notes.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# ODE's\n",
    "Differential equations solutions are a function or a set of functions\n",
    "\n",
    "## linear\n",
    "\n",
    "it needs to have this form\n",
    "\n",
    "$y' + p(t)y = g(t)$\n",
    "\n",
    "The way you solve it is by using the integrating factor\n",
    "\n",
    "$u(t) = e^{\\int p(t)dt}$\n",
    "\n",
    "where $u(t)$ is the integrating factor\n",
    "\n",
    "## seperable\n",
    "\n",
    "has the form\n",
    "\n",
    "$N(y) \\frac{dy}{dx} = M(x)$\n",
    "\n",
    "or \n",
    "\n",
    "$N(y) \\cdot dy = M(x) \\cdot dx$\n",
    "\n",
    "to solve these you simply integrate both sides\n",
    "\n",
    "$\\int N(y) \\cdot dy =\\int M(x) \\cdot dx$\n",
    "\n",
    "## homogenous\n",
    "\n",
    "the strategy to solve these is \n",
    "\n",
    "$v = \\frac{y}{x}$\n",
    "\n",
    "## exact\n",
    "\n",
    "takes the form\n",
    "\n",
    "$M(x,y) + N(x,y)y' = 0$\n",
    "\n",
    "which is\n",
    "\n",
    "$\\frac{\\partial M}{\\partial y} = \\frac{\\partial N}{\\partial y}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Im going to solve a differenctial equation using pyhton\n",
    "\n",
    "the equations is \n",
    "\n",
    "$\\tau \\frac{dy}{dt} = -y + ku$\n",
    "\n",
    "where $\\tau$ is the time constant and k is the gain u is the unit step \n",
    "\n",
    "I will use $\\tau = 5$ and $k=2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.integrate import odeint\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def first_order(y, t, tau, K, u):\n",
    "    dydt = (-y + K*u(t))/tau\n",
    "    return dydt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I need to define the unit step function, it returns 1 if time is greater than 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "u = lambda t: 0 if t<0 else 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "next I need to define the other constants"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "tau = 5\n",
    "K = 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "lastly i need to define the time. this will be from 0 to 10 in increments of 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = np.linspace(0,10,100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = odeint(first_order,0,t,args=(tau,K,u))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "odeint is a library that solves first order differential equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'amplitude')"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t,y)\n",
    "plt.xlabel(\"seconds\")\n",
    "plt.ylabel(\"amplitude\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "we can even change the step function. lets say it starts at 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "u = lambda t: 0 if t < 3 else 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = odeint(first_order,0,t,args=(tau,K,u))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'amplitude')"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t,y)\n",
    "plt.xlabel(\"seconds\")\n",
    "plt.ylabel(\"amplitude\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# systems of differential equations\n",
    "\n",
    "This led me to thinking about systems of differential equations. that takes the form\n",
    "\n",
    "$$x' = Ax$$\n",
    "\n",
    "for example\n",
    "\n",
    "$$\\begin{pmatrix} x_1' \\\\ x_2' \\end{pmatrix} = \\begin{pmatrix} 1 & 2 \\\\ 2 & 4 \\end{pmatrix} \\begin{pmatrix} x_1 \\\\ x_2 \\end{pmatrix}$$\n",
    "\n",
    "This is the first time I've done something like this\n",
    "\n",
    "## determine the eigenvalues\n",
    "\n",
    "first determine the eigenvalues of A via $|A - \\lambda I| = 0$\n",
    "\n",
    "we need to find the values of lambda where the determinant equals 0\n",
    "\n",
    "$$\\begin{vmatrix}\n",
    "1-\\lambda & 2 \\\\\n",
    "2 & 4-\\lambda\n",
    "\\end{vmatrix} = 0 $$\n",
    "\n",
    "$$ (1-\\lambda) (4 -\\lambda) - 2^2 = 0$$\n",
    "\n",
    "$$\\lambda (\\lambda -5) $$\n",
    "\n",
    "$\\lambda = 0$ or $\\lambda = 5$\n",
    "\n",
    "## determine the eigenvectors\n",
    "\n",
    "$$(A - \\lambda I)v=0 $$\n",
    "solve for v\n",
    "\n",
    "### case $\\lambda = 0$\n",
    "\n",
    "$$\\begin{pmatrix} 1 & 2 \\\\ 2 & 4 \\end{pmatrix} \\begin{pmatrix} v_1 \\\\ v_2 \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix}$$\n",
    "\n",
    "\n",
    "$$ v_1 + 2v_2 = 0$$ \n",
    "\n",
    "\n",
    "I'll choose $v_2 = 1$ and $v_1 = -2$\n",
    "\n",
    "so the eigenvector for $\\lambda =0$ will be $\\begin{pmatrix} -2 \\\\ 1 \\end{pmatrix}$ of course you can have any multiple of the eigenvector which will also be an eigenvector\n",
    "\n",
    "### case $\\lambda = 5$\n",
    "\n",
    "$$\\begin{pmatrix} -4 & 2 \\\\ 2 & -1 \\end{pmatrix} \\begin{pmatrix} v_1 \\\\ v_2 \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix}$$\n",
    "\n",
    "so $2v_1 = v_2$\n",
    "\n",
    "the eigenvector in this case is $\\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix}$\n",
    "\n",
    "## general solution\n",
    "\n",
    "for distinct real eigenvalues $\\lambda_1, \\lambda_2, ... \\lambda_n$ and eigenvectors $\\eta_1, \\eta_2, ... \\eta_n$ the general solution takes the form\n",
    "\n",
    "$$ x = c_1 e^{\\lambda_1 t} \\eta_1 + c_2 e^{\\lambda_2 t} \\eta_2 + ... + c_n e^{\\lambda_n t} \\eta_n$$\n",
    "\n",
    "plugging in the known values\n",
    "\n",
    "$$x(t) = c_1 \\begin{pmatrix} -2 \\\\ 1 \\end{pmatrix} e^{0t} + c_2 \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix} e^{5 t}$$\n",
    "\n",
    "which finally is \n",
    "\n",
    "$$x(t) = c_1 \\begin{pmatrix} -2 \\\\ 1 \\end{pmatrix} + c_2 \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix} e^{5 t} $$\n",
    "\n",
    "## linear independence \n",
    "\n",
    "to check if they are linearly independent you take the two sides of the equation and find their determinant if its non-zero then they are linearly independent\n",
    "\n",
    "$$\\begin{vmatrix}\n",
