Fast Consensus Tracking of Multiagent Systems with Diverse Communication Delays and Input Delays

1 Faculty of Chemical Engineering, Ministry of Education and Faculty of Metallurgical and Energy Engineering, Kunming University of Science and Technology, Kunming 650500, China 2 Faculty of Land Resource Engineering, Ministry of Education and Faculty of Metallurgical and Energy Engineering, Kunming University of Science and Technology, Kunming 650500, China 3 Engineering Research Center of Metallurgical Energy Conservation & Emission Reduction, Ministry of Education and Faculty of Metallurgical and Energy Engineering, Kunming University of Science and Technology, Kunming 650500, China


Introduction
As an effective method to solve decentralized multiagent cooperative control which is widely applied into many fields such as flocking [1,2], formation control [3,4], and unmanned air vehicles [5], consensus algorithms designing of multiagent systems has attracted great attention in recent years.A key task for consensus algorithms is to achieve a global common behavior through designing a distributed protocol based on local information.Reference [6] proposed a simple model for phase transition of self-driven of the model.A simple consensus protocol to solve the average consensus problem was discussed in [7].Furthermore, two survey papers which introduce the basic concepts of consensus of multiagent systems, the methods of convergence, and performance analysis for the protocols and recent development can be seen in [8,9].In real applications, when local information data travel along channels in a multiagent network, a communication delay exists due to the physical characteristics of the medium transmitting the information.And at the same time, each agent also needs computing time to process its information.As a result, time-delay problem including communication delays and computing delays (also called input delays) is not avoided in designing consensus algorithms.Reference [10] discusses the consensus problem for multiagent systems with input and communication delays based on the frequency-domain in discrete-time formulation and a conclusion where the consensus condition is dependent on input delays but independent of communication delays is obtained.
Convergence rate or speed is an important performance index in the analysis of consensus problems.For example, sensors need to reach fast consensus on the estimates between sensor observing intervals in distributed estimation problem.In this field, main research works focus on fast consensus [11][12][13] and finite-time consensus [14][15][16].Reference [11] proposes a new consensus protocol which considers the average information of the agents' states in a certain time interval and increases consensus speed of multiagent systems through determining suitable upper limit of time interval based on the frequently domain analysis and matrix theory.Reference [12] proposes a class of pinning predictive controllers for consensus networks to substantially increase their convergence speed towards consensus.In [13], an optimal synchronization protocol was designed for the fastest convergence speed when the protocol is perturbed by an additive measurement and process noise.As to finite-time consensus algorithms design, [14] designed continuous distributed control algorithms for double integrators leaderless and leader-follower multiagent systems with external disturbances based on the finite-time control technique.Based on positive or negative values of errors between their neighbor's values and their own state values in multiagent systems, a simple distributed continuoustime protocol is introduced by [15] that guarantees finitetime consensus in networks of autonomous agents when the network has directed switching network topologies and timedelayed communications.
Although fast or finite-time consensus without a virtual leader is interesting, it is sometimes more meaningful and interesting to study consensus tracking problem when the virtual leader's state (also called reference) may represent the state of interest for these systems.In [17], a coordinated tracking algorithm with a time-varying leader for first-order dynamics is proposed and bounded control and directed switching interaction topologies are considered when a timevarying consensus reference state is available to only a subset of a team.However, this algorithm requires the estimates of the neighbors' velocities.In [18], distributed coordinated tracking algorithms are studied when only partial measurements of the states of the virtual leader and the followers are available.Reference [19] studies the issues associated with distributed coordinated tracking for multiple networked Euler-Lagrange systems where only a subset of the followers has access to the leader.As to discrete-time formulation, [20] considers consensus tracking problem when location information of the active leader is completely known but the acceleration information may not be measured; a neighborbased pinning control law and a neighbor-based state estimation rule are proposed.Although consensus tracking problem is discussed widely, few works focus on fast consensus tracking with time-delays in discrete-time formulation.
Motivated by these topics, fast consensus tracking problem of discrete-time multiagent systems with communication delays and input delays is discussed in this paper.The main contribution of this paper is to establish a simple protocol in order to guarantee consensus tracking convergence in general directed network topology based on the frequency-domain analysis.Then, an increment PID algorithm is introduced to improve the convergent speed and an inequity condition which can describe relations of controller gain, input delays, communications delays and topology structure is obtained.Furthermore, genetic algorithm [21] is introduced to construct increment PID based on genetic algorithm for obtaining optimization consensus tracking performance.This makes the proposed protocol more practical for application to real-time applications.This paper is organized as follows.In the next section, preliminary notions and multiagent systems model are provided.A conventional -like discrete-time consensus tracking algorithm for a single-integrator systems is stated in Section 3 and a fast discrete-time consensus tracking algorithm based on increment PID is established in Section 4. By applying to genetic algorithms, the fast consensus tracking algorithm mentioned above is optimized in order to obtain an optimal cost in Section 5. Simulation example is shown in Section 6.Finally, concluding remarks are stated in Section 7.

