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Inverted Pendulum

The inverted pendulum is a classic example in control system literature due to its inherent instability when open loop. In this system, a pendulum is mounted on a motorized cart. The control objective is to balance the pendulum by applying a force to the cart.

InvertedPendulumVideo

1. Problem Setup and Design Requirements

  • System Description:
    A pendulum is mounted on a cart that can move horizontally. The system is unstable without control (the pendulum falls if not actively balanced).
    Inputs and Outputs:
  • Input: Force ( F ) applied to the cart.

  • Outputs: Pendulum angle ( \theta ) and cart position ( x ).

  • Given Parameters:

  • ( M ): Mass of the cart (0.5 kg)
  • ( m ): Mass of the pendulum (0.2 kg)
  • ( b ): Friction coefficient for the cart (0.1 N/m/s)
  • ( l ): Length to the pendulum's center of mass (0.3 m)

  • ( I ): Moment of inertia of the pendulum (0.006 kg·m²)

  • Design Criteria:

  • For single-input, single-output (SISO) control (pendulum angle control):
    • Settling time for ( \theta ) less than 5 seconds.
    • After an impulse of 1 N·s, the pendulum deviates no more than 0.05 radians from vertical.
  • For state-space design (SIMO) control:
    • A 0.2 m step in cart position: settling time under 5 seconds and rise time under 0.5 seconds.
    • The pendulum angle deviation remains within 20° (0.35 radians) of the vertical.
    • Steady-state error less than 2% for both outputs.

2. State-Space

State-Space Diagram

State-Space Diagram 1

Note: It is recommended to use PID tuning to further refine the control performance of the inverted pendulum system.

Reference

Purpose:
This reference serves as an educational resource for understanding the modeling of an inverted pendulum, a well-known unstable system. It explains how to derive the nonlinear equations of motion, linearize the system about its unstable equilibrium, and represent the system using both transfer functions and state-space models. Additionally, MATLAB code examples are provided for simulating the system and designing controllers, with PID tuning recommended as a means to improve control performance.