Synchronous control method of hidden Markov hopping system based on sampling data

Abstract

The invention discloses a synchronous control method of a hidden Markov hopping system based on sampling data, and belongs to the technical field of Markov hopping systems. The invention introduces a novel HMM (Hidden Markov Model) designed for a complex network in a continuous time domain, which is different from the dependence of controllers described in most literatures on the current system mode, the HMM proposed in the research can determine the current system mode through an observer, thereby providing higher flexibility; in comparison with some existing SOF documents, integral inequality techniques are also used herein to eliminate peer constraints, rank conditions of output matrices, and the use of certain inequality. In addition, how to reduce the use of redundant data while eliminating the limitation is also a key point in the invention.

Description

Technical Field

[0001] The present invention relates to the technical field of Markov jump systems, and in particular to a synchronous control method for a hidden Markov jump system based on sampled data. Background Art

[0002] Since Albert and Barabási introduced random networks, the theory of scale-free networks has been widely studied, providing an important theoretical framework and method for understanding and analyzing various complex system networks in the real world. Complex dynamic networks (CDNs) are interconnected through edges or links between various vertices or nodes, and this model has important practical and theoretical significance in fields such as medicine, physics, and economics. For example, on the existing basis, scholars have studied fixed-time synchronization in complex networks and discussed exponential synchronization in complex networks. Obviously, the synchronization phenomenon in complex networks is very common and has attracted extensive attention from scholars. It is worth noting that the topological connection of complex networks may randomly change in a dynamic environment, posing a challenge to synchronous control. Fortunately, the Markov process has proven its effectiveness in modeling sudden changes in network structure. Therefore, it is of great significance to study the synchronous control problem of complex dynamic networks based on the Markov process.

[0003] Most of the existing research on Markov jump systems assumes that the modal information of the system is completely available. However, in practical applications, various adverse factors such as incomplete observations and modal errors often lead to inconsistent modal information between the original system and the controller. To solve this problem, scholars have introduced the hidden Markov model (HMM). Its basic idea is to use a constructed observer to monitor the jump modal information of the original system, and then use this observed information to design a controller or filter. For example, on the existing technology, scholars have studied the asynchronous H ∞ control problem of a specific type of hidden Markov jump system (HMJS) in the continuous time domain under actuator saturation conditions. Another group of scholars has studied the H ∞ control problem of discrete-time HMJS under denial-of-service attacks. In addition, scholars have explored the asynchronous H ∞ control problem of discrete-time HMJS characterized by complex mode transition probabilities. Obviously, the hidden Markov jump complex dynamic network (MJCDN) has received increasing attention. It is worth noting that although the HMM has been widely studied, the application research of continuous asynchronous HMM in CDNs is very few. Therefore, combining complex dynamic networks with HMM is one of the main motivations of this paper.

[0004] On the other hand, in practical applications, directly measuring the system state is often challenging. Therefore, using output information for feedback control is undoubtedly a superior choice. Output feedback control mainly includes: static output feedback (SOF), dynamic output feedback, and observer-based output feedback. Considering the structure and economy, SOF control has received more attention. For example, experts have proposed a linear matrix inequality strategy for SOF controllers based on the full row rank of the output matrix. Some researchers have also achieved SOF synchronization of neural networks, where the Lyapunov matrix is in a diagonal structure. The above methods impose strict requirements on the structure and rank of the input and output matrices. Fortunately, some scholars have considered the case without an input matrix when designing SOF controllers, thus eliminating these limitations and the need for inequalities. At the same time, with the rapid development of communication technology, the application of sampled-data control (SDC) in CDN has received extensive attention. Compared with previous periodic sampling, SDC can effectively reduce the generation of redundant data. To sum up, one of the research motivations of this paper is how to eliminate such constraints without using inequality scaling techniques when designing a SOF controller based on sampled control. For this purpose, a synchronization control method for sampled-data based hidden Markov jump systems is proposed. Summary of the Invention

[0005] The technical problem to be solved by the present invention is: how to consider the SOF synchronization control problem of CDN while considering sampling delay, and through the combination of controller control and H ∞ performance index control, it is possible to achieve the synchronization control of the system through static output feedback, and a synchronization control method for sampled-data based hidden Markov jump systems is provided.

