# Nonlinear Uncertainties Canceling in Multi-Agent Systems Enabled by Cooperative Adaptation

Authors: Cyuansi Shih; Chengyu Cao
Journal: Journal of Control Science and Engineering
 (2018)

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## Abstract

This paper deals with uncertainties problem in multi-agent systems with novel cooperative adaptation approach. Since uncertainties in multi-agent systems are interconnected, local agent often faces uncertainties not only from itself but also from neighbors. The proposed approach is that a local agent estimates uncertainties from itself and neighboring agents and then changes control strategy. The uncertainties or the equivalences of neighbors can be estimated based on their available outputs; thus, the local agent can adapt to them to cancel out these effects. Stability analysis is also derived that characterizes the transient and steady state performance of multi-agent system. The simulation presents the details of the proposed cooperative adaptation mechanism by compared typical cooperative control.


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## Body

## 1. Introduction

The cooperative control in multi-agent system has attracted compelling attention in recent years from various applications including scheduling satellite formation, cooperative unmanned air vehicles, connected autonomous vehicles, and multiple underwater vehicle coordination. In these applications, the agents are required to collaborate with each other in order to finish cooperative mission. Consensus in multi-agent systems is a typical example that agents have common interest and then make negotiations to reach consensus by appropriate algorithmic methods. Consensus problems with various dynamic systems have already been successfully addressed in literature. The consensus in multi-agent systems with single-integrator and double-integrator kinematics is studied in [1–3]. The general linear dynamics of consensus in multi-agent systems is also discussed in [4–10]. In [11–16], the authors further extended to nonlinear dynamics for multi-agent systems.In practical applications, the models of agents are difficult to obtain because of sensing or measurement limitation. Also, disturbances often exist in interconnected network systems, such as channel noise and communication noise. Thus, it is necessary to consider uncertainty effects in multi-agent systems. The typical direct and indirect adaptive control [17] are two nonlinear schemes explicitly designed for signal system uncertainties. In [18], the model reference adaptive control approach is used to ensure asymptotic tracking of a defined reference model for a class of multi-agent systems with constant unknown parameters. Utilizing the universal approximation capability of artificial neural network [19], adaptive control could deal with nonlinear functions instead of linear parameters in multi-agent systems, and this leads to the neural network controller design. Consensus problem of second-order systems with Lipschitz nonlinearity is addressed [16]. The uncertainties of nonlinear multi-agent systems under jointly connected directed switching networks are also addressed in [20, 21]. The results in [9, 18] deal with high-order system with Lipschitz nonlinearity. In these literatures, the results are restricted to local stability and certain type of network structure.This paper proposed a novel cooperative adaptation approach to deal with uncertainties issue of network consensus. The cooperative law drives agents to collaborate and then cooperative adaptation law eliminates uncertainties from network. The idea is that the local agent adjusts his own control strategies based on the uncertainty estimation of itself and neighbors. The uncertainties from other agents can be treated as external disturbances and are not available to the local agent. Thus, other agents uncertainties can be estimated based on their available output and then local agent cancels out these uncertainties by adapting to them. This paper also provided transient and steady state performance analysis in the presence of uncertainties for multi-agent systems.The article is organized as follows. The problem formulation is given in Section2. The adaptive cooperation control scheme is introduced in Section 3. In Section 4, the stability analysis and performance bounds for multi-agent systems are presented in Section 4. The simulation results are presented in Section 5, while Section 6 concludes this paper.


## 2. Problem Formulation

There areN agents moving in m dimensional Euclidean space, where the dynamics of each agent is described by(1)x˙it=Axit+Buit+σixi,t where xi∈Rm is the state vector, such as position and velocity vector of agent i,  ui∈Rm are the control input, and σi:Rm×Rm×R↦Rm represents the unknown, nonlinear, and time-varying uncertainties.The objective is to design decentralized control inputs that ensure multi-agent systems with uncertainties achieve consensus. Consensus in this paper means all agents converge to common attitude or speed. Each agent is subjected to individual uncertainties, such as dynamics uncertainty, affected from neighborhood agents. The decentralized control input plays the role not only cooperating but also canceling uncertainty among agents. The network connectivity has a fairly general assumption that creates a spanning tree in the network connection. The uncertainties are restricted to being Lipschitz and allow less restriction on structure of nonlinearity.The connection among agents are specified by a graphG=(V,E) which consists of a set of vertices denoted by V={1,2,…,N} and a set of edges denoted by E⊆{(i,j):i,j∈V,i≠j}. A vertex represents an agent, and each edge represents a connection. The adjacency matrix A with elements aij from the graph denotes the connections, where aij=1 means a path from agent j to agent i; otherwise aij=0. The Laplacian matrix L is defined as lii=∑j=1Naij and lij=-aij when i≠j. M is defined as the normalized Laplacian matrix, and thus M=IN-1rT. The rank of M is N-1, and M has N-1 eigenvalues at 1 and one eigenvalue at 0. The connection graph is undirected if aij=aji and thus the adjacency matrix is symmetric; otherwise the graph is directed. We consider the undirected graph among agents communication in this paper.Assumption 1.
The uncertainties depended on agent’s state and its changes. For local agentxi>0, there exist L>0,  β0>0, and β>0 such that (2)σiξit1-σiξit2<Lξit1-ξit2∞+β0where ξi=xi-∑j∈Nirjxj and rj is normalized edge of graph. For overall network, we have (3)σξt1-σξt2<LMξt1-ξt2∞+βAssumption 2.
The nonlinear uncertaintiesσi(ξi,t) are bounded with respect to state ξi. This means the uncertainties do not become unbounded even system state keep changing. The partial derivatives of σi(ξi,t) with respect to ξi and t are bounded, i.e., ∂σi(ξi,t)/∂ξi≤dσiξi and ∂σi(ξi,t)/∂t≤dσit. We also further have ∂σ(ξ,t)/∂ξ≤dσξ and ∂σ(ξ,t)/∂t≤dσt.Assumption 3.
The undirected graphG is connected and thus creates a spanning tree in the graph. The Laplacian matrix L has a single eigenvalue at 0 and all other eigenvalues are with positive real part.


