This work presents an exhaustive analysis of the impact of time delay on the stability of dual-rate haptics controllers using an exact discrete-time method. The mathematical formulation of such controllers leads to higher order state-space models, in particular, for higher values of time delay and sampling rates. A balanced truncation based model order reduction framework is therefore utilized for obtaining reduced order models, while preserving the stability properties of the original higher order models. The likely order of the reduced models is selected on the basis of the Hankel singular values which represent the contribution of the system states towards the overall system energy. Using this framework, it is empirically found that third order reduced models yield exactly the same stability ranges as given by the allied full order models. This empirical finding is backed up by a rigorous analysis of the controller while considering a wide range of values for the time delay spanning both the realistic and worst-case application scenarios. This result is hitherto unknown in the haptics literature and is for the first time reported in this paper. For comparison purposes, the stability ranges of the dual-rate controller are also obtained using an equivalent continuous-time method, and numerical simulations. This work generalizes the results of previous works available in the literature for uniform-rate sampling scheme both for delayed and non-delayed haptics controllers. The results demonstrate that for a time-delayed dual-rate haptics controller, an increase in the sampling rate leads to an enhancement in the stable range of virtual wall parameters as long as the value of time delay relative to the sampling rate (non-dimensional time delay) remains small. For higher values of the non-dimensional time delay, higher sampling rates do not necessarily lead to performance enhancement.

这项工作使用精确的离散时间方法对时间延迟对双速率触觉控制器稳定性的影响进行了详尽的分析。这种控制器的数学公式导致更高阶的状态空间模型，特别是对于更高的时间延迟和采样率值。因此，基于平衡截断的模型降阶框架用于获得降阶模型，同时保留原始高阶模型的稳定性属性。根据代表系统状态对整个系统能量的贡献的 Hankel 奇异值选择简化模型的可能顺序。使用这个框架，根据经验发现，三阶简化模型产生与联合全阶模型给出的完全相同的稳定性范围。这一经验发现得到了对控制器的严格分析的支持，同时考虑了跨越现实和最坏情况应用场景的广泛时间延迟值。这一结果在触觉文献中是迄今为止未知的，并首次在本文中报道。为了比较，还使用等效连续时间方法和数值模拟获得了双速率控制器的稳定范围。这项工作概括了文献中可用于延迟和非延迟触觉控制器的均匀速率采样方案的先前工作的结果。结果表明，对于延时双速率触觉控制器，只要相对于采样率的时间延迟值（无量纲时间延迟）保持较小，采样率的增加会导致虚拟墙参数稳定范围的增强。对于更高的无量纲时间延迟值，更高的采样率不一定会导致性能增强。