P RACTICAL I SSUES WITH P REDICTIVE C URRENT C ONTROLLERS S.J. Henriksen† , R.E. Betz and B.J. Cook Department of Electrical and Computer Engineering University of Newcastle, Callaghan, NSW Australia, 2308. email: †
[email protected], A BSTRACT Predictive current controllers have been used for a wide variety of machine control applications. They offer a simple structure and high-bandwith response, but suffer difficulties in estimating the correct system state. This paper is concerned with implementation issues associated with using a constant-switching frequency digital predictive current controller on an induction machine. The stability and performance of a practical predictive controller is analysed, and compared to experimental results. New prediction methods are then presented to overcome the weaknesses of the basic control algorithm. 1. I NTRODUCTION In many applications, current control is useful as part of a larger control structure [6, 9]. In the case of a machine, a current controller may be used in conjunction with outer torque and speed control loops. The application that is the focus in this paper is the control of an induction machine, although the results have broader relevance.
chosen based on a prediction from a machine model. The model used for this controller is in a stationary d-q axis, as shown in Figure 1.
Ll
eq
In the first section, a basic description of the controller is presented, showing the calculations required per control cycle. A more detailed description appears in [3]. The description is followed by a theoretical analysis of the predictive current control algorithm. This shows the limitations of the basic predictive scheme, which are then addressed in the remainder of the paper. The main shortcoming of predictive current control is the prediction of the back-emf estimate. Typical extrapolation methods [4] incur stability problems. These may be overcome by adding filters to the feedback path [7], but this approach is only suitable for a fixed frequency output, as found on active rectifiers. Two new prediction methods are presented, which are suited to variable-frequency applications, and don’t compromise the controller stability. Results of practical tests with an outer loop controller are included. 2. T HE P REDICTIVE C ONTROL S TRUCTURE Constant switching frequency predictive controllers [5] involve the use of a fixed length control interval, which typically coincides with the PWM switching interval. For each iteration, the optimal output PWM voltage is
ed
Ll
Fig. 1 T HE TWO - PHASE EQUIVALENT INDUCTION MACHINE MODEL
This is a very simple model of an induction machine, but can be shown to be very accurate for the purposes of current control [1]. It consists of two elements in each of the two independent axes, the back-emf and the leakage inductance. For a control interval of length T , the change in current over the interval for each axis is given by, vk − ek =
Ll (ik − ik−1 ) , T
(1)
where vk and ek are the average voltage and back emf applied across the cycle respectively. Both the back-emf and leakage inductance parameters are slowly time-varying, and are estimated as part of the control calculation. For the following analysis, Symmetric space-vector PWM is used and, as a result each PWM cycle is di-
vided into two halves each with the same average voltage. Figure 2 shows a schematic current waveform for currents sampled at the end and midpoints of a switching interval.
Current
vk+1 Calculation time e k vk
ik−1
i*k
vk+1
3. C ONTROLLER A NALYSIS
ρ=1
ik
A discrete-time transfer function for the system is obtained by combining the controller, back-emf estimation, and system equations.
uk+1
ρ=2
To facilitate analysis, it is assumed that there is no prediction error when estimating the current at the start of a control interval. This is necessary to cast the controller in a standard shift operator form. By making the prediction assumption, each sample period has one voltage and one current associated with each axis of the machine.
Time Fig. 2 T HE VARIABLES USED IN THE CONTROLLER ..
The predictive control algorithm involves first estimating the model parameters. This is done from past measurements. The model is then used to determine the voltage necessary to meet the control objective for each control interval. Using discrete-time notation, where i∗ is the current at the midpoint, a set of estimation and control equations are: 2Ll ∗ (ik − ik−1 ) . (2) T Ll vk+1 = eest (ik+1 − 2i∗k + ik−1 ) . (3) k+1 + T The controller developed using these equations was able to successfully control the machine current, but the bandwidth was affected by the delay in the back-emf estimate. For typical operating conditions, an error of up to 4% can be expected in the current tracking. eest k = vk −
2.1. Inductance Estimation The inductance may be estimated from the measured currents and past applied voltages. In order to obtain independence from the back-emf estimation procedure, an additional constraint must be added. This is done through the assumption that the leakage inductance varies very slowly with time. This means that the change in back-emf can be assumed constant from one cycle to the next,
ek − ek−1 ≈ ek−1 − ek−2 ek − 2ek−1 + ek−2 ≈ 0.
