Elastic time-varying affine formation optimal tracking method and system for cluster system

Abstract

The invention discloses an elastic time-varying affine formation optimal tracking method and system for a cluster system, and relates to the field of cluster systems, the cluster system comprises a plurality of agents, the agents are divided into Byzantine agents and honesty agents after being attacked by the Byzantine, and the method comprises the following steps: establishing a dynamic model and an affine formation of the cluster system, determining the optimal tracking problem of the elastic time-varying affine formation: by setting the control input of each honesty agent at the current moment, minimizing the target function under the condition that the state of each honesty agent at the next moment meets the affine formation constraint and the global inequality constraint according to the dynamic model; and determining an elastic time-varying affine formation optimal tracking control protocol, obtaining control input of each honesty agent at the current moment, and enabling the affine formation to perform optimal tracking according to the control input of each honesty agent at the current moment. According to the invention, time-varying affine formation optimal tracking is realized under the Byzantine attack, and the anti-interference capability and robustness of the cluster system are improved.

Description

Technical Field

[0001] The present application relates to the field of cluster systems, and in particular to an optimal tracking method and system for a flexible time-varying affine formation for a cluster system. Background Art

[0002] In recent decades, swarm systems have attracted widespread attention due to their wider range of applications compared to single agents. These applications include but are not limited to formation tracking, resource allocation, and task scheduling. In these research areas, a large amount of research work focuses on the formation tracking problem under various conditions, such as finite-time formation tracking, robust formation tracking, and optimal formation tracking. However, these strategies and most existing control methods presuppose that the trajectory is known. This means that swarm systems lack the ability to autonomously set trajectories through negotiation, which is particularly critical in the following scenarios: 1) the swarm system is completely disconnected from the processor that publishes the trajectory; 2) the trajectory needs to be determined based on the specific tasks of all agents, especially time-varying tasks. In these cases, traditional formation control methods are no longer applicable, which leads to the problem of distributed time-varying formation optimal tracking. Among them, the time-varying optimal trajectory that maintains the formation configuration needs to be achieved through collaborative calculation and tracking.

[0003] In the formation optimal tracking problem, handling constraints is a core issue. For example, C. Wu et al. proposed an algorithm to achieve formation optimal tracking with affine formation constraints; L. Jiang et al. proposed an algorithm to handle the case of time-invariant formation constraints. F. Huang et al. proposed a primal-dual algorithm that can effectively deal with time-varying formation constraints and local inequality constraints. The above three methods are all designed for strong convex objective functions. It should be noted that the methods proposed by L. Jiang et al. and F. Huang et al. are limited to handling time-invariant optimization problems. For time-varying convex optimization problems, the primal-dual summation algorithm proposed by X. Li et al. solves the problem of coupled time-invariant inequality constraints. In addition, the online convex optimization algorithm proposed by C. Wang et al. solves the problem of coupled time-varying inequality constraints with set constraints. In addition, K. Lu et al. proposed another primal-dual method that handles strong pseudo-convex objective functions by introducing auxiliary optimization strategies. X. Cao et al. designed a distributed online algorithm with a central server that uses saddle point technology to handle convex objective functions. However, it should be pointed out that none of the above four methods considers formation constraints. In addition, the algorithms designed by C. Wu et al., L. Jiang et al., and F. Huang et al. for the continuous time domain cannot be directly applied to the discrete time domain.

[0004] All the above research results are based on a simplified assumption that the communication links of the cluster system can work normally as expected. However, this assumption does not always hold true in practice, especially when facing network attacks. Among them, when the behavior of an intelligent agent deviates arbitrarily from its expectations, it is said to have suffered a Byzantine attack, which is highly disguised and can easily imitate other types of network attacks, including replay attacks, denial of service attacks, and false data injection attacks. Over the years, distributed robust average consensus algorithms for Byzantine attacks have been widely studied, and their application scope has also expanded from consensus problems to the field of distributed optimization. In the context of distributed learning, defense algorithms against Byzantine attacks have also been studied, but the implementation of these algorithms requires the existence of a central processor. In a fully distributed environment, Z. Yang et al. proposed a robust coordinate descent algorithm, C. Fang et al. proposed a robust gradient descent algorithm, and J. Li et al. proposed a robust stochastic gradient descent algorithm. However, it is worth noting that the above three methods are currently only applicable to time-invariant optimization problems. For time-varying optimization problems, especially those that are also constrained, the theoretical guarantees or extensions of distributed robust algorithms are still relatively scarce. Summary of the invention

[0005] The purpose of this application is to provide a flexible time-varying affine formation optimal tracking method and system for cluster systems, which can achieve optimal tracking of time-varying affine formations under Byzantine attacks and improve the anti-interference ability and robustness of cluster systems.

