Reachable frequency response

reach_freq.ipynb

{
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  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
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     ]
    },
    {
     "data": {
      "text/plain": [
       "Plots.PlotlyJSBackend()"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import Pkg\n",
    "Pkg.activate(temp=true)\n",
    "Pkg.add([\"PlotlyJS\", \"Optim\", \"ControlSystems\", \"IntervalArithmetic\", \"LazySets\", \"Plots\"])\n",
    "\n",
    "import PlotlyJS\n",
    "import Optim\n",
    "using ControlSystems\n",
    "using IntervalArithmetic\n",
    "using LazySets\n",
    "using LinearAlgebra\n",
    "using Plots\n",
    "plotlyjs()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the Laplace variable `s`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = tf(\"s\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These will be our uncertain parameters"
   ]
  },
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   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 20 ± 0.05\n",
    "b = 1 ± 0.001;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define system transfer function using the uncertain parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "G = 7.6 * (s^2 + 0.1s + 0.55) / (s * (s+b) * (s+a) * (s^2 + 0.06s + 0.5));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate phase and gain of frequency response"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "ω = 10 .^ (-0.5:0.001:0.1)\n",
    "freq = freqrespv(G, ω)\n",
    "gain = @. 20log10(abs(freq))\n",
    "phase = @. atan(imag(freq)/real(freq))*180/π - 180;  # Manual hack to prevent wrapping"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create an interval box of phase and gain"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "phase_gain = phase .× gain;"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define requirements keepout zone ellipsoid. This is defined as a distance from the 'critical point' of -180° phase and 0 dB gain. Any frequency response that crosses this critical point will be unstable when the loop is closed. To ensure healthy robustness to unmodelled uncertainty, we require that our frequency response be outside of this ellipsoid."
   ]
  },
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   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "phase_req = 15.0\n",
    "gain_req = 3.0\n",
    "requirements_keepout = Ellipsoid([-180.0, 0.0], Diagonal([phase_req, gain_req]).^2);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate the intersection of the requirements keepout zone and the reach sets to get violations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "req_poly = overapproximate(requirements_keepout, HPolygon, 0.01)\n",
    "phase_gain_rect = convert.(Hyperrectangle, phase_gain)\n",
    "violations = intersect.(Ref(req_poly), phase_gain_rect);"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot frequency response reachable sets against requirements"
   ]
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  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
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