#### Abstract

Robust consensus control problems of linear swarm systems with parameter uncertainties and time-varying delays are investigated. In this literature, a linear consensus protocol for high-order discrete-time swarm systems is proposed. Firstly, the robust consensus control problem of discrete-time swarm systems is transformed into a robust control problem of a set of independent uncertain systems. Secondly, sufficient linear matrix inequality conditions for robust consensus analysis of discrete-time swarm systems are given by the stability theory, and a performance level is determined meanwhile. Thirdly, the convergence result is derived as a final consensus value of swarm systems. Finally, numerical examples are presented to demonstrate theoretical results.

#### 1. Introduction

As a hot topic of swarm systems, distributed cooperative control for swarm systems has gained an increasing attention from many communities because of its wide applications in various areas including rendezvous [1–3], filtering [4–6], formation control [7–9], synchronization of power-driven machine [10–12], and other areas [13–16]. It should be also pointed out that some relevant novel nearest results in aerospace have been reported in [17–19]. Furthermore, as an essential research aspect of distributed cooperative control for swarm systems, the consensus problem, which is referred to an agreement on some common values by all agents, has received extensive research in recent years [20–22].

According to the difference of the system dynamics, the research objective of the consensus problem for swarm systems can be classified into three types: first-order swarm systems, second-order swarm systems, and high-order swarm systems. For first-order swarm systems, the consensus problem with input and communication delays was studied by the frequency analysis in [23]. The consensus of second-order agents with multiple time-varying delays was investigated based on the graph theory in [24]. In combination with realistic applications, high-order swarm systems are more valuable and general than low-order swarm systems due to the practicality and complexity of the former. Some significant theoretical achievements about the consensus of continuous-time high-order swarm systems have been discovered in [25]. However, digital signals are processed rather than analog signals under engineering background, so discrete-time swarm systems should be paid more attention. The distributed output feedback consensus problem of general linear model discrete-time swarm systems with fixed topology was presented in [26, 27]. Based upon the leader-following model, the consensus problem of discrete-time swarm systems with directed graphs and external disturbances was investigated in [28]. Furthermore, the consensus problem of discrete-time swarm systems with switching topologies based on the matrix inequality theory was presented in [29–31].

As the industrial technology continues to develop, the research on the control theory becomes more mature. Due to operating state changes, external disturbances and model errors of swarm systems in the real world, robust consensus control problems of swarm systems were wildly explored by researchers, as shown in [32–34]. Robust consensus control problems of first-order swarm systems with external disturbances and model uncertainties were presented in [32]. The second-order robust consensus control problem of swarm systems with measurement noises and asymmetric delays was studied in [33]. Distributed and consensus control problems of high-order linear swarm systems with dependent noise were investigated in [34]. However, due to the general existence of parameter uncertainties and time delays of swarm systems in practical application, it is more meaningful and important to take these constraints into consideration in the theoretical analysis of robust consensus control problems of high-order linear discrete-time swarm systems, which can been deemed as inadequately available in current literature.

As is well known, precise dynamics models of swarm systems can be hardly established in practice, and many kinds of faults in systems can also contribute to uncertainties of the dynamics model. Besides, components of systems will be out of order or aged with the increase of working hours. Therefore, researches on the robust consensus problem of high-order linear discrete-time swarm systems with parameter uncertainties are very valuable and significant. Moreover, time delays cannot be avoided in nature and they take place so commonly in biological, physical, and electrical systems. The acquisition of parameters becomes more difficult when time delays exist in systems. If time delays are ignored when the consensus problem of swarm systems with parameter uncertainties is discussed, the stability of the system is hard to be guaranteed. Motivated by the above analysis, for high-order linear discrete-time swarm systems with parameter uncertainties to achieve consensus by the robust control method, time delays are necessary to be considered and the problem remains the most open as we know.

This paper studies robust control consensus problems of high-order linear discrete-time swarm systems with parameter uncertainties and time-varying delays, and the innovations of this paper are threefold: (i) a sufficient linear matrix inequality (LMI) condition such that swarm systems can achieve the robust consensus subject to parameter uncertainties and time-varying delays is derived; (ii) a performance index of swarm systems is obtained; and (iii) the final consensus value of swarm systems is developed. Compared with the previous work on the robust consensus of high-order swarm systems, the novelties of this literature are also threefold. Firstly, only the robust consensus of high-order linear swarm systems was investigated in [35–39], where the performance index was not obtained. However, in order to show the performance of the robust consensus of high-order linear discrete-time swarm systems, the performance index, which is necessary and important, was involved in this paper. Secondly, the final consensus value was obtained by LMI theory in this paper, which could better show the advantages of the proposed consensus protocol, but most existing works for the robust consensus of high-order linear swarm systems subject to parameter uncertainties and time delays, including [35–38], did not consider this useful point. Thirdly, to better correspond with the practical application, influences of time delays on the robust consensus of swarm systems were sufficiently considered in this paper, which did not appear in [37–39].

