IEEE Industry Application Society Annual Meeting New Orleans, Louisiana, October 5-9, 1997
NEURAL NETWORK BASED SELF-TUNING CONTROL OF A SWITCHED RELUCTANCE MOTOR DRIVE TO MAXIMIZE TORQUE PER AMPERE A. V. Rajaratham Student Member, E E E
B. Fahimi Student Member, E E E
M Ehsai Fellow, E E E
Texas A&M University Department of Electrical Engineering College Station, TX 77843 phone (409) 845-7582 fax (409) 862-1976 email:
[email protected] Abstract - On-line self-tuning control is essential to
optimize the performance of a Switched Reluctance Motor (SRM) Drive in the presence of parameter variations. This paper introduces an advanced adaptive Neural Network (NN) based control to maximize torque per ampere in the low speed re@on. The proposed control technique utilizes a heuristic search method to find the change in the optimal excitation instances in case of parameter variations. Based on the results of this heuristic search, the NN employs an incremental learning to adapt its network weights. Computer simulations are performed to verify the applicability of the proposed algorithm. Experimental results are provided to demonstrate the working of the self-tuning controller. I. INTRODUCTION
The main advantage of SRM is in it's simple yet rugged rotor construction. Though it has a simple structure, it's control is very complicated due to the highly nonlinear characteristics of the machine. The advances made in the field of Digital signal Processors (DSPs) can be utilized in developing advanced digital controllers which can handle complicated control strategies. Several control strategies have been developed to improve the performance of the S M drives. The performance indices that are usually considered were maximum torque, torque ripple and drive efficiency. Previous work on optimization of SRM drive performance consisted mainly of off-line calculations to find the excitation instances to optimize performance indices like efficiency and torque output [1,2]. These control strategies are based on the assumption that there can be no occurrence of parameter variations that can change the electrical
characteristics of the machine. However, sigmticant SRh4 parameter variations occur in its mass production or with motor aging. Control techniques with self-tuning capability are essential to maintain an optimal performance of a SRM drive, in the presence of parameter variations. It has been shown that parameter variations can alter the inductance profile to a significant extent [3]. Since the control of SRM is essentially based on the inductance profile, it necessitates an on-line self-tuning control strategy for optimum performance. Only recently, work has been done on on-line optimization of SRM drive efficiency [4] and torque per ampere [ 5 ] . Tandon et al. has developed a self-tuning control method which takes into account the variations in the inductance profile due to parameter variations. But this method has finite accuracy, restricted self-learning capability and a limited dynamic response. This gaper introduces an improved control technique that overcomes the above shortcomings and optimizes the drive performance as measured by torque per ampere. Torque per ampere is defined as the ratio of the average torque to the phase current amplitude. A novel method is presented which combines an Adaptive Artificial Neural Network based control with a heuristic search method which periodically updates the weights of the NN in accordance with the parameter variations. The NN is trained with the experimental data obtained from the heuristic search based self-tuning setup. Hence it includes the effect of saturation and has a very good accuracy. In addition, it offers a good dynamic response and has an excellent self-learning capability. Computer simulations are performed to prove the validity of this algorithm. Experimental results are provided to demonstrate the working of the proposed self-tuning controller.
0-7803-4067-1 /97/$10.00 0 1997 IEEE. 548
11. THE SELF-TUNING CONTROL PRINCIPLE
Assuming a linear magnetic circuit, the instantaneous torque
is given by A . Basic Principle Of Operation SRM is a doubly salient, singly excited reluctance machine. A typical SRM of 8/6 configuration is shown in Fig. 1. Fig. 2 shows the classical power converter circuit, with two switches and two diodes per phase, typically used with a SRM. The measured phase inductance profile, under unsaturated condition, of an 816, 0.6 kW S R M drive is shown in Fig. 3. This profile can be approximated With a Fourier series. For simulation purposes, this s e ~ e scan be simplified by considering only the first two terms
Motoring torque in a SIRM is produced when the phase winding is energized during the positive slope of the phase inductance variation. Itleally, the maximum torque per ampere is produced when the winding is energized with a 1
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Fig 3. Inductance (Measured and Fourier Series Approximation) vs. Angle Phase A
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rectangular pulse of maximum allowable current during this period. But due to the effects of winding inductance, motional emf and magnetic saturation, the current has a definite rise and fall tinie. This introduces notable negative torque which has to be taken into consideration to optimize the performance of the S R M drive.
Fig. 1. 8/6 S R M
B. Fundamentals of the Seelf-tuning Control principle In the low speed region ( below base speed ), where the motional emf is not enough to limit the current, some kind of current control is required to limit the current below the rated value. In the hysteresis type current control, the chopping current band hias to be optimally chosen as there is a trade-off between the width of the band and the chopping frequency. Assuming the band chosen is optimal, maximum torque per ampere can be obtained by aptly tuning the turnon angle (e,) and tRe turn-off angle (eo$ of the phase current excitation.
