# Distributed Adaptive Output Consensus for High-Order Multiagent Systems with Input Saturation and Uncertain Nonlinear Dynamics

Authors: Ming-Can Fan; Wen Qin
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper deals with the leader-following output consensus problem for a class of high-order affine nonlinear strict-feedback multiagent systems with unknown control gains and input saturation under a general directed graph. Nussbaum gain function technique is used to handle the unknown control gains, and the uncertain nonlinear dynamics of each agent is approximated by radial basis function neural networks. Distributed adaptive controllers are designed via the backstepping technique as well as the dynamic surface control approach. It is proved that the closed-loop multiagent systems are semiglobally uniformly ultimately bounded, and the output consensus error can converge to a small region around the origin. Finally, the theoretical results are supported by a numerical simulation.


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## Body

## 1. Introduction

Consensus control of multiagent systems (MAS) has drawn considerable attention in the past two decades due to its broad applications in multiple ground-moving robots [1], unmanned aerial vehicles (UAVs) [2], unmanned surface vessels [3], sensor networks [4], smart grids [5], and synchronization and flocking models [6–8], for instance. The investigation on this topic has been carried out from different perspectives, to mention a few, such as single-integrator or double-integrator MAS [9, 10], general linear MAS [11–13], nonlinear MAS [14–16], fractional-order MAS [17, 18], and high-order MAS [19, 20].Consensus problems of MAS with unknown control gains, including unknown amplitudes of control gains or unknown signs of control gains (the so-called control directions), are gaining researchers’ increasing attention in recent years. It is necessary and beneficial to study this topic since the control gains are often unknown in many practical control systems, such as the autopilot design of time-varying ships and unmanned sailboat heading control. There are many results of consensus control for MAS with unknown control gains, for instance, [21–26]. In [21], Nussbaum function-based adaptive control was developed to handle the consensus problem for a first-order MAS provided that the communication digraph was balanced and weakly connected. By using a Nussbaum-like switching function, the authors in [22] also addressed the first-order consensus problem under a strongly connected communication digraph. In [23], using a novel Nussbaum gain function, consensus for a second-order linearly parameterized MAS was realized under the assumption that the control directions were the same and the communication graph was undirected and connected. Some progress has been made with respect to high-order MAS. In [24], adaptive backstepping-based controllers were designed to guarantee the consensus for high-order MAS under an undirected and connected communication graph. In [25], the authors investigated high-order linearly parameterized MAS, while external disturbances were not considered. In [26], high-order MAS with nonlinear dynamics and partially unknown nonidentical time-varying control directions, as well as bounded external disturbances, was considered, where the communication digraph was assumed to be strongly connected. Note that most of the existing works including those mentioned above did not take input saturation into account, which is often encountered in practical applications due to physical limitations of actuators and may cause the instability or damage the control systems’ performance [27–31].Motivated by the limitation of the existing literatures, this paper studies the leader-following output consensus problem for a class of high-order nonlinear MAS subject to input saturation and bounded external disturbances. Moreover, the control gains are assumed to be time varying and unknown for the controller design. This issue has not been mentioned in any existing references to the best of the authors knowledge. Based on the backstepping technique combined with the dynamic surface control (DSC) method, and under the assumption that the communication digraph has a spanning tree, a distributed adaptive neural controller is proposed with the aid of the well-known Nussbaum gain function. It is proved that the closed-loop MAS is semiglobally uniformly ultimately bounded (SUUB), and the output consensus error can converge to a small region around the origin.Compared with the existing results, the primary contributions of this paper can be summarized as follows. (1) First, the high-order MAS model discussed in this paper is a class of affine nonlinear systems in the strict-feedback form with external disturbances, which is more general than most existing systems regarding output consensus control with unknown control gains [26, 32, 33], where the system dynamics was described with the Brunovsky form or the input saturation was not considered. Hence, the consensus schemes in these references could not be applied. (2) Second, unlike some existing results where the signs of control gains were assumed to be known in advance and the derivative of each control gain function was assumed to be bounded [34] or the unknown nonlinear dynamics satisfied global Lipschitz conditions [35, 36], the control gains (including their amplitudes and signs) and the nonlinear dynamics in this paper are completely unknown.The rest of this paper is organized as follows. In Section2, some preliminaries are introduced and the problem of this paper is formulated. Then, a backstepping-based control algorithm is proposed in Section 3, and the closed-loop stability is proved in Section 4. In Section 5, a simulation example is given to verify the proposed control algorithm. Finally, some remarks of this paper are concluded in Section 6.


