Proceedings of the 33rd Chinese Control Conference July 28-30, 2014, Nanjing, China
Sliding Mode-Based Observer Design for Field-Oriented Control of Induction Machine Drive for Applications in Hybrid Electric Vehicles Athar Hanif1 , Aamer Iqbal Bhatti2 , Abdul Rehman Yassin1 , Ghulam Murtaza2 , Qadeer Ahmed3 1. Department of Electrical Engineering, The University of Lahore, Lahore 54600, I. R. Pakistan E-mail:
[email protected],
[email protected] 2. Department of Electronic Engineering, Muhammad Ali Jinnah University,Control and Signal Processing Research Group (CASPR), Islamabad 75400, I. R. Pakistan E-mail:
[email protected],
[email protected] 3. Cenetr for Automotive Research (CAR), The Ohio State University, Ohio 43210, America E-mail:
[email protected] Abstract: The feedback (direct) eld-oriented (vector) control is the most commonly adopted instantaneous speed/torque control method for the hybrid electric vehicle’s drive system. The feedback (direct) eld-oriented (vector) control of induction machines primarily depends upon precise ux estimation. However, an accurate estimation of ux is hard due to the deviations in the machine’s electrical parameters. Both the resistances (stator and rotor), which produces the imprecision in the ux estimation and in the generation of unit vectors, increase linearly with temperature, depending upon the temperature co-efcient of the resistance of the material. The unit vectors are used to guarantee correct alignment of stator direct-axis current with the ux vector and stator quadrature-axis current perpendicular to it. This provides the decoupled control as in separately-excited DC machines. Secondly, unit vectors are used for the purpose of control. Therefore, the imprecision of the estimated ux will degrade the performance of speed control. In this paper, a sliding mode based observer is presented subject to machine’s electrical parameters variations. The mathematics and derivations of the proposed sliding mode observer is expressed. The performance and stability of the proposed observer is then worked out through the regulation problem of the 3-φ induction machine used for the implementation of eldoriented control for the application in hybrid electric vehicle.The results of the proposed observer shows the good performance in estimating the actual ux which is necessary for the accurate and exact performance of eld-oriented control in hybrid electric vehicles. The proposed observer is also robust against the variations in rotor and stator resistance. Key Words: Field-oriented control, Hybrid electric vehicles, Induction machine, Rotor ux, and Sliding mode based observer
1 Introduction
tor drive exact knowledge of rotor ux as well as speed is required. Rotor ux and speed is sensitive to the variation in rotor resistance, stator resistance, and load torque. Therefore robust observer is required. There are several methods are available in the existing literature for the estimation of rotor ux and speed. The current model ux estimator [8]-[9]-[10] and voltage model ux estimator [11] are extensively used for the ux estimation using the terminal quantities of the induction machine. But , the performance of current model ux estimator is deteriorated at high speed and the performance of voltage model ux observer is deteriorated at low speed due to the variation in the rotor resistance and stator resistance respectively. A hybrid model ux observer is suggested in [12], to use the current model ux estimator at low speed and voltage model ux estimator at high speed to overcome the problem of rotor and stator resistance variations. But, it does not completely overcome the dependency of observer on stator and rotor resistance. Model referencing adaptive techniques are suggested in [13]-[14]-[15], two ux estimators are considered. One operates as a reference model, and other performs as a adaptive observer. But the exact estimation of ux is still remaining as a problem. Another technique for the ux estimation is Extended Kalman lter is proposed in [16], where only motor terminal quantities are used. However, this technique has some inherent problem of computation expense and it has no detailed design and tuning criterion. Kubota, Matsuse and Nankano [17], used the full-order adaptive Luenberger ux observer to estimate the stator currents and rotor uxes
In order to diminish the emissions to the environment and to reduce the fuel consumption, the automobile industries are decided on the growth of innovative technologies such as hybrid electric vehicle [1]-[2]-[3]. This attractive technology (HEVs) is a combination of Atkinson engine propulsion system and electric machine traction system [4]. These two components of propulsion system are either connected in parallel or in series. Parallel coupling of both (ICE and EM) components is common in hybrid electric vehicles, in which both components delivered repulsion torque to move the vehicle. Therefore in order to achieve the fuel economy and emission reduction, only electric motor provides the traction force for the HEV within city (upto speed 40 Km/h). Electrical machine plays an important role for the traction system of hybrid electric vehicles. Among the various electrical machines are available, induction machine is used for the traction system of hybrid electric vehicle because of reasonable low cost, easier control, better power density and efciency, reliable operation over wide speed range, high initial torque and technological maturity.Induction machines are also very robust, has rugged construction, and require little maintenance [5]-[6]-[7]. In hybrid electric vehicle, to achieve high performance, eldoriented control of induction motor drive is adopted. To implement the effective eld-oriented control of induction moThis work is supported by The University of Lahore, Lahore, 54600, Pakistan .
