阅读说明：本技术 一种调控电机速度环的分数阶自抗扰控制器的设计方法 (Design method of fractional order active disturbance rejection controller for regulating and controlling motor speed loop ) 是由 罗映 陈鹏冲 于 2020-05-28 设计创作，主要内容包括：本发明属于控制器设计相关技术领域,并公开了一种调控电机速度环的分数阶自抗扰控制器的设计方法。该方法包括：S1构建电机速度环与分数阶自抗扰控制器的关系,构建待调控电机速度环系统的控制率u、分数阶PD控制器的输出u<Sub>0</Sub>以及待调控电机速度环系统中总扰动f的关系式；S2设计扩张状态观测器；S3构建包括扩张状态观测器、PD控制器、增益-相位裕度测试器和补偿后的速度环控制对象的闭环控制系统,求解扩张状态观测器和PD控制器中的未知参数,以此实现分数阶自抗扰控制器的设计。通过本发明,实现控制器的设计,鲁棒性强,精度高。(The invention belongs to the technical field related to controller design and discloses a design method of a fractional order active disturbance rejection controller for regulating and controlling a motor speed loop. The method comprises the following steps: s1, establishing the relation between the motor speed loop and the fractional order active disturbance rejection controller, and establishing the control rate u of the motor speed loop system to be regulated and controlled and the output u of the fractional order PD controller 0 And a relational expression of total disturbance f in a motor speed loop system to be regulated; s2 designing an extended state observer; s3, a closed-loop control system comprising the extended state observer, the PD controller, the gain-phase margin tester and the compensated speed loop control object is constructed, unknown parameters in the extended state observer and the PD controller are solved, and therefore the design of the fractional order active disturbance rejection controller is achieved. The invention realizes the design of the controller, and has strong robustness and high precision.)

1. A design method of a fractional order active disturbance rejection controller for regulating and controlling a motor speed ring is characterized by comprising the following steps:

s1 construction of relation between motor speed loop and fractional order active disturbance rejection controller

For a motor speed loop system to be regulated and controlled, a fractional order active disturbance rejection controller is adopted to regulate and control a speed loop of the motor, and the fractional order active disturbance rejection controllerThe device comprises a fractional order PD controller and an extended state observer, and the control rate u of a motor speed loop system to be regulated and controlled and the output u of the fractional order PD controller are constructed₀The relation formula (I) of the total disturbance f in the motor speed loop system to be regulated and controlled is shown, wherein the total disturbance f in the motor speed loop system to be regulated and controlled is estimated through the extended state observer;

s2 design extended state observer

For a motor speed loop system to be regulated and controlled, acquiring a differential equation and a state space model of a motor speed loop control object, and designing an extended state observer according to the effect of the extended state observer on estimating the total disturbance;

s3 solving unknown parameters in extended state observer and PD controller

S31, according to the relation (I), compensating the motor speed loop object to be regulated and controlled by the total disturbance estimated by the extended state observer, so as to obtain a compensated motor speed loop control object;

s32, constructing a closed-loop control system comprising an extended state observer, a PD controller, a gain-phase margin tester and a compensated speed loop control object;

s33, calculating unknown parameters in the extended state observer and the fractional order PD controller by using the constraint conditions and the boundary conditions of the closed-loop control system, and designing the fractional order active disturbance rejection controller.

2. The method of claim 1, wherein in step S1, the relation (one) is expressed by the following expression:

where b is the motor speed loop gain.

3. The method as claimed in claim 1, wherein in step S1, the transfer function of the fractional order PD controller is:

C_FOPD(s)＝K_p+K_ds^μ

wherein, C_FOPD(s) is the transfer function of the fractional order PD controller, K_pAnd K_dAre unknown parameters, proportional gain and differential gain, respectively, μ is the order of differentiation, μ ∈ (0,2), and s is the laplacian.

