An elastic time-varying affine formation optimal tracking method and system for a swarm system

Abstract

The application discloses an elastic time-varying affine formation optimal tracking method and system for a swarm system, relates to the field of swarm systems, and the swarm system comprises a plurality of intelligent agents, is divided into a Byzantine intelligent agent and an honest intelligent agent after being attacked by a Byzantine attack, and the method comprises the following steps: establishing a dynamic model and an affine formation of the swarm system, determining an elastic time-varying affine formation optimal tracking problem, setting the control input of each honest intelligent agent at the current moment, minimizing a target function according to the dynamic model so that the state of each honest intelligent agent at the next moment meets the affine formation constraint and the global inequality constraint, determining an elastic time-varying affine formation optimal tracking control protocol, obtaining the control input of each honest intelligent agent at the current moment, and making the affine formation perform optimal tracking according to the control input of each honest intelligent agent at the current moment. The application realizes time-varying affine formation optimal tracking under the Byzantine attack, and improves the anti-interference capability and robustness of the swarm system.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of swarm systems, and in particular, to an elastic time-varying affine formation optimal tracking method and system for swarm systems. BACKGROUND

[0002] In recent decades, swarm systems have attracted extensive attention due to their wider application range compared to single agents. These applications include, but are not limited to, formation tracking, resource allocation, and task scheduling. In these research fields, a large number of research works focus on formation tracking problems 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 the swarm system lacks the ability to autonomously set the trajectory through negotiation, which is particularly crucial in the following scenarios: 1) the swarm system is completely disconnected from the processor that issues the trajectory; 2) the trajectory needs to be determined according to the specific tasks of all agents, especially time-varying tasks. In these cases, traditional formation control methods are no longer applicable, thereby raising the problem of distributed time-varying formation optimal tracking. The time-varying optimal trajectory that maintains the formation configuration needs to be achieved through collaborative computation and tracking.

[0003] In the problem of formation optimal tracking, handling constraints is a core issue. For example, C. Wu et al. proposed an algorithm aimed at achieving formation optimal tracking with affine formation constraints; L. Jiang et al. proposed an algorithm for handling 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 designed for strongly convex objective functions. It is worth noting 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 sum algorithm proposed by X. Li et al. solves the coupled time-invariant inequality constraint problem. In addition, the online convex optimization algorithm proposed by C. Wang et al. solves the coupled time-varying inequality constraint problem with set constraints. Furthermore, K. Lu et al. proposed another primal-dual method that handles strongly pseudo-convex objective functions by introducing an auxiliary optimization strategy. X. Cao et al. designed a distributed online algorithm involving a central server that utilizes saddle point technology to handle convex objective functions. However, it is important to note that the above four methods do not consider formation constraints. In addition, the algorithms designed by C. Wu et al., L. Jiang et al., and F. Huang et al. for continuous time domains cannot be directly applied to discrete time domains.

[0004] All the above research results are based on a simplified assumption that the communication links of the swarm system can work as expected. However, this assumption does not always hold in practice, especially when facing network attacks. Among them, when the behavior of a certain agent deviates arbitrarily from its expectation, it is called to suffer from a Byzantine attack, which has a high degree of camouflage and can easily imitate other types of network attacks, including replay attacks, denial of service attacks and false data injection attacks. For many years, distributed robust average consensus algorithms against Byzantine attacks have been widely studied, and their application range has been extended from the consensus problem to the distributed optimization field. In the context of distributed learning, defense algorithms against Byzantine attacks have also been studied, but the implementation of these algorithms requires the presence 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 subject to constraints, theoretical guarantees or extensions of distributed robust algorithms are still relatively scarce. SUMMARY

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

[0006] To achieve the above-mentioned purpose, the present application provides the following solutions:

[0007] In a first aspect, the present application provides a resilient time-varying affine formation optimal tracking method for a swarm system, the swarm system comprising a plurality of agents; the plurality of agents being divided into Byzantine agents and honest agents after being attacked by Byzantine attacks; the resilient time-varying affine formation optimal tracking method for the swarm system comprising:

[0008] establishing a dynamic model of the swarm system; the dynamic model being used to determine the state of each agent at the next time according to the state of each agent at the current time and the control input of each agent;

[0009] establishing an affine formation of the swarm system, and determining a resilient time-varying affine formation optimal tracking problem based on the affine formation and the dynamic model; the resilient time-varying affine formation optimal tracking problem being to minimize an objective function by setting the control input of each honest agent at the current time according to the dynamic model, so that the state of each honest agent at the next time satisfies the affine formation constraint and the global inequality constraint; the objective function being the sum of the local objective functions accumulated by the honest agents from the initial time to the current time;

[0010] determine an 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 agent at the current time 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 time.

