Backstepping is a mature and powerful Lyapunov-based design approach for a specific set of systems. Throughout the development over three decades, innovative theories and practices have extended backstepping to stabilization and tracking problems for nonlinear systems with growing complexity. The attractions of the backstepping-like approach are the recursive design processes and modularized design. A nonlinear system can be transferred into a group of simple problems and solved it by a sequential superposition of the corresponding approaches for each problem. To handle the complexities, backstepping designs always come up with adaptive control and robust control. The survey aims to review the milestone theoretical achievements among thousands of publications making the state-feedback backstepping designs of complex ODE systems to be systematic and modularized. Several selected elegant methods are reviewed, starting from the general designs, and then the finite-time control enhancing the convergence rate, the fuzzy logic system and neural network estimating the system unknowns, the Nussbaum function handling unknown control coefficients, barrier Lyapunov function solving state constraints, and the hyperbolic tangent function applying in robust designs. The associated assumptions and Lyapunov function candidates, inequalities, and the deduction key points are reviewed. The nonlinearity and complexities lay in state constraints, disturbance, input nonlinearities, time-delay effects, pure feedback systems, event-triggered systems, and stochastic systems. Instead of networked systems, the survey focuses on stand-alone systems.

翻译：反步法是一种成熟且强大的基于Lyapunov的设计方法，适用于特定类型的系统。经过三十年的发展，创新性的理论与实践已将反步法扩展至日益复杂的非线性系统的镇定与跟踪问题。反步类方法的吸引力在于其递归设计过程与模块化设计：非线性系统可转化为一组简单子问题，并通过依次叠加各子问题的相应方法加以求解。为处理复杂性，反步设计常涉及自适应控制与鲁棒控制。本综述旨在梳理数千篇文献中具有里程碑意义的理论成果，这些成果使得复杂常微分方程系统的状态反馈反步设计趋于系统化与模块化。本文回顾了若干精选的优雅方法，从通用设计出发，逐步涵盖：提升收敛速度的有限时间控制、估计系统未知量的模糊逻辑系统与神经网络、处理未知控制系数的Nussbaum函数、解决状态约束的障碍Lyapunov函数，以及应用于鲁棒设计的双曲正切函数。综述还回顾了相关假设、Lyapunov函数候选、不等式及推导要点。非线性与复杂性体现在状态约束、扰动、输入非线性、时滞效应、纯反馈系统、事件触发系统及随机系统中。本综述聚焦于独立系统，而非网络化系统。