4. The method of claim 1, wherein in step S2, the model of the extended state observer is expressed by the following expression:

in the formula (I), the compound is shown in the specification,

L＝[β₁β₂β₃]^Tis the gain of the extended state observer, β₁，β₂And β₃Are all unknown parameters, z ═ z₁z₂z₃]^TWherein z is₁，z₂And z₃Are all the outputs of the extended state observer, z₁And z₂For estimating the derivatives of y and y, respectively, z₃And estimating the total disturbance f, A, B and C to be intermediate variables respectively, wherein u is the control rate of the motor speed loop system to be regulated and controlled, and y is the output of the motor speed loop system to be regulated and controlled.

5. The method as claimed in claim 1, wherein in step S3, the compensated motor speed loop control objects are:

β₁＝3ω_o

β₂＝3ω_o ²

β₃＝ω_o ³

wherein, P_c(s) is the compensated motor speed loop control object, a is the motor speed loop model parameter, β₁＝3ω_o，β₂＝3ω_o ²And β₃＝ω_o ³，ω_oIs the bandwidth of the extended state observer.

6. The method of claim 5, wherein in step S32, the transfer function of the closed-loop control system is:

the characteristic equation of the transfer function is:

D(K_p,K_d,μ,A,φ；s)＝(s⁵+(a+β₁)s⁴+(aβ₁+β₂)s³+(aβ₂+β₃)s²)+Ae^-jφ(K_p+K_ds^μ)(s³+β₁s²+β₂s+β₃)

where A is the amplitude margin, φ is the phase margin, and G(s) is the transfer function of the closed-loop control system.

7. The method of claim 6, wherein the step of calculating the unknown parameters in the extended state observer and the PD controller by using the constraints and boundary conditions of the closed-loop control system in the step S33 is performed according to the following steps:

s331, constructing constraint conditions of phase margin, crossing indexes and ITAE indexes;

s332 determining K according to the real root boundary, the infinite root boundary and the complex root boundary_pAnd K_dAnd K_pAnd K_dWith respect to μ and ω, respectively_oThe relational expression of (1);

s333 in mu and omega_oWithin the range of [ mu ] and [ omega ] respectively_oAssigning values of all mu and omega satisfying the above constraint conditions and boundary conditions_oForming a feasible solution set;

s334, carrying out simulation calculation on points in the feasible solution set to calculate ITAE values, and carrying out simulation calculation on mu and omega corresponding to the minimum value in the ITAE values obtained_oI.e. the desired value, mu and omega are used_oCalculating to obtain unknown parameters K_p、K_d、β₁、β₂And β₃The value of (c).

8. The method as claimed in claim 7, wherein in step S331, the phase margin, the crossing index and the ITAE index are respectively expressed as follows:

wherein, ω is_gcIs the crossover frequency, dB is the unit of amplitude, t is the real-time simulation time, and e (t) is the error at time t.

9. The method of claim 7, wherein in step S332, K is the integer multiple of K_pAnd K_dThe boundaries of (a) are:

K_p＝0，K_dno boundary exists;

K_pand K_dWith respect to μ and ω, respectively_oThe relationship of (A) is as follows:

wherein the content of the first and second substances,

F₂E₅is F₂And E₅Product between, F₁F₇Is F₁And F₇Product between, E₆E₇Is E₆And E₇Product between, E₅E₈Is E₅And E₈The product between them.

Technical Field

The invention belongs to the technical field related to controller design, and particularly relates to a design method of a fractional order active disturbance rejection controller for regulating and controlling a motor speed loop.

Background

The permanent magnet synchronous motor is widely applied in various fields, most of various automation equipment and robots adopt the permanent magnet synchronous motor, the permanent magnet synchronous motor has the advantages of high response speed, high precision, high power and the like, and a speed ring is the most important ring in motor vector control, so that the precise control of the speed ring of the permanent magnet synchronous motor is very critical.

