CHAPTER 3 WAVELET TRANSFORM BASED CONTROLLER FOR INDUCTION MOTOR DRIVES

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1 49 CHAPTER 3 WAVELET TRANSFORM BASED CONTROLLER FOR INDUCTION MOTOR DRIVES 3.1 INTRODUCTION The wavelet transform is a very popular tool for signal processing and analysis. It is widely used for the analysis of non-stationary, non-periodic and transient signals. The popularity of the wavelet transforms is mainly due to their ability to concentrate the energy of the processed signal into a finite number of coefficients. They are capable of providing the time-frequency localization of the signal. Wavelet Transform uses multi-resolution technique by which different frequency components are analyzed with different resolutions. It is realized through successive stages of filters followed by down sampling operation. Currently, there is a tremendous increase in the use of wavelet transform for real time applications. Wavelet transform has been applied in diverse fields such as image denoising, video signal compression, optics, climate change analysis, financial market analysis, fault analysis and condition monitoring of rotating machines and power system analysis. Part of the thesis work reported in this chapter has been published as detailed below: Febin Daya, J. L. and Subbiah V. Implementation of a Hybrid Wavelet Fuzzy based Controller for Speed Control of Induction Motor Drives, Iranian Journal of Electrical and Computer Engineering, Vol. 11, No. 1, pp , 2012

2 50 Parvez and Gao (2005) developed a wavelet based PID controller for motion control application. Khan and Rahman (2008) implemented a wavelet based PID controller for interior permanent magnet synchronous motor. The wavelet based controller was also proposed for dc motor speed control application (Yousef et al 2010). This chapter presents a wavelet based speed controller for IFOC of induction motor drive. The criteria for selecting the appropriate wavelet function and optimal level of decomposition for the proposed wavelet based speed controller are also presented. The structure of a wavelet based controller, implementation and the simulation results for induction motor drives are discussed in detail. 3.2 WAVELET BASED MULTIRESOLUTION Wavelet transform is a powerful statistical tool which can be used for parsimonious representation of signal. It can be used to perform multiresolution analysis (MRA), which can extract and localize frequency components of a signal at a time. MRA represents a function as a successive limit of approximations, at different stages. Each stage consists of an approximate version and detailed version. In general the discrete wavelet representation of a signal ) is defined in terms of its orthonormal bases, that is scaling function and wavelet function as (Khan and Rahman 2008) ( ) = ( ) + ) (3.1) where = ( ) ) (3.2) = ( ) ) (3.3)

3 51 Discrete Signal Approximation Coefficients at Level 1 (L) a 1 Detail Coefficients at Level 1 (H) d 1 1 st Level Decomposition Approximation Coefficients at Level 2 (LL) a 2 Detail Coefficients at Level 2 (LH) d 2 2 nd Level Decomposition Figure 3.1 Two level decomposition tree of discrete wavelet transform where )and ) are the conjugate functions of scaling function and wavelet function respectively. The multiresolution can be realized using quadrature mirror filter banks. The appropriate wavelet function is used to generate the quadrature mirror filter coefficients. The filter coefficients of the filter banks are derived using the dilation equation and is mathematically represented as (Stang and Nguyen 1997) ( ) = 2 ( ) (2 ) (3.4) The quadrature mirror filter banks have different stages of filter with each stage having a low pass and high pass filter. In order to ensure perfect reconstruction, the filter banks scheme must satisfy the property of perfect energy conservation. This can be represented mathematically as

4 52 + = (3.5) where is the decimated version of the finite energy signal filtered through the low pass filter. Therefore can be expressed as (2 ), where ). Therefore is given by (2 ), where ). The coefficients of the low pass filter and high pass filter are related by the following expression (Stang and Nguyen 1997) ( ) = ( 1) ( ) = 0,1,2, 1 (3.6) where g(k) represents the low pass filter coefficients and h(k) represents the high pass filter coefficients. The filter bank structure must also satisfy the orthogonality property represented as (Stang and Nguyen 1997) ) + ) = (3.7) where ( ) and ) are the transfer function of high pass and low pass filter respectively. The quadrature mirror filter bank must satisfy the conditions of equation (3.5)-(3.7). The filters of Daubechies wavelet family satisfy all these properties and they come under the category of quadrature mirror filter. The two level decomposition tree of a discrete signal is shown in Figure 3.1. It is called as Mallet-tree decomposition (Addison 2002). L and H represent the low frequency and high frequency components of the discrete signal respectively. LL and LH represents the low frequency and high frequency components of the approximate coefficients at level 1 after down sampling. The resolution of the signal is changed by filtering operation and the scale is changed by up sampling and down sampling operation. Up sampling corresponds to increase in the sampling rate of the signal by adding new samples and down sampling corresponds to decrease in the sampling rate of the signal by removing some samples from the signal.

