A data-driven online cooperative control method for nonlinear multi-agent systems

Abstract

The application discloses a data-driven nonlinear multi-agent system online cooperative control method, relates to the technical field of multi-agent system cooperative control, considers an affine nonlinear multi-agent system with external disturbance and unknown system model under a non-directional connected graph, and realizes consistent terminal value bounded control of the system. The scheme comprises the following steps: considering a discrete time affine nonlinear multi-agent system with external disturbance and unknown system model, a distributed state feedback controller is designed; based on an online updating data collection mechanism, state data and input data of the system are collected, and a data-based system parameterized representation is constructed; a state-dependent model is constructed, a data-based distributed robust controller is solved online, and consistent terminal value bounded control of the unknown model affine nonlinear multi-agent system is realized.

Description

Technical Field

[0001] This invention relates to the field of cooperative control technology for multi-agent systems, and more specifically to a data-driven online cooperative control method for nonlinear multi-agent systems. Background Technology

[0002] Multi-agent systems (MASs) consist of multiple simple, individual agents with certain perception, computation, communication, and execution capabilities. These agents collaborate through information sharing and coordination to perform complex tasks that a single agent cannot accomplish. As a fundamental problem in the field of MASs, cooperative control of MASs has broad application prospects in smart grids, smart manufacturing, and drone swarming, making it a current research hotspot. Taking smart manufacturing as an example, intelligent collaborative production and flexible discrete manufacturing processes can be achieved through mutual cooperation between humans, machines, and materials on the production line.

[0003] In real-world environments, low measurement accuracy and poor environmental awareness are common problems. This inevitably makes the modeling process susceptible to various factors such as strong nonlinearity, uncertainty, and interference, leading to difficulties in obtaining accurate mathematical models of the system. Therefore, traditional control methods based on accurate models are no longer sufficient for the analysis and design of complex controls. How to achieve cooperative control of multi-agent systems in the face of nonlinearity and unknowns has become an urgent problem to be solved.

[0004] In recent years, inspired by the fundamental lemma proposed by Willems et al., direct data-driven control methods have gained widespread attention by providing an end-to-end paradigm for controlling and analyzing unknown systems using only data, thus addressing the problem of unknown system models. Compared with traditional model-based control methods, this approach has certain advantages, such as avoiding the challenges posed by model inaccuracies and high computational costs associated with system identification. With the rapid development of this field, data-driven methods have been extensively studied, ranging from state feedback control to model predictive control. A major challenge in data-driven control is constructing a data-based parameterized representation of the system from disturbed data, as disturbances can disrupt the sustained excitation conditions required by the fundamental lemma. The S-lemma, once proposed, has become a general framework for handling disturbances and has been widely applied in data-driven control methods. However, most existing related studies assume that disturbances only exist during offline data collection, while they do not exist during online closed-loop control. On the other hand, existing control strategies are mainly limited to linear systems, and the design and analysis of control strategies for nonlinear systems remain challenging. Although some scholars have conducted research on data-driven control methods based on sum-of-squares optimization for polynomial systems, this method is computationally expensive and limited to rational nonlinearity. Meanwhile, the aforementioned studies mainly focus on centralized settings for single systems, and there is currently no in-depth exploration of distributed cooperative control of multi-agent systems simultaneously subjected to offline and online disturbances. Therefore, achieving cooperative control of nonlinear multi-agent systems with unknown models using only data under unknown external disturbances while ensuring good robustness has become an urgent problem to be solved. Summary of the Invention

[0005] In view of this, the present invention provides a data-driven online cooperative control method for nonlinear multi-agent systems, which considers an affine nonlinear multi-agent system with external disturbances and an unknown system model under an undirected connected graph, and achieves consistent final value bounded control of the system.

[0006] To achieve the above objectives, the technical solution of the present invention includes the following steps:

[0007] S100, Considering a discrete-time affine nonlinear multi-agent system with external disturbances and an unknown system model, design a distributed state feedback controller.

