Structure preserving integrators for solving linear quadratic optimal control problems with applications to describe the flight of a quadrotor

TL;DR

This paper introduces structure-preserving exponential integrators based on Magnus series for solving matrix Riccati equations in linear quadratic control, ensuring positivity and stability, with applications to quadrotor flight stabilization.

Contribution

It develops second order exponential Magnus integrators that unconditionally preserve positivity in Riccati equations, improving stability in control applications.

Findings

Unconditionally preserve positivity of the solution

Effective stabilization of a quadrotor UAV

Higher order Magnus integrators analyzed

Abstract

We present structure preserving integrators for solving linear quadratic optimal control problems. This problem requires the numerical integration of matrix Riccati differential equations whose exact solution is a symmetric positive definite time-dependent matrix which controls the stability of the equation for the state. This property is not preserved, in general, by the numerical methods. We propose second order exponential methods based on the Magnus series expansion which unconditionally preserve positivity for this problem and analyze higher order Magnus integrators. This method can also be used for the integration of nonlinear problems if they are previously linearized. The performance of the algorithms is illustrated with the stabilization of a quadrotor which is an unmanned aerial vehicle.