Fixed-time non-periodic intermittent control synchronization method based on event trigger containment

Abstract

The invention belongs to the field of complex network synchronous control, and discloses a complex network fixed time event trigger pinning synchronous control method based on aperiodic intermittent control, which aims at solving the problem that a pinning strategy of a traditional fixed pinning node set requires that the node set is fixed and cannot flexibly change along with a network state. Constructing a dynamical model of the master system, the slave system and the synchronization error; designing a fixed-time non-periodic intermittent containment controller based on an event triggering containment mechanism; constructing a Lyapunov function, and solving the synchronization completion time of the complex network; the controller is applied to the slave system, so that the synchronization error between the slave system and the master system is reduced to 0 within the fixed time, and the fixed time synchronization control of the complex network is completed; according to the method, the control condition is controlled based on event triggering, the communication cost is reduced, and meanwhile, the non-periodic intermittent control of the complex network is completed within fixed time. The effectiveness of the control method is verified through numerical simulation.

Description

Technical Field

[0001] The present invention belongs to the field of complex network synchronization control, and more specifically, relates to a fixed-time non-periodic intermittent control synchronization method based on event triggering and containment. Background Art

[0002] Complex networks intuitively demonstrate the interactions between different units and are an effective modeling tool for studying natural, social, and engineering systems in the real world. Synchronization is an important and meaningful issue in the cooperative behavior of network nodes, which means that all nodes in the network eventually reach a common state through a period of coupling. In fact, complex network synchronization has attracted great attention due to its potential applications in various fields such as image processing, secure communication, neural networks, etc.

[0003] Intermittent control is a representative non-continuous control strategy with the advantages of low cost and less information transmission. In the intermittent control strategy, the controller is active during the working time interval and inactive during the rest time interval. Among them, periodic intermittent control requires the duration of the control cycle to be the same. However, this control method has a small scope of application. In reality, most systems, such as wind power generation systems, are non-periodic intermittent. Therefore, it is necessary to study non-periodic intermittent control, which can eliminate periodic constraints and use different control intervals and control frequencies, which enables it to eliminate the imposed restrictions on periodic intermittent control systems and provide researchers with greater flexibility.

[0004] On the other hand, convergence speed is one of the important criteria for measuring the performance of a controller. The goal of finite time control is to make the state of the system reach the desired target state within a predetermined finite time. However, its convergence speed is related to the size of the initial value. In order to eliminate the influence of the initial state of the system on the convergence time, the concept of fixed time stability came into being. Therefore, the fixed time non-periodic intermittent control strategy has good practicality and can save network resources.

[0005] In addition, for a large-scale network, it is unreasonable to impose the controller on all nodes. In order to solve this problem, a series of different containment control strategies have been proposed. The traditional containment strategy of fixed containment node set requires that the node set is fixed and cannot be flexibly changed with the network state. Therefore, it is necessary to introduce an event-triggered containment control strategy, which can transform the containment node set according to the triggering of the designed event, effectively reduce the number of containment nodes, and thus save network communication resources. So far, there is still little research on non-periodic intermittent fixed-time control of complex networks with event-triggered containment control.

[0006] Based on the problems described above, it is an urgent problem to develop an event-triggered containment strategy to reduce communication costs and enable complex networks to complete non-periodic intermittent synchronization within a fixed time. Summary of the invention

[0007] In order to develop an event-triggered containment strategy to reduce communication costs and solve the problem of how to enable complex networks to complete non-periodic intermittent synchronization within a fixed time, the present invention provides a fixed-time non-periodic intermittent control synchronization method based on event-triggered containment, introduces an event-triggered containment control mechanism, and adjusts the containment node set in real time according to the network status. The fixed-time non-periodic intermittent controller only needs to work part of the time, while breaking through the limitations of periodic intermittent control.

