Abstract: In this study, an indirect Field-Oriented Control (IFOC) induction machine drive with a conventional PI and sliding mode controllers is presented. The robustness of ac machine drive speed performance with these controllers is checked in terms of variation of machine parameters. The design includes rotor speed estimation from measured stator terminal voltages and currents. The estimated speed is used as feedback in an indirect vector control system, such that the speed control is performed without the use of shaft mounted transducers. The high performance of the proposed control schemes under load disturbances is studied via simulation cases. The components of the speed controlled indirect field-oriented induction machine with the both controllers are simulated using SIMULINK, while the dynamic of induction machine is simulated using the potential of S-function block and its attached script file.
INTRODUCTION
Field Orientation Control (FOC) or vector control of induction machine achieves decoupled torque and flux dynamics leading to independent control of the torque and flux as for a separately excited DC motor (Denai and Attia, 2002). This is achieved by orthogonal projection of the stator current into a torque-producing component and flux-producing component. This technique is performed by two basic methods: direct and indirect vector control. With direct field orientation, the instantaneous value of the flux is required and obtained by direct measurement using flux sensors or flux estimators, whereas indirect field orientation is based on the inverse flux model dynamics and there are three possible implementation based on the stator, rotor, or air gap flux orientation. The rotor flux indirect vector control technique is the most widely used due to its simplicity (Attia and Denai, 2002). FOC methods are attractive but suffer from one major disadvantage. They are sensitive to parameter variations such as rotor time constant and incorrect flux measurement or estimation at low speeds. Consequently, performance deteriorates and a conventional controller such as a PI is unable to maintain satisfactory performance under these conditions (Denai and Attia, 2002).
A Sliding Mode Control (SMC) is basically an adaptive control scheme that gives robust performance of a drive with parameter variations. The control is nonlinear and applied to linear and nonlinear plant. In SMC, the drive response is forced to slide along a predefined trajectory in a phase plane by a switching control algorithm, irrespective drive`s parameter variation and load disturbance (Edwards and Spurgeon, 2002; Barambones et al., 2002).
Controlled induction motor drives without mechanical speed sensors at the motor shaft have the attractions of low cost and high reliability. To replace the sensor, the information on the rotor speed is extracted from measured stator voltages and currents at the motor terminals. Vector-controlled drives require estimating the magnitude and spatial orientation of the fundamental magnetic flux waves in the stator or in the rotor (Barambones et al., 2002; Bose, 2001).
In this study, two control strategies are considered to adjust the speed of the drive system: PI and sliding mode controller. The robustness of these suggested controllers are checked in terms of motor parameter variations. The speed controller is performed under no mechanical speed sensors and speed observer, based on the software program, is adopted for this purpose.
Dynamic Model of Induction Machine
Induction Machine (IM) equations in arbitrary rotating reference frame
can be represented in stator and rotor dq voltage equations (Bose, 2001;
Leonhard, 1996):
(1) |
where, v is voltage; λ is the flux linkage; i is the current; ω is the arbitrary speed of the reference frame; r is the resistance and p is the time derivative. The subscript r and s denotes the rotor and stator values, respectively referred to the stator and the subscripts d and q denote the dq-axis components in the arbitrary reference frame.
The equations of the machine in the stationary and synchronously rotating reference frame can be obtained from (1) by setting ω to zero and ω = ωe, respectively. To distinguish these two frames from each other, an additional superscript will be used; s for stationary frame variables and e for synchronously rotating frame variables.
The electromagnetic torque equation can be given by (Bose, 2001; Chee-Mun Ong, 1998):
(2) |
where, P denotes the number of machine pole pairs. Using Eq. 1 and 2, one can obtain the state-space model for induction motor developed in stationary reference frame as given below (Bose, 2001; Ouhrouche and Volat, 2000; Akin, 2003):
(3) |
(4) |
where,
| Fig. 1: | Phasor diagram explaining indirect vector control |
Indirect Field Orientation Control (IFOC)
Indirect vector control is very popular in industrial applications. Figure
1 explains the fundamental principle of indirect vector control with
the help of a phasor diagram. The axes are fixed on the stator, but the
dr-qr axes, which are fixed on the rotor, are ds-qs
moving at speed ωr. Synchronously rotating axes de-qe
are rotating ahead of the dr-qr axes by the positive
slip angle θsl corresponding to slip frequency ωsl.
