# Robust Time-Varying Output Formation Control for Swarm Systems with Nonlinear Uncertainties

Authors: Xiaofeng Chai; Qing Wang; Yao Yu; Changyin Sun
Journal: Complexity
 (2020)

---

## Abstract

Time-varying output formation control problems for high-order time-invariant swarm systems are studied with nonlinear uncertainties and directed network topology in this paper. A robust controller which consists of a nominal controller and a robust compensator is applied to achieve formation control. The nominal controller based on the output feedback is designed to achieve desired time-varying formation properties for the nominal system. And the robust compensator based on the robust signal compensator technology is constructed to restrain nonlinear uncertainties. The time-varying formation problem is transformed into the stability problem. And the formation errors can be arbitrarily small with expected convergence rate. Numerical examples are provided to illustrate the effectiveness of the proposed strategy.


---

## Body

## 1. Introduction

The past decades have witnessed an increasing attention in distributed cooperative control of large-scale swarm systems due to their broad applications in various fields, such as physics, biology, mobile robots, sensor networks, and unmanned aerial vehicles [1–5]. Compared with the individual agent, swarm systems have great benefits in high efficiency, low cost, robust, and easy maintenance. The research involves a variety of branches, including consensus, flocking, synchronization, formation, and containment [6–11]. As one of the critical problems in distributed cooperative control systems, the formation problem is to find control laws that drive states or outputs of all agents to reach a predefined configuration. Many approaches have been proposed to achieve formation control [12], to name a few, leader-follower [13], virtual structure [14], and behavior-based approaches [15].Consensus is one of the fundamental problems of swarm systems [16, 17]. Inspired by the development of consensus control theory, more and more researchers are interested in realizing the predefined formation via consensus approach without a central controller [18]. A consensus protocol was applied to achieve distributed formation control by Ren [16]. Olfati-Saber, Fax, and Murray [19] proposed a consensus-based theoretical framework for swarm systems with fixed or dynamic network topology. Xie and Wang [20] presented sufficient conditions for second-order swarm systems to realize time-invariant formation via local neighboring information interaction. Lafferriere et al. [21] studied the time-invariant formation stability problem for integrator-type high-order multiagent systems.The formation mentioned above in [19–21] is time-invariant. In practical applications, the formation may change with the environment and tasks. Dong et al. [22] presented necessary and sufficient conditions for linear swarm systems with time delay to achieve time-varying formation. Xiao et al. [23] proposed a finite-time formation control framework which can realize a variety of complex formations and greatly reduce the data exchange. Generally speaking, only the output information of each agent can be obtained rather than state information. As a result, it is significant to study the formation control problem only using output information. Fax and Murry [23] studied output formation stability problems of high-order linear swarm systems. In [24], a dynamic output approach was applied to solve a fully distributed time-varying formation-tracking problem for linear swarm systems. Zuo et al. [25] studied the adaptive output formation-tracking problem of linear heterogeneous swarm systems whose followers only received relative output information from their neighbors, and leaders’ dynamics were only known to their neighboring followers. Dong et al. [26] proposed a consensus-based approach for swarm systems with directed interaction topologies to achieve time-varying output formations.All these results listed above were for linear swarm systems. Practically, there are many cases where each agent involves uncertainties such as unknown time-varying parameters, unknown functions, and bounded external disturbances. Considering uncertain leader dynamics and uncertain local dynamics, Peng et al. [27] presented a neural network-based adaptive formation control approach for swarm systems. Li et al. [28] provided sufficient and necessary conditions to achieve the leader-following formation for second-order swarm systems with time-varying delay and nonlinear dynamics. Liu et al. [29] proposed a iterative learning-based method to deal with the formation control problem of swarm systems with unknown dynamics. Lu [30] proposed a robust controller consisting of a nominal controller and a robust compensator which is linear, time-invariant, and easy to implement. Because of great advantages, robust control can well deal with nonsmooth and discontinuous uncertainties compared to other approaches.Compared with the approaches on formation control mentioned above, the main contributions of this paper are as follows. First, the dynamics of each agent is high-order with nonlinear uncertainties, and an output feedback control approach is proposed to deal with the time-varying formation problem. Second, the output formation error can be as small as desired with arbitrarily specified convergence rate under the proposed controller. Meanwhile, the proposed controller can be easily applied to practical situations.The rest of the paper are arranged as follows. In Section2, some basic concepts on graph theory and the problem description are introduced, respectively. Meanwhile, some assumptions, definitions, and useful lemmas are presented. In Section 3, sufficient and necessary conditions for swarm systems to achieve time-varying formation are given, and the robust compensator is introduced to suppress the nonlinear part. In Section 4, simulation examples are shown to verify the analytical results. Finally, Section 5 concludes the whole work.Notation 1.
Let0N and 1 denote the matrix with all elements being 0 and column vectors of ones with dimension N, respectively. Use the superscripts T and H to represent the transpose and Hermitian adjoint of a matrix, respectively. ⋅ and ⊗ denote the Euclidean norm and Kronecker product, respectively. ℒ and ℒ−1 represent Laplace transform and inverse transformation, respectively. And ∗ represents the convolution operator.


## 2. Problem Formulation

### 2.1. Basic Concept on Graph Theory

A directed graph of orderN can be denoted by G=V,E,W, where V=v1,v2,…,vN is the set of nodes, E⊆vi,vj,vi,vj∈V is the set of edges, and W=wij∈ℝN×N is the adjacency matrix with nonnegative weights wij. For an edge vij=vi,vj, node vi is the parent node, node vj is the child node, and vi is a neighbour of vj. The set of neighbours of node vi is denoted by Ni=vj∈V:vj,vi∈E. For the adjacency matrix W, wij>0 if vi,vj∈E and wij=0, otherwise. The Laplacian matrix L=lij∈ℝN×N is defined as lii=∑j=1,j≠iNwij and lij=−wiji≠j for all i,j∈1,2,…,N. If at least one node has a directed path to all the other nodes, the graph G is said to have a spanning tree. More details on graph theory can be found in [31].