## 3. Uncertainties Canceling Algorithm

In this section, we introduce a novel cooperative adaptation law to drive multi-agent systems as (1) achieve global objective, xi toward neighbor agents xj. The algorithm first deals with the agent’s collaboration by the cooperative law that is based on local neighbors’ relative information. The algorithm further handles uncertainties issue among network through cooperative adaptation law. The overall control law is represented as below:(4)ui=uic+uiawhere uic is cooperative control law and uia is cooperative adaptation law.First, we introduced the state transformation to describe the relation between individual agent and network as follows:(5)ξi=xi-∑j∈Nrjxj The network dynamics between local agent and neighbors is described as(6)ξ˙i=Aξi+Buic-∑j∈Nrjujc+Buia-∑j∈Nrjuja+Bσi-∑j∈NrjσjThe overall network transformation follows(7)ξ=x-1rT⊗Inx=M⊗BInxCooperative Law. In order to achieve the network consensus, the cooperative control law design is moving to the average state of neighbors. In other words, it uses neighbors’ relative information as feedback and is presented as(8)uic=-kxi-∑j∈Nrjxj where the selection of k will make sure to stabilize the overall network system that is described as(9)ξ˙=IN⊗A-L⊗Bkξ+M⊗Bua+σwhere ξ=ξ1…ξn,  ua=u1…un,  σ=σ1…σn,  Am=(IN⊗A-L⊗Bk), and Bm=M⊗B1n. Thus, Am is Hurwitz matrix by k selection.Cooperative Adaptation Law. We further design the decentralized cooperative adaptation law to deal with uncertainty issue among network. This disturbance rejection includes state predictor, adaptive law, and control as below. Since uncertainty for dynamical systems is made in real time, all uncertainties from network can be estimated based on their available output. Hence, a local agent can adjust its own control law based on the estimation of network uncertainty. Thus, the local agent estimates these uncertainties through transferred dynamics. With network dynamics from (9), we consider the following state predictor:(10)ξ^˙=Amξ^+Bmua+σ^where ξ^∈Rm is the estimated state vector of ξ in (9) and σ^ is the estimated uncertainty of σ. These parameters vectors are updated by adaptive law below.Fast adaptation mechanism is applied in here as a high-gain feedback for state predictor. It is noted that both state predictor and adaptation work together and there is no time-delay or at most one integration time-step in the loop. Here, the predicted state,ξ^i, is used to generate the adaptive parameter, σ^i, based on the network error dynamics. The piece-wise constant adaptive law is used in here. In order to drive ξ^ to track ξ^. The update law is given by(11)σ^t=-Φ-1TeAmTξ~kT,t∈kT,k+1T where Φ(T)=∫0TeAm(T-τ)dτ,  T>0 is the adaptation step size, and ξ~=ξ^-ξ. The adaptive law drives ξ~ to be small and as a result, σ^ will approach σ. The information of σ^ will be used in the control law design. The error dynamics for network system is described as(12)ξ~˙=Amξ~+Bmσ^-σTo prevent aggressive signals in real application, a low-pass filtering mechanism is added in control law, which filters away aggressive signals and recovers robustness. The adaptive control law design is(13)uat=-Csσ^twhere C(s) is low-pass filter to filter the control signal. The introduction of low-pass filter at this point in the control loop allows us to decouple control and estimation by canceling the overall network uncertainty. Furthermore, the bandwidth of filter can be chosen to attenuate frequencies based on the variation of uncertainties. The closed-loop network system from (1) and (13) becomes(14)ξ˙=Amξ-CsBmσ^+BmσRemark 4.
The objective of cooperative adaptation law is to make the adaptive parameterσ^(t) approach the network disturbance σ(t). The effects of the control law ua(t) cancel the effects of σ(t) and as a result ξ^(t) in (12) will become very small. Thus, the agents can achieve consensus without uncertainties.


