Stability of a Consensus Protocol for Second Order Agents With Multiple Time Delays

2011 ◽  
Author(s):  
Rudy Cepeda-Gomez ◽  
Nejat Olgac
Keyword(s):  
Information Exchange ◽  
Time Delays ◽  
Second Order ◽  
Multiple Time ◽  
State Transformation ◽  
Consensus Protocol ◽  
Position Information ◽  
Characteristic Roots ◽  
Velocity Information ◽  
The Stability

In this study we consider the consensus problem for a group of second order agents interacting under a fixed, undirected communication topology. Communication lines are affected by two rationally independent delays. The first delay is assumed to be in the position information channels whereas the second one is in the velocity information exchange. The delays are assumed to be uniform throughout the entire network. We first reduce the complexity of the problem, by performing a state transformation that allows the decomposition of the characteristic equation of the system into a set of second order factors. The stability of the resulting subsystems is analyzed exactly and exhaustively in the domain of the time delays using the Cluster Treatment of Characteristic Roots (CTCR) paradigm. CTCR is a recent method which declares the stability features of the system for any composition of the time delays. Example cases are provided to verify the analytical conclusions.

Preserving Stability Under Communication Delays in Multi Agent Systems

2013 ◽  
Author(s):  
Hossein Rastgoftar ◽  
Suhada Jayasuriya
Keyword(s):  
Time Delays ◽  
Communication Delays ◽  
Multi Agent System ◽  
Multi Agent Systems ◽  
Continuum Approach ◽  
Position Information ◽  
Characteristic Roots ◽  
Multi Agent ◽  
Velocity Information ◽  
The Stability

The effect of time delays on the stability of a recently proposed continuum approach for controlling a multi agent system (MAS) evolving in n-D under a special local inter-agent communication protocol is considered. There a homogenous map determined by n+1 leaders is learned by the follower agents each communicating with n+1 adjacent agents. In this work both position and velocity information of adjacent agents are used for local control of follower agents whereas in previous work [1, 2] only position information of adjacent agents was used. Stability of the proposed method under a time delay h is studied using the cluster treatment of characteristic roots (CTCR) [3]. It is shown that the stability of MAS evolution can be preserved when (i) the velocity of any follower agent is updated using both position and velocity of its adjacent agents at time (t-h); and (ii) the communication matrix has real eigenvalues. In addition, it is shown that when there is no communication delay, deviations from a selected homogenous map during transients may be minimized by updating only the position of a follower using both position and velocity of its adjacent agents.

Kernel and Offspring Concepts for the Stability Robustness of Multiple Time Delayed Systems (MTDS)

2006 ◽  
Vol 129 (3) ◽  
pp. 245-251 ◽  
Author(s):  
Rifat Sipahi ◽  
Nejat Olgac
Keyword(s):  
Time Delays ◽  
Linear Time ◽  
Characteristic Root ◽  
Invariance Property ◽  
Multiple Time ◽  
Delayed Systems ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
Stability Robustness ◽  
The Stability

A novel treatment for the stability of linear time invariant (LTI) systems with rationally independent multiple time delays is presented in this paper. The independence of delays makes the problem much more challenging compared to systems with commensurate time delays (where the delays have rational relations). We uncover some wonderful features for such systems. For instance, all the imaginary characteristic roots of these systems can be found exhaustively along a set of surfaces in the domain of the delays. They are called the “kernel” surfaces (curves for two-delay cases), and it is proven that the number of the kernel surfaces is manageably small and bounded. All possible time delay combinations, which yield an imaginary characteristic root, lie either on this kernel or its infinitely many “offspring” surfaces. Another hidden feature is that the root tendencies along these surfaces exhibit an invariance property. From these outstanding characteristics an efficient, exact, and exhaustive methodology results for the stability assessment. As an added uniqueness of this method, the systems under consideration do not have to be stable for zero delays. Several example case studies are presented, which are prohibitively difficult, if not impossible to solve using any other peer methodology known to the authors.

Kernel and Offspring Concepts for the Stability Robustness of Multiple Time Delayed Systems (MTDS)

2005 ◽  
Author(s):  
Rifat Sipahi ◽  
Nejat Olgac
Keyword(s):  
Time Delays ◽  
Linear Time ◽  
Characteristic Root ◽  
Test Point ◽  
Invariance Property ◽  
Multiple Time ◽  
Delayed Systems ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
The Stability

A novel treatment for the stability of linear time invariant (LTI) systems with rationally independent multiple time delays is presented. The independence of delays makes the problem much more challenging compared to the systems with commensurate time delays (where the delays have rational relations). It is shown that the imaginary characteristic roots can all be found along a set of curves in the domain of the delays. They are called the “kernel curves”, and it is proven that their number is small and bounded. All possible time delay combinations, which yield an imaginary characteristic root, lie on a curve so called the offspring of the kernel curves within the domain of the delays. We also claim that the root tendencies show a very interesting invariance property as a test point crosses these curves. An efficient, exact and exhaustive methodology results from these outstanding characteristics. It is unique to the new methodology that, the systems under consideration do not have to possess stable behavior for zero delays. Several example case studies are presented, which are prohibitively difficult, if not impossible to solve using any other peer methodology.