[0008] S200, based on an online updated data collection mechanism, collects system status data and input data, and constructs a data-based parameterized representation of the system.

[0009] S300 constructs a state-dependent model and uses a data-driven distributed robust controller for online solution, achieving consistent final value bounded control of an affine nonlinear multi-agent system with an unknown model.

[0010] Furthermore, for a discrete-time affine nonlinear multi-agent system S100 with external disturbances and an unknown system model, containing N agents, the dynamic model of each agent is as follows:

[0011] x i (t+1)=f(x i (t))+g(x i (t))u i (t)+w i (t)

[0012] Where t is time, i is the i-th agent, i = 1, 2, ..., N, x i (t) and u i (t) represents the state and control input of agent i, respectively, and f and g represent vector fields whose functions are unknown. and and They are the sets of n-dimensional and n×m-dimensional real numbers, respectively; w i (t) is a bounded external disturbance;

[0013] In a discrete-time affine nonlinear multi-agent system, the connectivity between agents is represented by an undirected connected graph. To describe;

[0014] Furthermore, the S100 distributed state feedback controller specifically includes:

[0015] u i (t)=K i (t)z i (t)

[0016]

[0017] Among them, u i (t) represents the distributed state feedback controller. Let z be the feedback gain matrix to be designed. i (t) represents the combined measurement variables of the i-th agent at time t; a ij It is a connected graph of a discrete-time affine nonlinear multi-agent system. adjacency matrix The element in the i-th row and j-th column; d i It is a scalar that satisfies d when i = 1, 2, ..., q i When d > 0, i = q + 1, 2, ..., N i =0, where q is any natural number between 1 and N.

[0018] Furthermore, the S200 is based on an online data collection mechanism, and its design steps are as follows:

[0019] S211, the affine nonlinear multi-agent system is represented as:

[0020] x i (y+1)=ΛF(x i (t))+ΘG(x i (t))u i (t)+w i (t)

[0021] Where Λ and Θ are unknown coefficients, and F and G are known basis functions; w i (t) is a bounded external disturbance; u i (t) represents a distributed state feedback controller; x i (t) represents the state of agent i;

[0022] S212, based on an online data collection mechanism, is designed as follows:

[0023] First, a buffer of size T is defined on the controller side; at each time t, the buffer records the latest input and state samples of each agent i with a step size of T, which are collected into the first to third data matrices X respectively. i,t+ X i,t- U i,t- Specifically:

[0024] X i,t+ :=[x i (t-T+1)…x i (t)],

[0025] X i,t- :=[F(x i (tT))…F(x i (t-1))],

[0026] U i,t- :=[G(x i (tT))u i (tT)…G(x i (t-1))u i (t-1)].

[0027] Specifically, when t∈[0,T-1], the initial data was obtained through offline experiments. Replace t with T in the data matrix above:

[0028] X i,t+ :=[x i (1)…x i (T)],

[0029] X i,t- :=[F(x i (0))…F(x i (T-1))],

[0030] U i,t- :=[G(x i (0))u i (0)…G(x i (T-1))u i (T-1)].

[0031] When t≥T, the data matrix is ​​updated at each time step, that is, the buffer window moves forward one step, the oldest sample is deleted, that is, the first column of the data matrix, and the latest sample is added to the buffer.

[0032] Furthermore, S200 constructs a data-based parameterized representation of the system, specifically as follows:

[0033]

[0034] Where, Σ i Let Λ be the set of system parameters for agent i; Θ and Λ are unknown coefficients; Ω i For matrices related to the data matrix, W i,t- For an unknown interference matrix, Q d R d S d All are known matrices, satisfying Q d <0, < is the less than sign for matrices, and ≥ is the greater than or equal to sign for matrices.

[0035] Furthermore, the S300 state dependency model is specifically as follows:

[0036] x i (t+1)=ΛA(x i (t))x i (t)+ΘB(x i (t))u i (t)+w i (t).

