CN117254735B - Position-sensor-free active disturbance rejection control method based on high-frequency square wave injection - Google Patents
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
- H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
- H02P21/24—Vector control not involving the use of rotor position or rotor speed sensors
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
- H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
- H02P21/13—Observer control, e.g. using Luenberger observers or Kalman filters
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
- H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
- H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
- H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
- H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
- H02P21/18—Estimation of position or speed
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
- H02P25/00—Arrangements or methods for the control of AC motors characterised by the kind of AC motor or by structural details
- H02P25/02—Arrangements or methods for the control of AC motors characterised by the kind of AC motor or by structural details characterised by the kind of motor
- H02P25/022—Synchronous motors
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
- H02P2207/00—Indexing scheme relating to controlling arrangements characterised by the type of motor
- H02P2207/05—Synchronous machines, e.g. with permanent magnets or DC excitation
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Abstract
The invention relates to the technical field of position-free sensors, in particular to a position-free sensor active disturbance rejection control method based on high-frequency square wave injection, which comprises the following steps: 1. injecting high-frequency square waves into an estimated rotation coordinate system of an interior permanent magnet synchronous motor (IPSM), and obtaining a rotor position estimation error through signal extraction and processing; 2. an adaptive extended state observer (ASEO) based on differential algebra spectrum theory is designed by utilizing an error signal, and the total disturbance and state variable of the system are estimated to obtain an accurate rotor position; 3. an ADRC controller (ASEO ADRC) that builds an adaptive extended state observer replaces the conventional PI controller in the speed loop. The method can ensure that the system still has good dynamic performance and rotor position estimation precision when being influenced by external interference, unmodeled dynamic, parameter uncertainty and other interference factors.
Description
Technical Field
The invention relates to the technical field of position-free sensors, in particular to an active disturbance rejection control method of a position-free sensor based on high-frequency square wave injection.
Background
The built-in permanent magnet synchronous motor (Interiorpermanent magnet synchronous motor, IPSM) is a high-efficiency motor with the advantages of compact structure, high power density, wide speed regulation range and the like, and is widely applied to the fields of electric automobiles, aerospace, household appliances and the like. The sensorless control technique of IPMSM is replacing the traditional sensor mode with increased system reliability due to reduced motor volume and cost. However, high performance control of IPMSM is still very challenging, especially when the motor is running at low speed, the dynamic performance of the IPMSM is reduced due to the influence of low signal to noise ratio and external interference caused by modeling uncertainty, inverter nonlinearity and other factors, so research on improving the dynamic performance and anti-interference capability of the IPMSM system without a position sensor has become an urgent problem to be solved.
The method based on signal injection has good position estimation precision and dynamic performance due to low sensitivity to motor parameters, and is an effective method for realizing the control of the IPSM low-speed domain position-free sensor. The method extracts a specific frequency component containing rotor position information from a high-frequency response current by injecting a high-frequency signal in a stationary or rotating reference coordinate system and using the saliency of the IPMSM to acquire a rotor position and a rotational speed. However, the conventional injection sine-type rotation high frequency signal and pulse vibration high frequency signal require the use of a large number of filters in the signal processing process, which makes the system bandwidth limited and dynamic performance lowered. In order to overcome this disadvantage, yoon et al propose a square wave signal injection method to increase the signal frequency to half of the switching frequency or even to the switching frequency, increasing the bandwidth of the controller and improving the system robustness. However, the control performance of IPMSM may be seriously deteriorated due to the influence of uncertain factors such as unmodeled dynamics inside the model and external disturbances. In order to achieve high performance control of IPMSM, it is also necessary to improve the anti-interference capability of the system on the basis of no position sensor control.
Active disturbance rejection control (Activedisturbancerejectioncontrol, ADRC) was proposed by the korean clear teaching service in 1998. ADRC is independent of an accurate mathematical model, has strong anti-interference performance, combines unknown dynamics of a system and external disturbance into total disturbance, is represented by an extended state, and estimates and compensates the total disturbance. There are literature on comprehensive theoretical analysis of ADRC; there is literature comparing the performance of disturbance estimation for DOB, a special PID and ADRC, giving a demonstration of the stability of ADRC for single input single output nonlinear systems with unknown dynamics and disturbances. In the design of nonlinear ADRC, due to its complex structure, there are a large number of parameters to be set, and stability and estimation error analysis are difficult to perform. In order to simplify the controller design, a linearized active disturbance rejection control (ladc) method is proposed in the literature, a complete stability analysis is performed on the ladc, and a parameter adjustment method is proposed, wherein only bandwidth needs to be adjusted. However, laccs may sacrifice their flexibility to improve performance. In addition, a large observer gain may exceed its bandwidth. For high gain observers, a peak phenomenon may occur when the initial real state and the estimated state do not match.
Disclosure of Invention
The invention provides a position-sensor-free active disturbance rejection control (ASEO _ADRC) method based on high-frequency square wave injection, which can improve the anti-disturbance performance of an IPSM position-sensor-free control system.
The invention relates to a position-sensor-free active disturbance rejection control method based on high-frequency square wave injection, which comprises the following steps of:
1. injecting high-frequency square waves into an estimated rotation coordinate system of the IPSM, and obtaining a rotor position estimation error through signal extraction and processing;
2. an Adaptive Extended State Observer (AESO) based on differential algebra spectrum theory is designed by utilizing an error signal, and the total disturbance and state variable of the system are estimated to obtain an accurate rotor position;
3. An ADRC controller (ASEO ADRC) that builds an adaptive extended state observer replaces the conventional PI controller in the speed loop.