scholarly journals Stability Analysis of a Predecessor-Following Platoon of Vehicles With Two Time Delays

2015 ◽  
Vol 27 (1) ◽  
pp. 35-46 ◽  
Author(s):  
Ali Ghasemi ◽  
Saeed Rouhi
Keyword(s):  
Stability Analysis ◽  
Time Delays ◽  
Disturbance Attenuation ◽  
One Dimension ◽  
String Stability ◽  
Characteristic Roots ◽  
Vehicle Platoon ◽  
Velocity Information ◽  
The Stability ◽  
Lead Vehicle

The problem of controlling a platoon of vehicles moving in one dimension is considered so that they all follow a lead vehicle with constant spacing between successive vehicles. The stability and the string stability of a platoon of vehicles with two independent and uncertain delays, one in the inter-vehicle distance and the other in the relative velocity information channels, are considered. The main objectives of this paper are: (1) using a simplifying factorization procedure and deploying the cluster treatment of characteristic roots (CTCR) paradigm to obtain exact stability boundaries in the domain of the delays, and (2) for the purpose of disturbance attenuation, the string stability analysis is examined. Finally, a simulation example of multiple vehicle platoon control is used to demonstrate the effectiveness of the proposed method.

scholarly journals Stability and stabilizability of dynamical systems with multiple time-varying delays: delay-dependent criteria

2003 ◽  
Vol 2003 (4) ◽  
pp. 137-152 ◽  
Author(s):  
D. Mehdi ◽  
E. K. Boukas
Keyword(s):  
Time Delays ◽  
Uncertain Systems ◽  
Sufficient Conditions ◽  
Linear Matrix ◽  
Multiple Time ◽  
Multiple Time Delays ◽  
The Stability ◽  
Delay Dependent ◽  
Bounded Uncertainties ◽  
Norm Bounded Uncertainties

This paper deals with the class of uncertain systems with multiple time delays. The stability and stabilizability of this class of systems are considered. Their robustness are also studied when the norm-bounded uncertainties are considered. Linear matrix inequality (LMIs) delay-dependent sufficient conditions for both stability and stabilizability and their robustness are established to check if a system of this class is stable and/or is stabilizable. Some numerical examples are provided to show the usefulness of the proposed results.

Stability Analysis of Multiple Time Delayed Systems Using the Direct Method

2003 ◽  
Author(s):  
Rifat Sipahi ◽  
Nejat Olgac
Keyword(s):  
Stability Analysis ◽  
Time Delays ◽  
Linear Time ◽  
Direct Method ◽  
Multiple Time ◽  
Delayed Systems ◽  
Practical Applications ◽  
Linear Time Invariant ◽  
Multiple Time Delays ◽  
The Stability

A novel treatment for the stability of a class of linear time invariant (LTI) systems with rationally independent multiple time delays using the Direct Method (DM) is studied. Since they appear in many practical applications in the systems and control community, this class of dynamics has attracted considerable interest. The stability analysis is very complex because of the infinite dimensional nature (even for single delay) of the dynamics and furthermore the multiplicity of these delays. The stability problem is much more challenging compared to the TDS with commensurate time delays (where time delays have rational relations). It is shown in an earlier publication of the authors that the DM brings a unique, exact and structured methodology for the stability analysis of commensurate time delayed cases. The transition from the commensurate time delays to multiple delay case motivates our study. It is shown that the DM reveals all possible stability regions in the space of multiple time delays. The systems that are considered do not have to possess stable behavior for zero delays. We present a numerical example on a system, which is considered “prohibitively difficult” in the literature, just to exhibit the strengths of the new procedure.

Hopf Bifurcation of a Type of Neuron Model with Multiple Time Delays

2019 ◽  
Vol 29 (12) ◽  
pp. 1950163 ◽  
Author(s):  
Suqi Ma
Keyword(s):  
Hopf Bifurcation ◽  
Parameter Space ◽  
Time Delays ◽  
Neuron Model ◽  
Multiple Time ◽  
Multiple Time Delays ◽  
The Stability ◽  
Geometrical Scheme ◽  
Delay Differential ◽  
Two Parameter

By applying a geometrical scheme developed to tackle the eigenvalue problem of delay differential equations with multiple time delays, Hopf bifurcation of Hopfield neuron model is analyzed in two-parameter space. By the introduction of two new angles, the calculation of imaginary roots is carried out analytically and effectively. By increasing the parameter to cross over the Hopf bifurcation lines, the stability switching direction is confirmed. The method is a useful tool to show the partition of stable and unstable regions in two-parameter space and detect double Hopf bifurcation further. The typified dynamical behaviors based on nearby double Hopf points are analyzed by applying the normal form technique and center manifold method.

Sign in / Sign up

Export Citation Format

Share Document