Fixed / time-varying neural network adaptive containment consensus method for heterogeneous nonlinear multi-leader systems

Abstract

The present application relates to a kind of fixed / assigned time neural network adaptive inclusive consistency method of heterogeneous nonlinear multi-leader system. With the intelligent agent system as the research object (unmanned aerial vehicle formation, unmanned aerial vehicle driving, underwater vehicle), for fixed time and assigned time inclusive consistency problem, wherein the dynamic behavior of intelligent agent is heterogeneous, first, a radial basis function neural network is proposed to approximate unknown nonlinear dynamics, in addition, a new fixed time adaptive distributed strategy is proposed to solve the heterogeneous nonlinear multi-agent inclusive consistency problem with unknown external disturbance, finally, by designing the assigned time adaptive control method with bounded control gain, the assigned time inclusive consistency is explored, wherein the convergence time can be arbitrarily predefined according to task requirements.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of robot formation, traffic vehicle management, unmanned aerial vehicle driving, underwater vehicle, and particularly relates to a fixed / assigned time neural network adaptive containment consensus method for heterogeneous nonlinear multi-leader systems. BACKGROUND

[0002] Multi-agent systems are large-scale systems composed of independent entities connected to each other, which perceive, decide and act in a shared environment through a network, and each agent is an independent entity that performs tasks autonomously and interacts with other agents. The main feature of multi-agent systems is to complete complex tasks through the coordination of each agent. This is more in line with the actual situation, because many tasks cannot be completed by a single system, such as robot formation, military surveillance and reconnaissance, unmanned aerial vehicle driving, underwater vehicle, etc. require the coordinated control of multi-agent systems to achieve the expected task.

[0003] In order to realize the effectiveness of cooperative control, distributed consensus control is particularly critical. It is worth emphasizing that previous research has mainly focused on the consensus problem of linear multi-agent systems. However, actual physical systems often have inherent nonlinear dynamics, or even nonlinear dynamics cannot be accurately modeled. To address this challenge, intelligent control methods based on neural networks and fuzzy systems with function approximation capabilities have been proposed. Based on the nonlinear dynamic behavior of the system, multi-agent systems can be divided into homogeneous and heterogeneous systems. Due to the differences between individuals, they may exhibit diversity in terms of capabilities, characteristics, behaviors or tasks. This diversity has advantages in terms of flexibility and complexity of task decomposition. Therefore, it is very necessary to study heterogeneous nonlinear systems.

[0004] However, most of the existing results focus on the case that the time tends to infinity, while the convergence in finite time is more practical in real applications, and the problem of finite-time consensus has attracted much attention. Unfortunately, the estimation of settling time depends on the initial value of the individual, and as the uncertainty of the initial condition increases, the convergence time of the system can become unpredictable, which directly affects the stability of the system. To solve this problem, Polyakov [A. Polyakov, Nonlinear feedback design for fixed-time stabilization of linear control systems [J], IEEE Transactions on Automatic Control, vol. 57, no. 8, pp. 2106-2110, 2012.] proposed a fixed-time control strategy to remove the restriction of the initial value on the estimation of the settling time. Consensus is one of the key performance indicators to improve the convergence speed in control, and it is essential to evaluate the dynamic behavior of the agent. In recent years, the fixed-time control strategy for heterogeneous nonlinear multi-agent systems has become a research hotspot. The main goal of the problem of fixed-time containment consensus of multi-leaders is to make the followers enter and remain in the convex hull spanned by the state variables or output variables of the leaders. However, the existing research mainly adopts homogeneous nonlinear dynamics modeling. Therefore, it is necessary to consider the fixed-time consensus of heterogeneous nonlinear containment.