## 4. Network Stability Analysis

The analysis first verifies the network stability by introducing a reference network system. The reference system will asymptotically track desired system when the bandwidth of low-pass filter is large enough. The network convergence analysis also proved the stability conditions. Further, the real network system is proven to track the reference system and its stability condition is provided. Finally, the estimation errors between real network and predict network system are shown to be bounded and the uncertainties canceling is achieved by decreasing the size of time-step.


### 4.1. Nonadaptive Reference System

We consider a reference network dynamics for (9) as(15)ξ˙r=Amξr+Bmura+σrwhere ξr=ξr1,…,ξrnT,ξri=xri-∑j∈Nirjxrj,σr=σr1,…,σrnT and the reference control law(16)ura=-CsσrThe closed-loop reference network system from (15) and (16) becomes(17)ξ˙r=Amξr+1-CsBmσrLemma 5.
For the closed-loop network reference system in (15) and (16) subject to the L1 gain upper bound(18)Θr1β+ξr0∞1-Θr1Lρr+γξ<ρr(19)Θr1Lρr+γξ<1where Θr1=(sIN-Am)-1(IN-C(s)IN)BmL1.(20)ξr0<ρrThen(21)ξr∞<ρr(22)ura∞<ρurwhere ρr is introduced in (18) and (19)(23)ρur=CsMLρrρr+βProof.
Becauseξr0∞<ρr and ξr(t) is continuous, to prove ξr∞<ρr, assuming the opposite is true, it implies there exists a time t′ such that(24)ξrt′=ρr,ξrt′∞≤ρrTake Laplace transform for (17); we have(25)ξrs=1-CssIN-Am-1Bmσrs+ξr0and its upper bound has(26)ξrt′∞≤1-CssIN-Am-1BmL1σrs∞+ξr0∞Using Assumption 1 and upper bound in (24), we arrived at(27)σrt′∞≤Lρrξrt′∞+βSubstituting (27) into (26), we obtain(28)ξrt′∞≤Θr1β+ξr0∞1-Θr1Lρrwhere Θr1=(sIN-Am)-1(IN-C(s)IN)BmL1. The stability condition in (18), together with (28), implies that ξrt′∞<ρr, which clearly contradicts assumption in (24) since there is no t′ to make assumption true. This proves (21). From (16), the network control law is(29)urat′=-CsMσrIt follows from (29), (21), and (27) that we have(30)urat′∞<CsML1Lρrρr+β<ρurwhich proves (22) and concludes the proof.Lemma 6.
Consider the reference system in (17). If the bandwidth of low-pass filter C(s) is large enough, the control law in (16) can make ξr asymptotically track the desired network system described by(31)ξ˙d=Amξdwhere ξd=ξd1,…,ξdn and ξdi=xdi-∑j∈Nirjxdj are desired network state, Am=IN⊗A-L⊗Bk.Proof.
For a given network, if the bandwidth of low-pass filterC(s)=a/s+a can be chosen large enough, the disturbance effect in (17) will be canceled and (17) become as(32)ξ˙r=AmξrThis further implies(33)ξrs=sIN-Am-1ξr0We further obtain that(34)lims→0ξrs=lims→0ξdsThus, the reference system will track the desired network system, which concludes the proof.


### 4.2. Network Convergence Analysis

Since networkG is connected, it follows from Assumption 3 that 0 is a simple eigenvalue of L and the other eigenvalues have positive real part. Thus, there exists a nonsingular matrix T such that T-1LT is in the Jordan canonical form [9]. Because the left and right eigenvectors of L corresponding to zero eigenvalue are r and 1, respectively, the following can be chosen:(35)T=1U,T-1=rTVwhere U∈CN×(N-1) and V∈CN×(N-1) such that(36)T-1LT=J=000Δwhere Δ∈CN×(N-1) is an upper-triangular matrix, where the diagonal entries are the nonzero eigenvalues of L. Let ζd=(T-1⊗In)ξd. Then, (31) can be presented as(37)ζ˙d=IN⊗A-J⊗BkζdLemma 7.
Suppose Assumption3 holds. The distributed control law in (16) with agents in (15) solves consensus problem if and only if all the matrices A-λiB,i=2,…,N, are Hurwitz, where λi are the nonzero eigenvalues of Laplacian matrix L.Proof.
The elements of the state matrix in (37) are either block diagonal or block upper-triangular. Thus, ξi,i=2,…,N, converge asymptotically to zero if and only if the N-1 subsystems along the diagonal as(38)ζ˙di=A-λiBζdi,i=2,…,Nare asymptotically stable. The stability of closed-loop system in (32) and (31) is equal to the system in (37) that converges asymptotically to zero. The system consensus is achieved. This concludes the proof.Lemma 8.
Suppose Assumption3 holds and the cooperative control law (8) satisfies Lemma 7; then(39)limt→∞xdi=rT⊗eAtrxd10⋮xdN0where r∈RN such that rTL=0 and rT1=1.Proof.
The desired network dynamics (31) can transform as(40)x˙d=IN⊗A-L⊗Bkxd where ξd=Lxd. The solution of (40) with Lemma 7 can be obtained as(41)xdt=eIN⊗A-L⊗Bktxd0=T⊗IneIN⊗A-J⊗BkT-1⊗Inxd0=T⊗IneAt00eIN-1⊗A-Δ⊗BktT-1⊗Inxd0From Lemma 7, IN⊗A-Δ⊗Bk is Hurwitz. Thus,(42)limt→∞eIN⊗A-L⊗Bkt=1⊗IneAtrT⊗In=1rT⊗eAtThus, the final consensus value is(43)limt→∞xdt=1rT⊗eAtxd0This concludes the proof.


