Heterogeneous four-pendulum system fixed time synchronization method based on pulse control

Abstract

The invention discloses a heterogeneous four-pendulum system fixed time synchronization method based on pulse control. The method comprises the following steps: firstly, designing a fixed time controller based on pulse control for a four-pendulum system with heterogeneous characteristics; and designing a novel Lyapunov functional, partitioning pulse intervals, and establishing a fixed time synchronization sufficient condition represented by a matrix inequality by using a convex combination technology to realize synchronization of the four-pendulum system. A comparison system is constructed to estimate the stabilization time under synchronous pulse, inactive pulse and asynchronous conditions. When the method is applied to the heterogeneous four-pendulum system, the synchronization stabilization time of the system is shortened, and the actual efficiency is improved.

Description

Technical Field

[0001] The present invention relates to a method for realizing fixed time synchronization of a four-pendulum system, and in particular to a fixed time synchronization method for a heterogeneous four-pendulum system based on pulse control. Background Art

[0002] Many dynamic models for four-pendulum systems only consider homogeneous network models. In reality, heterogeneous networks composed of individuals with different dynamics are ubiquitous. For example, the generator in the power grid is controlled by an oscillation equation with different inertia coefficients, different viscous damping, and different natural frequencies. The manipulator with Lagrangian dynamics in multiple manipulators is described by a second-order differential equation with different inertia matrices that characterize its Coriolis force and centrifugal force. Since heterogeneity may degrade the synchronization performance of the system, studying the synchronization behavior in heterogeneous dynamic networks is more challenging. Based on the above ideas, researchers have carried out extensive research on the output synchronization of four-pendulum systems and achieved a series of results.

[0003] Furthermore, the fixed-time controller used in the current dynamic model of the four-pendulum system is a sum of signed power functions. This not only has numerous parameters and a complex structure, but can also lead to chattering. Therefore, it is crucial to use a hyperbolic sine function as the fixed-time controller to reduce parameters, simplify the structure, and avoid chattering. Summary of the Invention

[0004] The purpose of this invention is to propose a fixed-time synchronization method for a heterogeneous four-pendulum system based on pulse control, which can effectively improve the stability and convergence time of the system.

[0005] The specific technical solution of the present invention is as follows: A fixed time synchronization method for a heterogeneous four-pendulum system based on pulse control, comprising the following steps:

[0006] A fixed-time controller based on impulse control is designed for a four-pendulum system with heterogeneous characteristics.

[0007] The four-pendulum system with heterogeneous characteristics and the desired target dynamic model are as follows:

[0008]

[0009] Where i = 1, 2, ..., N, N represents the total number of nodes, g(·) represents the nonlinear activation function, w i (t) represents the i-th system state variable, w0(t) represents the desired target system state variable, u i (t) represents the control input variable of the i-th system, Γ represents the coupling matrix within the system, S = (s ij ) N×N represents the system external coupling matrix, Ci , H i , C, H represent known parameter matrices of appropriate dimensions; g(·) has the following properties:

[0010] |g i (ζ1)-g i (ζ2)|≤δ i |ζ1-ζ2|,ζ1≠ζ2,δ i >0, where |·| represents the 1-norm of the vector.

[0011] in

[0012] Define the synchronization error of the i-th node as e i (t) = w i (t)-w0(t), we can get the error system

[0013] Construct the following controller:

[0014] Where η i >0 indicates the control gain constant, and is satisfied A positive odd number, sh(·) represents the hyperbolic sine function: r0 is a non-negative integer, represents the pulse gain matrix, Among them I n is the n-dimensional identity matrix, is the pulse gain, which depends on the pulse time t r and node i, δ(·) is the Dirac function, δ(x)=0, (x≠0), t r Has the following properties: τ min <t r -t r-1 ≤τ max , r∈R0,t0≥0,τ min >0,τ max >0, R0={1,…,r0}

