CN109799697A - Method for controlling permanent magnet synchronous motor based on robust fractional order PI strategy - Google Patents

Abstract

The present invention provides a kind of method for controlling permanent magnet synchronous motor based on robust fractional order PI strategy, this method is directed to control system for permanent-magnet synchronous motor, according to the constraint of the Real Number Roots boundary value of closed-loop pole and the constraint of σ stability boundaris value, parameter variation range of the fractional order PI controller when system keeps bounded input-output bounds to stablize is obtained；Take the gain cross-over frequency criterion and phase margin criterion in frequency domain into consideration, on the basis of the parameter variation range is further constrained, system output response can realize every specific performance indicator；Finally by smoothing pseudorange criterion is considered, the overshoot of system output response and regulating time is enabled to remain constant in the case where open-loop gain variations, to ensure that the robustness of permanent magnet synchronous motor system.

Description

Method for controlling permanent magnet synchronous motor based on robust fractional order PI strategy

Technical field

The present invention relates to a kind of method for controlling permanent magnet synchronous motor, in particular to the permanent magnetism based on robust fractional order PI strategy Synchronous motor control method belongs to the scopes such as movement control technology and optimal method.

Background technique

For given stabilization single-input-single-output control system, fractional order control device has more than integer rank controller The conclusion of good characteristic has obtained the research demonstration of lot of documents.In addition, it is contemplated that possessed by permanent magnet synchronous motor system The complex characteristics such as multivariable, close coupling can more precisely portray it using fractional calculus theory.How to design Fractional order control device has obtained increasingly wider as an important topic recently so that system reaches specific expected performance index General concern.

In existing controller design method, thought of a part research based on ideal open-loop transfer function passes through zero Originally complicated controller and control object are equivalent to one and preferably have the pure of fractional order order by the form that pole offsets Integral element, it is advantageous in that the form for greatlying simplify system open loop and closed loop transfer function, only configures fractional order rank Secondary and open-loop gain can realize the control to system, but the shortcomings that this method is to have lost fractional order control device to be had originally Some flexibilities.Another part is studied based on the Domain Design criterion in classical control theory, in presetting gain cross-over frequency On the basis of phase margin, the parameter in fractional order control device can be gone out by analytical method solving, but pre- in such methods Setting value is not easy to determine, it usually needs by a large amount of trials of trial-and-error method, can just obtain desired result.

In conclusion current method can not also effectively determine gain cross-over frequency and phase margin, for permanent-magnet synchronous Electric system designs that overshoot is small, stable fractional order control device.

Summary of the invention

In order to overcome existing method that can not effectively determine gain cross-over frequency and phase margin, for permanent magnet synchronous motor system System designs the technological deficiency of small, the steady fractional order control device of overshoot.The present invention provides one kind to be based on robust fractional order PI The method for controlling permanent magnet synchronous motor of strategy, this method is directed to control system for permanent-magnet synchronous motor, according to the real number of closed-loop pole The constraint of root boundary value and the constraint of σ stability boundaris value, show that fractional order PI controller keeps bounded input-output bounds steady in system The parameter variation range of timing；It takes the gain cross-over frequency criterion and phase margin criterion in frequency domain into consideration, becomes in the parameter On the basis of change range is further constrained, system output response can realize every specific performance indicator；Finally by consideration Smoothing pseudorange criterion enables the overshoot of system output response and regulating time to remain permanent in the case where open-loop gain variations It is fixed, to ensure that the robustness of permanent magnet synchronous motor system.

The technical solution adopted by the present invention to solve the technical problems: a kind of permanent magnetism based on robust fractional order PI strategy is same Motor control method is walked, its main feature is that the following steps are included:

Step 1: the voltage characteristic equation and mechanical characteristic equation for providing permanent magnet synchronous motor are respectively as follows:

Wherein, u_q,i_qFor the stator voltage and electric current of motor, R, L are stator resistance and inductance, and E is induced electromotive force, C_e For induced electromotive force coefficient, n is motor speed, T_eFor electromagnetic torque, T_LFor load torque, C_mFor torque coefficient, i_LFor equivalent negative Carry electric current, GD²For conversion to the flywheel inertia on motor shaft, B_mFor friction factor.Two characteristic equations can be characterized with attached drawing 1.

Merge each module in attached drawing 1, obtain the transmission function of permanent magnet synchronous motor model are as follows:

C in denominator_eC_m/B_mR is the number of a very little, and enabling it is approximately zero, then according to two lesser inertia of time constant Combinable link is an inertial element, while the time lag characteristic in view of inherently including in reality system, permanent magnet synchronous electric Machine model can be equivalent to the single order controlled device P (s) containing Time Delay, and given fractional order PI controller C (s):

Wherein, K_pFor proportional gain, K_iFor integral gain, λ is fractional order order, and K is loop gain, and T is time constant, L For hysteresis factors.

