# Finite-Time Formation of Unmanned Aerial Vehicles with Switching Topologies and Disturbances: An Average Dwell Time Approach

Authors: Chao Ma; Wei Wu
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

This paper investigates the finite-time formation problem of unmanned aerial vehicles (UAVs) with switching topologies and external disturbances. The formation problem is first transformed into the finite-time stability and bounded problems of a switched system, respectively. In particular, the finite-time unachievable topology is concerned for unreliable information exchanges. By employing the average dwell time (ADT) method, sufficient criteria are established to deal with the switching topologies such that the desired time-varying formation can be achieved in finite time. Then, the topology-dependent controller can be designed in terms of matrix technique. Finally, an illustrative example is given to verify the effectiveness of our proposed formation strategy.


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## Body

## 1. Introduction

During the past decade, the unmanned aerial vehicles (UAVs) have grown from academic researches into practical applications capable of various missions in both civilian and military areas [1–3]. Recently, with the substantial progresses in swarm intelligence, cooperation issues of multiple UAVs and robots have been receiving increasing attention for distinguishing advantages. Based on information exchanges according to the communication topology, a group of UAVs can work collectively to meet the need for high robustness, reliability, flexibility, and efficiency [4–10]. In particular, the formation problem is one of the most hot research topics for UAVs, which means that the UAVs can keep a certain flying configuration while accomplishing the tasks. Note that the communication among the UAVs can play a very significant role in formation since the information exchanges are conducted via the communication networks according to the communication topology [11–15]. However, it is worth mentioning that the communication topology may change at time due to unreliable communication networks or security reason, and this may lead to unpredictable mission failure or performance degradation. As a consequence, the UAVs with switching topologies are of theoretical importance and practical background. So far, there are different remarkable approaches reported in the literature and the references therein [16–18].On another research front line, the switched systems have been receiving increasing attention due to their novel ability in describing practical systems with mode switchings [19–22]. In particular, as one of the most effective analysis and synthesis methods for switched systems, the average dwell time (ADT) has been widely utilized. In comparison to the dwell time method, the ADT can further reduce the conservatism for each subsystem [23–25]. Therefore, a natural question arises from the prior discussions: can we introduce the merit of ADT of switched systems to deal with the switching topologies? The answer to this question is challenging, but the efforts can lead to encouraging results. Furthermore, in most applications, the stability is required to be achieved over a finite-time interval, which gives rises to the researches on finite-time stability, and burgeoning investigations on finite-time consensus problems of multiagent systems with switching topologies have been made [26–28]. However, it is worth mentioning that the unachievable switching topologies are not concerned in most existing results. It is noted that there is also a need for the finite-time formation of UAVs by taking into account its realistic background. Until now, to the best of the authors’ knowledge, there still exists a considerable gap in solving the formation problem of UAVs with switching topologies based on the switched system theory, let alone those with unachievable topologies. The aim of this article is to make one of the first attempts to shorten such a gap.Inspired by the ADT method in switched systems, in this paper, the finite-time formation of UAVs with switching topologies and disturbances is studied in terms of the ADT approach. Compared with the existing results, the main contributions of our paper can be summarized as follows. (1) The finite-time formation problem of UAVs with unachievable switching topologies is investigated for the first time. More precisely, the switching topologies are categorized into two types: the achievable topologies and the unachievable topologies. The standard of category relies on whether the finite-time formation can be reached for a fixed topology. For example, it can be verified that the directed topology without a directed spanning tree is unachievable topology. This can further extend the applications of switching topologies and is more practical for UAVs. It can be found that the established results with unachievable topologies are more practical in the applications. (2) The disturbances of UAVs are also considered for formation robustness with the finite-time boundedness problem. On the basis of ADT with the multiple Lyapunov function (MLF) method, sufficient conditions are established for ensuring the finite-time formation and the desired topology-dependent controller gains are obtained accordingly.The remainder of our paper is given as follows: Section2 introduces the finite-time formation control problem of UAVs with some essential preliminaries. In Section 3, the main analysis and synthesis procedures are provided with the ADT method. Section 4 presents the numerical example to demonstrate the effectiveness of our proposed formation strategies. Finally, the paper is concluded in Section 5.Notation 1.
Rn and Rm×n denote the n dimensional Euclidean space and the space of m×n real matrices, respectively. A−B≻0A−B≺0 denotes that A−B is positive definite (negative definite). λmaxA and λminA denote the maximum and minimum eigenvalues of matrix A, respectively. ℒ20,∞ denotes the space of square-integrable vector functions over 0,∞. A⊗B stands for the Kronecker product. ∗ represents the ellipsis symmetry terms in symmetric block matrices. All matrices are compatible for algebraic operations.


## 2. Problem Formulation and Preliminaries

### 2.1. The UAV Dynamics

Consider a group of UAVs with the following kinematics and dynamics:(1)ẋi=vicosaicosbi,ẏi=vicosaisinbi,żi=visinai,(2)v̇i=Ti−Dimi−gsinai−d1i,ȧi=gvingicosϕi−cosai+d2i,ḃi=Lisinϕimivicosai+d3i,where xi,yi,ziT denotes the position of the ith UAV, vi represents the ground speed, ai stands for the flight-path angle, bi is the heading angle, Ti denotes the engine thrust, Di represents the drag, mi denotes the mass, g stands for the acceleration due to gravity, Li is the lift force, ngi=Li/mig denotes the g-load value, ϕi represents the banking angle, and dki,k=1,2,3 stands for the vortex disturbances.By applying the feedback linearization method as(3)x¨i=uxi+wxi,y¨i=uyi+wyi,z¨i=uzi+wzi,where uxi,uyi,uziT denotes the virtual acceleration control input and wxi,wyi,wziT represents the disturbances, the relationships between virtual control variables and the actual control variables are given by
(4)Ti=miuzi+gsinai+uxyicosai+Di,ngi=uzi+gcosai−uxyisinaigcosϕi,ϕi=arctanuyicosbi−uxisinbiuzi+gcosai−uxyisinai,with uxyi=uxicosbi+uyisinbi, ai=arcsinżi/vi, and bi=arctanẏi/ẋi.Consequently, by denoting(5)ξit=xi,yi,zi,ẋi,ẏi,żiT,uit=uxi,uyi,uziT,wit=wxi,wyi,wziT,the UAV dynamics can be obtained as follows:
(6)ξ̇it=Aξit+Buit+Bwit,where
(7)A=0I300,B=0I3.


