Sensorless passivity based control for induction motor via an adaptive observer

Accepted Manuscript Sensorless passivity based control for induction motor via an adaptive observer Ramzi Salim, Abdellah Mansouri, Azeddine Bendiabdellah, Sofyane Chekroun, Mokhtar Touam

PII: DOI: Reference:

S0019-0578(18)30371-9 https://doi.org/10.1016/j.isatra.2018.10.002 ISATRA 2903

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ISA Transactions

Received date : 6 March 2018 Revised date : 28 July 2018 Accepted date : 1 October 2018 Please cite this article as: Salim R., et al. Sensorless passivity based control for induction motor via an adaptive observer. ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.10.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Sensorless Passivity Based Control for Induction Motor via an Adaptive Observer Ramzi Salim*1, Abdellah Mansouri *2, Azeddine Bendiabdellah**3, Sofyane Chekroun***4, Mokhtar Touam*5 *

Laboratoire d’Automatique et Analysis des Systems (L.A.A.S.), Department of Electrical Engineering, Ecole Nationale Polytechnique -Maurice Audin- Oran. BP 1523 El’ M’naouer, Oran, Algeria.

** Laboratoire de Développement des Entrainements Electriques (LDEE), University of Sciences and Technology Mohamed Boudiaf, Oran, Alegria. El M’naouer, BP 1505, Bir El Djir 31000, Oran, Algeria. *** Department of Electrical Engineering, University Mustapha Stambouli of Mascara. BP.305, Route El Mamounia, 29000 Mascara-Algeria.

Corresponding Author: Ramzi Salim Polytechnic school Maurice Audin of Oran No: B.P 1523 El M’naouer 31000, Oran, Algeria Tel.: +213 554 334 765

E-mail: [email protected] Running Title: Sensorless Passivity Based Control for Induction Motor via an Adaptive Observer

Sensorless Passivity Based Control for Induction Motor via an Adaptive Observer Abstract The work of this paper addresses the study and application of control strategies based on the passivity of a sensorless induction motor (IM) in order to guarantee a high performance operation and to increase reliability at a lower cost. This control approach based on the passivity or the energy formulation is generally simple and physically meaningful. It achieves the control objective by reshaping the system natural energy and then injecting a damping term. A full-order adaptive observer is also considered to estimate the IM rotor flux and mechanical speed. These estimated quantities are then used in the control scheme. The observer gain is synthesized in the way that it minimizes the instability zone in the regenerative mode to a line in the torque‐speed plane. The control-observer set is tested on the trajectories of the various operating modes (motor mode, regenerating mode and low speed mode). Keywords: Induction motor, passivity control, sensorless, adaptive observer, regenerating mode. I. Introduction Induction motor (IM) sensorless speed control has become in recent years a major concern for many industrialists and an important research topic for the scientific community; this is justified by the large number of publications on this subject. Nevertheless, for the majority of IM control strategies, the precise knowledge of the rotational speed or the rotor axis position is more than indispensable, not only for the control but also for the observation, the identification, the diagnosis or the protection of the electrical drives. To achieve this requirement, the use of a mechanical speed sensor or an incremental encoder is a practical solution, but the inherent drawbacks of the use of this type of sensors are numerous. Indeed, they generate an increase of equipment complexity, an additional cost and a regular maintenance; hence, decreasing the reliability in many applications. Moreover, they are sensitive to electromagnetic interference, vibration and temperature; they also require a careful connection and an additional space for their installation which can cause difficulties for small machines. A large number of research work dedicated to the IM has been carried out in order to develop sensorless speed control techniques that perform well in terms of accuracy and robustness of the estimate. It is also to note that currently the designers of control techniques have at their disposal many techniques of estimation to realize variable speed drives without mechanical sensor. These techniques make use of the machine behavior model based on the observation techniques to synthesize linear or nonlinear observers allowing the reconstitution of the speed starting from the measured electrical quantities. Most techniques used being the Luenberger observer [1-3], the extended Kalman filter [4],

the adaptive observers [5-8], with high gain [9-10], with sliding modes [11-12], or the estimators with Model Reference Adaptive System [13-14]. It is to note that most of these control approaches found in the existing literature are based on the vector control technique. The Vector control technique in its various implementations is the de facto standard for high dynamic performance control of AC drives. Such a technique posses a superior dynamic performance compared to the use of classical linear methods. One of the most common implementations of vector control is the rotor-flux oriented control. Unfortunately, the main drawbacks of this technique are the assumption of the full state measurement (the flux measurement) and the problem of the only asymptotic decoupling between torque and flux control. The second problem can be solved with linearizing the controllers, while the approach to solve the first drawback has been the use of a nonlinear separation principle for which no rigorous theoretical justification is given. The present work investigates the control based on the passivity without a speed sensor dedicated to three-phase IM. The passivity-based control; also defined by the energy; has been successful in several control areas. This approach has not evolved from a mathematically motivated procedure like the feedback linearization but by considering physical properties such as energy conservation and passivity [15-18]. Briefly, it is aimed at reshaping the energy of the system in a way leading to the desired asymptotic output tracking properties. The main objective is to drive the system to the desired dynamics; leaving the closed loop system nonlinear; without cancelling the dynamics or introducing the controller singularities. In fact, several interesting results are obtained by applying this approach in various fields ranging from robot control [19-21], electrochemical system to electromechanical system such as induction and synchronous machines [18], [22-26]. The main objective of this paper work is therefore to verify the merits and effectiveness of this proposed approach; that is to see if the contribution of this proposed approach brings significant improvements to the control without speed sensor under various operating conditions such as the motor mode, the regenerating mode and the low speed operation. A full-order adaptive observer is also used to estimate the rotor flux and the mechanical speed of the IM [1-15]. This observer consists of a state observer associated with a speed adaptation loop [5-8]. The speed adaptation law uses the error of the stator currents with the estimated rotor fluxes; where the synthesis of the gain of the full-order adaptive flux observer with the speed adaptation loop allows the minimization of the operating instability zone of this observation to a line in the torque-speed plane in the regenerating mode. The stability is studied by using the necessary condition of stability based on the state matrix determinant of the linearized error between both the motor and the observer models [5-6]. The paper is structured as follows: after an introduction of the state of art in section one, the second section presents an Euler-Lagrange Model (ELM) of the IM which is to be compatible with the requirements of the synthesis of the control law based on passivity [18], [26]. Some definitions related to the concept of passivity that will be applied to the IM are also given in this section. The third

