{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "639f45ae-0ee8-426e-9d52-a7b9bb95d45a",
   "metadata": {},
   "source": [
    "# System Representation Gallery\n",
    "\n",
    "This Jupyter notebook creates different types of systems and generates a variety of representations (`__repr__`, `__str__`) for those systems that can be used to compare different versions of python-control.  It is mainly intended for uses by developers to make sure there are no unexpected changes in representation formats, but also has some interesting examples of different choices in system representation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "c4b80abe-59e4-4d76-a81c-6979a583e82d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0.10.1.dev324+g2fd3802a.d20241218'"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "import control as ct\n",
    "import control.flatsys as fs\n",
    "\n",
    "ct.__version__"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "035ebae9-7a4b-4079-8111-31f6c493c77c",
   "metadata": {},
   "source": [
    "## Text representations\n",
    "\n",
    "The code below shows what the output in various formats will look like in a terminal window."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "eab8cc0b-3e8a-4df8-acbd-258f006f44bb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "============================================================================\n",
      "                                Default repr\n",
      "============================================================================\n",
      "\n",
      "StateSpace: sys_ss, dt=0:\n",
      "-------------------------\n",
      "StateSpace(\n",
      "array([[ 0.,  1.],\n",
      "       [-4., -5.]]),\n",
      "array([[0.],\n",
      "       [1.]]),\n",
      "array([[-1.,  1.]]),\n",
      "array([[0.]]),\n",
      "name='sys_ss', states=2, outputs=1, inputs=1)\n",
      "----\n",
      "\n",
      "StateSpace: sys_dss, dt=0.1:\n",
      "----------------------------\n",
      "StateSpace(\n",
      "array([[ 0.98300988,  0.07817246],\n",
      "       [-0.31268983,  0.59214759]]),\n",
      "array([[0.00424753],\n",
      "       [0.07817246]]),\n",
      "array([[-1.,  1.]]),\n",
      "array([[0.]]),\n",
      "dt=0.1,\n",
      "name='sys_dss', states=2, outputs=1, inputs=1)\n",
      "----\n",
      "\n",
      "StateSpace: stateless, dt=None:\n",
      "-------------------------------\n",
      "StateSpace(\n",
      "array([], shape=(0, 0), dtype=float64),\n",
      "array([], shape=(0, 2), dtype=float64),\n",
      "array([], shape=(2, 0), dtype=float64),\n",
      "array([[1., 0.],\n",
      "       [0., 1.]]),\n",
      "dt=None,\n",
      "name='stateless', states=0, outputs=2, inputs=['u0', 'u1'])\n",
      "----\n",
      "\n",
      "TransferFunction: sys_ss$converted, dt=0:\n",
      "-----------------------------------------\n",
      "TransferFunction(\n",
      "array([ 1., -1.]),\n",
      "array([1., 5., 4.]),\n",
      "name='sys_ss$converted', outputs=1, inputs=1)\n",
      "----\n",
      "\n",
      "TransferFunction: sys_dss_poly, dt=0.1:\n",
      "---------------------------------------\n",
      "TransferFunction(\n",
      "array([ 0.07392493, -0.08176823]),\n",
      "array([ 1.        , -1.57515746,  0.60653066]),\n",
      "dt=0.1,\n",
      "name='sys_dss_poly', outputs=1, inputs=1)\n",
      "----\n",
      "\n",
      "TransferFunction: sys[3], dt=0:\n",
      "-------------------------------\n",
      "TransferFunction(\n",
      "array([1]),\n",
      "array([1, 0]),\n",
      "outputs=1, inputs=1)\n",
      "----\n",
      "\n",
      "TransferFunction: sys_mtf_zpk, dt=0:\n",
      "------------------------------------\n",
      "TransferFunction(\n",
      "[[array([ 1., -1.]), array([0.])],\n",
      " [array([1, 0]), array([1, 0])]],\n",
      "[[array([1., 5., 4.]), array([1.])],\n",
      " [array([1]), array([1, 2, 1])]],\n",
      "name='sys_mtf_zpk', outputs=2, inputs=2)\n",
      "----\n",
      "\n",
      "FrequencyResponseData: sys_ss$converted$sampled, dt=0:\n",
      "------------------------------------------------------\n",
      "FrequencyResponseData(\n",
      "array([[[-0.24365959+0.05559644j, -0.19198193+0.1589174j ,\n",
      "          0.05882353+0.23529412j,  0.1958042 -0.01105691j,\n",
      "          0.0508706 -0.07767156j]]]),\n",
      "array([ 0.1       ,  0.31622777,  1.        ,  3.16227766, 10.        ]),\n",
      "name='sys_ss$converted$sampled', outputs=1, inputs=1)\n",
      "----\n",
      "\n",
      "FrequencyResponseData: sys_dss_poly$sampled, dt=0.1:\n",
      "----------------------------------------------------\n",
      "FrequencyResponseData(\n",
      "array([[[-0.24337799+0.05673083j, -0.18944184+0.16166381j,\n",