    "-2 & e^{5t} \\\\\n",
    "1 & 2e^{5t}\n",
    "\\end{vmatrix} = -4e^{5t} - e^{5t} = -5e^{5t}$$\n",
    "which is not zero"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# using Laplace transform to solve Differential equations\n",
    "\n",
    "this is just a quick review on how to solve differential equations using Laplace\n",
    "\n",
    "## problem\n",
    "\n",
    "$$y'' + 5y' + 6y = 0 ~~~~~\\text{where:}~~ y(0) =2 ~~~ y'(0) = 3$$\n",
    "\n",
    "first take the Laplace transform of both sides\n",
    "\n",
    "$$\\mathcal{L}\\{y''\\} + 5\\mathcal{L}\\{y'\\} + 6\\mathcal{L}\\{y\\} = \\mathcal{L}\\{0\\}$$\n",
    "\n",
    "$$ \\mathcal{L}\\{y^{n}\\} = s^{n} \\mathcal{L}\\{y\\} - s^{n-1}y(0) - ... - sy^{n-2}(0) - y^{n-1}(0)$$\n",
    "\n",
    "$$ s^2 \\mathcal{L}\\{y\\}-2s-3 + 5s\\mathcal{L}\\{y\\} - 2 + 6 \\mathcal{L}\\{y\\} = 0$$\n",
    "\n",
    "factor the Laplace\n",
    "\n",
    "$$ \\mathcal{L}\\{y\\} (s^2 + 5s + 6)- 2s -13 = 0$$\n",
    "\n",
    "get $\\mathcal{L}\\{y\\}$ by itself\n",
    "\n",
    "$$ \\mathcal{L}\\{y\\} = \\frac{2s + 13}{s^2 + 5s +6}$$\n",
    "\n",
    "inverse Laplace both sides\n",
    "\n",
    "$$\\mathcal{L^{-1}}\\{\\mathcal{L}\\{y\\}\\} = \\mathcal{L^{-1}}\\left\\{\\frac{2s + 13}{s^2 + 5s +6}\\right\\}$$\n",
    "\n",
    "$$y = e^{-3t}(9e^t -7)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Non homogeneous equations\n",
    "\n",
    "$$ y'' + y = \\text{sin} 2t ~~~~~\\text{where}~~ y(0) = 2~~~y'(0) = 1$$\n",
    "\n",
    "Laplace\n",
    "$$s^2 Y(s) - 2s - 1 + Y(s) = \\frac{2}{s^2 + 4}$$\n",
    "\n",
    "$$Y(s)(s^2 + 1) - 2s -1 = \\frac{2}{s^2 + 4}$$\n",
    "\n",
    "$$Y(s) = \\frac{\\frac{2}{s^2 + 4} + 2s + 1}{s^2 + 1}$$\n",
    "\n",
    "inverse Laplace\n",
    "\n",
    "$$ y = \\frac{1}{3}(5 \\text{sin}(t) - \\text{sin}(2t) + 6 \\text{cos}(t))$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## step function\n",
    "\n",
    "solving a differential equation with a step function using Laplace\n",
    "\n",
    "$$ y'' + 4y = \\text{sin}(t) - \\mu_{2\\pi} (t) - \\text{sin} (t-2\\pi) ~~~~~ \\text{where} ~~ y(0) = 0 ~~ y'(0) = 0$$\n",
    "\n",
    "Laplace transform\n",
    "\n",
    "$$s^2 \\mathcal{L}\\{y\\} - sy(0) - y'(0) + 4\\mathcal{L}\\{y\\} = \\frac{1}{s^2 + 1} - e^{-2\\pi s} \\frac{1}{s^2 + 1}$$ \n",
    "\n",
    "$$s^2 \\mathcal{L}\\{y\\}+ 4\\mathcal{L}\\{y\\} = \\frac{1}{s^2 + 1} - e^{-2\\pi s} \\frac{1}{s^2 + 1}$$ \n",
    "\n",
    "$$ \\mathcal{L}\\{y\\} ( s^2 + 4) = \\frac{1}{s^2 + 1} - \\frac{e^{-2\\pi s}}{s^2 + 1} $$\n",
    "\n",
    "$$\\mathcal{L}\\{y\\} = \\frac{\\frac{1}{s^2 + 1} - \\frac{e^{-2\\pi s}}{s^2 + 1}}{s^2 + 4}$$\n",
    "\n",
    "$$ y = \\frac{-2}{3} ( \\theta (t - 2 \\pi) - 1) \\text{sin}^2 \\bigl ( \\frac{t}{2} \\bigr ) \\text{sin} (t) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# water tank notes\n",
    "![alt text](tank.png)\n",
    "\n",
    "first how do we find y(t)\n",
    "\n",
    "I know that the pressure at the bottom of a tank can be simulated as $\\rho g h$ where $\\rho$ is the density g is gravity and h is the depth of the water\n",
    "\n",
    "so y(t) will depend on $\\rho ghR$ and R will be the resistance and that can the cross sectional area or the material of the outlet. it can be lots of things. \n",
    "\n",
    "next i want to know what the rate of change in the volume stored\n",
    "\n",
    "$$\\frac{dV}{dt} = A \\frac{dh}{dt} = f_{in} - f_{out}$$\n",
    "\n",
    "i know what flow out is\n",
    "\n",
    "$$A \\frac{dh}{dt} + R \\rho g h = f_{in}$$\n",
    "\n",
    "which can be rearranged as \n",
    "\n",
    "$$\\frac{A}{R \\rho g} \\frac{dh}{dt} + h = \\frac{1}{R \\rho g} f_{in}$$\n",
    "\n",
    "this is time constant form. therefore\n",
    "\n",
    "$$\\tau \\frac{dh}{dt} + h = K f_{in} ~~~~~ \\text{where}~~ \\tau = \\frac{A}{R \\rho g} ~~~ K = \\frac{1}{R \\rho g}$$\n",
    "\n",
    "notice that if the cross sectional area is increased the time constant increases, if the resistance increases then $\\tau$ decreases"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# state spaces\n",
    "\n",
    "\n",
    "the state based system\n",
    "\n",
    "$$\\dot{\\pmb{q}}(t) =  \\pmb{A}\\pmb{q}(t) + \\pmb{B}u(t)$$\n",
    "$$ y(t) = \\pmb{C} \\pmb{q}(t) + Du(t)$$\n",
    "\n",
    "now i can take the laplace transform of this\n",
    "\n",
    "$$ s\\pmb{Q}(s) = \\pmb{A}\\pmb{Q}(s) + \\pmb{B}U(s)$$ \n",
    "\n",
    "$$ Y(s) = \\pmb{C} \\pmb{Q}(s) + DU(s)$$\n",
    "\n",
    "I want the transfer function $H(s) = \\frac{Y(s)}{U(s)}$ so now i need to remove Q(s) form the output so frist solve for q\n",
    "\n",
    "$$ s\\pmb{Q}(s) = \\pmb{A}\\pmb{Q}(s) + \\pmb{B}U(s)$$ \n",
    "\n",
    "$$ s\\pmb{Q}(s) -\\pmb{A}\\pmb{Q}(s) =  \\pmb{B}U(s)$$ u8\n",
    "\n",
    "$$ (s\\pmb{I}- \\pmb{A} ) \\pmb{Q}(s) = \\pmb{B}U(s)$$\n",
    "\n",
    "\n",
    "$$ \\pmb{Q}(s) = (s\\pmb{I} - \\pmb{A})^{-1} \\pmb{B}U(s)$$\n",
    "\n",
    "let $\\phi(s) =  (s\\pmb{I} - \\pmb{A})^{-1}$ \n",
    "\n",
    "leaving \n",
    "$$\\pmb{Q}(s) = \\pmb{\\phi}(s) \\pmb{b}U(s) $$\n",
    "\n",
    "now that I have $\\pmb{Q}(s)$ i can put it into y(s)\n",
    "\n",
    "original problem\n",
    "$$ Y(s) = \\pmb{C} \\pmb{Q}(s) + DU(s)$$\n",
    "replaced \n",
    "$$Y(s) = \\pmb{C}\\pmb{\\phi}(s) \\pmb{B}U(s) + DU(s)$$\n",
    "$$Y(s) = U(s)(\\pmb{C}\\pmb{\\phi}(s) \\pmb{B} + D)$$\n",
    "\n",
    "now simply divide U(s) to get the transfer function\n",
    "\n",
    "$$H(s) = \\frac{Y(s)} {U(s)}  =\\pmb{C}\\pmb{\\phi}(s) \\pmb{B} + D $$\n",
    "\n",
    "## example from the midterm\n",
    "\n",
    "\n",
    "$$ \\dot{x} = \\begin{pmatrix}\\dot{x_1} \\\\ \\dot{x_2} \\end{pmatrix} = \\begin{pmatrix} 0 & 1 \\\\ -2 & -3 \\end{pmatrix}  \\begin{pmatrix} x_1 \\\\ x_2\\end{pmatrix} + \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} u $$\n",
    "$$ y = \\begin{pmatrix} 1 & \\alpha \\end{pmatrix} \\begin{pmatrix} x_1 \\\\ x_2\\end{pmatrix}$$\n",
    "\n",