## 2. Preliminaries and Problem Statement

In this paper, a class of high-order affine nonlinear MAS withN followers is considered. The dynamics of each follower is described as follows:(1)x˙i,j=hi,jx¯i,j+ℓi,jx¯i,jxi,j+1+di,jt,x˙i,ni=hi,nix¯i,ni+ℓi,nix¯i,niσvi+di,nit,yi=xi,1,j=1,2,…,ni−1,where x¯i,j=xi,1,xi,2,…,xi,jT∈ℝj, vi∈ℝ, and yi∈ℝ are the state vectors, actual controllers, and outputs of the MAS, respectively, i=1,2,…,N, di,jt are the unknown bounded time-varying external disturbances, the unknown continuous function hi,j⋅ represents the uncertain nonlinear dynamics, ℓi,j⋅ represents a continuous and unknown time-varying control gain function, and σ⋅:ℝ⟶ℝ is a standard saturation function defined as σx=sgnxminx,v∗, where v∗>0 is a saturation level constant. The output of the leader is y0t, which satisfies the following assumption.Assumption 1.
y0t, y˙0t, and y¨0t are bounded for t≥0.Remark 1.
The dynamics of the uncertain MAS (1) is affine nonlinear in the strict-feedback form with external disturbances, which cannot be converted to the Brunovsky form studied in most existing literatures [26, 32, 33] due to the unknown control gains. Hence, the MAS is more general, and the controller design is more challenging.Remark 2.
The strict-feedback nonlinear MAS (1) may have great potential for practical applications since it can describe many dynamical behaviors, such as robotic systems, flight systems, and biochemical process [37].
In order to use the backstepping technique, the discontinuous saturation nonlinearityσ⋅ is replaced by a smooth function [38]: σx=v∗×erfpx, where p=π/2v∗ and erfx=2/π∫0xe−t2dt is a Gaussian error function, which is a real-valued and continuous differentiable function. Defining ηvi=v∗×erfpvi−qvi, q>0, and then we have σvi=qvi+ηvi.
The purpose of this paper is to present an adaptive neural output consensus controller for MAS (1) such that the semiglobal uniform ultimate boundedness for all the signals in the closed-loop system is ensured, and each follower’s output yii=1,2,…,N synchronically tracks the leader’s output y0. Moreover, the output consensus error can converge to a small region around the origin. To this end, the following assumptions and preliminaries are required.Assumption 2.
There exist constantsη¯, q¯, q¯, and d¯i,j, such that ηvi≤η¯, q∈q¯,q¯, and di,jt≤d¯i,j, respectively, i=1,2,…,N and j=1,2,…,ni.Assumption 3.
There exist positive constantsℓ¯i,j and ℓ¯i,j such that the unknown control gains ℓi,jx¯i,j satisfy ℓ¯i,j≤ℓi,jx¯i,j≤ℓ¯i,j,∀x¯i,j∈Ω⊂ℝj.Definition 1 (see [39]).
A continuous functionN⋅:ℝ⟶ℝ is called a Nussbaum gain function if it satisfies(2)lims⟶+∞sup1s∫0sNκdκ=+∞,lims⟶+∞inf1s∫0sNκdκ=−∞.
In this paper, a Nussbaum gain functionκ2cosπ/2κ is chosen.Lemma 1 (see [40]).
LetΓt and κit be smooth functions defined on 0,T with Γt≥0, ∀t∈0,T, and N⋅ be an even smooth Nussbaum gain function. If the following inequality holds:(3)Γt≤A+e−Bt∑i=1N∫0tℓisNκis+1κ˙iseBsds,where A represents some suitable constant, B>0 is a positive constant, and the value of the time-varying parameter ℓi⋅ is located in the unknown intervals I=l−,l+ with 0∉I, then Γt, κit, and ∑i=1N∫0tℓisNκis+1κ˙isds are bounded on 0,T.
The communication topology in MAS (1) with N followers is expressed via a digraph G=V,ℰ,A, where N followers are denoted by a node set V=1,2,…,N, ℰ⊂V×V is an edge set, and A=aij∈ℝN×N is an adjacency matrix. aij>0 if the i-th follower can receive information from the j-th follower, otherwise aij=0. Self-loop is not considered, i.e., aii=0. D=diagd1,d2,…,dN is defined as an in-degree matrix with di=∑j=1Naij being the in-degree of the i-th follower. The Laplacian matrix of G is denoted by ℒ=D−A=lij∈ℝN×N. A digraph contains a spanning tree provided that there exists a directed path from one node (called root node) to every other node in the graph. A leader adjacency matrix ℛ=diagr1,r2,…,rN is used to demonstrate the communication between the leader and the follower, where ri>0 represents that the leader is a neighbor of the i-th follower, and ri=0 otherwise.Assumption 4.
The communication digraphG among the N followers contains a spanning tree, and the leader is a neighbor of the root node.
Since the uncertain nonlinear dynamics of MAS (1) is unknown and continuous, many function approximators can be used, such as neural networks, spline functions, polynomials, and fuzzy systems. Gaussian radial basis function neural network (RBFNN) approach is easy to implement due to its small number of control parameters; therefore, it is widely used in nonlinear function approximation. In this paper, we employ an RBFNN to approximate the unknown function ΨZ:ℝm⟶ℝ on a prescribed compact set Ω⊂ℝm, i.e.,(4)ΨZ=ϱ∗TϑZ+ϖZ,Z∈Ω,where Z=z1,z2,…,zmT is the input vector, ϱ∗=ϱ1∗,ϱ2∗,…,ϱs∗T∈ℝs is the ideal weight vector with s>1 being the number of neurons, ϑZ=ϑ1Z,ϑ2Z,…,ϑsZT∈ℝs is the Gaussian basis function vector with ϑiZ=exp−Z−ci2/2wi2, where ci and wi are the center and the width of the Gaussian function, respectively, i=1,2,…,s, and ϖZ denotes the approximation error. It is known that given any positive constant ε, if s is large enough, there exist suitable vectors ϱ∗ and ϑZ such that ϖZ≤ε.


