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

Authors: Hui Luo; Jin Zhao; Quan Yin
Journal: Complexity
 (2022)

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## Abstract

This 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 system matrix 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.


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## Body

## 1. Introduction

With the discovery of the swarm intelligence and rapid development of the computer science [1–5], consensus of MASs has gained considerable attention [6–10], 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 [13–15]. 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. Thus, 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. Then, 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 [20–23] 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 full-order 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. This 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.The rest of this paper is organized as follows: Problem formulation and some useful preliminaries used in this paper are introduced in Section2. The main results are given in Section 3, and numerical simulations are presented in Section 4. Finally, conclusion is presented in Section 5.


## 2. Preliminaries and Problem Statement

### 2.1. Notation

ℝ,ℕ,ℝm×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, for instance, va<≤vb (va=va1⋯van and vb=vb1⋯vbn) if for all i=1,…,n, one has vai<≤vbi. For any symmetricA∈ℝn×n, λiA denotes its eigenvalues, which is arranged as λ1A≤λ2A≤…≤λnA. For any matrix A=aij, A denotes the 2-norm of A, A+=aij+ (with aij+=maxaij,0), A−=A+−A (similarly for vectors), and A=aij=A++A−. AT denotes its transpose matrix. IN and 1N 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). diagA1,…,AN denotes a block diagonal matrix, in which all the off-diagonal matrices are zeros and Aii=1,…,N is the i-th diagonal block.


### 2.2. Graph Theory

Let a tripleG=V,E,G be an undirected network, where V=v1,…,vN and E⊆V×V are the node set and edge set, respectively, and G=gij∈ℝN×N is the adjacency matrix. If the information can be communicated between node vi and node vj, then vi,vj∈E, that is, the edge vi,vj exists in G=V,E,G. The neighbor set of node i is Ni=vj|vi,vj∈E. A path is a sequence of edges in G=V,E,G of the form i1,i2…i2,i3. 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.The adjacency matrixG=gij∈ℝN×N is defined as gij=1 when vi,vj∈E and gij=0 otherwise. The Laplacian matrix is defined as L=lij∈ℝN×N, where lij=−gij for j≠i and lij=∑j≠igij for j=i. For this symmetric matrix L, in [9, 25, 26], it has exactly one zero eigenvalue with an associated eigenvector 1N, and all the other ones are positive, if and only if G=V,E,G is connected.


### 2.3. Problem Statement

Consider a continuous-time MAS withN 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:(1)x˙it=A+ΔAtxit+Buit,yit=Cxit,i=1,2,…,N,where xit∈ℝn×1,uit∈ℝm×1, and yit∈ℝp×1 are the state, control input, and output of agent i, respectively. The matrices A,B,C are with compatible dimensions, while ΔAt (the uncertainty in system matrix) is a matrix-valued function of the time variable t. Moreover, ΔAt is time-varying interval uncertainty, which satisfies Assumption 1.Assumption 1.
There existΔA¯∈ℝn×n and ΔA¯∈ℝn×n with ΔA¯≤ΔA¯, such that ΔAt∈ΔA¯,ΔA¯ for all t≥0.
Moreover, two technical assumptions are given.Assumption 2.
A,B is stabilizable.Assumption 3.
C,A is detectable.
Denote(2)wit=∑j=1Nlijxjt,as the sum of relative state between the i-th agent and its neighbors. The main objective of this work is to realize the interval estimation on wit on the basis of only the interval bound information of ΔAt 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 wit. Again by [27], for C, there exists a matrix D∈ℝn−p×n to get a nonsingular matrix P=C/D∈ℝn×n. Denote P−1=Q=Q1Q2 with Q1∈ℝn×p and Q2∈ℝn×n−p. With these matrices, one has CQ1=Ip and CQ2=0. For i=1,…,N, let w˜i≜Pwi=w˜iy/w˜iu=∑j=1Nlijx˜jy/∑j=1Nlijx˜ju with w˜iy∈ℝp and w˜iu∈ℝn−p. Intuitively, w˜iy=∑j=1Nlijx˜jy=∑j=1Nlijyj, so that there is no need to estimate w˜iy but we have to estimate w˜iu. For simplicity, denote PAP−1=A˜11A˜12A˜21A˜22 and PB=B˜1/B˜2 with A˜11∈ℝp×p,A˜12∈ℝp×n−p,A˜21∈ℝn−p×p,A˜22∈ℝn−p×n−p,B˜1∈ℝp×m, and B˜2∈ℝn−p×m. 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(3)z˙iu=A˜22−KA˜12ziu+A˜22−KA˜12K+A˜21−KA˜11∑j=1Nlijyj+B˜2−KB˜1∑j=1Nlijuj+DΔAwi−KCΔAwi,where K∈ℝn−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(4)ziu=w˜iu−Kw˜iy=w˜iu−K∑j=1Nlijyj.
However, in case thatΔA 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.Definition 1.
Consider two dynamical systems in the form of(5)z¯˙iu=f¯1A,B,C,ΔA¯,ΔA¯,∑j=1Nlijyj,∑j=1Nlijuj,z¯iu,z¯˙i=f¯1A,B,C,ΔA¯,ΔA¯,∑j=1Nlijyj,∑j=1Nlijuj,z¯iu,z¯iu0=f¯2w¯i0,w¯i0,z¯iu0=f¯2w¯i0,w¯i0,with(6)ui=fz¯iu,z¯iu,yj,j∈Ni,where f¯1,f¯1,f¯2,f¯2, and f are some differentiable continuous functions. Define(7)w¯i=f¯3z¯iu,z¯iu,yj,w¯i=f¯3z¯iu,z¯iu,yj,with j∈Ni; if w¯i≤wi≤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≤wi≤w¯i for t≥0, we also have(8)limt⟶∞w¯i−∑j=1Nlijxjt=0,limt⟶∞w¯i−∑j=1Nlijxjt=0,i=1,…,N.
It is said thatz¯iu and z¯iu in (4) constitute a reduced NIO for (1).