[0006] The present invention solves the above technical problems through the following technical solutions, which specifically include the following steps:

[0007] S1: Use a hidden Markov process to model the random mutation behavior in complex dynamic network topologies;

[0008] S2: Design a sampling controller, and combine it with the system model in S1 to obtain a sampled-data based hidden Markov jump dynamic network closed-loop control system;

[0009] S3: Give the linear matrix inequality conditions for the hidden Markov jump complex dynamic network system to be stochastically stable and satisfy the H ∞ performance index under disturbances;

[0010] S4: Verify whether the linear matrix inequality conditions in S3 are correct through Lyapunov functions and performance index functions;

[0011] S5: Use integral inequality techniques to solve the gain matrix of the sampled feedback controller;

[0012] S6: Implement system synchronization control based on the calculated gain matrix and the known system parameters.

[0013] Furthermore, the specific way to model the stochastic mutation behavior in a complex dynamic network topology is as follows:

[0014] S11: According to the Markov chain In this paper, the hidden MJCDNs are selected as follows, where the hidden state Observed state

[0015]

[0016] Among them, and represent the state variable and control input of the i-th node respectively, is the control output, is the measurement output, A = (a ij ) n×n ∈R n×n is the internal coupling matrix between nodes, B = (B ij ) N×N is the external coupling matrix of the complex dynamic network, is the additional disturbance belonging to the interval L[0, ∞), is the measurement matrix of the sensor, J is the output matrix of the Markov jump complex dynamic network system, define and

[0017] S12: is a homogeneous Markov process, take value, where the conversion rate is satisfies the following properties:

[0018]

[0019] Among them, and The conversion rate of (n, l), (m, k) in Q×Y satisfies the following conditions:

[0020]

[0021] S13: The term represents when and jump simultaneously, satisfying the condition For λ mn ≥0 and m = n, it means The conversion rate from the modality m of time to the modality n; the expression used to represent when the modality remains unchanged the jump; where It should be noted that is directly obtained by the detector, and the detector provides the estimated value of.

[0022] S14: Let be the synchronization error, where is the state trajectory of the unforced isolated node Derive the synchronization error system expression as follows:

[0023]

[0024] where

[0025] Furthermore, in step S2, the specific way to obtain the closed-loop control system of the hidden Markov jump dynamic network is:

[0026] S21: Construct a sampled-data feedback controller model:

[0027]

[0028] where K k is the gain matrix of the controller, and the expression represents the value of at the sampling time Assume that for any non-negative integer τ, the condition holds, where η represents the maximum time interval between samplings;

[0029] S22: Substitute the sampled-data feedback controller model in step S21 into the system model in step S11 to obtain the closed-loop system model of the static output feedback control as follows:

[0030]

[0031] where

[0032] S23: Simplify the above closed-loop system model as follows:

[0033]

[0034] where

[0035]

[0036] denotes the Kronecker product; diag{·} denotes the diagonal matrix; denotes the mathematical expectation; * denotes the symmetric block in the matrix.

[0037] Furthermore, in step S3, the specific manner in which the linear matrix inequality condition is given is as follows:

[0038] S31: Define the following symbols:

[0039]

[0040] S32: For given scalars δ, η, the known matrices and If there exist invertible matrices ∩, P nk > 0, W > 0, Γ > 0, such that the following inequalities hold:

[0041]

[0042] Δ 1 ≥ 0,

[0043] where:

[0044]

[0045]

[0046] Furthermore, in step S4, the manner of verifying whether the linear matrix inequality condition is correct is as follows: S41: Select the Lyapunov function as follows:

[0047]

[0048] S42: Introduce the free weight matrix as follows:

[0049]

[0050] S43: Select the performance index function as follows:

[0051]

[0052] S44: Using the above Lyapunov function, free weight matrix, and performance index function, based on the Lyapunov stability theory, it can be proven that the synchronization error system is stochastically stable and satisfies the specified H ∞ performance index.

