Adaptive event‐triggered control of multi‐agent systems with state constraints and unknown disturbances
2021; Institution of Engineering and Technology; Volume: 15; Issue: 17 Linguagem: Inglês
10.1049/cth2.12183
ISSN1751-8652
AutoresXi‐Zi Zhang, Jie Lan, Yan‐Jun Liu, Lei Liu,
Tópico(s)Adaptive Dynamic Programming Control
ResumoIET Control Theory & ApplicationsVolume 15, Issue 17 p. 2171-2182 ORIGINAL RESEARCH PAPEROpen Access Adaptive event-triggered control of multi-agent systems with state constraints and unknown disturbances Xi-Zi Zhang, Xi-Zi Zhang College of Science, Liaoning University of Technology, Jinzhou, 121001 ChinaSearch for more papers by this authorJie Lan, Corresponding Author Jie Lan lanjiecz@163.com orcid.org/0000-0002-8919-8425 College of Science, Liaoning University of Technology, Jinzhou, 121001 China Correspondence Jie Lan, College of Science, Liaoning University of Technology, Jinzhou 121001, China. Email: lanjiecz@163.comSearch for more papers by this authorYan-Jun Liu, Yan-Jun Liu orcid.org/0000-0003-3724-0596 College of Science, Liaoning University of Technology, Jinzhou, 121001 ChinaSearch for more papers by this authorLei Liu, Lei Liu College of Science, Liaoning University of Technology, Jinzhou, 121001 ChinaSearch for more papers by this author Xi-Zi Zhang, Xi-Zi Zhang College of Science, Liaoning University of Technology, Jinzhou, 121001 ChinaSearch for more papers by this authorJie Lan, Corresponding Author Jie Lan lanjiecz@163.com orcid.org/0000-0002-8919-8425 College of Science, Liaoning University of Technology, Jinzhou, 121001 China Correspondence Jie Lan, College of Science, Liaoning University of Technology, Jinzhou 121001, China. Email: lanjiecz@163.comSearch for more papers by this authorYan-Jun Liu, Yan-Jun Liu orcid.org/0000-0003-3724-0596 College of Science, Liaoning University of Technology, Jinzhou, 121001 ChinaSearch for more papers by this authorLei Liu, Lei Liu College of Science, Liaoning University of Technology, Jinzhou, 121001 ChinaSearch for more papers by this author First published: 17 August 2021 https://doi.org/10.1049/cth2.12183AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract The leader-following consensus problem of a class of non-strict feedback multi-agent systems with unknown disturbances, this paper develops a novel adaptive event-triggered control strategy ground on the consideration of the full state constraints. Combining the dynamic surface control and event triggering mechanism, the unknown external disturbances are estimated by designing a disturbance observer. In order to deal with unknown functions, this paper employs the neural networks. On purpose of keeping the output variable within a constraint boundary, the barrier Lyapunov function is chosen during the design course. Then, through the backstepping method and the Lyapunov stability theorem, it is proved that all signals in the closed-loop systems are bounded, and the control performance of the closed-loop system is ensured by selecting design parameters reasonably. Eventually, the availability of the control strategy is guaranteed through experimental example. 