A Distributed Fixed-Time and Prescribed-Time Cooperative Optimal Consensus Control Method for Multi-Agent Systems

Abstract

The present invention discloses a distributed fixed-time and pre-specified-time cooperative optimal consensus control method for a multi-agent system, comprising the steps of: Step 1, constructing the dynamic model of each agent; Step 2, constructing a global cost function based on the dynamic model and describing the optimization problem; Step 3, designing a two-stage distributed cooperative optimization consensus algorithm based on exponential functions, including a fixed-time cooperative optimization consensus algorithm and a pre-specified-time cooperative optimization consensus algorithm; Step 4, constructing Lyapunov functions for the fixed-time cooperative optimization consensus algorithm and the pre-specified-time cooperative optimization consensus algorithm respectively, and performing consensus analysis on the two optimization consensus algorithms respectively to achieve optimal consensus. The present invention ensures that the multi-agent system can achieve consensus at the global optimal point within the estimated or pre-set time in the case of unknown initial states and changing communication topology networks, while avoiding chattering phenomena.

Description

Technical Field

[0001] The present invention relates to the field of agent control, and particularly to a distributed fixed-time and pre-specified-time cooperative optimal consensus control method for multi-agent systems. Background Art

[0002] A multi-agent system refers to a system composed of multiple independent agents and their corresponding organizational rules and information interaction protocols. Compared with a single agent, a multi-agent system can complete more complex specific tasks through cooperative operations and exhibit the advantage of "1 + 1 > 2". This unique advantage has sparked a research boom on the cooperative control of multi-agent systems with the development of computer network technology. As a fundamental problem in the cooperative control of multi-agent systems, the theoretical results of the consensus problem can be widely applied to fields such as source localization, UAV formation, intelligent robots, intelligent transportation management, resource exploration, and power grid economic dispatch. In these applications, the control task often requires the multi-agent system to meet some optimization criteria while achieving consensus, that is, minimizing the sum of its local cost functions while achieving consensus. The core goal of distributed optimal consensus in multi-agent systems is to design an effective control protocol so that each agent in the system continuously adjusts its behavior according to the information received from its neighbors, and finally makes the states of all agents consistent at the global optimal point. In addition, factors such as changes in communication distance, signal interference, and failures of agents and networks in practical applications can lead to the interruption or increase of the communication links in the multi-agent system, making the system consistency control difficult.

[0003] Most distributed optimal control schemes can only make the states of agents converge to the global optimal point within a limited time. Based on this method, estimating the optimal consensus time requires prior knowledge of the initial state of the system, and different initial states may lead to different optimal consensus times. However, the initial state of the actual system is difficult to obtain or unavailable. In addition, considering different environmental and operating factors, the initial state of the system may not be the same in each experiment, and the system communication topology will change due to factors such as communication distance and interference, which undoubtedly increases the difficulty of the system to achieve optimal consensus and the computational burden of estimating the optimal consensus time. Although there are some algorithms that consider achieving optimal consensus control within a fixed / pre-specified time, they have drawbacks such as too many parameters, overly complex algorithm design and time estimation, and prone to chattering phenomena. Summary of the Invention

[0004] Aiming at the deficiencies of the prior art, the present invention aims to provide a more flexible and simple distributed fixed-time and pre-specified-time optimal consensus control method to ensure that the multi-agent system can achieve consensus at the global optimal point within the estimated or pre-set time in the case of unknown initial state and changing communication topology network, while avoiding chattering phenomena.

[0005] The technical solution for achieving the object of the present invention is as follows: A distributed fixed-time and pre-determined-time cooperative optimal consensus control method for a multi-agent system, comprising the steps of:

[0006] Step 1, construct the dynamic model of each agent;

[0007] Step 2, based on the dynamic model, construct a global cost function and describe the optimization problem;

[0008] Step 3, design a two-stage distributed cooperative optimization consensus algorithm based on exponential functions, including a fixed-time cooperative optimization consensus algorithm and a pre-determined-time cooperative optimization consensus algorithm;

[0009] Step 4, for the fixed-time cooperative optimization consensus algorithm and the pre-determined-time cooperative optimization consensus algorithm, respectively construct Lyapunov functions, conduct consensus analysis on the two optimization consensus algorithms respectively, and achieve optimal consensus.

[0010] Further, the agent is an omnidirectional wheeled unmanned vehicle with an undirected communication topology.