263
min
with the help of fourth order part of the fth-order induction machine model in stationary reference frame with constant rotor speed. Then, estimation of speed is done through the proportional plus integral formula. Sliding mode has advantages of robustness and parameter invariance and order reduction [18] . Sliding mode observer of Derdiyok et al. [19] is used for the ux estimation with the same portion of induction machine model used in Kubota’s observer. Sliding mode observer proposed by Utkin, Guldner, and Shi [18] is also used for the ux and speed estimation of induction machine. Utkin’s observer is also based on the fourth order part of the fthe-order induction machine model in stationary reference frame. In Verghese and Sander’s ux observer [8], only ux estimation is focused by the authors. A fth-order observer is also proposed in [20], to overcome the parameter’s variation due to the abrupt change in operating conditions. Due to inherent robustness and parameter invariance propoerty of the sliding mode control, the aim of this research is to design the robust sliding mode observer for the applications in hybrid electric vehicles, where parameters variations in fth-order model of induction machine are more prominent than other industrial applications. The objective of the sliding mode based observer is to complete the current loop as well as speed and ux control loop in eld-oriented control of induction machine drive for applications in hybrid electric vehicles. The simulation results show good performance of the proposed robust sliding mode based observer. The rest of the paper is structured as follows. Section II illustrates the HEVs needs for the eld-oriented control of induction machine drive and formulation of control problem. Section III describes the mathematical model of induction machine both in synchronously rotating reference frame (SRRF) and in stationary reference frame (SRF). Section IV describes the proposed observer for the induction machine, its structure after some transformation and stability proof of the proposed observer. Section V elaborates the induction machine drive’s depiction used in simulation as well as simulation results. Concluding comments are presented in section VI.
ids ,iqs ,ωe
3
ieds ieqs e vds e vqs isds isqs s vds s vqs e ψds e ψqs s ψds s ψqs ωe ωr np Rs Rr Ls Lr Lm τe τL B J
direct-axis current in SRRF quadrature-axis current in SRRF direct-axis voltage in SRRF quadrature-axis voltage in SRRF direct-axis current in SRF quadrature-axis current in SRF direct-axis voltage in SRF quadrature-axis voltage in SRF direct-axis ux in SRRF quadrature-axis ux in SRRF direct-axis ux in SRF quadrature-axis ux in SRF stator supply frequency rotor speed no. of pole pairs stator resistance rotor resistance stator inductance rotor inductance mutual inductance electromagnetic generated torque load torque damping inertia
3.1 Model in Synchronously Rotating Reference Frame Induction machine model based on d-q axis coordinate in SRRF is presented in [21] dieds dt
(L2m Rr + L2r Rs ) e Lm Rr e ids + ωe ieqs + ψ σLs L2r σLs L2r dr np Lm ωr e 1 e ψ + v (4) σLs Lr qr σLs ds
=
−
+
dieqs dt
(L2m Rr + L2r Rs ) e Lm Rr e iqs − ωe ieds + ψ σLs L2r σLs L2r qr np Lm ωr e 1 e ψ + v (5) σLs Lr dr σLs qs
=
−
+
ids ,iqs ,ωe
ids ,iqs ,ωe
Mathematical Model
Table 1: Model Description in Induction Machine
In hybrid electric vehicle applications, the key problem is the performance of the electric traction system. Vector control, used for speed and ux control of induction machine, is sensitive to parameter variations. The induction machine parameters changes as the operating condition changes. Operating conditions for HEV propulsion system will change constantly. Trafc situations, driving cycles, etc.are the reason of variation in speed. Also temperature has the effect on parameters, which is inuenced by ambient season, loading, etc..Inspite of all these, induction machine must track the reference ux, which is desired to reduce the energy consumption in HEVs. Also torque demanded by the controller must be exact and efcient. Therefore, HEV constraints for led-orientd control can be formulated as: (1) min τe(t) − τeref (t) ψdr(t) − ψref (t)
(3)
This section describes the time domain model of the induction machine in synchronously rotating reference frame (SRRF) and stationary refernce frame (SRF) based on d-q axis coordinates.