### 2.4. The Theory 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 described byx˙=Ax+B, where A is a Metzler matrix and B≥0 for t≥0. Then, x≥0 for t>0, provided that x0≥0.Lemma 2 (see [29]).
Letx∈ℝn×1 be a vector variable, x¯≤x≤x¯ for some x¯,x¯∈ℝn×1. Then,(1)
ForA∈ℝm×n, A+x¯−A−x¯≤Ax≤A+x¯−A−x¯(2)
ForA∈ℝm×n satisfying A¯≤A≤A¯ for some A¯,A¯∈ℝm×nA¯+x¯+−A¯+x¯−−A¯−x¯++A¯−x¯−≤Ax≤A¯+x¯+−A¯+x¯−−A¯−x¯++A¯−x¯−In the following,t will be omitted in all variables without confusion for notational simplicity. Similarly, we denote x=x1T…xNTT and give similar manners for y, u as well as other variables.


## 2.1. Notation

ℝ,ℕ,ℝm×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, for instance, va<≤vb (va=va1⋯van and vb=vb1⋯vbn) if for all i=1,…,n, one has vai<≤vbi. For any symmetricA∈ℝn×n, λiA denotes its eigenvalues, which is arranged as λ1A≤λ2A≤…≤λnA. For any matrix A=aij, A denotes the 2-norm of A, A+=aij+ (with aij+=maxaij,0), A−=A+−A (similarly for vectors), and A=aij=A++A−. AT denotes its transpose matrix. IN and 1N 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). diagA1,…,AN denotes a block diagonal matrix, in which all the off-diagonal matrices are zeros and Aii=1,…,N is the i-th diagonal block.


## 2.2. Graph Theory

Let a tripleG=V,E,G be an undirected network, where V=v1,…,vN and E⊆V×V are the node set and edge set, respectively, and G=gij∈ℝN×N is the adjacency matrix. If the information can be communicated between node vi and node vj, then vi,vj∈E, that is, the edge vi,vj exists in G=V,E,G. The neighbor set of node i is Ni=vj|vi,vj∈E. A path is a sequence of edges in G=V,E,G of the form i1,i2…i2,i3. 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.The adjacency matrixG=gij∈ℝN×N is defined as gij=1 when vi,vj∈E and gij=0 otherwise. The Laplacian matrix is defined as L=lij∈ℝN×N, where lij=−gij for j≠i and lij=∑j≠igij for j=i. For this symmetric matrix L, in [9, 25, 26], it has exactly one zero eigenvalue with an associated eigenvector 1N, and all the other ones are positive, if and only if G=V,E,G is connected.


