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    "### PID Controller\n",
    "\n",
    "A control system aims to keep the pendulum balanced using feedback.\n",
    "\n",
    "Main Elements\n",
    "\n",
    "- **Plant**: The physical system to be controlled (e.g., pendulum)\n",
    "- **Controller**: Algorithm producing control signals\n",
    "- **Sensor**: Measures the system state\n",
    "- **Actuator**: Applies control input (motor)\n",
    "- **Feedback**: Uses sensor data to adjust behavior\n",
    "\n",
    "Feedback Control\n",
    "\n",
    "Continuously reduces error by adjusting inputs based on outputs.\n",
    "\n",
    "- **Bang-bang**: All-or-nothing control; fast but unstable.\n",
    "- **PID**: Combines:\n",
    "  \n",
    " $$ \n",
    "  u(t) = K_p e(t) + K_i \\int e(t) \\, dt + K_d \\frac{de(t)}{dt}\n",
    " $$   \n",
    "\n",
    "- **LQR**: Uses optimal gains to minimize cost function:\n",
    "- \n",
    "   $$ \n",
    "  u = -Kx\n",
    "   $$ \n",
    "\n",
    "  where $x$ is the system state and $K$ is the gain matrix."
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