A FAS approach for stabilization of generalized chained systems: multi-vector case

In this paper, the type of general nonholonomic systems proposed in part 1 and part 2 is generalized into a multi-vector form, for which the nonsingular condition is not assumed. First, by differentiating the first equation in the system, and through stability analysis, the stabilization of the proposed system is firstly converted into the stabilization of a sub-strict feedback system (sub-SFS), that is, a system in the form of a strict feedback system but with the gain matrices not satisfying the nonsingular conditions. Second, using the fully actuated system (FAS) approach, the formulated sub-SFS is further converted into a sub-FAS, for which the concepts of singular and feasible sets are defined. Finally, the problem is solved by designing a sub-stabilizing controller of the obtained sub-FAS, which drives all the system responses starting from a so-called region of exponential attraction to the origin exponentially, and the closed-loop response of the formulated sub-SFS controlled by the designed controller is also explicitly provided. Technically, an external parameter is introduced in the designed controller. In general, the designed controller contains an integral term. In the case that the introduced two sets of matrix functions in the proposed system are time-invariant, the controller turns out to be a continuous time-varying one, but when the external parameter is particularly chosen, the controller takes a simpler form but becomes a discontinuous time-invariant one.