## 2.3. Problem Statement

Consider a continuous-time MAS withN 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:(1)x˙it=A+ΔAtxit+Buit,yit=Cxit,i=1,2,…,N,where xit∈ℝn×1,uit∈ℝm×1, and yit∈ℝp×1 are the state, control input, and output of agent i, respectively. The matrices A,B,C are with compatible dimensions, while ΔAt (the uncertainty in system matrix) is a matrix-valued function of the time variable t. Moreover, ΔAt is time-varying interval uncertainty, which satisfies Assumption 1.Assumption 1.
There existΔA¯∈ℝn×n and ΔA¯∈ℝn×n with ΔA¯≤ΔA¯, such that ΔAt∈ΔA¯,ΔA¯ for all t≥0.
Moreover, two technical assumptions are given.Assumption 2.
A,B is stabilizable.Assumption 3.
C,A is detectable.
Denote(2)wit=∑j=1Nlijxjt,as the sum of relative state between the i-th agent and its neighbors. The main objective of this work is to realize the interval estimation on wit on the basis of only the interval bound information of ΔAt 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 wit. Again by [27], for C, there exists a matrix D∈ℝn−p×n to get a nonsingular matrix P=C/D∈ℝn×n. Denote P−1=Q=Q1Q2 with Q1∈ℝn×p and Q2∈ℝn×n−p. With these matrices, one has CQ1=Ip and CQ2=0. For i=1,…,N, let w˜i≜Pwi=w˜iy/w˜iu=∑j=1Nlijx˜jy/∑j=1Nlijx˜ju with w˜iy∈ℝp and w˜iu∈ℝn−p. Intuitively, w˜iy=∑j=1Nlijx˜jy=∑j=1Nlijyj, so that there is no need to estimate w˜iy but we have to estimate w˜iu. For simplicity, denote PAP−1=A˜11A˜12A˜21A˜22 and PB=B˜1/B˜2 with A˜11∈ℝp×p,A˜12∈ℝp×n−p,A˜21∈ℝn−p×p,A˜22∈ℝn−p×n−p,B˜1∈ℝp×m, and B˜2∈ℝn−p×m. 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(3)z˙iu=A˜22−KA˜12ziu+A˜22−KA˜12K+A˜21−KA˜11∑j=1Nlijyj+B˜2−KB˜1∑j=1Nlijuj+DΔAwi−KCΔAwi,where K∈ℝn−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(4)ziu=w˜iu−Kw˜iy=w˜iu−K∑j=1Nlijyj.
However, in case thatΔA 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.Definition 1.
Consider two dynamical systems in the form of(5)z¯˙iu=f¯1A,B,C,ΔA¯,ΔA¯,∑j=1Nlijyj,∑j=1Nlijuj,z¯iu,z¯˙i=f¯1A,B,C,ΔA¯,ΔA¯,∑j=1Nlijyj,∑j=1Nlijuj,z¯iu,z¯iu0=f¯2w¯i0,w¯i0,z¯iu0=f¯2w¯i0,w¯i0,with(6)ui=fz¯iu,z¯iu,yj,j∈Ni,where f¯1,f¯1,f¯2,f¯2, and f are some differentiable continuous functions. Define(7)w¯i=f¯3z¯iu,z¯iu,yj,w¯i=f¯3z¯iu,z¯iu,yj,with j∈Ni; if w¯i≤wi≤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≤wi≤w¯i for t≥0, we also have(8)limt⟶∞w¯i−∑j=1Nlijxjt=0,limt⟶∞w¯i−∑j=1Nlijxjt=0,i=1,…,N.
It is said thatz¯iu and z¯iu in (4) constitute a reduced NIO for (1).


## 2.4. The Theory 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 described byx˙=Ax+B, where A is a Metzler matrix and B≥0 for t≥0. Then, x≥0 for t>0, provided that x0≥0.Lemma 2 (see [29]).
Letx∈ℝn×1 be a vector variable, x¯≤x≤x¯ for some x¯,x¯∈ℝn×1. Then,(1)
ForA∈ℝm×n, A+x¯−A−x¯≤Ax≤A+x¯−A−x¯(2)
ForA∈ℝm×n satisfying A¯≤A≤A¯ for some A¯,A¯∈ℝm×nA¯+x¯+−A¯+x¯−−A¯−x¯++A¯−x¯−≤Ax≤A¯+x¯+−A¯+x¯−−A¯−x¯++A¯−x¯−In the following,t will be omitted in all variables without confusion for notational simplicity. Similarly, we denote x=x1T…xNTT and give similar manners for y, u as well as other variables.


