16th international conference on Sciences and Techniques of Automatic control & computer engineering - STA'2015, Monastir, Tunisia, December 21-23, 2015
STA'2015-PID3889-SMC
Second order sliding mode observer based on twisting algorithm for induction motor Wajdi Hammouda, Taoufik Ladhari, Faouzi M’sahli 0
Abstract—This work is devoted to the synthesis of an observer for induction motor. Indeed, most control laws require a precise knowledge on model at each sampling time. However, the inaccessibility of some states and large variations of some parameters such as the stator and the rotor resistances, during induction motor operating, represent major challenges for performances of this system. In this sense, a second order sliding mode observer based on twisting algorithm is proposed for recovering induction machine internal states. This observer presents the interest of second order sliding mode in terms of convergence in finite time and robustness against parameters variation. Simulation results show performances of this observer. Keywords— Induction motor; Second order sliding mode observer; Twisting algorithm.
I. INTRODUCTION Most control laws of induction motor (IM) require an accurate knowledge of its state variables and model parameters. In practice, the access to these variables using sensors increases the complexity and cost of installation. Moreover, the problem of the inaccessibility of some states and the unobservability of the IM in some operating modes make the operation of measurement very hard. For these reasons, several methods have been developed and have attracted the large attention for observing and estimating some variables of IM such as mechanical speed and rotor flux. Among the observation techniques used, we can cite the Luenberger observer [1], the high gain observer [2], the Kalman filter [3], the model reference adaptive system (MRAS) observer and adaptive observers [4] [5], techniques based on the signal injection and neural networks [6] and sliding mode observer (SMO) which is proposed by Utkin in [7]. These techniques have attracted the most attention for observing the flux and speed. Each technique has its advantages and disadvantages. Compared to other observers, sliding mode technique has some benefits such as the convergence in finite time and the robustness against parameters variation. But, its drawback is the chattering effect which is characterized by high frequency W. HAMMOUDA, Research Unit of industrial systems Study and renewable energy (ESIER), National Engineering School of Monastir, Avenue Ibn ElJazzar, 5019 Monastir, University of Monastir, Tunisia (e-mail:
[email protected] ). T. LADHARI, Research Unit of industrial systems Study and renewable energy (ESIER), National Engineering School of Monastir, Avenue Ibn ElJazzar, 5019 Monastir, University of Monastir, Tunisia (e-mail:
[email protected]) F. M’SAHLI, Research Unit of industrial systems Study and renewable energy (ESIER), National Engineering School of Monastir, Avenue Ibn ElJazzar, 5019 Monastir, University of Monastir, Tunisia (e-mail:
[email protected])
978-1-4673-9234-1/15/$31.00 ©2015 IEEE
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oscillations around the sliding surface. This commutation can solicit actuators and sensors dynamics, during the synthesis of the control law, affecting system performances [8]. To overcome this problem the high order sliding mode (HOSM) technique, based on some convergence algorithms, has been designed ([9]-[13]). Indeed, this technique reduces this phenomenon of chattering which has been always an obstacle to practical applications [14][15] and it assures the convergence of estimated components in finite time. In the present paper we present a design of a second order sliding mode observer (SOSMO) for an IM test bench, followed by a test of robustness against parameters and load torque variation. The rest of paper is organized in the following manner: Section II presents the induction motor model in a fixed frame. Section III is devoted to the synthesis of a SOSMO based on twisting algorithm for IM. After that, we will look at the presentation of some simulation results in section IV. Finally, this work is fenced by conclusions in section V. II. MODEL OF INDUCTION MOTOR (IM) The design of different mathematical models of IM allows the analysis of the evolution of its electromechanical variables and also the development of control and observation algorithms. The equivalent two-phase model of induction motor, under assumptions of linear magnetic circuits and balanced operating conditions, is presented in the fixed (α, β) frame [16]: 𝑑𝑖𝑠𝛼 𝐾 1 = −𝛾𝑖𝑠𝛼 + 𝜑𝑟𝛼 + 𝑝Ω𝐾𝜑𝑟𝛽 + 𝑣 𝑑𝑡 𝑇𝑟 𝜎𝐿𝑠 𝑠𝛼 𝑑𝑖𝑠𝛽 𝐾 1 = −𝛾𝑖𝑠𝛽 − 𝑝Ω𝐾𝜑𝑟𝛼 + 𝜑𝑟𝛽 + 𝑣 𝑑𝑡 𝑇𝑟 𝜎𝐿𝑠 𝑠𝛽 𝑑𝜑𝑟𝛼 𝑀 1 = 𝑖𝑠𝛼 − 𝜑𝑟𝛼 − 𝑝Ω𝜑𝑟𝛽 𝑑𝑡 𝑇𝑟 𝑇𝑟 𝑑𝜑𝑟𝛽 𝑀 1 = 𝑖𝑠𝛽 + 𝑝Ω𝜑𝑟𝛼 − 𝜑𝑟𝛽 { 𝑑𝑡 𝑇𝑟 𝑇𝑟 with, 𝛾=
1 𝜎𝐿𝑠
(𝑅𝑠 + 𝑅𝑟
𝑀2 𝐿2𝑟
),𝜎 = 1 −
𝑀2
𝐿𝑠 𝐿𝑟
, 𝐾=
𝑀 𝜎𝐿𝑠 𝐿𝑟
(1)
and 𝑇𝑟 =
𝑅𝑟 𝐿𝑟
p : number of pole pairs. Ls : stator inductance. Rs : stator resistance. Lr : rotor inductance. Rr : rotor resistance. M: mutual inductance. ( 𝑣𝑠𝛼 , 𝑣𝑠𝛽 ), ( 𝑖𝑠𝛼 , 𝑖𝑠𝛽 ) and (𝜑𝑟𝛼 , 𝜑𝑟𝛽 ) represent respectively stator voltages, stator currents and rotor fluxes.