Fig 2. Classic Power Converter Circuit Fig. 3 also shows the approximated inductance plot where 0" corresponds to the unaligned position and 180" corresponds to the aligned position, measured in electrical degrees.
Computer simulations,, based on a simple mathematical model, are performed to1 prove the existence of a unique (eon,
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sufEcient to give the optimal torque per ampere. Therefore, the optimization problem reduces to calculation of OOff online that gives maximum torque per ampere. It is shown by theoretical analysis that there exists a unique solution for 8 , ~ for each operating in the (o,T) region. The simulations results obtained for the machine operating at 800 rpm with rated current are shown in figures 4 and 5. Fig. 4 shows the current waveforms with and without optimization obtained from the simulation result. Fig. 5 shows the torques for the correspondingcurrent waveforms in Fig. 4. 111. ADVANCED SELF-TUNING CONTROL The proposed self-tuning control technique incorporates a W hewistic search method dong with a novel adaptive type I based method.
A. Heuristic Search method In the closed loop speed control, the transfer function is approximated by
Fig 4. Comparison of Phase Currents
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100 150 200 250 300 Rotor angle (electricaldegrees)
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Fig 5. Comparison of Torque's It has been shown that the optimal values 651 of e,, and €Ioff are bounded within the following limits
where is the turn-on angle such that the current reaches the desired value at 0" and q r is the turn-off angle such that the current reduces to zero at 180". The optimal 8, is not very susceptible to the change in inductance due to the parameter variations because of the large airgap at the unaligned position. Hence optimal 8 , calculated off-line based on the linear model will be
The current I is set to an appropriate value to obtain the desired torque. Now, maximum torque per ampere can be obtained when the current I is minimum, with the speed (0) and Torque (T) remaining constant. With this setup, the control variables 8 , and OOff are varied to obtain the maximum torque per ampere at the set speed and torque. Since the 8 , is not very susceptible to parameter variations in the low speed region, its optimal value can be directly calculated from the linear model of the SRM. Now, the problem is reduced to calculatingjust the optimal In the heuristic search method, the controller undergoes the following steps to obtain the optimal 1. First the is initialized to maximum value of 180" as obtained in (4). 2. Then 8 , is~ decremented in small steps of A8. The step size has to be chosen in such a way that there is a compromise between the speed of convergence and the accuracy, 3 . After each step change in angle, the controller allows a small delay for the speed to settle down to the set value. 4. Now a comparison is made between the new current value with the previous value. 5. If the new current value is less than the previous value then the controller repeats step 2, otherwise it changes the sign of AB. 6. When the 8 , ~is near optimal, the search will be localized to around 2-3 A8 steps, in the neighborhood of the optimal 8,s. The controller halts the search when the sign of A8 changes for the second time. 550
B. Data Generation for the NN The data to train the NN is obtained based on the above heuristic search method. The drive is operated at some arbitrary operating point and the tuning process is activated to find the optimal OOff. This process is repeated for v ~ o u s operating points to obtain an array of data in the low speed where k is the index of the next layer and m is number of region of the (o,T)plane. These operating points are neurons in the next layer and f ‘(t,) is given by appropriately chosen such that these are distributed in the f‘(tj) = (1 -- f(tJ))(l f(tJ)) entire low speed region. For each of these operating points, the optimal 8 , is~ obtained. These optimal values of 8 , are ~ stored with their given operating points. The data obtained Now the weight correction term is calculated as above are then used to train the NN.
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C, NN Based Self-Tuning The proposed control technique is based on a multilayer feedforward NN with a differentiable type threshold function. The NN topology has optimal number of neurons with an input layer with 2 inputs, 1 hidden layer with 6 neurons and an output layer with a single output which calculates the optimal 8B , on-line. The inputs are I and o and the output being optimal 0,s. Back propagation l e d n g aIgorithm has been employed for training the NN [6]. The activation function of neurons in the hidden layers is chosen to be tan-hyperbolicfunction as given by:
f(sj) = tanh(sj) = t j n
+ &wjj
Ab,j=
(13)
Finally the weights anti biases are updated as
(6)
and
sj = b,
where E: is the learning rate. The bias correction term is calculated as
(7)
i=%
where s is the input to the current layer, t is the output of the current layer, i is the index of the previous layer, 3 is index of the current layer, n is the number of neurons in the current layer, wij is the weight matrix between current and previous layer and bi is the bias for the current layer.