[0011] Further, the dynamic model of the unmanned vehicle is:

[0012]

[0013] where x i1 , x i2 , v i1 , v i2 respectively represent the position and linear velocity of the i-th unmanned vehicle moving on a two-dimensional plane; θ i represents the orientation angle of the i-th unmanned vehicle, that is, the angle with the x-axis; represents the angular velocity of the i-th unmanned vehicle;

[0014] Since θ i = sinθ i = 0, cosθ i = 1, the dynamic model of the i-th unmanned vehicle is simplified to:

[0015]

[0016] Let x i (t) = [x i1 , x i2 T , u i (t) = [v i1 , v i2 T , and the dynamic equation of the i-th unmanned vehicle is obtained as:

[0017] ​​

[0018] Furthermore, in step 2, it is considered that each agent i has a local cost function C i (x i ). The global cost function is the sum of all local cost functions, which is:

[0019]

[0020] The optimization problem is described as:

[0021]

[0022] where x i (t) and x j (t) respectively represent the position states of the i-th and j-th agents.

[0023] denotes finding the minimum point x of the function , and x i (t) = x j (t), i = 1, 2, …, N, i ≠ j means that the position states of the N agents are the same.

[0024] Furthermore, the fixed-time cooperative optimization consensus algorithm designed in step 3 is:

[0025]

[0026] where the control parameters λ1 and λ2 are positive constants; and q are positive constants and satisfy represents the gradient of the global cost function C i (x i (t)) with respect to time t, represents the Hessian matrix of the global cost function C i (x i (t)) with respect to time t; a ij (t) represents the element of the adjacency matrix A = (a ij (t)) N×N where a ij (t) > 0 means that at time t, agent i can obtain the state information from agent j, otherwise a ij (t) = 0; represents the set of agents that have a communication connection with the i-th agent at time t; T1 is the time estimate for all agents to reach their respective local optimal solutions , and x i (t) and x j (t) respectively represent the positions of agent i and agent j moving in the two-dimensional plane.

[0027] Furthermore, the predefined time collaborative optimization consensus algorithm designed in step 3 is as follows:

[0028]

[0029] In the formula, the control parameters q, and δ are positive constants and satisfy: 0 < q < 1 / 2, 0 < δ < 1; denotes the natural logarithm with base e ; is the algebraic connectivity of the undirected connected communication topology graph of the multi-agent system, where represents the communication topology graph of the multi-agent system, which changes over time, represents the global cost function C i (x i (t)) with respect to the gradient of time t, represents the Hessian matrix of the global cost function C i (x i (t)) with respect to time t; a ij (t) represents the element of the adjacency matrix A = (a ij (t)) N×N where a ij (t) > 0 means that at time t, agent i can obtain the state information from agent j, otherwise a ij (t) = 0; represents the set of agents that have communication connections with the i-th agent at time t; the constants M and M i satisfy the inequality: M ≥ max{M i} and T p is the preset time for the multi-agent system to achieve global optimal consensus, x i (t), x j (t) respectively represent the positions of agent i and agent j moving on the two-dimensional plane.

[0030] Furthermore, in step 4, a Lyapunov function is constructed to conduct a consensus analysis on the fixed-time collaborative optimization consensus algorithm, including: According to the two stages of the algorithm, the analysis is carried out in two steps. The time of the first stage is 0 < t ≤ T1, corresponding to the first stage of the algorithm, and the goal is to enable all agents to reach their respective local optimal solutions within a fixed time The time of the second stage is t > T1, corresponding to the second stage of the algorithm, and the goal is to enable all agents to move from their respective local optimal solutions to the global optimal solution x * .

[0031] Furthermore, construct a Lyapunov function and conduct consensus analysis on the fixed-time cooperative optimization consensus algorithm, specifically including:

[0032] Step 4.1.1, analyze the fixed-time local optimality of the multi-agent system and give an estimate of the required time, including:

[0033] For the i-th agent, construct a positive semi-definite and radially unbounded Lyapunov function as:

[0034]

[0035] The derivative with respect to time is:

[0036]

[0037] where the coefficient

[0038] Based on the fixed-time stability theory, the agents can reach their respective local optimal solutions from the initial states within a fixed time The estimate of the required time is:

[0039]

[0040] Step 4.1.2, analyze the fixed-time global optimality of the multi-agent system and give an estimate of the required time, including:

[0041] Based on the zero-gradient sum algorithm, prove the fixed-time global optimality of the multi-agent system under the action of the second stage of the algorithm, and construct a Lyapunov function as:

[0042]

[0043] where x * represents the global optimal solution of the optimization problem. Since the local cost function C i (x) is m i strongly convex, that is, for any vectors x, y and m i > 0, there is Using the constructed Lyapunov function, we can obtain:

[0044]

[0045] is positive definite and radially unbounded, and if and only if x i (t) = x * ;

[0046] For Taking the derivative gives:

