VSB – Technical University of Ostrava
Faculty of Electrical Engineering and Computer Science
Department of Electronics
Application of Sliding Mode Control in Induction Motor Drive
PHD THESIS-SUMMARY Ing. Chau Si Thien Dong
Supervisor: Prof. Ing. Pavel Brandstetter, CSc.
Ostrava, December 2017
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Abstract
Induction motors, as well as electrical drives, are widely used in industry applications but they consume a large number of electrical energy in the world. Energy saving, torque fast response and speed accuracy are main area in controlling induction motors. Together with the development of semiconductor technologies and digital signal processing (DSP), software instruments have been used to estimate speed, reducing hardware complexity and cost of a mechanical speed sensor. However, due to the nonlinearity, high order and multivariable properties of induction motor dynamics, the development of advanced induction motor control is still a challenging task.
In this research proposal, basic description of the torque and flux control, as well as the theory and application of sliding mode algorithms are reviewed in details. From that, a sliding mode control algorithms for speed control is proposed to implement the pulse width modulation with a constant switching frequency. In addition, the sliding mode observer for speed estimation is investigated. The parameter sensitivity of the observer and controller are analyzed. Furthermore, the robustness of control and observer algorithms are also proved by Lyapunov’s criterion. Simulation models and control structures in MATLAB – Simulink environment are developed to verify the performance of the proposed algorithms. Finally, the experimental work in an induction motor drive controlled by eZdspTMF28335 is presented to compare with theoretical assumptions.
Key words: induction motor, sliding mode control, sliding mode observer, Lyapunov’s theory, vector control, rotor resistance estimation, rotor resistance estimation, sensorless control.
Dissertation Thesis Objectives
Applications of sliding mode control in induction motor drives are the main objectives of this dissertation thesis. Expected usages of sliding mode algorithms for the speed control and speed estimation of the induction motor are illustrated. The control algorithms will be tested on the laboratory stand with the induction motor drive. To achieve the work objectives, the following specific tasks were determined:
Theoretical analysis of the sliding mode control.
Choice of suitable methods for the speed estimation and speed control of the AC drive with induction motor.
Creation of simulation models and the simulation of the control structures in the Matlab- Simulink software environment.
Choice of the proper control microcomputer system with a digital signal processor.
Development and realization of the sliding mode control algorithms.
Design of the laboratory stand with the induction motor drive for the experimental verification of the chosen control algorithms.
Experimental measurements and evaluation of laboratory model properties and their comparison with theoretical assumptions.
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Content
ABSTRACT ... II DISSERTATION THESIS OBJECTIVES ... II CONTENT ... III
1. INTRODUCTION ... 1
2. SLIDING MODE OBSERVER IN CONTROL OF INDUCTION MOTOR DRIVES ... 2
2.1 Sliding Mode Observer ...2
2.2 Robust Sliding Mode Observer...5
2.3 Simulation Results ...6
2.3.1 Speed Control of Induction Motor Drives ... 6
2.3.2 Sensorless Control of Induction Motor Drives ... 7
2.3.3 Parameter Sensitivity ... 8
2.3.4 Robust Sliding Mode Observer ... 9
3. SLIDING MODE CONTROL OF INDUCTION MOTOR DRIVES ... 14
3.1 Sliding Mode Controller Design ...14
3.2 Gain Adaptation Sliding Mode Controller ...15
3.3 Simulation Results ...16
3.3.1 Sliding Mode Control of Induction Motor Drives ... 16
3.3.2 Sensorless Sliding Mode Control of Induction Motor Drives ... 17
4. EXPERIMENTAL RESULTS ... 19
4.1 Speed Control of Induction Motor ...19
4.2 Speed Control of Induction Motor with Load...20
4.3 Sensorless Speed Control of Induction Motor ...20
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4.4 Sensorless Speed Control of Induction Motor with Load ... 21
4.5 Sliding Mode Control of Induction Motor ... 21
4.6 Sliding Mode Control of Induction Motor Drives with Load ... 22
4.7 Sensorless Sliding Mode Control of Induction Motor ... 22
4.8 Sensorless Sliding Mode Control of Induction Motor with Load ... 23
5. CONCLUSION ... 24
REFERENCES ... 26
LIST OF OWN PUBLICATIONS ... 31
LIST OF PROJECTS ... 33
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1. Introduction
Induction motors, a type of AC electrical drives, are robust and sturdy. Because of their advantages, induction motors are widely used in industry such as pumps and fans, paper and textile mills, locomotive propulsion, electric vehicles, machine tools, heat pumps, rolling mills, wind generation systems, etc. [1] – [33], [73], [85]. However, AC drives with induction motors have complex relationships between frequency and stator voltages implemented by frequency converters, complex dynamics of AC machines, machine parameters’ variation, and signal processing of feedback signals.
