Explicit feedbacks stabilizing the attitude of a rigid spacecraft with two control torques
The full system which describes position and angular velocity of a rigid body is considered. By assuming that the attitude of the rigid body can be controlled by means of two independent torques, the authors address the problem of local asymptotic stabilizability at the origin. It is well known that this system cannot be stabilized in the above sense by using time-invariant smooth feedback. In fact, this negative result remains true even if one allows discontinuous time-invariant feedback. On the other hand, early work seems to suggest that stabilizability can be performed by time-varying feedback. This can be actually proved to be true by virtue of certain non-constructive arguments and using the fact that the system is small time locally controllable. Explicit constructions have been proposed in other papers. However, they work only in the particular case where the control torques act along the principal axes of inertia. The contribution of this paper consists in the construction of an explicit time-varying (periodic) almost-continuous feedback law which locally stabilizes the full system at the origin. Note that since the closed loop system is not continuous, solutions should be intended in Filippov's sense. The first step of the construction is a change of coordinate which carries the system in the form \[ \left\{\begin{aligned} \dot x_1&= x_5x_6+R_1(x),\\ \dot x^2&= x_1+cx_3x_6+R_2(x),\\ \dot x_3&= x_5+R_3(x),\\ \dot x_4&= x_6+R_4(x),\\ \dot x_5&=u_1,\\ \dot x_6&=u_2,\end{aligned} \right. \] where \(c\) is a positive constant. This system exhibits some interesting homogeneity properties with respect to a suitable dilation. In particular, as far as we are interested in local results, it is possible to prove that the terms \(R_1\), \(R_2\), \(R_3\), \(R_4\) are ``small'' and can be neglected. The second step consists in the construction of a stabilizing feedback for the subsystem formed by the first three equations, where \(x_5\) and \(x_6\) are reviewed as inputs. The expression of the feedback law is explicitly given, and the proof is based on an (explicit) Lyapunov function. Then the authors consider the system formed in the first, second, third, fifth and sixth equation as a cascade connection of the first three equations and two independent integrators. For this new system, a Lyapunov control function is explicitly constructed, starting with a homogeneous desingularizing procedure applied to the feedback law determined at the previous step. Hence, a new feedback is computed according to the well-known Artstein-Sontag universal formula. Finally, the feedback law is modified in order to guarantee stability for the forth equation, as well. This can be done by exploiting the periodicity issue. For a certain time interval, the final feedback law coincides with the feedback defined at the previous step. For a subsequent time interval, we can take for instance many linear feedback which stabilizes the subsystem formed by the fourth and sixth equation. Although it is not clear whether this construction may be feasible for practical applications, the paper is very interesting: it represents a very clever and elegant compound of different methodologies and approaches for stabilization of nonlinear systems.
position and angular velocity of a rigid body
scientific article; zbMATH DE number 896676
scientific article