线性时滞系统的相对稳定性分析方法

Relative Stability Analysis Method of LTI System with Time Delay

实际控制过程中,时滞的引入常常导致系统性能的下降,也使得系统的稳定性分析变得困难。从一类具有时滞项的线性时不变(LTI)系统的特征根求解出发,研究了系统的稳定性分析问题。复平面上系统特征根的位置不仅决定了系统的绝对稳定性,还决定系统的相对稳定性-瞬态性能。由于时滞的引入,系统特征方程变成超越方程,其解的数量为无穷。并提出一种从超越方程实部和虚部系数中提取出双向量并结合二维向量旋转的解决方法,可以准确简洁地求出超越方程在复平面上指定区域边界上的根。最后给出仿真实例表明了算法的正确性和有效性。

Time-delay is frequently encountered in many control problems. The existence of time-delay induces difficulty in stability a- nalysis and may thereby lead to oscillations or degraded performance in closed-loop systems. This paper investigates the stability analysis of linear time-invariant（LTI） systems with time delay depending upon the solution of the characteristic equation. Not only the absolute stability but also the relative stabilities the transient performances are depended on the locations of the characteristic equation roots. The existence of time delay transforms the closed-loop characteristic equation into a transcendental equation with an infinite number of roots. By investigating the real part and the imaginary part of the transcendental equation, the characteristic equation can be transformed into a rotation equation with two vectors related to the system parameters. The characteristic roots on the given region boundaries can be deter- mined exactly and concisely. Numerical examples are provided to illustrate the proposed algorithm.