A novel distributed proportional-integral temperature control method based on directed graph theory

Abstract

The application relates to a novel distributed proportional integral temperature control method based on a directed graph theory, which comprises theoretical modeling, construction of a mathematical model of a multi-agent network system under a heterogeneous disturbance condition, and then construction of a leader-following multi-agent system for a cluster system composed of several temperature control units; a distributed proportional integral controller containing a matrix type proportional integral gain is designed, controller parameters are designed, integral state feedback of each follower temperature control unit is constructed, control input of each follower temperature control unit is solved, the system state is updated, cyclic control of the temperature control unit is carried out, and then leader-following consistency of the temperature control unit is realized. The application can still ensure consistency under the condition of limited actual communication, asymmetric network topology, and existence of uneven heterogeneous disturbance environment, compared with a platform requiring complete controllability, the threshold of model and actuator completeness is reduced, the application object is wider, and the application is more suitable for large-scale deployment and industrialized scenes existing complex noise interference.

Description

Technical Field

[0001] This invention relates to the field of distributed cooperative control technology, and in particular to a novel distributed proportional-integral temperature control method based on directed graph theory. Background Technology

[0002] With the accelerated advancement of intelligent manufacturing and smart society, distributed control systems have shown broad application prospects in various fields such as industrial production, agricultural environmental regulation, building energy management, and smart services. Multi-agent consensus, as an important theoretical foundation for achieving coordinated regulation, has always been a research hotspot in the field of distributed control. Temperature control systems, as a typical distributed coordinated control object, have significant application value in various scenarios such as industrial heat treatment, greenhouse environmental regulation, and building HVAC systems. Compared to traditional single-point temperature control methods, distributed temperature control systems, through the coordinated regulation among multiple control units, can achieve higher environmental adaptability and regulatory robustness, and can complete tasks such as regional temperature balancing, coordinated regulation of local heat sources, and dynamic temperature tracking under constraints such as complex disturbances. In distributed control problems, consensus control is one of the fundamental and crucial core issues. Consistency refers to the gradual convergence of the states of each temperature control unit to a common target value under conditions relying only on local information interaction. In practical applications, the temperature of each region usually starts from a non-uniform initial state, updates its state through information exchange between adjacent regions, and gradually forms the desired coordinated temperature regulation mode under the action of the control algorithm to adapt to changes in the external environment and adjustments to task requirements.

[0003] Generally, distributed consensus algorithms for multi-agent systems are mostly implemented under the assumption of an undirected graph context. This implies that information flow within the system is symmetrical, and symmetrical information exchange requires high communication bandwidth and incurs high costs, which is impractical in general industry. Furthermore, in reality, information flow is generally directed. Therefore, undirected graphs can be considered a special case of directed graphs, and each undirected edge in an undirected graph can be decomposed into two directed edges pointing in opposite directions to describe its characteristics.

[0004] In real-world systems, not all systems are stable. A stabilized linear system is one where the system may not be fully controllable, but all unstable modes that lead to divergent responses can be controlled through inputs, and feedback can help the closed-loop system reach asymptotic stability. Distributed proportional (P) control is a common type of consensus control. While this method is simple in structure and easy to implement, and consensus convergence can be proven under certain connectivity conditions, it has limitations in steady-state accuracy. When the system is subject to disturbances, model uncertainties, external bias inputs, or long-term accumulated errors, the proportional term alone often fails to further improve steady-state accuracy, potentially leading to problems such as difficulty in eliminating steady-state errors and limited convergence accuracy. In classical control, integral action can accumulate long-term errors, significantly reducing or eliminating steady-state errors. Applying this idea to multi-agent consensus problems yields a distributed proportional-integral (PI) controller, which promises better control performance. However, under the directed communication topology assumption, no relevant technical solutions have yet been proposed. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a novel distributed proportional-integral (PI) temperature control method based on directed graph theory. This method solves the challenge of distributed control of temperature control systems under heterogeneous disturbances on a directed graph and proposes a novel distributed PI controller. Unlike the scalar gain widely used in existing controllers, the proposed controller constructs a matrix-type PI gain based on Riccati-stabilized gain and, based on directed graph theory, transforms the consistency problem into a stability problem of divergent dynamics.

[0006] To achieve the above technical objectives, the present invention provides the following technical solution: a novel distributed proportional-integral temperature control method based on directed graph theory, comprising the following steps: The cluster system consisting of several temperature control units is modeled as a multi-agent network system. The temperature control unit is divided into one leader temperature control unit and several follower temperature control units, and the multi-agent network system is constructed as a leader-follower multi-agent system. In the leader-follower multi-agent system, the communication graph between the temperature control units is a directed graph, and the directed graph contains at least one directed support tree; the directed support tree has the leader temperature control unit as the root node; Based on the communication diagram between each temperature control unit, a distributed proportional-integral controller with matrix proportional-integral gain is designed for each follower temperature control unit. Design the parameters of a distributed proportional-integral controller; Communication between temperature control units is achieved based on a directed support tree, and the integral status feedback of each follower temperature control unit is updated. The proportional and integral errors between the system states of each temperature control unit are calculated. The control input of each follower temperature control unit is solved using a distributed proportional-integral controller, and then the system state of each temperature control unit is updated to achieve leader-follower consistency of the temperature control units.