section is devoted to the synthesis of a control law based on the passivity of the IM and also dedicated to the synthesis of a full-order adaptive observer to estimate the rotor quantities such as the flux and the rotational speed that are required for the control. In the fourth section, the merits and effectiveness of the proposed approach are investigated through several simulation works conducted using the Matlab / Simulink environment; the proposed benchmarks are also designed to illustrate the different operating modes of the machine. The last section presents a conclusion of this paper work. II. Modeling and passivity of induction motor II.1 Induction motor Euler-Lagrange model The state representation requires a choice of the state variables, inputs and the outputs of the system; this choice depends on the objective related to the control or the observation. The state model of the IM is a nonlinear multivariable system. The ELM of the IM is one of the mathematical models given to this machine that allows us to design control law based on passivity [15], [18]. The dynamical equations of the IM combined with electrical and mechanical subsystems are written as follow:  De  qm  qe  W1  qm  qm qe  Re qe  M eU   1 T  Dm qm  Rm qm   L    qe W1  qm  qe 2 

(1)

Where

qe   qsT 

 qs1   q  T qr T    s 2  , U  usT    qr1     qr 2 

0T  

T

And qe is the currents vector, qm the rotor angular velocity, De  qm  the inductance matrix, U the voltage input, us the stator voltages,  L the load torque, Dm  0 the rotor inertia, Rm  0 the mechanical viscous damping constant.  Ls I 2 Lsr e Jpqm  R I De    , Re   s 2  Jpqm Lr I 2   0  Lsr e  cos  pqm  e Jpqm    sin  pqm 

0  I  0 , Me   2  , J    Rr I 2  1 0 

 sin  pqm    Jpqm  e Jpqm  e cos  pqm  



The flux vector   sT

T

r T   s1



, W1  qm  

s 2

r1

T

1  J T , 0 

dDe  qm  dqm T

r 2  is related to the current vector

through the following relations: s  Ls I 2 q s  Lsr e Jpqm qr   Jpqm qs  Lr I 2 qr r  Lsr e

(2)

From which the second expression of eq. (2) gives: qr 



1 r  Lsr e Jpqm q s Lr



(3)

Where Ls , Lr , Rs , Rr are the stator and rotor inductances and resistors respectively, Lsr the mutual inductance, p the pair pole number and I 2 the identity matrices  2  2  .

The system of equations (1) becomes:  W1  qm  qm qe   D( q )q   1 T Rq   Mu       2 qe W1  qm  qe  dissipated forces external forces  

(4)

workless forces

To perform the design of the controller, an appropriate linear factorization of its workless forces is necessary and is given as:  W1  qm  qm qe   1 T   C  q,q  q   2 qe W1  qm  qe 

(5)

More precisely, C  q,q  is given such that D  q   2C  q,q   C T  q,q   C  q,q  is the skew-symmetric and the third and fourth row of C  q,q  are independent of qe . These conditions are necessary for the following stability analysis. Using the transposed expression of W1  qm  , it is clear that the objectives can be achieved with the choice:  0  C  q,q     pLsr Je Jpqm  T   f  q,q 

0 0 0

f  q,q    0   0 

(6)

With f  q,q   pLsr Je Jpqm qr

(7)

The system can be then presented in the following compact form: D( q )q  C( q,q )q  Rq  Mu  

(8) T

T

Where D  q   diag  De  q m  , Dm  , R  diag  Re , Rm  , q  qeT ,qm  , M   M eT , 0 ,    0,   L  . T

From eq. (1) and W1  qm  , it is found that: 

1 T qe W1  qm  qe  pLsr qsT Je Jpqm qr 2

(9)

Eq. (9) is rewritten by substituting eq. (3) in it as follows: p

Lsr T Jpqm q s Je r Lr

(10)

II.2 Passivity of the IM II.2.1 Definition: The basic idea of the passivity consists in shaping the total energy of the system and then adding a damping term. The ELM equation makes it easy to obtain the formulation after expressing the total

energy of the system; it is modified to the desired minimum value. The system converges to this minimum also if the control capable of injecting an additive dissipative term into the system, then the convergence to the derived state can be improved compared to that obtained by the natural dissipation given by system. In other words, the passivity is a property that some physical systems possess and is related to the concept of input-output stability [15-18]. A system is said to be passive if there exists a non-negative function V : R n  R , called a storage function, with V  0   0 , as for all u  U , x  0   x0  X and t  0 , it is satisfied t