### 2.2. Graph Theory

The directed graphG=V,E,A is adopted to describe the communication topology of the UAVs. A=aij∈RN×N represents the weighted adjacency matrix with
(8)aij>0vi,vj∈E,aij=0,otherwise.VG=v1,⋯,vN and E denote the sets of nodes and edges, respectively. The L=lijN×N denotes the Laplacian matrix with
(9)lij=−aij,i≠j,lii=∑j=1,i≠jNaij.IfG has a directed spanning tree, then 0 is the eigenvalue of L. More details of the algebraic graph theory can be found in [29].The switching topologies are studied in this paper. LetGσt:σt∈S=1,2,…,n denote all possible switching graphs, where σt represents the index set for all switching graphs. The switching signal σt:0,∞→p, and its value is the index of the graph at time t. The switching sequence is t1,t2,…, where th,th+1, h=1,2,…, is uniformly bounded nonoverlapping [30].Moreover, for any possibleLσt∈RN×N of graph Gσt and a full row rank matrix E∈RN−1×N defined as
(10)E=1−10⋯001−1⋯0⋮⋮⋱⋮⋮0001−1,there exists a matrix Mσt∈RN×N−1 such that Lσt=MσtE.Remark 1.
It is worth mentioning that whenGσt is achievable (Lσt has a directed spanning tree), Mσt is of full column rank and the eigenvalues of EMσt are the same as that of Lσt. When Gσt is unachievable, the above conditions of Mσt will not be satisfied.
Without loss of generality, suppose that there arer achievable topologies with σt=1,2,…,r which means that the formation can be achieved with the Lσt and n−r unachievable topologies with r+1 and r+2,…,n, otherwise. All the possible communication topologies can be divided into the achievable set denoted as AT and the unachievable set denoted as UAT.


### 2.3. Finite-Time Formation

For the time-varying formation problem, denoteht=h1t,h2t,…,hNtT∈R6N, where hit=hxit,hyit,hzit,hvxit,hvyit,hvyit,hvzitT is a piecewise continuously differentiable, and et=e1t,e2t,…,eNtT∈R6N, where eit=ξit−hit.Consequently, the formation controller for each UAV can be designed as follows:(11)uit=K1σtξit−hit+K2σt∑j=1Naijσtξit−hit−ξit−hjt+μiht,i=1,2,…,N,where ḣit=Ahit+Bμiht and K1σt and K2σt are topology-dependent controller gains to be determined, respectively.As a result, the closed-loop dynamics of the UAVs can be obtained by(12)ξ̇t=I⊗Aξt+I⊗BK1σtξt−ht+Lσt⊗BK2σt×ξt−ht+I⊗Bμht+I⊗Bwt,where μht=μ1hTt,μ2hTt,…,μNhTtT.In addition, note thatet=ξt−ht; then, it follows that
(13)ėt=I⊗Aet+I⊗BK1σtet+Lσt⊗BK2σtet+I⊗Bwt.Premultiplying both sides of (13) by E⊗I yields that
(14)ε̇t=I⊗Aεt+I⊗BK1σtεt+EMσt⊗BK2σtεt+I⊗Bωt,where εt=E⊗Iet denotes the disagreement of et and ωt=E⊗Iwt, respectively.To this end, the following definitions are given for later use.Definition 1 (average dwell time, ADT) [31].
For anyT>t≥0, denote NσT,t as the number of switching over t,T. σt is said to have an average dwell time (ADT) τa>0, if there exist positive numbers N0 such that
(15)NσT,t≤N0+T−tτa.Definition 2 (finite-time formation, FTF) [32].
Given constantsc1>0, c2>0, and T>0, with c1<c2, matrix R>0, and a switching signal σt∈S, the finite-time formation is said to be achieved with c1,c2,T,R,σt, if it holds that
(16)εT0Rε0≤c1⇒εTtRεt≤c2,∀t∈0,T.Definition 3 (uniformly finite-time formation, UFTF) [32].
Given constantsc1>0, c2>0, and T>0, with c1<c2 and matrix R>0, the uniformly finite-time formation is said to be achieved with c1,c2,T,R, if it holds that
(17)εT0Rε0≤c1⇒εTtRεt≤c2,∀t∈0,T.Definition 4 (finite-time bounded formation, FTBF) [32].
Given constantsc1>0, c2>0, dw≥0, and T>0, with c1<c2, matrix R>0, and a switching signal σt∈S, the finite-time bounded formation is said to be achieved with c1,c2,T,R,σt,dw, if it holds that
(18)εT0Rε0≤c1⇒εTtRεt≤c2,∀t∈0,T,∀wt:∫0TwTtwtdt≤dw.Definition 5 (uniformly finite-time bounded formation, UFTBF) [32].
Given constantsc1>0, c2>0, dw≥0, and T>0, with c1<c2 and matrix R>0, the uniformly finite-time bounded formation is said to be achieved with c1,c2,T,R,dw, if for any switching signal σt∈S it holds that
(19)εT0Rε0≤c1⇒εTtRεt≤c2,∀t∈0,T,∀wt:∫0TwTtwtdt≤dw.Remark 2.
It should be pointed out that the concept of finite-time stability is different from Lyapunov asymptotic stability. A system can be finite-time stable but not Lyapunov asymptotically stable and vice versa.