[0007] Optionally, in the multi-agent network system, the linear dynamic equations of each temperature control unit are defined as follows: ; in, Indicates the first Each temperature control unit at time The system status, including the temperature and humidity of the temperature control unit. for About time The first derivative describes Changes over time; Indicates the first Each temperature control unit at time The control input to be designed; Indicates the first Heterogeneous unknown constant disturbance on the actuator side of each temperature control unit; The system matrix for the temperature control unit; This is the input matrix for the temperature control unit; Number the temperature control unit. This represents the total number of temperature control units.

[0008] Optionally, dividing the temperature control unit into a leader temperature control unit and several follower temperature control units, and constructing the multi-agent network system as a leader-follower multi-agent system, includes: The temperature control unit numbered 1 is designated as the leader temperature control unit, and the remaining temperature control units are designated as follower temperature control units. A leader-follower multi-agent system is constructed by combining the aforementioned multi-agent network system. The linear dynamic equation of the leader temperature control unit in the leader-follower multi-agent system is defined as follows: ; in, This indicates the external setting input for the leader's temperature control unit; The linear dynamic equation of the follower temperature control unit in the leader-follower multi-agent system is defined as follows: , .

[0009] Optionally, the leader-follower multi-agent system includes a system matrix for its temperature control unit. With input matrix The system dynamics matrix consists of pairs It is sedative and there exists a feedback matrix. This makes the closed-loop matrix The real parts of all eigenvalues ​​are less than 0.

[0010] Optionally, the aforementioned design of a distributed proportional-integral controller with matrix proportional-integral gain for each follower temperature control unit, based on the communication diagram between each temperature control unit, includes: The mathematical representation of the distributed proportional-integral controller of each follower temperature control unit is as follows: ; in, , These represent the proportional feedback gain matrix and the integral feedback gain matrix, respectively. The adjacency matrix of the communication graph between each temperature control unit is the first... Line 1 Column element, indicating the first Can the temperature control unit send to the first...? Each temperature control unit transmits information. Number the temperature control unit. This indicates that the current temperature control unit is the leader temperature control unit. This indicates that the current temperature control unit is the follower temperature control unit; This indicates that for the expression From 0 Within range Integral Let be the integral variable, and its physical meaning is time.

[0011] Optionally, the proportional feedback gain matrix satisfies the following condition: ; in, Represents the positive scalar adjustment coefficient of the proportional feedback gain matrix; Indicates the transpose operation; Let be any symmetric positive definite matrix, denoted as , express The inverse matrix; The Laplace matrix represents the communication diagram between the various temperature control units; This indicates the operation of taking the real part; Expressing the request The second smallest eigenvalue; Let be any symmetric positive definite matrix, and satisfy... It is an algebraic Riccati equation The only solution Let be any symmetric positive definite matrix, denoted as ; The integral feedback gain matrix satisfies the following condition: ; in, This represents the positive scalar adjustment coefficient of the integral feedback gain matrix.

[0012] Optionally, the parameters for designing the distributed proportional-integral controller include: According to the conditions Select ; Select matrix , According to the equation Solve ; Deriving the proportional feedback gain matrix With integral feedback gain matrix .

[0013] Optionally, the directed support tree, its Laplace difference matrix, is defined as follows: ; in, Represents the Laplace difference matrix The Middle Line 1 Column elements; In a directed support tree, the first... The set of nodes in the subtree rooted at node n. This is the node index in the node set; In a directed support tree, the first... The node index of the unique parent node of each node; , The Laplace matrix representing the communication diagram between each temperature control unit. The Line 1 Column element, first Line 1 Column elements; The Laplace difference matrix is ​​an asymmetric square matrix, and its symmetric part is calculated as follows: ; in, Represents the Laplace difference matrix The symmetrical part, and It is a symmetric positive definite matrix; This indicates the transpose operation.

[0014] The present invention also provides an electronic device comprising: at least one processor; and a memory communicatively connected to the at least one processor; wherein the memory stores a computer program executable by the at least one processor, the computer program being executed by the at least one processor to enable the at least one processor to execute the novel distributed proportional-integral temperature control method based on directed graph theory.

[0015] The present invention also provides a computer-readable storage medium storing computer instructions for causing a processor to execute the novel distributed proportional-integral temperature control method based on directed graph theory.