5 SELECTION OF APPROPRIATE WAVELET FUNCTION Before applying wavelet, it is required to select appropriate wavelet function. The choice of mother wavelet and scaling function is application dependent. The best selected wavelet function exactly parameterizes and expands the signal. It also decomposes and reconstructs the signal using the shifted and dilated version of the wavelet function. Some of the desirable properties of the wavelet function are compactness, orthogonality, linear phase, low approximation error etc (Stang and Nguyen 1997). The compactness property of the wavelet function has the advantage of lesser computational efforts. It also detects the frequency components present in the signal, which can be used in the design of speed controller for the motor drive (Parvez and Gao 2005). Different methods are available in the literature, but the minimum description length (MDL) data criterion (Hamid and Kawasaki 2002) is best suited for the selection of the optimum wavelet function. The MDL criterion selects the best wavelet filter for the signal decomposition. According to MDL criterion, the best model within group will have the shortest description of data model itself. The MDL criterion can be defined as (Hamid and Kawasaki 2002) ) = ) 0 k < N; 1 n M (3.8) where k and n are the indices. The integer N and M denote respectively, the length of the signal and the wavelet filters used. The is the vector of wavelet transformed coefficients of the signal using the wavelet filter and ) denotes the vector containing k non-zero elements. The threshold parameter keeps k number of largest element of and sets all the other elements to zero. The number of coefficients k, for which the MDL criterion gives the minimum value is considered as the optimum one.

6 54 In the proposed work, the objective is to apply wavelet transform technique to the speed error signal of the induction motor drive. The IFOC of the induction motor drive is simulated in Matlab version 12b using the configuration shown in Figure 2.1. The PI controller is used as the controller in the speed control loop. The gain values of the PI controller are tuned used Ziegler-Nichols method (Nichols and Ziegler 1993) in order to obtain better performance from the controller. This method of tuning is selected since this is a well accepted standard method for tuning the controllers. Moreover, the tuning is simple, used in real time control system design process and is used in literatures for comparing the controller performance with emerging control techniques (Nasir Uddin et al 2002). The actual speed, command speed and the speed error of the induction motor drive are stored in the workspace of the Matlab during simulation. The induction motor drive is simulated for different command speed and the corresponding data were stored. The drive is also simulated at different load conditions to get different sets of data. The MDL criterion is applied to the stored data set in order to select the optimum wavelet function for the wavelet based speed controller. The orthogonal wavelets available in the Matlab wavelet tool box are tested by the MDL criterion. The mother wavelets tested are db1, db2, db3, db4, db5, db6, db7 of Daubechies wavelet family, sym2, sym3, sym4, sym5, sym6 of Symlets wavelet family and coif1, coif2, coif3, coif4 and coif 5 of Coiflets wavelet family. The MDL criterion is applied on the data stored in the workspace obtained during simulation of the induction motor drive. The speed error is decomposed using DWT up to second level of resolution. The MDL indices are obtained for all the mother wavelets mentioned above and tabulated.

7 55 Table 3.1 The MDL Indices of the speed error signal when the command speed of the induction motor drive is set as 180 rad/sec Sl.No Wavelet Function MDL Indices 1 st level decomposition MDL Indices 2 nd level decomposition 1 db db db db db db db sym sym sym sym sym coif coif coif coif coif

8 56 Table 3.2 The MDL Indices of the speed error signal when the command speed of the induction motor drive is changed from 0 to 120 rad/sec and again increased to 180 rad/sec Sl.No Wavelet Function MDL Indices 1 st level decomposition MDL Indices 2 nd level decomposition 1 db db db db db db db sym sym sym sym sym coif coif coif coif coif

