PID Controller

A control system aims to keep the pendulum balanced using feedback.

Main Elements

• Plant: The physical system to be controlled (e.g., pendulum)

• Controller: Algorithm producing control signals

• Sensor: Measures the system state

• Actuator: Applies control input (motor)

• Feedback: Uses sensor data to adjust behavior

Feedback Control

Continuously reduces error by adjusting inputs based on outputs.

• Bang-bang: All-or-nothing control; fast but unstable.

• PID: Combines:

\[u(t) = K_p e(t) + K_i \int e(t) \, dt + K_d \frac{de(t)}{dt}\]

• LQR: Uses optimal gains to minimize cost function:

• \[u = -Kx\]

  where \(x\) is the system state and \(K\) is the gain matrix.