Smooth time-varying pure feedback control for chained non-holonomic systems with exponential convergent rate

The feedback stabilisation problem of non-holonomic chained systems and a novel feedback design scheme is proposed, which renders a smooth, time-varying, aperiodic, pure feedback control with exponential convergence rates. There are three main advantages with the proposed design. (i) In general, time-varying designs are mostly periodic and render asymptotic stability, whereas the proposed approach is aperiodic and have exponential convergent rates. (ii) A novel state scaling transformation is proposed. It shows that even though u₁ vanishes in regulation problems, intrinsic controllability of chained systems can be regained by judiciously designing the input u₁ and by applying the state scaling transformations. (iii) A class of memory functions is introduced into the control design, the controller dependency on the system´s initial conditions in our previous work is removed and the control is a pure feedback. Moreover, the design is shown to be inversely optimal. Simulations and comparisons are conducted to verify the effectiveness of the proposed approach.