A fixed-time synchronization method for heterogeneous four-pendulum system based on pulse control

Abstract

The application discloses a heterogeneous four-pendulum system fixed-time synchronization method based on pulse control. The method first designs a fixed-time controller based on pulse control for a four-pendulum system with heterogeneous characteristics. Then, a novel Lyapunov functional is designed, the pulse interval is partitioned, and a convex combination technique is used to establish a fixed-time synchronization sufficient condition expressed by a matrix inequality, thereby realizing the synchronization of the four-pendulum system. A comparison system is constructed to estimate the stable time under the conditions of a synchronization pulse, an inactive pulse and non-synchronization. When the method is applied to the heterogeneous four-pendulum system, the stable time for realizing synchronization of the system is reduced, and the actual efficiency is improved.

Description

Technical Field

[0001] This invention relates to a method for achieving fixed-time synchronization of a four-pendulum system, specifically a method for fixed-time synchronization of a heterogeneous four-pendulum system based on pulse control. Background Technology

[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, generators in a power grid are controlled by pendulum equations with different inertia coefficients, different viscous damping, and different natural frequencies. Multiple manipulators with Lagrangian dynamics are described by second-order differential equations with different inertia matrices characterizing their Coriolis and centrifugal forces. Since heterogeneity can degrade the synchronization performance of a system, studying synchronization behavior in heterogeneous dynamic networks is more challenging. Based on these ideas, researchers have conducted extensive research on the output synchronization of four-pendulum systems and achieved a series of results.

[0003] Furthermore, the fixed-time controller currently used in the dynamic model of the four-pendulum system is a sum of signed power functions, which not only has numerous parameters and a complex structure, but also leads to chattering. Therefore, using a hyperbolic sine function as the fixed-time controller component to reduce parameters, simplify the structure, and avoid chattering is particularly crucial and important. 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 effectively improves the system's stability and convergence time.

[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] Design a fixed-time controller based on pulse control for a four-pendulum system with heterogeneous characteristics.

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

[0008]

[0009] In the formula, i = 1, 2, ..., N, N represents the total number of nodes, g(·) represents the nonlinear activation function, and w i (t) represents the i-th system state variable, w0(t) represents the desired target system state variable, and u i (t) represents the i-th system control input variable, Γ represents the system coupling matrix, and S = (s ij ) N×N C represents the external coupling matrix of the system.i H i C and 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) gives the error system.

[0013]

[0014] Construct the controller as follows:

[0015]

[0016] In the formula, η i >0 indicates the control gain constant. and It is to satisfy A positive odd number, sh(·) denotes the hyperbolic sine function: r0 is a non-negative integer. Represents the pulse gain matrix. Where I n It is an n-dimensional identity matrix. It 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 It has the following properties:

[0017] τ min <t r -t r-1 ≤τ max , r∈R0, t0≥0, τ min >0, τ max >0, R0={1,…,r0}

[0018] Where ξ = 0.5, η i =3.2, τ min =0.04, τ max =0.06,

[0019] At this point, the error system can be rewritten as

[0020]

[0021] 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

[0022]

[0023] A novel Lyapunov functional was designed, and the pulse interval was partitioned and convex combination techniques were used to establish the necessary and sufficient conditions for fixed-time synchronization in matrix inequality representation, thereby realizing the synchronization of a four-pendulum system. The specific steps are as follows:

[0024] C001: Select the energy function in the following form:

[0025] V(t) = V(e(t)) = e T (t)P(t)e(t),

[0026] Where P(t) represents a positive definite diagonal matrix and T represents the transpose.

[0027] C002: Considering a positive integer Θ>0, the pulse interval [t] r , t r+1 The interval is divided into Θ+1 subintervals. The starting time of the (i+1)th subinterval is defined. Where i = 0, 1, ..., Θ-1, then when i = 0, t r，0 =t r That is, the starting time of the first sub-interval is the same as the starting time of the original pulse interval. When i = Θ, t r，Θ =t r +τ min That is, at a specific time after dividing the data into Θ subintervals,

[0028] C003: Define the (i+1)th subinterval Then for P(t) = P(t) r,i +ρF)=(1-ρ)P i +ρP i+1 =P i (ρ), where P i It is a positive definite matrix. It is used to describe the sub-interval The parameter representing the relative position within time t normalizes the position of time t within the sub-interval, and its value ranges from [0, 1], reflecting the relative progress of time t within the sub-interval. It is a minimum time interval T min The parameters related to the partitioning parameter Θ are the minimum time intervals for each subinterval on the normalized time scale. Therefore, 0 ≤ ρ ≤ 1.