      "          0.07043649+0.23113479j,  0.19038528-0.04416494j,\n",
      "          0.00286505-0.09595906j]]]),\n",
      "array([ 0.1       ,  0.31622777,  1.        ,  3.16227766, 10.        ])\n",
      "dt=0.1,\n",
      "name='sys_dss_poly$sampled', outputs=1, inputs=1)\n",
      "----\n",
      "\n",
      "FrequencyResponseData: sys_mtf_zpk$sampled, dt=0:\n",
      "-------------------------------------------------\n",
      "FrequencyResponseData(\n",
      "array([[[-0.24365959 +0.05559644j, -0.19198193 +0.1589174j ,\n",
      "          0.05882353 +0.23529412j,  0.1958042  -0.01105691j,\n",
      "          0.0508706  -0.07767156j],\n",
      "        [ 0.         +0.j        ,  0.         +0.j        ,\n",
      "          0.         +0.j        ,  0.         +0.j        ,\n",
      "          0.         +0.j        ]],\n",
      "\n",
      "       [[ 0.         +0.1j       ,  0.         +0.31622777j,\n",
      "          0.         +1.j        ,  0.         +3.16227766j,\n",
      "          0.        +10.j        ],\n",
      "        [ 0.01960592 +0.09704931j,  0.16528926 +0.23521074j,\n",
      "          0.5        +0.j        ,  0.16528926 -0.23521074j,\n",
      "          0.01960592 -0.09704931j]]]),\n",
      "array([ 0.1       ,  0.31622777,  1.        ,  3.16227766, 10.        ]),\n",
      "name='sys_mtf_zpk$sampled', outputs=2, inputs=2)\n",
      "----\n",
      "\n",
      "NonlinearIOSystem: sys_nl, dt=0:\n",
      "--------------------------------\n",
      "<NonlinearIOSystem sys_nl: ['u[0]'] -> ['y[0]']>\n",
      "----\n",
      "\n",
      "NonlinearIOSystem: sys_dnl, dt=0.1:\n",
      "-----------------------------------\n",
      "<NonlinearIOSystem sys_dnl: ['u[0]'] -> ['y[0]'], dt=0.1>\n",
      "----\n",
      "\n",
      "InterconnectedSystem: sys_ic, dt=0:\n",
      "-----------------------------------\n",
      "<InterconnectedSystem sys_ic: ['r[0]', 'r[1]'] -> ['y[0]', 'y[1]']>\n",
      "----\n",
      "\n",
      "LinearICSystem: sys_ic, dt=0:\n",
      "-----------------------------\n",
      "<LinearICSystem sys_ic: ['r[0]', 'r[1]'] -> ['y[0]', 'y[1]']>\n",
      "----\n",
      "\n",
      "FlatSystem: sys_fs, dt=0:\n",
      "-------------------------\n",
      "<FlatSystem sys_fs: ['u[0]'] -> ['y[0]']>\n",
      "----\n",
      "\n",
      "FlatSystem: sys_fsnl, dt=0:\n",
      "---------------------------\n",
      "<FlatSystem sys_fsnl: ['u[0]'] -> ['y[0]']>\n",
      "----\n",
      "\n",
      "============================================================================\n",
      "                            Default str (print)\n",
      "============================================================================\n",
      "\n",
      "StateSpace: sys_ss, dt=0:\n",
      "-------------------------\n",
      "<StateSpace>: sys_ss\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "States (2): ['x[0]', 'x[1]']\n",
      "\n",
      "A = [[ 0.  1.]\n",
      "     [-4. -5.]]\n",
      "\n",
      "B = [[0.]\n",
      "     [1.]]\n",
      "\n",
      "C = [[-1.  1.]]\n",
      "\n",
      "D = [[0.]]\n",
      "----\n",
      "\n",
      "StateSpace: sys_dss, dt=0.1:\n",
      "----------------------------\n",
      "<StateSpace>: sys_dss\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "States (2): ['x[0]', 'x[1]']\n",
      "dt = 0.1\n",
      "\n",
      "A = [[ 0.98300988  0.07817246]\n",
      "     [-0.31268983  0.59214759]]\n",
      "\n",
      "B = [[0.00424753]\n",
      "     [0.07817246]]\n",
      "\n",
      "C = [[-1.  1.]]\n",
      "\n",
      "D = [[0.]]\n",
      "----\n",
      "\n",
      "StateSpace: stateless, dt=None:\n",
      "-------------------------------\n",
      "<StateSpace>: stateless\n",
      "Inputs (2): ['u0', 'u1']\n",
      "Outputs (2): ['y[0]', 'y[1]']\n",
      "States (0): []\n",
      "dt = None\n",
      "\n",
      "A = []\n",
      "\n",
      "B = []\n",
      "\n",
      "C = []\n",
      "\n",
      "D = [[1. 0.]\n",
      "     [0. 1.]]\n",
      "----\n",
      "\n",
      "TransferFunction: sys_ss$converted, dt=0:\n",
      "-----------------------------------------\n",
      "<TransferFunction>: sys_ss$converted\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "\n",
      "      s - 1\n",
      "  -------------\n",
      "  s^2 + 5 s + 4\n",
      "----\n",
      "\n",
      "TransferFunction: sys_dss_poly, dt=0.1:\n",
      "---------------------------------------\n",
      "<TransferFunction>: sys_dss_poly\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "dt = 0.1\n",
      "\n",
      "   0.07392 z - 0.08177\n",
      "  ----------------------\n",
      "  z^2 - 1.575 z + 0.6065\n",
      "----\n",
      "\n",
      "TransferFunction: sys[3], dt=0:\n",
      "-------------------------------\n",