## 3. Consensus Controller Design

The distributed adaptive output consensus controller design procedure consists ofni steps using the backstepping technique. To begin with, the following error surfaces for the j-th follower are defined:(5)ei,1=∑j=1Naijyi−yj+riyi−y0,(6)ei,j=xi,j−γ¯i,j,j=2,…,ni,where γ¯i,j is the filtered virtual controller, which is acquired through a first-order filter with the virtual controller γi,j as the input. The boundary layer error between γ¯i,j and γi,j is defined as γ˜i,j=γ¯i,j−γi,j.Step 1.
The time derivative ofei,1 in equation (5) is obtained as(7)e˙i,1=ri+dihi,1xi,1+ℓi,1xi,1xi,2+di,1−∑j=1Naijhj,1xj,1+ℓj,1xj,1xj,2+dj,1−riy˙0.
DefineΨi,1ζi,1:=hi,1xi,1−1/ri+di∑j=1Naijhj,1xj,1+ℓj,1xj,1xj,2−riy˙0, ζi,1=xi,1,xj,1,xj,2,y˙0T. Throughout this paper, let x^ be the estimates of x∗, and the corresponding estimation error be x˜=x^−x∗.
Then, the unknown smooth functionΨi,1ζi,1 can be approximated by an RBFNN as(8)Ψi,1ζi,1=ϱi,1∗Tϑi,1+ϖi,1=ϱ^i,1Tϑi,1−ϱ˜i,1Tϑi,1+ϖi,1,ϖi,1≤εi,1.
Then, we have(9)e˙i,1=ri+diℓi,1xi,1xi,2+ϱ^i,1Tϑi,1−ϱ˜i,1Tϑi,1+ϖi,1+di,1−∑j=1Naijdj,1.
The virtual controllersγi,2 and adaptation laws κi,1 and ϱ^i,1 are designed as follows:(10)γi,2=Nκi,1αi,1ei,1+ϱ^i,1Tϑi,1,(11)κ˙i,1=αi,1ei,12+ei,1ϱ^i,1Tϑi,1,(12)ϱ^˙i,1=Λi,1ϑi,1ei,1−βi,1ϱ^i,1,where Λi,1 is a positive-definite matrix, αi,1>0 and βi,1>0 are the design parameters. Adjusting the value of αi,1 properly will make the error ei,1 converge to a small neighborhood of the origin. βi,1ϱ^i,1 is a modification term with βi,1 being a small positive constant, such that ϱ^i,1 will not drift to very large values.
Under Assumption4, we have ri+di>0∀i∈V. Hence, we can construct the Lyapunov function candidate ϒi,1=1/2ri+diei,12. Note that γ¯i,2=γ˜i,2+γi,2, and then we have(13)ϒ˙i,1=1ri+diei,1e˙i,1=ei,1ℓi,1ei,2+γ˜i,2+γi,2+ϱ^i,1Tϑi,1−ϱ˜i,1Tϑi,1+ϖi,1+di,1−1ri+di∑j=1Naijdj,1=ei,1ℓi,1ei,2+γ˜i,2+Nκi,1αi,1ei,1+ϱ^i,1Tϑi,1+ϱ^i,1Tϑi,1−ϱ˜i,1Tϑi,1+ϖi,1+di,1−1ri+di∑j=1Naijdj,1=−αi,1ei,12+ℓi,1Nκi,1+1κ˙i,1+ℓi,1ei,1ei,2+ℓi,1ei,1γ˜i,2−ei,1ϱ˜i,1Tϑi,1+ei,1ϖi,1+di,1−1ri+di∑j=1Naijdj,1≤−αi,1ei,12+ℓi,1Nκi,1+1κ˙i,1+ℓi,1ei,1ei,2+ℓi,1ei,1γ˜i,2+ξi,1ei,1−ei,1ϱ˜i,1Tϑi,1,where ξi,1=εi,1+d¯i,1+1/ri+di∑j=1Naijd¯j,1>0. Using Young’s inequality [41], the following inequalities hold:(14)ℓi,1ei,1ei,2≤ei,12+ℓi,12ei,224,(15)ℓi,1ei,1γ˜i,2≤ei,12+ℓi,12γ˜i,224,(16)ξi,1ei,1≤ei,12+ξi,124.