Graph Theory Notions
Notations.The notation used in this paper for graph theory is quite standard.For a system with  agents, the communication graph among these agents is modeled by a directed weighted graph  = (, , ), where  = {V 1 , V 2 , . . ., V  },  ⊆  2 , and  = {  } ∈ R × represent the set of agents, the edge set, and the weighted adjacency matrix, respectively.The agent indexes belong to a finite index set Φ = {1, 2, . . ., }.An edge denoted as (V  , V  ) means that the V  th agent can access the information of the V  th agent.We assume that the adjacency elements associated with the edges of the digraph are positive.That means   > 0, if agent  receives information from agent  otherwise   = 0.Moreover, we assume feedback gain   > 0 for all  ∈ Φ, if th agent has feedback control loop and   = 0 otherwise.Define the set of neighbors of agent V  as   = {V  ∈  : (V  , V  ) ∈ }.
For the directed digraph the outdegree of agent  is defined as deg out (V  ) = ∑  =1   .Let  be the degree matrix of , which is defined as a diagonal matrix with the degree of each agent along its diagonal.The Laplacian matrix of the weighted digraph is defined as  =  −  satisfying zero-row sum.
In multiagent systems, each agent can be considered as a node in a digraph, and the information flow between two agents can be regarded as a directed path between the nodes.Thus, a directed graph has a directed spanning tree if there exists at least one agent called a globally reachable agent that has a directed path to all other agents.

Multiagent System Modeling.
Consider agents with a single-integrator kinematics in discrete-time formulation given by where   ∈ R and   ∈ R denote the state and the control input of agent , respectively.The following consensus tracking protocol for the multiagent systems ( 1) is a classical formulation mentioned by literature [10] which can be described by where   () is a time-varying reference state or a virtual leader with the states, named agent 0 and the other agents indexed by 1, 2, . . .,  are referred to as followers without loss of generality.(Especially, if   () =   , this reference state can be simplified to a constant one). (0) is 1 if agent  has access to   () and 0 otherwise.  denotes the neighbors of agent  and   > 0 is the adjacency element of  in the directed digraph  = (, , ).  > 0 denotes the feedback control gain of agent  and   = 0 otherwise.When agent  is subjected to a time-varying input delay   (), system (1) can be rewritten as follows: Consider the total delay where an agent receives data from its neighbors is sum of time-varying input delay and time-varying diverse communication delay, so the consensus tracking protocol becomes where   (),   () denotes time-varying input delays of agent ,  and   () denotes time-varying communication delay from agent  to agent , respectively.It is assumed that each agent has similar computer capacity, so the time-varying input delay of each agent can be treated with the same time-delay value; that is,   () =   ().To simplify the complexity of calculation, we assume that agent  needs to possess memory capability such that   ( −   () −   ()) can be used in the consensus tracking protocol.Substitute state   ( −   ()) in coupling terms of (4) for   ( −   () −   ()) and let the total delay T () =   () +   () which satisfies   () ≤ T ().As a result, (4) can be rewritten as Moreover, these two classes of delays can be approximated by   () = (  −1)+  and T () = (  −1)+  , respectively.Where   (), T () ≥ 0,  denotes sample period of this discrete-time system,   ,   are all nonnegative integers and   ,   are unknown-but-bounded variables which belong to interval (0, ).So it is reasonable that   () and T () are approximated by   and   although some artificial delays are included.In the end, (4) can be rewritten as Substituting protocol (6) to the system (3), we have Using algorithm (6), each agent essentially updates its next state based on its past state with limited time delay and its neighbors' current as well as the reference's current if the reference is a neighbor of the agent.As a result, ( 6) can be easily implemented in practice.

