Abstract: This study deals with the problem of stator currents coupling effect in induction motor drives. In field oriented control, d-axis stator current and q-axis stator current should be regulated independently. However, the stator currents in d-q synchronously rotating reference frame are not naturally decoupled, therefore, a new control scheme for current control is proposed that utilizes Sliding Mode Control (SMC) and Radial Basis Function Neural Network (RBFNN) to achieve the decoupling without considerable chattering. In this control strategy, a RBFNN controller replaces the discontinuous part of the sliding mode controller to eliminate undesired chattering of conventional sliding mode controller. The decoupling method in this study uses two RBFNN sliding mode controllers to regulate d-axis stator current and q-axis stator current, respectively. Finally, simulation results of the proposed scheme have presented perfect performances, such as perfect decoupling, strong robustness and reduced chattering, in comparison with proportional-integral controller and sliding mode controller.
INTRODUCTION
In recent years, much research has been done on the field oriented control for high performance of induction motor drives. It is well-known that the current loop is essential to the overall performance of motor drives. Consequently, fast and precise current control methods are research interests at all times.
In general, the coupling action between d-axis stator current and q-axis stator current is not significant in low speed operation and its effect can be ignored. However, in the high speed operation, the effect of this coupling action becomes so obvious that the performance of motors degrades. Several current control methods have been proposed to regulate stator currents. Some typical control strategies include hysteresis control, synchronous PI control, feedforward control and internal mode control (Holtz et al., 2002; Harnefors and Nee, 1998; Jung et al., 1997). The hysteresis control scheme is easier than other schemes, but the performance of induction motors would deteriorate, due to high harmonic contents of the stator currents. The synchronous PI control scheme can not guarantee the complete decoupling of stator currents dynamics. The feedforward control scheme and internal mode control scheme depend highly on the accuracy of the motor parameters.
Sliding Mode Control (SMC) has many good properties, such as strong robustness, disturbance rejection and easy implementation (Utkin, 1993; Young et al., 1999). Sliding mode control is one type of variable-structure control scheme. Normally, two steps, namely the reaching and sliding phases, are necessary in the design of sliding mode controller. Therefore, the sliding mode controller usually consists of an equivalent law and a switching law. The equivalent law is given so that the states can stay on sliding surface. The switching law is used to drive the state trajectory to the sliding manifold. However, usually the switching law is discontinuous part and the frequency of switching in control system is finite high, so undesired chattering exists in control system.
In this study, a new sliding mode control method that can achieve perfect decoupling is proposed. This method combines the merits of sliding mode control and Radial Basis Function Neural Network (RBFNN), in order to eliminate the undesired chattering. In this scheme, we use two RBFNN-sliding mode controller to regulate the d-axis stator current and q-axis stator current respectively. In addition, RBFNN-sliding mode controller is deeply investigated, in which the radial basis function neural network is employed to replace the discontinuous switching law.
FIELD ORIENTED MODEL OF INDUCTION MOTOR
The dynamic model of induction motor in d-q synchronously rotating reference frame is presented as (Bose, 2002; Lasaad et al., 2007):
| (1) |
In above equations, Rs, Rr, Lm, Ls and Lr denote stator resistance, rotor resistance, mutual induction, stator induction and rotor induction, respectively. Tr is rotor time constant; ωe is motor synchronous angular frequency; ωr is motor rotor electrical angular frequency; np is number of pole pairs; J is rotor moment of inertia; Tl is load torque; usd, usq denote d-axis stator current and q-axis stator voltages; isd, isq denote d-axis stator current and q-axis stator currents; ørd, ørq denote d-axis stator current and q-axis rotor flux linkages. Moreover, the total leakage induction coefficient is σ = 1-Lm2/LrLs.