Currently, the most common speed loop controller is a PID, but its resistance to external disturbances is limited. The active disturbance rejection control method is proposed by professor kyoto, han scholars, and has been widely paid attention and applied in recent years. The fractional order controller is a general form of the integer order controller, and the control performance of the fractional order controller is proved to be superior to that of the integer order controller. Therefore, the present invention provides a fractional auto-disturbance-rejection controller. Regarding the setting method of the active disturbance rejection controller, the method based on the frequency domain index has not been taken into consideration, but the frequency domain method is important in engineering application, so the invention provides the parameter setting method of the active disturbance rejection controller based on the frequency domain index and the time domain index.

Disclosure of Invention

Aiming at the defects or improvement requirements of the prior art, the invention provides a design method of a fractional order active disturbance rejection controller for regulating and controlling a motor speed loop, wherein the design method of the fractional order active disturbance rejection controller is realized by adopting the fractional order active disturbance rejection controller to control the motor speed loop and adopting a gain-phase margin tester to solve the solution of unknown parameters in the fractional order active disturbance rejection controller.

To achieve the above object, according to the present invention, there is provided a design method of a fractional order active disturbance rejection controller for regulating a speed loop of a motor, the design method comprising the steps of:

s1 construction of relation between motor speed loop and fractional order active disturbance rejection controller

For a motor speed loop system to be regulated and controlled, a fractional order active disturbance rejection controller is adopted to regulate and control a speed loop of the motor speed loop system to be regulated and controlled, the fractional order active disturbance rejection controller comprises a fractional order PD controller and an extended state observer, and the control rate u of the motor speed loop system to be regulated and controlled and the output of the fractional order PD controller are constructedu₀The relation formula (I) of the total disturbance f in the motor speed loop system to be regulated and controlled is shown, wherein the total disturbance f in the motor speed loop system to be regulated and controlled is estimated through the extended state observer;

s2 design extended state observer

For a motor speed loop system to be regulated and controlled, acquiring a differential equation and a state space model of a motor speed loop control object, and designing an extended state observer according to the effect of the extended state observer on estimating the total disturbance;

s3 solving unknown parameters in extended state observer and PD controller

S31, according to the relation (I), compensating the motor speed loop object to be regulated and controlled by the total disturbance estimated by the extended state observer, so as to obtain a compensated motor speed loop control object;

s32, constructing a closed-loop control system comprising an extended state observer, a PD controller, a gain-phase margin tester and a compensated speed loop control object;

s33, calculating unknown parameters in the extended state observer and the fractional order PD controller by using the constraint conditions and the boundary conditions of the closed-loop control system, and designing the fractional order active disturbance rejection controller.

Further preferably, in step S1, the relation (one) is performed according to the following expression:

where b is the motor speed loop gain.

Further preferably, in step S1, the transfer function of the fractional order PD controller is:

C_FOPD(s)＝K_p+K_ds^μ

wherein, C_FOPD(s) is the transfer function of the fractional order PD controller, K_pAnd K_dAre unknown parameters, proportional gain and differential gain, respectively, μ is the order of differentiation, μ ∈ (0,2), s is lapraAnd (5) a gaussian operator.

Further preferably, in step S2, the model of the extended state observer is performed according to the following expression:

in the formula (I), the compound is shown in the specification,

C＝[1 0 0]

L＝[β₁β₂β₃]^Tis the gain of the extended state observer, β₁，β₂And β₃Are all unknown parameters, z ═ z₁z₂z₃]^TWherein z is₁，z₂And z₃Are all the outputs of the extended state observer, z₁And z₂For estimating the derivatives of y and y, respectively, z₃And estimating the total disturbance f, A, B and C to be intermediate variables respectively, wherein u is the control rate of the motor speed loop system to be regulated and controlled, and y is the output of the motor speed loop system to be regulated and controlled.

Further preferably, in step S3, the compensated motor speed loop control objects are:

β₁＝3ω_o

β₂＝3ω_o ²

β₃＝ω_o ³

wherein, P_c(s) is the compensated motor speed loop control object, a is the motor speed loop model parameter, β₁＝3ω_o，β₂＝3ω_o ²And β₃＝ω_o ³，ω_oIs the bandwidth of the extended state observer.