1 INTRODUCTION The study about multi-agent systems (MASs) can not only enhance people's understanding of natural phenomena, but also lay a theoretical foundation for the development of various industries. The agents of distributed MASs can cooperate with each other through appropriate strategies to complete the global goals. Therefore, the distributed coordination of MASs acquires further thoughts in many fields includes biology, communication, computer science and other fields [1-7]. MASs also encounter some problems in distributed cooperative control, wherein the consensus problem, as the basis of cooperation and coordinated control between agents, is a research hot issue in many fields [8]. So far, there are so many papers have proposed suitable multi-agent models for various systems. The consensus control protocols were obtained under the following systems such as first-order [9-11], second-order [12-14] or higher-order [15-18] multi-agent system models. Among the rest, there were also many research outcomes about how MASs to address systems with disturbance problems. Aiming at the finite time consensus problem of second-order nonlinear multi-agent systems, a new convergent discontinuous disturbance observer is proposed in ref. [17], which finally realises the finite-time robust consensus of nonlinear MASs with bounded disturbances. In ref. [19], for a general higher order controllable linear system, a distributed consensus control protocol is constructed, under the same network connectivity assumption that the union of the directed interaction graphs contains a spanning tree. It is noteworthy that the research outcomes mentioned above all have a common feature. The aforementioned method does not consider the influence of unknown parameters and uncertain functions on the system, hence it is necessary to use neural networks [20-23] or fuzzy logic systems to deal with MASs with unknown functions. On account of the wide applications of the intelligent control theory, the intelligent control of MASs has drew further and further people's attention [24-26]. Adaptive control technology is particularly important, and there have been a lot of research results in this area [27-31]. In ref. [32], the problem of active fault-tolerant control (FTC) of large-scale nonlinear systems in the form of non-strict feedback was studied, so that a new neural network adaptive output feedback FTC method is proposed, in which the neural network is used to approximate the unknown nonlinear function. The problems of structure uncertainty and unknown correlation coefficients were solved. In ref. [33], on the strength of fuzzy neural network, an intelligent control method of robot football system is constructed. The proposed fuzzy neural network structure can adapt to all possible field configurations flexibly, and is supposed to coordinate multi-agent system by selecting sensitive actions. All the above research results ignore the constraint problems. In actual engineering, the systems usually perform complex works and are blocked by internal conditions and external environments. Accordingly, the variables in the system need to be restricted. That is, the states need to be limited to a certain range, otherwise the stability of the systems may be affected. Consequently, how to handle the state constraints of the system has received extensive attention from researchers [34-38]. In ref. [39], the tracking control problem of a robot system with full-state constraints was researched, and the Moore–Penrose inverse term was used to restrict violating the full-state constraints. In ref. [40], the authors considered ships with time-varying constraints, assured transient response and unknown dynamics, then the ship motion was modelled by introducing a series of constraint functions. In ref. [41], the