## 3. Main Results

Define(9)z¯˙iu=A˜22−KA˜12z¯iu+A˜22−KA˜12K+A˜21−KA˜11∑j=1Nlijyj+B˜2−KB˜1∑j=1Nlijuj+D+h¯i−D−h¯i+−KC+h¯i−−KC−h¯i,z¯˙iu=A˜22−KA˜12z¯iu+A˜22−KA˜12K+A˜21−KA˜11∑j=1Nlijyj+B˜2−KB˜1∑j=1Nlijuj+D+h¯i−D−h¯i+−KC+h¯i−−KC−h¯i,z¯iu0=D+w¯i0−D−w¯i0+−KC+w¯i0−−KC−w¯i0,z¯iu0=D+w¯i0−D−w¯i0+−KC+w¯i0−−KC−w¯i0,where(10)h¯i≜ΔA¯+w¯i+−ΔA¯+w¯i−−ΔA¯−w¯i++ΔA¯−w¯i−,h¯i≜ΔA¯+w¯i+−ΔA¯+w¯i−−ΔA¯−w¯i++ΔA¯−w¯i−.Then, one has Theorem1.Theorem 1.
Under Assumptions1 and 3, if w¯i0 and w¯i0 in (8) are chosen to satisfy w¯i0≤wi0≤w¯i0 and there exists an observer gain K∈ℝn−p×p to make A˜22−KA˜12 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), where w¯i≤wi≤w¯i holds for i=1,…,N with(11)w¯i≜Q+∑j=1Nlijyjz¯iu+K∑j=1Nlijyj−Q−∑j=1Nlijyjz¯iu+K∑j=1Nlijyj,w¯i≜Q+∑j=1Nlijyjz¯iu+K∑j=1Nlijyj−Q−∑j=1Nlijyjz¯iu+K∑j=1Nlijyj.Proof.
1 By Lemma2 (1), if the following holds:(12)∑j=1Nlijyjz¯iu+K∑j=1Nlijyj≤∑j=1Nlijyjw˜iu≤∑j=1Nlijyjz¯iu+K∑j=1Nlijyj,for t≥0, then one can get w¯i≤wi≤w¯i for t≥0. Therefore, the proof of Theorem 1 will be completed if the inequality in (11) holds for t≥0, that is to prove the establishment of.(13)z¯iu≤w˜iu−K∑j=1Nlijyj≤z¯iu.
By (3), (12), and (13) is equivalent to,(14)z¯iu≤ziu≤z¯iu,where ziu is given in (2).
In order to prove the relationship in (13), let e¯iu=z¯iu−ziu and e¯iu=ziu−z¯iu. By (2) and (8), one has(15)e¯˙iu=A˜22−KA˜12e¯iu+D+h¯i−D−h¯i+−KC+h¯i−−KC−h¯i−DΔAwi+KCΔAwi=A˜22−KA˜12e¯iu+D+h¯i−ΔAwi+D−ΔAwi−h¯i+−KC+h¯i−ΔAwi+−KC−ΔAwi−h¯i,e¯˙iu=A˜22−KA˜12e¯iu+DΔAwi−KCΔAwi−D+h¯i+D−h¯i−−KC+h¯i+−KC−h¯i=A˜22−KA˜12e¯iu+D+ΔAwi−h¯i+D−h¯i−ΔAwi+−KC+ΔAwi−h¯i+−KC−h¯i−ΔAwi.
Under Assumption1, similar to the proof of Lemma 5 in [21], there hold h¯i≥ΔA+w¯i−ΔA−w¯i and h¯i≤ΔA+w¯i−ΔA−w¯i, so that(16)e¯˙iu≥A˜22−KA˜12e¯iu+D+ΔA+w¯i−ΔA−w¯i−ΔAwi+D−ΔAwi−ΔA+w¯i−ΔA−w¯i+−KC+ΔA+w¯i−ΔA−w¯i−ΔAwi+−KC−ΔAwi−ΔA+w¯i−ΔA−w¯i=A˜22−KA˜12e¯iu+D+ΔA+w¯i−wi+ΔA−wi−w¯i+D−ΔA+wi−w¯i+ΔA−w¯i−wi+−KC+ΔA+w¯i−wi+ΔA−wi−w¯i+−KC−ΔA+wi−w¯i+ΔA−w¯i−wi,e¯˙iu≥A˜22−KA˜12e¯iu+D+ΔAwi−ΔA+w¯i−ΔA−w¯i+D−ΔA+w¯i−ΔA−w¯i−ΔAwi+−KC+ΔAwi−ΔA+w¯i−ΔA−w¯i+−KC−ΔA+w¯i−ΔA−w¯i−ΔAwi=A˜22−KA˜12e¯iu+D+ΔA+wi−w¯i+ΔA−w¯i−wi+D−ΔA+w¯i−wi+ΔA−wi−w¯i+−KC+ΔA+wi−w¯i+ΔA+w¯i−wi+−KC−ΔA+w¯i−wi+ΔA−wi−w¯i.
