Indirect Vector Controlled Induction Motor with four Hybrid P+Fuzzy PI Controllers J.A. Cortajarena, J. De Marcos
P. Alvarez, F.J.Vicandi, P.Alkorta
EUITI UPV-EHU EIBAR, SPAIN Email:
[email protected],
[email protected]
EUITI UPV-EHU EIBAR, SPAIN Email:
[email protected],
[email protected],
[email protected]
Abstract— In this paper, a new high performance induction motor drive is presented. The induction motor is controlled with four proportional plus fuzzy PI controllers (P+FUZZY PI). This hybrid controller replaces the conventional PI controllers traditionally used for indirect vector control of induction motors. The hybrid indirect vector control using the fuzzy controllers offers enhanced performance both in mathematical simulations and during actual test utilizing a 7.5 kW induction motor. The results demonstrate the superior performance and robustness of the fuzzy logic controller over the conventional controller when there are mismatched motor parameters. Notably, the performance of the fuzzy logic controller is retained when a new different motor replaces the test motor.
I.
This paper shows the application of an indirect vector control with four P+fuzzy PI controllers. First the simulation process will be validated against the real system. After that the fuzzy and conventional vector controlled by PIs will be compared by means of simulation. It will be shown that motor control performance using the fuzzy controllers is at least as good as conventional controllers, and is more robust. II. CONTROL STRUCTURE AND MOTOR DYNAMICS A schematic diagram of the induction motor indirect vector control with the P+fuzzy PI controllers is shown in Fig 1.
INTRODUCTION
An induction motor is normally controlled using traditional PI or PID devices. In practice these conventional controllers are often developed via crude system models that satisfy basic and necessary assumptions before being tuned by using established methods. These techniques are traditionally solved using a mathematical model of the motor with fixed parameters. However, in a real system a machine’s nonlinearities and parameter variations degrade the system performance over the full range of motor operation and in extreme conditions this can lead to instability [1]. To solve this problem the controller parameters have to be continuously adapted. This adaptation can be achieved using different techniques such as MRAC or model reference adaptive control [2], sliding mode [3], or self tuning PIDs [4]. For some of these techniques the motor parameters and load inertia must be calculated in real time, so there is a high processing requirement for the used processors. In the conventional controller design process, heuristics enter into the implementation and tuning of the final design. Consequently, successful controller design can in part be attributable to the clever heuristic tuning of a control engineer. An advantage of fuzzy control is that it provides a method of manipulating and implementing a human’s heuristic knowledge to control such a system [5]. Because the fuzzy logic approach is based on linguistic rules, the controller design does not need to use any motor parameters to make a controller adjustment, so the controller robustness is high [6].
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Fig.1. Indirect FOC with P+PI fuzzy speed controller.
The configuration of the drive consists of four P+fuzzy PI controllers, the speed controller, the torque producing stator current controller, the flux controller and the flux producing stator current controller. The decoupling unit in a current controlled VSI (voltage source inverter) is required to produce the appropriated stator voltages [1], and the flux weakening block is included to increase the response speed beyond the nominal speed. The mathematical equations for a 3 phase, Y connected squirrel cage induction motor in a general rotating reference system are d\ s ,dq (1) v s ,dq Rs is ,dq jZ g\ s ,dq dt
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0
Rr ir ,dq
\ s ,dq \ r ,dq
d\ r ,dq
Ls i s ,dq
j Z g Z r \ r ,dq dt Lm ir ,dq
Lr ir ,dq Lm i s ,dq
(2) (3) (4)
Te
3 Lm P \ rd I sq \ rq I sd 2 Lr
Te TL
J
(5)
d Zm BZm dt
(6)
Where: dq , DE - Axis of the generic reference system and of the system fixed to the stator. v s ,dq , i s ,dq , \ s,dq - Stator voltage, current and flux vectors.
ir ,dq ,\ r ,dq - Rotor current and flux vectors.
Z r , Z g , Z s , Zm
- Rotor electrical, generic reference
system, slip and rotor mechanical speed. Lm , Ls , Lr - Mutual, stator and rotor inductances.
Rs , Rr - Stator and rotor resistances. Te , TL - Motor and load torque. J , B - Inertia of the system and friction coefficient.