### 2.2. Problem Description

A swarm system consists ofN agents with the directed graph G which is used to describe the interaction topology. The dynamics of each agent ii∈1,2,…,N is described by(1)x˙it=Axit+B1uit+BωωiΔi,xi,yit=Cxit,where xi∈ℝn is the state of agent i, xt=x1Tt,x2Tt,…,xnTtT, uit∈ℝm, and yit∈ℝq denote the control input and output of agent i with ut=u1Tt,u2Tt,…,unTtT and yt=y1Tt,y2Tt,…,ynTtT. Δi is the external disturbance, and ωiΔi,xi is the nonlinear uncertainty. Simply, the nonlinear uncertainty ωiΔi,xi is represented as ωit∈ℝp with ωt=ω1Tt,ω2Tt,…,ωnTtT.Assumption 1.
The matrixB1 is of full-column rank, i.e., rankB1=m. The matrix C is of full-row rank, i.e., rankC=q. Moreover, the output dimension q and input dimension m satisfy q≥m.Assumption 2.
There are positive constantsξx1, ξu1, and ξ1 such that ωt∞≤ξx1xt∞η1+ξu1ut∞+ξ1, where η1≥1.Assumption 3.
There are positive constantsξx2, ξu2, and ξ2 such that N0−1sLsωt∞≤ξx2xt∞η2+ξu2ut∞+ξ2, where η2≥1 and ξu2<1.Remark 1.
D
0
−
1
s
N
0
s and D0−1sLs can be seen in the following in equation (4). N0−1sLs=D0−1sN0s−1⋅D0−1sLs. The nonlinear uncertainties associated with the ith agent are assumed to be bounded, and the assumption is widely used in literature studies.Definition 1.
(see [26]). The swarm system consisting of (1) is said to achieve time-varying output formation ht if for any given bounded initial states, there exists a vector-valued function rt∈ℝq satisfying(2)limt⟶∞yit−hit−rt=0i=1,2,…,N,where rt is called an output formation reference function.Definition 2.
(see [26]). If the swarm system can achieve time-varying output formation ht under control input uii=1,2,…,N with ht=h1Tt,h2Tt,…,hNTtT∈ℝNq, then ht is said to be feasible.Definition 3.
(see [26]). Swarm system (1) is said to achieve output consensus if there exists a function ct∈ℝq satisfying(3)limt⟶∞yit−ct=0i=1,2,…,N,where ct is called an output consensus function.Remark 2.
From Definitions1 and 3, one sees that the output formation reference function is equal to the output consensus function if ht≡0, and output formation problems are converted into state formation problems if C=I.Lemma 1 (see [32]).
L
∈
ℝ
N
×
N is the Laplacian matrix of a directed graph G; then,(i)
L has at least one zero eigenvalue, and 1 is the associated eigenvector, that is, L1=0N(ii)
IfG has a spanning tree, then 0 is a simple eigenvalue of L, and all the other N−1 eigenvalues have positive real parts
Consider the following system:(4)x˜˙t=A˜x˜ty˜˙t=C˜x˜t,where A˜=A˜11A˜12A˜21A˜22 and C˜=I0.Lemma 2 (see [33]).
IfA˜22,A˜12 is completely observable, then system (2) is asymptotically stable with respect to y˜t if and only if A˜ is Hurwitz.


## 2.1. Basic Concept on Graph Theory

A directed graph of orderN can be denoted by G=V,E,W, where V=v1,v2,…,vN is the set of nodes, E⊆vi,vj,vi,vj∈V is the set of edges, and W=wij∈ℝN×N is the adjacency matrix with nonnegative weights wij. For an edge vij=vi,vj, node vi is the parent node, node vj is the child node, and vi is a neighbour of vj. The set of neighbours of node vi is denoted by Ni=vj∈V:vj,vi∈E. For the adjacency matrix W, wij>0 if vi,vj∈E and wij=0, otherwise. The Laplacian matrix L=lij∈ℝN×N is defined as lii=∑j=1,j≠iNwij and lij=−wiji≠j for all i,j∈1,2,…,N. If at least one node has a directed path to all the other nodes, the graph G is said to have a spanning tree. More details on graph theory can be found in [31].


## 2.2. Problem Description

A swarm system consists ofN agents with the directed graph G which is used to describe the interaction topology. The dynamics of each agent ii∈1,2,…,N is described by(1)x˙it=Axit+B1uit+BωωiΔi,xi,yit=Cxit,where xi∈ℝn is the state of agent i, xt=x1Tt,x2Tt,…,xnTtT, uit∈ℝm, and yit∈ℝq denote the control input and output of agent i with ut=u1Tt,u2Tt,…,unTtT and yt=y1Tt,y2Tt,…,ynTtT. Δi is the external disturbance, and ωiΔi,xi is the nonlinear uncertainty. Simply, the nonlinear uncertainty ωiΔi,xi is represented as ωit∈ℝp with ωt=ω1Tt,ω2Tt,…,ωnTtT.Assumption 1.
The matrixB1 is of full-column rank, i.e., rankB1=m. The matrix C is of full-row rank, i.e., rankC=q. Moreover, the output dimension q and input dimension m satisfy q≥m.Assumption 2.
There are positive constantsξx1, ξu1, and ξ1 such that ωt∞≤ξx1xt∞η1+ξu1ut∞+ξ1, where η1≥1.Assumption 3.
There are positive constantsξx2, ξu2, and ξ2 such that N0−1sLsωt∞≤ξx2xt∞η2+ξu2ut∞+ξ2, where η2≥1 and ξu2<1.Remark 1.
D
0
−
1
s
N
0
s and D0−1sLs can be seen in the following in equation (4). N0−1sLs=D0−1sN0s−1⋅D0−1sLs. The nonlinear uncertainties associated with the ith agent are assumed to be bounded, and the assumption is widely used in literature studies.Definition 1.
(see [26]). The swarm system consisting of (1) is said to achieve time-varying output formation ht if for any given bounded initial states, there exists a vector-valued function rt∈ℝq satisfying(2)limt⟶∞yit−hit−rt=0i=1,2,…,N,where rt is called an output formation reference function.Definition 2.
(see [26]). If the swarm system can achieve time-varying output formation ht under control input uii=1,2,…,N with ht=h1Tt,h2Tt,…,hNTtT∈ℝNq, then ht is said to be feasible.Definition 3.
(see [26]). Swarm system (1) is said to achieve output consensus if there exists a function ct∈ℝq satisfying(3)limt⟶∞yit−ct=0i=1,2,…,N,where ct is called an output consensus function.Remark 2.
From Definitions1 and 3, one sees that the output formation reference function is equal to the output consensus function if ht≡0, and output formation problems are converted into state formation problems if C=I.Lemma 1 (see [32]).
L
∈
ℝ
N
×
N is the Laplacian matrix of a directed graph G; then,(i)
L has at least one zero eigenvalue, and 1 is the associated eigenvector, that is, L1=0N(ii)
IfG has a spanning tree, then 0 is a simple eigenvalue of L, and all the other N−1 eigenvalues have positive real parts
Consider the following system:(4)x˜˙t=A˜x˜ty˜˙t=C˜x˜t,where A˜=A˜11A˜12A˜21A˜22 and C˜=I0.Lemma 2 (see [33]).
IfA˜22,A˜12 is completely observable, then system (2) is asymptotically stable with respect to y˜t if and only if A˜ is Hurwitz.