Substituting (14)–(16) into (13), one has(17)ϒ˙i,1≤−αi,1−3ei,12+ℓi,1Nκi,1+1κ˙i,1+14ℓi,12ei,22+14ℓi,12γ˜i,22+14ξi,12−ei,1ϱ˜i,1Tϑi,1.
In order to avoid differentiatingγi,2, the so-call explosion of the complexity problem is inherent in the backstepping technique, and let γi,2 pass through a first-order filter such that a filtered virtual controller γ¯i,2 is acquired:(18)τi,2γ¯˙i,2+γ¯i,2=γi,2,γ¯i,20=γi,20,where τi,2>0 is a small time constant.
A Lyapunov function candidateΓi,1t is given as(19)Γi,1t=ϒi,1+12γ˜i,22+12ϱ˜i,1TΛi,1−1ϱ˜i,1.
Using (18), we have γ˜˙i,2=γ¯˙i,2−γ˙i,2=−γ˜i,2/τi,2+Bi,2, where Bi,2=−∂γi,2/∂κi,1κ˙i,1−∂γi,2/∂ei,1e˙i,1−∂γi,2/∂ϱ^i,1ϱ^˙i,1 is a continuous function. Since for any B0>0 and Q>0, the sets Π:y0,y˙0,y¨0:y02+y˙02+y¨02≤B0 and Πi:=∑j=1i1/rj+djej,12+∑j=1iγj,22+∑j=1iϱ˜j,1TΛj,1−1ϱ˜j,1≤2Q, i=1,2,…,N, are compact in R3 and R3i, respectively, and Π×Πi is also compact in R3i+3. Therefore, Bi,2 has a maximum B¯i,2 on Π×Πi, i.e., Bi,2≤B¯i,2. Similar analysis can be found in [42]. Hence, we have(20)γ˜i,2γ˜˙i,2≤−γ˜i,22τi,2+γ˜i,2B¯i,2≤−γ˜i,22τi,2+γ˜i,22+14B¯i,22.
By taking the time derivative ofΓi,1, one has(21)Γ˙i,1t=ϒ˙i,1+γ˜i,2γ˜˙i,2+ϱ˜i,1TΛi,1−1ϱ˜˙i,1≤−αi,1−3ei,12+ℓi,1Nκi,1+1κ˙i,1+14ℓi,12ei,22+ℓ¯i,124−1τi,2+1γ˜i,22+14ωi,12−βi,1ϱ˜i,1Tϱ^i,1,where ωi,12=ξi,12+B¯i,22. Note that(22)−ϱ˜i,1Tϱ^i,1=−ϱ˜i,1Tϱ˜i,1+ϱi,1∗≤−ϱ˜i,12+ϱ˜i,1⋅ϱi,1∗≤−12ϱ˜i,12+12ϱi,1∗2,then we have(23)Γ˙i,1t≤−αi,1−3ei,12−1τi,2−ℓ¯i,124−1γ˜i,22−βi,12ϱ˜i,12+ℓi,1Nκi,1+1κ˙i,1+βi,12ϱi,1∗2+14ωi,12+14ℓi,12ei,22.
Stepj (2≤j≤ni−1). The derivative of the error surface ei,j=xi,j−γ¯i,j+1 in (6) is obtained as(24)e˙i,j=hi,jx¯i,j+ℓi,jxi,j+1+di,j−γ¯˙i,j+1.
DefineΨi,jζi,j:=hi,jx¯i,j−γ¯˙i,j+1, ζi,j=x¯i,j,γ¯˙i,jT∈ℝj+1. This unknown smooth function Ψi,jζi,j can be approximated by an RBFNN as(25)Ψi,jζi,j=ϱi,j∗Tϑi,1+ϖi,j=ϱ^i,jTϑi,j−ϱ˜i,jTϑi,j+ϖi,j,ϖi,j≤εi,j.
The virtual controllersγi,j+1, j=2,…,ni−1, and adaptation laws κi,j and ϱ^i,j are designed as follows:(26)γi,j+1=Nκi,jαi,jei,j+ϱ^i,jTϑi,j,(27)κ˙i,j=αi,jei,j2+ei,jϱ^i,jTϑi,j,(28)ϱ^˙i,j=Λi,jϑi,jei,j−βi,jϱ^i,j,where Λi,j is a positive-definite matrix, αi,j>0 and βi,j>0 are the design parameters.