[0047]

[0048] where denotes the Kronecker product, and I m denotes the m-dimensional identity matrix. Based on the fixed-time stability theory, the agents can reach their respective local optimal values from their respective local optimal values within a fixed time to the global optimal solution x * , and the estimate of the time required is:

[0049]

[0050] Combining Step 4.1.1 and Step 4.1.2, under the action of the first stage of the algorithm, the agents can first converge from the initial state to their respective local optimal values within a fixed time T1, and at this time, the local cost functions corresponding to each agent are minimized; based on the zero-gradient sum algorithm, under the action of the second stage of the algorithm, the position states of all agents converge to the global optimal point, and at this time, the global cost function corresponding to the entire system is minimized, and the entire optimization consensus process is completed within a fixed time T1 + T2.

[0051] Furthermore, in Step 4, a Lyapunov function is constructed to conduct a consensus analysis on the pre-determined time cooperative optimization consensus algorithm, including: according to the two stages of the algorithm, it is analyzed in two steps, where the time of the first stage is 0 < t ≤ δT p , corresponding to the first stage of the algorithm, and the goal is to enable all agents to reach their respective local optimal solutions within a pre-set time δT p ; The time of the second stage is t > δT p , corresponding to the second stage of the algorithm, and the goal is to enable all agents to reach the global optimal solution x of the optimization problem from their respective local optimal solutions within a pre-set time (1 - δ)T p ; ; * .

[0052] Furthermore, a Lyapunov function is constructed to conduct a consensus analysis on the pre-determined time cooperative optimization consensus algorithm, specifically including:

[0053] Step 4.2.1: Analyze the pre-determined time local optimality of the multi-agent system, including:

[0054] For the i-th agent, construct a semi-positive definite and radially unbounded Lyapunov function as:

[0055]

[0056] Then the derivative with respect to time is;

[0057]

[0058] Based on the theory of pre-defined time stability, intelligent agents can reach their respective local optimal solutions from the initial state within the pre-defined time δT p and satisfy the basic requirements of the zero-gradient sum algorithm, that is, the initial sum of the local gradients is zero; At this time, it meets the basic requirements of the zero-gradient sum algorithm, that is, the initial sum of the local gradients is zero;

[0059] Step 4.2.2: Analyze the pre-defined time global optimality of the multi-agent system, including:

[0060] Based on the zero-gradient sum algorithm, the constructed positive definite and radially unbounded Lyapunov function is:

[0061]

[0062] Deriving gives:

[0063]

[0064] where

[0065] Based on the theory of pre-defined time stability, intelligent agents can reach the global optimal solution x p from their respective local optimal values within the pre-defined time (1 - δ)T ; * ;

[0066] Combining Step 4.2.1 and Step 4.2.2, under the action of the first stage of the algorithm, each intelligent agent can first converge from the initial state to its respective local optimal value within the pre-defined time δT p , at which time the local cost function corresponding to each intelligent agent is minimized; then, based on the zero-gradient sum algorithm, under the action of the second stage of the algorithm, the position states of all intelligent vehicles can converge to the global optimal point within the time (1 - δ)T p , at which time the global cost function corresponding to the entire system is minimized, and the entire optimization consensus process is completed within the pre-set time T p .

[0067] Compared with the prior art, the remarkable effects of the present invention are:

[0068] (1) The patented technology designs a two-stage distributed fixed-time optimization consensus algorithm and a preset-time optimization consensus algorithm based on the exponential function. The control in the first stage can converge any initial state of the agents to the local optimal point, thus meeting the basic requirement of zero gradient sum (which facilitates the subsequent use of the zero gradient sum algorithm to prove that the states of the agents can converge to the unique global optimal point under the second-stage control). Therefore, the proposed algorithm solves the optimization consensus control problem of multi-agent systems under any initial state;

[0069] (2) Under the fixed-time optimization algorithm, the time for the agents to achieve consensus at the optimal point does not depend on the initial state, and a fixed upper bound of the time to achieve optimal consensus can be estimated; under the preset-time optimization algorithm, the time for the agents to achieve consensus at the optimal point does not depend on the initial state and system parameters, and the preset optimization consensus time can be used as a control parameter, so that the agents can achieve optimization consensus within the preset time;

[0070] (3) The designed algorithm is applicable to the case of changes in the system communication topology network, has strong robustness and anti-interference ability, and does not use discontinuous sign functions, thus avoiding the chattering phenomenon that may affect the control effect in actual control;

[0071] (4) Compared with the distributed optimization consensus algorithm designed based on the traditional fixed / preset-time stability theorem with two terms in the discriminant condition, the algorithm designed in this patented technology is based on the fixed / preset-time stability theorem of the exponential function type condition, has fewer parameters, a simple structure, and simpler time estimation. Description of the Drawings

[0072] Figure 1 is the communication topology diagram of the unmanned vehicle, where (a) is the communication topology diagram (b) is the communication topology diagram (c) is the communication topology diagram

[0073] Figure 2 is the schematic diagram of the change of the position state of the unmanned vehicle under the fixed-time optimization algorithm.