(4) (5)
This may be related to the model currents and inductance by Equation (1). By substituting into (5), the estimator is formed, vk − 2vk−1 + vk−2 =
Using this equation, a recursive least-squares estimator may be used to obtain accurate estimates of the leakage inductance parameter.
Under these assumptions, the basic controller equations are: ρLl ∆L (uk+1 − ik ) + epred k+1 T Ll ∆L (ik+1 − ik ) . = vk+1 − T
vk+1 =
(7)
eest k+1
(8)
In these equations, the real machine leakage inductance is Ll , while the estimated inductance is Ll ∆L. ∆L is the multiplicative estimation error. ρ is a controller parameter, where ρ = 1 denotes endpoint control, and ρ = 2 represents deadbeat control of the average current across each control interval. In order to maintain stability, typically ρ = 1 is used. In addition to the controller equations, the back-emf prediction is taken as the estimate from the previous interval. The prediction is necessary in the controller for causality. The simple assumption of a nearly constant back-emf has shortcomings. Alternatives such as linear extrapolation are considered later. For the first analysis, the back-emf prediction is given by: est epred k+1 = ek .
(9)
The behaviour of the coupled controller and machine may be found by simultaneously solving these controller equations with the machine model in Equation (1). The resulting update equation is, ρLl ∆L (uk+1 − ik ) + vk T Ll ∆L − (ik − ik−1 ) . T
vk+1 =
(10)
Lest l (ik − 3ik−1 + 3ik−2 − ik−3 ) After . converting the system expression to the forward T (6) shift operator form (zxk ≡ xk+1 ), the transfer function
Gain From Back−emf to Current, ρ=1
0.01
0.008
(11)
The variation in the back-emf is treated by including the back-emf as a disturbance term. For a constant backemf this term is zero. Controller stability may be determined by the location of the poles of the closed loop transfer function from u to i. In this case, they are the solutions of z to,
Current due to e
from the reference, u, to the output current, i is, i z 2 + (∆L(ρ + 1) − 2)z + (1 − ∆L) T = z 2 ρ∆Lu − (z − 1)z e. Ll
0.006
0.004
0.002
0 0
0.5
z 2 + (∆L(ρ + 1) − 2)z + (1 − ∆L) = 0
1 1.5 Inductance Error ∆L
2
(12)
The existence of poles outside the unit circle centred at the origin indicate an unstable syste. The location of these poles may be plotted for varying inductance estimation error. This root-locus plot appears in Figure 3. The poles must remain within the unit circle indicated 1
Fig. 4 D ISTURBANCE DUE TO BACK - EMF ERROR AT 50H Z .
Parameter Electrical Frequency Control Period Leakage Inductance
Value 50Hz 0.3ms 10mH
TABLE I T EST MACHINE PARAMETERS .
0
∆ L=1.3
∆ L=1
∆ L=0
machine parameters. The parameters used for this analysis are shown in Table I. −1 −1
0
1
Fig. 3 ROOT- LOCUS PLOT FOR THE CONTROLLER .
by the broken line,, and this does occur for all estimated values of inductance up to 1.3 times the true value. This is adequate, as the inductance can safely be estimated to within this range. 4. E FFECT OF C HANGING BACK - EMF 4.1. Error Due to Back-emf Assumption Although this controller operates in a stable manner, the performance is affected by the assumption of constant back-emf across each control cycle in Equation (9). The magnitude of the error may be found by considering the gain of the transfer function from e to i in (11). The magnitude response for a fixed fundamental electrical frequency, and varied controller parameters is shown in Figure 4. Unlike the the transfer functions involving only the current, the results here depend highly on the
With endpoint control and the correct value for inductance, the gain from the back-emf to the output is approximately 3mS. For a sinusoidal back-emf of amplitude 200V, this corresponds to a peak current error of 600mA. Under normal machine operation, this could easily represent a 10% error in the machine current. 4.2. Prediction The control errors due the time-changing back-emf may be addressed by predicting the back-emf with linear extrapolation. This is expressed as: est est epred k+1 = 2ek − ek−1 .
(13)
The closed loop discrete-time transfer function is obtained using the method described for the simple controller. In forward shift operator this is, i z 3 + (∆L(ρ + 2) − 3)z 2 + 3(1 − ∆L)z + ∆L − 1 T = z 3 ρ∆Lu − (z − 1)2 z e. (14) Ll The controller error due to the change in back-emf may again be calculated for this system. Again using the parameters in Table I, the error gain is now about 0.25mS.
