control of a quadrotor using the linear quadratic regulator

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control of a quadrotor using the linear quadratic regulator

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Control of a quadrotor using the Linear Quadratic Regulator (LQR) involves designing a state feedback controller to stabilize the system and achieve desired performance. The quadrotor, being an underactuated and highly nonlinear system, can be linearized around a hover condition to apply LQR effectively.

### Key Steps in LQR Control for Quadrotors

Modeling and Linearization The quadrotor dynamics are derived using Newton-Euler equations, capturing translational and rotational motion. A simplified state-space model is obtained by linearizing the nonlinear dynamics around the equilibrium (hover) point.

State-Space Representation The linearized model is expressed in state-space form, with states typically including position, velocity, attitude (roll, pitch, yaw), and angular rates. The control inputs correspond to motor thrusts.

Cost Function Selection LQR minimizes a quadratic cost function that balances state deviations and control effort. The designer selects weighting matrices to prioritize tracking accuracy or energy efficiency.

Solving the Algebraic Riccati Equation The optimal feedback gain matrix is computed by solving the Riccati equation, ensuring stability and optimal performance.

Implementation and Tuning The derived gain matrix is applied in closed-loop control. Fine-tuning weighting matrices helps achieve desired response characteristics, such as fast settling time or minimal overshoot.

LQR provides an efficient and systematic approach for stabilizing quadrotors, though it relies on accurate modeling and may require additional robustness techniques for real-world disturbances.

This method is widely used in autonomous drones due to its computational efficiency and stability guarantees near the linearization point.