Since the rotor pole is directed on the de axis and ωe
= ωr + ωsl, one can write:
(5) |
The phasor diagram suggests that for decoupling control, the stator flux
component of current
(6) |
(7) |
(8) |
To implement the indirect vector control strategy, it is necessary to
use the condition in Eq. 6 -8 in order
to satisfy the condition for proper orientation. Figure
2 shows an indirect field-oriented control scheme for a current controlled
PWM induction machine motor drive.
| Fig. 2: | Indirect field-Oriented control of a current regulated pwm inverter induction motor drive |
The command values for the abc stator currents can then be computed as follows:
(9) |
(10) |
Motor Speed Calculation
Figure 2 shows the connection of the speed estimator
with IFOC IM. The block calculates the synchronous and rotor speed based
on the measurements of line voltages and currents. Starting from the flux
equations (Denai and Attia, 2002; Barambones et al., 2002):
(11) |
the expressions for
(12) |
Substituting of (12) in the drive voltage equations, Eq. 1, gives:
(13) |
Hence,
(14) |
(15) |
where,
(16) |
Differentiating (16) and substituting (15) leads to the drive speed:
(17) |
Therefore, given a complete knowledge of the motor parameters, the instantaneous speed ωr can be calculated from Eq. 17, where the voltage model of Eq. 14 is used to estimate the rotor flux amplitude.
Sliding Mode Control
With the proper field orientation and with rated rotor flux, the torque
equation, Eq. 6, can be rewritten as (Denai and Attia,
2002):
(18) |
where, KT is the torque constant and is defined as follows:
(19) |
where,
(20) |
where, ωm = (2/P)ωr is the rotor mechanical speed. Using Eq. 18, one can obtain:
(21) |
Where:
a = B/J, b = KT/J, f
= TL/J |
(22) |
Equation 21 can be written with uncertainties Δa, Δb and Δf in the terms a, b and f, respectively, as follows:
(23) |
The tracking speed error can be defined as:
(24) |
where,
(25) |
where, the following terms have been collected in the signal u(t):
(26) |
and the uncertainty terms have been collected in the signal d(t):
(27) |
The sliding variable S(t) can be defined with an integral component as:
(28) |
where, k is a constant gain.
When the sliding mode occurs on the sliding surface of Eq.
28, then
(29) |
In order to obtain the speed trajectory tracking the following assumption should be formulated (Barambones et al., 2002):
| • | The gain k must be chosen so that the term (k-a) is strictly negative, therefore |
(k<0). Then the sliding surface is defined as:
(30) |
The variable structure speed controller is designed as:
(31) |
where, k is the gain defined previously, β is the switching gain, sgn(.) is signum function.
| • | The gain β must be chosen so that β≥|d(t)|. |
The current command
(32) |
Therefore, the proposed variable structure speed control resolves the speed-tracking problem for the induction motor, with some uncertainties in mechanical parameters and load torque.
Modeling of PI and Sliding Mode Control-Based IFOC IM
Referring to Eq. 6, the stator quadrature-axis current
reference
(33) |
The stator direct-axis current reference
(34) |
The rotor flux position θe is generated from synchronous speed ωe integration, which estimated from the estimator block.
Equation 33 and 34 represent the main equations responsible for field oriented-control, which is represented by the Field Oriented block diagram shown in Fig. 3.
The conversion of quantities from dqe to abc reference frames are executed by dq_abc block which is shown in Fig. 4 (Humaidi, 2006).
The quantities in abc reference frame are converted to dqs frame using Fig. 5.
The speed controller with the proportional-integral type can be implemented using Simulink blocks shown in Fig. 6 (Humaidi, 2006).
The modeling of the elements of the sliding mode controller is shown
if Fig. 7.
| Fig. 3: | Inside the block diagram of field oriented control |
| Fig. 4: | Conversion block from dqe to abc |
| Fig. 5: | Conversion block from dqe to abc |
| Fig. 6: | Block of the PI speed controller modeling |
| Fig. 7: | Block of sliding mode speed controller modeling |
The current regulator indicated in Fig. 8 consists of three hysteresis controllers, is built with Simulink blocks. The motor currents are provided by the multiplexer output of the induction machine block (Humaidi, 2006).
