This note discusses common errors on using convex modelling and linear matrix inequalities for nonlinear control, a methodology that has become increasingly popular due to its systematicness and numerical implementability. Illustrations on common problems are made from existing literature: they are classified and discussed; advices are given to prevent them. Convex modelling is employed in linear parameter varying, Takagi-Sugeno models, and other convex structures in order to subsume or rewrite a nonlinear system for analysis or design via the direct Lyapunov method. Convexity plays a central role in allowing a finite set of vertex conditions in the form of linear matrix inequalities to be sufficient for the corresponding task. In contrast with other nonlinear methodologies, this one produces expressions resembling linear results, which makes it easier to grasp while often inducing subtle mistakes.
In this work, a novel family of exact nonlinear control laws is developed for trajectory tracking of unmanned aerial vehicles. The proposed methodology exploits the cascade structure of the dynamic equations of most of these systems. In a first step, the vehicle position in Cartesian coordinates is controlled by means of fictitious inputs corresponding to the angular coordinates, which are fixed to a combination of computed torque and proportional-derivative elements. In a second step, the angular coordinates are controlled as to drive them to the desired fictitious inputs necessary for the first part, resulting in a double-integrator 3-input cascade control scheme. The proposal is put at test in two examples: 4-rotor and 8-rotor aircrafts. Numerical simulations of both plants illustrate the effectiveness of the proposed method, while real-time results of the first one confirm its applicability.
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