#### Abstract

This paper considers the consensus control problem of nonlinear spatial-temporal hyperbolic partial difference multiagent systems and parabolic partial difference multiagent systems with time delay. Based on the system’s own fixed topology and the method of generating the desired trajectory by introducing virtual leader, using the consensus tracking error between the agent and the virtual leader agent and neighbor agents in the last iteration, an iterative learning algorithm is proposed. The sufficient condition for the system consensus error to converge along the iterative axis is given. When the iterative learning number approaches infinity, the consensus error in the sense of the norm between all agents in the system will converge to zero. Furthermore, simulation results illustrate the effectiveness of the algorithm.

#### 1. Introduction

As sensor technology advances, communication and network technology, when solving complex practical problems, the ability of communication and coordination between various agents in a multiagent system (MAS) is used to solve the problem that a single agent cannot complete complex task [1, 2]. In recent years, MASs’s cooperative control has been widely used in the industry, such as formation control for drones [3], satellite attitude control [4], and congestion control of communication networks [5]. At present, the main problems are collaborative control of MASs cluster control [6], formation control [7], and consensus control [8]. Among them, the consensus problem of MASs is its most fundamental problem. The research on the MASs consensus problem is mainly through designing appropriate controllers for agents, so that all agents must be consensus in some given state or output [9].

In recent years, the consensus of MASs has been extensively studied from different perspectives [10]. In [11], Vicsek et al. first designed a classic consensus model and verified its effectiveness through simulation examples. In [12, 13], Ren studied the consensus problem of discrete and continuous models of MASs. In [14], Olfati-Saber et al. handled the progressive consensus of the leaderless MASs. Subsequently, based on the characteristics of the MASs, scholars designed more control algorithms to achieve consensus control of the MASs. In practical applications, there are many tasks that need to be executed repeatedly, and coordination should be carried out among several independent individuals [15]. For this reason, we need to study MASs. For example, in the coordinated control of a multimanipulator system with repeated operations on a production line, it is impossible to fully realize the control of the tracking desired trajectory within a given finite-time control [16]. For example, the periodic operation of multisatellite systems cannot ignore the influence of changes in the internal parameters of the system and external disturbances, and the application of the sliding mode control strategy cannot guarantee the stability of such systems [17]. Due to the limitation of information obtained by each agent, the expected information (or trajectory) of the leader specified by the topology diagram is only available to some agents. For each agent, this requires that the designed distributed algorithm only allows the use of local information obtained by certain agents to track the desired trajectory as perfectly as possible. Iterative learning control (ILC) has great advantages in dealing with MASs that continuously repeat tasks within the finite time interval [18].

Other control methods [19–21] (such as feedback control, adaptive control, and robust control) are compared with ILC. Many formers consider a continuous process system; that is, the operation of the system will continue to run over time. ILC is to complete a given task in a period of time and repeat it continuously. Therefore, compared with other control methods, in designing ILC algorithms, not only the time axis but also the iteration axis must be considered. The main control goal is to achieve convergence along the iteration axis, which is different from other control methods that only consider the time axis. In other words, ILC is a typical 2D process, and this characteristic also causes the theoretical analysis method of ILC algorithm to be different from other control methods. The controller of the ILC algorithm has a simple structure, and the ILC algorithm is effective for complex dynamic systems (such as uncertain, nonlinear, difficult to model, and missing information). It was proposed by Arimoto et al. in 1984 and after more than 30 years of development. There are many results in the research on ILC [22–25]. Because ILC has the above advantages, it provides an effective and feasible method for solving the consensus control problem of networked MASs in the finite-time interval. For example, in [26], the ILC method is used to solve the consensus problem that the leader with packet loss follows the nonlinear MASs. In [27], in order to solve the consensus tracking problem of the heterogeneous MASs, a corresponding ILC protocol was designed. In [28], P-type and PI-type ILC update algorithms are designed to solve the consensus tracking problem of a class of nonlinear fractional MASs. In [29], ILC technology is used to solve the consensus tracking problem in singular MASs. In [30], the problem of iterative learning consensus control in a continuous-time MAS is studied, and, during the learning process, the test length of each iteration is allowed to change randomly.

With the rapid development of digital sensors and digital controllers, in practical applications, although the system itself is a continuous process, since the channel bandwidth is limited in the communication system, the controller can only apply the sampled data obtained at discrete times [31]. Therefore, related research on discrete systems is necessary. Many scholars have begun to study the issues related to discrete-time MASs [32–36]. In [34], a discrete form of ILC algorithm was designed for the discrete MASs consensus tracking problem with random noise. In [35], the consensus tracking problem of a class of nonlinear discrete MASs is studied when the iteration length changes randomly. In [36], with the help of ILC algorithm, the consensus tracking problem of discrete-time MASs with cooperative competition network is solved.

