Backstepping Control for Induction Machine with High ... - IEEE Xplore

Backstepping Control for Induction Machine with High Order Sliding Mode Observer and Unknown Inputs Observer : A Comparative Study 1

A. Guezmil, 2 H. Berriri, 3A. Sakly, 4M. F. Mimouni

Electrical Department Monastir National Engineering School Ibn Eljazzar City, 5019, Monastir, Tunisia Email: [email protected], [email protected], [email protected], [email protected]

Abstract — In this paper, a synthesis of a speed and flux control approach for an induction machine based on backstepping strategy is designed. Indeed, the lyapunov theory is used to prove the stability of the proposed approach. To avoid the use of speed and flux sensors, a high order sliding mode observer and unknown input observer are developed to estimate the flux and the speed. In addition, the purpose of this study is to compare and evaluate the performance of each observer proposed. Numerous simulations are also shown to test the robustness of the proposed control scheme in low and high speed and presence of rotor resistance variations. Keywords—Induction machine; backstepping control; high order sliding mode observer; unknown inputs observer; parameter variations

I.

INTRODUCTION

The produced electric energy from renewable energy sources is becoming increasingly important in the aftermath of growing world energy demand. Thus, green power, and particularly wind and hydroelectric power plants are intensively used today. As regards the latter, electric machine is the hub of electric generation system [1]. Due to its high performances, low cost and reliability, induction machine (IM) has a great importance in such applications [2]. Indeed, IM control is particularly challenging as these required a variables state acknowledgment, which are not usually measured, and less sensitivity to parameters variations. Indeed, in real operating condition, there are temperature increasing that involve parameters change and magnetic saturation [3]. So, conventional linear IM control can't maintain the system transient stability. To overcome this limitation, the nonlinear control has been recognized. Numerous approaches, used for nonlinear IM control, have been proposed. Such as, sliding mode [4], high order sliding mode [5] [6], the input-output linearization [7] [8] and the passivity based control [9] [10]. In last few years, a special control design known as “backstepping” has been tremendous amount of research works [11] [12]. A special attention has been paid to IM control. A backstepping control for sensoreless speed of IM is presented in [13], based on rotor flux estimator and model reference adaptative system. In [14], a sensoreless control for IM by using both backstepping strategy and lyapunov stability theory

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is reviewed. The proposed structure consists in the backstepping rotor speed and rotor flux observer associated to an online adaptative rotor resistance estimator. A backstepping control for IM rotor flux and speed tracking control has been proposed by [15], using sliding mode observer in order to reconstruct the IM speed and rotor flux. A robust adaptative backstepping control for efficiency optimization is developed by Ho Hwang [16], is based on adaptative sliding mode rotor flux observer and robust speed control, using a function approximation for mechanical uncertainties. In [17], a field oriented control using integral backstepping technique has been designed for IM drive without mechanical sensors. Also, an adaptative interconnected observer is designed for estimating rotor flux, rotor speed, load torque and stator resistance. Most of the IM backstepping control schemes require rotor flux observation. A research activity has been carried out aimed at developing the observer, such as luenberger observer [18], extended kalman filter [19], high gain observer [20], sliding mode observer (SMO) [21] and unknown inputs observer (UIO) [22]. SMO, proposed by Utkin [23] is widely used due to their finite time convergence, robustness opposite to uncertainties and disturbances. It is used to estimate rotor flux and speed. Its drawback is the "chattering" effect at high frequency. To eschew chattering effect, the high order sliding mode observer has been designed based on twisting or super twisting algorithm [24] [25]. UIO has also draws a great attention in the research area. Indeed, it can estimate simultaneously the system state and the unknown input just using input and output measurements [26]. Also, it does not require the output differentiation. It assumes that the dynamics of unknown inputs are bounded without making hypothesis on inputs variation [27]. The outline of the paper is organized as follows. In section 2, a backstepping approach for induction machine control is designed. Section 3 is devoted to expand the high order sliding mode observer. In section 4, unknown inputs observer design is developed for IM. Simulation results are illustrated and discussed in section 5. Finally, some conclusions are presented in section 6.

II.

BACKSTEPPING CONTROL DESIGN

A. Induction Machine Equations Under assumptions of linear magnetic circuits and balanced operating conditions, the three-phase induction machine equations are considered here in a rotating reference frame (d-q) for field-oriented control [28]. Therefore, the rotor flux r is orientated on the d-axis. Consequently, q-component rotor flux rq is equal to zero. ⎧ I sq 2 + cV sd ⎪ Isd = aI sd + a1ϕ rd + ω N p I sq + b ϕ rd ⎪ ⎪ I I ⎪ Isq = aI sq − a 2 N p ω ϕ rd − N p ω I sd − b sq sd + cV sq (1) ϕ rd ⎪ ⎪ ⎨ϕ r = ϕ rd = b1ϕ rd + bI sd ⎪ ⎪ ω = 1 ⎡ mϕ I − C − f ω ⎤ r rd sq ⎦ ⎪ j⎣ ⎪ ⎪θ = ω + b I sq r ⎪ s ϕ rd ⎩ is the rotor flux Where Isd and Isq are the stator currents, and is the electrical rotor speed. The control variables are the stator voltage Vsd and Vsq, and θs, the stator field angle. Expressions of a, a1, a2, b, b1, c and m are depend on IM parameters:

⎛ Rs

a = −⎜

⎝ σ Ls

b1 = −

1 Tr

+

;c =

M M M ⎞ ; a1 = ; a2 = ;b = ⎟ σ Lr LsTr ⎠ σ Lr LsTr σ Lr Ls Tr M

1

σ Ls

2

;σ

= 1−

M

2

Lr Ls

;T

s

=

Ls Rs

;T

s

=

Lr Rr

;m =

NpM Lr

The electromagnetic torque expressed in terms of stator current Isq and rotor flux as follow:

