Pendulum exercise

1. Introduction & System Modeling

1.1 System Description and Parameters

This exercise demonstrates Active Disturbance Rejection Control (ADRC) for a pendulum system. The controller must regulate the pendulum angle to a setpoint while rejecting disturbances and handling sensor noise.

System Parameters:

• Pendulum tip mass: M = 0.4 kg
• Rod length: l = 0.3 m
• Viscous friction coefficient: c = 0.1
• Gravitational acceleration: g = 9.81 m/s²

Control Specifications:

• Settling time (10% of setpoint): < 5 seconds (initial), < 2 seconds (final)
• Disturbance rejection: oscillations within 10% of setpoint
• Control effort: torque oscillations within 1 Nm peak-to-peak
• Setpoint: θ_setpoint = 1 rad

Disturbance Configuration:

• Eccentric mass: m_e = 0.2 kg at radius r_e = 0.02 m
• Disturbance frequency: ω_m = 20 rad/s
• Centrifugal force: Fc = m_e * r_e * ω_m²

2. ADRC Design & Implementation

2.1 Controller Tuning Parameters

The ADRC controller consists of a state-feedback controller and an Extended State Observer (ESO) that estimates both system states and total disturbance.

2.2 Debug Mode: Validation with Ideal Plant

Before controlling the actual pendulum, we validate our ADRC implementation on a simplified double-integrator plant where the ESO model perfectly matches the plant dynamics.

Objective: Verify that the simulated response matches the theoretical second-order response, confirming correct implementation of the ESO and control law.

Expected Result: Perfect overlap between simulated and theoretical responses, with ESO accurately estimating all states and zero total disturbance (no model mismatch).

Observation: The responses match perfectly. Furthermore, the ESO's state estimates (theta_hat, theta_hat_dot) align exactly with the plant's true states, and the estimated total disturbance f_hat is zero. This is expected because the plant model and the ESO's internal model are identical, leaving no "unmodeled" dynamics to reject.

3. Pendulum Model Validation

3.1 Open-Loop Response Verification

Test 1: Free Oscillation from Non-Zero Angle

• Initial condition: θ = π/2 rad
• Control: Manual mode with u_manual = 0
• Disturbance: Enabled at t = 5s

Expected Behavior: Damped oscillations settling to vertical position (θ = 0) due to gravity and friction, with visible disturbance effects after t = 5s.

Observation: The pendulum swings and settles as expected, confirming the basic dynamics (gravity, inertia, friction) are implemented correctly. The disturbance at t=5s creates a clear oscillation.

Test 2: Static Torque Balance

• Initial condition: θ = π/2 rad
• Control: Constant torque u_manual = Mgl
• Disturbance: Disabled

Expected Behavior: Pendulum remains steady at π/2 rad because the applied torque balances gravitational torque.

Test 3: Dynamic Torque Balance

• Initial condition: θ = 0 rad
• Control: Constant torque u_manual = Mgl/2
• Disturbance: Disabled

Expected Behavior: Pendulum settles at non-zero angle where applied torque balances gravitational torque.

Observation: The pendulum moves and settles at a non-zero angle, confirming that the input (motor torque) correctly influences the plant model.

We also check the states estimated by the ESO.

Observation: Note that now the total disturbance term is different from zero because the actual plant differes from the ideal double integrator plant. However, the ESO is doing a good job in properly estimating the states of the actual plant.

4. Closed-Loop Performance Analysis

4.1 Nominal Case with ADRC Control

Configuration:

• Control mode: ADRC enabled
• Initial conditions: θ = 0 rad, θ_dot = 0 rad/s
• Disturbance: Enabled
• Observer bandwidth: keso = 20

Performance Assessment:

• Verify settling within 5 seconds to 10% of setpoint
• Check disturbance rejection within 10% bounds
• Monitor control torque within 1 Nm peak-to-peak

keso = 20

4.2 Effect of Sensor Noise

Real sensors introduce measurement noise that affects control performance, particularly through the high-gain observer.

Problem: High observer bandwidth (keso=20) amplifies measurement noise into the control signal, causing excessive torque oscillations that violate the 1 Nm specification.

4.3 Noise-Robust Tuning Trade-off

Solution: Reduce observer bandwidth to filter high-frequency noise while maintaining adequate disturbance rejection.

Trade-off Analysis:

• Lower bandwidth (keso=7): Smoother control action, better noise rejection
• Higher bandwidth (keso=20): Faster state estimation, better disturbance rejection
• Compromise: keso=7 provides acceptable performance with compliant control effort

keso = 7

5. Performance Specification Change

5.1 Faster Set-Point Tracking

New Specification: Settling within 10% of setpoint in less than 2 seconds.

Approach: Increase controller bandwidth while maintaining observer ratio for noise immunity.

Validation: Verify that the faster response meets the new settling time requirement while maintaining acceptable control effort and disturbance rejection.

keso = 5

6. Conclusion

6.1 Summary of Results

The ADRC controller successfully meets all control specifications:

1. Setpoint Tracking: Achieves required settling times (5s and 2s versions)
2. Disturbance Rejection: Maintains oscillations within 10% of setpoint despite eccentric mass disturbance
3. Control Effort: Maintains torque oscillations within 1 Nm peak-to-peak specification
4. Noise Robustness: Proper observer tuning provides immunity to sensor noise

6.2 Key Insights

• Observer Bandwidth Trade-off: Critical balance between estimation speed and noise sensitivity
• Model Independence: ADRC effectively handles unmodeled dynamics and disturbances
• Practical Tuning: Systematic approach to meeting multiple competing specifications

6.3 Final Controller Parameters

For nominal performance with noise:

• Controller bandwidth: ω_c = 3 rad/s
• Observer ratio: keso = 7
• Settling time: ~2 seconds

For enhanced set-point racking performance:

• Controller bandwidth: ω_c = 4 rad/s

• Observer ratio: keso = 5

• Settling time: ~1.5 seconds