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   "cell_type": "markdown",
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   "source": [
    "#Problem 3.g\n",
    "\n",
    "Use the following code to simulate the system. Plug in your solutions for $A_{CL}$ and $B_{CL}$ and your values of K.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
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   "source": [
    "SIM_LENGTH = 100\n",
    "\n",
    "# YOUR CODE HERE#\n",
    "k1 = \n",
    "k2 = \n",
    "Ts = .1\n",
    "mu = -.2\n",
    "\n",
    "A = np.matrix([[1, Ts], [0, 2]])\n",
    "B = np.matrix([[.01], [.2]])\n",
    "K = np.matrix([k1, k2])\n",
    "\n",
    "x = np.zeros((2,SIM_LENGTH))\n",
    "xd = np.zeros((2,SIM_LENGTH))\n",
    "xd[0,:] = np.linspace(0 , SIM_LENGTH*Ts, SIM_LENGTH)\n",
    "xd[1,:] = np.ones(SIM_LENGTH)\n",
    "\n",
    "# YOUR CODE HERE #\n",
    "ACL = \n",
    "BCL = \n",
    "\n",
    "\n",
    "for i in range(0,SIM_LENGTH-1):\n",
    "    x[:,i+1] = list(np.add(np.dot(ACL, np.matrix(x[:,i]).T), np.dot(BCL,xd[:,i]).T))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now check your eigenvalues to make sure that your system will be stable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "print('ACL:')\n",
    "print(ACL)\n",
    "e = np.linalg.eig(ACL)\n",
    "print('eigenvalues:')\n",
    "print(e[0])\n",
    "print('eigenvectors:')\n",
    "print(e[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, let's look at the simultion and see how quickly it converges."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plt.plot(x[0,:].T)\n",
    "plt.plot(xd[0,:].T)\n",
    "plt.xlim([0,SIM_LENGTH-2])\n",
    "plt.legend(['simulation', 'desired'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plt.plot(x[1,:].T)\n",
    "plt.plot(xd[1,:].T)\n",
    "plt.xlim([0,SIM_LENGTH-2])\n",
    "plt.legend(['simulation', 'desired'])"
   ]
  }
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