[0053] Furthermore, in step S5, the specific manner of solving the gain matrix of the sampled feedback controller is as follows:

[0054] S51: Let \(K = \cap\) -T \(\Omega\) k , combined with the freedom matrix;

[0055] S52: For given scalars \(\delta\), \(\eta\), given matrices and If there exist invertible matrices \(\Gamma>0\), \(W>0\), \(P\) nk \(>0\), \(\Omega\) k \(>0\), \(\cap\), such that the following inequalities hold:

[0056]

[0057] \(\Delta\) 1 \(\geq0\)

[0058] where

[0059]

[0060] S53: Through simulation software, calculate the gain matrix \(K\) according to the known matrix parameters k , and then realize the synchronization of the complex dynamic network and the isolated node.

[0061] The present invention has the following advantages compared with the prior art: Based on the insights mentioned above, this paper explores the SOF synchronization of the hidden MJCDN through sampled data and introduces a new type of HMM designed for complex networks in the continuous time domain. Different from the dependence of the controllers described in most documents on the current system mode, the HMM proposed in this study can determine the current system mode through an observer, thus providing higher flexibility; compared with some existing SOF literatures, this paper also uses integral inequality techniques to eliminate the use of equality constraints, rank conditions on the output matrix, and certain inequalities. In addition, how to reduce the use of redundant data while eliminating the above limitations is also the focus of this paper. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 is the control system block diagram of the image encryption method for hidden Markov jump synchronization control based on sampled data in the embodiment of the present invention;

[0063] Figure 2 is the Markov chain diagram introduced in the present invention;

[0064] Figure 3 is the Markov chain diagram introduced in the present invention;

[0065] Figure 4 is the system error diagram under the controller gain matrix solved in the present invention;

[0066] Figure 5It is the trajectory diagram of the controller under the gain matrix of the controller solved in the present invention;

[0067] Figure 6 They are the original images and corresponding histograms of Channels 1, 2, and 3 in the present invention;

[0068] Figure 7 They are the encrypted images and corresponding histograms of Channels 1, 2, and 3 in the present invention;

[0069] Figure 8 They are the fully decrypted images and corresponding histograms of Channels 1, 2, and 3 in the present invention;

[0070] Figure 9 They are the images and corresponding histograms of Channels 1, 2, and 3 after being decrypted by the key v(t) in the present invention;

[0071] Figure 10 They are the keys of Channels 1, 2, and 3 in the present invention i (t) and the corresponding histograms of the decrypted images. Specific embodiments

[0072] The following details the embodiments of the present invention. These embodiments are implemented on the premise of the technical solution of the present invention, and detailed implementation manners and specific operation processes are given. However, the protection scope of the present invention is not limited to the following embodiments.

[0073] As Figures 1-10 shown, an image encryption method based on sampling data for hidden Markov jump synchronization control applies the synchronization control method of the hidden Markov jump system based on sampling data of the present invention to the image encryption scenario, and specifically includes the following steps:

[0074] Step S1: Use the hidden Markov process to model the random mutation behavior in the complex dynamic network topology;

[0075] The hidden Markov jump complex dynamic network system model is as follows:

[0076]

[0077] Among them, and respectively represent the state variable and control input of the i-th node, is the control output, is the measurement output, A = (a ij ) n×n ∈R n×n is the internal coupling matrix between nodes, B = (B ij ) N×N is the external coupling matrix of the complex dynamic network, is the additional interference belonging to the interval L[0, ∞), is the measurement matrix of the sensor, is the output matrix of the Markov jump complex dynamic network system, defined as and

[0078] is a homogeneous Markov process, taking values, where the transition rate is satisfies the following properties:

[0079]

[0080] where, and the transition rate of (n, l), (m, k) in Q×Y satisfies the following conditions.

[0081]

[0082] The term represents that when and jump simultaneously, satisfying the condition For λ mn ≥0 and m = n, it represents the transition rate from the mode m at time to the mode n; the expression is used to represent the jump when the mode remains unchanged where It should be noted that is directly obtained by the detector, and the detector provides an estimated value of

[0083] and the transition rate matrix Υ λ and Q q , and the detection probability matrix Σ α are respectively selected as

[0084]

[0085] where α = λ = q = {1, 2}. Then the final transition rate is further determined.