2 HEV Constraints for Field-oriented Control
min
ψqr(t)
e dψdr dt
=
−
Rr e Lm R r e e ψ + np (ωe − ωr )ψqr + i (6) Lr dr Lr ds
e dψqr dt
=
−
Rr e Lm R r e e ψ − np (ωe − ωr )ψdr + i (7) Lr qr Lr qs
dωr dt
(2)
264
=
P (τe − τL − Bωr ) 2J
(8)
variations in ux degrades the performance of eld-oriented control of 3-φ induction machine drive for the applications in hybrid electric vehicles. Because of inherent robustness and invarianve to the uncertainties in the parametrs properties of the sliding mode control make it usefull for the applications like hybrid electric vehicles. Induction machine model represented in equations (10)-(13) is further represented in the suitable form required for the design of the estimator.
In equation(8), the electromagnetic generate torque (τe ) is given as
where σ =1− 3.2
3 P Lm e e e e ( ) (ψ i − ψqr ids ) 2 2 Lr dr qs
=
τe L2m Ls Lr ;
(9)
is the leakage factor
Model in Stationary Reference Frame
To achieve the induction machine model based on d-q axis coordinate in stationary reference frame, sets the stator supply frequency zero in the SRRF model presented in equations (4)-(8) i.e ωe = 0. disds dt
σ α
⎡ ˙s ⎤ ⎡ ψds −ζ s ⎥ ⎢ψ˙qs ⎢ np ω ⎢ ⎥=⎢ ⎣ is˙ ⎦ ⎣ ζγ dr ˙ −np γω isqr
(L2m Rr + L2r Rs ) s Lm R r s iqs + ψ 2 σLs Lr σLs L2r qr np Lm ωr s 1 s ψdr + v (11) σLs Lr σLs qs
=
dt
γ ζ
−
+
−
+ s dψdr dt
=
−
Rr s Lm R r s s ψ − np ωr ψqr + i (12) Lr dr Lr ds
s dψqr dt
=
−
Rr s Lm R r s s ψqr + np ωr ψdr + i (13) Lr Lr qs
2 (L2 m Rr +Lr Rs ) σLs L2 r Lm σLs Lr Rr Lr L2 1 − Lsm Lr 3 P Lm ( ) 2 2 Lr
β
(L2m Rr + L2r Rs ) s Lm R r s ids + ψ σLs L2r σLs L2r dr np Lm ωr s 1 s ψqr + v (10) σLs Lr σLs ds
=
disqs
Table 2: Model Description in Induction Machine
−np ω −ζ np γω ζγ ⎡
ζLm 0 β 0
0 ⎢0 +⎢ ⎣b 0
⎤ 0
s 0⎥ ⎥ . vds s ⎦ vqs 0 b
x˙
Ax + Bvs
=
⎤ ⎡ s ⎤ ψds 0 s ⎥ ⎢ψqs ζLm ⎥ ⎥.⎢ s ⎥ 0 ⎦ ⎣ idr ⎦ β isqr
(16)
(17)
The output equation is dωr dt
=
P (τe − τL − Bωr ) 2J
(14)
y=
In equation (14), the electromagnetic generate torque (τe ) is given as τe
=
3 P Lm s s s s ( ) (ψ i − ψqr ids ) 2 2 Lr dr qs
0 0 1 0 x 0 0 0 1
y
(15)
=
Cx
(18)
(19)
In equation (16) the measurement of the speed, ω is available. Pair (A,C) is observable. As the outputs are needed in the proposed sliding mode observer, therefore a transformation is required in such a way that outputs become visible in the state vector.