[0016] By employing the above technical solution, the present invention provides a novel distributed proportional-integral temperature control method based on directed graph theory, which has at least the following beneficial effects: This invention focuses on solving the challenge of distributed proportional-integral (PI) consistency control in temperature control systems implemented on directed graphs. The designed PI controller can still ensure leader-follower consistency even in environments with limited communication, asymmetric network topology, and non-uniform heterogeneous disturbances. Compared with platforms requiring complete controllability, it lowers the threshold for model and actuator completeness, has a wider range of applications, and is more suitable for large-scale deployment and industrial scenarios with complex noise interference. At the same time, due to the addition of the integrator, the proposed controller has higher robustness than traditional distributed proportional controllers, providing an efficient solution for achieving distributed asymptotic consistency control of industrial temperature control systems. Attached Figure Description

[0017] The accompanying drawings, which are included to provide a further understanding of this application and form part of this application, illustrate exemplary embodiments and are used to explain this application, but do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a flowchart of a novel distributed proportional-integral temperature control method based on directed graph theory according to the present invention. Figure 2 This is a flowchart of the control algorithm in an embodiment of the present invention; Figure 3 This is a schematic diagram of the leader-follower agent information topology in an embodiment of the present invention; Figure 4 This is a comparison diagram of the temperature status of each temperature control unit and the trajectory of the set temperature over time under the control of the winter temperature control system in scenario one of the embodiments of the present invention, under the control of the distributed proportional controller and the distributed proportional integral controller. Figure 5This is a comparison diagram of the temperature status of each temperature control unit and the trajectory of the set temperature over time under the control of the summer temperature control system in scenario one of the present invention. Figure 6 This is a comparison of the temperature status of each temperature control unit with the set temperature over time in scenario two of this invention, under the distributed proportional controller. Figure 7 This is a comparison of the temperature status of each temperature control unit with the set temperature over time in scenario two of the embodiments of the present invention, under the distributed proportional-integral controller. Detailed Implementation

[0018] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. This will allow for a full understanding of how the present application uses technical means to solve technical problems and achieve technical effects, and to facilitate its implementation.

[0019] Those skilled in the art will understand that all or part of the steps in the implementation of the methods of the embodiments can be implemented by a program instructing related hardware. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Moreover, this application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0020] Please refer to Figures 1-7 This embodiment illustrates a specific implementation of the present invention. Through theoretical modeling, a mathematical model of a multi-agent network system under heterogeneous disturbance conditions is constructed. The cluster system composed of several temperature control units is then constructed as a leader-follower multi-agent system. A distributed proportional-integral controller with matrix proportional-integral gain is designed, and controller parameters are designed to construct the integral state feedback of each follower temperature control unit. The control input of each follower temperature control unit is calculated, the system state is updated, and cyclic control of the follower temperature control units is performed to achieve leader-follower consistency of the temperature control units. Simultaneously, this embodiment provides a disturbance rejection consistency verification of the matrix proportional-integral gain, analyzes the feasibility of the control method of the present invention under a leader-follower network topology with heterogeneous disturbances, proves that the control method of the present invention can still achieve asymptotic consistency under a leader-follower network topology with heterogeneous disturbances, and verifies the effectiveness and reliability of the method in practical applications using simulation experiments.

[0021] Please refer to Figure 1This embodiment proposes a novel distributed proportional-integral temperature control method based on directed graph theory, which includes the following steps: S1. Model the cluster system consisting of several temperature control units as a multi-agent network system.

[0022] Consider by A cluster system model consisting of temperature control units is presented. For ease of theoretical analysis, this actual system model is modeled as a multi-agent network system, and its distributed anti-heterogeneous disturbance consistency control problem is studied based on this model. Generally, the influence of heterogeneous constant disturbances is completely eliminated by introducing an integral term. As a preferred implementation of step S1, it specifically includes: For higher-order linear multi-agent systems (i.e., first-order or second-order), assuming the disturbance is matched with the control input channel, the linear dynamic equations of each temperature control unit in the multi-agent network system are defined as follows: ; in, Indicates the first Each temperature control unit at time The system status, including the temperature and humidity of the temperature control unit. Represents the space of real numbers. Dimensions representing system state for About time The first derivative describes Changes over time; Indicates the first Each temperature control unit at time The control input to be designed, Indicates the dimension controlling the input; Indicates the first Heterogeneous unknown constant disturbance on the actuator side of each temperature control unit; The system matrix for the temperature control unit; This is the input matrix for the temperature control unit; Number the temperature control unit. This represents the total number of temperature control units.

[0023] S2. Divide the temperature control unit into one leader temperature control unit and several follower temperature control units, and construct the multi-agent network system as a leader-follower multi-agent system.

[0024] In temperature control systems, the various temperature control units are not always in a completely equal position within the control structure. In practical industrial scenarios, a leader-follower control architecture is often adopted. Some temperature control units act as leaders (reference nodes or master nodes), responsible for providing the global temperature setpoint or adjustment benchmark, while the remaining temperature control units act as followers, achieving state coordination and collaborative adjustment through information exchange with the leaders and neighboring followers. This structure retains the advantages of distributed control while introducing a unified control objective and scheduling mechanism, thereby improving the system's controllability, stability, and operational efficiency in complex industrial environments.

[0025] As a preferred embodiment of step S2, it specifically includes: Temperature control unit numbered 1 is designated as the leader temperature control unit, and the remaining temperature control units are designated as follower temperature control units. The current temperature control unit is the leader temperature control unit. The current temperature control unit is a follower temperature control unit. A leader-follower multi-agent system is constructed by combining the aforementioned multi-agent network system. The linear dynamic equations of each temperature control unit in the leader-follower multi-agent system are defined as follows: ; ; in, This indicates that the external setting input of the leader's temperature control unit is not available to the follower's temperature control unit. The information relies solely on the coordinated operation of the distributed proportional-integral controller designed subsequently, with the control objective being: ; in, The system status of the leader's temperature control unit. Indicates time When the expression approaches infinity, take the limit. This indicates the calculation of the Euclidean norm.

