Lecture Notes: Introduction to Linear Quadratic Regulator (LQR) Control

Overview of the Inverted Pendulum System

• System Components: Inverted pendulum on a cart.
• Steps Taken:
  • Simulated the full nonlinear system without control.
  • Linearized the system around the fixed point where the pendulum is upright.
  • Obtained the A matrix (system dynamics) and D matrix.
  • Verified system controllability by checking the rank of the controllability matrix, confirming it spans the four-dimensional state space.

Introduction to Control Design

• Objective: Stabilize the unstable inverted pendulum configuration using full state feedback control (u = -Kx).
• Key Concept: Ability to place eigenvalues of the closed-loop system anywhere desired by designing an appropriate gain matrix K.

The Problem of Pole Placement

• Challenge: Determining the optimal placement of poles or eigenvalues for system stability.
• Solution: Utilize the Linear Quadratic Regulator (LQR) for deciding the best placement of poles to achieve desired performance.

Linear Quadratic Regulator (LQR) Control

• Purpose: Optimally chooses control actions to minimize a defined cost function over time.
• Components:
  • Cost Function (J): Integrates penalties for state deviation and control effort over time.
  • Q Matrix: Penalizes deviation of state from desired values. Should be positive semi-definite.
  • R Matrix: Penalizes magnitude of control effort.
• Operation: MATLAB lqr function computes the optimal gain matrix K that minimizes the cost function.
• Q and R Matrices Configuration: The choice of Q and R reflects the trade-off between state stability and control effort.

Practical Implementation in MATLAB

• Procedure:
  • Start with initial Q and R values, adjusting based on desired performance.
  • Apply the lqr function with system dynamics (A, B matrices) and desired Q, R matrices to obtain the optimal gain matrix K.
  • Simulate the nonlinear system with the designed controller to observe stabilization performance.
• Results: Successful stabilization of the inverted pendulum with adjustable aggressiveness of control based on Q, R configurations.

Insights and Applications

• Fine-tuning Q and R allows control over system responsiveness and energy consumption.
• The LQR approach provides a systematic way to achieve optimal control with respect to a defined cost function, making it a powerful tool in control theory.
• Real-world Applicability: Despite being computationally expensive for high-dimensional systems, LQR is highly effective for systems of manageable size and complexity, like the four-dimensional inverted pendulum system illustrated in the example.

Conclusion

• LQR offers a mathematically rigorous, yet intuitively accessible method for designing controllers that balance system performance with control effort, demonstrating its value in practical control system design.