Method and device for constructing proportional-integral synchronization controller of complex dynamic network

Abstract

The application discloses a proportional-integral synchronization controller construction method and device for a complex dynamic network, and the method is as follows: a mathematical model of the complex dynamic network with a Markov switching topology, a dynamic event triggering mechanism and a proportional-integral control strategy is established; an exponential synchronization target in the complex dynamic network in the mean square sense is designed; a Lyapunov function is constructed, and a sufficient condition for the exponential synchronization of the system in the mean square sense is deduced; by solving a linear matrix inequality, a gain matrix of the proportional-integral controller required to ensure the synchronization performance of the system and parameters of an event triggering condition are designed; Zeno behavior is excluded, and the effectiveness of the dynamic event triggering mechanism is verified. The device comprises the following modules: a mathematical model establishment module, an exponential synchronization target design module, a condition deduction module, a proportional-integral controller design module and an effectiveness verification module. The application improves the stability performance of the complex dynamic network system, reduces the steady-state error of the system, and has the advantages of simplicity, easy realization and strong applicability.

Description

Technical Field

[0001] This invention relates to the field of proportional-integral synchronous control technology, and in particular to a method and apparatus for constructing a proportional-integral synchronous controller for complex dynamic networks. Background Technology

[0002] Complex dynamic networks (CMNs) are networks with complex dynamic characteristics, formed by a large number of dynamically connected nodes coupled according to a certain topology. Many real-world networks, such as transportation networks, gene regulatory networks, and social networks, can be incorporated into the architecture of a complex dynamic network. Synchronization, as a fundamental characteristic of collective behavior in complex dynamic networks, has profound applications in engineering, sociology, and many other fields. Research on the synchronization problem of complex dynamic networks has yielded rich results, and designing effective control strategies for synchronization in complex dynamic networks has always been a research hotspot. However, due to the complexity of the dynamic characteristics of complex dynamic networks, the use of proportional-integral (PI) synchronization controllers to analyze the synchronization problem of complex networks has not been fully studied.

[0003] With the rapid development of modern technology and productivity, data transmission and exchange among nodes in a network are handled by communication networks. This presents both new opportunities and challenges for the development of complex networks. One significant issue is the insufficient bandwidth caused by the massive number of network nodes. Limited communication resources and a large number of nodes can have unpredictable impacts on the control performance of complex dynamic networks, leading to deterioration of expected control performance or even failure to achieve synchronization. If this occurs in practical engineering, it can have serious consequences. Therefore, designing effective information exchange strategies to maximize the use of limited bandwidth resources and reduce information transmission is of significant theoretical and practical value. Summary of the Invention

[0004] The purpose of this invention is to provide a method and apparatus for constructing a proportional-integral synchronous controller for complex dynamic networks that features low network resource consumption, strong system control performance, and wide applicability.

[0005] The technical solution to achieve the purpose of this invention is: a method for constructing a proportional-integral synchronization controller for complex dynamic networks, comprising the following steps:

[0006] Step 1: Establish a mathematical model of a complex dynamic network with Markov switching topology, dynamic event triggering mechanism and proportional-integral control strategy;

[0007] Step 2: Design an exponential synchronization objective in the mean square sense for complex dynamic networks;

[0008] Step 3: By constructing the Lyapunov function, derive the sufficient condition for exponential synchronization of the system in the mean-square sense;

[0009] Step 4: By solving the linear matrix inequalities, jointly design the gain matrix of the proportional-integral controller and the parameters of the event triggering conditions required to ensure the system's synchronization performance;

[0010] Step 5: Verify the effectiveness of the dynamic event triggering mechanism by excluding Zeno behavior.

[0011] A device for constructing a proportional-integral (PI) synchronization controller for a complex dynamic network is provided. This device is used to implement the method for constructing a PI synchronization controller for the complex dynamic network. The device includes a mathematical model establishment module, an exponential synchronization target design module, a condition derivation module, a PI controller design module, and an effectiveness verification module, wherein:

[0012] The mathematical model building module is used to build a mathematical model of a complex dynamic network with Markov switching topology, dynamic event triggering mechanism and proportional-integral control strategy.

[0013] The exponential synchronization target design module is used to design exponential synchronization targets in the mean square sense for complex dynamic networks.

