Privacy-Preserving Adaptive Tracking Control for Multiagent Systems

Authors: siyu  guo
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-16 10:12:06

Abstract: This paper investigates the privacy-preserving adaptive tracking control problem for nonlinear multiagent systems with unknown disturbances. Firstly, based on the phenomenon of Lorenz system, a masking function endowed with uncertainty and unpredictability is proposed. Meanwhile, a time-assist function is designed to achieve privacy protection within a predefined period of time, which enhances the flexibility of the mechanism. In addition, a dynamically self-adjustable gain parameter is designed in the disturbance observer. The introduction of this parameter enables the proposed disturbance observer to not only handle unknown disturbances but also achieve better results in improving system performance. Afterward, the Lyapunov stability theorem is utilized to prove that all signals of the closed-loop system are semi-globally uniformly ultimately bounded. Finally, simulation results confirm the effectiveness of the proposed control scheme.

Main paper: 
 i. introduction: 
 In recent years, multiagent systems (MASs) have garnered ongoing attention across diverse fields [1, 2] . In the control problems of MASs, effective communication between agents is crucial to accomplish common control tasks. Most communication channels frequently lack protection, leaving communication networks susceptible to malicious attacks. Additionally, each agent has sensitive information that it prefers to keep private. Hence, there arises an urgent necessity for integrating privacy protection attributes into MASs.Given the hidden risks in information interaction, some scholars have increasingly directed their attention to researching privacy protection in MASs [3, 4] . At present, the mainstream method is to design a protection function with masking ability. Notably, the parameters in these masking functions are fixed, which increases the risk of the specific form of the masking function being deciphered. Furthermore, most of the current masking functions are applied in infinite time domains, which may not only affect the stability of the system but also lack the ability to flexibly choose protection time. The question of enhancing the protective effectiveness of masking functions and achieving privacy protection within user-defined time frames is worth exploring, and it has inspired the research presented in this paper.Unknown disturbances are prevalent in industrial applications, posing a threat to the stability of the system. To effectively detect and mitigate the impact of these disturbances, disturbance observer has been extensively researched [5, 6] . Most of the design methods involved in the disturbance observers lack certain adaptability. In order to improve the performance of the disturbance observers, some scholars have delved into the correlation between the disturbance observers and system response [7] . This prompts us to further investigate the interplay between the disturbance observers and system errors, with the goal of detecting unknown disturbances while optimizing system performance.Motivated by the above discussion, this paper proposes a preassigned-time hyperchaotic privacy-preserving mechanism and an optimization-based adaptive disturbance observer. The contributions are summarized as follows.1) Based on the characteristics of Lorenz system, the designed masking function is endowed with unpredictable stochastic properties, which enhances the masking effectiveness and protective functionality. Furthermore, by designing a time-assist function, privacy protection within predefined time-frames is achieved, which mitigates the adverse impact of protection-induced uncertainty on system performance. 2) An adaptive gain parameter is designed in the disturbance observer based on the gradient descent method. The introduced adaptive gain parameter enables the disturbance observer not only to detect and estimate unknown disturbances but also to achieve the effect of improving the control performance of the MASs.Consider a directed graph G = (V, E, A) with M nodes. V = (1, 2, . . . , M) is a non-empty set of agents. E ∈ V × V is an edge set, where (V h , V j ) ∈ E is an edge from node j to node h. A = [p h,j ] ∈ R M×M represents the adjacency matrix. If (V h , V j ) ∈ E, p h,j = 1, otherwise p h,j = 0. The Laplacian matrix is defined as L = D -A, including the indegree matrices D = diag (d 1 , . . . , d M ) with d h = M j=1 p h,j . Define B = diag (b 1 , . . . , b M ). The positive and negative of b h represent whether the node h is able to obtain information directly from the leader node or not, respectively. Lemma 1: [1] There exists a spanning tree in the graph G, where the leader node 0 is designated as the root node. With this, L + B is nonsingular.In this paper, we examine nonlinear MASs comprising M followers and a leader, where the dynamic of the hth (h = 1, . . . , M) follower is characterized byEQUATIONwhere m represents the order of the system and satisfies1 ≤ m < n -1, xh,m = [x h,1 , x h,2 , . . . , x h,m ] T ∈ R m and xh,n = [x h,1 , x h,2 , . . . , x h,n ] T ∈ R n indicatethe state vectors of the hth follower. u h is the control input signal, and the output signal of the hth follower is expressed as y h . ω h (t) is the unknown disturbance.  