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Abstract - This paper presents a new approach for sensorless vector control of induction motor using singularly perturbed sliding mode observer. This approach ...
Adaptive Speed Sensorless Vector Control of Induction Motor using Singularly Perturbed Sliding Mode Observer A. Mezouar 1, M. K. Fellah 2, S. Hadjeri, Y. Sahali Intelligent Control and Electrical Power Systems Laboratory, Department of Electrical Engineering, Faculty of Sciences Engineering, Djillali Liabès University, (22 000) Sidi Bel Abbès, ALGERIA. 1 [email protected] ([email protected]), 2 [email protected] ([email protected]) In other hand, singular perturbation theory provides the mean to decompose two-time-scale systems into slow and fast subsystems of lower order described in separate timescales, which greatly simplify their structural analysis and any subsequent control design. Then, the control (and/or observer) design may be done for each lower order subsystem, and the combined results yield to a composite control (and/or observer) for the original system [22, 23]. So, the idea of combining singular perturbation theory and sliding mode technique constitutes a good possibility to achieve classical control objectives for systems having unmodeled or parasitic dynamics and parametric uncertainties [24, 25, 26, 27]. In this paper, an adaptive sliding mode observer is developed, using singular perturbation theory, for the simultaneous estimation of the rotor flux components, rotor resistance and rotor speed for induction motor under the assumption that only the stator currents and voltages are available for measurement. The rotor flux observer stability is insured through the Lyapunov theory. This paper is organized as follows: In Section II, we briefly review the indirect field oriented sliding mode control of induction motors. The design of a two-time-scale sliding-mode-observer for the model of the induction motor is presented in Section III. In that section, a study of stability analysis of this observer is made via singular perturbation method with sliding mode concept and Lyapunov stability theory. Details of the drive system configuration are given in Section IV. In Section V, and through simulation, the studied observer is associated to the indirect field oriented sliding-mode-control where rotor flux, rotor resistance and rotor speed are replaced by those delivered by the observer. Finally, in Section VI, we give some comments and conclusions.

Abstract  This paper presents a new approach for sensorless vector control of induction motor using singularly perturbed sliding mode observer. This approach is based on the singular perturbation theory which decomposes the original system of the observer error dynamics into separate slow and fast subsystems and permits a simple design and sequential determination of the observer gains. In this way, the flux observer accuracy (slow dynamics) is guaranteed through the current observer (fast dynamics). The rotor speed and the rotor resistance are estimated by adaptive laws based on measured and estimated stator currents and estimated rotor flux. The effectiveness of this new approach has been successfully verified through computer simulations, where the control algorithm is based on the indirect field oriented sliding mode control.

I. INTRODUCTION The vector control technique has been widely used for high-performance inductance motor drives, where the knowledge of the rotor speed is necessary [1, 2]. This requires additional speed sensor on the machine which adds to the cost and the complexity of the drive system. Recently, sensorless vector control has received much attention in AC drives. The elimination of speed sensor reduces the hardware complexity, size and cost, and increases the reliability of the drive system. The requirement is to have high performances over the entire speed range. Indeed, the estimation of the rotor speed has constituted a subject of interest for many research teams, and several solutions for estimating the state variables and parameters of the induction motor have been proposed in the literature such as: the extended Luenberger observer (ELO) [3, 4], the extended Kalman filter (EKF) [5, 6, 7], Model reference adaptive systems (MRAS) [8, 9], adaptive observer (AO) [10, 11], sliding mode observer (SMO), [12, 13, 14], and newly fuzzy logic and neural networks observers [15]. In the last years, the variable structure control (VSC) strategy using the sliding mode concept has been widely studied and developed for the control and state estimation problems since the works of Utkin [16]. This control technique has many good properties to offer such as insensitivity to parameter variations, external disturbance rejection and fast dynamic response [17, 18]. Several methods of applying sliding mode control and/or observer to induction motor drives have been presented [19, 20, 21]. All of these methods have a common feature: the analysis and design of the sliding mode controller or observer are based on the mathematical model of the induction motor as used in indirect vector control. 1-4244-0136-4/06/$20.00 '2006 IEEE

