Research On The Initial Characteristics Of The Spectrum Of Time-delay Systems Based On Frequency-sweeping Approach

Time-delay is a common phenomenon that exists in various real industrial system, or we may say that any dynamic systems are time-delay systems, and time-delay is an important reason for the instability of the systems. Therefore, the stability analysis of the systems has received considerable attentions in the area of control over the decades. In the previous studies we know that the frequency-sweeping approach not only can be used to detect all the critical imaginary roots, but also can analyze the complete stability of the time-delay systems by studying the algebraic properties of frequency-sweeping curves. So, using the algebraic properties of frequency-sweeping curves, We study the number of unstable roots of the time-delay systems in this thesis. The major contributions of this thesis are as follows:1. Based on the frequency-sweeping curves, discussing the Retarded time-delay systems x=A0x(t)+∑A1x(t-1(?), as the time-delay r increasing form 0 to+ε (i.e. (?)= 0 has a tiny disturbance+ε), we study the number of unstable root of the Retarded time-delay systems, namely to study the number of unstable roots NU(+ε). In this thesis, we only study a type of the Retarded time-delay systems, which n,g are both odd. We proved that there have the specific properties between the number of unstable roots NU(+ε) of the Retarded time-delay systems and the index NF+(+(?)), NF-(-(?)) of the frequency-sweeping curves of the time-delay systems, and we give an example to verify this method. In previous studies we have been using the Newton Polygon method for defining the structure of the Puiseux series with the explicit computation of the corresponding coefficients to compute the number of unstable roots, which is much more complicated compared with the frequency-sweeping approach.2. We will solve the number of unstable roots NU(+ε) of the Neutral time-delay systems via the frequency-sweeping approach which is used in the Retarded time-delay systems. χ(t)= Ax(t)+ Bχ(t-(?))+Cχ(t-(?)), So, when the Neutral operator stable, we can obtain the number of unstable roots NU(+ε) of the Neutral time-delay systems by the same approach. And we may directly verify the stability of the Neutral operator.