The mechanical speed Ω, taken into account of the rotor velocity, is written by equation (2) [16]: 𝑑Ω 𝑀 𝑓 1 =𝑝 (𝜑 𝑖 − 𝜑𝑟𝛽 𝑖𝑠𝛼 ) − Ω − 𝑇𝑙 𝑑𝑡 𝐽𝐿𝑟 𝑟𝛼 𝑠𝛽 𝐽 𝐽
Estimation errors of current and flux are given by: 𝑒𝑖 = 𝑖̂𝑠 − 𝑖𝑠 { ̂ 𝑟 − ∅𝑟 𝑒∅ = ∅
(2)
Dynamics of the observation error obtained from (4) and (5) are defined by:
f: viscous friction coefficient. J: inertia. Tl: load torque.
The IM is considered as a nonlinear system presented in the following form: 𝑋̇ = 𝑓(𝑋, 𝑈)
(3)
Where 𝑓: 𝑋 × 𝑈 → ℝ𝑛 represent a continuous function, 𝑋 ∈ ℝ𝑛 and 𝑈 ∈ ℝ𝑚 are respectively the state and input system. This work has an objective the design of a SOSMO based on twisting algorithm for an IM. This observer is devoted to recover unavailable system state (rotor flux). The outputs supposed to be measured are the speed and stator currents.
𝑒̇𝑖 = 𝐾𝐴𝑒∅ { 𝑒̇∅ = 𝑀𝛼𝑟 𝐼2 𝑒𝑖 − 𝐴𝑒∅ + 𝛤
In this section, our interest focuses on the synthesis of a SOSMO based on twisting algorithm for an IM. It is supposed that the induction motor has a mechanical speed sensor, so we consider only electromagnetic dynamic. The stator current and the rotor flux equations of system (1) can be rewritten in the frame(α, βas follows
𝑆 = {𝑋 ∈ ℝ𝑛 / S = Ṡ = 0}
The region S is considered as a sliding surface which represents a relationship between the system state variables [17]. So one defines a sliding surface for IM as follow: 𝑆(𝑋) =
𝑇
𝑇
where 𝑖𝑠 = [𝑖𝑠𝛼 𝑖𝑠𝛽 ] ∈ ℝ2 , ∅𝑟 = [∅𝑟𝛼 ∅𝑟𝛽 ] ∈ ℝ2 𝑝Ω 1 ], 𝐼2 = [ 𝛼𝑟 0
(9)
A is supposed an invertible matrix. The derivative of the surface S is given by equation (10): 𝑆̇(𝑋) =
1 −1 𝑑(𝐴−1 ) 𝐴 𝑒̇𝑖 + 𝑒𝑖 𝐾 𝑑𝑡
(10)
can consider that:
𝑑(𝐴−1 ) 𝑑𝑡
= 0.
