Strict feedback nonlinear multi-body system sampling type distributed aggregation optimization method

Abstract

The invention discloses a sampling type distributed aggregation optimization method for a strict feedback nonlinear multi-body system, which comprises the following steps of: designing an aggregation regulation variable for integrating neighbor information in each sampling period, and updating the regulation variable in combination with an auxiliary sampling function, therefore, a distributed aggregation optimization problem on a complex nonlinear system with strict feedback dynamics is converted into a variable adjustment problem, and effective adjustment of aggregation variables can be realized by using a classical control method. And proposing a control law fusing a performance function and an aggregation regulation variable so as to cope with the distributed aggregation optimization problem with state time lag and time-varying directed graph communication topology. According to the method provided by the invention, the distributed aggregation optimal control of the complex multi-body system can be realized under the conditions of unknown dynamic state, state delay and directed balance time-varying topology.

Description

Technical Field

[0001] The present invention belongs to the field of multi-agent distributed optimization control, and specifically relates to a sampling-type distributed aggregation optimization method for a strict feedback nonlinear multi-body system. Background Art

[0002] With technological advancements and growing industrial demand, multi-agent systems have become a hot topic in research and application. These systems consist of multiple interactive agents with autonomous decision-making capabilities, working together to achieve overall optimization. Distributed optimization, as a core approach, is widely used to address problems such as resource allocation, task allocation, and data analysis. In such systems, each agent typically only has partial information and relies on communication networks for collaborative optimization to improve overall performance. This approach offers excellent scalability and robustness, significantly reducing reliance on centralized control while effectively alleviating communication pressure and improving computational efficiency. Consequently, distributed optimization has been widely used in scenarios such as power dispatching, coordinated control of unmanned systems, and network protection.

[0003] In multi-body systems, strict feedback nonlinear models are widely used to enhance system modeling capabilities in order to adapt to complex and changing dynamic environments. This type of model can deeply characterize the nonlinear interactions between intelligent agents, encompassing features such as adaptive regulation, learning behavior, and multi-level control strategies. By introducing a strict feedback dynamic structure, the system can more fully reflect the uncertainty and time-varying characteristics of the environment, thereby improving control accuracy and predictive performance. In addition, strict feedback nonlinear models have the ability to handle high-dimensional system states, adapting to more complex system requirements without sacrificing modeling accuracy. At the same time, this type of modeling approach also enhances the system's adaptability to external disturbances, allowing it to maintain operational efficiency and stability even in unstable or dynamic environments.

[0004] Most existing multi-agent distributed optimization methods assume that each agent's cost function depends solely on its own decision variables. However, in real-world applications such as resource competition, traffic scheduling, and load balancing, an agent's cost is not only affected by its own behavior but also coupled with the behaviors of other agents. This coupling is often manifested in some form of aggregation, such as the average or sum of all agent states, resulting in what is known as an aggregated optimization problem. Traditional methods struggle to effectively address such structures, especially in environments with dynamically changing communication topologies and resource constraints.

[0005] Existing patent CN 119472776 A proposes a method for pre-set time ratio consistency control of a multi-agent system. The consistency control problem studied in this patent is essentially different from the aggregation optimization problem studied by the method of the present invention. Its switching network topology and preset time convergence requirements are significantly different from the control technology used in this article. In addition, patent CN 116088531A discloses a finite time consistency control method for a multi-agent system under a directed continuous time-varying communication topology. The problem studied in this patent is also essentially different from the aggregation optimization problem of the present invention, and the research of this patent is only aimed at first-order continuous algorithms, while the present invention considers more complex high-order nonlinear strict feedback systems.