V  xt   V  x0    yT  s  u  s  ds

(11)

0

Where V is a continuous storage function with continuous derivatives of order r. Condition (11) can also be expressed as V  yT  t  u  t  [15]. The system is said to be lossless when V  yT  t  u  t  . II.2.2 Application to the IM The Hamiltonian of IM equals the kinetic energy E plus the potential energy P of the system. If the potential energy equals zero then H E

1 T 1 qe De qe  Dm qm 2 2 2

Hence, H 

(12)

1 T De qe qm qe  qeT De qe  Dm qm qm qm 2 

(13)

W1

Replacing W1  qm  and eq. (4) in eq. (13), it is found that: 1 H  qeT W1qmqe  qeT Mus W1qmqe  Re qe    L  Rmqm qm 2

H  qT Rq   qsT 

T

qmT  us 

(14)

 L 

H  q T Rq  yT u

(15)

By integrating eq. (15), H  t   H  0       Stored Energy

t



 q T Rqdt

t



y

T

udt

0   

0  

Dissipated Energy

Supplied Energy

t

t

t

t

0

0

0

0

   yT udt  H  0  , so that H  t   0    q T Rqdt    yT udt  H  0   0 That is H  t     q T Rqdt

When t

t

t

t

0

0

0

0

T T   H  0    yT uds   min  R  q ds  H  0   0  y uds   q Rqds 2

This outcome proves that the system (IM) is passive.

(16)

III. Passivity based control design for IM The IM model of eq. (8) has as outputs the parameters  and r which are to be controlled [18]. Assume that: 

 L is an unknown constant.



q s1 , qs 2 are available for measurement.



All motor parameters are exactly known and Rm  0 .

Given the desired torque  d to be a bounded and differentiable function with known bounded firstorder derivative, and the desired rotor-flux norm is a strictly positive bounded and twice-differentiable function  with known bounded first and second-order derivatives. Under these conditions, let us design a control law that will ensure internal stability, asymptotic torque and rotor flow norm tracking, i.e. the closed-loop system must give lim     d  0 , lim  r t 

t 

2

   0 , from all the initial

conditions and with all the signals uniformly bounded. By injecting a damping term, the control law is then defined as [13]: ˆ ˆ u  Ls qsd  Lsr e Jpqm qrd  pLsr Je Jpqm qˆr qmd  Rs qsd  K1  qmd  qs

(17)

Where qs  qs  qsd , the index d stands for the desired variable, with

qrd  e J sld

      Rr  , q  1 e J ( pqˆ m  sld ) sd  d  Lsr    p  

Lr      R   r    Lr  d    p 

The expression relating the slip speed to the torque can be written as: sld 

Rr p r

2

d

(18)

Therefore, it allows controlling the torque via controlling the rotor flux norm and the slip speed. qmd 





1 ˆ  pLsr qˆrT Je Jpqm qsd  Rm qmd  L  K2  qd  qm , qm  qˆm  qmd and q md  0   qˆm  0  Dm

The gains K1  qmd  , K 2  qd  are given by the following form: K1  qmd  

p 2 L2sr 2 qmd  k1 , 41

k1  0

p 2 L2sr 2 K 2  qd   q1d  q22d  k2 , 41





R 0  1  r , k2  0 2

(19)

With qˆr , qˆ m and qˆm are respectively the estimated rotor flux vector, the position and the speed. Under these conditions, the closed-loop system tracks the overall torque and the rotor flux norm with all the uniformly bounded signals. It is well known that the quality of the control laws for controlling the IM requires a good knowledge of the necessary state quantities. Technically and economically, the states quantities such as the rotor flux and the rotor speed in the context of a control without

mechanical sensor are not easily accessible to measurement. So, it is essential to use soft sensors through the design of estimators and observers [1-15]. Several types of these exist in the literature and among these state observers is the full-order adaptive observer. The adaptive observer is an observer constructed from a deterministic model of the considered process [8]. The solely objective of this adaptive observer used in this paper work is in fact the reconstruction of the rotor flux and the speed without a mechanical sensor; that is to compute the corresponding estimation values of these state quantities. In order to present the adaptive observer dynamic as existing in literature [5-8], the next step to follow is to express the state representation of the IM in the dq frame. It is common practice to present the model in the dq frame in terms of the fictitious stator currents idq and the rotor flux linkages dq [18], [26]. It is also to note that the following three equations are necessary in order to achieve the required dq transformation. rd   J   pq J   pq rdq     e  s m  r  r  e  s m rdq   rq 

(20.a)

isd  isdq     e Js qs  qs  e Js isdq  isq 

(20.b)

usd  usdq     e  J s us  us  e J s usdq  usq 

(20.c)

From eq. (1), it is found that: r  Rr qr  0

 qr  

1  r Rr

(21)

Substituting the derivative of eq. (20.a) in eq. (21), the obtained expression along with eq. (20.b) are then substituted into the eq. (3) and rearranging terms, multiplying from the left by e J s  pqm  , gives: rdq 

Lsr 1 isdq  s  pqm  J rdq   rdq Tr Tr

(22)

The part of the stator of eq. (1) is written as:





d Ls q s  Lsr e Jpqm qr  Rs qs  us dt

(23)

