Prescribed Performance-Based Consensus Tracking for Multiagent Systems

Authors: yihang  chen
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-16 10:42:53

Abstract: This paper investigates the prescribed performance-based cooperative control problem for a class of uncertain strict-feedback nonlinear multiagent systems with input saturation. Compared with the existing results, the new performance function is constructed with a larger initial value and faster convergence speed to meet the feasible condition of prescribed performance control, while ensuring that tracking errors converge to the prescribed range in a configurable time frame. On this basis, the constrained system is converted into an equivalent totally unconstrained one, which indicates that the convergence of the logarithmic error transformation function is mapped to the convergence of the tracking error. Meanwhile, the problem of input saturation is considered, which can be addressed by constructing an auxiliary system with the same order as the dynamics for compensating the effect of input saturation. Moreover, for the topology communication redundancy problem, a topology optimization scheme is designed to eliminate unnecessary edges and save communication resources. The results of numerical simulation on the control systems verify the approach.

Main paper: 
 i. introduction: 
 D URING the past few decades, people gradually pay attention to the research on multiagent systems (MASs) due to the ability to attack more complex, realistic, and large-scale problems which are beyond the capabilities of an individual agent, such as extensive search and rescue [1] , traffic vehicle [2] , environmental monitor [3] . A research goal is to transform large complex systems (hardware and software systems) into easy-to-manage systems in which individual agents can communicate and coordinate with each other and eventually complete control objective precisely. Accomplishing precise control has always been an essential control requirement for the smooth operation of multiagent systems. Researchers have achieved abundant results; see, for instance [4] . The control outcomes of formation control, consensus control, and containment control are commonly demonstrated by the convergence of tracking errors to a small vicinity around the origin. Extensive investigation has been undertaken regarding the design of stable controllers especially for systems with nonlinear uncertain terms, including fuzzy systems [5] , adaptive backstepping [6] , adaptive feedback [7] .The purpose of this paper is to reconstruct the performance function for a particular class of tracking error initial values that exceed its boundaries. The reconstruction aims to meet the following requirements 1) The initial value ρ i (0) of the new performance function satisfies ρ i (0) > |e i (0)|, where e i (0) represents the error initial value.2) The new prescribed performance function should converge within the boundaries of the original performance function after a designable time τ mp .With the above observations, this work investigates consensus tracking problem of multiagent systems via a performance function reconstruction approach and topology optimization method. The advancements of this paper are delineated as follows 1) A new performance function is designed to satisfy the feasible condition of PPC and larger convergence speed of tracking errors. Compared to the performance function delineated in [8] , the new performance function expedites the convergence rate of the tracking error and mitigates tracking error overshoots by incorporating a specifically designed logarithmic term. 2) A new topology optimization method is developed based on the minimum communication distance, which can eliminate redundant edges while ensuring that the communication graph contains a spanning tree. The structure of the remaining sections of this paper is as follows. Nonlinear dynamics model and some preliminaries including graph theory, fuzzy logic system, topology optimization as well as prescribed performance scheme are introduced in Section II. Controller and adaptive laws based backstepping approach are designed in Section III. Numerical examples are provided in provided in Section IV.The dynamics of the ith (i = 1, ..., N ) follower is described in the strict-feedback formEQUATIONwhere x i = [x i,1 , ..., x i,n ] T ∈ R n and y i represent state vector and system output of the ith agent with xi,k =[x i,1 , ..., x i,k ] T ∈ R k (k = 1, ..., n), f i,k (x i,k ) is unknown smooth nonlinear function, ω i,k (t) is unknown time-varying disturbance, The system input signal u i (v i (t)) is affected by saturation nonlinearity, as expressed byEQUATIONwhere u i,M > 0 represents the known upper bounds of u i (v i (t)), and v i (t) denotes the actuator input to be designed. The leader's signal y r is assumed to be generated by a linear autonomous system of the formEQUATIONwhere x l = [x l,1 , ..., x l,n ] T ∈ R n is the state vector of leader node, S ∈ R n×n is the constant coefficient matrix, we assume that S is detectable.Consider the direct graph Υ = (U, E, P), where U = {1, ..., N } are the sets of nodes, E = {(i, j) : i ∈ U, j ∈ U} are edges between ith agent and jth agent, P = [a i,j ] N ×N stands for the adjacency matrix. If a = 0, there is no information interaction between ith and jth agents; otherwise a i,j > 0. The Laplacian matrix is defined as L = D-P, whereD = diag{d 1 , ..., d N } with d i = N j=1 a i,jis the in-degree matrix of the ith follower. In order to achieving subsequent study, the following assumptions are imposed.The augmented graph is denoted as Ῡ = Ū, Ē with Ū = {0, 1, ..., N } and Ē = {(i, j) : i ∈ Ū, j ∈ Ū}, a i,0 > 0 represent ith agent maintains communication with the leader, A = diag{a 1,0 , a 2,0 , ..., a N,0 } ∈ R N ×N .In order to approximate the unknown continue function, we introduce the following definitions and lemma. Consider the fuzzy rule base consisting of M rules in the following formR j :If x 1 is A j 1 , x 2 is A j 2 , ..., x n is A j n , Then z is B j , with j = 1, 2, ..., M , X = [x 1 , x 2 , ...,x n ] T ∈ R n and z as the input vector and the output variable of the fuzzy system, respectively. A j i and B j represent linguistic terms defined by fuzzy membership functions µ A j i (x i ) and µ B j (z), respectively. The fuzzy logic system has the formEQUATIONwhere µ A j i (x i ) is the Gaussian membership function, denoted asEQUATIONherein, a j i , xj i , and σ j i are real-valued parameters with 0 < a j i ≤ 1 and zj = max z∈R µ B j (z).Throughout this work, the cooperative control objective is to confine e i (t) = N j=1 a i,j (y i -y j ) + a i,0 (y i -y r ) within a predefined arbitrarily small residual set.EQUATIONwhere the candidate function could be ρ i (t) = (ρ i (0)ρ i (∞))e -lt + ρ i (∞) which is a strictly positive and monotonically decreasing function satisfying ρ i (0) > lim t→∞ ρ i (t) > 0, l > 0 is a constant representing the required exponential convergence rate. Clearly, ρ i (t) is bounded and continuously differentiable. ρ i (0) is the initial performance bound, ρ i (∞) is the maximum steady state error. The prerequisite for achieving the above control objective is to select suitable initial value ρ i (0) and satisfy ρ i (0) > |e i (0)|. We consider the case that the precise value of initial error e i (0) can be obtained, a performance function ρ * i (t) is expressed byEQUATIONwhereρ * i (0) = |e i (0)| with > 1 is a constant, τ mp > 0 is an arbitrarily specified constant which satisfies ρ * i (τ mp ) = ρ i (τ mp ).Then, a new performance function is designed asEQUATIONStep m(1 ≤ m ≤ n -1): The time derivative of z i,m can be denoted asEQUATIONwhere ˙ i,m-1 is denoted as Define the following Lyapunov function:EQUATIONwhere κ i,m > 0 and γ i,m > 0 are designed constant, θi,m = ϑ i,m -θi,m and Γi,m = Γ i,m -Γi,m are the estimation errors. Then, the derivative ofEQUATIONgiven that fi,m = f i,m -˙ i,m-1 is an unknown function, a fuzzy logic systemEQUATION, θi,1 , ..., θi,m-1 , Γi,1 , ..., Γi,m-1 , ς i,1 , ..., ς i,m-1 , y r ] T . Thus, fi,m can be expressed asEQUATIONDefineEQUATIONEQUATIONwhere σ i,m > 0. Combining Young's inequality, substituting (12)-( 14) into (11) yieldsVi,m ≤z i,m (z i,m+1EQUATIONThe virtual controller i,m is designed asEQUATIONit is interesting to note that Ξ i,m exhibits the following properties: 1) Ξ i,m = λ i,1 (d i + a i,0 ) if and only if m = 2; 2) Ξ i,m = 1 as m = 3, 4, ..., n -1 and c i,m > 0 is a constant. Substituting (12)-( 16) into (11) yieldsEQUATIONIV. STABILITY ANALYSIS Proof : We first define a Lyapunov functionEQUATIONwhereEQUATIONEQUATIONwherez * ,1 (t) = [z 1,1 , z 2,1 , ..., z N,1 ] T .The result in (20) illustrates that e i is bounded and e i is constrained by performance function (8) , which implies that the synchronization error e i satisfy -ρ * i < e i < ρ * i with t ∈ [0, τ mp ) and -ρ i ≤ -ρ * i < e i < ρ * i ≤ ρ i with t ∈ [τ mp , +∞). Meanwhile, one has We propose a performance function reconstruction approach for reconstructing the initial performance function into a new performance function, thereby ensuring that the initial values of the tracking error align with the newly defined performance bounds. Contrasted with existing methods for prescribed performance, the proposed scheme offers two primary advantages. On the one hand, it reduces the overshoot of the tracking error, on the other hand, it ensures that the tracking error converges to the initial performance function bound within the specified time. 
 