[0086] Assume that the system parameters have the following two modes:

[0087]

[0088] Let be the synchronization error, where, Non-forced isolated nodes The state trajectory. Based on this, the expression of the synchronization error system can be derived as follows:

[0089]

[0090] Where, Specifically expressed as follows:

[0091]

[0092] In this embodiment, other parameter selections are as follows:

[0093] δ = 20, η = 0.01

[0094] Step S2: Construct a sampled-data feedback controller model:

[0095]

[0096] Where K k Is the gain matrix of the controller, and the expression Represents the value of At the sampling time Assume that for any non-negative integer τ, the condition Holds, where η represents the maximum time interval between samplings.

[0097] Based on the sampled-data feedback controller model and the system model in Step S1, a closed-loop system model of the hidden Markov jump complex dynamic network based on sampled data is obtained, and a closed-loop system of static output feedback control of the hidden Markov jump complex dynamic network based on sampled data is obtained. Substituting the sampling controller model into the system model gives:

[0098]

[0099] Where,

[0100] The system model can be further written in the following form:

[0101]

[0102] Step S3: Give the linear matrix inequality conditions for the hidden Markov jump dynamic network to satisfy the requirements of stochastic stability and H ∞ Performance index under the action of disturbances.

[0103] Define the following symbols:

[0104]

[0105] For given scalars δ, η, the known matrix and If there exists an invertible matrix ∩, P nk >0, W>0, Γ>0, such that the following inequality holds:

[0106]

[0107] Δ 1 ≥0

[0108] where:

[0109]

[0110]

[0111] then the synchronization error system is stochastically stable and satisfies the specified H ∞ performance index γ.

[0112] Step S4: Verify whether the linear matrix inequality conditions in S3 are correct through the Lyapunov function and the performance index function.

[0113] According to the definition of the weak infinitesimal operator, it can be calculated that:

[0114]

[0115] Lemma 1: For a given symmetric positive definite matrix Λ, and any differentiable function ν(s) in [a, b] → R n then the following inequality holds:

[0116]

[0117] It can be seen from using Lemma 1 that:

[0118]

[0119] According to it can be obtained that:

[0120]

[0121] where,

[0122] By calculation, if the matrices and take the following values, the nonlinear function can be satisfied:

[0123]

[0124] Introduce free weight matrices:

[0125]

[0126] Substituting the above formula, we can get:

[0127]

[0128] where

[0129]

[0130] When satisfies the condition, under zero initial conditions, the following formula holds:

[0131]

[0132] If

[0133]

[0134] If at this time, under zero initial conditions, we can get:

[0135]

[0136] Therefore, the synchronization error system is stochastically stable and satisfies the specified H ∞ performance index γ.

[0137] Step S5: Based on the inequality conditions in Step S3, use integral inequality techniques to eliminate the requirements for equality conditions, reduce the dependence on inequalities, and relax the restrictions on the rank of the input matrix, and solve for the gain matrix of the sampled feedback controller.

[0138] Let K k = ∩ -T Ω k ;

[0139] K k is the control gain. The free weight matrix combines the auxiliary matrices ∩ and Ω to solve for the control gain. Through Matlab simulation, the control gain is finally obtained:

[0140] For given scalars δ, η, the known matrices and If there exist invertible matrices Γ > 0, W > 0, P nk > 0, Ω k > 0, ∩, such that the following inequalities hold:

[0141]

[0142] Δ 1 ≥ 0

[0143] Among them,

[0144]

[0145] Then, for the network to achieve synchronization, the gain matrix of the controller is as follows:

[0146] K 1 = diag{K 1，1 , K 1，2 , K 1，3}, K 2 = diag{K 2，1 , K 2，2 , K 2，3},

[0147] Among them,

[0148]

[0149] According to the derived gain matrix K of the controller k and the given system parameters, assuming the disturbance term

[0150]

[0151] The initial value is selected as

[0152] Figure 2 , 3 represents the stochastic process of the Markov chain. It can be seen from Figure 4 that the controller designed in the present invention successfully realizes the synchronization of the isolated node and the CDN, verifying the feasibility of the controller designed in the present invention.