𝑃-Like Discrete-Time Consensus Tracking with Input Delays and Communication Delays
In this section we consider consensus tracking problem of multiagent systems with both communication delays and input delays.Firstly, two lemmas related to this topic need to be introduced.Then, a sufficient consensus tracking condition of multiagent systems (7) with conventional like algorithmn is proposed based on the frequency-domain analysis and matrix theory.
Lemma 1 (Gershgorin's disk theorem).Let Λ = (  ) be a  ×  complex matrix; then all eigenvalues of matrix Λ belong to the union set of  circular disc on the complex plane; that is, where Lemma 2 (see [10]).The following inequality: holds for all nonnegative integers  and all  ∈ [−, ].
In the following, we apply Lemmas 1 and 2 to derive our main result.Theorem 3. Consider multiagent systems (3) with algorithm (6).Assume that the interconnection topology digraph  = (, , ) of the system has no less than a globally reachable agent and at least one globally reachable agent can receive reference information.Then the system achieves a consensus tracking asymptotically if where   =  (0)   denotes the feedback control gain of agent ,   = ∑ V  ∈    denotes outdegree of agent .
Remark 4. If we rewrite (19) as (  +  )(2  +1)+2  (  −   ) < 1, it is easy to know that consensus tracking problem of multiagent systems is more sensitive to input delays than communication delays.Moreover, Theorem 3 is also suitable for the case when it only has input delays if we let   =   .

Fast Discrete-Time Consensus Tracking Algorithm Based on Increment 𝑃𝐼𝐷
Considering feedback control gains   ,  = 1, 2, . . .,  is the similar with conventional -like controllers, an increment PID algorithm is proposed consequently to accelerate the convergence speed of multiagent systems (7).Discrete PID algorithm is described by where ℎ  () is feedback control signal of agent  at time interval .  () denotes the error of current state between agent  and current reference state and   ( − 1) denotes the error of the value between agent  and reference state at time interval  − 1, respectively.  denotes proportional coefficient,   denotes integral time,   denotes derivative time, and sampling period is described by .Through ℎ  () − ℎ  ( − 1), we obtain increment PID algorithm as where  = (1 + /  +   /),  =   (1 + 2(  /)), and  =   (  /).
Remark 5. From (21) we know that if sampling period  and coefficient , ,  are chosen, control signal Δℎ  () will be obtained by only using three adjacent deviation values.Because this algorithm is easily realized in the agent with limited computing capacity, it is very suitable for multiagent system.

Remark 7.
Because an inequality is used in (28), the result of Theorem 6 is considerable conservatism.If  =   , ,  = 0, we can obtain Theorem 3.That is to say, this conservatism can be ignored if ,  are very small.Remark 8.By using increment PID algorithm, the maximum allowable time delay of consensus tracking of multiagents system become larger; even input delays   and communication delay   in Theorem 6 are chosen to disobey the inequality; this multiagent systems is still converged to its reference in many cases.