[0015] Where ξ=0.5, η i =3.2, τ min =0.04, τ max =0.06,

[0016] At this point, the error system can be rewritten as

[0017] Define e(t)=(e1 T (t), e2 T (t), ..., e N T (t)) T , ,G(e(t))=(g T (e1(t)), g T (e2(t)),...,g T (e N (t))) T , Λ=diag(n1,n2,...,n N ), The error system can then be rewritten as

[0018] A novel Lyapunov functional is designed. The pulse interval is partitioned and convex combination techniques are used to establish the necessary and sufficient conditions for fixed-time synchronization expressed as matrix inequalities. The synchronization of the four-pendulum system is achieved. The specific steps are as follows:

[0019] C001: Select the following energy function:

[0020] V(t)=V(e(t))=e T (t)P(t)e(t), Where P(t) represents a positive definite diagonal matrix and T represents the transpose.

[0021] C002: Considering a positive integer Θ>0, the pulse interval [t r , t r+1 ) is divided into Θ+1 subintervals. Define the starting time of the i+1th subinterval Where i = 0, 1, ..., Θ-1, then when i = 0, t r，0 =t r , that is, the starting time of the first subinterval is the same as the starting time of the original pulse interval. When i=Θ, t r，Θ =t r +τ min , that is, after Θ sub-interval division, the specific moment obtained is,

[0022] C003: Define the i+1th subinterval Then for P(t)=P(tr,i +ρF)=(1-ρ)P i +ρP i+1 =P i (ρ), where P i is a positive definite matrix, is a term used to describe the subinterval The parameter of the relative position of time t in the subinterval normalizes the position of time t in the subinterval. Its value range is [0, 1], which reflects the relative progress of time t in the subinterval. is a minimum time interval T min The parameter related to the partition parameter Θ is the minimum time interval of each subinterval in the normalized time scale. It follows that 0≤ρ≤1,

[0023] C004: When t∈[t r ,Θ,t r+1 ), P(t) is fixed as a positive definite matrix P θ . It follows that for r∈N + ,

[0024] C005: For β i >0, i=0, 1,..., Θ, define β(t)=(1-ρ)β i +ρβ i+1 =β i (ρ), i=0, 1, ..., Θ-1. Thus, for r∈N + ,

[0025] C006: Using the above specific time-varying variables P(t) and β(t) and the convex combination technique, we can obtain:

[0026]

[0027] C007: For When t≠t r , calculate the first-order derivative of V(t):

[0028]

[0029] C008: Since when i=0, 1, ..., Θ-1, β i >0 and β i (ρ)>0, so using the global Lipschitz condition, we can get:

[0030] β i (ρ)G T (e(t))G(e(t))≤βi (ρ)Δ 2 e T (t)e(t),

[0031] C009: Due to Combining C007-C008, we can get:

[0032]

[0033] in

[0034] C010: When When, based on P i (ρ) and β i Definition of (ρ), Ω i (ρ)=(1-ρ)ζ i +ρζ i+1 <0 is established,

[0035] C011: Combining C009-C010, we can get:

[0036]

[0037] C012: For t∈[t r +τ min , t r+1 ), when t≠t r When , we can get:

[0038]

[0039] C013: When When , we can get:

[0040]

[0041] C014: Combining C011 and C013, for t≠t r , we can get:

[0042]

[0043] C015: Using Taylor expansion We can get:

[0044]

[0045] C016: From the Taylor expansion of sh(·), we know that for a given There exists J>0 such that definition We can get:

[0046]

[0047] in Therefore, q m >q>1,

[0048] C017: Using the properties of inequality, we can get:

[0049]

[0050] C018: Combining C001-C017, we can get:

[0051]

[0052] in λ min (P(t)) and λ max (P(t)) represents the minimum and maximum eigenvalues ​​of P(t), inf{·} represents the maximum lower bound of a set or function, sup{·} represents the minimum upper bound of a set or function,

[0053] C019: Since qm>q, when V(t)>1, we can get:

[0054]

[0055] C020: For t=t r , r∈N + ,when When , we can get:

[0056]

[0057] C021: Combined with C018 and C020, the system achieves fixed time synchronization according to the fixed time stability definition.