Analysis of the above literatures reveals that the current results of using ILC algorithms to solve the consensus problems of MASs are mostly studies on ordinary differential MASs, which study the convergence of all agents in the system in the time domain. However, in practice, MASs have spatial dynamic behaviors, such as biological systems, chemical reactions, food networks, and other processes that depend on time and space [37]. The distributed parameter MASs consider the convergence of both the time domain and space domain, which has been of certain practical significance for the research of MASs related problems. Recently, there have been reports on related research on distributed parameter MASs. For example, in [38], the consensus of the continuous distributed parameter model MASs was achieved based on the Lyapunov function method. In [39], an ILC algorithm was proposed for continuous distributed parameter model MASs. In [40], the consensus problem of continuous distributed parameter MASs with time delay is studied. In [41], a second-order ILC algorithm is proposed for a class of consensus problems with continuous distributed parameter MASs with time delay.

Considering these factors, in this paper, a class of nonlinear first-order hyperbolic partial difference MASs and parabolic partial difference MASs with time delay are studied in the finite-time interval. Based on the fixed topology for the MASs, an ILC algorithm is proposed with a method for the virtual leader agent to generate the desired trajectory. This algorithm uses the tracking error between any agent with the virtual leader agent and neighbor agents in the last iteration and combines the characteristics of the MASs communication topology to continuously modify the previous control law to obtain the ideal control law. When the iterative learning index approaches infinity, in the sense of the norm, the consensus error between all agents in the MASs will converge to zero.

The main contributions of this paper are summarized as follows:(1)It can be noted that the works in [37–41] present a type of consensus problem in the study of continuous distributed parameter MASs. However, due to the limited channel bandwidth in the communication system, the controller can only apply the sampling data obtained at discrete moments. Therefore, it can be considered that the consensus control of discrete distributed parameter MASs is more practical.(2)In the works in [38–41], the consensus control problem of parabolic distributed parameter MASs is studied. Since hyperbolic system is an important system type, a kind of consensus control of hyperbolic distributed parameter MASs has been studied. When the leader agent number is unknown, complete consensus control of all agents in the system is achieved within a finite time.(3)From the analysis of the works in [38–42], in reality, the MASs have relatively complex nonlinearities and time delay. Therefore, an ILC algorithm is designed, which contains consensus errors between any two agents. After the system has been iteratively learned enough times, the consensus problem of nonlinear hyperbolic partial difference MASs and parabolic partial difference MASs with time delay is better solved.

The remainder of the article is organized as follows. In Section 2, we consider the iterative learning consensus control for nonlinear first-order hyperbolic partial difference MASs with time delay. In Section 3, we discuss iterative learning consensus control of the nonlinear parabolic partial difference MASs with time delay. In Section 4, two examples illustrate the results. Section 5 gives the conclusion.

*Notations*. This article uses the following symbols: is the dimension unit matrix, is a column vector of order, and each element is 1. denotes the Kronecker product. For a function , is expressed as , where , and . Let be a positive constant, and, for the binary function , its norm is expressed as .

Using the knowledge of graph theory to describe the interrelationship between any two agents, the corresponding graph theory knowledge is as follows: let represent a weighted directed graph, where is a collection of nodes, and is the set of edges. represents the adjacency matrix of graph ; if node to node has an edge, then , and this indicates that agent can accept the information of agent ; otherwise, . Define the Laplacian matrix of the graph as , degree matrix , be node .

#### 2. Iterative Learning Consensus Control for Nonlinear Hyperbolic Partial Difference Mass with Time Delay

Consider a class of nonlinear hyperbolic partial difference MASs composed of homogeneous discrete multiple agents, which run repeatedly in the finite-time interval. The dynamic model of the -th agent iswhere are discrete variables of space and time, respectively. and are given integers, and , . is the number of iterations, and indicates the -th agent. are uncertain bound real sequences, and . , respectively, denote the state of the -th agent in system (1), control input, and control output. is the time delay. Let the initial state function be ; when , . are both nonlinear functions.

The differential form of the first expression in system (1) is defined as follows:

*Assumption 1. *Graph contains the spanning tree, and the virtual leader agent is the root of the spanning tree.

*Assumption 2. *For all , the initial boundary condition of the -th agent in system (1) is given as

*Assumption 3. *The dynamic characteristics of system (1) are reversible, ensuring that there is a single desired control input such that the state and output of system (1) are expected.

*Assumption 4. *The nonlinear functions , respectively, satisfy the following uniform global Lipschitz condition:where and are constants.

*Remark 1. *Agents can only get information from virtual leader and neighbor agents, while virtual leaders cannot get information from other agents.

Lemma 1 (see [43]). *Let , , and be real sequences defined for satisfying**Then,*

In this paper, the trajectory of the virtual leader agent is the desired trajectory of the discrete distributed parameter MAS (see (1)) . Because MAS has a distributed structure, only a few agents in the system are allowed to directly obtain the required trajectory information. The agents in system (1) which will directly obtain the desired trajectory information depend on the communication topology between the multiple agents.

The ultimate goal of this paper is that each agent in the MAS is able to completely track the motion trajectory of the virtual leader agent in the finite-time interval and finally achieve the consensus control of the MAS. However, due to the uncertainty of system (1), it is not easy to directly find the appropriate control input . Therefore, we use the ILC method to gradually find the appropriate control input to satisfywhere is the output of the -th agent in the -th iteration of system (1). This equation means that after the MAS learns with an ILC algorithm, in system (1), the consensus error between any two agents is able to converge to zero.