Applying the Lyapunov functionϒi,j=1/2ei,j2 and noting that γ¯i,j+1=γ˜i,j+1+γi,j+1, we have(29)ϒ˙i,j=ei,je˙i,j=ei,jℓi,jei,j+1+γ˜i,j+1+γi,j+1+ϱ^i,jTϑi,j−ϱ˜i,jTϑi,j+ϖi,j+di,j=−αi,jei,j2+ℓi,jNκi,j+1κ˙i,j+ℓi,jei,jei,j+1+ℓi,jei,jγ˜i,j+1−ei,jϱ˜i,1Tϑi,1+ei,jϖi,j+di,j≤−αi,jei,j2+ℓi,jNκi,j+1κ˙i,j+ℓi,jei,jei,j+1+ℓi,jei,jγ˜i,j+1+ei,jξi,j−ei,jϱ˜i,jTϑi,j,where ξi,j=ϖi,j+d¯i,j. Using Young’s inequality, we have(30)ℓi,jei,jei,j+1≤ei,j2+ℓi,j2ei,j+124,ℓi,jei,jγ˜i,j+1≤ei,j2+ℓi,j2γ˜i,j+124,ei,jξi,j≤ei,j2+ξi,j24.
Substituting the above equations into (29) gives(31)ϒ˙i,j≤−αi,j−3ei,j2+ℓi,jNκi,j+1κ˙i,j+14ℓi,j2ei,j+12+14ℓi,j2γ˜i,j+12+14ξi,j2−ei,jϱ˜i,jTϑi,j.
Letγi,j+1 pass through a first-order filter with a small time constant τi,j+1>0 to acquire γ¯i,j+1, j=2,…,ni−1:(32)τi,j+1γ¯˙i,j+1+γ¯i,j+1=γi,j+1,γ¯i,j+10=γi,j+10.
From (32), we have(33)γ˜˙i,j+1=−γ˜i,j+1τi,j+1+Bi,j+1≤−γ˜i,j+12τi,j+1+γ˜i,j+12+14B¯i,j+12,where Bi,j+1=∂γi,j+1/∂κi,jκ˙i,j+∂γi,j+1/∂ei,je˙i,j+∂γi,j+1/∂ϱ^i,jϱ^˙i,j is a continuous and bounded function satisfying Bi,j+1≤B¯i,j+1.
Applying the Lyapunov function,(34)Γi,jt=ϒi,j+12γ˜i,j+12+12ϱ˜i,jTΛi,j−1ϱ˜i,j.
Similar to the previous step, we have(35)Γ˙i,jt=ϒ˙i,j+γ˜i,j+1γ˜˙i,j+1+ϱ˜i,jTΛi,j−1ϱ˜˙i,j≤−αi,j−3ei,j2+ℓi,jNκi,j+1κ˙i,j+14ℓi,j2ei,j+12+ℓ¯i,j24−1τi,j+1+1γ˜i,j+12+14ωi,j2−βi,jϱ˜i,jTϱ^i,j,where ωi,j2=ξi,j2+B¯i,j+12. Similar to (22), one has(36)−ϱ˜i,jTϱ^i,j≤−12ϱ˜i,j2+12ϱi,j∗2,then we have(37)Γ˙i,jt≤−αi,j−3ei,j2−1τi,j+1−ℓ¯i,j24−1γ˜i,j+12−βi,j2ϱ˜i,j2+ℓi,jNκi,j+1κ˙i,j+βi,j2ϱi,j∗2+14ωi,j2+14ℓi,j2ei,j+12.
Stepni. At this final step, the actual controller vit is designed. The derivative of the error surface ei,ni=xi,ni−γ¯i,ni is obtained as(38)e˙i,ni=hi,nix¯i,ni+ℓi,nix¯i,nicvi+ηvi+di,ni−γ¯˙i,ni.
DefineΨi,niζi,ni:=hi,nix¯i,ni−γ¯˙i,ni, ζi,ni=x¯i,ni,γ¯˙i,niT∈ℝni+1. This unknown smooth function Ψi,niζi,ni can be approximated by an RBFNN as(39)Ψi,niζi,ni=ϱi,ni∗Tϑi,ni+ϖi,ni=ϱ˜i,niTϑi,niζi,ni−ϱ˜i,niTϑi,niζi,ni+ϖi,ni,ϖi,ni≤εi,ni.
The actual controllersvi and adaptation laws κi,ni and ϱ^i,ni are designed as follows:(40)vi=Nκi,niαi,niei,ni+ϱ^i,niTϑi,ni,(41)κ˙i,ni=αi,niei,ni2+ei,niϱ^i,niTϑi,ni,(42)ϱ^˙i,ni=Λi,niϑi,niei,ni−βi,niϱ^i,ni,where Λi,ni is a positive-definite matrix, αi,ni>0 and βi,ni>0 are the design parameters.
Applying the Lyapunov functionΓi,nit=1/2ei,ni2+1/2ϱ˜i,niTΛi,ni−1ϱ˜i,ni; similar to the aforementioned step, we have(43)Γ˙i,nit=ei,niϱ^i,niTϑi,niζi,ni−ϱ˜i,niTϑi,niζi,ni+ϖi,ni+ℓi,nicvi+ηvi+di,ni−γ¯˙i,ni≤−αi,ni−1ei,ni2+ℓi,nicNκi,ni+1κ˙i,ni−βi,ni2ϱ˜i,ni2+βi,ni2ϱi,ni∗2+14ωi,ni2,where ωi,ni=ℓ¯i,niη¯+d¯i,ni+εi,ni.