## 2.1. The UAV Dynamics

Consider a group of UAVs with the following kinematics and dynamics:(1)ẋi=vicosaicosbi,ẏi=vicosaisinbi,żi=visinai,(2)v̇i=Ti−Dimi−gsinai−d1i,ȧi=gvingicosϕi−cosai+d2i,ḃi=Lisinϕimivicosai+d3i,where xi,yi,ziT denotes the position of the ith UAV, vi represents the ground speed, ai stands for the flight-path angle, bi is the heading angle, Ti denotes the engine thrust, Di represents the drag, mi denotes the mass, g stands for the acceleration due to gravity, Li is the lift force, ngi=Li/mig denotes the g-load value, ϕi represents the banking angle, and dki,k=1,2,3 stands for the vortex disturbances.By applying the feedback linearization method as(3)x¨i=uxi+wxi,y¨i=uyi+wyi,z¨i=uzi+wzi,where uxi,uyi,uziT denotes the virtual acceleration control input and wxi,wyi,wziT represents the disturbances, the relationships between virtual control variables and the actual control variables are given by
(4)Ti=miuzi+gsinai+uxyicosai+Di,ngi=uzi+gcosai−uxyisinaigcosϕi,ϕi=arctanuyicosbi−uxisinbiuzi+gcosai−uxyisinai,with uxyi=uxicosbi+uyisinbi, ai=arcsinżi/vi, and bi=arctanẏi/ẋi.Consequently, by denoting(5)ξit=xi,yi,zi,ẋi,ẏi,żiT,uit=uxi,uyi,uziT,wit=wxi,wyi,wziT,the UAV dynamics can be obtained as follows:
(6)ξ̇it=Aξit+Buit+Bwit,where
(7)A=0I300,B=0I3.


## 2.2. Graph Theory

The directed graphG=V,E,A is adopted to describe the communication topology of the UAVs. A=aij∈RN×N represents the weighted adjacency matrix with
(8)aij>0vi,vj∈E,aij=0,otherwise.VG=v1,⋯,vN and E denote the sets of nodes and edges, respectively. The L=lijN×N denotes the Laplacian matrix with
(9)lij=−aij,i≠j,lii=∑j=1,i≠jNaij.IfG has a directed spanning tree, then 0 is the eigenvalue of L. More details of the algebraic graph theory can be found in [29].The switching topologies are studied in this paper. LetGσt:σt∈S=1,2,…,n denote all possible switching graphs, where σt represents the index set for all switching graphs. The switching signal σt:0,∞→p, and its value is the index of the graph at time t. The switching sequence is t1,t2,…, where th,th+1, h=1,2,…, is uniformly bounded nonoverlapping [30].Moreover, for any possibleLσt∈RN×N of graph Gσt and a full row rank matrix E∈RN−1×N defined as
(10)E=1−10⋯001−1⋯0⋮⋮⋱⋮⋮0001−1,there exists a matrix Mσt∈RN×N−1 such that Lσt=MσtE.Remark 1.
It is worth mentioning that whenGσt is achievable (Lσt has a directed spanning tree), Mσt is of full column rank and the eigenvalues of EMσt are the same as that of Lσt. When Gσt is unachievable, the above conditions of Mσt will not be satisfied.
Without loss of generality, suppose that there arer achievable topologies with σt=1,2,…,r which means that the formation can be achieved with the Lσt and n−r unachievable topologies with r+1 and r+2,…,n, otherwise. All the possible communication topologies can be divided into the achievable set denoted as AT and the unachievable set denoted as UAT.


## 2.3. Finite-Time Formation

For the time-varying formation problem, denoteht=h1t,h2t,…,hNtT∈R6N, where hit=hxit,hyit,hzit,hvxit,hvyit,hvyit,hvzitT is a piecewise continuously differentiable, and et=e1t,e2t,…,eNtT∈R6N, where eit=ξit−hit.Consequently, the formation controller for each UAV can be designed as follows:(11)uit=K1σtξit−hit+K2σt∑j=1Naijσtξit−hit−ξit−hjt+μiht,i=1,2,…,N,where ḣit=Ahit+Bμiht and K1σt and K2σt are topology-dependent controller gains to be determined, respectively.As a result, the closed-loop dynamics of the UAVs can be obtained by(12)ξ̇t=I⊗Aξt+I⊗BK1σtξt−ht+Lσt⊗BK2σt×ξt−ht+I⊗Bμht+I⊗Bwt,where μht=μ1hTt,μ2hTt,…,μNhTtT.In addition, note thatet=ξt−ht; then, it follows that
(13)ėt=I⊗Aet+I⊗BK1σtet+Lσt⊗BK2σtet+I⊗Bwt.Premultiplying both sides of (13) by E⊗I yields that
(14)ε̇t=I⊗Aεt+I⊗BK1σtεt+EMσt⊗BK2σtεt+I⊗Bωt,where εt=E⊗Iet denotes the disagreement of et and ωt=E⊗Iwt, respectively.To this end, the following definitions are given for later use.Definition 1 (average dwell time, ADT) [31].
For anyT>t≥0, denote NσT,t as the number of switching over t,T. σt is said to have an average dwell time (ADT) τa>0, if there exist positive numbers N0 such that
(15)NσT,t≤N0+T−tτa.Definition 2 (finite-time formation, FTF) [32].
Given constantsc1>0, c2>0, and T>0, with c1<c2, matrix R>0, and a switching signal σt∈S, the finite-time formation is said to be achieved with c1,c2,T,R,σt, if it holds that
(16)εT0Rε0≤c1⇒εTtRεt≤c2,∀t∈0,T.Definition 3 (uniformly finite-time formation, UFTF) [32].
Given constantsc1>0, c2>0, and T>0, with c1<c2 and matrix R>0, the uniformly finite-time formation is said to be achieved with c1,c2,T,R, if it holds that
(17)εT0Rε0≤c1⇒εTtRεt≤c2,∀t∈0,T.Definition 4 (finite-time bounded formation, FTBF) [32].
Given constantsc1>0, c2>0, dw≥0, and T>0, with c1<c2, matrix R>0, and a switching signal σt∈S, the finite-time bounded formation is said to be achieved with c1,c2,T,R,σt,dw, if it holds that
(18)εT0Rε0≤c1⇒εTtRεt≤c2,∀t∈0,T,∀wt:∫0TwTtwtdt≤dw.Definition 5 (uniformly finite-time bounded formation, UFTBF) [32].
Given constantsc1>0, c2>0, dw≥0, and T>0, with c1<c2 and matrix R>0, the uniformly finite-time bounded formation is said to be achieved with c1,c2,T,R,dw, if for any switching signal σt∈S it holds that
(19)εT0Rε0≤c1⇒εTtRεt≤c2,∀t∈0,T,∀wt:∫0TwTtwtdt≤dw.Remark 2.
It should be pointed out that the concept of finite-time stability is different from Lyapunov asymptotic stability. A system can be finite-time stable but not Lyapunov asymptotically stable and vice versa.


