# Distributed Control Design for Uncertain Multiagent Systems with Heterogenous High Powers

Authors: Hanqiao Huang; Hantong Mei; Di Zhang; Yunhe Guo; Ge Zhou
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

In this paper, a distributed asymptotic tracking control strategy is investigated by establishing filters and barrier function-based consensus control scheme to address the control of heterogenous power-chained multiagent systems (MASs) under a directed graph subject to the unknown input deadzone nonlinearities and unknown control coefficients. First, to generate estimation information from the leader, a two-order filter is exploited for every agent which solves the difficultly of the time-varying control coefficients in multiagent systems with a directed topology. Then, based on the two-order filters, prescribed performance method and barrier functions are utilized to establish the distributed tracking protocol to handle the power-chained deadzone input nonlinearities, such that the MAS can reach the global consensus while guaranteeing the prescribed tracking error performance. Using the Lyapunov stability theorem, the proof of the convergence is accomplished rigorously. Ultimately, the efficacy and advantage of the devised method are validated by two simulation examples.


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## Body

## 1. Introduction

The problem of distributed control for nonlinear MASs has been investigated for several decades, and a lot of significant achievements have been achieved [1–4]. Recently, various control methodologies, including the function approximation control techniques [5], adaptive control [6], and robust control [7], have been studied extensively for nonlinear MASs. In [8, 9], a consensus tracking protocol for uncertain pure feedback MASs is addressed. The distributed tracking problem based on observer is studied for nonlinear MASs with uncertain networks in [10]. In [11], on the strength of Nussbaum-type gain technique, both leaderless and leader-following consistent tracking control methodologies can make the errors converge be guaranteed asymptotically to zero for a second-order MAS with unknown control directions, without any information about the global communication graph. Under directed topologies or undirected topologies, a distributed adaptive control approach is applied to the leader-following networked Lagrangian systems with unknown control directions in [12]. However, the leader-following system assumes that there is only one leader in the system, and the communication graph among the followers is an undirected connected topology. Different from adaptive designs, [13] proposes a prescribed performance control method for an uncertain multiagent system. For resolving the consensus difficulty in MASs with unknown nonlinearities, neural networks or fuzzy logic systems are used as universal approximators in [14–16]. To settle the issue of unknown control directions, [17] presents a novel Nussbaum-type function. Although these above studies can obtain good command tracking performance, there is still a lack of research on fault-tolerant control design.From a practical perspective, input deadzone nonlinearity is one of the key problems in nonlinear control system design, which results in the unaccomplished desired control efforts [18]. In the process of multiagent control design, if the control input constraints are not considered, the closed-loop system may be limited by performance and even lose stability. Therefore, the control investigations on restricted MASs has already attracted extensive attention in recent years [19]. On the basis of meeting preset trigger conditions, in [20], a distributed adaptive control approach is investigated for the uncertain MAS to deal with intermittent actuator faults. By introducing a predictor to obtain each error surface estimation, a distributed adaptive fuzzy consensus tracker design is used for MASs with input saturations [21]. In [22], the time-varying formation tracking control protocol for multileader MASs is studied in order to address the problem of actuator failure and saturation nonlinearity simultaneously. Whereas, the augmented plant of [22] is constructed complicatedly, and the first-order low-pass filter greatly increases the uncertainty of numerical simulation results. To compensate for both actuator bias fault and loss of actuator effectiveness fault, a fully distributed control scheme is constructed in [23]. In [24], a fully distributed protocol based on prescribed performance control is proposed to set predefined overshoot range, guarantee convergence rates, and limit neighbourhood synchronization steady-state errors for uncertain nonlinear MASs with deadzone inputs. Despite the workload of [24] is very sufficient and it is greatly improved compared with other aforementioned papers, the proposed strategy is still limited in the scope to the unity low power of MASs. As a matter of fact, there are a lot of practical heterogenous power-chained MASs, such as the coupled underactuated unstable 2-DOF machinery model (i.e., equation (38)) of [25] and the roll dynamic model of multiple axisymmetric skid to turn missiles as Example 2 of [26]. Thus, it is of necessity to do the research on MASs with heterogenous high powers. However, most control approaches applied to date have been limited in their boundary to the unity low power of MASs owing to the uncontrollable linearization of nonlinear systems caused by the high-power chains of control variables.Contrary to these, the distributed consensus control for power-chained MASs with limited communication networks is seldom studied. Based on the controller of single nonlinear power-chained systems, aiming at the networked nonlinear power-chained MASs, [27] proposes a robust cooperative output tracking control protocol. Adding a power integrator technique and adaptive disturbance compensator are introduced to reduce the degradation of system performance for complex nonlinear MASs with input noise in [28]. By utilizing the distributed integrator backstepping approach, the boundness of the closed-loop system for multiple strict-feedback stochastic nonlinear systems under a directed leader-followers type communication graph is achieved in [29]. In [30], a neural adaptive tracking control scheme is employed to nonstrict feedback nonlinear MASs. A distributed low-complexity control strategy is proposed in [26] to guarantee consensus tracking performance of uncertain heterogenous power-chained MASs. We note that considerable progress has been made in the distributed control of MASs, but there are still some nonnegligible issues that need to be solved. First, for MASs with high powers, there are some restrictions on the heterogenous high powers. Specifically, the existing design methods usually require the high powers to be identical or to be known for establishing the control protocols. For example, in order to enable the control design for high power systems, [26] assumes the powers of the first order to be known (i.e., see Assumption 3 in [26]). Second, for MASs with a directed communication graph, the existing approaches do not consider the control coefficients or require that the control coefficients must be constants. The reason making it difficult is that the directed graph Laplace matrix is asymmetric. Considering the time-varying control coefficients of MASs are unknown, traditional adaptive control techniques [3, 20] used to address the time-varying coefficients will make the unknown matrix P inestimable, where P is defined by the Laplacian matrix and the communication directed matrix (i.e., see Lemma 1 in [5]). To circumvent the difficulties, the symmetric positive definite property of Laplacian matrix between followers is implicitly used in most theoretical analyses [24]. Whereas, this means that these methods are not applicable to the MAS with a directed topology. Third, there is no performance envelope scheme to predetermine the distributed consensus tracking error in the controller design of MASs with heterogeneous unknown uncertain powers. Moreover, it should be stressed out that the global control of heterogenous power-chained MASs with unknown input deadzone nonlinearities and unknown control coefficients is still an unsolved problem.Inspired by the aforementioned discussion, the paper proposes the design of a distributed control for uncertain MASs with a directed graph. Compared to the relevant existing researches in the literature, the main contributions of this paper can be concluded as follows.(1)
The investigated control scheme is able to relax the aforementioned four restrictions in the existing consensus researches [26–30] for power-chained MASs with input deadzone nonlinearities(2)
To deal with the time-varying difficulty of multi-agent control coefficients, a second-order filter is exploited to make each agent generate the estimations of the leader signals, thus avoiding to use the asymmetric Laplacian matrix of the directed topology(3)
To solve the unknown nonlinearities in MAS, barrier functions are applied to the prescribed tracking distributed protocol, which make the specified tracking performance guaranteed; thus, the convergence of the controlled system is proved, and all the closed signals are globally bounded


## 2. Problem Formulation

### 2.1. Problem Statement

In this subsection, the analysis on control system with nonlinear deadzone is given first. Then, the new mathematical model under nonlinear model with asymmetric deadzone is constructed.In practical physical systems, certain abnormal events and emergencies may happen in time of operation, which will result in the failure of the actuator. Besides, as a result of the practical constraints on manufacturing and assembly process, the actual output of actuators often presents the characteristic of asymmetric deadzone nonlinearity. The actual control input nonlinear model can be modelled as follows.(1)Dt,u=hrtu−br,0,hltu−bl,u≥br,bl<u<br,u≤bl,where u∈ℝ is the control input, Dt,u:ℝ⟶ℝ is the control output with nonlinear deadzone, hrt and hlt are unknown time-varying positive continuous functions denoting the left and right slopes of the deadzone, bl<0 and br>0 are unknown parameters representing the break-points of deadzone nonlinearity, as shown in Figure 1.Figure 1
A typical example of thinking about deadzone classes.According to (1), for the purpose of facilitating the control law design, the deadzone function can be modelled as
(2)Dt,u=htu+dt,u,with(3)ht=hltifu≤0hrtifu>0,dt,u=−hrtbrifu≥br,−htuifbl<u<br,−hltblifu≤bl.The uncertain high-order nonlinear MASs withNn-th order agents can be modelled by the following dynamic equations
(4)ẋk,i=fk,it,x¯k,i+gk,it,x¯k,ixk,i+1pk,i,i=1,⋯,n−1,ẋk,n=fk,nt,x¯k,n+gk,nt,x¯k,nhktuk+dkt,upk,n,yk=xk,1,k=1,⋯,N,where x¯k=xk,1,⋯,xk,nT∈ℝn are the states with initial conditions x¯k0=xk,10,⋯xk,n0T, x¯k,i=xk,1,⋯,xk,iT∈ℝi, xi=x1,i,⋯,xN,iT∈ℝN, xipi=x1,ipi,⋯,xN,ipiT∈ℝN, uk∈ℝ is the actual control input, yk∈ℝ is the output; pk,i≥1 is all unknown odd integers. The system nonlinearities fk,it,x¯k,i,gk,it,x¯k,i:ℝ×ℝi⟶ℝ and i=1,⋯,n are unknown continuous functions of x¯k,it. Without loss of generality, the unknown control coefficients gk,it,x¯k,i are supposed to be strictly positive.The desired trajectory for the subsystem outputydt is bounded and known only to some subsystems, and ẏdt is bounded and unknown to all subsystems.The goal of the paper is to design a distributed tracking protocol for the multiagent system (4) with heterogenous high powers under the directed graph, so that all the agents’ output tracking errors (i.e., yk−yd, k=1,⋯,N) converge to the predesigned transient and steady state performance boundaries, and simultaneously, make sure all signals of the closed loop system bounded.


