Consensus of Time-Varying Interval Uncertain Multiagent Systems via Reduced-Order Neighborhood Interval Observer

'is work focuses on a multiagent system (MAS) with time-varying interval uncertainty in the system matrix, where multiple agents interact through an undirected topology graph and only the bounding matrices on the uncertainty in the systemmatrix are known. A reduced-order interval observer (IO), which is named the reduced-order neighborhood interval observer (NIO), is designed to estimate the relative state of each agent and those of its neighbors. It is shown that the reduced-order IO can guarantee the consensus of the uncertain multiagent system. Finally, simulation examples are proposed to verify the theoretical findings.


Introduction
With the discovery of the swarm intelligence and rapid development of the computer science [1][2][3][4][5], consensus of MASs has gained considerable attention [6][7][8][9][10], which means all the agents attain an agreement upon a common quantity of interest via distributed communications. Readers interested in consensus of MASs are referred to the great literature reviews on this topic, such as [11,12].
Uncertainty exists widely in existing practical engineering, which can affect the stability of the control systems [13][14][15]. Most of the exisiting related works on MASs with uncertanities are carried out to eliminate the impact of uncertainties and achieve the consensus of the MASs [16,17]. However, it is difficult or even impossible to get the specific information of the uncertainties.
us, the acquirement of the bounding information on the uncertainties (BIU) is easier than that of the uncertainties. On the other hand, the states of the MASs cannot be achieved in some situations. Taking these two facts into consideration, IO is firstly proposed to single-agent systems [18,19] to implement the state interval estimation and stabilization, where only the outputs and the BIU are related. An IO consists of two dynamical systems which are both in the form of Luenberger observer, where one is used to estimate the upper bound of the system state, while the other one aims at estimating the lower bound of the system state. en, Wang et al. extended the IO to uncertain MASs and proposed some interesting results [20,21]. According to the estimation objective, two kinds of IOs are defined for uncertain multiagent systems, the local IO [20][21][22][23] and the NIO [23,24]. To be specific, the local IO is designed to do the estimation which relates only to the output information of the associated agent. Yet, the NIO is designed to estimate the relative states between agents and its neighbors, which relates to the sum of the relative outputs between each agent and its neighbors. In [20,24], coordination control of MASs with uncertain disturbances is solved by introducing IO in MASs, including the local IO [20,22] and the NIO [24]. In [21,23], the IO-based consensus of MASs with time-varying interval uncertainties (TIUs) in the system dynamics is considered, by using only the outputs and the BIU, where the local fullorder IO and neighborhood full-order IO are designed in [23], while local reduced-order IO is given in [21].
As stated above, the reduced-order NIO design problem of MASs with TIUs is not solved. is work pays attention to the reduced-order NIO design of MASs with TIUs and aims at estimating the sum of relative states between each agent and its neighbors and simultaneously achieving consensus. In this paper, the definition of reduced-order NIO is proposed in detail. It shows that the consensus is a by-part of the reduced-order NIO. e rest of this paper is organized as follows: Problem formulation and some useful preliminaries used in this paper are introduced in Section 2. e main results are given in Section 3, and numerical simulations are presented in Section 4. Finally, conclusion is presented in Section 5.

Preliminaries and Problem Statement
2.1. Notation. R, N, R m×n , and R m×n + are denoted as the sets of real numbers, natural numbers, m × n real matrices, and m × n real matrices in which each element is a matrix with nonnegative entries, respectively. A square matrix is called to be Metzler when all its off-diagonal elements are nonnegative. All vector inequalities are understood element-wise, similarly for vectors), and |A| � (|a ij |) � A + + A − . A T denotes its transpose matrix. I N and 1 N denote an N-dimensional identity matrix and an N × 1 vector with all the entries being 1, respectively. 0 denotes the number zero (or the zero matrix with compatible dimensions). diag A1, . . . , A N denotes a block diagonal matrix, in which all the off-diagonal matrices are zeros and A i (i � 1, . . . , N) is the i-th diagonal block.

