CN111293941B - Permanent magnet synchronous motor finite time dynamic surface control method considering iron loss - Google Patents

Abstract

The invention belongs to the technical field of position tracking control of permanent magnet synchronous motors, and specifically discloses a finite time dynamic surface control method of permanent magnet synchronous motors considering iron loss. Aiming at the iron loss and input saturation problems existing in the permanent magnet synchronous motor, the method of the invention introduces the dynamic surface technology in the traditional backstepping method to solve the problem of "computational complexity" in the calculation process, and uses the fuzzy logic system to approximate the permanent The unknown nonlinear term in the magnetic synchronous motor drive system; the method of the invention adopts the finite time control technology to speed up the response speed of the system and reduce the tracking error, and the method of the invention can ensure that the tracking error of the system converges to a sufficiently small neighborhood of the origin , thereby improving the response speed of the permanent magnet synchronous motor drive system. The method of the invention has better robustness, stronger anti-load disturbance capability, and achieves ideal control effect.

Description

A finite-time dynamic surface control method for permanent magnet synchronous motors considering iron loss

technical field

The invention belongs to the technical field of position tracking control of permanent magnet synchronous motors, and in particular relates to a finite time dynamic surface control method of permanent magnet synchronous motors considering iron loss.

Background technique

Permanent magnet synchronous motors have been widely used in industry, manufacturing and other fields due to their simple structure, small size, high efficiency, and high power factor. However, the permanent magnet synchronous motor has the characteristics of multi-variable, highly nonlinear and strong coupling, so how to overcome the above control difficulties and realize the accurate and effective control of the permanent magnet synchronous motor is of great significance.

At present, researchers have proposed many control methods for nonlinear systems, such as backstepping control, direct torque control, Hamiltonian control and sliding mode control. However, the permanent magnet synchronous motor has problems of iron loss and input saturation in actual operation.

The iron loss problem mainly refers to that when the permanent magnet synchronous motor is in the working state of light load for a long time, the system will generate a large amount of iron core loss, which will have an adverse effect on the entire control system; and the input saturation problem mainly refers to the implementation of the engineering system. The controller is limited by non-smooth and nonlinear conditions, which will seriously affect the control performance of the system and the stability of the control system. Therefore, it has certain practical significance to consider iron loss and input saturation in the position control process of permanent magnet synchronous motor.

In addition, the backstepping method has been widely used in the permanent magnet synchronous motor control system, and has achieved good control results. However, there is a problem of "computational complexity" in the traditional backstepping method due to repeated derivation of the virtual control variables; in addition, the traditional backstepping method also requires that some functions in the permanent magnet synchronous motor drive system must be linear.

SUMMARY OF THE INVENTION

The purpose of the present invention is to propose a finite-time dynamic surface control method of permanent magnet synchronous motor considering iron loss, so as to solve the adverse effect of iron loss and input saturation problem on the system, and ensure the tracking control effect of permanent magnet synchronous motor.

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

A finite-time dynamic surface control method for a permanent magnet synchronous motor considering iron loss, comprising the following steps:

a. Establish a dynamic mathematical model of the permanent magnet synchronous motor considering iron loss on the d-q axis, as shown in formula (1):

Where, Θ is the rotor angle, ω is the rotor angular velocity, n _p is the magnetic logarithm, J is the moment of inertia, _TL is the load torque, id is the _d -axis current, i _q is the q-axis current, and ud is the _d -axis Voltage, u _q represents the q-axis voltage, i _od represents the d-axis excitation current component, i _oq represents the q-axis excitation current component, L _d represents the d-axis stator inductance, L _q represents the q-axis stator inductance, and L _ld represents the d-axis leakage inductance , L _lq represents the q-axis leakage inductance, L _md represents the d-axis excitation inductance, L _mq represents the q-axis excitation inductance, R ₁ represents the stator resistance, R _c represents the iron core loss resistance, λ _PM is the rotor permanent magnet excitation flux;

In order to simplify the dynamic mathematical model of permanent magnet synchronous motor considering iron loss, new variables are defined as follows:

Then the dynamic mathematical model of permanent magnet synchronous motor considering iron loss is expressed by formula (2), namely:

b. According to the finite-time dynamic surface technology and the principle of the adaptive backstepping method, a finite-time dynamic surface control method of permanent magnet synchronous motor considering iron loss is designed. The specific process is as follows:

Assuming that f(Z) is a continuous function in the compact set Ω _Z , for any constant ε>0, there is always a fuzzy logic system W ^T S(Z) satisfying:

q是模糊输入维数，R^q为实数向量集；W∈R^l是模糊权向量，模糊节点数l为正整数，且l＞1，R^l为实数向量集；S(Z)＝[s₁(Z),...,s_l(Z)]^T∈R^l为基函数向量，s₁(Z),...,s_l(Z)分别表示S(Z)的基向量；选取基函数s_j(Z)为如下的高斯函数：where, the input vector

q is the fuzzy input dimension, R ^q is a real vector set; W∈R ^l is a fuzzy weight vector, the number of fuzzy nodes l is a positive integer, and l>1, R ^l is a real vector set; S(Z)=[s ₁ (Z),...,s _l (Z)] ^T ∈R ^l is the basis function vector, s ₁ (Z),...,s _l (Z) represent the basis vector of S(Z) respectively; The basis function s _j (Z) is a Gaussian function as follows:

其中，μ_j＝[μ_j1,...,μ_jq]^T是Gaussian函数分布曲线的中心位置，而η_j则为其宽度；μ_j1,...,μ_jq分别表示μ_j的基向量；

Among them, μ _j =[μ _j1 ,...,μ _jq ] ^T is the center position of the Gaussian function distribution curve, and η _j is its width; μ _j1 ,...,μ _jq represent the basis vectors of μ _j respectively ;

其中，V(x)表示系统的Lyapunov函数；系统的收敛时间T_r通过T_r≤t₀+[1/λ₁(1-γ)]ln[(λ₁V^1-γ(t₀)+λ₂)/λ₂]来估计，t₀表示初始时间；Define finite time: for any real numbers λ ₁ >0, λ ₂ >0, 0<γ<1, the extended Lyapunov condition for finite time stability is expressed as:

Among them, V(x) represents the Lyapunov function of the system; the convergence time T _r of the system is determined by T _r ≤t ₀ +[1/λ ₁ (1-γ)]ln[(λ ₁ V ^1-γ (t ₀ )+ λ ₂ )/λ ₂ ] to estimate, t ₀ represents the initial time;

Considering the input voltage constraint problem of the permanent magnet synchronous motor as follows: u _min ≤v≤u _max , where u _min and u _max represent the minimum and maximum values of the known stator input voltage, namely:

Among them, u _max >0 and u _min <0 are unknown constants limited by input constraints, and v is the actual input signal;

Use the piecewise smooth function g(v) to approximate the constraint function, and define g(v) as follows:

u=sat(v)=g(v)+d(v), d(v) is a bounded function whose bounds are:

|d(v)|=|sat(v)-g(v)|≤max{u _max [1-tanh(1)}]u _min [tanh(1)-1]=D;

Among them, D represents D _d and D _q on the d-axis and q-axis, respectively, and both D _d and D _q are constants greater than 0;

Using the mean value theorem, we know that there is a constant μ such that

v_μ＝μv+(1-μ)v₀；in,

v _μ = μv+(1-μ) v ₀ ;

因此，

则有

其中存在一个未知常数g_m，使得

Selecting v ₀ =0, the above function can be written as:

therefore,

then there are

where there is an unknown constant g _m such that

α_id(0)＝α_i(0)，i＝1,2,3,4；Define a new variable α _id and a time constant ∈ _i ,

α _id (0)=α _i (0), i=1, 2, 3, 4;

Among them, α _id (0) represents the initial value of α _id , and α _i (0) represents the initial value of α _i ;

The virtual control law α _i obtains α _id through a first-order filter, wherein, the virtual control laws α ₁ , α ₂ , α ₃ , α ₄ are the input signals of the first-order filter, α _1d , α _2d , α _3d , α _4d is the output signal of the first-order filter;

The tracking errors z ₁ , z ₂ , z ₃ , z ₄ , z ₅ and z ₆ are defined as:

Among them, x _1d is the desired position signal, and x _4d is the desired rotor flux linkage signal;

In the above finite-time dynamic surface control method of permanent magnet synchronous motor considering iron loss, a Lyapunov function is selected for each step to construct a virtual control law or a real control law. The specific steps are as follows:

对V₁求导得到：b.1. According to the first equation z ₁ =x ₁ -x _1d in formula (3), choose the Lyapunov function:

Derivation with respect to V1 gives _:

Choose a virtual control law:

Wherein, the control gain k ₁ >0, the constant s ₁ >0, the constant 0<γ<1;

Based on the above formula (4) and formula (5), we get:

对V₂求导并将公式(6)代入，得到公式(7)：b.2. According to the second equation z ₂ =x ₂ -α _1d in formula (3), choose the Lyapunov function:

Differentiating V ₂ and substituting Equation (6) yields Equation (7):

Among them, the load torque T _L is an unknown constant and the upper limit is d, that is, |T _L |≤d, where d>0;

其中，ε₁是一个任意小的正数，则：By Young's inequality we have

where ε ₁ is an arbitrarily small positive number, then:

由万能逼近定理，对于任意小的正数ε₂，选取模糊逻辑系统

使得：

其中，δ₂(Z)为逼近误差，并满足不等式|δ₂(Z)|≤ε₂，||W₂||是向量W₂的范数；in,

According to the universal approximation theorem, for any small positive number ε ₂ , choose the fuzzy logic system

makes:

Among them, δ ₂ (Z) is the approximation error and satisfies the inequality |δ ₂ (Z)|≤ε ₂ , and ||W ₂ || is the norm of the vector W ₂ ;

Choose a virtual control law:

为θ的估计值，θ的定义在下文给出；Among them, the control gain k ₂ >0, the constant s ₂ >0, the constant l ₂ >0,

is an estimate of θ, the definition of θ is given below;

表示为：According to the third equation z ₃ =x ₃ -α _2d in formula (3), then

Expressed as:

对V₃求导并将公式(10)代入，得到公式(11)：b.3. According to the third equation in formula (3): z ₃ =x ₃ -α _2d , choose the Lyapunov function:

Differentiating V ₃ and substituting Equation (10) yields Equation (11):

由万能逼近定理，对于任意小的正数ε₃，选取模糊逻辑系统

使得：

其中δ₃(Z)为逼近误差，并满足不等式|δ₃(Z)|≤ε₃，||W₃||是向量W₃的范数；从而：in,

According to the universal approximation theorem, for any small positive number ε ₃ , choose the fuzzy logic system

makes:

where δ ₃ (Z) is the approximation error and satisfies the inequality |δ ₃ (Z)|≤ε ₃ , and ||W ₃ || is the norm of the vector W ₃ ; thus:

Choose a virtual control law:

Wherein, the control gain k ₃ >0, the constant s ₃ >0, and the constant l ₃ >0;

表示为：According to the fourth equation z ₄ =x ₄ -α _3d in formula (3), then

Expressed as:

对V₄求导并将公式(14)代入，得到公式(15)：b.4. According to the fourth equation z ₄ =x ₄ -α _3d in formula (3), choose the Lyapunov function:

Differentiating V ₄ and substituting Equation (14) yields Equation (15):

由万能逼近定理，对于任意小的正数ε₄，选取模糊逻辑系统

使得

其中，δ₄(Z)为逼近误差，并满足不等式|δ₄(Z)|≤ε₄，||W₄||是向量W₄的范数；从而：in,

According to the universal approximation theorem, for any small positive number ε ₄ , choose the fuzzy logic system

make

where δ ₄ (Z) is the approximation error and satisfies the inequality |δ ₄ (Z)|≤ε ₄ , and ||W ₄ || is the norm of the vector W ₄ ; thus:

Build the real control law:

Wherein, the control gain k ₄ >0, the constant s ₄ >0, and the constant l ₄ >0;

From the input saturation formula u _q =sat(v _q )=g(v _q )+d(v _q ), we get: d ₁ z ₄ u _q =d ₁ z ₄ g(v _q )+d ₁ z ₄ d( v _q ),

其中，常数D_q＞0，得到：From Young's inequality we get

where the constant D _q > 0, we get:

对V₅求导得到：b.5. According to the fifth equation z ₅ =x ₅ in formula (3), choose the Lyapunov function:

Derivation with respect to _V5 gives:

Build a virtual control law:

Wherein, the control gain k ₅ >0, the constant s ₅ >0; according to the sixth equation z ₆ =x ₆ -α _4d in formula (3), we can obtain:

对V₆求导得到：

b.6. According to the sixth equation z ₆ =x ₆ -α _4d in formula (3), select the Lyapunov function:

Derivating V ₆ gives:

由万能逼近定理，对于任意小的正数ε₆，选取模糊逻辑系统

使得

其中δ₆(Z)为逼近误差，并满足不等式|δ₆(Z)|≤ε₆，||W₆||是向量W₆的范数；从而：in,

According to the universal approximation theorem, for any small positive number ε ₆ , choose the fuzzy logic system

make

where δ ₆ (Z) is the approximation error and satisfies the inequality |δ ₆ (Z)|≤ε ₆ , and ||W ₆ || is the norm of the vector W ₆ ; thus:

Build the real control law:

Wherein, the control gain k ₆ >0, the constant s ₆ >0, and the constant l ₆ >0;

By entering the saturation formula ud =sat(v _d )=g(v _d )+d(v _d ), we get: d ₂ z ₆ u _d = _{d 2} z ₆ g(v _d )+d ₂ z ₆ d( v _d ),

为θ的估计值，定义

definition

is the estimated value of θ, define

其中，常数D_d＞0，得到：By Young's inequality

where the constant D _d > 0, we get:

b.7. Define y _i =α _id -α _i , i=1, 2, 3, 4, and get:

选择系统的Lyapunov函数：

in,

Choose the Lyapunov function for the system:

where r ₁ is a positive number, and derivation with respect to V yields:

Among them, the control gain k ₆ >0; the adaptive law is constructed as follows:

Among them, m ₁ is a positive number;

c. Stability analysis of the finite-time dynamic surface control method of permanent magnet synchronous motor considering iron loss in step b;

Substituting formula (27) into formula (26) yields:

常数τ＞0；where |B _i | has a maximum value |B _iM | on the compact set |Ω _i |,i=1,2,3,4 and |B _i |≤B _iM , then:

constant τ>0;

From Young's inequality we get:

From inequality scaling we get:

Scaling according to the above inequality, Equation (28) is written as:

in,

b ₀ =min{2, 2s ₁ , 2s ₂ , 2s ₃ , 2g _m s ₄ , 2s ₅ , 2g _m s ₆ , m ₁ };

It is obtained by formula (29):

From formula (30), if a ₀ -(c/2V)>0 and b ₀ -(c/2V ^[(γ+1)/2] )>0;

跟踪误差z₁将在有限时间内收敛到原点的一个小邻域内。According to the definition of finite time, in the convergence time _Tr of the system,

The tracking error z ₁ will converge to within a small neighborhood of the origin in finite time.

The present invention has the following advantages:

(1) The method of the present invention considers the adverse effects of iron loss and input saturation on the permanent magnet synchronous motor, and avoids the safety problem that may be caused by the input saturation; (2) The present invention adopts the dynamic surface technology, which effectively avoids the The "computational complexity" problem caused by the continuous derivation of the virtual function in the traditional backstepping method, and the fuzzy logic system is used to approximate the unknown nonlinear term in the permanent magnet synchronous motor drive system; (3) the present invention adopts limited The time control technology enables the tracking error to converge to a sufficiently small neighborhood of the origin within a limited time, and improves the response speed of the permanent magnet synchronous motor drive system; (4) the method of the present invention has better robustness and resistance to The load disturbance ability is strong, and the ideal control effect is realized.