h,m (x h,m ) and  h,n (x h,n ) are unknown nonlinear functions. y r (t) stands for the leader signal of the MASs. Then, the synchronization error of the hth follower is given by zh,1 = M j=1 p h,j (y h -y j ) + b h (y h -y r ). Lemma 2: [7] Define źh,1 = (z 1,1 , . . . , z M,1 ) T , ýh = (y 1 , . . . , y M ) T , ýr = (y r , . . . , y r ) T . Then, it follows that ýh -ýr ≤ źh,1 M(L+B), where M(L + B) denotes the minimum singular value of L + B.Lemma 3: [8] The second-order sliding mode integral filter (SSMIF) is established as   Kh,k10 = - K h,k10 -w(t) k h,k10 - R h,k10 (K h,k10 -w(t)) K h,k10 -w(t) +ι h,k10Kh,k20 = -EQUATIONwhere K h,k10 and K h,k20 denote the states of the filter (2). k h,k10 , k h,k20 , ι h,k10 and ι h,k20 are positive design parameters.Lemma 4: [5] One can approximate an unknown function P (s) by using Radial Basis Function Neural Networks (RBF NNs). P (s) = ζ * T φ(s) + ε(s), where φ(s) = [φ 1 (s), φ 2 (s), . . . , φ r (s)] T denotes the basis function vector. ε(s) represents the approximation error. The ideal weight vectorζ * = [ζ * 1 , ζ * 2 , . . . , ζ * r ] T ∈ R r can be chosen as ζ * = arg min ζ∈R r sup s∈Ω P (s) -ζ T φ(s) .In this paper, a class of Lorenz-Stenflo hyperchaotic systems are considered, which is modelled as       ẋ = a(y -x) + µw ẏ = cx -y -xz ż = -βz + xy ẇ = -x -aw (3)where a is the Prandtl number, c is the generalized Rayleigh parameter,β = 4k 2 1 /k 2 2 and µ = 4Ω 2 k 2 1 /k 2 h k 6 2, Ω is the angular frequency of the earth rotation, k h is the heat dissipation coefficient, k 1 and k 2 are design constants. In this paper, we choose a = 3, µ = 30, c = 30 and β = 8/3.Next, a time-assist function Π(t) is designed to achieve privacy protection within a specified timeframe. The specific expression of the function is as follows:EQUATION)where T is the user-defined protection time with arbitrary settings, and 2n + 1 is the designed function order. Correspondingly, it is concluded thatEQUATIONBased on the above analysis, a masking function is designed for the leader signal, which is expressed asy π = y r + κ( (-x -aw)dt)e -ς( (ay-ax+sw)dt)t Π(t) (6)where κ and ς > 0 are designed parameters.The optimization-based adaptive disturbance observer is designed in the following form [5] :EQUATIONwhere ζh,n and φ h,n (x h,n ) denote the approximate parameter vector and basis function vector, respectively. κ h means a positive design parameter. ωh is the estimated value of ωh with ωh = ωh -ωh , and the specific form of ωh will be given later. h is an iterative parameter that can be varied adaptively.After that, it can be deduced thatEQUATIONwith ωh being a bounded value. The controller u h is designed in the form ofu h = -(ϑ h,n + 1 2 )δ h,n + K h,n20 -δ h,n-1 - h ωh -ζh,n φ h,n (x h,n ).The detailed design process of u h will be analyzed later. Then, according to the error transformation relation δ h,n = x h,n -α h,n-1 , the derivative of the error variable δh,n satisfies δh,n = u h + f h,n + ω h (t) -αh,n-1 .From this, a gradient expression for the derivative of the error variable δh,n with respect to h is obtained as∂ δh,n ∂ h =ωh , which reveals a potential link between δh,n and h , inspiring us to improve system performance by controlling the variation of h . The gradient descent method is applied to realize the selfregulation function of h and improve the performance of the MASs. The specific iterative expression is given ash = Pr h + s L h Ξ h (δ h,n ) ωh, where Pr h is the position of h after the last iteration, and the initial position of h is artificially designed. s L h is the step size of each iteration. Ξ h is a sign function related to δ h,n , when δ h,n > 0, Ξ h = 1; when δ h,n < 0, Ξ h = -1.In order to accomplish the proposed control objective, the following common error transformation is implemented, specified asEQUATIONwhere α h,g-1 is the virtual control signal.Step 1. The Lyapunov function is selected asV h,1 = 1 2 δ 2 h,1 + 1 2 h,1 ζ2 h,1, where h,1 > 0 is a design constant. In view of the unknown function( Xh ) = ph  h,1 - M j=1 p h,j (x j,2 +  j,1 ) with Xh = [x h,1 , x j,1 , x j,2 ]T , the RBF NN is introduced to approximate ( Xh ).To achieve system stability, the following