[0005] Generally, fixed-time controllers need to be carefully parameterized to ensure good performance under specific conditions. In view of this background, fixed-time adaptive control protocols are proposed. The adaptive containment control problem for a class of stochastic nonlinear heterogeneous multi-agent systems with unobservable states is solved, however, the fixed-time adaptive containment control problem has not been fully considered. Therefore, the bipartite fixed-time output formation-containment tracking problem for heterogeneous systems with multiple leaders is studied, where the system is linear and the controller is static [Y. Cai, H. Zhang, Y. Wang, Z. Gao and Q. He, "Adaptive Bipartite Fixed-Time Time-Varying Output Formation-Containment Tracking of Heterogeneous Linear Multiagent Systems[J], bIEEE Transactions on Neural Networks and Learning Systems, vol. 33, no. 9, pp. 4688-4698, 2022,]. Therefore, it is necessary to study the fixed-time adaptive containment control problem for heterogeneous nonlinear multi-agent systems. Fixed-time control algorithms can adjust the parameters of the controller according to the actual operating conditions and changes of the system, and obtain the desired convergence time. However, the problem of pushing the followers into a specific convex hull within a specified time has not been fully solved. To overcome this difficulty, a prescribed-time containment control algorithm is proposed. Liu et al. studied the prescribed-time containment control for uncertain nonlinear multi-agent systems [D. Liu, Zhi, Liu a. b, C. Chen, Y. Zhang, Prescribed-time containment control with prescribed performance for uncertain nonlinear multi-agent systems[J], Journal of the Franklin Institute, vol. 358, pp. 1782-1811, 2021.].Zhao et al. studied the prescribed-time containment control of high-order nonlinear multi-agent systems, including a distributed observer [G. Zhao, Q. Liu, C. Hua, Prescribed-time containment control of high-order nonlinear multi-agent systems based on distributed observer[J], Journal of the Franklin Institute, vol. 360, pp. 6736-6756, 2023.]. The prescribed-time controller mentioned above has time-varying and unbounded control gain. Obviously, for infinite control gain, this control strategy is unacceptable and difficult to implement in practical applications. Therefore, how to design some simple and effective finite gain control protocol to solve the prescribed-time containment control of heterogeneous nonlinear multi-agent is a meaningful and challenging problem. SUMMARY

[0006] The purpose of the present application is to provide a fixed / prescribed-time neural network adaptive containment consensus method for heterogeneous nonlinear multi-leader systems. The method is aimed at the containment consensus problem of multi-agent systems (robot formation, traffic vehicle management, unmanned aerial vehicle driving, underwater vehicle field), and for this purpose, a fixed\prescribed-time heterogeneous nonlinear containment consensus method based on adaptive control is proposed. The complex dynamics of individual agents are trained to better approximate the dynamics of independent individuals, so as to realize the entry of followers into the convex hull of leaders.

[0007] To achieve the above purpose, the technical scheme of the present application is as follows: a fixed / prescribed-time neural network adaptive containment consensus method for heterogeneous nonlinear multi-leader systems, taking the agent system as the research object, establishing a fixed-time and prescribed-time containment consensus problem, wherein the dynamics of the agent is heterogeneous. Specifically, first, a radial basis function neural network is proposed to approximate the unknown nonlinear dynamics, in addition, a new fixed-time adaptive distributed strategy is proposed to solve the containment consensus problem of heterogeneous nonlinear multi-agent with unknown external disturbance, and finally, a prescribed-time adaptive control method with bounded control gain is designed to explore the prescribed-time containment consensus, wherein the convergence time can be arbitrarily predefined according to the task requirement.

[0008] In an embodiment of the present application, the multi-agent system is mathematically modeled, and χ r (t) represents the state of the rth agent.

[0009] In an embodiment of the present application, based on the adaptive distributed control strategy and algebraic graph theory, a radial basis function neural network is established to approximate the unknown nonlinear dynamics.

[0010] In an embodiment of the present application, the method specifically comprises the following steps:

[0011] Step one, construction of a heterogeneous nonlinear multi-leader system;

[0012] Step two, establishment of the fixed-time containment consensus problem of the heterogeneous nonlinear multi-leader system;

[0013] Step three, establishment of the specified-time containment consensus problem of the heterogeneous nonlinear multi-leader system.

