**5. Proofs of the boundedness and convergence properties of smooth sliding mode controllers**

In this section, the proofs of the boundedness and convergence properties of the smooth sliding mode controllers are introduced. In particular, the convergence analysis of smooth sliding mode controllers will be explained and discussed to the finest detail. Lyapunov's direct method is used to handle the finite-time convergence of the tracking error vector to the boundary layer. Also, once in the boundary layer, the tracking error vector is said to have exponentially converged to a bounded region, as proven analytically.