[0006] Existing patent CN118567226A proposes a high-order nonlinear multi-agent distributed optimization method based on proportional-integral regulation. However, the problem type addressed in this patent is the optimal consistency problem, and its objective function structure is fundamentally different from the aggregate optimization problem that this invention focuses on, and therefore cannot be directly applied to aggregate optimization scenarios. In addition, this method is based on a fixed communication topology, the communication connection is not time-varying, and its control strategy relies on continuous communication, without considering the system performance and implementation cost under limited communication resources. Therefore, the application of this solution in dynamic communication environments and resource-constrained systems has certain limitations.

[0007] To address these issues, this paper introduces aggregated control variables, transforming the aggregate optimization problem into a distributed variable control problem. Furthermore, a sampling-based information exchange mechanism is constructed within a time-varying directed graph communication structure. This significantly reduces communication frequency while ensuring the system's effective perception and coordination of global aggregate information and its gradients, thereby achieving an efficient and practical optimization control strategy. Therefore, the study of distributed aggregate optimization methods for nonlinear multi-body systems with strict feedback structures under a time-varying directed equilibrium communication topology has important practical significance and application value. Summary of the Invention

[0008] To address these issues, the present invention discloses a sampling-based distributed aggregation optimization method for strict feedback nonlinear multibody systems. This method proposes an aggregation control variable based on sampling technology and independent of communication topology information. This variable is used to reconstruct the original distributed aggregation optimization problem into an equivalent variable control problem. This aggregation control variable is then effectively regulated using classical control methods, making the proposed optimization method applicable to nonlinear multibody systems with strict feedback structures, thereby achieving distributed aggregation optimization control of complex systems.

[0009] To achieve the above objectives, the present invention provides the following technical solutions:

[0010] A sampling-based distributed aggregation optimization method for a strict feedback nonlinear multi-body system includes the following steps:

[0011] Step A: Define the dynamic model of the strict feedback nonlinear multibody system.

[0012] Preferably, the dynamic model of the strict feedback nonlinear multi-body system described in step A is as follows:

[0013]

[0014] in, Indicates the system status, The vector form of the state, captures the time delay effect of the system, where γ∈[-τ,0],τ>0. In addition, z i and u i Denote the measurable control output and input, respectively, and the function and ω ij Assume an unknown nonlinear function. Each agent has a local cost function d represents bounded interference. For all j=1,…,m.

[0015] Preferably, the auxiliary function q(t) used in the sampling technique is defined as follows:

[0016]

[0017] in represents the largest integer less than or equal to this number, T represents the sampling period; τ d ≥0 indicates the maximum allowable delay; q0=(2m+1)! / [(m!) 2 (T-τ d ) 2m+1 ] is the regularization constant.

[0018] The ultimate goal of the proposed algorithm is to make the system state satisfy the inequality:

[0019]

[0020] Where ε is a positive constant that can be adjusted to an arbitrarily small value; Ψ is a constant that is independent of ε; Indicates the system status; Indicates a problem The optimal solution of Represents aggregate information.

[0021] Step B: Model the directed time-varying balanced communication topology between agents as a set of time-varying directed graph sequences (jointly connected graphs), each of which consists of an agent set V = {1, 2, ..., N}, an edge set and the weighted adjacency matrix W k =[w ij,k ], the edge set describes the information transmission relationship between each agent; for any time, its communication structure is composed of the weighted adjacency matrix W k =[w ij,k ] represents that when agent j can send information to agent i at time k, w is satisfied. ij,k >0, otherwise w ij,k = 0. The weighted adjacency matrix is ​​used to describe the communication pattern and supports the construction of the product of communication weight matrices in multiple consecutive moments to characterize the time-varying nature of the communication topology. The weighted adjacency matrix W k Satisfy W k 1 N =1 N , and all non-zero elements have a lower bound of a positive number In addition, there exists a positive integer So that the continuous time steps, their joint communication graph remains strongly connected over this time window.