Now the derivative of eq. (20.a) substituted in eq. (21). The obtained expression with eq. (20.b) and eq. (20.c) are substituted into eq. (23) and rearranging terms, multiplying from the left by e  J s , gives:  d 1  1 isdq   s J   I  isdq  K  pqm J  I  rdq  usdq  dt T Ls r  

With Tr 

R R L 2 L L 2 Lr , σ  1  sr , K  sr , γ  s  r sr2 σLs σLs Lr Ls Lr σLs Lr Rr

(24)

From eq. (22) and eq. (24), and by considering  the motor angular speed, such that   pqm is a constant,  s the synchronous speed, the system of equ. (1) can also be written as follows K 1 d  dt isd   isd  s isq  T rd  K rq   L usd r s  d K 1 usq  isq  s isd   isq  K rd  rq  Tr  Ls  dt  d Lsr 1 isd   rd  sl rq  rd  dt T T r r  d Lsr 1 isq  sl rd  rq  rq  Tr Tr  dt d  0  dt

(25)

Where isd , isq are he stator currents and, rd ,  rq the rotor fluxes in the dq frame, sl the slip speed expressed as sl  s   . The conventional full-order model of the adaptive observer where the state matrix is found depends on the speed written in the following form:



 xˆ  Aˆ ˆ s , ˆ  xˆ  Bus  G isdq  iˆsdq   iˆsdq  Cxˆ

Where xˆ  iˆsd

iˆsq

ˆrd



(26) T

T

isq  , G  G1

ˆrq  , idq  isd

G2 

T

The speed adaptation law is given in [8] where:



 Lsr d ˆ  eisd ˆrq  eisq ˆrd dt c



(27)

With  a positive constant, eisd  isd  iˆsd , eisq  isq  iˆsq and c   Ls Lr . In practice, the proportionalintegral action is used in order to improve the dynamic behavior of the estimator, then:



 

d d ˆ  k p ei ˆrq  eisq ˆrd  ki eisd ˆrq  eiq ˆrd dt dt sd



(28)

By choosing the particular form of the gain G ; where G1  022 and G2  g 2 I 22 ; then from eq. (26) and eq. (28), the proposed observer can be written as: K 1 d ˆ ˆ ˆ ˆ ˆrq  usd  dt isd   isd  ˆ s isq  T ˆrd  K  Ls  r  d ˆ K 1 ˆ ˆ ˆ ˆrd  ˆrq  usq  isq  ˆ s isd   isq  K  Tr  Ls  dt  d Lsr ˆ 1 isd  ˆrd  ˆ sl ˆrq  g 2 eisd  ˆrd  dt T T r r  d Lsr ˆ 1 isq  ˆ sl ˆrd  ˆrq  g 2 eisq  ˆrq  Tr Tr  dt d d  ˆ  k p ei ˆrq  eisq ˆrd  ki eisd ˆrq  eisq ˆrd dt sd  dt



 

(29)



By linearizing eq. (25) and eq. (29) around an equilibrium operating point and subtracting both the linearized systems as depicted in Fig.1 or Appendix 1, the error dynamic is obtained and written as follows:  e  Aˆ1 e  B1u1

(30)

Where  e   eisd

 eisq



 erd

 erq

T

 e  , u1  s 

  .

To obtain the analytical conditions of the local stability via the necessary conditions for stability based on the matrix determinant Aˆ1 such that:     s 0   Lsr  g2 Aˆ1   Tr  0     k p s 0ˆ0 

K Tr  K 0

s 0 



0



1 Tr

Lsr  g2 sl 0 Tr ki   k p ˆ0  k p K 0ˆ0



K 0

0

K Tr

 K ˆ0

sl 0

0

1 Tr K k p ˆ0 Tr

ˆ0



k p K ˆ0

           

(31)

Then the following propriety is used when: 5

  

det Aˆ1 

(32)

i

1

With  i are the Eigen-values of Aˆ1 . Giving the determinant of this matrix as:

 

det Aˆ1   Lsr ˆ0 2 ki s 0

 

s0

 0   Ls Lr Tr  Lsr 2 0  Lsr 0 g 2 Tr  s 0 Ls Lr

   L L  T  2

s

r

(33)

 

Under the condition when det Aˆ1  0 , two solutions are found: s 0  0     g 2 Lsr  Rs Lr 0  s 0 Rr Ls  Rs Lr

(34)

Under these conditions, the stability of this observer may be expressed as a line in the torque-speed plane for regenerating mode [5-6]. From eq. (1), if the torque due to the friction is supposed negligible compared to the load torque then the mechanical equation is given as: p p d   l dt Dm Dm

(35)

In the steady state we have:   l

(36)

From eq. (18) and eq. (36), we obtain sl 0 

Rr p ˆr

2

l0

(37)

This implies that s 0  0 

Rr p ˆr

2

l0

(38)

Replacing s 0 found in eq. (34), the two lines defining the instability regions in the regenerating mode can be determined as: l0  

l0  

p ˆr

2

0

Rr p ˆr Rr

2

(39.a)

  g 2 Lsr  Rs Lr 1     Rr Ls  Rs Lr

   0 

(39.b)

For the lines obtained from eq. (39.a) and eq. (39.b) to be superposed (i.e. with the same slope), then in this case, the zone of unobservability (instability) will be minimal. This enables us to find the necessary gain g 2 of the matrix G such that:   g 2 Lsr  Rs Lr 1     Rr Ls  Rs Lr

    1 

Then 

g 2 Lsr  Rs Lr  0 , from which Rr Ls  Rs Lr

g2  

Rs Lr Lsr

.

