Heterogeneous generator system finite time output synchronization method based on pulse control

Abstract

The invention discloses a finite time output synchronization method for a heterogeneous generator system based on pulse control. The method comprises the following steps: firstly, designing a finite time controller based on pulse control for a generator system with heterogeneous characteristics; and then, by virtue of a finite time stability definition established by a differential inequality with a pulse effect based on a Lyapunov functional, a global Lipschitz condition and an inequality property, obtaining sufficient and necessary conditions for realizing finite time output synchronization, and realizing synchronization of a generator system network. A finite time stability definition established by a differential inequality with a pulse effect is used for estimating different pulse times and the number of pulses under an initial value, so that stable time estimation is obtained. When the method is applied to a heterogeneous generator system, the stabilization time of the system for realizing output synchronization is reduced, and the power supply efficiency of the generator is improved.

Description

Technical Field

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

[0002] Many dynamic models for generator systems only consider homogeneous network models. In reality, heterogeneous networks composed of individuals with different dynamics are ubiquitous. For example, the generators in the power grid are controlled by swing equations with different inertia coefficients, different viscous damping, and different natural frequencies. The manipulators with Lagrangian dynamics in multiple manipulators are described by second-order differential equations with different inertia matrices characterizing their Coriolis and centrifugal forces. Since heterogeneity may degrade the synchronization performance of the system, it is more challenging to study the synchronization behavior in heterogeneous dynamic networks. Based on the above ideas, researchers have carried out a lot of research on the output synchronization of heterogeneous generator systems and achieved a series of results.

[0003] In addition, the control matrix before the control input in the current dynamic model of the generator system is a unit matrix, which means that each control input only affects one corresponding component of the state variable, and the system cannot flexibly use the combination of different control inputs to affect the entire state vector, which may limit the selection and optimization of the control strategy. Therefore, it is particularly critical and important to use an asymmetric control matrix to more accurately describe the dynamic model of the generator system. Summary of the invention

[0004] The purpose of the present invention is to propose a finite-time output synchronization method for a heterogeneous generator 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 finite time output synchronization method for a heterogeneous generator system based on pulse control comprises the following steps:

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

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

[0008] in ξ=0.1, μ=0.8, c=1.6, c1=c2=0.2, t r=t0+0.4r, r=1,..., r0, t0=0

[0009] Where i = 1, 2, ..., N, N represents the number of nodes, g(·) represents the nonlinear activation function, m i (t) represents the i-th system state variable, y i (t) represents the i-th system output variable, m0(t)>0 represents the desired target system state variable, y0(t)>0 represents the desired target system output variable, u i (t) represents the control input variable of the ith system, c represents the coupling strength of the system, Γ represents the coupling matrix within the system, A = (a ij ) N×N represents the system external coupling matrix, D i , B i , C i , H i , D, B, H represent known parameter matrices of appropriate dimensions, where matrix D i , B i , H i Satisfy D i ≤D,B i ≤B,H i ≤H. g(·) and C i It has the following properties:

[0010] |g i (s1)-g i (s2)|≤l i |s1-s2|, s1≠s2, l i >0, where |·| represents the 1-norm of the vector, C i =Q i +S i , Q i is a real symmetric matrix and its minimum eigenvalue λ min (Q i )>0,S i is an upper triangular matrix and

[0011] Define the synchronization error of the i-th node as e i (t) = m i (t)-m0(t), the output synchronization error of the i-th node is z i (t) = y i (t)-y0(t), we can get the error system

[0012] Construct the following controller:

[0013]

[0014] In the formula, sign(z i (t)) = diag(sign(z i |(t)), sign(z i2 (t)), ..., sign(z iq (t))), r0 is a non-negative integer, ξ>0 and K∈R q×q is the gain constant and pulse gain matrix, and δ(·) is the Dirac function. r It has the following properties:

[0015] 0 <t r -t r-1 ≤τ max , r∈R0, t0≥0, τ max >0, R0={1,...,r0}

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

[0017] The necessary and sufficient conditions for finite-time output synchronization based on Lyapunov functional and finite-time stability are established to achieve synchronization of the generator system. The specific steps are as follows:

[0018] First, based on the finite-time stability, the finite-time stability based on the Lyapunov functional is given, namely:

[0019] Right now: Then there is Stabilization time

[0020] In the formula, α1>0, α2>0, q∈(0,1).

[0021] Then, a finite-time stability definition based on the differential inequality with impulse effect under Lyapunov functional is given, namely:

[0022] Right now Then there is

[0023] Where α1>0, α2>0, q∈(0,1), η∈(0,1), ε∈[0,1), 0 <V(t0)<χ, Impulse degree β=ηexp(α1(1-q)τ max ) Number of pulses

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

[0025] Where P represents a positive definite diagonal matrix and T represents the transpose.