[0029] C004: When t∈[t] r ,Θ,t r+1 P(t) is fixed as a positive definite matrix P θ Therefore, for r∈N + ,

[0030]

[0031] 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. Therefore, for r ∈ N + ,

[0032] C006: Using the above-mentioned specific time-varying variables P(t) and β(t) with convex combination techniques, we can obtain:

[0033]

[0034] C007: Regarding When t≠t r At this point, calculate the first derivative of V(t):

[0035]

[0036] C008: Because when i = 0, 1, ..., Θ-1, β i >0 and β i Since (ρ) > 0, we can obtain the following using the global Lipschitz condition:

[0037] β i (ρ)G T (e(t))G(e(t))≤β i (ρ)Δ 2 e T (t)e(t),

[0038] C009: Due to Combining C007-C008, we can obtain:

[0039]

[0040] in

[0041]

[0042] C010: When At that time, based on P i (ρ) and β i The definition of (ρ), Ω i (ρ)=(1-ρ)ζ i +ρζ i+1 <0 is true,

[0043] C011: Combining C009-C010, we can obtain:

[0044]

[0045] C012: For t∈[t r +τ min , t r+1 When t≠t r At that time, we can obtain:

[0046]

[0047] C013: When At that time, we can obtain:

[0048]

[0049] C014: Combining C011 and C013, for t≠t r We can obtain:

[0050]

[0051] C015: Using Taylor expansion We can obtain:

[0052]

[0053] C016: From the Taylor expansion of sh(·), we know that for a given There exists J>0 such that definition We can obtain:

[0054]

[0055] in Therefore, q m >q>1,

[0056] C017: Using the properties of inequalities, we can obtain:

[0057]

[0058] C018: Combining C001-C017, we can obtain:

[0059]

[0060]

[0061] in λ min (P(t)) and λ max (P(t)) represent the minimum and maximum eigenvalues ​​of P(t), respectively; inf{·} represents the maximum lower bound of a set or function; sup{·} represents the minimum upper bound of a set or function.

[0062] C019: Since qm > q, when V(t) > 1, we can obtain:

[0063]

[0064] C020: For t=t r , r∈N + ,when At that time, we can obtain:

[0065]

[0066] C021: Combining C018 and C020, the system achieves fixed-time synchronization based on the definition of fixed-time stability.

[0067] The following steps are taken to construct a comparison system and estimate the optimal settling time under synchronous, inactive, and asynchronous conditions:

[0068] Construct the following system for comparison:

[0069]

[0070] 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:

[0071]

[0072] where At this time, Φ(t) → 1 + is equivalent to j(t) → 1 - .

[0073] 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:

[0074]

[0075] where At this time, Φ(t) → 0 + is equivalent to j(t) → 0 + .

[0076] Estimated stable times in the case of synchronous pulses, inactive pulses, and asynchronous conditions: 0 < α < 1, α = 1, and α > 1.

[0077] Consider case 1: 0 < α < 1. For this case, there are and

[0078] For 0 < j(t) < 1, it can be obtained that: where

[0079] Since and then when 0 < t < T1, there exists T1 such that and j(t) ∈ (0, 1). Therefore, it can be obtained that:

[0080] Furthermore, it can be obtained that:

[0081] Considering it can be obtained that:

[0082] Solving the above inequality, it can be obtained that:

[0083] Will Substituting into the above inequality, we get:

[0084] Next, calculate T2 to change j(t) from 1 to 0. Similar to calculating T1, we get:

[0085]

[0086] j(t) is monotonically decreasing on [0, ∞), since We can obtain: Let j(0) = 1, j(t) = 0, then we can obtain:

[0087]

[0088] definition because and Therefore, there exists a unique T2>0 such that

[0089] in Therefore, the optimal settling time

[0090] Consider case 2: a = 1. In this case, we have...