### 2.2. Algebraic Graph Theory

Let the telecommunication network among agents be denoted by a directed graphG=V,E,A, where V=v1,v2,⋯,vn represents the index set corresponding to each node with vi being the i-th agent, E⊆V×V represents the set of the edges, A=aij represents the adjacency matrix. i,j denotes an edge of the graph G, and i,j∈E indicates that there is an agent j-to-agent i communication. The element aij of A corresponding to the edge i,j represents the quality of communication between the agents i and j (i.e., i,j∈E⇔aij>0), otherwise, aij=0. G is referred to as an undirected graph if and only if aij=aji. Stipulate that the diagonal elements aii=0. Clearly, for a directed graph, A is unsymmetrical. The in-degree matrix D=diagDi∈ℝN×N, with Di=∑j=1aij being the i-th row sum of A. Moreover, the Laplacian matrix L=lij∈ℝN×N for the directed digraph G can be defined as L=D−A. Besides, defining B=diagbi∈ℝN×N to denote the communication directed topology between the agents and the desired trajectory yd, where bi=1 means that yd is directly accessible by agent i, otherwise, bi=0. A sequence of edges in a graph G of the form i,i1,i1,i2,i2,i3,i3,i4 is called a path. The following notation is utilized over all the paper. Have a∈ℝn and b∈ℝn being two vectors, next, a vector operator .∗ is defined as a.∗b=a1b1,⋯,anbnT.Assumption 1.
The directed graphG=V,E,A includes a directed spanning tree, and the desired trajectory ydt is accessible to at least one subsystem.Assumption 2.
There are unknown and continuous functionsh⌢k⋅:ℝ⟶0,∞ and nonnegative d⌢k⋅:ℝ⟶0,∞ such that hkt,u≤h⌢ku and dkt,u≤d⌢ku.Lemma 3 (see [1]).
For any real numbery, time-varying function at in field of real number and any odd integer p≥1, the following inequality is always true
(5)y−y+atp−atp≤−yp+12p−1.Lemma 4 (see [31]).
Letq and w be real variables. Then, for any integers a>0, b>0 and any real number ε>0, the following inequality is always true
(6)qawb≤aa+bεqa+b+ba+bε−a/bwa+b.Remark 5.
Assumption1 and Assumption 2 are not violated conditions. Under a directed topology, the expected trajectory ydt can only be accessible to a subset of followers (i.e., ∑i=1Nbi>0) in Assumption 1. In Assumption 2, hkt denotes the left and right slopes of the deadzone when u≤bl and u≥br (i.e., when u≤bl and u≥br, hkt=∂Dkt,uk/∂uk). In engineering practice, the slope is usually bounded, because for any real physical system, the slope of the control input cannot be infinite, which means that there must be an upper bound of hkt. Simultaneously, the unknown input nonlinearity function dkt,u is bounded obviously due to the upper boundness of hkt. Besides, it is usually supposed that ∂Dt,u/∂u>0 (i.e., see Assumption 2 in [32]), but this assumption cannot be satisfied when there are unknown deadzone nonlinearities because the derivative between the breakpoints is zero [24], (i.e., ∂Dkt,uk/∂uk=0, for bl<uk<br). In this work, when bl≤u≤br, hkt≠∂Dkt,uk/∂uk (i.e., the aforementioned positive nature requirement on hrt, hlt will not restrict a zero value appearing in the dead-band as ∂Dkt,uk/∂uk=0). Therefore, Assumption 2 is realistic and reasonable.Remark 6.
In this paper, the time-varying system nonlinearitiesfk,i and gk,i are unknown continuous functions, such that little knowledge can be used to establish the control scheme. To handle the difficulty, neural networks and fuzzy logic [14–16] are utilized for approximating the unknown functions as a consequence of the system nonlinearities fk,it,x¯k,i and gk,it,x¯k,i. But only semiglobal results can be got due to the employment of the approximators. It is a challenge problem to establish a distributed protocol for the unknown nonlinear MASs with global consensus, which will be solved skillfully by cooperating two-order filters and barrier functions as follows.


## 2.1. Problem Statement

In this subsection, the analysis on control system with nonlinear deadzone is given first. Then, the new mathematical model under nonlinear model with asymmetric deadzone is constructed.In practical physical systems, certain abnormal events and emergencies may happen in time of operation, which will result in the failure of the actuator. Besides, as a result of the practical constraints on manufacturing and assembly process, the actual output of actuators often presents the characteristic of asymmetric deadzone nonlinearity. The actual control input nonlinear model can be modelled as follows.(1)Dt,u=hrtu−br,0,hltu−bl,u≥br,bl<u<br,u≤bl,where u∈ℝ is the control input, Dt,u:ℝ⟶ℝ is the control output with nonlinear deadzone, hrt and hlt are unknown time-varying positive continuous functions denoting the left and right slopes of the deadzone, bl<0 and br>0 are unknown parameters representing the break-points of deadzone nonlinearity, as shown in Figure 1.Figure 1
A typical example of thinking about deadzone classes.According to (1), for the purpose of facilitating the control law design, the deadzone function can be modelled as
(2)Dt,u=htu+dt,u,with(3)ht=hltifu≤0hrtifu>0,dt,u=−hrtbrifu≥br,−htuifbl<u<br,−hltblifu≤bl.The uncertain high-order nonlinear MASs withNn-th order agents can be modelled by the following dynamic equations
(4)ẋk,i=fk,it,x¯k,i+gk,it,x¯k,ixk,i+1pk,i,i=1,⋯,n−1,ẋk,n=fk,nt,x¯k,n+gk,nt,x¯k,nhktuk+dkt,upk,n,yk=xk,1,k=1,⋯,N,where x¯k=xk,1,⋯,xk,nT∈ℝn are the states with initial conditions x¯k0=xk,10,⋯xk,n0T, x¯k,i=xk,1,⋯,xk,iT∈ℝi, xi=x1,i,⋯,xN,iT∈ℝN, xipi=x1,ipi,⋯,xN,ipiT∈ℝN, uk∈ℝ is the actual control input, yk∈ℝ is the output; pk,i≥1 is all unknown odd integers. The system nonlinearities fk,it,x¯k,i,gk,it,x¯k,i:ℝ×ℝi⟶ℝ and i=1,⋯,n are unknown continuous functions of x¯k,it. Without loss of generality, the unknown control coefficients gk,it,x¯k,i are supposed to be strictly positive.The desired trajectory for the subsystem outputydt is bounded and known only to some subsystems, and ẏdt is bounded and unknown to all subsystems.The goal of the paper is to design a distributed tracking protocol for the multiagent system (4) with heterogenous high powers under the directed graph, so that all the agents’ output tracking errors (i.e., yk−yd, k=1,⋯,N) converge to the predesigned transient and steady state performance boundaries, and simultaneously, make sure all signals of the closed loop system bounded.