## 4. Stability Analysis

Theorem 1.
Suppose that the initial conditions are bounded and the design parameters fulfill the following inequalities (44), then the output consensus problem for MAS (1) can be addressed by the controllers (10), (26), and (40) combining with adaptation laws (11), (12), (9), (28), (41), and (42), as well as the filters (18) and (32) under Assumptions 1–4. Moreover, all signals in the closed-loop MAS are SUUB, and the output consensus error can converge to a small region around the origin provided that the design parameters are appropriately selected:(44)αi,1>3,αi,j+1>ℓ¯i,j24+3,j=1,2,…,ni−2,αi,ni>1,1τi,j+1>ℓ¯i,j24+1,j=1,2,…,ni−1.Proof.
Select the following constantsμi,1>0 and νi,1>0 as(45)μi,1=min2ri+diαi,1−3,βi,1λmaxΛi,1−1,21τi,2−ℓ¯i,124−1,νi,1=βi,1ϱi,j∗22+ωi,124.
From (23), one has(46)Γ˙i,1t≤−μi,1Γi,1t+ℓi,1Nκi,1+1κ˙i,1+νi,1+14ℓi,12ei,22.
Multiplying (46) by eμi,1t and then integrating it with respect to t over 0,t, one has(47)Γi,1t≤νi,1μi,1+Γi,10+e−μi,1t∫0tℓi,1Nκi,1+1κ˙i,1eμi,1sds+14e−μi,1t∫0tℓi,12ei,22eμi,1sds.
Sincee−μi,1t∫0tℓi,12ei,22eμi,1sds≤1/μi,1ℓ¯i,12sups∈0,tei,22s, we have(48)Γi,1t≤ρi,1+e−μi,1t∫0tℓi,1Nκi,1+1κ˙i,1eμi,1sds,where ρi,1=νi,1/μi,1+Γi,10+1/4μi,1ℓ¯i,12sups∈0,tei,22s. Then, according to Lemma 1, we can conclude that Γi,1t, ei,1t, κi,1t, ϱ˜i,1t, and ∫0tℓi,1Nκi,1+1κ˙i,1ds are all SUUB on 0,T. Let ςi,1>0 be the upper bound of ∫0tℓi,1Nκi,1+1κ˙i,1ds, i.e., ∫0tℓi,1Nκi,1+1κ˙i,1ds≤ςi,1.
Forj=2,…,ni−1, select the following constants μi,j>0 and νi,j>0 as(49)μi,j=min2αi,j−3,βi,jλmaxΛi,j−1,21τi,j+1−ℓ¯i,j24−1,νi,j=βi,jϱi,j∗22+ωi,j24.
From (37), we have(50)Γ˙i,jt≤−μi,jΓi,jt+ℓi,jNκi,j+1κ˙i,j+νi,j+14ℓi,j2ei,j+12.