That is,(17)e¯˙iue¯˙iu=A˜22−KA˜1200A˜22−KA˜12e¯ie¯i+D+ΔA++−KC+ΔA+D+ΔA−+−KC+ΔA−D−ΔA−+−KC−ΔA−D−ΔA++−KC−ΔA+w¯i−wiwi−w¯i.
By (3) and (10), one has(18)w¯i−wi=Q2+z¯iu−ziu+Q2−ziu−z¯iu,wi−w¯i=Q2−z¯iu−ziu+Q2+ziu−z¯iu,i.e.,(19)w¯i−wiwi−w¯i=Q2+Q2−Q2−Q2+e¯iue¯iu.
Then, one has(20)e¯˙iue¯˙iu≥A˜22−KA˜1200A˜22−KA˜12e¯ie¯i+D+ΔA++−KC+ΔA+D+ΔA−+−KC+ΔA−D−ΔA−+−KC−ΔA−D−ΔA++−KC−ΔA+Q2+Q2−Q2−Q2+e¯iue¯iu≜A˜22−KA˜1200A˜22−KA˜12e¯ie¯i+Πe¯ie¯i.
It is apparent thatΠ≥0.
On the other hand, by (3), one has wi=D−KCwi, so that there hold(21)e¯iu0=z¯iu0−ziu0=D+w¯i0−D−w¯i0+−KC+w¯i0−−KC−w¯i0−D−KCwi0=D+w¯i0−wi0+−KC+w¯i0−wi0+D−wi0−w¯i0+−KC−wi0−w¯i0=D++−KC+D−+−KC−w¯i0−wi0wi0−w¯i0,e¯iu0=ziu0−z¯iu0=D−KCwi0−D+w¯i0+D−w¯i0−−KC+w¯i0+−KC−w¯i0=D+wi0−w¯i0+D−w¯i0−wi0+−KC+wi0−w¯i0+−KC−w¯i0−wi0=D−+−KC−D++−KC+w¯i0−wi0wi0−w¯i0,which further result in(22)e¯iu0e¯iu0=D++−KC+D−+−KC−D−+−KC−D++−KC+w¯i0−wi0wi0−w¯i0.
Sincew¯i0≤wi0≤w¯i0, by (18), one has e¯iu0e¯iu0≥0.
Therefore, by Lemma1, 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).
This completes the proof.
Forw¯i and w¯i in (10), construct(23)ui=−BTP1w¯i+w¯i,where P1≻0 is the solution of the algebraic Riccati equation(24)ATP1+P1A−λ0P1BBTP1+ϵI=0,with λ0≥2λ2L and ϵ>0.
Define(25)E¯i=w¯i−wi,E¯i=wi−w¯i.
By (16), one has(26)E¯iE¯i=Q2+Q2−Q2−Q2+e¯iue¯iu.
Forui in (19), by (1), (21), and (22), one has(27)x˙i=A+ΔAxi−BBTP1w¯i+w¯i=A+ΔAxi−BBTP12wi+E¯i−E¯i=A+ΔAxi−2BBTP∑j=1Nlijxj−BBTP1Q2e¯iu+BBTP1Q2e¯iu.
On the other hand, by (14), there hold(28)e¯˙iu=A˜22−KA˜12e¯iu+D++−KC+D−+−KC−h¯i−ΔAwiΔAwi−w¯i,e¯˙iu=A˜22−KA˜12e¯iu+D−+−KC−D++−KC+h¯i−ΔAwiΔAwi−w¯i.
Since(29)h¯i−ΔAwi=ΔA¯+ΔA¯−E¯iE¯i+−ΔA¯+ΔA¯−E¯iE¯i−+ΔA¯−ΔAwi++ΔA−ΔA¯wi−,ΔAwi−h¯i=ΔA¯−ΔA¯+E¯iE¯i+−ΔA¯−ΔA¯+E¯iE¯i−+ΔA−ΔA¯wi++ΔA¯−ΔAwi−,that is,(30)h¯i−ΔAwiΔAwi−w¯i=ΔA¯+ΔA¯−ΔA¯−ΔA¯+E¯iE¯i+−ΔA¯+ΔA¯−ΔA¯−ΔA¯+E¯iE¯i−+ΔA¯−ΔAΔA−ΔA¯wi++ΔA−ΔA¯ΔA¯−ΔAwi−,with (22) and (24), one has(31)e¯˙iue¯˙iu=A˜22−KA˜1200A˜22−KA˜12e¯iue¯iu+D++−KC+D−+−KC−D−+−KC−D++−KC+ΔA¯+ΔA¯−ΔA¯−ΔA¯+Q2+Q2−Q2−Q2+e¯iue¯iu+−ΔA¯+ΔA¯−ΔA¯−ΔA¯+Q2+Q2−Q2−Q2+e¯iue¯iu−+ΔA¯−ΔAΔA−ΔA¯wi++ΔA−ΔA¯ΔA¯−ΔAwi−.