[0011] In a second aspect, the present application provides an elastic time-varying affine formation optimal tracking system for a swarm system, which applies the elastic time-varying affine formation optimal tracking method for a swarm system as described above, and includes:

[0012] a dynamics model establishing module, configured to establish a dynamics model of the swarm system; the dynamics model is configured to determine the state of each agent at the next time according to the state of each agent at the current time and the control input of each agent;

[0013] an elastic time-varying affine formation optimal tracking problem determining module, connected with the dynamics model establishing module, configured to establish an affine formation of the swarm system, and determine an elastic time-varying affine formation optimal tracking problem based on the affine formation and the dynamics model; the elastic time-varying affine formation optimal tracking problem is to minimize an objective function by setting the control input of each honest agent at the current time, so that the state of each honest agent at the next time meets the affine formation constraint and the global inequality constraint according to the dynamics model; the objective function is the sum of local objective functions accumulated by the honest agents from the initial time to the current time;

[0014] a tracking module, connected with the elastic time-varying affine formation optimal tracking problem determining module, configured to determine an elastic time-varying affine formation optimal tracking control protocol based on the elastic time-varying affine formation optimal tracking problem, obtain the control input of each honest agent at the current time according to the elastic time-varying affine formation optimal tracking control protocol, and make the affine formation perform optimal tracking according to the control input of each honest agent at the current time.

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

[0016] The application provides an elastic time-varying affine formation optimal tracking method and system for a swarm system, determines the state of each intelligent agent at the next moment according to the control input of each intelligent agent at the current moment through a dynamic model, reflects the position of each intelligent agent and all formation configurations through an affine formation, and can be adjusted over time, determines the elastic time-varying affine formation optimal tracking problem through the dynamic model and the affine formation, designs an elastic time-varying affine formation optimal tracking control protocol for the elastic time-varying affine formation optimal tracking problem to obtain the control input of each honest intelligent agent at the current moment, and makes the swarm system still realize time-varying affine formation optimal tracking under a Byzantine attack, thereby improving the anti-interference ability and robustness of the swarm system. BRIEF DESCRIPTION OF DRAWINGS

[0017] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed in the embodiments will be briefly introduced as follows. Obviously, the drawings in the following description only constitute some embodiments of the present application, and for those skilled in the art, other drawings can also be obtained without creative labor on the basis of these drawings.

[0018] Figure 1 A flowchart of an elastic time-varying affine formation optimal tracking method for a swarm system according to the present application;

[0019] Figure 2 A schematic diagram of an elastic communication topology under a Byzantine attack according to the present application;

[0020] Figure 3 A block diagram of an elastic time-varying affine formation optimal tracking control protocol;

[0021] Figure 4 A schematic diagram of the result output by the norm-based aggregation rule K 1l = 0 of an honest intelligent agent under the elastic time-varying affine formation optimal tracking control protocol according to the present application;

[0022] Figure 5 A schematic diagram of the result output by the norm-based aggregation rule K 3l = 0 of an honest intelligent agent under the elastic time-varying affine formation optimal tracking control protocol according to the present application;

[0023] Figure 6 A schematic diagram of the result output by the norm-based aggregation rule K 4l = 0 of an honest intelligent agent under the elastic time-varying affine formation optimal tracking control protocol according to the present application;

[0024] Figure 7 A schematic diagram of the result output by the norm-based aggregation rule K 5lA diagram illustrating the output when =0;

