Global-Stabilizing Near-Optimal Control Design for Nonholonomic Chained Systems

It is well known that a nonlinear optimal control requires the solution to a two-point boundary value problem or to a nonlinear partial differential equation and that such a solution can only be obtained off line by numerical iteration. In this paper, a new and near-optimal control design framework is proposed for controlling any nonholonomic system in the chained form. The proposed design is based upon thorough study of uniform complete controllability of the corresponding linear time varying nominal system. It is shown that, no matter whether the first component u
_1d
(t) of reference input vector is uniformly nonvanishing or simply nonconvergent to zero or vanishing or identically zero, uniform complete controllability of the (nominal) system can be recovered by employing the proposed time-folding/unfolding technique. Upon establishing the common property of uniform complete controllability, the proposed framework can be used to design both trajectory tracking control and regulation control in a systematic and unified manner. Using duality, uniform complete observability can also be established, a closed-form and exponentially convergent observer can be synthesized, and the controls designed using the proposed framework can be either state-feedback or output-feedback. The tracking controls are designed using the same three-step process. That is, design of the proposed controls starts with optimal control solutions to two linear nominal subsystems, one time-invariant and the other time varying. The two solutions combined together render a globally stabilizing suboptimal control for the overall system. Then, the optimality condition is invoked to determine the distance between the suboptimal control and the optimal one. Consequently, an improved control can be obtained by introducing a nonlinear additive control term in such a way that the distance aforementioned is minimized as much as possible in closed form. An example is used to show that regulation control can be designed similarly. All the controls designed are in simple closed forms and hence real-time implementable, they are time varying and smooth, globally and exponentially/asymptotically stabilizing, and they are near optimal since their closeness to the optimal control (attainable only offline) can be measured, monitored on line, and has been minimized