II. SLIDING-MODE-CONTROL DESIGN FOR INDUCTION MOTORS Assuming that the induction model system is controllable and observable, the sliding mode control consists into two phases [18, 19]: • Designing an equilibrium surface, called sliding surface, such that any state trajectory of the plant restricted to the sliding surface is characterized by the desired behavior; • Designing a discontinuous control law to force the system to move on the sliding surface in a finite time.

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A. Dynamic model of induction motor

Rλ µ 1   f3 = − σL isd + ωsisq + σL T φrd s s r  R µ λ  f = −ωsi − isq − ωφ sd  4 σLs σLs rd

Under the assumptions of linearity of the magnetic circuit and neglecting iron losses, the state space model of three-phase induction motor expressed in the synchronous rotating reference frame (d-q) is [1]:

(5.a)

and

Rλ µ 1 µ 1 d dt isd = − σL isd + ωsisq + σL T φrd + σL ωφrq + σL vsd s r s s s  R d µ µ 1 1  λ dt isq = −ωsisd − σL isq − σL ωφrd + σL T φrq + σL vsq s s s r s  Lm 1 d (1) i − φ + (ωs − ω)φrq  φrd = Tr sd Tr rd dt d Lm 1 dt φrq = T isq − (ωs − ω)φrd − T φrq r r  f dω p  dt = J (Tem −TL ) − J ω 

kc =

p 2Lm JLr

C. Speed and flux sliding-mode- controllers Using the reduced non-linear induction motor model of equation (3), it is possible to design both a speed and a rotor flux sliding mode controllers. Let us define the sliding surfaces d *  * S c1 = S c1 (ω) = λ ω (ω − ω) + dt (ω − ω)   S = S (φ ) = λ ( φ * − φ ) + d (φ * − φ ) φ c2 c2 r r dr r dr dt 

where the state variables are the stator currents (isd,isq) , the rotor fluxes (φrd , φrq ) and the rotor speed ω . Stator

(6)

where λ ω > 0 , λ φ > 0 , ω* and φ*r are the reference speed and the reference rotor flux. To determine the control law that leads the sliding functions (6) to zero in finite time, one has to consider the dynamics of S c = (S c1 , S c 2 ) T , described by

voltages (vsd,vsq) and slip frequency ω sl ( ωsl = ωs − ω ) are the control variables. The developed electromagnetic torque expressed in terms of these state variables is pLm Tem = (φrd isq − φrq isd ) (2) Lr

Sc = F + D V S

with constants defined as follows L2 L2 L Rλ = Rs + m2 Rr , σ = 1 − m , µ = m , Lr Ls Lr Lr

(7)

where f  f  *  * (ω + λωω + J TL ) + (−λω + J )f1 − kc (isq f2 + φrd f4 ) F=  (φr* + λ φr* ) + (−λ + 1 )f − Lm f  φ φ 2 3   Tr Tr

B. Induction motor model with oriented rotor field Among the various sliding mode control solutions for induction motor proposed in the literature, the one based on indirect field orientation can be regarded as the simple one. Its purpose is to directly control the inverter switching by use of two switching surfaces. The induction motor equations in the synchronous rotating reference frame (d − q ) , oriented in such a way that the rotor flux vector points into d − axis direction, are the following d  dt ω = f1  d φ = f 2  dt rd (3) d 1  isd = f3 + vsd σLs  dt d 1 vsq  isq = f4 + σLs  dt with the synchronous electrical angular speed L i (4) ωs = ω + m sq Tr φrd where p f   f1 = kcφrdisq − J TL − J ω (5.a)  M 1  f2 = isd − φrd Tr Tr 