    "$$s \\bar{I} - \\bar{A} = \\begin{pmatrix} s & 0 \\\\ 0 & s \\end{pmatrix} - \\begin{pmatrix} 0 & 1 \\\\ -2 & -3\\end{pmatrix} = \\begin{pmatrix} s & -1 \\\\ 2 & s + 3 \\end{pmatrix} \\Leftarrow sa$$\n",
    "\n",
    "$$\\phi = \\frac{1}{\\text{det} ~ sa} \\begin{pmatrix} d & -b \\\\ -c & a\\end{pmatrix} $$ \n",
    "\n",
    "$$ \\text{det}~ sa = s(s+3) - 2 (-1) = s^2 + 3s + 2$$\n",
    "\n",
    "$$ \\phi = \\frac{\\begin{pmatrix} s+3 & 1 \\\\ -2 & s \\end{pmatrix}}{s^2 + 3s + 2}$$\n",
    "\n",
    "$$ H(s) = \\begin{pmatrix} 1 & \\alpha \\end{pmatrix} \\frac{\\begin{pmatrix} s+3 & 1 \\\\ -2 & s \\end{pmatrix}}{s^2 + 3s + 2} \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}$$\n",
    "\n",
    "$$ H(s) = \\frac{s + 3 - 2 \\alpha}{s^2 + 3s + 2}$$\n",
    "\n",
    "\n",
    "\n",
    "$$ \\dot{x} = \\begin{pmatrix}\\dot{x_1} \\\\ \\dot{x_2} \\\\ \\dot{x_3}\\end{pmatrix} = \\begin{pmatrix}0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\0 & 0 & 1 \\end{pmatrix} \\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}  +\\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\end{pmatrix} w $$\n",
    "$$ y = \\begin{pmatrix} 0&0&1 \\end{pmatrix} \\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}$$\n",
    "\n",
    "$$s \\bar{I} - \\bar{A} = \\begin{pmatrix} s & 0 & 0  \\\\ 0 & s & 0 \\\\ 0 & 0 & s\\end{pmatrix}  - \\begin{pmatrix}0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\0 & 0 & 1 \\end{pmatrix} = \\begin{pmatrix}s & 0 & 0 \\\\ 0 & s & 0 \\\\0 & 0 & s-1 \\end{pmatrix}$$\n",
    "\n",
    "\n",
    "this inverse can be done using row reduction\n",
    "\n",
    "$$\\phi = \\left[\\begin{array}{rrr|rrr} s & 0 & 0 & 1 & 0 & 0 \\\\ 0 & s & 0 & 0 & 1 & 0 \\\\0 & 0 & s-1 & 0 & 0 & 1 \\end{array}\\right] $$ \n",
    "$$\\frac{R_1}{s} \\rightarrow R_1 $$\n",
    "$$\\frac{R_2}{s} \\rightarrow R_2 $$\n",
    "$$\\frac{R_3}{s-1} \\rightarrow R_3 $$\n",
    "\n",
    "$$\\phi = \\begin{pmatrix} \\frac{1}{s} & 0 & 0 \\\\ 0 & \\frac{1}{s} & 0 \\\\0 & 0 & \\frac{1}{s-1} \\end{pmatrix}$$\n",
    "\n",
    "$$ H(s) = \\begin{pmatrix} \\frac{1}{s} & 0 & 0 \\\\ 0 & \\frac{1}{s} & 0 \\\\0 & 0 & \\frac{1}{s-1} \\end{pmatrix} \\begin{pmatrix}0 \\\\ 0 \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\\\ \\frac{1}{s-1}\\end{pmatrix}$$\n",
    "\n",
    "$$ \\begin{pmatrix} 0 & 0 & 1 \\end{pmatrix} \\begin{pmatrix} 0 \\\\ 0 \\\\ \\frac{1}{s-1}\\end{pmatrix} = \\frac{1}{s-1} $$\n",
    "\n",
    "\n",
    "### since the system is in series you can multiply the transfer function\n",
    "\n",
    "$$\\frac{s + 3 - 2 \\alpha}{s^2 + 3s + 2} \\cdot \\frac{1}{s-1}$$\n",
    "\n",
    "$$H(s)_{\\text{total}}= \\frac{s + 3 - 2 \\alpha}{s^2 + 3s + 2(s-1)}$$\n",
    "\n",
    "### find zeros \n",
    "\n",
    "$$ s = 2\\alpha - 3$$\n",
    "\n",
    "### find poles\n",
    "\n",
    "$$(s+2)(s+1)(s-1)= 0$$\n",
    "\n",
    "$$s = -2,~~-1,~~1$$\n",
    "\n",
    "Therefore the system is unstable and alpha has no effect on the stability of the system"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# class notes\n",
    "\n",
    "## feedback systems\n",
    "\n",
    "![fsf](fsf.jpeg)\n",
    "\n",
    "### these are the equations \n",
    " $$\\dot{x_1} = x_2 $$\n",
    "\n",
    "\n",
    " $$\\dot{x_2} = -0.125x_1 - 0.75x_2 + \\mu (t)$$\n",
    "\n",
    "\n",
    " $$y = x_1$$\n",
    "\n",
    " $$\\ddot{x} + 0.75\\dot{x} + 0.125x = \\mu(t)$$\n",
    "\n",
    "\n",
    " $$x_1 = x$$\n",
    "\n",
    " $$x_2 = \\dot{x}$$\n",
    "\n",
    "### the equation in state space form\n",
    "\n",
    " $$\\dot{x} = \\begin{pmatrix}\n",
    "  \\dot{x_1} \\\\\n",
    "  \\dot{x_2} \n",
    "  \\end{pmatrix} = \\begin{pmatrix}\n",
    "  0 & 1 \\\\\n",
    "  \\frac{1}{8} & \\frac{-3}{4} \n",
    "  \\end{pmatrix} \\begin{pmatrix}\n",
    "  x_1 \\\\\n",
    "  x_2 \n",
    "  \\end{pmatrix}$$\n",
    " given that\n",
    " $$ |\\lambda I - A | = 0 $$\n",
    "\n",
    "\n",
    " this is th\n",
    " $$ \\lambda^2 + \\frac{3}{4} \\lambda + \\frac{1}{8} =0 $$\n",
    "\n",
    " which gives the eigenvalues\n",
    " $$ \\lambda_2 = \\frac{1}{2}, ~~ \\lambda_2 = \\frac{-1}{4} $$\n",
    "\n",
    "A' is the digonal of eigen values\n",
    "\n",
    "$$ b' = P^{-1} b $$\n",
    "\n",
    "diagonalization of a state space\n",
    "\n",
    "\n",
    "\n",
    "the input signal\n",
    "$$ \\ddot{x} + 0.75\\dot{x} + 0.125x = \\mu $$\n",
    "\n",
    "\n",
    "\n",
    "$$ A' = p^{-1} A p $$\n",
    "$$ b' = p^{-1} b $$\n",
    "$$ c' = cp $$\n",
    "\n",
    "$$ \\dot{z} = \\Lambda z $$\n",
    "\n",
    "where \n",
    "\n",
    "$$ \\Lambda = \\begin{pmatrix}\n",
    " \\lambda_1 & 0 & 0 \\\\\n",
    " 0 & \\ddots & 0 \\\\\n",
    " 0 & 0 & \\lambda_n \n",
    " \\end{pmatrix} $$\n",
    "\n",
    "$$ \\Lambda = p^{-1} A p $$\n",
    "\n",
    "where p is a vector of eigenvalues\n",
    "\n",
    "$$ p = [v_1 \\dots v_n] $$\n",
    "\n",
    "remember that \n",
    "\n",
    "$$ e^{A t} = I + At + A^2 \\frac{t^2}{2!}\\dots $$\n",
    "\n",
    "we can derive that \n",
    "\n",
    "$$ A = p \\Lambda p^{-1}  $$\n",
    "$$ =  pI^{-1}p^{-1} + p\\Lambda p^{-1} t + p \\Lambda ^2 p^{-1} \\frac{t^2}{2!}+\\dots $$\n",
    "$$ =  p[I^{-1} + \\Lambda t +  \\Lambda ^2 \\frac{t^2}{2!}+\\dots]p^{-1} $$\n",
    "$$ = \\begin{pmatrix}\n",
    " e^{\\lambda_1 t} & &  \\\\\n",
    "   & \\ddots & \\\\\n",
    "  & & e^{\\lambda_n t} \n",
    " \\end{pmatrix}$$\n",
    "\n",
    "$$ A \\rightarrow e^{At} $$\n",
    "\n",
    "\n",
    "$$e^{At} = p\\begin{pmatrix}\n",
    " e^{\\lambda_1 t} & &  \\\\\n",
    "   & \\ddots & \\\\\n",
    "  & & e^{\\lambda_n t} \n",
    " \\end{pmatrix}p^{-1}$$\n",
    "\n",
    "### equilibrium\n",
    "\n",
    "$$ \\dot{x} = Ax+ bu ~~~~ \\text{if}~~ \\mu = 0 ~~ \\text{then}~~  x = 0,~~ \\dot{x} = 0$$\n",
    "\n",
    "equilibrium by definition \n",
    "\n",
    "$$ x = -A^{-1} b \\mu_0 $$\n",
    "\n",
    "we have eigenvalues \n",
    "$$ \\lambda_1,\\lambda_2,\\dots \\lambda_n $$\n",
    "\n",
    "$$ \\text{Re}(\\lambda_1)\\geq \\text{Re}(\\lambda_2)\\geq\\dots\\text{Re}(\\lambda_n)    $$\n",
    "$$ \\dot{x} = \\Lambda x $$\n",
    "\n",