As the NN is trained with the experimental data obtained from the self-tuning setup, it includes the effect of magnetic saturation. Hence the accuracy of the optimal 8,E obtained from the NN is quite high. B. Adaptive NN Based Self-Tuning
The control technique incorporates a periodic heuristic search of optimal to venfy the accuracy of the OOE obtained from the PJN, If there is a variation in the inductance profile due to parameter drift, the optima9 8,ff The intermediate weight correction term for the weight obtained from the NN will no longer hold good. This matrix between the output layer and it’s previous layer is prompts the control1e:r to activate the heuristic search by obtained from m o w n g the 8,E in small steps till the current I reaches the minimum value. This new optimal 8,E at that particular 6 = ef’(ti) Operating point is now used to adapt the weights of the NN. Hence this novel NN based control technique coupled with where ti is the input to the previous layer e is the error term the heuristic search learns and adapts to any parameter drift obtained as to give the optimal BOff. The procedure followed in adapting the weights of the NTlJ is discussed in detail in Section IV. e=y-t (9) Fig. 6 shows the flowchart of the above algorithm. Since the NN directly provides the optimal 8,ff on-line, it has a very where y is the desired optimal 8,ffand t is the calculated 8,ff good dynamic response limited only by the speed of the DSP from the NN output. For all other layers the intermediate and the size of the NN. Also, since the control is based on NN, it is not restricted by finite resolution problem as in the weight correction term is calculated as ‘look-uptable approach’. 55 1
IV, ADAPTIVE LEARNING OF THE NN
NNs have been successfully used for many applications in control systems. But the NN learning algorithm perform remarkably well when used off-line i.e., they have to be fully trained before being applied. NNs with incremental learning capability with stable adaptation of network parameters are essential for on-line adaptive control. A simple, modelindependent method was suggested by Foslien et al [7]. Their approach is based on the assumption that the NN to start with is well trained in such a way that it can perform input/output mapping for the initial training set with high degree of accuracy. This can be achieved by training the NN with sufficient amount of data to a very low error rate. In this application, this training can be done off-line as it may require more time. Now when a new training data is obtained, the already trained NN is used to generate additional examples. These additional examples with the newly obtained Vaining data are then used to retrain the current NN. This ensures that the 0rigind NN mapping is retained with only a change localized around the neighborhood of the new training data. This makes the network to gradually adapt to the new data. The above method ensures the stability of the network weight variations by slow adaptation as the new optimal 0 , ~ will be in the neighborhood of the old value. The main disadvantage in multilayer NN for incremental learning is that the interference by which future training disrupts the traces of previous training [SI. But this doesn’t affect this application as the previous information of optimal Ooff is not required in case of parameter variations.
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V. RESULTS
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A. Simulation Results Computer simulations are used to verify the accuracy of the above algorithm. A 816, 12V, 0.6kW SRM of the laboratory experimental test set up is modeled using MATLAB. Two different models, a linear model as discussed in Section I1 and a nonlinear model which utilizes the magnetic data obtained from the Finite Element Analysis, were developed. Fig. 7 shows the torque profiles, obtained from the nonlinear model, before and after OOff optimization. This plot clearly shows the improvement in torque per ampere with optimization. It was found that the torque per ampere increased by 13.6%.
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Increment e,,, angle by one step
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A. 1. Verification of incremental NN [earning Fig. 8 shows the optimal OoE vs. speed at I=30A obtained from the linear and the nonlinear model. To veri& adaptive NN algorithm, the NN was initially trained with linear model data of Fig. 8. Fig. 9 shows the output of the trained
Fig 6. Flowchart of the NN Algorithm
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Fig. 9 Optimal O,ff vs. speed at I=30A output of NN after initial training
Fig 7. Developed torque before and after optimization NN. This plot shows that the NN has a very good accuracy. The new optimal 0 , ~s after parameter varia~onscan be approximated by the data obtained from the nonlinear model. To demonstrate the working of the adaptive W, this new data is randomly chosen to train the NN adaptively. This procedure corresponds to the on-line adaptive training in the experimental set up. Fig 10. shows the output of Np9 after first adaptive training. The plot also shows the new data that are added and the additional data generated by NN. The above procedure is repeated whenever the heuristic search method finds new set of data. Fig. 11 shows the output of NN after 20 such adaptive training's. It can be seen that the I"has adapted to the new data with very good accuracy.