[0037] Where, F(x) i (t))=A(x i (t))x i (t), G(x) i (t))=B(x i (t)).

[0038] Furthermore, the S300 constructs a state-dependent model, which is solved online based on a data-driven distributed robust controller, specifically as follows:

[0039] Consider the connectivity graph of discrete-time affine nonlinear multi-agent systems and discrete-time affine nonlinear multi-agent systems with external perturbations and unknown system models; for all [Λ,Θ]∈Σ i For i = 1, 2, ..., N, if there exists a matrix Q > 0 and R > 0, H i (t) and L i (t), and parameters τ≥0 and η>0, satisfy the following linear matrix inequality:

[0040] min trace(QP i (t))+trace(RL i (t))

[0041]

[0042]

[0043] Among them, A is used i,t Replace A(x) i (t)), B i,t Replace B(x) i (t)), It is the Laplace matrix Eigenvalues, where I is the identity matrix, R, P i (t), H i (t) and L i (t) is a real matrix; Q d R d S d All are known matrices, satisfying Q d <0, V i (t) is a Lyapunov function;

[0044] For any initial state, the multi-agent system achieves consistent final value bounded control under the action of a distributed state feedback controller.

[0045] In addition, the feedback gain matrix is ​​designed as follows Laplace matrix The largest eigenvalue.

[0046] Beneficial effects:

[0047] This invention provides a data-driven online cooperative control method for nonlinear multi-agent systems, solving the cooperative control problem of discrete-time affine nonlinear multi-agent systems with external disturbances and unknown system models under undirected communication topologies. First, a state-dependent model is constructed, and a distributed state feedback controller is designed. Further, an online-updated data collection mechanism is proposed, utilizing collected state and input data to construct a data-based parameterized representation of the system. Finally, a data-based online solution method for distributed robust controllers is proposed, achieving consistent final value bounded control of affine nonlinear multi-agent systems with unknown models. In summary, the main contributions of this invention are: First, considering the complex situation of unknown external disturbances during both the data collection phase and closed-loop operation, an innovative online data collection mechanism based on online updates and an online design method for distributed robust controllers are proposed. Second, this invention is not limited to rational nonlinearity and simplifies computational complexity by establishing low-dimensional linear matrix inequalities, making it comparable to the linear case and more suitable for real-time operating systems, with a wide range of application scenarios. Finally, this invention eliminates the dependence of traditional control on the system model, achieving for the first time consistent final value bounded control of the system using only data, with good robustness. Attached Figure Description

[0048] Figure 1 This is a flowchart illustrating a data-driven online collaborative control method for a nonlinear multi-agent system provided by the present invention. Detailed Implementation

[0049] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0050] like Figure 1 The diagram shown illustrates the process of this invention, a data-driven online cooperative control method for a nonlinear multi-agent system, comprising the following steps:

[0051] S100. Consider a discrete-time affine nonlinear multi-agent system with external disturbances and an unknown system model. Design a distributed state feedback controller.

[0052] In one embodiment, consider a discrete-time affine nonlinear multi-agent system with N agents, where the dynamics model of each agent is:

[0053] x i (t+1)=f(x i (t))+g(x i (t))u i (t)+w i (t)

[0054] Where t is time. Let i be the set of non-negative integers, i = 1, 2, ..., N. and These represent the state and control input of agent i, respectively. Let be the set of n-dimensional real numbers. and Represents a vector field. The functions of f and g are unknown. w i (t) represents a bounded external perturbation. The connectivity between agents in the system can be represented by an undirected connected graph. To describe.