consensus tracking control of a class of nonlinear multi-agent systems with state constraints and unknown disturbances is examined. It is worth considering that most of the works that have been done before focused on the consensus of multi-agent systems about time triggering. But in actual operation, in order to take into account the stability of the systems, a relatively small value is generally selected for a fixed sampling period [42]. It will inevitably cause controller updates frequently and data transfers continually, thereby results in excessive system energy consumption and network congestion [43]. Hence, from the perspective of computing and communication resource consumption, the event-triggered control may achieve better results, and more and more investigators have begun to conduct in depth research on distributed event-triggered control of multi-agent systems [44-47]. The event-triggered strategy in ref. [48] can collect information by embedded microprocessor to form a self-trigger setting, thus it is not necessary to track the state error between two consecutive update instants. Based on the containment control of the second-order nonlinear multi-agent system, the authors in ref. [49] study the centralised and distributed event triggered control ways. In ref. [50], a class of nonlinear multi-agent systems with multi-dimensional dynamics, uncertain disturbances and non-zero control input of leader are considered, and a novel triggering mechanism and the fixed-time event-triggered time-varying formation containment are proposed. In the cause of reducing the event trigger sampling frequency and weakening the communication load, a distributed event trigger strategy using a more conservative trigger function is proposed in ref. [51]. In ref. [52], a consensus strategy for multiagent networked systems with time varying delays is studied by aiming at leader-following consensus strategy. The methods mentioned above have a common feature that they do not take into account the impact of constraints on event-triggered control. Among the existing research results, there is rare research on the constraint problem of MASs' event-triggered control. The available methods for solving the problem of event-triggered consensus control are relatively simple. The existing methods do not consider a difficult point which restrict system variables within the bounds of constraints. Consequently, the tangent barrier Lyapunov function (BLF-Tan) is utilised during the design process, such that all the design parameters in the closed-loop systems are all bounded by the proof of the Lyapunov stability theorem. The innovations are as follows: 1) Different from ref. [53], this paper selects an unknown nonlinear function as the interference, so it is difficult to design the disturbance observer that can estimate the external interference. The disturbance observer can affect the interference on the control accuracy of system, ensure that the state of leader and followers are consistent, and achieve anti-interference consistent control. 2) For non-strict feedback multi-agent systems, a distributed event-triggered control based on the BLF-Tan is proposed to undertake the stable closed-loop systems within constraint range and the reference signal converges to a small neighborhood of zero. The paper is divided into the following parts, certain preliminaries are assumed during Section 2. Then, the adaptive dynamic surface control and stability analysis which design by associating backstepping technique and neural network are recommended during Section 3. Section 4 demonstrates certain experimental results. In the last section, the conclusion is presented. 