Multi-agent consistency method with input saturation and unknown disturbance

Abstract

The invention discloses a multi-agent system consistency method with input saturation and unknown disturbance, and relates to the field of multi-agents. The method does not need to know an accurate boundary or model of disturbance in advance, and controller design can be carried out only by assuming that disturbance is bounded. The disturbance range is obtained online through adaptive estimation, and the robustness of the algorithm to disturbance and uncertainty is enhanced. By properly selecting the value of the low-gain parameter, for any bounded initial state, the method can ensure that all follower states gradually track the leader state, and the consistency convergence in a semi-global range is realized. And a low-gain feedback technology is adopted to ensure that the consistency of the system is realized under the condition that the input of the actuator is limited. Compared with a traditional method which needs to design a control law according to the worst disturbance condition, the method has the advantages that the control input is adaptively adjusted according to the actual disturbance condition, excessive conservative control is avoided, and therefore the energy consumption of the actuator is remarkably reduced.

Description

Technical Field

[0001] This invention relates to the field of multi-agent systems, specifically to a method for achieving consistency between low-gain feedback control and adaptive disturbance estimation in multi-agent systems with actuator saturation and unknown disturbances. Background Technology

[0002] With the continuous development of communication and computing technologies, networked systems have become increasingly prominent in engineering applications, becoming a research hotspot in the field of modern control. Among them, multi-agent systems (MASs), as typical networked systems, have been widely applied in traffic management, power systems, and UAV formations due to their distributed structure, good scalability, and collaborative capabilities. In the research of MASs, the consistency problem is a fundamental and core issue, aiming to achieve convergence among all agents in terms of state, output, or trajectory through local interactions. This is a prerequisite for achieving group collaborative control and task division. Formation control, on the other hand, further maintains spatial structure and formation requirements on the basis of consistency, reflecting a higher level of system coordination capabilities.

[0003] In recent years, some scholars have explored the problem of consensus control under disturbance and input-constrained conditions. Zhao et al. proposed an adaptive distributed state feedback control protocol with performance guarantees, which can achieve leader-follower consensus control under known disturbance models and boundary conditions. However, this method relies on accurate modeling and is difficult to apply to disturbance environments with unknown structural parameters in practice. Yang et al. designed a control scheme combining low-gain feedback and sliding mode control, which can handle actuator saturation and disturbance input problems. However, its control law design relies on prior information of the disturbance boundary, which significantly increases control energy consumption and affects system performance when the boundary estimation error is large. Summary of the Invention

[0004] This invention addresses the problems of existing consensus control methods for multi-agent systems with input saturation and unknown disturbances, which require disturbance boundary information, disturbance dynamic information, and excessive actuator energy consumption. It proposes a semi-global consensus control method based on low-gain feedback control and adaptive disturbance estimation.

[0005] To address the aforementioned technical problems, the specific technical solution of the multi-agent consensus method with input saturation and unknown perturbations of the present invention is as follows:

[0006] Step 1: Construction of the communication topology of the multi-agent system. The multi-agent system includes a leader and multiple followers. Matrix M is obtained based on the Laplace matrix of the followers and the diagonal matrix of communication between the leader and followers.

[0007] Step 2: Construct dynamic models of followers and leaders with input saturation;

[0008] One agent is controlled by m actuators. First, define the saturation function of the agent:

[0009] σ(α)=[σ1(α1),σ2(α2),…,σ m (α m )]

[0010] Where, σ m (α m () represents the saturation function of the m-th actuator state, where m represents the number of actuators. The input variables of the saturation function are represented by , where

[0011]

[0012] Where j = 1, 2, ..., m, β j It is a positive number;

[0013] Consider a multi-agent system with N followers and one leader, where the actuator velocity is fluctuated. The input variable of the saturation function is v. i (t), the dynamic model of the follower is represented by the following formula:

[0014]

[0015] in, x represents the i-th follower. i At time t, the state is i = 1, 2, ..., N. x represents i The derivative; This represents the ideal output of the i-th follower's actuator at time t. It has m components, and its j-th state component is denoted as v. ij j = 1, 2, ..., m Indicates v i The derivative; This represents the control law of the i-th follower at time t. It has m components, and its j-th state component is denoted as u. ij j = 1, 2, ..., m; This represents an unknown disturbance, which affects the actuator speed, i.e., the actuator's derivative. It has m components, and its j-th state component is denoted as d. ij j = 1, 2, ..., m, the perturbation is bounded but the specific boundary values ​​are unknown; Given the system matrix, Represents a known input matrix;

[0016] The follower agent has n states. The dynamic model of the leader system can be represented by the following formula:

[0017]

[0018] in This represents the leader's state at time t, which has n components. The j-th executor state component is denoted as x. 0j j = 1, 2, ..., n Let x0 be the derivative.