design is implemented for both the virtual control signal and the adaptive law.EQUATIONwhere ϑ h,1 > 0 and χ h,1 > 0 are design constants.Step s. For step s with s = 2, . . . , n -1, an SSMIF is used to estimate αh,s-1 . The Lyapunov function is selected asV h,s = V h,s-1 + 1 2 δ 2 h,s + 1 2 h,s ζ2 h,s, where h,s > 0 is a design constant. Then, the virtual control signal and the adaptive law are designed asEQUATIONwhere ϑ h,s and χ h,s are positive constants.Step n. Similar to the previous treatment, through SSMIF, we have αh,n-1 = K h,n20 -K h,fn-1 , where |K h,fn-1 | < Kh,fn-1 with Kh,fn-1 > 0. Accounting for the effects of unknown disturbances, the Lyapunov function is chosen asV h,n = V h,n-1 + 1 2 δ 2 h,n + 1 2 h,n ζ2 h,n + 1 2 ω2 h , where h,n > 0 is a design constant.With the aim of ensuring that all signals in the system are SGUUB, the final controller u h and the adaptive law ζh,n are devised asEQUATIONwhere ϑ h,n and χ h,n are positive constants. Following the properties of the proposed optimization-based adaptive disturbance observer involved in (7) and ( 8), we derived Vh,n ≤ -n o=1 ϑ h,o δ 2 h,o + n-1 o=1 ε2 h,o 2 + n-1 o=2 Kh,f o-1 2 - n-1 o=1 χ h,o ζ2 h,o 2 h,o -( χ h,o 2 h,o -|κ h | 2 ) ζ2 h,n + n o=1 χ h,o ζ 2 h,o 2 h,o + 1 2 ω2 h - 1 2 2κ h h -2 h -1 -|κ h |φ T h,n (x h,n )φ h,n (x h,n ) ω2 h .Theorem 1: If the controller and adaptive law are formulated as (12), then it can be concluded that all signals within the MASs (1) exhibit SGUUB behavior, and the synchronization errors of all agents have the capability to converge to a narrow region around the origin.Proof : Construct the Lyapunov function V * = M h=1 V h,n , it is known that Vh,n can be restated asEQUATION)where Th = min{ϑ h,o , ϑ h,n ,χ h,o 2 h,o , ( χ h,n 2 h,n -|κ h | 2 ), 2κ h h - 2 h -|κ h |φ T h,n (x h,n )φ h,n (x h,n ) -1 } with o = 1, . . . , n -1 and Ph = n-1 o=1 ε2 h,o 2 + n-1 o=2 Kh,f o-1 2 + n o=1 χ h,o ζ 2 h,o 2 h,o + 1 2 ω2 h .After that, it can be inferred that V * ≤ -T V * + P, where T = min{ Th } with h = 1, . . . , M and P = M h=1 Ph . With mathematical calculations, it follows thatEQUATIONBased on the above analysis and Lemma 2, it can be justified that the synchronization errors between agents are SGUUB. In the given example, an MAS comprising five nodes is examined. This MAS is composed of four followers (nodes 1-4) and a leader (node 0) arranged on a directed graph, with the communication topology depicted in Fig. 1 .Consider a mass-spring-damper system [9] with dynamic M Ẍ + B Ẋ + RX = F . Let x h,1 = X, x h,2 = Ẋ, and u h = F , then the dynamic of hth (h = 1, 2, 3, 4) follower subject to external disturbance is modelled asEQUATIONThe actual parameters are specified as M = 1 kg, R = 3 N/m and B = 0.5 Ns/m. The masked leader signal is established as y π = 2 sin(t)+2w(t) exp(-0.1x(t)t)Π(t) with T = 3. The initial values of the states within MASs are selected asx 1,1 (0) = 2, x 2,1 (0) = 0.3, x 3,1 (0) = -0.1, x 4,1 (0) = -0.2, x 1,2 (0) = 0.5, x 2,2 (0) = 1, x 3,2 (0) = 1, x 4,2 (0) = 1.For hth (h = 1, 2, 3, 4) follower, θh,1 (0) = θh,2 (0) = 0.1. The initial values of the second-order sliding film filter are assigned as K h,k10 (0) = 0.1 and K h,k20 (0) = 0.1 for h = 1, 2.The relevant parameters of the adaptive law are given by h,1 = h,2 = 12 and χ i,1 = χ i,2 = 2. The controller parameters are designed as ϑ 1,1 = 39.5, ϑ 2,1 = 19.5, ϑ 3,1 = 39.5, ϑ 4,1 = 39.5, ϑ 1,2 = 79.5, ϑ 2,2 = 79.5, ϑ 3,2 = 79.5, ϑ 4,2 = 79.5. The informations of the disturbance observer are configured as κ 1 = κ 2 = κ 3 = κ 4 = 0.01, Figs. 2 and 3 show the simulation results. There is seen in Fig. 2 that after 4 seconds the trajectories of all the agents are able to fulfil the cooperative goal of validly following the leader for which a communication link exists. Before 3 seconds it can be observed that the privacy-preserving mechanism is effective, and the followers are biased by the information they receive to the extent that the trajectories fluctuate significantly. Moreover, this fluctuation is bounded and controllable and does not lead to the collapse of the MASs. Information about the synchronisation errors is demonstrated in Fig. 3 , where it can be found that when the system is relatively stable, the overall error distribution is basically in the range of -0.1 to 0.1. This illustrates that the method proposed in this paper works effectively in achieving the established control objectives.1 (0) = 2 (0) = 3 (0) = 4 (0) = 1, s L 1 = s L 2 = s L 3 = s L 4 = 1.In this paper, the privacy-preserving adaptive tracking control problem has been researched for MASs subject to unknown external disturbances. The future goal of our work is to apply the preassigned-time privacy-preserving mechanism to more control methods and control systems. 
 