[0026] C002: When t≠t r , then calculate the first-order derivative of V(t):

[0027]

[0028] C003: Using the global Lipschitz condition and inequality properties, we can obtain:

[0029] where λ max [(PH i ) 2 ] and λ max [D i 2 +(B i L) 2 ] are (PH i ) 2 and D i 2 +(B i L) 2 The maximum eigenvalue of .

[0030] C004: Order z v (t)=(z 1v (t), z 2v (t), ..., z Nv (t)) T , due to P, H i and Γ i is diagonal, further, we can get:

[0031] in λ max (A) is the largest eigenvalue of A.

[0032] C005: According to matrix C i The properties of , further, can be obtained:

[0033] where p minand p max denote the smallest and largest diagonal elements in the matrix P, respectively.

[0034] C006: Combining B004-B005, we can get:

[0035]

[0036] C007: When t = t r , we can get:

[0037]

[0038] C008: When Sometimes, there are

[0039]

[0040] C009: Combining C006 and C008, the system achieves finite-time output synchronization according to a finite-time stability definition established by a differential inequality with impulse effects under the Lyapunov functional.

[0041] Estimate the number of pulses under different pulse degrees and initial values ​​to obtain the estimated stabilization time. The specific steps are as follows:

[0042] The finite time stability definition established based on the differential inequality with impulse effect is used to obtain the number of pulses with different pulse times and initial values.

[0043] In the formula

[0044] Taking ε∈[0,1) as a parameter to be determined, for any small initial value That is, when r0=0 and there is no pulse control, according to the definition of finite-time stability based on Lyapunov functional, we can get V(t0+T1)=0.

[0045] in

[0046] For any large initial value That is, when r0>0, there is an impulse control situation. According to a finite-time stability definition based on the Lyapunov functional established by a differential inequality with impulse effect, V(t0+T2)=0 can be obtained.

[0047] in

[0048] Solution

[0049] We get a unique ε=ε0, and Therefore, we get min 0≤ε＜1 T2(ε)=T2(ε0).

[0050] Furthermore, the best stabilization time can be obtained

[0051] in BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 The output trajectory diagram of the generator system without control input using the method proposed by the present invention is shown in the embodiment;

[0053] Figure 2 The output synchronization error diagram of the generator system without control input using the method proposed by the present invention is shown in the embodiment;

[0054] Figure 3 The output trajectory diagram of the generator system using the method proposed by the present invention when there is a control input and r0=0 is shown in the embodiment;

[0055] Figure 4 The output synchronization error diagram of the generator system using the method proposed by the present invention when there is a control input and r0=0 is shown in the embodiment;

[0056] Figure 5 The output trajectory diagram of the generator system using the method proposed by the present invention under the condition of control input is shown in the embodiment;

[0057] Figure 6 The output synchronization error diagram of the generator system with control input using the method proposed in the present invention is shown in the embodiment. DETAILED DESCRIPTION

[0058] The present invention is further explained below in conjunction with specific embodiments. It should be understood that these embodiments 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, various equivalent forms of modifications to the present invention by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0059] A finite-time output synchronization method for a heterogeneous generator system based on pulse control comprises the following steps:

[0060] Step 1: Set various system parameters;

[0061] Step 2: Solve the positive definite matrix P and parameter η that meet the conditions through the finite time stability condition;

[0062] Step 3: Set the initial state of the system;

[0063] Step 4: Calculate the required number of pulses r0 and the stabilization time T according to the set system parameters and the initial state of the system;

[0064] Step 5: Remove the control input of the system and verify whether the system can achieve output synchronization without control input;

[0065] Step 6: When the control input of the system has only limited time control input and no pulse control input, verify whether the system can achieve output synchronization without pulse control input;

[0066] Step 7: Verify whether the system achieves output synchronization within the stable time T under the combined effect of finite time and pulse control input;

[0067] An embodiment of the present invention is described below:

[0068] Considering a generator system with heterogeneous characteristics, its corresponding dynamic model is:

[0069]

[0070] in ξ=0.1, μ=0.8, c=1.6, c1=c2=0.2, t r =t0+0.4r, r=1,..., r0, t0=0,

[0071] The output trajectory diagram and output synchronization error diagram of the generator system without control input are shown in Figure 1 and Figure 2 As shown in the figure, the output trajectory diagram and output synchronization error diagram of the generator system with control input and r0 = 0 are as follows Figure 3 and Figure 4 As shown, according to the calculation, the required number of pulses r0 = 7 and the stabilization time T = 3.2956 can be obtained. The output trajectory diagram of the generator system with control input is as follows Figure 5 As shown in the figure, the output synchronization error of the generator system with control input is as follows Figure 6 As shown. It can be seen that the generator system can achieve output synchronization within the stabilization time T, and the generator system can achieve output synchronization faster than the case without pulse control.