[0055] Furthermore, the distributed state feedback controller in step S100 is specifically designed as follows:

[0056] u i (t)=K i (t)z i (t)

[0057]

[0058] in, The feedback gain matrix to be designed, A is an m×n dimensional real matrix; ij It is the adjacency matrix of the system's connected graph. The element in the i-th row and j-th column; d i It is a scalar that satisfies d when i = 1, 2, ..., q i When d > 0, i = q + 1, 2, ..., N i =0, where q is any natural number between 1 and N. z i (t) represents the combined measurement variable of the i-th agent at time t; j represents the j-th agent.

[0059] S200 proposes a data collection mechanism based on online updates to collect system state data and input data, and construct a data-based parameterized representation of the system.

[0060] In one embodiment, step S200, based on an online data collection mechanism, is designed as follows:

[0061] S211, rewriting the nonlinear multi-agent system as follows:

[0062] x i (t+1)=ΛF(x i (t))+ΘG(x i (t))u i (t)+w i (t),

[0063] in, and It is an unknown coefficient. For n×n g n-dimensional real matrix g It is a positive integer. For n×n f n-dimensional real matrix f Let F be a positive integer, and let F and G be known basis functions.

[0064] S212, based on an online data collection mechanism, is designed as follows:

[0065] First, define a controller with a size of... The buffer, where T is the buffer length. It is a positive integer. At each time step... The set of non-negative integers is used. The buffer records the latest T-step input and state samples of each agent i and collects them in the first to third data matrices X. i,t+ X i,t- U i,t- In the middle, as shown below:

[0066] X i,t+ :=[x i (t-T+1)…x i (t)],

[0067] X i,t- :=[F(x i (tT))…F(x i (t-1))],

[0068] U i,t- :=[G(x i (tT))u i (tT)…G(x i (t-1))u i (t-1)].

[0069] Specifically, when t∈[0,T-1], the initial data was obtained through offline experiments. Replace t with T in the data matrix above:

[0070] X i,t+ :=[x i (1)…x i (T)],

[0071] X i,t- :=[F(x i (0))…F(x i (T-1))],

[0072] U i,t- :=[G(x i (0))u i(0)…G(x i (T-1))u i (T-1)].

[0073] When t≥T, the data matrix is ​​updated at each time step, that is, the buffer window moves forward one step, the oldest sample (the first column of the data matrix) is deleted, and the latest sample is added to the buffer.

[0074] Furthermore, the system parameterization representation of S200 based on data is constructed as follows:

[0075]

[0076] Where, Σ i Let i be the set of system parameters for agent i. Q d <0, and S d Let X be a known matrix. i,t- X i,t+ For the data matrix, W i,t- For the interference matrix, It is R d The transpose matrix, Q d R d S d The selection should be based on the actual scenario and meet Q. d <0, That's it. < is the less-than sign for matrices, and ≥ is the greater-than-equal-than sign for matrices.

[0077] Specifically, the steps for constructing a data-based system parameterized representation are as follows:

[0078] S221, for external interference, w is defined. i (t), corresponding to T input and state samples, and the interference matrix W is defined. i,t- :=[w i (tT)…w i (t-1)], and by constructing the following noise model, the unknown interference W i,t- The upper bound is restricted, specifically as follows:

[0079]

[0080] Among them, Q d <0, and S d Given a matrix, Let be the noise set of agent i.

[0081] S222, Define data (X) that can explain agent i. i,t+ ,X i,t- U i,t-W i,t- The set containing all coefficient matrices Λ and Θ of ) is

[0082] Σ i ={[Λ,Θ]|X i,t+ =ΛX i,t- +ΘU i,t- +W i,t-}

[0083] S223, based on the above steps S221 and S222, the data-based system parameterization representation can be obtained as follows:

[0084]

[0085] Where, Σ i Let Λ be the set of system parameters for agent i; Θ and Λ are unknown coefficients; Ω i For matrices related to the data matrix, W i,t- For an unknown interference matrix, Q d R d S d All are known matrices, satisfying Q d <0, < is the less than sign for matrices, and ≥ is the greater than or equal to sign for matrices.