C em =

3 N pM 2 Lr

sr

ϕ rd I s q

(2)

The control aim is to make electrical rotor speed and the , track wanted references. This tracking can be rotor flux performed through a backstepping technique establishing a direct relation among variables and stator voltages Vsd and Vsq. B. Backstepping Control The fundamental idea of the backstepping approach is the use of the so-called “virtual control” quantities to break down a complex non-linear control design problem into simpler and smaller ones [29]. Backstepping design is divided into two-step scheme. In each one it deal with an easier and Single Input Single Output (SISO) design problem, and first step provides a reference for the next design step. Lyapunov function acquires the overall stability and performance for the whole system. 1) Step1: Stator currents reference determination The goal of the first step is to provide the stator current and , given by (3), be reference Isq-ref and Isd-ref. Let respectively the rotor flux magnitude and speed tracking error. According for (1), the dynamic errors (3) are given by (4).

⎧⎪ eω = ω ref − ω ⎨ ⎪⎩ eϕ = ϕ ref − ϕ rd

(3)

⎧ ⎡1 ⎤ ⎪ eω = ω ref − ω = ω ref − ⎢ ( m ϕ rd I sq − f ω − C r ) ⎥ ⎨ ⎣j ⎦ ⎪ e = ϕ − ϕ = ϕ − b ϕ − bI 1 rd ref rd ref sd ⎩ ϕ

(4)

In order to satisfy the tracking performances, the first lyapunov function associated with the rotor flux and speed errors is defined as (5). Using (4), the dynamic of lyapunov function is written as (6).

v1 =

eω 2 + eϕ 2

(5)

2

v1 = eω eω + eϕ eϕ ⎛

= eω ⎜ ω ref −

mϕrd I sq

+

fω

+

Cr ⎞

⎟ + eϕ (ϕref − b1ϕrd − bI sd )

(6)

j j j ⎠ ⎝ Equation (6) can be, also, written such as:

v1 = − k1eω 2 − k 2 eϕ 2

(7)

Where k1 and k2 must be positive parameters, in array to guarantee a stable tracking, so:

⎧eω = ω ref − ω = −k1eω ⎨ ⎩eϕ = ϕref − ϕrd = −k2 eϕ

(8)

Equation (8) allow for generate current references ensure that the lyapunov stability condition. These current references are represented as follows: ⎧ C ⎤ j ⎡ fω k e + ω r e f + + r ⎥ ⎪ I sq − ref = (9) ⎪ ϕ r d m ⎢⎣ 1 ω j j ⎦ ⎨ 1 ⎪I  ⎪⎩ s d − r e f = b ⎡⎣ k 2 e ϕ + ϕ r e f − b 1ϕ r d ⎤⎦ The virtual controls in (9) are chosen to satisfy the aim of stator current regulation and are considered as references for the next step. 2) Step 2: stator voltages reference determination Let us define the current errors as follows:

⎧ e Isq = I sq − ref − I sq ⎨ ⎩ e Isd = I sd − ref − I sd

(10)

Then, accounting to (9), (10) will take the following form:

⎧ j ⎛ fω C ⎞ + r ⎟ − I sq ⎪ e Is q = ⎜ k 1 e ω + ω ref + ⎪ j j ⎠ ϕ rd m ⎝ ⎨ 1 ⎪  ⎪⎩ e Is d = b ( k 2 eϕ + ϕ ref − b1ϕ r d ) − I s d

(11)

Current errors dynamic is:

⎧⎪ eIsq = Isq − ref − Isq ⎨ ⎪⎩ eIsd = Isd − ref − Isd

(12)

Following the substitutes of (1) in (12), current errors dynamic can be written as: I sq I sd ⎧  ⎪ eIsq = I sq − ref − aI sq + a 2ω N pϕ rd + ω N p I sd + b ϕ − cVsd ⎪ rd (13) ⎨ 2 I sq ⎪ e = I − aI sd − a1ϕ rd − ω N p I sq − b − cVsq sq − ref ⎪⎩ Isd ϕ rd Stator voltages Vsd and Vsq are included in (13). These could be establishing a new lyapunov function based on the errors of speed, rotor flux magnitude and of the stator currents defined as (14). Using (13), the dynamic of lyapunov function is written as (15).

v2 =

⎡ − k1 ⎡ eω ⎤ ⎡ eω ⎤ ⎢ ⎢ ⎢ e ⎥ ⎢ ⎥ ⎢ ϕ ⎥ = E . ⎢ eϕ ⎥ = ⎢ 0 ⎢ e Isd ⎥ ⎢ e Isd ⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎣⎢ eIsq ⎦⎥ ⎣⎢ e Isq ⎦⎥ ⎢ − m ϕ rd ⎢ ⎣ j

(14)

2

v2 = eω eω + eϕ eϕ + e Isq e Isq + e Isd e Isd

(15)

− k2 b1ϕ rd

b1ϕ rd − k3

0

0

m ⎤ ϕ rd ⎥ ⎡e ⎤ j ⎥ ⎢ ω ⎥ (20) 0 ⎥ ⎢ eϕ ⎥ . 0 ⎥⎥ ⎢ e Isd ⎥ ⎢ ⎥ ⎥ e − k 4 ⎥ ⎣⎢ Isq ⎦⎥ ⎦

Figure 1 shows the block diagram of the proposed control design. Iqs-ref Calculation

φref+

2 2 − k 4 e Isd = − k1eω2 − k 2 eϕ2 − k 3 e Isq + f1e Isq + f 2 e Isd

0

The stability of the control scheme is insured if the matrix E (regression matrix) is Hurwitz what is verified by a good choice of Ki.