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

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

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

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

[0029] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0030] To make the above-mentioned objectives, features and advantages of this application more apparent and understandable, the application will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0031] In one exemplary embodiment, such as Figure 1 As shown, a method for optimal tracking of a resilient time-varying affine formation in a cluster system is provided. This method is executed by a computer device, specifically by a terminal or server alone, or by both a terminal and a server. The cluster system includes multiple agents; after being subjected to a Byzantine attack, the agents are divided into Byzantine agents and honest agents; the method for optimal tracking of a resilient time-varying affine formation in a cluster system includes 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 based on 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 an 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 minimize a target function by setting a control input of each honest agent at a current time, so that a state of each honest agent at a next time meets an affine formation constraint and a global inequality constraint according to the dynamic model; the target function is a sum of local target functions accumulated by the honest agents from an initial time to the current time.

[0034] Step 103, determine an 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 agent at the current time 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 time.

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

[0036] The communication topology of the cluster system under the Byzantine attack has N = H + B agents and is represented by a directed graph , where represents a node set, represents an edge set. It is assumed that there is a malicious attacker who can select a node subset to destroy and control, and this attack is called a Byzantine attack. All agents in the cluster system that have suffered the Byzantine attack, i.e., Byzantine agents, constitute a node subset with a cardinality (the number of elements in a set) of B. The remaining agents that have not suffered the Byzantine attack, i.e., honest agents, constitute a node subset with a cardinality of H. max Note that in the cluster system, any agent cannot determine which agents belong to the set of Byzantine agents max , and only knows that the upper limit of the cardinality B is B . For any agent (e,j)∈E represents that information can be transmitted from agent j to agent e; represents a neighbor set of agent e; Figure 2 represents an in-degree of agent e, i.e., the cardinality of the neighbor set of agent l. As shown in max , the nodes and edges of the Byzantine agents in the communication topology structure are represented by dashed lines, and the nodes and edges of the honest agents are represented by solid lines. Let B

[0037] Let be a directed graph The associated row stochastic matrix, i.e., for any has is a set of N x N real-valued matrices. For any if (e, j) E E, then w ej > 0, otherwise w ej = 0. Let be the normalized left eigenvector of the row stochastic matrix W, i.e., p T W = p T and p T 1 = 1, is a set of N real-valued vectors, (·) T denotes the transpose.

[0038] The degenerated topology is a subgraph of , where denotes the edge set of the degenerated topology, denotes the subset of E consisting of all edges whose transmitting information node or receiving information node is a Byzantine agent, is an arbitrary subset of and its cardinality satisfies E H,e denotes the subset of E consisting of all edges whose transmitting information node is an honest agent.

[0039] In an exemplary embodiment, consider a swarm system whose communication network is the communication topology For each time instant where denotes the set of integers greater than or equal to a, for the lth agent, The dynamic model of the swarm system is specified as follows:

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

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

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

[0043] Secondly, in the optimal tracking problem of elastic time-varying affine formations, at time... Any intelligent agent The local objective function is derived from Given, among which It is a local set constraint; the local inequality constraint function is... Give, For n l A set of dimensional real-valued vectors. Note the local objective function f. l,k Local inequality constraint g l,k Local set constraints It can only be acquired by agent l at the upcoming time k, and cannot be acquired by other agents. The local objective function and local inequality constraint functions are local information of agent l. At each time... The optimal tracking problem of elastic time-varying affine formation is for designing controllers in cluster systems. in, Indicates the relationship between (·) and The new vector or matrix formed by the related elements or row vector stack, u 1,k u is the control input for the first agent at the current moment. 2,k u is the control input for the second agent at the current moment. 1,k u is the control input for the first agent at the current moment. N,k This represents the control input for the Nth agent at the current time. Therefore, the expression for 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 variable xk+1 minimize global objective function f k (x k+1 ) subject to constraints imposed by global objective function f k (x k+1 ) at current time k, c is the c-th time, x k+1 is the state of the agent at k+1, x represents that for k+1, x k+1 belongs to and is a subset of this set of Hm-dimensional real-valued vectors, is the local set constraint of the state of the honest agent at 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 k+1, T is the sampling time, u k is the control input at current time, is the sum of the local objective functions of all honest agents from the initial time to the current time, g k (x k+1 ) is the global inequality constraint of the state of the honest agent at k+1, is the affine formation of the swarm system, is the affine formation about the honest agent.