Preferably, in the first step, when the IPMSM is operated at a low speed under the synchronous rotation coordinate system, the counter electromotive force and the stator resistance voltage drop are ignored, and the high-frequency voltage equation after the high-frequency signal is injected is expressed as follows:
Wherein, the subscript h represents a corresponding high-frequency signal; p is a differential operator; u dh、uqh is the d and q axis high frequency stator voltage respectively; i dh、iqh is d and q axis high frequency current respectively; l dh、Lqh is d and q axis inductance respectively;
injecting a high-frequency square wave signal in the following formulas (2) - (3) at the d e axis of the estimated coordinate system:
Where u inj is the vector of the injected high frequency square wave in the estimated rotational coordinate system, For injecting high-frequency square waves into components of the d e axis and the q e axis of the estimated rotating coordinate system, V in represents the injected high-frequency square waves, t is a time variable, ts is positive and negative voltage duration, V h is injection voltage amplitude, and n is the number of sequences;
From equation (1), when the high-frequency square wave signal is injected, the high-frequency current response in the stationary coordinate system αβ is obtained as Δi αh、Δiβh:
where θ err is the estimated rotor position error, defined as Theta r and/>The actual value and the estimated value of the rotor position are represented as a coordinate transformation matrix R (x) represented by a variable x:
solving equation (4) to obtain a high-frequency current response, expressed as equation (6):
Where the high-frequency current response in the stationary coordinate system αβ can be obtained by a time delay, Δi αh、Δiβh is represented by I sin、Icos, and when the estimation error θ err tends to 0, the envelope of the high-frequency current response with the rotor position information can be reduced from formula (6) to formula (7):
from equation (7), a suitable position observer is designed to obtain rotor position information.
Preferably, in the synchronous rotation coordinate system, the electromagnetic torque equation of the IPMSM is expressed as:
Wherein, T e is electromagnetic torque, n p is pole pair number of the three-phase permanent magnet synchronous motor, psi f is permanent magnet flux linkage, and L d、Lq is d and q axis inductances respectively; i s is the stator current amplitude, and gamma is the included angle between the stator flux linkage vector and the permanent magnet flux linkage vector; based on equation (8), the electrical angular velocity is expressed as follows:
wherein, B is a damping coefficient, T L is a load torque, J is a moment of inertia, omega r is a mechanical angular velocity of the motor, f is a total disturbance, and the total disturbance is expressed as:
Where b is an intermediate variable, b 0 is a compensation factor, the total disturbance f is defined as an extended state, h (x, w) is an unknown and bounded function, and f can be observed and compensated; the superscript ≡represents the estimated value, and the superscript ≡represents the derivative; expanding the state equation of IPMSM to equation (14):
Preferably, the ADRC is composed of four parts of a fastest tracking differentiator TD, a nonlinear state error feedback control law NLSEF, an extended state observer ESO and disturbance compensation;
a tracking differentiator in ADRC tracks an input instruction rapidly and obtains differentiation of a tracking signal; let the input signal be v (t), the discrete representation of the fastest tracking differentiator for a second order system is:
Where k represents the number of discretization times, x 1 (k) is the tracking signal of the k times discretization v (k) of the input signal v (t); x 2 (k) is the differential signal of the k times discretized v (k) of the input signal v (t); e (k) is k times the estimation error, and the maximum value of x 2 differential is r 0; h is a sampling period; r 0 is a speed factor, and the larger r 0 is, the better the tracking performance is; h 0 is a filtering factor, and the larger h 0 is, the stronger noise filtering is; r 0 and h 0 are adjustable parameters; sign is a sign function; fhan (·) is the fastest tracking synthesis function of the discrete system, expressed as:
an Extended State Observer (ESO) is a core part of the ADRC, which expands the total disturbance of the system into a new state variable for estimating the state variable of the system and the total disturbance in real time and compensating and eliminating the disturbance; the algorithm of ESO is shown in formula (17):
where fal (·) is a nonlinear function expressed as:
Wherein epsilon is the output of the control object and the error of the observed value thereof; the output signal z 1、z2、z3 of the ESO is the output estimated value of the system, the estimated value of the output differential and the estimated value of the total disturbance respectively; beta 01、β02、β03 is the observer gain of the ESO; d. a 0、y、sy、sa and a are functional factors, and a 1、a2、a3 is a nonlinear factor; delta is a filtering factor; u is input; f 0 is the disturbance;
The input of the nonlinear state error feedback control is the error between the TD output tracking signal x 1 and tracking differential signal x 2 and ESO estimated signal z 1、z2, and the algorithm is as follows:
u0＝β1fal(e01,a4,δ0)+β2fal(e02,a5,δ0) (19)
β 1、β2 is a proportional gain, u 0 is a controller control amount, e 01、e02 is an output of a control object of two nonlinear functions fal (·) in the formula (19) and an observed value error thereof, a 4、a5 is a power of the two nonlinear functions fal (·) respectively, and δ 0 is a filter factor.
Based on the disturbance estimation signal output by the ESO and the known portion of the controlled object, the compensation process for the disturbance can be obtained as follows:
Where u is the input signal of the controlled object and b 0 is the compensation factor.
Preferably, by equation (17), the adaptive extended state observer AESO is designed to:
Wherein e 1、e2、e3 is the estimated error of rotor position, rotational speed and total disturbance; definition: obtaining a third-order error dynamic equation:
m i(e1) is a fal (·) function, l 1(t)、l2(t)、l3 (t) is a nonlinear function, the modified AESO selects m i(e1)＝e1, which is in a linear form like the LESO, and the observer gain l i (t) is not a constant but a time-varying parameter, and the adaptive characteristic is obtained by distributing the time-varying PD eigenvalues according to differential algebraic spectrum theory, so as to reduce the estimation error of the state and the interference of the total disturbance.