As a result, controlling of induction motor drives is very complicated.
Among these control methods, the field - oriented control (FOC) or the vector control, was originated in Germany at the end of 1960s [18]. In FOC, the stator or rotor currents are presented by vectors [2] – [6], [8] –[11], [14], [15], [19] – [25], [28] – [30], [34] – [39], [74], [78], [79], [81], [86] – [88], [92]. This control is based on projections which transform a three – phase time and speed dependent system into a two co-ordinate time invariant system. This projections lead to a structure similar to that of a DC machine. Because of advantages of FOC, FOC is applicable to both induction and synchronous motors. In FOC control, classic PI controllers are normally used in control loops.
However, the control performance depends on the induction motor parameters and unknown disturbances and controller parameters are chosen by “trial and error” [2], [36]. Once motor parameter and speed reference change, the performance is not good [20], [21], [23], [25] – [27], [32], [44] – [46], [77]. To improve the control performance, many advanced control methods have been developed, such as back – stepping control, neural network [30], [45], [50], [51], fuzzy logic [9], [22], [33], [36], sliding mode control [2] – [4], [8], [9], [14], [15], [22], [25], [26], [28], [30], [31], [33] – [35], [38], [39], [45], [50], [52] – [59], [79], [88]. Among these control methods, the sliding mode control proves the superior performance when parameter variations and external disturbances occur. Although sliding mode control has some disadvantages, combined methods have developed during years to overcome these disadvantages [9], [11], [14], [15], [22], [25], [28], [30], [31], [33], [35], [36], [38], [39], [45], [50], [56], [57], [59], [76], [79] – [81], [90] – [92].
In controlling induction machine drives, stator voltages and currents, rotor fluxes and mechanical speed are required. However, sensors, especially flux sensor, increase the complexity and high cost of hardware and decrease reliability of the system. In practice, rotor flux is estimated from stator currents and voltages by a current model or a voltage model and the rotor speed is estimated by an observer, called sensorless control of induction motor drives [2], [3], [5] – [7], [9], [13], [17], [19] – [21], [23], [24], [26], [27], [35], [37] – [39], [42] – [48], [74], [75], [78] – [80], [83], [86], [89]. Popular methods can be listed as model reference adaptive system (MRAS) [27], [29], [36], [61], [63], [64], [75], [84], [89], extended Kalman filter (EKF) [2], [36], [39], [65], sliding mode observer (SMO) [2], [5] – [8], [13], [17], [19] – [21], [23], [25] – [27], [32], [35], [36], [38], [42] – [47], [74], [86] and Luenberger observer (LO) [66] – [68]. Among these sliding mode observers, MRAS and EKF observers are easy to design but exact model parameters of AC drives are needed. They are suitable for applications with a medium speed. In contrast, the Luenberger observer and sliding mode observer are robust to parameter variation and noise and can work well at low speed and very low speed ranges. Moreover, some other methods are proposed to increase the robustness of the system [36], [40], [42], [43], [46], [49] – [51], [60], [62], [64], [69], [82].
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2. Sliding Mode Observer in Control of Induction Motor Drives
2.1 Sliding Mode Observer
In this section, a sliding mode observer is proposed. The robustness of the proposed sliding mode observer is also discussed in detail in this section. From that, the adaptation algorithm for the rotor speed is derived.
The sliding mode observer is based on the knowledge of an induction motor model. A mathematical description of induction motor by a state space can be rewritten as in Equation (2.1)
x C y
u B x A x
(2.1)
Where x iS iS R RT is the state vector, u uS uST is the input vector, yis the output vector, and A, B, and C are state matrix, input matrix, output matrix, respectively. Matrixes A, B, C are defined in Equations (2.2) and (2.3).