 Appendix:  



Reviews: review: title: title: Privacy-Preserving Adaptive Tracking Control for Multiagent Systems
main_review: main_review: The article presents a privacy-preserving adaptive tracking control method for nonlinear multi-agent systems (MASs) subject to unknown disturbances. The key innovation is the introduction of a masking function based on the Lorenz system, which provides unpredictability and ensures privacy protection within a user-defined timeframe. Additionally, an adaptive disturbance observer is designed using a dynamically adjustable gain parameter to handle unknown disturbances and improve system performance. The effectiveness of the proposed control scheme is validated through simulations, demonstrating its ability to maintain system stability while preserving privacy. On the whole, the paper is good, but there are still some problems that need to be corrected.

Comments:

1.The paper introduces a novel privacy-preserving mechanism and an adaptive disturbance observer. However, the distinction between these contributions and existing methods could be more explicitly articulated to highlight the innovation.

2.While the simulation results effectively demonstrate the proposed method's performance, a more detailed comparison with other existing methods would strengthen the paper's argument for the proposed approach's superiority.

3.Have you considered using other diagrams to illustrate your method? I think that adding a block diagram could help in understanding the principles of the method used.

4.The research content is well constructed. But there are grammatical errors and incorrect English writing in the current manuscript. Please make the necessary corrections.

5.The formula symbol is more complex, it is recommended to carefully check and modify the full text.
title: title: Lack of innovation
main_review: main_review: First， the main controbation of this paper is not clear. Second, the literature review in the Introduction surveys some previous work on the solutions and methods to address privacy-preserving adaptive tracking control, while comparation among these methods which is the most important part is neglected. So, this article is not recommended for acceptance.
title: title: Lack innovation, not sufficiently important, not align
main_review: main_review: The findings of this work lack innovation or are not sufficiently important to the academic community. This requires reconsideration of the research question and results to ensure that a substantive contribution to the academy and the field can be made.  Meanwhile, The topic of the manuscript does not align with the scope or interests of the journal. On balance, the article will be rejected.