(40)

(41)

Now, instead of the real rotor flux and speed for the control synthesis, the observed values of the latter which allows eliminating the speed sensor are used. In the following, the general flowchart describing the various steps involved in applying the proposed passivity control approach is illustrated in Fig.1.

Sensoless speed control based on passivity via an adaptive observer 

Euler Lagrange Model expressed in generalized coordinate 

D( q )q  C( q,q )q  Rq  Mu  

Transformation from Generalized coordinate to

Motor passivity demonstration t

t

0

0

T  y uds   min R  q ds  H  0   0

From dq to

dq frame

2

qˆr , qˆ m , qˆm  

generalized coordinate

u

Control based on passivity law

Estimated rotor components such as flux, position and speed 

The observer gain is synthesized in such a way that the instability zone is minimum in the torque/speed plane; there the necessary condition of stability study is used based on the determinant of the state matrix of the linearized

Linearizing both systems and making a subtraction between them, the linearized error dynamics is obtained and written in the following form

 e  Aˆ1 e  B1u1

error Aˆ1 . 

Obtained gain observer g 2  

Fig.1 Flowchart of the proposed passivity control approach

K 1 d  dt isd    isd   s isq  T  rd  K  rq   L u sd r s  d K 1  rq  u  isq   s isd   isq  K  rd   Ls sq Tr  dt  d Lsr 1 isd   rd   sl  rq   rd  dt T T r r  d L 1  rq   rq  sr isq   sl  rd  dt T T  r r d  0  dt

d  dt  d   dt  d   dt d   dt d   dt

K 1 ˆ ˆ rq  ˆ rd  K  iˆsd    iˆsd  ˆ s iˆsq  u  L s sd Tr K 1 ˆ ˆ rd  ˆ rq  iˆsq   ˆ s iˆsd   iˆsq  K  u  L s sq Tr

ˆ rd 

L sr ˆ 1 ˆ rd  ˆ sl ˆ rq  g 2 eisd isd  Tr Tr

ˆ rq 

L sr ˆ 1 ˆ rq  g 2 eisq isq  ˆ sl ˆ rd  Tr Tr

ˆ  k p







d eisd ˆ rq  eisq ˆ rd  k i eisd ˆ rq  eisq ˆ rd dt

In the next section, the theoretical results for the control and the observer presented above are simulated and discussed. VI. Simulations Results and Discussion In this section, the passivity-based control approach without speed sensor associated with an adaptive observer is simulated under the software Matlab / Simulink environment to study and validate the proposed approach. The performance of the control strategy is investigated by introducing a variation both in the load torque and the reference speed. The study is carried out for both the IM operating modes; that is the motor mode and the regenerating mode and the IM flux norm reference is taken as constant. The parameters and constants of the IM are shown in the appendix 2. Two benchmarks are proposed and commented in this paper as two case studies where the flux standard reference is kept constant in both cases. The objective is to impose the trajectories with profiles taking into account the observability problems of the IM when the speed information is not accessible to the measurement. 

The first, designated "case 1", shows the operation of the machine in both directions at high speeds.



The second, called "case 2", discusses different operating modes: the motor mode, the regenerating mode and the low speed operation under different load torque conditions.

Case 1: Motor mode In this first case, the IM speed is supposed to vary from 0 to 100 rad/s for an interval of [0-1] sec; it remains constant for the interval of [1-3] sec. The speed then decreases from 100 to -100 rad/s during an interval of [3-5] sec; then the speed remains again constant at -100 rad/s until it reaches 7 sec, then increases to 100 rad/s for an interval of [7-9,5] sec and remains constant until 12 sec, see Fig.2 (a). The load torque is taken constant varying from 7 N.m to -7 N.m within the following intervals respectively: [1.5-2.5] sec, [5.3-6.5] sec, see Fig.2 (d). The reference flux standard is constant taken equal to 0.85 wb, see Fig.2 (e). 150

100

des mes

speed [rad/s]

50

est 0

-50

-100

-150 0

2

4

6 time [s]

(a)

8

10

12

Tracking speed error

Estimation speed error

0.8

0.6

0.6 0.4

0.4 error [rad/s]

0 -0.2

0

-0.4

-0.6

2

4

6 time [s]

8

10

12

-0.6 0

2

4

(b)

6 time [s]

8

10

(c)

8

L 

6 4

torque [N.m]

-0.8 0

0.2

-0.2

-0.4

2 0 -2 -4 -6 -8 0

2

4

6 time [s]

8

10

12

(d) 1 0.9 0.8

des

0.7 flux Norm [wb]

error [rad/s]

0.2

mes

0.6

est

0.5 0.4 0.3 0.2 0.1 0 0

2

4

6 time [s]

(e)

8

10

12

12

Tracking flux norm error 0.06

0.02

0.04

0.015

0.02

0.01

0 error [wb]

error [wb]

Estimation flux norm error 0.025

0.005

-0.02

0

-0.04

-0.005

-0.06

-0.01

-0.08

-0.015 0

2

4

6 time [s]

8

10

12

-0.1 0

2

4

(f)

6 time [s]

8

10

12

(g)

Fig.2: IM speed, torque and flux variations under operating motor mode By observing the results obtained under the operating motor mode in Fig.2, the following findings can be highlighted: 1) The IM speed benchmark is taken as follows: 

When the reference speed is constant, the error is noticed to be zero. At start-up, there is small oscillatory estimation error around a value of 0.3 rad/s due to the imperfection of the flux estimation in the low speed range,



When inverting the reference speed by passing from the positive values to negative values and vice versa, strong oscillations can be observed due to the zero crossing regions where the motor operates at very low speeds as depicted in Fig.2 (b). Similarly, there is a small error when the speed varies,



2proposed controller has succeeded in tracking the reference continuously with an error due to the unobservable regions (zeros crossing, very low speed) and a steady-state error of the order of 0.3rad / s due to the load torque as seen in Fig.2 (c).