The remainder of the paper is organized as follows. Section 2 states several important conclusions of the graph theory and the problem description in which the robust consensus protocol is proposed. Section 4 shows a sufficient LMI condition that can guarantee the robust consensus of swarm systems and a final consensus value of swarm systems. The effectiveness of theoretical results is illustrated via simulation examples in Section 5. Finally, Section 5 draws the conclusion.

*Notations. *A matrix can be defined to be positive if each entry of the matrix is positive. A square matrix can be known as Schur stable if each eigenvalue of the matrix lies in the open unit disk . denotes the diagonal matrix, where are diagonal entries. refers to the expectation. The symbol represents the Kronecker product. The transpose of a matrix is denoted as . stands for the identity matrix with appropriate dimension. Let be the vector of ones with all components 1.

#### 2. Preliminaries and Problem Description

##### 2.1. Graph Theory

In general, exchanges of information among agents in swarm systems can be shown as a directed graph , where denotes the node set, represents the edge set, and refers to the weighted adjacency matrix with the element if while if . Define the node index of graph as a finite index set . Let stand for the set of all neighbours of node , where the information flows from node to node . Let subset of the nodes in graph be a cluster. Suppose that represents the neighbour set of a cluster . It can be defined that the in-degree matrix of node is , and the out-degree matrix of node is . Note that a diagonal matrix can be provided to describe the degree matrix of digraph , which is defined as follows:

The definition of the Laplacian matrix in connection with digraph can be shown in matrix form as .

##### 2.2. Problem Description

Consider a linear swarm system composed of homogeneous agents marked as , in which the th agent’s dynamics can be expressed as a state update equation where indicates the state vector; and , respectively, represent the input matrix and the external disturbance; , , , and are the given constant matrices; is the control output; and stand for uncertainties in the system matrix and input channel, respectively, such that where denotes a positive constant matrix and refers to an unknown matrix function which meets constraints as follows:

The parameter uncertainties and are considered permissible when equations (3) and (4) hold meanwhile.

*Remark 1. *When there is the external disturbance in system (2), external disturbance belongs to signal shown in [33, 40], which satisfies , that is, the energy of the external disturbance in system (2) is bounded on any finite interval. In this case, the robust control problem can be analysed in an appropriate and meaningful way.

To achieve consensus for linear swarm system (2), the following control protocols are adopted: where , is a constant gain matrix with appropriate dimensions, and denotes time-varying delays of swarm systems. Then, to show the consensus analysis and design for swarm system (2), the definition is presented as follows:

*Definition 1. *For the given gain matrix and any given bounded initial states, swarm system (2) with consensus protocol (5) are assured to arrive consensus if there exists a vector-valued depending on the initial states such that , where denotes the final consensus value.

Let , , and , and we can write swarm system (2) with equation (5) into a compact form as follows:
where and is an initial value at .

The suboptimal robust consensus control problem of system (6) is stated to find a distributed protocol (5) such that (i)with , system (6) is asymptotically stable for all acceptable uncertain matrices (ii)with interpreted as deterministic signal, the transfer function from to of system (6), which is denoted by , satisfies for all acceptable uncertain matrices and a given allowable scalar , where , is the norm of , defined by

In order to make an in-depth analysis for the robust consensus control problem of system (6), we presume henceforth that the communication graph is connected and give the following lemma about the graph theory.

Lemma 1 (see [41]). *For the Laplacian matrix associated with an undirected graph , the following description holds: (i) 0 is an eigenvalue of , and its corresponding eigenvector is , and (ii) 0 is a simple eigenvalue of , and all the other eigenvalues are positive and real if is connected.*

Assume that denotes eigenvalues of matrix of an undirected graph . Moreover, , which satisfies , is related to its corresponding eigenvector . Thus, an orthogonal matrix can be found in the form such that .

Theorem 1. *For a given , system (6) is asymptotically stable and , if and only if the following systems are simultaneously and asymptotically stable and all of the norms of their transfer function matrices are less than :
where and .*

*Proof. *Let be the eigenvalues of matrix for an undirected topology , where with the corresponding eigenvector , and . An orthogonal matrix can be found in the form
such that . Assume that
Then, system (6) can be rewritten in terms of as
Furthermore, reformulate the disturbance variable and the performance variable via
Subsequently, substituting (12) and (13) into (11) gives
In fact, system (14) consists of individual systems in (6). Denote by and the transfer function matrices of system (14) and (6), respectively. Afterwards, we can deduce from equations (6), (12), (13), and (14) that
which implies that
Therefore, the proof of Theorem 1 is completed.

In addition, it is worth mentioning that, let

By Lemma 1, we can convert system (14) into subsystems as follows:

Obviously, swarm system (6) can achieve consensus if subsystem (19) is asymptotically stable. Note that subsystem (18) determines the final consensus value of swarm system (6), and the details of it will be discussed below.