44 43 42 41
34 33 32 31
24 23 22 21
14 13 12 11
a a a a
a a a a
a a a a
a a a a
A ;
1 0 1 0 1
0 0 0 0 LS
B ;
0 0 1 0
0 0 0
C 1 (2.2)
2 2
11 22 2
m R R S
S R
L R L R a a
L L
; a12a210;
13 24 2
m R
S R
L R
a a
L L
; 14 23 m R
S R
a a L
L L
31 42 m R; 32 41 0; 33 44 R; 34 43 R
R R
L R R
a a a a a a a a
L L
(2.3)
The sliding mode observer can be given as in Equation (2.4) [74], [86]
ˆ ˆ ˆ ( )
ˆ ˆ
sign
x A x B u G S y C x
(2.4)
The gain matrix is defined as
3
42 41
32 31
22 21
12 11
g g
g g
g g
g g
G (2.5)
Firstly, the sliding surface S is defined as in Equation (2.6) ˆ
ˆ ˆ
S S
S S
i i
i i
S y y (2.6)
The first derivation of the sliding mode surface is x C x C y y
Sˆ ˆ (2.7)
ˆ ˆ sign ˆ
S C A x A A x G y y (2.8)
The positive definite Lyapunov function as follows
V aR
2
2 ~
2
1 S (2.9)
It is clearly that the Lyapunov function in Equation (2.9) is a positive definite function. The time derivation of Lyapunov function Equation (2.9) is written by
T 1
R R
VS S a (2.10)
In order to satisfy the global asymptotic stability, the time derivation of Lyapunov function, V must be a negative definite function. Thus, the estimated rotor speed is updated by using PI controller as in Equation (2.11)
ˆR KP z KI z dt
(2.11)
ˆ
ˆ
ˆ
ˆ S S R S S R
z i i i i (2.12)
And the gain matrix Gis chosen as:
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1 2
2 2
2
3 2 2 2
2 2
4 2 2 2
2 0
2 m S R R R m S R R R S b m
m m R b R
b R R R m S S R
R b R m
g b
g
L R R w L
L L L
g L b R L L L
L L R w L
b w L R b L L R b L L
g
R w L L
(2.13)
By choosing appropriate values of b, the desired performance of sliding mode observer can be derived. In our simulation, b and wbare chosen to be -0.52 and 1000.
The block diagram of vector control combined with the sliding mode observer is proposed in Figure 2-1.
Figure 2-1. Control structure of induction motor drive with SMO
In the proposed block (Figure 2-1), the current sensors are used to measure two phase currents of the induction motor drive iSa and iSb. Then, they are transformed into two components of the stator current vector in stator coordinate system iS,iS. These current components are the inputs of the sliding mode observer (SMO), which estimates the rotor speed ˆm. They are also inputs of a current model, which estimates rotor flux components ˆR,ˆR, magnetizing current magnitude ˆimand synchronous position ˆ . The current components iS,iS are also used to calculate the current components in the rotor coordinate system iSd,iSqand the current components in the oriented coordinate system iSx,iSy. The estimated speed replaces the real speed in PI control with encoder.
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2.2 Robust Sliding Mode Observer
As presenting in section 2.1, sliding mode observer is based on induction motor parameters.
Traditionally, parameters of an induction motor can be derived from manufacturers. Practically, parameters, like resistances, vary in a rather wide range. For example, both stator and rotor resistances increase linearly with temperature, depending on the temperature coefficients of the resistance material.
Especially, stator and rotor resistances vary so much at low speed range. In this thesis, variations of inductances are bypased. However, an online identifiation method for both stator and rotor resistance at the same time is proposed. The range of stator and rotor resistance variation is from -50 % to +50 % of their nominal values.
The stator and rotor resistances are estimated online by
ˆR R PRr Rr IRr Rr
R R K z K
z dt (2.14)ˆS S PRs Rs IRs Rs
R R K z K
z dt (2.15)2 2
2 ˆ 2 ˆ 2 ˆ 2 ˆ
1 ˆ 1 ˆ
m m m m
Rr S S R S S R
S R S R S R S R
Rs S S S S
s s
L L L L
z i i i i
L L L L L L L L
z i i i i
L L
(2.16)
The block diagram of vector control combining with the sliding mode observer as well as resistance estimation can be proposed in Figure 2-2.
Figure 2-2. Control structure with SMO and resistance estimation
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In Figure 2-2, rotor and stator resistances are estimated from the actual stator current components iS, iS , the estimated stator current components ˆiS, ˆiS and the estimated rotor flux components ˆR and ˆR of rotor flux vector in the stator coordinate system [ , ] , which are estimated by the current model. After that, these stator and rotor resistances are updated in the sliding mode observer to estimate the stator current components ˆiS, ˆiS as well as the rotor speed ˆm.
2.3 Simulation Results
2.3.1 Speed Control of Induction Motor Drives
In this subsection, traditional control method of induction motor drive is presented, with PI controllers and speed sensor.