[0026] For a leader-follower multi-agent system, the communication graph between agents (i.e., the temperature control unit) is a directed graph, denoted as . ;in This represents the set of nodes in the communication diagram between temperature control units, that is, the set of all temperature control units. The actual physical meaning of each node is the system state of the temperature control unit. ; This represents the set of edges in the communication graph between the various temperature control units. Indicates from the first The node (temperature control unit) points to the first The edges of each node (temperature control unit), Number the temperature control unit. This indicates that the current temperature control unit is the leader temperature control unit. This indicates that the current temperature control unit is the follower temperature control unit; This is the adjacency matrix of the communication diagram between the temperature control units. Representing the adjacency matrix The Middle Line 1 Column elements, if edge ,but ( (represents the space of positive real numbers), otherwise If the side Then the first The node is the _th The incoming neighbors of each node. The Laplace matrix of a communication graph (directed graph). Its elements , .

[0027] For a leader-follower multi-agent system, consider the following assumptions: Assumption 1, Assumption 2, and Assumption 3: Assumption 1: The system dynamics matrix pairs It can be calmed down.

[0028] Under Assumption 1, there exists a feedback matrix. This makes the closed-loop matrix It is Hurwitz stable (i.e.) The real part of all eigenvalues ​​is less than 0.

[0029] Assumption 2: Communication Graph (Directed Graph) It contains a directed support tree (DST).

[0030] A directed support tree is an important concept in graph theory. It is a subgraph without cycles, consisting of all the vertices and some edges of a directed graph. In this subgraph, there is a node with no incoming neighbors, which is the root node of the directed support tree; all other nodes have one and only one incoming neighbor.

[0031] Assumption 2 is general and is a necessary condition for achieving consensus among multiple agents. (Choose any one...) A directed support tree (DST) is denoted as ,in , , These represent directed support trees. Given the set of nodes, the set of edges, and the adjacency matrix. Without loss of generality, number the root node as 1, and let... In a directed support tree, the first... The node index of the unique parent node of each node. , This indicates that the range of values ​​is [ A set of positive integer indices. Based on a directed support tree. Define matrix for: .

[0032] Further define the Laplace difference matrix as follows: ; in, Represents the Laplace difference matrix The Middle Line 1 Column elements; In a directed support tree, the first... The set of nodes in the subtree rooted at node n. This is the node index in the node set; In a directed support tree, the first... The node index of the unique parent node of each node; , The Laplace matrix representing the communication diagram between each temperature control unit. The Line 1 Column element, first Line 1 Column elements. The key "class exchange" relationship that satisfies the above construction is as follows: ; And there are: ; ; in This indicates taking the null space of the matrix; Represents the span of vectors; Represents an N-dimensional unit column vector whose elements are all 1s; The first eigenvalue in the ascending sorted sequence of matrix eigenvalues Each feature value.

[0033] S3. Based on the communication diagram between each temperature control unit, design a distributed proportional-integral controller with matrix proportional-integral gain for each follower temperature control unit.

[0034] Existing distributed proportional controllers typically only utilize real-time measured local cooperative error signals, without taking advantage of the historical accumulated information of these error signals. Therefore, they are highly susceptible to steady-state errors when heterogeneous constant disturbances or external bias inputs are present. To fill this gap, a distributed proportional-integral control strategy is considered based on existing distributed proportional consistency control. As a preferred implementation of step S3, it specifically includes: The mathematical representation of the distributed proportional-integral controller of each follower temperature control unit is as follows: ; in, , These represent the proportional feedback gain matrix and the integral feedback gain matrix, respectively. This indicates that for the expression From 0 Within range Integral Let be the integral variable, and its physical meaning is time.

[0035] First, define It is a vector sequence. Indicates vectorization operation, This represents the transpose operation. Definition As the first The integral state feedback of each temperature control unit is defined. Therefore, the compact form of the distributed proportional-integral controller can be obtained as follows: ; in , = ; This represents the Kronecker product.

[0036] Therefore, combining the linear dynamic equations of each temperature control unit with the distributed proportional-integral controller, the dynamics of the closed-loop network nodes can be obtained as follows: ; in, The heterogeneous unknown constant disturbance vector of the temperature control unit is defined as follows: ; , System status Integral status feedback The first-order time derivatives describe respectively Integral status feedback Changes over time.

[0037] In the following text, for any real matrix , show It is a symmetric positive definite matrix; if If it is a square matrix but not symmetrical, then it is called... It is a matrix The symmetrical part.