[0022] Step C: Propose an aggregate adjustment variable based on sampling technology and independent of communication topology information; reconstruct the distributed aggregate optimization problem into a variable adjustment problem through the aggregate adjustment variable; use a classical control method (preset performance control) to adjust the aggregate adjustment variable, so that the optimization method is applicable to nonlinear multi-body systems with strict feedback structures, thereby realizing aggregate optimization control under complex systems. Preferably, the aggregate adjustment variable based on sampling technology and independent of communication topology information proposed in step C includes:

[0023]

[0024] c i (t) represents the aggregated adjustment variable, where

[0025]

[0026] Among them, the parameters Determine the adjacency matrix W k The positive constant α represents the learning rate of the algorithm. The discrete form auxiliary function for the construction, the local estimator and The role of is to enable each agent to estimate the aggregate variable σ(z) and its gradient

[0027] Step D: In conjunction with Step C, and based on the convergence analysis of distributed preset performance control, a distributed control algorithm based on aggregated adjustment variables and a preset performance function was designed to ensure that all agents can collaboratively find the optimal aggregate state of the system under a time-varying equilibrium communication topology. Specifically, the aggregated adjustment variable is constructed using the neighbor information received by each agent during each sampling period, and this variable is adjusted to approximate the aggregated optimal solution. A preset performance function is also introduced to constrain the system's error evolution process, allowing each agent's state to achieve aggregated optimal control while meeting performance requirements.

[0028] Preferably, the control algorithm combining the aggregated adjustment variable and the preset performance function described in step D is described as follows:

[0029] First, we introduce the logarithmic transformation in

[0030] Secondly, design the virtual control input: δ ij (ρ ij )=-v ij λ(ρ ij ),j=1,…,m and actual control input u i (ρ im )=-v im λ(ρ im ). Parameter υ ij Indicates a positive control gain. ρ ij represents the standardized error variable, where the first error variable γ i1 (t) represents a positive performance function, c i (t) is the aggregation adjustment variable defined previously.

[0031] For the intermediate step, the standardized error variable is defined as ρ ij :=(x ij (t)-δ i,j-1 (ρ i,j-1 )) / k ij (t), where

[0032] Step E: The distributed control algorithm based on aggregated adjustment variables and preset performance functions is loaded into each intelligent agent through programming, and then reasonable variable initialization is performed, and finally the distributed operation and real-time response of the control algorithm are realized.

[0033] The beneficial effects of the present invention are:

[0034] Compared with existing nonlinear optimization strategies that rely on embedded reference trajectories or auxiliary systems, the present invention does not rely on global reference signals. It only constructs aggregate adjustment variables through sampling of neighbor information, thereby converting the aggregate optimization problem into a local variable adjustment problem, effectively reducing the implementation complexity.

[0035] The sampling technology introduced in the present invention not only avoids the need for continuous communication, but also allows a certain degree of communication delay, significantly reducing the communication burden of the system and improving the adaptability of the algorithm in actual resource-constrained environments.

[0036] Furthermore, the system targeted by this invention is a strict feedback nonlinear structure with state delays, which presents higher dynamic complexity and modeling challenges than traditional delay-free systems. By combining aggregated control variables and performance function constraints, this method achieves stable optimal control in complex systems while ensuring convergence.

[0037] At the same time, this method is applicable to time-varying directed balanced communication topologies, breaking through the traditional method's reliance on static undirected graphs. It has stronger topological adaptability and engineering versatility, and is suitable for deployment in practical application scenarios such as collaborative control of multi-agent systems and resource optimization. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 It is a time-varying directed communication topology network of the multi-body system in this embodiment.

[0039] Figure 2 is the agent state x i1 evolution curve.

[0040] Figure 3 is the agent state x i2 evolution curve.

[0041] Figure 4 is the aggregate adjustment variable ξ i (t) evolution curve. DETAILED DESCRIPTION

[0042] The present invention will be further described below with reference to the accompanying drawings and specific embodiments. It should be understood that the following embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the scope of protection of the present invention.