[0014] In an embodiment of the present application, step one is specifically implemented as follows:

[0015] Consider a multi-agent model with Q followers and R leaders:

[0016]

[0017] where Θ = {1, 2,..., Q, Q+1,..., Q+R}, χ r (t) represents the state of the rth agent, is a heterogeneous kinetic behavior ( represents a set of Euclidean space, represents the set of real numbers), κ r (t) is an unknown external disturbance, ω r (t) is the control input of the rth agent;

[0018] Assumption 1: For unknown external disturbance |κ r (t)|≤κ * , where κ * is a constant;

[0019] Assumption 2: For there exists a constant h>ζ>0 such that the following equation holds

[0020]

[0021] Based on the approximation of neural networks, the unknown nonlinear function is approximated, denoted as

[0022]

[0023] where represents the optimal weight vector, is the number of neurons, is a Gaussian kernel function, where

[0024]

[0025] where τ r = [τ r1 ,τ r2 ,…,τ rq ] denotes the field center, and p,q are constants, denotes the width of the Gaussian function, denotes the approximation error bounded by a constant Φ * .

[0026] In one embodiment of the present application, step two is implemented as follows:

[0027] The adaptive distributed timing controller is constructed as follows:

[0028]

[0029] where d r (t), α r (t) denote the adaptive control gains, θ>1, ξ>0,

[0030] sign(t) denotes the sign function, and the update rate of the controller parameters is:

[0031]

[0032] where ρ1, ρ2, d ＊ , α ＊ , δ1, δ2 denote constants greater than 0;

[0033] Substituting equation (5) into equation (1) gives

[0034]

[0035] The tight vector form of equation (8) is obtained as:

[0036]

[0037] E Q and E R denote the identity matrix, denotes the leader set and denotes the follower set and denotes the leader controller, denotes the follower controller.

[0038] Define and calculate:

[0039]

[0040] where En Represents an n-dimensional identity matrix. Describe a Q×Q dimensional Euclidean space. Represents a Q×R dimensional Euclidean space;

[0041] Error system:

[0042]

[0043] in

[0044] Theorem 1: Based on Assumption 1 and the fixed-time containment control controller (5) under the distributed protocol, if there exist α > 0, d > 0, satisfying

[0045]

[0046] The convergence time T1 is estimated to be

[0047]

[0048] in α * ,d * It is the optimal control gain, κ * >0 indicates the boundary of the perturbation, csc(x) represents the cocut, λ r express eigenvalues;

[0049] Proof: Construct the following Lyapunov function.

[0050]

[0051] Where d > 0 and α > 0 are constants; let Indicates that it does not include the zero point, 0 nQ (representing an n×Q dimensional column vector with all zero elements) has:

[0052] right Differentiation

[0053]

[0054] right Differentiation

[0055]

[0056] right Differentiation

[0057]

[0058] Combining (13)-(16), there exists an orthogonal matrix. such that defined there is

[0059]

[0060] Therefore, there always exists d, a such that A T + 2d ＊ lambda r E n < 0, Therefore, we have

[0061]

[0062] Because

[0063]

[0064] where lambda min denotes the minimum eigenvalue;

[0065] we have

[0066]

[0067] From (19), we have

[0068]

[0069] Therefore,

[0070]

[0071] From (18)-(22), we have

[0072]

[0073] where

[0074]

[0075] The heterogeneous nonlinear multi-agent model (1) is fixed-time containment convergent, and the settling time satisfies

[0076]

[0077] In an embodiment of the present application, step three is specifically implemented as follows:

[0078] Adaptive specified-time controller:

[0079]

[0080] Controller parameter adaptive update rate:

[0081]

[0082] wherein wherein represents the minimum eigenvalue;

[0083] Theorem two: based on assumption 1 and the specified time containment control controller (5) under the distributed protocol, if there is alpha>0, d>0, satisfies

[0084]

[0085] reach containment consensus within a specified time;

[0086] Proof:

[0087]

[0088] wherein

[0089]