### 4.3. Network Transient Performance

We discussed the relation between reference network system and desired network system and then perform the network convergence analysis in last section. We further prove the real network system can track reference network system in this section. Before this proof, we will establish the error bounds between the network state predictor and real network system and between the adaptive law and the signal it estimates.Lemma 9.
Given the network system in (9) with L1 adaptive control law in (13) and the reference system in (15) with (16) subject to stability conditions in (18) and (19), if ξr0<ρr and we choose T in (11) and low-pass filter C(s) in (13) to ensure(44)γ0T<γξ~(45)γξeT<γξwhere γξ~ is arbitrary positive constant and γξ is introduced in (19); then we have(46)ξ~∞<γξ~(47)ξ-ξr∞<γξ(48)u-ura∞<γuSuppose there exist upper bounds for the truncated L∞ norms of ξ(t) and σ(t)-σ^(t) at some time, t1(49)ξt1∞<ρDefine(50)β1=λmaxAmTAm(51)β2=∫0TλmaxΦT-τTΦT-τBmdτΦ-1TeAT(52)β3=∫0TλmaxΦT-τTΦT-τBmdτProof.
Becauseξ(t),ξ~(t) is continuous, to prove ξ~∞<γξ~, assume that the opposite is true assuming ξ~∞≥γξ~. This implies that there exists a time t′ such that(53)ξ~t′∞=γξ~(54)ξt′-ξrt′∞=γξ,or(55)uat′-urat′=γu,orThe solution of network error dynamics (12) over the interval jT,(j+1)T is described as(56)ξ~t=ΦTξ~jT+∫jTtΦt-τBmσ^tdτ-∫jTtΦt-τBmσtdτ,j=0,1,2,⋯where Φ(T)=eAmT,  (j+1)T<t′ and τ is a dummy variable. At t=(j+1)T, it becomes(57)ξ~j+1T=ΦTξ~jT+∫0TΦT-τBmσ^jTdτ-∫0TΦT-τBmσjT+τdτWe can choose σ^ such that its effects on the system derive x~ to zero at the next time-step. Then, we have(58)ξ~j+1T=-∫0TΦT-τBmσjT+τdτIt follows from Assumption 1, Lemma 5, and (54) that(59)σt∞≤Lρr+γξ+β≤bσMoreover, following Lemma 5 and (55), we have(60)uat∞≤ρur+γu≤buThe upper bound in (59) and (60) allows for the following upper bound:(61)ξ~j+1T∞≤β3bσ≤κTfor all j such that (j+1)T<t′. From (56), we arrived at the following upper bound:(62)ξ~jT+t∞≤β1κ+β2κ+β3bσ≤γ0Twhere β1, β2, and β3 are defined in (50), (51), and (52), respectively. Then, for all t∈[0,t′], following from (44), we have(63)ξ~t≤γ0T<γξ~The results clearly contradict assumption in (53) since there is no t′ to make assumption true. This proves (46).
Next, we will drive the bounds for the estimation error of adaptive law,σ~=σ^-σ, which are used to further obtain the bounds of ξ-ξr. We derive the bound for σ˙(x,t) first and then get(64)σ˙t=∂σ∂ξTξ˙tFrom (14), we have(65)ξ˙=Amξ-CsBmσ^+BmσTo establish the bound for this signal, we need to derive a bound for ξ(s). Using (11) and (53) for all t<t1, then we have(66)σ^t1∞≤Φ-1TeAmTγξ~≤bσ^Using (64), we can now write the(67)σ˙t∞≤LAmL1ρr+γξ+CsBmL1bσ^+BmL1Lρr+γξ+β≤bdσThen, control signal upper bound(68)u˙∞≤sCsM⊗1nbσ^≤bduNote that the adaptive law of (11) is designed such that(69)ΦTξ~jT+∫0TΦT-τdτBmσ^jT=0For the time t∈[(j-1)T,jT], we also have(70)ξ~jT=-∫0TΦT-τBmσj-1T+τdτ Following from the definition of adaptive law, we obtain(71)ξ~jT=-∫0TΦT-τAmξ~jTdτ-∫0TΦT-τBmσ^jTdτThus, (70) and (71) give us(72)∫0TΦT-τBmσj-1T+τdτ=∫0TΦT-τAmξ~jT+Bmσ^jTdτwhich imply there exist a time, tp∈[(j-1)T,jT], such that(73)Bmσ^jT+Amξ~jT=BmσtpThis imply that for any t∈[jT,(j+1)T](74)Bmσ^t-σt≤Bmσ^jT-σtp+σt-σtp≤β1ξ~jT+∫tptσ˙τdτ≤β1γ0+2bdσT≤bσ~Let σ~=σ^-σ; then we have(75)Bmσ~∞≤bσ~T,ρb,ρξ Next, we derive the bounds of ua-ura. From network control law in (8) and (13), we have(76)uas=-Csσ^ From (29), we have(77)uras=-Csσr Thus, apply (76) and (77), we arrived(78)uas-uras=-Csσ-σr-Csσ~ where σ^=σ+σ~. Let ξe(t)=ξ(t)-ξr(t); from Assumption 1 and (45), we can arrive at the following upper bound:(79)σ-σr∞≤Lξe∞+β
Now, we derive the bounds ofξ-ξr and then we have(80)ξ˙et=Amξe+Bm1-Csσ-σr-Csσ~The solution of (80) is(81)ξes=sIN-Am-1Bm1-Csσ-σr-Csσ~Substituting (79) and (55) to (81), we further get(82)γξe=Θe1β+Θe2bσ~1-Θe1Lρr+γξand(83)ξet∞≤Θe1Lξe∞+β+Θe2bσ~≤γξe
whereΘe1=(sIN-Am)-1Bm(1-C(s))L1  Θe2=(sIN-Am)-1C(s).(84)ξet∞≤γξe≤γξ
Substituting (75) and (79) to (78), the upper bound is(85)uas-uras≤CsLγξe+β+bσ~<γu
The stability condition in (18), together with (84), implies that ξ-ξr∞<γξ, which clearly contradicts assumption in (55) since there is no t′ to make assumption true. This proves (47). If we choose T=0 and the bandwidth of low-pass filters C(s) large enough, (84) with the condition become ξet∞<γξ. Thus, it always exists γξ to satisfy (84). This completes the proof of Lemma 9.Lemma 10.
The error bounds from Lemma9 depend on the time-step T. When T approaches 0, the error approaches 0 as well.(86)limT→0γξ~T,ρb→0(87)limT→0γξT,ρb→0(88)limT→0γuT,ρb→0Proof.
The inside of integration in (61) is a bounded function; so when T→0, then γξ~→0. Since (61) holds, the first term in (74) goes to 0 as T→0 and the second term clearly goes to 0 as well. Thus, γξ→0 when T→0. The last statement is γu. Because (74) holds, we can clearly see from (48) that when T→0,  γu→0. This completes the proof.


