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.

通过扩展 Lin-Sontag 的通用公式，并引入一种通用的（状态相关的）缩放项，本文提出了一种统一的控制器设计方法，该方法具有多种有利特性的替代通用公式的推导，适用于满足特定要求的定制控制设计，并可在不同控制场景中具备灵活性。此外，我们还提出了一种构造性方法来确定最优的缩放项，从而得到了一个显式解决方案的优化问题，命名为基于优化的通用公式。最终得到的控制器确保了渐近稳定性，满足了范数有界输入限制，并优化了预定义的成本函数。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、稳定裕度和反向最优性。仿真验证了该方法，在解决非线性系统的挑战性稳定控制问题方面的有效性。