ωref +

2 2 eω 2 + eϕ 2 + eIsd + eIsq

0

Ids-ref Calculation

-

Iqs-ref +

Vqs-ref Calculation

Ids-ref

+

-

Vds-ref Calculation

Vqs-ref

Vds-ref

Va Three dq to Vb level SVM abc Vc

Where:

Ia

f1 = k3eIsq + Isq − ref − aI sq + a2ω N pϕrd + ω N p I sd + b

I sq I sd

ϕrd

Calculation

− cVsd

2

f2 = k4 eIsd + Isd − ref − aI sd − a1ϕrd − ω N p I sq − b

I sq

Observer

Vqs

(16)

Isq Isd ⎧  ⎪k3eIsq + Isq−ref − aIsq + a2ωNpϕrd + ωN p I sd + b ϕ − cVsd = 0 ⎪ rd ⎨ 2 ⎪k e + I − aI − a ϕ − ωN I − b I sq − cV = 0 sd p sq sq 1 rd ⎪⎩ 4 Isd sd −ref ϕrd

(17)

I sq I sd ⎞ ⎧ 1⎛  ⎟ ⎪Vsd −ref = ⎜ k3eIsq + Isq−ref − aIsq + a2ωNpϕrd + ωNp Isd + b ϕrd ⎠ c⎝ ⎪ (18) ⎨ 2 ⎪V = 1 ⎛ k e + I − aI − a ϕ − ωN I − b I sq ⎞ ⎟ 1 rd sd p sq ⎪ sq−ref c ⎜ 4 Isd sd −ref ϕrd ⎠ ⎝ ⎩ To guarantee a faster dynamic of the stator current, rotor flux and rotor speed, k3 and k4 must be positive parameters. Then (13) can be expressed as:  e ⎧ Isd = − k 3 e Isd + b1ϕ rd e ϕ

From the equations (8) and (19), we can deduce:

Ic

ω

Fig.1. Bloc Diagram of Backstepping Control Design for IM

The backstepping control considers that the speed and the rotor flux are accessible for feedback and the IM parameters are known exactly. Unfortunately, the rotor flux is inaccessible for measurement [29]. To avoid this problem, it is necessary to devise and implement an observer, such as: high order sliding mode observer and unknown input observer. III. HIGH ORDER SLIDING MODE OBSERVER DESIGN

The stator voltages control can be deduced as follows:

⎪ m ⎨ ⎪ e Is q = − k 4 e Isq − j e ω ⎩

Ib

Vds

− cVsq

ϕrd The dynamic lyapunov function v2 could be negative definite, if f1 and f2 are chosen equal to zero.

abc to dq

(19)

Sliding mode observer is widely used for IM control to estimate the speed ω and the flux φrd. The drawback of this method is the chattering phenomenon especially at high frequency oscillations [30]. To eliminate this undesirable phenomenon, high order sliding mode approach has been introduced by levant [31]. It merits of strong robustness to parameter variation and disturbance rejection. To simplify and clarify the development of HOSMO, the IM equations according to (1) are transform to (α-β) reference frame such as the following set of state variables equations:

⎧ ⎪ Is = aI 2 I s + a 2 Aϕ r + cU s ⎪ ⎨ϕ rd = ϕ r = bI 2 I s − Aϕ r ⎪ 1 ⎪ ω = ⎡⎣ m J ϕ r I sT − C r − f ω ⎤⎦ J ⎩

(21)

Where

⎛0 J =⎜ ⎝1

−1⎞ ⎛ − b1 ; A = ⎜⎜ ⎟ 0 ⎠ ⎝ − N pω

N pω − b1

⎞ ⎛1 ⎟⎟ ; I 2 = ⎜ ⎝0 ⎠

0⎞ ⎟ 1⎠

[

I s = I sα

I sβ

]

T

; ϕ r = [ϕ rα

ϕ r β ] ; U s = [U sα T

U sβ

]

T

The corresponding HOSMO for the system (21) is based on the method as proposed in [32]. It can be written as:

⎧ ⎪ Iˆs = aI 2 I s + a 2 Aϕˆ r + cU s ⎪ (22) ⎨ ϕˆ r = bI 2 Iˆs − Aϕˆ r + χ ⎪ 1 ⎪ ωˆ = ⎡⎣ m J ϕˆ r I sT − C r − f ω ⎤⎦ J ⎩ and are the estimated stator current and the rotor flux components, respectively, in the stationary reference frame. is the observer matrix gains to be designed. Currents and flux estimation errors are defined as follow:

⎧⎪ e I = Iˆs − I s ⎨ ⎪⎩ eϕ = ϕˆ r − ϕ r

(23)

The dynamic error observation is obtained from (21) and (22): ⎧⎪ eI = a 2 Aeϕ (24) ⎨ ⎪⎩ eϕ = bI 2 eI − Aeϕ + χ Let us select a sliding surface as follow:

⎡ S1 ⎤ 1 −1 Sobs = ⎢ ⎥ = A eI ⎣ S2 ⎦ a2

(25)

The dynamic sliding surface is:

1 −1 1  −1 S obs = A e I + A eI a2 a2

(26)

The derivative of ω is supposed constant compared to the derivative of Is and r. Consequently, term is considered null and the dynamic of sliding surface become: 1 ⎧  −1 d e I ⎪ S o b s = a A d t = eϕ (27) 2 ⎨ ⎪ S = e = b I e − A e + χ ϕ ϕ 2 I ⎩ obs define χ as

Let us the following control laws, called the twisting algorithm [33].

⎧⎪−λm sign ( Sobs ) χi = ⎨ ⎪⎩−λM sign ( Sobs ) i

i

⎧ λm ; eϕ ⎪

Where ⎨

SS ≤ 0 SS ; 0

element in feedback control scheme. Compared to HOSMO is different in that use of sign function of current error in the feedback correction item. Indeed, this section is devoted to detail the UIO design.