[0091] Similar to case 1, we can obtain: Therefore, the optimal settling time

[0092] Consider case 3: α > 1. In this case, we have... and

[0093] because We can obtain:

[0094]

[0095] Due to τ min <τ max We can conclude that as t→+∞, the right side of the inequality approaches infinity; therefore, a fixed time cannot be guaranteed. Make j(t) approach 0 from 1. Attached Figure Description

[0096] Figure 1 The following is a state trajectory diagram of a four-pendulum system using the method proposed in this invention under no control input conditions, as an example.

[0097] Figure 2The following is a state trajectory diagram of a four-pendulum system using the method proposed in this invention under controlled input conditions, as an example.

[0098] Figure 3 The following is a diagram showing the state trajectory of a four-pendulum system using the method proposed in this invention with control input and r0 = 0.

[0099] Figure 4 The synchronization error diagram of a four-pendulum system using the method proposed in this invention under control input is shown in the example.

[0100] Figure 5 This is a comparison chart of the synchronization error under the action of the method proposed in this invention and two different fixed-time controllers in an embodiment.

[0101] Figure 6 This is a comparison chart of control inputs under the action of two different fixed-time controllers using the method proposed in this invention as an example. Detailed Implementation

[0102] The present invention will be further illustrated below with reference to specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. After reading the present invention, any modifications of the present invention in various equivalent forms by those skilled in the art will fall within the scope defined by the appended claims.

[0103] A fixed-time synchronization method for a heterogeneous four-pendulum system based on pulse control includes the following steps:

[0104] Step 1: Set the various system parameters;

[0105] Step 2: Solve for the positive definite matrix P(t), parameters β(t) and α that satisfy the fixed-time stability condition;

[0106] Step 3: Set the initial state of the system;

[0107] Step 4: Calculate the settling time T based on the set system parameters;

[0108] Step 5: Remove the system's control inputs and verify whether the system achieves synchronization without control inputs;

[0109] Step 6: When the system's control input consists only of fixed-time control input and no pulse control input, verify whether the system achieves synchronization without pulse control input.

[0110] Step 7: Verify whether the system achieves synchronization within a steady time T under the combined action of a fixed time and a pulse control input;

[0111] Step 8: Compare the synchronization error and control input of the method proposed in this invention with those of two different fixed-time controllers to verify its advantages;

[0112] An embodiment of the present invention is described below:

[0113] Consider a four-pendulum system with heterogeneous characteristics. Its corresponding dynamic model and the designed controller are as follows:

[0114]

[0115]

[0116]

[0117] in c1 = c2 = 0.2,

[0118] ξ = 0.5, η i =3.2, τ min =0.04, τ max =0.06,

[0119] The state trajectory diagram of the four-pendulum system under no control input is as follows: Figure 1 As shown, the state trajectory diagram of the four-pendulum system under 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, the synchronization error diagram of the four-pendulum system under control input is as follows: Figure 4 As shown, the settling time T = 14.7851 can be calculated based on the method proposed in this invention and the controller. Below and in the controller The synchronization error is compared with the control input as follows: Figure 5 and 6 As shown, the method proposed in this invention can converge the error to zero more quickly, verifying its high convergence performance, and requires only low control input, thus saving a lot of cost.