Further preferably, in step S32, the transfer function of the closed-loop control system is:

the characteristic equation of the transfer function is:

D(K_p,K_d,μ,A,φ；s)＝(s⁵+(a+β₁)s⁴+(aβ₁+β₂)s³+(aβ₂+β₃)s²)+

Ae^-jφ(K_p+K_ds^μ)(s³+β₁s²+β₂s+β₃)

where A is the amplitude margin, φ is the phase margin, and G(s) is the transfer function of the closed-loop control system.

Further preferably, in step S33, calculating the unknown parameters in the extended state observer and the PD controller using the constraints and boundary conditions of the closed-loop control system is performed according to the following steps:

s331, constructing constraint conditions of phase margin, crossing indexes and ITAE indexes;

s332 determining K according to the real root boundary, the infinite root boundary and the complex root boundary_pAnd K_dAnd K_pAnd K_dWith respect to μ and ω, respectively_oThe relational expression of (1);

s333 in mu and omega_oWithin the range of [ mu ] and [ omega ] respectively_oAssigning values of all mu and omega satisfying the above constraint conditions and boundary conditions_oForming a feasible solution set;

s334, carrying out simulation calculation on points in the feasible solution set to calculate ITAE values, and carrying out simulation calculation on mu and omega corresponding to the minimum value in the ITAE values obtained_oI.e. the desired value, mu and omega are used_oCalculating to obtain unknown parameters K_p、K_d、β₁、β₂And β₃The value of (c).

Further preferably, in step S331, the phase margin, the crossing index and the ITAE index are performed according to the following expressions, respectively:

wherein, ω is_gcIs the crossover frequency, dB is the unit of amplitude, t is the real-time simulation time, and e (t) is the error at time t.

Further preferably, in step S332, K is_pAnd K_dThe boundaries of (a) are:

K_p＝0，K_dno boundary exists;

K_pand K_dWith respect to μ and ω, respectively_oThe relationship of (A) is as follows:

wherein the content of the first and second substances,

F₂E₅is F₂And E₅Product between, F₁F₇Is F₁And F₇Product between, E₆E₇Is E₆And E₇Product between, E₅E₈Is E₅And E₈The product between them.

Generally, compared with the prior art, the technical scheme of the invention has the following beneficial effects:

1. the invention provides a fractional order active disturbance rejection controller applied to the speed control of a permanent magnet synchronous motor, which can realize the accurate control of the motor speed, and simultaneously considers the frequency domain and time domain indexes, so that the control system has strong robustness and optimal dynamic control performance;

2. the invention adopts the frequency domain index phase margin and the crossing frequency for constructing the control system, so that the control system can meet the frequency domain index given by a user, and the dynamic performance of the control system is optimal by constructing the time domain constraint index of the control system.

Drawings

Fig. 1 is a schematic structural diagram of a motor speed loop system to be regulated and controlled connected with a fractional order active disturbance rejection controller according to a preferred embodiment of the present invention;

FIG. 2 is a schematic block diagram of a closed loop control system constructed in accordance with a preferred embodiment of the present invention;

FIG. 3 is a depiction of a stable region and an unstable region in accordance with a preferred embodiment of the present invention;

FIG. 4 is a diagram of controller parameters that meet a given frequency domain criterion, constructed in accordance with a preferred embodiment of the present invention;

FIG. 5 is a graphical three-dimensional visualization of controller parameters meeting a given frequency domain metric constructed in accordance with a preferred embodiment of the present invention;

FIG. 6 is a J constructed in accordance with a preferred embodiment of the present invention_ITAEAnd μ;

FIG. 7 is a diagram of all controller parameters that satisfy the frequency domain criterion constructed in accordance with the preferred embodiment of the present invention:

FIG. 8 is a simulation calculated J for all controller parameters constructed in accordance with a preferred embodiment of the present invention_ITAE。

Detailed Description

In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

S1 construction of relation between motor speed loop and fractional order active disturbance rejection controller