2) The IM torque benchmark is taken as follows: 

When the speed is constant, it is noticed that there is a good continuation of the torque. However, the torque seems to take more time to return back to the load torque value. During the transient response of the speed, the error is seen to have a significant value. Another small error also appears due to the existence of the friction torque. Important errors are exhibited when the speed varies, with oscillations during the inversion of the direction of the speed. There are also some oscillations due to the zero crossing of the estimated speed as seen in Fig. 2 (d). 

3) The IM flux benchmark is taken as follows: 

From an observation of the flux estimation in Fig.2, better results can be noticed. There is a small error at the startup with oscillations in the flux that do not exceed the value of 0.005wb and are manifested only when passing through the zero speed or during the inversion of the IM speed direction as shown in Fig.2 (f). During the inversion of the speed, errors which do

not exceed 5% are seen. Outside these areas, the error is practically zero. When applying the load torque as depicted in Fig.2 (g), it can be noted that the error is not much influenced.

Case 2: Regenerating mode In this second case, the IM speed is taken as zero for the interval of [0-0.5] sec; it then increases to 20 rad/s during the interval of [0.5-1] sec and then remains constant until it reaches 3 sec. The speed varies from 20 to 100 rad/s for the interval of [3-4] sec and remains constant until it reaches 6 sec; then it decreases from 100 to -20 rad/s in the period of [6-7] sec and remains constant until it attains 9 sec. During the period of [9-10] sec, the speed increases to 20 rad/s where it remains constant until it attains 12 sec. Similarly, the load torque is varied for various intervals: [0-1.5] sec, [1.5-2.5] sec, [2.55] sec and [5-12] sec corresponding to various values of 0 N.m, 7 N.m, 0 N.m and 7 N.m respectively, see Fig.3 (d). The reference flux norm is constant taken equal to 0.85 wb, see Fig.3 (e). It is to notice that through the given tracks, one remarks that the induction machine operates during the period of [0-7] sec as a motor for which the torque and the speed have the same signs and during the period [7-9.5] sec the machine works as a generator such that the induction machine torque and speed have not from the same signs. Then the induction machine returns to work again as a motor from 9.5s to the end of the benchmark time. All this information are illustrated in the following graphs in Fig.3.

140

des

120

mes est

speed [rad/s]

100 80 60 40 20 0 -20 -40 0

2

4

6 time [s]

(a)

8

10

12

Tracking speed error

Estimation speed error

1.2

0.6

1 0.4

0.8

error [rad/s]

0.6

0 -0.2

0.4 0.2 0 -0.2

-0.4

-0.4 -0.6

-0.6 2

4

6 time [s]

8

10

12

-0.8 0

2

4

6 time [s]

8

(b)

10

(c)

9

L

8



7 6 torque [N.m]

-0.8 0

5 4 3 2 1 0 -1 0

2

4

6 time [s]

8

10

12

(d) 1 0.9 0.8

des

0.7 flux norm [wb]

error [rad/s]

0.2

mes

0.6

est

0.5 0.4 0.3 0.2 0.1 0 0

2

4

6 time [s]

(e)

8

10

12

12

Estimation flux norm error

Tracking flux norm error

0.025

0.08

0.02

0.06

0.015

0.04

0.01 error [wb]

error [wb]

0.02

0.005 0

-0.02

-0.005

-0.04

-0.01

-0.06

-0.015 -0.02 0

0

2

4

6 time [s]

8

10

12

-0.08 0

(f)

2

4

6 time [s]

8

10

12

(g)

Fig.3: IM speed, torque and flux variations under operating regenerating mode Similarly, by observing the results obtained under regenerating mode in Fig.3, the following findings can be presented: 1) The IM speed benchmark is taken as follows: 

At the IM start-up, there is an error of the order 0.15 rad/s at very low speed operation. Similarly, when the speed varies, it is noted that the error is of the order of 0.2 rad/s. In the generator mode, there is a fairly large error in the form of oscillations of the order of 3% until the passage to the motor mode is reached which itself has some oscillations at low frequency.



The same errors are obtained as those of the estimation error, except at the moment of the appearance of the load torque; under these conditions, there is an error equals to 0.3 rad/s in the motor mode and some strong oscillations in the generator mode as seen in Fig.3 (b).



The errors mentioned previously in the estimated speed error depict their effects on the track of the measured speed of the reference, which is reflected by the tracking speed error as shown in Fig.3 (c).

2) The IM torque benchmark is taken as follows: 

When the speed is constant, a good torque tracking is obtained and only a small error exists due to friction torque as depicted in Fig.3 (d). This is not the case when working in the generator mode where the oscillations are clearly apparent; when the speed is variable, an error appears proportional to the value of the speed.