D=

1 σLs

kcφrd  0 

0  vsq  , VS =   Lm /Tr  vsd  

If the Lyapunov theory of stability is used to ensure that S c is attractive and invariant, the following condition has to be satisfied S c ⋅ S c < 0 (8) So, it is possible to choose the switching control law for stator voltages as follows K ω  v sq  −1 −1   = −D F − D  0 v sd  

0   sign (S c1 )   K φ  sign( S c 2 )

(9)

where Kω > 0 , Kφ > 0 (10) The sliding mode causes drastic changes of the control variables introducing high frequency components (chattering phenomenon). To reduce this phenomenon a saturation function sat ( S c ) instead of the switching one sign( S c ) has been introduced Sc i if (S c i ) ≤ ρ i  sat (S c i ) =  ρ i (11)  sign(S c i ) if (S c i ) > ρ i where ρ i > 0 and i = 1,2. 933

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where ωˆ , xˆ i and zˆ j are the estimation of ω , xi and z j

Remark (1): In the following, we will assume to operate with constant reference speed, constant reference rotor flux and constant load torque.

for i ∈ {1,2} and j ∈ {1,2}. G x 1 , G x 2 , G z 1 and G z 2 are the observer gains.

III. SINGULARLY PERTURBED SLIDING MODE OBSERVER DESIGN FOR INDUCTION MOTOR

Remark (2): • All parameters of induction motor will be considered as constant except rotor resistance. The rotor resistance Rr will be treated as uncertain parameter with Rrn as its nominal value. An addition assumption is that Rr varies slowly (practical assumption), so that Rr ≈ 0 .

A. Dynamic model of induction motor Using the model of equation (1), the state space model of induction motor, without mechanical equation, expressed in the fixed stator reference frame ( α , β ) is

• The motor speed will be treated as an unknown bounded time-varying variable.

 d i = − Rsr i + Lm 1 φ + Lm ωφ + 1 v sα rα sα dt sα σLs σLs Lr Tr σLsLr rβ σLs   d i = − Rsr i − Lm ωφ + Lm 1 φ + 1 v rα dt sβ σLs sβ σLsLr σLsLr Tr rβ σLs sβ (12)   d φrα = Lm isα − 1 φrα − ωφrβ dt Tr Tr d Lm 1  φrβ = isβ + ωφrα − φrβ dt Tr Tr

The switching vector Γs is chosen as sign(s1 )  Γs =   sign(s 2 ) with s1  z 1 − zˆ1  e z  S = =  = e  s  2  z 2 − zˆ2   z  1

Voltage, current and flux transformation from the synchronous to the stationary reference frame and vice versa is made by the rotational transformation as [1, 2]:  xα   xd    = ℜ(θ s )   xβ   xq 

(17)

2

Setting ∆ω = ωˆ − ω, ex i = xˆi − x i and ez j = zˆj − z j for i ∈ {1,2} and j ∈ {1,2}, and subtracting (14) from (15), the estimation error dynamics are

(13)

with

εez1  εez2 Σe :   ex1  e  x2

 cos(θ s ) − sin(θ s )   and [ℜ(θ s )] −1 = ℜ(−θ s ) ℜ(θ s ) =  θ θ sin( ) cos( ) s s   where x = v, i, φ , and θ s is defined by (4).

B. Singularly perturbed induction motor model

= +ηrex1 + ωex2 + ∆ωxˆ2 + ∆ηr (xˆ1 − Lmz1) + Gz1Γs = −ωex1 + ηr ex2 − ∆ωxˆ1 + ∆ηr (xˆ2 − Lmz 2 ) + Gz 2 Γs = −ηrex1 − ωex2 − ∆ωxˆ2 − ∆ηr (xˆ1 − Lmz1) + Gx1Γs

(18)

= +ωex1 − ηr ex2 + ∆ωxˆ1 − ∆ηr (xˆ2 − Lmz 2 ) + Gx 2 Γs

Equation (18) can be expressed in the matrix form as

From singular perturbation theory and based on the well-known of the induction machine model dynamics [25, 26, 27], the slow variables are (φrα , φrβ ) and the fast