      "<TransferFunction>: sys[3]\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "\n",
      "  1\n",
      "  -\n",
      "  s\n",
      "----\n",
      "\n",
      "TransferFunction: sys_mtf_zpk, dt=0:\n",
      "------------------------------------\n",
      "<TransferFunction>: sys_mtf_zpk\n",
      "Inputs (2): ['u[0]', 'u[1]']\n",
      "Outputs (2): ['y[0]', 'y[1]']\n",
      "\n",
      "Input 1 to output 1:\n",
      "\n",
      "       s - 1\n",
      "  ---------------\n",
      "  (s + 1) (s + 4)\n",
      "\n",
      "Input 1 to output 2:\n",
      "\n",
      "  s\n",
      "  -\n",
      "  1\n",
      "\n",
      "Input 2 to output 1:\n",
      "\n",
      "  0\n",
      "  -\n",
      "  1\n",
      "\n",
      "Input 2 to output 2:\n",
      "\n",
      "         s\n",
      "  ---------------\n",
      "  (s + 1) (s + 1)\n",
      "----\n",
      "\n",
      "FrequencyResponseData: sys_ss$converted$sampled, dt=0:\n",
      "------------------------------------------------------\n",
      "<FrequencyResponseData>: sys_ss$converted$sampled\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "\n",
      "Freq [rad/s]  Response\n",
      "------------  ---------------------\n",
      "       0.100     -0.2437   +0.0556j\n",
      "       0.316      -0.192   +0.1589j\n",
      "       1.000     0.05882   +0.2353j\n",
      "       3.162      0.1958  -0.01106j\n",
      "      10.000     0.05087  -0.07767j\n",
      "----\n",
      "\n",
      "FrequencyResponseData: sys_dss_poly$sampled, dt=0.1:\n",
      "----------------------------------------------------\n",
      "<FrequencyResponseData>: sys_dss_poly$sampled\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "dt = 0.1\n",
      "\n",
      "Freq [rad/s]  Response\n",
      "------------  ---------------------\n",
      "       0.100     -0.2434  +0.05673j\n",
      "       0.316     -0.1894   +0.1617j\n",
      "       1.000     0.07044   +0.2311j\n",
      "       3.162      0.1904  -0.04416j\n",
      "      10.000    0.002865  -0.09596j\n",
      "----\n",
      "\n",
      "FrequencyResponseData: sys_mtf_zpk$sampled, dt=0:\n",
      "-------------------------------------------------\n",
      "<FrequencyResponseData>: sys_mtf_zpk$sampled\n",
      "Inputs (2): ['u[0]', 'u[1]']\n",
      "Outputs (2): ['y[0]', 'y[1]']\n",
      "\n",
      "Input 1 to output 1:\n",
      "\n",
      "  Freq [rad/s]  Response\n",
      "  ------------  ---------------------\n",
      "         0.100     -0.2437   +0.0556j\n",
      "         0.316      -0.192   +0.1589j\n",
      "         1.000     0.05882   +0.2353j\n",
      "         3.162      0.1958  -0.01106j\n",
      "        10.000     0.05087  -0.07767j\n",
      "\n",
      "Input 1 to output 2:\n",
      "\n",
      "  Freq [rad/s]  Response\n",
      "  ------------  ---------------------\n",
      "         0.100           0      +0.1j\n",
      "         0.316           0   +0.3162j\n",
      "         1.000           0        +1j\n",
      "         3.162           0    +3.162j\n",
      "        10.000           0       +10j\n",
      "\n",
      "Input 2 to output 1:\n",
      "\n",
      "  Freq [rad/s]  Response\n",
      "  ------------  ---------------------\n",
      "         0.100           0        +0j\n",
      "         0.316           0        +0j\n",
      "         1.000           0        +0j\n",
      "         3.162           0        +0j\n",
      "        10.000           0        +0j\n",
      "\n",
      "Input 2 to output 2:\n",
      "\n",
      "  Freq [rad/s]  Response\n",
      "  ------------  ---------------------\n",
      "         0.100     0.01961  +0.09705j\n",
      "         0.316      0.1653   +0.2352j\n",
      "         1.000         0.5        +0j\n",
      "         3.162      0.1653   -0.2352j\n",
      "        10.000     0.01961  -0.09705j\n",
      "----\n",
      "\n",
      "NonlinearIOSystem: sys_nl, dt=0:\n",
      "--------------------------------\n",
      "<NonlinearIOSystem>: sys_nl\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "States (2): ['x[0]', 'x[1]']\n",
      "Parameters: ['a', 'b']\n",
      "\n",
      "Update: <function nl_update at 0x1201863e0>\n",
      "Output: <function nl_output at 0x120186480>\n",
      "----\n",
      "\n",
      "NonlinearIOSystem: sys_dnl, dt=0.1:\n",
      "-----------------------------------\n",
      "<NonlinearIOSystem>: sys_dnl\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "States (2): ['x[0]', 'x[1]']\n",
      "dt = 0.1\n",
      "\n",
      "Update: <function dnl_update at 0x120186520>\n",
      "Output: <function dnl_output at 0x1201865c0>\n",
      "----\n",
      "\n",
      "InterconnectedSystem: sys_ic, dt=0:\n",