## 3. Main Results

In this section, the FTF and the FTBF problems of UAVs are solved with details.


### 3.1. Finite-Time Formation without Disturbances

Theorem 1.
Consider the group of UAVs (1) with switching topologies, for giving constants αi≥0, βi≥0, i∈S, μ≻1, and the topology-dependent controller gains K1i and K2i, if there exist matrices Pi≻0, ∀i,j∈S×S, i≠j, such that
(20)∏i−aiI⊗Pi≺0,i∈AT,(21)∏iβiI⊗Pi≺0,i∈UAT,(22)Pi≺¯μPj,i≠j,where
(23)∏i≔symI⊗PiA+I⊗PiBK1i+EMi⊗PiBK2i,and the total dwell time of the unachievable topologies T+ satisfies
(24)T+0,t<lnc2λminPiλminR−lnc1λmaxRλmaxPi−Nlnμ−aTβ−a.
Then, the FTF can be achieved with respect toc1,c2,T,R,σt.
Furthermore, whenT+ can be fixed, the switching signal has an ADT satisfying
(25)τa≥τa∗=Tlnμlnc2λminPiλminR−lnc1λmaxRλmaxPi−aT−β−aT+0,t.Proof 1.
Choose the multiple Lyapunov-like functions as follows:(26)Vσtt=εTtI⊗Pσtεt.
Then, the LMIs (20) and (21) mean that
(27)V̇it≺αiVit,i∈AT,V̇it≺βiVit,i∈UAT.
Fort∈tk,tk+1, it can be obtained by (27) that
(28)Vt<eaσtkt−tkVσttk,σtk∈AT,Vt<eβσtkt−tkVσttk,σtk∈UAT.
Without loss of generality, denoteσtk=i and σtk−=j with N0=0. Then, for any i≠j, if (22) holds then
(29)Vσtktk≤μVσtk−tk−.
Thus, fort ∈0,T, letting a=maxi∈Sai and β=maxi∈Sβi yields that
(30)Vt<eaT−0,t+βT+0,tVσt00=μNeaT−T+0,t+βT+0,tVσ00,where N is the switching number of σt over 0,T and T+0,t is the total dwell time of UAT.
On the other hand, it holds that(31)λminPσtεTtεt≤Vσtt=εTtPσtεt≤λmaxPσtεTtεt,which implies that when the FTF in Definition 2 can be achieved it can be verified that
(32)εTtRεt≤λmaxRεTtεt≤λmaxRVσttλminPσt<λmaxRλminPσtμNeaT−T+0,t+βT+0,tVσt00≤λmaxRλmaxPσtλminPσtμNeaTeβ−aT+0,tεT0ε0≤c1μNλmaxRλmaxPσtλminPσtλminReaTeβ−aT+0,t<c2.
Moreover, the ADTτa can be obtained for a fixed T+0,t according to Definition 1, which completes the proof.Based on the established conditions in Theorem1, Theorem 2 is given to calculate the topology-dependent controller gains.Theorem 2.
Consider the group of UAVs (1) with switching topologies, for giving constants αi≥0, βi≥0, i∈S, and μ>1, if there exist matrices Pi≻0 and W2i, ∀i,j∈S×S, i≠j, such that
(33)Ξi−αiI⊗P~i<0,i∈AT,Ξi−βiI⊗P~i<0,i∈UAT,P~i≺¯μP~j,i≠j,where
(34)Ξi≔symI⊗AP~i+I⊗BK1iP~i+EMi⊗BW2i,and the total dwell time of the unachievable topologies T+ satisfies
(35)T+0,t<lnc2λminP~i−1λminR−lnc1λmaxRλmaxP~i−1−Nlnμ−aTβ−a.
Then, the FTF can be achieved with respect toc1,c2,T,R,σt.
Furthermore, whenT+ can be fixed, the switching signal has an ADT satisfying
(36)τa≥τa∗=Tlnμlnc2λminP~i−1λminR−lnc1λmaxRλmaxP~i−1−aT−β−aT+0,t.
In addition, when the topology-dependent controller gainK1i is configured, K2i can be obtained by
(37)K2i=W2iP~i−1.Proof 2.
By lettingP~i=Pi−1 and W2i=K2iP~i and performing matrix congruent transformation, the conditions then follow directly from Theorem 1.Remark 3.
It is noted thatK1i is related with the motion modes of the formation center, such that K1i is generally separately designed in advance of K2i. When K1i is configured appropriately, then the desired K2i can be obtained by solving the conditions in Theorem 2 accordingly. For the large-scale UAVs, since the formation configuration ht is prescribed, the controllers of UAVs can be designed according to uit, where the information exchanges among the UAVs are also considered.
Note that if a common Lyapunov-like function can be found for (6), Corollary 1 can be derived.Corollary 1.
Consider the group of UAVs (1) with switching topologies, for giving constants αi≥0 and βi≥0, i∈S, if there exist matrices Pi≻0 and W2i, ∀i,j∈S×S, i≠j, such that
(38)Ξi−αI⊗P~i<0,i∈AT,Ξi−βI⊗P~i,i∈UAT,and the total dwell time of the unachievable topologies T+ satisfies
(39)T+0,t<lnc2λminP~i−1λminR−lnc1λmaxRλmaxP~i−1−aTβ−a.
Then, the UFTF can be achieved with respect toc1,c2,T,R.
In addition, when the topology-dependent controller gainK1i is configured, K2i can be obtained by
(40)K2i=W2iP~i−1.