## 2.2. Algebraic Graph Theory

Let the telecommunication network among agents be denoted by a directed graphG=V,E,A, where V=v1,v2,⋯,vn represents the index set corresponding to each node with vi being the i-th agent, E⊆V×V represents the set of the edges, A=aij represents the adjacency matrix. i,j denotes an edge of the graph G, and i,j∈E indicates that there is an agent j-to-agent i communication. The element aij of A corresponding to the edge i,j represents the quality of communication between the agents i and j (i.e., i,j∈E⇔aij>0), otherwise, aij=0. G is referred to as an undirected graph if and only if aij=aji. Stipulate that the diagonal elements aii=0. Clearly, for a directed graph, A is unsymmetrical. The in-degree matrix D=diagDi∈ℝN×N, with Di=∑j=1aij being the i-th row sum of A. Moreover, the Laplacian matrix L=lij∈ℝN×N for the directed digraph G can be defined as L=D−A. Besides, defining B=diagbi∈ℝN×N to denote the communication directed topology between the agents and the desired trajectory yd, where bi=1 means that yd is directly accessible by agent i, otherwise, bi=0. A sequence of edges in a graph G of the form i,i1,i1,i2,i2,i3,i3,i4 is called a path. The following notation is utilized over all the paper. Have a∈ℝn and b∈ℝn being two vectors, next, a vector operator .∗ is defined as a.∗b=a1b1,⋯,anbnT.Assumption 1.
The directed graphG=V,E,A includes a directed spanning tree, and the desired trajectory ydt is accessible to at least one subsystem.Assumption 2.
There are unknown and continuous functionsh⌢k⋅:ℝ⟶0,∞ and nonnegative d⌢k⋅:ℝ⟶0,∞ such that hkt,u≤h⌢ku and dkt,u≤d⌢ku.Lemma 3 (see [1]).
For any real numbery, time-varying function at in field of real number and any odd integer p≥1, the following inequality is always true
(5)y−y+atp−atp≤−yp+12p−1.Lemma 4 (see [31]).
Letq and w be real variables. Then, for any integers a>0, b>0 and any real number ε>0, the following inequality is always true
(6)qawb≤aa+bεqa+b+ba+bε−a/bwa+b.Remark 5.
Assumption1 and Assumption 2 are not violated conditions. Under a directed topology, the expected trajectory ydt can only be accessible to a subset of followers (i.e., ∑i=1Nbi>0) in Assumption 1. In Assumption 2, hkt denotes the left and right slopes of the deadzone when u≤bl and u≥br (i.e., when u≤bl and u≥br, hkt=∂Dkt,uk/∂uk). In engineering practice, the slope is usually bounded, because for any real physical system, the slope of the control input cannot be infinite, which means that there must be an upper bound of hkt. Simultaneously, the unknown input nonlinearity function dkt,u is bounded obviously due to the upper boundness of hkt. Besides, it is usually supposed that ∂Dt,u/∂u>0 (i.e., see Assumption 2 in [32]), but this assumption cannot be satisfied when there are unknown deadzone nonlinearities because the derivative between the breakpoints is zero [24], (i.e., ∂Dkt,uk/∂uk=0, for bl<uk<br). In this work, when bl≤u≤br, hkt≠∂Dkt,uk/∂uk (i.e., the aforementioned positive nature requirement on hrt, hlt will not restrict a zero value appearing in the dead-band as ∂Dkt,uk/∂uk=0). Therefore, Assumption 2 is realistic and reasonable.Remark 6.
In this paper, the time-varying system nonlinearitiesfk,i and gk,i are unknown continuous functions, such that little knowledge can be used to establish the control scheme. To handle the difficulty, neural networks and fuzzy logic [14–16] are utilized for approximating the unknown functions as a consequence of the system nonlinearities fk,it,x¯k,i and gk,it,x¯k,i. But only semiglobal results can be got due to the employment of the approximators. It is a challenge problem to establish a distributed protocol for the unknown nonlinear MASs with global consensus, which will be solved skillfully by cooperating two-order filters and barrier functions as follows.


## 3. Main Results

In this section, first, we design a filterqk,1,qk,2 for each agent to generate estimate information from the leader. Subsequently, a distributed asymptotic tracking controller for an uncertain MAS with deadzone input nonlinearities and heterogenous high powers will be designed. Finally, we shall demonstrate that it results in the solution for the problem of predesigned performance for (4).


### 3.1. Filter Design

To facilitate the distributed control design, a filterqk,1,qk,2 is utilized for each agent k, where k=1,⋯,N.Denote(7)zk,p=bkqk,p−ydp−1+∑j=1Nakjqk,p−qj,p,p=1,2,where yd0=yd, yd1=ẏd. Then, design the filters as
(8)q̇k,1=qk,2,q̇k,2=αk,with
(9)αk=−c2zk,1−c1zk,2−c0qk,2−c0sgnz¯k∑p=12F^k,p,(10)F^̇k,1=bkFk,1−yd+∑j=1NakjF^k,1−F^j,1,F^̇k,2=bkFk,2−ẏd+∑j=1NakjF^k,2−F^j,2,where c0, c1, and c2 are positive constant parameters selected as c0≥1, c1>c0+1, and c2=c0c1, Fk,1 is the upper boundary of yd, Fk,2 is the upper boundary of ẏd, k=1,⋯,N.Theorem 7.
Consider a closed-loop system composed ofN filters (8) under Assumption 1 with control design (9). The asymptotic and consistent tracking of all outputs of the filter to ydt can be realized (i.e., limt⟶+∞qk,1−ydt=0, k=1,⋯,N). In addition, qk,1 and qk,2 are bounded.Proof.
Based on Assumption1, the matrix L+B is nonsingular. Same as [5], define θ=θ1,⋯,θNT=L+B−11,⋯,1T, P=diagP1,⋯,PN=diagθ1−1,⋯,θN−1, Q=PL+B+L+BTP, where θk>0 for k=1,⋯,N. It can be summarized from [5] that Q is positive definite.
Construct the following Lyapunov function(11)Vz=12zTPz+12γ∑p=12F~pTPF~p,where z=z¯1,z¯2,⋯,z¯NT∈ℝN with z¯k=c0zk,1+zk,2, F^p=F∧1,p,F∧2,p,⋯,F∧N,pT∈ℝN, Fp=F1,p,F2,p,⋯,FN,pT∈ℝN, F~p=F^p−Fp∈ℝN, p=1,2, and γ>0 is a constant satisfying γ<2λmin2Q/φ2 with φ=PL+B∈ℝ. Denote zp=z1,p,z2,p,⋯,zN,pT∈ℝN, α=α1,α2,⋯,αNT∈ℝN, and qp=q1,p,q2,p,⋯,qN,pT∈ℝN, p=1,2. Then, the following equation can be obtained
(12)ż=L+Bc0q2+α−c0ẏd−y¨d.
Through (11) and (12), by denoting sgnz¯=sgnz¯1,⋯,sgnz¯NT, the time derivative of Vz is
(13)V̇z=zTPL+B−c1z−c0∑p=12sgnz¯.∗Fp+c0∑p=12εz¯.∗Fp−c0ẏd−y¨d−1γ∑p=12F~pTPL+BF~p≤−c1zTQz−1γ∑p=12F~pTQF~p−c0∑p=12zTPDsgnz¯.∗Fp+c0∑p=12zTPAsgnz¯.∗Fp−c0∑p=12zTPBsgnz¯.∗Fp−∑p=12zTPL+Bc0ẏd+y¨d+c0∑p=12zPL+BF~p.
It should be noticed thatc0∑p=12zTPDsgnz.∗Fp=c0∑p=12Fp∑j=1Npjaijz¯j, c0∑p=12zTPAsgnz.∗Fp≤c0∑p=12Fp∑j=1Npjaijz¯j, c0∑p=12zTPBsgnz.∗Fp=c0∑p=12Fp∑j=1Npjbjz¯j, ∑p=12zTPL+Bc0Ẋd+X¨d≤c0∑p=12Fp∑j=1Npjμjz¯j, and ∑p=12zPL+BF~p2+∑p=12φ2F~p/2λminQ, with λminQ representing the minimum eigenvalue of Q.
Combing these above inequality, (13) becomes
(14)V̇z≤−c2z2−γ∗F~p,where c2=λminQc1−c0, γ∗=λminQ1/γ−φ2/2λmin2Q. It is easy to verify that c2>0 and γ∗>0. Thus, it can be obtained from (14) that limt⟶+∞z=0 and limt⟶+∞qk,1t−ydt=0 accordingly, k=1,⋯,N. The boundedness of qk,1 and qk,2 can be deduced on the basis of the boundedness of Vz and z. The proof is completed.


### 3.2. Control Scheme

In this subsection, the distributed tracking control scheme is constructed cooperating with the filter (8). We define the error variables ek,i∈ℝ, k=1,⋯,N, and i=1,⋯,n as
(15)ek,1=xk,1−qk,1,(16)ek,i=xk,i−vk,i,i=2,⋯,n,where vk.i is the intermediate control signal, which will be defined later.Defineρi=ρ1,1,⋯,ρN,1T∈ℝN and select an output performance function for t≥0(17)ρk,it=ρk,i0−ρk,i∞exp−lk,it+ρk,i∞,where k=1,⋯,N, i=1,⋯,n and the constant design parameters lk,i>0, ρk,i0=ρk,i0>0, and ρk,i∞=limt⟶∞ρk,it>0 are chosen appropriately to satisfy ρk,i0>ρk,i∞ and ek,i0<ρk,i0 with any given initial condition x¯k,i0.Define the barrier functionst↦rk,it as
(18)rk,it=ln1+ξk,it1−ξk,it,where ξk,i=ek,i/ρk,i and i=1,2,⋯,n are the normalized errors. The i+1-th virtual control signals vk,i+1, k=1,⋯,N, and i=1,⋯,n−1 are designed as
(19)vk,i+1=−Kk,irk,i,where Kk,i is a positive control parameter. At this stage, the actual tracking control signal uk is designed as
(20)uk=−Kk,nrk,n,where Kk,n is a positive control parameter.Remark 8.
It is worth pointing out that the output performance functionρk,it is applied to limit the error ek,it within the imposed time-varying constraints. Given any initial condition, the constants ρk,i0, i=1,⋯,n is selected satisfying ρk,i0>ek,i0, and ρk,i∞ represents the maximum allowable value of the steady-state ek,it, which can be assigned to an arbitrarily small value. Hence, the convergence of ek,it to a predefined set of arbitrary small residuals is realized. Furthermore, the rate of ρk,it decline is influenced by lk,i and results in the lower bound of the required rate of ek,it convergence. Therefore, ek,it is restricted to a preassigned boundary by using the performance function.Remark 9.
Under transient and steady state responses, for the construction of (17), the proposed control scheme (19) and (20) merely made up of nonlinear transformation error surfaces and design constants can be designed to preallocate the tracking performance boundary.