[0074] Figure 3 is the schematic diagram of the change of the position state of the unmanned vehicle under the preset-time optimization algorithm.

[0075] Figure 4 is the flowchart of the method of the present invention. Detailed Implementation Manner

[0076] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0077] In the case of arbitrary initial states and changes in the communication topology network of a multi-agent system, how to design a distributed control method that achieves consensus at the global optimal point within a fixed / predetermined time based on the local cost functions of the agents, and estimate the upper bound of the time for the controlled target to achieve fixed-time optimal consensus is a technical problem to be solved currently. Based on this, this patent fully considers practical applications and proposes a two-stage distributed optimal consensus control method that is effective for arbitrary initial states and switched communication topologies, where the time to achieve the control target can be estimated or preset in advance, and the estimation of this time is independent of the initial state of the system. The proposed method has the advantages of fast convergence speed, strong anti-interference ability, and can avoid chattering phenomena. Combining Figure 4 , the method specifically includes the following steps:

[0078] Step 1: Construct the dynamic model of multiple unmanned vehicles.

[0079] Consider a multi-unmanned vehicle system composed of N Mecanum wheel unmanned vehicles with an undirected communication topology structure, where the dynamic model of the i-th unmanned vehicle is as follows:

[0080]

[0081] In the formula, x i1 , x i2 , v i1 , v i2 respectively represent the position and linear velocity of the i-th unmanned vehicle moving in the two-dimensional plane; θ i represents the orientation angle of the i-th unmanned vehicle, that is, the angle with the x-axis; represents the angular velocity of the i-th unmanned vehicle. Since the Mecanum wheel unmanned vehicle can achieve translational motion at any angle without any rotation, that is, the orientation angle of the unmanned vehicle does not change (i.e., θ i =sinθ i =0, cosθ i =1), the dynamic model of the i-th unmanned vehicle can be simplified to the following dynamic model

[0082]

[0083] Let x i (t)=[x i1 , x i2 T ​, u i (t) = [v i1 , v i2 T , the general form of the dynamic equation of the i-th autonomous vehicle can be obtained as

[0084]

[0085] Step 2: Construct the global cost function and describe the optimization problem.

[0086] Considering that the i-th autonomous vehicle has a local cost function C i (x i ), the global cost function is formed by summing all local cost functions, and its expression is:

[0087]

[0088] The goal of all autonomous vehicles is to reach the same position state within a fixed time or a preset time, while minimizing the global cost function C(x), that is, the position states of all agents converge to the optimal solution of the global cost function. This distributed optimization problem can be described as:

[0089]

[0090] In the formula, x i (t), x j (t) represent the position states of the i-th and j-th autonomous vehicles respectively,

[0091] denotes finding the minimum point x of the function , x i (t) = x j (t), i = 1, 2, …, N, i ≠ j means that the position states of N autonomous vehicles are the same.

[0092] Step 3: Design a two-stage distributed cooperative optimization consensus algorithm based on the exponential function, including a fixed-time cooperative optimization consensus algorithm and a preset-time cooperative optimization consensus algorithm.

[0093] Design an algorithm to solve the above distributed optimization problem within a fixed time or a preset time. The two-stage distributed fixed-time cooperative optimization consensus algorithm based on the exponential function is as follows:

[0094]

[0095] Among them, the control parameters λ1 and λ2 are positive constants; and q are positive constants and satisfy denotes the function C i (x​i (t)) Gradient with respect to time t, represents the function C i (x i (t)) Hessian matrix with respect to time t; a ij (t) represents the adjacency matrix A = (a ij (t)) N×N of which the element a ij (t) > 0 means that at time t, the autonomous vehicle i can obtain the state information from the autonomous vehicle j, otherwise a ij (t) = 0;

[0096] represents the set of autonomous vehicles that have a communication connection with the i-th autonomous vehicle at time t; T1 is the time estimate for all agents to reach their respective local optimal solutions respectively.