3) The IM flux benchmark is taken as follows: 

There is an insignificant estimation error at the IM start-up. During the inversion of the speed and the passage to the generator mode, strong oscillations are noticed. Similarly, during the inversion of the speed and the passage of the machine from the generator mode to the motor mode, some oscillations appear when the motor is operating at low speed as shown in the Fig.3 (f).



It can be observed that in the motor mode when the speed variation occurs, there is a very small error and a much smaller error when the torque appears. Peaks appear during the speed reversal with oscillations in the generator mode as illustrated in Fig.3 (g).

At the end of this section, the passivity-based control without speed sensor associated with an adaptive observer is simulated under the software Matlab / Simulink environment to study and analyze different operating modes such as the motor mode, the regenerating mode and the low speed operation mode under different load torque conditions. The performance results obtained for the proposed control strategy are acceptable and useful. The present paper work has the peculiarity of using the control based on the passivity compared to other existing works [2-3] [13-14] which mainly use the vector control technique. VI. Conclusion The present paper is addressing a very interesting issue related to the study and application of control strategies based on the passivity of a speed sensorless IM. The paper proposes essentially a control approach that enables the realization of a drive system without speed sensor, associated with a control based on the passivity for the induction machine. The particularity of this approach compared to the other existing approaches; implemented within the framework of the control of induction machine without speed sensor; is to make use of its physical concept (based on the system total energy formulation) rather than the mathematical concept (based on the system mathematical formulation). The proposed observer has advantages as long as the machine operates in the motor mode. The performance results obtained are acceptable and they show well the merits and effectiveness of the proposed control strategy. Unfortunately, when the machine operates in the regenerating mode, it exhibits instability in some areas. This requires introducing the concept of passivity for stability in a near future. Some perspectives of this work can also be oriented towards two main axes; the first related to the study of the robustness with respect to the machine parameters variation. The second axis can be directed towards real-time implementation under the Dspace 1104 card. Appendix 1:  Both linearized systems (motor and observer) K 1 d  dt  isd   isd  s 0  isq  T rd  K 0 rq   L  usd  isq 0 s r s  d 1 K  usq  isd 0 s  K 0    isq  s 0  isd   isq  K 0  rd  rq   dt T Ls r   d Lsr 1  isd  rd  sl rq  rd  Tr Tr  dt d L 1  rq  sr  isd   rd  sl 0 rq  0 sl dt T T  r r d    0  dt

d K 1   iˆsd   iˆsd  ˆ s0 iˆsq  ˆrd  Kˆ0ˆrq   u  iˆ ˆ  Ls sd sq0 s Tr  dt   d  iˆsq  ˆ s0 iˆsd   iˆsq  Kˆ0ˆrd  K ˆrq  1  usq  iˆsd 0ˆ s  Kˆ0ˆ  dt  Ls Tr  Lsr ˆ 1 d ˆ  isd  ˆrd  ˆ sl ˆrq  g1 eisd  rd  Tr Tr  dt d Lsr ˆ 1  ˆrq   isd  ˆrd  ˆ sl 0ˆrq  ˆ0ˆ sl  g1 eisq T dt T  r r    d ˆ  k  ˆ  iˆ  ˆ  iˆ  K ˆ ˆ  K ˆ ˆ  Kˆ ˆ   k e ˆ  e ˆ  ˆ  e p  s 0 0 sd 0 sq 0 0 rd 0 rq 0 s i isd 0 rq isq 0 rd 0 isq  dt Tr   





Appendix 2: Induction machine data Parameters Values Rr

4.3047 Ω

Rs

9.65 Ω

Lsr

0.4475 H

Ls

0.4718 H

Lr

0.4718 H

Dm

0.0293 kg / m 2

p

2

Rm

0.0038 Nms / rd

References [1] A. Bennassar, A. Abbou, M. Akherraz, M. barara, "Fuzzy Logic Based Adaptation Mechanism for Adaptive Luenberger Observer Sensorless Direct Torque Control of Induction Motor", Journal of Engineering Science and Technology, Vol. 11, No. 1, (2016), 046 – 059. [2] M. Jouili, Y. Agrebi, Y. Koubaa and M. Boussak, "A Luenberger state observer for simultaneous estimation of speed and stator resistance in sensorless IRFOC induction motor drives", 16th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA), Monastir, 2015, pp. 898-904. [3] Zorgani YA, Koubaa Y, Boussak M. MRAS state estimator for speed sensorless ISFOC induction motor drives with Luenberger load torque estimation. ISA Trans, 2016 Mar, 61:308-317. [4] Z. Yin, G. Li, Y. Zhang, J. Liu, X. Sun and Y. Zhong, "A Speed and Flux Observer of Induction Motor Based on Extended Kalman Filter and Markov Chain," in IEEE Transactions on Power Electronics, vol. 32, no. 9, pp. 7096-7117, Sept. 2017.