      "-----------------------------------\n",
      "<InterconnectedSystem>: sys_ic\n",
      "Inputs (2): ['r[0]', 'r[1]']\n",
      "Outputs (2): ['y[0]', 'y[1]']\n",
      "States (2): ['proc_nl_x[0]', 'proc_nl_x[1]']\n",
      "\n",
      "Subsystems (2):\n",
      " * <NonlinearIOSystem proc_nl: ['u[0]', 'u[1]'] -> ['y[0]', 'y[1]']>\n",
      " * <NonlinearIOSystem ctrl_nl: ['u[0]', 'u[1]'] -> ['y[0]', 'y[1]']>\n",
      "\n",
      "Connections:\n",
      " * proc_nl.u[0] <- ctrl_nl.y[0]\n",
      " * proc_nl.u[1] <- ctrl_nl.y[1]\n",
      " * ctrl_nl.u[0] <- -proc_nl.y[0] + r[0]\n",
      " * ctrl_nl.u[1] <- -proc_nl.y[1] + r[1]\n",
      "\n",
      "Outputs:\n",
      " * y[0] <- proc_nl.y[0]\n",
      " * y[1] <- proc_nl.y[1]\n",
      "----\n",
      "\n",
      "LinearICSystem: sys_ic, dt=0:\n",
      "-----------------------------\n",
      "<LinearICSystem>: sys_ic\n",
      "Inputs (2): ['r[0]', 'r[1]']\n",
      "Outputs (2): ['y[0]', 'y[1]']\n",
      "States (2): ['proc_x[0]', 'proc_x[1]']\n",
      "\n",
      "Subsystems (2):\n",
      " * <StateSpace proc: ['u[0]', 'u[1]'] -> ['y[0]', 'y[1]']>\n",
      " * <StateSpace ctrl: ['u[0]', 'u[1]'] -> ['y[0]', 'y[1]'], dt=None>\n",
      "\n",
      "Connections:\n",
      " * proc.u[0] <- ctrl.y[0]\n",
      " * proc.u[1] <- ctrl.y[1]\n",
      " * ctrl.u[0] <- -proc.y[0] + r[0]\n",
      " * ctrl.u[1] <- -proc.y[1] + r[1]\n",
      "\n",
      "Outputs:\n",
      " * y[0] <- proc.y[0]\n",
      " * y[1] <- proc.y[1]\n",
      "\n",
      "A = [[-2.  3.]\n",
      "     [-1. -5.]]\n",
      "\n",
      "B = [[-2.  0.]\n",
      "     [ 0. -3.]]\n",
      "\n",
      "C = [[-1.  1.]\n",
      "     [ 1.  0.]]\n",
      "\n",
      "D = [[0. 0.]\n",
      "     [0. 0.]]\n",
      "----\n",
      "\n",
      "FlatSystem: sys_fs, dt=0:\n",
      "-------------------------\n",
      "<FlatSystem>: sys_fs\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "States (2): ['x[0]', 'x[1]']\n",
      "\n",
      "Update: <bound method FlatSystem._flat_updfcn of <FlatSystem sys_fs: ['u[0]'] -> ['y[0]']>>\n",
      "Output: <bound method FlatSystem._flat_outfcn of <FlatSystem sys_fs: ['u[0]'] -> ['y[0]']>>\n",
      "\n",
      "Forward: <function fs_forward at 0x120186ca0>\n",
      "Reverse: <function fs_reverse at 0x120186d40>\n",
      "----\n",
      "\n",
      "FlatSystem: sys_fsnl, dt=0:\n",
      "---------------------------\n",
      "<FlatSystem>: sys_fsnl\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "States (2): ['x[0]', 'x[1]']\n",
      "\n",
      "Update: <function nl_update at 0x1201863e0>\n",
      "Output: <function nl_output at 0x120186480>\n",
      "\n",
      "Forward: <function fs_forward at 0x120186ca0>\n",
      "Reverse: <function fs_reverse at 0x120186d40>\n",
      "----\n",
      "\n",
      "============================================================================\n",
      "                             repr_format='info'\n",
      "============================================================================\n",
      "\n",
      "StateSpace: sys_ss, dt=0:\n",
      "-------------------------\n",
      "<StateSpace sys_ss: ['u[0]'] -> ['y[0]']>\n",
      "----\n",
      "\n",
      "StateSpace: sys_dss, dt=0.1:\n",
      "----------------------------\n",
      "<StateSpace sys_dss: ['u[0]'] -> ['y[0]'], dt=0.1>\n",
      "----\n",
      "\n",
      "StateSpace: stateless, dt=None:\n",
      "-------------------------------\n",
      "<StateSpace stateless: ['u0', 'u1'] -> ['y[0]', 'y[1]'], dt=None>\n",
      "----\n",
      "\n",
      "TransferFunction: sys_ss$converted, dt=0:\n",
      "-----------------------------------------\n",
      "<TransferFunction sys_ss$converted: ['u[0]'] -> ['y[0]']>\n",
      "----\n",
      "\n",
      "TransferFunction: sys_dss_poly, dt=0.1:\n",
      "---------------------------------------\n",
      "<TransferFunction sys_dss_poly: ['u[0]'] -> ['y[0]'], dt=0.1>\n",
      "----\n",
      "\n",
      "TransferFunction: sys[3], dt=0:\n",
      "-------------------------------\n",
      "<TransferFunction sys[3]: ['u[0]'] -> ['y[0]']>\n",
      "----\n",
      "\n",
      "TransferFunction: sys_mtf_zpk, dt=0:\n",
      "------------------------------------\n",
      "<TransferFunction sys_mtf_zpk: ['u[0]', 'u[1]'] -> ['y[0]', 'y[1]']>\n",
      "----\n",
      "\n",
      "FrequencyResponseData: sys_ss$converted$sampled, dt=0:\n",
      "------------------------------------------------------\n",
      "<FrequencyResponseData sys_ss$converted$sampled: ['u[0]'] -> ['y[0]']>\n",
      "----\n",
      "\n",
      "FrequencyResponseData: sys_dss_poly$sampled, dt=0.1:\n",
      "----------------------------------------------------\n",
      "<FrequencyResponseData sys_dss_poly$sampled: ['u[0]'] -> ['y[0]'], dt=0.1>\n",