[0038] This invention does not simply apply conventional proportional-integral (PI) controllers to temperature control systems. Instead, it addresses leader-follower temperature control networks composed of multiple temperature control units. In a directed communication topology, with heterogeneous unknown constant disturbances and a stabilized linear temperature control unit model, it constructs matrix-type proportional and integral feedback gains based on the algebraic Riccati equation. This leads to the construction of a matrix-type distributed PI controller. Furthermore, it combines directed support trees and Laplace difference matrices to establish bifurcation dynamics stability analysis, enabling the system to achieve leader-follower asymptotic consistency under conditions of asymmetric communication topology and heterogeneous constant disturbances. Traditional temperature control PI control typically targets a single controlled object or employs centralized control, with the error signal often set as the deviation between the controlled object and the setpoint. In contrast, the proportional and integral terms in this invention are constructed from the distributed consistency error between adjacent temperature control units. Each follower temperature control unit controls only based on its neighboring node information, without relying on global information. Furthermore, this invention does not employ ordinary scalar PI gain, but instead constructs a matrix-type distributed PI gain based on the temperature control unit dynamics model and algebraic Riccati equations. It also combines directed support trees to prove the stability of the bifurcation dynamics, thus ensuring that even with directed topological asymmetry and heterogeneous unknown constant steady-state error disturbances on the actuator side, the follower temperature control unit can still achieve asymptotic consistency tracking of the leader temperature control unit. Compared to traditional centralized PI temperature control methods, this invention is more suitable for multi-region, multi-temperature control unit collaborative temperature regulation scenarios; compared to methods using only distributed proportional control, this invention can effectively reduce or eliminate steady-state errors caused by heterogeneous constant disturbances.

[0039] S4. Design the parameters of the distributed proportional-integral controller.

[0040] As a preferred embodiment of step S4, the specific process includes: According to the conditions Select ; in This indicates the operation of taking the real part; Represents the positive scalar adjustment coefficient of the proportional feedback gain matrix; The Laplace matrix represents the communication diagram between the various temperature control units; Expressing the request The second smallest eigenvalue.

[0041] Select matrix , According to the algebraic Riccati equation (ARE) , Solve , ,in It represents the base gain, which is not directly added to the distributed proportional-integral control law as an independent control term, but is used as the base matrix for constructing a matrix-type proportional feedback gain.

[0042] Deriving the proportional feedback gain matrix With integral feedback gain matrix .

[0043] Theorem 1: Under the conditions that Assumptions 1 and 2 hold, for a multi-agent network system, for the th The temperature control unit is designed as described above with a distributed proportional-integral controller, and the matrix can be chosen arbitrarily. , ,make It is an algebraic Riccati equation The unique solution, where , , .

[0044] The parameters for designing a distributed proportional-integral controller are as follows: Proportional feedback gain matrix ; in, Represents the positive scalar adjustment coefficient of the proportional feedback gain matrix; express The inverse matrix; Integral feedback gain matrix ; in, Represents the positive scalar adjustment coefficient of the integral feedback gain matrix; If based on directed support tree The constructed Laplace difference matrix This makes its symmetrical part In this way, the system states of all agents (temperature control units) in the multi-agent network system will achieve asymptotic consistency, that is, there are .

[0045] Theorem Proof: Consider coordinate transformation and , for 3D identity matrix , These represent the state divergence variables and the integral feedback divergence variables, respectively. ,but If and only if the state This means that the state of a multi-agent system is consistent. Based on the properties of the Kronecker mixture product, it can be deduced that... and The resulting system dynamics (bifaction dynamics) are as follows: ; in, , They are respectively , The first-order time derivative describes the change of both over time; the Laplace difference matrix. Depend on and The Laplace difference matrix is ​​defined and readily known. It is reversible.

[0046] Define coordinate transformation ,but: (Formula 1); This represents the integral divergence state vector after perturbation offset compensation.

[0047] Consider the following Lyapunov candidate function : ; Obviously, It is positive definite and radially unbounded. Differentiating the above equation along the trajectory of Equation 1, we get: .

[0048] for The derivative; the designed , Substituting into the above formula, we get: ; in, eigenvalues Therefore when At that time, matrix It is Hurwitz stable, therefore: .

[0049] because , can be obtained ,thus , It is bounded. Note that... If and only if According to LaSalle's invariance theorem, the system states of all agents (temperature control units) in a multi-agent network system achieve asymptotic consistency. Thus, the proof of Theorem 1 is complete.

[0050] Assumption 3: Communication Diagram It contains a directed support tree Furthermore, the leader's temperature control unit is the directed support tree. The root node.

[0051] The leader-follower consistency tracking problem can be viewed as a special case of the leaderless consistency problem, the only difference being that the leader does not enter neighbor nodes.

[0052] Corollary 1: Under the condition that Assumption 1 and Assumption 3 are true, choose , ,make It is the unique solution to the algebraic Riccati ARE equation; based on Define matrix and .like In the leader-follower multi-agent system, the agent (temperature control unit) will achieve asymptotic leader-follower consistency under the action of the distributed proportional-integral controller and the parameters of the above design.

[0053] Proof Corollary 1: Since the leader agent (leader temperature control unit) has no neighboring nodes, under the action of the distributed proportional-integral controller, the leader-follower multi-agent system can be regarded as a leaderless multi-agent system (the multi-agent network system), which is equivalent to making the leader-follower multi-agent system... ( (This indicates that it is always equal to). If the conditions of Theorem 1 hold, the intelligent agent (temperature control unit) will achieve asymptotic uniformity. Specifically, there is... Corollary 1 has been proven.