[5] Ouyang Y, Maerz M. "Stable analysis on speed-adaptive observer in regenerating mode". Sci China Tech Sci, 2012, 55: 213-222. [6] E. Etien, C. Chaigne and N. Bensiali, "On the Stability of Full Adaptive Observer for Induction Motor in Regenerating Mode," in IEEE Transactions on Industrial Electronics, vol. 57, no. 5, pp. 1599-1608, May 2010. [7] A. Dib, "Sensorless indirect adaptive control with parameters and load-torque estimation of induction motor", CCCA12, Marseilles, Pages 1-6, 2012. [8] H. Kubota, I. Sato, Y. Tamura, K. Matsuse, H. Ohta and Y. Hori, "Regenerating-mode low-speed operation of sensorless induction motor drive with adaptive observer", in IEEE Transactions on Industry Applications, vol. 38, no. 4, Pages 1081-1086, Jul/Aug 2002. [9] Y. Wang, L. Zhou, S. A. Bortoff, A. Satake and S. Furutani, "High gain observer for speedsensorless motor drives: Algorithm and experiments", IEEE International Conference on Advanced Intelligent Mechatronics (AIM), Banff, AB, Pages 1127-1132, 2016. [10] Abdelaziz Maouche, Mohammed M’Saad, Bachir Bensaker, and Mondher Farza, "High Gain Adaptive Observer Design for Sensorless State and Parameter Estimation of Induction Motors", International Journal of Control, Automation, and Systems, vol.13, no. 6, Pages 1-12, 2015. [11] S. Wu, Z. Liu and H. Su, "Adaptive sliding mode sensorless vector control of induction motor using sliding mode MRAS observer", 36th Chinese Control Conference (CCC), Dalian, Pages 91249129, 2017. [12] Dong, C., Vo, H., Tran, T., Brandstetter, P., Simonik, P. "Application of Sensorless Sliding Mode Observer in Control of Induction Motor Drive". Advances in Electrical and Electronic Engineering, North America, 15, Jan. 2018. [13] N. Bensiali, E. Etien, N. Benalia, Convergence analysis of back-EMF MRAS observers used in sensorless control of induction motor drives, Mathematics and Computers in Simulation, Volume 115, 2015,Pages 12-23. [14] Korzonek, M., & Orlowska-Kowalska, T. (2016). Stability analysis of MRAS SCC speed estimator in motoring and regenerating mode, Power Electronics and Drives, 1(2), 113-131. [15] C. I. Byrnes, A. Isidori and J. C. Willems, "Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems", in IEEE Transactions on Automatic Control, vol. 36, no. 11, Pages 1228-1240, Nov 1991. [16] Mizumoto, Ikuro & Ohdaira, Satoshi & Iwai Z, "Output feedback strict passivity of discrete-time nonlinear systems and adaptive control system design with a PFC, Automatica, vol. 46, 1503-1509, 2010. [17] Leipo Liu n, Zhumu Fu, Xiaona Song, "Passivity-based sliding mode control for a polytopic stochastic differential inclusion system", ISA Transactions, vol. 52, no. 6, 775–780, November 2013.

[18] P. J. Nicklasson, R. Ortega, G. Espinosa-Perez and C. G. J. Jacobi, "Passivity-based control of a class of Blondel-Park transformable electric machines", in IEEE Transactions on Automatic Control, vol. 42, no. 5, Pages 629-647, May 1997. [19] Hsin-Chen Hu, Yen-Chen Liu, "Passivity-based control framework for task-space bilateral teleoperation with parametric uncertainty over unreliable networks", ISA Transactions, vol. 70, 2017. [20] Z. Lin, J. Liu, W. Zhang and Y. Niu, "Passivity and feedback design of nonlinear Markovian jump systems", 25th Chinese Control and Decision Conference (CCDC), Guiyang, Pages 154-158, 2013. [21] Jin Erdong, Sun Zhaowei, "Passivity based control for a flexible spacecraft in the presence of disturbances", International Journal of Non-Linear Mechanics, vol. 45, Pages 348–356, 2010. [22] S. Aranovskiy, R. Ortega and R. Cisneros, "Robust PI passivity-based control of nonlinear systems: Application to port-Hamiltonian systems and temperature regulation", American Control Conference (ACC), Chicago, IL, Pages 434-439, 2015. [23] Reine Talj, Romeo Ortega, Alessandro Astolfi, "Passivity and robust PI control of the air supply system of a PEM fuel cell model", Automatica, vol. 47, Issue 12, Pages 2554-2561, 2011. [24] J. Linares-Flores, J. L. Barahona-Avalos, H. Sira-Ramirez and M. A. Contreras-Ordaz, "Robust Passivity-Based Control of a Buck–Boost-Converter/DC-Motor System: An Active Disturbance Rejection Approach", in IEEE Transactions on Industry Applications, vol. 48, no. 6, Pages. 23622371, Nov.-Dec. 2012. [25] R. Mocanu and A. Onea, "Robust control of externally excited synchronous machine based on passivity theory", 24 th Mediterranean Conference on Control and Automation (MED), Athens, Pages 473-478, 2016. [26] Delaleau, Emmanuel, Louis, Jean-Paul, and Ortega Romeo, "Modeling and control of induction motors", International Journal of Applied Mathematics and Computer Science, vol. 11, no.1, Pages 105-129, 2001.

 

Highlights

 The control based on the passivity without a speed sensor dedicated to the Induction motor, was not mathematically motivated procedure but by considering physical properties such as energy conservation and passivity,  Our goal is to submit the system behavior to a desired dynamics; leaving the closed loop system nonlinear; without cancelling the dynamics or introducing the controller singularities.  An adaptive observer is proposed for estimate the rotor flux that is inaccessible to measurement and speed ,  The stability is studied using necessary condition to determine the observer gain in the line torque/speed in regenerative mode.  throughout this approach we overcame the need to use a mechanical speed sensor.