εez = +Αex + ∆ωJ 2xˆ + ∆ηr (x − Lm z ) + Gz Γs (19) Σe :   ex = −Αex − ∆ωJ 2xˆ − ∆ηr (x − Lm z ) + Gx Γs

variables are (isα , isβ ) . Therefore, the corresponding

with

Α = (η r I 2 − ωJ 2 ) (20) where I 2 is the (2 × 2) identity matrix and J 2 is the (2 × 2) skew symmetric matrix defined by 0 −1 J2 =  1 0    Using singular perturbation theory, the stability analysis of the above system consists of determining G z1 and G z 2 to ensure the attractiveness of the sliding surface S (τ) = 0 in the fast time scale. Thereafter G x1 and G x 2 are determined, such that the reduced-order system obtained when S (τ ) ≅ S(τ ) ≅ 0 is locally stable.

singularly perturbed model of (12), with ε = σ Ls Lr / Lm ,

x = (φrα , φrβ )T , z = (isα , isβ )T and ηr = Rr / Lr is Lr Lr   εz1 = − Lm Rsr z1 + ηr x1 + ωx 2 + Lm vsα  Lr Lr  Σ : εz2 = − L Rsr z 2 − ωx1 + ηr x 2 + L vsβ m m  x1 = Lmηr z1 − ηr x1 − ωx 2 x = L η z + ωx − η x m r 2 r 2 1  2

(16)

(14)

C. Singularly Perturbed Sliding Mode Observer The observer equations of the above model based on the sliding mode concept are the following εzˆ = − Lr R ˆsrz + ηˆr xˆ + ωˆxˆ + Lr vsα + G Γs 1 1 2 z1  1 Lm Lm  Lr Lr ˆ   ˆ ˆ ˆ : εzˆ2 = − L Rsr z 2 − ωxˆ1 + ηr xˆ2 + L vsβ + Gz 2Γs (15) Σ m m  xˆ1 = Lmηˆr z1 − ηr xˆ1 − ωˆxˆ2 + Gx1Γs  xˆ2 = Lmηˆr z 2 + ωˆxˆ1 − ηˆr xˆ2 + Gx 2Γs 934

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D. Fast reduced-order error dynamics

E. Slow reduced-order error dynamics

From singular perturbation theory, the fast reducedorder system of the observation errors can be obtained by introducing the fast time scale τ = (t − t 0 ) / ε , and thereafter setting ε = 0 . So, System of (19) gives  d e = Ae + ∆ωJ xˆ + ∆η (xˆ − L z ) + G Γ z x r m z s 2 dτ (21) Σe, f :   d ex = 0 dτ By appropriate choice of the observer gain terms G z1

For slow error dynamics (when S ≡ 0 , i.e. e z = 0 and e z = 0 ), we use the system (19) and setting ε = 0 . So, we can write  0 = +Αex + ∆ωJ 2xˆ + ∆ηr (xˆ − Lmz ) + Gz Γs ˆe,s :  (28) Σ ex = −Αex − ∆ωJ 2xˆ − ∆ηr (xˆ − Lmz ) + Gx Γs F. Stability analysis of the slow reduced-order error dynamics In this time-scale, system of (28) becomes 0 = +Αex + ∆ωJ 2xˆ + ∆ηr (xˆ − Lmz ) + H ˆe,s :  (29) Σ ex = −Αex − ∆ωJ 2xˆ − ∆ηr (xˆ − Lmz ) + LH where H = G z Γs , L = G xG z−1 (30)

and G z2 sliding mode occurs in (21) along the manifold ez = 0 . Proposition (1): Assume that e x1 and e x2 are bounded in this time-scale (practical assumption) and ω varies slowly, and consider the first equation of (21) with the following observer gains matrix 0  −δ 1 Gz =  (22) 0 − δ2   Then, the attractivity condition of the sliding surface S (τ) = 0 given by dS ST ( ) < 0 (23) dτ is verified with the following inequalities δ 1 > ηr ex 1 + ωex 2 − ∆ωxˆ2 + ∆ηr (xˆ1 − Lm z1 ) (24) δ 2 > ηr ex 2 − ωex 1 + ∆ωxˆ1 + ∆ηr (xˆ2 − Lm z 2 )