[0054] Finding a directed support tree from a directed graph typically involves breadth-first search (BFS) or depth-first search (DFS). This invention employs BFS to achieve this goal. The method first selects a root node in the directed graph and establishes a queue structure using this root node as the starting point for the search, while simultaneously initializing the node visit status. Then, starting from the root node, a breadth-first search is performed: first, the root node is marked as visited and added to the directed support tree queue; then, nodes are popped from the queue sequentially, and all reachable neighbors of the searched node are traversed. If an unvisited neighbor is found, it is marked as visited, its corresponding directed edge is added to the directed support tree edge set, and the node is added to the queue; then the search continues. This process is repeated until the queue is empty, thus obtaining a directed support tree.

[0055] Based on the above method, this invention uses the leader temperature control unit as the root node and the communication graph (directed graph) between temperature control units. Constructing a directed support tree .

[0056] S5. Calculate the proportional error and integral error between the system states of each temperature control unit, use a distributed proportional-integral controller to solve the control input of each follower temperature control unit, and then update the system state of each temperature control unit to achieve leader-follower consistency of the temperature control unit.

[0057] As a preferred embodiment of step S5, it specifically includes: For the number is The follower temperature control unit, in which Based on the system status received from the neighboring temperature control unit, the proportional error is calculated as follows: ; in, This indicates that for the number is The proportional error is calculated by the follower temperature control unit.

[0058] Therefore, based on the proportional error, the integral error is constructed as follows: ; in, This indicates that the number is The integral error is calculated by the follower temperature control unit.

[0059] Based on the obtained proportional error and integral error, the solution number is... The control input of the follower temperature control unit : .

[0060] Substitute the control input into the numbered... The linear dynamic equation of the follower temperature control unit: ; Thus, the system state derivative is obtained. The system state of the follower temperature control unit is updated based on the state derivative.

[0061] Repeat the proportional error calculation, integral error update, control input calculation, and system state update process for all follower temperature control units until the state of each follower temperature control unit meets the requirements. At this time, the leader-follower consistency of the temperature control unit is thus achieved.

[0062] The control algorithm flow in this embodiment can be referred to Figure 2 First, system modeling and parameter initialization are performed. Then, a directed tree communication topology is constructed, which involves the following steps: inputting a directed graph, selecting the root node, traversing reachable nodes, recording valid edges, and constructing a directed spanning tree communication network topology. Simultaneously, a positive definite matrix is ​​selected... , Solve the ARE equation (i.e., the algebraic Riccati equation) to calculate the fundamental gain. Then, the proportional feedback gain matrix is ​​derived. With integral feedback gain matrix The steps involve designing a matrix-type PI gain, designing a distributed proportional-integral (PI) control law, dynamically updating the temperature control unit system state, then conducting interactive communication between neighboring temperature control units, and using a directed support tree to calculate distributed consistency errors (proportional error and integral error), ultimately achieving asymptotic convergence and output of the temperature control system state.

[0063] This embodiment also includes experiments designed to verify the effectiveness of the proposed method.

[0064] This invention is based on a type of temperature control system, considering the leader-follower multi-agent system. The temperature and humidity states of the temperature control units are used as the two system states of the temperature control system. The system has a total of eight temperature control units in operation, including one leader temperature control unit. The system parameters are as follows: .

[0065] The communication diagram (directed graph) of the temperature control system is as follows: Figure 3 As shown. Figure 3 The diagram shows temperature control units numbered 1-8, where unit 1 is the leader and the rest are follower units. Clearly, assumption 3 is satisfied; the directed graph itself is a suitable directed support tree, and the Laplace difference matrix can be calculated. for: .

[0066] Therefore, we can obtain the result. The smallest eigenvalue ,thus .design ( (Representing a 1×1 matrix of all ones), solve the algebraic Riccati equation. : .

[0067] To highlight the superiority of the distributed proportional-integral (PI) control method designed in this invention, this embodiment further designs the following two temperature control system control scenarios.

[0068] Scenario 1: In a workshop operating with automated equipment, the presence of precision instruments necessitates maintaining a baseline temperature of 25°C. The external setting input for the leader temperature control unit is then configured. Because winter temperatures in this region are below 25°C while summer temperatures are above 25°C, and due to differences in the spatial location of the temperature control units, heterogeneous errors typically exist, varying with the control input. Enter the temperature control system. Assume the unknown constant disturbance vector between indoor and outdoor materials during winter is... The heterogeneous unknown constant disturbance vector for indoor and outdoor environments in summer is: .

[0069] Figure 4 This displays the temperature status of each temperature control unit (numbered 1-8) under the control of a distributed proportional (P) controller and a distributed proportional-integral (PI) controller in winter, as shown in Scenario 1. (Unit: °C) Comparison of the trajectory of the set temperature (i.e., the leader's external setting input) over time.

[0070] Figure 5 This displays the temperature status of each temperature control unit (numbered 1-8) under the control of a distributed proportional (P) controller and a distributed proportional-integral (PI) controller in summer, as shown in Scenario 1. (Unit: °C) Comparison of the trajectory of the set temperature (i.e., the leader's external setting input) over time.