For the same 200V back-emf, the resulting error in current is only 50mA. This is quite acceptable, and only about one tenth of the original error.
Im
Unfortunately extrapolation also has a harmful impact on the controller stability and noise sensitivity. For a controller with extrapolation, the maximum pole location magnitude is shown in Figure 5. The system is
eq
2.5
de dφ ek
∆φ
3
Maximum root magnitude
e k+1
φk
Extrapolation No Extrapolation
Re
ed
2
Fig. 6 T HE BACK - EMF TRAJECTORY.
1.5 1 0.5 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Inductance Error ∆L
Fig. 5 M AXIMUM ROOT MAGNITUDE AS A FUNCTION OF THE INDUCTANCE ESTIMATION ERROR .
to be available from an outer-loop controller, such as a field-oriented torque controller. If the electrical rotation speed is not known, the observer method, described later, may be used instead.
5.1.1. Implementing Feed-forward only stable for ∆L in the range of 0.5 < ∆L < 1.2. The consequence of this is that the inductance estimate must be close to the true value, and large errors in either direction result in instability. In fixed frequency applications, such as PWM rectifiers, filters may be used to provided prediction with better stability properties [7], but these methods cannot be used for a variable frequency drive using a stationary reference frame.
It is assumed that the expected change in back-emf angle is known for the given control cycle, ∆φk = φk+1 − φk .
(17)
With this knowledge, the trigonometric transformation may be used to re-map the estimated back-emf coordinates by the adjustment angle. In order to remove the need for trigonometric evaluations, small angle assumptions may be made, as ∆φ will be much less than π.
5. I MPROVED P REDICITON M ETHODS 5.1. Rotation Feed-forward
cos(∆φ) ≈ 1, sin(∆φ) ≈ ∆φ.
The extrapolation method described in the previous section operated purely on each axis separately. However, it is known that the back-emf space vector will generally follow a circular trajectory at the electrical rotation frequency. This information may be used to better predict the path of the space vector over the control interval. Figure 6 shows the assumed back-emf geometry. The direct and quadrature axis back-emf components may be derived from the diagram to be, edk = ek cos(ωT k) eqk = ek sin(ωT k).
(15) (16)
If the value of ∆φ is known, the value of ek+1 can be quite accurately predicted from ek . ∆φ is likely
(18)
These approximations are very close for the range of ∆φ under consideration. For the example of a 50Hz fundamental with a 3kHz control rate, ∆φ will be of the order of 0.1. In this case, the approximations are accurate to less than 1% error, and the final effect on the current will be much less than this. With the approximation, the transformations are simply, "
epred dk+1 epred qk+1
#
≈
1 −∆φ ∆φ 1
eest dk eest qk
.
(19)
In this form, the only additional calculation, over the original controller, is a multiplication of the back-emf estimates with the supplied ∆φ, and the associated additions.
5.1.2. Performance Analysis The performance of this predictor will depend heavily on the value of ∆φ supplied. However, it is reasonable to expect that it is a slowly-changing quantity with respect to the current controller dynamics. In this case, the stability of the controller will be unaffected, as the adjustment could be considered an open-loop disturbance. This controller adjustment could be considered analogous to the rotating frame PID controller operating in stationary coordinates [8]. It offers the steady-state tracking advantages of a rotating-frame controller, but without the requirement for the coordinate transformations. 5.2. Observer Based Rotational Adjustment The principal drawback of the feed-forward method is that knowledge of the electrical rotation speed is required. At the cost of additional complexity, this information can instead be provided by an observer within the current controller. The observer may be developed from equation 19. Rearranging this to form an estimate for ∆φ, ∆φest k =
est −eest dk + edk−1 eest qk
(20)
∆φest k =
est eest qk − eqk−1 . est edk
(21)
This offers two different estimates of the change in angle of the e. Due to the division operation, a good approach would be to use the equation which involves division by the larger quantity. Although noisy, ∆φest k is slowly-changing with respect to the control frequency, and so may be filtered. The filter used in the analysis is, ilt ∆φfk+1 = (1 − )∆φfk ilt + ∆φest k ,
(22)
where controls the amount of data filtering. = 1 represents no filtering of the estimate. This filtered estimate may be used in place of the torque controller supplied value that was used in the feed-forward controller. 5.2.1. Observer Transfer Function Unlike the basic feed-forward controller, the introduction of the observer introduces additional dynamics into the system. These dynamics may be analysed by forming a transfer function for the revised system. The primary difficulty is to find a linear model of the new estimator, preferably decoupled between the d and q axes.