### 3.2. Finite-Time Formation with Disturbances

When the external disturbances of UAVs are considered, the theorems of FTBF and UFTBF are presented.Theorem 3.
Consider the group of UAVs (1) with switching topologies and disturbances, for giving constants αi≥0, βi≥0, i∈S, μ>1, κ1i>0, κ2i>0, and the topology-dependent controller gains K1i and K2i, if there exist matrices Pi≻0, ∀i,j∈S×S, i≠j, such that
(41)Ψ1iI⊗PiB∗−κ1iI<0,i∈AT,(42)Ψ2iI⊗PiB∗−κ2iI<0,i∈UAT,(43)Pi≺¯μiPj,where
(44)Ψ1i≔symI⊗PiA+I⊗PiBK1i+EMi⊗PiBK2i−αiI⊗Pi,Ψ2i≔symI⊗PiA+I⊗PiBK1i+EMi⊗PiBK2i−βiI⊗Pi,and the total dwell time of the unachievable topologies T+ satisfies
(45)T+0,t<lnc2λminPiλminR−μNλmaxRλmaxPiλminRκeβTdwβ−a−Nlnμ−lnc1λmaxRλmaxPi−aTβ−a.
Then, the FTBF can be achieved with respect toc1,c2,T,R,σt,dw.
Furthermore, whenT+ can be fixed, the switching signal has an ADT satisfying
(46)τa>τa∗=Tlnμlnc2λminPiλminR−lnλmaxRλmaxPieaT−T+0,t+βT+0,tc1+λminRκeβTdw.Proof 3.
Select the multiple Lyapunov-like function as follows:(47)Vit=εTtI⊗Piεt.
Then, the LMIs (41) and (42) mean that
(48)V̇it≺αiVit+κ1iωTtωt,i∈AT,V̇it≺βiVit+κ2iωTtωt,i∈UAT.
Following same lines in the proof of Theorem1, for t∈tk,tk+1, it holds that
(49)Vt<eaσtkt−tkμVσtk−tk−+∫tkteaσtkt−sκ1σtkωTsωsds,σtk∈AT,Vt<eβσtt−tkμVσtk−tk−+∫tkteβσtkt−sκ2σtkωTsωsds,σtk∈UAT.
By lettingπσt=aσt, κσt=κ1σt, and σt∈AT and πσt=βσt, κσt=κ2σt, and σt∈UAT, one has
(50)Vt≤eπσtkt−tkμVσtk−tk−+∫tkteπσtkt−sκσtkωTsωsds≤eπσtkt−tkμeπσtk−1tk−tk−1μVσtk−1−tk−1−+∫tk−1teπσtk−1tk−sκσtk−1ωTsωsds+∫tkteπσtkt−sκσtkωTsωsds≤⋯≤μNeaT−0,t+βT+0,tVσt00+∫0teπσtt−sμNκσtωTsωsds≤μNeaT−T+0,t+βT+0,tVσt00+κeβTdw,where a=maxi∈Sai, β=maxi∈Sβi, and κ=maxi∈Sκi.
Consequently, it can be obtained that(51)εTtRεt≤λmaxRεTtεt≤λmaxRVσttλminPσt<λmaxRλminPσtμNeaT−T+0,t+βT+0,tVσt00+κeβTdw≤λmaxRλmaxPσtλminPσtμNeaT−T+0,t+βT+0,tεT0ε0+κeβTdw≤λmaxRλmaxPσtλminPσtλminRμNeaT−T+0,t+βT+0,tc1+λminRκeβTdw<c2,which means that
(52)T+0,t<lnc2λminPiλminR−μNλmaxRλmaxPiλminRκeβTdwβ−a−Nlnμ−lnc1λmaxRλmaxPi−aTβ−a.
In addition, the ADTτa can be obtained for a fixed T+0,t according to Definition 1, which completes the proof.Theorem 4.
Consider the group of UAVs (1) with switching topologies and disturbances, for giving constants αi≥0, βi≥0, i∈S, μ>1, κ1i>0, and κ2i>0, if there exist matrices P~i≻0 and W2i, ∀i,j∈S×S, i≠j, such that
(53)Φ1iI⊗B∗−κ1iI<0,i∈AT,Φ2iI⊗B∗−κ2iI<0,i∈UAT,P~i≺¯μiP~j,where
(54)Φ1i≔symI⊗AP~i+I⊗BK1iP~i+EMi⊗BW2i−αiI⊗AP~i,Φ2i≔symI⊗AP~i+I⊗BK1iP~i+EMi⊗BW2i−βiI⊗AP~i,and the total dwell time of the unachievable topologies T+ satisfies
(55)T+0,t<lnc2λminP~i−1λminR−μNλmaxRλmaxP~i−1λminRκeβTdwβ−a−Nlnμ−lnc1λmaxRλmaxP~i−1−aTβ−a.
Then, the FTBF can be achieved with respect toc1,c2,T,R,σt,dw.
Furthermore, whenT+ can be fixed, the switching signal has an ADT satisfying
(56)τa>τa∗=Tlnμlnc2λminP~i−1λminR−lnλmaxRλmaxP~i−1eaT−T+0,t+βT+0,tc1+λminRκeβTdw.
In addition, when the topology-dependent controller gainK1i is configured, K2i can be obtained by
(57)K2i=W2iP~i−1.Proof 4.
By applying the same matrix congruent transformation in Theorem2, the results can be directly obtained.Based on the above results, Algorithm1 can be given as follows.Algorithm 1: FTBF algorithm.
Input: A, B, Lσt
Output: K2σt
1: Given constantsασ(t) > 0, βσ(t) > 0, μ > 1, σt∈S, κ1i > 0, κ2i > 0;
2: Configure the formation centre byA + BK1σ(t) with desired eigenvalues;
3: ObtainingMσ(t) by Lσ(t) = Mσ(t)E;
4: Solving the LMIs in Theorem4;
5: Obtaining feasible solutions ofP~σt≻0 and W2σ(t);
6: Obtaining the total dwell time of the unachievable topologiesT+;
7:return K2σt=W2σtP~σt−1.Corollary 2.
Consider the group of UAVs (1) with switching topologies and disturbances, for giving constants αi≥0 and βi≥0, i∈S, if there exist matrices P~i≻0 and W2i, ∀i,j∈S×S, i≠j, such that
(58)Φ1iI⊗B∗−κ1iI<0,i∈AT,Φ2iI⊗B∗−κ2iI<0,i∈UAT,and the total dwell time of the unachievable topologies T+ satisfies
(59)T+0,t<lnc2λminP~i−1λminR−λmaxRλmaxP~i−1λminRκeβTdwβ−a−lnc1λmaxP~i−1−aTβ−a.
Then, the UFTBF can be achieved with respect toc1,c2,T,R,dw.
In addition, when the topology-dependent controller gainK1i is configured, K2i can be obtained by
(60)K2i=W2iP~i−1.