## 4.1. Nonadaptive Reference System

We consider a reference network dynamics for (9) as(15)ξ˙r=Amξr+Bmura+σrwhere ξr=ξr1,…,ξrnT,ξri=xri-∑j∈Nirjxrj,σr=σr1,…,σrnT and the reference control law(16)ura=-CsσrThe closed-loop reference network system from (15) and (16) becomes(17)ξ˙r=Amξr+1-CsBmσrLemma 5.
For the closed-loop network reference system in (15) and (16) subject to the L1 gain upper bound(18)Θr1β+ξr0∞1-Θr1Lρr+γξ<ρr(19)Θr1Lρr+γξ<1where Θr1=(sIN-Am)-1(IN-C(s)IN)BmL1.(20)ξr0<ρrThen(21)ξr∞<ρr(22)ura∞<ρurwhere ρr is introduced in (18) and (19)(23)ρur=CsMLρrρr+βProof.
Becauseξr0∞<ρr and ξr(t) is continuous, to prove ξr∞<ρr, assuming the opposite is true, it implies there exists a time t′ such that(24)ξrt′=ρr,ξrt′∞≤ρrTake Laplace transform for (17); we have(25)ξrs=1-CssIN-Am-1Bmσrs+ξr0and its upper bound has(26)ξrt′∞≤1-CssIN-Am-1BmL1σrs∞+ξr0∞Using Assumption 1 and upper bound in (24), we arrived at(27)σrt′∞≤Lρrξrt′∞+βSubstituting (27) into (26), we obtain(28)ξrt′∞≤Θr1β+ξr0∞1-Θr1Lρrwhere Θr1=(sIN-Am)-1(IN-C(s)IN)BmL1. The stability condition in (18), together with (28), implies that ξrt′∞<ρr, which clearly contradicts assumption in (24) since there is no t′ to make assumption true. This proves (21). From (16), the network control law is(29)urat′=-CsMσrIt follows from (29), (21), and (27) that we have(30)urat′∞<CsML1Lρrρr+β<ρurwhich proves (22) and concludes the proof.Lemma 6.
Consider the reference system in (17). If the bandwidth of low-pass filter C(s) is large enough, the control law in (16) can make ξr asymptotically track the desired network system described by(31)ξ˙d=Amξdwhere ξd=ξd1,…,ξdn and ξdi=xdi-∑j∈Nirjxdj are desired network state, Am=IN⊗A-L⊗Bk.Proof.
For a given network, if the bandwidth of low-pass filterC(s)=a/s+a can be chosen large enough, the disturbance effect in (17) will be canceled and (17) become as(32)ξ˙r=AmξrThis further implies(33)ξrs=sIN-Am-1ξr0We further obtain that(34)lims→0ξrs=lims→0ξdsThus, the reference system will track the desired network system, which concludes the proof.