[0097] The two-stage distributed pre-determined time cooperative optimization consensus algorithm based on the exponential function is as follows:

[0098]

[0099] In the formula, the control parameters q, and δ are positive constants and satisfy 0 < q < 1 / 2, 0 < δ < 1; represents the natural logarithm with base e ; is the algebraic connectivity of the undirected connected communication topology graph of the multi-autonomous vehicle system, where represents that the communication topology graph of the multi-autonomous vehicle system changes with time. represents the function C i (x i (t)) Gradient with respect to time t, represents the function C i (x i (t)) Hessian matrix with respect to time t; a ij (t) represents the adjacency matrix A = (a ij (t)) N×N of which the element a ij (t) > 0 means that at time t, the autonomous vehicle i can obtain the state information from the autonomous vehicle j, otherwise a ij (t) = 0; represents the set of autonomous vehicles that have a communication connection with the i-th autonomous vehicle at time t; the constants M and M i satisfy the inequality M ≥ max{M i} and T pThe time to achieve global optimal consensus for a pre-set multi-unmanned vehicle system.

[0100] Step 4: Judge the optimal consensus of the system under the designed algorithm.

[0101] Construct Lyapunov functions respectively, and analyze the two-stage distributed fixed-time cooperative optimization consensus algorithm and the pre-determined time cooperative optimization consensus algorithm based on exponential functions respectively;

[0102] Step 4.1: Conduct consensus analysis on the two-stage distributed fixed-time cooperative optimization consensus algorithm

[0103] According to the two stages of the algorithm, analyze in two steps. The time of the first stage is 0 < t ≤ T1, corresponding to the first stage of the algorithm, and the goal is to enable all unmanned vehicles to reach their respective local optimal solutions within a fixed time The time of the second stage is t > T1, corresponding to the second stage of the algorithm, and the goal is to enable all unmanned vehicles to move from their respective local optimal solutions to the global optimal solution x * .

[0104] Step 4.1.1: Analyze the fixed-time local optimality of the multi-unmanned vehicle system and give an estimated value of the required time

[0105] For the i-th unmanned vehicle, construct a semi-positive definite and radially unbounded Lyapunov function as:

[0106]

[0107] The derivative with respect to time is:

[0108]

[0109] where,

[0110] Based on the fixed-time stability theory, it can be known that the unmanned vehicle can reach its respective local optimal solution from the initial state within a fixed time (at this time, the zero gradient and the basic requirements of the algorithm are satisfied, that is, the initial sum of the local gradients is zero), and the estimated value of the required time is:

[0111]

[0112] Step 4.1.2: Analyze the fixed-time global optimality of the multi-unmanned vehicle system and give an estimated value of the required time

[0113] Since the zero-gradient and the basic requirements of the algorithm have been satisfied in the previous step, the Lyapunov function is constructed next, and based on the zero-gradient and algorithm, the fixed-time global optimality of the multi-unmanned vehicle system under the action of the second stage of the algorithm is proved. The Lyapunov function is constructed as follows:

[0114]

[0115] where x * represents the global optimal solution of the optimization problem. Since the local cost function C i (x) is m i strongly convex, that is, for any vectors x, y and m i > 0, there is Using the constructed Lyapunov function, we can get:

[0116] Therefore is positive definite, radially unbounded, and if and only if x i (t) = x * (the states of all unmanned vehicles converge to the global optimal value).

[0117] Taking the derivative of gives:

[0118]

[0119] where represents the Kronecker product, and I m represents the m-dimensional identity matrix. Based on the fixed-time stability theory, it can be known that the unmanned vehicle can reach the global optimal solution x from its respective local optimal values * in a fixed time. The estimate of the required time is:

[0120]

[0121] Combining Step 4.1.1 and Step 4.1.2, it can be seen that under the action of the first stage of the algorithm, the unmanned vehicles can first converge from the initial state to their respective local optimal values within a fixed time T1. At this time, the local cost functions corresponding to each unmanned vehicle are the smallest. Therefore, further optimization is required. Next, based on the zero-gradient and algorithm, under the action of the second stage of the algorithm, the position states of all unmanned vehicles can converge to the global optimal point, and at this time, the global cost function corresponding to the entire system is the smallest. The entire optimization consensus process can be completed within a fixed time T1 + T2.

[0122] Step 4.2. Consensus analysis of the two-stage distributed pre-determined time cooperative optimization consensus algorithm

[0123] According to the two stages of the algorithm, the analysis is carried out in two steps. The time of the first stage is 0 < t ≤ δT p , corresponding to the first stage of the algorithm, the goal is to enable all unmanned vehicles to reach their respective local optimal solutions within the preset time δT p The time of the second stage is t > δT p , corresponding to the second stage of the algorithm, the goal is to enable all unmanned vehicles to reach the global optimal solution x of the optimization problem from their respective local optimal solutions within the preset time (1 - δ)T p * .