[0008] In order to achieve the above object, the present invention adopts the following technical solutions:

[0009] A fixed-time non-periodic intermittent control synchronization method based on event-triggered pinning is proposed. The fixed-time non-periodic intermittent pinning controller is designed by using the event-triggered pinning control mechanism. The pinning node set is adjusted in real time according to the network status of the complex network. The user can accurately design the completion time of data transmission and obtain the optimal control cost. The method specifically includes the following steps:

[0010] Step 1: Based on the complex network, construct the dynamic model of the master system, the slave system and the synchronization error;

[0011] Step 2: Based on the event-triggered control mechanism, design a fixed-time non-periodic intermittent control controller;

[0012] Step 3: Construct a Lyapunov function to solve the time required for the complex network to complete synchronization;

[0013] Step 4: applying the non-periodic intermittent controller to the slave system so that the synchronization error between the slave system and the master system is reduced to 0 within a fixed time, thereby completing the complex network fixed time synchronization control;

[0014] Step 5: Use numerical simulation to verify the effectiveness of the fixed-time non-periodic intermittent control controller designed in step 2 for complex network synchronization control.

[0015] Furthermore, step 1 constructs the dynamic model of the master system, the slave system and the synchronization error based on the complex network, including:

[0016] The dynamic model of the main system is described as:

[0017]

[0018] The dynamic model of the system is described as:

[0019]

[0020] Assume the synchronization error is e i (t) = s i (t)-x i (t), then the dynamic model of synchronization error is described as:

[0021]

[0022] Among them, x i (t) = (x i1 (t),x i1 (t),...,x in (t)) T ∈R n Represents the state variable of the i-th node; f:R n →R n represents the nonlinear function of node dynamics; c>0 represents the coupling strength; Γ∈R n×n , A=(a ij ) N×N denote the inner coupling matrix and the outer coupling matrix respectively; if there is an edge between node i and node j (j≠i), then a ij >0, otherwise a ij = 0, and the diagonal elements are expressed as From the system solution i (t)=(s i1 (t),s i1 (t),...,s in (t)) T ∈R n ;u i (t) represents the fixed-time non-periodic intermittent control controller to be designed; nonlinear function

[0023] Furthermore, in step 2, based on the event-triggered containment mechanism, a fixed-time non-periodic intermittent controller is designed as follows:

[0024] For a positive constant T f >0, design a fixed-time event-triggered control controller based on non-periodic intermittent control so that the complex network meets the following requirements: And when t ≥ T f Time||e i (t)||≡0, the complex network realizes the fixed-time event-triggered control based on non-periodic intermittent control, where T f represents the convergence time;

[0025] Introduce an event-triggered containment mechanism, that is, the controller contains the node set D tThe moment of node replacement in will satisfy the following event triggering conditions:

[0026]

[0027] Among them, the time t ∈ [t r , t r+1 ); The node θ1 is a node that does not belong to the pinning node set D t , and the node θ2 is a node that belongs to the pinning node set D t . The synchronization error The control precision c0>0, σ>0, 0 < ι < 1 are all parameters in network synchronization control;

[0028] Then the fixed-time aperiodic intermittent controller based on the event-triggered pinning mechanism is:

[0029]

[0030] Among them, the gain parameter d of the fixed-time aperiodic intermittent controller i >0, the positive odd numbers w1 and w2 satisfy w1 < w2, and w = w1 / w2 is the exponential parameter of the fixed-time aperiodic intermittent controller; The hyperbolic sine function t0 = 0, the natural number k = 0, 1, 2,...,[t k , s k and (s k , t k+1 ) are the working period and the rest period respectively;

[0031] The fixed-time aperiodic intermittent controller only needs to work at [t k , s k and only needs to control the nodes in the pinning node set D t ; The -d i e i (t) term in the controller makes the synchronization error between the master-slave systems monotonically decreasing, The term makes the synchronization error be reduced to 0 within a fixed time.