Description of drawings

1 is a schematic diagram of a composite controlled object composed of a finite-time dynamic surface position tracking controller, coordinate transformation, and SVPWM inverter in the embodiment of the present invention;

2 is a tracking simulation diagram of the actual value of the rotor position signal and the given value of the rotor position signal after being controlled by the method of the present invention;

3 is a simulation diagram of the tracking error of the rotor position signal after being controlled by the method of the present invention;

4 is a simulation diagram of the q-axis stator voltage after being controlled by the method of the present invention;

FIG. 5 is a simulation diagram of the d-axis stator voltage after being controlled by the method of the present invention.

Detailed ways

The basic idea of the invention is as follows: by combining the adaptive backstepping method and the dynamic surface technology in the position tracking control of the permanent magnet synchronous motor, it can solve the parameter uncertainty and external load change existing in the permanent magnet synchronous motor drive system. The problem and the "computational complexity" problem in the traditional backstepping method; the adverse effects of iron loss and input saturation on the permanent magnet synchronous motor are considered; the finite-time control technology is introduced, so that the tracking error can be controlled within a limited time. The control method of the present invention has higher engineering practice value by converging to a field with a very small origin, and obtains an ideal tracking effect.

The present invention is described in further detail below in conjunction with the accompanying drawings and specific embodiments:

As shown in Figure 1, a finite-time dynamic surface control method for permanent magnet synchronous motors considering iron loss, the components used include a finite-time dynamic surface position tracking controller 1, a coordinate transformation unit 2, an SVPWM inverter 3 and a rotational speed. Detection unit 4 and current detection unit 5 . Among them, the finite-time dynamic surface position tracking controller 1 is a controller designed according to the finite-time dynamic surface control method of the permanent magnet synchronous motor considering iron loss in the present invention. U _α and U _β represent the voltage in the two-phase rotating coordinate system, and U, V and W represent the three-phase voltage. The rotational speed detection unit 4 and the current detection unit 5 are mainly used to detect the variables related to the rotational speed and current value of the permanent magnet synchronous motor, and use the actual measured current and rotational speed variables as input, and perform the voltage control through the finite-time dynamic surface position tracking controller 1, The final conversion is to three-phase electric control of the rotor position of the permanent magnet synchronous motor. In order to design a more effective finite-time dynamic surface position tracking controller 1, it is necessary to establish a dynamic model of permanent magnet synchronous motor. The concrete steps of the method of the present invention are described in detail below:

A finite-time dynamic surface control method for a permanent magnet synchronous motor considering iron loss, comprising the following steps:

a. Establish a dynamic mathematical model of the permanent magnet synchronous motor considering iron loss on the d-q axis, as shown in formula (1):

Where, Θ is the rotor angle, ω is the rotor angular velocity, n _p is the magnetic logarithm, J is the moment of inertia, _TL is the load torque, id is the _d -axis current, i _q is the q-axis current, and ud is the _d -axis Voltage, u _q represents the q-axis voltage, i _od represents the d-axis excitation current component, i _oq represents the q-axis excitation current component, L _d represents the d-axis stator inductance, L _q represents the q-axis stator inductance, and L _ld represents the d-axis leakage inductance , L _lq represents the q-axis leakage inductance, L _md represents the d-axis excitation inductance, L _mq represents the q-axis excitation inductance, R ₁ represents the stator resistance, R _c represents the iron core loss resistance, and λ _PM is the rotor permanent magnet excitation flux.

In order to simplify the dynamic mathematical model of permanent magnet synchronous motor considering iron loss, new variables are defined as follows:

Then the dynamic mathematical model of permanent magnet synchronous motor considering iron loss is expressed by formula (2), namely:

b. According to the finite-time dynamic surface technology and the principle of the adaptive backstepping method, a finite-time dynamic surface control method of permanent magnet synchronous motor considering iron loss is designed. The specific process is as follows:

Assuming that f(Z) is a continuous function in the compact set Ω _Z , for any constant ε>0, there is always a fuzzy logic system W ^T S(Z) satisfying:

q是模糊输入维数，R^q为实数向量集；W∈R^l是模糊权向量，模糊节点数l为正整数，且l＞1，R^l为实数向量集；S(Z)＝[s₁(Z),...,s_l(Z)]^T∈R^l为基函数向量，s₁(Z),...,s_l(Z)分别表示S(Z)的基向量；选取基函数s_j(Z)为如下的高斯函数：where, the input vector

q is the fuzzy input dimension, R ^q is a real vector set; W∈R ^l is a fuzzy weight vector, the number of fuzzy nodes l is a positive integer, and l>1, R ^l is a real vector set; S(Z)=[s ₁ (Z),...,s _l (Z)] ^T ∈R ^l is the basis function vector, s ₁ (Z),...,s _l (Z) represent the basis vector of S(Z) respectively; The basis function s _j (Z) is a Gaussian function as follows:

其中，μ_j＝[μ_j1,...,μ_jq]^T是Gaussian函数分布曲线的中心位置，而η_j则为其宽度；μ_j1,...,μ_jq分别表示μ_j的基向量。

Among them, μ _j =[μ _j1 ,...,μ _jq ] ^T is the center position of the Gaussian function distribution curve, and η _j is its width; μ _j1 ,...,μ _jq represent the basis vectors of μ _j respectively .