## 4.2. Network Convergence Analysis

Since networkG is connected, it follows from Assumption 3 that 0 is a simple eigenvalue of L and the other eigenvalues have positive real part. Thus, there exists a nonsingular matrix T such that T-1LT is in the Jordan canonical form [9]. Because the left and right eigenvectors of L corresponding to zero eigenvalue are r and 1, respectively, the following can be chosen:(35)T=1U,T-1=rTVwhere U∈CN×(N-1) and V∈CN×(N-1) such that(36)T-1LT=J=000Δwhere Δ∈CN×(N-1) is an upper-triangular matrix, where the diagonal entries are the nonzero eigenvalues of L. Let ζd=(T-1⊗In)ξd. Then, (31) can be presented as(37)ζ˙d=IN⊗A-J⊗BkζdLemma 7.
Suppose Assumption3 holds. The distributed control law in (16) with agents in (15) solves consensus problem if and only if all the matrices A-λiB,i=2,…,N, are Hurwitz, where λi are the nonzero eigenvalues of Laplacian matrix L.Proof.
The elements of the state matrix in (37) are either block diagonal or block upper-triangular. Thus, ξi,i=2,…,N, converge asymptotically to zero if and only if the N-1 subsystems along the diagonal as(38)ζ˙di=A-λiBζdi,i=2,…,Nare asymptotically stable. The stability of closed-loop system in (32) and (31) is equal to the system in (37) that converges asymptotically to zero. The system consensus is achieved. This concludes the proof.Lemma 8.
Suppose Assumption3 holds and the cooperative control law (8) satisfies Lemma 7; then(39)limt→∞xdi=rT⊗eAtrxd10⋮xdN0where r∈RN such that rTL=0 and rT1=1.Proof.
The desired network dynamics (31) can transform as(40)x˙d=IN⊗A-L⊗Bkxd where ξd=Lxd. The solution of (40) with Lemma 7 can be obtained as(41)xdt=eIN⊗A-L⊗Bktxd0=T⊗IneIN⊗A-J⊗BkT-1⊗Inxd0=T⊗IneAt00eIN-1⊗A-Δ⊗BktT-1⊗Inxd0From Lemma 7, IN⊗A-Δ⊗Bk is Hurwitz. Thus,(42)limt→∞eIN⊗A-L⊗Bkt=1⊗IneAtrT⊗In=1rT⊗eAtThus, the final consensus value is(43)limt→∞xdt=1rT⊗eAtxd0This concludes the proof.