[0071] To demonstrate the advantages of the distributed proportional-integral controller of the present invention, we take... , At this time, the controller is a distributed proportional (P) controller, and the temperature state trajectories of each temperature control unit in winter and summer are as follows: Figure 4 , Figure 5 As shown on the left. (By...) Figure 4 , Figure 5It is evident that the follower temperature control unit failed to effectively track the leader temperature control unit, and the temperature control system not only exhibited significant steady-state error, but this error was also greater in winter than in summer. Under the same initial conditions, further... , At this time, the controller is a distributed proportional-integral (PI) controller, and the temperature state trajectories of each temperature control unit in winter and summer are as follows: Figure 4 , Figure 5 As shown on the right. By comparison, it can be found that the introduction of the integral feedback loop in the method of the present invention effectively eliminates the steady-state error of the system, enabling the temperature of each temperature control unit to gradually reach uniformity, thereby achieving asymptotic uniformity.

[0072] Scenario 2: In a precision equipment component processing workshop, the workshop begins operation at 8:00 AM, with equipment startup, personnel entry, and testing tasks proceeding simultaneously. To prevent thermal deformation from affecting processing accuracy, the leader temperature control unit of the temperature control system issues a unified reference temperature of 20°C to all areas. Each area's temperature control unit, acting as a follower, coordinates its adjustments based on information from neighboring areas, gradually bringing the entire workshop to a consistent and stable temperature control state. After 8:00 PM, the equipment shuts down, and the workshop enters standby mode. Maintaining the same stringent temperature control accuracy as during the day is no longer required, and the leader temperature control unit switches the reference temperature to 26°C. Each area's follower temperature control units then readjust, reducing the cooling load and operating energy consumption while still meeting basic environmental requirements. The external settings input for the temperature control system at this time are as follows: ; The heterogeneous unknown constant perturbation vector is set as .

[0073] Figure 6 This displays the temperature status of each temperature control unit (numbered 1-8) under the distributed proportional (P) controller scenario 2. (Unit: °C) Comparison of the trajectory of the set temperature (i.e., the leader's external setting input) over time.

[0074] Figure 7 This displays the temperature status of each temperature control unit (numbered 1-8) under the distributed proportional-integral (PI) controller scenario in scenario two. (Unit: °C) Comparison of the trajectory of the set temperature (i.e., the leader's external setting input) over time.

[0075] Pick , At this time, the controller is a distributed proportional (P) controller, and the temperature status trajectory of each temperature control unit is as follows: Figure 6As shown. Clearly, the follower temperature control unit failed to track the leader temperature control unit, and the steady-state error remained significant before the two switching of the external setting input at 8:00 and 20:00.

[0076] Under the same initial conditions, take At this time, the controller is a distributed proportional-integral (PI) controller, and the temperature state trajectory of each temperature control unit is as follows: Figure 7 As shown, it is obvious that from Figure 7 It can be concluded that the steady-state error was eliminated by the integral action before the two switching of the external setting input at 8:00 and 20:00.

[0077] This invention focuses on solving the challenge of distributed proportional-integral (PI) consistency control in temperature control systems implemented on directed graphs. The designed PI controller can still ensure leader-follower consistency even in environments with limited communication, asymmetric network topology, and non-uniform heterogeneous disturbances. Compared with platforms requiring complete controllability, it lowers the threshold for model and actuator completeness, has a wider range of applications, and is more suitable for large-scale deployment and industrial scenarios with complex noise interference. At the same time, due to the addition of the integrator, the proposed controller has higher robustness than traditional distributed proportional controllers, providing an efficient solution for achieving distributed asymptotic consistency control of industrial temperature control systems.

[0078] This application also provides an electronic device, which includes: at least one processor; and a memory communicatively connected to the at least one processor; wherein the memory stores a computer program executable by the at least one processor, and the computer program is executed by the at least one processor to enable the at least one processor to execute the novel distributed proportional-integral temperature control method based on directed graph theory.

[0079] This application also provides a computer-readable storage medium storing computer instructions that are used to cause a processor to implement the novel distributed proportional-integral temperature control method based on directed graph theory.

[0080] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this application. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of those different embodiments or examples.

[0081] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus or device (such as a computer-based system, a processor-included system or other system that can fetch and execute instructions from, an instruction execution system, apparatus or device).

[0082] The above embodiments provide a detailed description of the present invention. Specific examples have been used to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of the present invention. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A novel distributed proportional-integral temperature control method based on directed graph theory, characterized in that, include: The cluster system consisting of several temperature control units is modeled as a multi-agent network system. The temperature control unit is divided into one leader temperature control unit and several follower temperature control units, and the multi-agent network system is constructed as a leader-follower multi-agent system. In the leader-follower multi-agent system, the communication graph between the temperature control units is a directed graph, and the directed graph contains at least one directed support tree; the directed support tree has the leader temperature control unit as the root node; Based on the communication diagram between each temperature control unit, a distributed proportional-integral controller with matrix proportional-integral gain is designed for each follower temperature control unit. Design the parameters of a distributed proportional-integral controller; Communication between temperature control units is achieved based on a directed support tree, and the integral status feedback of each follower temperature control unit is updated. The proportional and integral errors between the system states of each temperature control unit are calculated. The control input of each follower temperature control unit is solved using a distributed proportional-integral controller, and then the system state of each temperature control unit is updated to achieve leader-follower consistency of the temperature control units.

