The concept of fixed-time stability (FxTS) has been studied widely in the past decade as it allows system trajectories to converge to an equilibrium point within a fixed time. For a control system, it is not possible to guarantee fixed-time stability from any arbitrary initial conditions in the presence of input constraints. In this paper, we study the effect of control input constraints on the domain of attraction of an FxTS equilibrium point. We first present a new result on FxTS, where we allow a positive term in the time derivative of the Lyapunov function. We provide analytical expressions for the domain of attraction and the settling time to the equilibrium in terms of the coefficients of the positive and negative terms that appear in the time derivative of the Lyapunov function. We show that this result serves as a robustness characterization of FxTS equilibria in the presence of additive, vanishing disturbances. We use the new FxTS result in formulating a provably feasible quadratic program (QP) that computes control inputs that drive the trajectories of a class of nonlinear, control-affine systems to a goal set, in the presence of control input constraints.

中文翻译：

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控制输入​​约束下固定时间稳定性域的刻画

固定时间稳定性 (FxTS) 的概念在过去十年中得到了广泛研究，因为它允许系统轨迹在固定时间内收敛到平衡点。对于控制系统，在存在输入约束的情况下，不可能保证任意初始条件下的固定时间稳定性。在本文中，我们研究了控制输入约束对 FxTS 平衡点的吸引力域的影响。我们首先在 FxTS 上提出一个新结果，其中我们允许 Lyapunov 函数的时间导数中有一个正项。我们根据 Lyapunov 函数的时间导数中出现的正项和负项的系数，提供了吸引力域和达到平衡的稳定时间的解析表达式。我们表明，在存在加性、消失干扰的情况下，该结果可作为 FxTS 平衡的稳健性表征。我们使用新的 FxTS 结果来制定可证明可行的二次规划 (QP)，该规划计算控制输入，在存在控制输入约束的情况下，这些控制输入将一类非线性控制仿射系统的轨迹驱动到目标集。