2 PRELIMINARIES 2.1 System formulation Think about the following nonstrict-feedback systems, we can depict ith agent as: x ̇ i , m = x i , m + 1 + f i , m x i + d i , m t x ̇ i , n = u i + f i , n x i + d i , n t y i = x i , 1 (1)where m = 1 , 2 , … , n − 1 , i = 1 , 2 , … , N , f i , m ( x i ) and f i , n ( x i ) are unknown nonlinear functions, d i , m and d i , n are unknown bounded functions which called external disturbances, x i = [ x i , 1 , x i , 2 , … , x i , n ] T ∈ R n is the state vector. There are positive constants k c r , and the inequality | x i , r | ≤ k c r is satisfied, where r = 1 , 2 , … , n ,. Assumption 1. ([[54]])The external disturbance d i , m and its derivatives up to nth order are bounded, i.e. There is a positive constant D i , m with | d ̇ i , m | ≤ D i , m . Assumption 2. ([[55]])The following inequality holds: m n ≤ 1 p m p + 1 q n q where m , n , p and q are positive real number satisfying 1 p + 1 q = 1 . Assumption 3. ([[56]])For i = 1 , 2 , … , n , it is assumed that λ i , r − 1 and its i t h derivative of λ i , r − 1 are continuous and bounded, Y 1 , Y 2 , … , Y n are the positive constants, where λ i , r − 1 is required to satisfy | λ i , r − 1 | ≤ Y 0 < k c r and | λ i , r − 1 ( i ) | ≤ Y r . Lemma 1. ([[57]])In the first place, we define x ∼ 1 = [ x ∼ 1 , 1 , … , x ∼ N , 1 ] T , y ¯ = [ y 1 , … , y N ] T , y ¯ 0 = 1 N ⊗ y 0 , so there will be ∥ y ¯ − y ¯ 0 ∥ ≤ ∥ x ∼ 1 ∥ ς ( L + C ) , where ς ( L + C ) is the minimum singular value of matrix ( L + C ) . 2.2 Graph theory In order to facilitate the description of the communication relationship between agents, the relevant knowledge of algebraic graph theory is introduced as follows. Consider a directed graph ζ = ( V , E , A ) which represents the directed communication topology diagram of the multi-agent system, each node in the graph corresponds to an agent. V = ( 1 , 2 , … , N ) represents a collection of all nodes, the set of edges is E ⊆ V × V , the edge from node i to node j is defined as an ordered pair ( i , j ) ∈ E , indicates that agent i can receive information from agent j. It is also said that node i is adjacent to node j. Now we define N i = { j ∈ V ∥ ( i , j ) ∈ E , i ≠ j } as the set of adjacent edges of agent i. A = [ a i , j ] ∈ R N × N denotes the weighted adjacency matrix, if ( i , j ) ∈ E then a i , j > 0 , otherwise a i , j = 0 . Assuming that there is no self-loop in the topology, then a i , i = 0 , ∀ i ∈ V . The in-degree of node i is b i = ∑ j ∈ N i a i , j , define B = diag { b 1 , b 2 , … , b N } to be an in-degree diagonal matrix, then the Laplace matrix of graph G is L = B − A . Since only some agents in the multi-agent system can directly receive the tracking trajectory signal, the given tracking signal can be regarded as the output of the leader. 2.3 RBF NNs A continuous function f ( z ) can be approximated by the radial basis function neural networks (RBFNN) as f z = W T S z where W = [ W 1 , W 2 , … , W k ] T ∈ R k denote the adjustable weight vector, S ( z ) = [ s 1 ( z ) , s 2 ( z ) , … , s k ( z ) ] T signify the basis function vector within k expresses the number of neuron. There is a smooth vector function f ( z ) ∈ R and ideal weights W ∗ , hence the smooth function f ( z ) can be approximated by the RBFNN as follows f z = W ∗ T S z + δ z , ∀ z ∈ Ω ∈ R q We choose W ∗ as follows W ∗ Δ = arg min sup f z − W T S z in which the error δ ( z ) fulfills | δ ( z ) | ≤ ε within ε > 0 . During the thesis, undermentioned Gaussian basis function S i ( z ) will be employed S i z = exp − z − ι i T z − ι i w i 2 where ι i = [ ι i 1 , ι i 2 , … , ι q ] T depicts the center of the receptive field and w i represents the width of the Gaussian function within i = 1 , 2 , … , k . 