[0006] To achieve the above objectives, this application provides the following solutions:

[0007] In a first aspect, the present application provides an optimal tracking method for a flexible time-varying affine formation for a cluster system, wherein the cluster system includes multiple agents; the multiple agents are divided into Byzantine agents and honest agents after being attacked by Byzantine agents; the optimal tracking method for a flexible time-varying affine formation for a cluster system includes:

[0008] Establishing a dynamic model of the cluster system; the dynamic model is used to determine the state of each agent at the next moment according to the state of each agent at the current moment and the control input of each agent;

[0009] Establish an affine formation of the cluster system, and determine the optimal tracking problem of the elastic time-varying affine formation based on the affine formation and the dynamic model; the optimal tracking problem of the elastic time-varying affine formation is to set the control input of each honest agent at the current moment, and according to the dynamic model, minimize the objective function while satisfying the affine formation constraint and the global inequality constraint. The objective function is the sum of the local objective functions accumulated by the honest agents from the initial moment to the current moment;

[0010] Based on the elastic time-varying affine formation optimal tracking problem, an elastic time-varying affine formation optimal tracking control protocol is determined, and the control input of each honest agent at the current moment is obtained according to the elastic time-varying affine formation optimal tracking control protocol, so that the affine formation performs optimal tracking according to the control input of each honest agent at the current moment.

[0011] In a second aspect, the present application provides a flexible time-varying affine formation optimal tracking system for a cluster system, applying the above-mentioned flexible time-varying affine formation optimal tracking method for a cluster system, including:

[0012] A dynamic model building module is used to build a dynamic model of the cluster system; the dynamic model is used to determine the state of each intelligent agent at the next moment according to the state of each intelligent agent at the current moment and the control input of each intelligent agent;

[0013] The elastic time-varying affine formation optimal tracking problem determination module is connected to the dynamic model establishment module, and is used to establish an affine formation of the cluster system, and determine the elastic time-varying affine formation optimal tracking problem based on the affine formation and the dynamic model; the elastic time-varying affine formation optimal tracking problem is to set the control input of each honest agent at the current moment, and according to the dynamic model, minimize the objective function so that the state of each honest agent at the next moment satisfies the affine formation constraint and the global inequality constraint; the objective function is the sum of the local objective functions accumulated by the honest agent from the initial moment to the current moment;

[0014] A tracking module is connected to the elastic time-varying affine formation optimal tracking problem determination module, and is used to determine the elastic time-varying affine formation optimal tracking control protocol based on the elastic time-varying affine formation optimal tracking problem, and obtain the control input of each honest intelligent agent at the current moment according to the elastic time-varying affine formation optimal tracking control protocol, so that the affine formation performs optimal tracking according to the control input of each honest intelligent agent at the current moment.

[0015] According to the specific embodiments provided in this application, this application has the following technical effects:

[0016] The present application provides a method and system for optimal tracking of a flexible time-varying affine formation for a cluster system. A dynamic model is used to determine the state of each intelligent agent at the next moment according to the control input of each intelligent agent at the current moment. The position of each intelligent agent and all formation configurations are reflected by the affine formation, and can be adjusted over time. The optimal tracking problem of the flexible time-varying affine formation is determined by the dynamic model and the affine formation. An flexible time-varying affine formation optimal tracking control protocol is designed for the optimal tracking problem of the flexible time-varying affine formation to obtain the control input of each honest intelligent agent at the current moment, so that the cluster system can still achieve optimal tracking of the time-varying affine formation under Byzantine attacks, thereby improving the anti-interference ability and robustness of the cluster system. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0018] Figure 1 A schematic diagram of a flow chart of an optimal tracking method for a flexible time-varying affine formation of a cluster system for this application;

[0019] Figure 2 This is a schematic diagram of the elastic communication topology structure of this application under Byzantine attacks;

[0020] Figure 3 The block diagram of the optimal tracking control protocol for elastic time-varying affine formation;

[0021] Figure 4 In this application, the norm-based aggregation rule κ is used to optimize the tracking control protocol of the honest agent in the elastic time-varying affine formation. 1l = 0 output result diagram;

[0022] Figure 5 In this application, the norm-based aggregation rule κ is used to optimize the tracking control protocol of the honest agent in the elastic time-varying affine formation. 3l = 0 output result diagram;

[0023] Figure 6 In this application, the norm-based aggregation rule κ is used to optimize the tracking control protocol of the honest agent in the elastic time-varying affine formation. 4l = 0 output result diagram;

[0024] Figure 7 In this application, the norm-based aggregation rule κ is used to optimize the tracking control protocol of the honest agent in the elastic time-varying affine formation. 5l= 0 output result diagram;

[0025] Figure 8 The dynamic regret value Reg obtained by this application under the elastic time-varying affine formation optimal tracking control protocol h,k Schematic diagram of the curve result divided by time;

[0026] Fig. 9 The dynamic regret value Reg obtained by this application under the elastic time-varying affine formation optimal tracking control protocol f,k Schematic diagram of the curve result divided by time;

[0027] Fig.10 The dynamic regret value Reg obtained by this application under the elastic time-varying affine formation optimal tracking control protocol g,i,k Schematic diagram of the curve result divided by time;

[0028] Fig.11 This is a schematic diagram of the trajectory curve results obtained by the cluster system of this application under the elastic time-varying affine formation optimal tracking control protocol. DETAILED DESCRIPTION

[0029] The following will be combined with the drawings in the embodiments of the present application to clearly and completely describe the technical solutions in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0030] In order to make the above-mentioned objects, features and advantages of the present application more obvious and easy to understand, the present application is further described in detail below in conjunction with the accompanying drawings and specific implementation methods.