Figure 2-3. Simulation result: Speed control Speed
In Figure 2-3, at first, when the reference speed is 0, the actual speed is 0, too. Then, the reference speed rise up to 100 rpm, the actual speed approach to reference speed after 0.1 sec. Moreover, at the point where the load torque is changed, there are some spikes of 5.5 rpm (at the time of 0.3 sec, 0.6 sec in clock wise rotation and at the time of 1.4 sec and 1.7 sec in reverse rotation). The maximum overshoot is about 4 rpm (percentage of overshoot is 4%, compared to the reference speed of 100 rpm).
From the above results, the transient characteristics is very good although the reference speed is rather small (from 50 rpm to 100 rpm). The transient characteristics depends on the parameters of PI controllers.
To evaluate the robustness of PI controllers to external noise, the amplitude of the load torque increases up to the nominal torque 14.8 Nm. When the load torque is increased to the value of 12 Nm, the same responses are done. However, at the time load torque occur (from 0.3 s to 0.6s in forward rotation and 1.4 s to 1.7 s in reverse rotation), the peaks increase up to 14.5 rpm and once in the rated load torque (14.8 Nm), the system is under control.
White noise with amplitude of about 5% of the reference speed (100 rpm), step time 0.001 s is added to measured speed to simulate the measurement noise of encoder. When measurement noise exists, there is an oscillation in speed response with amplitude of 6 rpm. Besides that, the peak is up to
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8 rpm at load torque added. In summary, the performance characteristics of speed control is very good under nominal conditions (all parameters of induction motor are nominal values). In the presence of load torque and external noise, the response of speed is acceptable.
2.3.2 Sensorless Control of Induction Motor Drives
Figures (from Figure 2-4 to Figure 2-6) show the estimated speed and actual speed which are compared to the reference speed in PI control of induction motor with the proposed sliding mode observer.
Figure 2-4. Simulation result: Sensorless control with nominal values Reference speed and estimated speed
Figure 2-5. Simulation result: Sensorless control with nominal values Reference speed and actual speed
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Figure 2-6. Simulation result: Sensorless control with nominal values Absolute speed error between actual speed and estimated speed
From Figure 2-4 to Figure 2-6, it is clearly that the speed response of PI control with encoder is nearly the same with PI control with encoder. In Figure 2-4, the absolute error between the reference speed and the estimated speed is 0.8 rpm when having load and 0.5 rpm with no load. These absolute error is rather small. With this estimated speed, the absolute error between the reference speed and the actual speed is higher, 2 rpm when having load and 1 rpm with no load. Results of the simulation also show good agreement between the estimated speed and the actual one, which the absolute error is maximum at 1.3 rpm (1.3% in term of a relative error, compared to reference speed of 100 rpm) (Figure 2-6). It can imply that the proposed sliding mode observer demonstrates very good performance. To verify the performance of sliding mode observer, some simulation results are presented as follow.
If sign function in Equation (2.4) is replaced by saturation function with bound of 0.1, it is easy to realize that the relative settling error between the estimated speed and the actual speed is nearly 0 rpm. The relative settling error is about 0.1 rpm at the time when load torque occur (from 0.3s to 0.6s and from 1.4s to 1.7s). It is really smaller than the error in Figure 2-6. However, relative error at the time of increasing or decreasing motor speed is higher, nearly 1 rpm. To compare the performance of sliding mode observer, the Luenberger observer is also simulated. The results show that response of Luenberger observer is similar to the response of sliding mode observer with saturation function.
Through this subsection (subsection 2.3.2), simulation results shows that the proposed sliding mode observer has good response under nominal condition and can be used in speed estimation in low speed. The saturation function is used to reduce the relative settling error between the estimated speed and the actual speed. The response of Luenberger observer are similar to sliding mode observer when using the same gain parameters. The effects of parameter variations, such as rotor and stator resistance, are considered in next subsection (subsection 2.3.3)
2.3.3 Parameter Sensitivity
In section 2.2, clearly, both stator and rotor resistances, inductances affect characteristics of sliding mode observer. Variation of stator and rotor resistances and inductiances increase overshoot, settling time and settling error, even make the system unstable. In practice, stator and rotor resistances vary in a wide range value, linearly with temperature, depending on the temperature coefficients of the
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resistance material [6, 20, 26, 27, 45, 46, 62, 63, 69]. Thus, in this thesis, simulations are done under assumption that the stator and rotor resistances vary 30% of their value, the stator and rotor inductances are nominal values. In case of increasing stator and rotor resistances to 30 % of their nominal value, the transient characteristics of speed is worse: the percentage of overshoot increases, the settling error is very large (up to 18 rpm), and the settling increases, too. When the stator and rotor resistances decrease, the system becomes unstable.