Similar to (48), one has(51)Γi,jt≤ρi,j+e−μi,jt∫0tℓi,jNκi,j+1κ˙i,jeμi,jsds,where ρi,j=νi,j/μi,j+Γi,j0+1/4μi,jℓ¯i,j2sups∈0,tei,j+12s. Then, according to Lemma 1, one can conclude that Γi,jt, ei,jt, κi,jt, ϱ˜i,jt, and ∫0tℓi,jNκi,j+1κ˙i,jds are all SUUB. Let ςi,j>0 be the upper bound of ∫0tℓi,jNκi,j+1κ˙i,jds, i.e., ∫0tℓi,jNκi,j+1κ˙i,jds≤ςi,j.
Finally, applying the Lyapunov function,(52)Γt=∑i=1N∑j=1niΓi,jt=∑i=1N12ri+diei,12+12∑j=2niei,j2+12∑j=1ni−1γ˜i,j+12+12∑j=1niϱ˜i,jTΛi,j−1ϱ˜i,j.
Denote(53)0<δ≤mini=1,2,…,Nj=1,2,…,ni−1μi,j,2αi,j+1−3−ℓ¯i,j24,2αi,ni−1βi,niλmaxΛi,ni−1,χ=∑i=1N∑j=1niβi,j2ϱi,j∗2+14∑i=1N∑j=1niωi,j2,ℓ^i,j=ℓi,j,j=1,2,…,ni−1,cℓi,ni,j=ni,then substitute (23), (37), and (43) into the time derivative of (52), and based on (44), we have(54)Γ˙t=−∑i=1Nαi,1−3ei,12−∑i=1N∑j=2ni−1αi,j−3−ℓ¯i,j−124ei,j2−∑i=1Nαi,ni−1ei,ni2−∑i=1N∑j=1ni−11τi,j+1−∑i=1Nℓ¯i,j24−1γ˜i,j+12−∑i=1N∑j=1niβi,j2ϱ˜i,j2+∑i=1N∑j=1niℓ^i,jNκi,j+1κ˙i,j+∑i=1N∑j=1niβi,j2ϱi,j∗2+14∑i=1N∑j=1niωi,j2≤−δΓt+∑i=1N∑j=1niℓ^i,jNκi,j+1κ˙i,j+χ.
Multiplying the above inequality byeδt and then integrating it with respect to t over 0,t, we have(55)Γt≤χδ+e−δtΓ0−χδ+e−δt∑i=1N∑j=1ni∫0tℓ^i,jNκi,j+1κ˙i,jeδsds,(56)≤χδ+Γ0+∑i=1N∑j=1ni∫0tℓ^i,jNκi,j+1κ˙i,je−δt−sds,(57)≤χδ+Γ0+∑i=1N∑j=1ni−1ςi,j+e−δt∑i=1N∫0tℓ^i,niNκi,ni+1κ˙i,nieδsds.
By Lemma1, it can be concluded that Γt, ei,nit, κi,nit, ϱ˜i,nit, and ∑i=1N∫0tℓ^i,niNκi,ni+1κ˙i,nids are all SUUB; i.e., all the signals in the closed-loop system are SUUB. Besides, from (52) and (55), one has(58)∑i=1N12ri+diei,12≤Γt≤χδ+e−δtΓ0−χδ+e−δt∑i=1Nℓ^i,jNκi,j+1κ˙i,jeδsds.
Since the matrixℒ+ℛ is nonsingular under Assumption 4, denote its minimum singular value as σ¯, which is a positive constant [10]. Define M=maxiri+di and z1=z1,1,z2,1,…,zN,1T=ℒ+ℛy−y01 with y=y1,y2,…,yNT and 1=1,1,…,1T∈ℝN. Then, we have(59)limt⟶∞y−y01≤2Mχσ¯δ.
Hence, the output consensus error can converge to a small region around the origin provided that the design parametersαi,j, βi,j, and τi,j+1 are appropriately selected.