1 § · (8) Kp ¨ en en 1 enTs ¸ Ti © ¹ The controller output is an increment to the control signal. It is an advantage that the controller output is driven directly from an integrator, as it is then is easier to deal with windup and noise [8]. The fuzzy PI controller output, CU2, is called the change in output, and U2 is defined by, (9) U 2 n ¦ cu 2i GCU 2 Ts 'u n
The measured motor speed is compared with the reference speed to get a speed error and a rate of change of the speed error. The speed error is then the input of a P controller and the error and the change in error are the inputs of the fuzzy PI controller. The output of the P+fuzzy PI controller will generate the torque producing stator current component command iq*. The flux controller generates the flux producing stator current component id* according to the flux-speed profile. Both currents are the input of two controllers to produce the stator voltages in the synchronous reference and then transformed to the stationary reference system to generate in the inverter the voltage vector for the motor. III. FUZZY LOGIC CONTROL
saturation level. So, as the error becomes smaller the integral action gains in importance as does the proportional action of the fuzzy PI controller. This second proportional action is used for fine tuning and to correct the response to sudden reference changes, helping the proportional controller when needed. As a convention, signals are written in lower case before gains and upper case after gains. The gains are mainly for tuning the response but since there are three gains, they can also be used for scaling the input signal [7]. In the fuzzy PI controller the error and change in error are the inputs to the rule base. Because the inputs of the fuzzy controller are the error and change of error it is useful to configure it as an incremental controller. This incremental controller adds a change to the current control signal of ǻu. (7) un un 1 'un
STRUCTURE
The proposed P+fuzzy PI structure can be seen in Fig.2
i
Where Ts is the sample or control period. The adder will integrate only if the output is inside the limits. (10) cu 2 n f GE en , GCE cen The function f is the fuzzy input-output map of the fuzzy controller. If it were possible to take the function f as a linear approximation, considering equation (9), the gains related to the conventional PI would be, Kp GCE GCU (11) 1 GE GCE Ti The overall controller output is: OUTn LIM H if U 1 U 2 t LIM H (12)
OUTn or,
LIM L
OUTn
KP en ¦ f GE en , GCE cen GCU 2 Ts
if n
U 1 U 2 d LIM L (13)
i
Fig.2. Proposed P+fuzzy PI.
The proportional gain KP is able to make the fast corrections when a sudden reference change occurs. There is a sustained error whilst the system is in steady state so, integral action is necessary, for which a fuzzy PI is used as can be seen in Fig.2. When the error is large, if the controller tries to obtain a larger output value than the limits, the integral action will remain in pause until the correction level drops below the
if the value of U1+U2 is inside limits. The proportional gain hence depends on GCE and GCU and the integral action of the ratio between the two input gains. The final value of the proportional and integral gain will depend on the fuzzy membership, and the rules. The integrator windup is limited according to the maximum value of the P and fuzzy PI controllers. The limit value will be the maximum in each controller of: the torque producing stator current controller for the speed loop; the flux producing stator current controller for the flux controller; and the maximum permissible voltage for the current controllers.
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Fig.4. Surface plot of the fuzzy PI speed controller. Fig.3. Membership functions for error change of error and fuzzy PI output.
For a practical implementation of the fuzzy controllers on a DSP the fuzzification and defuzzification membership functions are triangular and trapezoidal types. The number and position of the membership functions were selected after a number of different simulations, because there are no straightforward procedures for the tuning of fuzzy controllers. Fig 3 details the fuzzy PI controller input and output membership functions. The fuzzy rule based matrix for the PI fuzzy controller is a 49 rule table (Table I) and the fuzzy inference used is Mandani type using max-min composition [9]. The results of the fuzzy inference are defuzzificated using the centre of gravity or the centre of area equation [7] [10], equation (14). n
¦ z P(z ) k
z *COG
k
(14)
k 1 n
¦ P(z ) k
k 1
Where P ( zk ) are the k = 1,2,…n sampled values of the aggregated output membership function. The start values of the constants were set close to values for conventional PIs. The initial membership functions where then configured to obtain a linear input-output fuzzy function [8]. Subsequently, the constants and the positions of the membership functions for the inputs and outputs were adjusted using multiple Matlab simulations in order to obtain the best performance [11]. TABLE I FUZZY PI RULE BASE WITH 49 RULES
Fig.5. Surface plot of the conventional PI speed controller.
The surface plot of figure 4 belongs to the fuzzy PI speed controller, while figure 5 belongs to a conventional PI speed controller. The fuzzy surface for the torque and flux current controllers and the flux controller have the same form but different scale according to the adjusted gains. As can be seen by comparing the two plots, for the PI controller there are differences in the change of output against the change of error that result in a non linear surface characteristic. IV. EXPERIMENTAL RESULTS Simulations of the full system were carried out using various operating conditions and parameter mismatching. The real system is based on a DS1103 board (Fig. 6). The board controls the inverter and the chopper generating the SVPWM pulses for the inverter [12]. The speed is measured with a 4096 impulse encoder via a FPGA connected to the DS1103 using the multiple period method [13]. The board was programmed whit Matlab/Simulink. This gave full access to the programming variables and allowed them to be visualised and modified. The ratings and parameters of the induction motor used for the real system and simulation are shown in Table II. As mentioned previously, the adjustments to the gain and membership functions and rules were finalised after conducting simulations. To guarantee that the simulated and real systems were closely matched, a number of different simulations were performed.