[0058] Construct a comparison system to estimate the optimal settling time under synchronous pulse, inactive pulse and asynchronous conditions. The specific steps are as follows:

[0059] Construct the following systems for comparison:

[0060]

[0061] When Φ(t) ≥ 1, define j(t) = [Φ(t)] 1-q , from the comparison system, it can be seen that when Φ(0) → ∞, j(0) → 0, and when Φ(t) → 1, j(t) → 1. Therefore, it can be obtained that:

[0062]

[0063] where At this time, Φ(t) → 1 + is equivalent to j(t) → 1 - .

[0064] When 0 ≤ Φ(t) < 1, define j(t) = [Φ(t)] 1-p , from the comparison system, it can be seen that when Φ(t) → 1, j(t) → 1, and when Φ(t) → 0, j(t) → 0. Therefore, it can be obtained that:

[0065]

[0066] where At this time, Φ(t) → 0 + is equivalent to j(t) → 0 + .

[0067] Estimated stable times in the case of synchronous pulses, inactive pulses, and asynchronous conditions: 0 < α < 1, α = 1, and α > 1.

[0068] Consider case 1: 0 < α < 1. For this case, there are and

[0069] For 0 < j(t) < 1, it can be obtained that: where

[0070] Since and [[ID=5३]]Then when 0 < t < T1, there exists T1 such that and j(t) ∈ (0, 1). Therefore, it can be obtained that:

[0071] Furthermore, it can be obtained that:

[0072] Considering [[ID=६5]]it can be obtained that:

[0073] Solving the above inequality, it can be obtained that:

[0074] Substitute into the above inequality, it can be obtained that:

[0075] Next, calculate T2 so that j(t) changes from 1 to 0. Similar to the calculation of T1, we can get:

[0076]

[0077] j(t) is monotonically decreasing on [0,∞) due to We can get: Let j(0) = 1, j(t) = 0, we can get:

[0078]

[0079] definition because and Therefore there exists a unique T2>0 such that

[0080] in Therefore, the optimal stabilization time

[0081] Consider case 2: ɑ=1. For this case, we have

[0082] Similar to case 1, we can get: Therefore, the optimal stabilization time

[0083] Consider case 3: α>1. For this case, we have and

[0084] because We can get:

[0085] Since τ min <τ max , we can get: when t→+∞, the right side of the inequality tends to infinity; therefore, there is no guarantee that there is a fixed time Make j(t) approach 0 from 1. BRIEF DESCRIPTION OF THE DRAWINGS

[0086] Figure 1 The state trajectory diagram of the four-pendulum system using the method proposed in the present invention without control input is shown in the following example;

[0087] Figure 2 The state trajectory diagram of the four-pendulum system using the method proposed in the present invention with control input is shown in the following example;

[0088] Figure 3: This is a state trajectory diagram of a four-pendulum system using the method proposed in the embodiment of the present invention when there is a control input and r0=0;

[0089] Figure 4 Graph showing the synchronization error of a four-pendulum system with control input using the method proposed in the present invention;

[0090] Figure 5 This is a comparison diagram of synchronization errors using the method proposed in the present invention and two different fixed time controllers;

[0091] Figure 6 This is a comparison diagram of the control input using the method proposed in the present invention and two different fixed-time controllers. DETAILED DESCRIPTION

[0092] The present invention is further illustrated below with reference to specific examples. It should be understood that these examples are only used to illustrate the present invention and are not used to limit the scope of the present invention. After reading the present invention, modifications of various equivalent forms of the present invention made by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0093] A fixed-time synchronization method for a heterogeneous four-pendulum system based on pulse control includes the following steps:

[0094] Step 1: Set various system parameters;

[0095] Step 2: Solve the positive definite matrix P(t) and parameters β(t) and α that meet the conditions through the fixed time stability condition;

[0096] Step 3: Set the initial state of the system;

[0097] Step 4: Calculate the stabilization time T based on the set system parameters;

[0098] Step 5: Remove the control input of the system and verify whether the system can achieve synchronization without the control input.

