pwsode.ipynb

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  "metadata": {
    "colab": {
      "name": "pwsode.ipynb",
      "provenance": [],
      "collapsed_sections": [],
      "toc_visible": true,
      "authorship_tag": "ABX9TyOyklpWmGQ52l/R7uAuMbjI",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/slashssab/BST/blob/master/pwsode.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "fFPGf9mlRz0d"
      },
      "source": [
        "# Introduction\n",
        "\n",
        "The following paper shows several ways to analyze and vizualize pierce-wise smooth problems given by LTI systems. We will use well known python packages like:\n",
        "- scipy\n",
        "- numpy\n",
        "\n",
        "to provide tools concerning stability and periodic behaviors of concidered systems. Let us bring a following example of the LTI system with switching:\n",
        "$$\\ddot{\\theta}=A\\theta + \\mathcal{T}$$\n",
        "where\n",
        "$$\\mathcal{T} = \\begin{cases}\n",
        "     B \\theta + C \\dot{\\theta},& \\text{if }  |\\theta|\\geq \\theta_0\\\\\n",
        "    0,              & \\text{otherwise}\n",
        "\\end{cases},$$\n",
        "where $\\theta_0$ is fixed border value of $\\theta$ where switching is trigered. Most software tools offering solving ODEs accepts problems defined as the first order differential equations in a following form:\n",
        " $$\\dot{x} = f(x),$$\n",
        " where $x$ is a function of time. Having hnown that fact we have to transform equation:\n",
        " $$\\ddot{\\theta}=A\\theta + \\mathcal{T}$$\n",
        " to the first order representation using Jacobian:\n",
        " $$\n",
        "    \\ddot{\\theta}=A\\theta \\Rightarrow \\dot{x}=L_1 x,$$ \n",
        "\n",
        "where $x$ is a state vector:\n",
        "$$x = \\begin{bmatrix}\n",
        "\\theta \\\\\n",
        "\\dot{\\theta}\n",
        "\\end{bmatrix},$$\n",
        "and $L_1$ is Jacobian:\n",
        "$$L_1=\\begin{bmatrix}\n",
        "0 & 1\\\\\n",
        "A & 0 \n",
        "\\end{bmatrix}.$$\n",
        "The same can be applied for the system with control:\n",
        "$$\n",
        "    \\ddot{\\theta}=A\\theta+B\\theta+C\\dot{\\theta} \\Rightarrow \\dot{x}=L_2 x, \n",
        "$$\n",
        "where\n",
        "$$L_2=\\begin{bmatrix}\n",
        "0 & 1\\\\\n",
        "A+B & C \n",
        "\\end{bmatrix}.$$\n",
        "Problem defined as above will be an example for our consideration."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cAFF6CXHiCBR"
      },
      "source": [
        "# Implementation\n",
        "Before we get started we have to import necessary packages"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "o7kWt2t3idgA"
      },
      "source": [
        "import numpy as np\n",
        "from scipy.integrate import odeint"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "O_gPjn51RzKn"
      },
      "source": [
        "\n",
        "## defining variables\n",
        "We define constants $A$, $B$ and $C$ form our example:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "wKhiE90ZhnS8"
      },
      "source": [
        "A = 0.5\n",
        "B = -0.7\n",
        "C = 0"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "769Ht8gFiqLG"
      },
      "source": [
        "and Jacobians $L_1$ and $L_2$:\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "PRTLf_DRh4rt"
      },
      "source": [
        "L1 = np.array([[0,1],[A, 0]])\n",
        "L2 = np.array([[0,1],[A + B , C]])"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "lhpQzhkajoRP"
      },
      "source": [
        "Having defined those elements we can define our problm with switching by using particular jacobian in dependence of a defined condition. In our case: \n",
        "$$|\\theta| \\ge \\theta_0,$$\n",
        "functionlooks as follows:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Y3Tde7a9k_oy"
      },
      "source": [
        "def func(y, t, Ain, Aout, theta_0): \n",
        "    if abs(y[0])>theta_0:\n",
        "        return np.dot(Aout,y)\n",
        "    else:\n",
        "        return np.dot(Ain,y)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ygwZhQadlP1H"
      },
      "source": [
        "## Initial value problem\n",
        "Now we define the initial problem for our system:\n",
        "- IVP $$\\begin{bmatrix}\n",
        "\\theta \\\\\n",
        "\\dot{\\theta}\n",
        "\\end{bmatrix}= \\begin{bmatrix}\n",
        "1 \\\\\n",
        "1\n",
        "\\end{bmatrix}$$\n",
        "- time of simuation $T=20$, $N=1000$ samples\n",
        "- $\\theta_0 = 1$\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "kYdVZCqrlbAv"
      },
      "source": [
        "ivp = np.array([1,1])\n",
        "theta_0 = 1\n",
        "T = 20\n",
        "N = 1000\n",
        "tspan = np.linspace(0, T, N) "
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3xN3EFiylZ5P"
      },
      "source": [
        "and run the simulation"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "zNxU3xEMi1o-"
      },
      "source": [
        "args = (L1,L2,theta_0)\n",
        "y = odeint(func,ivp,tspan,args=args)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7h53FAMLoSHH"
      },
      "source": [
        "## Results\n",
        "Now we can observe simulation results on a phase portrait "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 302
        },
        "id": "yjiJiJ8WoOgr",
        "outputId": "babe4a22-b90c-4541-e158-55bcbcabc9d2"
      },
      "source": [
        "from matplotlib import pyplot\n",
        "pyplot.title('Phase portrait, T = {0}'.format(T), fontsize=16)\n",
        "pyplot.xlabel(r'$\\theta$', fontsize=14)\n",
        "pyplot.ylabel(r'$\\dot{\\theta}$', rotation = 0, fontsize=14)\n",
        "pyplot.plot(y[:,0], y[:,1])\n",
        "pyplot.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    }
  ]
}