[0049] In another exemplary embodiment of the present application, the elastic time-varying affine formation optimal tracking control protocol is determined based on the elastic time-varying affine formation optimal tracking problem, specifically comprising:

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

[0051]

[0052] s.t. Ps k+1 = Ps k + Tu k .

[0053]

[0054] wherein s kaffine formation related variables of all honest agents at time k, s k+1 affine formation related variables of all honest agents at time k + 1, local consensus constraint of the ith honest agent on affine formation related variables, intersection of local consensus constraints of all honest agents, s.t. is the optimization problem subjected to constraint conditions, augmented configuration matrix on expected formation configuration , is a set of N x (m + 1) -dimensional real matrices, h1is the position of the first agent, h N is the position of the Nth agent, horzis the horizontal concatenation of vectors or matrices, vertis the vertical concatenation of vectors or matrices, q is a column vector with all elements being 1 with appropriate dimension, augmented configuration matrix on affine formation related variables, is a set of Nm x (m + 1) m-dimensional real matrices, I m is an identity matrix, is a set of m x m-dimensional real matrices, is the sub-matrix in P related to the ith honest agent, is a set of m x (m + 1) m-dimensional real matrices, is the sub-matrix in P related to the ith honest agent, is a set of Hm x (m + 1) m-dimensional real matrices, g k (·) is a global inequality constraint.

[0055] In another exemplary embodiment of the present application, the resilient time-varying affine formation optimal tracking control protocol is based on the following ideas for each time each honest agent : first, design a norm-based aggregation rule to filter out the Byzantine agents in the neighbor set; second, construct a resilient weighted average consensus mechanism to ensure that the whole swarm system can reach an agreement; third, design a gradient descent-based update mechanism to achieve online estimation of the time-varying optimal formation reference trajectory; and finally, develop a tracking controller that can track the local estimated value of the optimal formation reference trajectory while maintaining the affine formation configuration. As shown in Figure 3 , the resilient time-varying affine formation optimal tracking control protocol specifically includes:

[0056] The norm-based aggregation rule and the cluster system obtain a Byzantine judgment factor of each honest agent to any agent, based on the Byzantine judgment factor of each honest agent to any agent, an elastic weighted average consensus mechanism is used to iterate the affine formation related variable of each honest agent at the current time for a certain number of times, to obtain the estimated value of the affine formation related variable of each honest agent at the current time;

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

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

[0059] The norm-based aggregation rule and the cluster system obtain a Byzantine judgment factor of each honest agent to any agent, and the specific expression is:

[0060]

[0061] Wherein, k is the current time, κ il is the Byzantine judgment factor of the i th honest agent to the l th agent, if κ il = 1, it means that the i th honest agent considers that the l th agent belongs to the honest agent, if κ il = 0, it means that the i th honest agent considers that the l th agent belongs to the Byzantine agent, is the neighbor set of the i th honest agent, is the sub-set of the neighbor set of the i th honest agent with the smallest estimated value deviation from the i th honest agent, is the sub-set of the neighbor set of the i th honest agent with the largest estimated value deviation from the i th honest agent.

[0062]

[0063] Wherein, 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, the greater the absolute value of e il,k , the greater the deviation, is the sub-neighbor set corresponding to e il,k < 0, that is, is the sub-neighbor set corresponding to e il,k > 0, that is, If the number of elements in is less than or equal to B max , then is If the number of elements in is less than or equal to B max , then is Otherwise, is composed of B elements in max , and the sum of e il,k corresponding to the elements is minimum, is composed of B elements in max , and the sum of e il,k corresponding to the elements is maximum, Λ represents (or ) is a subset, and the cardinality is equal to (or ) and B max is the minimum value.

[0064] The estimated value deviation e il,k between the ith honest agent and the lth agent obtained by using the aggregation rule is specifically expressed as:

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

[0066]

[0067] Wherein, 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 ith honest agent at time k, is the random vector estimate value inconsistent with the affine formation related variable of the ith honest agent, is a set of m (m+1) m-dimensional real value matrices, sgn is a sign function, ||·|| is a Euclidean norm, (·) T is a transpose.