[0099] Step 6: When the system's control input only has fixed-time control input and no pulse control input, verify whether the system achieves synchronization without pulse control input;

[0100] Step 7: Verify whether the system achieves synchronization within the settling time T under the combined action of fixed time and pulse control input;

[0101] Step 8: Compare the synchronization error and control input of the method proposed by the present invention with those of two different fixed-time controllers to verify its advantages;

[0102] An embodiment of the present invention is described below:

[0103] Considering a four-pendulum system with heterogeneous characteristics, its corresponding dynamic model and the designed controller are:

[0104]

[0105] in c1=c2=0.2, ξ=0.5, η i =3.2, τ min =0.04, τ max =0.06,

[0106] The state trajectory diagram of the four-pendulum system without control input is as follows Figure 1 As shown, the state trajectory diagram of the four-pendulum system with control input is as follows Figure 2 As shown, the state trajectory diagram of the four-pendulum system with control input and r0 = 0 is as follows Figure 3 As shown in the figure, the synchronization error diagram of the four-pendulum system with control input is as follows Figure 4 As shown, the stabilization time T=14.7851 can be obtained by calculation. The method proposed by the present invention is used in the controller. Under and in controller The synchronization error under the control input is compared as follows Figure 5 and 6 As shown, it can be seen that the method proposed in the present invention can converge the error to zero faster, verifying its high convergence performance, and only requires lower control input, thereby saving a lot of cost.

