Systems & Control Letters 36 (1999) 181–192 Output feedback stabilization of the angular velocity of a rigid body A. Astol * Department of Electrical and Electronic Engineering and Centre for Process Systems Engineering, Imperial College, London SW7 2BT, UK Received 25 November 1997; accepted 5 May 1998 Abstract The problem of stabilization of the angular velocity of a rigid body using only two control signals and partial state information is addressed. It is shown that if any two (out of three) states are measured the system is not asymptotically stabilizable with (continuous) dynamic output feedback. Nevertheless, we prove that practical stability is achievable if the measurable states fulll a certain structural property, and that, under the same structural condition, a hybrid control law yielding exponential convergence can be constructed. Finally, we also study some geometric features of the Euler’s equations and the connection between local strong accessibility and local observability. c  1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Rigid body; Euler’s equations; Output feedback stabilization; Nonlinear systems 1. Introduction The problem of asymptotic stabilization of the angular velocity of a rigid body has been studied by several researchers. In the papers [1, 7] it was shown that the zero solution of Euler’s angular velocity equations can be made asymptotically stable by means of two control torques, whereas in the works [2, 3, 13, 14] the same problem has been addressed and solved in the case of only one control torque. Robust stabilization has been studied in [4, 11]. In the rst work, robustness with respect to parameters variation has been discussed, whereas in the second one robustness against exogenous disturbances has been dealt with. All the aforementioned works assume that the whole state vector is available for feedback, whereas, to the best of our knowledge, no result is available for the case of partial state information. In this work we assume that only two out of three states are available for feedback. This problem, of intrinsic mathematical diculty, is also relevant from the applications point of view, as it models the situation of sensor failure. The paper is organized as follows. In Section 2 we present the equations of the system we are dealing with and we formulate precisely the problem under investigation. Section 3 contains a digression devoted to some geometric considerations related to the observability of the Euler’s equations with two measured states. The stabilization problem is then addressed. In Section 4 we prove a negative result, namely that asymptotic stabilization by measurement feedback is not achievable with any dynamic smooth control law, whereas in * Tel.: +44 171 594 6289; fax: +44 171 594 6282; e-mail: a.astol@ic.ac.uk. 0167-6911/99/$ – see front matter c  1999 Published by Elsevier Science B.V. All rights reserved. PII: S0167-6911(98)00089-9