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

[0069] In another exemplary embodiment of this application, the expression for the elastic weighted average consensus mechanism is:

[0070]

[0071] Where, γ k s is the weighting factor, L is the number of iterations, and s i,k[L+1] Let κ be the affine formation-related variable for the i-th honest agent in the (L+1)-th iteration at the current time. il Let κ be the Byzantine judgment factor of the i-th honest agent on the l-th agent. il =1, which means that the i-th honest agent believes that the l-th agent is an honest agent. If κ il =0, which means that the i-th honest agent believes that the l-th agent belongs to the Byzantine agent category. l,k[L] Let s be the affine formation-related variables for the l-th agent in the L-th iteration at the current time. i,k[L] Let be the affine formation-related variables for the i-th honest agent in the L-th iteration at the current time.

[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 time through D > 0 iterations. Among them, s i,k[1] =s i,k s i,k[1] Let s be the affine formation related variables for the i-th honest agent in the first iteration at the current time. i,k Let be the affine formation-related variables for the i-th honest agent at the current moment. s i,1 Let be the affine formation related variables for the i-th honest agent at time 1.

[0073] like Figure 3 As shown, where s l[L] Let s be the affine formation related variables for the l-th agent in the L-th iteration. i[L+1] For the affine formation related variables of the i-th honest agent in the (L+1)-th iteration, Let s be the estimated value of the affine formation-related variables for the i-th honest agent. i Let s be the affine formation related variables for the i-th honest agent. i[1] For the affine formation related variables of the i-th honest agent in the first iteration.

[0074] In another exemplary embodiment of this application, the expression for the gradient descent update mechanism is:

[0075]

[0076] where s i,k+1 is the affine formation related variable of the i-th honest agent at time k + 1, is the estimate of the affine formation related variable of the i-th honest agent at time k, λ i,k is the dual variable of the i-th honest agent at time k + 1 for handling the constraints, λ i,k+1 is the dual variable of the i-th honest agent at time k + 1 for handling the constraints, is the projection onto the set M si , [·] + is the projection onto the non-negative quadrant, α k and δ k are positive diminishing steps, 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 i-th honest agent, the local objective function of the i-th honest agent and the local inequality constraint function of the i-th honest agent are the local information of the i-th honest agent, is the subgradient value of the local objective function of the i-th honest agent with respect to the affine formation related variable, 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 with respect to the affine formation related variable and the dual variable, is the subgradient with respect to the affine formation related variable, is the subgradient with respect to the dual variable.

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

[0078]

[0079] where 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. As Figure 3 shown, x i is the state of the i-th agent, u i is the control input of the i-th honest agent, is the subgradient 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, the local set constraint of the i-th honest agent.

[0080] In another exemplary embodiment of the present application, the resilient time-varying affine formation optimal tracking method for the swarm system further comprises:

[0081] determining a performance evaluation condition of the resilient time-varying affine formation optimal tracking control protocol based on the resilient time-varying affine formation optimal tracking problem, and determining a parameter range of the resilient time-varying affine formation optimal tracking control protocol according to the performance evaluation condition of the resilient time-varying affine formation optimal tracking control protocol.

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

[0083]

[0084] wherein Reg f,k is the cumulative error of all honest agents with respect to the optimal solution, Reg g,i,k is the cumulative violation value of the i-th honest agent with respect to the inequality constraint, Reg h,k is the cumulative violation value of all honest agents with respect to the affine formation constraint, and k is the current time.

[0085] To ensure that Reg h,k , Reg f,k and Reg g,i,k converge sublinearly, for each time the parameter of the resilient time-varying affine formation optimal tracking control protocol of each honest agent can be designed as follows: a decreasing step size is selected, and the step size is decreased by a weighting factor is set to a certain number of times wherein 2τ2<τ1<1-2τ2,τ1<τ3,vτ1<τ4, denotes the maximum number of the neighbor set of all honest agents, B max is a priori known upper limit of the Byzantine agent set cardinality, denotes the total number of degenerate topologies, a degenerate topology represents a sub-topology obtained by removing all nodes and edges related to Byzantine agents from the original topology, and then removing at most B max edges arbitrarily, and H is the number of honest agents.