[0014] The condition derivation module derives sufficient conditions for exponential synchronization of the system in the mean-square sense by constructing a Lyapunov function;

[0015] The proportional-integral controller design module, by solving linear matrix inequalities, jointly designs the gain matrix and event triggering condition parameters of the proportional-integral controller required to ensure the system's synchronization performance.

[0016] The validity verification module verifies the validity of the dynamic event triggering mechanism by excluding Zeno behaviors.

[0017] A mobile terminal includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, it implements the proportional-integral synchronization controller construction method for the complex dynamic network.

[0018] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps in a method for constructing a proportional-integral synchronous controller for a complex dynamic network.

[0019] Compared with the prior art, the present invention has the following significant advantages: (1) The external coupling configuration matrix adopts the Markov switching form instead of a fixed topology structure, which has strong applicability; (2) The dynamic event triggering mechanism is adopted to determine when to transmit the control signal generated by the proportional-integral controller, and to eliminate Zeno behavior and verify the effectiveness of the dynamic event triggering mechanism; (3) The proportional-integral controller structure is simple and easy to implement, which improves the stability performance of the system and reduces the steady-state error of the system; (4) By combining the dynamic event triggering mechanism and the proportional-integral control strategy, not only can the control performance of the system be improved, but unnecessary data transmission can also be reduced. Attached Figure Description

[0020] Figure 1 This is a flowchart illustrating a method for constructing a proportional-integral synchronous controller for a complex dynamic network according to the present invention.

[0021] Figure 2 This is a Markov switching signal curve diagram in an embodiment of the present invention.

[0022] Figure 3 This is a state curve diagram of each node in an embodiment of the present invention without the use of a proportional-integral control strategy.

[0023] Figure 4 This is a state curve diagram of each node under the proportional-integral control strategy in this embodiment of the invention. Figure 5 This is a curve showing the triggering time of each node under the proportional-integral control strategy in this embodiment of the invention. Detailed Implementation

[0024] Combination Figure 1 The present invention discloses a method for constructing a proportional-integral synchronization controller for a complex dynamic network, comprising the following steps:

[0025] Step 1: Establish a mathematical model of a complex dynamic network with Markov switching topology, dynamic event triggering mechanism and proportional-integral control strategy;

[0026] Step 2: Design an exponential synchronization objective in the mean square sense for complex dynamic networks;

[0027] Step 3: By constructing the Lyapunov function, derive the sufficient condition for exponential synchronization of the system in the mean square sense; Step 4: By solving the linear matrix inequality, jointly design the gain matrix of the proportional-integral controller and the parameters of the event triggering condition required to ensure the synchronization performance of the system.

[0028] Step 5: Verify the effectiveness of the dynamic event triggering mechanism by excluding Zeno behavior.

[0029] As a specific implementation method, in step 1, a mathematical model of a complex dynamic network with Markov switching topology, dynamic event triggering mechanism, and proportional-integral control strategy is established, as follows:

[0030] Step 1.1: Establish a complex dynamic network with Markov switching topology consisting of N nodes:

[0031]

[0032] in This represents the state variable of the i-th node. f(·) represents the control input of the i-th node. Represents a nonlinear function; Let a represent the external coupling configuration matrix of the network, and switch according to a Markov process. ij (r(t)) is defined as follows: if there is a connection between node i and node j (i ≠ j), then a ij (r(t))>0; otherwise, a ij (r(t))=0, and Γ=(γ ij ) n×n It is the internal coupling matrix; the set of all nodes is denoted as .

[0033] {r(t), t≥0} is the complete probability space. A right-continuous Markov process in a finite state space. Take values ​​from , where s is a positive integer, and its generating function is . as follows:

[0034]

[0035] Where Δ>0, Indicates from state to state The transition probability, and

[0036] Setting 1: Nonlinear function f(·): Satisfying the Lipschitz condition:

[0037] ||f(ι)-f(j)||≤θ||ι-j|| (2)

[0038] Where ι, θ is a known positive real number;

[0039] Setting 2: The continuous-time Markov process {r(t)} with the transition rate matrix Π is ergodic;

[0040] It is an isolated node system The solution is given by ε. i (t)=x i (t)-s(t), where ε i (t) represents the synchronization error, and the following error system can be obtained:

[0041]

[0042] in,

[0043] Step 1.2: For the i-th node, adopt the following proportional-integral control strategy:

[0044]

[0045] in and It is the gain matrix of the node i to be designed, and it depends on the change of the Markov process r(t);