By the structure of controller and controlled device, can obtain system open loop transmission function G (s), closed loop transfer function, Φ (s) and The equation of proper polynomial D (s) is respectively as follows:

D (s)=K (K_p+K_is^-λ)e^-Ls+ Ts+1=0

Step 2: for the closed loop proper polynomial D (s) of system, it is contemplated that if wanting to make system closed-loop stabilization, closed loop pole Point all needs the left-half positioned at complex plane, thus provides the constraint of Real Number Roots boundary value and the constraint of σ stability boundaris value, is respectively as follows:

Wherein, σ is the real part of complex number type pole, and ω is the imaginary part of complex number type pole, defines phase angle theta, form are as follows:

It enables the real and imaginary parts of σ stability boundaris value constraint formulations be respectively equal to zero, and abbreviation, then has:

Enabling λ and θ is definite value, then K_pAnd K_iIt can be uniquely determined by ω_。λ is traversed in range [0,2] again, then can be closed In (K_p,K_i, λ) three Parameters variations three-dimension curved surface, be defined as with reference to stabilising surface.

Step 3: for the open-loop transfer function G (s) of system, the gain cross-over frequency criterion and phase in consideration frequency domain are abundant Criterion is spent, following formula can be obtained:

Wherein, ω_cFor gain cross-over frequency, φ_mFor phase margin, M₁,M₂,N₁,N₂Meet following relationship:

M₁=cos (ω_cL)-ω_cTsin(ω_cL)

M₂=ω_cTcos(ω_cL)+sin(ω_cL)

The formula that abbreviation is obtained by two criterion, then have:

K_p=ZK_i

Wherein,

Enable λ and φ_mFor definite value, then K_pAnd K_iIt can be by ω_cIt uniquely determines.λ is traversed in range [0,2] again, then can be obtained About (K_p,K_i, λ) three Parameters variations three-dimension curved surface, be defined as absolute stability face.

Step 4: observation refers to the intersection of stabilising surface and absolute stability face, and further considers smooth phase criterion, then has:

Meet the point of above-mentioned two formula in selection intersection, which is then the parameter of desired fractional order PI controller, referred to as Smooth phase point.

Detailed description of the invention

Fig. 1 is permanent magnet synchronous motor model equivalent structure figure.

Specific embodiment

The following further describes the present invention with reference to the drawings.

The specific implementation steps are as follows by the present invention:

Step 1: the voltage characteristic equation and mechanical characteristic equation for providing permanent magnet synchronous motor are respectively as follows:

Wherein, u_q,i_qFor the stator voltage and electric current of motor, R, L are stator resistance and inductance, and E is induced electromotive force, C_e For induced electromotive force coefficient, n is motor speed, T_eFor electromagnetic torque, T_LFor load torque, C_mFor torque coefficient, i_LFor equivalent negative Carry electric current, GD²For conversion to the flywheel inertia on motor shaft, B_mFor friction factor.Two characteristic equations can be characterized with attached drawing 1.

Merge each module in attached drawing 1, obtain the transmission function of permanent magnet synchronous motor model are as follows:

C in denominator_eC_m/B_mR is the number of a very little, and enabling it is approximately zero, then according to two lesser inertia of time constant Combinable link is an inertial element, while the time lag characteristic in view of inherently including in reality system, permanent magnet synchronous electric Machine model can be equivalent to the single order controlled device P (s) containing Time Delay, and given fractional order PI controller C (s):

Wherein, K_pFor proportional gain, K_iFor integral gain, λ is fractional order order, and K is loop gain, and T is time constant, L For hysteresis factors.

By the structure of controller and controlled device, can obtain system open loop transmission function G (s), closed loop transfer function, Φ (s) and The equation of proper polynomial D (s) is respectively as follows:

D (s)=K (K_p+K_is^-λ)e^-Ls+ Ts+1=0

Step 2: for the closed loop proper polynomial D (s) of system, it is contemplated that if wanting to make system closed-loop stabilization, closed loop pole Point all needs the left-half positioned at complex plane, thus provides the constraint of Real Number Roots boundary value and the constraint of σ stability boundaris value, is respectively as follows:

Wherein, σ is the real part of complex number type pole, and ω is the imaginary part of complex number type pole, defines phase angle theta, form are as follows:

It enables the real and imaginary parts of σ stability boundaris value constraint formulations be respectively equal to zero, and abbreviation, then has:

Enabling λ and θ is definite value, then K_pAnd K_iIt can be uniquely determined by ω.λ is traversed in range [0,2] again, then can be closed In (K_p,K_i, λ) three Parameters variations three-dimension curved surface, be defined as with reference to stabilising surface.