For a motor speed loop system to be regulated and controlled, a fractional order active disturbance rejection controller is adopted to regulate and control a speed loop of the motor speed loop system to be regulated and controlled, the fractional order active disturbance rejection controller comprises a fractional order PD controller and an extended state observer, as shown in FIG. 1, the right side in the figure is the motor speed loop system to be regulated and controlled, the left side is the fractional order active disturbance rejection controller, and the fractional order active disturbance rejection controller is used for regulating and controlling the speed loop of the motor speed loop system to be regulated and controlled;

reference input u for constructing motor speed loop system to be regulated and output u of fractional order PD controller₀And a relational expression (I) of the total disturbance f in the motor speed loop system to be regulated, wherein the total disturbance f in the motor speed loop system to be regulated is estimated by an extended state observer, and the relational expression (I) is as follows:

s2 design extended state observer

(1) Extended state observer

The motor speed loop control objects are as follows:

the structure of the fractional order active disturbance rejection controller consists of a fractional order PD controller and an ESO, wherein the ESO is an extended state observer;

the motor speed control object can be written as:

wherein y is the system output, u is the reference input, d is the external disturbance, f is the total disturbance, and the total disturbance comprises the model unknown dynamics and the external disturbance.

ESO is used to estimate the total disturbance f, defined assuming f is guidedThe state space model of equation (1) is:

wherein

According to the above formula, the ESO can be designed as:

wherein L is [ β ]₁β₂β₃]^TIs the gain of ESO, z ═ z₁z₂z₃]^T,z₁,z₂,z₃Is the output of the ESO: z is a radical of₁,z₂Estimating y and its derivative, z, respectively₃Estimating a total disturbance f, the total disturbance being compensated by:

wherein u is₀Is the output of the fractional order PD controller.

(2) Fractional order PD controller

The transfer function of the fractional order PD controller is:

C_FOPD＝K_p+K_ds^μ(6)

wherein, K_pAnd K_dIs the proportional and derivative gain, μ is the derivative order, μ ∈ (0,2)

S3 solving unknown parameters in extended state observer and PD controller

S31 performs laplace transform on equation (4) to solve the equation set:

according to the formulas (1), (5) and (7), the speed loop control object after disturbance compensation is obtained as follows:

wherein the content of the first and second substances,

β₁＝3ω_o,β₂＝3ω_o ²,β₃＝ω_o ³(9)

s32, constructing a closed-loop control system comprising an extended state observer, a PD controller, a gain-phase margin tester and a compensated speed loop control object;

as shown in fig. 2, in which M is_TRepresenting a gain-phase margin tester, D_FOPD(s) stands for PD controller, P_cAnd(s) represents a compensated speed loop control object, and the three are connected in sequence to form a closed-loop control system.

S33 calculates the unknown parameters in the extended state observer and the PD controller using the constraint conditions and the boundary conditions of the closed-loop control system as follows:

(1) constraint conditions

In the invention, two frequency domain indexes and one time domain index are used for designing the fractional order active disturbance rejection controller. The two frequency domain indicators are:

phase margin:

the crossing frequency:

the time domain index is:

ITAE index

(2) Boundary condition

MT is a Gain-Phase Margin Tester, which can provide boundary information of a stable region, and has a transfer function of:

M_T(A,φ)＝Ae^-jφ(10)

the transfer function of the closed-loop control system is as follows:

the closed loop transfer function has the following characteristic equation:

determining the parameter boundary specifically as follows:

range of: for the order μ of a fractional order PD controller, its range is defined as μ e (0, 2);

② solid root boundary is defined as D (K)_p,K_dμ, A, φ; s-0) 0, hence K_pThe boundaries of (a) are:

K_p＝0 (13)

③ infinite root boundary since the f relative order between the numerator and denominator of control object equation (8) is 2, K_dNo boundary exists;

④ multiple root boundary s ═ j ω is substituted into (10), and the multiple root boundary can be defined as D (K)_p,K_dμ, A, φ; s ═ j ω) ═ 0, i.e.

The real and imaginary parts of the above equation are each equal to 0, i.e.