Based on the characteristics of information transfer between agents, the following ILC algorithm is designed:where is the learning gain, is the number of iterations. is the tracking error of the -th agent, and is the tracking error of the -th agent. reflects the communication relationship between the -th agent and virtual leader agents. indicates that the -th agent can communicate with virtual leader agent, and is the opposite.

For the simplicity of the subsequent convergence analysis, the compact form of system (1) is as follows:where

Define the consensus error of system (10) aswhere is the trajectory of virtual leader agent and .

Further, the compact form corresponding to ILC law (see (9)) is can be written aswhere is the Laplacian matrix of graph and .

After system (10) iterates times, the corresponding difference form is expressed as

Theorem 1. *If in the ILC law (see (13)) satisfies and satisfies Assumptions 1–4 at the same time, and when the number of iterations , the consensus error of system (10) can converge to zero in the following sense:*

*Proof. *From equations (12) and (13), after the *k*-th iteration, the consensus error of system (10) can be written aswhereImmediately after the two sides of equation (17) are squared, one obtainsSumming from to on both sides of inequality (19), one obtainswhereIt can be seen from inequality (20) that, to prove the convergence of consensus error of system (10), it is necessary to estimate . Thus, by the first equation in system (10) and equation (14), one can deriveFurther, during the -th iteration, the agents state can also be written asNext, after subtracting equation (22) from equation (23) and combining with equation (15), we have the following results:The two sides of equation (24) are squared at the same time. Since , equation (24) can be rewritten asSumming from to on both sides of inequality (25) and using Lipschitz condition (5), one obtainswhere . Combining the boundary condition (3), in inequality (26) can be written asSubstituting (27) into (26), one haswhere . Combining Lemma 1 and the initial value condition (4), we have the following results:So, we can see the relationship between and . Next, choosing ILC algorithm (13), one obtainswhere .

In addition, by inequality (30), inequality (29) can be further written asMultiplying both sides of inequality (31) by , (31) can be rewritten as follows:Estimating state with time delay in inequality (32) and letting , from the known conditions, when , there is , and one hasBy the initial value condition , inequality (33) can be written asSubstituting inequality (32) into inequality (34), one hasSince is sufficiently small, can be obtained. Inequality (35) can be written asMultiplying both sides of inequality (20) by , one obtainsSubstituting inequality (37) into inequality (36), one obtainsLetting can be obtained from the condition , of Theorem 1. Therefore, . After recursion by inequality (38), one getsThen, by the contraction mapping principle, one obtains

Further, we have the following results:Thus, one hasNext, combining equation (22), the -th agent consensus tracking error can be rewritten as follows:Further, due to , one getsFinally, based on equation (42) and inequality (43), one obtainsThe proof of Theorem 1 is completed.

#### 3. Iterative Learning Consensus Control for Nonlinear Parabolic Partial Difference Mass with Time Delay

In this section, consider a nonlinear parabolic partial difference MAS (45) composed of homogeneous discrete multiple agents, which run repeatedly within the finite-time interval, where the -th agent’s dynamic equation iswhere are discrete variables of space and time, respectively. and are given integers, and , . is the number of iterations, and indicates the *i*-th agent. are uncertain bound real sequences, and . , respectively, denote the state of the -th agent in system (41), control input, and control output. is the time delay; let the initial state function be ; when , . are both nonlinear functions.

The differential form of the first expression in system (45) is defined as

Here, it is still necessary to achieve consensus error between any two agents in system (45) to satisfy . Under the conditions of Assumptions 1–4, we use ILC algorithm (13) to achieve this goal.

For the simplicity of the subsequent convergence analysis, the compact form of system (45) is as follows:where

After system (45) iterates times, the corresponding difference form is expressed as

Theorem 2. *If is in use, ILC algorithm (13) satisfies and satisfies Assumptions 1–4 at the same time; then when the iteration index , in the following sense, consensus error for the nonlinear parabolic partial difference system (47) can converge to zero; that is,*

*Proof. *To prove the convergence for system (47) consensus error , we need to estimate . By equation (49) and the first equation in system (47), one hasSimilar to equation (52), during the -th iteration, the agent’s state can also be written asNext, subtracting equation (52) from equation (53) and combining with equation (50), one hasHere, squaring both sides of equation (54) and by , one obtainsSumming from to on both sides of inequality (55) and using Lipschitz condition (5), we have the following results:whereBy the boundary condition (4), the following inequality holds:Substituting (58) into (56), one has where . Combining Lemma 1 and the initial value condition (4), one hasHere, we found the relationship between and . Following on, by the ILC law (see (13)), one getswhere . It is observed that the output equation of system (45) is the same as that of system (1), which gives the following results:whereIn addition, by inequality (61), inequality (60) can be further written asMultiplying both sides of inequality (64) by , one obtainsEstimating state with time delay in inequality (65) and letting , from the known conditions, when , there is , and one has