[0046] Step 1.3: To reduce the number of controller updates and improve network resource utilization, a dynamic event triggering mechanism is designed to determine when to send control signals to the actuator. The event triggering time sequence of the i-th node is defined as follows: And given in Determined by the following triggering rules:

[0047]

[0048] in This represents the control signal of node i at the latest triggering time. δ i and μ i It is a given positive scalar; the control signal is updated and sent to the actuator when the event triggering condition is met; in addition, the dynamic variable ξ of the event triggering condition. i (t) satisfies:

[0049]

[0050] Where the initial value ξ i,0 >0, γ i >0 is a given scalar, which indicates that ≤(1 / μ i )ξ i (t), that is Therefore, we can obtain

[0051] Step 1.4: Design the control input u of node i i (t) is:

[0052]

[0053] in Indicates the time of triggering node i The controller output;

[0054] Therefore, the synchronization error system can be rewritten as:

[0055]

[0056] definition

[0057]

[0058]

[0059]

[0060] Therefore, the synchronization error system can be expressed as:

[0061]

[0062] As a specific implementation method, in step 2, an exponential synchronization target in the mean square sense of complex dynamic networks is designed, as follows:

[0063] Positive scalar exists and For any initial conditions, if

[0064]

[0065] If true, then complex dynamic networks are exponentially synchronized in the mean square sense.

[0066] As a specific implementation method, in step 3, the Lyapunov function is constructed, and the sufficient condition for exponential synchronization of the system in the mean-square sense is derived, as follows:

[0067] Step 3.1: Given matrix ρ, and positive real number δ i μ i γ i , If a symmetric positive definite matrix exists sum matrix Make:

[0068]

[0069] If this holds true, then the complex dynamic network is exponentially synchronous in the mean-square sense, where...

[0070]

[0071]

[0072]

[0073]

[0074] Υ 44 =-I-Δ2,Δ3=diag{δ1I,δ2I,…,δ N I},

[0075]

[0076]

[0077] Step 3.2, let Construct the following Lyapunov function:

[0078]

[0079] By calculating the infinitesimal operator of V(t) Furthermore, by employing Lyapunov stability theory, free weighted matrices, and linear matrix inequalities, we can obtain:

[0080]

[0081] in

[0082] Therefore, complex dynamic networks can achieve exponential synchronization in the mean square sense.

[0083] As one specific implementation, in step 4, by solving linear matrix inequalities, the gain matrix of the proportional-integral controller and the parameters of the event triggering conditions required to ensure the system's synchronization performance are jointly designed, as follows:

[0084] Step 4.1: Set a given positive real number If a symmetric matrix exists If the linear matrix inequality (13) holds, then the complex dynamic network can achieve mean squared exponential synchronization.

[0085]

[0086] in

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093]

[0094]

[0095] The controller gain matrix corresponding to node i is obtained as follows:

[0096]

[0097] Step 4.2, let Pick and Then multiply both sides of inequality (13) by the left and right sides respectively. It can be obtained

[0098]

[0099] in

[0100]

[0101]

[0102]

[0103]

[0104]

[0105] Therefore, inequality (13) holds.

[0106] As a specific implementation method, in step 5, the effectiveness of the dynamic event triggering mechanism is verified by excluding Zeno behavior, as follows:

[0107] For complex dynamic networks with dynamic event triggering mechanisms, when At that time, through analysis, we can obtain the following:

[0108]

[0109] in

[0110] For node i, the Zeno behavior is set to occur during... Right now Then you can get Therefore:

[0111]

[0112] We can obtain:

[0113]

[0114] Right now:

[0115]

[0116] This means ξ i,0 ≤0, and ξ i,0 The result is greater than 0, which contradicts the Zeno behavior, thus ruling it out.

[0117] This invention also provides a device for constructing a proportional-integral (PI) synchronization controller for complex dynamic networks. This device is used to implement the aforementioned method for constructing a PI synchronization controller for complex dynamic networks. The device includes a mathematical model establishment module, an exponential synchronization target design module, a condition derivation module, a PI controller design module, and an effectiveness verification module, wherein:

[0118] The mathematical model building module is used to build a mathematical model of a complex dynamic network with Markov switching topology, dynamic event triggering mechanism and proportional-integral control strategy.

[0119] The exponential synchronization target design module is used to design exponential synchronization targets in the mean square sense for complex dynamic networks.