## 3. Main Results

In this section, a robust control approach is introduced to deal with time-varying output formation problems with nonlinear uncertainties in directed networks. The robust controller consists of a nominal controller and a robust compensator. Section3.1 designs the robust compensator, and Section 3.2 is to design the nominal controller and provide time-varying output formation analysis.


### 3.1. Robust Compensator Design

The robust controller is constructed with two parts:(5)ui=uinomt+uirobt,where uinomt stands for the nominal controller for the nominal model to obtain desired time-varying formation and uirobt stands for the robust compensator to restrain the influence of nonlinear uncertainty ωit.Based on Laplace transformation, the following frequency domain equation can be found from (1):(6)yis=D0−1sN0suis+D0−1sLsωis,where D0−1sN0s=CsI−A−1B1 and D0−1sLs=CsI−A−1Bω.To restrain uncertainties, a robust compensator is designed as(7)uirob∗s=−N0−1sLsωis.Substituting (5) and (7) into (6), we can get(8)yis=D0−1sN0suinoms,which means robust compensator (7) can restrain the nonlinear uncertainties perfectly.As a matter of fact that nonlinear uncertaintyωis cannot be measured directly, substituting ωis into (7), one can further express the robust compensator by(9)uirob∗s=−N0−1sD0syis+uis.To avoid the high-order derivative terms of the output, a robust filter is considered which can be described as(10)uirobs=Fsuirob∗s=−FsN0−1sD0syis−uis,where the robust filter is .Fs=f/s+fd. f is a positive constant and d is a positive integer greater than or equal to the relative degree of the elements in N0−1sD0s.Substituting (5) into (10), the robust compensator uirobs is finally represented as(11)uirobs=−1−Fs−1FsN0−1sD0syis−uinoms.Thus, one can obtain that(12)uirobt=−ℒ−11−Fs−1FsN0−1sD0sℒyit−ℒuinomt.