The state space of the error dynamic equation is expressed by equation (22):
in the formula, the state error e= [ e 1,e2,e3]T, matrix Matrix b= [0, 1] T; x is a state variable, w is external disturbance;
definition: observer gain matrix L (t) = [ -L 1(t) -l2(t) -l3(t)]T;
Equation (23) is a controllable linear time-varying LTV system with unknown but finite input, and the stability of equation (23) is guaranteed by designing a gain matrix A (t) and good estimation performance is obtained; firstly, lyapunov transformation is carried out, and an AESO error dynamic equation is converted into a controllable standard form by taking z as a state variable, wherein the controllable standard form is represented by a formula (24):
Wherein the matrix Matrix B c＝[0,0,1]T;
The controllable standard form (24) is an implementation of a linear scalar differentiation system (25), and the elements in A c (t) are coefficients of formula (25);
the observer gain matrix L (t), denoted by ζ, can be converted into:
Converting the observer gain l i (t) design into a design time-varying parameter g i (t) to stabilize AESO;
the homogeneity of the linear scalar differential equation (25) is expressed as:
SPDO of the LTV system (27) using the scalar polynomial differentiation operator SPDO is denoted as As shown in formula (28):
Where delta=d/dt is the derivative operator, Representing a scalar polynomial differential operator; the unified theory of LTV is established by using classical factorization of Floque pair SPDO, the Floque decomposition formula is:
In the theory of spectra, the set in formula (29) Called/>SD spectrum of (a); aggregationCalled/>Wherein λ 1,k (t) is n special solutions where λ 1 (t) satisfies some nonlinear independent constraints; a c (t) is called AND/>An associated companion matrix; the diagonal matrix diag [ ρ 1(t),ρ2(t),ρ3 (t) ] is called/>And a parallel spectral standard form of a c (t);
Using the transformation matrix V (t), the associated matrix a c (t) is reduced to its associated canonical form η (t), wherein:
To calculate g i (t) from ρ i (t), the following two theorem are introduced:
Theorem 1: order the For/>V k (t) is the determinant of the k-th order transformation matrix in equation (30), thenSD spectrum/>The method comprises the following steps:
Where k=1,..p and V 0 (t) =1;
theorem 2: order the For/>Will/>Is defined as g p,j(t),gk,0(t)＝0、gk,k+1 (t) =1; g p,j (t) is calculated recursively as:
wherein j=1, …, p and k=1, …, p-1;
According to theorem 1 and 2, g i (t) is obtained by assigning an appropriate characteristic value ρ i (t).
In order to improve the anti-interference capability of the IPSM sensorless control method, the invention provides an improved IPSM sensorless control method based on high-frequency square wave injection of an AESO_ADRC controller, and the method can ensure that the system still has good dynamic performance and rotor position estimation precision when being influenced by interference factors such as external interference, unmodeled dynamics, parameter uncertainty and the like.
Drawings
FIG. 1 is a flow chart of an active disturbance rejection control method of a position-free sensor (adaptive extended state observer) based on high frequency square wave injection in an embodiment;
FIG. 2 is a schematic diagram illustrating injection of a high-frequency square wave signal according to an embodiment;
FIG. 3 is a block diagram of position error signal demodulation in an embodiment;
FIG. 4 is a schematic diagram of an embodiment of an active disturbance rejection controller;
FIG. 5 (a) is a diagram showing the ADRC control structure of the IPSM no-position sensor based on high-frequency square wave injection in the embodiment;
FIG. 5 (b) is a block diagram of an AESO_ADRC control structure in an embodiment;
FIG. 6 (a), (b), (c) and (d) are simulated waveform diagrams of LADRC, NADRC, AESO _ADRC and conventional PI control in the example at a rotation speed of 200r/min, and given torques of 0 N.m, 20 N.m and 10 N.m, respectively;
Fig. 7 (a), (b), (c) and (d) are simulation waveform diagrams of LADRC, NADRC, AESO _adrc and conventional PI control in the example at a torque of 15n·m, and a given rotation speed of 10r/min, 200r/min and 100r/min, respectively;
Fig. 8 (a), (b), (c) and (d) are schematic diagrams of rotor estimation accuracy of the conventional PI control at a rotational speed of 200r/min and given torques of 0n·m, 20n·m and 10n·m in the embodiment LADRC, NADRC, AESO _adrc, respectively;
Fig. 9 (a), (b), (c) and (d) are schematic diagrams of rotor estimation accuracy of LADRC, NADRC, AESO _adrc and conventional PI control in the embodiment, where the torque is 15n·m and the given rotational speeds are 20r/min, 200r/min and 100r/min, respectively.
Detailed Description
For a further understanding of the present invention, the present invention will be described in detail with reference to the drawings and examples. It is to be understood that the examples are illustrative of the present invention and are not intended to be limiting.
Examples
As shown in fig. 1, the present embodiment provides a position-sensor-free (adaptive extended state observer) active disturbance rejection control method based on high-frequency square wave injection, which includes the following steps:
1. injecting high-frequency square waves into an estimated rotation coordinate system of the IPSM, and obtaining a rotor position estimation error through signal extraction and processing;
2. An adaptive extended state observer AESO based on differential algebra spectrum theory is designed by utilizing an error signal, and the total disturbance and state variable of the system are estimated to obtain an accurate rotor position;
3. An ADRC controller (AESO_ADRC) that builds an adaptive extended state observation replaces the conventional PI controller in the speed loop.
