Algebraic Research On A Class Of Delayed Functional Differential Equations

The state change of time-delay systems depends not only on the current state of the system,but also on the state of some time in the past.The stability of time-delay systems is a hot issue at present.It has attracted much attention in the fields of circuit,optics,biomedicine and so on.Among the existing research methods,a frequency-sweeping framework for stability analysis of time-delay systems which is based on analytical curves can be uesd to detect all the critical imaginary roots.The asymptotic behavior of critical imaginary roots can be detected by frequency-sweeping curves,but this method requires a lot of manpower in practical operation,and is not suitable for large-scale systems.The results obtained by sweeping also have some inevitable error in numerical calculation.In the background of Computer assisted technology having developed rapidly,we want to hand over the complex operation to the machine,so we propose different methods of parameter detection based on the characristic equation method of time-delay system,and solve the stability problem of time-delay system by combining the graphical properties of frequency-sweeping curves.The characteristic equation of a time-delay system is a transcendental equation with infinite characteristic roots.The existing methods are not automated enough and have not small enough errors.In this paper,we discuss how to transform a problem of transcendental equation into polynomial equations through Mobius transformation in the case of a given Retarded time-delay system.The related mechanized algorithm is given and the calculation process is simplified and the accuracy of the results is ensured.Under the new frequency-sweeping framework,we can clearly judge the change state of critical virtual roots when the critical time delay changes slightly,and the value of ANF can be obtained according to the shape of curves.Then,we can establish the relationship between critical imaginary roots and the number of unstable roots(NU)of the system,and give the relationship between ?NU and ?NF,so as to quantitatively analyse the change of stability due to the asymptotic behavior of critical imaginary roots.So as the goal of solving the complete stability problem of time-delay systems is achieved.