A. Problem formulation Consider systems presented as follow form: ⎧ x = f ( x , u , v ) (30) ⎨ 1 ⎩ y = Cx = x With ⎛ x1 ⎞ ⎛ f 1 ( x1 , x 2 , u , v ) ⎞ ⎜ 2⎟ ⎜ ⎟ 2 1 2 3 x ; C = ⎡In1 0n1×n2 "0n1×nq ⎤ x = ⎜ ⎟ f ( x, u , v ) = ⎜ f ( x , x , x , u , v ) ⎟ ⎣ ⎦ ⎜ ⎟ ⎜ .... ⎟ ...... ⎜ ⎟ ⎜ ⎟ ⎜ xq ⎟ ⎜ f q ( x, u , v ) ⎟ ⎝ ⎠ ⎝ ⎠ Where: with , 1, … , and • The state and ∑ . • The inputs , the set of bounded absolutely continuous functions with bounded intoW a compact subset derivatives from of . In such way the subvector is known while the remaining subvector is unknown. • The output and , , with , , . The UIO design need the adoption of some hypotheses which will be adopted in due courses. At this step, it is convenient to assume the followings assumptions [27]: (H1) The state x(t), the control u(t)and the unknown input v are bounded, i.e. ., and where , and are compacts sets. (H2)

There

,

exist

0,

such

that

for

all

k ∈ {1, ..., q − 1} , ∀ x ∈ R n , ∀ ( u , v ) ∈ U × V , T

⎛ ∂f k ⎞ ⎛ ∂f k ( x, u, v) ⎟ ⎜ k +1 k +1 ⎝ ∂x ⎠ ⎝ ∂x

α 2f I n ≤ ⎜ 1

⎞

( x, u , v )⎟ ⎠

≤ β f2 I n1 .

Another point should be to assume for 1… q-1, for all , , the map ,…., , , , into is one to one. from 1

(H3) The output x can be partitioned as follow:

(28)

with

,

and

. Such a partition ,

induces the following one

, ,

, ,

, , , ,

that as to satisfy the following two conditions: max

⎪⎩ λM ; λm + 2 eϕ

(29) max

The system evolves featuring a high order sliding mode, after a finite time. Therefore, Sobs = Sobs = 0 .

IV. UNKNOWN INPUTS OBSERVER DESIGN IM control requires a perfect knowledge about the state and sensitive parameter. Therefore, to design an UIO is a principle

(i) There exist

,

0, such that T

⎛ ⎞ ⎛ α v2 I m ≤ ⎜ ( x1 , x 2 , u , v ) ⎟ ⎜ v ∂ ⎝ ⎠ ⎝ ∂v ∂ f11

∂ f11

(x

, 1

,

⎞ , x 2 , u , v ) ⎟ ≤ β v2 I m ⎠

,

⎛ ⎜ (ii) R ank ⎜ ⎜ ⎜ ⎝

∂ f 11 ∂x

2

∂v

⎞

⎛ 2 I m1 ⎞ ⎛ K1 ⎞ ⎟ ; K1 = ⎜ ⎟; K ⎝ 2⎠ ⎝ I m1 ⎠

( x1 , x 2 , u , v ) ⎟

,

C = d ia g {C 1 , C 2 } ; C 1 = ⎡⎣ I m 0 ⎤⎦ ; C 2 = ⎡⎣ I p − m 0 ⎤⎦ ; 1

1

⎛ f 1 ( x 1 , x 2 , u , v ) ⎞ f ( u , x ) ⎞ ;  1 f ( u , x ) = ⎜ 1 ⎟ ⎟ f 2 ( u , x ) ⎟⎠ ⎝0 ⎠ 1

⎛ f21 ( x 1 , x 2 , u , v ) ⎞ ⎜ 1 1 2 3 ⎟ ⎛0⎞ ⎛ε ⎞  ⎜ f ( x , x , x , u , v) ⎟ ; ε = ⎜ ⎟ ;ε = ⎜ ⎟ f 2 ( u , x ) = ⎜ 2 ⎟ ⎝ε ⎠ ⎝0⎠ ⎜ .... ⎟ ⎜ f q ( x, u , v) ⎟ ⎝ ⎠

f 1 ( x , u ) + ε f 2 ( x, u )

(31)

⎡ x11 ⎤ y = Cx = ⎢ 1 ⎥ ⎣⎢ x2 ⎦⎥

Δ

1

!

!

Δ 2 (θ ) = b l o c k d i a g (

With

(32)

Therefore, the equation of the observer is given by the following form:  + −1 xˆ = f u (t ), xˆ (t ) − ( Λ ( x, u , v ) ) Δ (θ ) K Cxˆ − y (33)

(

)

Where: , , the left inverse of bloc diagonal matrix Λ with Λ Λ Λ . ; Λ ⎛ ⎞ ∂f 1 1 2 Λ 1 = b lo ckd ia g ⎜ I m 1 , 1 ( x , x , u , v ) ⎟ v ∂ ⎝ ⎠

⎛ ∂f 1 ⎞ , I ⎜ p − m ∂x 2 ( x , u ) , ⎟ ⎜ 1 ⎟ ∂f 2 ⎜ ∂f ⎟ = blockdiag ⎜ 2 ( x , u ) 3 ( x , u , v ) , ⎟ ∂x ∂x ⎜ ⎟ q −1 ∂f 1 ∂f k ⎜ ⎟ ......, , , , x u x u v ( ) ( ) ∏ ∂ x k +1 2 ⎜ ⎟ ∂ x k =1 ⎝ ⎠ 1

Λ2

1

θ

1 q

I(

p − m

1

)

C

2 q

I(

p − m

1

)

( p − m

1

)

.... q q

C

I

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

;

1

θ I

I

q −1

p − m1

,

m1

1

θ

2

1

,

θ

I

2 ( q −1)

p − m1

," ,

0 is the observer design parameter.