*Remark 2. *By Theorem 1, the distributed consensus control problem of swarm system (6) is transformed into the robust control problems of subsystems (8), so that a set of independent systems which have the same dimensions as a single agent in (2) is obtained. Therefore, the computational complexity is decreased evidently and the stability theory of swarm systems can be used. The essential tools producing this result by right of the state space decomposition approach.

#### 3. Main Results

In this section, the robust stability and existence conditions of the consensus control problem for swarm system (2) with protocol (5) is provided by using a basis-dependent Lyapunov function and the LMI approach. Above all, we should introduce the following two lemmas.

Lemma 2 (Jensen inequality [42]). *For any constant matrix , , and , positive integers and satisfying , an inequality holds as follows:
*

Lemma 3. *Suppose that equal positive integer time-varying delays exist in whole communication network of swarm system (2). Meanwhile, the undirected graph is fixed and connected. Then, by use of control protocol (5), the robust consensus of swarm system (2) with -norm consensus performance bound can be achieved globally asymptotically if we can prove that, there are a feedback matrix with suitable dimensions and four positive-definite matrices , , , and with dimensions satisfying
where
*

*Proof. *For purpose of consensus analysis, we construct a Lyapunov function as follows:
where
for .

Then, we can define the derivative of as
From the solution of (8), can be rewritten as follows:
can be reformulated as
can be rewritten as
can be reformulated as
can be rewritten as
Furthermore, the term can be shown that
At the same time, according to Lemma 2, the following inequality holds:
Let . Then, it follows from (8) and (25)–(32) that
if system (8) is asymptotically stable. This implies that the robust consensus of swarm system (2) with -norm consensus performance bound can be achieved globally asymptotically by the use of control protocol (5) if matrix inequality (21) holds.

Therefore, the conclusion of Lemma 3 is drawn.

*Remark 3. *In Lemma 3, for making sure that system (8) can achieve robust consensus with a performance index , a sufficient condition is obtained. Nevertheless, it is not difficult to find that (21) is a nonlinear matrix inequality (NMI) and therein lies parameter uncertainties.

To cope with the uncertain matrices and the nonlinear terms of (21), the following two lemmas are given.

Lemma 4 (see [43]). *For given matrices , , and of appropriate dimensions where is symmetric, a matrix inequality holds as follows:
for all subject to , if and only if there exists a scalar such that
*

Lemma 5 (Schur complement [44]). *The linear matrix inequality
where , , and are dependent on , is the identical meaning as anyone of the following two descriptions:
*(1)*, *(2)*, *

Theorem 2. *Consider swarm system (2) with a fixed, undirected, and connected communication topology. The robust consensus of swarm system (2) with -norm consensus performance bound can be achieved globally asymptotically by use of control protocol (5) if we can find two scaling factors and , a feedback matrix with suitable dimensions and six positive-definite matrices , , , , , and with dimensions which are a settlement of the minimization problem as follows:
subject to
where and . It can also be said if the minimization problem (37)–(40) has an available settlement , , , , , , and , the robust consensus of swarm system (2) with -norm consensus performance bound can be achieved globally asymptotically by the use of control protocol (5).*

*Proof. *Inequality (21) can be expressed as
According to Lemma 5, one can obtain that inequality (41) can be rewritten as follows:
By replacing and , inequality (42) is equivalently written as follows:
In virtue of Lemma 4 and Lemma 5, inequality (43) comes into existence for all acceptable uncertain matrices if and only if we can prove that there are two scaling factors and satisfying
By the Cone Complementarity Linearization (CCL) algorithm in [45], we set and . Then, the optimization problem (37)–(40) can be reasonably settled by transforming the nonlinear matrix inequality (NMI) (21).

Thus, the proof of Theorem 2 is completed.

*Remark 4. *According to Theorem 2, notice that the optimization problem (40)–(45) transformed from the NMI (21) can be reasonably solved based upon Algorithm 1. Hereafter, the linear swarm system (2) with the control protocol (5) achieves robust consensus.

It should be pointed out that the final outcome of first “exit” in Step 1 is that available initial values of , , , and cannot be obtained. In this case, we set and repeat Step 1. As long as the available values are obtained, we switch to Step 2. Then, second “exit” in Step 3 means that suitable values of , , , and are obtained.

Theorem 3. *With interpreted as deterministic signal, when swarm system (6) achieves robust consensus with parameter uncertainties and time-varying delays, the final consensus value satisfies
where can be the following form:
for .*

*Proof. *In the following, we consider two cases for the proof.*Case 1*. ().

In this case, system (18) can be equivalently written as follows:
Assume that and , then from (10), we can uniquely decompose into . Considering the mentioned above, we can also conclude that if the robust consensus of system (6) is achieved, the subsystem (19) should be *Schur stable*, which implies that the response of system (19) due to complies with . Therefore, the final consensus value depends totally on . Since , we have , and because , then we can obtain , that is to say
Likewise, let and , then by (12), can be uniquely decomposed as . If system (6) achieves the robust consensus, the response of system (19) due to also should comply with . Since , we have