This may be achieved by noting that ∆φeqk is a slowly varying quantity, and so the change in ∆φeqk across one control cycle will be small. On this basis, it is reasonable to approximate the change in ∆φeqk as being equal to its true physical value [2]. This is, est ∆φ eest (23) qk − eqk−1 ≈ edk − 2edk−1 + edk−2 . Again, because the change in γ is small compared to γ itself, it is reasonable to approximate the true value of ∆φ with the filtered estimate, ∆φf ilt . The controller and machine equations may be solved for i, u, m and e to obtain the discrete-time transfer function. The result is, u ρ∆Lz 2 (z + − 1) T 3 + e −z + (3 − )z 2 + (2 − 3)z + (1 − ) Ll = i z 3 + ((∆L − 1)(1 + ) + + ρ∆L − 2) z 2 + ((∆L − 1)(− − 2) + ( − 1)ρ∆L + 1 − ) z +(∆L − 1)] . (24) One useful observation about this transfer function is that with = 1, it reduces to the transfer function in the back-emf extrapolation case. This means for that value of the parameter, both the performance and stability attributes will be the same as those of the extrapolation controller. This is not surprising, as the controllers are of the same order, and use the same information for the predictions. The advantage of the new predictor is that the filtering may be used to alter the stability and performance characteristics. 5.2.2. Stability Analysis The primary aim of developing the observer was to improve the stability over the back-emf extrapolation case, while still maintaining the same prediction performance. The incorporation of the filter is aimed at decoupling the prediction dynamics from the controller dynamics. Figure 7 shows the magnitude of the maximum root as a function of the error in the leakage inductance parameter. Three cases are shown in this plot. The original controller is shown for reference, and behaviour for = 0.1 and = 0.5 is plotted. The system becomes unstable for estimation values where this plot is above unity. It can be seen that the range of stability is less than that of the original controller but, as the parameter is reduced, the stable range approaches that of the original controller. In the case of the = 0.1, the near-flat section in the centre of the plot is due to the pole located on the real axis. This pole, at approximately (1 − ) represents the filtering mode. As this is not subject to uncertainty, it does not represent a risk to stability.
Pole Location as a Function of ∆L
Machine Direct Axis Current
Maximum root magnitude
2.5
10
2
Current (A)
ε=0.1 ε=0.5 Original Control
1.5
1
5
0
−5
0.5
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Inductance Error ∆L
0
0.005
0.01
0.015
0.02
0.025
0.03
Time (s)
Fig. 7 M AXIMUM ROOT MAGNITUDE AS A FUNCTION OF THE INDUCTANCE ESTIMATION ERROR .
Fig. 9 R ESPONSE TO A SET- POINT TRANSIENT.
6. P RACTICAL R ESULTS The performance of the back-emf prediction on a 7kW machine under test is shown in Figure 8. This plot shows the estimated and predicted back-emf quantities for each control interval. The predictor acts perfectly when it matches the estimated value of the subsequent cycle. The main feature of this plot is the performance
Reference Measured
−10
Machine Back−emf
100
back-emf plus leakage inductance model is quite adequate, provided care is taken with the parameter estimation. Analysis shows that the effect of the winding resistance is very small, but that variations in the backemf across a control cycle need to be predicted for the best performance. Simple extrapolation methods provide correct tracking, but incur stability and sensitivity costs. The rotation feed-forward and observer based methods shown here overcome these stability issues in a computationally simple manner.
Estimated Predicted
R EFERENCES
Back−emf (V)
50
0
−50
−100 0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
Time (s)
Fig. 8 BACK - EMF PREDICTION PERFORMANCE .
of the prediction algorithm. Except in the case of disturbances, it performs well at predicting the future value. The transient performance of the controller was evaluated using the 7kW induction machine. A plot of the d − axis current is shown in Figure 9 for a step change in torque demand. There is a small amount of overshoot and oscillation, which is consistent with parameter estimation errors. The endpoint tracking performs well, with no noticable steady-state error. 7. C ONCLUSION This paper presents a number of implementation issues associated with predictive current controllers. A simple
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