    "$$e^{\\Lambda t} = \\begin{pmatrix}\n",
    " e^{\\Lambda_1 t} & &  \\\\\n",
    "   & \\ddots & \\\\\n",
    "  & & e^{\\Lambda_n t} \n",
    " \\end{pmatrix}$$\n",
    "\n",
    "\n",
    "$$ x(t) = e^{\\lambda_1 t} x_1(0) + \\dots + e^{\\lambda_n t} x_n(0) $$\n",
    "\n",
    "### example\n",
    "\n",
    "$$ \\dot{x} = Ax + bu $$\n",
    "\n",
    "where\n",
    "\n",
    "$$ A = \\begin{pmatrix}\n",
    " 0 & 1 & 0 \\\\\n",
    " 0 & 0 & 1 \\\\\n",
    " -1 & -5 & -6 \n",
    " \\end{pmatrix}\\begin{pmatrix}\n",
    " 0 \\\\\n",
    " 0 \\\\\n",
    " 1 \n",
    " \\end{pmatrix} $$\n",
    "\n",
    "which gives the characteristic equation \n",
    "$$ \\lambda^3 + 5\\lambda^2 + 6\\lambda + 1 = 0 $$\n",
    "\n",
    "$$ \\mu = kx $$\n",
    "$$ \\mu = \\begin{pmatrix}\n",
    " k_1 & k_2 & k_3 \n",
    " \\end{pmatrix}\\begin{pmatrix}\n",
    " x_1 \\\\\n",
    " x_2 \\\\\n",
    " x_3 \n",
    " \\end{pmatrix} $$\n",
    "\n",
    "$$ \\mu = k_1x_1 + k_2x_2 + k_3x_3 $$\n",
    "\n",
    "$$ \\dot{x} = Ax+bu = Ax+bkx $$\n",
    "$$ \\dot{x} = (A + bk)x $$\n",
    "\n",
    "$$ bk = \\begin{pmatrix}\n",
    " 0 \\\\\n",
    " 0 \\\\\n",
    " 1 \n",
    " \\end{pmatrix}\\begin{pmatrix}\n",
    " k_1 & k_2 & k_3 \n",
    " \\end{pmatrix} $$\n",
    "\n",
    "$$ bk = \\begin{pmatrix}\n",
    " 0 & 0 & 0 \\\\\n",
    " 0 & 0 & 0 \\\\\n",
    " k_1 & k_2 & k_3 \n",
    " \\end{pmatrix} $$\n",
    "\n",
    "$$ A + bk = \\begin{pmatrix}\n",
    " 0 & 1 & 0 \\\\\n",
    " 0 & 0 & 1 \\\\\n",
    " -1+k_1 & -5+k_2 & -6+k_3 \n",
    " \\end{pmatrix} $$\n",
    "\n",
    "the characteristic equation\n",
    "\n",
    "$$ \\lambda^3 + (6 - k_3)\\lambda^2 + (5-k_2)\\lambda +(1-k) = 0$$\n",
    "\n",
    "lets say i want the eigenvalues to be $-2\\pm j4$ and -10\n",
    "\n",
    "$$ (\\lambda^3 + 10) + (\\lambda - (-2+j4)) (\\lambda - (-2-j4))$$\n",
    "\n",
    "$$ \\lambda^3 + 14\\lambda^2 + 60\\lambda+200 $$\n",
    "\n",
    "the open loop equation is\n",
    "\n",
    "$$ x^3 + 6x^2 + 5x^2 + 1 $$\n",
    "\n",
    "$$ k_3 = 8, ~~ k_2 = 55 ~~ k_1 = 199 $$\n",
    "\n",
    "if the system is completely controllable then the eigenvalues of the closed loop system with full state feedback can be placed anywhere\n",
    "\n",
    " "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## discrete time\n",
    "\n",
    "the state of a linear, time-invariant physical discrete-time system is assumed to satisfy\n",
    "\n",
    "$$ x(k + 1) = Ax(k) + bu(k) $$\n",
    "$$ y(k) = Cx(k) + Du(k) $$\n",
    "\n",
    "where at time k, x(k) is the plants state $\\mu(k)$ is its inputs and y(k) is its outputs. \n",
    "\n",
    "\n",
    "now the observer model\n",
    "\n",
    "$$ \\hat{x}(k+1) = A \\hat{x}(k) + L[y(k) - \\hat{y}(k)] +Bu(k)$$\n",
    "\n",
    "$$ \\hat{y}(k) C\\hat{x}(k) + Du(k) $$\n",
    "\n",
    "note that the observer model has a hat to differentiate it from the normal one. L is some matrix \n",
    "\n",
    "the observer is called asymptotically stable if hte observer error $e(k) = \\hat{x}(k) - x(k)$ converges to zero when $k\\rightarrow \\infty$ for a luenberger observer, the observer error satisfies $e(k+1) = (A - LC) e(k)$. the luenberger observer for this discrete-time system is therefore asymptotically stable when the matrix A - LC has all the eigenvalues inside the unit circle.\n",
    "for control purposes the output of the observer system is fed back to the input of both the observer and the plant through the gains matrix k\n",
    "\n",
    "$$ u(k) = -k \\hat{x}(k) $$\n",
    "\n",
    "the observer equations then become:\n",
    "\n",
    "$$ \\hat{x} (k+1) = A \\hat{x} (k) + L(y(k) - \\hat{y}(k)) - BK \\hat{x} (k) $$\n",
    "\n",
    "$$ \\hat{y}(k) = C \\hat{x}(k) - DK \\hat{x}(k) $$\n",
    "\n",
    "or more simply\n",
    "\n",
    "$$ \\hat{x}(k+1) = (A - BK) \\hat{x}(k)+L(y(k) - \\hat{y}(k)) $$\n",
    "$$ \\hat{y}(k) = (C - DK) \\hat{x}(k) $$\n",
    "\n",
    "due to separation  principle we know that we can choose k and l independently without harm to the overall stability of the systems. as a rule of thumb the poles of the observer A - LC are usually chosen to converge 10 times faster than the poles of the system A - BK\n",
    "\n",
    "## continuous-time\n",
    "\n",
    "The processes is similar\n",
    "\n",
    "$$ \\dot{x} = Ax + Bu $$\n",
    "$$ y = Cx + Du $$\n",
    "\n",
    "where $x \\in R^n ~~ u \\in R^m ~~ y \\in R^r$\n",
    "\n",
    "$$\\dot{x} = A \\hat{x} + Bu + L(y - \\hat{y})$$\n",
    "$$ \\hat{y} = C \\hat{x} + Du $$\n",
    "\n",
    "The observer error $e = x -\\hat{x}$\n",
    "\n",
    "$$ \\dot{e} = (A - LC) e$$\n",
    "\n",
    "the eigenvalues of the matrix A - LC can be chosen arbitrarily by appropriate choice of the observer gain L when the pair [A,C] is observability condition holds. in particular it can be made Hurwitz so the observer error $$e(t) \\rightarrow 0 ~~ \\text{when} ~~ t\\rightarrow \\infty$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# controllability and observability\n",
    "\n",
    "## controllability\n",
    "\n",
    "A system is controllable if there exists a control that can transition a system from an arbitrary initial condition to an arbitrary final condition in finite time\n",
    "\n",
    "if your system is linear $\\dot{x}(t) = Ax(t) + Bu(t)$\n",
    "\n",
    "You have a matrix A that is nxn then the controllability matrix is \n",
    "\n",
    "$$ C = [B~ AB~ A^2B~ A^3B~ \\dots ~A^{n-1}B] $$\n",
    "\n",
    "The system is controllable iff rank(c) = n\n",
    "\n",
    "### example\n",
    "\n",
    "$$ \\dot{x}(t) = \\begin{pmatrix}\n",
    " 0 & 3 \\\\\n",
    " 2 & 4 \n",
    " \\end{pmatrix} x(t) + \\begin{pmatrix}\n",
    " -2 \\\\\n",
    " 1 \n",
    " \\end{pmatrix}u $$\n",
    "\n",
    "$$C = \\begin{pmatrix}\n",
    " B & AB \n",
    " \\end{pmatrix}$$\n",
    "\n",
    "$$ C = \\begin{pmatrix}\n",
    " -2 & 3 \\\\\n",
    " 1 & 0 \n",
    " \\end{pmatrix} $$\n",
    "\n",
    "check the rank of this matrix and it equals 2 therefore this system is controllable\n",