2.4 Disturbance observer design The RBFNN is introduced to estimate f i , m ( x i ) . For the sake of estimating the unknown nonlinear function f i , m ( x i ) , it can be represented as f i , m x i = M i , m ∗ T S i , m x i + v i , m x i . ∀ x i ∈ Ω x i (2)where M i , m ∗ is a desired constant weight vector, S i , m ( x i ) is Gaussian basis function, meanwhile v i , m ( x i ) is an approximation error satisfying | v i , m ( x i ) | ≤ v ¯ i , m ( x i ) within v ¯ i , m ( x i ) > 0 . The multiagent systems could be written as x ̇ i , m = x i , m + 1 + M i , m ∗ T S i , m x i + v i , m x i + d i , m t x ̇ i , n = u i + M i , n ∗ T S i , n x i + v i , n x i + d i , n t y i = x i , 1 (3) In system (3) the disturbance observer could be regarded as follows d ̂ i , m = z ̂ i , m + ξ i , m x i , m z ̂ ̇ i , m = − ξ i , m x i , m + 1 + M ̂ i , m T S i , m x i + d ̂ i , m where i = 1 , 2 , … , N . m = 1 , 2 , … , n . And ξ i , m > 0 can be regarded as designed parameter, x i , n + 1 = u . z ̂ i , m is the state of the auxiliary system of the interference observer. Define M ∼ i , m = M i , m ∗ − M ̂ i , m , in which M ̂ i , m are the estimation of M i , m ∗ , d ̂ i , m are the estimate value of d i , m within d ∼ i , m = d i , m − d ̂ i , m . We can obtain d ∼ ̇ i , m = d ̇ i , m − ξ i , m d ∼ i , m − ξ i , m M ∼ i , m S i , m x i + v i , m x i (4) The control goal of this research is to devise an adaptive dynamic surface controller based on multi-agent systems containing the radial basis function neural networks, as a consequence, the controller could ensure that all the signals in the closed-loop system are bounded, the tracking error converges to a bounded compact interval of the origin and time-varying full state constraints would never be violated. 3 EVENT-TRIGGERED ADAPTIVE CONTROLLER DESIGN AND STABILITY ANALYSIS 3.1 Adaptive controller design Carry out adaptive dynamic surface control design for the system, and define the error surface of agent i as follows x ∼ i , 1 = ∑ j ∈ N i a i , j y i − y j + a i , 0 y i − y 0 (5) x ∼ i , r = x i , r − λ i , r − 1 , r = 2 , 3 , … , n (6)where formula (5) is derived from the knowledge of graph theory [58], y 0 is the output signal of the leader, λ i , k is the output of the virtual controller α i , k with k = 1 , 2 , … , n − 1 through the first-order low-pass filter, so the output error of the first-order filter is η i , k = λ i , k − α i , k (7) Step 1: Construct Lyapunov function candidate V i , 1 = k b 1 2 π tan π x ∼ i , 1 2 2 k b 1 2 + 1 2 d ∼ i , 1 2 + 1 2 M ∼ i , 1 T Γ i , 1 − 1 M ∼ i , 1 (8)where Γ i is a positive definite symmetric matrix. | x ∼ i , 1 | ≤ k b 1 , in which k b r are constants and we'll define them later. Take the derivation of both sides of Equation (8), then substitute Equations (4) and (5), it is hold that V ̇ i , 1 = sec 2 π x ∼ i , 1 2 2 k b 1 2 x ∼ i , 1 x ∼ ̇ i , 1 + M ∼ i , 1 T Γ i , 1 − 1 M ∼ ̇ i , 1 + d ∼ i , 1 d ∼ ̇ i , 1 = sec 2 π x ∼ i , 1 2 2 k b 1 2 x ∼ i , 1 a i , 0 + b i x ∼ i , 2 + M i , 1 ∗ T S i , 1 + η i , 1 + α i , 1 + v i , 1 + d i , 1 − a i , 0 y ̇ 0 − ∑ j ∈ N i a i , j x j , 2 + M j , 1 ∗ T S j , 1 + v j , 1 + d j , 1 + d ∼ i , 1 d ̇ i , 1 − ξ i , 1 d ∼ i , 1 − ξ i , 1 M ∼ i , 1 T S i , 1 + v i , 1 + M ∼ i , 1 T Γ i , 1 − 1 M ∼ ̇ i , 1 = p 1 a i , 0 + b i x ∼ i , 2 + M i , 1 ∗ T S i , 1 + η i , 1 + α i , 1 + v i , 1 + d ∼ i , 1 + d ̂ i , 1 − a i , 0 y ̇ 0 − ∑ j ∈ N i a i , j x j , 2 + M ∼ j , 1 T S j , 1 + M ̂ j , 1 T S j , 1 + v j , 1 + d ∼ j , 1 + d ̂ j , 1 − M ∼ i , 1 T Γ i , 1 − 1 M ̂ ̇ i , 1 + d ∼ i , 1 d ̇ i , 1 − ξ i , 1 d ∼ i , 1 − ξ i , 1 M ∼ i , 1 T S i , 1 + v i , 1 (9)where p 1 = sec 2 ( π x ∼ i , 1 2 2 k b 1 2 ) x ∼ i , 1 . On the basis of Young's inequality, it obtains d ∼ i , 1 d ̇ i , 1 ≤ 1 2 D i , 1 2 + 1 2 d ∼ i , 1 2 (10) b i + a i , 0 p 1 d ∼ i , 1 + η i , 1 + v i , 1 ≤ 3 2 b i + a i , 0 2 p 1 2 + 1 2 d ∼ i , 1 2 + 1 2 η i , 1 2 + 1 2 v ¯ i , 1 2 (11) − p 1 ∑ j ∈ N i a i , j M ∼ j , 1 T S j , 1 + d ∼ j , 1 + v j , 1 ≤ ∑ j ∈ N i a i , j 2 M ∼ j , 1 T M ∼ j , 1 S j , 1 2 + d ∼ j , 1 2 + v ¯ j , 1 2 + ∑ j ∈ N i a i , j 2 p 1 2 (12) − ξ i , 1 d ∼ i , 1 M ∼ i , 1 T S i , 1 + v i , 1 ≤ ξ i , 1 d ∼ i , 1 2 ρ i , 1 2 + 1 2 ξ i , 1 ρ i , 1 2 M ∼ i , 1 T M ∼ i , 1 S i , 1 2 + 1 2 ξ i , 1 ρ i , 1 2 v ¯ i , 1 2 (13)where ρ i , 1 is positive design parameter. From Equations (10)–(13), the derivative of V 1 could be computed as V ̇ i , 1 ≤ p 1 a i , 0 + b i x ∼ i , 2 + α i , 1 + M ̂ i , 1 T S i , 1 + d ̂ i , 1 − ∑ j ∈ N i a i , j x j , 2 + M ̂ j , 1 T S j , 1 + d ̂ j , 1 − a i , 0 y ̇ 0 + 3 2 b i + a i , 0 2 + ∑ j ∈ N i a i , j 2 p 1 2 + 1 2 η i , 1 2 + ∑ j ∈ N i a i , j 2 M ∼ j , 1 T M ∼ j , 1 S j , 1 2 + d ∼ j , 1 2 + v ¯ j , 1 2 − ξ i , 1 d ∼ i , 1 2 + 1 2 D i , 1 2 + d ∼ i , 1 2 + ξ i , 1 d ∼ i , 1 2 ρ i , 1 2 + 1 2 v ¯ i , 1 2 + 1 2 ξ i , 1 ρ i , 1 2 M ∼ i , 1 T M ∼ i , 1 S i , 1 2 + 1 2 ξ i , 1 ρ i , 1 2 v ¯ i , 1 2 + M ∼ i , 1 T − Γ i , 1 − 1 M ̂ ̇ i , 1 + p 1 a i , 0 + b i S i , 1 (14) Construct the following virtual controller α i , 1 and adaptive law of M ̂ i , 1 as α i , 1 = − M ̂ i , 1 T S i , 1 − d ̂ i , 1 − 3 2 b i + a i , 0 p 1 + 1 b i + a i , 0 a i , 0 y ̇ 0 − c i , 1 p 1 + 1 b i + a i , 0 ∑ j ∈ N i a i , j x j , 2 + M ̂ j , 1 T S j , 1 + 1 b i + a i , 0 ∑ j ∈ N i a i , j d ̂ j , 1 − p 1 2 (15) M ̂ ̇ i , 1 = Γ i , 1 a i , 0 + b i S i , 1 p 1 − σ 1 M ̂ i , 1 (16)where c i , 1 > 0 is a constant. Substituting Equations (15) and (16) into Equation (14), we get V ̇ i , 1 ≤ − c i , 1 p 1 2 + a i , 0 + b i p 1 x ∼ i , 2 + 1 2 η i , 1 2 + 1 2 D i , 1 2 + ∑ j ∈ N i a i , j 2 M ∼ j , 1 T M ∼ j , 1 S j , 1 2 + d ∼ j , 1 2 + v ¯ j , 1 2 + 1 2 v ¯ i , 1 2 + 1 2 ξ i , 1 ρ i , 1 2 M ∼ i , 1 T M ∼ i , 1 S i , 1 2 + v ¯ i , 1 2 − d ∼ i , 1 2 ξ i , 1 − ξ i , 1 ρ i , 1 2 − 1 + σ 1 M ∼ i , 1 T M ̂ i , 1 (17)Step r: In accordance with what the first-order low-pass filter does, then cause a virtual signal α i , r − 1 to pass this device with a time constant of τ i , r − 1 , it gets τ i , r − 1 λ ̇ i , r − 1 + λ i , r − 1 = α i , r − 1 , λ i , r − 1 0 = α i , r − 1 0 (18) Then from Equations (7) and (18), we can get η ̇ i , r − 1 = − η i , r − 1 τ i , r − 1 − α ̇ i , r − 1 = − η i , r − 1 τ i , r − 1 + ψ i , r − 1 (19)For simplicity, let ψ i , r − 1 = − α ̇ i , r − 1 . Consider the Lyapunov function as V i , r = V i , r − 1 + k b r 2 π tan π x ∼ i , r 2 2 k b r 2 + 1 2 d ∼ i , r 2 + 1 2 η i , r − 1 2 + 1 2 M ∼ i , r T Γ i , r − 1 M ∼ i , r (20)where p r = sec 2 ( π x ∼ i , r 2 2 k b r 2 ) x ∼ i , r , and | x ∼ i , r | ≤ k