I

m1

1

θ

q

) I

p − m1

)

Observer (33) can take the following develop form: ⎧ xˆ 11 = f 1 1 ( xˆ 1 , xˆ 2 , u , vˆ ) − 2 θ q − 1 ( xˆ 11 − x 11 ) ⎪ ⎨  ∂f 1 q −1 ( 1 ( xˆ 1 , xˆ 2 , u , vˆ ) ) + ( xˆ 11 − x 11 ) ⎪ vˆ = − θ ∂v ⎩ 1 1  ⎡ xˆ 2 ⎤ ⎡ f 2 ( xˆ 1 , xˆ 2 , u , vˆ ) ⎤ ⎢ 2 ⎥ ⎢ 2 ⎥ 1 2 3  ⎢ xˆ ⎥ ⎢ f ( xˆ , xˆ , xˆ , u , vˆ ) ⎥ = ⎢ .... ⎥ ⎢ .... ⎥ ⎢ ⎥ ⎢ ⎥ 1 2 q q ⎢ xˆ q ⎥ ⎣⎢ f ( xˆ , xˆ , " , xˆ , u , vˆ ) ⎦⎥ ⎣ ⎦ −

(Λ

× Δ

−1 2

2

( xˆ 1 , xˆ 2 , . . . . , xˆ q , u , vˆ ) )

(θ ) K

2

(34)

+

( C 2 xˆ 21 − y 2 )

B. Rotor flux and resistance estiamtions The goal of this step is to estimate the rotor flux and the sensitive parameter Rr. In order to achieve this, the UIO is developed here for the IM. According to the IM model (21), the following suitable distributions are considered as follow: 5

⎪⎧ x = f ( x , u ) + ε ⎨ ⎪⎩ y = Cx

)

; for 1

(θ ) = b l o c k d i a g (

The above system can be condensed as follows:

(

2

C

⎛ Δ 1 (θ ) ⎞ ⎜ ⎟ ⎝ Δ 2 (θ ) ⎠

Δ =

Using these notations, system (30) can be written as follows:

⎧⎪ x1 = ⎨ 1 ⎪⎩ x =

!

With

(H4) Dynamic unknown input v(t) is completely unknown | | function, which is uniformly bounded that is where 0 is a real number. In fact, a simple notation change has been introduced: ⎛ x1 ⎞ ⎛ x 1 ⎞ ⎛ x1 ⎞ x = ⎜ 2 ⎟ ; x 1 = ⎜ 1 ⎟ ; x 2 = ⎜ 22 ⎟ ; ⎜ x ⎟ ⎜x ⎟ ⎝v ⎠ ⎝ ⎠ ⎝ ⎠

⎛ f ( u , x ) = ⎜ ⎜ ⎝

K

∂ f 21

,

⎛ ⎜ ⎜ = ⎜ ⎜ ⎜ ⎝

K = ⎜

⎟ = n2 + m ⎟ 1 2 1 2 ( x , x , u, v) ( x , x , u, v) ⎟ 2 ⎠ ∂x ∂v ∂ f 21

,

For all

∂ f 11

( x1 , x 2 , u , v )

is the system state. Thus, x is 3 ,

defined such as: for k=1, 2 and

,∑

3,

.

3 denotes a vector of directly measured outputs. The input vector , a compact set of 3 . The subvector of the input w is known while the remaining subvector 1 is unknown. According to the appropriate choice of two parts state, the control and the unknown input, it is easy to verify the availability of the assumptions (H1), (H2), (H3) and (H4). Thus, the different elements of UIO design can be distinguish: 5; 1;

3;

2; ;

2;

3; ;

3; ;

2; ;

C = d iag {C 1 , C 2 } ; C 1 = ⎡⎣ I 2 0 ⎤⎦ ; C 2 = [1 0 ] ;

θ

I2,

1

θ

2

I2 )

⎧ 1 ⎡ Is ⎤ ⎡ a ( R r ) I 2 I s + a 2 A ( R r )ϕ r + cU s ⎤ ⎪ x = ⎢  ⎥ = ⎢ ⎥ ⎦ ⎣ Rr ⎦ ⎣ 0 ⎪ ⎪ 1 ⎪⎪ 2 ⎡ ω ⎤ ⎡ ⎡⎣ m J ϕ r I sT − C r − f ω ⎤⎦ ⎤ ⎢ ⎥  ⎨ x = ⎢ ⎥ = ⎢J ⎣ϕ r ⎦ ⎢ b ( R ) I I − A ( R )ϕ ⎥⎥ ⎪ 2 s r r r ⎣ ⎦ ⎪ ⎪ ⎡Is ⎤ y = Cx = ⎢ ⎥ ⎪ ⎣ω ⎦ ⎪⎩

⎧⎪ Iˆs = a ( Rˆ r ) I 2 Iˆs + a 2 A ( Rˆ r )ϕˆ r + cU s − 2θ ( Iˆs − I s ) ⎨ + Rˆ r = − θ 2 ( a 2 I 2 ϕˆ r − a 2 Iˆs ) ( Iˆs − I s ) ⎪⎩ ⎡1 ⎤ ⎡ ωˆ ⎤ ⎢ ⎡⎣ m J ϕˆ r IˆsT − C r − f ωˆ ⎤⎦ ⎥ − ⎢ ⎥ = ⎢J ⎣ ϕˆ r ⎦ ˆ ) I Iˆ − A ( Rˆ )ϕˆ ⎥ ( b R ⎣ ⎦ 2 s r r r 0

0

m I sβ

0

m I sα

1

0 ⎤ ⎡ 1 − 1⎥ ⎥ 0 ⎥⎦

⎢θ 0 ⎢ 1 ⎢0 ⎣⎢ θ2

t (s)

1

(c)

⎤ ⎥ ⎥ ⎥ ⎦⎥

0

0

0.5

4

(35)

(e)

2 0

t (s) 1

3 2.5 0.5

0

0.5045

0.5

-4

x 10

5

(b)

0

-5

0

-4

0.5

t (s)

1

x 10

(d)

2 0 -2 -4

0 0.01

0.5

t (s) 1 (f)

0

-0.01

t(s)

1

0

0.5

t (s)

1

Fig.2. System performance using HOSMO at low speed (a) Real and estimated electromagnetic and load torque, (b) Torque error, (c) Reference, real and observer magnitude rotor flux (d) Flux error, (e) Reference, real and estimated rotor speed, (f) Speed error.

