曲洪成,畢永利

(黑龍江大學 機電工程學院，哈爾濱 150080)

Permanent magnet synchronous motor has the advantages of low cost, simple structure and high energy utilization[1-2]. The research on its speed regulation system has become an inevitable trend[3]. Permanent magnet synchronous motor is a very important power module in the speed regulation system. At present, the way to obtain the position angle of motor and rotor is to use the traditional photoelectric encoder, but the application occasions and installation methods limit the development of mechanical sensors. In order to solve the above problems, the research on speed sensorless has become an inevitable trend. Under multiple working conditions, no speed sensor may cause the deviation of identification results, resulting in insufficient adaptive ability and strong anti-interference ability, which will affect the whole speed regulation system. In order to solve the above problems and increase the reliability and anti-interference ability of the system, an improved extended Kalman filter speed sensorless control method is adopted. Firstly, the model reference adaptive method is applied to identify the stator resistance, inductance and rotor flux, and the identification results are fed back to the extended Kalman filter algorithm for speed identification in time[4- 6]. At the same time, the traditional PI controller of the speed loop is improved, auto disturbance rejection control is used to enhance the dynamic performance of the system.

1 Extended Kalman filter speed identification

The extended Kalman filter is divided into two parts: filtering and gain calculation. The filter loop estimates and corrects the state variables, and the gain calculation loop updates the square difference matrix and gain matrix. The performance of the prediction correction algorithm is better than that of other Kalman filtering algorithms.

In the extended Kalman filter recursion of PMSM sensorless control, the prediction equation is

(1)

The prediction error variance matrix is

Pk/k-1=φk/k-1Pkφk/k-1T+Q/k-1

(2)

Pk/k-1represents the covariance prediction value estimated from the previous time,Q/k-1represents the matrix expression of the uncertainty at the last moment in the prediction process.

The optimal gain matrix equation is

Kk=Pk/k-1HkT(HkPk/k-1HkT+R)-1

(3)

Kkrepresents the Kalman gain,Hkrepresents the observation transfer matrix,Rrepresents the matrix expression of the uncertainty at the last moment.

The filtering equation is

(4)

The filtering error variance matrix is

Pk=Pk/k-1-KkHkPk/k-1

(5)

Let the linear discrete stochastic system be

(6)

xkandxk-1are the state vectors of the current time and the previous time respectively,φk/k-1xk-1is a state transition matrix,Gk/k-1uk-1is the input matrix,uk-1is the input vector,wk-1is an n-dimensional random vector, it represents system disturbance and uncertainty,Hk-1is the observation matrix,zk-1is an m-dimensional observation vector,vk-1is the m-dimensional random measurement noise vector.

The equation of state of PMSM is

(7)

Lis the stator inductance,RSis the stator resistance,ψfis the permanent magnet flux linkage,θeis the rotor position angle,idis the d-axis current,iqis the q-axis current,uqis the q-axis voltage.

The mathematical model of PMSM is discretized by the first-order antecedent Euler method

(8)

Tsis the sampling period of the system.

Select state vector: [id,iq,ω,θ]T, input variable: [ud,uq]T, output equation:y=id,y=iq.

The relevant Jacobian matrix is as follows

(9)

2 On line parameter identification of model reference adaptive method

MRAS is composed of a reference model and an adjustable model. The reference model does not contain the variables to be observed, and the adjustable model contains the variables to be observed. The two models output the same variable. Make a difference between the output variables of the two models, and use the difference to quickly adjust the variables to be observed in the adjustable model. Through an appropriate adaptive rate, the position variables can be observed in a short time.

The mathematical model of PMSM is expressed as the equation of stator current model

(10)

Rewrite equation (10) into the standard form of MRAS

(11)

Take equation (11) as the reference model, and its adjustable model can be written as

(12)

Make a difference between equation (11) and equation (12) and write it into a state space expression

(13)

Where

Define generalized error

(14)

Then equation (14) can be expressed as

(15)

Let

(16)

(17)

(18)

(19)

3 ADRC speed regulator design

ADRC consists of three parts: tracking differentiator (TD), extended state observer (ESO) and nonlinear state error feedback (NLSEF)[7- 8].

When the system is a first-order system, the following formula

(20)

TD mathematical modelfhan(x1-v0,x2,r,h0) under the first-order system is the fastest synthesis function, which is defined as

(21)

ris the tracking speed factor.