Graph eory.
Let a triple G � (V, E, G) be an undirected network, where V � v 1 , . . . , v N and E ⊆ V × V are the node set and edge set, respectively, and G � (g ij ) ∈ R N×N is the adjacency matrix. If the information can be communicated between node v i and node v j , then If there exists a path from every node to every other node [9,25,26], it is said that G � (V, E, G) is connected. e adjacency matrix G � (g ij ) ∈ R N×N is defined as g ij � 1 when (v i , v j ) ∈ E and g ij � 0 otherwise. e Laplacian matrix is defined as L � (l ij ) ∈ R N×N , where l ij � − g ij for j ≠ i and l ij � j≠i g ij for j � i. For this symmetric matrix L, in [9,25,26], it has exactly one zero eigenvalue with an associated eigenvector 1 N , and all the other ones are positive, if and only if G � (V, E, G) is connected.

Problem Statement. Consider a continuous-time MAS
with N agents and time-varying uncertainty in system matrix, where each agent moves in an n-dimensional Euclidean space and regulates itself based on the following dynamics: where x i (t) ∈ R n×1 , u i (t) ∈ R m×1 , and y i (t) ∈ R p×1 are the state, control input, and output of agent i, respectively. e matrices A, B, C are with compatible dimensions, while ΔA(t) (the uncertainty in system matrix) is a matrix-valued function of the time variable t. Moreover, ΔA(t) is timevarying interval uncertainty, which satisfies Assumption 1.
Moreover, two technical assumptions are given.
as the sum of relative state between the i-th agent and its neighbors. e main objective of this work is to realize the interval estimation on w i (t) on the basis of only the interval bound information of ΔA(t) given in Assumption 1, by using as few integrators as possible. Motivated by [27], a reduced-order NIO will be designed for system (1) to realize the interval estimation on w i (t). Again by [27], for C, there exists a matrix D ∈ R (n− p)×n to get a nonsingular matrix P � [C/D] ∈ R n×n . Denote P − 1 � Q � Q 1 Q 2 with Q 1 ∈ R n×p and Q 2 ∈ R n×(n− p) . With these matrices, one has CQ 1 � I p and l ij x jy / N j�1 l ij x ju ] with w iy ∈ R p and w iu ∈ R n− p . Intuitively, w iy � N j�1 l ij x jy � N j�1 l ij y j , so that there is no need to estimate w iy but we have to If ΔA in (1) is known, under Assumption 3, motivated by [27] and the full-order NIO constructed in [23], a neighborhood reduced-order observer can be designed as

Complexity
where K ∈ R (n− p)×p is chosen to make A 22 − KA 12 Hurwitz. On the other hand, for w iu and w iy defined above, by (1), one can get that However, in case that ΔA is unknown, the neighborhood reduced-order observer in (3) cannot be designed. Motivated by the local reduced-order IO given in [21], in this paper, we will solve the reduced-order NIO design for MASs steered by (1), where only the bounding information on ΔA is known. For better understanding, Definition 1 is given. with where f 1 , f 1 , f 2 , f 2 , and f are some differentiable continuous functions. Define with j ∈ N i ; if w i ≤ w i ≤ w i holds for t ≥ 0, it is said that z iu and z iu in (4) constitute a neighborhood reduced framer for (1). Beyond the holding w i ≤ w i ≤ w i for t ≥ 0, we also have It is said that z iu and z iu in (4) constitute a reduced NIO for (1).

e eory of Positive Systems.
In order to realize the main objective of this paper, two lemmas about the positive systems theory are introduced.
Lemma 1 (see [28]). Given a nonautonomous system de- Lemma 2 (see [29]). Let x ∈ R n×1 be a vector variable, In the following, t will be omitted in all variables without confusion for notational simplicity. Similarly, we denote x �

Main Results
Define Complexity 3 where en, one has eorem 1.

