两能级量子系统的固定时间控制及鲁棒性

Fixed-time control and robustness of two-level quantum systems

本文考虑刘维尔方程描述的两能级量子系统对于目标平衡态的固定时间控制问题.首先通过相干矢量和复数的指数形式对系统模型及控制目标进行等价变换,并借助合适的Lyapunov函数设计一个含分数幂的连续非光滑控制律,然后利用固定时间稳定性定理给出了系统实现固定时间收敛的条件.针对某些分数幂取值导致控制性能较差的情况,提出了两种非光滑切换策略.此外,论文也分析了实际量子系统中可能存在的多种不确定性对固定时间控制的影响.最后,通过数值仿真实验验证了本文方案的有效性,同时展示了非光滑控制比标准Lyapunov控制具有更强的鲁棒性.

The problem of fixed-time control is considered for a target equilibrium state of two-level quantum systems described by the Liouville equation. Equivalent transformations are performed on the system model and the control goal by the coherence vector representation and the exponential forms of complex numbers. A continuous non-smooth control law with a fractional power is designed by a candidate Lyapunov function and the fixed-time stability theorem is used to obtain the conditions for the fixed-time convergence of quantum systems. For the case of bad control performance caused by some values of the fractional power, two non-smooth switching control strategies are proposed. Besides, the effect of various uncertainties that may exist in actual quantum systems on fixed-time control is also analyzed. Finally, the effectiveness of the control scheme in this paper is verified by numerical simulation experiments. At the same time, it is demonstrated that the fixed-time control has better robustness than the standard Lyapunov control.