      "----\n",
      "\n",
      "FrequencyResponseData: sys_mtf_zpk$sampled, dt=0:\n",
      "-------------------------------------------------\n",
      "<FrequencyResponseData sys_mtf_zpk$sampled: ['u[0]', 'u[1]'] -> ['y[0]', 'y[1]']>\n",
      "----\n",
      "\n",
      "NonlinearIOSystem: sys_nl, dt=0:\n",
      "--------------------------------\n",
      "<NonlinearIOSystem sys_nl: ['u[0]'] -> ['y[0]']>\n",
      "----\n",
      "\n",
      "NonlinearIOSystem: sys_dnl, dt=0.1:\n",
      "-----------------------------------\n",
      "<NonlinearIOSystem sys_dnl: ['u[0]'] -> ['y[0]'], dt=0.1>\n",
      "----\n",
      "\n",
      "InterconnectedSystem: sys_ic, dt=0:\n",
      "-----------------------------------\n",
      "<InterconnectedSystem sys_ic: ['r[0]', 'r[1]'] -> ['y[0]', 'y[1]']>\n",
      "----\n",
      "\n",
      "LinearICSystem: sys_ic, dt=0:\n",
      "-----------------------------\n",
      "<LinearICSystem sys_ic: ['r[0]', 'r[1]'] -> ['y[0]', 'y[1]']>\n",
      "----\n",
      "\n",
      "FlatSystem: sys_fs, dt=0:\n",
      "-------------------------\n",
      "<FlatSystem sys_fs: ['u[0]'] -> ['y[0]']>\n",
      "----\n",
      "\n",
      "FlatSystem: sys_fsnl, dt=0:\n",
      "---------------------------\n",
      "<FlatSystem sys_fsnl: ['u[0]'] -> ['y[0]']>\n",
      "----\n",
      "\n",
      "============================================================================\n",
      "                        iosys.repr_show_count=False\n",
      "============================================================================\n",
      "\n",
      "StateSpace: sys_ss, dt=0:\n",
      "-------------------------\n",
      "StateSpace(\n",
      "array([[ 0.,  1.],\n",
      "       [-4., -5.]]),\n",
      "array([[0.],\n",
      "       [1.]]),\n",
      "array([[-1.,  1.]]),\n",
      "array([[0.]]),\n",
      "name='sys_ss')\n",
      "----\n",
      "\n",
      "StateSpace: sys_dss, dt=0.1:\n",
      "----------------------------\n",
      "StateSpace(\n",
      "array([[ 0.98300988,  0.07817246],\n",
      "       [-0.31268983,  0.59214759]]),\n",
      "array([[0.00424753],\n",
      "       [0.07817246]]),\n",
      "array([[-1.,  1.]]),\n",
      "array([[0.]]),\n",
      "dt=0.1,\n",
      "name='sys_dss')\n",
      "----\n",
      "\n",
      "StateSpace: stateless, dt=None:\n",
      "-------------------------------\n",
      "StateSpace(\n",
      "array([], shape=(0, 0), dtype=float64),\n",
      "array([], shape=(0, 2), dtype=float64),\n",
      "array([], shape=(2, 0), dtype=float64),\n",
      "array([[1., 0.],\n",
      "       [0., 1.]]),\n",
      "dt=None,\n",
      "name='stateless', inputs=['u0', 'u1'])\n",
      "----\n",
      "\n",
      "LinearICSystem: sys_ic, dt=0:\n",
      "-----------------------------\n",
      "<LinearICSystem sys_ic: ['r[0]', 'r[1]'] -> ['y[0]', 'y[1]']>\n",
      "----\n",
      "\n",
      "============================================================================\n",
      "                  xferfcn.display_format=zpk, str (print)\n",
      "============================================================================\n",
      "\n",
      "TransferFunction: sys_ss$converted, dt=0:\n",
      "-----------------------------------------\n",
      "<TransferFunction>: sys_ss$converted\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "\n",
      "       s - 1\n",
      "  ---------------\n",
      "  (s + 1) (s + 4)\n",
      "----\n",
      "\n",
      "TransferFunction: sys_dss_poly, dt=0.1:\n",
      "---------------------------------------\n",
      "<TransferFunction>: sys_dss_poly\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "dt = 0.1\n",
      "\n",
      "   0.07392 z - 0.08177\n",
      "  ----------------------\n",
      "  z^2 - 1.575 z + 0.6065\n",
      "----\n",
      "\n",
      "TransferFunction: sys[3], dt=0:\n",
      "-------------------------------\n",
      "<TransferFunction>: sys[3]\n",
      "Inputs (1): ['u[0]']\n",
      "Outputs (1): ['y[0]']\n",
      "\n",
      "  1\n",
      "  -\n",
      "  s\n",
      "----\n",
      "\n",
      "TransferFunction: sys_mtf_zpk, dt=0:\n",
      "------------------------------------\n",
      "<TransferFunction>: sys_mtf_zpk\n",
      "Inputs (2): ['u[0]', 'u[1]']\n",
      "Outputs (2): ['y[0]', 'y[1]']\n",
      "\n",
      "Input 1 to output 1:\n",
      "\n",
      "       s - 1\n",
      "  ---------------\n",
      "  (s + 1) (s + 4)\n",
      "\n",
      "Input 1 to output 2:\n",
      "\n",
      "  s\n",
      "  -\n",
      "  1\n",
      "\n",
      "Input 2 to output 1:\n",
      "\n",
      "  0\n",
      "  -\n",
      "  1\n",
      "\n",
      "Input 2 to output 2:\n",
      "\n",
      "         s\n",