[0031] In an exemplary embodiment, Figure 1 As shown, a method for optimal tracking of a flexible time-varying affine formation for a cluster system is provided. The method is executed by a computer device, and specifically can be executed by a computer device such as a terminal or a server alone, or can be executed by a terminal and a server together. The cluster system includes multiple agents; after being attacked by Byzantine, the multiple agents are divided into Byzantine agents and honest agents; the method for optimal tracking of a flexible time-varying affine formation for a cluster system includes the following steps 101 to 103.

[0032] Step 101, establish a dynamic model of the cluster system; the dynamic model is used to determine the state of each agent at the next moment according to the state of each agent at the current moment and the control input of each agent.

[0033] Step 102, establish an affine formation of the cluster system, and determine the optimal tracking problem of the elastic time-varying affine formation based on the affine formation and the dynamic model; the optimal tracking problem of the elastic time-varying affine formation is to set the control input of each honest agent at the current moment, and according to the dynamic model, make the state of each honest agent at the next moment minimize the objective function while satisfying the affine formation constraint and the global inequality constraint; the objective function is the sum of the local objective functions accumulated by the honest agents from the initial moment to the current moment.

[0034] Step 103, based on the elastic time-varying affine formation optimal tracking problem, determine the elastic time-varying affine formation optimal tracking control protocol, obtain the control input of each honest agent at the current moment according to the elastic time-varying affine formation optimal tracking control protocol, so that the affine formation performs optimal tracking according to the control input of each honest agent at the current moment.

[0035] In an exemplary embodiment, the resilient communication topology of the cluster system under Byzantine attacks is as follows:

[0036] The communication topology of a cluster system with N = H + B agents under Byzantine attack is represented by a directed graph. Indicates that represents a node set, represents the edge set. Assume that there is a malicious attacker who can select a node subset This attack is called a Byzantine attack. All agents in the cluster system that have been attacked by Byzantine, namely Byzantine agents, constitute a node subset with a cardinality (the number of elements in a set) of B. The remaining agents that have not been attacked by Byzantine, i.e., honest agents, constitute a subset of nodes of cardinality H Note that in a cluster system, no agent can determine which agents belong to the Byzantine agent set. We only know that the upper limit of its cardinality B is B max , that is, 0≤B≤B max For any agent (e,j)∈E means that information can be transferred from agent j to agent e; represents the neighbor set of agent e; represents the in-degree of agent e, that is, the cardinality of the neighbor set of agent l. Figure 2 As shown in the figure, the nodes and edges of the Byzantine agents in the communication topology are represented by dashed lines, and the nodes and edges of the honest agents are represented by solid lines. max =1.

[0037] set up For a directed graph The associated row random matrix, that is, for any Both is a set of N×N dimensional real-valued matrices. If (e,j)∈E then w ej > 0, otherwise w ej = 0. Assume is the normalized left eigenvector of the row random matrix W, i.e., π T W=π T and π T 1=1, is a set of N-dimensional real-valued vectors, (·) T Indicates transpose.

[0038] Degenerate topology yes A subgraph of , where represents the edge set of degenerate topology, Indicates that the transmitting or receiving information nodes in E are a subset of all the edges of the Byzantine agents. yes Any subset of whose cardinality satisfies E H,e It means that the nodes transmitting information in E are a subset of all edges consisting of honest agents.

[0039] In an exemplary embodiment, consider a cluster system whose communication network is the above communication topology For every moment in represents a set of integers greater than or equal to a. For the lth agent, The dynamic model of the cluster system is as follows:

[0040] x l,k+1 =x l,k +Tu l,k .

[0041] Among them, k is the current time, x l,k is the state of the lth agent at the current moment, u l,k is the control input of the lth agent at the current moment, is the control input, is a set of m-dimensional real-valued vectors, T is the sampling time, T>0.

[0042] In an exemplary embodiment, an affine formation of the cluster system is defined. Given a desired formation configuration in is a set of Nm-dimensional real-valued vectors, where Nm represents N m-dimensional vectors, h1 is the position of the first agent, and h N is the position of the Nth agent, and N is the number of agents in the cluster system. Affine formation of the cluster system It is the set of all possible formation configuration affine transformations, which can describe the rotation, scaling, translation, shearing or any combination of the above for a given formation configuration h.