2.3.4 Robust Sliding Mode Observer
In this part, stator and rotor resistances are estimated by using Equations (2.14) and (2.15). When the stator and rotor resistances increase or decrease to 50%, the resistance estimation work well. The PI parameters for stator resistance estimation areKP Rs_ 0.002, TI Rs_ 0.001, PI parameters for rotor resistance estimateion are KP Rr_ 0.021, TI Rr_ 0.001. Simulation results are presented in Figure 2-7 to Figure 2-11 for the case that the resistance decrease 50%.
Figure 2-7. Simulation result: Robust sensorless control with decreased resistances of 50%
Reference speed and actual speed
Figure 2-8. Simulation result: Robust sensorless control with decreased resistances of 50%
Reference speed and estimated speed
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Figure 2-9. Simulation result: Robust sensorless control with decreased resistances of 50%
Absolute speed error between actual speed and estimated speed
Figures from Figure 2-7 to Figure 2-9 demonstrate the responses similar to figures from Figure 2-4 to Figure 2-6 when the rotor and stator resistances of induction motor are nominal values. The error of speed estimation is about 1 rpm. In this case, the response of speed is similar to nominal case, the absolute error between the reference speed and the actual speed is about 2 rpm with load and 0.5 rpm without load.
Figure 2-10. Simulation result: Robust sensorless control with decreased resistances of 50%
The stator resistance estimation response
Figure 2-10 shows the response of stator resistance, the real value of resistances are 50% of the nominal value. Thus, in Figure 2-10, stator resistance, which starts from the nominal value of 2.59 Ω, down to actual value after the settling time of 0.1 s. An error estimation of the stator resistance is about 0.015 Ω.
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Figure 2-11. Simulation result: Robust sensorless control with decreased resistances of 50%
The rotor resistance estimation response
Figure 2-11 shows the rotor resistance estimation. Similar to stator resistance estimation, the rotor resistance decreases from the nominal value of 1.7178 Ω to the actual value of 0.8589 Ω. The error of rotor estimation is about 0.01 Ω in positive speed and 0.03 Ω in reverse speed. Next, when both stator and rotor resistances increase 50% from the nominal values. Simulation results show very good property (figures from Figure 2-12 to Figure 2-16).
Figure 2-12. Simulation result: Robust sensorless control with increased resistances of 50%
Reference speed and actual speed
Figure 2-13. Simulation result: Robust sensorless control with increased resistances of 50%
Reference speed and estimated speed
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Figure 2-14. Simulation result: Robust sensorless control with increased resistances of 50%
Absolute speed error between actual speed and estimated speed
The speed response of the induction motor is nearly similar to response in PI control with sliding mode observer under nominal condition.
Figure 2-15. Simulation result: Robust sensorless control with increased resistances of 50%
The stator resistance estimation response
Figure 2-16. Simulation result: Robust sensorless control with increased resistances of 50%
The rotor resistance estimation response
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In the above figures, the stator and rotor resistances start from the nominal value and approach to the actual values. The settling time is about 0.1 s and the absolute error between the actual resistance and the estimated resistance of stator and rotor resistance are about 0.01 Ω and 0.02 Ω (percentage of 0.2% and 0.77%), respectively.
In a summary, the stator and rotor estimation works rather well, the estimation error is about 0.02 Ω for stator resistance and 0.04 Ω for rotor resistance. With these errors, the transient response of speed is still good. In the cases that the resistance changed up to 80%, the stator and rotor resistance estimation are not good, the error of speed increases up to 5 rpm.
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3. Sliding Mode Control of Induction Motor Drives
3.1 Sliding Mode Controller Design
In this section, a step-by-step design method for the integral sliding mode control is presented.
Moreover, the stability of sliding mode controller is proved by using Lyapunov’s stability theory. From that, the parameters of sliding mode controller are derived.
The SMC design consists of two steps, designing a sliding surface S in the state space to represent the desired system dynamic and designing desired control vectors. Firstly, the sliding mode surface is defined as [70], [79], [88]
*
1
*
x m m x m m
S
i i dtc i i (3.1)
*
1
*
y m m y m m
S
dtc (3.2)To prove the stability of the proposed sliding mode control with the Lyapunov’s stability theory, the Lyapunov function is chosen as
2 2
1
2 x y
V S S (3.3)
The Lyapunov function in Equation (3.3) is a positive definite function. The first derivation of Equation (3.3) is in following equation.
x x y y
VS S S S (3.4)
Depending on the Lyapunov’s stability criterion, the system is asymptotic stable if the first derivation of Equation (3.4) is a negative definite function or
x x y y 0
VS S S S (3.5)
To satisfy Equation (3.5), the first derivation of Sx and Sy are chosen as
i i ( )i
S K sign S (3.6)
The first derivation of Equation (3.1) is in following equation.