[0124] Step 4.2.1. Analyze the prespecified-time local optimality of the multi-unmanned vehicle system

[0125] For the i-th unmanned vehicle, construct a positive semi-definite and radially unbounded Lyapunov function as The derivative with respect to time is

[0126]

[0127] Based on the prespecified-time stability theory, it can be known that the unmanned vehicle can reach its respective local optimal solution from the initial state within the prespecified time δT p It can be known that the zero gradient and the basic requirements of the algorithm are satisfied at this time, that is, the initial sum of the local gradients is zero.

[0128] Step 4.2.2. Analyze the prespecified-time global optimality of the multi-unmanned vehicle system

[0129] Based on the zero gradient and the algorithm, prove the prespecified-time global optimality of the multi-unmanned vehicle system under the action of the second stage of the algorithm. The selected positive definite and radially unbounded Lyapunov function is

[0130]

[0131] For Derivation gives

[0132]

[0133] Among them,

[0134] Based on the prespecified-time stability theory, it can be known that the unmanned vehicle can reach the global optimal solution x from its respective local optimal values within the prespecified time (1 - δ)T p * .

[0135] ​​​​​​Combining Step 4.2.1 and Step 4.2.2, it can be seen that under the action of the first stage of the algorithm, each unmanned vehicle can first converge from the initial state to its respective local optimal value within the predetermined time δT p At this time, the local cost function corresponding to each unmanned vehicle is the smallest. Therefore, further optimization is required. Next, based on the zero-gradient sum algorithm, under the action of the second stage of the algorithm, the position states of all unmanned vehicles can converge to the global optimal point within the time (1 - δ)T p At this time, the global cost function corresponding to the entire system is the smallest. The entire optimization consensus process can be completed within the preset time T p

[0136] Step 5: Conduct simulation experiments on the proposed method.

[0137] Taking the problem of multiple unmanned vehicles system searching for an unknown sound source as an example, the two-stage distributed fixed and predetermined time optimization method based on exponential function is illustrated. Assume that the multiple unmanned vehicles system consists of 4 identical omnidirectional-wheel unmanned ground vehicles (UGVs), and its communication topology is as Figure 1 shown. Define the communication topology switching signal ω(t) as

[0138]

[0139] where s = 0, 1, …, when s ≤ t < s + 0.1, the communication topology graph of the unmanned vehicle is as Figure 1 (a) shown; when s + 0.1 ≤ t < s + 0.2, the communication topology graph is as Figure 1 (b) shown; when s + 0.2 ≤ t < s + 1, the communication topology graph is as Figure 1 (c) shown. From the topology graph it can be obtained that Then

[0140] The goal of this example is to use the designed algorithm and the position information of 4 reference anchor points around the unknown sound source to enable the unmanned vehicles to collaboratively locate the position of the unknown sound source within a fixed time or a predetermined time.

[0141] Define the matrix B = [b ij (i, j = 1, 2, 3, 4). If the i-th unmanned vehicle can obtain the position information of the j-th anchor point, then b ij = 1; otherwise b ij = 0. The local cost function corresponding to the i-th unmanned vehicle is which indicates that each unmanned vehicle tries to get closer to its respective reference anchor point. The corresponding global cost function is ​The essence of the source localization problem is to find an intersection point as an estimate of the unknown source localization by minimizing the sum of the squares of the distances to the reference anchor points;

[0142] Set the initial positions of the unmanned vehicles as x1(0) = (3, 5) T , x2(0) = (2, 1.8) T , x3(0) = (-2, 1) T , x4(0) = (1, 4) T , and the positions of the four anchor points around the sound source are R1 = (3.2, 4.2) T , R2 = (1, 2) T , R3 = (-1, 2.4) T , R4 = (-0.5, 5) T , and through calculation, the position of the unknown sound source is x * = (0.675, 3.4) T ;

[0143] Set the parameters λ = 2, q = 0.4, The variation of the positions of each unmanned vehicle with time Figure 2 is shown. Each unmanned vehicle can first reach its local optimal position within the time T1 = 0.625 i.e., the position of the reference anchor point R i , and then reach the global optimal intersection point x i from R * within the fixed time T2 = 1.25. That is, within the fixed time T = 1.875, each unmanned vehicle can locate the position of the unknown sound source.

[0144] Let the expected time T p = 1 to complete source localization, set the parameters δ = 0.5, q = 0.4, Calculation shows that the curves of the position states of each unmanned vehicle are as Figure 3 shown. It can be seen that each unmanned vehicle can first reach the position R of the reference anchor point within the predetermined time δT p = 0.5 i , and then reach the global optimal intersection point x p from R i within the predetermined time (1 - δ)T * .