Step 3: for the open-loop transfer function G (s) of system, the gain cross-over frequency criterion and phase in consideration frequency domain are abundant Criterion is spent, following formula can be obtained:

Wherein, ω_cFor gain cross-over frequency, φ_mFor phase margin, M₁,M₂,N₁,N₂Meet following relationship:

M₁=cos (ω_cL)-ω_cTsin(ω_cL)

M₂=ω_cTcos(ω_cL)+sin(ω_cL)

The formula that abbreviation is obtained by two criterion, then have:

K_p=ZK_i

Wherein,

Enable λ and φ_mFor definite value, then K_pAnd K_iIt can be by ω_cIt uniquely determines.λ is traversed in range [0,2] again, then can be obtained About (K_p,K_i, λ) three Parameters variations three-dimension curved surface, be defined as absolute stability face.

Step 4: observation refers to the intersection of stabilising surface and absolute stability face, and further considers smooth phase criterion, then has:

Meet the point of above-mentioned two formula in selection intersection, which is then the parameter of desired fractional order PI controller, referred to as Smooth phase point.

Claims (1)

1. a kind of method for controlling permanent magnet synchronous motor based on robust fractional order PI strategy, its main feature is that the following steps are included:

Step 1: the voltage characteristic equation and mechanical characteristic equation for providing permanent magnet synchronous motor are respectively as follows:

Wherein, u_q,i_qFor the stator voltage and electric current of motor, R, L are stator resistance and inductance, and E is induced electromotive force, C_eFor sense Power coefficient is answered, n is motor speed, T_eFor electromagnetic torque, T_LFor load torque, C_mFor torque coefficient, i_LFor equivalent load electricity Stream, GD²For conversion to the flywheel inertia on motor shaft, B_mFor friction factor.Two characteristic equations can be characterized with attached drawing 1.

Merge each module in attached drawing 1, obtain the transmission function of permanent magnet synchronous motor model are as follows:

C in denominator_eC_m/B_mR is the number of a very little, and enabling it is approximately zero, then according to two lesser inertial elements of time constant Combinable is an inertial element, while the time lag characteristic in view of inherently including in reality system, permanent magnet synchronous motor mould Type can be equivalent to the single order controlled device P (s) containing Time Delay, and given fractional order PI controller C (s):

Wherein, K_pFor proportional gain, K_iFor integral gain, λ is fractional order order, and K is loop gain, and T is time constant, and L is stagnant Postfactor.

By the structure of controller and controlled device, system open loop transmission function G (s), closed loop transfer function, Φ (s) and feature can be obtained The equation of multinomial D (s) is respectively as follows:

D (s)=K (K_p+K_is^-λ)e^-Ls+ Ts+1=0

Step 2: for the closed loop proper polynomial D (s) of system, it is contemplated that if wanting to make system closed-loop stabilization, closed-loop pole is all The left-half of complex plane need to be located at, the constraint of Real Number Roots boundary value and the constraint of σ stability boundaris value is thus provided, be respectively as follows:

Wherein, σ is the real part of complex number type pole, and ω is the imaginary part of complex number type pole, defines phase angle theta, form are as follows:

It enables the real and imaginary parts of σ stability boundaris value constraint formulations be respectively equal to zero, and abbreviation, then has:

Enabling λ and θ is definite value, then K_pAnd K_iIt can be uniquely determined by ω.Again by λ in the range [0,2] traversal, then can be obtained about (K_p,K_i, λ) three Parameters variations three-dimension curved surface, be defined as with reference to stabilising surface.

Step 3: for the open-loop transfer function G (s) of system, considering the gain cross-over frequency criterion and phase margin standard in frequency domain Then, following formula can be obtained:

Wherein, ω_cFor gain cross-over frequency, φ_mFor phase margin, M₁,M₂,N₁,N₂Meet following relationship:

M₁=cos (ω_cL)-ω_cTsin(ω_cL)

M₂=ω_cTcos(ω_cL)+sin(ω_cL)

The formula that abbreviation is obtained by two criterion, then have:

K_p=ZK_i

Wherein,

Enable λ and φ_mFor definite value, then K_pAnd K_iIt can be by ω_cIt uniquely determines.Again by λ in the range [0,2] traversal, then can be obtained about (K_p,K_i, λ) three Parameters variations three-dimension curved surface, be defined as absolute stability face.

Step 4: observation refers to the intersection of stabilising surface and absolute stability face, and further considers smooth phase criterion, then has:

Meet the point of above-mentioned two formula in selection intersection, which is then the parameter of desired fractional order PI controller, referred to as smoothly Phase point.