Theorem 1. Under Assumptions 1 and 3, if
Metzler, then z iu and z iu given in (8) with this K are considered a neighborhood reduced-order framer for the uncertain MAS described by (1), Proof. 1 By Lemma 2 (1), if the following holds: for t ≥ 0, then one can get w i ≤ w i ≤ w i for t ≥ 0. erefore, the proof of eorem 1 will be completed if the inequality in (11) holds for t ≥ 0, that is to prove the establishment of.
By (3), (12), and (13) is equivalent to, where z iu is given in (2). In order to prove the relationship in (13), let e iu � z iu − z iu and e iu � z iu − z iu . By (2) and (8), one has

Complexity
Under Assumption 1, similar to the proof of Lemma 5 in [21], there hold at is, By (3) and (10), one has i.e., en, one has Complexity 5 It is apparent that Π ≥ 0.
On the other hand, by (3), one has w i � (D − KC)w i , so that there hold which further result in Since w i (0) ≤ w i (0) ≤ w i (0), by (18), one has e iu (0)e iu (0)] ≥ 0. erefore, by Lemma 1, if A 22 − KA 12 is a Metzler matrix, then z iu and z iu given in (8) are considered a neighborhood reduced-order framer for the uncertain MAS described by (1).
is completes the proof. For w i and w i in (10), construct where P 1 ≻0 is the solution of the algebraic Riccati equation with λ 0 ≥ 2λ 2 (L) and ϵ > 0. Define By (16), one has For u i in (19), by (1), (21), and (22), one has On the other hand, by (14), there hold

Complexity
Since that is, with (22) and (24), one has Let η i � x T i e T iu e T iu ] T . It follows from (15) and (23) that where Γ 0 � I n 0 0 which induces where Since L is with an undirected graph, there is an or- Let η u � (U T ⊗ I (3n− 2p) )η and η uu � (U T 2 ⊗ I (3n− 2p) )η � η T u2 · · · η T uN T , and then one has (37) Construct a Lyapunov function: where P � diag P 1 , P 2 , P 2 with P 1 in (20), P 2 ≻0 being the solution of the Lyapunov equation where , then z iu and z iu given in (8) with the control algorithm (19) constitute a reduced-order NIO for (1), provided that w i (0) ≤ w i (0) ≤ w i (0).
Proof. 2 For the Lyapunov function given in (29), its derivative according to (28) yields where c > 0 is constant to be determined and with i � 2, 3, . . . , N.

Remark 2.
e main results are provided under the premise that A 22 − KA 12 is Hurwitz and Metzler. If there exists a K to make A 22 − KA 12 Hurwitz and Metzler, it can be acquired according to Lemma 4 in [21]. However, if such K does not exist, the time-invariant transformation and time-varying transformation in [20,21], respectively, can be introduced to carry out the problem of reduced-order NIO design.

Numerical Simulation
Some numerical simulations are proposed to verify the theoretical results in this section. Similar to [21], the system matrices are given as Obviously, (C, A) is observable, and (A, B) is stabilizable. e time-varying uncertainty is  us, the relationship w i (0) ≤ w i (0) ≤ w i (0) holds.
For multiagent system with above details, choose ϵ � 9; then, (31) is satisfied. at is, the premises in eorem 2 are satisfied. Figure 1 is given to verify eorem 2. As shown in Figure 1(a), N j�1 l ij x j converges to 0 for i � 1, . . . , N, and simultaneously, Figure 1(b) shows that the control input u i can also converge to 0 for i � 1, . . . , N. Both figures imply the consensus. Figures 2(a) and 2(b) display the trajectories of w i − w i and w i − w i , respectively. As shown in these two figures, w i − w i and w i − w i are guaranteed to be nonnegative, provided that w i (0) ≤ w i (0) ≤ w i (0) holds. erefore, eorem 1 is established. Further, as shown in Figure 2, w i − w i and w i − w i approach 0 as time goes to ∞.
at is, the interval on which the sum of the relative information of each agent associated with the uncertain multiagent system in this example is located can be estimated by the reduced-order NIO given in this paper. Consequently, Remark 1 holds.

Conclusions
In this paper, the reduced-order NIO is designed for MASs with TIUs in system matrix to implement the interval estimation, by using only the outputs and the bounding information of the uncertain system matrix. Consensus of this kind of uncertain multiagent systems can be achieved as a by-part of the reduced-order NIO design. is work is an important complement to the IO design for MASs with TIUs.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.