### 3.2. Nominal Controller Design and Feasibility Analysis

The nominal system without uncertainties is shown as(13)x˙nit=Axnit+B1uinomt,ynit=Cxnit.To achieve time-varying output formation, the nominal controller can be designed as(14)uinomt=K1ynit+K2ynit−hit+K3∑j∈Niwijynit−hit−ynjt−hjt.Under protocol (14), swarm system (13) can be written as follows:(15)x˙nt=IN⊗A+B1K1C+B1K2C+L⊗B1K3Cxnt−IN⊗B1K2+L⊗B1K3ht,ynt=IN⊗Cxnt,where xnt=xn1Tt,xn2Tt,…,xnNTtT and ynt=yn1Tt,yn2Tt,…,ynNTtT.According to Assumption1, one can find C¯∈ℝn−q×n such that T=CT,C¯TT is nonsingular. Let y¯ni=C¯xniti=1,2,…,N, y¯nt=y¯n1Tt,y¯n2Tt,…,y¯nNTtT and(16)TAT−1=A¯11A¯12A¯21A¯22,TB1=B¯11B¯12,TBω=B¯ω1B¯ω2.Using nonsingular transformationIN⊗T, swarm system (15) can be rewritten as(17)y˙nt=IN⊗A¯11+B¯11K1+B¯11K2+L⊗B¯11K3ynt+IN⊗A¯12y¯nt−IN⊗B¯11K2+L⊗B¯11K3ht,y¯˙nt=IN⊗A¯21+B¯12K1+B¯12K2+L⊗B¯12K3ynt+IN⊗A¯22y¯nt−IN⊗B¯12K2+L⊗B¯12K3ht.Letθit=ynit−hiti=1,2,…,N and θt=θ1Tt,θ2Tt,…,θNTtT; it is easy to find(18)θ.t=IN⊗A¯11+B¯11K1+B¯11K2+L⊗B¯11K3θt+IN⊗A¯12y¯nt+IN⊗A¯11+B¯11K1ht−IN⊗Ih˙t,y¯˙nt=IN⊗A¯21+B¯12K1+B¯12K2+L⊗B¯12K3θt+IN⊗A¯22y¯nt+IN⊗A¯21+B¯12K1ht.According to Definition1, time-varying output formation means swarm system (18) achieves output consensus. Let λii=1,2,…,N be the eigenvalues of the Laplacian matrix L and λ1=0 with the associated eigenvector u¯1=1. J is the Jordan canonical form of L, where U=u¯1,u¯2,…,u¯N and U−1=u˜1,u˜2,…,u˜NH such that U−1LU=J. Let ck∈ℝqk=1,2,…,q be linearly independent vectors and pj=u¯1⊗ckj=i−1q+k;i=1,2,…,N;k=1,2,…,q. The subspace ℂU spanned by pk=u¯1⊗ck=1⊗ckk=1,2,…,q is an output consensus subspace, and the subspace ℂ¯U spanned by pq+1,pq+2,…,pqN is a complement output consensus subspace. Because pjj=1,2,…,qN are linearly independent, Lemma 3 can be obtained as follows.Lemma 3 (see [33]).
ℂ
U
⊕
C
¯
U
=
ℂ
q
N.J can be denoted as J=diag0,J¯ by Lemma 1, and J¯ consists of Jordan blocks corresponding to λii=2,3,…,N. Let U˜=u˜2,u˜3,…,u˜NH, ξt=u˜1H⊗Iθt, ξ¯t=u˜1H⊗Iy¯nt, ζt=U˜⊗Iθt, and ζ¯t=U˜⊗Iy¯nt; then, swarm system (18) can be converted into(19)ξ˙t=A¯11+B¯11K1+B¯11K2ξt+A¯12ξ¯t+u˜1H⊗A¯11+B¯11K1ht−u˜1H⊗Ih˙t,ξ¯˙t=A¯21+B¯12K1+B¯12K2ξt+A¯22ξ¯t+u˜1H⊗A¯21+B¯12K1ht,(20)ζ˙t=IN−1⊗A¯11+B¯11K1+B¯11K2+J¯⊗B¯11K3ζt+IN−1⊗A¯12ζ¯t+U˜⊗A¯11+B¯11K1ht−U˜⊗Ih˙t,ζ¯˙t=IN−1⊗A¯21+B¯12K1+B¯12K2+J¯⊗B¯12K3ζt+IN−1⊗A¯22ζ¯t+U˜⊗A¯21+B¯12K1ht.Lemma 4 (see [26]).
Swarm system (15) achieves time-varying output formation ht if and only if(21)limt⟶∞ζt=0.
Lemma4 shows that the two subsystems with states ξt and ζt determine the output consensus and complement output consensus parts of system (18), respectively. From Lemma 2, it is easy to find that only observable components of A¯22,A¯12 affect the subsystem with state ζt. Let T˜ be a nonsingular matrix, and the observability decomposition of A¯22,A¯12 is given as follows:(22)T˜−1A¯22T˜,A¯12T˜=D˜10D˜2D˜3,E˜10,where D˜1,E˜1 is completely observable. Denote ζ˜it=T˜−1ζ¯it=ζ˜ioTt,ζ˜io¯TtTi=1,2,…,N, ζ˜ot=ζ˜1oTt,ζ˜2oTt,…,ζ˜NoTtT, ζ˜o¯t=ζ˜1o¯Tt,ζ˜2o¯Tt,…,ζ˜No¯TtT, T˜−1A¯21=F˜1T,F˜2TT, T˜−1B¯12=B˜121T,B˜122TT, and T˜−1B¯ω2=B˜ω21T,B˜ω22TT. Then, system (20) can be transformed into(23)ζ˙t=IN−1⊗A¯11+B¯11K1+B¯11K2+J¯⊗B¯11K3ζt+IN−1⊗E˜1ζ˜ot+U˜⊗A¯11+B¯11K1ht−U˜⊗Ih˙t,ζ˜˙ot=IN−1⊗F˜1+B˜121K1+B˜121K2+J¯⊗B˜121K3ζt+IN−1⊗D˜1ζ˜ot+U˜⊗F˜1+B˜121K1ht,ζ˜˙o¯t=IN−1⊗F˜2+B˜122K1+B˜122K2+J¯⊗B˜122K3ζt+IN−1⊗D˜2ζ˜ot+IN−1⊗D˜3ζ˜o¯t+U˜⊗F˜2+B˜122K1ht.
ForB¯11 and B˜121, there exist nonsingular matrices T¯=T¯11T¯12T¯21T¯22 and T^ with T^−1=F^1F^2 such that T¯B¯11B˜121T^=I000.Let(24)xet=xt−xnt,(25)et=yt−ynt,(26)qt=ℒ−11−Fss+f∗ωt,(27)zt=xet−IN⊗Bωqt.Theorem 1.
Suppose that the system described by (1), satisfying Assumptions 1–3. If the controller given by (5), (12), and (14) is applied, for any given positive constant ϵ and any initial conditions, one can find T≥t0 that et2≤ϵ, t≥T if and only if the following conditions hold simultaneously:(i)
For∀i∈1,2,…,N and j∈Ni,(28)limt⟶∞A¯11+B¯11K1F˜1+B˜121K1hit−hjt−I0h˙it−h˙jt=0.(ii)
The followingN−1 matrices are Hurwitz:(29)Γi=A¯11+B¯11K1+B¯11K2+λiB¯11K3E˜1F˜1+B˜121K1+B˜121K2+λiB˜121K3D˜1,where i=2,3,…,N.Proof.
From Corollary 1 in [26], one can see that (28) and (29) are sufficient and necessity conditions for nominal system (13) with nominal controller (14) to achieve time-varying output formation ht.
Then,unomt=u1nomtT,u2nomtT,…,unnomtTT. And unomt can be described as(30)unomt=Dyyt+Drht.
From (1), (5), and (10), one has(31)x˙t=IN⊗A+B1K1C+B1K2C+L⊗B1K3Cxt−IN⊗B1K2+L⊗B1K3ht+IN⊗Bωωt∗ℒ−11−Fs,yt=IN⊗Cxt.
From (15), (24), (26), and (31), one can obtain the error system given by(32)x˙et=Anxet+IN⊗Bωωt∗ℒ−11−Fset=Cxet,where An=IN⊗A+B1K1C+B1K2C+L⊗B1K3C.
Combining (26), (27), and (32), one can get that(33)z˙t=x˙et−Bωq˙t=Anxe+IN⊗Bωωt∗ℒ−11−Fs−IN⊗Bωℒ−1sqs=Anzt+AnIN⊗Bωqt+IN⊗Bωℒ−1ωs1−Fs−IN⊗Bωℒ−1ss+fωs1−Fs=Anzt+AnIN⊗Bωqt+IN⊗Bωℒ−1fs+fωs1−Fs=Anzt+AnIN⊗Bωqt+fIN⊗Bωqt=Anzt+An+fInNIN⊗Bωqt.
Moreover, from (5), (10), (31), and Assumption 3, one gets(34)ut∞=unomt+urobt∞≤unomt∞+FsN0−1sLsωt∞≤ξyxxt∞+ξx2xt∞η2+ξu2ut∞+ξu.
Combining (34) and Assumption 3, one can obtain that(35)ut∞≤ζxxt∞η2+ζu,where ζx and ζu are positive constants.
From Assumption2, (24), and (35), positive constants ζ and μ can be found satisfying(36)ωt∞≤ζxet∞η+μ,where η=maxη1,η2.
From (26), (27), and (35), positive constants ζq and μq can be found such that(37)qt∞≤q∞≤1s+f1−Fs1ζxet∞η+μ≤ζqfzt∞η+q∞η+μq,where(38)qt∞=max1≤i≤nqit,q∞=max1≤i≤nsuptqit=max1≤i≤nqi∞.
Iff is large enough and satisfies ζqq∞η−1≤f/2, from (37), one can get(39)qt∞≤q∞≤2ζqfzt∞η+μq.
Choose the Lyapunov function candidate(40)Vt=zTtPzt,where P is a symmetric positive definite constant matrix and satisfies(41)PAn+AnTP=−Q,Q=QT>0.
Then, the derivative ofV is as follows:(42)V˙t=z˙TtPzt+zTtPz˙t=−ztTQzt+2ZTtPAn+fInNIN⊗Bωqt≤−zTtQzt+ζzzt2qt2≤−λqzt22+ζzzt2qt2,where ζz=2PAn+fInNIN⊗Bω∞ and λq=minλiQ.
Noticing thatqt2≤nqt∞≤nq∞, ∀qt∈Rp, from (39) and (42), one can obtain(43)V˙≤−λqzt22+ζzzt22ζqnfzt∞η+μq≤−λqzt2zt2−ζzλq2ζqnfzt∞η+μq.
Iff is large enough to satisfy(44)zt∞η≤λqf2nζzζq−μq,then one can obtain(45)V˙≤−λqzt2zt2−1f≤−λqzt2VλmaxP−1f.
From (45), one can get that if Vz0≤λmaxP/f, then Vzt≤λmaxP/f; hence, zt2≤λmaxP/fλminP. Else, if Vz0>λmaxP/f, then Vzt≤Vz0 so that zt2≤Vz0/λminP. That means zt is bounded.
Because (39) and zt are bounded, it follows that if f is sufficiently large, then(46)qt∞≤λf,where λ is a positive constant.
As a result,xet is bounded and converges for any initial conditions. And for any given positive constant ϵ,(47)et2≤ϵ,∀t≥T,where T is a positive constant depending on initial conditions. Therefore, the time-varying output formation is achieved.Remark 3.
From equation (28), we can see that the condition is very conservative and restrictive. Only a few formations ht can satisfy the condition. The contribution of this paper is to find sufficient and necessary conditions to achieve time-varying formation for general high-order linear systems. Especially, when the formation ht can be described as sin function, cos function, or exponential function, it is very likely to satisfy condition (28). Moreover, the trial-and-error method can be used to find K1. C¯ and T˜ can be adjusted to find proper K1.Remark 4.
From controllers (11) and (14), one can see that only gain matrices K1, K2, and K3 and positive constants f and d are required. K1 and K2 can be found based on the pole place method. According to He and Wang [34], iterative linear matrix inequality algorithm can be applied to find K3. d is the relative degree of the elements in N0−1sD0s. The value of f can be chosen by using the trial-and-error method. In general, the proposed controller can be easily applied to practical situations.