[0032] Further, step 3 constructs a Lyapunov function, and the time to solve the complex network to reach synchronization includes:

[0033] The following definitions are given:

[0034] For any k = 0, 1, 2,...,[t

[0035]

[0036] Then Ψ represents the ratio of the non-control interval to the control period, and 0 ≤ Ψ < 1;

[0037] Establishing the semi-intermittent fixed-time stability lemma:

[0038] Suppose there exists an unbounded positive definite function V(t) such that

[0039]

[0040] The constant but And V(t)≡0 when

[0041]

[0042] Establishing the completely intermittent fixed-time stability lemma:

[0043] Assume that the function V(t)≥0 satisfies

[0044]

[0045] in, constant Assume that there exists a positive constant ε>0 such that b-εv<0, then And V(t)≡0 when

[0046]

[0047] Compared with the prior art, the present invention has the following beneficial effects:

[0048] The present invention realizes synchronous control within a fixed time for complex network information transmission, ensures the consistency of event occurrence time and data record on different networks, and contributes to the accuracy and economy of data transmission between various networks. The present invention introduces an event-triggered containment control mechanism, and adjusts the containment node set in real time according to the network status, so that the present invention has less conservatism. The fixed-time non-periodic intermittent controller of the present invention only needs to work part of the time and does not need to rely on the initial value of the complex network. At the same time, it breaks through the limitations of periodic intermittent control, improves practicality and saves network costs. The present invention determines the time for complex network synchronization completion by a fixed-time stability lemma with non-periodic intermittent characteristics. Compared with the existing lemmas of this type, the present invention has fewer stable lemma parameters for solving the time for complex network synchronization completion, a larger range of parameter values, and a given set time that changes according to the range of parameter values, which can effectively avoid the waste of network resources during data transmission. At the same time, it enables users to design the data transmission completion time more conveniently and accurately, obtain the optimal control cost, and improve the completion efficiency of complex network synchronization control. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 is a flow chart of the present invention;

[0050] Figure 2It is the dynamic trajectory of the master system (left figure) and the slave system (right figure) provided by the embodiment of the present invention;

[0051] Figure 3 is a diagram of the convergence process of the synchronization error system provided by an embodiment of the present invention;

[0052] Figure 4 This is a diagram of an event triggering example of a dynamic containment mechanism provided by an embodiment of the present invention;

[0053] Figure 5 This is an example diagram of the total event triggering times of each node provided by an embodiment of the present invention;

[0054] Figure 6 It is a comparison chart of the convergence speed of the control method of the present invention provided by an embodiment of the present invention (left figure) and the common non-periodic intermittent control method in the prior art (right figure). DETAILED DESCRIPTION

[0055] The following describes several preferred embodiments of the present invention with reference to the drawings in the specification, and clearly and in detail describes the technical solutions in the embodiments of the present invention. The present invention can be embodied in many different forms of embodiments, and the protection scope of the present invention is not limited to the embodiments mentioned in the text.

[0056] like Figures 1 to 6 As shown, the present invention discloses a fixed-time non-periodic intermittent control synchronization method based on event-triggered pinning, and designs a fixed-time non-periodic intermittent pinning controller by using the event-triggered pinning control mechanism. Fixed-time control is a control method that does not rely on the initial value of the system, and can complete network synchronization within a fixed time. The pinning node set is adjusted in real time according to the network status of the complex network. The user can accurately design the completion time of data transmission and obtain the optimal control cost. Specifically, the following steps are included:

[0057] Step 1: Based on the complex network, a dynamic model of the master system, the slave system and the synchronization error is constructed; in the present invention, the complex network uses the basic form of the complex network, and based on the complex network, the dynamic model of the master system, the slave system and the synchronization error is constructed, including:

[0058] The dynamic model of the main system is described as:

[0059]

[0060] The dynamic model of the system is described as:

[0061]

[0062] Assume the synchronization error is e i (t) = s i (t)-xi (t), then the dynamic model of synchronization error is described as:

[0063]

[0064] Among them, x i (t) = (x i1 (t),x i1 (t),...,x in (t)) T ∈R n Represents the state variable of the i-th node; f:R n →R n represents the nonlinear function of node dynamics; c>0 represents the coupling strength; Γ∈R n×n , A=(a ij ) N×N denote the inner coupling matrix and the outer coupling matrix respectively; if there is an edge between node i and node j (j≠i), then a ij >0, otherwise a ij = 0, and the diagonal elements are expressed as From the system solution

[0065] s i (t)=(s i1 (t),s i1 (t),...,s in (t)) T ∈R n ;u i (t) represents the controller to be designed; nonlinear function

[0066]

[0067] The nonlinear function f(x i (t))Satisfy:

[0068]

[0069] Among them, γ∈{1,2,...,n}, a positive constant l iγ >0, let L i =diag{l i1 ,...,l in}.