    "\n",
    "## observability\n",
    "\n",
    "Are all the states measurable?\n",
    "\n",
    "the procedure to find this is relatively the same you create a matrix and find its rank. if it is rank deficient then the system is not observable.\n",
    "\n",
    "the Observability matrix is \n",
    "\n",
    "$$ o = \\begin{pmatrix}\n",
    " C \\\\\n",
    " CA \\\\\n",
    " CA^2 \\\\\n",
    " \\vdots \\\\\n",
    " CA^{n-1} \n",
    " \\end{pmatrix} $$\n",
    "\n",
    "if rank(o) is rank deficient then the system is not observable"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Pole Placement\n",
    "\n",
    "For this I used [full state feedback](https://en.wikipedia.org/wiki/Full_state_feedback#:~:text=Full%20state%20feedback%20(FSF)%2C,locations%20in%20the%20s%2Dplane.)\n",
    "\n",
    "lets  start with the linear state space \n",
    "\n",
    "$$ \\dot{x}(t) = Ax(t) + Bu(t) $$\n",
    "$$ y(t) = Cx(t) + Du(t) $$\n",
    "\n",
    "\n",
    "lets assume that C = I meaning C is equal to the identity matrix and D = 0\n",
    "\n",
    "Therefore y(t) = x(t)\n",
    "\n",
    "![regulator](regulator.jpg)\n",
    "\n",
    "If we have this system lets consider a control law that is $u(t) = -K x(t)$\n",
    "\n",
    "This is the control law for full state feedback (FSF) K is considered a regulator because it regulates non zero inputs back to the origin \n",
    "\n",
    "we can rewrite the linear state space equation as $\\dot{x}(t) = [A - BK]x(t)$ The eigenvalues of [A-BK] control the system\n",
    "\n",
    "we can move the eigenvalues of [A - Bk] by changing K iff A,B are controllable. \n",
    "\n",
    "## Example:\n",
    "\n",
    "take this controllable system\n",
    "\n",
    "$$ \\dot{x} (t) = \\begin{pmatrix}\n",
    " 0 & 3 \\\\\n",
    " 2 & 4 \n",
    " \\end{pmatrix}x(t) + \\begin{pmatrix}\n",
    " -2 \\\\\n",
    " 1 \n",
    " \\end{pmatrix}u(t) $$\n",
    "\n",
    "I've discussed controllability in previous sections. In short the controllability matrix is full rank.\n",
    "\n",
    "look at the control law $u(t) = - kx(t)$ k is a matrix that looks like $[k_1 k_2]$ now I'll do A-bk\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    " 0 & 3 \\\\\n",
    " 2 & 4 \n",
    " \\end{pmatrix} - \\begin{pmatrix}\n",
    " -2 \\\\\n",
    " 1 \n",
    " \\end{pmatrix}  (k_1 k_2) = \\begin{pmatrix}\n",
    " 2k_1 & 3+2k_2 \\\\\n",
    " 2-k_1 & 4-k_2 \n",
    " \\end{pmatrix}$$\n",
    "\n",
    "as you can see the k's get into every part of the matrix. meaning that if you change the k's you can change the eigenvalues of the matrix. to get the eigenvalues use $\\text{det}(\\lambda I - [A-BK]) = 0$ doing this on the equation above ill get the eigenvalues of $\\lambda^2 + (-4 -2k_1 +k_2)\\lambda + 11k_1 - 6 - 4k_2 = 0$ this is the characteristic equation of the closed loop system under full state feedback using k_1 k_2. now its just a question of where should the poles go?\n",
    "\n",
    "Suppose i want the closed loop eigenvalues at $-5 \\pm 2i$. that means i want the characteristic equation to look like $(\\lambda + 5-2i) (\\lambda + 5 +2i)$ which expands $\\lambda^2 + 10\\lambda + 29$ all i need to do is compare the coefficients of lambda and the constants. for lambda $-4-2k_1 +k_2 = 10$ for the constant $11k_1 - 6 - 4k_2 = 29$ which gives $k_1 = 30.33$ and $k_2 = 74.67$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Root locus\n",
    "A root locus is a graph that shows the output on the s plane with varying levels of K\n",
    "\n",
    "\n",
    "There are a few rules for root locus.\n",
    "\n",
    "consider an open loop transfer function\n",
    "$$ G(s) = \\frac{a(s)}{b(s)} = \\frac{\\alpha (s+z_1)(s+z_2)\\dots (s+z_m)}{(s+p_1)(s+p_2)\\dots(s+p_n)} $$\n",
    "\n",
    "m = number of zeros and n = number of poles, assume $\\alpha > 0$ $n \\geq m$ and $k \\in [0,+\\infty]$ \n",
    "\n",
    "Rule 1: number of poles is unchanged, number of poles = n\n",
    "\n",
    "Rule 2: closed loop poles start at open poles and go to either one of two scenarios\n",
    "\n",
    "a) end at open loop zeros (m poles will go here)\n",
    "\n",
    "B) end at zeros at infinity (n - m poles go here)\n",
    "\n",
    "Rule 3: valid regions on the real axis are to the left of odd numbered poles/zeros (start numbering from right to left)\n",
    "\n",
    "Rule 4: angle of asymptotes\n",
    "\n",
    "angle of asymptotes is equal to\n",
    "$$ \\phi _a = \\frac{1 \\pm 2k \\pi}{n - m} ~~ \\text{where} ~~ k=0,1,\\dots n-m-1$$ \n",
    "\n",
    "centroid of asymptotes is equal to\n",
    "\n",
    "$$ \\sigma_a = \\frac{\\sum^n_{k=1} (-p_k) - \\sum^m_{k=1}(-z_k)}{n-m} $$\n",
    "\n",
    "Rule 5: angle of departure from a complex pole\n",
    "\n",
    "$$ \\theta _j = \\angle(s_j + p_j) = \\pi \\pm 2\\pi k + \\sum^m_{i=1} \\angle(p_j + z_j) - \\sum^n_{i=1,i\\neq j} \\angle(p_j+p_i)  $$\n",
    "\n",
    "however drawing root locus by hand is vary rarely done. I will end out this section a python example.\n",
    "\n",
    "## example\n",
    "\n",
    "take the transfer function \n",
    "$$ \\frac{s-1}{s+3}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import control as ctl\n",
    "\n",
    "num = [1, -1]\n",
    "den = [1, 3]\n",
    "\n",
    "tf = ctl.tf(num,den)\n",
    "ctl.rlocus(tf)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This shows how the poles move through the S plane as you move k. The lines in the map are the lines of constant damping ratio"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PID control\n",
    "\n",
    "![pid controller](PID_en.svg.png)\n",
    "\n",
    "\n",
    "e(t) is the error value, as the difference between a desired setpoint SP = r(t) and measured process variable PV = y(t): e(t) = r(t) - y(t)\n",
    "\n",
    "PID is made up of 3 parts\n",
    "\n",
    "## Term P \n",
    "\n",
    "P is proportional to the current value of the SP-PV error e(t) for example if the error is large and positive the control output will be proportionately large and positive, taking into account the gain factor \"K\". using proportional control alone will result in an error between the setpoint and the actual process value.\n",