### 3.3. Stability and Performance Analysis

Theorem 10.
Consider system (4) obeying Assumption 1 and Assumption 2 controlled by the virtual control signals (19) and the proposed distributed controller (20), all the signals in the closed-loop system are globally bounded. Then, we have the following properties:
(1)
The normalized errorξk,it, k=1,⋯,N, and i=1,⋯,n satisfies ξk,it<1, which can be guaranteed by prespecified tracking performance(2)
The outputs of each agent eventually satisfylimt⟶+∞ykt−ydt<ρk,1∞, where k=1,⋯,NProof.
From the definition of the errors, the statesx1,⋯,xn can be rewritten as
(21)xk,1=ek,1+qk,1,xk,i=ek,i+vk,i,i=2,⋯,n.
Letξit=ξ1,it,⋯,ξN,itT∈ℝN, from the definition of ξit, we can get that
(22)ξ̇k,1=fk,1t,x¯k,1+gk,1t,x¯k,1xk,2pk,1−qk,2−ξk,1ρ̇k,1ρk,1,(23)ξ̇k,i=fk,it,x¯k,i+gk,it,x¯k,ixk,i+1pk,i−v̇k,i−ξk,iρ̇k,iρk,i,i=2,⋯,n−1,(24)ξ̇k,n=fk,nt,x¯k,n+gk,nt,x¯k,nhktuk+dktpk,n−v̇k,n−ξk,nρ̇k,nρn.
By denotingrit=r1,it,⋯,rN,itT∈ℝN, i=1,⋯,n, the time derivative of rit can be given by
(25)ṙk,1=μk,1fk,1t,x¯k,1+gk,1t,x¯k,1xk,2pk,1−qk,2−ξk,1ρ̇k,1,(26)ṙk,i=μk,ifk,it,x¯k,i+gk,it,x¯k,ixk,i+1pk,i−v̇k,i−ξk,iρ̇k,i,i=2,⋯,n−1,(27)ṙk,n=μk,nfk,nt,x¯k,n+gk,nt,x¯k,nhktuk+dktpk,n−v̇k,n−ξk,nρ̇k,n,where μk,i=2/ρk,i1−ξk,i2∈ℝ.
The performance functionsρk,it have been chosen to meet requirements as ρk,i0>ek,i0, which equals to ξ¯0∈ϒ where ϒ=ϒ1×⋯×ϒn an open set with ϒi=−1,1, i=1,⋯,n. Additionally, the fact that from (17), the desired trajectory yd and the performance functions ρk,it are bounded and continuously differentiable with respect to time. The intermediate control signals vk,i, i=2,⋯,n, and the control law uk,i are smooth over the set ϒ. It is deduced that ξ̇kt is bounded and piecewise continuous in t and locally Lipschitz on ξkt over ϒ. According to Theorem 54 of [33], the conditions on ξ̇kt ensure the existence and uniqueness of a maximal solution ξkt of (22)–(24) over the set ϒ, for ∀t∈0,τmax, k=1,⋯,N,i=1,⋯,n such that ξkt∈ϒ, ∀t∈0,τmax or equivalently that for ∀t∈0,τmax, the following formula holds
(28)ξk,it∈−1,1.
Sincerk,i is well established.Next,τmax=+∞ will be proved by looking for a contradiction. It is supposed that τmax<+∞, thus, the following analysis is performed. And a systematic procedure for the proof of the aforementioned statements is given below for t∈0,τmax.Step 1.
From (17) and (28), for any t∈0,τmax, xk,1t is bounded as
(29)xk,1t=ξk,1tρk,1t+qk,1t≤ρk,i0+q¯k,1,where q¯k,1>0 is a constant such that qk,1<q¯k,1. Moreover, the boundedness of xk,1 results in the boundedness of the functions fk,1t,xk,1 as
(30)fk,1t,xk,1≤βk,1xk,1t,where βk,1xk,1 is a positive C1 nonlinear function.By denotingrit=r1,it,⋯,rN,itT∈ℝN, establish the first Lyapunov function candidate as
(31)V1=12r1Tr1.Substituting (18) and (22), the time derivative of V1 is derived as follows
(32)V̇1=∑k=1Nrk,1μk,1fk,1t,xk,1+gk,1t,xk,1xk,2p1−qk,2−ρ̇k,1ξk,1.Using (30) and yields
(33)∑k=1Nfk,1xk,1−qk,2−ξk,1ρ̇k,1≤∑k=1Nι⌣k,1≤∑k=1NCk,1,where ι⌣k,1≜βk,1xk,1+qk,2+ςk,1ρ̇k,1 and Ck,1 is a positive constant.Applying Lemma3 to the terms rk,1μk,1gk,1t,xk,1xk,2p1, using ek,2=xk,2−vk,2 and ek,2=ξk,2ρk,2, one has
(34)rk,1μk,1gk,1t,xk,1ξk,2ρk,2+vk,2pk,1≤−Kk,1pk,1g¯k,12pk,1−1μk,1rk,1pk,1+1+μk,1rk,1g¯k,1ξk,2ρk,2pk,1,where g¯k,1 and g¯k,1 are both positive constants satisfying g¯k,1≤gk,1t,xk,1≤g¯k,1.Combining (33) and (34), we obtain
(35)V̇1=∑k=1N−κ¯k,1μk,1rk,1pk,1+1+g¯k,1ρk,20pk,1+Ck,1μk,1rk,1,where κ¯k,1=Kk,1pk,1g¯k,1/2pk,1−1.Applying Lemma4 to the terms κ¯k,1μk,1rk,1pk,1+1, one has
(36)κ¯k,1μk,1rk,1≤κ¯k,1μk,1pk,1+1rk,1pk,1+1+pk,1pk,1+1κ¯k,1μk,1.Therefore, (35) becomes
(37)V̇1=∑k=1Nμk,1−κ~k,1rk,1+ιk,1,where κ~k,1=κ¯k,1pk,1+1−g¯k,1ρk,20pk,1−Ck,1, ιk,1=pk,1κ¯k,1. From (37), for all t∈0,τmax, it follows that V̇1 is negative when rk,1≥ιk,1/κ⌢1, where κ⌢1=min0≤k≤N,k∈N+κ~k,1 and subsequently that rk,1t<r¯k,1≤r¯1≜max0≤k≤N,k∈N+ιk,1/κ⌢1, which implies that the normalized error is bounded as
(38)−1<e−r¯1−1e−r¯1+1=ξk,1,low<ξk,1t<ξk,1upper=er¯1−1er¯1+1<1,for k=1,⋯,N. According to (19), for all t∈0,τmax, the boundedness of r1 results in the boundedness of v2t. Additionally, from ξk,i=ek,i/ρk,i, we conclude that
(39)−ρk,1t<e−r¯1−1e−r¯1+1ρk,1t≤ek,1t≤er¯1−1er¯1+1ρk,1t<ρk,1t.Stepii=2,⋯,n−1: construct the i−th Lyapunov function as
(40)Vi=12riTri.Differentiating along (40) with respect to time, using (26), yields
(41)V̇i=∑k=1Nrk,iμk,ifk,it,x¯k,i+gk,it,x¯k,ixk,i+1pk,i−v̇k,i−ρ̇k,iξk,i.According to the boundedness ofvit in the last step, using ek,i=xk,i−vk,i, ek,i=ξk,iρk,i, and (28), xk,it is bounded as xk,it=ξk,itρk,it+vk,it≤ρk,i0+vk,i, k=1,⋯,N, and i=1,⋯,n. Furthermore, the boundedness of x¯k,it results in the boundedness of fk,i. From (38), for all t∈0,τmax, μk,i−1=2/ρk,i−11−ξk,i−12∈ℝ is also bounded. Moreover, v̇it is bounded. Similar to (30) and (33), there is a C1 nonlinear function ι⌣k,i, and we have
(42)V̇i≤∑k=1Nrk,iμk,igk,it,x¯k,ixk,i+1pk,i+ι⌣k,i≤∑k=1Nrk,iμk,igk,it,x¯k,ixk,i+1pk,i+Ck,i,where ι⌣k,i≜fk,it,x¯k,i+v̇k,i+ξk,iρ̇k,i and Ck,i is a positive constant.Applying Lemma3 to the terms gk,it,x¯k,ixk,i+1pk,i, using ek,i+1=ξk,i+1ρk,i+1, (42) becomes
(43)V̇i≤∑k=1N−κ¯k,iμk,irk,ipk,i+1+κ⌣k,i+Ck,iμk,irk,i,where κ¯k,1=Kk,1pk,1g¯k,1/2pk,1−1, κ⌣k,i=g¯k,iρk,i0pk,i, g¯k,i>0, and g¯k,i>0 are both constants satisfying g¯k,i≤gk,it,x¯k,i≤g¯k,i.Then, applying Lemma4 to the terms κ¯k,iμk,irk,ipk,i+1 yields
(44)κ¯k,iμk,irk,i≤μk,iκ¯k,ipk,i+1rk,ipk,i+1+μk,ipk,ipk,i+1κ¯k,i.Thus, (43) becomes
(45)V̇i=∑k=1Nμk,i−κ~k,irk,i+ιk,i,where κ~k,i=κ¯k,ipk,i+1−κ⌣k,i−Ck,i, ιk,i=pk,iκ¯k,i. From (45), we conclude that V̇i<0 when rk,i≥ιk,i/κ⌢i, where κ⌢i=min0≤k≤N,k∈N+κ~k,i and subsequently that rk,it<r¯k,i≤r¯i≜max0≤k≤N,k∈N+ιk,i/κ⌢i for all t∈0,τmax. Additionally, for all t∈0,τmax by using (18), we get
(46)−1<e−r¯i−1e−r¯i+1=ξi,low<ξk,it<ξi,upper=er¯i−1er¯i+1<1,Fromξk,i=ek,i/ρk,i, (46) implies that
(47)−ρk,it<e−r¯i1−1e−r¯i+1ρk,it≤ek,it≤er¯i−1er¯i+1ρk,it<ρk,it.In addition, according to (19), the boundedness of ri results in the boundedness of vi+1t, i=2,⋯,n−1 for all t∈0,τmax.Stepn: construct the final Lyapunov function as
(48)Vn=12rnTrn.Differentiating along (48) with respect to time and invoking (18) and (24), one has
(49)V̇n=∑k=1Nrk,nμk,nfk,nt,x¯k,n−v̇k,n−ξk,nρ̇k,n+gk,nt,x¯k,nhktuk+dktpk,n.It can be deduced that for allt∈0,τmax, μk,n−1 is bounded from (46). Based on the boundedness of vk,nt in the n−1 step. Additionally, from (17), (28), it is obviously that xk,nt is bounded as xk,nt=ξk,ntρk,nt+vk,nt≤ρk,n0+vk,n for all t∈0,τmax. From (19), v̇k,n is also bounded. Then, the boundedness of x¯k,n results in the boundedness of fk,n−1t,x¯k,n−1. Therefore, there is a constant Ck,n>0 satisfying fk,nt,x¯k,n+v̇k,n+ξk,nρ̇k,n≜ι⌣k,n≤Ck,n, such that
(50)V̇k,n≤∑k=1Nμk,nrk,ngk,nt,x¯k,nhktuk+dktpk,n+μk,nrk,nCk,n.Under Assumption2 and Lemma 3, invoking (20) and the previous analysis yields
(51)μk,ngk,nt,x¯k,nrk,nhktuk+dktpk,n=μk,ngk,nrk,n−Kk,nhkrk,n+dkpk,n−dkpk,n+dkpk,n≤−μk,nKk,npk,ngk,n2pk,n−1h¯kpk,nrk,npk,n+1+μk,ngk,nrk,nd⌢kpk,n,where h¯k is an unknown positive constant satisfying h¯k≤h⌣ku≤hku, with an unknown positive continuous function h⌣ku. Invoking (51) and Assumption 2, we have
(52)V̇n≤∑k=1Nμk,n−Knpk,ng¯k,n2pk,n−1h¯kpk,nrk,npk,n+1+rk,nι⌢k,n,where ι⌢k,n=Ck,n+g¯k,nd¯kpk,n with an unknown positive constant d¯k satisfying d¯k≤d⌢ku, g¯k,n and g¯k,n are both positive constants satisfying g¯k,n≤gk,nt,x¯k,n≤g¯k,n.Substituting (20) into the third term of (52) yields
(53)V̇n≤∑k=1N−Knpk,ng¯k,nh¯kpk,n2pk,n−1μk,nrk,npk,n+1+μk,nι⌢k,nrk,n.Applying Lemma4 to the term g¯k,nh¯kpk,nμk,nrk,npk,n+1, we obtain
(54)g¯k,nh¯kpnμk,nrk,n≤g¯k,nh¯kpnμk,npk,n+1rk,npn+1+pk,npk,n+1g¯k,nh¯kpnμk,n.Thus, (53) becomes
(55)V̇n≤∑k=1Nμk,n−κ~k,nrk,n+ιk,n,where κ~k,n=Knpk,ng¯k,nh¯kpk,npk,n+1/2pk,n−1−ι⌢k,n and ιk,n=Knpk,npk,ng¯k,nh¯kpk,n/2pk,n−1. Then, where V̇n is negative when rk,nt≥ιk,n/κ⌢n, where κ⌢n=min0≤k≤N,k∈N+κ~k,n, we obtain rk,nt<r¯k,n≤r¯n≜max0≤k≤N,k∈N+ιk,n/κ⌢n for all t∈0,τmax. Invoking (18), for k=1,⋯,N, we get
(56)−1<e−r¯n−1e−r¯n+1=ξn,low<ξk,nt<ξn,upper=er¯n−1er¯n+1<1.Fromξk,n=ek,n/ρk,n, (56) implies that
(57)−ρk,nt<e−r¯n1−1e−r¯n1+1ρk,nt≤ek,nt≤er¯n−1er¯n+1ρk,nt<ρk,nt.As a result, due to (20), the control signal uk,nt is bounded from the boundedness of rk,nt. Moreover, for all t∈0,τmax, (38), (46), and (56) imply that ξk,it∈ϒξ⊂ϒ, where the set ϒξ=ξi,low,ξi,upper×⋯×ξn,low,ξn,upper is nonempty and compact. Thus, it is an obvious contradiction to assume that τmax<+∞ determines the existence of a temporal instant tξ∈0,τmax, which means that ek,itξ∉ϒξ. Therefore, τmax=+∞. Finally, from (39), (47), and (57), we can draw the conclusion that ek,it<ρk,it for all t≥0. Based on the exponential decay property of ρk,i, the following inequality holds limt⟶∞ek,i<ρk,i∞, k=1,⋯,N, and i=1,⋯,n. Then, in view of (15), we have ykt−ydt=ykt−qk,1t+qk,1t−ydt≤ek,1t+qk,1t−ydt. Based on Theorem 7, it can be derived that
(58)limt⟶+∞ykt−ydt<ρk,1∞.In the Lyapunov sense,ek,it is kept within the preassigned boundary of transient and steady state range, and this is the end of the proof.Remark 11.
From Theorem10, it should be noticed that this memoryless control tracker is recursively constructed by virtue of the specified performance design method, and the transient and steady state performance boundaries of ek,i are up to the performance functions ρk,i.


