A Geometric Approach to Feedback Stabilization of Nonlinear Systems with Drift

where, the state x(t) evolves on R n , u def = [u 1 , . . . , u m ], u i ∈ R are the control inputs, m < n, and f i , i = 0, 1, . . . , m, are real analytic vector fields on R n .

Feedback stabilization of systems of this type can be very challenging in the case when (1) fails to satisfy Brockett’s necessary condition for the existence of continuous stabilizing state feedback laws, see [2]. Systems of this type are encountered in a number of applications, for example, the control of rigid bodies in space, control of systems with acceleration constraints, and control of a variety of underactuated dynamical systems. Hence, the development of general stabilization approaches for such systems deserves further attention.

As compared with driftless systems, relatively few approaches have so far been proposed for the stabilization of systems with drift. The difficulty of steering systems with drift arises from the fact that, in the most general case of non-recurrent or unstable drift, the system motion along the drift vector field needs to be counteracted by enforcing system motions along adequately chosen Lie bracket vector fields in the system’s underlying controllability Lie algebra. Such indirect system motions are complex to design for and can be achieved only through either time-varying open-loop controls or discontinuous state feedback.

The majority of relevant feedback stabilization methods for systems with drift found in the literature, contemplate systems of specific structure and deliver feedback laws which apply exclusively to the particular models considered, see for example [1,3,9,13,17,20,22,23].

More general feedback design methodologies have been proposed in [10,7,15,18,19] and apply to systems in general form (1). The first systematic procedure for the construction of stabilizing piece-wise constant controls was presented in [10] and clearly demonstrates the difficulty of the problem. The method of [10] has been applied in [7] to asymptotically steer the the attitude and angular velocity of an underactuated spacecraft to the origin. General averaging techniques on Lie groups have been successfully applied to attitude control in [15]. The Lie algebraic approach outlined in [18] requires an analytic solution of a trajectory interception problem for the flows of the original system and its Lie algebraic extension in terms of the logarithmic coordinates on the associated Lie group. Such analytic solutions are generally difficult to derive. The approach in [19] draws on the ideas of [5,6], which apply to systems without drift. The method of [19] combines the construction of a periodic time-varying critically stabilizing control with the on-line calculation of an additional corrective term to provide for asymptotic convergence to the origin.

In the above context, the contributions of this paper can be described as follows.

• An approach to stabilization of general systems of the form (1) is presented which is based on the explicit calculation of a reachable set of desirable states for the controlled system. Such reachable set is determined when the system Σ is reformulated as a right-invariant system on an analytic, simply connected, nilpotent Lie group and once the stabilization problem is restated accordingly. The reformulation allows for the time-varying part of the stabilizing feedback control to be derived as the solution of a nonlinear programming problem for steering the open-loop system Σ to the given reachable set of states. The construction of the feedback law does not require numerical integration of the model differential equation and is independent of the choice of a Lyapunov type function. • Unlike other approaches, [10,7,14,18,19], point-to-point steering is not required to achieve asymptotic stability. Instead, the flow of the controlled system is required to intersect periodically a certain reachable set in the space of the logarithmic coordinates. In contrast to [10,7,14], the approach presented here offers a proof for Lyapunov asymptotic stability of the controlled system to the equilibrium and applies to systems whose linearization may be uncontrollable. • The approach presented might prove useful for the construction of feedback laws with a reduced number of control discontinuities and for the development of computationally feasible methods toward the design of smooth time-varying stabilizing feedback. This is because the system is shown to remain asymptotically stable provided that the state of the system traverses the sets of desirable states periodically in time. Since the sets of desirable states are typically large, the latter condition leaves much freedom for improved design, which is not the case for previous methods based on point-to-point steering. • A fairly complex example is presented which clearly demonstrates the computational feasib