Inverted Pendulum Control Systems project
Overview
Designed and compared three control strategies (PID, LQR, and MPC) to stabilize an inverted pendulum on a moving cart under external disturbances. This classic control problem has real-world applications in humanoid robotics, self-balancing vehicles, rocket stabilization, and robotic arms.
Challenge: Maintain balance at an inherently unstable equilibrium point while responding to unpredictable forces like vibrations, wind gusts, and impulses.
View Full Project Report (PDF)
System Modeling
Created a comprehensive mathematical model of the pendulum-cart system:
- Physical Simulation: Developed CAD model in SOLIDWORKS and implemented rigid body dynamics in Simscape
- Mathematical Framework: Derived nonlinear equations of motion using Newton’s laws, then linearized around equilibrium (θ = π) for controller design
- State-Space Representation: Enabled modern optimal control techniques with 4-state model (cart position/velocity, pendulum angle/velocity)
System Parameters:
| Parameter | Value |
|---|---|
| Cart Mass | 8.44 kg |
| Pendulum Mass | 0.79 kg |
| Pendulum Length | 0.25 m |
| Max Actuator Force | 40 N |
| Disturbance | 10 N sinusoidal impulse (50ms period) |
Three Controller Approaches
| Controller | Design Approach | Key Strengths | Limitations |
|---|---|---|---|
| PID (Cascaded) | Dual-loop: inner loop stabilizes pendulum, outer loop controls cart position | Simplest design, minimal computation | Time-intensive manual tuning, no constraint handling |
| LQR | Optimal state feedback minimizing J = ∫(X^T·Q·X + U^T·R·U)dt Gain Matrix: K = [-10.00, -16.79, 228.09, 35.81] | Best overall performance, intuitive weight-based tuning | High actuator effort (38N peak) |
| MPC | Predictive optimization over future horizon with Kalman filter-based state estimation | Lowest actuator effort (28N), guaranteed constraint satisfaction | High computational cost (2.45s vs 0.02s for PID) |
Results & Performance Comparison
| Performance Metric | PID | LQR | MPC | Winner |
|---|---|---|---|---|
| Settling Time (θ) | 3.2s | 1.8s | 2.5s | LQR |
| Settling Time (x) | 4.1s | 2.3s | 3.2s | LQR |
| Overshoot (θ) | 12% | 5% | 8% | LQR |
| Overshoot (x) | 8% | 3% | 6% | LQR |
| Max Actuator Force | 35N | 38N | 28N | MPC |
| Computation Time | 0.02s | 0.15s | 2.45s | PID |
Key Insights
| Controller | Best Use Case |
|---|---|
| PID | Simple, resource-constrained applications requiring basic stabilization |
| LQR | Real-time control demanding best performance with moderate computational resources |
| MPC | Offline systems requiring minimal actuator effort and strict constraint enforcement |
Technical Skills Demonstrated
- Control Theory: PID tuning, optimal control (LQR), predictive control (MPC)
- System Modeling: Nonlinear dynamics, linearization, state-space representation
- Simulation: MATLAB/Simulink, Simscape multibody dynamics, SOLIDWORKS CAD
- Analysis: Performance metrics, comparative evaluation, constraint optimization
Future Directions
- Hardware implementation on physical inverted pendulum system
- Hybrid controller combining PID simplicity with MPC robustness
- Adaptive control for time-varying dynamics