Synthesis of Stabilizing Control of a Quadcopter Based on Linear Matrix Inequalities

Abstract

The relevance of the work is due to the widespread introduction of unmanned aerial vehicles, including quadcopters, in various areas of both civil and military applications. A review of various methods of controlling quadcopters, considering their features as nonlinear objects of high dimensionality, is performed. The work is devoted to stabilizing a quadcopter on a complex trajectory defined by functional coordinate relationships in 3D space. A nonlinear dynamic model of the quadcopter in a coupled coordinate system is constructed. The quadcopter control is based on a combination of two control actions. When solving the inverse dynamics problem, program control provides motion along a given trajectory. Stabilization of motion along the desired trajectory is provided by phase coordinate feedback. The stabilizing regulator ratios are found by the modal control method based on the solution of a linear matrix inequality using a linearized model. The found feedback ratios provide the required degree of stability of the closed-loop system, ensuring the quadcopter robustness to parametric perturbations. The legitimacy of this approach to the synthesis of control of a nonlinear system is substantiated by the topological equivalence theorem for the nonlinear system and the linearized model in that the nonlinear system has stable or unstable manifolds, which are analogs of the stable or unstable spaces of the linearized system. The results of computational experiments to estimate the error in reproducing a given quadcopter trajectory are presented. A simulation of the quadcopter behavior was performed, and the trajectory reproduction error was calculated to confirm the effectiveness of the synthesized stabilizing control and the optimal control based on the Pontryagin maximum principle. According to this criterion, the stabilizing control synthesized based on linear matrix inequalities is more effective for the quadcopter. Computational experiments were performed using the MATLAB application software package.