9 57 Table 3.3 The MDL Indices of the speed error signal when the induction motor drive is applied a load torque of 2 Nm and the command speed is set as 150 rad/sec Sl.No Wavelet Function MDL Indices 1 st level decomposition MDL Indices 2 nd level decomposition 1 db db db db db db db sym sym sym sym sym coif coif coif coif coif

10 58 The MDL indices of all the tested mother wavelets when the motor in driven at no load and with a command speed of 180 rad/sec are shown in Table 3.1. The evaluation of the MDL indices shows that the db4 wavelet of the Daubechies wavelet family has the lowest MDL index of the speed error during first level and second level of decomposition. Table 3.2 shows the MDL when the command speed of the motor is changed from 0 to 120 rad/sec and again increased to 180 rad/sec. In this particular case, the db3, db4 and db6 wavelets of Daubechies wavelet family has the lowest and almost the same value of MDL index of the speed error for first level and second level of decomposition. The MDL indices of the speed error when the motor is applied a load torque of 2Nm and the command speed is set as 150 rad/sec is shown in Table 3.3. The db4 wavelet once again outperformed all the other tested wavelet functions in terms of the lowest MDL indices for first level and second level of decomposition. Therefore, the db4 wavelet of Daubechies wavelet family is selected as the optimum wavelet function for the proposed wavelet based speed controller for induction motor drive. The speed error is decomposed using the selected db4 wavelet function in order carry out the multiresolution analysis of the speed error signal. 3.4 LEVEL OF DECOMPOSITION The appropriate level of decomposition of the error signal has to be selected before applying DWT. The number of level of decomposition decides the number of tuning gains required for the wavelet based controller. The level of decomposition depends on the signal as well as the wavelet used for decomposition. The Shannon entropy based criterion best suits to find the optimum level of decomposition of the speed error signal for motor drive applications. The entropy of a signal ( ) = {. } of length N can be represented as (Hamid and Kawasaki 2002)

11 59 ( ) ) ) (3.9) The entropy calculated at every level of decomposition for both the approximate and the detailed coefficients of the transformed signal in order to find the optimum level of decomposition. According to Shannon entropy based criterion, if the entropy of the signal at a new level ) is higher than the previous level -1), that is if ) ) (3.10) then the decomposition of the signal can be stopped at level ( ) and ( ) represents the optimum level decomposition. Speed error x x x10-11 Figure 3.2 Entropy values of the decomposed speed error signal, when the motor in driven on no load and with a command speed of 180 rad/sec

12 60 Speed error x x x10-9 Figure 3.3 Entropy values of the decomposed speed error signal, when the command speed of the motor is changed from 0 to 120 rad/sec and again increased to 180 rad/sec In the proposed work, the entropy based criterion is used to find the optimum level of decomposition. The entropy values are calculated for the speed error signal after decomposing it using db4 wavelet which is selected as the optimum wavelet function using MDL criterion. Figure 3.2 shows the entropy values at each subspace up to third level of decomposition for the speed error signal, when the motor in driven on no load and with a command speed of 180 rad/sec. It can be observed that the entropy values at level two and three is higher than the entropy values at level one. Figure 3.3 shows the entropy values at each subspace up to third level of decomposition for the speed error signal, when the command speed of the motor is changed from 0 to 120 rad/sec and again increased to 180 rad/sec. It is observed from the figure that the entropy values at level one and two are almost the same and

13 61 lower than the entropy values at level three. Figure 3.4 shows the entropy values at each subspace up to third level of decomposition for the speed error signal, when the motor is applied a load torque of 2 Nm and the command speed is set as 150 rad/sec. It can be observed from the figure that the entropy values at level three is higher than the entropy values at level one and two. It can be concluded that the first case gives the optimum level of decomposition as level one and the second and the third case gives the optimum level of decomposition as level two. Hence, the optimum level of decomposition is concluded as level two. Therefore the speed error has to be decomposed up to second level using db4 wavelet for the wavelet based speed controller for induction motor drive. Speed error x x x10-9 Figure 3.4 Entropy values of the decomposed speed error signal, when the motor is applied a load torque of 2Nm and the command speed is set as 150 rad/sec