In order to achieve field oriented control, the d-axis in the synchronously rotating reference frame should be aligned with the rotor flux linkages. Therefore, we have:
| (2) |
where, ør is the amplitude of the rotor flux linkages. Taking Eq. 2 into account, the rotor flux equation and the electromagnetic torque equation can be described as:
| (3) |
| (4) |
| Fig. 1: | Block diagram of induction motor |
where, p is the derivative operator. And the stator voltage equations can be written as (Ma et al., 2005):
| (5) |
| (6) |
Based on the above analysis, a block diagram is reasonably presented in Fig. 1. It is obvious that there is coupling action between d-axis stator current and q-axis stator current of induction motor.
DESIGN OF SLIDING MODE CONTROL
The sliding mode control strategy has been widely used as a nonlinear control method. In general, It utilizes a high frequent switch to drive the phase trajectory towards a predetermined sliding surface. Theoretically, the system states can gradually approximate the sliding mode surface, finally they can approach the control objective. In order to design a sliding mode controller, two essential steps should be carefully investigated, namely, the selection of sliding mode surface and the design of control law.
The selection of sliding mode surface is based on desired motion of the system. Considering the simplicity of design, we define sliding mode surfaces s = [s1 s2] as follows (Ma et al., 2005):
| (7) |
| (8) |
where isd* and isq* are the reference values of d-axis stator current and q-axis stator current. If the system states are on the sliding mode surfaces, equations s1 = 0 and s2 = 0 can be obtained. Substituting Eq. 7 and 8 into s1 = 0, s2 = 0, we can get as follows:
| (9) |
| (10) |
The above equations can ensure that the actual values of the stator currents equate the reference values, if the system states stay on the sliding mode surfaces.
In general, there are several forms of the sliding mode control law. One type of control law consists of an equivalent control law uequ and a switch control law us. The control law can be described as u = uequ+us. The equivalent control law represents the linear part of the control force, which is usually derived from sliding mode surface and the differential of sliding mode surface. So, it highly depends on the parameters of the control model. The switch control law is a discontinuous control law which enforces the system states towards the sliding mode surface. A possible choice of the switching law is us = k*sign(s), where k is a constant, which is used to represent the maximum of switching control law. And function sign(s) is defined as:
| (11) |
In this study, to define the equivalent control law, we assume that the sliding mode surface is constant, i.e.,
| (12) |
From Eq. 5, 6, 9, 10 and 12, we can define the equivalent control laws of d-axis and q-axis stator current controllers, respectively. They can be defined as follows:
| (13) |
| (14) |
where, the rotor flux linkage can be derived from:
| (15) |
The switching control laws of two stator current controllers are designed as follows:
| (16) |
| (17) |
where, ksd and ksq are positive constants, which effect greatly the performance of control system.
Therefore, two stator current controllers can be described as follow:
| (18) |
| (19) |
To guarantee the reachability of sliding mode control, the mathematic form of this condition can be described as:
| (20) |
| (21) |
We can be informed that if a larger ksd or ksq is in switching control law, the speed of reaching sliding mode surface will become faster and the control system has stronger robust property. However, amplified chattering will exist in the sliding phase. On the contrary, a smaller ksd or ksq will slow down the reaching speed and there is reduced chattering. The selection of ksd and ksq is relative to the boundary of uncertainties to keep the trajectory on sliding surface. Taking into account parameter disturbances, since the boundary of these disturbances are often unknown, to trade off the robustness, ksd and ksq are usually chosen to be large enough. Consequently, this conservative control law will enlarge undesired chattering and degrade the performance of overall system.
THE PROPOSED METHOD
The sliding mode controller has a serious drawback that undesired chattering often exists when system states are moving on sliding surface, due to discontinuous switching control law. In order to reduce the system chattering, many methods have been proposed (Wai, 2000; Lin and Hsu, 2004; Lin et al., 2007). In this study, a neural network sliding mode control method is introduced to reduce the undesired chattering phenomenon.
Neural Networks (NN) have the ability to approximate nonlinear functions. A NN can approximate any smooth function to any desired accuracy, provided that the number of hidden-layer neuron is large enough. In addition, Radial Basis Function Neural Network (RBFNN) has an advantage of faster learning ability and less chance of falling into local minimum, in comparison with standard BP neural network (Tsoi, 1989). Therefore, RBFNN is quite suitable to design of the switching control law.