F₁+A(K_pE₅+K_dE₆)＝0,

F₂+A(K_pE₇+K_dE₈)＝0

Wherein the content of the first and second substances,

F₁＝(a+β₁)ω⁴-(aβ₂+β₃)ω²,F₂＝ω⁵-(aβ₁+β₂)ω³,

E₃＝-ω³+β₂ω,E₄＝-β₁ω²+β₃,E₅＝E₄cosφ+E₃sinφ,

E₇＝E₃cosφ+E₄sinφ,

solving the equation yields, from the above equation:

thus, given a 1, 0, fixed μ, ω_oPlane of parameters (K)_p,K_d) The device is divided into a stable region and an unstable region by a solid root boundary and a multiple root boundary.

And the fractional order active disturbance rejection controller meets the frequency domain index:

the characteristic equation of the closed loop transfer function (11) is:

1+M_T(A,φ)C_FOPD(s)P_c(s)|_s＝jω＝0

meaning that the open loop transfer function t(s) is equal to-1,

T(s)_s＝jω＝M_T(A,φ)C_FOPD(s)P_c(s)|_s＝jω＝-1

thus, it is possible to obtain an amplitude and phase of:

|M_T(A,φ)C_FOPD(s)P_c(s)|_s＝jω＝1,

arg[M_T(A,φ)C_FOPD(s)P_c(s)]_s＝jω＝-π

therefore, given a 1, all ω satisfying equation (14) can be considered to be the crossover frequency of the control system.

(3) Solving for unknown parameters

For practical application scenarios, the crossing frequency omega required to be satisfied is given_gcAnd a phase margin phi_mFix mu, omega_oOther two parameters K of fractional order auto-disturbance rejection controller_pAnd K_dCan be determined as a point of the parameter plane, and then scanning all μ ∈ (0,2), a curve can be obtained in the three-dimensional parameter space, where all points satisfy ω_gcAnd phi_m。

Then all omega are scanned_o∈(ω_gc,ω_max)(ω_oThe larger the disturbance, the stronger the ability to estimate the disturbance, but at the same time the noise will be amplified, therefore ω is_maxShould be valued according to the actual environment), K can be obtained_p、K_dMu three-dimensional graph, the controller parameter corresponding to each point on the graph satisfies the crossing frequency omega_gcAnd a phase margin phi_mThe points obtained above are simulated, ITAE is calculated, and omega can be obtained_o,μ,J_ITAESelecting the smallest J_ITAECorresponding μ and ω_oUsing mu and omega_oCalculating to obtain unknown parameters K_p、K_d、β₁、β₂And β₃To determine the unknown parameters in the fractional order PD controller and the extended state observer, i.e. to complete the design of the controller.

The simulation procedure is to calculate ITAE by the following formula,

where T is the simulation time and Δ T is the discrete time.

The present invention will be further illustrated with reference to specific examples.

1) Setting specific parameters of motor speed loop26.08 for a and 383.635 for b, the designed control system satisfies the crossing frequency of ω_gc50rad/s, phase margin phi_m＝80°；

2) Select fractional order μ ═ 0.9 and ω_oLet a be 1 and phi be 0 °, real root boundaries and complex root boundaries are drawn according to equations (13) and (15), and as shown in fig. 3, points are randomly selected and subjected to a simulation test to obtain a stable region and an unstable region. Let omega become omega_gc＝50,φ＝φ_m80 °, according to (15), the controller parameter may be determined as "× a";

3) scanning all λ ∈ (0,2) to get a series of ". about." as shown in FIG. 4, showing it in three-dimensional space as shown in FIG. 5, all points in the figure satisfy the given crossover frequency and phase margin_ITAEMu and J_ITAEThe corresponding diagram is shown in fig. 6;

4) scanning all omega_o∈(ω_gc,ω_max) (in the motor control system herein, ω is_max800), all parameters satisfying a given frequency domain criterion can be obtained and visualized in the three-dimensional graph 7, all parameters simulating the calculated J_ITAEAs shown in FIG. 8, the smallest J is selected_ITAEThe corresponding controller parameter is the designed fractional order active disturbance rejection controller.

It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.