## 3.1. Robust Compensator Design

The robust controller is constructed with two parts:(5)ui=uinomt+uirobt,where uinomt stands for the nominal controller for the nominal model to obtain desired time-varying formation and uirobt stands for the robust compensator to restrain the influence of nonlinear uncertainty ωit.Based on Laplace transformation, the following frequency domain equation can be found from (1):(6)yis=D0−1sN0suis+D0−1sLsωis,where D0−1sN0s=CsI−A−1B1 and D0−1sLs=CsI−A−1Bω.To restrain uncertainties, a robust compensator is designed as(7)uirob∗s=−N0−1sLsωis.Substituting (5) and (7) into (6), we can get(8)yis=D0−1sN0suinoms,which means robust compensator (7) can restrain the nonlinear uncertainties perfectly.As a matter of fact that nonlinear uncertaintyωis cannot be measured directly, substituting ωis into (7), one can further express the robust compensator by(9)uirob∗s=−N0−1sD0syis+uis.To avoid the high-order derivative terms of the output, a robust filter is considered which can be described as(10)uirobs=Fsuirob∗s=−FsN0−1sD0syis−uis,where the robust filter is .Fs=f/s+fd. f is a positive constant and d is a positive integer greater than or equal to the relative degree of the elements in N0−1sD0s.Substituting (5) into (10), the robust compensator uirobs is finally represented as(11)uirobs=−1−Fs−1FsN0−1sD0syis−uinoms.Thus, one can obtain that(12)uirobt=−ℒ−11−Fs−1FsN0−1sD0sℒyit−ℒuinomt.