[0043] Secondly, in the optimal tracking problem of elastic time-varying affine formation, the time Any agent The local objective function is given by Given, where is a local set constraint; the local inequality constraint function is given by Given, n l dimensional real-valued vectors. Note that the local objective function f l,k , local inequality constraints g l,k , local collection constraints can only be obtained by agent l at the upcoming time k, and cannot be obtained by other agents. The local objective function and the local inequality constraint function are the local information of agent l. The elastic time-varying affine formation optimal tracking problem is to design a controller for the cluster system. in, Indicates that (·) The new vector or matrix formed by stacking related element or row vectors, u 1,k is the control input of the first agent at the current moment, u 2,k is the control input of the second agent at the current moment, u 1,k is the control input of the first agent at the current moment, u N,k is the control input of the Nth agent at the current moment. Thus, the expression of the optimal tracking problem of the elastic time-varying affine formation is:

[0044]

[0045] stx k+1 =x k +Tu k .

[0046]

[0047] g k (x k+1 )≤0.

[0048] in, To solve for the variable xk+1 So that the global objective function f k (x k+1 ) is minimized, st is the global objective function f k (x k+1 ) is subject to the constraints, k is the current moment, c is the cth moment, x k+1 is the state of the agent at time k+1, It means that for time k+1, x k+1 belong and is a subset of This Hm-dimensional real-valued vector set, is the local set constraint of the state of the honest agent at time k+1, f k (x k+1 ) is the global objective function, f i,c (x i,k+1 ) is the local objective function, is the set of honest agents, i is the serial number of the honest agent, x i,k+1 is the state of the i-th honest agent at time k+1, T is the sampling time, u k is the control input at the current moment, is the sum of the local objective functions accumulated by all honest agents from the initial moment to the current moment, g k (x k+1 ) is the global inequality constraint of the state of the honest agent at time k+1, is the affine formation of the cluster system, is an affine formation for honest agents.

[0049] In another exemplary embodiment of the present application, determining an optimal tracking control protocol for an elastic time-varying affine formation based on the elastic time-varying affine formation optimal tracking problem specifically includes:

[0050] The optimal tracking problem of the elastic time-varying affine formation is equivalently transformed, and the optimal tracking control protocol of the elastic time-varying affine formation is determined according to the optimal tracking problem of the elastic time-varying affine formation after the equivalent transformation; the expression of the optimal tracking problem of the elastic time-varying affine formation after the equivalent transformation is:

[0051]

[0052] sP k+1 =Ps k +Tu k .

[0053]

[0054] Among them, s kis the affine formation related variable of all honest agents at time k, s k+1 is the affine formation related variable of all honest agents at time k+1, is the local set constraint of the ith honest agent on the variables related to the affine formation, is the intersection of the local set constraints of all honest agents, and st is the optimization problem The constraints imposed, For the expected formation configuration The augmented configuration matrix of is a set of N×(m+1)-dimensional real-valued matrices, h1 is the position of the first agent, h N is the position of the Nth agent, horz is the horizontal connection of the vector or matrix, vert is the vertical connection of the vector or matrix, q is a column vector with all elements set to 1 and of appropriate dimension, is the augmented configuration matrix of the variables related to the affine formation, is a set of Nm×(m+1)m-dimensional real-valued matrices, I m is the identity matrix, is a set of m×m dimensional real-valued matrices, is the submatrix in P related to the i-th honest agent, is a set of m×(m+1)m-dimensional real-valued matrices, is the submatrix associated with the i-th honest agent in P, is a set of Hm×(m+1)m-dimensional real-valued matrices, g k (·) is a global inequality constraint.

[0055] In another exemplary embodiment of the present application, the elastic time-varying affine formation optimal tracking control protocol is for each moment Each honest agent The control protocol is mainly based on the following ideas: first, design a norm-based aggregation rule to filter out Byzantine agents in the neighbor set; second, build a flexible weighted average consensus mechanism to ensure that the cluster system can reach a consensus; then, design an update mechanism based on gradient descent to achieve online estimation of the time-varying optimal formation reference trajectory; finally, formulate a tracking controller that can track the local estimate of the optimal formation reference trajectory while maintaining the affine formation configuration. Figure 3 As shown, the elastic time-varying affine formation optimal tracking control protocol specifically includes:

[0056] Based on the aggregation rule and cluster system of the norm, the Byzantine judgment factor of each honest agent to any agent is obtained. Based on the Byzantine judgment factor of each honest agent to any agent, the elastic weighted average consensus mechanism is used to iterate the affine formation related variables of each honest agent at the current moment for a set number of times, and the estimated value of the affine formation related variables of each honest agent at the current moment is obtained;

[0057] Based on the estimated value of the affine formation related variables of each honest agent at the current moment and the local information of each honest agent at the current moment, a tracking control protocol is used to obtain the control input of each honest agent at the current moment;

[0058] Based on the estimated value of the affine formation related variables of each honest agent at the current moment, the local information of each honest agent at the current moment and the sub-gradient value, a gradient descent update mechanism is adopted to obtain the affine formation related variables of each honest agent at the next moment.