[0070] Step 2: Based on the event-triggered control mechanism, a fixed-time aperiodic intermittent control controller is designed; the control node set of the fixed-time aperiodic intermittent control controller (i.e., the index set in Definition 1) is defined as follows:

[0071] Definition 1: Index set D tis defined as the set of restraining nodes at time t, then: Apply control to node θ2; No control is applied to node θ1.

[0072] Definition 2: Let t b The set of nodes D is t The start time of Assume t z The set of nodes D is t The end time of That is t z -t b Indicates D t The maintenance time, for any l∈[t b ,t z ) has D l =D tb .

[0073] If the fixed-time non-periodic intermittent controller satisfies the following conditions, then the controller can achieve complex network synchronization in fixed time: For a positive constant T f >0, design a fixed-time event-triggered control controller based on non-periodic intermittent control so that the complex network meets the following requirements: And when t ≥ T

[0074] ||e i (t)||≡0, the complex network realizes the fixed-time event-triggered control based on non-periodic intermittent control, where T f Indicates the convergence time.

[0075] Introduce an event-triggered containment mechanism, that is, the controller contains the node set D t The node replacement moment in will meet the following event triggering conditions:

[0076]

[0077] Among them, time t∈[t r ,t r+1 );Node θ1 does not belong to the restraining node set D t , node θ2 belongs to the restraining node set D t Node, synchronization error Control accuracy c0>0, σ>0, 0<ι<1 are all parameters that need to be designed in network synchronization control;

[0078] Then the fixed-time non-periodic intermittent controller based on the event-triggered containment mechanism is:

[0079]

[0080] Among them, the gain parameter d of the controller (i.e., the fixed-time non-periodic intermittent controller) i > 0, the positive odd numbers w1 and w2 satisfy w1 < w2, and w = w1 / w2 is the controller exponential parameter; the hyperbolic sine function t0 = 0, the natural number k = 0, 1, 2,...,[t k , s k and (s k , t k+1 ) are the working period and the rest period respectively, and s k represents the moment when the working period ends;

[0081] This controller only needs to work within the time period [t k , s k , and only needs to control the nodes in the pinning node set D t ; the -d i e i (t) term makes the synchronization error between the master-slave systems monotonically decreasing, and the term makes the synchronization error be reduced to 0 within a fixed time.

[0082] The selection of D t is determined by the following rules:

[0083] Rule 1:

[0084] S1: Let t, P1, P2, and establish linked lists [Λ], [p],

[0085] S2:

[0086] S3: Arrange [Λ] in descending order (by the sorting rule):

[0087] S4:

[0088] S5: Find the largest term in that satisfies

[0089]

[0090] Store its subscript as △:

[0091]

[0092] Among them, the control accuracy Let P1 and P2 be the minimum and maximum proportions of the number of pinned nodes to the total number of nodes respectively, then 0 < P1 < P2 < 1, and all of the above are parameters required in the synchronization control.

[0093] Step 3: Construct a Lyapunov function and solve the time for the complex network to reach synchronization;

[0094] Definition 3: For any k = 0, 1, 2,..., assume

[0095]

[0096] Then Ψ represents the ratio of the non-control interval to the time period, and 0 ≤ Ψ < 1.

[0097] Definition 4: For the controller of formula (6), there exists 0 < v < 1, and the elasticity number T v ≥ 0 such that

[0098]

[0099] where [t y , t x represents a time period, T(t x , t y ) represents the maximum control time period in [t y , t x , and v represents the average control rate.

