Department of Electronics Engineering, Ramrao Adik Institute of Technology, Nerul, Navi Mumbai 400 706, India.
Department of Instrumentation Engineering, Bharati Vidyapeeth College of Engineering, Navi Mumbai 400 614, India.
ISA Trans. 2014 Mar;53(2):380-90. doi: 10.1016/j.isatra.2013.09.002. Epub 2013 Oct 7.
Recently fractional-order (FO) differential equations are widely used in the areas of modeling and control. They are multivalued in nature hence their stability is defined using Riemann surfaces. The stability analysis of FO linear systems using the technique of Root Locus is the main focus of this paper. Procedure to plot root locus of FO systems in s-plane has been proposed by many authors, which are complicated, and analysis using these methods is also difficult and incomplete. In this paper, we have proposed a simple method of plotting root locus of FO systems. In the proposed method, the FO system is transformed into its integer-order counterpart and then root locus of this transformed system is plotted. It is shown with the help of examples that the root locus of this transformed system (which is obviously very easy to plot) has exactly the same shape and structure as the root locus of the original FO system. So stability of the FO system can be directly deduced and analyzed from the root locus of the transformed IO system. This proposed procedure of developing and analyzing the root locus of FO systems is much easier and straightforward than the existing methods suggested in the literature. This root locus plot is used to comment about the stability of FO system. It also gives the range for the amplifier gain k required to maintain this stability. The reliability of the method is verified with analytical calculations.
近年来，分数阶（FO）微分方程在建模和控制领域得到了广泛的应用。它们本质上是多值的，因此它们的稳定性是使用黎曼曲面定义的。本文主要关注使用根轨迹技术对 FO 线性系统进行稳定性分析。许多作者已经提出了在 s 平面上绘制 FO 系统根轨迹的程序，这些程序很复杂，使用这些方法进行分析也很困难且不完整。在本文中，我们提出了一种绘制 FO 系统根轨迹的简单方法。在提出的方法中，将 FO 系统转换为其整数阶对应系统，然后绘制这个转换系统的根轨迹。通过示例表明，该转换系统的根轨迹（显然非常容易绘制）与原始 FO 系统的根轨迹具有完全相同的形状和结构。因此，可以直接从转换后的 IO 系统的根轨迹推导出和分析 FO 系统的稳定性。与文献中提出的现有方法相比，这种开发和分析 FO 系统根轨迹的方法更加简单和直接。该根轨迹图用于说明 FO 系统的稳定性，并给出了保持这种稳定性所需的放大器增益 k 的范围。该方法的可靠性通过分析计算得到验证。