𝑆̇(𝑋) =
1 −1 𝐴 𝑒̇𝑖 = 𝑒∅ 𝐾
(11)
If we derive again this expression we obtain the form given by the following equation:
1 1 0 ], 𝑏 = and𝛼𝑟 = 𝜎𝐿𝑠 𝑇𝑟 1
𝑆̈(𝑋) = 𝑒̇∅ = 𝑀𝛼𝑟 𝐼2 𝑒𝑖 − 𝐴𝑒∅ + 𝛤
The observer given below, without considering the changes of parameters, is a copy of the original system with addition of a discontinuous correcting term Γ: 𝑑𝑖̂𝑠 ̂ 𝑟 + 𝑏𝑣𝑠 = −𝛾𝐼2 𝑖𝑠 + 𝐾𝐴∅ 𝑑𝑡 ̂𝑟 𝑑∅ ̂ { 𝑑𝑡 = 𝑀𝛼𝑟 𝐼2 𝑖̂𝑠 − 𝐴∅𝑟 + Γ
1 −1 𝐴 𝑒𝑖 𝐾
The dynamic of sliding surface becomes:
𝑋 = [𝑖𝑠 ∅𝑟 ]𝑇 ∈ ℝ4 and 𝑈 = [ 𝑣𝑠𝛼 𝑣𝑠𝛽 ] ∈ ℝ2 are the state and input of IM. 𝛼𝑟 −𝑝Ω
(8)
It is supposed that the dynamic of the angular speed Ω is constant compared to dynamics of current and flux then we
𝑑𝑖𝑠 = −𝛾𝐼2 𝑖𝑠 + 𝐾𝐴∅𝑟 + 𝑏𝑣𝑠 { 𝑑𝑡 𝑑∅𝑟 = 𝑀𝛼𝑟 𝐼2 𝑖𝑠 − 𝐴∅𝑟 𝑑𝑡
𝐴=[
(7)
To generate a second order sliding mode the state system must be confined, in a finite time, to a set appropriately selected defined by:
III. SOSMO DESIGN FOR IM
𝑇
(6)
Assuming that errors estimations 𝑒𝑖 and 𝑒∅ converge asymptotically to zero, given a time t0 such that: ∀ 𝑡 > 𝑡0
𝑇 ̂ 𝑟𝛼 ∅ ̂ 𝑟𝛽 ]𝑇 are considered as 𝑖̂𝑠 = [𝑖̂𝑠𝛼 𝑖̂𝑠𝛽 ] and ̂∅𝑟 = [∅ current and flux estimated components, in the stationary frame (α, β), which constitute the estimated state vector ̂ 𝑟 ]𝑇 . 𝑋̂ = [𝑖̂𝑠 ∅
Γ = [Γ1 Γ2 ]T is the gain to be conceived.
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(12)
|𝑀𝛼𝑟 𝐼2 𝑒𝑖 − 𝐴𝑒∅ | < 𝑐0
(13)
In this case, we can define the twisting algorithm presented in the following form: −𝜆 𝑠𝑖𝑔𝑛(𝑆) 𝛤={ 𝑚 −𝜆𝑀 𝑠𝑖𝑔𝑛(𝑆)
𝑖𝑓 𝑠𝑠̇ ≤ 0 𝑖𝑓 𝑠𝑠̇ > 0
(14)
Taking account of conditions given in (15) 𝜆𝑚 > |𝑀𝛼𝑟 𝐼2 𝑒𝑖 − 𝐴𝑒∅ |𝑚𝑎𝑥 { 𝜆𝑀 > 𝜆𝑚 + 2|𝑀𝛼𝑟 𝐼2 𝑒𝑖 − 𝐴𝑒∅ |𝑚𝑎𝑥
(15)
2 Flux r (wb)
The trajectories of the system evolve, after a finite time, on 1 the surfaces: 𝑆 = 𝑆̇ = 0, i.e. 𝑆 = 𝐴−1 𝑒𝑖 = 𝑆̇ = 𝑒∅ = 0 𝐾 This leads to estimation errors equal to zero (𝑒𝑖 = 𝑒∅ = 0). The structure of the flux observer is given by the following figure:
r r obs
1 0 -1 -2 0
3 phases
0.1
0.2
0.3
0.4
0.5
Time (s)
̂ 𝑟𝛼 ∅
̂ 𝑟𝛽 ∅
Flux r (wb)
2
Display Flux & current
Flux observer PWM Inverter
𝑢𝑐
𝑢𝑠𝛼 𝑢𝑠𝛽
𝑖̂𝑠𝛼 𝑖̂𝑠𝛽
-1 -2 0
0.1
0.2
0.3
0.4
0.5
Time (s)
To check the robustness of the developed observer against parameters variation, many tests are carried out by modifying some parameters of induction motor such that rotor and stator resistances. The following figures will show the simulation results by increasing and decreasing rotor and stator resistances (±50% of Rs and Rr). We note that small errors have been occurred. These errors will be removed in finite time and the observed flux and currents converge again to the real components.