[0043] To achieve the above objectives, the present invention proposes a sampling-based distributed aggregation optimization method for a strict feedback nonlinear multi-body system, comprising the following steps:

[0044] Step A: Define the dynamic model of the strict feedback nonlinear multibody system.

[0045] This type of high-order nonlinear multi-body system with strict feedback contains a total of N agents, where the dynamic model of the i-th agent is:

[0046]

[0047] in, Indicates the system status, The vector form of the state, captures the time delay effect of the system, where γ∈[-τ,0],τ>0. In addition, z i and u i Denote the measurable control output and input, respectively, and the function and ω ij Assume an unknown nonlinear function. Each agent has a local cost function d represents bounded interference. For all j = 1,…,m.

[0048] The auxiliary function q(t) used in the sampling technique is defined as follows:

[0049]

[0050] in represents the largest integer less than or equal to this number, T represents the sampling period; τ d ≥0 indicates the maximum allowable delay; q0=(2m+1)! / [(m!) 2 (T-τ d ) 2m+1 ] is the regularization constant.

[0051] The ultimate goal of the proposed algorithm is to make the system state satisfy the inequality:

[0052]

[0053] Where ε is a positive constant that can be adjusted to an arbitrarily small value; Ψ is a constant that is independent of ε; Indicates the system status; Indicates a problem The optimal solution of Represents aggregate information.

[0054] Step B: Model the directed time-varying balanced communication topology between agents as a set of time-varying directed graph sequences, each of which consists of an agent set V = {1, 2, ..., N}, an edge set and the weighted adjacency matrix W k =[w ij,k ], the edge set describes the information transmission relationship between each agent; for any time, its communication structure is composed of the weighted adjacency matrix W k =[wij,k ] represents that when agent j can send information to agent i at time k, w is satisfied. ij,k >0, otherwise w ij,k = 0. Adjacency matrix W k Meet W k 1 N =1 N , and all non-zero elements have a lower bound of a positive number In addition, there exists a positive integer So that the continuous Within a time step, its joint communication graph remains strongly connected in that time window. Compared with a fixed communication topology, the time-varying directed graph structure can more realistically reflect the dynamically changing communication connections in the actual system, thereby improving the adaptability and practicality of the algorithm under complex network conditions.

[0055] Step C: A sampling-based aggregation control variable, independent of communication topology information, is proposed. This variable is used to reconstruct the original distributed aggregation optimization problem into an equivalent variable control problem. Classical control methods are then combined to effectively regulate this aggregation control variable, making the proposed optimization method applicable to nonlinear multibody systems with strict feedback structures, thereby achieving distributed aggregation optimization control of complex systems. It can be seen that the auxiliary function ψ(t) in this design plays a crucial role in linking discrete-time decision updates with continuous system evolution, significantly reducing the algorithm's communication burden. Furthermore, the function's design structure also indicates that it allows for a certain amount of communication delay in the system, which increases the robustness of the algorithm to a certain extent.

[0056] The proposed aggregate moderator variables based on sampling techniques are as follows:

[0057]

[0058] c i (t) represents the aggregated adjustment variable, where

[0059]

[0060] Among them, the parameters Determine the adjacency matrix W k The positive constant α represents the learning rate of the algorithm. The discrete form auxiliary function for the construction, the local estimator and The role of is to enable each agent to estimate the aggregate variable σ(z) and its gradient

[0061] Step D: Combined with step C, under the distributed preset performance control convergence analysis, a distributed control algorithm based on aggregated adjustment variables and preset performance functions is designed to ensure that under the time-varying equilibrium communication topology structure, all intelligent agents can collaboratively find the optimal aggregation state of the system.

[0062] Preferably, the control algorithm combining the aggregated adjustment variable and the preset performance function described in step D is described as follows:

[0063] First, we introduce the logarithmic transformation in

[0064] Secondly, design the virtual control input: δ ij (ρ ij )=-υ ij λ(ρ ij ),j=1,…,m and actual control input u i (ρ im )=-v im λ(ρ im ). Parameter v ij Indicates a positive control gain. ρ ij represents the standardized error variable, where the first error variable γ i1 (t) represents a positive performance function, c i (t) is the aggregation adjustment variable defined previously.

