Mathematical and Computational Tools for the Stability Analysis of Time-Varying Delay Systems and Applications in Mechanical Engineering

Abstract

Summary. An overview of eigenvalue based tools for the stability analysis of linear periodic systems with delays is presented. It is assumed that both the system matrices and the delays are periodically varying. First the situation is considered where the time-variation of the periodic terms is fast compared to the system's dynamics. Then averaging techniques are used to relate the stability properties of the time-varying system with these of a time-invariant one, which opens the possibility to use frequency domain tools. As a special characteristic the averaged system exhibits distributed delays if the delays in the original system are time-varying. Both analytic and numerical tools for the stability analysis of the averaged system are discussed. Special attention is paid to the characterization of situations where a variation of a delay has a stabilizing effect. Second, the assumption underlying the averaging approach is dropped. It is described how exact stability information of the original, periodic system can be directly computed. The two approaches are briefly compared with respect to generality, applicability and computational efficiency. Finally the results are illustrated by means of two examples from mechanical engineering. The first example concerns a model of a variable speed rotating cutting tool. Based on the developed theory and using the described computational tools, both a theoretical explanation and a quantitative analysis are provided of the beneficial effect of a variation of the machine speed on enhancing stability properties, which was reported in the literature. The second example concerns the stability analysis of an elastic column, subjected to a periodic force.

On characteristic roots and stability charts of delay differential equations

“…We now proceed to the analysis of the CRs of (19) by discussing the solutions of (22), keeping in mind that we shall eventually eliminate those solutions corresponding to CRs of (23), see Section 3.4.…”

“…The reason is simple to understand because the characteristic equation...is so complicated. It is therefore , worthwhile to obtain methods for determining approximations to the region of stability ’, in [2, p.305]:‘... but for characteristic equations in general , the rule seems to be that they are not at all amenable to analysis .’or more recently in [11, p.1671]: ‘... checking the eigenvalue conditions is much harder than for ODEs .’This nevertheless gave the start to an ample range of successful numerical investigations 12–16, also with regards to the ‘specular’ situation for characteristic multipliers when (1) has periodic coefficients17–23 as well as for more general classes involving multiple and distributed delays 24–26 and neutral dynamics 27, 28. The problem acquires even more importance when the question of robust stability is addressed, i.e.…”

“…or more recently in [11, p. 1671]: '...checking the eigenvalue conditions is much harder than for ODEs.' This nevertheless gave the start to an ample range of successful numerical investigations [12][13][14][15][16], also with regards to the 'specular' situation for characteristic multipliers when (1) has periodic coefficients [17][18][19][20][21][22][23] as well as for more general classes involving multiple and distributed delays [24][25][26] and neutral dynamics [27,28]. The problem acquires even more importance when the question of robust stability is addressed, i.e.…”

Multi-dimensional Taylor network adaptive predictive control for single-input single-output nonlinear systems with input time-delay

“…Non-linearity and time-delay are common among control systems, such as the chemical reaction system (Jia et al, 2017) , biological engineering system (Niu et al, 2015), mechanical engineering system (Michiels et al, 2007), etc. These control systems result in larger overhead, longer adjusting time and undesirable stability (Shu and Pi, 2000).…”