According to (33), the appropriate observer for IM can be designed in the following form:

⎡1 ⎢0 ⎢ ⎢⎣ 0

0.5

0.5

Then, the model (30) can be written under the following form:

+

0 1

Speed Error (%)

Δ 2 (θ ) = b lo ckd ia g (

1

-10

)

(a)

5

Cr Cem

0 -5

1

0

0.5

t (s)

0

−1

1

(c)

⎡2⎤ ⎢ 1 ⎥ × ( ωˆ − ω ) ⎣ ⎦

By choosing a real , representing the only design parameter of the observer, the whole state and the unknown input are estimated [26]. This parameter should be chosen as a compromise between the speed of convergence and noise convergence.

V. SIMULATION RESULTS In order to illustrate the performances and the robustness of HOSMO and UIO, many tests are carried out by considering parameter sensitivity. Both observers are tested with backstepping IM control for low and high speed.

A. Low Speed Operation With application of a load torque at 0.5s equal to 5 N.m (50% of rated torque), both HOSMO and UIO can operate at

0.5

t (s) 1 (e)

2 0

0 3 2.5 0.5

0

0.51

0.5

t (s)

-3

x 10

1 0 -1

0.5

(36)

Flux Error (%) Torque Error (%)

θ θ

2

Cr Cem

1

(b)

0 -3 x 10

1 0 -1

Speed Error (%)

,

⎠

| (Weber)

1

)

|

Δ 1 (θ ) = b lo ckd ia g (

1

(

(a)

0

Speed (rd/s)

∂v

Torque (N.m)

⎝

⎞

x1 , x 2 , u , v ⎟

| (Weber)

∂ f11

|

⎛

= blockdiag ⎜ 1,

10

Speed (rd/s)

Λ1

Torque (N.m)

⎡ f 11 ( x 1 , x 2 , u , v ) ⎤ ⎢ ⎥ ⎛2⎞ ⎛ 2I2 ⎞ 0 ⎥ ; K1 = ⎜ ⎟ ; K 2 = ⎜ ⎟ f (x, u ) = ⎢ 1 1 2 ⎝1 ⎠ ⎝ I2 ⎠ ⎢ f2 ( x , x , u , v)⎥ ⎢ ⎥ 2 ⎣ f ( x, u , v) ⎦ ⎛ ∂f 1 ⎞ Λ 2 = blockdiag ⎜ I 2 , ( x, u )⎟ 2 x ∂ ⎝ ⎠

Flux Error (%) Torque Error (%)

low speed of 3rd/s, as illustrated in figure 2 and 3, respectively. This shows the electromagnetic torque, rotor flux and rotor speed. Compared to figure 3 (a) and (c), figure 2 (a) and (c) indicates a good response for electromagnetic torque and rotor flux. But, the rotor speed level error, shown in figure 3 (f) is higher than that shown in figure 2 (f). Overall results, the proposed backstepping control can track the reference command accurately and quickly at low speed.

0.5

t (s) 1 (d)

0 0.01

0.5

t (s) 1 (f)

0

-0.01

0

0.5

t (s) 1

Fig.3. System performance using UIO at low speed (a) Real and estimated electromagnetic and load torque, (b) Torque error, (c) Reference, real and observer magnitude rotor flux (d) Flux error, (e) Reference, real and estimated rotor speed, (f) Speed error.

B. High Speed Operation The backstepping control, including HOSMO or UIO, illustrated in figure 4 and 5, respectively, has been also tested at high speed. A load torque is applied at 0.5s equal to 5 N.m (50% of rated torque). Compared to HOSMO, UIO present a low level error for electromagnetic torque, rotor flux and rotor speed. So, it is more performant at high speed.

0.5

t (s)

1

0.2 0 -0.2

| (Weber)

1

|

0

0

0.5

0

t (s)

0.5

1

(c)

0.5

Speed (rd/s)

0 400 340 0.5

0

t (s) (e)

350

200 0

0.5

0.52

0.5

-5

5 0 -5

1

t (s) 1

2 0 -2

3 0 -3

0 -5 x 10

(b)

0.5

t (s)

0

314 312 0.2

0

t (s)

0.5

t (s)

1

(f)

0

0.5

t (s)

10 5 0

Cr Cem

0

1

0

0.5

(a)

t (s)

1

(c)

0

0.5

Fig.5. System performance using UIO at high speed (a) Real and estimated electromagnetic and load torque, (b) Torque error, (c) Reference, real and observer magnitude rotor flux (d) Flux error, (e) Reference, real and estimated rotor speed, (f) Speed error.

C. Parameter Sensitivity To check the robustness of the developed observers, the following test amount to increase the rotor resistance with application of load torque at 0.2s equal to 5 N.m (50% of rated torque). Figure 6 and 7 show the simulation results under conditions of increasing rotor resistance by 50% (1.5 Rrn). For the proposed HOSMO, the levels errors for electromagnetic torque, rotor flux and rotor speed are higher than the UIO which have a decrease of those errors as shown in figure 7 (b), (d) and (f). Therefore, the UIO is more robust against parametric variations.

t (s) (e)

200

1

0,001 0 -0,001

t (s)1 (d)

0

0.5

0.25

t (s) 1 (f)

0

-0.25

1

0.5

0 -5 x 10

0.5

1

0.2045

0.5

0

0

0.5

t (s) 1

Fig.6. System performance using HOSMO with Rr variation (a) Real and estimated electromagnetic and load torque, (b) Torque error, (c) Reference, real and observer magnitude rotor flux (d) Flux error, (e) Reference, real and estimated rotor speed, (f) Speed error.