[0059] Based on the aggregation rule and cluster system of the norm, the Byzantine judgment factor of each honest agent to any agent is obtained. The specific expression is:

[0060]

[0061] Among them, k is the current time, κ il is the Byzantine judgment factor of the ith honest agent to the lth agent, if κ il = 1, it means that the i-th honest agent believes that the l-th agent is an honest agent. If κ il = 0, it means that the i-th honest agent believes that the l-th agent belongs to the Byzantine agent. is the neighbor set of the i-th honest agent, is the subset of the neighbors of the ith honest agent that has the smallest estimated value deviation from the ith honest agent, is the subset of the neighbors of the i-th honest agent that has the largest estimated value deviation from the i-th honest agent.

[0062]

[0063] Among them, e il,k is the estimated value deviation between the i-th honest agent and the l-th agent obtained by using the aggregation rule, e il,k The larger the absolute value of , the greater the deviation. For the corresponding e il,k <0, that is, For the corresponding e il,k >0, that is, if The number of elements of Less than or equal to B max ,So for if The number of elements of Less than or equal to B max ,So for otherwise, Depend on B max elements, and the corresponding e il,k The sum is the smallest, Depend on B max elements, and the corresponding e il,k The sum of is the largest, Λ represents (or ) has a cardinality equal to (or ) and B max The minimum of the two.

[0064] The estimated value deviation e between the i-th honest agent and the l-th agent obtained by the aggregation rule is il,k , the specific expression is:

[0065] e il,k =sgn(cos(θ il,k ))||s l,k -s i,k ||.

[0066]

[0067] Among them, s l,k is the affine formation-related variable of the lth agent at time k, s i,k is the affine formation-related variable of the i-th honest agent at time k, The random vector estimates that are inconsistent with the variables associated with the affine formation of the i-th honest agent, is a set of m(m+1)m-dimensional real-valued matrices, sgn is the sign function, ||·|| is the Euclidean norm, (·) T is transposed.

[0068] like Figure 4 , Figure 5 , Figure 6 and Figure 7 As shown in the figure, the cluster system outputs the result based on the aggregation rule of norm under the elastic time-varying affine formation optimal tracking control protocol, where κil =0 is represented by a solid black dot.

[0069] In another exemplary embodiment of the present application, the elastic weighted average consensus mechanism is expressed as:

[0070]

[0071] Among them, γ k is the weighting factor, L is the number of iterations, s i,k[L+1] is the affine formation-related variable of the i-th honest agent in the L+1th iteration at the current moment, κ il is the Byzantine judgment factor of the ith honest agent to the lth agent, if κ il = 1, it means that the i-th honest agent believes that the l-th agent is an honest agent. If κ il = 0, it means that the i-th honest agent believes that the l-th agent belongs to the Byzantine agent, s l,k[L] is the affine formation-related variable of the lth agent in the Lth iteration at the current moment, s i,k[L] It is the affine formation related variables of the i-th honest agent at the current moment in the L-th iteration.

[0072] Among them, the i-th honest agent calculates the estimated value of the affine formation related variables of the i-th honest agent at the current moment through D>0 iterations Among them, s i,k[1] =s i,k ,s i,k[1] is the affine formation related variable of the i-th honest agent in the first iteration at the current moment, s i,k is the affine formation related variable of the i-th honest agent at the current moment. s i,1 is the affine formation related variable of the i-th honest agent at the first moment.

[0073] like Figure 3 As shown, where s l[L] is the affine formation-related variable of the lth agent in the Lth iteration, s i[L+1] is the affine formation related variable of the i-th honest agent in the L+1th iteration, is the estimated value of the affine formation related variables of the i-th honest agent, s i is the affine formation related variable of the i-th honest agent, s i[1] are the affine formation related variables of the i-th honest agent in the first iteration.

[0074] In another exemplary embodiment of the present application, the update mechanism of the gradient descent is expressed as:

[0075]

[0076] Among them, s i,k+1 is the affine formation-related variable of the i-th honest agent at time k+1, is the estimated value of the affine formation related variables of the i-th honest agent at time k, λ i,k is the dual variable used to handle constraints for the i-th honest agent at time k+1, λ i,k+1 is the dual variable used to handle constraints for the i-th honest agent at time k+1, For the set M si Projection on [·] + is the projection on the non-negative quadrant, α k and δ k is a positive decreasing step, f i,k (·) is the local objective function of the i-th honest agent, g i,k (·) is the local inequality constraint function of the ith honest agent, the local objective function of the ith honest agent and the local inequality constraint function of the ith honest agent are the local information of the ith honest agent, is the sub-gradient value of the local objective function of the i-th honest agent with respect to the affine formation related variables, is the subgradient value of the local inequality constraint function of the i-th honest agent with respect to the dual variable, is the Lagrangian function of the dependent and dual variables of the affine formation, for With respect to the subgradients of the variables associated with the affine formation, for Subgradient with respect to the dual variable.

[0077] In another exemplary embodiment of the present application, the tracking control protocol is:

[0078]

[0079] Among them, u i,k is the control input of the i-th honest agent at time k, x i,k is the state of the i-th agent at time k. Figure 3 As shown, x i is the state of the ith agent, u i is the control input of the i-th honest agent, is the sub-gradient value of the local objective function of the i-th honest agent, is the subgradient value of the local inequality constraint function of the i-th honest agent, h i is the position of the i-th honest agent, is the local set constraint of the i-th honest agent.