## 3.1. Finite-Time Formation without Disturbances

Theorem 1.
Consider the group of UAVs (1) with switching topologies, for giving constants αi≥0, βi≥0, i∈S, μ≻1, and the topology-dependent controller gains K1i and K2i, if there exist matrices Pi≻0, ∀i,j∈S×S, i≠j, such that
(20)∏i−aiI⊗Pi≺0,i∈AT,(21)∏iβiI⊗Pi≺0,i∈UAT,(22)Pi≺¯μPj,i≠j,where
(23)∏i≔symI⊗PiA+I⊗PiBK1i+EMi⊗PiBK2i,and the total dwell time of the unachievable topologies T+ satisfies
(24)T+0,t<lnc2λminPiλminR−lnc1λmaxRλmaxPi−Nlnμ−aTβ−a.
Then, the FTF can be achieved with respect toc1,c2,T,R,σt.
Furthermore, whenT+ can be fixed, the switching signal has an ADT satisfying
(25)τa≥τa∗=Tlnμlnc2λminPiλminR−lnc1λmaxRλmaxPi−aT−β−aT+0,t.Proof 1.
Choose the multiple Lyapunov-like functions as follows:(26)Vσtt=εTtI⊗Pσtεt.
Then, the LMIs (20) and (21) mean that
(27)V̇it≺αiVit,i∈AT,V̇it≺βiVit,i∈UAT.
Fort∈tk,tk+1, it can be obtained by (27) that
(28)Vt<eaσtkt−tkVσttk,σtk∈AT,Vt<eβσtkt−tkVσttk,σtk∈UAT.
Without loss of generality, denoteσtk=i and σtk−=j with N0=0. Then, for any i≠j, if (22) holds then
(29)Vσtktk≤μVσtk−tk−.
Thus, fort ∈0,T, letting a=maxi∈Sai and β=maxi∈Sβi yields that
(30)Vt<eaT−0,t+βT+0,tVσt00=μNeaT−T+0,t+βT+0,tVσ00,where N is the switching number of σt over 0,T and T+0,t is the total dwell time of UAT.
On the other hand, it holds that(31)λminPσtεTtεt≤Vσtt=εTtPσtεt≤λmaxPσtεTtεt,which implies that when the FTF in Definition 2 can be achieved it can be verified that
(32)εTtRεt≤λmaxRεTtεt≤λmaxRVσttλminPσt<λmaxRλminPσtμNeaT−T+0,t+βT+0,tVσt00≤λmaxRλmaxPσtλminPσtμNeaTeβ−aT+0,tεT0ε0≤c1μNλmaxRλmaxPσtλminPσtλminReaTeβ−aT+0,t<c2.
Moreover, the ADTτa can be obtained for a fixed T+0,t according to Definition 1, which completes the proof.Based on the established conditions in Theorem1, Theorem 2 is given to calculate the topology-dependent controller gains.Theorem 2.
Consider the group of UAVs (1) with switching topologies, for giving constants αi≥0, βi≥0, i∈S, and μ>1, if there exist matrices Pi≻0 and W2i, ∀i,j∈S×S, i≠j, such that
(33)Ξi−αiI⊗P~i<0,i∈AT,Ξi−βiI⊗P~i<0,i∈UAT,P~i≺¯μP~j,i≠j,where
(34)Ξi≔symI⊗AP~i+I⊗BK1iP~i+EMi⊗BW2i,and the total dwell time of the unachievable topologies T+ satisfies
(35)T+0,t<lnc2λminP~i−1λminR−lnc1λmaxRλmaxP~i−1−Nlnμ−aTβ−a.
Then, the FTF can be achieved with respect toc1,c2,T,R,σt.
Furthermore, whenT+ can be fixed, the switching signal has an ADT satisfying
(36)τa≥τa∗=Tlnμlnc2λminP~i−1λminR−lnc1λmaxRλmaxP~i−1−aT−β−aT+0,t.
In addition, when the topology-dependent controller gainK1i is configured, K2i can be obtained by
(37)K2i=W2iP~i−1.Proof 2.
By lettingP~i=Pi−1 and W2i=K2iP~i and performing matrix congruent transformation, the conditions then follow directly from Theorem 1.Remark 3.
It is noted thatK1i is related with the motion modes of the formation center, such that K1i is generally separately designed in advance of K2i. When K1i is configured appropriately, then the desired K2i can be obtained by solving the conditions in Theorem 2 accordingly. For the large-scale UAVs, since the formation configuration ht is prescribed, the controllers of UAVs can be designed according to uit, where the information exchanges among the UAVs are also considered.
Note that if a common Lyapunov-like function can be found for (6), Corollary 1 can be derived.Corollary 1.
Consider the group of UAVs (1) with switching topologies, for giving constants αi≥0 and βi≥0, i∈S, if there exist matrices Pi≻0 and W2i, ∀i,j∈S×S, i≠j, such that
(38)Ξi−αI⊗P~i<0,i∈AT,Ξi−βI⊗P~i,i∈UAT,and the total dwell time of the unachievable topologies T+ satisfies
(39)T+0,t<lnc2λminP~i−1λminR−lnc1λmaxRλmaxP~i−1−aTβ−a.
Then, the UFTF can be achieved with respect toc1,c2,T,R.
In addition, when the topology-dependent controller gainK1i is configured, K2i can be obtained by
(40)K2i=W2iP~i−1.