[0019] Step 3: To address the input saturation problem, a low-gain feedback control algorithm is used for suppression, resulting in a system feedback matrix based on the low-gain feedback matrix. First, the algebraic Riccati equation is solved, as shown in the following formula:

[0020] A T P(ε)+PA-γP(ε)BB T P(ε)+εI=0

[0021] Where, γ≤λ min (M), λ min (M) represents the smallest eigenvalue of matrix M, and I is the identity matrix. The low-gain feedback matrix P(ε) is obtained by solving the ARE equation. P(ε) is simplified to P, and the feedback matrix K = B is constructed. T P.

[0022] Step 4: For disturbance suppression, estimate the upper and lower bounds for unknown disturbances;

[0023] For the bounded perturbation d of the i-th agent i The maximum and minimum values ​​are denoted as and respectively. The estimated values ​​of its maximum and minimum values ​​are denoted as follows: With d i ∈R m , d ij This represents the perturbation estimate corresponding to the j-th actuator; express d ij The derivative of d is used to realize the perturbation d. i The adaptive estimates of the upper and lower bounds are constructed as follows:

[0024]

[0025] Where, μ i1 ,μ i2 δ is a positive constant, representing the rate of perturbation estimation; ij The parameters for estimating the direction of the disturbance are:

[0026]

[0027] ξ i As an auxiliary variable, it is obtained through the following formula:

[0028]

[0029] Where ξ i ∈R m ,ξ ij Represents ξ i The j-th element; a ij Representing the adjacency matrix The elements corresponding to follower agent i and follower agent j represent their communication state, h. i Let be the i-th element of the communication diagonal matrix, representing the communication state between follower agent i and leader; K is the system feedback matrix.

[0030] Step 5: Combine the system feedback matrix and the upper and lower bound estimates of the unknown disturbances to construct the system's control law and achieve consensus among multiple agents.

[0031] The control law u of the i-th agent i The following formula is used to obtain:

[0032]

[0033] in Indicates the second intermediate control law The control law for the j-th actuator, For the first intermediate control law, ε∈(0,1] is the low gain parameter.

[0034] The consensus proof for the multi-agent system is as follows:

[0035] Define the error between followers and leaders as structure

[0036] The Lyapunov function is specifically divided into two parts:

[0037] V = V1 + V2

[0038] in

[0039]

[0040]

[0041] Where P is the low-gain feedback matrix. This represents the true maximum value of the disturbance. This represents the true minimum value of the disturbance.

[0042] Only proof is needed This can satisfy the system's semi-global consistency requirement.

[0043] The beneficial effects of this invention are as follows:

[0044] 1. Strong robustness: Controller design can be performed simply by assuming the disturbance is bounded, without needing prior knowledge of the precise boundaries or model of the disturbance. The disturbance range is obtained online through adaptive estimation, enhancing the algorithm's robustness to disturbances and uncertainties.

[0045] 2. Low energy consumption: Compared with the traditional method that requires designing control laws based on the worst disturbance conditions, the control algorithm of this invention adaptively adjusts the control input according to the actual disturbance conditions, avoiding overly conservative control, thereby significantly reducing the energy consumption of the actuator.

[0046] 3. No Prior Disturbance Boundary Information Required: An adaptive disturbance estimation law is designed, which can estimate the upper and lower bounds of the disturbance in real time without needing to know the disturbance boundaries, and use this estimate for control compensation. Effective disturbance compensation can be performed without pre-given upper and lower bounds, simplifying the prior knowledge assumptions required for controller design and broadening the applicability of this method. Furthermore, a composite control law including consistency control components and disturbance compensation components is constructed. By introducing a sign function-based switching mechanism, the controller can automatically adjust according to the disturbance sign, providing targeted compensation for disturbances in different directions.

[0047] 4. Achieves semi-global consistency: By appropriately selecting the value of the low-gain parameter, this method can guarantee that all follower states asymptotically track the leader state for any bounded initial state, achieving semi-global consistency convergence. Low-gain feedback technology is employed to ensure system consistency even when actuator inputs are constrained.