## 3.2. Nominal Controller Design and Feasibility Analysis

The nominal system without uncertainties is shown as(13)x˙nit=Axnit+B1uinomt,ynit=Cxnit.To achieve time-varying output formation, the nominal controller can be designed as(14)uinomt=K1ynit+K2ynit−hit+K3∑j∈Niwijynit−hit−ynjt−hjt.Under protocol (14), swarm system (13) can be written as follows:(15)x˙nt=IN⊗A+B1K1C+B1K2C+L⊗B1K3Cxnt−IN⊗B1K2+L⊗B1K3ht,ynt=IN⊗Cxnt,where xnt=xn1Tt,xn2Tt,…,xnNTtT and ynt=yn1Tt,yn2Tt,…,ynNTtT.According to Assumption1, one can find C¯∈ℝn−q×n such that T=CT,C¯TT is nonsingular. Let y¯ni=C¯xniti=1,2,…,N, y¯nt=y¯n1Tt,y¯n2Tt,…,y¯nNTtT and(16)TAT−1=A¯11A¯12A¯21A¯22,TB1=B¯11B¯12,TBω=B¯ω1B¯ω2.Using nonsingular transformationIN⊗T, swarm system (15) can be rewritten as(17)y˙nt=IN⊗A¯11+B¯11K1+B¯11K2+L⊗B¯11K3ynt+IN⊗A¯12y¯nt−IN⊗B¯11K2+L⊗B¯11K3ht,y¯˙nt=IN⊗A¯21+B¯12K1+B¯12K2+L⊗B¯12K3ynt+IN⊗A¯22y¯nt−IN⊗B¯12K2+L⊗B¯12K3ht.Letθit=ynit−hiti=1,2,…,N and θt=θ1Tt,θ2Tt,…,θNTtT; it is easy to find(18)θ.t=IN⊗A¯11+B¯11K1+B¯11K2+L⊗B¯11K3θt+IN⊗A¯12y¯nt+IN⊗A¯11+B¯11K1ht−IN⊗Ih˙t,y¯˙nt=IN⊗A¯21+B¯12K1+B¯12K2+L⊗B¯12K3θt+IN⊗A¯22y¯nt+IN⊗A¯21+B¯12K1ht.According to Definition1, time-varying output formation means swarm system (18) achieves output consensus. Let λii=1,2,…,N be the eigenvalues of the Laplacian matrix L and λ1=0 with the associated eigenvector u¯1=1. J is the Jordan canonical form of L, where U=u¯1,u¯2,…,u¯N and U−1=u˜1,u˜2,…,u˜NH such that U−1LU=J. Let ck∈ℝqk=1,2,…,q be linearly independent vectors and pj=u¯1⊗ckj=i−1q+k;i=1,2,…,N;k=1,2,…,q. The subspace ℂU spanned by pk=u¯1⊗ck=1⊗ckk=1,2,…,q is an output consensus subspace, and the subspace ℂ¯U spanned by pq+1,pq+2,…,pqN is a complement output consensus subspace. Because pjj=1,2,…,qN are linearly independent, Lemma 3 can be obtained as follows.Lemma 3 (see [33]).
ℂ
U
⊕
C
¯
U
=
ℂ
q
N.J can be denoted as J=diag0,J¯ by Lemma 1, and J¯ consists of Jordan blocks corresponding to λii=2,3,…,N. Let U˜=u˜2,u˜3,…,u˜NH, ξt=u˜1H⊗Iθt, ξ¯t=u˜1H⊗Iy¯nt, ζt=U˜⊗Iθt, and ζ¯t=U˜⊗Iy¯nt; then, swarm system (18) can be converted into(19)ξ˙t=A¯11+B¯11K1+B¯11K2ξt+A¯12ξ¯t+u˜1H⊗A¯11+B¯11K1ht−u˜1H⊗Ih˙t,ξ¯˙t=A¯21+B¯12K1+B¯12K2ξt+A¯22ξ¯t+u˜1H⊗A¯21+B¯12K1ht,(20)ζ˙t=IN−1⊗A¯11+B¯11K1+B¯11K2+J¯⊗B¯11K3ζt+IN−1⊗A¯12ζ¯t+U˜⊗A¯11+B¯11K1ht−U˜⊗Ih˙t,ζ¯˙t=IN−1⊗A¯21+B¯12K1+B¯12K2+J¯⊗B¯12K3ζt+IN−1⊗A¯22ζ¯t+U˜⊗A¯21+B¯12K1ht.Lemma 4 (see [26]).
Swarm system (15) achieves time-varying output formation ht if and only if(21)limt⟶∞ζt=0.
Lemma4 shows that the two subsystems with states ξt and ζt determine the output consensus and complement output consensus parts of system (18), respectively. From Lemma 2, it is easy to find that only observable components of A¯22,A¯12 affect the subsystem with state ζt. Let T˜ be a nonsingular matrix, and the observability decomposition of A¯22,A¯12 is given as follows:(22)T˜−1A¯22T˜,A¯12T˜=D˜10D˜2D˜3,E˜10,where D˜1,E˜1 is completely observable. Denote ζ˜it=T˜−1ζ¯it=ζ˜ioTt,ζ˜io¯TtTi=1,2,…,N, ζ˜ot=ζ˜1oTt,ζ˜2oTt,…,ζ˜NoTtT, ζ˜o¯t=ζ˜1o¯Tt,ζ˜2o¯Tt,…,ζ˜No¯TtT, T˜−1A¯21=F˜1T,F˜2TT, T˜−1B¯12=B˜121T,B˜122TT, and T˜−1B¯ω2=B˜ω21T,B˜ω22TT. Then, system (20) can be transformed into(23)ζ˙t=IN−1⊗A¯11+B¯11K1+B¯11K2+J¯⊗B¯11K3ζt+IN−1⊗E˜1ζ˜ot+U˜⊗A¯11+B¯11K1ht−U˜⊗Ih˙t,ζ˜˙ot=IN−1⊗F˜1+B˜121K1+B˜121K2+J¯⊗B˜121K3ζt+IN−1⊗D˜1ζ˜ot+U˜⊗F˜1+B˜121K1ht,ζ˜˙o¯t=IN−1⊗F˜2+B˜122K1+B˜122K2+J¯⊗B˜122K3ζt+IN−1⊗D˜2ζ˜ot+IN−1⊗D˜3ζ˜o¯t+U˜⊗F˜2+B˜122K1ht.
ForB¯11 and B˜121, there exist nonsingular matrices T¯=T¯11T¯12T¯21T¯22 and T^ with T^−1=F^1F^2 such that T¯B¯11B˜121T^=I000.Let(24)xet=xt−xnt,(25)et=yt−ynt,(26)qt=ℒ−11−Fss+f∗ωt,(27)zt=xet−IN⊗Bωqt.Theorem 1.
Suppose that the system described by (1), satisfying Assumptions 1–3. If the controller given by (5), (12), and (14) is applied, for any given positive constant ϵ and any initial conditions, one can find T≥t0 that et2≤ϵ, t≥T if and only if the following conditions hold simultaneously:(i)