[0080] In another exemplary embodiment of the present application, the elastic time-varying affine formation optimal tracking method for a cluster system further includes:

[0081] Based on the elastic time-varying affine formation optimal tracking problem, the performance evaluation conditions of the elastic time-varying affine formation optimal tracking control protocol are determined, and according to the performance evaluation conditions of the elastic time-varying affine formation optimal tracking control protocol, the parameter range of the elastic time-varying affine formation optimal tracking control protocol is determined.

[0082] In another exemplary embodiment of the present application, the performance evaluation conditions of the elastic time-varying affine formation optimal tracking control protocol specifically include:

[0083]

[0084] Among them, Reg f,k is the cumulative error of all honest agents about the optimal solution, Reg g,i,k is the cumulative violation value of the inequality constraint by the i-th honest agent, Reg h,k is the cumulative violation value of all honest agents regarding the affine formation constraint, and k is the current moment.

[0085] To ensure that Reg h,k , Reg f,k and Reg g,i,k Sublinear convergence, for each moment Each honest agent The parameters of the optimal tracking control protocol for the elastic time-varying affine formation can be designed as follows: Select the decreasing step size Decrease step size Weighting Factor Setting times in 2τ2<τ1<1-2τ2, τ1<τ3, vτ1<τ4, represents the maximum number of neighbors of all honest agents, B max is the upper limit of the cardinality of the Byzantine agent set known in advance, represents the total number of degenerate topologies. Degenerate topologies are those that remove all nodes and edges related to Byzantine agents from the original topology and then arbitrarily remove up to B max The subtopology after the edges, H is the number of honest agents.

[0086] Among them, the cumulative error Reg of all honest agents about the optimal solution f,k, the cumulative violation value Reg of the inequality constraint of the i-th honest agent g,i,k and the cumulative violation value Reg of all honest agents regarding the affine formation constraint h,k , are the dynamic regret values ​​of the elastic time-varying affine formation optimal tracking problem. The specific expression of the dynamic regret value is:

[0087]

[0088] in, is the optimal solution of the equivalent problem at time c, s c* is the optimal solution at time c, x i,c* is the optimal solution of the equivalent problem of the i-th honest agent at time c, is the weighted average of the estimated values ​​of the affine formation-related variables of the i-th honest agent at time c, is the weighting factor of the i-th honest agent, is a set of H-dimensional real-valued vectors, x i,c is the state of the i-th honest agent at time c, x i,c+1 is the state of the i-th honest agent at time c+1, satisfying ρ T 1=1,f i,c (x i,c+1 ) is the local objective function of the i-th honest agent in x i,c+1 The value on f i,c (x i,c* ) is the local objective function of the i-th honest agent in x i,c* The value on g i,c (x i,c+1 ) is the local inequality constraint function of the i-th honest agent in x i,c+1 The value on .

[0089] like Figure 8 As shown in the figure, the dynamic regret value Reg obtained by the cluster system under the elastic time-varying affine formation optimal tracking control protocol is shown. h,k Divide the curve result by time. Fig. 9 As shown in the figure, the dynamic regret value Reg obtained by the cluster system under the elastic time-varying affine formation optimal tracking control protocol is shown. f,k Divide the curve result by time. Fig.10 As shown in the figure, the dynamic regret value Reg obtained by the cluster system under the elastic time-varying affine formation optimal tracking control protocol is shown. g,i,k Divide the curve result by time. Fig.11As shown in the figure, the trajectory curve results obtained by the cluster system under the elastic time-varying affine formation optimal tracking control protocol. In the figure, the diamond represents the initial position of the agent, the five-pointed star represents the final position of the agent, and the line represents the trajectory curve of the agent. At time t = 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20 seconds, the actual positions of all honest agents and the formation configuration composed of the ideal positions of the Byzantine agents are displayed. The gray area represents the threat area.

[0090] This application describes the elastic communication topology of the cluster system under Byzantine attacks; establishes a dynamic model of the cluster system; focuses on the optimal tracking problem of elastic time-varying affine formations and clearly defines this problem; derives the specific expression of the optimal tracking control protocol of the time-varying affine formation; and determines the parameters of the control protocol. This application designs an elastic time-varying affine formation optimal tracking control protocol for the optimal tracking problem of elastic time-varying affine formations to obtain the control input of each honest agent at the current moment, ensuring that the affine formation can achieve optimal tracking of the time-varying affine formation according to the control input of each honest agent at the current moment, thereby improving the anti-interference ability and robustness of the cluster system.

[0091] Based on the same inventive concept, the embodiment of the present application further provides an elastic time-varying affine formation optimal tracking system for a cluster system, applying the above-mentioned elastic time-varying affine formation optimal tracking method for a cluster system, and the elastic time-varying affine formation optimal tracking system for a cluster system includes:

[0092] The dynamic model building module is used to build a dynamic model of the cluster system; the dynamic model is used to determine the state of each intelligent agent at the next moment according to the state of each intelligent agent at the current moment and the control input of each intelligent agent.

