An official website of the United States government
It is already well-understood that many delay differential equations with only a single constant delay exhibit a change in stability according to the value of the delay in relation to a critical delay value. Finding a formula for the critical delay is important to understanding the dynamics of delayed systems and is often simple to obtain when the system only has a single constant delay. However, if we consider a system with multiple constant delays, there is no known way to obtain such a formula that determines for what values of the delays a change in stability occurs. In this paper, we present some single-delay approximations to a multidelay system obtained via a Taylor expansion as well as formulas for their critical delays which are used to approximate where the change in stability occurs in the multidelay system. We determine when our approximations perform well and we give extra analytical and numerical attention to the two-delay and three-delay settings.
more »
« less
On Robustness of Pre-Asymptotic Stability to Delayed Jumps in Hybrid Systems
We show that pre-asymptotic stability of a compact set for a hybrid system is semiglobally and practically robust in the presence of delayed jumps under mild conditions on the data. More precisely, when the delay-free system has a pre-asymptotically stable compact set, it is shown that for small enough delays, solutions of the delayed system converge to a neighborhood of a set of interest related to the aforementioned compact set. Unlike prior work, this notion of practical stability also holds for time-varying delays in the presence of Zeno solutions. Simulation results of a state estimator with intermittent and delayed information validate the findings.
more »
« less
- Award ID(s):
- 1710621
- PAR ID:
- 10094105
- Date Published:
- Journal Name:
- 2018 American Control Conference
- Page Range / eLocation ID:
- 2204 to 2209
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
In our previous work, we developed a multirate observer design method in linear systems with asynchronous sampling based on a Luenberger observer design coupled with inter‐sample predictors. In this article, the problem of multirate multidelay observer design is addressed where both asynchronous sampling and possible measurement delays are accounted for. The proposed observer adopts an available multirate observer design in the time interval between two consecutive delayed measurements. A dead time compensation approach is developed to compensate for the effect of delay and update past estimates when a delayed measurement arrives. The stability and robustness properties of the multirate observer will be preserved under nonconstant, arbitrarily large measurement delays. A mathematical example and a gas‐phase polyethylene reactor example demonstrate good performance of the proposed observer in the presence of nonuniform sampling and nonconstant measurement delays. © 2018 American Institute of Chemical EngineersAIChE J, 65: 562–570, 2019more » « less
-
Fear of predation may assert privilege to prey species by restricting their exposure to potential predators, meanwhile it can also impose costs by constraining the exploration of optimal resources. A predator–prey model with the effect of fear, refuge, and hunting cooperation has been investigated in this paper. The system’s equilibria are obtained and their local stability behavior is discussed. The existence of Hopf-bifurcation is analytically shown by taking refuge as a bifurcation parameter. There are many ecological factors which are not instantaneous processes, and so, to make the system more realistic, we incorporate three discrete time delays: in the effect of fear, refuge and hunting cooperation, and analyze the delayed system for stability and bifurcation. Moreover, for environmental fluctuations, we further modify the delayed system by incorporating seasonality in the fear, refuge and cooperation. We have analyzed the seasonally forced delayed system for the existence of a positive periodic solution. In the support of analytical results, some numerical simulations are carried out. Sensitivity analysis is used to identify parameters having crucial impacts on the ecological balance of predator–prey interactions. We find that the rate of predation, fear, and hunting cooperation destabilizes the system, whereas prey refuge stabilizes the system. Time delay in the cooperation behavior generates irregular oscillations whereas delay in refuge stabilizes an otherwise unstable system. Seasonal variations in the level of fear and refuge generate higher periodic solutions and bursting patterns, respectively, which can be replaced by simple 1-periodic solution if the cooperation and fear are also allowed to vary with time in the former and latter situations. Higher periodicity and bursting patterns are also observed due to synergistic effects of delay and seasonality. Our results indicate that the combined effects of fear, refuge and hunting cooperation play a major role in maintaining a healthy ecological environment.more » « less
-
Several sources of delay in an epidemic network might negatively affect the stability and robustness of the entire network. In this paper, a multi-delayed Susceptible-Infectious-Susceptible (SIS) model is applied on a metapopulation network, where the epidemic delays are categorized into local and global delays. While local delays result from intra-population lags such as symptom development duration or recovery period, global delays stem from inter-population lags, e.g., transition duration between subpopulations. The theoretical results for a network of subpopulations with identical linear SIS dynamics and different types of time-delay show that depending on the type of time-delay in the network, different eigenvalues of the underlying graph should be evaluated to obtain the feasible regions of stability. The delay-dependent stability of such epidemic networks has been analytically derived, which eliminates potentially expensive computations required by current algorithms. The effect of time-delay on the H2 norm-based performance of a class of epidemic networks with additive noise inputs and multiple delays is studied and the closed form of their performance measure is derived using the solution of delayed Lyapunov equations. As a case study, the theoretical findings are implemented on a network of United States’ busiest airports.more » « less
-
Train routing is sensitive to delays that occur in the network. When a train is delayed, it is imperative that a new plan be found quickly, or else other trains may need to be stopped to ensure safety, potentially causing cascading delays. In this paper, we consider this class of multi-agent planning problems, which we call Multi-Agent Execution Delay Replanning. We show that these can be solved by reducing the problem to an any-start-time safe interval planning problem. When an agent has an any-start-time plan, it can react to a delay by simply looking up the precomputed plan for the delayed start time. We identify crucial real-world problem characteristics like the agent's speed, size, and safety envelope, and extend the any-start-time planning to account for them. Experimental results on real-world train networks show that any-start-time plans are compact and can be computed in reasonable time while enabling agents to instantly recover a safe plan.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.23919/ACC.2018.8431127
