Robust backstepping stabilization of uncertain Cartpole underactuated systems with full state constraints: A strict-feedback-like system design approach

Abstract

This article proposes a robust backstepping controller for the stabilization of uncertain Cartpole systems with full state constraints. We rely on existing results on transformed cascaded nonstrict feedback normal form with a feedforward term. Unlike previous research which treats the feedforward term as disturbance, this paper exploits its structure and integrates it as part of the virtual control variable, thus admitting a strict-feedback-like system with uncertain gains. Fractional type barrier functions are introduced to satisfy full-state constraints of the system. This not only ensures practicality of the method, but also induces upper bounds of all the uncertainties of the system, enabling the implementation of robust backstepping method. Besides, we explicitly relax the commonly-used assumption that the pole uniformly lies in the upper half-plane. Unlike existing literature which only demonstrate the convergence of the virtual control variables, we prove in addition that all state variables are Uniformly Ultimately Bounded (UUB) with the proposed controller. We also prove that the vanilla filtering-based backstepping controller is sufficient for stabilization without extra compensation schemes with proper parameters. Simulation results illustrate the convergence performance of the proposed controller, complemented by ablation studies and discussion on parameter tuning for practical consideration. The proposed strict-feedback-like perspective with full-state constraints treatment can potentially be extended to larger classes of nonstrict feedback systems.