Positiveness of matrix <i>M</i>.

<div><p>This paper deals with the problem of robust regulation of a Planar Vertical Takeoff and Landing (PVTOL) aircraft affected by crosswind disturbances. The proposed approach simultaneously exploits the advantages of passivity-based control and the integral sliding mode (ISM) control technique. We demonstrate that the passivity-based controller accomplishes the PVTOL regulation within a guaranteed stability region. At the same time, the ISM controller can effectively counteract the undesirable disturbances effect from the beginning of the operation since the disturbances are never seen in the measurable output. Using the passivity approach with LaSalle’s Invariance Theorem, we construct the required Lyapunov function to establish the closed-loop system stability. The proposed combination of controllers results in a robust system that satisfactorily solves the regulation problem. Numerical simulations are provided to show that the proposed approach allows the tracking of different reference signals: pulse-train shaped trajectory and ellipse-shaped trajectory, considering the presence of perturbations (crosswind). Furthermore, to illustrate the effectiveness of the proposed technique, a comparative analysis considering two robust control strategies is provided.</p></div>