High-frequency square wave injection estimation position principle
In the synchronous rotation coordinate system, when the IPMSM operates at a low speed, the back electromotive force and the stator resistance voltage drop are ignored, and the high-frequency voltage equation after the high-frequency signal is injected is expressed:
Wherein, the subscript h represents a corresponding high-frequency signal; p is a differential operator; u dh、uqh is the d and q axis high frequency stator voltage respectively; i dh、iqh is d and q axis high frequency current respectively; l dh、Lqh is d and q axis inductance respectively;
injecting a high-frequency square wave signal in the following formulas (2) - (3) at the d e axis of the estimated coordinate system:
Where u inj is the vector of the injected high frequency square wave in the estimated rotational coordinate system, For injecting high-frequency square waves into components of the d e axis and the q e axis of the estimated rotating coordinate system, V in represents the injected high-frequency square waves, T is a time variable, T s is positive and negative voltage duration, V h is injection voltage amplitude, and n is the number of sequences; the schematic diagram of the injection square wave signal is shown in fig. 2, and the arrow above V in in fig. 2 indicates the injection direction.
From equation (1), when the high-frequency square wave signal is injected, the high-frequency current response in the stationary coordinate system αβ is obtained as Δi αh、Δiβh:
where θ err is the estimated rotor position error, defined as Theta r and/>The actual value and the estimated value of the rotor position are represented as a coordinate transformation matrix R (x) represented by a variable x:
solving equation (4) to obtain a high-frequency current response, expressed as equation (6):
Where the high frequency current response in the stationary coordinate system αβ can be obtained by a time delay, and when the estimation error θ err approaches 0, the envelope of the high frequency current response with rotor position information can be reduced from equation (6) to equation (7):
In the formula, the high-frequency current Δi αh、Δiβh in the stationary coordinate system αβ is represented by I sin、Icos, and as can be seen from the formula (7), a suitable position observer is designed to obtain rotor position information. As shown in fig. 3, a demodulation block diagram of the position error signal of the high-frequency square wave injection method is shown.
IPSM mathematical model containing perturbation
In the synchronous rotation coordinate system, the electromagnetic torque equation of IPMSM is expressed as:
Wherein, T e is electromagnetic torque, n p is pole pair number of the three-phase permanent magnet synchronous motor, psi f is permanent magnet flux linkage, and L d、Lq is d and q axis inductances respectively; i s is the stator current amplitude, and gamma is the included angle between the stator flux linkage vector and the permanent magnet flux linkage vector; based on equation (8), the electrical angular velocity is expressed as follows:
wherein, B is a damping coefficient, T L is a load torque, J is a moment of inertia, omega r is a mechanical angular velocity of the motor, f is a total disturbance, and the total disturbance is expressed as:
Where b is an intermediate variable, b 0 is a compensation factor, the total disturbance f is defined as an extended state, h (x, w) is an unknown and bounded function, and f can be observed and compensated; the superscript ≡represents the estimated value, and the superscript ≡represents the derivative; expanding the state equation of IPMSM to equation (14):
ADRC controller composition
ADRC is composed of four parts, namely a fastest Tracking Differentiator (TD), a nonlinear state error feedback control law (NLSEF), an Extended State Observer (ESO) and disturbance compensation; a schematic of the structure of the second-order ADRC is shown in fig. 4.
A tracking differentiator in ADRC tracks an input instruction rapidly and obtains differentiation of a tracking signal; let the input signal be v (t), the discrete representation of the fastest tracking differentiator for a second order system is:
Where k represents the number of discretization times, x 1 (k) is the tracking signal of the k times discretization v (k) of the input signal v (t); x 2 (k) is the differential signal of the k times discretized v (k) of the input signal v (t); e (k) is k times the estimation error, and the maximum value of x 2 differential is r 0; h is a sampling period; r 0 is a speed factor, and the larger r 0 is, the better the tracking performance is; h 0 is a filtering factor, and the larger h 0 is, the stronger noise filtering is; r 0 and h 0 are adjustable parameters; sign is a sign function; fhan (·) is the fastest tracking synthesis function of the discrete system, expressed as:
The extended state observer is a core part of ADRC, expands the total disturbance of the system into a new state variable, and is used for estimating the state variable of the system and the total disturbance in real time and compensating and eliminating the disturbance; the algorithm of ESO is shown in formula (17):
where fal (·) is a nonlinear function expressed as:
/>
Wherein epsilon is the output of the control object and the error of the observed value thereof; the output signal z 1、z2、z3 of the ESO is the output estimated value of the system, the estimated value of the output differential and the estimated value of the total disturbance respectively; beta 01、β02、β03 is the observer gain of the ESO; d. a 0、y、sy、sa and a are functional factors, and a 1、a2、a3 is a nonlinear factor; delta is a filtering factor; u is input; f 0 is the disturbance;
The input of the nonlinear state error feedback control is the error between the TD output tracking signal x 1 and tracking differential signal x 2 and ESO estimated signal z 1、z2, and the algorithm is as follows:
u0＝β1fal(e01,a4,δ0)+β2fal(e02,a5,δ0) (19)
β 1、β2 is a proportional gain, u 0 is a controller control amount, e 01、e02 is an output of a control object of two nonlinear functions fal (·) in the formula (19) and an observed value error thereof, a 4、a5 is a power of the two nonlinear functions fal (·) respectively, and δ 0 is a filter factor;
based on the disturbance estimation signal output by the ESO and the known portion of the controlled object, the compensation process for the disturbance can be obtained as follows:
Where u is the input signal of the controlled object and b 0 is the compensation factor.
