Explicit feedbacks stabilizing the attitude of a rigid spacecraft with two control torques
type: Item
instance of: scholarly article
title: Explicit feedbacks stabilizing the attitude of a rigid spacecraft with two control torques
zbMATH Open document ID: 0858.93058
DOI: 10.1016/0005-1098(95)00194-8
author: Jean-Michel Coron
author: El-Yazid Keraï
published in: Automatica
publication date: 1997-03-25T00:00:00Z
review text: 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 input…
reviewed by: Andrea Bacciotti
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 93C95
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 93C99
zbMATH DE Number: 896676
zbMATH Keywords: time-varying feedback
zbMATH Keywords: position and angular velocity of a rigid body
zbMATH Keywords: local asymptotic stabilizability
zbMATH Keywords: Lyapunov control function
zbMATH Keywords: periodicity
zbMATH Keywords: stabilization of nonlinear systems
MaRDI profile type: Publication
full work available at URL: 0005-1098(95)00194-8
OpenAlex ID: W2127966006
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description: scientific article; zbMATH DE number 896676