This paper revisits a classical challenge in the design of stabilizing controllers for nonlinear systems with a norm-bounded input constraint. By extending Lin-Sontag's universal formula and introducing a generic (state-dependent) scaling term, a unifying controller design method is proposed. The incorporation of this generic scaling term gives a unified controller and enables the derivation of alternative universal formulas with various favorable properties, which makes it suitable for tailored control designs to meet specific requirements and provides versatility across different control scenarios. Additionally, we present a constructive approach to determine the optimal scaling term, leading to an explicit solution to an optimization problem, named optimization-based universal formula. The resulting controller ensures asymptotic stability, satisfies a norm-bounded input constraint, and optimizes a predefined cost function. Finally, the essential properties of the unified controllers are analyzed, including smoothness, continuity at the origin, stability margin, and inverse optimality. Simulations validate the approach, showcasing its effectiveness in addressing a challenging stabilizing control problem of a nonlinear system.

翻译：本文重新审视了具有范数有界输入约束的非线性系统镇定控制器设计中的经典难题。通过扩展Lin-Sontag通用公式并引入一个通用的（状态依赖）缩放项，提出了一种统一的控制器设计方法。该通用缩放项的引入不仅给出了统一的控制器形式，还促成了具有多种优良特性的替代通用公式的推导，使其能够满足特定需求的定制化控制器设计，并在不同控制场景中展现出多功能性。此外，我们提出了一种确定最优缩放项的构造性方法，从而得到了一个优化问题的显式解，称为基于优化的通用公式。由此得到的控制器能够保证渐近稳定性，满足范数有界输入约束，并优化预定义的代价函数。最后，分析了统一控制器的基本特性，包括光滑性、原点处的连续性、稳定裕度和逆最优性。仿真结果验证了该方法在处理非线性系统具有挑战性的镇定控制问题中的有效性。