Adaptive ESO (AESO) design
From equation (17), the adaptive extended state observer ESO (AESO) is designed to:
Wherein e 1、e2、e3 is the estimated error of rotor position, rotational speed and total disturbance; definition: obtaining a third-order error dynamic equation:
m i(e1) is a fal (·) function, l 1(t)、l2(t)、l3 (t) is a nonlinear function, modified AESO selects m i(e1)＝e1; the system is in a linear form like LESO, the observer gain l i (t) is a time-varying parameter, and the time-varying characteristic value is distributed according to the differential algebraic spectrum theory to obtain the self-adaptive characteristic, so that the estimation error of the state and the interference of the total disturbance are reduced;
The state space of the error dynamic equation is expressed by equation (22):
in the formula, the state error e= [ e 1,e2,e3]T, matrix Matrix b= [0, 1] T; x is a state variable, w is external disturbance;
Definition: observer gain matrix L (t) = [ -L 1(t)-l2(t)-l3(t)]T;
Equation (23) is a controllable linear time-varying (LTV) system with unknown but finite inputs, by designing the gain matrix a (t) to ensure the stability of equation (23) and obtain good estimation performance; firstly, lyapunov transformation is carried out, and an AESO error dynamic equation is converted into a controllable standard form by taking z as a state variable, wherein the controllable standard form is represented by a formula (24):
Wherein the matrix Matrix B c＝[0,0,1]T;
The controllable standard form (24) is an implementation of a linear scalar differentiation system (25), and the elements in A c (t) are coefficients of formula (25);
the observer gain matrix L (t), denoted by ζ, can be converted into:
Converting the observer gain l i (t) design into a design time-varying parameter gi (t) to stabilize the AESO and to improve its performance as much as possible;
the homogeneity of the linear scalar differential equation (25) is expressed as:
SPDO of the LTV system (27) using the scalar polynomial differentiation operator SPDO is denoted as As shown in formula (28):
Where delta=d/dt is the derivative operator, Representing a scalar polynomial differential operator; the unified theory of LTV is established by using classical factorization of Floque pair SPDO, the Floque decomposition formula is:
In the theory of spectra, the set in formula (29) Called/>Sequence D spectrum (SD spectrum) of (a); aggregationCalled/>Wherein λ 1,k (t) is n special solutions for λ 1 (t) satisfying some nonlinear independent constraints; a c (t) is called AND/>An associated companion matrix; the diagonal matrix diag [ ρ 1(t),ρ2(t),ρ3 (t) ] is called/>And a parallel spectral standard form of a c (t);
Using the transformation matrix V (t), the associated matrix a c (t) is reduced to its associated canonical form η (t), wherein:
To calculate g i (t) from ρ i (t), the following two theorem are introduced:
Theorem 1: order the For p-th SPDO/>V k (t) is determinant of the k-th order transformation matrix in equation (30), then/>SD spectrum/>The method comprises the following steps:
Where k=1,..p and V 0 (t) =1;
theorem 2: order the For p-th SPDO/>Will/>Is defined as g p,j(t),gk,0(t)＝0、gk,k+1 (t) =1; g p,j (t) is calculated recursively as:
wherein j=1, …, p and k=1, …, p-1;
According to theorem 1 and 2, g i (t) is obtained by assigning an appropriate characteristic value ρ i (t).
ADRC parameter design
Parameters that this embodiment requires design and tuning include relevant parameters for TD, AESO, and NLSEF.
(1) TD parameter design: the speed factor r 0 and the filter factor h 0.r0 determine the tracking speed of the signal, and the larger r 0 is, the faster the transition process is tracked, the smaller the phase lag is, the smaller the amplitude attenuation is, and the closer the amplitude is to the true value. However, r 0 cannot be too large due to the physical characteristics and control amount of the controlled object, and excessive r 0 may exacerbate the oscillation. h 0 is generally larger than the calculation step length of a discrete system, the proper value of h 0 can eliminate overshoot, static error is reduced, h 0 is generally 2 times of the step length, and excessive h 0 can cause phase lag and amplitude attenuation and reduce rapidity.
(2) ESO parameter design: parameters of ESO include β 01、β02、β03、a1、a2、a3, δ. Generally, a 1、a2、a3 and delta are designed as fixed parameters, and if 0<a i≤1,ai is smaller, the tracking effect is better, and vice versa. In this example, a 1＝0.75,a2＝0.5,a3 =0.25 was taken. Delta is a filtering factor, the greater delta, the better the filtering effect, but this also increases the tracking delay. Generally, δ should be within the interval of [5T,10T ], where T is the sampling period, and δ=0.01 is selected in this embodiment. For LADRC, the fal (·) nonlinear function is constant, so only the observer gain β 01、β02、β03 needs to be set, which determines the accuracy of the observer's estimation. The parameters are set according to the observer bandwidth ω 0, and the setting of the observer parameters is simplified to the tuning of one parameter ω 0, i.e. β 01＝3ω0,β02＝3ω0 2,β03＝ω0 3,ω0 should be maximized as much as possible within the range allowed by noise sensitivity and sampling delay. Too large β 01 can cause oscillations and even divergences, β 01 is typically an order of magnitude greater than 1/h. Too small β 02 may cause divergence and too large β may produce high frequency noise; too small a β 03 reduces the tracking speed and too large a β 03 produces oscillations.