*
1 m
*
1 1
x m m x m m x Sx m
R
S i i c di i i c i i
dt T
(3.7)
And the first derivation of Sxis chosen such as
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*
1 1
( )x m m x Sx m x x
R
S i i c i i K sign S
T
(3.8)
So, the the reference flux current component is derived as
* *
1 1
( )
R R
Sx m m m x x
x x
T T
i i i i K sign S
c c
(3.9)
Similarly, the first derivation of Sy in Equation (3.2) is
*
1
* '
1
*
1
1 1
( )
L
y m m y e L
m m y T Sy L
m m Sy T y y
S c T T
J
c K i T
J
w i d K sign S
(3.10)
By maintaining the amplitude of the rotor flux at a fix value, let KT' KTR and
' 1
y1 T
c K
w J
, L
y1
T L
d c T
J , the desired torque current component iSy is written by
*
*
1 1
( )
m m y
Sy y
i K sign S
w w
(3.11)
In our simulation, Kx and Ky are chosen as Kx7 Ky 3, cx10.09, cy10.0172
3.2 Gain Adaptation Sliding Mode Controller
An adaptive method for Kx and Ky which are changed to external disturbance and parameter variations, is proposed as follows
* 0
R
x x m
R R
K K R i
R R
(3.12)
0 y1
y y L
K K c T
J (3.13)
The sliding mode control structure of induction motor, together with sliding mode observer and rotor and stator resistance estimation, is described in the following figure (Figure 3-1)
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Figure 3-1. Sliding mode control structure
3.3 Simulation Results
In order to demonstrate the effectiveness of the proposed control system, two analyses are done:
the first is conventional PI speed control, the second is proposed integral sliding mode technique. The conventional PI speed control in section 2.3 of chapter 2. Thus, in this section, the simulation results of integral sliding mode technique are shown. The simulations are done in following cases. Firstly, the sliding mode control with speed sensor under nominal condition is shown in subsection 3.3.1. The sliding mode control with sliding mode observer for speed estimation is presented in subsection 3.3.2.
3.3.1 Sliding Mode Control of Induction Motor Drives
Figure 3-2. Simulation result: Sliding mode control with saturation function Actual speed
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In Figure 3-2, the response of speed in sliding mode control is better than PI control with encoder, shown in the speed transient characteristics. The settling time is about 0.5 s, compared to 1 s in PI control. The percentage of overshoot is just 1.5 % (compared to 4%), the settling error with load is 0.3 rpm (compared to 0.5 rpm) and the settling error without load is 0.2 rpm (compared to 0.3 rpm).
Especially, when having load torque, in PI controller, some spikes of 6 rpm occur but in sliding mode control, the value of these spikes is about 2.2 rpm. So, sliding mode control is more robust to external noise (the load torque) than PI control. It is shown more clearly in case of load torque of 12 Nm. With this value of load torque, in PI control, some spikes of 14 rpm occur but in sliding mode control, the peak of these pulses is at least 6 rpm.
Some of simulation results are done with a traditional sliding mode control, which was stated in [52]. In traditional SMC, the time response of the torque current component is about 0.12 s. Thus, the time response of the speed is about 0.12 s, with almost no overshoot. The peak of the current components in the proposed SMC is higher than in the traditional SMC and as a result, the settling time of speed response in the proposed SMC is shorter than in the traditional SMC. In traditional SMC, the highest difference between reference speed and actual speed is 1 rpm. Among three controllers, including PI controller, traditional SMC and ISMC, the settling time in the proposed SMC is the smallest (0.05 s), the traditional SMC is largest (0.12 s). Both TSMC and ISMC are more robust to external noise than PI controller.
3.3.2 Sensorless Sliding Mode Control of Induction Motor Drives
In this subsection, speed estimation from sliding mode observer is used. In case of nominal parameters of induction motor, the transient response of the system, which is controlled by sliding mode controller with the estimated speed, are nearly the same to the transient response of the system with speed sensor.
