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Partial stabilizability of cascaded systems applications to partial attitude control

In this work, the problem of partial stabilization of nonlinear control cascade systems with integrators is considered. The latter systems present an anomaly, which is the non complete stabilization via continuous purestate feedback, this is due to Brockett necessary condition. To cope with this difficulty we propose the partial stabilization. For a given motion of a dynamical system, say x(t, x0, t0) = (y(t, y0, t0), z(t, z0, t0)), the partial stabilization is the qualitative behavior of the y-component of the motion (i.e the asymptotic stabilization of the motion with respect to y) and thez-component converges, relative to the initial vector x(t0) = x0 = (y0, z0). In the present work, we establish a new results for the adding integrators for partial stabilization, we show that if the control systems is partially stabilizable, then the augmented cascade system is partially stabilizable. Two applications are considered. The first one is devoted to partial attitude stabilization of rigid spacecraft. The second application is intended to the study of underactuated ship. Numerical simulations are given to illustrate our results.