(3) Parameters design of AESO: according to the AESO design method described above, to obtain the observer gain l 1(t)～l3 (t), the eigenvalue is set to:
Wherein, Representing the reference eigenvalue, ω (t) is a time-varying multiplier. Selecting nominal eigenvalues in LESO, i.eAnd/>The time-varying bandwidth of the AESO is obtained to be omega n(t)＝ω0 ω (t), which is represented by formulas (31) and (32): /(I)
The multiplier ω (t) is designed by defining a time-based function as shown in equation (35):
Omega (t) and its derivatives are generated by a Butterworth filter with omega (t) as the filter input. ω (T) is set to ω 0 when T is set to ω 0/3,t>Tω when T is equal to or less than T ω, which is advantageous in resolving the conflict between the rapidity and the overshoot.
(4) NLSEF parameter design: parameters of NLSEF include β 1、β2、a4、a5、δ0. The parameters required to be set for a linear SEF are β 1、β2, corresponding to the proportional gain K p and the differential gain K d of the PD control. For NLSEF, the choice of parameter β 1、β2 is similar to linear SEF. In order to avoid excessive gain causing speed fluctuations, a 4 is generally smaller than a 5,0≤a4≤1≤a5, and a 4＝0.25,a5 =1.5 is generally taken. The parameter δ 0 is a parameter related to the range of the controlled variable and the control accuracy, and δ 0 =0.22 is generally taken.
(5) Disturbance compensation parameter design: b 0 is a compensation factor, which is an estimated value of a compensation coefficient, and is an important parameter affecting system stability. b 0 is selected to compromise stability and response speed, and the dynamic performance of the motor can be improved by adjusting the proper b 0. Increasing b 0 within a certain range increases the stability domain of the system, but too large b 0 will make the control signal u too small, thereby slowing down the system response; the smaller b 0, the faster the disturbance compensation response, the better the dynamic performance of the motor, but too small b 0 can cause overshoot and ripple. The error (b-b 0) of the compensation coefficient estimate is included in the total disturbance, and the intermediate variable b is related to the model of the controlled object, which can be calculated as an estimate of the compensation coefficient b 0.
In summary, the structure of the IPMSM sensorless ADRC control system based on high-frequency square wave injection is shown in fig. 5 (a) and 5 (b).
Simulation verification and analysis
In order to verify the effectiveness and feasibility of the IPMSM position-free sensor aeso_adrc control strategy based on high-frequency square wave injection, simulation verification and analysis are performed. And a permanent magnet synchronous motor sensorless control system based on LADRC, NADRC and AESO_ADRC controllers is built on a MATLAB/SIMULINK platform. Simulation parameters of IPMSM are shown in table 1. In the simulation, the switching frequency of the PWM was 10kHz, the dead time was set to 2.0 μs, and the current sampling frequency was the same as the PWM switching frequency. The frequency of the injected high-frequency square wave voltage signal is 5KHz, and the amplitude is 20V.
Table 1 IPMSM motor parameters
| Parameters of the motor | Parameter value |
| Number of phases m | 3 |
| Polar logarithm n p | 4 |
| Rated rotation speed/(r/min) | 1500 |
| Rated power/kW | 22 |
| Stator winding resistance R s/mΩ | 839 |
| D-axis inductance L d/mH | 5.05 |
| Q-axis inductance L q/mH | 15.35 |
| DC side voltage U d/V | 500 |
| Moment of inertia J/(kg.m 2) | 0.00376 |
The parameters of each part of the ADRC controller are selected as follows:
Sampling step length: h=0.01;
Arranging transition process parameters: r 0 = 20000;
nonlinear state error feedback parameters: β 1＝1.0,β2＝1.5,a4＝0.25,a5＝1.5,δ0 = 0.22;
LESO parameters: beta 01＝100,β02＝30000,β03＝106;
NESO parameters ：β01＝100,β02＝30000,β03＝106,a1＝0.75,a2＝0.5,a3＝0.25,δ＝0.01;
Disturbance compensation parameters: b 0 = 0.5b.
In order to compare and analyze the dynamic performance and anti-interference performance of LADRC, NADRC, AESO _ADRC and the traditional PI control, simulation research is carried out under the conditions of different operation conditions and additional disturbance of the IPSM.
As shown in (a), (b), (c) and (d) of fig. 6, the IPMSM is a waveform of the change in the rotational speed and stator current of the IPMSM when the IPMSM is suddenly increased from 0n·m to 20n·m at 0.2s and suddenly decreased to 10n·m again at 0.4s at a given rotational speed command of 200r/min with a random disturbance applied. When the load torque is suddenly increased at the moment of 0.2s, the stability time of LADRC, NADRC, AESO _ADRC and the stability time of the traditional PI control are respectively 0.022s, 0.018s, 0.012s and 0.1s; when the load torque is suddenly reduced at the time of 0.4s, the stabilizing time is respectively 0.0032s, 0.0074s, 0.0010s and 0.08s; the traditional PI control overshoot is the largest and AESO_ADRC overshoot is the smallest.
The (a), (b), (c) and (d) in FIG. 7 are waveforms of changes in the rotational speed and stator current of the IPSM when the rotational speed of the IPSM is suddenly increased from 10r/min to 200r/min at 0.3s and suddenly decreased to 100r/min at 0.6s at the given torque command of 15 N.m with random disturbance applied. After the rotational speed is suddenly increased at the moment of 0.3s, the stability time of LADRC, NADRC, AESO _ADRC and the stability time of the traditional PI control are respectively 0.012s, 0.016s, 0.014s and 0.335s; after the rotation speed is reduced at the moment of 0.6s, the stabilizing time is respectively 0.009s, 0.014s, 0.009s and 0.01s; NADRC and AESO_ADRC have almost no overshoot, LADRC has a small amount of overshoot, and the traditional PI control overshoot is larger. The comprehensive observation shows that the control system has better dynamic performance than the traditional PI speed regulation after adopting the active disturbance rejection controller, and the AESO_ADRC has better dynamic performance than LADRC and NADRC under the condition of abrupt change of load torque and rotating speed, and the tracking speed is also faster.