[0100] Lemma 1: For any k = 0, 1, 2,..., let

[0101]

[0102] Then Ψ(t) is a strictly monotonically increasing function and

[0103] Lemma 2: There exist positive constants a i ≥ 0 (i ∈ N) such that

[0104] Lemma 3: Assume that there exists a non-negative scalar function Ο(t) satisfying:

[0105] (1)

[0106] (2) For any t ≥ 0, Ο(t) ≥ 0 and Ο(t) is differentiable with respect to t;

[0107] (3) When Ο ≥ 0 and χ(0) = 0, χ(Ο) > 0;

[0108] where χ(·) is a function, let Ο0 = Ο(0), then there is:

[0109] If is finite, then when t ≥ T1(Ο0), Ο(t) = 0;

[0110] If is finite, then when t ≥ T2 and Ο0 > 0, Ο(t) = 0.

[0111] Establish the semi - batch fixed - time stability lemma:

[0112] Suppose there exists an unbounded positive definite function V(t). If V(t) satisfies

[0113]

[0114] where is the derivative of V(t), and the constant then when

[0115]

[0116] there is and V(t) ≡ 0;

[0117] First, suppose When t ∈ [t k , s k , there is:

[0118]

[0119] Therefore, when t ∈ [t k , s k , let V0 = V(0). According to Lemma 3, it only needs to be proved that is finite.

[0120] Case 1 When t ∈ [t k , s k , 0 < V0 < 1,

[0121]

[0122] Suppose Then There is:

[0123]

[0124] When t ∈ [t k , s k , V0 ≥ 1,

[0125]

[0126] Therefore, when t ∈ [t k , s k , the synchronization error system of formula (3) is fixed-time stable, and there is

[0127]

[0128] When 0 < V0 < 1 and t ∈ [0, s0], let Θ0(t) = J(t) - H0, then Θ0(0) = 0. Taking the derivative of Θ0(t) and calculating, we get:

[0129]

[0130] From this, it can be further deduced that Θ0(t) is monotonically decreasing on t ∈ [0, s0], and J(t) ≤ H0. When t ∈ [s0, t1) and 0 < V0 < 1, let Taking the derivative of Θ0(t):

[0131]

[0132] Therefore,[[]]

[0133] When t ∈ (s k , t k+1 ), 0 < V0 < 1, using mathematical induction and it can be deduced that:

[0134]

[0135]

[0136] Among them, t ∈ (s k , t k+1 ), 0 < V0 < 1.[[]]

[0137] When V0 ≥ 1 and t ∈ [0, s0], let Θ0(t) = J(t) - H0, then Θ0(0) = 0. Taking the derivative of Θ0(t):

[0138]

[0139] Therefore, Θ0(t) is monotonically decreasing on t ∈ [0, s0], and J(t) ≤ H0. When t ∈ [s0, t1) and V0 ≥ 1, let Taking the derivative of Θ0(t) and calculating, we get:

[0140]

[0141] Therefore,[[]]

[0142] When \(V_0\geq1\) and \(t\in(s k ,t k+1 ), the mathematical induction and can be deduced that:

[0143]

[0144] Wherein, t\in(s k ,t k+1 ), \(V_0\geq1\).

[0145] When \(t\in(s k ,t k+1 ), Let

[0146] Combined with \(T f11 and \(T 31 , when \(t\in[0, +\infty)\), the convergence time is:

[0147]

[0148] Case 2 When \(t\in[t k ,s k , there is:

[0149]

[0150] Then the convergence time at this time is

[0151] Similarly to the derivation in Case 1, it can be obtained that when \(0 < V_0 < 1\), let When \(t\in[s_0, t_1)\), let Then there is

[0152] When \(t\in(s k ,t k+1 ), the mathematical induction and can be deduced that:

[0153]

[0154] When \(V_0\geq1\), let Derive \(\Theta(t)\):

[0155]

[0156] Therefore

[0157] When \(t\in(s k ,t k+1 ), using mathematical induction and it can be deduced that:

[0158]

[0159] When t\in(s k ,t k+1 ), the convergence time is From this, it can be obtained that when t\in[0, +\infty), the convergence time is

[0160] Case 3 When \(t\in[t k ,s k ,

[0161]

[0162] Similarly to the derivation in Case 1, it can be obtained that:

[0163] Let Then there is Furthermore, there is:

[0164]