14 WAVELT BASED SPEED CONTROLLER All physical systems are subjected to some type of extraneous signals or noise during operation. Therefore, in the design of a control system, consideration has to be made that the system provides greater insensitivity to noise and disturbance. The effect of feedback on noise and disturbance greatly depends on where these signals occur in the system. In practice, the disturbance and commands are often low frequency signals, where sensor noises are high frequency signals. This makes it difficult to minimize the effect of these uncertainties simultaneously. Under these conditions, the wavelet based controller can perform extremely well by discriminating the signals into different frequency bands. In a conventional PID controller, the control output is generated making use of the error signal and further processing on it. The output of the PID controller is given by = + (3.11) where and are the proportional, integral and derivative gain constants respectively. These gain constants acts on the error signal as shown in (3.11). In terms of frequency, the proportional term corresponds to the low frequency information, the integral term corresponds to medium frequency information and the derivative term corresponds to high frequency information of the given error signal (Parvez and Gao 2005). Discrete wavelet transform (DWT) performs the same operation of decomposing a signal into low frequency (detail) and high frequency (approximate) coefficients at different levels of resolution. This feature of the

15 63 wavelet transform can be made use of in developing a wavelet based controller for the expected control actions. The control signal for the wavelet based controller can be calculated from the detail and approximate coefficients of wavelet transform as (Khan and Rahman 2008) = + + (3.12) where,,, corresponds to detail components of the error signal and is the approximate component of the error signal. The gains,, are used to tune the high and medium frequency components of the error signal. Gain is used for tuning the low frequency component of the error signal (Parvez and Gao 2005). While dealing with motor drives, the command and disturbance are low frequency signals. The sensor noises are high frequency signals. Therefore, the gain which corresponds to low frequency components of the error signal can be used to improve the disturbance rejection of the system. The gain which corresponds to high frequency components of the error signal can be set to minimum to eliminate the effect of noise on the system (Nejadpak et al 2011). The optimum level of decomposition is estimated as two using the Shannon entropy based criterion as explained in section 3.4. The control signal of the wavelet based controller can now be represented as (Parvez and Gao 2005) = + (3.13)

16 64 The wavelet based controller decomposes the speed error between the command speed and the actual speed into approximate and detailed coefficients up to second level using db4 wavelet function. The db4 wavelet Command Speed 2 Level DWT Decomposition + - IFOC IM Actual Speed Figure 3.5 Schematic of the wavelet based speed controller for IFOC of Induction motor drive function selected as the optimum wavelet function and is used to generate the low pass and high pass filter coefficients. The DWT coefficients can be represented as (Khan and Rahman 2010) = ] (3.14) = ] (3.15) = [2 ] (3.16)

17 65 = [2 ] (3.17) where the and represents the low pass and high pass filter coefficients and is generated using the db4 wavelet function. The schematic of the wavelet based speed controller (Parvez and Gao 2005) for induction motor drive is shown in Figure 3.5. The error signal is decomposed up to second level of decomposition using DWT. The decomposed signal is multiplied with their corresponding gain values and summed up together to generate the command signal as represented by equation (3.13). The command is used as the torque component current signal for the indirect field oriented control of the induction motor drive. The gain which corresponds to the low frequency components of the error signal can be kept high in order to improve the disturbance rejection and to reduce the settling time. The gain represents the medium frequency components of the error signal and it can be used to adjust the steady state behavior of the system. The gain can be kept high during steady state operating region so as to reduce the steady state error. Similarly the gain which corresponds to the high frequency components of the error signal can be used to improve the transient response and to reduce the overshoot of the drive system in order to produce smooth control of the induction motor drive. The gain can be used accordingly during the transient period, to achieve smooth control performance of the induction motor drive.

18 SIMULATION OF THE WAVELET BASED SPEED CONTROLLER The simulation of the wavelet based speed controller for induction motor drives is done using Matlab/Simulink. The IFOC speed control scheme incorporating the wavelet based speed controller is shown in Figure 3.6. The torque component command current signal is generated by the wavelet based speed controller. The command current signal is used by the IFOC to generate the switching pulses for the three phase six pulse inverter switches using PWM technique. V DC Command Speed + - WAVELET BASED SPEED CONTROLLER VOLTAGE CONVERSION AND TRANSFORMATION PWM THREE- PHASE INVERTER TRANSFORMATION AND CURRENT MODEL Actual Speed LPF SPEED SENSOR IM Figure 3.6 Schematic of the IFOC speed control scheme incorporating the wavelet based speed controller Once the input, the level of decomposition and the optimal wavelet function of the wavelet based speed controller are selected, it is also necessary to select the scaling gains of the wavelet based speed controller. The scaling gains, and is used for tuning the low frequency, medium