[0048] 5. The control scheme of the present invention ensures the semi-global asymptotic stability of the closed-loop system, that is, under certain parameter conditions, a multi-agent system with any bounded initial state can achieve leader-follower consistency and avoid continuous saturation of the actuator. Attached Figure Description

[0049] Figure 1 This is a communication topology diagram for the example.

[0050] Figure 2 The first state of the leader and followers x i1 Trajectory diagram.

[0051] Figure 3 The second state of leaders and followers x i2 Trajectory diagram.

[0052] Figure 4 Leaders and followers control signals u i Trajectory diagram.

[0053] Figure 5 In existing technologies, the leader and follower control signals u i Trajectory diagram. Detailed Implementation

[0054] To better understand the purpose, structure, and function of this invention, the method of this invention will be described in further detail below with reference to the accompanying drawings.

[0055] Example 1

[0056] A multi-agent consensus method with input saturation and unknown perturbations specifically includes the following steps:

[0057] Step 1: Construct a multi-agent system and its communication topology.

[0058] A multi-agent system consists of N follower agents and one leader agent; the communication topology among the followers is an undirected graph. Description, node set is Let represent the i-th agent, where i = 1, 2, ..., N, and N represents the total number of follower agents. The edge set is... Let q = 1, 2, ..., N. The edge set represents the communication between agents. In an undirected graph, if an edge exists between two agents, it means that the two agents can receive each other's state information. Adjacency matrix. If This indicates that agent i and agent q can receive each other's state information, and a iq =a qi =1, otherwise a iq =a qi =0, and a ii =0.

[0059] Define the Laplace matrix in And i≠q, l iq Let represent the element in the i-th row and q-th column of the Laplace matrix L. Communication between followers and the leader is represented by the diagonal matrix H = diag{h1, h2, ..., h...}. N} represents diag{h1,h2,…,h N} is the element on the diagonal that is h. i h i Let represent the elements in the i-th row and i-th column of a diagonal matrix, with all other elements being 0. This matrix reflects the communication status between the follower and the leader. If follower i can access the leader's information h... i =1, otherwise h i=0. Define matrix M = L + H.

[0060] Define the saturation function of the actuator as σ(α) = [σ1(α1), σ2(α2), ..., σ m (α m )], The input variables of the saturation function are represented by , where

[0061]

[0062] σ j (α j ), j = 1, 2, ..., m represents the saturation function for a single actuator state, m represents the number of actuators, β j It is a positive number.

[0063] For a multi-agent system with actuator input saturation, where actuator input saturation refers to the constraint of actuator states by a saturation function, consider a multi-agent system with N followers and one leader, where one follower agent has m actuators. The dynamic model of the followers can be expressed by the following formula:

[0064]

[0065] in, x represents the i-th follower. i The state vector at time t, i = 1, 2, ..., N, has n components, and its j-th state component is represented as x. ij j = 1, 2, ..., n x represents i The derivative; This represents the ideal output of the i-th follower's actuator at time t. It has m components, and its j-th state component is denoted as v. ij j = 1, 2, ..., m Indicates v i The derivative; This represents the control law of the i-th follower at time t. It has m components, and its j-th state component is denoted as u. ij j = 1, 2, ..., m; This represents an unknown disturbance, which affects the actuator speed, i.e., the actuator's derivative. It has m components, and its j-th state component is denoted as d. ij j = 1, 2, ..., m, the perturbation is bounded but the specific boundary values ​​are unknown. Given the system matrix, This represents a known input matrix.

[0066] The follower agent has n actuators. The dynamic model of the leader system can be represented by the following formula:

[0067]

[0068] in This represents the leader's state at time t, which has n components. The j-th executor state component is denoted as x. 0j j = 1, 2, ..., n Let x0 be the derivative.

[0069] Step 2: Construct the control law for the multi-agent system to achieve consistent control of the system.

[0070] The following assumptions must be satisfied before constructing the control law:

[0071] Assumption 1: The eigenvalues ​​of the system matrix A are all negative real parts, and the system matrix A and the input matrix B are controllable.

[0072] Assumption 2: The topological graph is connected and the edges between followers are undirected.

[0073] Step 2.1: First, to address the input saturation problem, a low-gain feedback control algorithm is used for suppression; this step is achieved by solving the algebraic Riccati equation (ARE), as shown in the following formula:

[0074] A T P(ε)+PA-γP(ε)BB T P(ε)+εI=0

[0075] Where ε∈(0,1] is the low-gain parameter, and γ≤λ min (M), λ min (M) represents the smallest eigenvalue of matrix M, and I is the identity matrix. The low-gain feedback matrix P(ε) is obtained by solving the ARE equation. P(ε) is simplified to P, and the feedback matrix K = B is constructed. T P.