## 3.1. Filter Design

To facilitate the distributed control design, a filterqk,1,qk,2 is utilized for each agent k, where k=1,⋯,N.Denote(7)zk,p=bkqk,p−ydp−1+∑j=1Nakjqk,p−qj,p,p=1,2,where yd0=yd, yd1=ẏd. Then, design the filters as
(8)q̇k,1=qk,2,q̇k,2=αk,with
(9)αk=−c2zk,1−c1zk,2−c0qk,2−c0sgnz¯k∑p=12F^k,p,(10)F^̇k,1=bkFk,1−yd+∑j=1NakjF^k,1−F^j,1,F^̇k,2=bkFk,2−ẏd+∑j=1NakjF^k,2−F^j,2,where c0, c1, and c2 are positive constant parameters selected as c0≥1, c1>c0+1, and c2=c0c1, Fk,1 is the upper boundary of yd, Fk,2 is the upper boundary of ẏd, k=1,⋯,N.Theorem 7.
Consider a closed-loop system composed ofN filters (8) under Assumption 1 with control design (9). The asymptotic and consistent tracking of all outputs of the filter to ydt can be realized (i.e., limt⟶+∞qk,1−ydt=0, k=1,⋯,N). In addition, qk,1 and qk,2 are bounded.Proof.
Based on Assumption1, the matrix L+B is nonsingular. Same as [5], define θ=θ1,⋯,θNT=L+B−11,⋯,1T, P=diagP1,⋯,PN=diagθ1−1,⋯,θN−1, Q=PL+B+L+BTP, where θk>0 for k=1,⋯,N. It can be summarized from [5] that Q is positive definite.
Construct the following Lyapunov function(11)Vz=12zTPz+12γ∑p=12F~pTPF~p,where z=z¯1,z¯2,⋯,z¯NT∈ℝN with z¯k=c0zk,1+zk,2, F^p=F∧1,p,F∧2,p,⋯,F∧N,pT∈ℝN, Fp=F1,p,F2,p,⋯,FN,pT∈ℝN, F~p=F^p−Fp∈ℝN, p=1,2, and γ>0 is a constant satisfying γ<2λmin2Q/φ2 with φ=PL+B∈ℝ. Denote zp=z1,p,z2,p,⋯,zN,pT∈ℝN, α=α1,α2,⋯,αNT∈ℝN, and qp=q1,p,q2,p,⋯,qN,pT∈ℝN, p=1,2. Then, the following equation can be obtained
(12)ż=L+Bc0q2+α−c0ẏd−y¨d.
Through (11) and (12), by denoting sgnz¯=sgnz¯1,⋯,sgnz¯NT, the time derivative of Vz is
(13)V̇z=zTPL+B−c1z−c0∑p=12sgnz¯.∗Fp+c0∑p=12εz¯.∗Fp−c0ẏd−y¨d−1γ∑p=12F~pTPL+BF~p≤−c1zTQz−1γ∑p=12F~pTQF~p−c0∑p=12zTPDsgnz¯.∗Fp+c0∑p=12zTPAsgnz¯.∗Fp−c0∑p=12zTPBsgnz¯.∗Fp−∑p=12zTPL+Bc0ẏd+y¨d+c0∑p=12zPL+BF~p.
It should be noticed thatc0∑p=12zTPDsgnz.∗Fp=c0∑p=12Fp∑j=1Npjaijz¯j, c0∑p=12zTPAsgnz.∗Fp≤c0∑p=12Fp∑j=1Npjaijz¯j, c0∑p=12zTPBsgnz.∗Fp=c0∑p=12Fp∑j=1Npjbjz¯j, ∑p=12zTPL+Bc0Ẋd+X¨d≤c0∑p=12Fp∑j=1Npjμjz¯j, and ∑p=12zPL+BF~p2+∑p=12φ2F~p/2λminQ, with λminQ representing the minimum eigenvalue of Q.
Combing these above inequality, (13) becomes
(14)V̇z≤−c2z2−γ∗F~p,where c2=λminQc1−c0, γ∗=λminQ1/γ−φ2/2λmin2Q. It is easy to verify that c2>0 and γ∗>0. Thus, it can be obtained from (14) that limt⟶+∞z=0 and limt⟶+∞qk,1t−ydt=0 accordingly, k=1,⋯,N. The boundedness of qk,1 and qk,2 can be deduced on the basis of the boundedness of Vz and z. The proof is completed.