      "  ---------------\n",
      "  (s + 1) (s + 1)\n",
      "----\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Grab system definitions (and generate various representations as text)\n",
    "from repr_gallery import *"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19f146a3-c036-4ff6-8425-c201fba14ec7",
   "metadata": {},
   "source": [
    "## Jupyter notebook (HTML/LaTeX) representations\n",
    "\n",
    "The following representations are generated using the `_html_repr_` method in selected types of systems.  Only those systems that have unique displays are shown."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "16ff8d11-e793-456a-bf27-ae4cc0dd1e3b",
   "metadata": {},
   "source": [
    "### Continuous time state space systems"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "c1ca661d-10f3-45be-8619-c3e143bb4b4c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&lt;StateSpace sys_ss: ['u[0]'] -&gt; ['y[0]']&gt;\n",
       "$$\n",
       "\\left[\\begin{array}{rllrll|rll}\n",
       "0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "-4\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&-5\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\hline\n",
       "-1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "$$"
      ],
      "text/plain": [
       "StateSpace(\n",
       "array([[ 0.,  1.],\n",
       "       [-4., -5.]]),\n",
       "array([[0.],\n",
       "       [1.]]),\n",
       "array([[-1.,  1.]]),\n",
       "array([[0.]]),\n",
       "name='sys_ss', states=2, outputs=1, inputs=1)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sys_ss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b10b4db3-a8c0-4a2c-a19d-a09fef3dc25f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&lt;StateSpace sys_ss: ['u[0]'] -&gt; ['y[0]']&gt;\n",
       "$$\n",
       "\\begin{array}{ll}\n",
       "A = \\left[\\begin{array}{rllrll}\n",
       "0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "-4\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&-5\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "&\n",
       "B = \\left[\\begin{array}{rll}\n",
       "0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "\\\\\n",
       "C = \\left[\\begin{array}{rllrll}\n",
       "-1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "&\n",
       "D = \\left[\\begin{array}{rll}\n",
       "0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "\\end{array}\n",
       "$$"
      ],
      "text/plain": [
       "StateSpace(\n",
       "array([[ 0.,  1.],\n",
       "       [-4., -5.]]),\n",
       "array([[0.],\n",
       "       [1.]]),\n",
       "array([[-1.,  1.]]),\n",
       "array([[0.]]),\n",
       "name='sys_ss', states=2, outputs=1, inputs=1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "with ct.config.defaults({'statesp.latex_repr_type': 'separate'}):\n",
    "    display(sys_ss)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "0537f6fe-a155-4c49-be7c-413608a03887",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&lt;StateSpace sys_ss: ['u[0]'] -&gt; ['y[0]']&gt;\n",
       "$$\n",
       "\\begin{array}{ll}\n",
       "A = \\left[\\begin{array}{rllrll}\n",
       "  0.&\\hspace{-1em}0000&\\hspace{-1em}\\phantom{\\cdot}&  1.&\\hspace{-1em}0000&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       " -4.&\\hspace{-1em}0000&\\hspace{-1em}\\phantom{\\cdot}& -5.&\\hspace{-1em}0000&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "&\n",
       "B = \\left[\\begin{array}{rll}\n",
       "  0.&\\hspace{-1em}0000&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "  1.&\\hspace{-1em}0000&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "\\\\\n",
       "C = \\left[\\begin{array}{rllrll}\n",
       " -1.&\\hspace{-1em}0000&\\hspace{-1em}\\phantom{\\cdot}&  1.&\\hspace{-1em}0000&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "&\n",
       "D = \\left[\\begin{array}{rll}\n",
       "  0.&\\hspace{-1em}0000&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "\\end{array}\n",
       "$$"
      ],
      "text/plain": [
       "StateSpace(\n",
       "array([[ 0.,  1.],\n",
       "       [-4., -5.]]),\n",
       "array([[0.],\n",
       "       [1.]]),\n",
       "array([[-1.,  1.]]),\n",
       "array([[0.]]),\n",
       "name='sys_ss', states=2, outputs=1, inputs=1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "with ct.config.defaults({\n",
    "    'statesp.latex_repr_type': 'separate',\n",
    "    'statesp.latex_num_format': '8.4f'}):\n",
    "    display(sys_ss)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fa75f040-633d-401c-ba96-e688713d5a2d",
   "metadata": {},
   "source": [
    "### Stateless state space system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "5a71e38c-9880-4af7-82e0-49f074653e94",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&lt;StateSpace stateless: ['u0', 'u1'] -&gt; ['y[0]', 'y[1]'], dt=None&gt;\n",