## 3.2. Finite-Time Formation with Disturbances

When the external disturbances of UAVs are considered, the theorems of FTBF and UFTBF are presented.Theorem 3.
Consider the group of UAVs (1) with switching topologies and disturbances, for giving constants αi≥0, βi≥0, i∈S, μ>1, κ1i>0, κ2i>0, and the topology-dependent controller gains K1i and K2i, if there exist matrices Pi≻0, ∀i,j∈S×S, i≠j, such that
(41)Ψ1iI⊗PiB∗−κ1iI<0,i∈AT,(42)Ψ2iI⊗PiB∗−κ2iI<0,i∈UAT,(43)Pi≺¯μiPj,where
(44)Ψ1i≔symI⊗PiA+I⊗PiBK1i+EMi⊗PiBK2i−αiI⊗Pi,Ψ2i≔symI⊗PiA+I⊗PiBK1i+EMi⊗PiBK2i−βiI⊗Pi,and the total dwell time of the unachievable topologies T+ satisfies
(45)T+0,t<lnc2λminPiλminR−μNλmaxRλmaxPiλminRκeβTdwβ−a−Nlnμ−lnc1λmaxRλmaxPi−aTβ−a.
Then, the FTBF can be achieved with respect toc1,c2,T,R,σt,dw.
Furthermore, whenT+ can be fixed, the switching signal has an ADT satisfying
(46)τa>τa∗=Tlnμlnc2λminPiλminR−lnλmaxRλmaxPieaT−T+0,t+βT+0,tc1+λminRκeβTdw.Proof 3.
Select the multiple Lyapunov-like function as follows:(47)Vit=εTtI⊗Piεt.
Then, the LMIs (41) and (42) mean that
(48)V̇it≺αiVit+κ1iωTtωt,i∈AT,V̇it≺βiVit+κ2iωTtωt,i∈UAT.
Following same lines in the proof of Theorem1, for t∈tk,tk+1, it holds that
(49)Vt<eaσtkt−tkμVσtk−tk−+∫tkteaσtkt−sκ1σtkωTsωsds,σtk∈AT,Vt<eβσtt−tkμVσtk−tk−+∫tkteβσtkt−sκ2σtkωTsωsds,σtk∈UAT.
By lettingπσt=aσt, κσt=κ1σt, and σt∈AT and πσt=βσt, κσt=κ2σt, and σt∈UAT, one has
(50)Vt≤eπσtkt−tkμVσtk−tk−+∫tkteπσtkt−sκσtkωTsωsds≤eπσtkt−tkμeπσtk−1tk−tk−1μVσtk−1−tk−1−+∫tk−1teπσtk−1tk−sκσtk−1ωTsωsds+∫tkteπσtkt−sκσtkωTsωsds≤⋯≤μNeaT−0,t+βT+0,tVσt00+∫0teπσtt−sμNκσtωTsωsds≤μNeaT−T+0,t+βT+0,tVσt00+κeβTdw,where a=maxi∈Sai, β=maxi∈Sβi, and κ=maxi∈Sκi.
Consequently, it can be obtained that(51)εTtRεt≤λmaxRεTtεt≤λmaxRVσttλminPσt<λmaxRλminPσtμNeaT−T+0,t+βT+0,tVσt00+κeβTdw≤λmaxRλmaxPσtλminPσtμNeaT−T+0,t+βT+0,tεT0ε0+κeβTdw≤λmaxRλmaxPσtλminPσtλminRμNeaT−T+0,t+βT+0,tc1+λminRκeβTdw<c2,which means that
(52)T+0,t<lnc2λminPiλminR−μNλmaxRλmaxPiλminRκeβTdwβ−a−Nlnμ−lnc1λmaxRλmaxPi−aTβ−a.
In addition, the ADTτa can be obtained for a fixed T+0,t according to Definition 1, which completes the proof.Theorem 4.
Consider the group of UAVs (1) with switching topologies and disturbances, for giving constants αi≥0, βi≥0, i∈S, μ>1, κ1i>0, and κ2i>0, if there exist matrices P~i≻0 and W2i, ∀i,j∈S×S, i≠j, such that
(53)Φ1iI⊗B∗−κ1iI<0,i∈AT,Φ2iI⊗B∗−κ2iI<0,i∈UAT,P~i≺¯μiP~j,where
(54)Φ1i≔symI⊗AP~i+I⊗BK1iP~i+EMi⊗BW2i−αiI⊗AP~i,Φ2i≔symI⊗AP~i+I⊗BK1iP~i+EMi⊗BW2i−βiI⊗AP~i,and the total dwell time of the unachievable topologies T+ satisfies
(55)T+0,t<lnc2λminP~i−1λminR−μNλmaxRλmaxP~i−1λminRκeβTdwβ−a−Nlnμ−lnc1λmaxRλmaxP~i−1−aTβ−a.
Then, the FTBF can be achieved with respect toc1,c2,T,R,σt,dw.
Furthermore, whenT+ can be fixed, the switching signal has an ADT satisfying
(56)τa>τa∗=Tlnμlnc2λminP~i−1λminR−lnλmaxRλmaxP~i−1eaT−T+0,t+βT+0,tc1+λminRκeβTdw.
In addition, when the topology-dependent controller gainK1i is configured, K2i can be obtained by
(57)K2i=W2iP~i−1.Proof 4.
By applying the same matrix congruent transformation in Theorem2, the results can be directly obtained.Based on the above results, Algorithm1 can be given as follows.Algorithm 1: FTBF algorithm.
Input: A, B, Lσt
Output: K2σt
1: Given constantsασ(t) > 0, βσ(t) > 0, μ > 1, σt∈S, κ1i > 0, κ2i > 0;
2: Configure the formation centre byA + BK1σ(t) with desired eigenvalues;
3: ObtainingMσ(t) by Lσ(t) = Mσ(t)E;
4: Solving the LMIs in Theorem4;
5: Obtaining feasible solutions ofP~σt≻0 and W2σ(t);
6: Obtaining the total dwell time of the unachievable topologiesT+;
7:return K2σt=W2σtP~σt−1.Corollary 2.
Consider the group of UAVs (1) with switching topologies and disturbances, for giving constants αi≥0 and βi≥0, i∈S, if there exist matrices P~i≻0 and W2i, ∀i,j∈S×S, i≠j, such that
(58)Φ1iI⊗B∗−κ1iI<0,i∈AT,Φ2iI⊗B∗−κ2iI<0,i∈UAT,and the total dwell time of the unachievable topologies T+ satisfies
(59)T+0,t<lnc2λminP~i−1λminR−λmaxRλmaxP~i−1λminRκeβTdwβ−a−lnc1λmaxP~i−1−aTβ−a.
Then, the UFTBF can be achieved with respect toc1,c2,T,R,dw.
In addition, when the topology-dependent controller gainK1i is configured, K2i can be obtained by
(60)K2i=W2iP~i−1.