Letηi=xiTe¯iuTe¯iuTT. It follows from (15) and (23) that(32)η˙i=A−BBTP1Q2BBTP1Q20A˜22−KA˜12000A˜22−KA˜12ηi−2BBTP100000000∑j=1Nlijηj+Γ2ℳ1Γ1ηi+−Γ2ℳ2Γ1ηi−+Γ2ℳ3∑j=1Nlijηj++Γ2Γ0ℳ3∑j=1Nlijηj−,where(33)Γ0=In0000In0In0,Γ1=In000Q2+Q2−0Q2−Q2+,Γ2=In000D++−KC+D−+−KC−0D−+−KC−D++−KC+,ℳ1=ΔA000ΔA¯+ΔA¯−0ΔA¯−ΔA¯+,ℳ2=ΔA000ΔA¯+ΔA¯−0ΔA¯−ΔA¯+,ℳ3=000ΔA¯−ΔA00ΔA−ΔA¯00,which induces(34)η˙=IN⊗A−L⊗ℬη+IN⊗Γ2ℳ1IN⊗Γ1η+−IN⊗Γ2ℳ2IN⊗Γ1η−+IN⊗Γ2ℳ3L⊗I3n−2pη++IN⊗Γ2Γ0ℳ3L⊗I3n−2pη−,where(35)A=A−BBTP1Q2BBTP1Q20A˜22−KA˜12000A˜22−KA˜12,ℬ=2BBTP100000000.
SinceL is with an undirected graph, there is an orthogonal matrix U=1/N1NU2 with U2∈ℝN×N−1 such that UTLU=diagλ1L,…,λNL, where λiL>0 for i=2,…,N, if G=V,E,W is connected. For better understanding, denote(36)Λ=diagλ2L,…,λNL=U2TLU2.
Letηu=UT⊗I3n−2pη and ηuu=U2T⊗I3n−2pη=ηu2T⋯ηuNTT, and then one has(37)η˙uu=U2T⊗A−U2TL⊗ℬη+U2T⊗Γ2ℳ1IN⊗Γ1η+−U2T⊗Γ2ℳ2IN⊗Γ1η−+U2T⊗Γ2ℳ3L⊗I3n−2pη++U2T⊗Γ2Γ0ℳ3L⊗I3n−2pη−.
SinceU2TU2=IN−1, U2U2T=IN−1/N1N1NT and L1N=0, one has(38)η˙uu=IN−1⊗AU2T⊗I3n−2pη−U2TLU2U2T+1N1N1NT⊗ℬη+U2T⊗Γ2ℳ1IN⊗Γ1η+−U2T⊗Γ2ℳ2IN⊗Γ1η−+U2T⊗Γ2ℳ3L⊗I3n−2pη++U2T⊗Γ2Γ0ℳ3L⊗I3n−2pη−=IN−1⊗A−Λ⊗ℬηuu+U2T⊗Γ2ℳ1IN⊗Γ1η+−U2T⊗Γ2ℳ2IN⊗Γ1η−+U2T⊗Γ2ℳ3L⊗I3n−2pη++U2T⊗Γ2Γ0ℳ3L⊗I3n−2pη−.
Construct a Lyapunov function:(39)V=ηuuTIN−1⊗Pηuu,where P=diagP1,P2,P2 with P1 in (20), P2≻0 being the solution of the Lyapunov equation(40)A˜22−KA˜12TP2+P2A˜22−KA˜12+ϵIn−p+Q2TP1BBTP1Q2=0.Theorem 2.
Consider an uncertain MAS (1) communicating through an undirected networkG=V,E,G and suppose that Assumptions 1, 2, and 3 hold. If K∈ℝn−p×p is given to make A˜22−KA˜12 Hurwitz and Metzler, and(41)θ1θ2<ϵ216,where(42)θ1=Γ12+λNL,θ2=γ2+2ΔA¯−ΔA¯2Γ22maxλnP1,λ3n−2pP2,with γ=maxΔA¯,ΔA¯,ΔA¯+ΔA¯−ΔA¯−ΔA¯+, then z¯iu and z¯iu given in (8) with the control algorithm (19) constitute a reduced-order NIO for (1), provided that w¯i0≤wi0≤w¯i0.Proof.
2 For the Lyapunov function given in (29), its derivative according to (28) yields(43)V˙=∑i=2Nηuu,iTΦiηuu,i+IN⊗Γ1η+TU2T⊗Γ2ℳ1TIN−1⊗Pηuu+ηuuTIN−1⊗PU2T⊗Γ2ℳ1IN⊗Γ1η+−IN−1⊗Γ1η−TU2T⊗Γ2ℳ2TIN−1⊗Pηuu−ηuuTIN−1⊗PU2T⊗Γ2ℳ2IN⊗Γ1η−+L⊗I3n−2pη+TU2T⊗Γ2ℳ3TIN−1⊗Pηuu+ηuuTIN−1⊗PU2T⊗Γ2ℳ3L⊗I3n−2pη++L⊗I3n−2pη−TU2T⊗Γ2Γ0ℳ3TIN−1⊗Pηuu+ηuuTIN−1⊗PU2T⊗Γ2Γ0ℳ3L⊗I3n−2pη−≤∑i=2Nηuu,iTΦiηuu,i+cIN⊗Γ1η+TU2U2T⊗I3n−2pIN⊗Γ1η++λ3n−2pPcΓ22ℳ12ηuuTηuu+cIN⊗Γ1η−TU2U2T⊗I3n−2pIN⊗Γ1η−+λ3n−2pPcΓ22ℳ22ηuuTηuu+cL⊗I3n−2pη+TU2U2T⊗I3n−2pL⊗I3n−2pη++λ3n−2pPcΓ22ℳ32ηuuTηuu+cL⊗I3n−2pη−TU2U2T⊗I3n−2pL⊗I3n−2pη−+λ3n−2pPcΓ22Γ02ℳ32ηuuTηuu,where c>0 is constant to be determined and(44)Φi=A−λiLℬTP+PA−λiLℬ=ATP1+P1A−4λiLP1TBBTP1−P1TBBTP1Q2P1TBBTP1Q2−Q2TP1TBBTP1A˜22−KA˜12TQ+QA˜22−KA˜120Q2P1TBBTP10A˜22−KA˜12TQ+QA˜22−KA˜12,withi=2,3,…,N.