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Fig.6. Real system used to test the implemented algorithms. TABLE II RATINGS AND PARAMETERS OF INDUCTION MOTOR
Voltage 380V III-Y Frequency 50Hz Nominal current 14A Rated Torque 50Nm Rated speed 1440r.p.m.
J = 0.038Kg*m2 B = 0.008Kg*m2/s P = 2 pole-pairs Rr = 0.57ȍ Rs = 0.81ȍ Lm = 0.117774H Lr = 0.121498H Ls = 0.120416H
Selected results are shown in figures 7 and 8. The upper of each pair of comparator charts belongs to the real system whilst the lower charts belong to the simulated systems. In figure 7, the speed reference is a square 200 r.p.m. signal. In the acceleration time from -200 r.p.m. to +200 r.p.m. the controller is saturated to 20 A. When the error is close to zero the P+fuzzy PI controllers are working keeping the error very close to zero as can be seen in the close up. As can be seen, the speed response between the real and simulated figures compares favourably. Like, the alpha and beta components of the stator current and the torque producing stator current controller compare favourably. Figure 8 shows the real and simulated signals when the speed reference is sinusoidal. The minimum value is zero when the motor is stopped, and the maximum value is 3000 r.p.m., double of nominal speed. This test is used to check the flux controller and the speed regulation when the motor is working in the flux weakening region. It can be seen how the flux is changing from a nominal 1 Wb to a lower value according to the speed of the motor. The flux reference and the real flux demonstrate a good dynamic for the flux controller. The last graphic in figure 8 shows the real system and simulated iq reference and real values. Both are very similar. These tests validated the simulated system against the real system. They also presented the opportunity for a further study of the comparative robustness when there are induction motor parameters variations. Simulations were performed by varying a number of different motor parameters. For example, a doubling of the rotor or stator resistance from its nominal value slightly degraded the speed response but it was still good whit booth the fuzzy or the conventional PI controller.
Fig.7. Real (top) and simulated (bottom) signals whit a square speed reference.
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Fig.9. Left fuzzy controllers. Right conventional PI controllers when the rotor and stator leakage inductances increase.
In figure 9 the speed response is shown when the rotor and stator leakage inductance is three times bigger than the nominal value. The left figure shows the good performance for motor working whit the proposed fuzzy controllers. In contrast, the right-hand chart shows a marked degradation of performance when the motor is working with conventional PIs. The conventional control could be improved by further adjustment of the again. Another situation where the fuzzy control is superior to the conventional controller is when the motor is replaced with a new or different model. Table III shows the parameters of the new test motor. The left-hand chart in the figure 10 shows the response of the new motor without making any changes to the fuzzy controller. The right-hand chart in the figure shows the results for the conventional controller. The conventional system controlled by PIs has not been readjusted, so as expected, the system is unstable and the motor speed increases without control. In marked contrast, the motor working with the fuzzy controller exhibits good speed control performance. TABLE III RATINGS AND PARAMETERS OF INDUCTION MOTOR 2
Voltage 380V III-Y Frequency 50Hz Nominal current 3.5A Rated Torque 15Nm
J = 0.058Kg*m2 P = 2 pole-pairs Rr = 1.522ȍ Rs = 2.229ȍ Lm = 0.2385H Lr = 0.2497H Ls = 0.2444H
Rated speed 1450r.p.m.
Fig.10. Left, fuzzy controllers. Right, conventional PI controllers. New motor replaces the old one.
Fig.8. Real (top) and simulated (bottom) signals in flux weakening region.
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The simulations and real tests presented demonstrate that effective and robust induction motor control was achieved using a hybrid P+fuzzy PI controller. V. CONCLUSION In this paper, a hybrid P+fuzzy PI controller for indirect vector control of an induction motor was proposed. Four hybrid controllers replaced conventional PIs and simulations were carried out to validate the system and to make final adjustments. It has been shown that system parameters are no required for implementation of the proposed hybrid controller and the drive performance characteristics were robust even after testing the controller on a different motor. ACKNOWLEDGMENT The authors wish to thank to the Programa Red Guipuzcoana de Ciencia, Tecnología e Innovación, for their support of this research.
Fig.11. Motor speed and stator Į and ȕ currents after a load step of 30Nm
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Fig.12. Motor speed, stator Į and ȕ currents and iq reference and real values when there is noise in current measurement.
Fig. 11 shows a good speed regulation using the fuzzy PI controller when a sudden load change occurs. In Fig. 12 the satisfactory speed regulation can also be seen when the measured currents have high frequency white noise. The speed regulation performance is still good, even if the noise is in the speed measurement signal.
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