[0145] Through simulation experiments, it is shown that a more flexible and simple distributed fixed-time and predetermined-time optimal consensus control method provided by the present invention can ensure that the multi-agent system can achieve consensus at the global optimal point within the estimated or preset time under the conditions of unknown initial state and changing communication topology network, while avoiding chattering phenomena.

[0146] It should be noted that in this text, relational terms such as first and second are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the term "comprising", "including" or any other variant thereof is intended to cover non-exclusive inclusion, such that a process, method, article or device comprising a series of elements not only includes those elements but also includes other elements not expressly listed, or elements inherent to such process, method, article or device.

[0147] Although the embodiments of the present invention have been shown and described, those of ordinary skill in the art can understand that various changes, modifications, substitutions and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A distributed fixed-time and scheduled-time collaborative optimal consensus control method for a multi-agent system, characterized in that: Includes steps: Step 1, construct the dynamic model of each agent; Step 2: Based on the dynamic model, a global cost function is constructed and the optimization problem is described; Step 3, design a two-stage distributed collaborative optimization consensus algorithm based on exponential function, including a fixed-time collaborative optimization consensus algorithm and a scheduled-time collaborative optimization consensus algorithm; Step 4: construct Lyapunov functions for the fixed-time collaborative optimization consensus algorithm and the scheduled-time collaborative optimization consensus algorithm, respectively, and perform consensus analysis on the two optimization consensus algorithms to achieve the optimal consensus; The fixed-time collaborative optimization consensus algorithm designed in step 3 is: Among them, the control parameter λ 11 and λ 21 is a positive constant; and q are positive constants and satisfy: 0<q<1 / 2, Denotes the global cost function C i (x i (t)) with respect to the gradient of time t, Denotes the global cost function C i (x i (t)) Hessian matrix with respect to time t; a ij (t) represents the adjacency matrix A = (a ij (t)) N×N elements of ij (t)>0 means that at time t, agent i can obtain state information from agent j, otherwise a ij (t) = 0; represents the set of agents that have communication connections with the i-th agent at time t; T1 is the local optimal solution reached by all agents The time estimate, x i (t), x j (t) represents the positions of agent i and agent j on the two-dimensional plane; The scheduled time collaborative optimization consensus algorithm designed in step 3 is: In the formula, the control parameter q. and δ are positive constants and satisfy: 0<q<1 / 2, 0<δ<1; Indicates that the base is e The logarithm of is the undirected connected communication topology graph of the multi-agent system The algebraic connectivity of Represents the communication topology of a multi-agent system, which changes over time. Denotes the global cost function C i (x i (t)) with respect to the gradient of time t, Denotes the global cost function C i (x i (t)) Hessian matrix with respect to time t; a ij (t) represents the adjacency matrix A = (a ij (t)) N×N elements of ij (t)>0 means that at time t, agent i can obtain state information from agent j, otherwise a ij (t) = 0; represents the set of agents that have communication connections with the ith agent at time t; constants M and M i Satisfies the inequality: M≥max{M i }and T p is the time for the pre-set multi-agent system to achieve the global optimal consensus, x i (t), x j (t) represent the positions of agent i and agent j moving on the two-dimensional plane respectively.

2. According to claim 1, a distributed fixed-time and scheduled-time collaborative optimal consensus control method for a multi-agent system is characterized in that: The intelligent agent is a Mecanum wheel unmanned vehicle with an undirected communication topology structure.

3. According to claim 2, a distributed fixed-time and scheduled-time collaborative optimal consensus control method for a multi-agent system is characterized in that: The dynamic model of the unmanned vehicle is: In the formula, x i1 、x i2 、v i1 、v i2 Respectively represent the position and linear velocity of the i-th unmanned vehicle moving on the two-dimensional plane; θ i represents the orientation angle of the i-th unmanned vehicle, that is, the angle with the x-axis; represents the angular velocity of the i-th unmanned vehicle; Since θ i = sinθ i =0, cosθ i =1, simplifying the dynamic model of the i-th unmanned vehicle to: Let x i (t) = [x i1 ,x i2 ] T ,u i (t) = [v i1 ,v i2 ] T , the dynamic equation of the i-th unmanned vehicle is:

4. According to claim 1, a distributed fixed-time and scheduled-time collaborative optimal consensus control method for a multi-agent system is characterized in that: Step 2 considers that each agent i has a local cost function C i (x i ), the global cost function is the sum of all local cost functions, which is: The optimization problem is described as: Where x i (t), x j (t) represents the position status of the i-th and j-th agents respectively, Represents the function The minimum point x, x i (t) = x j (t),i=1,2,…,N,i≠j means that the position states of N agents are the same.