Optimization Consensus Tracking PID Algorithm Based on Genetic Algorithm
The main result in Section 4 gives a consensus tracking range whose multiagent systems can converge to reference.However, our interesting is how fast these multiagent systems can converge to reference or are there optimal PID parameters which make these systems track reference with optimal performances?Here, a new optimization increment PID algorithm based on genetic algorithm (GA-PID) is proposed for optimization cost including rise time, output energy of controllers, and tracking error.
Here, the fast consensus tracking problem of multiagent system is described as follows: finding the optimal PID parameters   ,   ,   of Theorem 6 which can make the system achieve faster consensus tracking.It is well known that the genetic algorithm is an effective method that can find the global optimal solution, so we improve the conventional genetic algorithm in order to solve this optimization tracking problem.The basic design steps of self-adjusting PID controller based on the genetic algorithm are as follows.
(1) To ascertain parameters.To ascertain the values of PID parameters according to the mathematic model of the system so as to narrow the searching scope and improve the efficiency of optimization, here, let   ,   ,   ≤ 1.
(2) To select the initial population.Here, 50 initial populations are chosen at random, so populations size  is equal to 50.
(3) To ascertain the adaptation parameter.Combing three control performances stability, raPIDity, and accuracy, the target functions shown as below can be used as the optimal index for the selection of parameters: where   is the global optimal index,   is the local optimal index of agent  for tracking reference.  (),   (), and    are the error, controller input, and rising time   In fact, the multiagent systems can converge to the reference on the condition that at least one globally reachable agent can receive the constant reference.Similar results can also be obtained in different topologies.Details could be seen in Table 1.Comparing these three algorithms we can see that -like algorithm has slowest convergence speed, increment PID algorithm has strongest robustness performance, and increment PID based on GA has fastest convergence speed in almost all cases.What is more, a very interesting thing  is also deduced from Table 1.That is, convergence speed is similar between all agents receiving the reference and only globally reachable agents receiving the reference.This means that only globally reachable agents instead of all agents receive reference and a similar convergence speed can also be obtained.
From [9] we know that conventional -like algorithm is not sufficient for consensus tracking when all agents receive a time-varying consensus reference.However, if our increment PID algorithm is adopted, consensus tracking can be achieved through choosing suitable PID parameters.Comparing Figure 6 with Figure 7 we can see that suitable PID parameters not only decrease errors between reference and current states of agents, but also increase the convergence   speed.Regretfully, these three algorithms cannot be used to track a time-varying reference, that is, available to only a subset of the team members for only receiving timevarying reference state.If time-varying reference changes in a piecewise constant case, increment PID algorithm based on genetic algorithm can track this time-varying reference whether the reference is available to all team members or to a subset of the team member.From Figure 8 we can see that Re, X 1 , X 2 , X 3 , X 4 , X 5 , X the multiagent systems can converge to a piecewise constant reference within about 40 seconds when only agents (2,3,6) receive this time-varying reference.This characteristic can be applied to many fields such as synchronizing a network of clocks [13].

Conclusions
In this paper, three consensus tracking algorithms named -like algorithm, increment PID algorithm, and increment PID algorithm based on genetic algorithm, respectively, for discrete-time multiagent systems with time-varying input delays and communication delays are proposed based on the frequency-domain analysis.Firstly, a consensus tracking sufficient condition of conventional -like algorithm is obtained.Secondly, a new increment PID algorithm based on similar frequency-domain method is designed for improving consensus convergence speed and an inequality condition is also deduced.Finally, considering three control performances stability, rapidity, and accuracy, an increment PID algorithm based on genetic algorithm is designed to find optimal PID parameters within an inequality allowable span for achieving optimization cost.These three algorithms can solve tracking problem of multiagent systems with a constant reference effectively when reference state is available to all the team members.If the reference state might only be available to a portion of the agents in the team, the convergence speed may increase in the same condition.As for a time-varying reference case, if the reference state has a directed path to all team agents, increment PID algorithm and increment PID algorithm based on genetic algorithm can realize consensus tracking through choosing suitable PID parameters while conventional -like algorithm fails to track the reference.
Mathematical Problems in Engineering However, these three algorithms cannot be used to track a time-varying reference state when the reference is available to only a subset of team members.In the future research, we will focus on more complex issues in the controller design such as actuator delay and fault,  ∞ controller design with control delay, quantized control, and global consensus problem with saturated control [22][23][24][25][26][27][28][29].

Figure 2 :
Figure 2: States trajectory achieving consensus by -like algorithm.

Figure 3 :
Figure 3: States trajectory achieving consensus by increment PID algorithm.

Figure 6 :
Figure 6: States trajectory with time-varying reference by -like algorithm when  = 0.1.