其中，V(x)表示系统的Lyapunov函数；系统的收敛时间T_r通过T_r≤t₀+[1/λ₁(1-γ)]ln[(λ₁V^1-γ(t₀)+λ₂)/λ₂]来估计，t₀表示初始时间。Define finite time: for any real numbers λ ₁ >0, λ ₂ >0, 0<γ<1, the extended Lyapunov condition for finite time stability is expressed as:

Among them, V(x) represents the Lyapunov function of the system; the convergence time T _r of the system is determined by T _r ≤t ₀ +[1/λ ₁ (1-γ)]ln[(λ ₁ V ^1-γ (t ₀ )+ λ ₂ )/λ ₂ ] to estimate, t ₀ represents the initial time.

Considering the input voltage constraint problem of the permanent magnet synchronous motor as follows: u _min ≤v≤u _max , where u _min and u _max represent the minimum and maximum values of the known stator input voltage, namely:

Among them, u _max >0 and u _min <0 are unknown constants limited by input constraints, and v is the actual input signal.

Use the piecewise smooth function g(v) to approximate the constraint function, and define g(v) as follows:

u=sat(v)=g(v)+d(v), d(v) is a bounded function whose bounds are:

|d(v)|=|sat(v)-g(v)|≤max{u _max [1-tanh(1)}]u _min [tanh(1)-1]=D;

Among them, D represents D _d and D _q on the d-axis and q-axis, respectively, and both D _d and D _q are constants greater than 0;

Using the mean value theorem, we know that there is a constant μ such that

v_μ＝μv+(1-μ)v₀。in,

v _μ = μv+(1−μ)v ₀ .

因此，

则有

其中存在一个未知常数g_m，使得

Selecting v ₀ =0, the above function can be written as:

therefore,

then there are

where there is an unknown constant g _m such that

α_id(0)＝α_i(0)，i＝1,2,3,4。Define a new variable α _id and a time constant ∈ _i ,

α _id (0)=α _i (0), i=1,2,3,4.

Among them, α _id (0) represents the initial value of α _id , and α _i (0) represents the initial value of α _i .

The virtual control law α _i obtains α _id through a first-order filter, wherein, the virtual control laws α ₁ , α ₂ , α ₃ , α ₄ are the input signals of the first-order filter, α _1d , α _2d , α _3d , α _4d is the output signal of the first-order filter.

The tracking errors z ₁ , z ₂ , z ₃ , z ₄ , z ₅ and z ₆ are defined as:

Among them, x _1d is the desired position signal, and x _4d is the desired rotor flux linkage signal.

In the above finite-time dynamic surface control method of permanent magnet synchronous motor considering iron loss, a Lyapunov function is selected for each step to construct a virtual control law or a real control law. The specific steps are as follows:

对V₁求导得到：b.1. According to the first equation z ₁ =x ₁ -x _1d in formula (3), choose the Lyapunov function:

Derivation with respect to V1 gives _:

Choose a virtual control law:

Among them, the control gain k ₁ >0, the constant s ₁ >0, and the constant 0<γ<1.

Based on the above formula (4) and formula (5), we get:

对V₂求导并将公式(6)代入，得到公式(7)：b.2. According to the second equation z ₂ =x ₂ -α _1d in formula (3), choose the Lyapunov function:

Differentiating V ₂ and substituting Equation (6) yields Equation (7):

Wherein, the load torque _TL is an unknown constant and the upper limit is d, ie | _TL |≤d, where d>0.

其中，ε₁是一个任意小的正数，则：By Young's inequality we have

where ε ₁ is an arbitrarily small positive number, then:

由万能逼近定理，对于任意小的正数ε₂，选取模糊逻辑系统

使得：

其中，δ₂(Z)为逼近误差，并满足不等式|δ₂(Z)|≤ε₂，||W₂||是向量W₂的范数。in,

According to the universal approximation theorem, for any small positive number ε ₂ , choose the fuzzy logic system

makes:

Among them, δ ₂ (Z) is the approximation error, and satisfies the inequality |δ ₂ (Z)|≤ε ₂ , and ||W ₂ || is the norm of the vector W ₂ .

Choose a virtual control law:

为θ的估计值，θ的定义在下文给出。Among them, the control gain k ₂ >0, the constant s ₂ >0, the constant l ₂ >0,

is an estimate of θ, the definition of θ is given below.

表示为：According to the third equation z ₃ =x ₃ -α _2d in formula (3), then

Expressed as:

对V₃求导并将公式(10)代入，得到公式(11)：b.3. According to the third equation in formula (3): z ₃ =x ₃ -α _2d , choose the Lyapunov function:

Differentiating V ₃ and substituting Equation (10) yields Equation (11):

由万能逼近定理，对于任意小的正数ε₃，选取模糊逻辑系统W₃ ^TS₃(Z)，使得：

其中δ₃(Z)为逼近误差，并满足不等式|δ₃(Z)|≤ε₃，||W₃||是向量W₃的范数。从而：in,

According to the universal approximation theorem, for any small positive number ε ₃ , the fuzzy logic system W ₃ ^T S ₃ (Z) is selected such that:

where δ ₃ (Z) is the approximation error and satisfies the inequality |δ ₃ (Z)|≤ε ₃ , and ||W ₃ || is the norm of the vector W ₃ . thereby:

Choose a virtual control law:

Wherein, the control gain k ₃ >0, the constant s ₃ >0, and the constant l ₃ >0.