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

[0087]

[0088] in, For the optimal solution of the equivalent problem at time c, s c* x is the optimal solution at time c. i,c* Let be the optimal solution to the equivalent problem of the i-th honest agent at time c. Let c be the weighted average of the estimates of the affine formation-related variables for the i-th honest agent at time c. Let be the weighting factor for the i-th honest agent. Let x be a set of H-dimensional real-valued vectors. i,c Let x be the state of the i-th honest agent at time c. i,c+1 Let be the state of the i-th honest agent at time c+1, satisfying ρ T 1 = 1, f i,c (x i,c+1 Let x be the local objective function of the i-th honest agent. i,c+1 The value of f i,c (x i,c* Let x be the local objective function of the i-th honest agent. i,c* The value of g i,c (x i,c+1 Let be 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, the dynamic regret value Reg obtained by the cluster system under the elastic time-varying affine formation optimal tracking control protocol is illustrated. h,k The result of dividing by time. For example... Figure 9 As shown, the dynamic regret value Reg obtained by the cluster system under the elastic time-varying affine formation optimal tracking control protocol is illustrated. f,k The result of dividing by time. For example... Figure 10 As shown, the dynamic regret value Reg obtained by the cluster system under the elastic time-varying affine formation optimal tracking control protocol is illustrated. g,i,k The result of dividing by time. For example... Figure 11As shown, the trajectory curve results obtained by the swarm system under the elastic time-varying affine formation optimal tracking control protocol. In the figure, the rhombus represents the initial position of the agent, the pentagram represents the end position of the agent, the line represents the trajectory curve of the agent, and the formation configuration composed of the actual position of all honest agents and the ideal position of the Byzantine agent is displayed at time t = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 seconds, and the gray area represents the threat area.

[0090] The application describes the elastic communication topology of the swarm system under the Byzantine attack; establishes the dynamic model of the swarm system; focuses on the elastic time-varying affine formation optimal tracking problem and clearly defines the problem; derives the specific expression form of the time-varying affine formation optimal tracking control protocol; and determines the parameters of the control protocol. The application designs the elastic time-varying affine formation optimal tracking control protocol for the elastic time-varying affine formation optimal tracking problem to obtain the control input of each honest agent at the current time, ensures that the affine formation can realize the time-varying affine formation optimal tracking according to the control input of each honest agent at the current time, and improves the anti-interference ability and robustness of the swarm system.

[0091] Based on the same inventive concept, the embodiment of the application also provides an elastic time-varying affine formation optimal tracking system for a swarm system, which applies the above-mentioned elastic time-varying affine formation optimal tracking method for a swarm system. The elastic time-varying affine formation optimal tracking system for a swarm system comprises:

[0092] A dynamic model establishment module is configured to establish a dynamic model of the swarm system. The dynamic model is configured to determine the state of each agent at the next time according to the state of each agent at the current time and the control input of each agent.

[0093] An elastic time-varying affine formation optimal tracking problem determination module is connected with the dynamic model establishment module and is configured to establish an affine formation of the swarm system and determine an 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 minimize the objective function by setting the control input of each honest agent at the current time and making the state of each honest agent at the next time satisfy the affine formation constraint and the global inequality constraint according to the dynamic model. The objective function is the sum of the local objective functions accumulated by the honest agents from the initial time to the current time.

[0094] The tracking module is connected with the elastic time-varying affine formation optimal tracking problem determination module, and is configured to determine an elastic time-varying affine formation optimal tracking control protocol based on the elastic time-varying affine formation optimal tracking problem, to obtain a control input of each honest intelligent agent at a current time according to the elastic time-varying affine formation optimal tracking control protocol, and to make the affine formation perform optimal tracking according to the control input of each honest intelligent agent at the current time.

[0095] The technical features of the above embodiments can be combined in any manner. To make the description concise, all possible combinations of the technical features in the above embodiments are not described, but as long as there is no contradiction, any combination of the technical features should be considered within the scope of the present disclosure.

