Efficient Stability Chart Computation for General Delayed Linear Time Periodic Systems

Abstract

The determination of the stability of systems with time delay is of high importance in many industrial and research applications, like cutting processes, wheel shimmy, traffic jams and even in neural systems, human balancing. A user friendly numerical method was implemented to analyse the general form of delayed linear time periodic systems with time periodic coefficients. The goal is to create a freeware Matlab package which is able to determine automatically the so-called stability chart, which illustrates the parameter range for which the given linear system is stable. The user has to define the governing equation by the time periodic coefficient matrices, the corresponding time delays, the orders of time derivatives of the general coordinate vector, as well as the range of the parameters and the resolution of the stability chart. The method is optimized for 2 parameters, which is a typical case in engineering applications, but 1 and 3 parameter stability charts are also supported and tested. The stability is analysed in frequency domain based on the Nrth order approximation of Hill’s infinite determinant. The parameter points where the number of unstable Floquet multipliers changes are computed by the Multi Dimensional Bisection Method. From these parameter points, another algorithm selects the stability boundary lines. The algorithm is tested by means of numerous examples.