19 67 frequency and high frequency components of the error signal respectively. Since there is no standard procedure available in the literature for selecting the scaling gains of the wavelet based controller, they are selected using trial and error method in order to get optimum performance from the induction motor drive (Parvez and Gao 2005). The gain value are selected as = 2.6, = 0.8 and = for the wavelet based speed controller Simulation Results The effectiveness of the wavelet based speed controller is validated by several simulations under various operating conditions. The IFOC of the induction motor drive with the wavelet based speed controller is simulated in Matlab/ Simulink. The simulation studies are carried out on a 1.47 kw squirrel cage induction motor. The motor parameters are given in Appendix 2. The switching frequency of the PWM signal is selected as 2 MHz and hence the sampling time is set as 2 sec. The complete induction motor drive system is simulated with different command speed, step increase and decrease of command speed, change of load torque and variation of system parameters. The performance of the wavelet based controller is compared with the speed response obtained from PI, PID and fuzzy based controller given in Chapter 2 The induction motor drive with the wavelet based speed controller is simulated at no load with the command speed of rad/sec, the rated speed of the induction motor. The speed, the line current and the q axis command current under this operating condition are shown in Figure 3.7. The induction motor drive is also started with a load of 2.5 Nm at rated speed of rad/sec and the simulation results are shown in Figure 3.8. Compared to speed response of PI, PID and fuzzy based controller for the same operating condition given in Figure 2.5 and 2.6, the wavelet based controller responded to the command speed quickly and settled to the steady state in less than 0.1 sec.

20 68 Figure 3.7 Simulated starting response of the wavelet controller based induction motor drive at no load with a command speed of rad/sec (rated speed). (a) Speed (b) Line current (c) q-axis command current

21 69 Figure 3.8 Simulated starting response of the wavelet controller based induction motor drive with a load of 2.5 Nm and a command speed of rad/sec (rated speed). (a) Speed (b) Line current (c) q-axis command current

22 70 Figure 3.9 Simulated starting response of the wavelet controller based induction motor drive at no load with a command speed of 91.7 rad/sec (50% of rated speed). (a) Speed (b) Line current (c) q-axis command current

23 71 Figure 3.10 Simulated starting response of the wavelet controller based induction motor drive at no load with a command speed of rad/sec (125% of rated speed). (a) Speed (b) Line current (c) q-axis command current

24 72 Figure 3.11 Simulated response of the wavelet controller based induction motor drive at no load for step increase in command speed from 120 rad/sec to 180 rad/sec. (a) Speed (b) Line current (c) q-axis command current

25 73 Figure 3.12 Simulated response of the wavelet controller based induction motor drive at no load for step decrease in command speed from 150 rad/sec to 90 rad/sec. (a) Speed (b) Line current (c) q-axis command current

26 74 Figure 3.13 Simulated response of the wavelet controller based induction motor drive for a command speed of 180 rad/sec and 25% of the rated load is applied at t = 0.5 sec. (a) Speed (b) Line current (c) q-axis command current

27 75 Figure 3.14 Simulated response of the wavelet controller based induction motor drive started with 25% of rated load and the load removed at t = 0.5 sec. (a) Speed (b) Line current (c) q-axis command current

28 76 Figure 3.15 Simulated response of the wavelet controller based induction motor drive for doubled rotor inertia, at no load with a command speed of 180 rad/sec. (a) Speed (b) Phase current (c) q-axis command current

29 77 Figure 3.16 Simulated response of the wavelet controller based induction motor drive for doubled stator resistance, at no load with a command speed of 180 rad/sec. (a) Speed (b) Phase current (c) q-axis command current

