Bounded backstepping control and robustness analysis for time-varying systems under converging-input-converging-state conditions

We provide new bounded backstepping results that ensure global asymptotic stability for a large class of partially linear systems with an arbitrarily large number of integrators. We use a dynamic extension that contains one artificial delay, and a converging-input-converging-state assumption. When the nonlinear subsystem is control affine, we provide sufficient conditions for our converging-input-converging-state assumption to hold. We also show input-to-state stability with respect to a large class of model uncertainties, and robustness to delays in the measurements of the state of the nonlinear subsystem. We illustrate our result in a first example that has a nondifferentiable vector field and so is beyond the scope of classical backstepping, and then in a nonlinear example that illustrates how one can combine Lyapunov and trajectory based methods to check our assumptions.