x ⇒ n−p (20) Tc = y ⇒ p
4 Description of the Estimator As explained earlier, a sliding mode based observer is desired for the exact estimation of ux vector which necessary for the implementation of eld-oriented control of 3 − φ induction machine drive for the applications in hybrid electric vehicles. Therefore, induction machine model represented in equations (10)-(13) is used to construct the estimator in this segment of paper. The structure of the estimator is motivated from the Utkin’s observer [18]. Parameters that appears in the architecture of the estimator is given in Table II. In the denition of propose observer structure, it is assumed that measurement of the rotor speed (ωr ) is available from the speed sensor. In hybrid electric vehicles, operating conditions are different than industrial applications like trafc situations, driving cycles, and different environmental effects. Due to these effect behaviour of the stator resistance (Rs ) and rotor resistance (Rr ) varies. Therefore, stator resistance (Rs ) and rotor resistance (Rr ) are the uncertain parameters which produces the variations in the ux. In result these
Where n is the number of states and p is the number of outputs. The transformation matrix for the observer used in the application of hybrid electric vehicle is given as ⎡ ⎤ x1 ⎢x2 ⎥ ⎥ (21) Tc = ⎢ ⎣ y1 ⎦ y2 ⎡
1 ⎢0 Tc = ⎢ ⎣0 0
265
0 1 0 0
0 0 1 0
⎤ 0 0⎥ ⎥ 0⎦ 1
(22)
Plant matrices after the transformation are written as ⎡ ⎤ A11 A12 A13 A14 ⎢A21 A22 A23 A24 ⎥ ⎥ Tc ATc−1 = ⎢ ⎣A31 A32 A33 A34 ⎦ A41 A42 A43 A44
4.3
To accomplish the convergence of the proposed sliding mode observer, the sliding surface is choosen as (23)
s = (yˆ1 − y1 ) + (yˆ2 − y2 )
4.1
(24)
Structure of the Induction Machine Model after Transformation
μ = M sign(s)
Structure of the induction machine model which act as a plant for this application is given in equation (24) after the transformation, done using equation (21)-(23). x˙1 (t)
x˙2 (t)
y˙1 (t)
y˙2 (t)
=
A11 x1 (t) + A12 x2 (t) + A13 y1 (t) + A14 y2 (t)
+
B1 v(t)
(28)
Let e(t) = x ˆ(t) − x(t) and ey (t) = yˆ(t) − y(t) be the error in measured and estimated values. The error in d-axis and q-axis rotor ux of the 3-φ induction machine is dened as e1
=
s − ψs ψˆdr dr
(29)
e2
=
s − ψs ψˆqr qr
(30)
= +
A21 x1 (t) + A22 x2 (t) + A23 y1 (t) + A24 y2 (t) B2 v(t)
and
= +
A31 x1 (t) + A32 x2 (t) + A33 y1 (t) + A34 y2 (t) B3 v(t) (25)
The error in d-axis and q-axis stator current of the 3-φ induction machine is dened as
= +
A41 x1 (t) + A42 x2 (t) + A43 y1 (t) + A44 y2 (t) B4 v(t)
Structure of the Observer for 3-φ Induction Machine The structure of the observer for 3-φ induction machine used in the hybrid electric vehicle acts as a propulsion system is written as given in equation (25). =
A11 xˆ1 (t) + A12 xˆ2 (t) + A13 yˆ1 (t) + A14 yˆ2 (t)
+
B1 v(t) + G1 μ
xˆ˙2 (t)
= +
A21 xˆ1 (t) + A22 xˆ2 (t) + A23 yˆ1 (t) + A24 yˆ2 (t) B2 v(t) + G2 μ
yˆ˙1 (t)
= +