## 5. Numerical Simulation

To test the availability of the proposed control algorithm, consider the following uncertain MAS with 3 followers:(60)x˙i1=hi1xi1+ℓi1xi1xi2+di1t,x˙i2=hi2x¯i2+ℓi2x¯i2σvi+di2t,yi=xi1,i=1,2,3,where h11x11=−2x11+x112, ℓ11x11=−3+2sinx112, h12x¯12=x11x122, ℓ12x¯12=−4+2cosx11x122, h21x21=−x21+sinx21, ℓ21x21=5−3cosx21, h22x¯22=−cosx21x22, ℓ22x¯22=−2−sinx21x223, h31x31=−x31cosx312, ℓ31x31=2+sinx312, h32x¯32=−0.5x31x32, and ℓ32x¯32=−2+cosx31x32 are the unknown continuous functions and d11t=d21t=−0.01sint, d12t=d22t=0.02cost, d31t=−0.02cost, and d32t=0.01sint are the unknown disturbances. The leader’s output is given by y0t=0.4sint. It is easy to verify that the conditions in Assumptions 1–3 are all satisfied. Figure 1 depicts the communication digraph of the leader and the 3 followers.Figure 1
Communication graph, where 0 denotes the leader and 1–3 denote the 3 followers, respectively.In simulation, RBFNNsϱi,1Tϑi,1ζi,1 and ϱi,2Tϑi,2ζi,2, which contain 11 neurons with centers spaced evenly in −1,1 and widths being equal to w1=0.5 and w2=5, respectively, are used to approximate the unknown continuous functions hi1xi1 and hi2x¯i2, i=1,2,3. The initial conditions are chosen as(61)x110,x120,x210,x220,x310,x320T=0.3,0.01,0.2,0.01,0.1,0.01T,v∗=5,αi,1=20,αi,2=10,Λi,j=0.1I,τi,j=0.01,βi,j=0.01,κi,j0=0,ϱ^i,j0=0,i=1,2,3;j=1,2.Figures2–4 display the evolution of the output trajectories of the 3 followers and the leader, the corresponding output consensus errors, and the actual controllers vi, respectively. One can see that the outputs of the 3 followers can track the leader’s output since the output consensus error converges to a small neighborhood of zero. Additionally, it can be seen that there exist large differences between vi and the saturation level constant v∗, which indicates the significant saturation effect. The simulation results illustrate that the proposed control algorithm is effective.Figure 2
Evolution of the output trajectories.Figure 3
Evolution of the output consensus errors.Figure 4
Evolution of the actual controllersvi.Remark 3.
The reason that why there exist large fluctuations in the output trajectories ofyi lies in the large fluctuations of the actual controllers vi (see Figure 4). Because the Nussbaum gain function Nκi,2=κi,22cosπ/2κi,2 will be very large (positively or negatively) if κi,2 becomes very large, correspondingly the actual controllers vi will be very large according to equation (40). To reduce the values of κi,2, the design parameters αi,2 should be turned smaller. However, from the proof of Theorem 1, we know that smaller αi,2 may lead to a smaller value of δ and thereby causes a larger output consensus error (see equation (59)). In a word, there is a tradeoff between smaller output consensus errors and smooth output trajectories. This is another problem worthy to explore, which may be our future work.


## 6. Conclusion

In this paper, an output consensus scheme for a class of high-order affine nonlinear MAS with unknown control gains and input saturation under a directed communication graph is proposed. Combining the adaptive RBFNN approximation and the DSC methodology, consensus control laws are developed to guarantee the semiglobal uniform ultimate boundedness for all signals in the closed-loop system. Moreover, the output consensus error can converge to a small region around the origin provided that the design parameters are appropriately selected. The effectiveness of the proposed control algorithm is revealed by a numerical simulation.



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*Source: 8098421-2020-08-12.xml*