Control of a Rotary Inverted Pendulum (Furuta Pendulum)

Problem Statement 

Design and implement a robust control system to stabilize an inverted pendulum in the upright position and track motor angle setpoints using a Quanser Qube-Servo 2 platform. 

Key Results: Successfully achieved upright stabilization and ± 10° square-wave tracking using an LQR controller and a custom state observer implemented on both Simulink and Arduino hardware.

System Dynamics & Modeling 

To control the system, I first developed a mathematical model using the nonlinear dynamics of the Furuta Pendulum.

• Mathematical Model: The system consists of a horizontal rotary arm and a vertical pendulum. The motion is governed by coupled nonlinear differential equations accounting for inertia (J_r, J_p), damping (D_r, D_p), and motor constants (k_m).

• System Identification: Conducted experimental step responses to find the system gain (K) and time constant (tau) for the motor, ensuring the model accurately reflected the physical hardware.

• Linearization: For controller design, the system was linearized about two equilibrium points: the stable "pendulum down" position and the unstable "upright" position.

State Estimation & Observer Design

The hardware only provides measurements for the motor and pendulum angles (theta, alpha), but the control laws require angular velocities.

• Luenberger Observer/Kalman Filter: Designed and implemented a state estimator to derive the full 4-state vector in real-time.

• Signal Processing: Compared the observer performance against a simpler derivative with a Low-Pass Filter (LPF) approach to determine the most efficient estimation method for the Arduino's processing constraints.

LQR Control Implementation

• Controller Design: Utilizing a Linear Quadratic Regulator (LQR) approach, I optimized the Q and R weighting matrices to balance tracking performance with control effort (motor voltage).

• Simulink/MATLAB: Developed a nonlinear simulation to estimate the "catch range" - the maximum angle from which the pendulum could be successfully recovered.

• Arduino/C++: Ported the state-space matrices to C++ for standalone hardware execution.

• Swing-up & Catch: Implemented a "triggered subsystem" that enabled the LQR controller only when the pendulum was swung into the up right vicinity.

Results & Performance Analysis

Tracking: The system successfully tracked 0.2 Hz square waves with a 10° magnitude in the motor angle while maintaining the pendulum's vertical stability.

Sampling Rate Sensitivity: Analyzed the limits of the digital implementation by testing various sampling rates. I identified the critical frequency at which the system lost stability, comparing these findings with simulated quantization effects.

Discrete vs. Continuous: Evaluated the performance difference between controllers designed in the continuous-time domain versus those designed directly in discrete-time at various frequencies.

Technical Skills Demonstrated

• Tools: MATLAB, Simulink, Arduino (C++), Quanser Qube-Servo 2.

• Control Theory: LQR Control, State-Space Representation, Luenberger Observers, PID Tuning, System Identification.

• Embedded Systems: Real-time signal processing, discretization of continuous systems, and hardware-in-the-loop (HIL) testing.