For∀i∈1,2,…,N and j∈Ni,(28)limt⟶∞A¯11+B¯11K1F˜1+B˜121K1hit−hjt−I0h˙it−h˙jt=0.(ii)
The followingN−1 matrices are Hurwitz:(29)Γi=A¯11+B¯11K1+B¯11K2+λiB¯11K3E˜1F˜1+B˜121K1+B˜121K2+λiB˜121K3D˜1,where i=2,3,…,N.Proof.
From Corollary 1 in [26], one can see that (28) and (29) are sufficient and necessity conditions for nominal system (13) with nominal controller (14) to achieve time-varying output formation ht.
Then,unomt=u1nomtT,u2nomtT,…,unnomtTT. And unomt can be described as(30)unomt=Dyyt+Drht.
From (1), (5), and (10), one has(31)x˙t=IN⊗A+B1K1C+B1K2C+L⊗B1K3Cxt−IN⊗B1K2+L⊗B1K3ht+IN⊗Bωωt∗ℒ−11−Fs,yt=IN⊗Cxt.
From (15), (24), (26), and (31), one can obtain the error system given by(32)x˙et=Anxet+IN⊗Bωωt∗ℒ−11−Fset=Cxet,where An=IN⊗A+B1K1C+B1K2C+L⊗B1K3C.
Combining (26), (27), and (32), one can get that(33)z˙t=x˙et−Bωq˙t=Anxe+IN⊗Bωωt∗ℒ−11−Fs−IN⊗Bωℒ−1sqs=Anzt+AnIN⊗Bωqt+IN⊗Bωℒ−1ωs1−Fs−IN⊗Bωℒ−1ss+fωs1−Fs=Anzt+AnIN⊗Bωqt+IN⊗Bωℒ−1fs+fωs1−Fs=Anzt+AnIN⊗Bωqt+fIN⊗Bωqt=Anzt+An+fInNIN⊗Bωqt.
Moreover, from (5), (10), (31), and Assumption 3, one gets(34)ut∞=unomt+urobt∞≤unomt∞+FsN0−1sLsωt∞≤ξyxxt∞+ξx2xt∞η2+ξu2ut∞+ξu.
Combining (34) and Assumption 3, one can obtain that(35)ut∞≤ζxxt∞η2+ζu,where ζx and ζu are positive constants.
From Assumption2, (24), and (35), positive constants ζ and μ can be found satisfying(36)ωt∞≤ζxet∞η+μ,where η=maxη1,η2.
From (26), (27), and (35), positive constants ζq and μq can be found such that(37)qt∞≤q∞≤1s+f1−Fs1ζxet∞η+μ≤ζqfzt∞η+q∞η+μq,where(38)qt∞=max1≤i≤nqit,q∞=max1≤i≤nsuptqit=max1≤i≤nqi∞.
Iff is large enough and satisfies ζqq∞η−1≤f/2, from (37), one can get(39)qt∞≤q∞≤2ζqfzt∞η+μq.
Choose the Lyapunov function candidate(40)Vt=zTtPzt,where P is a symmetric positive definite constant matrix and satisfies(41)PAn+AnTP=−Q,Q=QT>0.
Then, the derivative ofV is as follows:(42)V˙t=z˙TtPzt+zTtPz˙t=−ztTQzt+2ZTtPAn+fInNIN⊗Bωqt≤−zTtQzt+ζzzt2qt2≤−λqzt22+ζzzt2qt2,where ζz=2PAn+fInNIN⊗Bω∞ and λq=minλiQ.
Noticing thatqt2≤nqt∞≤nq∞, ∀qt∈Rp, from (39) and (42), one can obtain(43)V˙≤−λqzt22+ζzzt22ζqnfzt∞η+μq≤−λqzt2zt2−ζzλq2ζqnfzt∞η+μq.
Iff is large enough to satisfy(44)zt∞η≤λqf2nζzζq−μq,then one can obtain(45)V˙≤−λqzt2zt2−1f≤−λqzt2VλmaxP−1f.
From (45), one can get that if Vz0≤λmaxP/f, then Vzt≤λmaxP/f; hence, zt2≤λmaxP/fλminP. Else, if Vz0>λmaxP/f, then Vzt≤Vz0 so that zt2≤Vz0/λminP. That means zt is bounded.
Because (39) and zt are bounded, it follows that if f is sufficiently large, then(46)qt∞≤λf,where λ is a positive constant.
As a result,xet is bounded and converges for any initial conditions. And for any given positive constant ϵ,(47)et2≤ϵ,∀t≥T,where T is a positive constant depending on initial conditions. Therefore, the time-varying output formation is achieved.Remark 3.
From equation (28), we can see that the condition is very conservative and restrictive. Only a few formations ht can satisfy the condition. The contribution of this paper is to find sufficient and necessary conditions to achieve time-varying formation for general high-order linear systems. Especially, when the formation ht can be described as sin function, cos function, or exponential function, it is very likely to satisfy condition (28). Moreover, the trial-and-error method can be used to find K1. C¯ and T˜ can be adjusted to find proper K1.Remark 4.
From controllers (11) and (14), one can see that only gain matrices K1, K2, and K3 and positive constants f and d are required. K1 and K2 can be found based on the pole place method. According to He and Wang [34], iterative linear matrix inequality algorithm can be applied to find K3. d is the relative degree of the elements in N0−1sD0s. The value of f can be chosen by using the trial-and-error method. In general, the proposed controller can be easily applied to practical situations.