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Reviews: review: title: title: This paper lacks sufficient innovation, therefore it is recommended to reject the submission.
main_review: main_review: This paper lacks sufficient innovation, therefore it is recommended to reject the submission. The reasons for rejection are as follows：
1.While the newly designed performance function aims to enhance the feasible condition of PPC and accelerate the convergence rate of tracking errors by incorporating a specific logarithmic term, the paper lacks rigorous theoretical proofs to substantiate these claims.  Moreover, the comparison with the performance function outlined in [8] is qualitative and lacks quantitative evaluations through experiments or simulations across multiple scenarios.  Consequently, it is difficult to comprehensively assess the practical value and generality of the proposed function.
2.The novel topology optimization method based on the minimum communication distance, which claims to eliminate redundant edges while ensuring a spanning tree in the communication graph, suffers from inadequate experimental validation. The lack of detailed experimental setups, parameter tuning processes, and test results across varying scenarios hinders the assessment of the method's effectiveness and stability in practical applications. Furthermore, the absence of comparisons with other topology optimization approaches limits a comprehensive evaluation of its novelty.
3. There are many errors in the format of the article and the content is simple, which is not suitable for publication.
title: title: This paper investigates the prescribed performance-based cooperative control problem for a class of uncertain strict-feedback nonlinear multiagent systems with input saturation. However, the manuscript lacks clarity and organization, making it difficult for readers to understand the proposed control scheme and its implementation.
main_review: main_review: The manuscript does not offer significant advancements over existing techniques or provide a thorough comparative analysis, limiting its contribution to the field. As such, I do not think this paper is suitable for ICIST 2024.
title: title: Recommended rejection
main_review: main_review: The innovation points of this manuscript are insufficient and the logic is unclear. It is not recommended to publish this paper in ICIST 2024.