[0086] S300 constructs a state-dependent model and proposes a data-based online solution method for distributed robust controllers, enabling consistent final value bounded control of affine nonlinear multi-agent systems with unknown models.

[0087] In one embodiment, the state dependency model described in step S300 is specifically as follows:

[0088] x i (t+1)=ΛA(x i (t))x i (t)+ΘB(x i (t))u i (t)+w i (t).

[0089] Where, F(x) i (t))=A(x i (t))x i (t), G(x) i (t))=B(x i (t)).

[0090] Furthermore, step S300, a data-based online solution method for distributed robust controllers, is designed as follows:

[0091] Consider topology graph And discrete-time affine nonlinear multi-agent systems with external perturbations and unknown system models. For all [Λ,Θ]∈Σ i For i = 1, 2, ..., N, if there exist matrices Q > 0, R > 0, P i (t)=P i (t) T >0, H i (t) and L i (t), Q, R, P i (t), H i (t) and L i Let (t) be a real matrix of appropriate dimension, with parameters τ≥0 and η>0, satisfying the following linear matrix inequalities:

[0092] min trace(QP i (t))+trace(RL i (t))

[0093]

[0094]

[0095]

[0096]

[0097] For simplicity, A is used. i,t Replace A(x) i (t)), B i,t Replace B(x) i (t)), It is the Laplace matrix The eigenvalues ​​of , where I is the identity matrix.

[0098] Therefore, for any initial state, under the action of the distributed state feedback controller in step S100, the system can achieve consistent final value bounded control. Furthermore, the feedback gain matrix is ​​designed as follows: Laplace matrix The largest eigenvalue.

[0099] Specifically, the design steps of the data-based distributed robust controller online solution method are as follows:

[0100] S311, define the consistency error of agent i as: Let i be the number of the neighbors of agent i. Let be the set of neighbors of agent i, and j be agent j. The consensus problem of a nonlinear multi-agent system is transformed into a stability problem of consensus error. A Lyapunov function is designed as follows:

[0101] S312, The model-based controller can be obtained by solving the following linear quadratic regulation problem:

[0102] min trace(QP i (t))+trace(RL i (t))

[0103]

[0104]

[0105] V i (t-1)-V i (t)>0

[0106] in, and It is a positive definite symmetric matrix; It is the Laplace matrix The eigenvalues; trace represents the trace of the matrix.

[0107] S313, combining S311, S312, and the data-based system parameterization representation of step S200, using the S-lemma, the following data-based stability conditions can be obtained, including a data-based online solution method for distributed robust controllers, specifically:

[0108] Consider topology graph And discrete-time affine nonlinear multi-agent systems with external perturbations and unknown system models. For all [Λ,Θ]∈Σ i For i = 1, 2, ..., N, if there exists a matrix Q > 0 and R > 0, H i (t) and L i Given (t), and parameters τ≥0 and η>0, satisfying the following linear matrix inequality, then for any initial state, under the action of the distributed state feedback controller in step S100, the system can achieve consistent final value bounded control. The feedback gain matrix is ​​designed as follows:

[0109] min trace(QP i (t))+trace(RL i (t))

[0110]

[0111]

[0112]

[0113]

[0114] In one embodiment, simulation experiments are used to verify the feasibility and effectiveness of the online design method for data-driven controllers in this invention.

[0115] Consider a multi-inverted pendulum system consisting of four agents. The dynamic model of each inverted pendulum is as follows:

[0116]

[0117] Among them, T c It is a scalar, T c =0.1. Choose basis functions. and G(x) i (t))=1. Choose Q=I, R=I, V i (0) = 1000. For external disturbance w... i The previous one of (t), choose Q. d =-I, and S d =0. The initial state and input of each agent follow a normal distribution, and the perturbation follows a Gaussian distribution with a standard deviation of 0.