The simulation block diagram for a three-phase, two-level PWM inverter is illustrated in Fig. 9 (Humaidi, 2006).
Each leg of the inverter is represented by a switch which has three input terminals and one output terminal. The output of the switch oscillates between (+0.5Vdc) and (-0.5 Vdc), which is characteristic of a pole of an inverter.
(35) |
The SIMULINK modeling of speed observer based on Eq. 17 is illustrated in Fig. 10.
Block diagrams of Fig. 3-10 can
be assembled to yield the block diagram of PI or sliding mode controller-based
IFOC I.M., which is given in Fig. 11.
| Fig. 8: | Current regulator modeling |
| Fig. 9: | Modeling of the two-level PWM inverter |
| Fig. 10: | Inside the speed estimator block |
| Fig. 11: | SIMULINK modeling of PI and Slide Mode Control-Based IFOC IM |
RESULTS
The speed regulation performance of the proposed sliding mode is compared to that of the PI controller of the field oriented control IM. The performance is checked in terms of load torque variations. The simulation is performed in SIMULINK, while the IM model is simulated using s-function block after dicretizing the IM model defined in Eq. 3 and 4 (Humaidi, 2006). The IFOC IM of Fig. 11 is run at sampling period Ts = 2e-6. The IM used in this case study has the parameters listed in Table 1.
In the following study, the variation of the motor parameters is confined to load change only, while the other parameters, e.g., viscous friction B, magnetizing inductance Lm and rotor inductance Lr are held constant.
In the first test, the PI controller is used as a speed controller and a cyclic change of different load torque levels are subjected to the machine at certain times and as follows:
Time = [0 0.75 0.75 1.0 1.0 1.25 1.25 1.5 1.5 2];
Torque = [0 0 -10 -10 -5 -5 5 5 0 0];
The responses of speed, developed torque and stator current are shown
in Fig. (12). It is evident from the figure that the
PI controller shows bad speed response at these applied changes of loads
and lacks the ability to hold the speed at the required value. Therefore,
it is true to say that the PI controller is not robust against the changes
of IM parameters. Retuning of the PI parameters may slightly reduce the
change of speed due to its corresponding change of load.
| Table 1: | Induction motor parameter |
Fig. 12: |
PI controller-Based IFOC |
| Fig.13: | Sliding mode control-Based Induction Machine IFOC Induction Machine |
| Fig.14: | Sliding mode control-based IFOC induction machine when higher levels of torque load are exerted |
In the second test, the sliding mode controller is proposed and a change of ±20 N.m of load applied for a period of time. For this value of load level change β is found to be 8 and the value of k is set to be 100. The value of β limits the value of exerted load, while the value of k determines the speed of tracking according to Eq. 34. The form of this repeated load change can be clarified as follows:
Time = [0 0.75 0.75 1.0 1.0 1.25 1.25 1.5 1.5 2];
Torque = [0 0 -20 -20 0 0 20 20 0 0];
where, the load is exerted in time ranges [0.75 - 1.0] and [1.25 - 1.5]. The speed response of Fig. 13 shows no detected change in the rotor speed at these ranges, meaning that the sliding mode controller does well in rejecting the applied torque; therefore the rotor speed is by now riding over the sliding trajectory and the controller is robust against IM parameter variations.
To which extent the controller can reject the applied load depends on the value β, which has been determined based on the load level. In the above case, the value of β has been calculated to be greater than 8 for the sliding mode controller to reject the load levels less (greater) than 20 (-20). Otherwise the controller begins to lack the ability to reject the values of load level greater than 20 (or less than -20) unless the boundary of β is increased over the previous one (8). The strong evident of this argument is clearly shown in Fig. 14, where the exerted load level exceeded 20 N.m while the value of β is held constant at the value 8. The speed response deteriorates and a change in the rotor speed will appear at time of exerting ±25 N.m. Therefore the boundary of β should be released and should have a new value other than 8 for the controller to manage the new level of load change.
CONCLUSION
From the simulated results the following points can be concluded:
| • | The PI controller is not robust against IM parameter variations and as has been shown the speed response degrades to different level of load changes |
| • | The sliding mode controller can overcome the problem of system degradation, encountered in PI controller, on the condition that the sliding mode controller is well designed for the specified level of load changes |
If the load level exceeds the design specification, degradation in the speed response is observed unless another design procedure is adopted.