表示为：According to the fourth equation z ₄ =x ₄ -α _3d in formula (3), then

Expressed as:

对V₄求导并将公式(14)代入，得到公式(15)：b.4. According to the fourth equation z ₄ =x ₄ -α _3d in formula (3), choose the Lyapunov function:

Differentiating V ₄ and substituting Equation (14) yields Equation (15):

由万能逼近定理，对于任意小的正数ε₄，选取模糊逻辑系统

使得

其中，δ₄(Z)为逼近误差，并满足不等式|δ₄(Z)|≤ε₄，||W₄||是向量W₄的范数。从而：in,

According to the universal approximation theorem, for any small positive number ε ₄ , choose the fuzzy logic system

make

Among them, δ ₄ (Z) is the approximation error and satisfies the inequality |δ ₄ (Z)|≤ε ₄ , and ||W ₄ || is the norm of the vector W ₄ . thereby:

Build the real control law:

Among them, the control gain k ₄ >0, the constant s ₄ >0, and the constant l ₄ >0.

From the input saturation formula u _q =sat(v _q )=g(v _q )+d(v _q ), we get: d ₁ z ₄ u _q =d ₁ z ₄ g(v _q )+d ₁ z ₄ d( v _q ),

其中，常数D_q＞0，得到：From Young's inequality we get

where the constant D _q > 0, we get:

对V₅求导得到：b.5. According to the fifth equation z ₅ =x ₅ in formula (3), choose the Lyapunov function:

Derivation with respect to _V5 gives:

Build a virtual control law:

Wherein, the control gain k ₅ >0, the constant s ₅ >0; according to the sixth equation z ₆ =x ₆ -α _4d in formula (3), we can obtain:

对V₆求导得到：

b.6. According to the sixth equation z ₆ =x ₆ -α _4d in formula (3), select the Lyapunov function:

Derivating V ₆ gives:

由万能逼近定理，对于任意小的正数ε₆，选取模糊逻辑系统

使得

其中δ₆(Z)为逼近误差，并满足不等式|δ₆(Z)|≤ε₆，||W₆||是向量W₆的范数。从而：in,

According to the universal approximation theorem, for any small positive number ε ₆ , choose the fuzzy logic system

make

where δ ₆ (Z) is the approximation error, and satisfies the inequality |δ ₆ (Z)|≤ε ₆ , and ||W ₆ || is the norm of the vector W ₆ . thereby:

Build the real control law:

Among them, the control gain k ₆ >0, the constant s ₆ >0, and the constant l ₆ >0.

By entering the saturation formula ud =sat(v _d )=g(v _d )+d(v _d ), we get: d ₂ z ₆ u _d = _{d 2} z ₆ g(v _d )+d ₂ z ₆ d( v _d ),

为θ的估计值，定义

definition

is the estimated value of θ, define

其中，常数D_d＞0，得到：By Young's inequality

where the constant D _d > 0, we get:

b.7. Define y _i =α _id -α _i , i=1, 2, 3, 4, and get:

选择系统的Lyapunov函数：

in,

Choose the Lyapunov function for the system:

where r ₁ is a positive number, and derivation with respect to V yields:

Wherein, the control gain k ₆ >0. The adaptive law is constructed as follows:

where m ₁ is a positive number.

c. Stability analysis of the finite-time dynamic surface control method of permanent magnet synchronous motor considering iron loss in step b.

Substituting formula (27) into formula (26) yields:

where |B _i | has a maximum value |B _iM | on the compact set |Ω _i |,i=1,2,3,4 and |B _i |≤B _iM , then:

From Young's inequality we get:

From inequality scaling we get:

Scaling according to the above inequality, Equation (28) is written as:

in,

b ₀ =min{2, 2s ₁ , 2s ₂ , 2s ₃ , 2g _m s ₄ , 2s ₅ , 2g _m s ₆ , m ₁ }.

It is obtained by formula (29):

From formula (30), if a ₀ -(c/2V)>0 and b ₀ -(c/2V ^[(γ+1)/2] )>0;

跟踪误差z₁将在有限时间内收敛到原点的一个小邻域内。According to the definition of finite time, in the convergence time _Tr of the system,

The tracking error z ₁ will converge to within a small neighborhood of the origin in finite time.

In the virtual environment, the established finite-time dynamic surface control method of permanent magnet synchronous motor considering iron loss is simulated, and its feasibility in permanent magnet synchronous motor drive system is verified. The permanent magnet synchronous motor and load parameters are as follows:

R ₁ =2.21Ω, R _c =200Ω, L _d =L _q =0.00977H, L _ld =L _lq =0.00177H;

L _md =L _mq =0.008H, J = 0.00379 kg·m ² , λ _PM =0.0844, n _p =3.

The control law parameters are selected as:

k ₁ =100,k ₂ =200,k ₃ =200,k ₄ =8000,k ₅ =100,k ₆ =800,∈ ₁ =∈ ₂ =∈ ₄ =0.00005;

∈ ₃ =0.001, r ₁ =0.05, m ₁ =0.5, l ₂ =l ₃ =l ₄ =100,l ₆ =10.