[0093] The elastic time-varying affine formation optimal tracking problem determination module is connected to the dynamic model establishment module, and is used to establish an affine formation of the cluster system, and determine the elastic time-varying affine formation optimal tracking problem based on the affine formation and the dynamic model; the elastic time-varying affine formation optimal tracking problem is to set the control input of each honest agent at the current moment, and according to the dynamic model, make the state of each honest agent at the next moment meet the affine formation constraint and the global inequality constraint, and minimize the objective function; the objective function is the sum of the local objective functions accumulated by the honest agents from the initial moment to the current moment.

[0094] A tracking module is connected to the elastic time-varying affine formation optimal tracking problem determination module, and is used to determine the elastic time-varying affine formation optimal tracking control protocol based on the elastic time-varying affine formation optimal tracking problem, and obtain the control input of each honest intelligent agent at the current moment according to the elastic time-varying affine formation optimal tracking control protocol, so that the affine formation performs optimal tracking according to the control input of each honest intelligent agent at the current moment.

[0095] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0096] This article uses specific examples to illustrate the principles and implementation methods of this application. The description of the above embodiments is only used to help understand the method and core ideas of this application. At the same time, for those skilled in the art, according to the ideas of this application, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting this application.

Claims

1. An optimal tracking method for elastic time-varying affine formations of cluster systems, characterized in that: The cluster system includes a plurality of intelligent agents; After being attacked by Byzantine agents, multiple agents are divided into Byzantine agents and honest agents; The elastic time-varying affine formation optimal tracking method for cluster systems includes: Establishing a dynamic model of the cluster system; the dynamic model is used to determine the state of each agent at the next moment according to the state of each agent at the current moment and the control input of each agent; Establish an affine formation of the cluster system, and determine the optimal tracking problem of the elastic time-varying affine formation based on the affine formation and the dynamic model; the optimal tracking problem of the elastic time-varying affine formation is to set the control input of each honest agent at the current moment, and according to the dynamic model, minimize the objective function while satisfying the affine formation constraint and the global inequality constraint. The objective function is the sum of the local objective functions accumulated by the honest agents from the initial moment to the current moment; Based on the elastic time-varying affine formation optimal tracking problem, an elastic time-varying affine formation optimal tracking control protocol is determined, and the control input of each honest agent at the current moment is obtained according to the elastic time-varying affine formation optimal tracking control protocol, so that the affine formation performs optimal tracking according to the control input of each honest agent at the current moment.

2. The elastic time-varying affine formation optimal tracking method for cluster systems according to claim 1, characterized in that: The expression of the optimal tracking problem of the elastic time-varying affine formation is: stx k+1 =x k +You k ; g k (x k+1 )≤0; in, To solve for the variable x k+1 So that the global objective function f k (x k+1 ) is minimized, st is the global objective function f k (x k+1 ) is subject to the constraints, k is the current moment, c is the cth moment, x k+1 is the state of the agent at time k+1, f k (x k+1 ) is the global objective function, f i,c (x i,k+1 ) is the local objective function, is the set of honest agents, i is the serial number of the honest agent, x i,k+1 is the state of the i-th honest agent at time k+1, is the local set constraint of the state of the honest agent at time k+1, T is the sampling time, u k is the control input at the current moment, is the sum of the local objective functions accumulated by all honest agents from the initial moment to the current moment, g k (x k+1 ) is the global inequality constraint of the state of the honest agent at time k+1, is the affine formation of the cluster system, is an affine formation for honest agents.

3. The elastic time-varying affine formation optimal tracking method for cluster systems according to claim 2, characterized in that: Based on the elastic time-varying affine formation optimal tracking problem, an elastic time-varying affine formation optimal tracking control protocol is determined, specifically including: The optimal tracking problem of the elastic time-varying affine formation is equivalently transformed, and the optimal tracking control protocol of the elastic time-varying affine formation is determined according to the optimal tracking problem of the elastic time-varying affine formation after the equivalent transformation; the expression of the optimal tracking problem of the elastic time-varying affine formation after the equivalent transformation is: please k+1 =Ps k +You k ; Among them, s k is the affine formation related variable of all honest agents at time k, s k+1 is the affine formation related variable of all honest agents at time k+1, is the local set constraint of the ith honest agent on the variables related to the affine formation, is the intersection of the local set constraints of all honest agents, and st is the optimization problem The constraints imposed, For the expected formation configuration The augmented configuration matrix of N is the position of the Nth agent, N is the number of agents in the cluster system, horz is the horizontal connection of the vector or matrix, vert is the vertical connection of the vector or matrix, q is a column vector with all elements set to 1 of the appropriate dimension, is the augmented configuration matrix of the variables related to the affine formation, I m is the identity matrix, is the submatrix in P related to the i-th honest agent, is the submatrix related to the i-th honest agent in P, g k (·) is a global inequality constraint.