Claims

1. A finite time output synchronization method for heterogeneous generator systems based on pulse control, characterized in that: The following steps are involved: Design a finite-time controller based on impulse control for a generator system with heterogeneous characteristics; Establish necessary and sufficient conditions for finite-time output synchronization based on Lyapunov functional and finite-time stability to achieve synchronization of generator systems; Estimate the number of pulses at different pulse degrees and initial values ​​to obtain the estimated stabilization time.

2. According to claim 1, a pulse-controlled finite-time output synchronization method for heterogeneous generator systems is characterized in that: Design a finite-time controller based on pulse control for a generator system with heterogeneous characteristics. The specific steps are as follows: The generator system with heterogeneous characteristics and the desired target dynamic model are as follows: in ξ=0.1,μ=0.8,c=1.6,c1=c2=0.2,t r =t0+0.4r,r=1,…,r0,t0=0, Where i = 1, 2, ..., N, N represents the total number of nodes, g(·) represents the nonlinear activation function, m i (t) represents the i-th system state variable, y i (t) represents the i-th system output variable, m0(t)>0 represents the desired target system state variable, y0(t)>0 represents the desired target system output variable, u i (t) represents the control input variable of the ith system, c represents the coupling strength of the system, Γ represents the coupling matrix within the system, A = (a ij ) N×N represents the system external coupling matrix, D i , B i , C i , H i , D, B, H represent known parameter matrices of appropriate dimensions, where matrix D i , B i , H i Satisfy D i ≤D,B i ≤B,H i ≤H; g(·) and C i It has the following properties: |g i (s1)-g i (s2)|≤l i |s1-s2|, s1≠s2, l i >0, where |·| represents the 1-norm of the vector, C i =Q i +S i , Q i is a real symmetric matrix and its minimum eigenvalue λ min (Q i )>0,S i is an upper triangular matrix and Define the synchronization error of the i-th node as e i (t) = m i (t)-m0(t), the output synchronization error of the i-th node is z i (t) = y i (t)-y0(t), we can get the error system Construct the following controller: where sign(z i (t)) = diag(sign(z i1 (t)), sign(z i2 (t)),..., sign(z iq (t))), r0 is a non-negative integer, ξ>0 and K∈R q×q is the gain constant and pulse gain matrix, and δ(·) is the Dirac function. r Has the following properties: 0 <t r -t r-1 ≤τ max , r∈R0, t0≥0, τ max >0, R0={1,...,r0}. At this point, the error system can be rewritten as 3. The method for finite-time output synchronization of a heterogeneous generator system based on pulse control according to claim 1, characterized in that: The necessary and sufficient conditions for finite-time output synchronization based on Lyapunov functional and finite-time stability are established to achieve synchronization of the generator system. The specific steps are as follows: First, based on the finite-time stability, the finite-time stability based on the Lyapunov functional is given, namely: Then there is Stabilization time In the formula, α1>0, α2>0, q ∈ (0, 1). Then, a finite-time stability definition based on the differential inequality with impulse effect under Lyapunov functional is given, namely: Then there is Where α1>0, α2>0, q∈(0,1), η∈(0,1), ε∈[0,1), 0 <V(t0)<χ, Impulse degree β=ηexp(α1(1-q)τ max ) Number of pulses B001: Select the energy function in the following form: Where P represents a positive definite diagonal matrix and T represents the transpose. B002: When t≠t r When , the first-order derivative of V(t) is calculated: B003: Using the global Lipschitz condition and inequality properties, we can get: where λ max [(PH i ) 2 ] and λ max [D i 2 +(B i L) 2 ] are (PH i ) 2 and D i 2 +(B i L) 2 The maximum eigenvalue of . B004: Order z v (t)=(z 1v (t), z 2v (t), ..., z Nv (t)) T , due to P, H i and Γ i is diagonal, further, we can get: in λ max (A) is the largest eigenvalue of A. B005: According to matrix C i The properties of , further, can be obtained: where p min and p max denote the smallest and largest diagonal elements in the matrix P, respectively. B006: Combining B004-B005, we can get: B007: When t = t r , we can get: B008: When Sometimes, there are B009: Combining B006 and B008, the system achieves finite-time output synchronization according to a finite-time stability definition established by a differential inequality with impulse effects under the Lyapunov functional.

4. According to the novel method for realizing finite-time output synchronization of a generator system by a hybrid controller as described in claim 1, the number of pulses under different pulse degrees and initial values ​​is estimated to obtain an estimated stabilization time, and the specific steps are as follows: The finite time stability definition established based on the differential inequality with impulse effect is used to obtain the number of pulses under different pulse times and initial values. In the formula Taking ε∈[0,1) as a parameter to be determined, for any small initial value That is, when r0=0 and there is no pulse control, according to the definition of finite-time stability based on Lyapunov functional, we can get V(t0+T1)=0, where For any large initial value That is, when r0>0, there is an impulse control situation. Based on a finite time stability definition established by a differential inequality with impulse effect under the Lyapunov functional, V(t0+T2)=0 can be obtained, where Solution We get a unique ε=ε0, and Therefore, the optimal stabilization time in

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