LQR Hover

Problem Statement

LQR hover control optimizes stabilization around a hover equilibrium by balancing state error and control effort. It gives a principled gain matrix and clean performance trade-offs through weighting matrices.

Model and Formulation

Linearized dynamics around hover:

$$\dot{x}=Ax+Bu$$

Objective:

$$J={\int }_0^{\mathrm{\infty }}(x^{\mathrm{\top }}Qx+u^{\mathrm{\top }}Ru)dt$$

Optimal policy:

$$u=-Kx,K=R^{-1}{B}^{\mathrm{\top }}P$$

where P solves the continuous-time algebraic Riccati equation.

Algorithm Procedure

1. Linearize UAV dynamics around trim hover.
2. Choose Q and R according to tracking vs effort priorities.
3. Solve Riccati equation for P, then compute feedback gain K.
4. Apply full-state feedback in the hover envelope.

Tuning Guidance

• Increase position-state weights in Q for tighter hover.
• Increase R to smooth actuation and reduce aggressive commands.
• Re-linearize if operating point drifts far from hover assumptions.

Failure Modes and Diagnostics

• Performance degrades in strongly nonlinear/aggressive regimes.
• State-estimation latency can destabilize high-gain solutions.
• Poorly scaled units in Q/R create unintuitive controller behavior.

Implementation and Execution

python -m flybots.simulations.path_tracking.lqr_hover

Evidence

References

• Anderson and Moore, Optimal Control: Linear Quadratic Methods
• Bertsekas, Dynamic Programming and Optimal Control

Related Algorithms

• PID Hover
• NMPC