Since the eigenvalues ofU2U2T are no less than 0, one has(45)V˙≤∑i=2Nηuu,iTΦiηuu,i+cIN⊗Γ1ηTU2U2T⊗I3n−2pIN⊗Γ1η+cIN⊗Γ1ηTU2U2T⊗I3n−2pIN⊗Γ1η+cL⊗I3n−2pηTU2U2T⊗I3n−2pL⊗I3n−2pη+cL⊗I3n−2pηTU2U2T⊗I3n−2pL⊗I3n−2pη+λ3n−2pPcΓ22ℳ12+ℳ22+ℳ32+Γ02ℳ32ηuuTηuu=∑i=2Nηuu,iTΦiηuu,i+2cηuuTIN−1⊗Γ1TΓ1ηuu+2cηTLU2U2TL⊗I3n−2pη+λ3n−2pPcΓ22ℳ12+ℳ22+ℳ32+Γ02ℳ32ηuuTηuu.
Since(46)2cηTLU2U2TL⊗I3n−2pη=2cηTU2U2T+1N1N1NTLU2U2TL⊗I3n−2pU2U2T+1N1N1NTη=2cηuuTΛ2ηuuV˙≤∑i=2Nηuu,iTΦiηuu,i+2cΓ12+λN2L+λ3n−2pPcΓ22ℳ12+ℳ22+ℳ32+Γ02ℳ32ηuuTηuu.there holds(47)Φi≤¯ATP1+P1A−2λiLP1TBBTP100−0A˜22−KA˜12TQ+QA˜22−KA˜12+1λiLQ2TP1BBTP1Q2000A˜22−KA˜12TQ+QA˜22−KA˜12+1λiLQ2TP1BBTP1Q2,with i=2,3,…,N.
Therefore, one can get that(48)V˙≤−ϵ−2cΓ12+λN2LηuuTηuu−λ3n−2pPcΓ22ℳ12+ℳ22+ℳ32+Γ02ℳ32ηuuTηuu.
SinceΓ0−1=Γ0 and Γ0=1,ℳ2=Γ0ℳ1Γ0 and(49)ℳ1=maxΔA¯,ΔA¯,ΔA¯+ΔA¯−ΔA¯−ΔA¯+=γ,ℳ32≤2ΔA¯−ΔA¯2, so that(50)V˙≤−ϵ−2θ1c+θ2cηuuTηuu.
Choosec=θ2/θ1, and under (31), one has ϵ−2θ1c+θ2/c>0, which induces that V˙≤0, where the equality sign holds if and only if ηuu=0. Thus, with ui in (19), uncertain MAS (1) can achieve consensus.Remark 1.
By Theorem2, one has limt⟶∞wi+=0 and limt⟶∞wi−=0 for i=1,…,N. Thus, by (25), as t⟶∞, one can get that(51)e¯˙iue¯˙iu=Ϝ1e¯iue¯iu+Ϝ2Q2+Q2−Q2−Q2+e¯iue¯iu,where(52)Ϝ1=A˜22−KA˜1200A˜22−KA˜12+12D++−KC+D−+−KC−D−+−KC−D++−KC+ΔA¯+ΔA¯−ΔA¯−ΔA¯++ΔA¯+ΔA¯−ΔA¯−ΔA¯+Q2+Q2−Q2−Q2+,Ϝ2=12D++−KC+D−+−KC−D−+−KC−D++−KC+ΔA¯+ΔA¯−ΔA¯−ΔA¯+−ΔA¯+ΔA¯−ΔA¯−ΔA¯+.
Consequently, by Lyapunov stability theory,e¯iu/e¯iu may approach 0 as t goes to ∞. This result will be established, if Ϝ1, Ϝ2, and Q2+/Q2−Q2−/Q2+ in (33) meet some conditions. In this case, the interval on which the sum of the relative information of each agent is located can be estimated by w¯i and w¯i in (10).Remark 2.
The main results are provided under the premise thatA˜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.