    "\n",
    "## Term I\n",
    "\n",
    "I Accounts for the past values of SP - PV error and integrates them over time to produce the I term. For example, if there is a residual SP - PV error after the application of proportional control, the integral term seeks to eliminate the residual error by adding a control effect due to the historic cumulative value of the error. When the error is eliminated, the integral term will cease to grow. This is will result in the proportional effect diminishing as the error decreases, but this is compensated for by the growing integral effect.\n",
    "\n",
    "## Term D\n",
    "\n",
    "D is a best estimate of the future trend of the SP-PV error, based on its current rate of change. It is sometimes called \"anticipatory control\" as it is effectively seeking to reduce the effect of SP-PV error by exerting a control influence generated by the rate of error change. The more rapid the change the greater the controlling or dampening effect\n",
    "\n",
    "\n",
    "## tuning\n",
    "The balance of these effects is achieved by looping to produce the optimal function. The tuning constants K are derived for each application. meaning we can change the K values to get the proper response we desire.\n",
    "\n",
    "## Math\n",
    "\n",
    "the control function for PID is \n",
    "\n",
    "$$ u(t) = K_p e(t) + k_i \\int^t_0 e(t') dt' + K_d \\frac{de(t)}{dt}$$ \n",
    "\n",
    "In the s domain \n",
    "\n",
    "$$ G(s) = k_p + \\frac{k_i}{s} + k_d s$$\n",
    "\n",
    "Where $k_p, k_i, k_d$ are all non negative. \n",
    "\n",
    "\n",
    "I will now show the effect of each element on a signal\n",
    "\n",
    "This signal is $\\frac{1}{s^2 + 3s + 2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ORIGINAL transfer function\n",
      "\n",
      "\n",
      "      1\n",
      "-------------\n",
      "s^2 + 3 s + 2\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f51b22e1c40>]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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AkYOlQiApJRJxnnqrlFueW8bO8ioumjCA7552ON3z24UdTSRlqRBIyvhg815u+L9FLFizk8KBXXl4ynEc2adz2LFEUp4KgSS9yppa7pi5kjtmFdOhXRa/mXoUU8f104lgkVaiQiBJbVHpbv7zyXdYsbmMc47pw0/OGqVmIJFWpkIgSam6NsJtLxdz28xieua348+XH8ukEb3CjiXSJqkQSNIp2VrGt4veYdH63Zw7ti83feFIOudlhx1LpM1SIZCk4e48tbCUn05fTHZmBndePI4zxvQOO5ZIm6dCIElhX2UNP3xmEX9/ZwPHDe7GHy84RiOCiRwiKgQSuuIte7n6kbco2VrG904/nGsnDVPfQCKHkAqBhOrZ9zby/afeJTc7k798bQInDusRdiSRtJPQDlnMbLKZLTezYjO7vpHlvmhmbmbjE5lHkkck4tz6/DKufewtRhzWkX9+6yQVAZGQJOyIwMwygduB04FSYL6ZTXf3JQcs1xH4NjA3UVkkuZRV1vCdond4celmLjy2PzdPGU1OljqJEwlLIv/1HQcUu3uJu1cBRcCUepb7b+AWoCKBWSRJrN+1ny/e8Tozl2/hpi+M4lfnjVEREAmZuXtiNmw2FZjs7lcG05cCE9x9Wswy44AfufsXzWwWcJ27L6hnW1cBVwEUFBQUFhUVNStTWVkZ+fn5zVo3DKmUN56sa/bU8oeFlVTWOtcek8voHuH1FNrWvttkkkp5UykrtCzvpEmTFrp7/c3v7p6QBzAVuC9m+lLgtpjpDGAWMCiYngWMb2q7hYWF3lwzZ85s9rphSKW8TWWdtXyLj/rJc37CL1/0ZRv3HJpQjWhL322ySaW8qZTVvWV5gQXewH41kcfk64H+MdP9gnl1OgKjgVlmtho4HpiuE8Ztz9/eXs8VD85nYPcOPHPtiYw4rGPYkUQkRiIvH50PDDezwUQLwIXARXUvuvtu4MPLRBprGpLU9fAbq7nx74s5YUh37vlKIR1z1VWESLJJWCFw9xozmwY8D2QCD7j7YjO7meghyvREvbeEz9257eVifvfCCk4fVcD/fnmsRg8TSVIJvaHM3WcAMw6Yd2MDy05MZBY5dNydW59fzh2zVnLeuL785otHaQxhkSSmO4ulVbk7t/xrOXfNXsnFEwbw31NGawAZkSSnQiCtxt359XPLuPuVEi49fiA3TzkSMxUBkWSn43VpNbc+v5y7XynhqyeoCIikEh0RSKv4x8oqnv5gJRdNGMBNZ6sIiKQSHRFIi/15ziqe/qCac8f25edTRqsIiKQYFQJpkacXlvKzfyyhsCCTW6cepRPDIilITUPSbLOWb+H7T7/HicO6c9mQ/bpEVCRF6V+uNMu763ZxzaNvMaKgI3ddUki2jgREUpYKgRy0Ndv3ccWD8+nWIYcHrzhW3UaIpDgVAjkou8urufzB+UTcefiK4+jVMTfsSCLSQjpHIHGrro1wzWMLWbejnEevPJ4hPVOnH3cRaZgKgcTF3bnx74uZU7yd355/NMcN7hZ2JBFpJWoakrg8+PpqHp+3lmsmDmVqYb+w44hIK1IhkCa9vnIbP392KaePKuC6z44IO46ItDIVAmnU+l37mfbY2wzqnsfvv3S0bhgTaYNUCKRBFdW1XP2XhVTVRLjnK+N1mahIG6WTxdKgn/ztfRat3829XxnPUF0hJNJm6YhA6vXUwlKeXFjKtEnDOH1UQdhxRCSBVAjkE1Zs3suP/7aI44d047unHx52HBFJMBUC+Zh9lTVc8+hb5LfL5k8XjiVTJ4dF2jydI5CPufHvi1m5tYxHvjaBXp3UfYRIOtARgXxo+rsbePqtUr45aRgnDusRdhwROURUCASA0p3l/OiZRYwd0IVvnTo87DgicgipEAi1Eed7T7yLO/zPBWM1wIxImtE5AuHOWcXMW72D33/paAZ0zws7jogcYvrpl+beX7+bP774AWcd1Ztzx/YNO46IhECFII1V1tRy3ZPv0rVDDj8/ZzRmulRUJB2paSiN/emlD1i2aS/3f3U8XfJywo4jIiHREUGaemfdLu6ctZLzC/tx6hHqQkIknakQpKG6JqHDOuXyky+MCjuOiIRMTUNp6PaZKyneUsaDlx9LJ3UtLZL2EnpEYGaTzWy5mRWb2fX1vH61mS0ys3fM7DUz08/TBFu+aS93zirm3LF9mTiiV9hxRCQJxFUIzOxwM3vJzN4Ppo8ysx83sU4mcDtwBjAK+HI9O/rH3H2Mux8D/Ab4/cF+AIlfbcT5wdPv0TE3m5+cpZorIlHxHhHcC9wAVAO4+3vAhU2scxxQ7O4l7l4FFAFTYhdw9z0xkx0AjzOPNMNDr6/mnXW7+OkXRtGtg64SEpEoc29632tm8939WDN7293HBvPeCX7JN7TOVGCyu18ZTF8KTHD3aQcsdy3wPSAH+Iy7f1DPtq4CrgIoKCgoLCoqivfzfUxZWRn5+akz0lZr5t2+P8IPX9vPiK6ZfLewXavfM5DO322ipVJWSK28qZQVWpZ30qRJC919fL0vunuTD+A5YCjwVjA9FXiuiXWmAvfFTF8K3NbI8hcBDzWVpbCw0Jtr5syZzV43DK2Z9+q/LPDDfzTD127f12rbjJXO322ipVJW99TKm0pZ3VuWF1jgDexX471q6FrgHmCkma0HVgGXNLHOeqB/zHS/YF5DioA748wjB2Hm8i089/4mrvvs4fTvpr6EROTj4ioE7l4CnGZmHYAMd98bx2rzgeFmNphoAbiQ6K/+D5nZcP+oKehM4BPNQtIyFdW1/PTvixnSswP/8ekhYccRkSTUaCEws+81MB8Ad2/wKh93rzGzacDzQCbwgLsvNrObiR6iTAemmdlpRE9C7wS+2qxPIQ26Y9ZK1u4o57ErJ9AuKzPsOCKShJo6IugY/B0BHAtMD6a/AMxrauPuPgOYccC8G2OefzvupHLQ1mzfx12zVnL20X34lEYcE5EGNFoI3P1nAGb2CjCurknIzG4Cnk14OmmRnz+7lKxM40dnHhF2FBFJYvHeR1AAVMVMVwXzJEm9smIrLyzZzLTPDKNAg9CLSCPivWroYWCemT0TTJ8DPJSQRNJi1bURbv7nEgZ2z+NrJw0OO46IJLl4rxr6hZk9B5wczLrc3d9OXCxpiYffWEPxljLu/cp4nSAWkSbFVQjMbACwDXgmdp67r01UMGmeHfuq+OOLKzh5eA9OO0KdyolI0+JtGnqWj/oBag8MBpYDRyYilDTf/7y4gvKqWm48a5SGnhSRuMTbNDQmdtrMxgHXJCSRNNvKrWU8OnctFx7bn+EFHZteQUSEZo5H4O5vARNaOYu00K+fW0a7rAy+c9rhYUcRkRQS7zmC2DuMM4BxwIaEJJJmebNkOy8s2cx/fW4EPTu2CzuOiKSQeM8RxLYz1BA9Z/B068eR5ohEnF/OWErvzrm6XFREDlq8hWCJuz8ZO8PMzgeebGB5OYT+uWgj75Xu5nfnH01uti4XFZGDE+85ghvinCeHWHVthN/9ezkjD+vIuWP7hh1HRFJQU72PngF8HuhrZn+KeakT0SYiCdkT89exZns5D1w2nowMXS4qIgevqaahDcAC4GxgYcz8vcB3ExVK4rO/qpb/eekDjh3UlUkjdPOYiDRPU72Pvgu8a2aPuruOAJLMA3NWsXVvJXdePE43j4lIszXVNPRXd/8S8LaZfWKUe3c/KmHJpFG7yqu4a/ZKTjuiF+MHdQs7joiksKaahuoGjjkr0UHk4Nz7aglllTVc97kRYUcRkRTXVNPQxuDvmkMTR+KxvaySP89ZzVlH9WHkYZ3CjiMiKa6ppqG9fNTZHIAF0wa4u2svFIK7Zq+korqW75w2POwoItIGNHVEoJ7LksyWPRU8/MYazhnbl6E988OOIyJtQLx3Ftf1OHoS0SOC1zQwTTjumLWSmojz7VN1NCAirSOuO4vN7EaiQ1N2B3oAD5rZjxMZTD5pw679PDZ3LecX9mNg9w5hxxGRNiLeI4KLgaPdvQLAzH4NvAP8PEG5pB53zV6J40z7zLCwo4hIGxJvX0MbgNyY6XbA+taPIw3ZtLuConnrmFrYj35d88KOIyJtSLxHBLuBxWb2AtFzBKcD8+r6H3L3byUonwTumr2SiDvXTNTRgIi0rngLwTPEDFwPzGr9KNKQLXsqeHzeWs4b15f+3XQ0ICKtK94xix9KdBBp2N2vlFATca6dpKMBEWl98V41dJaZvW1mO8xsj5ntNbM9iQ4nsHVvJY/OXcM5x/TVlUIikhDxNg39ETgPWOTun+h8ThLn/tdWUVUT4dpJQ8OOIiJtVLxXDa0D3lcROLT2VTuPvLmGM4/qwxDdRSwiCRLvEcH3gRlmNhuorJvp7r9PSCoB4KW11ZRV1nDNRB0NiEjixHtE8AugnOi9BB1jHo0ys8lmttzMis3s+npe/56ZLTGz98zsJTMbeDDh27Lyqhr+vbqaU0f24oje6ttPRBIn3iOCPu4++mA2bGaZwO1E7zkoBeab2XR3XxKz2NvAeHcvN7NvAL8BLjiY92mrHpu7lrJquEZXColIgsV7RDDDzD57kNs+Dih29xJ3rwKKgCmxC7j7THcvDybfBPod5Hu0SZU1tdz7agkju2VQOLBr2HFEpI2zeM7/BuMSdCB6fqCaOMYjMLOpwGR3vzKYvhSY4O7TGlj+NmCTu3+i/yIzuwq4CqCgoKCwqKioycz1KSsrIz8/+U+6zlpXzYOLq5g22hnfL/nzQup8t3VSKW8qZYXUyptKWaFleSdNmrTQ3cfX91q8N5R1NLNuwHA+3udQqzCzS4DxwCkNvP89wD0A48eP94kTJzbrfWbNmkVz1z1UaiPOz34/mzF921PYtzrp89ZJhe82VirlTaWskFp5UykrJC5vXIXAzK4kOn5xP6K9jh4PvA6c2shq64H+MdP9qKejOjM7DfgRcIq7Vx74erp5YckmVm3bx+0XjcN2LA87joikgXjPEXwbOBZY4+6TgLFEO6JrzHxguJkNNrMc4EJgeuwCZjYWuBs42923HFTyNsjduXN2CQO75zF59GFhxxGRNBFvIaiIGYugnbsvA0Y0toK71wDTgOeBpcBf3X2xmd1sZmcHi90K5ANPmtk7Zja9gc2lhTdLdvDuul38x8lDyMywsOOISJqI9/LRUjPrAvwNeMHMdgJrmlrJ3WcAMw6Yd2PM89PiTpoG7pq9kh75OUwt1MVTInLoxHuy+Nzg6U1mNhPoDPwrYanS0NKNe5i9YivXffZwcrMzw44jImkk7sHr67j77EQESXf3vlJCXk4mlx4/KOwoIpJm4j1HIAm0cfd+pr+7gQuPHUDnvOyw44hImlEhSAIPzlmNA5efOCjsKCKShlQIQra3oprH5q7ljNGHaRhKEQmFCkHInpi/jr2VNVz16SFhRxGRNKVCEKLq2ggPvLaKCYO7cVS/LmHHEZE0pUIQohmLNrJhd4WOBkQkVCoEIXF37n21hCE9OzBpRK+w44hIGlMhCMncVTt4f/0erjxpCBnqTkJEQqRCEJL7Xl1F17xszhvXN+woIpLmVAhCsGrbPl5atplLjx+o7iREJHQqBCH485xVZGdkcMkJA8OOIiKiQnCo7Sqv4skFpZx9TB96dWz1wd5ERA6aCsEh9ti8teyvruVrJw0OO4qICKBCcEhV10Z46PXVnDisO0f07hR2HBERQIXgkJqxaCOb91Ry5Um6gUxEkocKwSHi7tz/2iqG9OzAKYf3DDuOiMiHVAgOkYVrdvJe6W4uP3GwbiATkaSiQnCI3P/aKjq3z+aLuoFMRJKMCsEhsG5HOc8v3sRFEwaQl3PQo4OKiCSUCsEh8NDrq8kw4yu6gUxEkpAKQYKVVdbwxPx1fH5Mb3p3bh92HBGRT1AhSLAnF0RHILtCN5CJSJJSIUig2ojz5zmrKRzYlWP6dwk7johIvVQIEuilpZtZu6Nc3UmISFJTIUig+19bRd8u7fnsqIKwo4iINEiFIEHeX7+buat2cNmnBpGVqa9ZRJKX9lAJ8sCcVeTlZPKlY/uHHUVEpFEqBAmwZU8F/3h3A+cX9qNz++yw44iINCqhhcDMJpvZcjMrNrPr63n902b2lpnVmNnURGY5lB5+Yw01EdcloyKSEhJWCMwsE7gdOAMYBXzZzEYdsNha4DLgsUTlONT2V9Xy6Nw1nH5EAQO7dwg7johIkxLZ8c1xQLG7lwCYWREwBVhSt4C7rw5eiyQwxyH1f2+XsrO8WpeMikjKMHdPzIajTT2T3f3KYPpSYIK7T6tn2QeBf7r7Uw1s6yrgKoCCgoLCoqKiZmUqKysjPz+/WevGI+LOD1/bT26m8dMTcjFrWXfTic7bmlIpK6RW3lTKCqmVN5WyQsvyTpo0aaG7j6/vtZToCtPd7wHuARg/frxPnDixWduZNWsWzV03HjOXbWHTvvn88YKjmTS25d1NJzpva0qlrJBaeVMpK6RW3lTKConLm8iTxeuB2Gsn+wXz2qz7XivhsE65fH5M77CjiIjELZGFYD4w3MwGm1kOcCEwPYHvF6rFG3Yzp3g7X/nUQHKydFWuiKSOhO2x3L0GmAY8DywF/urui83sZjM7G8DMjjWzUuB84G4zW5yoPIl27ysldMjJ5OIJGnNARFJLQs8RuPsMYMYB826MeT6faJNRStuwaz//eG8jXz1hkG4gE5GUozaMVvDnOasAuPzEQeEGERFpBhWCFtpTUc3j86IjkPXvlhd2HBGRg6ZC0EJF89ZSVlnDVScPCTuKiEizqBC0QFVNhAdeW80JQ7ozpl/nsOOIiDSLCkEL/O3t9WzaU8FVp+hoQERSlwpBM9VGnLteWcmo3p2YeHjPsOOIiDSbCkEz/XvxJkq27uMbE4e2uE8hEZEwqRA0g7tzx6yVDOqep+4kRCTlqRA0w2vF21i0fjdfP2UomRk6GhCR1KZC0Ax3zlpJQad2nDeu5T2MioiETYXgIC1YvYPXV27nypOG0C4rM+w4IiItpkJwkP7w4gp65Odw8fEDwo4iItIqVAgOwrxVO5hTvJ2rTxlKXk5KjOkjItIkFYKD8IcXVtAjv526mhaRNkWFIE5vlmznjZLtfGPiUNrn6NyAiLQdKgRx+sMLK+jVsR0XT9C5ARFpW1QI4jB7xVbmrtrBNyYOJTdbRwMi0raoEDShNuL88tmlDOyep3MDItImqRA04amF61i+eS8/mDxSg9KLSJukPVsj9lXW8Lt/r2DcgC6cMfqwsOOIiCSECkEj7n21hC17K/nRmaPUw6iItFkqBA3YuHs/d88u4cwxvSkc2DXsOCIiCaNCUA935yd/ex/H+cHkkWHHERFJKBWCejy7aCMvLt3Cf54+ggHd88KOIyKSUCoEB9hVXsVN0xczpm9nLj9xUNhxREQSTj2nHeDnzy5lZ3k1D18xgaxM1UkRafu0p4vx/OJNPLWwlK9/egij+nQKO46IyCGhQhBYsXkv33viHY7u34VvnTo87DgiIoeMCgHR8wL/8fAC8tplcfclhepPSETSStoXgpraCN98/G027qrgrksKOaxzbtiRREQOqbQuBLv3V3PFQwt49YNt/Pc5R+rGMRFJSwktBGY22cyWm1mxmV1fz+vtzOyJ4PW5ZjYokXlirdxaxrm3z+GNldv41XljuOBYjTMgIukpYZePmlkmcDtwOlAKzDez6e6+JGaxrwE73X2YmV0I3AJckKhMEXfmr97Bi0s389jcteRkZvDolcdz3OBuiXpLEZGkl8j7CI4Dit29BMDMioApQGwhmALcFDx/CrjNzMzdvbXDFM1by89fLqes+g2yMowTh/XgF+eOpl9X3TksIunNErDPjW7YbCow2d2vDKYvBSa4+7SYZd4PlikNplcGy2w7YFtXAVcBFBQUFBYVFR10nve21vDq2gqO7ZPL6B6Z5GUnf2+iZWVl5Ofnhx0jLqmUFVIrbyplhdTKm0pZoWV5J02atNDdx9f3WkrcWezu9wD3AIwfP94nTpx40NuYCBw1axbNWTcss1IobyplhdTKm0pZIbXyplJWSFzeRJ4sXg/0j5nuF8yrdxkzywI6A9sTmElERA6QyEIwHxhuZoPNLAe4EJh+wDLTga8Gz6cCLyfi/ICIiDQsYU1D7l5jZtOA54FM4AF3X2xmNwML3H06cD/wFzMrBnYQLRYiInIIJfQcgbvPAGYcMO/GmOcVwPmJzCAiIo1L6zuLRUREhUBEJO2pEIiIpDkVAhGRNJewO4sTxcy2AmuauXoPYFuTSyWPVMqbSlkhtfKmUlZIrbyplBValnegu/es74WUKwQtYWYLGrrFOhmlUt5UygqplTeVskJq5U2lrJC4vGoaEhFJcyoEIiJpLt0KwT1hBzhIqZQ3lbJCauVNpayQWnlTKSskKG9anSMQEZFPSrcjAhEROYAKgYhImkubQmBmk81suZkVm9n1YedpjJk9YGZbghHckpqZ9TezmWa2xMwWm9m3w87UEDPLNbN5ZvZukPVnYWeKh5llmtnbZvbPsLM0xsxWm9kiM3vHzBaEnacpZtbFzJ4ys2VmttTMTgg7U33MbETwndY99pjZd1r1PdLhHIGZZQIrgNOBUqJjJXzZ3Zc0umJIzOzTQBnwsLuPDjtPY8ysN9Db3d8ys47AQuCcZPxuzcyADu5eZmbZwGvAt939zZCjNcrMvgeMBzq5+1lh52mIma0Gxh841GyyMrOHgFfd/b5gzJQ8d98VcqxGBfuy9USH9G3ujbWfkC5HBMcBxe5e4u5VQBEwJeRMDXL3V4iOz5D03H2ju78VPN8LLAX6hpuqfh5VFkxmB4+k/iVkZv2AM4H7ws7SlphZZ+DTRMdEwd2rkr0IBE4FVrZmEYD0KQR9gXUx06Uk6c4qlZnZIGAsMDfkKA0KmlneAbYAL7h70mYN/BH4PhAJOUc8HPi3mS00s6vCDtOEwcBW4M9Bs9t9ZtYh7FBxuBB4vLU3mi6FQBLMzPKBp4HvuPuesPM0xN1r3f0YomNoH2dmSdv0ZmZnAVvcfWHYWeJ0kruPA84Arg2aOJNVFjAOuNPdxwL7gGQ/d5gDnA082drbTpdCsB7oHzPdL5gnrSBob38aeNTd/y/sPPEImgFmApNDjtKYE4Gzg7b3IuAzZvZIuJEa5u7rg79bgGeINskmq1KgNOaI8CmihSGZnQG85e6bW3vD6VII5gPDzWxwUFUvBKaHnKlNCE7A3g8sdfffh52nMWbW08y6BM/bE714YFmooRrh7je4ez93H0T0/9mX3f2SkGPVy8w6BBcLEDSxfBZI2qve3H0TsM7MRgSzTgWS7gKHA3yZBDQLQYLHLE4W7l5jZtOA54FM4AF3XxxyrAaZ2ePARKCHmZUCP3X3+8NN1aATgUuBRUHbO8APg/Gqk01v4KHgyosM4K/untSXZKaQAuCZ6O8CsoDH3P1f4UZq0jeBR4MfhyXA5SHnaVBQXE8Hvp6Q7afD5aMiItKwdGkaEhGRBqgQiIikORUCEZE0p0IgIpLmVAhERNKcCoFIiMxsYrL3KiptnwqBiEiaUyGQtBbcEftsMEbB+2Z2gZkVmtnsoPO054OutjGzYWb2YrDsW2Y21KJuDdZdZGYXBMtONLNZMf3dPxrchV03NsYyM3sLOC8myykxfc6/XXenrkiipcWdxSKNmAxscPcz4cPuiZ8Dprj71mDH/gvgCuBR4Nfu/oyZ5RL9IXUecAxwNNADmG9mrwTbHgscCWwA5gAnBgO23At8BigGnojJch1wrbvPCTrxq0jcxxb5iI4IJN0tAk43s1vM7GSinROOBl4Iusz4MdAv+HXe192fAXD3CncvB04CHg96Nd0MzAaODbY9z91L3T0CvAMMAkYCq9z9A4/e1h/bidwc4Pdm9i2gi7vXJPSTiwR0RCBpzd1XmNk44PPAz4GXgcXu/rFhC5vZTFMZ87yWJv69ufuvzezZIMscM/ucuydtp3jSduiIQNKamfUByt39EeBWYALQs278WjPLNrMjg9HXSs3snGB+OzPLA14FLggGvOlJdNSreY285TJgkJkNDaa/HJNlqLsvcvdbiPaYO7JVP6xIA3REIOluDHCrmUWAauAbQA3wp+B8QRbRUcIWE+1l9W4zuzlY9nyi/e6fALxLdISu77v7JjOrdyfu7hXB6F3Pmlk50UJSd7TxHTObRHQ0ssVEz1WIJJx6HxURSXNqGhIRSXMqBCIiaU6FQEQkzakQiIikORUCEZE0p0IgIpLmVAhERNLc/wMRnHzsO9LUqQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import control as ctl\n",
