1](case-1) and with backtracking algorithm (case-2): communication loss occurs on 3% of the total area. 35 3.9 Position-Based Consensus of Switching Topologies Without Memory (Example 1) [2] and with Memory-Enabled Agents (Example 2): Communication loss occurs at 5% of the total area.
N OMENCLATURE AND M ATHEMATICAL
N OTATIONS
C HAPTER 1
I NTRODUCTION
- Background
- A Survey of Previous Works
- Motivation
- Problem statement
- Contribution of the Thesis
- Thesis Organization
In [62], linear and non-linear consensus protocols are considered in a multi-agent system with time delay. A second-order multi-agent system with time-varying and multiple asynchronous time-delays is considered in [82, 83].
C HAPTER 2
P RELIMINARIES
Algebraic Graph Theory
- Basics
- Matrix representation of graphs
Connected and strongly connected graphs: An undirected graph is called connected if there exists a path between any pair of nodes(vi,vj),∀(i,j)∈[1,n] andi6= j. A directed graph is called strongly connected if there exists a directed path between any pair of nodes(vi,vj),∀(i,j)∈[1,n]andi6= j[103].
C HAPTER 3
C ONSENSUS OF M ULTI - AGENT S YSTEMS IN P RESENCE C OMMUNICATION L OSS
Introduction
The presence of time delays due to processing and communication can affect the stability of agents. Simulation and implementation on a set of robots are performed to demonstrate the operation of the proposed algorithms.
System Model
Static obstacles are considered to be present in the agents' paths, which requires an obstacle avoidance algorithm to be included. 3.2) where, feedback inputs, uix, uiy, corresponding to local interactions with respect to each agent, are given by Eqns. 3.4) The feedback information about the velocity of the agent is given by Eqn.
Back-tracking Algorithm
This can result in agents not being able to decide to stop or use the backtracking algorithm if the topology does not have a spanning tree. If 1 History Following with Memory Enabled Agents Sensor Based Obstacle Avoidance It is considered that for the first stepmin = 10 steps, all the agents broadcast data to enable backtracking without failure. Here, it is considered that the switching of communication topology is based on position and agents have the ability to use backtracking algorithm. It can be observed that the agents eventually reach consensus using backtracking even with communication loss at multiple instances. In the comparison shown in Table 3.1, the different cases have the following parameters. Example 1: Position-based switching without backtracking using the algorithm in the figure. From the simulation results shown in Table 3.1, it can be observed that increasing acceleration and velocity with backtracking will not directly reduce the convergence time. As mentioned earlier, we can see that the root mean square error for the backtracking algorithm is the lowest at both 5% and 3% communication loss. For example, some pins on BBB can be defined as GPIO or PWM output or PRU IO based on the mode definition given in the system reference manual. Sometimes, pulses can be counted incorrectly if a process blocks the processor for a longer duration. The measurements can be further used to estimate the position (using dead reckoning with initial conditions) and velocity of the robot. PRUs can be programmed using assembly-level coding or C script compiled in the TI-CGT compiler provided by Texas Instruments. Using the left and right wheel velocities (viL, viR), the linear velocity of the robot can be calculated using Eq. 3.7) and the angular velocity is calculated using Eq. It can be observed that the time taken to reach consensus in this scenario is less than that of backtracking. Conclusion C HAPTER 4 To the best of our knowledge, frequency domain analysis is not used for the case of unequal input and communication time delays. The control laws that enable the use of frequency domain analysis with unequal delays are considered. In this chapter, we discuss the consensus problem of a distributed multi-agent system with homogeneous nonlinear agents under a fixed directed graph using two control laws. Conditions for a stable interval of time delays are derived using the Nyquist stability criterion in the linear block and their effectiveness for the nonlinear system is tested. Stability analysis of a nonlinear multi-agent system with the help of function analysis description and nyquist stability criterion. Obtaining the necessary and sufficient conditions for the stability of the multiagent system with the proposed control laws. Since the nonlinearity is single-valued, it can be approximated with a descriptive function as given in [108] and always lies on real axis. The approximate gain N(A) of the saturation nonlinearity is given by Eq. 4.3) Going further, it can be found that the non-linear system considered here gives rise to limit cycles only if the linear system is either marginally stable or unstable [108]. Prediction of the existence of limit cycles is done using stability analysis of approximate linear system as shown in sections 4.2.1 and 4.2.2 for different control laws. Approximate linear system given in Figure 4.4) Stability analysis of the linear system for various inputs is performed using frequency domain analysis and the Nyquist stability criterion discussed in Sections 4.2.1 and 4.2.2. Assume that the input time delay is τ1 and the communication delay is τ2 in seconds (sec) with τ1≤τ2. Fornagenti, the above system can be represented as,. 