[0118] First, utilizing the online-updated data collection mechanism proposed in step S200, the first 20 time steps of the open-loop multi-inverted pendulum system are offline-excited, i.e., T = 20. Then, for t ∈ [20, 120], the controller gain of each agent is calculated at each time step by solving the linear matrix inequality in step S300, and applied to the multi-inverted pendulum system. Furthermore, to emphasize the effectiveness of the proposed method, the linear quadratic regulation problem in step S312 is solved, yielding a model-based controller at each time step. Under both the proposed controller and the model-based controller, the multi-inverted pendulum system achieves bounded consistency of final state values ​​and exhibits similar performance. Moreover, since the proposed method does not require system model information during implementation, it possesses certain advantages.

[0119] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A data-driven online cooperative control method for a nonlinear multi-agent system, characterized in that, Includes the following steps: S100, considering a discrete-time affine nonlinear multi-agent system with external disturbances and an unknown system model, design a distributed state feedback controller; which contains There are [number] intelligent agents, and the dynamic model of each agent is as follows: Where t represents time t, and i represents the i-th agent. , and Representing intelligent agents respectively Status and control inputs, and Let represent a vector field whose function is unknown, where and , and These are the sets of real numbers in n-dimensional and n×m-dimensional respectively; It is a bounded external disturbance; The connectivity between agents in the discrete-time affine nonlinear multi-agent system is represented by an undirected connected graph. To describe; The distributed state feedback controller is as follows: in, This refers to the distributed state feedback controller. The feedback gain matrix to be designed, For the first An intelligent agent in Combined measurement variables at time; It is the connected graph of the discrete-time affine nonlinear multi-agent system. adjacency matrix The Middle line, number Column elements; It is a scalar, satisfying hour , hour q is any natural number between 1 and N; S200, based on an online updated data collection mechanism, collects system status data and input data, and constructs a data-based parameterized representation of the system. S300 constructs a state-dependent model and uses a data-driven distributed robust controller for online solution, achieving consistent final value bounded control of an affine nonlinear multi-agent system with an unknown model. Specifically: The state dependency model is specifically as follows: in, , ; Consider the connectivity graph of the discrete-time affine nonlinear multi-agent system. and discrete-time affine nonlinear multi-agent systems with external disturbances and unknown system models; for all and If a matrix exists , R , , and and parameters and It satisfies the following linear matrix inequalities: s.t Among them, use replace , replace , It is the Laplace matrix eigenvalues, It is the identity matrix. R , , and It is a real matrix; , , All are known matrices, satisfying , ; It is a Lyapunov function; For any initial state, the multi-agent system achieves consistent final value bounded control under the action of the distributed state feedback controller. In addition, the feedback gain matrix is ​​designed as follows , Laplace matrix The largest eigenvalue.

2. The data-driven online cooperative control method for a nonlinear multi-agent system according to claim 1, wherein the data collection mechanism based on online updates described in S200 is designed with the following steps: S211, the affine nonlinear multi-agent system is represented as: in, , It is an unknown coefficient. and These are known basis functions; It is a bounded external disturbance; This refers to the distributed state feedback controller; Represents intelligent agents The state; S212, based on an online data collection mechanism, is designed as follows: First, define a controller with a size of... Buffer; at each time step The buffer records each agent. The latest The step size input and state samples are collected in the first to third data matrices, respectively. ， ， Specifically: Specifically, when At that time, the initial data was obtained through offline experiments, and the data matrix above was used to... Replace with : when At each time step, the data matrix is ​​updated, meaning the buffer window moves forward one step, the oldest sample (the first column of the data matrix) is removed, and the latest sample is added to the buffer.

3. The data-driven online cooperative control method for a nonlinear multi-agent system according to claim 1, wherein the construction of the data-based system parameterization representation in step S200 specifically comprises: in, For intelligent agents The system parameter set; , These are unknown coefficients; For matrices related to the data matrix, , For unknown interference matrix, , , All are known matrices, satisfying , ; It is a matrix less than sign. It is the matrix greater than or equal to sign.

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