Modelling and control of nonlinear negative imaginary systems

This article investigates mathematical modelling and control of a nonlinear negative imaginary system using an illustrative benchmark physical example of a quadrotor dynamic model. A generalized methodology based on Euler-Lagrange equation is applied to obtain nonlinear negative imaginary dynamic model for the quadrotor. In this method, the Kronecker product is employed to formulate the Coriolis matrix, which is then used to construct a mathematical model of a quadrotor. This article further presents nonlinear negative imaginary systems theory-based analysis and synthesis framework to find the control solution for quadrotor's attitude stability problem while hovering. The multi-loop control scheme is applied to the quadrotor that directly uses the Euler angles (angular positions) instead of angular velocity measurements. Numerical simulation results of this paper show that the investigated control strategy ensures the asymptotic stabilization of quadrotor attitude system model in the presence of external disturbance.