## 4. Illustrative Example

In the following, a numerical example is given to verify the effectiveness of our proposed formation design.Consider a group of four UAVs with the following switching communication topologies depicted in Figure1.Figure 1
The switching communication topologies of UAVs.
(a)
Topology 1
(b)
Topology 2
(c)
Topology 3The corresponding Laplacian matrices can be given as follows:(61)L1=100−100000−1100000,L2=0000−1100000000−11,L3=0000−11000−11000−11.It can be verified that the formation cannot be achieved since topology 1 and topology 2 do not have directed spanning trees, which implies that they are unachievable.Accordingly, it can be obtained that(62)M1=1110000−10000,M2=000−10000000−1,M3=000−1000−1000−1.The time-varying formation configuration is given ashit=3sint,0.6cost,ϱicost,3cost,−0.6sint,−sintT, ϱ1=1, ϱ2=0.8, ϱ3=0.6, and ϱ4=0.4, and the external disturbances are set by 0.005sint.The design parameters are chosen asα=0.1, β=0.5, μ=1.3, κ13=0.06, κ21=0.02, and κ22=0.03. For the FTBF, the parameter c1 is chosen as 0.5, the parameter c2 is set as 50, T is given by 10 s, and dw is supposed to be 0.0025.The eigenvalues ofI⊗A+BK are configured by (−1±i, −1±i, and −1±i), one has
(63)K1i=−200−2000−200−2000−200−2.Based on the established results of Theorem4, it can be obtained by solving the LMIs that
(64)K21=0.3424000.54930000.3424000.54930000.3424000.5493,K22=0.4996000.76770000.4996000.76770000.4996000.7677,K23=0.4310000.51330000.4310000.51330000.4310000.5133,and the total dwell time of the unachievable topologies T+ is calculated as 4.04 s.In the simulation, the initial conditions of the UAVs are given as0.1;0.2;0.3;0;0;0;−0.1;−0.2;0.3;0;0;0;0.1;−0.3;0.2;0;0;0;0.1;−0.2;0.1;0;0;0T and T+ is set as 2.5 s. Figures 2–4 show the desired time-varying formation trajectory, the open-loop FTBF trajectory, and the closed-loop FTBF trajectory, respectively. It can be seen that the designed formation controllers for each UAV can achieve the prescribed formation with unachievable topologies and external disturbances. Figure 5 depicts the formation errors of the UAVs with the finite-time interval, and Figure 6 indicates the corresponding FTBF conditions. It can be found that the formation errors are finite-time bounded with switching topologies according to Definition 4 with εT0Rε0=0.32 and εT10Rε10=0.002, which supports our theoretical analysis.Figure 2
State trajectories of desired time-varying formation.Figure 3
State trajectories of the open-loop FTBF trajectory.Figure 4
State trajectories of the closed-loop FTBF trajectory.Figure 5
State trajectories of formation errors.Figure 6
State trajectories ofεTtRεt.


## 5. Conclusion

In this paper, the formation problem of UAVs with switching topologies and disturbance is concerned with an ADT approach. In particular, the concept of FTBF is introduced to deal with the finite-time stability problem and disturbances. Moreover, both achievable and unachievable topologies are taken into account for a more practical background of communications among the UAVs. Based on model transformation, sufficient conditions are established for the FTBF and the desired topology-dependent controller can be designed accordingly. In the end, a numerical example is given to demonstrate the effectiveness of our proposed control method. Our future study will focus on the cases with model uncertainties and communication delays.



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