Figure 3-3. Simulation result: Sensorless sliding mode control Estimated speed
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Figure 3-4. Simulation result: Sensorless sliding mode control Actual speed
Figure 3-5. Simulation result: Sensorless sliding mode control Speed error between the actual speed and the estimated speed
The response of speed is very well, the absolute error between the actual speed and the estimated speed is maximum value of 0.2 rpm (percentage of 0.2%, compared to the reference speed of 100 rpm).
When the stator and rotor resistances increase 50% and the SMO are used, the response of the proposed SMC is similar to PI controller.
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4. Experimental Results
A laboratory stand with an induction motor drive is designed for the experimental verification of the chosen control algorithms. The experimental platform is shown in Figure 4-1. In this thesis, experiments just focus on the sliding mode observer with PI controllers and sliding mode controllers.
[71], [72]
Figure 4-1. Experimental structure
4.1 Speed Control of Induction Motor
In this section, PI control with encoder are presented. The experimental results show a good response in controlling of the induction motor, the time response of speed is 0.1 s with the overshoot of 1.5 rpm (the percentage overshoot of speed is 1.5% at reference speed of 100 rpm). In addition, the absolute average settling error is about 0.5 rpm (0.5%, compared to reference speed of 100 rpm). The osciallation of torque is about 0.5 Nm and the oscillation of the current component is about 0.5 A.
Compared to simulation result, the time response is equal, the absolute settling error is larger (0.5 rpm compared to 0.3 rpm) but the percentage of overshoot is smaller (1.5% compared to 4%).
Figure 4-2. Experimental result: Speed control Speed
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4.2 Speed Control of Induction Motor with Load
In this section, some experimental results for speed control of induction motor using encoder with load are done. These results are similar to results in simulation. When having load torque, the torque current component iSy increases and the motor torque follows the load torque with oscillation of 0.6 Nm. At that time, the speed slows down 98 rpm but the controller drive the motor to desired value again (100 rpm) . In addition, the absolute settling error between the reference speed and the actual speed with load is higher than the absolute error without load (0.8 rpm compared to 0.5 rpm if having no load).
Figure 4-3. Experimental result: Speed control with load Speed
4.3 Sensorless Speed Control of Induction Motor
In this section, PI control with sliding mode observer is done. The response of speed is shown in following figure.
Figure 4-4. Experimental result: Sensorless control Speed
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The sliding mode observer works rather well, the estimated speed follows the reference speed with the absolute settling error is 1 rpm (a percentage of 1 %, compared to the reference speed of 100 rpm). The overshoot percentage of estimated speed is 9 % (compared to the reference speed of 100 rpm).
Because of the error of estimated speed, the settling error between the reference speed and the actual speed (from encoder) increases to 2 rpm, compared to 0.5 rpm when controlling induction motor with encoder. Compared to simulation results, the experimental results are worse, the settling time, absolute settling error and the percentage of overshoot are larger.
4.4 Sensorless Speed Control of Induction Motor with Load
When having load torque, PI control with sliding mode observer still work well: the overshoot percentage is 17%, the settling time is about 1 s and the settling absolute error is 2 rpm. Compared to PI control with encoder, these values are larger (response characteristics of PI control with encoder are 4%, 0.25 s and 1 rpm, respectively). Moreover, when having load, the decreasing of load is larger than in PI control with encoder (3 rpm compared to 2 rpm). And clearly, these parameters are larger than simulation results. In simulation, the setting time, the percentage of overshoot and the absolute error between the reference speed and the actual speed are 4%, 0.1s and 0.5 rpm, respectively.
Figure 4-5. Experimental result: Sensorless control with load Speed
4.5 Sliding Mode Control of Induction Motor
The speed of sliding mode control with encoder is shown in Figure 4-6. Compared to the same situation in subsection 4.1, in sliding mode control, it is clearly that the absolute error between the reference speed and the actual speed is smaller (0.3 rpm compared to 0.5 rpm), the percentage of overshoot is also decreased 0.86% (compared to 1.5%). From these results, the sliding mode control is obviously better than PI controller
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Figure 4-6. Experimental result: Sliding mode control Speed
4.6 Sliding Mode Control of Induction Motor Drives with Load
In this section, sliding mode control of induction motor drive is tested with load. The experimental result is shown in following figure.
Figure 4-7. Experimental result: Sliding mode control with load Speed
Compared to the speed response of PI control in subsection 4.2, the speed response of sliding mode control is better. In SMC controllers, the settling time is smaller (0.1 s compared to 0.25 s), the percentage of overshoot is smaller (1% compared to 4%, with reference speed of 100 rpm), and when having load torque, the speed reduces just 0.8 rpm (compared to 2 rpm in PI control). So, it can be said that the sliding mode control is more robust than PI controller.