       "$$\n",
       "\\left[\\begin{array}{rllrll}\n",
       "1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "$$"
      ],
      "text/plain": [
       "StateSpace(\n",
       "array([], shape=(0, 0), dtype=float64),\n",
       "array([], shape=(0, 2), dtype=float64),\n",
       "array([], shape=(2, 0), dtype=float64),\n",
       "array([[1., 0.],\n",
       "       [0., 1.]]),\n",
       "dt=None,\n",
       "name='stateless', states=0, outputs=2, inputs=['u0', 'u1'])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sys_ss0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "7ddbd638-9338-4204-99bc-792f35e14874",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&lt;StateSpace stateless: ['u0', 'u1'] -&gt; ['y[0]', 'y[1]'], dt=None&gt;\n",
       "$$\n",
       "\\begin{array}{ll}\n",
       "D = \\left[\\begin{array}{rllrll}\n",
       "1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "\\end{array}\n",
       "$$"
      ],
      "text/plain": [
       "StateSpace(\n",
       "array([], shape=(0, 0), dtype=float64),\n",
       "array([], shape=(0, 2), dtype=float64),\n",
       "array([], shape=(2, 0), dtype=float64),\n",
       "array([[1., 0.],\n",
       "       [0., 1.]]),\n",
       "dt=None,\n",
       "name='stateless', states=0, outputs=2, inputs=['u0', 'u1'])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "with ct.config.defaults({'statesp.latex_repr_type': 'separate'}):\n",
    "    display(sys_ss0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "06c5d470-0768-4628-b2ea-d2387525ed80",
   "metadata": {},
   "source": [
    "### Discrete time state space system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "e7b9b438-28e3-453e-9860-06ff75b7af10",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&lt;StateSpace sys_dss: ['u[0]'] -&gt; ['y[0]'], dt=0.1&gt;\n",
       "$$\n",
       "\\left[\\begin{array}{rllrll|rll}\n",
       "0.&\\hspace{-1em}983&\\hspace{-1em}\\phantom{\\cdot}&0.&\\hspace{-1em}0782&\\hspace{-1em}\\phantom{\\cdot}&0.&\\hspace{-1em}00425&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "-0.&\\hspace{-1em}313&\\hspace{-1em}\\phantom{\\cdot}&0.&\\hspace{-1em}592&\\hspace{-1em}\\phantom{\\cdot}&0.&\\hspace{-1em}0782&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\hline\n",
       "-1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "$$"
      ],
      "text/plain": [
       "StateSpace(\n",
       "array([[ 0.98300988,  0.07817246],\n",
       "       [-0.31268983,  0.59214759]]),\n",
       "array([[0.00424753],\n",
       "       [0.07817246]]),\n",
       "array([[-1.,  1.]]),\n",
       "array([[0.]]),\n",
       "dt=0.1,\n",
       "name='sys_dss', states=2, outputs=1, inputs=1)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sys_dss"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7719e725-9d38-4f2a-a142-0ebc090e74e4",
   "metadata": {},
   "source": [
    "### \"Stateless\" state space system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "938fd795-f402-4491-b2c9-eb42c458e1e1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&lt;StateSpace stateless: ['u0', 'u1'] -&gt; ['y[0]', 'y[1]'], dt=None&gt;\n",
       "$$\n",
       "\\left[\\begin{array}{rllrll}\n",
       "1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "$$"
      ],
      "text/plain": [
       "StateSpace(\n",
       "array([], shape=(0, 0), dtype=float64),\n",
       "array([], shape=(0, 2), dtype=float64),\n",
       "array([], shape=(2, 0), dtype=float64),\n",
       "array([[1., 0.],\n",
       "       [0., 1.]]),\n",
       "dt=None,\n",
       "name='stateless', states=0, outputs=2, inputs=['u0', 'u1'])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sys_ss0"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c620f1a1-40ff-4320-9f62-21bff9ab308e",
   "metadata": {},
   "source": [
    "### Continuous time transfer function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "e364e6eb-0cfa-486a-8b95-ff9c6c41a091",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&lt;TransferFunction sys_ss\\$converted: ['u[0]'] -&gt; ['y[0]']&gt;\n",
       "$$\\dfrac{s - 1}{s^2 + 5 s + 4}$$"
      ],
      "text/plain": [
       "TransferFunction(\n",
       "array([ 1., -1.]),\n",
       "array([1., 5., 4.]),\n",
       "name='sys_ss$converted', outputs=1, inputs=1)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sys_tf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "af9959fd-90eb-4287-93ee-416cd13fde50",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&lt;TransferFunction sys_ss\\$converted: ['u[0]'] -&gt; ['y[0]']&gt;\n",
       "$$\\dfrac{s - 1}{(s + 1) (s + 4)}$$"
      ],
      "text/plain": [