The ESO model of the first-order system is as follows

(22)

z1is the tracking signal ofy,z2is the estimated value under disturbance,α1andα2are nonlinear factors,δare filter factors,β1andβ2are observer error gain, andfal(e,α2,δ) is nonlinear function. And the expression offal(e,α2,δ) is

(23)

The first-order NLSEF mathematical model is as follows

u0(t)=∑βifal(ei,αi,δi)

(24)

The speed loop is a first-order system. In order to make up for the contradiction between system rapidity and overshoot, a first-order linear differentiator is added to the speed feedωref, and the excessive process is arranged for the input to obtain the controller input signalωtd

(25)

When designing the observer, the load disturbance term and viscous friction coefficient term are taken as internal disturbancefi, and the sum of internal disturbance and external disturbancefeis recorded asfc,

(26)

uis the control quantity andfcis the total disturbance of speed loop.

The speed loop observer is designed as follows

(27)

The system is equivalent to an integral link to form a proportional control rate

u0=kp(ωtd-z1)

(28)

4 Sensorless control algorithm of PMSM Based on ADRC

In the traditional vector control, PI controller is generally used in the speed loop, but its robustness is poor and can not meet the requirements of rapidity and small overshoot at the same time. Therefore, auto disturbance rejection control technology is adopted to make the system achieve the expected control accuracy[9-10]. In the above control system, the basic model of PMSM is transformed through coordinates, and the stator resistance, inductance and rotor flux of the system are identified online according to MRAS. The obtained parameter values are fed back to the extended Kalman filter algorithm in real time for speed online identification. The combined algorithm has strong control accuracy, but there is a certain overshoot. The speed loop controller is optimized through ADRC algorithm to replace the traditional PI controller for adjustment.The purpose of controlling PMSM speed is achieved.The principle diagram of the sensorless control algorithm of PMSM Based on ADRC is shown in Fig.1.

Fig.1 Principle diagram of the sensorless control algorithm

5 Simulation verification

The simulation model of PMSM vector control system based on speed sensorless is built in Matlab / Simulink. It is composed of ADRC speed regulation module, coordinate transformation module, SVPWM module, inverter module and motor parameter identification module. Using the vector control principle of id = 0, the double closed-loop control theory is that the current loop is the inner loop and the speed loop is the outer loop. The PMSM simulation parameters are as follows: rated speed 600 r·min-1; Polar logarithmnp=4; Stator resistanceRs=2.85 Ω; Stator inductanceL=0.002 mH; Permanent magnet flux linkageψf=0.175 Wb.

Simulation response curves are presented in Fig.2～7.

Fig.2 Response curve of stator resistance

Fig.3 Response curve of stator inductance

Fig.4 Response curve of permanent magnet flux

Fig.5 EKF Speed identification curve

Fig.6 ADRC Speed identification curve

Fig.7 ADRC Tracking capability test curve

It can be seen from Fig.2～5 that when MRAS is used for resistance, inductance and flux identification, there is a certain overshoot due to the dynamic characteristics of the system, but its response is fast. The parameter identification can be completed in 0.04 s, and the identification result is accurate. The identified threemain parameters are fed back to the extended Kalman filter algorithm, when using the traditional PI control for speed identification, the identification results are shown in Fig.5, with certain overshoot, and the peak value is 726 m·min-1, the steady state is reached in 0.065 s, so ADRC controller is used to replace the traditional PI controller for its speed loop. The identification results are shown in Fig.6, the improved algorithm basically has no overshoot and reaches the steady state in 0.03 s. In order to verify its tracking performance, the load torque is suddenly changed to 1 N·m in 0.2 s. The identification results are shown in Fig.8, the results show that the algorithm has good tracking performance.

6 Conclusions

During the operation of PMSM, the dynamic performance changes. In order to deal with the parameter changes, MRAS is used for parameter value identification, which has good identification accuracy. The identification results are fed back to the extended Kalman filter algorithm, which can quickly identify the speed, with small amount of calculation and high accuracy. At the same time, ADRC algorithm perfectly balances the contradiction between identification rapidity and small overshoot. The results of parameter identification can be used for the parameter configuration of PMSM control system, It provides a solution to improve the performance of PMSM control system.