Claims

1. A fixed-time synchronization method for a heterogeneous four-pendulum system based on pulse control, characterized in that, Includes the following steps: A fixed-time controller based on pulse control is designed for a four-pendulum system with heterogeneous characteristics, specifically as follows: The heterogeneous four-pendulum system and the dynamic model of the desired target are as follows: ， ， In the formula , Indicates the total number of nodes. Represents a non-linear activation function. Indicates the first One system state variable, Represents the desired target system state variables. Indicates the first A system control input variable, Represents the coupling matrix within the system. Represents the external coupling matrix of the system. , , , Represents a known parameter matrix of appropriate dimensions; It has the following properties: , , ,in Denotes the 1 norm of a vector. , , , , , ， ， ， ， ， ， ， ， ； Definition of the first The synchronization error of each node is Then the error system can be obtained. ； Construct the controller as follows: ， In the formula, Represents the control gain constant. , and It is to satisfy Positive odd numbers, Represents the hyperbola sine function number: pulse number It is a non-negative integer. Represents the pulse gain matrix. ,in It is an n-dimensional identity matrix. It is the pulse gain, which depends on the pulse timing. and nodes ; It is the Dirac function, for hour And there are ; satisfy: , , , , , ,in , , , , , At this point, the error system can be rewritten as: when hour ,when hour ; definition , , , , , , , Then the error system can be rewritten as ； A novel Lyapunov functional is designed, and the pulse interval is partitioned and convex combination techniques are used to establish sufficient conditions for fixed-time synchronization expressed in matrix inequalities, thereby achieving synchronization of a four-pendulum system. Specifically: B001: Select the energy function in the following form: ， in Represents a positive definite diagonal matrix. Indicates transpose; B002: Considering a positive integer , pulse interval Divided into The number of subintervals is defined as the first... The start time of each sub-interval ,in Then there is a time hour That is, the start time of the first sub-interval is the same as the start time of the original pulse interval, when hour, That is, after The specific time obtained after dividing the intervals into sub-intervals; B003: Definition of the first Sub-intervals For ,have ,in It is a positive definite matrix. , It is used to describe the sub-interval Internal time The parameter of relative position, time The position within the sub-interval was normalized, and its value range is... It reflects time Relative progress within sub-intervals It is a minimum time interval and partition parameters The relevant parameters are the minimum time interval for each sub-interval on the normalized time scale, from which the following is derived. , ; B004: When , Fixed as a positive definite matrix Therefore, for , ； B005: Regarding , ,definition , Therefore, for , ; B006: Utilizing the aforementioned specific time-varying variables and Using the convex combination technique, we can obtain: ； B007: Regarding ,when At this time, the calculation is performed. First derivative: ； B008: Due to when hour, and Therefore, using the global Lipschitz condition, we can obtain: ; B009: Due to Combining B007-B008, we can obtain: ， in , ； B010: When , At that time, based on and Definition, Established; B011: Combining B009-B010, we can obtain: , , ; B012: Regarding ,when At that time, we can obtain: ; B013: When At that time, we can obtain: ， ； B014: Combining B011 and B013, for We can obtain: ; B015: Using Taylor expansion , We can obtain: ； B016: By From Taylor's expansion, we can see that for a given... ,exist ,make , ,definition We can obtain: ， in , , , , , , , ,therefore, ; B017: Using the properties of inequalities, we can obtain: ； B018: Combining B001-B017, we can obtain: ， in , , , , , and They represent The minimum and maximum eigenvalues, Denotes the maximum lower bound of a set or function. It represents the least upper bound of a set or function; B019: Due to Therefore, when At that time, we can obtain: ； B020: Regarding , ,when At that time, we can obtain: ； B021: Combining B018 and B020, the system achieves fixed-time synchronization based on the definition of fixed-time stability. Construct a comparison system to estimate the settling time under synchronous pulse, inactive pulse, and asynchronous conditions, specifically as follows: Construct the following systems for comparison: ； when When, define From the comparative system, we can see that when hour, ,when hour, Therefore, we can conclude that: ， in ,at this time, Equivalent to ; when When, define From the comparative system, we can see that when hour, ,when hour, Therefore, we can conclude that: ， in ,at this time, Equivalent to ; Estimated settling time for synchronous pulses, inactive pulses, and asynchronous pulses: , and ; Consideration 1: Corresponding to the synchronization pulse, there is and ; for We can obtain: , in ; because and Then when At that time, there exists , making and Therefore, we can conclude that: ; Furthermore, we can obtain: ; Considering We can obtain: ; Solving the above inequality, we get: ; Will Substituting into the above inequality, we get: ; Next, calculate to make Transforming from 1 to 0, and calculation Similarly, we can obtain: ， exist The upper part is monotonically decreasing, because We can obtain: ,make , We can obtain: ； definition ,because , and Therefore, there exists a unique Make ,in Therefore, the optimal settling time ; Consideration 2: For inactive pulses, there are ; Similar to the case of synchronization pulses, we can obtain: , Therefore, the optimal settling time ; Consideration 3: For asynchronous pulses, there are and ; because We can obtain: ； because We can obtain: when When the right side of the inequality approaches infinity, it cannot be guaranteed that there is a fixed time. make It approaches 0 from 1.

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