the values\n",
    "for ii in range(1, n_dt):\n",
    "    neighbor_position = np.array([1, 2, 3, 0])\n",
    "    current_velocities = np.zeros((4, 2))\n",
    "    \n",
    "    current_velocities = current_positions[neighbor_position] - current_positions\n",
    "    current_positions = current_positions  + current_velocities*dt  # Implicit Euler Integration sufficient here\n",
    "    all_positions[ii] = current_positions\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this cell the interactive plot for the path visualization of the robots is called."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax16 = plt.subplots(figsize=custom_figsize_square)\n",
    "line, = ax16.plot([], [])\n",
    "ax16.set_xlim(( -1, 1))\n",
    "ax16.set_ylim((-1, 1))\n",
    "\n",
    "# Setup interactive plot function\n",
    "def interactive_robots(timestep):\n",
    "    # Only reset when set back to zero\n",
    "    if timestep['new'] == 0:\n",
    "        ax16.clear()\n",
    "        ax16.set_xlim(( -1, 1))\n",
    "        ax16.set_ylim((-1, 1))\n",
    "    line, = ax16.plot(all_positions[timestep['new'],0:4,0], all_positions[timestep['new'],0:4,1], 'b.', markersize=2)\n",
    "    return None\n",
    "\n",
    "# Widget\n",
    "# If this cell is executed twice we are making sure in the following, that the previous widget instances are all closed\n",
    "try:\n",
    "    [c.close() for c in widget16.children]  # Note: close_all() does also affect plot, thus list compr.\n",
    "except NameError:  # Only want to except not defined variable error\n",
    "    pass\n",
    "widget16 = lib.create_widgets_play_slider(fnc=interactive_robots, minv=0, maxv=n_dt-1, step=1, play_speed=500)\n",
    "\n",
    "display(widget16)"
   ]
  },
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