1

(d)

t (s) (e)

200

x 10

1

0.5

Torque (N.m)

t (s)

Speed Error (%) Flux Error (%) Torque Error (%)

Torque (N.m)

10 0

(a)

Cr Cem

0

Flux Error (%) Torque Error (%)

Torque (N.m)

(f)

(c)

0.5

t (s)1

Fig.4. System performance using HOSMO at high speed (a) Real and estimated electromagnetic and load torque, (b) Torque error, (c) Reference, real and observer magnitude rotor flux (d) Flux error, (e) Reference, real and estimated rotor speed, (f) Speed error.

20

0

t (s) 1

Speed Error (%)

0

0.5

1

0.5

(b)

0.01 0 -0.01

(a)

314 0.2

0

1

-3

2 0 -2

x 10

(b)

0 -4 x 10 2 0 -2 0 0.01

0.204

0.5

t (s)

0.5

t (s)

1

(d)

0.5

t (s) 1 (f)

0

-0.01

312

0

Speed Error (%) Flux Error (%) Torque Error (%)

0.506

0

| (Weber)

1

(e) 350 348 0.5

200

t (s)

(d)

|

0.5

-5

0.5

0

Speed (rd/s)

0 400

5 0

0 -3 x 10

t (s) 1

| (Weber)

|

0

Speed (rd/s)

t (s) (c)

0.5

0

1

0,2 0 -0,2

Cr Cem

10 5 0

|

0.5

(b)

Speed (rd/s)

0

Torque Error (%)

| (Weber)

1

Flux Error (%)

10 0

(a)

Cr Cem

Speed Error (%)

Torque (N.m)

20

0

0.5

t (s)

1

1

Fig.7. System performance using UIO with Rr variation (a) Real and estimated electromagnetic and load torque, (b) Torque error, (c) Reference, real and observer magnitude rotor flux (d) Flux error, (e) Reference, real and estimated rotor speed, (f) Speed error.

VI. CONCLUSION In this paper, the performance of backstepping control is shown. Also, two strategies of speed and rotor flux estimation are exposed, these concern the high order sliding mode observer and unknown inputs observer. The aim of this paper was to compare performances and features of these two kinds of observers. Moreover, the robustness of these observers against parameter variations of IM was investigated. From the simulation results, at low speed operation, HOSMO is the best. However, UIO provides excellent performance in high speed. The proposed UIO has proved to be more robust than the HOSMO when parameter variations of the IM occur. VII. APPENDIX : IM AND CONTROL PARAMETERS Induction machine parameters and its nominal values are given as follow: P=1.5Kw, Np=2, Ls=Lr=0.464H, Rs=5.72Ω, Rr=4.2Ω, M=8.99H, J=0.0049Kg.m² and f=0.003N.m/rd/s.

The backstepping control parameters are k1=2000, k2=1500, k3=1250 and k4=800.

[17]. D. Traoré, J. De Leon, and A. Glumineau, ‘‘Sensoreless induction motor

The HOSMO and UIO parameters are λm=0.3, λM=0.5 and θ=100.

Vol. 4, Iss. 10, pp. 1989 --- 2002, Mexico, October 2010. [18]. T-S. Kuen, M-H. Shin and D-S. Hyun, ‘‘Speed sensoreless stator flux-

VIII. REFERENCES [1]. C. Petrea Ion and C. Marinescu, ‘‘Three-phase induction generators for single-phase power generation: An overview’’, Renewable and Sustainable Energy Reviews 22, pp. 73---80, 2013. [2]. P.S. Mayurappriyan, J. Jeromeb and T. Centhil Raj, ‘‘Performance

improvement in an Indian wind farm by implementing design modifications in yaw and hub hydraulic systems: a case study’’, Renewable and Sustainable Energy Reviews 32, pp. 67-75, 2014. [3]. K. T. Chau, W. L. Li and C. H. T. Lee, ‘‘Challenges and opportunities of electric machines for renewble energy’’, Progress In Electromagnetic Research, Vol.2, No. 3, pp. 45-74, June 2012. [4]. A. Gouichiche and al, ‘‘Sensorless Sliding Mode Vector Control of Induction Motor Drives’’, International Journal of Power Electronics and Drive System, pp. 277-284, September 2012. [5]. A. Msaddek, A. Gaaloul and F. M’sahli, ‘‘A Novel Higher Order Sliding Mode Control: Application to an Induction Motor’’, International Conference on Control, Engineering & Information Technology, Vol. 1, pp. 13-21, Tunisia, 2013. [6]. A. Rhif, ‘‘A high order sliding mode control with PID sliding surface: simulation on a torpedo’’, International Journal of Information Technology, Control and Automation, Vol. 2, No. 1, pp.1-13, Australia, January 2012. [7]. A. Tohidi, A. Shamsaddinlou, and A.K.Sedigh, ‘‘Multivariable input-

output linearization sliding mode control of DFIG based wind energy conversion system‘‘, 9th Control Conference, pp. 1 - 6, June 2013. [8]. E. Stanislav, ‘‘Discrete Time Modeling and Input- Output Linearization of Current Fed Induction Motors for Torque and Stator Flux Decoupled Control’’, Cybernetics and information technologies, Vol. 14, pp. 128140, 2014. [9]. S. Aissi, L. Saidi and A. Rachid, ‘‘Passivity Based Control of Doubly Fed Induction machine Using a Fuzzy Controller’’, International Journal of Advanced Science and Technology, Vol. 36, pp. 51-64, November 2011. [10]. J-Y. Chen, ‘‘Passivity-Based Parameter Estimation and Position Control of Induction Motors via Composite Adaptation’’, Transactions of the Canadian Society for Mechanical Engineering, pp. 559-569, 2013. [11]. A.L, Nemmour and al., ‘‘Advanced Backstepping controller for induction generator using multi-scalar machine model for wind power purposes’’, Renewable Energy 36, pp. 2375---2380, October 2010. [12]. R. Trabelsi and al, ‘‘Backstepping control for an induction motor using an adaptive sliding rotor-flux observer’’, Electric Power Systems Research 93, pp. 1-15, December 2012. [13]. S. Chaouch, L. Chrifi, A. Makouf and M.S.N. Said, ‘‘Backstepping