[0153] In this embodiment, the variables and the motion trajectory of the isolated node in the complex dynamic network are used to encrypt the three channels of the given image. We first set The steps of image encryption are as follows:

[0154] Input: Initial image…;

[0155] Output: Encrypted picture… J ;

[0156] Step 1: The size of the input image is 3*J*K;

[0157] Step 2: The state vector is processed by the equation i takes 1, 2, 3;

[0158] Step 3: Calculate the encryption matrix L z = mod(round(GE*10 4 ), 256) where

[0159] Step 4: Calculate the permutation matrix P according to the dimension of the initial image and the formula 1 ∈R J*J , P 2 ∈R K*K : j ∈ R J , k ∈ R J : NewIndexJ = sort(F 3 - F 1 ), NewIndexK = sort(F 3 - F 2 ), and fill || 3 with the index values returned in sort(F 1 - F 3 - F 2 ), sort(F 1 ; 2 ;

[0160] Step 5: Encrypt the image Τ using the formula ;

[0161] Step 6: Replace the obtained image with that in the previous step

[0162] In the above algorithm, F i is the i-th row vector of F; bitor represents the bitwise "AND-OR" operation; sort(s) represents sorting the data in s and returning the index; mod(s1, s2) represents mod s1 with respect to s2; flip(s) represents flipping the data in the vector x; norm(s) represents the 2 parameters of s; round(s) represents rounding x to the nearest integer.

[0163] According to the algorithm, encrypt and decrypt F i . The generated image is as Figures 6-10 shown. Figure 6 shows the original image and its histogram. a, b, and c are the original images of channels 1, 2, and 3 respectively, and d, e, and f are the histograms of a, b, and c respectively; Figure 7 is the encrypted image. a, b, and c are the encrypted images of channels 1, 2, and 3 respectively, and d, e, and f are the histograms of a, b, and c respectively. The histogram shows a uniform distribution, which is significantly different from the original image, indicating obvious encryption effect and verifying the effectiveness of the synchronization control method in image encryption. The decryption process is carried out through the inverse algorithm, but the complete image cannot be restored only by the system state vector or the state vector of an isolated node, and only partial information can be obtained, as Figures 9-10 shown, Figure 9Among them, a, b, and c are the decryption diagrams of the keys v(t) for channels 1, 2, and 3 respectively, and d, e, and f are the histograms of a, b, and c respectively. Figure 10 Among them, a, b, and c are the keys x for channels 1, 2, and 3 respectively. i (t) decryption diagrams, and d, e, and f are the histograms of a, b, and c respectively. The original image can be restored only when the controller is running and two keys are used, as Figure 8 shown. Figure 8 Among them, a, b, and c are the fully decrypted diagrams for channels 1, 2, and 3 respectively, and d, e, and f are the histograms of a, b, and c respectively.

[0164] In summary, for the synchronous control method of the hidden Markov jump system based on sampled data in the above embodiments, this paper explores the SOF synchronization of the hidden MJCDN through sampled data and introduces a new type of HMM designed for complex networks in the continuous time domain. Different from the dependence of the controllers described in most documents on the current system mode, the HMM proposed in this study can determine the current system mode through an observer, thus providing higher flexibility; compared with some existing SOF literatures, this paper also uses integral inequality techniques to eliminate the use of equality constraints, rank conditions on the output matrix, and certain inequalities. In addition, how to reduce the use of redundant data while eliminating the above limitations is also the focus of this paper.

[0165] Although the embodiments of the present invention have been shown and described above, it can be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those of ordinary skill in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention.