## 4. Numerical Simulations

In this section, a numerical simulation is presented to illustrate the effectiveness of the proposed control method. The designing process of nominal controller (12) is presented for swarm system (13) to achieve time-varying output formation.Step 1: solving feasible condition (28) for K1Step 2: choosingK2 to assign the eigenvalues of A+BK1C+BK2C at desired locations in the complex planeStep 3: designingK3 to make condition (29) satisfiedLet(48)A0=A¯11+B¯11K1+B¯11K2E˜1F˜1+B˜121K1+B˜121K2D˜1,B0=B¯11B˜121,C0=I0.Then,Γii=2,3,…,N in condition (ii) can be rewritten as(49)Γi=A0+λiB0K3C0i=2,3,…,N.From (49), one can see that if and only if K3 can stabilize subsystems A0,λiB0,C0 through the static output feedback (SOF), then Γi are Hurwitz. To find the gain matrix K3, an improved iterative linear matrix inequality (ILMI) algorithm [34] is employed which can solve the SOF problem without introducing any additional variables. And the dimensions of the LMIs need not be increased.Time-varying output formation control for a fifth-order system with eight agents is considered. The directed interaction topology is shown in Figure1, and its adjacency matrix is assumed to be 0-1 matrix. The dynamics of each agent described by (1) are(50)xi=xi1xi2xi3xi4xi5,A=−220.56.5−0.5012602−20−41000−20−21−12−1,B1=44−303,Bω=10100101−11211.20.50.8−0.33.25−2.521.75,C=2−101−101110001−11.Figure 1
Directed interaction topology G.Choose(51)C¯=0001000001.Then, we can get(52)T=2−101−101110001−110001000001,A11=−11−11020−10,A12=201010,A21=000−10−1,A22=−202−1,B¯11=110,B¯12=03.Obviously,A,B is stabilizable, and A¯22,A¯12 is not observable. The nonsingular matrix T˜ is selected as(53)T˜=T−1=I=1001.It can be obtained that(54)D˜1=−2,D˜2=2,D˜3=−1,E˜1=211,F˜1=000,F˜2=−10−1,B˜121=0,B˜122=3.According to the designing process of protocol (5), choose K1 to satisfy condition (i) in Theorem 1. K1 can be chosen as K1=0,0,0. Choose K2=−1.2,−3.3,14.1 to assign the eigenvalue of A+B1K1C+B2K2C at −1,−2.0773,−1.7114+3.5863j,−1.7114−3.5863j, and -2 with j2=−1. Finally, the gain matrix K3 is chosen as K3=−1.4457,−2.1366,0.4930.The initial state is(55)x10=−1−1.7500.2−0.7,x20=−0.50.60.61.12,x30=0.5−1.3−1.5−1.51,x40=1.610.9−32,x50=371.3−1.1−2.6,x60=−2.1−3−1.31.80.6,x70=1.21.21.3−1−2,x80=11.1−1.30.5−0.3.The eight agents need to achieve predefined time-varying output formations which are defined ashit=10sint+i−1π/410cost+i−1π/4−10sint+i−1π/4Ti=1,2,…,8. The nonlinear uncertainty is denoted as ωit=0.3sint+i−1π/40.4cost+i−1π/40.6xi10.4xi3Ti=1,2,…,8which is related to external disturbances and agents’ states. The robust compensator parameters are f=60 and d=2.Considering nonlinear uncertainty, Figures2 and 3 show the output formation snapshot of eight agents with the nominal controller and robust controller, respectively. The outputs of agents are denoted by the plus, triangle, diamond, square, asterisk, x-mark, circle, and point, and the output formation reference function is denoted by the pentagram. From the comparison of two figures, one can see that the robust controller has better control effect under nonlinear uncertainties. Figure 3 achieves the specified formation in ten seconds while Figure 2 does not.Figure 2
Output snapshots of eight agents without the robust compensator. (a)t = 0 s. (b) t = 10 s.
(a)
(b)Figure 3
Output snapshots of eight agents with the robust controller. (a)t = 0 s. (b) t = 10 s.
(a)
(b)Figures4 and 5 show the output formation error of eight agents with the nominal controller and robust controller, respectively. From Figure 4, we can see that output errors converge. Because of the existence of nonlinear uncertainty ωit, output errors can only converge to the small neighborhood of zero. Compared with Figure 4, output errors in Figure 5 can also converge to the small neighborhood of zero. However, the convergence speed is faster, and the convergence domain is smaller in Figure 5. From the comparison of two figures, one can see that the robust controller can restrain the nonlinear uncertainties well. Meanwhile, the formation errors can be made as small as desired with arbitrarily specified convergence rate.Figure 4
Output error of eight agents without the robust compensator.
(a)
(b)
(c)Figure 5
Output error of eight agents with the robust compensator.
(a)
(b)
(c)Figure6 shows the input signal of eight agents including uinormt2 and uirobt2. From Figure 6, we can see that two controllers tend to be stable. The robust compensator outputs change over time and are similar to nonlinear uncertainty ωit which also verifies the effectiveness of the proposed controller. Meanwhile, input constrain may be encountered considering actuator saturation in practical situations. As a result, it is meaningful to study the influence of input constrain on multiagent systems in future work.Figure 6
Input signal of eight agents.
(a)
(b)


## 5. Conclusion

Time-varying output formation control problems for multiagent systems with directed interaction topologies and nonlinear uncertainties are studied. The nonlinear uncertainties are related to external disturbances, parameter uncertainties, nonlinearities, and couplings. By extending the dimensions of the observation matrix, the formation control problem is transformed into a consensus problem and then into a stability problem of the subsystem. A novel robust controller is proposed which includes a nominal controller and a robust compensator. The nominal controller based on relative outputs of neighbour agents is designed to achieve expected performance for the nominal system. And the robust compensator is to restrain nonlinear uncertainties. Based on the proposed controller, the output formation error can be as small as desired. Future work will focus on swarm systems with switching topologies, input constrain, and time-delay group formation control problems.Some related works can be found in [35]. Compared with previous works, different problems are addressed using the similar method. First, time-varying formation is studied in this paper. However, it is time-invariant formation that was studied. Second, most of the formulas are different. And more assumptions and remarks are added to compare with other works. Third, the proof of Theorem 1 is different. The process of proof in this paper is more rigorous and complete.



---

*Source: 2069170-2020-04-10.xml*