5. According to claim 1, a distributed fixed-time and scheduled-time collaborative optimal consensus control method for a multi-agent system is characterized in that: In step 4, the Lyapunov function is constructed to perform consensus analysis on the fixed-time collaborative optimization consensus algorithm, including: according to the two stages of the algorithm, the analysis is divided into two steps, where the time of the first stage is 0<t≤T1, corresponding to the first stage of the algorithm, and the goal is to enable all agents to reach their respective local optimal solutions within a fixed time. The time of the second stage is t>T1, which corresponds to the second stage of the algorithm. The goal is to make all agents solve from their local optimal solutions. Reach the global optimal solution x * .

6. A distributed fixed-time and scheduled-time collaborative optimal consensus control method for a multi-agent system according to claim 5, characterized in that: Construct Lyapunov function and conduct consensus analysis on the fixed-time collaborative optimization consensus algorithm, including: Step 4.1.1, analyze the fixed-time local optimality of the multi-agent system and give an estimate of the time required, including: For the i-th agent, a semi-positive definite, radially unbounded Lyapunov function is constructed as: The derivative with respect to time is: Based on the fixed-time stability theory, the agents can reach their local optimal solutions from the initial state within a fixed time. The estimated time required is: Step 4.1.2, analyze the fixed-time global optimality of the multi-agent system and give an estimate of the time required, including: Based on the zero gradient and algorithm, the fixed-time global optimality of the multi-agent system under the second stage of the algorithm is proved, and the Lyapunov function is constructed as: Where x * Represents the global optimal solution of the optimization problem. Since the local cost function C i (x) is m i Strongly convex, that is, for any vector x, y and m i >0Yes Using the constructed Lyapunov function, we get: but is positive definite, radially unbounded, and If and only if x i (t) = x * ; right Taking the derivative we get: In the formula, the coefficient represents the Kronecker product, I m Represents an m-dimensional unit matrix. Based on the fixed-time stability theory, the agents can move from their local optimal values ​​within a fixed time. Reach the global optimal solution x * , the estimated time required is: Combined with step 4.1.1 and step 4.1.2, under the action of the first stage of the algorithm, the agents can first converge from the initial state to their respective local optimal values ​​within a fixed time T1, at which point the local cost function corresponding to each agent is minimized; based on the zero gradient and algorithm, under the action of the second stage of the algorithm, the position states of all agents converge to the global optimal point, at which point the global cost function corresponding to the entire system is minimized, and the entire optimization consensus process is completed within a fixed time T1+T2.

7. According to claim 1, a distributed fixed-time and scheduled-time collaborative optimal consensus control method for a multi-agent system is characterized in that: In step 4, a Lyapunov function is constructed to perform consensus analysis on the scheduled time collaborative optimization consensus algorithm, including: according to the two stages of the algorithm, the analysis is carried out in two steps, where the time of the first stage is 0<t≤δT p , corresponding to the first stage of the algorithm, the goal is to make all agents in the pre-set time δT p Reach their respective local optimal solutions The time of the second stage is t>δT p , corresponding to the second stage of the algorithm, the goal is to make all agents able to p From their respective local optimal solutions Reach the global optimal solution x of the optimization problem * .

8. A distributed fixed-time and scheduled-time collaborative optimal consensus control method for a multi-agent system according to claim 7, characterized in that: Construct Lyapunov function and conduct consensus analysis on the scheduled time collaborative optimization consensus algorithm, including: Step 4.2.1: Analyze the local optimality of the multi-agent system in terms of scheduled time, including: For the i-th agent, a semi-positive definite, radially unbounded Lyapunov function is constructed as: but The derivative with respect to time is; Based on the theory of scheduled time stability, the agent can p From the initial state to the local optimal solution At this time, the basic requirement of the zero gradient sum algorithm is met, that is, the initial sum of the local gradient is zero; Step 4.2.2: Analyze the global optimality of the scheduled time of the multi-agent system, including: Based on the zero gradient and algorithm, the positive definite and radially unbounded Lyapunov function is constructed as: right Taking the derivative we get: in Based on the theory of scheduled time stability, the agent can p From their respective local optimal values Reach the global optimal solution x * ; Combined with steps 4.2.1 and 4.2.2, in the first stage of the algorithm, each agent can first p The local cost function of each agent is minimized. Then, based on the zero gradient algorithm, the position states of all agents can be minimized in the second stage of the algorithm within the time (1-δ)T. p The internal convergence reaches the global optimal point, at which point the global cost function of the entire system is the smallest, and the entire optimization consensus process is completed within the preset time T p Completed within.

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