Control of Uncertain High-Order Fully Actuated Strict-Feedback Systems: A Backstepping Approach With High-Gain Observer-Based Derivative Approximation

In this article, a high-gain observer (HGO)-based differentiator is proposed to approximate the derivatives of the virtual control to ease the "explosion of complexity" problem in backstepping for the second-order strict-feedback systems (SOSFSs) and high-order strict-feedback systems (HOSFSs). Unlike the existing high-order command-filtered backstepping or extended dynamic surface control that needs to tune a large number of parameters, this proposed high-order HGO-based backstepping (HOHGOB) scheme can improve the derivative approximation performance by only tuning a single parameter, i.e., the observer gain. A further advantage of the proposed HOHGOB scheme, in addition to its simplicity and ease of implementation, is that the estimation error of the derivatives shrinks to zero as the observer gain grows to infinity. We have also rigorously established that the states of the closed-loop system achieve uniform ultimate boundedness under the designed high-order backstepping controller. Additionally, the output tracking error can be made arbitrarily small by the designer. The efficacy of the proposed scheme is numerically validated through a benchmark application to a single-link robot arm.