In this paper, we determine the approximation ratio of a linear-saturated control policy of a typical robust-stabilization problem. We consider a system, whose state integrates the discrepancy between the unknown but bounded disturbance and control. The control aims at keeping the state within a target set, whereas the disturbance aims at pushing the state outside of the target set by opposing the control action. The literature often solves this kind of problems via a linear-saturated control policy. We show how this policy is an approximation for the optimal control policy by reframing the problem in the context of quadratic zero-sum differential games. We prove that the considered approximation ratio is asymptotically bounded by 2, and it is upper bounded by 2 in the case of 1-dimensional system. In this last case, we also discuss how the approximation ratio may apparently change, when the system’s demand is subject to uncertainty. In conclusion, we compare the approximation ratio of the linear-saturated policy with the one of a family of control policies which generalize the bang–bang one.

中文翻译：

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通过二次零和微分博弈实现线性饱和控制的鲁棒次优性

在本文中，我们确定了典型鲁棒稳定问题的线性饱和控制策略的近似比。我们考虑一个系统，它的状态整合了未知但有界的扰动和控制之间的差异。控制旨在将状态保持在目标集内，而干扰旨在通过反对控制动作将状态推到目标集之外。文献通常通过线性饱和控制策略来解决此类问题。我们通过在二次零和微分博弈的背景下重构问题来展示该策略如何近似最优控制策略。我们证明了所考虑的近似比渐近地以 2 为界，并且在一维系统的情况下为 2 的上界。在最后一种情况下，我们还讨论了当系统的需求受不确定性影响时，近似比率如何明显变化。总之，我们将线性饱和策略的逼近率与推广 bang-bang 策略的一系列控制策略中的一个进行比较。