4. The elastic time-varying affine formation optimal tracking method for cluster systems according to claim 3, characterized in that: The elastic time-varying affine formation optimal tracking control protocol specifically includes: Based on the aggregation rule and cluster system of the norm, the Byzantine judgment factor of each honest agent to any agent is obtained. Based on the Byzantine judgment factor of each honest agent to any agent, the elastic weighted average consensus mechanism is used to iterate the affine formation related variables of each honest agent at the current moment for a set number of times, and the estimated value of the affine formation related variables of each honest agent at the current moment is obtained; Based on the estimated value of the affine formation related variables of each honest agent at the current moment and the local information of each honest agent at the current moment, a tracking control protocol is used to obtain the control input of each honest agent at the current moment; Based on the estimated value of the affine formation related variables of each honest agent at the current moment, the local information of each honest agent at the current moment and the sub-gradient value, a gradient descent update mechanism is adopted to obtain the affine formation related variables of each honest agent at the next moment.

5. The elastic time-varying affine formation optimal tracking method for cluster systems according to claim 4, characterized in that: The expression of the elastic weighted average consensus mechanism is: Among them, γ k is the weighting factor, L is the number of iterations, s i,k[L+1] is the affine formation-related variable of the i-th honest agent in the L+1th iteration at the current moment, κ il is the Byzantine judgment factor of the ith honest agent to the lth agent, s l,k[L] is the affine formation-related variable of the lth agent in the Lth iteration at the current moment, s i,k[L] It is the affine formation related variables of the i-th honest agent at the current moment in the L-th iteration.

6. The elastic time-varying affine formation optimal tracking method for cluster systems according to claim 5, characterized in that: The update mechanism of the gradient descent is expressed as: Among them, s i,k+1 is the affine formation-related variable of the i-th honest agent at time k+1, is the estimated value of the affine formation related variables of the i-th honest agent at time k, λ i,k is the dual variable used to handle constraints for the i-th honest agent at time k, λ i,k+1 is the dual variable used to handle constraints for the i-th honest agent at time k+1, For the set M si Projection on [·] + is the projection on the non-negative quadrant, α k and δ k is a positive decreasing step size, is the Lagrangian function of the dependent and dual variables of the affine formation, for The subgradients of the variables associated with the affine formation are, for Subgradient with respect to the dual variable.

7. The elastic time-varying affine formation optimal tracking method for cluster systems according to claim 6, characterized in that: The tracking control protocol is: Among them, u i,k is the control input of the i-th honest agent at time k, x i,k is the state of the i-th agent at time k.

8. The elastic time-varying affine formation optimal tracking method for cluster systems according to claim 1, characterized in that: The elastic time-varying affine formation optimal tracking method for cluster systems also includes: Based on the elastic time-varying affine formation optimal tracking problem, the performance evaluation conditions of the elastic time-varying affine formation optimal tracking control protocol are determined, and according to the performance evaluation conditions of the elastic time-varying affine formation optimal tracking control protocol, the parameter range of the elastic time-varying affine formation optimal tracking control protocol is determined.

9. The elastic time-varying affine formation optimal tracking method for cluster systems according to claim 8, characterized in that: The performance evaluation conditions of the elastic time-varying affine formation optimal tracking control protocol specifically include: Among them, Reg f,k is the cumulative error of all honest agents about the optimal solution, Reg g,i,k is the cumulative violation value of the inequality constraint by the i-th honest agent, Reg h,k is the cumulative violation value of all honest agents regarding the affine formation constraint, and k is the current moment.

10. An elastic time-varying affine formation optimal tracking system for a cluster system, applying the elastic time-varying affine formation optimal tracking method for a cluster system according to any one of claims 1 to 9, characterized in that: The elastic time-varying affine formation optimal tracking system for cluster systems comprises: A dynamic model building module is used to build a dynamic model of the cluster system; the dynamic model is used to determine the state of each intelligent agent at the next moment according to the state of each intelligent agent at the current moment and the control input of each intelligent agent; The elastic time-varying affine formation optimal tracking problem determination module is connected to the dynamic model establishment module, and is used to establish an affine formation of the cluster system, and determine the elastic time-varying affine formation optimal tracking problem based on the affine formation and the dynamic model; the elastic time-varying affine formation optimal tracking problem is to set the control input of each honest agent at the current moment, and according to the dynamic model, minimize the objective function so that the state of each honest agent at the next moment satisfies the affine formation constraint and the global inequality constraint; the objective function is the sum of the local objective functions accumulated by the honest agent from the initial moment to the current moment; A tracking module is connected to the elastic time-varying affine formation optimal tracking problem determination module, and is used to determine the elastic time-varying affine formation optimal tracking control protocol based on the elastic time-varying affine formation optimal tracking problem, and obtain the control input of each honest intelligent agent at the current moment according to the elastic time-varying affine formation optimal tracking control protocol, so that the affine formation performs optimal tracking according to the control input of each honest intelligent agent at the current moment.

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