[0090] Compared with the prior art, the present application has the following beneficial effects: the present application takes the robot formation as the research object, ensures that the information transmission of each intelligent body at different times is consistent, which is very important for the formation of intelligent bodies, in order to ensure the coordination between various devices. A fixed time and specified time multi-agent system containment consensus method based on adaptive distributed control is proposed, the distributed fixed time controller gets rid of the dependence on the initial value, overcomes the dependence of the original system on the initial value, optimizes the rest time, and obtains the optimal control cost. The distributed specified time controller is aimed at solving the delay and uncertainty in communication between different devices and nodes in the distributed system, so a mechanism is needed to ensure the consistency of each node at a certain time, so as to improve the task completion effect and improve the transmission efficiency of information. BRIEF DESCRIPTION OF DRAWINGS

[0091] Figure 1 It is a principle block diagram of the fixed / specified time neural network adaptive containment consensus method for the heterogeneous nonlinear multi-leader system of the embodiment of the present application;

[0092] Figure 2 It is a topology graph of the network of the embodiment of the present application;

[0093] Figure 3 It is an intelligent body dynamic behavior training graph of the embodiment of the present application;

[0094] Figure 4 It is a follower entering the leader's convex hull graph of the embodiment of the present application;

[0095] Figure 5State diagram of leader and follower for embodiments of the invention;

[0096] Figure 6 Controller parameter adaptation diagram for embodiments of the invention;

[0097] Figure 7 Fixed-time error diagram for embodiments of the invention;

[0098] Figure 8 Fixed-time controller evolution diagram for embodiments of the invention;

[0099] Figure 9 Designated-time leader and follower state diagram for embodiments of the invention;

[0100] Figure 10 Designated-time error diagram for embodiments of the invention;

[0101] Figure 11 Designated-time controller evolution diagram for embodiments of the invention. DETAILED DESCRIPTION

[0102] The technical solutions of the present invention will be described in detail below with reference to the accompanying drawings.

[0103] To introduce the present invention in detail, a fixed-time and designated-time heterogeneous nonlinear multi-agent containment consensus method based on adaptive control is described in detail below, which can be roughly divided into two parts: construction of heterogeneous multi-leader system model and establishment of fixed-time / designated-time containment consensus criteria.

[0104] Step one: construction of heterogeneous nonlinear multi-leader system

[0105] Consider the following multi-agent model with Q followers and R leaders:

[0106]

[0107] where is a heterogeneous kinetic behavior, κ r (t) is an unknown external disturbance, ω r (t) is the control input of the rth node.

[0108] Assumption 1: for unknown external disturbance |κ r (t)|≤κ * , where κ * is a constant.

[0109] Assumption 2: for there exists a constant h>ζ>0 such that the following equation holds

[0110]

[0111] Based on the approximation of neural networks, the unknown nonlinear function is approximated, denoted as

[0112]

[0113] where denotes the optimal weight vector, is the number of neurons, is the Gaussian basis function, where

[0114]

[0115] where τ r = [τ r1 ,τ r2 ,…,τ rq ] represents the field center, denotes the width of the Gaussian function. denotes the approximation error bounded by Φ * .

[0116] Step two: Establishment of the fixed-time containment consensus problem of heterogeneous nonlinear multi-leader systems

[0117] The adaptive distributed timing controller is constructed as follows:

[0118]

[0119] where d r (t), α r (t) are the adaptive control gains, θ>1, ξ>0. The update rate of the controller parameters is:

[0120]

[0121] Substituting equation (5) into equation (1) gives

[0122]

[0123] The tight vector form of equation (8) can be obtained:

[0124]

[0125] Define can be calculated:

[0126]

[0127] Error system:

[0128]

[0129] Theorem 1 : Based on Assumption 1 and the fixed-time containment controller (5) under the distributed protocol, if there exist a > 0, d > 0, and a constant such that

[0130]

[0131] Convergence time is estimated as

[0132]

[0133] where

[0134] Proof: Construct the Lyapunov function as follows

[0135]

[0136] where d > 0 and a > 0 are constants. Let For we have:

[0137] For take the derivative

[0138]

[0139] For take the derivative

[0140]

[0141] For take the derivative

[0142]

[0143] Combining (13) - (16), define we have

[0144]

[0145] Therefore, there always exist d and a such that A T + 2d * λ r E n ≤ 0, Thus we have

[0146]

[0147] Because

[0148]

[0149] We have

[0150]

[0151] From (19) we have

[0152]