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Fig. 8 Optimal 0 , ~ ~ speed s . at I=30A - from linear and nodinear model
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Fig. 11 Optimal 0 , ~ ~speed s . at I=30A output of NN after 20 adaptive training's
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B. Experimental Setup hhra experimental laboratory test setup was built to implement this algorithm. A 8/6, 12V, 0.6 kW S M was used. A position resolver with resolver to digital converter of 14 bit resolution was used for position information feedback and speed measurement. A resistor loaded permanent magnet dc machine operated as generator is used as load in this setup. A Classic 2 switches - 2 diodes per phase converter was used. The current was limited using hysteresis type current control. The digital control of the drive was achieved by a floating point Digital Signal Processor (DSP) TMS320C30. The 14 bit position information is read through the digital U 0 ports and appropriate gating signals are sent out through the U 0 ports. The entire control algorithm is implemented using assembly code.
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C. Experimental Results
Fig. 13 Current waveform after heuristic search based
The S M was initially operated at 500 rpm with no tuning i.e., e,, at the unaligned position and QOff at the aligned position. The current waveform for the above operating point is shown in Fig. 12. Then %he heuristic search based tuning is activated ancl Fig. 13 shows the current waveform &er tuning. The was decremented in steps of 0.1O mechanical. The o p e i d 8,s was found to be 28.13O . Figs 14 and 15 show the variation in speed and Bas, respectivelyy, with time during the above tuning procedure. The current was reduced by 1.5 %. Fig 16 shows the current waveform based on the activation of the NN based control. The optimal Oog, in this case, was found to be 28.72O . This optimal has a small error due to the tolerance in the output of the NN. These results demonstrate the applicability of tRe proposed algorithm.
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Fig. 12 Current waveform before tuning (5Ndiv)
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VI.CONCLUSION
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A novel ANN based conitrol method to maximize torque per ampere is presented. This method is highly accurate, robust and has a good dynamic response. This algorithm combines the ANN based control and a method of periodic search for the optimal €IoE based on the position and current feedback. This method has excellent self-learning capability as it tunes the weights of the ANN according to parameter variations that affects the inductance profile. This algorithm is implemented digitally with the help of a Digital Signal processor. This algorithm can be made flexible to optimize new performance index om-line.
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VII. ACKNOWLEDGEMENT Fig. 16 Current waveform after NN based tuning (5Ndiv)
D.Critical Issues Based on the experimental and simulation results, the following observationswere made 1. The- proposed control technique will learn adaptively with high accuracy when the parameter variations are slow with time. 2. This method requires initial set of training data obtained from the experimental set up. This problem is avoided by training the network with linear model data and allowing the NN controller to adapt itself during actual operation of the machine. 3. The accuracy of the output of the NN depends on the amount of new data used during adaptive training. From simulation results it is found the accuracy is acceptable when the adaptive training is done with number of new data > 4. 4. Time required for adaptive training will depend on the accuracy required. The controller will operate with the previous set of optimal 0,& until the adaptive training is completed. 5 . The criterion to include the new data for adaptive training must take into consideration the tolerance in the output of the NN. The &n advantage of the proposed method is that the software can be made very flexible so that this algorithm can be used for self tuning to optimize any of the performance indices like efficiency, torque ripple etc., . This can be easily done with minor modifications made to the software to adjust the structure of the NN to optimize the new performance index.
The support of Texas Higher Education Coordinating Board Advanced Technology Program, TRW Electronic Controlled Steering, Genieral Motors Research Laboratories and Texas Instruments:, for this research is greatfully acknowledged.
IX. :REFERENCES [I] D.A. Torrey and J.H. ILang, "Optimal-efficiency excitation of variable-reluctance motor drives," IEE hoc. B, Vol. 138, No. 1, pp. 1-14, Jan. 1991. [2] R. Orthmann and HI'. Schoner, "Turn-offangle control of switched reluctance motors for optimum torque output," Fifth European Conference on Power Electronics and Applications, pp. 20-25,1993. [3] M. Ehsani and K.R. Ramani, "Direct control strategies based on sensing inductance in switched reluctance motors," IEEE PESC Conference Record, pp. 493499, 1994. [4] P.C. Kjaer, P. Nielsen, L. Andersen and F. Blaabjerg, "A new energy optimizing control strategy for switched reluctance motors," IEEE-AF'EC Conference Record, pp. 48-55, 1994. [ 5 ] P. Tandon, A.V. Rajarathnam, and M. Ehsani, "Self-tuning control of a switcheid reluctance motor drive with shafl position sensor," EEE, IAS Conference Record, pp. 101-108, 1996. [6] L. Fausett, Fundamentkals of Neural Networks, Prentice Hall, 1994. 171 W.K.Foslien, T. Samad, "Incremental supervised learning : localized updates in nlonlocal networks," Sceinec of Artificial Neural Networks, Proc. SPIE 1710, pp. 608-617. 1992. [SI S. Grossberg, "competitive learning : From interactive activation to adaptive resoname," Cognit. Sci., vol. 1 1 , pp. 23-
63, 1987.
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