[0165] When \(t\in(s k ,t k+1 ), 0 < V0 < 1, similarly to the derivation in Case 1, let When \(t\in[s0, t1)\), let Derive \(\Theta(t)\):

[0166]

[0167] Therefore,

[0168] When \(t\in(s k ,t k+1 ), using mathematical induction and it can be deduced that:

[0169]

[0170] When V0 ≥ 1, let When \(t\in[s0, t1)\), let Derive \(\Theta(t)\):

[0171]

[0172] Therefore,

[0173] When t∈(s k ,t k+1 ), use mathematical induction and Can be pushed:

[0174]

[0175] Then when t∈(s k ,t k+1 )hour, Therefore, when When t>0, the convergence time is:

[0176] Combined with T f1 , T f2 , T f3 It can be obtained that the convergence time is

[0177] Establishing the completely intermittent fixed-time stability lemma:

[0178] Assume that the function V(t)≥0 satisfies

[0179]

[0180] in, constant Assume that there exists a positive constant ε>0 satisfying b-εv<0, then

[0181]

[0182] Sometimes, there are And V(t)≡0.

[0183] First, establish the auxiliary function Ξ(t):

[0184]

[0185] Among them, γ(t,ζ i )=min{εT v ,Ξ(s k )+ε(1-v)(tt k )}, then Ξ(t)≤εT v , and there are:

[0186]

[0187] When δ∈[t0,s0], Ξ(δ)=εT v , then for any t∈[0,+∞), there always exists Combining Definition 4 and Formula (51), we can obtain:

[0188]

[0189] Then we have:

[0190] 0≤Ξ(t)≤εT v , t∈[0,+∞)(54)Let Q(t)=e Ξ(t) V(t), when t∈[t k ,s k ], take the derivative of Q(t):

[0191]

[0192] Let T v →0, then:

[0193]

[0194] When t∈(s k ,t k+1 ), there are:

[0195]

[0196] Therefore, from formula (56) and formula (57), we can get:

[0197]

[0198] Combining the semi-intermittent fixed-time stability lemma and Q(t) = e Ξ(t) V(t) shows that the completely intermittent fixed-time stability lemma holds, and the convergence time is

[0199] Step 4: Apply the non-periodic intermittent controller to the slave system so that the synchronization error between the slave system and the master system is reduced to 0 within a fixed time, thereby completing fixed time synchronization; the fixed time in step 4 is the same as the fixed time in the fixed time non-periodic intermittent control controller in step 2.

[0200] From step 2, we know that the definition of complex network fixed time synchronization is: when the complex network satisfies: And when t ≥ T || e i (t)||≡0, the complex network achieves fixed-time synchronization. The lemma of constructing the Lyapunov function in step 3 to solve the time for the complex network to complete synchronization proves the effectiveness of the complex network fixed-time event triggering and pinning synchronization control method based on non-periodic intermittent control, and can exclude Zeno behavior.

[0201] First, suppose:

[0202]

[0203] Where ρ = λ max (Γ T Γ). If there exists b-εv<0, a positive constant k>0, (2k+1)ω>5, Then the synchronization error system of formula (3) completes fixed-time synchronization under the complex network fixed-time event-triggered pinning synchronization control method based on non-periodic intermittent control, and the convergence time is:

[0204]

[0205] set up

[0206]

[0207] When t∈[t k ,s k ]hour,

[0208]

[0209]

[0210] in, Expanded by Taylor's formula Can be pushed:

[0211]

[0212] in, μ=γ 2k+1 / ((2k+1)!), μ m =γ 2m+1 / ((2k+1)!), m>k.

[0213] set up Substituting formula (64) and formula (65) into formula (63), we obtain:

[0214]

[0215] Similarly, when t∈(s k ,t k+1 ), there are:

[0216]

[0217] Assume R i represents the cofactor of the ith diagonal element of A. Using the directed graph (G, A) and Kirchhoff's matrix tree theorem, we can get: R i >0(i∈N). Definition:

[0218]

[0219] When t∈[t k ,s k ], it can be deduced from formula (66) and formula (68):

[0220]

[0221] From the tree cycle characteristics, we can get:

[0222]

[0223] Then we have:

[0224]

[0225] in,

[0226] Similarly, when t∈(s k ,t k+1 ), it can be deduced from formula (64):

[0227]

[0228] Combining formula (71) and formula (72), according to the completely intermittent fixed-time stability lemma and the definition of fixed-time synchronization of complex networks, it can be seen that the synchronization error system can complete synchronization within a fixed time.