[0120] The condition derivation module derives sufficient conditions for exponential synchronization of the system in the mean-square sense by constructing a Lyapunov function;

[0121] The proportional-integral controller design module, by solving linear matrix inequalities, jointly designs the gain matrix and event triggering condition parameters of the proportional-integral controller required to ensure the system's synchronization performance.

[0122] The validity verification module verifies the validity of the dynamic event triggering mechanism by excluding Zeno behaviors.

[0123] The present invention also provides a mobile terminal, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the proportional-integral synchronization controller construction method for the complex dynamic network.

[0124] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps in the method for constructing a proportional-integral synchronous controller for a complex dynamic network.

[0125] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0126] Example

[0127] To further illustrate the effectiveness of the algorithm proposed in this invention, comparative simulations were conducted. The complex dynamic network was set to have N = 5 nodes, n = 2 node dimensions, and s = 2 switching modes. The inner coupling matrix and outer coupling configuration matrix were defined as follows:

[0128]

[0129]

[0130] The nonlinear function is

[0131]

[0132] That is, θ = 0.7.

[0133] Given parameters γ1=γ2=γ3=γ4=γ5=5, δ1=δ2=δ3=δ4=δ5=0.01, μ1=μ2=μ3=μ4=μ5=5, ρ=0.05, a set of gain matrices can be obtained using the Matlab toolbox. and

[0134]

[0135]

[0136]

[0137]

[0138]

[0139]

[0140] Simulation results are shown below Figures 2-5 . Figure 2 The Markov switching signal is described. Figure 3 and Figure 4The state trajectories of each node are shown under both no-control and proportional-integral (PI) control strategies. Without control, the state trajectories of each node do not converge to the state trajectories of individual nodes, and synchronization of the complex dynamic network is impossible. Despite the existence of complex dynamic networks, such as... Figure 2 The switching topology shown can effectively achieve synchronous control of the system by designing a control strategy with corresponding switching methods. Figure 5 The triggering time of the dynamic event triggering mechanism under the proportional-integral control strategy is shown. It can be seen that the interval between two adjacent triggering times of each node is greater than zero, which further indicates that Zeno behavior did not occur.

[0141] In summary, this invention employs a dynamic event triggering mechanism and a proportional-integral (PI) control strategy to solve the synchronization problem of complex networks with Markov switching topologies. It proposes a dynamic event triggering mechanism that maintains control performance while reducing communication costs; provides a sufficient condition to guarantee mean-square exponential synchronization of complex networks; designs the gain matrix of the PI controller and the parameters of the event triggering condition; and also eliminates Zeno behavior.

Claims

1. A method for constructing a proportional-integral synchronous controller for a complex dynamic network, characterized in that, Includes the following steps: Step 1: Establish a mathematical model of a complex dynamic network with Markov switching topology, dynamic event triggering mechanism and proportional-integral control strategy; Step 2: Design an exponential synchronization objective in the mean square sense for complex dynamic networks; Step 3: By constructing the Lyapunov function, derive the sufficient condition for exponential synchronization of the system in the mean-square sense; Step 4: By solving the linear matrix inequalities, jointly design the gain matrix of the proportional-integral controller and the parameters of the event triggering conditions required to ensure the system's synchronization performance; Step 5: Verify the effectiveness of the dynamic event triggering mechanism by excluding Zeno behavior; Step 1 includes: Step 1.1: Establish a system based on... A complex dynamic network with Markov switching topology consisting of nodes: (1) in Indicates the first The state variables of each node Indicates the first Control input for each node Represents a nonlinear function; This represents the external coupling configuration matrix of the network, and it is switched according to a Markov process. Defined as follows: If from node arrive If there is a connection, then ;otherwise, ,and ; It is the internal coupling matrix; the set of all nodes is denoted as . ; It is a complete probability space A right-continuous Markov process in a finite state space. Take the value from, where For positive integers, the generating function as follows: in , , Indicates from state to state The transition probability, and ; Setting 1: Nonlinear function Satisfying the Lipschitz condition: (2) in , , It is a known positive real number; Setting 2: Has a transfer rate matrix Continuous-time Markov process It is a traversal; It is an isolated node system The solution; let ,in To represent the synchronization error, we obtain the following error system: (3) in, ; Step 3 describes the construction of the Lyapunov function and the derivation of the sufficient condition for exponential synchronization in the mean-square sense of the system, as follows: Step 3.1, Given a matrix , Sum of positive real numbers , , , , If a symmetric positive definite matrix exists , sum matrix , so that: (10) If this holds true, then the complex dynamic network is exponentially synchronous in the mean-square sense, where... Step 3.2, let Construct the following Lyapunov functions: (11) Through calculation Infinitesimal operators Furthermore, using Lyapunov stability theory, free weighted matrices, and linear matrix inequalities, we obtained: (12) in , Therefore, complex dynamic networks can achieve exponential synchronization in the mean square sense.