[0076] Step 2.2: Then, for perturbation suppression, perturbation estimation is performed;

[0077] For a bounded perturbation d i The maximum and minimum values ​​are denoted as and respectively. The estimated values ​​of its maximum and minimum values ​​are denoted as follows: and d i ∈R m , d ij This represents the perturbation estimate corresponding to the j-th actuator. express d ij The derivative of d is used to realize the perturbation d.i The adaptive estimates of the upper and lower bounds are constructed as follows:

[0078]

[0079] Where, μ i1 ,μ i2 δ is a positive constant, representing the rate of perturbation estimation. ij The parameters for estimating the direction of the disturbance are determined according to the following formula:

[0080]

[0081] ξ i As an auxiliary variable, it is obtained through the following formula:

[0082]

[0083] Where ξ i ∈R m ,ξ ij Represents ξ i The j-th element. ij Representing the adjacency matrix Follower agent x i With follower agent x j The corresponding element indicates the communication status between the two parties.

[0084] Step 2.3: Combine the above two steps to construct the control law u for the i-th agent. i To achieve system consistency control;

[0085]

[0086] in express The control law of the j-th actuator.

[0087] Step 3: Proof of system consistency.

[0088] Define the error between followers and leaders as For i = 1, 2, ..., N, construct...

[0089] Due to the specific characteristics of the system, the Lyapunov function is divided into two parts:

[0090] V = V1 + V2

[0091] in

[0092]

[0093] This invention innovatively proposes another part of the Lyapunov function:

[0094]

[0095] in This represents the true maximum value of the disturbance. This represents the true minimum value of the disturbance.

[0096] Only proof is needed This can satisfy the system's semi-global consistency requirement.

[0097] Example 2

[0098] The following simulation example uses a multi-agent system consisting of one leader and five followers, with given system matrix A and output matrix B:

[0099]

[0100] Communication topology of multi-agent systems, such as Figure 1 As shown, the matrix M = L + H is:

[0101]

[0102] The smallest eigenvalue of the M matrix is ​​λ. min (M) = 0.2278, which requires γ < λ. min If (M), then take γ = 0.1 and take the low gain parameter ε = 0.1, then solve for the low gain feedback matrix P.

[0103]

[0104] The initial state of the leader is defined as follows:

[0105] x0(0) = [0.03, 0.49] T

[0106] Define the initial state of the 5 followers as follows:

[0107] x1(0) = [-0.07, 0.52] T

[0108] x2(0) = [0.57, -0.14] T

[0109] x3(0) = [0.2, -0.01] T

[0110] x4(0) = [0.3, -0.02] T

[0111] x5(0) = [0.2, 0.3] T

[0112] The initial state of the executor corresponding to the follower is:

[0113] v1(0)=1

[0114] v2(0)=2

[0115] v3(0)=3

[0116] v4(0)=4

[0117] v5(0)=5

[0118] Define the disturbance as

[0119] d = [d1, d2, d3, d4, d5] T

[0120] Pick

[0121] d=[2sin(4t-2),3sin(5t),-2.5sin(7t),3.5sin(t-3),-8sin(3t)] T

[0122] Under the control law obtained through this method, the first state x i1 The trajectory is as follows Figure 2 As shown, the second state x i2 The trajectory is as follows Figure 3 As shown, from Figure 2 , Figure 3 It can be seen that the system's state gradually reaches a consensus under the action of the control law. The system's control law is as follows: Figure 4 As shown, comparison Figure 5 Under the same conditions, the control law of the paper Semi-global consensus with position limited and rate disturbances via low gain feedback and integralsliding mode control is used. The control law proposed in this invention significantly reduces the peak energy of the control law while achieving system state consistency without the need for a consistent peak disturbance.

[0123] It is understood that the present invention has been described through some embodiments, and those skilled in the art will recognize that various changes or equivalent substitutions can be made to these features and embodiments without departing from the spirit and scope of the invention. Furthermore, under the teachings of the present invention, these features and embodiments can be modified to adapt to specific situations and materials without departing from the spirit and scope of the invention. Therefore, the present invention is not limited to the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are within the protection scope of the present invention.