[0229] Assume t r+1 -t r ≥κ, if the constant κ>0, then the event-triggered containment mechanism in the present invention can exclude Zeno behavior. From the above content, it can be seen that e(t) is bounded, so is bounded. Let e(t) and The upper bounds of are N1 and N2 respectively, then:

[0230]

[0231] set up From formula (75) and formula (76), we can deduce:

[0232]

[0233] From the event triggering condition of formula (5) and formula (77), we can get:

[0234]

[0235] From formula (78), we can see that κ>0.

[0236] Step 5: Use numerical simulation to verify the effectiveness of the fixed-time non-periodic intermittent control controller designed in step 2 for complex network synchronization control.

[0237] A single pendulum system is used to illustrate the effectiveness of the fixed-time non-periodic intermittent control controller of the present invention in achieving synchronous control of complex networks.

[0238] Assume a single pendulum system:

[0239]

[0240] Where ω(t) represents the angular displacement, represents the angular velocity, ξ=1, θ=0.25. Let x1=ω(t), Then we have:

[0241]

[0242] Further assume that a pendulum system with seven nodes is:

[0243]

[0244] Where i∈{1,2,...,7}, N=7. The external coupling matrix is:

[0245]

[0246] Assuming formula (81) as the master system, the slave system is:

[0247]

[0248] Assume e i (t) = s i (t)-x i (t), then the synchronization error system is:

[0249]

[0250] Select parameters Γ=diag{1,1},c=0.2,c0=0.01,γ=2,σ=4,ι=0.1, P1=0.2, P2=0.5,

[0251] d i =2.03,ω=0.2,L i =1, ε=1.67, k=12.1.

[0252] Assume the intermittent interval is:

[0253]

[0254] Then v=0.6,T v =0, ψ=0.4, which satisfies the conditions of Definition 4. By simple calculation, we can get: b-εv<0, k>0, (2k+1)ω>5, The proof condition of complex network synchronization in step 4 is met, and the convergence time T is calculated. f is 6.49 seconds. Figure 3 It can be obtained that the synchronization time in the simulation results is about 2.59, which is less than 6.49 seconds, that is, the control method of the present invention is effective.

[0255] The common non-periodic intermittent controller forms in the prior art are as follows:

[0256]

[0257] To emphasize the contrast, remove the control mechanism in the controller in step 2, and you have:

[0258]

[0259] In formula (85), select the following parameters: p = 0.8, q = 4, l1 = 8, l2 = l3 = 0.05; in formula (86), select the following parameters: d i =8, γ=4, ω=0.2; the complex network is synchronously controlled by the controller of formula (85) and the controller of formula (86), and the simulation results are as follows: Figure 6 Obviously, compared with the controller of formula (86), the complex network synchronization time obtained by the controller of the present invention is shorter and the control efficiency is higher.

[0260] The contents not described in detail in the specification of the present invention belong to the prior art known to the professional and technical personnel in the field. Although the illustrative specific embodiments of the present invention are described above to facilitate the understanding of the present invention by the technical personnel in the field, it should be clear that the present invention is not limited to the scope of the specific embodiments. For the ordinary technical personnel in the field, as long as various changes are within the spirit and scope of the present invention defined and determined by the attached claims, these changes are obvious, and all inventions and creations using the concept of the present invention are protected.