    "\n",
    "\n",
    "num = 1\n",
    "den = [1, 3, 2]\n",
    "original_tf = ctl.tf(num,den)\n",
    "\n",
    "print('ORIGINAL transfer function\\n')\n",
    "print(original_tf)\n",
    "T, yout = ctl.step_response(original_tf)\n",
    "plt.grid(True)\n",
    "plt.title('Original step response')\n",
    "plt.xlabel('seconds')\n",
    "plt.ylabel('amplitude')\n",
    "plt.plot(T,yout)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First i will show what effect $k_p$ has on the system\n",
    "\n",
    "The transfer function with the controller in it is \n",
    "\n",
    "$$T_f = \\frac{k_d s^2 + k_p s +k_i}{s^3+(k_d+3)s^2+(k_p + 2)s +k_i} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f51b2f5e340>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "kp_range = [3, 5, 10]\n",
    "ki = 0\n",
    "kd = 0\n",
    "\n",
    "for kp in kp_range:\n",
    "    num = [kd,kp,ki]\n",
    "    den = [1, kd+3,kp+2,ki]\n",
    "    final_tf = ctl.tf(num,den)\n",
    "    T, yout = ctl.step_response(final_tf)\n",
    "    plt.plot(T,yout,label=\"kp = \" +str(kp))\n",
    "plt.grid(True)\n",
    "plt.title('kp sweep')\n",
    "plt.xlabel('seconds')\n",
    "plt.ylabel('amplitude')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "as you can see when you increase kp it makes the gain higher but the output more wiggly\n",
    "\n",
    "now change ki while the others are constant"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f51b2301bb0>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "kp = 0\n",
    "ki_range = [0.5, 2, 3]\n",
    "kd = 0\n",
    "\n",
    "for ki in ki_range:\n",
    "    num = [kd,kp,ki]\n",
    "    den = [1, kd+3,kp+2,ki]\n",
    "    final_tf = ctl.tf(num,den)\n",
    "    T, yout = ctl.step_response(final_tf)\n",
    "    plt.plot(T,yout,label=\"ki = \" +str(ki))\n",
    "plt.grid(True)\n",
    "plt.title('ki sweep')\n",
    "plt.xlabel('seconds')\n",
    "plt.ylabel('amplitude')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This makes the signal a lot more wiggly. the values for ki have a big effect on the output. ki=10 makes the output oscillate violently. This indicates that the integral term is proportional to both magnitude of the error and the duration of the error\n",
    "\n",
    "Next we have the derivative term"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f51b226d7c0>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "kp = 0\n",
    "ki = 0\n",
    "kd_range = [0.5, 2, 3]\n",
    "\n",
    "for kd in kd_range:\n",
    "    num = [kd,kp,ki]\n",
    "    den = [1, kd+3,kp+2,ki]\n",
    "    final_tf = ctl.tf(num,den)\n",
    "    T, yout = ctl.step_response(final_tf)\n",
    "    plt.plot(T,yout,label=\"kd = \" +str(kd))\n",
    "plt.grid(True)\n",
    "plt.title('kd sweep')\n",
    "plt.xlabel('seconds')\n",
    "plt.ylabel('amplitude')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see the derivative value produces a spike then returns to the origin. this is used to smooth out the final state value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# system requirements\n",
    "\n",
    "The system requirements that we focused on in this class is\n",
    "\n",
    "Steady state error: the value at the final state of the system\n",
    "\n",
    "Overshoot: the percentage the signal goes over the target\n",
    "\n",
    "settling time: the time it takes for the system to settle. This can all be done with the step info command\n",
    "\n",
    "We we can change the values for the response of theses systems by either doing pole placement or PID control"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ORIGINAL transfer function\n",
      "\n",
      "\n",
      "      1\n",
      "-------------\n",
      "s^2 + 3 s + 2\n",
      "\n",
      "{'RiseTime': 2.571571571571572, 'SettlingTime': 4.51951951951952, 'SettlingMin': 0.44937395164782734, 'SettlingMax': 0.49908853379880513, 'Overshoot': 0.0, 'Undershoot': 0.0, 'Peak': 0.49908853379880513, 'PeakTime': 7.0, 'SteadyStateValue': 0.49908853379880513}\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "num = 1\n",
    "den = [1, 3, 2]\n",
    "original_tf = ctl.tf(num,den)\n",
    "\n",
    "print('ORIGINAL transfer function\\n')\n",
    "print(original_tf)\n",
    "T, yout = ctl.step_response(original_tf)\n",
    "plt.grid(True)\n",
    "plt.title('Original step response')\n",
    "plt.xlabel('seconds')\n",
    "plt.ylabel('amplitude')\n",
    "plt.plot(T,yout)\n",
    "print(ctl.step_info((original_tf)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'RiseTime': 2.4214192314733403, 'SettlingTime': 21.792773083260066, 'SettlingMin': 0.8160767402711524, 'SettlingMax': 1.3282062522996014, 'Overshoot': 32.894395513378825, 'Undershoot': 0.0, 'Peak': 1.3282062522996014, 'PeakTime': 4.55226815516988, 'SteadyStateValue': 0.9994448954515072}\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "kp_range = [2]\n",
    "ki_range = [3]\n",
    "kd_range = [2]\n",
    "\n",
    "for kd in kd_range:\n",
    "    for ki in ki_range:\n",
    "        for kd in kd_range:\n",
    "            num = [kd,kp,ki]\n",
    "            den = [1, kd+3,kp+2,ki]\n",
    "            final_tf = ctl.tf(num,den)\n",
    "            T, yout = ctl.step_response(final_tf)\n",
    "            plt.plot(T,yout,label=\"kd = \" +str(kd) + \" ki = \" +str(ki) +\" kd = \" +str(kd))\n",
    "plt.grid(True)\n",
    "plt.title('step_info proof plot')\n",
    "plt.xlabel('seconds')\n",
    "plt.ylabel('amplitude')\n",
    "plt.legend()\n",
    "print(ctl.step_info(final_tf))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "as you can see the step info is different proving you can change the step response by changing these values"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}