Claims

1. A multi-agent consensus method with input saturation and unknown perturbations, characterized in that, Includes the following steps: Step 1: Construction of the communication topology of the multi-agent system. The multi-agent system includes a leader and multiple followers. Matrix M is obtained based on the Laplace matrix of the followers and the diagonal matrix of communication between the leader and followers. Step 2: Construct dynamic models of followers and leaders with input saturation; Step 3: To address the input saturation problem, a low-gain feedback control algorithm is used to suppress it, resulting in a system feedback matrix based on the low-gain feedback matrix; Step 4: For disturbance suppression, estimate the upper and lower bounds for unknown disturbances; Step 5: Combine the system feedback matrix and the upper and lower bound estimates of the unknown disturbances to construct the system's control law and achieve consensus among multiple agents.

2. The multi-agent consensus method with input saturation and unknown perturbations according to claim 1, characterized in that, Step 2 is described in detail below: One agent is controlled by m actuators. First, define the saturation function of the agent: σ(α)=[σ1(α1),σ2(α2),…,σ m (a m )] Where, σ m (α m () represents the saturation function of the m-th actuator state, where m represents the number of actuators. The input variables of the saturation function are represented by , where Where j = 1, 2, ..., m, β j It is a positive number; Consider a multi-agent system with N followers and one leader, where the actuator velocity is fluctuated. The input variable of the saturation function is v. i (t), the dynamic model of the follower is represented by the following formula: in, x represents the i-th follower. i At time t, the state is i = 1, 2, ..., N. x represents i The derivative; This represents the ideal output of the i-th follower's actuator at time t. It has m components, and its j-th state component is denoted as v. ij j = 1, 2, ..., m Indicates v i The derivative; This represents the control law of the i-th follower at time t. It has m components, and its j-th state component is denoted as u. ij j = 1, 2, ..., m; This represents an unknown disturbance, which affects the actuator speed, i.e., the actuator's derivative. It has m components, and its j-th state component is denoted as d. ij j = 1, 2, ..., m, the perturbation is bounded but the specific boundary values ​​are unknown; Given the system matrix, Represents a known input matrix; The follower agent has n states. The dynamic model of the leader system can be represented by the following formula: in This represents the leader's state at time t, which has n components. The j-th executor state component is denoted as x. 0j j = 1, 2, ..., n Let x0 be the derivative.

3. The multi-agent consensus method with input saturation and unknown perturbations according to claim 2, characterized in that, Step 4 is as follows: For the bounded perturbation d of the i-th agent i The maximum and minimum values ​​are denoted as and respectively. The estimated values ​​of its maximum and minimum values ​​are denoted as follows: and d i ∈R m , d ij This represents the perturbation estimate corresponding to the j-th actuator; express d ij The derivative of d is used to realize the perturbation d. i The adaptive estimates of the upper and lower bounds are constructed as follows: Where, μ i1 ,μ i2 δ is a positive constant, representing the rate of perturbation estimation; ij The parameters for estimating the direction of the disturbance are: ξ i As an auxiliary variable, it is obtained through the following formula: Where ξ i ∈R m ,ξ ij Represents ξ i The j-th element; a ij Representing the adjacency matrix The elements corresponding to follower agent i and follower agent j represent their communication state, h. i Let be the i-th element of the communication diagonal matrix, representing the communication state between follower agent i and leader; K is the system feedback matrix.

4. The multi-agent consensus method with input saturation and unknown perturbations according to claim 3, characterized in that, Step 5 is described in detail below: The control law u of the i-th agent i The following formula is used to obtain: in Indicates the second intermediate control law The control law for the j-th actuator, For the first intermediate control law, ε∈(0,1] is the low gain parameter.

5. A multi-agent consensus method with input saturation and unknown perturbations according to claim 4, characterized in that, The consensus proof for the multi-agent system is as follows: Define the error between followers and leaders as For i = 1, 2, ..., N, construct... The Lyapunov function is specifically divided into two parts: V = V1 + V2 in Where P is the low-gain feedback matrix. This represents the true maximum value of the disturbance. This represents the true minimum value of the disturbance. Only proof is needed This can satisfy the system's semi-global consistency requirement.

6. A multi-agent consensus method with input saturation and unknown perturbations according to claim 5, characterized in that, The system feedback matrix is ​​obtained in the following way: First, solve the algebraic Riccati equation, as shown in the following formula: A T P(ε)+PA-γP(ε)BB T P(ε)+εI=0 Where, γ≤λ min (M), λ min (M) represents the smallest eigenvalue of matrix M, and I is the identity matrix. The low-gain feedback matrix P(ε) is obtained by solving the ARE equation. P(ε) is simplified to P, and the feedback matrix K = x is constructed. T P.

Images