[0096] The principles and implementation modes of the present application are described by using specific examples. The above description of the embodiments is only used to help understand the method and its core idea of the present application. For those skilled in the art, the specific implementation modes and application ranges can be changed according to the idea of the present application. In summary, the content of the present description should not be understood as a limitation of the present application.

Claims

1. A resilient time-varying affine formation optimal tracking method for a swarm system, characterized in that, The cluster system comprises a plurality of agents; The plurality of agents are divided into Byzantine agents and honest agents after being attacked by a Byzantine attack; The resilient time-varying affine formation optimal tracking method for the cluster system comprises: a dynamic model of the cluster system is established; the dynamic model is used to determine the state of each agent at the next time according to the state of each agent at the current time and the control input of each agent; an affine formation of the cluster system is established, and a resilient time-varying affine formation optimal tracking problem is determined based on the affine formation and the dynamic model; the resilient time-varying affine formation optimal tracking problem is to minimize a target function by setting the control input of each honest agent at the current time, so that the state of each honest agent at the next time meets the affine formation constraint and the global inequality constraint according to the dynamic model; the target function is the sum of local target functions accumulated by the honest agents from the initial time to the current time; a resilient time-varying affine formation optimal tracking control protocol is determined based on the resilient time-varying affine formation optimal tracking problem, and the control input of each honest agent at the current time is obtained according to the resilient time-varying affine formation optimal tracking control protocol, so that the affine formation optimally tracks according to the control input of each honest agent at the current time.

2. The elastic time-varying affine formation optimal tracking method for a swarm system according to claim 1, wherein, The expression of the resilient time-varying affine formation optimal tracking problem is: s.t.x k+1 = x k + Tu k ; g k (x k+1 )≤0; in, To solve for variable x k+1 Make the global objective function f k (x k+1 Minimize, where st is the global objective function f. k (x k+1 The constraints imposed by x, where k is the current time, c is the c-th time, and x k+1 Let f be the state of the agent at time k+1. k (x k+1 Let f be the global objective function. i,c (x i,k+1 () is a local objective function. Let i be the set of honest agents, and x be the index of the honest agent. i,k+1 Let i be the state of the i-th honest agent at time k+1. Let u be the local set constraint of the honest agent's state at time k+1, where T is the sampling time. k For the control input at the current moment, Let g be the sum of the local objective functions accumulated by all honest agents from the initial time to the current time. k (x k+1 Let be the global inequality constraints on the state of the honest agent at time k+1. For affine formations in a cluster system, For affine formations concerning honest intelligent agents.

3. The elastic time-varying affine formation optimal tracking method for a swarm system according to claim 2, wherein, The resilient time-varying affine formation optimal tracking control protocol is determined based on the resilient time-varying affine formation optimal tracking problem, and the expression of the resilient time-varying affine formation optimal tracking control protocol is: The resilient time-varying affine formation optimal tracking control protocol specifically comprises: s.t. Ps k+1 = Ps k + Tu k ; where 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 i-th honest agent with respect to the affine formation related variable, is the intersection of the local set constraints of all honest agents, s.t. is the optimization problem subject to the constraints, is the augmented configuration matrix with respect to the desired formation configuration h1is the position of the first agent, h N is the position of the N-th agent, N is the number of agents in the swarm system, horz is the horizontal concatenation of vectors or matrices, vert is the vertical concatenation of vectors or matrices, q is a column vector of ones with appropriate dimension, is the augmented configuration matrix with respect to the affine formation related variable, I m is the identity matrix, is the sub-matrix of P related to the i-th honest agent, is the sub-matrix of P related to the i-th honest agent, g k (·) is the global inequality constraint.