[0153] Therefore,

[0154]

[0155] From (18)-(22) we have

[0156]

[0157] where

[0158]

[0159] The heterogeneous nonlinear multi-agent system (1) is fixed-time containment convergent, and the settling time satisfies

[0160]

[0161] Step three: establishment of the fixed-time containment consensus problem of the heterogeneous nonlinear multi-leader system

[0162] Adaptive fixed-time controller:

[0163]

[0164] Controller parameter adaptive update rate:

[0165]

[0166] Theorem two: based on assumption 1 and the distributed protocol under the fixed-time containment control controller (5), if there exists α>0, d>0, satisfies

[0167]

[0168] The containment consensus is achieved within the fixed time.

[0169] Proof:

[0170]

[0171] where

[0172]

[0173] In order to introduce the present application in detail, a specific example is given below to embody the proposed fixed / fixed-time neural network adaptive containment consensus method of the heterogeneous nonlinear multi-leader system.

[0174] 1. Design of the basic task

[0175] Consider a multi-agent system (1) with three leaders and three followers, whose topology is described as Figure 2 .The Laplacian matrix of (1) is given by Figure 2

[0176]

[0177] where

[0178]

[0179] Dynamic behavior of agents 1-6

[0180]

[0181]

[0182]

[0183]

[0184]

[0185]

[0186] External disturbance

[0187] κ 11 (t) = 0.1 cos(t), κ 12 (t) = 0.5 cos(t);

[0188] κ 21 (t) = 0.5 sin(t), κ 22 (t) = 0.3 sin(t);

[0189] κ 31 (t) = 0.6 cos(t), κ 32 (t) = 0.8 cos(t);

[0190] κ 41 (t) = 0.7 cos(t), κ 42 (t) = 0.5 sin(t);

[0191] κ 51 (t) = 0.3 sin(t), κ 52 (t) = 0.7 cos(t);

[0192] κ 61 (t) = 0.3 cos(t), κ62 (t) = 0.4 cos(t).

[0193] The initial values of the six agents are chosen as

[0194] χ1(0) = (-5, 0), χ2(0) = (0, -5), χ3(0) = (1, -6), χ4(0) = (0.5, 0.5), χ5(0) = (-4, 0), χ6(0) = (0, -4).

[0195] where the approximation error of the nonlinear functions f1-f6 is * = 0.6, see Figure 3 and * = 1.

[0196] The consensus of the heterogeneous nonlinear multi-agent system (1) based on the fixed-time containment control strategy (5) will be verified. The parameters of the controller are

[0197] ρ1= ρ2= 10, δ1= δ1= 3, d = 10, α = 5,

[0198] From the simulation results, it can be seen that the follower enters the leader's convex hull as shown in Figure 4 and Figure 5 The error of the agents is shown in Figure 6 The evolution of the controller is shown in Figure 7 . Figure 8

[0199] 2. Construction of the fixed / time specified time synchronization task

[0200] The consensus of the heterogeneous nonlinear multi-agent system (1) based on the specified time containment control strategy will be verified. The parameters of the controller are

[0201] ρ1= ρ2= 10, δ1= δ1= 3, d = 10, α = 5, ξ = 4,

[0202] From the simulation results, it can be seen that the state trajectory of the multi-agent is shown in Figure 9 The state error of the agents is shown in Figure 10 The evolution of the specified time controller is shown in Figure 11 .

[0203] 3. Simulation analysis

[0204] In the above simulation case, first, the heterogeneous multi-agent system is modeled as Figure 2 ​The application is shown. Based on fixed / specified time containment control, a class of heterogeneous nonlinear multi-agent consensus problem is studied. Two kinds of distributed control strategies are constructed. By using fixed / specified time stability theory, the containment consensus of the leader and the follower of the multi-agent system is established, and the containment evolution is seen Figure 3 and 4 . Thus, the containment consensus is realized.

[0205] The above is the preferred embodiment of the application, any changes made according to the technical solutions of the application, as long as the function does not exceed the scope of the technical solutions of the application, belongs to the protection scope of the application.