We consider the rotor speed as a variable parameter [14], and let we chose the candidate Lyapunov function 1 1 1 1 V = eTxex + (∆ω )2 + (∆ηr )2 (31) 2 q1 2 q2 where q 1 > 0, q 2 > 0 . From singular perturbation theory, rotor speed ω is considered as constant or varies slowly. So, the t-timederivative of V can be expressed as dωˆ 1 d∆ηr 1 V = eTxex + ∆ω (32) + ∆ηr q1 dt q 2 dt We know that V is positive-definite. Now, looking for

ex and e x . From the first equation of (29), we get:

Proof (1): Let we use the definite positive Lyapunov functionV = S T S / 2 . In this time-scale, the derivative of V is given by d d V = S T ( S ) = S T [(ηr I 2 − ωJ 2 )ex + ∆ωJ 2xˆ (25) dτ dτ + ∆ηr (xˆ − Lm z ) + Gz Γs ] or d V = −s1 [δ 1sign (s1 ) − ηr ex 1 − ωex 2 dτ + ∆ωxˆ 2 − ∆ηr (xˆ1 − Lm z1 )] (26) − s 2 [δ 1sign (s1 ) − ηr ex 2 + ωex 1 − ∆ωxˆ 2 − ∆ηr (xˆ 2 − Lm z 2 )]

ex = − Α −1H − ∆ωΑ −1J 2xˆ − ∆ηr Α −1(xˆ − Lm z ) (33)

and with addition term to term of (29), it yields : ex = (I 2 + L)H

(34)

ex = ΛH

(35)

or with

Λ = (I 2 + L) Now, substituting (33) and (35) in (32), it yields: V = −H T ΛT Α −1H + Q1 + Q2

(36) (37)

where  1 dωˆ  Q1 = ∆ω  − H T ΛT Α −1J 2xˆ   q1 dt   1 d∆ηr  Q2 = ∆ηr  − H T ΛT Α−1 (xˆ − Lm z )   q 2 dt 

Taking into account that all states and parameters of induction motor are bounded, then, there exists sufficiently large positive numbers δ 1 and δ 2 (defined by (22)) such that dS (27) ST ( ) < 0 dτ □ Remark (3): Once the currents trajectory reaches the sliding surface S = e z = 0 , the error system of (19), in sliding mode, behaves as a reduced-order subsystem governed only by the rotor flux error e x assuming that ez = 0 and ez = 0 .

(38) (39)

Sufficient condition for (37) to be negative-definite will be satisfied if − H T ΛT Α−1H < 0 (40)  1 dωˆ   − H T ΛT Α −1J 2xˆ  = 0 (41)  q1 dt   1 d∆ηr   − H T ΛT Α−1 (xˆ − Lmz )  = 0 q dt  2 

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(42)

The condition of (40), is satisfied choosing T

Λ Α

−1

= q 0I 2

For the closed loop system, feedback signals of the rotor fluxes, rotor resistance and rotor speed are replaced with the estimated corresponding values of equations (14), (45) and (46), respectively. For the slip frequency estimation (S.F.E) of equation (4), the d − axis rotor flux is replaced with the rated rotor flux value φ r* . Due to the high harmonic content of the voltage provided by the inverter, the observer input voltages are the references v s*α and v s*β . Moreover, it is considered that the load torque is unknown and all the motor parameters are known and constant except for the rotor resistance which will change during the motor operating.