The RBF neural network is a three-layer forward network. The radial basis function which maps from input layer to hidden layer is nonlinear, but the output of RBF neural network is the linear combination of the output of hidden layer. Figure 2 is a block diagram of RBF neural network, which clearly describes the conventional structure of RBF network.
The output of this neural network takes the form as follow:
| Fig. 2: | Block diagram of RBF neural network |
| (22) |
where, ωi (i = 1 to n) is the weighting between hidden-layer neuron i and output-layer neuron, which is adjusted on the basis of an adaptive rule. hi (i = 1 to n) is the output of hidden-layer neuron i.
The output of hidden-layer neurons can be defined as a Guassian function, which can be described as:
| (23) |
where, ci is the central position of Guassian function in hidden-layer neuron i and bi is the spread factor of Guassian function in hidden-layer neuron i, which is a positive constant.
In this study, the RBF neural network sliding mode control is based on the combination of RBFNN and sliding mode control scheme. The conventional switching control law is replaced by the output of RBF neural network, i.e. y = us. The input of RBF neural network is the sliding function s. So the output of hidden-layer neurons can be defined as:
| (24) |
An adaptive rule is used to adjust the weightings for optimal weighting values
in the control operation. Because of the reaching condition of sliding mode
control, the adaptive rule is derived from the steep descent rule to minimize
the value of
| (25) |
| (26) |
where, E is the cost function and η is the adaptive rate.
In order to control the stator currents of the induction motor, two common PI current controllers are replaced by two RBF neural network sliding mode controllers. From above equations, The mathematic form of adjusting weightings in two currents controllers can be derived as:
| (27) |
| (28) |
| (29) |
| (30) |
| (31) |
| (32) |
where, ωisd and ωisq are the weighting between hidden-layer neuron i and output-layer neuron in d-axis stator and q-axis stator current controllers, respectively. ηsd and ηsq denote the adaptive rates in d-axis stator and q-axis stator current controllers, respectively. hisd and hisq are the output of hidden-layer neuron i in d-axis stator and q-axis stator current controllers, respectively. cisd and bisd are the central position and the spread factor of Guassian function in d-axis stator current controller, likewise, cisq and bisq are the central position and the spread factor of Guassian function in q-axis stator current controller. Therefore, the weightings between hidden-layer neurons and output-layer neurons can be adjusted online to achieve the learning ability.
SIMULATION RESULTS AND ANALYSIS
In order to validate the effectiveness of the proposed stator currents control strategy, the simulation of this strategy is performed in MATLAB/SIMULINK software.
The parameters of the induction motor used in this simulation are shown in Table 1. In addition, the overall structure of vector control technique in the induction motor can be shown in Fig. 3. In this simulation, the parameters of PI controllers are initially tuned by the Ziegler-Nichols method, then they are tuned through simulation to get satisfactory response.
| Table 1: | Parameter of induction motor |
| Fig. 3: | Block diagram of the simulation system |
The parameters of RBF neural network in the proposed control scheme are as follows: the adaptive rates ηsd = 0.1 and ηsq = 0.05; the central position vectors csd = [10.5 5.25 2.62 1.31 0 -1.31 -2.62 -5.25 -10.5] and csq = [30 15 7.5 3.75 0 -3.75 -7.5 -15 -30]; the spread factor vectors bsd = [0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8] and bsq = [14 14 14 14 14 14 14 14 14], the initial weightings are zero.
Response to a step change in q-axis reference current: This simulation compares the performance of RBF sliding mode controllers and conventional PI controllers, when a step change in q-axis reference current occurs. In this comparative simulation, the induction motor starts from a standstill state, the motor speed follows a command that starts from zero and begins to accelerate at time t = 1.0s until the rotor speed is 1200 r min-1, the d-axis reference current is 2A. Figure 4a-d show the response of motor using conventional PI current controllers. As may be observed, the step in iq* obviously disturbs the waveform of id at time t = 1.0 sec, when conventional PI controllers are used in current loop. Figure 5a-d show the stator current id is undisturbed under the same conditions, using the proposed control scheme.