2. The method for constructing a proportional-integral synchronization controller for a complex dynamic network according to claim 1, characterized in that, Step 1 further includes: Step 1.2, for the first Each node employs a proportional-integral control strategy in the following form: (4) in , and The node to be designed The gain matrix, and depends on the Markov process. Changes; Step 1.3: To reduce the number of controller updates and improve network resource utilization, a dynamic event triggering mechanism is designed to determine when to send control signals to the actuator. The event triggering time sequence of each node is defined as follows: And given ,in Determined by the following triggering rules: (5) in Represents a node The control signal at the latest trigger moment, , and It is a given positive scalar; the control signal is updated and sent to the actuator when the event triggering condition is met; in addition, the dynamic variable of the event triggering condition. satisfy: Where the initial value , Given a scalar, we know ,Right now Therefore, we obtain ; Step 1.4, Design Nodes control input for: (6) in , representing a node At the trigger time The controller output; Therefore, the synchronization error system can be rewritten as: (7) definition Therefore, the synchronization error system is expressed as: (8) 3. The method for constructing a proportional-integral synchronous controller for complex dynamic networks according to claim 2, characterized in that, The design of the exponential synchronization objective in the mean square sense for complex dynamic networks described in step 2 is as follows: Positive scalar exists and For any initial conditions, if (9) If true, then complex dynamic networks are exponentially synchronized in the mean square sense.

4. The method for constructing a proportional-integral synchronous controller for a complex dynamic network according to claim 3, characterized in that, Step 4 describes the process of solving linear matrix inequalities to jointly design the gain matrix of the proportional-integral controller and the parameters of the event triggering conditions required to ensure the system's synchronization performance. The details are as follows: Step 4.1: Set a given positive real number , , , , , If a symmetric matrix exists , , , If the linear matrix inequality (13) holds, then complex dynamic networks can achieve mean-square exponential synchronization: (13) in Get Node The corresponding controller gain matrix is: (14) Step 4.2, let , and ;Pick and Then multiply both sides of inequality (13) by the left and right sides respectively. ,get in Therefore, inequality (13) holds.

5. The method for constructing a proportional-integral synchronous controller for a complex dynamic network according to claim 4, characterized in that, Step 5, which involves excluding Zeno behavior to verify the effectiveness of the dynamic event triggering mechanism, is detailed below: For complex dynamic networks with dynamic event triggering mechanisms, analysis reveals the following: (15) in , , ; For nodes Zeno behavior is set to occur ,Right now Then we get Therefore: (16) get: (17) Right now: (18) This means ,and This contradicts the previous conclusion and rules out Zeno's behavior.

6. A device for constructing a proportional-integral synchronous controller for a complex dynamic network, characterized in that, This device is used to implement the proportional-integral synchronization controller construction method for complex dynamic networks as described in any one of claims 1 to 5. The device includes a mathematical model establishment module, an exponential synchronization target design module, a condition derivation module, a proportional-integral controller design module, and an effectiveness verification module, wherein: The mathematical model building module is used to build a mathematical model of a complex dynamic network with Markov switching topology, dynamic event triggering mechanism and proportional-integral control strategy. The exponential synchronization target design module is used to design exponential synchronization targets in the mean square sense for complex dynamic networks. The condition derivation module derives sufficient conditions for exponential synchronization of the system in the mean-square sense by constructing a Lyapunov function; The proportional-integral controller design module, by solving linear matrix inequalities, jointly designs the gain matrix and event triggering condition parameters of the proportional-integral controller required to ensure the system's synchronization performance. The validity verification module verifies the validity of the dynamic event triggering mechanism by excluding Zeno behaviors.

7. A mobile terminal, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the proportional-integral synchronization controller construction method for complex dynamic networks as described in any one of claims 1 to 5.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the steps in the method for constructing a proportional-integral synchronous controller for a complex dynamic network as described in any one of claims 1 to 5.

Images