The Study In The Limiting Case For A Class Of Transcendental Equation’s Related Properties By The Frequency Sweep Curve

Time-delay systems is very important in the control system,the time-delay systems is Universal and widespread in society.Time-delay exists not only can lead to system performance, can also lead to the unstability of the system,This makes the study of time-delay systems has been a very urgent and important.To the best of the author, the approach in most of the research results of calculating NU(+ε) is to calculate time delay system’s puiseux series, This makes calculation is cumbersome and inconvenience. Based on Li Xuguang teacher’s research results, this paper proposes a more simple method of calculating NU(+ε) of time-delay systems and Classify time-delay systems.The main work is as follows:This paper studies the transcendental equation of two limiting cases of system’s stability with the increase of time delay Ï„:0â†'ε.The two limiting cases are as follows:At the time delay Ï„ varying from 0 to 1 (delay to produce a tiny disturbance),the study of system unstable roots;At the time delay Ï„ tends to infinity (an infinite stability problem),the study of system unstable roots.1. Introduces the background of a class of time-delay systems; Frequency sweep curve properties; The Root’s distribution properties of y= (x)" and time-delay system consistency.2. On a limit case, when the system characteristic equation of delay varying from 0 to ε, We study the change of the time-delay systems unstable root. Proved that the time-delay systems unstable sweep curve is the relationship between unstable root number NU(+ε) and frequency domain when the delay varying from 0 to ε,meet the conditions: This method simplifies the calculation.By adopting the method of polygon based on Newton algorithm, we calculate the puiseux series of time-delay systems, to verify the correctness of the results.3. For another limiting cases, when the system characteristic equation of delay varying from 0 to oo, We study the change of the time-delay systems unstable root. Discuss when the delay tends to infinity,the unstable roots and related properties for time-delay systems. Classified situation of time-delay systems according to the frequency sweep curve.