(43)

where q 0 > 0 . Finally, the substitution of (43) in (41) and (42) permits to give a simple adaptive law for rotor speed and inverse of the rotor time constant estimations as dωˆ = q 0q 1 (G z Γs )T J 2 xˆ (44) dt d∆η r = q 0q 2 (G z Γs )T (xˆ − Lm z ) (45) dt □ Since ωˆ is a switching function, it will content high frequency components in addition of low frequency components. The low frequency component is equal to the speed [18]. A smoothed value of ωˆ can be found by passing it through a low-pass filter as :

ωˆeq =

1 ωˆ 1+ γ s

V. SIMULATION RESULTS The proposed estimation algorithm has been simulated for the induction motor whose data are given in Appendix 2. Sliding mode control and observer parameters are listed in Appendices 3 and 4, respectively. With the assumption that all states including rotor flux and all parameters are available, rotor flux, rotor resistance and rotor speed estimated by the proposed method are compared to their actual values. In order to verify the performance of the indirect field oriented control based on the proposed sliding mode observer, the control system has been tested in several operating conditions, where the rotor flux reference is kept at its rated value of 1.0 Wb. In this study, we make attention toward speed estimation and speed tracking performance for wide range of reference speed.

(46)

where γ is the time constant of the filter witch should be sufficiently small to preserve the real slow components and large enough to eliminate the high frequency components. IV. SYSTEM CONFIGURATION A block diagram of the proposed control-observer structure is shown in Fig. 1. The vector controlled drive is based on the indirect field oriented sliding mode control.

Vdc

φr* ω*

v

Sliding Mode Control Eq. (9)

ωˆeq TˆL φˆrd Rˆr

i sq

v

dq

v sq*

αβ

v s*β

is α

dq

isq

αβ

rd

* sα

αβ

PWM VSI

abc

θˆs

i sd isd

Load Torque Estimator ωˆ φˆ eq

* sd

isβ

αβ abc

isa isb

φˆrd φˆrq θˆs

dq

αβ φˆrα φˆrβ

v s*α v s*β

i sq

S.F.E Eq. (14)

ωˆeq

ˆr R isα

Adaptive S.M.O Eqs. (14), (44-46)

isβ

ωˆeq Fig. 1. Block diagram of the proposed control-observer structure.

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I.M

Reference signals of Rr(Ω) and TL(N.m)

Reference signals of Rr(Ω) and TL(N.m)

(g)

Time (s)

(f)

Time (s)

(h)

Time (s)

Stator currents error (A)

Time (s)

(e)

Time (s)

(g)

Time (s)

Rotor fluxes error (Wb)

Estimated rotor fluxes (Wb)

Rotor speed error (rpm)

Estimated rotor speed (rpm) Estimated rotor fluxes (Wb)

(c)

(b)

Time (s)

(d)

Time (s)

(f)

Time (s)

(h)

Time (s)

Stator currents error (A)

Time (s)

Time (s)

Time (s)

Rotor speed error (rpm)

(e)

(d)

(a)

Rotor fluxes error (Wb)

Time (s)

Time (s)

Actual rotor speed (rpm)

(c)

(b)

Estimated rotor speed (rpm)

Time (s)

Actual rotor speed (rpm)

(a)

Reference and estimated Rr (Ω)

The first test consists in increasing the rotor resistance at high reference speed. As shown in Fig. 2(a), the motor is started with its nominal rotor resistance value Rrn = 3.805Ω . Then, the rotor resistance of the motor model is suddenly set to 1.5Rrn at t = 1s, and to 2Rrn at t = 2s. The reference speed and reference rotor-flux are maintained at 1400 rpm and 1.0 Wb, respectively.

Reference and estimated Rr (Ω)

The state observer behavior is considered for the following perturbations: • Large error on the rotor resistance value (up to 100%), • Change in the load torque, • Running at low and high speeds, and reversion of rotating sense.