A31 xˆ1 (t) + A32 xˆ2 (t) + A33 yˆ1 (t) + A34 yˆ2 (t) B3 v(t) − μ (26)
yˆ˙2 (t)
= +
A41 xˆ1 (t) + A42 xˆ2 (t) + A43 yˆ1 (t) + A44 yˆ2 (t) B3 v(t) − μ
ey1
=
iˆsds − isds
(31)
ey2
=
iˆsqs − isqs
(32)
and
4.2
xˆ˙1 (t)
(27)
In sliding surface, yˆ1 and yˆ2 are the estimated values of the d-axis stator current and q-axis stator current in stationary reference frame(SRF) respectively. y1 and y2 are the actual values of the d-axis stator current and q-axis stator current in stationary reference frame(SRF) respectively. Let the discontinuous control is given as
⎡
⎤ B1 ⎢B2 ⎥ ⎥ Tc B = ⎢ ⎣B3 ⎦ B4
Stability of the Observer
With these denition of errors, the error dynamics are directed through the equations (32)-(34), e˙1 (t)
e˙2 (t)
ey1 ˙ (t)
ey2 ˙ (t)
= +
−A11 e1 (t) − A12 e2 (t) + A13 ey1 (t) (33) G1 μ
= +
A21 e1 (t) − A22 e2 (t) + A24 ey2 (t) G2 μ
= −
A31 e1 (t) + A32 e2 (t) − A33 ey1 (t) μ (35)
=
−A41 e1 (t) + A42 e2 (t) − A44 ey2 (t)
−
μ
(34)
(36)
When pair (A,C) is observable, then the pairs (A11 , A31 ) and (A21 , A41 ) are also observable. In this regards, G1 and G2 can be chosen with pole placement in such a way that eigenvalues are in the left-half plane. Another transformation is required to regularize the error system.
I G Tˆ = n−p (37) 0 Ip
In equation (25), xˆ1 and xˆ2 are the estimated values of the d-axis rotor ux and q-axis rotor ux in stationary reference frame (SRF) respectively. x1 and x2 are the actual values of the d-axis rotor ux and q-axis rotor ux in stationary reference frame (SRF) respectively.G1 and G2 are the gains of the sliding-mode estimator. μ is the discontinuous control.
266
The transformation matrix, Tˆ ,for the particular case of induction machine is given as ⎡ ⎤ 1 0 G1 0 ⎢0 1 0 G2 ⎥ ⎥ Tˆ = ⎢ (38) ⎣0 0 1 0⎦ 0 0 0 1
díaxis rotor flux (actual and measured)
díaxis rotor flux (webers)
0.02
If M is large enough then ey1 → 0 and ey2 → 0, we are left with A˜ =
TˆATˆ−1
0.01
0
í0.01
í0.02 0
(39)
qíaxis rotor flux (webers)
0 í0.01
í0.03 0
0.1
0.2
Time [seconds]
0.3
0.4
Fig. 2: Simulated results of measured and actual q-axis rotor ux for different initial conditions for the machine and estimator Error in actual and measured díaxis rotor flux
0.02
error: díaxis rotor flux
0.01 Error (webers)
After the substitution of numerical values into equation (16) yields ⎡ ⎤ −8.5106 −2ω 3.7447 0 ⎢ 2ω −8.5106 0 3.7447 ⎥ ⎥ A=⎢ ⎣ 137.1678 32.2344ω −198.0828 ⎦ 0 −32.2344ω 137.1678 0 −198.0828
0 í0.01 í0.02 í0.03 0
⎤
0 0 ⎥ ⎥ 0 ⎦ 17.2161
0.1
0.2
Time [seconds]
0.3
0.4
Fig. 3: Simulated result of error for d-axis rotor ux
rotor ux by utilizing the measurements of the stator currents and the rotor speed. Efcient and exact rotor ux is necessary for the implementation of eld-oriented control of induction machine for the applications in hybrid-electric vehicles. Simulation study is carried out to investigate the performance of the designed sliding mode based estimator and it does very good job.