Claims

1. A synchronization control method for a hidden Markov jump system based on sampled data, the steps of which are as follows: S1: Using Hidden Markov Processes to Model Random Mutation Behavior in Complex Dynamic Network Topologies; S2: Design a sampling controller and combine it with the system model of S1 to obtain a hidden Markov jump dynamic network closed-loop control system based on sampling data; S3: Given that under perturbations, the hidden Markov jump dynamic network satisfies random stability and H ∞ Linear matrix inequality conditions required by performance indicators; S4: Verify whether the linear matrix inequality conditions in S3 are correct through Lyapunov function and performance index function; S5: Solve the gain matrix of the sampling feedback controller using the integral inequality technique; S6: Implement synchronization control based on the obtained gain matrix and known system parameters.

2. According to the synchronous control method of hidden Markov jump system based on sampled data in claim 1, in step S1, the specific method of modeling random mutation behavior in complex dynamic network topology is: S11: According to Markov chain Select to hide MJCDNs as follows, where the hidden status Observation status In the formula, x i , and denote the state variable and control input of the ith node respectively, is the control output, is the measured output, A=(a ij ) n×n ∈R n×n is the internal coupling matrix between nodes, B = (B ij ) N×N is the external coupling matrix of the complex dynamic network, is the additional interference on the interval L[0,∞), is the sensor’s measurement matrix, is the output matrix of the Markov jump complex dynamic network system, defined as and S12: is a homogeneous Markov process taking values ​​of Q×Y, where the transition rate is Satisfies the following properties: in, (n,l)≠(m,k), and Conversion rate of (n,l), (m,k) in Q×Y The following conditions must be met: S13: Item Indicates when and When jumping at the same time, the conditions are met For λ mn ≥0 and m=n, indicating The mode m of time The conversion rate of mode n; the expression l≠k is used to represent When the mode remains unchanged of the jump; The value of is given by the detector; S14: Order is the synchronization error, Is an unforced isolated node The state trajectory of the synchronization error system is obtained as follows: in, 3. According to the synchronous control method of the hidden Markov jump system based on sampling data in claim 2, in step S2, the specific method of obtaining the hidden Markov jump dynamic network closed-loop control system is: S21: Constructing the sampled feedback controller model: in, K k is the controller gain matrix, the expression Indicates the sampling time Where For any non-negative integer τ, the condition holds, where η represents the maximum time interval between samples; S22: The sampling feedback controller model in step S21 is brought into the system model in step S11 to obtain a closed-loop system model of static output feedback control as follows: in, S23: The above closed-loop system model is simplified as follows: in, K k =diag{K 1,k ,K 2,k ,...,K N,k }, * denotes a symmetric block in a matrix; E{·} denotes mathematical expectation; diag{·} denotes a diagonal matrix; represents the Kronecker product.

4. According to the synchronous control method of the hidden Markov jump system based on sampling data in claim 1, in step S3, the specific manner in which the linear matrix inequality condition is given is: S31: Define the following symbols: S32: For a given scalar δ, η, given the matrix and If there exists an invertible matrix ∩, P nk >0,W>0,Γ>0, So that the following inequality holds: Δ1≥0, in:

5. According to the synchronous control method of the hidden Markov jump system based on sampling data in claim 1, in step S4, the method of verifying whether the linear matrix inequality condition is correct is: S41: Select the Lyapunov function as follows: S42: The free weight matrix is ​​introduced as follows: S43: Select the performance indicator function as follows: S44: Using Lyapunov function, free weight matrix, performance index function, based on Lyapunov stability theory, it is proved that the synchronization error system is stochastically stable and satisfies the specified H ∞ Performance indicators.

6. According to the synchronous control method of the hidden Markov jump system based on sampled data in claim 1, in step S5, the specific method of solving the gain matrix of the sampling feedback controller is: S51: Order K k =∩ -T Ω k , combined with the freedom weight matrix; S52: For a given scalar δ, η, the matrix and If there exists an invertible matrix Γ>0, W>0, P nk >0,Ω k >0,∩, So that the following inequality holds: Δ1≥0 in, In the formula, the. in D. represents a new unknown matrix obtained by multiplying two unknown matrices, ∩ -T K k =Ω k ; S53: Calculate the gain matrix K based on the known matrix parameters through simulation software k , thereby achieving synchronization of complex dynamic networks and isolated nodes.

Images