## 3.2. Control Scheme

In this subsection, the distributed tracking control scheme is constructed cooperating with the filter (8). We define the error variables ek,i∈ℝ, k=1,⋯,N, and i=1,⋯,n as
(15)ek,1=xk,1−qk,1,(16)ek,i=xk,i−vk,i,i=2,⋯,n,where vk.i is the intermediate control signal, which will be defined later.Defineρi=ρ1,1,⋯,ρN,1T∈ℝN and select an output performance function for t≥0(17)ρk,it=ρk,i0−ρk,i∞exp−lk,it+ρk,i∞,where k=1,⋯,N, i=1,⋯,n and the constant design parameters lk,i>0, ρk,i0=ρk,i0>0, and ρk,i∞=limt⟶∞ρk,it>0 are chosen appropriately to satisfy ρk,i0>ρk,i∞ and ek,i0<ρk,i0 with any given initial condition x¯k,i0.Define the barrier functionst↦rk,it as
(18)rk,it=ln1+ξk,it1−ξk,it,where ξk,i=ek,i/ρk,i and i=1,2,⋯,n are the normalized errors. The i+1-th virtual control signals vk,i+1, k=1,⋯,N, and i=1,⋯,n−1 are designed as
(19)vk,i+1=−Kk,irk,i,where Kk,i is a positive control parameter. At this stage, the actual tracking control signal uk is designed as
(20)uk=−Kk,nrk,n,where Kk,n is a positive control parameter.Remark 8.
It is worth pointing out that the output performance functionρk,it is applied to limit the error ek,it within the imposed time-varying constraints. Given any initial condition, the constants ρk,i0, i=1,⋯,n is selected satisfying ρk,i0>ek,i0, and ρk,i∞ represents the maximum allowable value of the steady-state ek,it, which can be assigned to an arbitrarily small value. Hence, the convergence of ek,it to a predefined set of arbitrary small residuals is realized. Furthermore, the rate of ρk,it decline is influenced by lk,i and results in the lower bound of the required rate of ek,it convergence. Therefore, ek,it is restricted to a preassigned boundary by using the performance function.Remark 9.
Under transient and steady state responses, for the construction of (17), the proposed control scheme (19) and (20) merely made up of nonlinear transformation error surfaces and design constants can be designed to preallocate the tracking performance boundary.