Simulation study of the designed estimator for the induction machine for the applications in hybrid-electric vehicle is carried out with different initial conditions for both the plant (machine) and estimator.Figure-1 and 2 shows the estimated ux tracks the actual ux within 0.25 seconds. The errors between the estimated and actual uxes are shown in gure-3 and 4.
6
0.01
í0.02
2 8Ω 4Ω 0.47 H 0.47 H 0.44 H 0.01 N.m.sec.rad−1 0.124 kg.m2
0 ⎢ 0 B=⎢ ⎣17.2161 0
0.4
actual measured
0.02
Table 3: Model Description in Induction Machine
⎡
0.3
qíaxis rotor flux (actual and measured)
After the completion of the estimator’s design, performance is measured with the help of computer simulation using a 3 − φ, 4-pole, 50 Hz induction machine whose parameters values are presented in Table III [21].
and
0.2
Time [seconds]
0.03
Induction Machine Drive’s Depiction Used In Simulation
np Rs Rr Ls Lr Lm B J
0.1
Fig. 1: Simulated results of measured and actual d-axis rotor ux for different initial conditions for the machine and estimator
With the help of appropriate values of G1 and G2 , the eigenvalues of A˜ can be placed in the left-half plane. This ensures that the estimation error goes to zero.This proves the stability of the observer.
5
actual measured
Acknowledgment
Conclusion
The authors would like to acknowledge The University of Lahore, Lahore, Pakistan for providing nancial support for this research work, and also acknowledge all the members of control and signal processing research group (CASPR) of Muhammad Ali Jinnah University (MAJU), who contributed
In this work, control problem is formulated for the application of hybrid-electric vehicle. A time domain model of 3−φ induction machine in both SRRF and SRF is presented. A sliding mode based estimator is designed to estimate the
267
Error in actual and measured qíaxis rotor flux
0.04
[14] H. Tajima and Y. Hori, “Speed sensorless eld-orientation control of the induction machine,” Industry Applications, IEEE Transactions on, vol. 29, no. 1, pp. 175–180, 1993. [15] C. Schauder, “Adaptive speed identication for vector control of induction motors without rotational transducers,” Industry Applications, IEEE Transactions on, vol. 28, no. 5, pp. 1054– 1061, 1992. [16] Y.-R. Kim, S.-K. Sul, and M.-H. Park, “Speed sensorless vector control of induction motor using extended kalman lter,” Industry Applications, IEEE Transactions on, vol. 30, no. 5, pp. 1225–1233, 1994. [17] K. Kubota, K. Matsuse, and T. Nakano, “Dsp-based speed adaptive ux observer of induction motor,” Industry Applications, IEEE Transactions on, vol. 29, no. 2, pp. 344–348, 1993. [18] V. I. Utkin, J. Guldner, and J. Shi, Sliding Mode Control in Electromechanical Systems. London: Taylor and Francis, 1999. [19] A. Derdiyok, M. Guven, H. Rehman, N. Inanc, and L. Xu, “Design and implementation of a new sliding-mode observer for speed-sensorless control of induction machine,” Industrial Electronics, IEEE Transactions on, vol. 49, no. 5, pp. 1177– 1182, 2002. [20] S. Zak, “On the stabilization and observation of nonlinear/uncertain dynamic systems,” Automatic Control, IEEE Transactions on, vol. 35, no. 5, pp. 604–607, 1990. [21] B. K. Bose, Modern power electronics and AC drives. Prentice Hall, 2002.
error:qíaxis rotor flux
0.03
Error (webers)
0.02 0.01 0 í0.01 í0.02 í0.03 0
0.1
0.2
Time [seconds]
0.3
0.4
Fig. 4: Simulated result of error for q-axis rotor ux
in this research work.
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