[0065] For the intermediate step, the standardized error variable is defined as ρ ij :=(x ij (t)-δ i,j-1 (ρ i,j-1 )) / k ij (t), where

[0066] Step E: The distributed control algorithm based on aggregated adjustment variables and preset performance functions is loaded into each intelligent agent through programming, and then reasonable variable initialization is performed, and finally the distributed operation and real-time response of the control algorithm are realized.

[0067] In this embodiment, a multi-stirred tank reactor system is used as an example to demonstrate the communication structure of the strict feedback multi-body system. Figure 1 As shown. The number of agents N = 5,

[0068] Among them, the dynamics of the i-th agent is:

[0069]

[0070] Among them, x i1 and xi2 Represents the concentration R C and R D The deviation from its equilibrium point, The nonlinearity representing the uncertainty of the system is ΥC i (t,x i1 (tD Ci ))=0.2x i1 (tD Ci )and The system state delay is D Ci =1.1+0.2(i-1),D Di =1+0.2(i-1). For each reactor, full state measurement is used, and the output of the i-th reactor is z i =x i1 The initial value in the aggregation optimization algorithm is set to σ i,0 =φ i (z i,0 ),

[0071] Preferably, the bounded interference is d i =0.09sin((1+i) / 2)t, the control gain is v i1 =6,v i2 =11. The preset performance function of exponential decay is designed as γ i1 (t) = 2.98e -0.1t +0.02,γ i2 (t)=5.97e -0.1t +0.03. The learning rate is α=0.006. The cost function of each agent is f i (z i ,σ(z))=e i ||z i -R i || 2 +||z i -σ(z)|| 2 , the importance trade-off parameter e i =75. The target concentration difference vector is [R1, R2, R3, R4, R5] = [0.5; -1.5; 1.1; -0.5; 2.5]. The aggregation function is selected as In addition, the sampling period is selected as T = 1s, and the maximum allowable delay is τ d =0.35s.

[0072] from Figure 2 and Figure 3 It can be seen that each intelligent reactor can converge to the target concentration difference. Figure 4It can be seen that the aggregation adjustment variable designed by the present invention is also successfully adjusted to near 0. This shows that the algorithm proposed by the present invention can solve the aggregation optimization problem of strict feedback nonlinear multi-body system.

[0073] It should be understood that the technical solutions disclosed in the present invention are not limited to the specific contents described in the above embodiments, but also include various variations and alternative solutions formed by combining the above technical features in any reasonable manner, which should all be regarded as the scope of protection of the present invention.

Claims

1. A sampling-based distributed aggregation optimization method for strict feedback nonlinear multi-body systems, characterized by: The following steps are involved: Step A: Define the dynamic model of the strict feedback nonlinear multi-body system; Step B: Model the directed time-varying balanced communication topology as a set of time-varying directed graphs, each of which consists of an agent set V = {1, 2, ..., N}, an edge set and the weighted adjacency matrix W k =[w ij,k ], where w ij,k >0 means that at time k, agent j sends information to agent i. If there is no information transmission, then w ij,k =0; the weighted adjacency matrix is ​​used to describe the communication pattern and supports the construction of the product of the communication weight matrices in multiple consecutive moments to characterize the time-varying nature of the communication topology; Step C: Propose an aggregate adjustment variable based on sampling technology and independent of communication topology information; The distributed aggregation optimization problem is restructured into a variable regulation problem by aggregating the regulation variables; The aggregate adjustment variable is adjusted using a preset performance control, so that the optimization method is applicable to a nonlinear multi-body system with a strict feedback structure, thereby realizing aggregate optimization control under a complex system; Step D: Design a distributed control algorithm based on aggregated control variables and preset performance functions to achieve aggregated optimal control of the multi-body system under performance constraints; Step E: The distributed control algorithm based on aggregated adjustment variables and preset performance functions is loaded into each intelligent agent through programming, and then reasonable variable initialization is performed, and finally the distributed operation and real-time response of the control algorithm are realized.