The desired position signal is: x _1d =sint, and the load torque is:

The fuzzy membership function is chosen as:

The simulation is carried out under the premise that the system parameters and nonlinear functions are unknown, and the corresponding simulation results are shown below. Among them, Fig. 2 is a tracking simulation diagram of the rotor position after being controlled by the method of the present invention. The simulation results show that the method of the present invention has a good tracking effect and a fast response speed; Fig. 3 is a rotor position and a given rotor position after being controlled by the control method of the present invention. The tracking error simulation diagram of the value; Figure 4 and Figure 5 are respectively the simulation diagrams of the permanent magnet synchronous motor q-axis stator and the permanent magnet synchronous motor d-axis stator voltage after the control of the method of the present invention. The simulation results show that the control method of the present invention can effectively Reduce the adverse effects of input saturation, the overall effect is better, the fluctuation is small, and the response speed is fast. The simulation signal clearly shows that the finite-time dynamic surface control method of the permanent magnet synchronous motor proposed by the present invention considering iron loss can efficiently track the reference signal.

Of course, the above descriptions are only the preferred embodiments of the present invention, and the present invention is not limited to the above-mentioned embodiments. , and obvious deformation forms, all fall within the essential scope of this specification, and should be protected by the present invention.

Claims (1)

1. a permanent magnet synchronous motor finite-time dynamic surface control method considering iron loss, is characterized in that, It includes the following steps: a. Establish a dynamic mathematical model of the permanent magnet synchronous motor considering iron loss on the d-q axis, as shown in formula (1):

Where, Θ is the rotor angle, ω is the rotor angular velocity, n _p is the magnetic logarithm, J is the moment of inertia, _TL is the load torque, id is the _d -axis current, i _q is the q-axis current, and ud is the _d -axis Voltage, u _q represents the q-axis voltage, i _od represents the d-axis excitation current component, i _oq represents the q-axis excitation current component, L _d represents the d-axis stator inductance, L _q represents the q-axis stator inductance, and L _ld represents the d-axis leakage inductance , L _lq represents the q-axis leakage inductance, L _md represents the d-axis excitation inductance, L _mq represents the q-axis excitation inductance, R ₁ represents the stator resistance, R _c represents the iron core loss resistance, and λ _PM is the rotor permanent magnet excitation flux; In order to simplify the dynamic mathematical model of permanent magnet synchronous motor considering iron loss, new variables are defined as follows:

Then the dynamic mathematical model of permanent magnet synchronous motor considering iron loss is expressed by formula (2), namely:

b. According to the finite-time dynamic surface technology and the principle of the adaptive backstepping method, a finite-time dynamic surface control method of permanent magnet synchronous motor considering iron loss is designed. The specific process is as follows: Assuming that f(Z) is a continuous function in the compact set Ω _Z , for any constant ε>0, there is always a fuzzy logic system W ^T S(Z) satisfying:

where, the input vector

q is the fuzzy input dimension, R ^q is a real vector set; W∈R ^l is a fuzzy weight vector, the number of fuzzy nodes l is a positive integer, and l>1, R ^l is a real vector set; S(Z)=[s ₁ (Z),...,s _l (Z)] ^T ∈R ^l is the basis function vector, s ₁ (Z),...,s _l (Z) represent the basis vector of S(Z) respectively; The basis function s _j (Z) is a Gaussian function as follows:

Among them, μ _j =[μ _j1 ,...,μ _jq ] ^T is the center position of the Gaussian function distribution curve, and η _j is its width; μ _j1 ,...,μ _jq represent the basis vectors of μ _j respectively ; Define finite time: for any real numbers λ ₁ >0, λ ₂ >0, 0<γ<1, the extended Lyapunov condition for finite time stability is expressed as:

Among them, V(x) represents the Lyapunov function of the system; the convergence time T _r of the system is determined by T _r ≤t ₀ +[1/λ ₁ (1-γ)]ln[(λ ₁ V ^1-γ (t ₀ )+ λ ₂ )/λ ₂ ] to estimate, t ₀ represents the initial time; Considering the input voltage constraint problem of permanent magnet synchronous motor as follows: u _min ≤v≤u _max ; Among them, u _min and u _max represent the minimum and maximum value of the known stator input voltage, namely:

Among them, u _max >0 and u _min <0 are unknown constants limited by input constraints, and v is the actual input signal; Use the piecewise smooth function g(v) to approximate the constraint function, and define g(v) as follows:

u=sat(v)=g(v)+d(v), d(v) is a bounded function whose bounds are: |d(v)|=|sat(v)-g(v)|≤max{u _max [1-tanh(1)}]u _min [tanh(1)-1]=D; Among them, D represents D _d and D _q on the d-axis and q-axis, respectively, and both D _d and D _q are constants greater than 0; Using the mean value theorem, we know that there is a constant μ such that

in,

v _μ = μv+(1-μ) v ₀ ; Selecting v ₀ =0, the above function can be written as:

therefore,

then there are

where there is an unknown constant g _m such that

Define a new variable α _id and a time constant ∈ _i ,

α _id (0)=α _i (0), i=1, 2, 3, 4; Among them, α _id (0) represents the initial value of α _id , and α _i (0) represents the initial value of α _i ; The virtual control law α _i obtains α _id through a first-order filter, wherein, the virtual control laws α ₁ , α ₂ , α ₃ , α ₄ are the input signals of the first-order filter, α _1d , α _2d , α _3d , α _4d is the output signal of the first-order filter; The tracking errors z ₁ , z ₂ , z ₃ , z ₄ , z ₅ and z ₆ are defined as:

Among them, x _1d is the desired position signal, and x _4d is the desired rotor flux linkage signal; In the above finite-time dynamic surface control method of permanent magnet synchronous motor considering iron loss, a Lyapunov function is selected for each step to construct a virtual control law or a real control law. The specific steps are as follows: b.1. According to the first equation z ₁ =x ₁ -x _1d in formula (3), choose the Lyapunov function:

Derivation with respect to V1 gives _:

Choose a virtual control law:

Wherein, the control gain k ₁ >0, the constant s ₁ >0, the constant 0<γ<1; Based on the above formula (4) and formula (5), we get:

b.2. According to the second equation z ₂ =x ₂ -α _1d in formula (3), choose the Lyapunov function:

Differentiating V ₂ and substituting Equation (6) yields Equation (7):

Among them, the load torque T _L is an unknown constant and the upper limit is d, that is, |T _L |≤d, where d>0; By Young's inequality we have

where ε ₁ is an arbitrarily small positive number, then:

in,

According to the universal approximation theorem, for any small positive number ε ₂ , choose the fuzzy logic system

makes:

Among them, δ ₂ (Z) is the approximation error and satisfies the inequality |δ ₂ (Z)|≤ε ₂ , and ||W ₂ || is the norm of the vector W ₂ ; Choose a virtual control law:

Among them, the control gain k ₂ >0, the constant s ₂ >0, the constant l ₂ >0,

is an estimate of θ, the definition of θ is given below; According to the third equation z ₃ =x ₃ -α _2d in formula (3), then

Expressed as:

b.3. According to the third equation in formula (3): z ₃ =x ₃ -α _2d , choose the Lyapunov function:

Differentiating V ₃ and substituting Equation (10) yields Equation (11):

in,

According to the universal approximation theorem, for any small positive number ε ₃ , choose the fuzzy logic system

makes:

where δ ₃ (Z) is the approximation error and satisfies the inequality |δ ₃ (Z)|≤ε ₃ , and ||W ₃ || is the norm of the vector W ₃ ; thus:

Choose a virtual control law:

Wherein, the control gain k ₃ >0, the constant s ₃ >0, and the constant l ₃ >0; According to the fourth equation z ₄ =x ₄ -α _3d in formula (3), then

Expressed as:

b.4. According to the fourth equation z ₄ =x ₄ -α _3d in formula (3), choose the Lyapunov function:

Differentiating V ₄ and substituting Equation (14) yields Equation (15):

in,

According to the universal approximation theorem, for any small positive number ε ₄ , choose the fuzzy logic system

make

where δ ₄ (Z) is the approximation error and satisfies the inequality |δ ₄ (Z)|≤ε ₄ , and ||W ₄ || is the norm of the vector W ₄ ; thus:

Build the real control law:

Wherein, the control gain k ₄ >0, the constant s ₄ >0, and the constant l ₄ >0; From the input saturation formula u _q =sat(v _q )=g(v _q )+d(v _q ), we get: d ₁ z ₄ u _q =d ₁ z ₄ g(v _q )+d ₁ z ₄ d( v _q ),

From Young's inequality we get

where the constant D _q > 0, we get:

b.5. According to the fifth equation z ₅ =x ₅ in formula (3), choose the Lyapunov function:

Derivation with respect to _V5 gives:

Build a virtual control law:

Wherein, the control gain k ₅ >0, the constant s ₅ >0; according to the sixth equation z ₆ =x ₆ -α _4d in formula (3), we can obtain:

b.6. According to the sixth equation z ₆ =x ₆ -α _4d in formula (3), select the Lyapunov function:

Derivating V ₆ gives:

in,

According to the universal approximation theorem, for any small positive number ε ₆ , choose the fuzzy logic system

make

where δ ₆ (Z) is the approximation error and satisfies the inequality |δ ₆ (Z)|≤ε ₆ , and ||W ₆ || is the norm of the vector W ₆ ; thus:

Build the real control law:

Wherein, the control gain k ₆ >0, the constant s ₆ >0, and the constant l ₆ >0; By entering the saturation formula ud =sat(v _d )=g(v _d )+d(v _d ), we get: d ₂ z ₆ u _d = _{d 2} z ₆ g(v _d )+d ₂ z ₆ d( v _d ),

definition

is the estimated value of θ, define

By Young's inequality

where the constant D _d > 0, we get:

b.7. Define y _i =α _id -α _i , i=1, 2, 3, 4, and get:

in,

Choose the Lyapunov function for the system:

where r ₁ is a positive number, and derivation with respect to V yields:

Among them, the control gain k ₆ >0; the adaptive law is constructed as follows:

Among them, m ₁ is a positive number; c. Stability analysis of the finite-time dynamic surface control method of permanent magnet synchronous motor considering iron loss in step b; Substituting formula (27) into formula (26) yields:

where |B _i | has a maximum value |B _iM | on the compact set |Ω _i |,i=1,2,3,4 and |B _i |≤B _iM , then:

constant τ>0; From Young's inequality we get:

By inequality scaling we get:

Scaling according to the above inequality, Equation (28) is written as:

in,

b ₀ =min{2, 2s ₁ , 2s ₂ , 2s ₃ , 2g _m s ₄ , 2s ₅ , 2g _m s ₆ , m ₁ };

It is obtained by formula (29):

From formula (30), if a ₀ -(c/2V)>0 and b ₀ -(c/2V ^[(γ+1)/2] )>0; According to the definition of finite time, in the convergence time _Tr of the system,

The tracking error z ₁ will converge to within a small neighborhood of the origin in finite time.

Images