2. The novel distributed proportional-integral temperature control method based on directed graph theory according to claim 1, characterized in that: The linear dynamic equations of each temperature control unit in the multi-agent network system are defined as follows: ； in, Indicates the first Each temperature control unit at time The system status, including the temperature and humidity of the temperature control unit. for About time The first derivative describes Changes over time; Indicates the first Each temperature control unit at time The control input to be designed; Indicates the first Heterogeneous unknown constant disturbance on the actuator side of each temperature control unit; The system matrix for the temperature control unit; This is the input matrix for the temperature control unit; Number the temperature control unit. This represents the total number of temperature control units.

3. The novel distributed proportional-integral temperature control method based on directed graph theory according to claim 2, characterized in that: The method of dividing the temperature control unit into one leader temperature control unit and several follower temperature control units, and constructing the multi-agent network system as a leader-follower multi-agent system, includes: The temperature control unit numbered 1 is designated as the leader temperature control unit, and the remaining temperature control units are designated as follower temperature control units. A leader-follower multi-agent system is constructed by combining the aforementioned multi-agent network system. The linear dynamic equation of the leader temperature control unit in the leader-follower multi-agent system is defined as follows: ； in, This indicates the external setting input for the leader's temperature control unit; The linear dynamic equation of the follower temperature control unit in the leader-follower multi-agent system is defined as follows: ， 。 4. The novel distributed proportional-integral temperature control method based on directed graph theory according to claim 3, characterized in that: The system matrix of the temperature control unit in the leader-follower multi-agent system. With input matrix The system dynamics matrix consists of pairs It is sedative and there exists a feedback matrix. This makes the closed-loop matrix The real parts of all eigenvalues ​​are less than 0.

5. A novel distributed proportional-integral temperature control method based on directed graph theory according to claim 3, characterized in that: The design of a distributed proportional-integral controller with matrix proportional-integral gain for each follower temperature control unit, based on the communication diagram between the various temperature control units, includes: The mathematical representation of the distributed proportional-integral controller of each follower temperature control unit is as follows: ； in, , These represent the proportional feedback gain matrix and the integral feedback gain matrix, respectively. The adjacency matrix of the communication graph between each temperature control unit is the first... Line 1 Column element, indicating the first Can the temperature control unit send to the first...? Each temperature control unit transmits information. Number the temperature control unit. This indicates that the current temperature control unit is the leader temperature control unit. This indicates that the current temperature control unit is the follower temperature control unit; This indicates that for the expression From 0 Within range Integral Let be the integral variable, and its physical meaning is time.

6. A novel distributed proportional-integral temperature control method based on directed graph theory according to claim 5, characterized in that: The proportional feedback gain matrix satisfies the following condition: ； in, Represents the positive scalar adjustment coefficient of the proportional feedback gain matrix; Indicates the transpose operation; Let be any symmetric positive definite matrix, denoted as , express The inverse matrix; The Laplace matrix represents the communication diagram between the various temperature control units; This indicates the operation of taking the real part; Expressing the request The second smallest eigenvalue; Let be any symmetric positive definite matrix, and satisfy... It is an algebraic Riccati equation The only solution Let be any symmetric positive definite matrix, denoted as ; The integral feedback gain matrix satisfies the following condition: ； in, This represents the positive scalar adjustment coefficient of the integral feedback gain matrix.

7. A novel distributed proportional-integral temperature control method based on directed graph theory according to claim 6, characterized in that: The parameters for designing the distributed proportional-integral controller include: According to the conditions Select ; Select matrix , According to the equation Solve ; Deriving the proportional feedback gain matrix With integral feedback gain matrix .

8. A novel distributed proportional-integral temperature control method based on directed graph theory according to claim 1, characterized in that: The Laplace difference matrix of the directed support tree is defined as follows: ； in, Represents the Laplace difference matrix The Middle Line 1 Column elements; In a directed support tree, the first... The set of nodes in the subtree rooted at node n. This is the node index in the node set; In a directed support tree, the first... The node index of the unique parent node of each node; , The Laplace matrix representing the communication diagram between each temperature control unit. The Line 1 Column element, first Line 1 Column elements; The Laplace difference matrix is ​​an asymmetric square matrix, and its symmetric part is calculated as follows: ； in, Represents the Laplace difference matrix The symmetrical part, and It is a symmetric positive definite matrix; This indicates the transpose operation.

9. An electronic device, characterized in that, The electronic device includes: at least one processor; and a memory communicatively connected to the at least one processor; wherein the memory stores a computer program executable by the at least one processor, the computer program being executed by the at least one processor to enable the at least one processor to execute the novel distributed proportional-integral temperature control method based on directed graph theory as described in any one of claims 1-8.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores computer instructions that, when executed by a processor, implement the novel distributed proportional-integral temperature control method based on directed graph theory as described in any one of claims 1-8.

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