Fig.2. Sensitivity of the system performance to changes on the rotor resistance and load torque at high speed (1400 rpm): (a) Reference signals of rotor resistance (solid) and load torque (dotted), (b) Reference (dotted) and estimated (solid) rotor resistance, (c) Actual speed, (d) Stator currents error: eisα (solid) and eisβ (dotted),

Fig.3. Sensitivity of the system performance to changes on the rotor resistance and load torque at low speed (100 rpm): (a) Reference signals of rotor resistance (solid) and load torque (dotted), (b) Reference (dotted) and estimated (solid) rotor resistance, (c) Actual speed, (d) Stator currents error: eisα (solid) and eisβ (dotted),

(e) Estimated speed, (f) Rotor speed error: ∆ω = ωˆeq − ω,

(e) Estimated speed, (f) Rotor speed error: ∆ω = ωˆeq − ω,

(g) Estimated rotor fluxes: φˆrd (solid) and φˆrq (dotted),

(g) Estimated rotor fluxes: φˆrd (solid) and φˆrq (dotted),

(h) Rotor fluxes error: eφrd (solid) and eφrq (dotted).

(h) Rotor fluxes error: eφrd (solid) and eφrq (dotted).

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The second test is related to the performance of the drive system at low speed. And again, we examine the same test as Fig. 2. In Fig. 3(b) are shown the actual and estimated rotor resistance. Fig. 3(c) presents very good performance for speed regulation. In Fig. 2(d) are shown the stator currents error. This error is very small and presents high frequency oscillations. This report is very clear since the stator currents are switching functions, which content high frequency components. Figs. 3(e) and 3(f) show the estimated rotor speed and the rotor speed error, respectively; the speed estimation is very accepted. Fig. 3(g) and 3(h) show that the completely decoupled control of rotor flux and torque is obtained. As shown in Figs. 3(b) to 3(h), the operating at low speed does not influence the estimation of the rotor speed and the adaptation of the rotor resistance. In the last test, we consider the speed tracking performance for wide variation range of the reference speed. The rotor flux reference is kept at its rated value of 1.0 Wb and the motor is operating without external load disturbances, under a change of +25% of the rotor resistance value. This practical error is made to test the efficacy of the adaptive law of equation (45). The observer performance for speed tracking is presented in Fig. 4(a). The actual and the estimated rotor resistance are shown in Fig. 4(b). The rotor resistance estimation is very good at high speed. At very low speed, the rotor resistance error is remarkable, but it remains in the accepted values range. It presents an estimation error of 2 %. Figs. 4(c) and 4(d) show the estimated rotor speed and the rotor speed error, respectively; the speed estimation is very satisfied. Fig. 4(e) and 4(f) show the estimation of the rotor fluxes and the error between the estimated rotor fluxes and the actual rotor fluxes, respectively. These results prove that the speed tracking is quite good and the rotor-field is always well-oriented.

Reference and estimated Rr (Ω)

Time (s)

(a)

(b)

Time (s)

Rotor speed error (rpm)

Estimated and reference speeds (rpm)

Actual and reference speeds (rpm)

Fig. 2(b) compares the estimated and actual rotor resistance. After a short convergence time, the estimated rotor resistance reaches the actual value. Fig. 2(c) shows the speed response of the motor; a very good speed regulation is obtained. In Fig. 2(d) are shown the stator currents error. Figs. 2(e) and 2(f) show the estimated rotor speed and the rotor speed error, respectively; the speed estimation is very satisfied. In Figs. 2(g) and 2(h), are shown the estimated rotor fluxes and the error between them and the actual values. It can be noticed the high flux tracking and the good rotor flux orientation. These results show that the sliding mode control with the proposed observer can track the reference command accurately and quickly at high speed. It is important to notice that the q − axis rotor flux error is greater than the d − axis rotor flux error in transient state. This report is very clear since the rotor resistance estimation error (in transient state) propagate on the slip frequency which directly affects the rotor field orientation (see: equation 4).