Claims

1. A fixed time synchronization method for heterogeneous four-pendulum system based on pulse control, characterized in that: The following steps are involved: A fixed-time controller based on pulse control is designed for a four-pendulum system with heterogeneous characteristics. Specifically: The four-pendulum system with heterogeneous characteristics and the desired target dynamic model are as follows: Where i = 1, 2, ..., N, N represents the total number of nodes, g(·) represents the nonlinear activation function, w i (t) represents the i-th system state variable, w0(t) represents the desired target system state variable, u i (t) represents the control input variable of the i-th system, Γ represents the coupling matrix within the system, S = (s ij ) N×N represents the system external coupling matrix, C i , H i , C, H represent known parameter matrices of appropriate dimensions; g(·) has the following properties: |g i (ζ1)-g i (ζ2)|≤δ i |ζ1-ζ2|,ζ1≠ζ2,δ i >0, where |·| represents the 1-norm of the vector. in c1=c2=0.2,, Define the synchronization error of the i-th node as e i (t) = w i (t)-w0(t), we can get the error system Construct the following controller: Where η i >0 indicates the control gain constant, and is satisfied A positive odd number, sh(·) represents the hyperbolic sine function: r0 is a non-negative integer, represents the pulse gain matrix, Among them I n is the n-dimensional identity matrix, is the pulse gain, which depends on the pulse time t r and node i, δ(·) is the Dirac function, δ(x)=0,(x≠0), t r Has the following properties: T min <t r -t r-1 ≤τ max ,r∈R0,t0≥0,τ min >0,τ max >0,R0={1,…,r0} where ξ = 0.5, η i = 3.2, τ min = 0.04, τ max = 0.06, At this point, the error system can be rewritten as Define e(t)=(e1 T (t), e2 T (t), ..., e N T (t)) T , ,G(e(t))=(g T (e1(t)), g T (e2(t)),...,g T (e N (t))) T , Λ=diag(n1,n2,…,n N ), The error system can then be rewritten as A novel Lyapunov functional is designed to partition the pulse interval and use convex combination technology to establish a fixed-time synchronization sufficient condition expressed as a matrix inequality to achieve the synchronization of the four-pendulum system. Specifically, B001: Select the following energy function: V(t)=V(e(t))=e T (t)P(t)e(t), Where P(t) represents a positive definite diagonal matrix and T represents the transpose. B002: Considering a positive integer Θ>0, the pulse interval [t r , t r+1 ) is divided into Θ+1 subintervals. Define the starting time of the ι+1th subinterval Where ι=0,1,...,Θ-1, then when ι=0, t r，0 =t r , that is, the starting time of the first subinterval is the same as the starting time of the original pulse interval. When ι=Θ, t r，Θ =t r +τ min , that is, the specific moment obtained after being divided into Θ subintervals. B003: Define the ι+1th subinterval Then for P(t)=P(t r，ι +ρF)=(1-ρ)Pι+ρP ι+1 =P ι (ρ), where P ι is a positive definite matrix, is a term used to describe the subinterval The parameter of the relative position of time t in the subinterval normalizes the position of time t in the subinterval. Its value range is [0, 1], which reflects the relative progress of time t in the subinterval. is a minimum time interval τ min The parameter related to the partition parameter Θ is the minimum time interval of each subinterval in the normalized time scale. It follows that 0≤ρ≤1, B004: When t∈[t r，Θ , t r+1 ), P(t) is fixed as a positive definite matrix P Θ . It follows that for r∈N + , B005: For β ι >0, ι=0, 1,..., Θ, define β(t)=(1-ρ)β ι +ρβ ι+1 =β ι (ρ), ι=0, 1, ..., Θ-1. From this, we can get that for r∈N + , B006: Using the above specific time-varying variables P(t) and β(t) and the convex combination technique, we can obtain: B007: For When t≠t r , calculate the first-order derivative of V(t): B008: Since when ι=0, 1, ..., Θ-1, β ι >0 and β ι (ρ)>0, so using the global Lipschitz condition, we can get: β ι (ρ)G T (e(t))G(e(t))≤β ι (ρ)Δ 2 e T (t)e(t), B009: Due to Combining B007-B008, we can get: among them κ(t)=(e T (t),G T (e(t))) T 。 B010: When When, based on P ι (ρ) and β ι Definition of (ρ), Ω ι (ρ)=(1-ρ)ζ ι +ρζ ι+1 <0 is established. B011: Combining B009-B010, we can get: B012: For t∈[t r +τ min , t r+1 ), when t≠t r When , we can get: B013: When When , we can get: B014: Combine B011 and B013, for t≠t r , we can get: B015: Using Taylor expansion We can get: B016: From the Taylor expansion of sh(·), we know that for a given There exists J>0 such that definition We can get: in Therefore, q m >q>1. B017: Using the properties of inequality, we can get: B018: Combining B001-B017, we can get: in λ min (P(t)) and λ max (P(t)) denotes the minimum and maximum eigenvalues ​​of P(t), inf{·} denotes the maximum lower bound of a set or function, and sup{·} denotes the minimum upper bound of a set or function. B019: Due to q m >q, therefore, when V(t)>1, we can get: B020: For t=t r , r∈N + ,when When , we can get: B021: Combined with B018 and B020, the system achieves fixed time synchronization according to the fixed time stability definition. Construct a comparison system and estimate the settling time under synchronous pulse, inactive pulse and asynchronous conditions, specifically: Construct the following systems for comparison: When Φ(t)≥1, define From the comparison system, we can see that when Φ(0)→∞, When Φ(t)→1, Therefore, we can get: in At this time, Φ(t)→1 + Equivalent to When 0≤Φ(t)<1, define From the comparison system, we can see that when Φ(t)→1, When Φ(t)→0, Therefore, we can get: in At this time, Φ(t)→0 + Equivalent to Estimated settling time in synchronous pulse, inactive pulse and non-synchronous cases: 0<α<1, α=1 and α>1. Consider case 1: 0<α<1. For this case, we have and for We can get: in Since and then when 0 < t < T1, there exists T1 such that and Therefore, it can be obtained that: Further we can get: Taking into account We can get: Solving the above inequality, we can get: Will Substituting into the above inequality, we can get: Next, calculate T2 so that From 1 to 0. Similar to the calculation of T1, we can get: is monotonically decreasing on [0,∞), due to We can get: make We can get: definition because and Therefore there exists a unique T2>0 such that in Therefore, the optimal stabilization time Consider case 2: α = 1. For this case, we have Similar to case 1, we can get: Therefore, the optimal stabilization time Consider case 3: α>1. For this case, we have and because We can get: Since τ min <τ max , we can get: when t→+∞, the right side of the inequality tends to infinity; therefore, there is no guarantee that there is a fixed time make From 1 to 0.

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