Time Delay Systems by Tamás Insperger Tulga Ersal & Gábor Orosz

Author:Tamás Insperger, Tulga Ersal & Gábor Orosz
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(8)

Equation (8) can be obtained directly by rearranging Eq. (4). The function can be determined from the history of the configurations and its inverse can be calculated numerically, for example, with Newton’s method. In [44] the calculation of the delay via Eqs. (7) and (8) are compared, with the result that the performance of the second method with Eq. (8) was better. In this paper, a third method is proposed. It is based on the solution of the equivalent system with constant delay Eq. (1). The performance of this method is much better than the other two methods because a numerical solver for DDEs with constant delays can be used, which is faster than solvers for DDEs with varying delay. In this case the only extra effort is the numerical calculation of the inverse function for the additional variable coefficient. Nevertheless, this is roughly the same effort compared to the calculation of the variable delay with Eq. (8). In fact, in many applications an explicit analytical expression can be derived for the additional state-dependent coefficient , which means that the analysis of the equivalent system Eq. (5) with constant delay is much simpler than the analysis of the original system Eq. (2) with state-dependent delay [26].

Another difficulty appears in the linearization of DDEs with state-dependent delay, which is not straightforward because the delayed argument depends itself on the state of the DDE [10, 13, 33]. This difficulty vanishes if the system is analyzed in the spatial domain where no state-dependent delay appears. In this case, the linearization can be done in the conventional way. For the interested reader, the linearization of a system with state-dependent delay as a model for a turning process is provided in [13]. In comparison, the linearization and the stability analysis for the equivalent system with constant delay and variable coefficient can be found in [26]. Of course, the results of both methods are completely equivalent. However, not only the linearization but also the resulting linearized dynamics is more complex for the representation with variable delay. For example, if a periodic solution appears in the DDE Eq. (2) with state-dependent delay, the coefficients in the linearized dynamics and similarly the time delay becomes time-periodic. In contrast, if the system is at first transformed to a DDE with constant delay, the linearized dynamics can be characterized by a system with periodic coefficients but constant delay. Typically, the methods and software packages for the linear stability analysis of DDEs with periodic coefficients are only applicable or at least optimized for systems with constant delays. A comparison of two numerical methods for the linear stability analysis of a DDE with variable delay and the analysis of the equivalent system with constant delays is given in the next Sect. 3.

Fig. 2 a Regenerative effect in turning with spindle speed variation . b Comparison of the error of the stability exponent between the semidiscretization method for constant delay and the semidiscretization method for time-varying delay

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