Overview
This lecture introduces the Linear Quadratic Regulator (LQR), an optimal control method using state-space models, covering its structure, concept of optimality, cost functions, and tuning using MATLAB examples.
LQR vs. Pole Placement Controllers
- Both LQR and pole placement use full state feedback and the same implementation structure: output = reference - K * state.
- The key difference is how the gain matrix K is chosen: pole placement picks desired pole locations, while LQR optimizes performance and control effort.
- Both approaches can achieve zero steady-state error through different feedback structures.*
The Concept of Optimality in LQR
- LQR defines "optimal" by balancing system performance with actuator effort via a cost function.
- Preferences (like speed vs. cost) are weighted using matrices Q (performance) and R (effort) in the cost function.
- There’s no universal optimal solution; the best one depends on user-defined weights.
The LQR Cost Function
- The cost function measures total system cost by summing (over time) weighted squared state errors and actuator efforts.
- States are penalized via the Q matrix, which can emphasize specific state variables by assigning higher values.
- Control effort is penalized via the R matrix, typically a diagonal matrix relating to each actuator.
- The cost function is quadratic, ensuring a single minimum.
Tuning LQR with Q and R
- Adjusting Q values increases penalty for state errors; increasing R penalizes actuator usage.
- Designers often start with identity matrices for Q and R and then tune by trial and error or intuition.
- Higher Q leads to faster system response (but more effort); higher R leads to less actuator usage (but slower response).
- LQR tuning is often more intuitive than pole placement, especially for complex systems.
MATLAB LQR Examples
- Example with a rotating mass: Q and R selected to balance rotation speed vs. fuel use.
- Increasing R lowered fuel use but increased maneuver time; increasing specific Q elements minimized overshoot.
- LQR easily accommodates actuator limitations by adjusting R without needing to manually reposition system poles.
Key Terms & Definitions
- LQR (Linear Quadratic Regulator) — Optimal control method using quadratic cost functions on linear state-space models.
- State-space representation — Mathematical model describing system dynamics with state variables.
- Gain matrix (K) — Matrix multiplying state vector in feedback for controlling system behavior.
- Cost function — Mathematical expression penalizing deviations in performance and excessive control effort.
- Q matrix — Weights state errors in the cost function.
- R matrix — Weights actuator effort in the cost function.
- Pole placement — Control design method where desired closed-loop pole locations are chosen directly.
Action Items / Next Steps
- Practice designing LQR controllers by adjusting Q and R matrices for different system goals in MATLAB.
- Read up on the mathematical derivation of LQR for deeper understanding.
- Explore additional feedback structures as mentioned for completeness.