Claims

1. A fixed / assigned time neural network adaptive containment consensus method for heterogeneous nonlinear multi-leader systems, characterized in that, Taking the multi-agent system as the research object, the fixed-time and specified-time containment consensus problems are established, in which the dynamic behaviors of agents are heterogeneous. Specifically, first, the radial basis function neural network is proposed to approximate the unknown nonlinear dynamics. In addition, a new fixed-time adaptive distributed strategy is proposed to solve the containment consensus problem of heterogeneous nonlinear multi-agent systems with unknown external disturbances. Finally, by designing a specified-time adaptive control method with bounded control gain, the specified-time containment consensus is explored, in which the convergence time can be arbitrarily predefined according to the task requirements. The mathematical modeling of the multi-agent system is performed by χ r (t) represents the state of the rth agent; the method specifically comprises the following steps: Step one, construction of heterogeneous nonlinear multi-leader system; Step two, establishment of fixed-time containment consensus problem of heterogeneous nonlinear multi-leader system; Step three, establishment of specified-time containment consensus problem of heterogeneous nonlinear multi-leader system; Step one is implemented as follows: Consider a multi-agent model with Q followers and R leaders: wherein χ r (t) denotes the state of the rth agent, is a heterogeneous kinetic behavior, denotes a set of Euclidean spaces, denotes the set of real numbers, κ r (t) is an unknown external disturbance, ω r (t) is the control input of the rth agent; Assumption 1 : For unknown external disturbance |κ r (t)|≤κ * where κ * is a constant; Assumption 2: For There exists a constant h > ζ > 0 such that the following equation holds Based on the approximation of neural network, the unknown nonlinear function is approximated as wherein denotes the optimal weight vector, is the number of neurons, is a Gaussian kernel, wherein where τ r = [τ r1 ,τ r2 ,…,τ rq ] denotes the field center, and p,q are constants, denotes the width of the Gaussian function, denotes the approximation error bounded by a constant Φ * ; Step two is implemented as follows: The adaptive distributed timing controller is constructed as follows: where d r (t), a r (t) denotes an adaptive control gain, θ > 1, ξ > 0, sign(t) denotes a sign function; The update rate of the controller parameters is: wherein p1, p2, d ＊ , a ＊ , d1, d2 represent constants greater than 0; Bring formula (5) into formula (1) to get Get the tight vector form of formula (8): E Q and E R denotes the identity matrix, denotes the set of leaders and denotes the set of followers and denotes a leader controller, denotes a follower controller; Definitions Calculated: wherein E n denotes the n-dimensional identity matrix, denotes the Q x Q-dimensional Euclidean space, denotes the Q x R-dimensional Euclidean space; Error system: wherein Theorem one: based on assumption 1 and the fixed-time containment controller (5) under the distributed protocol, if there exists α>0, d>0, satisfying The convergence time T1 is estimated as wherein α ＊ ,d ＊ is an optimal control gain, κ * > 0 denotes a boundary of the disturbance, csc(x) denotes the cosecant, λ r denotes an eigenvalue of Proof: construct the following Lyapunov function where d > 0, a > 0 is a constant; let For denotes not containing zero points, 0 nQ denotes an n x Q dimensional column vector with all zero elements, we have: right Differentiation for derivation for derivation In connection with (13) - (16), there is an orthogonal matrix such that Definition There are Thus, there always exists d, a such that A T + 2d ＊ λ r E n ≤ 0, Thus, we have Because where λ min denotes the smallest eigenvalue; It can be obtained that From formula (19), it can be obtained that Therefore, From formulas (18)-(22), it can be obtained that Where The heterogeneous nonlinear multi-agent model (1) is fixed-time containment convergence, and the settling time satisfies 2. The fixed / assigned-time neural network adaptive containment consensus method of heterogeneous nonlinear multi-leader systems of claim 1, wherein, Step three is implemented as follows: Adaptive specified-time controller: The adaptive update rate of the controller parameters is: wherein wherein denotes the smallest eigenvalue; Theorem Two: Based on assumption 1 and the designated time-inclusive controller (5) under the distributed protocol, if there exists satisfies Achieve containment consensus within specified time; Proof: Where

Images