30 78 The induction motor drive with the wavelet base controller is simulated with a command speed of 91.7 rad/sec, i.e. 50% of the rated speed at no load. The speed, the phase current and the q axis command current responses are shown in Figure 3.9. The drive system has followed the command signal with less overshoot and with less steady state error. Figure 3.10 shows the speed and the current response of the induction motor drive for a command speed of 229.1rad/sec which is 125% of the rated speed. The induction motor drive is able to track the command speed with less steady state error. The induction motor drive with the wavelet based speed controller is simulated with step change in command speed. The command speed is set at 120 rad/sec and increase to 180 rad/sec at t = 0.5 sec. The speed and current responses are shown in Figure Figure 3.12 show the speed and current response when the command speed is set as 150 rad/sec and decreased to 180 rad/sec at t =0.5 sec at no load. Simulation results shows that the wavelet speed controller based induction motor drive has followed the command speed with less over shoot and steady state error compared to the PI, PID and fuzzy based controller results given in Figure 2.7 and 2.8. The performance wavelet based speed controller is analyzed for the sudden impact of load. Figure 3.13 show the response of speed and current when the motor is started at 180 rad/sec without load and 25% of rated load applied at t =0.5 sec. The drive system has show less sensitive performance for this sudden application of load. The speed has dropped at the point of application of load and it came back to the steady state value after a period of time.

31 79 The induction motor drive is started with 25% of rate load at a command speed of 180 rad/sec and the load completely removed at t = 0.5 sec. The responses are shown in Figure The speed has jumped up at the point of removal of load and settled to the steady state value with a small steady state error compared to the results given in Figure The stating performance of the wavelet based controller is investigated for change in rotor inertia and stator resistance. The speed response, the line current and the q-axis command current at no load with a command speed 180 rad/sec are shown in Figure 3.15 and Figure 3.16 for doubled rotor inertia and doubled stator resistance respectively. The drive system has followed the command speed under these conditions. However, the drive system took slightly higher settling time to reach the steady state command speed. Table 3.4 Comparative RMSE Results Change in Speed PI Controller PID Controller Fuzzy based self-tuning Controller Wavelet Controller rad/sec rad/sec rad/sec rad/sec - 90 rad/sec rad/sec, Load applied at t = 0.5 sec

32 80 Table 3.5 Percentage improvement in RMSE for Wavelet Controller Change in Speed Wavelet Controller compared with PI Controller PID Controller Fuzzy based self -tuning Controller rad/sec 9.8 % 7.54% 0.1% rad/sec rad/sec rad/sec - 90 rad/sec rad/sec, Load applied at t = 0.5 sec 11.14% 4.04% 0.34% 7.7% 4.39% 0.43% 2.26% 0.64% 0.64% Comparing the performance of the wavelet based controller with the conventional PI, PID controller and the fuzzy based self-tuning PID controller discussed and simulated in chapter 2, the wavelet based speed controller in found to be better than the conventional PI and PID controllers for same motor parameters. The performance of the speed controller is compared in terms of time domain specifications such as rise time, peak time, settling time, over shoot and steady state error. However, there are only marginal changes in these time domain parameters for different controllers under consideration. Hence, root mean square error (RMSE) between the command speed and actual speed is computed in order to compare the performance. The RMSE is given by = ) (3.18)

33 81 where is the command speed, is the actual speed and is the number of samples. The comparative RMSE results are shown in Table 3.4. It can be observed that the performance of the wavelet based speed controller is almost to the fuzzy based self-tuning PID controller for difference speed conditions and load disturbances. The percentage improvement in the RMSE value for wavelet based controller compared to PI, PID and Fuzzy based self-tuning controller is tabulated in Table 3.5 for various speed changes. 3.7 SUMMARY The wavelet based speed controller for indirect field oriented control of induction motor drive is presented in detail. The wavelet based speed controller performs well compared to conventional PI and PID controller under various operating condition and load disturbance. However, the speed response is still sensitive to load disturbance especially when there is a sudden impact of load. There are still chances for improvement in peak overshoot, settling time and steady state error. The speed response of the induction motor drive can be improved by higher level of decomposition of the speed error using DWT. However, the computational complexities of the speed controller will be increased. The scaling gains of the wavelet based speed controller are selected by trial and error method. But the scaling gains have significant effect on the performance of the wavelet based speed controller (Parvez and Gao 2005). Therefore, a proper procedure has to be adopted for calculating and updating the scaling gains of the wavelet based speed controller. A self-tuning fuzzy logic is proposed for calculating the scaling gains of the wavelet based speed controller and it is presented in the next chapter.

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