Claims

1. A fixed-time non-periodic intermittent control synchronization method based on event triggering, characterized in that: The fixed-time non-periodic intermittent pinning controller is designed by using the event-triggered pinning control mechanism. The pinning node set is adjusted in real time according to the network status of the complex network. The user can accurately design the completion time of data transmission and obtain the optimal control cost. The specific steps include: Step 1: Based on the complex network, construct the dynamic model of the master system, the slave system and the synchronization error; Step 2: Based on the event-triggered control mechanism, design a fixed-time non-periodic intermittent control controller; Step 3: Construct a Lyapunov function to solve the time required for the complex network to complete synchronization; Step 4: applying the non-periodic intermittent controller to the slave system so that the synchronization error between the slave system and the master system is reduced to 0 within a fixed time, thereby completing the complex network fixed time synchronization control; Step 5: Use numerical simulation to verify the effectiveness of the fixed-time non-periodic intermittent control controller designed in step 2 for complex network synchronization control.

2. The fixed-time non-periodic intermittent control synchronization method based on event triggering control according to claim 1 is characterized in that: The step 1 constructs the dynamic model of the master system, the slave system and the synchronization error based on the complex network, including: The dynamic model of the main system is described as: The dynamic model of the system is described as: Assume the synchronization error is e i (t) = s i (t)-x i (t), then the dynamic model of synchronization error is described as: Among them, x i (t) = (x i1 (t),x i1 (t),...,x in (t)) T ∈R n Represents the state variable of the i-th node; f:R n →R n represents the nonlinear function of node dynamics; c>0 represents the coupling strength; Γ∈R n×n , A=(a ij ) N×N denote the inner coupling matrix and the outer coupling matrix respectively; if there is an edge between node i and node j (j≠i), then a ij >0, otherwise a ij = 0, and the diagonal elements are expressed as From the system solution i (t)=(s i1 (t),s i1 (t),...,s in (t)) T ∈R n ;u i (t) represents the fixed-time non-periodic intermittent control controller to be designed; nonlinear function 3. The fixed-time non-periodic intermittent control synchronization method based on event triggering control according to claim 1 is characterized in that: In step 2, based on the event-triggered control mechanism, a fixed-time non-periodic intermittent controller is designed as follows: For a positive constant T f >0, design a fixed-time event-triggered control controller based on non-periodic intermittent control so that the complex network meets the following requirements: And when t ≥ T f Time||e i (t)||≡0, the complex network realizes the fixed-time event-triggered control based on non-periodic intermittent control, where T f represents the convergence time; Introduce an event-triggered containment mechanism, that is, the controller contains the node set D t The node replacement moment in will meet the following event triggering conditions: Among them, time t∈[t r ,t r+1 );Node θ1 does not belong to the restraining node set D t , node θ2 belongs to the restraining node set D t Node, synchronization error The control accuracy c0>0, σ>0, 0<ι<1 are all parameters in network synchronization control; Then the fixed-time non-periodic intermittent controller based on the event-triggered containment mechanism is: Among them, the fixed-time aperiodic intermittent controller gain parameter d i > 0, the positive odd numbers w1 and w2 satisfy w1 < w2, and w = w1 / w2 is the fixed-time aperiodic intermittent controller exponential parameter; the hyperbolic sine function t0 = 0, the natural number k = 0, 1, 2,..., [t k , s k and (s k , t k+1 ) are the working time period and the rest time period, respectively; The fixed-time non-periodic intermittent controller only needs to k ,s k ] and only needs to control the set of pinned nodes D t nodes in the controller; -d in the controller i e i The term (t) makes the synchronization error between the master and slave systems monotonically decrease. The term enables the synchronization error to be reduced to 0 within a fixed time.

4. The fixed-time non-periodic intermittent control synchronization method based on event triggering control according to claim 1 is characterized in that: The step 3 constructs the Lyapunov function to solve the time for the complex network to reach synchronization, including: Given the following definition: For any k = 0, 1, 2, ..., assuming Then Ψ represents the ratio of the non-control interval to the control period, and 0≤Ψ<1; Establishing the semi-intermittent fixed-time stability lemma: Suppose there exists an unbounded positive definite function V(t) such that The constant but And V(t)≡0 when Establishing the completely intermittent fixed-time stability lemma: Assume that the function V(t)≥0 satisfies in, constant Assume that there exists a positive constant ε>0 such that b-εv<0, then And V(t)≡0 when

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