Gridding discretization-based multiple stability switching delay search algorithm: The movement of a human being on a controlled swaying bow

Delay represents a significant phenomenon in the dynamics of many human-related systems—including biological ones. It has i.a. a decisive impact on system stability, and the study of this influence is often mathematically demanding. This paper presents a computationally simple numerical gridding algorithm for the determination of stability margin delay values in multiple-delay linear systems. The characteristic quasi-polynomial—the roots of which decide about stability—is subjected to iterative discretization by means of pre-warped bilinear transformation. Then, a linear and a quadratic interpolation are applied to obtain the associated characteristic polynomial with integer powers. The roots of the associated characteristic polynomial are closely related to the estimation of roots of the original characteristic quasi-polynomial which agrees with the system′s eigenvalues. Since the stability border is crossed by the leading one, the switching root locus is enhanced using the Regula Falsi interpolation method. Our methodology is implemented on—and verified by—a numerical bio-cybernetic example of the stabilization of a human-being′s movement on a controlled swaying bow. The advantage of the proposed novel algorithm lies in the possibility of the rapid computation of polynomial zeros by means of standard programs for technical computing; in the low level of mathematical knowledge required; and, in the sufficiently high precision of the roots loci estimation. The relationship to the direct search QuasiPolynomial (mapping) Rootfinder algorithm and computational complexity are discussed as well. This algorithm is also applicable for systems with non-commensurate delays.时滞是诸多涉人系统（包括生物系统）动力学中的重要现象，其对系统稳定性具有决定性影响，而对该影响的研究往往对数学能力要求较高。本文提出一种计算简便的数值网格化算法，用于求解多时滞线性系统的稳定裕度时滞值。特征拟多项式（characteristic quasi-polynomial）的根决定了系统稳定性，本文采用预扭曲双线性变换对其进行迭代离散化；随后通过线性与二次插值，得到具有整数次幂的等价特征多项式。该等价特征多项式的根与原特征拟多项式的根估计值紧密相关，而原特征拟多项式的根即对应系统的特征值。由于主导根会穿越稳定边界，本文采用试位法（Regula Falsi）插值优化切换根轨迹。本文所提方法通过一项数值生物控制论示例完成实现与验证，该示例聚焦于受控摇摆船艏上的人体运动稳定问题。所提出的新型算法优势在于：可通过通用技术计算标准程序快速求解多项式零点；对所需数学知识门槛较低；且根轨迹估计精度足够高。本文还讨论了所提算法与直接搜索型拟多项式（映射）求根算法的关联，以及其计算复杂度。该算法亦可应用于非等时滞系统。