## 4.3. Network Transient Performance

We discussed the relation between reference network system and desired network system and then perform the network convergence analysis in last section. We further prove the real network system can track reference network system in this section. Before this proof, we will establish the error bounds between the network state predictor and real network system and between the adaptive law and the signal it estimates.Lemma 9.
Given the network system in (9) with L1 adaptive control law in (13) and the reference system in (15) with (16) subject to stability conditions in (18) and (19), if ξr0<ρr and we choose T in (11) and low-pass filter C(s) in (13) to ensure(44)γ0T<γξ~(45)γξeT<γξwhere γξ~ is arbitrary positive constant and γξ is introduced in (19); then we have(46)ξ~∞<γξ~(47)ξ-ξr∞<γξ(48)u-ura∞<γuSuppose there exist upper bounds for the truncated L∞ norms of ξ(t) and σ(t)-σ^(t) at some time, t1(49)ξt1∞<ρDefine(50)β1=λmaxAmTAm(51)β2=∫0TλmaxΦT-τTΦT-τBmdτΦ-1TeAT(52)β3=∫0TλmaxΦT-τTΦT-τBmdτProof.
Becauseξ(t),ξ~(t) is continuous, to prove ξ~∞<γξ~, assume that the opposite is true assuming ξ~∞≥γξ~. This implies that there exists a time t′ such that(53)ξ~t′∞=γξ~(54)ξt′-ξrt′∞=γξ,or(55)uat′-urat′=γu,orThe solution of network error dynamics (12) over the interval jT,(j+1)T is described as(56)ξ~t=ΦTξ~jT+∫jTtΦt-τBmσ^tdτ-∫jTtΦt-τBmσtdτ,j=0,1,2,⋯where Φ(T)=eAmT,  (j+1)T<t′ and τ is a dummy variable. At t=(j+1)T, it becomes(57)ξ~j+1T=ΦTξ~jT+∫0TΦT-τBmσ^jTdτ-∫0TΦT-τBmσjT+τdτWe can choose σ^ such that its effects on the system derive x~ to zero at the next time-step. Then, we have(58)ξ~j+1T=-∫0TΦT-τBmσjT+τdτIt follows from Assumption 1, Lemma 5, and (54) that(59)σt∞≤Lρr+γξ+β≤bσMoreover, following Lemma 5 and (55), we have(60)uat∞≤ρur+γu≤buThe upper bound in (59) and (60) allows for the following upper bound:(61)ξ~j+1T∞≤β3bσ≤κTfor all j such that (j+1)T<t′. From (56), we arrived at the following upper bound:(62)ξ~jT+t∞≤β1κ+β2κ+β3bσ≤γ0Twhere β1, β2, and β3 are defined in (50), (51), and (52), respectively. Then, for all t∈[0,t′], following from (44), we have(63)ξ~t≤γ0T<γξ~The results clearly contradict assumption in (53) since there is no t′ to make assumption true. This proves (46).
Next, we will drive the bounds for the estimation error of adaptive law,σ~=σ^-σ, which are used to further obtain the bounds of ξ-ξr. We derive the bound for σ˙(x,t) first and then get(64)σ˙t=∂σ∂ξTξ˙tFrom (14), we have(65)ξ˙=Amξ-CsBmσ^+BmσTo establish the bound for this signal, we need to derive a bound for ξ(s). Using (11) and (53) for all t<t1, then we have(66)σ^t1∞≤Φ-1TeAmTγξ~≤bσ^Using (64), we can now write the(67)σ˙t∞≤LAmL1ρr+γξ+CsBmL1bσ^+BmL1Lρr+γξ+β≤bdσThen, control signal upper bound(68)u˙∞≤sCsM⊗1nbσ^≤bduNote that the adaptive law of (11) is designed such that(69)ΦTξ~jT+∫0TΦT-τdτBmσ^jT=0For the time t∈[(j-1)T,jT], we also have(70)ξ~jT=-∫0TΦT-τBmσj-1T+τdτ Following from the definition of adaptive law, we obtain(71)ξ~jT=-∫0TΦT-τAmξ~jTdτ-∫0TΦT-τBmσ^jTdτThus, (70) and (71) give us(72)∫0TΦT-τBmσj-1T+τdτ=∫0TΦT-τAmξ~jT+Bmσ^jTdτwhich imply there exist a time, tp∈[(j-1)T,jT], such that(73)Bmσ^jT+Amξ~jT=BmσtpThis imply that for any t∈[jT,(j+1)T](74)Bmσ^t-σt≤Bmσ^jT-σtp+σt-σtp≤β1ξ~jT+∫tptσ˙τdτ≤β1γ0+2bdσT≤bσ~Let σ~=σ^-σ; then we have(75)Bmσ~∞≤bσ~T,ρb,ρξ Next, we derive the bounds of ua-ura. From network control law in (8) and (13), we have(76)uas=-Csσ^ From (29), we have(77)uras=-Csσr Thus, apply (76) and (77), we arrived(78)uas-uras=-Csσ-σr-Csσ~ where σ^=σ+σ~. Let ξe(t)=ξ(t)-ξr(t); from Assumption 1 and (45), we can arrive at the following upper bound:(79)σ-σr∞≤Lξe∞+β
Now, we derive the bounds ofξ-ξr and then we have(80)ξ˙et=Amξe+Bm1-Csσ-σr-Csσ~The solution of (80) is(81)ξes=sIN-Am-1Bm1-Csσ-σr-Csσ~Substituting (79) and (55) to (81), we further get(82)γξe=Θe1β+Θe2bσ~1-Θe1Lρr+γξand(83)ξet∞≤Θe1Lξe∞+β+Θe2bσ~≤γξe
whereΘe1=(sIN-Am)-1Bm(1-C(s))L1  Θe2=(sIN-Am)-1C(s).(84)ξet∞≤γξe≤γξ
Substituting (75) and (79) to (78), the upper bound is(85)uas-uras≤CsLγξe+β+bσ~<γu
The stability condition in (18), together with (84), implies that ξ-ξr∞<γξ, which clearly contradicts assumption in (55) since there is no t′ to make assumption true. This proves (47). If we choose T=0 and the bandwidth of low-pass filters C(s) large enough, (84) with the condition become ξet∞<γξ. Thus, it always exists γξ to satisfy (84). This completes the proof of Lemma 9.Lemma 10.
The error bounds from Lemma9 depend on the time-step T. When T approaches 0, the error approaches 0 as well.(86)limT→0γξ~T,ρb→0(87)limT→0γξT,ρb→0(88)limT→0γuT,ρb→0Proof.
The inside of integration in (61) is a bounded function; so when T→0, then γξ~→0. Since (61) holds, the first term in (74) goes to 0 as T→0 and the second term clearly goes to 0 as well. Thus, γξ→0 when T→0. The last statement is γu. Because (74) holds, we can clearly see from (48) that when T→0,  γu→0. This completes the proof.


