Global Stabilization for Systems Evolving on Manifolds

Abstract

We show that any globally asymptotically controllable system on any smooth manifold can be globally stabilized by a state feedback. Since we allow discontinuous feedbacks, we interpret the solutions of our systems in the "sample and hold" sense introduced by Clarke-Ledyaev-Sontag-Subbotin (CLSS). Our work generalizes the CLSS Theorem which is the special case of our result for systems on Euclidean space. We apply our result to the input-to-state stabilization of systems on manifolds relative to actuator errors, under small observation noise.

A dynamic estimator on SE(3) using range-only measurements

“…This is only possible when the state space is diffeomorphic to R n , which is not the case for SE (3). At most, one may expect to obtain an Almost Global Asymptotic Stability (AGAS) type of result as in [7] and show that the region of attraction is the entire state space except for a nowhere dense set of zero measure [13] [14].…”

“…Furthermore, and because the problem is directly formulated in SE(3), a non global result is expected if one draws an analogy with control problems. In fact, the results in [13] [14], [15] show that due to the non-Euclidean nature of SE(3) it is not possible to render a system evolving on this manifold globally asymptotically stable (GAS) by resorting to continuous feedback control laws. This is only possible when the state space is diffeomorphic to R n , which is not the case for SE (3).…”

An Experimentally Validated Attitude Observer Based on Range and Inertial Measurements*

“…Attitude observers rigorously formulated on non-Euclidean spaces, such as the set of rotation matrices SO(3) and the set of unit quaternions are presented in Kinsey and Whitcomb [2007], Mahony et al [2008], and Vasconcelos et al [2008]. The topological limitation of achieving global stabilization on the SO(3) manifold and some guidelines on observer design are discussed in Chaturvedi and McClamroch [2006], Fragopoulos and Innocenti [2004], and Malisoff et al [2006]. The work Hamel et al [2009] presents a study on the stability of attitude observers based on inertial and body fixed vector measurements and shows asymptotic stability for the case of one time-varying vector observation.…”

A Landmark Based Nonlinear Observer for Attitude and Position Estimation with Bias Compensation

“…The problem of formulating a stabilizing feedback law in non-Euclidean spaces, such as SO(3) and SE(3), has been addressed in several references, namely [Malisoff et al, 2006, Chaturvedi and McClamroch, 2006, Fragopoulos and Innocenti, 2004, Bhat and Bernstein, 2000, where the analysis of the topological properties of the SE(3) manifold provides important guidelines for the design of observers. Nonlinear attitude observers motivated by aerospace applications are found in [Salcudean, 1991, Thienel andSanner, 2003], yielding global exponential convergence of the attitude estimates in the Euler quaternion representation.…”