2. The method of sampling-based distributed aggregation optimization for a strict feedback nonlinear multi-body system according to claim 1, characterized in that: In step A: Define the dynamics model of a strict feedback nonlinear multibody system: in, Indicates the system status, The vector form of the state, captures the time delay effect of the system, where γ∈[-τ,0],τ>0; in addition, z i and u i Denote the measurable control output and input, respectively, and the function and ω ij Assume unknown nonlinear function; each agent has a local cost function d represents bounded interference, for all j = 1,…,m; The auxiliary function q(t) used in the sampling technique is defined as follows: in represents the largest integer less than or equal to this number, T represents the sampling period; τ d ≥0 indicates the maximum allowable delay; q0=(2m+1)! / [(m!) 2 (T-τ d ) 2m+1 ] is the canonical constant; The ultimate goal is to make the system state satisfy the inequality: Where ε is a positive constant that can be adjusted to an arbitrarily small value; Ψ is a constant that is independent of ε; Indicates the system status; Indicates a problem The optimal solution of Represents aggregate information.

3. The method of sampling-based distributed aggregation optimization for a strict feedback nonlinear multi-body system according to claim 2, characterized in that: In step B, the communication topology between agents is modeled as a set of time-varying directed graph sequences, each of which consists of an agent set V = {1, 2, ..., N}, an edge set and the weighted adjacency matrix W k =[w ij,k ], the edge set describes the information transmission relationship between each agent; for any time, its communication structure is composed of the weighted adjacency matrix W k =[w ij,k ] represents that when agent j can send information to agent i at time k, w is satisfied. ij,k >0, otherwise w ij,k =0; weighted adjacency matrix W k Satisfy W k 1 N =1 N , and all non-zero elements have a lower bound of a positive number In addition, there exists a positive integer So that the continuous time steps, their joint communication graph remains strongly connected over this time window.

4. The method of sampling-based distributed aggregation optimization for strict feedback nonlinear multi-body systems according to claim 3, characterized in that: In step C, an aggregate adjustment variable based on sampling technology and independent of communication topology information is proposed, including: c i (t) represents the aggregated adjustment variable, where Among them, the parameters Determine the adjacency matrix W k The elements in; the positive constant α represents the learning rate of the algorithm; The discrete form auxiliary function for the construction, the local estimator and The role of is to enable each agent to estimate the aggregate variable σ(z) and its gradient 5. The method of sampling-based distributed aggregation optimization for strict feedback nonlinear multi-body systems according to claim 4, characterized in that: In step D, a distributed control algorithm based on aggregated adjustment variables and preset performance functions is designed for the time-varying balanced communication topology. The aggregated adjustment variables are constructed using the neighbor information received by each agent in each sampling period. The aggregated adjustment variables are adjusted to approach the aggregated optimal solution. A preset performance function is introduced to constrain the error evolution process of the system, so that the states of each intelligent agent can achieve aggregated optimal control while meeting the performance indicators; specifically, it includes: First, we introduce the logarithmic transformation in Secondly, design the virtual control input: δ ij (ρ ij )=-v ij λ(ρ ij ),j=1,…,m and actual control input u i (ρ im )=-v im λ(ρ im ); parameter v ij represents a positive control gain; ρ ij represents the standardized error variable, where the first error variable γ i1 (t) represents a positive performance function, c i (t) is the aggregate adjustment variable defined previously; For the intermediate step, the standardized error variable is defined as ρ ij :=(x ij (t)-δ i,j-1 (ρ i,j-1 )) / k ij (t), where

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