adaptive observer-backstepping controller: experimental robustness tests on low frequencies benchmark’’, Control Theory & Applications, IET,

oriented control of induction motor in the field weakening region using Luenberger observer’’, 34th Annual Power Electronics Specialist Conference, Vol. 20, No. 4, pp. 1460 --- 1464, South Korea, June 2003. [19]. S. Meziane, R. Toufouti and H. Benalla, ‘‘Nonlinear Control of Induction Machines Using an Extended Kalman Filter’’, Acta Polytechnica Hungarica, Vol. 5, No. 4, pp. 41-58, 2008. [20]. I. Haj Brahim, S. Hajji and A. Chaari, ‘‘Backstepping Controller Design using a High Gain Observer for Induction Motor’’, International Journal of Computer Applications, Vol. 23, No.3, pp. 1-6, June 2011. [21]. M. Ghanes and G. Zheng, ‘‘On Sensorless Induction Motor Drives: Sliding-Mode Observer and Output Feedback Controller’’, Transactions on industrial electronics, Vol. 56, pp. 3405-3413, September 2009. [22]. H. Chehimi, S. HadjSaid and F. M'Sahli,, ‘‘A predictive controller for

induction motors using an unknown input observer for online rotor resistance estimation and an update technique for load torque’’, Journal of Control Theory and Applications, Vol. 11, pp 367-375, August 2013. [23]. V. I, Utkin, ‘‘Sliding mode control design principles and applications to electric drives’’, IEEE Transactions on Industrial Electronics, pp. 23-36, February 1993. [24]. N. Djeghali, M. Ghanes, S. Djennoune and J-P. Barbot, ‘‘Backstepping

Fault Tolerant Control Based on Second Order Sliding Mode Observer: Application to Induction Motors’’, 50th IEEE Conference on Decision

[25].

[26].

[27].

[28]. [29].

Backstepping Control Using an Adaptive Luenberger Observer with Three Levels NPC Inverter’’, International Journal of Electrical,

[30].

control analysis of two different speed sensoreless approaches for induction motor’’, 5th International Multi-Conference on Systems, Signals and Devices, pp. 1 --- 6, Amman, July 2008. [14]. R. Trabelsi, A. Kheder, M.F. Mimouni and F. M'Sahli, ‘‘Sensorless

[31].

Speed and Flux Control Scheme for an Induction Motor with an Adaptive Backstepping Observer’’, 7th International Multi-Conference

[32].

on Systems, Signals and Devices, pp. 1 - 7, Amman , June 2010. [15]. A. Lagrioui ; H. Mahmoudi, A. Abbou and M. Zazi, ‘‘Backstepping control of the induction machine with a sliding mode observer’’, International Conference on Multimedia Computing and Systems, pp. 1062 --- 1068, Tangier, May 2012. [16]. H-H. Young, K-K. Park and H-W. Yang, ‘‘Robust Adaptive

Backstepping Control for Efficiency Optimization of Induction Motors with Uncertainties’’, International Symposium on Industrial Electronics, pp. 878 --- 883, Cambridge, July 2008.

and Control and European Control, pp. 4598 --- 4603, Orlando, FL, December 2011. Y. Feng, M. Zhou, H. Shi and X. Yu, ‘‘Flux Estimation of Induction Motors Using High-order Terminal Sliding-Mode Observer’’, 10th World Congress on Intelligent Control and Automation, pp. 1860 --- 1863, Beijing, July 2012. H. Chehimi, S. HadjSaid and F. M'Sahli, ‘‘Unknown inputs observer based output feedback controller for rotor resistance estimation in IMs’’, International Journal of Sciences and Techniques of Automatic control & computer engineering, Vol. 5, pp. 1532- 1543, June 2011. H. Chehimi, S. HadjSaid and F. M'Sahli, ‘‘Unknown Input Observer Based Parameters Tuning for Induction Motors Drives’’, Eighth International Conference and Exhibition on Ecological Vehicles and Renewable Energies, pp. 1 --- 6, Monte Carlo, March 2013. J.P. Caron, J.P. Hautier. ‘‘Modeling and Control of Induction Machine’’, Technic Edition, 1995. A. Bennassar, A. Abbou, M. Akherraz, M. Barara, ‘‘Sensorless

[33].

Robotics, Electronics and Communications Engineering, Vol.7, No.8, pp. 612-618, 2013. Floquet, T ; Barbot, J.-P and Perruquetti, W, ‘‘Second order sliding mode control for induction motor’’, Proceedings of the 39th IEEE Conference on Decision and Control, pp. 1691 --- 1696, Sydney, December 2000. A. Levant, ‘‘Sliding order and sliding accuracy in sliding mode control’’, International journal on Control, pp. 1247-1263, 1993. H. Benderradji and et al., ‘‘Second order sliding mode induction motor control with a new Lyapunov Approach’’, 9th International MultiConference on Systems, Signals and Devices, pp. 1-6, Chemnitz, March 2012. J. Davila, L. Frid man and A. Levant, ‘‘Second-Order Sliding-Mode Observer for Mechanical Systems ‘‘, IEEE Transactions on automatic control, pp. 1785 - 1789, November 2005.