## 3.3. Stability and Performance Analysis

Theorem 10.
Consider system (4) obeying Assumption 1 and Assumption 2 controlled by the virtual control signals (19) and the proposed distributed controller (20), all the signals in the closed-loop system are globally bounded. Then, we have the following properties:
(1)
The normalized errorξk,it, k=1,⋯,N, and i=1,⋯,n satisfies ξk,it<1, which can be guaranteed by prespecified tracking performance(2)
The outputs of each agent eventually satisfylimt⟶+∞ykt−ydt<ρk,1∞, where k=1,⋯,NProof.
From the definition of the errors, the statesx1,⋯,xn can be rewritten as
(21)xk,1=ek,1+qk,1,xk,i=ek,i+vk,i,i=2,⋯,n.
Letξit=ξ1,it,⋯,ξN,itT∈ℝN, from the definition of ξit, we can get that
(22)ξ̇k,1=fk,1t,x¯k,1+gk,1t,x¯k,1xk,2pk,1−qk,2−ξk,1ρ̇k,1ρk,1,(23)ξ̇k,i=fk,it,x¯k,i+gk,it,x¯k,ixk,i+1pk,i−v̇k,i−ξk,iρ̇k,iρk,i,i=2,⋯,n−1,(24)ξ̇k,n=fk,nt,x¯k,n+gk,nt,x¯k,nhktuk+dktpk,n−v̇k,n−ξk,nρ̇k,nρn.
By denotingrit=r1,it,⋯,rN,itT∈ℝN, i=1,⋯,n, the time derivative of rit can be given by
(25)ṙk,1=μk,1fk,1t,x¯k,1+gk,1t,x¯k,1xk,2pk,1−qk,2−ξk,1ρ̇k,1,(26)ṙk,i=μk,ifk,it,x¯k,i+gk,it,x¯k,ixk,i+1pk,i−v̇k,i−ξk,iρ̇k,i,i=2,⋯,n−1,(27)ṙk,n=μk,nfk,nt,x¯k,n+gk,nt,x¯k,nhktuk+dktpk,n−v̇k,n−ξk,nρ̇k,n,where μk,i=2/ρk,i1−ξk,i2∈ℝ.
The performance functionsρk,it have been chosen to meet requirements as ρk,i0>ek,i0, which equals to ξ¯0∈ϒ where ϒ=ϒ1×⋯×ϒn an open set with ϒi=−1,1, i=1,⋯,n. Additionally, the fact that from (17), the desired trajectory yd and the performance functions ρk,it are bounded and continuously differentiable with respect to time. The intermediate control signals vk,i, i=2,⋯,n, and the control law uk,i are smooth over the set ϒ. It is deduced that ξ̇kt is bounded and piecewise continuous in t and locally Lipschitz on ξkt over ϒ. According to Theorem 54 of [33], the conditions on ξ̇kt ensure the existence and uniqueness of a maximal solution ξkt of (22)–(24) over the set ϒ, for ∀t∈0,τmax, k=1,⋯,N,i=1,⋯,n such that ξkt∈ϒ, ∀t∈0,τmax or equivalently that for ∀t∈0,τmax, the following formula holds
(28)ξk,it∈−1,1.
Sincerk,i is well established.Next,τmax=+∞ will be proved by looking for a contradiction. It is supposed that τmax<+∞, thus, the following analysis is performed. And a systematic procedure for the proof of the aforementioned statements is given below for t∈0,τmax.Step 1.
From (17) and (28), for any t∈0,τmax, xk,1t is bounded as
(29)xk,1t=ξk,1tρk,1t+qk,1t≤ρk,i0+q¯k,1,where q¯k,1>0 is a constant such that qk,1<q¯k,1. Moreover, the boundedness of xk,1 results in the boundedness of the functions fk,1t,xk,1 as
(30)fk,1t,xk,1≤βk,1xk,1t,where βk,1xk,1 is a positive C1 nonlinear function.By denotingrit=r1,it,⋯,rN,itT∈ℝN, establish the first Lyapunov function candidate as
(31)V1=12r1Tr1.Substituting (18) and (22), the time derivative of V1 is derived as follows
(32)V̇1=∑k=1Nrk,1μk,1fk,1t,xk,1+gk,1t,xk,1xk,2p1−qk,2−ρ̇k,1ξk,1.Using (30) and yields
(33)∑k=1Nfk,1xk,1−qk,2−ξk,1ρ̇k,1≤∑k=1Nι⌣k,1≤∑k=1NCk,1,where ι⌣k,1≜βk,1xk,1+qk,2+ςk,1ρ̇k,1 and Ck,1 is a positive constant.Applying Lemma3 to the terms rk,1μk,1gk,1t,xk,1xk,2p1, using ek,2=xk,2−vk,2 and ek,2=ξk,2ρk,2, one has
(34)rk,1μk,1gk,1t,xk,1ξk,2ρk,2+vk,2pk,1≤−Kk,1pk,1g¯k,12pk,1−1μk,1rk,1pk,1+1+μk,1rk,1g¯k,1ξk,2ρk,2pk,1,where g¯k,1 and g¯k,1 are both positive constants satisfying g¯k,1≤gk,1t,xk,1≤g¯k,1.Combining (33) and (34), we obtain
(35)V̇1=∑k=1N−κ¯k,1μk,1rk,1pk,1+1+g¯k,1ρk,20pk,1+Ck,1μk,1rk,1,where κ¯k,1=Kk,1pk,1g¯k,1/2pk,1−1.Applying Lemma4 to the terms κ¯k,1μk,1rk,1pk,1+1, one has
(36)κ¯k,1μk,1rk,1≤κ¯k,1μk,1pk,1+1rk,1pk,1+1+pk,1pk,1+1κ¯k,1μk,1.Therefore, (35) becomes
(37)V̇1=∑k=1Nμk,1−κ~k,1rk,1+ιk,1,where κ~k,1=κ¯k,1pk,1+1−g¯k,1ρk,20pk,1−Ck,1, ιk,1=pk,1κ¯k,1. From (37), for all t∈0,τmax, it follows that V̇1 is negative when rk,1≥ιk,1/κ⌢1, where κ⌢1=min0≤k≤N,k∈N+κ~k,1 and subsequently that rk,1t<r¯k,1≤r¯1≜max0≤k≤N,k∈N+ιk,1/κ⌢1, which implies that the normalized error is bounded as
(38)−1<e−r¯1−1e−r¯1+1=ξk,1,low<ξk,1t<ξk,1upper=er¯1−1er¯1+1<1,for k=1,⋯,N. According to (19), for all t∈0,τmax, the boundedness of r1 results in the boundedness of v2t. Additionally, from ξk,i=ek,i/ρk,i, we conclude that
(39)−ρk,1t<e−r¯1−1e−r¯1+1ρk,1t≤ek,1t≤er¯1−1er¯1+1ρk,1t<ρk,1t.Stepii=2,⋯,n−1: construct the i−th Lyapunov function as
(40)Vi=12riTri.Differentiating along (40) with respect to time, using (26), yields
(41)V̇i=∑k=1Nrk,iμk,ifk,it,x¯k,i+gk,it,x¯k,ixk,i+1pk,i−v̇k,i−ρ̇k,iξk,i.According to the boundedness ofvit in the last step, using ek,i=xk,i−vk,i, ek,i=ξk,iρk,i, and (28), xk,it is bounded as xk,it=ξk,itρk,it+vk,it≤ρk,i0+vk,i, k=1,⋯,N, and i=1,⋯,n. Furthermore, the boundedness of x¯k,it results in the boundedness of fk,i. From (38), for all t∈0,τmax, μk,i−1=2/ρk,i−11−ξk,i−12∈ℝ is also bounded. Moreover, v̇it is bounded. Similar to (30) and (33), there is a C1 nonlinear function ι⌣k,i, and we have
(42)V̇i≤∑k=1Nrk,iμk,igk,it,x¯k,ixk,i+1pk,i+ι⌣k,i≤∑k=1Nrk,iμk,igk,it,x¯k,ixk,i+1pk,i+Ck,i,where ι⌣k,i≜fk,it,x¯k,i+v̇k,i+ξk,iρ̇k,i and Ck,i is a positive constant.Applying Lemma3 to the terms gk,it,x¯k,ixk,i+1pk,i, using ek,i+1=ξk,i+1ρk,i+1, (42) becomes
(43)V̇i≤∑k=1N−κ¯k,iμk,irk,ipk,i+1+κ⌣k,i+Ck,iμk,irk,i,where κ¯k,1=Kk,1pk,1g¯k,1/2pk,1−1, κ⌣k,i=g¯k,iρk,i0pk,i, g¯k,i>0, and g¯k,i>0 are both constants satisfying g¯k,i≤gk,it,x¯k,i≤g¯k,i.Then, applying Lemma4 to the terms κ¯k,iμk,irk,ipk,i+1 yields
(44)κ¯k,iμk,irk,i≤μk,iκ¯k,ipk,i+1rk,ipk,i+1+μk,ipk,ipk,i+1κ¯k,i.Thus, (43) becomes
(45)V̇i=∑k=1Nμk,i−κ~k,irk,i+ιk,i,where κ~k,i=κ¯k,ipk,i+1−κ⌣k,i−Ck,i, ιk,i=pk,iκ¯k,i. From (45), we conclude that V̇i<0 when rk,i≥ιk,i/κ⌢i, where κ⌢i=min0≤k≤N,k∈N+κ~k,i and subsequently that rk,it<r¯k,i≤r¯i≜max0≤k≤N,k∈N+ιk,i/κ⌢i for all t∈0,τmax. Additionally, for all t∈0,τmax by using (18), we get
(46)−1<e−r¯i−1e−r¯i+1=ξi,low<ξk,it<ξi,upper=er¯i−1er¯i+1<1,Fromξk,i=ek,i/ρk,i, (46) implies that
(47)−ρk,it<e−r¯i1−1e−r¯i+1ρk,it≤ek,it≤er¯i−1er¯i+1ρk,it<ρk,it.In addition, according to (19), the boundedness of ri results in the boundedness of vi+1t, i=2,⋯,n−1 for all t∈0,τmax.Stepn: construct the final Lyapunov function as
(48)Vn=12rnTrn.Differentiating along (48) with respect to time and invoking (18) and (24), one has
(49)V̇n=∑k=1Nrk,nμk,nfk,nt,x¯k,n−v̇k,n−ξk,nρ̇k,n+gk,nt,x¯k,nhktuk+dktpk,n.It can be deduced that for allt∈0,τmax, μk,n−1 is bounded from (46). Based on the boundedness of vk,nt in the n−1 step. Additionally, from (17), (28), it is obviously that xk,nt is bounded as xk,nt=ξk,ntρk,nt+vk,nt≤ρk,n0+vk,n for all t∈0,τmax. From (19), v̇k,n is also bounded. Then, the boundedness of x¯k,n results in the boundedness of fk,n−1t,x¯k,n−1. Therefore, there is a constant Ck,n>0 satisfying fk,nt,x¯k,n+v̇k,n+ξk,nρ̇k,n≜ι⌣k,n≤Ck,n, such that
(50)V̇k,n≤∑k=1Nμk,nrk,ngk,nt,x¯k,nhktuk+dktpk,n+μk,nrk,nCk,n.Under Assumption2 and Lemma 3, invoking (20) and the previous analysis yields
(51)μk,ngk,nt,x¯k,nrk,nhktuk+dktpk,n=μk,ngk,nrk,n−Kk,nhkrk,n+dkpk,n−dkpk,n+dkpk,n≤−μk,nKk,npk,ngk,n2pk,n−1h¯kpk,nrk,npk,n+1+μk,ngk,nrk,nd⌢kpk,n,where h¯k is an unknown positive constant satisfying h¯k≤h⌣ku≤hku, with an unknown positive continuous function h⌣ku. Invoking (51) and Assumption 2, we have
(52)V̇n≤∑k=1Nμk,n−Knpk,ng¯k,n2pk,n−1h¯kpk,nrk,npk,n+1+rk,nι⌢k,n,where ι⌢k,n=Ck,n+g¯k,nd¯kpk,n with an unknown positive constant d¯k satisfying d¯k≤d⌢ku, g¯k,n and g¯k,n are both positive constants satisfying g¯k,n≤gk,nt,x¯k,n≤g¯k,n.Substituting (20) into the third term of (52) yields
(53)V̇n≤∑k=1N−Knpk,ng¯k,nh¯kpk,n2pk,n−1μk,nrk,npk,n+1+μk,nι⌢k,nrk,n.Applying Lemma4 to the term g¯k,nh¯kpk,nμk,nrk,npk,n+1, we obtain
(54)g¯k,nh¯kpnμk,nrk,n≤g¯k,nh¯kpnμk,npk,n+1rk,npn+1+pk,npk,n+1g¯k,nh¯kpnμk,n.Thus, (53) becomes
(55)V̇n≤∑k=1Nμk,n−κ~k,nrk,n+ιk,n,where κ~k,n=Knpk,ng¯k,nh¯kpk,npk,n+1/2pk,n−1−ι⌢k,n and ιk,n=Knpk,npk,ng¯k,nh¯kpk,n/2pk,n−1. Then, where V̇n is negative when rk,nt≥ιk,n/κ⌢n, where κ⌢n=min0≤k≤N,k∈N+κ~k,n, we obtain rk,nt<r¯k,n≤r¯n≜max0≤k≤N,k∈N+ιk,n/κ⌢n for all t∈0,τmax. Invoking (18), for k=1,⋯,N, we get
(56)−1<e−r¯n−1e−r¯n+1=ξn,low<ξk,nt<ξn,upper=er¯n−1er¯n+1<1.Fromξk,n=ek,n/ρk,n, (56) implies that
(57)−ρk,nt<e−r¯n1−1e−r¯n1+1ρk,nt≤ek,nt≤er¯n−1er¯n+1ρk,nt<ρk,nt.As a result, due to (20), the control signal uk,nt is bounded from the boundedness of rk,nt. Moreover, for all t∈0,τmax, (38), (46), and (56) imply that ξk,it∈ϒξ⊂ϒ, where the set ϒξ=ξi,low,ξi,upper×⋯×ξn,low,ξn,upper is nonempty and compact. Thus, it is an obvious contradiction to assume that τmax<+∞ determines the existence of a temporal instant tξ∈0,τmax, which means that ek,itξ∉ϒξ. Therefore, τmax=+∞. Finally, from (39), (47), and (57), we can draw the conclusion that ek,it<ρk,it for all t≥0. Based on the exponential decay property of ρk,i, the following inequality holds limt⟶∞ek,i<ρk,i∞, k=1,⋯,N, and i=1,⋯,n. Then, in view of (15), we have ykt−ydt=ykt−qk,1t+qk,1t−ydt≤ek,1t+qk,1t−ydt. Based on Theorem 7, it can be derived that
(58)limt⟶+∞ykt−ydt<ρk,1∞.In the Lyapunov sense,ek,it is kept within the preassigned boundary of transient and steady state range, and this is the end of the proof.Remark 11.
From Theorem10, it should be noticed that this memoryless control tracker is recursively constructed by virtue of the specified performance design method, and the transient and steady state performance boundaries of ek,i are up to the performance functions ρk,i.


