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This paper examines accelerating the well-known Laplacian average consensus algorithm by breaking its conventional delay-free input into two weighted parts and replacing one of these parts by an outdated feedback. We determine for what weighted sum there exists a range of time delay that leads to an increase in the rate of convergence of the algorithm. For such weights, using the Lambert W function, we obtain the rate increasing range of the time delay and also the maximum reachable rate and its corresponding maximizer delay. We also specify what combinations of the current and an outdated feedback increase the rate of convergence without increasing the control effort of the agents. Lastly, we determine the optimum combination of the current and the outdated feedback weights to achieve maximum increase in the rate of convergence without increasing the control effort. We demonstrate our results through a numerical example.
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On the Positive Effect of Delay on the Rate of Convergence of a Class of Linear Time-Delayed Systems
This paper is a comprehensive study of a long-observed phenomenon of an increase in the stability margin and so the rate of convergence of a class of linear systems due to time delay. We use Lambert W function to determine (a) in what systems the delay can lead to increase in the rate of convergence, (b) the exact range of time delay for which the rate of convergence is greater than that of the delay-free system, and (c) an estimate on the value of the delay that leads to the maximum rate of convergence. For the special case when the system matrix eigenvalues are all negative real numbers, we expand our results to show that the rate of convergence in the presence of delay depends only on the eigenvalues with minimum and maximum real parts. Moreover, we determine the exact value of the maximum rate of convergence and the corresponding maximizing time delay. We demonstrate our results through a numerical example on the practical application in accelerating an agreement algorithm for networked~systems by use of delayed feedback.
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- Award ID(s):
- 1653838
- PAR ID:
- 10158407
- Date Published:
- Journal Name:
- IEEE Transactions on Automatic Control
- ISSN:
- 0018-9286
- Page Range / eLocation ID:
- 1 to 1
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TAC.2019.2961348