## 4. Numerical Simulation

Some numerical simulations are proposed to verify the theoretical results in this section. Similar to [21], the system matrices are given as(53)A=−14003−101−2,B=−51−1,C=−1−21.Obviously,C,A is observable, and A,B is stabilizable. The time-varying uncertainty is(54)ΔAt=10−2sint10−cos2t1110sint/3501200cos22t20,which meets Assumption 1 and(55)ΔA¯=10−2110011101501200120,ΔA¯=10−2−110−100−15012000.ForP=−1−21010001, one has A˜21=0/0, A˜22=3−11−2, A˜11=−1,andA˜12=−111. Here, we can choose K=−1/1/4, so that A˜22−KA˜12 is Hurwitz and Metzler. A multiagent system consists of N=4 agents, which communicates through a connected graph with Laplacian matrix as follows:(56)L=2−10−1−12−100−12−1−10−12.The initial state of system is chosen randomly from−2020×−2020×−2020,w¯i0=wi0+α, and w¯i0=wi0−α, where α is a vector with each element choosing from 01. Thus, the relationship w¯i0≤wi0≤w¯i0 holds.For multiagent system with above details, chooseϵ=9; then, (31) is satisfied. That is, the premises in Theorem 2 are satisfied.Figure1 is given to verify Theorem 2. As shown in Figure 1(a), ∑j=1Nlijxj converges to 0 for i=1,…,N, and simultaneously, Figure 1(b) shows that the control input ui 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−wi and wi−w¯i, respectively. As shown in these two figures, w¯i−wi and wi−w¯i are guaranteed to be nonnegative, provided that w¯i0≤wi0≤w¯i0 holds. Therefore, Theorem 1 is established. Further, as shown in Figure 2, w¯i−wi and wi−w¯i approach 0 as time goes to ∞. That 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.Figure 1
Reduced-order NIO-based consensus. (a) Convergence of∑i=10Nlijxjt. (b) Convergence of ui.
(a)(b)Figure 2
Reduced-order NIO-based consensus. (a) Trajectory ofw¯i−wi. (b) Trajectory of w¯j−wj.
(a)(b)


## 5. 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. This work is an important complement to the IO design for MASs with TIUs.



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*Source: 5800097-2022-08-28.xml*