4.14). For a unity delay τ1 =τ2, a unique solution can be obtained if the magnitude is equal to unity and the phase is equal to z. It can be observed that the control input reaches zero as the agent reaches consensus and thus the goal is achieved. The control inputs of rest of the cases also follow a similar trend as in Fig. It can be observed that there is slight deviation in implementation results compared to simulation results as discussed earlier. When τ1=0.3 and τ2=0.9, the modes diverge faster towards the limit cycle behavior and can be observed in fig. The states slowly diverge towards exhibiting limit cycles and the corresponding results are shown in Fig. Similarly, the multi-agent system has better communication time delay (τ2) tolerance with ui2 in Eqn. Atτ1=0.237, both ui1 and ui2 have the communication time delay tolerance of 0.932, within which the multi-agent system converges. C HAPTER 5 A multi-agent system with double integrator dynamics and general nonlinear input without time delay is studied. Few conditions are derived to prove that a general nonlinear odd function g(x)symmetric with origin andg(x)>0, ∀x>0 is able to solve the consensus problem. In other words, limit cycles appear when a linear element is unstable in a multi-agent system. The approximate linear element given in Figure 5.4) Various control laws considered from the literature for analysis are given in Eqs. 5.8) where τ1 and τ2 represent the input and communication time delay, respectively. 5.5) and (5.6) produce a smaller magnitude of the control input, resulting in a slightly longer convergence time compared to those in Eqs. nj=1ai j in the control laws given by Eqs. 5.5) and (5.6) give a better tolerance for time lag due to the smaller Fiedler eigenvalue compared to the control laws in Eqs. A multi-agent system with a spanning tree communication topology and with inputs in equations (5.5) and (5.6) produces the balanced matrices E(A0+A1+A2), E A0, E A1, E A2. When the linear element is stable, the system reaches consensus and limit cycles are not shown. The LMIS feasibility proposed in Equations 5.31) to (5.34) determine the reachability of multi-agent system consensus affected by time delays. Similarly, some simulations and corresponding hardware validations are performed on a five-agent system with communication topology given in Fig. Using results in Theorems 1 to 3, stable regions with respect to time delays for both the . Lyapunov-Krasovskii approximation is more conservative with respect to time delay tolerance, while Nyquist approximation gives the full range of time delay. Compared to the work in [116], we have considered saturation and time delays in the system. Compared to the research presented by the authors in we have considered time delays along with saturation in second-order multi-agent systems. Compared to the work presented by Youet al.[96], asynchronous time delays are considered more than the single delay. If the higher rate agent is not receiving information from the lower rate agent, then consent cannot be guaranteed. Consider a function g which is an integrable nonlinear odd function and is symmetric about the origin with xg(x)>0; ∀|x|>0 and g(0) =0. If f1(uˆi,vi) =1,βi, k1,k2 are positive constants, then according to the functionality f1,. a) Graph of the odd function given in the equation b) Graph of the odd function given in the equation For an undirected network topology that is strongly connected, the control law ui in Equation (5.70) solves the consensus problem with xi→xjand vi→0as t→∞. That is, consensus is possible with xi→xj and vi→0 as t →∞, for a multi-agent system with an undirected connected graph topology using the control law in Eq. The convergence time is longer for lower values of k1 and k2 compared to higher values of k1 and k2. Simulation and Implementation Results Conclusion C HAPTER 6 The reasoning of the theoretical results is done with the help of simulations and corresponding implementations in hardware. Future Scope and Possible Directions A PPENDIX A Exponential stability implies that the system is asymptotically stable and that the states converge towards the origin with an exponential convergence rate of β. The states converge to the origin, regardless of the initial values, if the system is globally asymptotically (or exponentially) stable. If the Lyapunov function does not converge or diverge, but is bounded in a ball, then the system is stable. Let P be a symmetric positive definite matrix of suitable dimension, then the Lyapunov function can be of the form given in Eqn. For the system in Eq. A.6) must be considered globally asymptotically stable, ATP+PA=. Q must be negative definite or Q must be positive definite. A necessary and sufficient condition for the system in Eq. A.6) to be asymptotically stable is that there exists a unique symmetric positive definite Pmatrix for every symmetric positive definite Qmatrix. L IST OF P UBLICATIONS Distributed consensus of multi-agent systems with control input transmission error and time-varying delays. Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays. Consensus analysis of second-order continuous-time multi-agent systems with non-uniform time delays and switching topologies. Distributed consensus detection for nonlinear multi-agent systems with input saturation: an order-filtered backstepping approach. Consensus of heterogeneous first- and second-order multi-agent systems with directed communication topologies. Consensus of a class of second-order multi-agent systems with time delay and jointly-connected topologies.
Simulation Results
Implementation on Hardware
M ULTI - AGENT SYSTEM ANALYSIS : F REQUENCY D OMAIN A PPROACH
Introduction
System model and analysis
Simulation and Hardware implementation results
Conclusion
M ULTI - AGENT SYSTEM ANALYSIS : L YAPUNOV A PPROACH
Introduction
System model and analysis
Simulation and Implementation Results
Multi-agent Systems with General Nonlinear Input
C ONCLUSIONS AND F UTURE S COPE
Conclusions
S UPPLEMENTARY M ATERIALS
R EFERENCES
![Figure 3.9: Consensus of switching topologies based on position without memory(case-1) [2]](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10541903.0/50.892.132.808.123.1063/figure-consensus-switching-topologies-based-position-memory-case.webp)