## 4. Simulation Study

A numerical simulation study and a practical simulation study are presented in this section to give evidence of the efficacy and advantage of the constructed control design.Example 1.
Consider the uncertain MAS with unknown high powers and input deadzone with a communication topology depicted in Figure2 as follows
(59)ẋk,1=fk,1t,xk,1+gk,1t,xk,1xk,2pk,1,ẋk,2=fk,2t,x¯k,2+gk,2t,x¯k,2xk,3pk,2,ẋk,3=fk,3t,x¯k,3+gk,3t,x¯k,3hktuk+dkt,upk,3,yk=xk,1,k=1,⋯,4,where pk,1=pk,2=pk,3=3, gk,1=0.8+0.5x12, gk,2=1+0.4sinx3, gk,3=1, fk,1=sinxk,1, fk,2=xk,3cosxk,2, fk,3=xk,3sin2xk,2, and k=1,⋯,4. The desired signal is yd=cos0.5t. The initial conditions are selected as xk,10=0.5, xk,20=0, xk,30=0, and k=1,⋯,4.
The proposed tracking control scheme with prescribed performance is established as follows.(60)vk,2=−Kk,1rk,1,vk,3=−Kk,2rk,2,uk=−Kk,3rk,3,where the control parameters are Kk,1=2.8, Kk,2=5, Kk,3=18, and k=1,⋯,4.
Frome10=y0−yd0=−0.5, e20=x20−v20, and e30=x30−v30, set ρ0=ρ1,0,ρ2,0,ρ3,0T=3.5,4.5,2T such that ρ0>e0. For the preselected bound of e1, e2, and e3, design ρ1=3.5−0.35e−2t+0.35, ρ2=4.5−0.30e−3t+0.30, and ρ3=2−0.5e−t+0.5.
In addition, to emphasize the capacity of the proposed controller to deal with the deadzone with au−Du characteristic in the presence of nominal deviation from linear slopes, we consider the deadzone function (2) where hlt=0.80+0.2sint, hrt=0.75+0.2cost, bl=−0.7, and br=0.5.
As expected, from these simulation results shown in Figures3–5, it is indicated that the desired tracking performance can be realized, and all the closed loop signals are bounded under the proposed distributed controller.Example 2.
Consider the dynamics of four practical high maneuverer fighters with input deadzone as follows [34], and for simplicity, the subscripts of the following aerodynamic variables are omitted (i.e., ϕk=ϕ,αk=α,βk=β and so on).
(61)ẋk,1=fk,1t,xk,1+gk,1t,xk,1xk,2pk,1,ẋk,2=fk,2t,x¯k,2+gk,2t,x¯k,2xk,3pk,2,ẋk,3=fk,3t,x¯k,3+gk,3t,x¯k,3hktuk+dkt,upk,3,yk=xk,1,k=1,⋯,4,with
(62)fk,1t,xk,1=qtanθsinϕ+rtanθcosϕpβ+z0Δα+g0/Vcosθcosϕ−cosθ0yββ+psinα0+Δα+g0/Vcosθsinϕ,(63)gk,1t,xk,1=−−1000−1000cosα0,(64)fk,2t,x¯k,2=qcosϕ−rsinϕ,(65)gk,2t,x¯k,2=0,(66)fk,3t,x¯k,3=lββ+lpp+lqq+lrr+lβαβ+lrαrΔα−i1qrmαΔα+mqq+i2pr−mα̇g0Vcosθcosϕ−cosθ0nββ+nrr+npp+npαpΔα−i3pq+nqq,(67)gk,3t,x¯k,3=L,M,NT,(68)L=lδel,lδer,lδal,lδar,0,0,lδr,(69)M=mδel,mδer,mδal,mδar,mδlef,mδtef,mδr,(70)N=nδel,nδer,nδal,nδar,0,0,nδr,where xk,1=ϕ,α,βT is roll angle, attack angle, and sideslip angle, xk,2=θ is pitch angle, xk,3=p,q,r⌢T is roll angular velocity, pitching angular velocity, and yaw angular velocity, respectively. uk=δel,δer,δal,δar,δlef,δtef,δrT is left and right elevators, left and right ailerons, front and rear flaps, and rudder, respectively. The parameters and variables of this model are explained in detail in [34]. Assume that they are all flying at 40,000 feet, at 0.6 Mach. The expected roll angle, attack angle, and sideslip angle for the fighters are yd=ϕd,αd,βdT with ϕd=20°,αd=30°,βd=0°, respectively. The communication topology for the four fighters is shown in Figure 6.Figure 2
Communication topology for 4 subsystems.Figure 3
The output tracking performanceyk and yd.Figure 4
The errorek,1 of the closed-loop system.Figure 5
Control inputu with deadzone.Figure 6
Communication topology for 4 fighters.The tracking errors are defined asϕk,e=ϕk−qd,1, αk,e=αk−qd,2, and βk,e=βk−qd,3, where qd,1, qd,2, and qd,3 are the signals generated by (7), and yd,i is, respectively, the filter inputs. From ek,10=−16°,−25°,0°T, design ρk,10,ρk,20,ρk,30T=20°,33°,3°T to satisfy ρk,i0>ek,i0. ρk,1∞,ρk,2∞,ρk,3∞T=0.05°,0.02°,0.05°T, and lk,1,lk,2,lk,3T=0.25,0.25,0.40T.Defineξk,1,1=ϕk,e/ρ1,1, ξk,1,2=αk,e/ρ1,2, ξk,1,3=βk,e/ρ1,3, ξk,2,1=pk−vk,2,1/ρ2,1, ξk,2,2=qk−vk,2,2/ρ2,2, ξk,2,3=r⌢k−vk,2,3/ρ2,3, K¯k,i=diagKk,i1,Kk,i2,Kk,i3, rk,i,j=ln1+ξi,j/1−ξi,j, k=1,2,3,4, i=1,2,j=1,2,3, and r~k,i=rk,i,1,rk,i,2,rk,i,3T,i=1,2. The proposed control scheme with prescribed performance is established as follows.
(71)vk,2=−K¯k,1r~k,1,uk=−K¯k,2r~k,2,where the control parameters are set as Kk,11=8, Kk,12=5, Kk,13=12, Kk,21=4, Kk,22=5, Kk,23=1.8, and k=1,2,3,4.In addition, to emphasize the capacity of the proposed controller to deal with the deadzone with au−Du characteristic in the presence of nominal deviation from linear slopes, we consider the deadzone function (2) where hl=diag0.80+0.2sint,0.70+0.2cost,0.75+sin3t,0.85+cos0.5t,0.90+sint,0.70+0.2cost,0.80+0.4sinthr=diag0.75+0.2cost,0.80+0.4sint,0.70+0.1sin2t,0.85+0.2cos2t,0.90+0.4sint,0.70+0.2cos2t,0.80+0.4sint, bl=−0.7,0.5,0.3,0.6,0.8,0.4,0.5T, and br=0.5,0.5,0.8,0.6,0.3,0.4,0.7T.In order to facilitate comparison, we exploit the method [6] under the same conditions. From [6], define e~k,i=ek,i,1,ek,i,2,ek,i,3T,i=1,2, and the control protocols are presented as follows
(72)vk,2=−K¯k,1e~k,1,uk=−K¯k,2e~k,2.To ensure the validity of the comparative practical example,K¯k,i and i=1,2 are the same as our proposed methods.The results are indicated in Figures7–13. In Figures 7–9, the chain dotted curves represent the actuator responses under the method control of [6], and the solid curves represent the actuator responses under our proposed methods. It is observed from Figures 7–9 that control performance under our approach is fairly better than [6] since our roll angle, attack angle, and sideslip angle responses can be faster and more accurately trail the desired value. Figures 10–13 indicate the actions of actuators under the method of this paper. Figure 14 demonstrates the control performance of our approach and [6]. As shown in Figure 14, the blue curves represent the actuator efforts E1 under our methodology, and the red curves represent the actuator efforts E2 of [6], where E1 and E2 are defined as
(73)E=δel2+δer2+δal2+δar2+δlef2+δtef2+δr2,Figure 7
Roll angleϕ.Figure 8
Attack angleα.Figure 9
Sideslip angleβ.Figure 10
The responses of actuator of fighter 1.Figure 11
The responses of actuator of fighter 2.Figure 12
The responses of actuator of fighter 3.Figure 13
The responses of actuator of fighter 4.Figure 14
Actuator actions.It is clear from the result presented in Figure14 that at the beginning, the actuator actions of our approach are greater to that in [6], but eventually, the distinction of the efforts between our method and [6] is little. In other words, our method’s control performance is better with the almost same control force.The results that all the closed loop signals are bounded and the preset transient and steady state tracking performance are realized in the finite time can be concluded.


## 5. Conclusions

This paper investigates a distributed consensus control approach under a directed graph for a multiagent system with unknown input deadzone nonlinearities and time-varying control coefficients. To provide the required signals thus avoiding the estimation of the unknown matrix associated with the Laplace matrix, a two-order filter for each agent is established. Together with the two-order filter, a distributed prescribed performance control design was proposed for an uncertain heterogenous power-chained MAS with unknown input deadzone nonlinearities under a directed topology. This consensus control protocol is applied to two examples. Simulations results demonstrate the efficacy and advantage of the method.



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*Source: 6900904-2022-07-08.xml*