LQR Trajectory Control of a Quadcopter Drone

Context

Optimal Control for Multi-Phase Trajectory Tracking

This project designs and validates a Linear-Quadratic Regulator (LQR) to control a quadcopter drone through a two-phase autonomous mission: a vertical climb followed by a circular trajectory. The controller is built on a linearized state-space model of the drone's translational and rotational dynamics and validated in MATLAB/Simulink, with a 3D visualization in FlightGear.

Phase 1: vertical climb from 10 m to 200 m at a constant speed of 13 m/s
Phase 2: circular trajectory of 60 m radius at 0.5 m/s once target altitude is reached
Control method: Linear-Quadratic Regulator (LQR), state feedback u*(t) = -Kx(t)
Validation: MATLAB/Simulink simulation with FlightGear 3D visualization

My Role

Contributions

Project carried out with one teammate, covering the full chain from optimal-control theory to a validated flight simulation: system linearization, LQR controller design for all six degrees of freedom, and simulation/visualization in MATLAB and FlightGear.

System Modeling

Linearized the quadcopter's roll, pitch, yaw and translational dynamics into state-space form for controller design.

LQR Controller Design

Designed 6 independent LQR controllers (height, x/y position, roll/pitch, yaw) using MATLAB's lqr(A,B,Q,R) function.

Control Architecture

Implemented a dual-loop structure, an outer position loop feeding an inner attitude loop, for multi-phase trajectory tracking.

Simulation & Visualization

Validated the closed-loop system in MATLAB/Simulink and produced a 3D flight visualization in FlightGear.

Optimal Control Theory

LQR Fundamentals

The LQR is an optimal control method that finds the control input minimizing a quadratic cost function while respecting the system's linear dynamics, trading off state-tracking performance against control effort.

Cost Function

J = ∫[xᵀQx + uᵀRu] dt the term xᵀQx penalizes state deviations, while uᵀRu penalizes control effort. Q and R are tuned to trade off performance against actuation energy.

Optimization Problem

Minimize J subject to the linear dynamic constraint ẋ = Ax + Bu, yielding the optimal control law u*(t) = -Kx(t).

Riccati Equation

The optimal gain is obtained by solving the Algebraic Riccati Equation (ARE): AᵀP + PA - PBR⁻¹BᵀP + Q = 0, for a unique symmetric positive-definite P when (A,B) is stabilizable.

Optimal Gain

K = R⁻¹BᵀP a constant, optimal, stabilizing gain, computed in MATLAB via are(A, B*inv(R)*B', Q) or the lqr() function.

Stability is guaranteed when (A,B) is stabilizable and (Q,A) is detectable
Closed-loop matrix: Abf = A - BK, with all eigenvalues satisfying Re(λi) < 0
Solved numerically via Schur or Newton-Kleinman methods

System Modeling

Dynamics & Control Architecture

The quadcopter's nonlinear dynamics were linearized around hover into independent state-space subsystems for roll, pitch, yaw and translational motion (x, y, z), each driven by a corresponding control input. A dual-loop architecture was then implemented: an outer position-control loop generates attitude references, which feed an inner attitude-control loop producing the four motor commands.

Block diagram of the dual-loop position and attitude control architecture

Dual-Loop Control Architecture (Position & Attitude)

Controller Design

LQR Tuning

Six independent LQR controllers were tuned in MATLAB using diagonal Q and R weighting matrices, one pair per controlled axis, via the lqr(A,B,Q,R) function.

Controller	Q	R
Altitude (z)	diag([5, 1])	3
Position (x / y)	diag([15, 1])	1000
Roll / Pitch	diag([25, 1])	200
Yaw	diag([7, 1])	1

LQR Weighting Matrices

Results

Simulation Results

Position Trajectories

Simulated velocity profiles of the quadcopter throughout the mission

Velocities

3D and 2D visualization of the quadcopter's flight path

3D/2D Visualization

Euler angle response of the quadcopter under LQR control

Euler Angles — LQR Control

Motor rotation speeds and generated thrust during the mission

Motor Rotation Speeds & Generated Thrust

The controller tracked both mission phases accurately: a stable vertical climb to 200 m followed by a well-maintained 60 m radius circular trajectory, with bounded Euler angles and smooth motor commands throughout.

Conclusions

Takeaways

The LQR controller proved effective for multi-phase trajectory control of a quadcopter, combining a rigorous optimal-control formulation with a practical dual-loop implementation validated in simulation.

Accurate mission tracking: precise trajectory following through both the 200 m climb and the 60 m radius circular phase
Guaranteed stability: closed-loop system stable with good disturbance rejection
Efficient structure: dual-loop architecture (position + attitude) with independent controllers per axis
Full validation: MATLAB/Simulink simulation combined with realistic FlightGear visualization

View FlightGear Visualization Download Simulation Files