b r . Take the derivation of both ends of Equation (20), then substitute Equations (4), (6) and (19) into Equation (20), one obtains V ̇ i , r = V ̇ i , r − 1 + sec 2 π x ∼ i , r 2 2 k b r 2 x ∼ i , r x ∼ ̇ i , r + d ∼ i , r d ∼ ̇ i , r + η i , r − 1 η ̇ i , r − 1 + M ∼ i , r T Γ i , r − 1 M ∼ ̇ i , r = V ̇ i , r − 1 + p r x ∼ i , r + 1 + α i , r + M i , r ∗ T S i , r + p r v i , r + η i , r + d ∼ i , r + d ̂ i , r − λ ̇ i , r − 1 + d ∼ i , r d ̇ i , r − ξ i , r d ∼ i , r − ξ i , r M ∼ i , r T S i , r + v i , r + η i , r − 1 − η i , r − 1 τ i , r − 1 + ψ i , r − 1 − M ∼ i , r T Γ i , r − 1 M ̂ ̇ i , r (21)Based on Young's inequality, it has η i , r − 1 ψ i , r − 1 ≤ 1 2 ψ i , r − 1 2 + 1 2 η i , r − 1 2 (22) p r d ∼ i , r + η i , r + v i , r + d ∼ i , r d ̇ i , r ≤ 3 2 p r 2 + d ∼ i , r 2 + 1 2 η i , r 2 + 1 2 D i , r 2 + 1 2 v ¯ i , r 2 (23) − ξ i , r d ∼ i , r M ∼ i , r T S i , r + v i , r ≤ ξ i , r d ∼ i , r 2 ρ i , r 2 + 1 2 ξ i , r ρ i , r 2 M ∼ i , r T M ∼ i , r S i , r 2 + 1 2 ξ i , r ρ i , r 2 v ¯ i , r 2 (24)where ρ i , r is a positive design parameter. Combining Equation (22) to (24) into Equation (21), it gets V ̇ i , r ≤ V ̇ i , r − 1 + 1 2 η i , r 2 + 1 2 D i , r 2 + 1 2 v ¯ i , r 2 + 3 2 p r 2 + p r x ∼ i , r + 1 + M ̂ i , r T S i , r + d ̂ i , r − λ ̇ i , r − 1 + ξ i , r d ∼ i , r 2 ρ i , r 2 + 1 2 ξ i , r ρ i , r 2 v ¯ i , r 2 − η i , r − 1 2 τ i , r − 1 + 1 2 η i , r − 1 2 + p r α i , r + 1 2 ψ i , r − 1 2 + 1 2 ξ i , r ρ i , r 2 M ∼ i , r T M ∼ i , r S i , r 2 + M ∼ i , r T − Γ i , r − 1 M ̂ ̇ i , r + p r S i , r − ξ i , r − 1 d ∼ i , r 2 (25) Design such actual controller and adaptive law α i , r = − c i , r p r − 3 2 p r − M ̂ i , r T S i , r − d ̂ i , r + λ ̇ i , r − 1 − l i , r sec 2 π x ∼ i , r 2 2 k b r 2 (26) M ̂ ̇ i , r = Γ i , r S i , r p r − σ r M ̂ i , r (27)where l i , r = { p 1 ( a i , 0 + b i ) r = 2 p r − 1 r = 3 , 4 , … , n − 1 . Let the design parameter c i , r > 0 . Substituting Equations (26) and (27) into Equation (25), then we can deduce that V ̇ i , r ≤ − ∑ β = 1 r c i , β p β 2 − ∑ β = 1 r d ∼ i , β 2 ξ i , β − ξ i , β ρ i , β 2 − 1 + ∑ j ∈ N i a i , j 2 d ∼ j , 1 2 + M ∼ j , 1 T M ∼ j , 1 S j , 1 2 + v ¯ j , 1 2 + 1 2 ∑ β = 1 r ξ i , β ρ i , β 2 M ∼ i , β T M ∼ i , β S i , β 2 + v ¯ i , β 2 − ∑ β = 1 r − 1 η i , β 2 1 τ i , β − 1 + 1 2 ∑ β = 1 r D i , β 2 + 1 2 η i , r 2 + ∑ β = 1 r σ β M ∼ i , β T M ̂ i , β + p r x ∼ i , r + 1 + 1 2 ∑ β = 1 r − 1 ψ i , β 2 + 1 2 ∑ β = 1 r v ¯ i , β 2 (28)Step n: In accordance with what the first-order low-pass filter does, then cause a virtual signal α i , n − 1 to pass this device with a time constant τ i , n − 1 , therefore, it obtains τ i , n − 1 λ ̇ i , n − 1 + λ i , n − 1 = α i , n − 1 , λ i , n − 1 0 = α i , n − 1 0 (29)Then from Equations (7) and (29), it leads to η ̇ i , n − 1 = − η i , n − 1 τ i , n − 1 − α ̇ i , n − 1 = − η i , n − 1 τ i , n − 1 + ψ i , n − 1 (30)where ψ i , n − 1 = − α ̇ i , n − 1 is a continuous function. Select the following Lyapunov function V i , n = V i , n − 1 + k b n 2 π tan π x ∼ i , n 2 2 k b n 2 + 1 2 d ∼ i , n 2 + 1 2 η i , n − 1 2 + 1 2 M ∼ i , n T Γ i , n − 1 M ∼ i , n (31)where | x ∼ i , n | ≤ k b n . Take the derivation of both sides of Equation (31), then substitute Equations (4), (6) and (30) into Equation (31), it obtains V ̇ i , n = V ̇ i , n − 1 + sec 2 π x ∼ i , n 2 2 k b n 2 x ∼ i , n x ∼ ̇ i , n + d ∼ i , n d ∼ ̇ i , n + η i , n − 1 η ̇ i , n − 1 + M ∼ i
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