## 5. Simulation

The simulation will show the details of proposed adaptive cooperation algorithm by compared with typical cooperative control. The network systems are included uncertainties and disturbances in these examples. The agent dynamics under consideration is a vertical motion of a quadrotor. There are 10 quadrotors that try to maintain in the specified altitude and velocity for cooperation observation in simulation scenarios. The dynamics described by a second-order state space as(89)x˙i=0100xi+01mui+σiwhere xi=ziwiT are vertical position and velocity, m=1kg is the mass of quadrotor, ui is the thrust generated by rotors, and σi(xi,t) is the force produced by wind on quadrotor.The first example shows the cooperative control algorithm from (8). The integration step in simulation is set as T=10-2 seconds; the initial altitude and velocity x(0)i are chosen randomly within the range 0~10m and -1~1m/s individually. The control gain is chosen as k=11. The communication links are considered as undirected ring network topology as Figure 1. The Laplacian matrix is given by(90)L=2-100-1-1⋱000⋱⋱⋱⋱⋱⋱000⋱2-1-100-12.The eigenvalues of L only have one zero eigenvalues and other eigenvalues are positive that means Assumption 3 is satisfied. The uncertainties considered in agent dynamics are globally Lipschitz as(91)σi=c∗e-xisin⁡xi+cos⁡xiAlthough the cooperative control tries to drive quadrotors to common position, the uncertainties affect the effectiveness of algorithm (8). The cooperative control signals from Figure 3 show the fluctuation and diverge gradually. Thus, the consensus from Figure 2 cannot be achieved by cooperative control (8) for the group of quadrotors.Figure 1
Undirected ring network.Figure 2
The trajectory of quadrotors.Figure 3
Cooperative control input.The second example shows the proposed adaptive cooperation algorithm from (8) and (13). The integration step in simulation is set as T=10-2 seconds. The initial altitude and velocity x(0)i are chosen randomly within the range 0~10m and -1~1m/s individually. The filter of adaptive cooperation controller for agents is chosen as C(s)=10/s+10. The communication links are also considered as undirected ring network topology as Figure 1.Figure4 shows the consensus achieved with the same position and velocity for the group of quadrotors, and the uncertainties in network are also canceled in Figure 6. Figure 5 shows the adaptive cooperation control signal of quadrotors. Although network systems create uncertainties, the transient performance is still maintained well by (8) and (13). The simulation results with Figures 4 and 5 demonstrate Lemmas 6–8. Figure 6 shows the mismatched dynamics that means the state predictor tracks the real state well and adapts to uncertainty quickly. The results in Figure 6 thus demonstrate Lemma 10. Further, the results in Figures 4–6 also show the robustness of proposed adaptive cooperation control design.Figure 4
The trajectory of quadrotors.Figure 5
Adaptive cooperation control input.Figure 6
Estimation error of states.


## 6. Conclusion

This paper presents a novel cooperative adaptation approach to deal with uncertainty of multi-agent systems. The typical cooperative control was used to achieve collaboration in multi-agent systems but it also causes interconnected uncertainties among network. Then, the cooperative adaptation approach was proposed to cancel uncertainties in multi-agent systems. The cooperative adaptation algorithm features as fast adaptation scheme and can be treated as high-gain approach. To prevent aggressive changing in real control signal, a low-pass filtering mechanism is added in control law, which filters away aggressive signals and recovers robustness. The stability condition and tracking performance for closed-loop network system are presented. The estimation errors between predicted states and real state are proved to be arbitrarily small. Then, the network stability and performance bounds between real system and reference system can be made arbitrarily small by decreasing the integration time-step. The simulation results show the details of proposed scheme. This approach can be further extended to deal with uncertainties issue in multi-unmanned vehicle systems.



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*Source: 3927108-2018-09-02.xml*