Twisting algorithm
𝑢𝑏
𝑖𝑠𝛽
Current observer
Park transform
𝑢𝑎
0
Figure 3: Evolution of the real and estimated components of rotor flux r in the fixed frame (
𝛤
𝑖𝑠𝛼
r r obs
1
Ω SOSMO
IM
30 is is obs
current is (A)
3
TL Figure1. SOSMO design of IM
2.8
20
2.6 2.4 2.2 0.49
10 0 0
IV. SIMULATION RESULTS
0.2
0.495
0.5
0.505
0.4
0.51
0.515
0.52
0.6
0.8
1
0.6
0.8
1
Time (s)
0.05
Error (%)
To illustrate some performances of the SOSMO, many simulations are made under Matlab /Simulink using IM parameters listed in table 1. we have testing this observer with nominal conditions. The following curves will show good performances. Indeed, estimated components converge towards real components after a finite time even adding initial conditions (𝑖𝑠0= 1𝐴, ∅𝑟0= 0.5𝑊𝑏). The observer parameters were adjusted to 𝜆𝑀= 30 , 𝜆𝑚 = 10
0
-0.05 0
0.2
0.4 Time (s)
Figure 4: Influence of parameters variation (+50% of Rs and Rr) on the estimated stator current is
1.5
20 0
r r obs
1 1.17 Flux r (wb)
is is obs
Flux r (wb)
Current is (A)
40
0.5
1.16 1.15 1.14 0.5
-20 0
0.1
0.2
0.3
0.4
0 0
0.5
Time (s)
2 is is obs
20
Error (%)
Current is (A)
40
0 -20 -40 0
0.1
0.2
0.3
0.4
x 10
0.4
0.54 0.56 Time (s)
0.58
0.6
0.6
0.8
1
0.6
0.8
1
Time (s)
-3
0 -2 -4 -6 0
0.5
0.2
0.52
0.2
0.4 Time (s)
Time (s)
Figure 2: Evolution of the real and estimated components of stator current is in the fixed frame (
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Figure 5: Influence of parameters variation (+50% of Rs and Rr) on the observed rotor flux r
10
3 2.5 2 0.495
0 0
current is (A)
20
40
is is obs
3.5 current is (A)
current is (A)
30
0.2
0.5
0.505 Time (s)
0.4
0.51
0.515
0.6
0.8
is isobs 20 0 -20 0
1
0.2
0.4
current is (A)
Error (%)
0
-0.05 0
0.2
0.4
0.6
0.8
0 -20 -40 0
1
0.2
0.4
2 Flux r (wb)
1 1.22 1.2 1.18 1.16 0.5
0 0
0.2
0.55
0.4
0.6 0.65 Time (s)
0.6
0.7
0.75
0.8
1
Time (s)
0 -1 -2 0
0.2
0.4
0.6
0.8
1
Time (s) 2
0 -2 -4 -6 0
1
r r obs
1
-3
Flux r (wb)
Error (%)
x 10
0.8
Figure 10: Influence of load torque on the estimated current is
r r obs
Flux r (wb)
Flux r (wb)
1.5
0.5
0.6 Time (s)
Figure 6: Influence of parameters variation (-50% of Rs and Rr) on the estimated stator current is
0.2
0.4
0.6
0.8
1
Time (s)
Figure 7: Influence of parameters variation (-50% of Rs and Rr) on the observed rotor flux r
r robs
1 0 -1 -2 0
0.2
10
0.4
0.6
0.8
1
Time (s)
Figure 11: Influence of load torque on the estimated flux r TABLE I.
Furthermore, we have tested this observer by applying a load torque 𝑇𝑙 which decreases mechanical speed. Also, like previous, this test shows clearly the robustness of the proposed observer. Indeed, we always notice that the estimated components follow real components during all operating phases of the machine. (𝑇𝑙 = 10𝑁. 𝑚 at 𝑡 = 0.5𝑠) Load torque (N.m)
1
is isobs
20
Time (s)
INDUCTION MOTOR PARAMETERS
Rated power Rated voltage Lr Ls M Rs Rr p J f
1.5 KW 230/400V 0.464 H 0.464 H 0.4417 H 5.717Ω 4.282Ω 2 0.0049 kgm2 0.0029 N.m.s/rd
5
V. CONCLUSION 0 0
0.2
0.4
0.6
0.8
1
0.8
1
This work is concerned by a second order sliding mode observer for IM that is based on twisting algorithm. Likewise, this observer performed estimation concerning states which are unmeasured variables. We have provided a good global convergence and stability. Indeed, estimated components convergence in finite time towards real components after addition of initial conditions. Furthermore, this technique demonstrates very good performances, especially its robustness under rotor and stator resistances variation and in front of an application of a load torque. Simulations results prove successfully the validity of this observer which can be implemented in real time.
Time (s)
Figure 8: Load torque
200
speed (rad/s)
0.8
40
0.05
2
0.6 Time (s)
Time (s)
150 100 50 0 0
0.2
0.4
0.6 Time (s)
Figure 9: mechanical speed evolution
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