Fig. 8 (a), (b), (c) and (d) are comparison of rotor position estimation accuracy under variable load torque using the IPMSM sensorless control method of LADRC, NADRC, AESO _adrc and conventional PI control, respectively. The given rotation speed is 200r/min, under the condition of the external amplitude limiting random disturbance, the load torque is increased from 0 N.m to 20 N.m at the moment of 2s, and the load torque is reduced from 20 N.m to 10 N.m at the moment of 4 s. At steady state, the maximum error of rotor position estimation of LADRC, NADRC, AESO _ADRC is 0.18rad/s, 0.2rad/s and 0.17rad/s respectively; the conventional PI control has a large error in rotor position estimation when the load suddenly changes.
Fig. 9 (a), (b), (c) and (d) are comparison of rotor position estimation accuracy under variable rotational speed by IPMSM sensorless control method using LADRC, NADRC, AESO _adrc and conventional PI control, respectively. The given torque command is 15 N.m, under the condition of the additional amplitude limiting random disturbance, the rotating speed command is increased from 20r/min to 200r/min at the moment of 2s, and is reduced from 200r/min to 100r/min at the moment of 4 s. At steady state, the maximum estimated error of the rotor position estimation for LADRC, NADRC, AESO _ADRC and conventional PI control is 0.165rad/s, 0.200rad/s, 0.1rad/s, 0.12rad/s, respectively. Compared with the traditional PI control, the AESO_ADRC can keep good position estimation precision in steady state, and the position estimation error in dynamic state is smaller; aeso_adrc has a higher accuracy of position estimation than lacc and NADRC.
In summary, in order to improve the anti-interference capability of the IPMSM sensorless control method, an improved IPMSM sensorless control method based on an aeso_adrc controller is provided. The method can ensure that the system still has good dynamic performance and rotor position estimation precision when being influenced by external interference, unmodeled dynamic, parameter uncertainty and other interference factors. The superiority of aeso_adrc based on dynamic performance and immunity compared to conventional ladc, nadc and conventional PI controllers was verified by simulation.
The invention and its embodiments have been described above schematically, without limitation, and the actual construction is not limited to this, as it is shown in the drawings, which are only one of the embodiments of the invention. Therefore, if one of ordinary skill in the art is informed by this disclosure, the structural mode and the embodiments similar to the technical scheme are not creatively designed without departing from the gist of the present invention.
Claims (2)
1. The position-sensor-free active disturbance rejection control method based on high-frequency square wave injection is characterized by comprising the following steps of: the method comprises the following steps:
1. injecting high-frequency square waves into an estimated rotation coordinate system of the IPSM, and obtaining a rotor position estimation error through signal extraction and processing;
2. An adaptive extended state observer AESO based on differential algebra spectrum theory is designed by utilizing an error signal, and the total disturbance and state variable of the system are estimated to obtain an accurate rotor position;
3. An ADRC controller ASEO _ADRC of the self-adaptive extended state observer is constructed to replace a traditional PI controller in a speed loop;
In the first step, under the synchronous rotation coordinate system, when the IPMSM operates at a low speed, the counter electromotive force and the stator resistance voltage drop are ignored, and the high-frequency voltage equation after the high-frequency signal is injected is expressed:
Wherein, the subscript h represents a corresponding high-frequency signal; p is a differential operator; u dh、uqh is the d and q axis high frequency stator voltage respectively; i dh、iqh is d and q axis high frequency current respectively; l dh、Lqh is d and q axis inductance respectively;
injecting a high-frequency square wave signal in the following formulas (2) - (3) at the d e axis of the estimated coordinate system:
Where u inj is the vector of the injected high frequency square wave in the estimated rotational coordinate system, For injecting high-frequency square waves into components of the d e axis and the q e axis of the estimated rotating coordinate system, V in represents the injected high-frequency square waves, T is a time variable, T s is positive and negative voltage duration, V h is injection voltage amplitude, and n is the number of sequences;
From equation (1), when the high-frequency square wave signal is injected, the high-frequency current response in the stationary coordinate system αβ is obtained as Δi αh、Δiβh:
where θ err is the estimated rotor position error, defined as Theta r and/>The actual value and the estimated value of the rotor position are represented as a coordinate transformation matrix R (x) represented by a variable x:
solving equation (4) to obtain a high-frequency current response, expressed as equation (6):
Where the high-frequency current response in the stationary coordinate system αβ can be obtained by a time delay, and when the estimation error θ err tends to 0, the envelope of the high-frequency current response with rotor position information can be reduced from equation (6) to equation (7):
In the formula, the high-frequency current delta I αh、Δiβh in the static coordinate system alpha beta is represented by I sin、Icos, and as can be seen from the formula (7), a proper position observer is designed to obtain rotor position information;
ADRC is composed of a fastest tracking differentiator TD, a nonlinear state error feedback control law NLSEF, an extended state observer ESO and disturbance compensation;
a tracking differentiator in ADRC tracks an input instruction rapidly and obtains differentiation of a tracking signal; let the input signal be v (t), the discrete representation of the fastest tracking differentiator for a second order system is:
Where k represents the number of discretization times, x 1 (k) is the tracking signal of the k times discretization v (k) of the input signal v (t); x 2 (k) is the differential signal of the k times discretized v (k) of the input signal v (t); e (k) is the k times estimation error, and the maximum value of x 2 (k) differential is r 0; h is a sampling period; r 0 is a speed factor, and the larger r 0 is, the better the tracking performance is; h 0 is a filtering factor, and the larger h 0 is, the stronger noise filtering is; r 0 and h 0 are adjustable parameters; sign is a sign function; fhan (·) is the fastest tracking synthesis function of the discrete system, expressed as:
The algorithm of ESO is shown in formula (17):
where fal (·) is a nonlinear function expressed as:
Wherein epsilon is the output of the control object and the error of the observed value thereof; the output signal z 1、z2、z3 of the ESO is the output estimated value of the system, the estimated value of the output differential and the estimated value of the total disturbance respectively; beta 01、β02、β03 is the observer gain of the ESO; d. a 0、y、sy、sa and a are functional factors, and a 1、a2、a3 is a nonlinear factor; delta is a filtering factor; u is input; f 0 is the disturbance;
The input of the nonlinear state error feedback control is the error between the TD output tracking signal x 1 and tracking differential signal x 2 and ESO estimated signal z 1、z2, and the algorithm is as follows:
u0＝β1fal(e01,a4,δ0)+β2fal(e02,a5,δ0) (19)
β 1、β2 is a proportional gain, u 0 is a controller control amount, e 01、e02 is an output of a control object of two nonlinear functions fal (·) in the formula (19) and an observed value error thereof, a 4、a5 is a power of the two nonlinear functions fal (·) respectively, and δ 0 is a filter factor;
based on the disturbance estimation signal output by the ESO and the known portion of the controlled object, the compensation process for the disturbance can be obtained as follows:
Wherein u is an input signal of a controlled object, and b 0 is a compensation factor;
from equation (17), the adaptive extended state observer AESO is designed to:
Wherein e 1、e2、e3 is the estimated error of rotor position, rotational speed and total disturbance; definition: obtaining a third-order error dynamic equation:
m i(e1) is a fal (·) function, l 1(t)、l2(t)、l3 (t) is a nonlinear function, modified AESO selects m i(e1)＝e1;
The state space of the error dynamic equation is expressed by equation (22):
in the formula, the state error e= [ e 1,e2,e3]T, matrix Matrix b= [0, 1] T; x is a state variable, w is external disturbance;
definition: observer gain matrix L (t) = [ -L 1(t) -l2(t) -l3(t)]T;
Equation (23) is a controllable linear time-varying LTV system with unknown but finite input, and the stability of equation (23) is guaranteed by designing a gain matrix A (t) and good estimation performance is obtained; firstly, lyapunov transformation is carried out, and an AESO error dynamic equation is converted into a controllable standard form by taking z as a state variable, wherein the controllable standard form is represented by a formula (24):
Wherein the matrix Matrix B c＝[0,0,1]T;
The controllable standard form (24) is an implementation of a linear scalar differentiation system (25), and the elements in A c (t) are coefficients of formula (25);
the observer gain matrix L (t), denoted by ζ, can be converted into:
Converting the observer gain l i (t) design into a design time-varying parameter g i (t) to stabilize AESO;
the homogeneity of the linear scalar differential equation (25) is expressed as:
SPDO of the LTV system (27) using the scalar polynomial differentiation operator SPDO is denoted as As shown in formula (28):
Where delta=d/dt is the derivative operator, Representing a scalar polynomial differential operator; the unified theory of LTV is established by using classical factorization of Floque pair SPDO, the Floque decomposition formula is:
In the theory of spectra, the set in formula (29) Called/>SD spectrum of (a); aggregation/>Called asWherein λ 1,k (t) is n special solutions where λ 1 (t) satisfies some nonlinear independent constraints; a c (t) is called ANDAn associated companion matrix; the diagonal matrix diag [ ρ 1(t),ρ2(t),ρ3 (t) ] is called/>And a parallel spectral standard form of a c (t);
Using the transformation matrix V (t), the associated matrix a c (t) is reduced to its associated canonical form η (t), wherein:
To calculate g i (t) from ρ i (t), the following two theorem are introduced:
Theorem 1: order the For/>V k (t) is determinant of the k-th order transformation matrix in equation (30), then/>SD spectrum/>The method comprises the following steps:
Where k=1,..p and V 0 (t) =1;
theorem 2: order the For/>Will/>Is defined as g p,j(t),gk,0(t)＝0、gk,k+1 (t) =1; g p,j (t) is calculated recursively as:
wherein j=1, …, p and k=1, …, p-1;
According to theorem 1 and 2, g i (t) is obtained by assigning an appropriate eigenvalue ρ i (t).
2. The position sensor-less active disturbance rejection control method based on high frequency square wave injection according to claim 1, wherein: in the synchronous rotation coordinate system, the electromagnetic torque equation of IPMSM is expressed as:
Wherein, T e is electromagnetic torque, n p is pole pair number of the three-phase permanent magnet synchronous motor, psi f is permanent magnet flux linkage, and L d、Lq is d and q axis inductances respectively; i s is the stator current amplitude, and gamma is the included angle between the stator flux linkage vector and the permanent magnet flux linkage vector; based on equation (8), the electrical angular velocity is expressed as follows:
wherein, B is a damping coefficient, T L is a load torque, J is a moment of inertia, omega r is a mechanical angular velocity of the motor, f is a total disturbance, and the total disturbance is expressed as:
Where b is an intermediate variable, b 0 is a compensation factor, the total disturbance f is defined as an extended state, h (x, w) is an unknown and bounded function, and f can be observed and compensated; the superscript ≡represents the estimated value, and the superscript ≡represents the derivative; expanding the state equation of IPMSM to equation (14):
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