4. The elastic time-varying affine formation optimal tracking method for a swarm system according to claim 3, wherein, Based on the norm-based aggregation rule and the cluster system, a Byzantine judgment factor of each honest agent to any agent is obtained, and based on the Byzantine judgment factor of each honest agent to any agent, an elastic weighted average consensus mechanism is used to iterate the affine formation related variable of each honest agent at the current time for a certain number of times to obtain an estimated value of the affine formation related variable of each honest agent at the current time; Based on the estimated value of the affine formation related variable of each honest agent at the current time and the local information of each honest agent at the current time, a tracking control protocol is used to obtain the control input of each honest agent at the current time; Based on the estimated value of the affine formation related variable of each honest agent at the current time, the local information of each honest agent at the current time and the sub-gradient value, a gradient descent update mechanism is used to obtain the affine formation related variable of each honest agent at the next time. The expression of the elastic weighted average consensus mechanism is:

5. The elastic time-varying affine formation optimal tracking method for a swarm system according to claim 4, wherein, The expression of the gradient descent update mechanism is: where γ k is a weighting factor, L is the iteration number, s i,k[L+1] is the affine formation related variable of the i-th honest agent in the current moment at the L+1-th iteration, κ il is the Byzantine judgment factor of the i-th honest agent to the l-th agent, s l,k[L] is the affine formation related variable of the l-th agent in the current moment at the L-th iteration, s i,k[L] is the affine formation related variable of the i-th honest agent in the current moment at the L-th iteration.

6. The elastic time-varying affine formation optimal tracking method for a swarm system according to claim 5, wherein, The tracking control protocol is: where s i,k+1 is the affine formation related variable of the i-th honest agent at time k + 1, is the estimate of the affine formation related variable of the i-th honest agent at time k, λ i,k is the dual variable of the i-th honest agent at time k for handling the constraints, λ i,k+1 is the dual variable of the i-th honest agent at time k + 1 for handling the constraints, is the projection onto the set M si , [·] + is the projection onto the non-negative quadrant, α k and δ k are positive diminishing steps, is the Lagrangian function with respect to the affine formation related variable and the dual variable, is the subgradient with respect to the affine formation related variable, is the subgradient with respect to the dual variable.

7. The elastic time-varying affine formation optimal tracking method for a swarm system according to claim 6, wherein, The resilient time-varying affine formation optimal tracking method for the cluster system further comprises: where u i,k is the control input of the ith honest agent at time k, x i,k is the state of the ith agent at time k.

8. The elastic time-varying affine formation optimal tracking method for swarm systems according to claim 1, wherein, ​ The performance evaluation condition of the elastic time-varying affine formation optimal tracking control protocol is determined based on the elastic time-varying affine formation optimal tracking problem, and the parameter range of the elastic time-varying affine formation optimal tracking control protocol is determined according to the performance evaluation condition of the elastic time-varying affine formation optimal tracking control protocol.

9. The elastic time-varying affine formation optimal tracking method for a swarm system according to claim 8, wherein, The performance evaluation condition of the elastic time-varying affine formation optimal tracking control protocol specifically includes: where Reg f,k is the accumulated error of all honest agents with respect to the optimal solution, Reg g,i,k is the accumulated violation of the inequality constraint by the i-th honest agent, Reg h,k is the accumulated violation of the affine formation constraint by all honest agents, and k is the current time instant.

10. A resilient time-varying affine formation optimal tracking system for a swarm system, applying the resilient time-varying affine formation optimal tracking method for a swarm system of any one of claims 1-9, characterized in that, The elastic time-varying affine formation optimal tracking system for the swarm system includes: A dynamics model establishing module is configured to establish a dynamics model of the swarm system, and the dynamics model is configured to determine a state of each agent at a next time point according to a state of each agent at a current time point and a control input of each agent; An elastic time-varying affine formation optimal tracking problem determining module is connected with the dynamics model establishing module and configured to establish an affine formation of the swarm system and determine an elastic time-varying affine formation optimal tracking problem based on the affine formation and the dynamics model; the elastic time-varying affine formation optimal tracking problem is to minimize an objective function by setting a control input of each honest agent at the current time point and making a state of each honest agent at the next time point meet affine formation constraints and global inequality constraints according to the dynamics model; the objective function is a sum of local objective functions accumulated by the honest agents from an initial time point to the current time point; A tracking module is connected with the elastic time-varying affine formation optimal tracking problem determining module and configured to determine an elastic time-varying affine formation optimal tracking control protocol based on the elastic time-varying affine formation optimal tracking problem, obtain the control input of each honest agent at the current time point according to the elastic time-varying affine formation optimal tracking control protocol, and make the affine formation perform optimal tracking according to the control input of each honest agent at the current time point.

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