1

Time (s)

(e)

Time (s)

(d)

Time (s)

(f)

Time (s)

Rotor fluxes error (Wb)

Estimated rotor fluxes (Wb)

(c)

VI. CONCLUSION In this paper, a novel approach to adaptive sliding mode observer for sensorless speed control of an induction motor has been presented using singular perturbation theory. The proposed observer algorithm design consists of two stages, first, the fast part (stator currents) of the observer is studied so that in this time scale, the resulted model is stable, and then, the stability analysis of the slow part (rotor fluxes) is examined through Lyapunov theory. The accurate rotor flux and rotor speed obtained by this algorithm has been applied to indirect field sliding mode control. The effectiveness of the control-observer structure scheme with rotor speed estimation has been successfully verified by simulation. The proposed sliding mode observer-based control demonstrated very good performance, especially; it is robust under rotor resistance variation, external load disturbances and speed tracking.

Fig.4. Sensitivity of the system performance to change in the reference * = 1.0Wb : speed with Rr = 1.25Rrn and φrd

(a) Reference (dotted) and actual (solid) speeds, (b) Reference (dotted) and estimated (solid) rotor resistance, (c) Reference (dotted) and estimated (solid) speeds, (d) Rotor speed error: ∆ω = ωˆeq − ω,

[1]

(e) Estimated rotor fluxes: φˆrd (solid) and φˆrq (dotted),

[2]

(f) Rotor fluxes error: eφ

rd

REFERENCES

(solid) and eφrq (dotted).

[3]

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APPENDIXES Appendix 01: Notations In this paper, the following notations are used: I.M

Induction motor,

ω , ω*

Electrical rotor speed and reference rotor speed,

ωˆeq

Filtered rotor-speed estimation,

vsd , vsq

Stator voltages in the synchronous rotating frame,

i sd , i sq

Stator currents in the synchronous rotating frame,

φrd , φrq vsα , vsβ

Rotor fluxes in the synchronous rotating frame, Stator voltages in the stationary reference frame,

isα , isβ

Stator currents in the stationary reference frame,

φrα ,φrβ

Rotor fluxes in the stationary reference frame,

ωs , ωsl

Synchronous and slip frequencies,

Ls , Lr

Stator and rotor inductances,

Rs , Rr

Stator and rotor resistances,

Ts ,Tr ηr

Stator and rotor time-constants,

Lm , σ

Mutual inductance and leakage factor,

J f p

Moment of rotor inertia,

Tem ,TL

Electromagnetic and load torques,

Inverse of the rotor time-constant,

Coefficient of viscous friction, Number of pole pairs,

xˆ , x *

Estimated and reference value of x ,

S.M.C

Sliding Mode Control,

S.M.O

Sliding Mode Observer,

V.S.I

Voltage Source Inverter.

Appendix 02: Parameters of induction motor 1.5 kW ,

220 / 380 V ,

3.68 / 6.31 A,

N = 1420 rpm, Rs = 4.85 Ω, Rs = 4.805 Ω, Lr = 0.274, Lr = 0.274 H , Ls = 0.274 H , p = 2, J = 0.031 Kg .m 2 , f = 0.00114 N .m .s / rd .

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Appendix 03: Sliding mode control parameters

[21] A. Benchaib, A. Rachid, E. Audrezet, M. Tadjine, “Real-Time Sliding-Mode Observer and Control of an Induction Motor”, IEEE Trans. Ind. Electron., vol. 46, no. 1, pp. 128–138, 1999. [22] P.V. Kokotovic, H. Khalil, J. O‘Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, New York, 1986.

λω

λφ





ρω

ρφ

120

120

80

80

0.5

0.5

Appendix 04: Sliding mode observer parameters

[23] P.V. Kokotovic, J. O‘Mailley, P sannuti, “Singular Perturbation and Order Reduction in Control Theory: An overview”, Automatica, vol. 12, pp.123–132, 1976.

δ1

δ2

q0

q1

q2

100

100

0.002

7000

1000

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