4.7 Sensorless Sliding Mode Control of Induction Motor
The experimental results of sliding mode control with sliding mode observer are obtained in this section. The advantage of sliding mode control shows in response of speed. The estimated speed is rather good, the absolute error between the reference speed and the estimated speed is just 0.5 rpm (compared
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to 1 rpm in PI control). So the response of actual speed is better in PI control with sliding mode observer.
The settling time is smaller (0.7 s compared to 1 s) and the settling error is smaller, too (under 1 rpm compared to 2 rpm in PI control).
Figure 4-8. Experimental result: Sensorless sliding mode control Speed
4.8 Sensorless Sliding Mode Control of Induction Motor with Load
In this section, the sliding mode control with sliding mode observer still works well with load.
Although speed response is worse than having no load but still better than PI control with sliding mode observer with load.
Figure 4-9. Experimental result: Sensorless sliding mode control Speed
In above figure, the settling time is just 0.7 s (compared to 1s), the percentage of overshoot is 9% and the settling error between the reference speed and the actual speed is 1.5 rpm (larger than in no load 1 rpm and smaller than PI control using SMO with load 2 rpm). At the point of having load, the speed just goes down 1.5 rpm, smaller than PI control under the same condition
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5. Conclusion
In this thesis, some basic theories about the sliding mode observer and sliding mode control are presented. Furthermore, the methods of sliding mode controller and sliding mode observer for the induction motor are also developed. In chapter 3, the author presents an adaptive sliding mode observer, which is combined between a sliding mode observer and a stator and rotor resistance estimation. The algorithms for estimating stator and rotor resistance, as well as rotor speed, are derived from Lyapunov theory. So the stability of algorithms are believable. In chapter 4, an integral sliding mode control is proposed. It proves over advantages compared to traditional sliding mode controller such as a better transient characteristics and more robust to the load torque. An adaptive gain for sliding mode controller is also proposed.
Moreover, a simulation models and simulation of the control structures in the MATLAB/Simulink software environment as well as a laboratory stand with induction motor are built for the evaluation theoretical assumptions.
In simulation, firstly, PI control with encoder and with sliding mode observer are done. The robust of PI control with encoder is also tested with measuring noise and load torque. Sliding mode observer is checked robust to induction motor parameter by changing the stator and rotor resistances.
After that, the estimation algorithms for stator and rotor resistances are checked. Next, simulations are repeated with sliding mode control with or without encoder. The simulation results shows that PI control depends on parameter of induction motor and external noise. Choosing PI parameter depends on trial – and – error methods and knowledge of the system. Sliding mode observer, as well as resistance estimation algorithm, work rather well. Under nominal condition, PI control with sliding mode observer gives the same results in PI control with encoder. When having stator and rotor resistance variation, using estimation algorithms give good response of speed. The estimation algorithms can estimate resistances which are changed up to 50 % of their nominal values. During PI control, parameters of PI controllers are chosen by trial – and – error method. It depends so much on the knowledge of the system.
Next, similar to PI control, the sling mode control are verified with encoder and with sliding mode observer. In both cases with encoder and with sliding mode observer, the sliding mode control shows advantages over PI control, such as a better transient characteristics and robust to the load torque.
With and without encoder, sliding mode control has lower settling time, lower settling error and lower percentage of overshoot. When having load, the speed response in sliding mode control change lower than in PI control.
Finally, the laboratory stands are used to verify the simulation results and theoretical assumptions. The experimental results are similar to simulation results. The response of stator current vector components, rotor flux vector components have the same amplitude and frequency under the same condition. In experiments, the parameters of PI control differ from parameters of PI control in simulation. It can be explained that the parameters of the induction motor drive in practice is not the same with nominal parameters, which is given by the manufacturers and offline estimation. Parameters of sliding mode controller are easier to choose.
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In experiments, PI parameters need to be changed over the error between the reference speed and the actual speed to get better response. Especially, when control with the sliding mode observer, response of the actual speed is very low compared to control with encoder. The speed needs to be estimated as fast as possible and the controllers are fast, too. However, if it is too fast, the speed oscillates around the settling point and the settling time is too long. In sliding mode control with sliding mode observer, the settling time is shorter but is still much longer than in sliding mode control with encoder.
In the future, the works will focus on online estimation of parameters of induction motor drives.
Besides that, adaptive parameters of PI controllers and sliding mode controllers are interesting subjects.
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