       "TransferFunction(\n",
       "array([ 1., -1.]),\n",
       "array([1., 5., 4.]),\n",
       "name='sys_ss$converted', outputs=1, inputs=1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "with ct.config.defaults({'xferfcn.display_format': 'zpk'}):\n",
    "    display(sys_tf)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7bf40707-f84c-4e19-b310-5ec9811faf42",
   "metadata": {},
   "source": [
    "### MIMO transfer function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "b2db2d1c-893b-43a1-aab0-a5f6d059f3f9",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/murray/miniconda3/envs/python3.13-slycot/lib/python3.13/site-packages/scipy/signal/_filter_design.py:1112: BadCoefficients: Badly conditioned filter coefficients (numerator): the results may be meaningless\n",
      "  b, a = normalize(b, a)\n",
      "/Users/murray/miniconda3/envs/python3.13-slycot/lib/python3.13/site-packages/scipy/signal/_filter_design.py:1116: RuntimeWarning: invalid value encountered in divide\n",
      "  b /= b[0]\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "&lt;TransferFunction sys_mtf_zpk: ['u[0]', 'u[1]'] -&gt; ['y[0]', 'y[1]']&gt;\n",
       "$$\\begin{bmatrix}\\dfrac{s - 1}{(s + 1) (s + 4)}&\\dfrac{0}{1}\\\\\\dfrac{s}{1}&\\dfrac{s}{(s + 1) (s + 1)}\\\\ \\end{bmatrix}$$"
      ],
      "text/plain": [
       "TransferFunction(\n",
       "[[array([ 1., -1.]), array([0.])],\n",
       " [array([1, 0]), array([1, 0])]],\n",
       "[[array([1., 5., 4.]), array([1.])],\n",
       " [array([1]), array([1, 2, 1])]],\n",
       "name='sys_mtf_zpk', outputs=2, inputs=2)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sys_mtf  # SciPy generates a warning due to 0 numerator in 1, 2 entry"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ef78a05e-9a63-4e22-afae-66ac8ec457c2",
   "metadata": {},
   "source": [
    "### Discrete time transfer function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "365c9b4a-2af3-42e5-ae5d-f2d7d989104b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&lt;TransferFunction sys_dss_poly: ['u[0]'] -&gt; ['y[0]'], dt=0.1&gt;\n",
       "$$\\dfrac{0.07392 z - 0.08177}{z^2 - 1.575 z + 0.6065}$$"
      ],
      "text/plain": [
       "TransferFunction(\n",
       "array([ 0.07392493, -0.08176823]),\n",
       "array([ 1.        , -1.57515746,  0.60653066]),\n",
       "dt=0.1,\n",
       "name='sys_dss_poly', outputs=1, inputs=1)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sys_dtf"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b49fa8ab-c3af-48d1-b160-790c9f4d3c6e",
   "metadata": {},
   "source": [
    "### Linear interconnected system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "78d21969-4615-4a47-b449-7a08138dc319",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&lt;LinearICSystem sys_ic: ['r[0]', 'r[1]'] -&gt; ['y[0]', 'y[1]']&gt;\n",
       "$$\n",
       "\\left[\\begin{array}{rllrll|rllrll}\n",
       "-2\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&3\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&-2\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "-1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&-5\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&-3\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\hline\n",
       "-1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n",
       "\\end{array}\\right]\n",
       "$$"
      ],
      "text/plain": [
       "<LinearICSystem sys_ic: ['r[0]', 'r[1]'] -> ['y[0]', 'y[1]']>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sys_lic"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bee65cd5-d9b5-46af-aee5-26a6a4679939",
   "metadata": {},
   "source": [
    "### Non-HTML capable system representations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "e5486349-2bd3-4015-ad17-a5b8e8ec0447",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FrequencyResponseData(\n",
       "array([[[-0.24365959+0.05559644j, -0.19198193+0.1589174j ,\n",
       "          0.05882353+0.23529412j,  0.1958042 -0.01105691j,\n",
       "          0.0508706 -0.07767156j]]]),\n",
       "array([ 0.1       ,  0.31622777,  1.        ,  3.16227766, 10.        ]),\n",
       "name='sys_ss$converted$sampled', outputs=1, inputs=1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<NonlinearIOSystem sys_nl: ['u[0]'] -> ['y[0]']>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<InterconnectedSystem sys_ic: ['r[0]', 'r[1]'] -> ['y[0]', 'y[1]']>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<FlatSystem sys_fs: ['u[0]'] -> ['y[0]']>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for sys in [sys_frd, sys_nl, sys_ic, sys_fs]:\n",
    "    display(sys)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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