Response to a step change in d-axis reference current: We compare the performance of RBF sliding mode controllers and conventional PI controller when a step change in d-axis reference current is applied. The induction motor starts from a standstill state, the d-axis reference current is 2A and has a step change (from 2A to 1A at t = 1.5 sec), the motor speed follows a command that starts from zero and begins to accelerate at time t = 1.0 sec until the rotor speed is 1200 r min-1.
| Fig. 4: | Simulated response to a step change in q-axis reference current using PI controllers. (a) d-axis current response (b) q-axis current response (c) d-axis voltage response and (d) q-axis voltage response |
| Fig. 5: | Simulated response to a step change in q-axis reference current using RBF-sliding mode controllers. (a) d-axis current response (b) q-axis current response (c) d-axis voltage response and (d) q-axis voltage response |
| Fig. 6: | Simulated response to a step change in d-axis reference current using PI controllers. (a) d-axis current response (b) q-axis current response (c) d-axis voltage response and (d) q-axis voltage response |
Figure 6a-d depicts the response of motor using conventional PI current controllers. It is obvious that the step in id* disturbs the waveform of iq at time t = 1.5 sec, when conventional PI controllers are used in current loop. Figure 7a-d depict the stator current iq is undisturbed under the same conditions, using RBF sliding mode control scheme.
Response to a step change in stator resistance: This simulation validates the effectiveness of the proposed control scheme, concerning a step change in stator resistance (from 0.813 to 1.626 Ω at 1.4 sec). The induction motor starts from a standstill state, the d-axis reference current is 2A, the motor speed follows a command that starts from zero and begins to accelerate at time t = 1.0 sec until the rotor speed is 1200 r min-1. It is evident that the proposed control scheme offers a good response with little disturbance in stator current in Fig. 8 and 9.
Comparison of chattering: The objective of this simulation is the comparison of undesired chattering between RBF neural network sliding mode controllers and conventional sliding mode controllers. Conventional sliding mode controllers are used in controlling stator currents in this experiment. The parameters of conventional sliding mode controllers are as: ksd = 110 and ksq = 10. The induction motor starts from a standstill state, the d-axis reference current is 2A, the motor speed follows a command that starts from zero and begins to accelerate at time t = 1.0 sec until the rotor speed is 1200 r min-1.
| Fig. 7: | Simulated response to a step change in d-axis reference current using RBF-sliding mode controllers. (a) d-axis current response (b) q-axis current response (c) d-axis voltage response and (d) q-axis voltage response |
| Fig. 8: | Simulated response of d-axis current to a step change in stator resistance at 1.4 sec |
| Fig. 9: | Simulated response of q-axis current to a step change in stator resistance at 1.4 sec |
Figure 10a, b show little chattering is in RBF neural network sliding mode controllers in current loop. On the contrary, Fig. 11a and b show serious chattering obviously exists in conventional sliding mode controllers.
| Fig. 10: | Simulation results of sliding mode surface using RBF sliding mode controllers. (a) d-axis current sliding mode surface and (b) q-axis current sliding mode surface |
| Fig. 11: | Simulation results of sliding mode surface using conventional sliding mode controllers. (a) d-axis current sliding mode surface and (b) q-axis current sliding mode surface |
It shows clearly that RBF neural network sliding mode design can reduce undesired chattering considerably.
CONCLUSIONS
This study has investigated the coupling problem of stator currents in induction motor. A new method has been presented in this study to achieve the current decoupling, which is based on the combination of sliding mode control and RBF neural network. As with the conventional SMC, the proposed scheme can reduce chattering notably. The simulation results have validated the satisfactory performance of the proposed method, such as perfect decoupling, strong robustness and reduced chattering, in comparison with PI and conventional SMC.
ACKNOWLEDGMENT
Thanks for the support of the National Natural Science Foundation of China (No.60874047) Peng Kang and Zhao Jin developed the concept. Peng Kang performed the simulation using MATLAB software from April, 2009 to March, 2010 in Institute of Automation, Huazhong University of Science and Technology, Wuhan, China