Robust nonlinear receding-horizon control of induction motors

Electrical Power and Energy Systems 46 (2013) 353–365

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Robust nonlinear receding-horizon control of induction motors Ramdane Hedjar a,⇑, Patrick Boucher b, Didier Dumur b a b

Computer Engineering Department, College of Computer and Information Sciences, King Saud University, P.O. Box 51178, Riyadh 11543, Saudi Arabia Automatic Department, Supélec - Paris, F91192 Gif-Sur-Yvett, France

a r t i c l e

i n f o

Article history: Received 11 January 2012 Received in revised form 13 September 2012 Accepted 9 October 2012 Available online 27 November 2012 Keywords: Robust control Predictive control Induction motor Kalman filter Cascade structure

a b s t r a c t Nonlinear robust receding-horizon control is designed and applied to fifth-order model of induction motor in cascade structure. The controller is based on a finite horizon continuous time minimization of the predicted tracking errors and no online optimization is needed. The initial system is decomposed into two sub-systems (mechanical and electromagnetic sub-systems) in cascade form. An integral action is incorporated in external loop to increase the robustness of the controller with respect to unknown time-varying load torque and mechanical parameters uncertainties. The control uses only measurement of the rotor speed and stator currents. The rotor flux is estimated by Kalman filter. The proposed nonlinear controller permits to achieve asymptotic speed and flux tracking in presence of mechanical parameters uncertainties, unknown variable load torque and resistances variations. In addition, it assures asymptotic decoupling of the speed and flux subsystems. The controller is applied, via simulation, to a benchmark example.  2012 Elsevier Ltd. All rights reserved.

1. Introduction Induction machines are very attractive and widely used in many industrial applications owing to their size, low cost, simple structure and high reliability. Nevertheless, they represent a highly coupled and nonlinear multivariable system. Furthermore, the rotor flux is not usually measurable and there are uncertain critical parameters in addition to load torque, which is typically unknown in all electric drives. Thus, the induction motors constitute an important area of application for nonlinear robust control theory. The field-oriented control technique has been widely used in industry for high performance induction motor drive because it gives control characteristics similar to separately excited DC motor. However, the performance is sensitive to the variation of motor parameters, especially the rotor time constant which varies with temperature [1–3]. To increase the performance of field-oriented control, much attention has been given to the possibility of identifying the changes in the motor parameters of an induction motor while the drive is in normal operation. Indeed, most of the proposed nonlinear control strategies use online rotor resistance estimator to achieve better tracking performance and to maintain the decoupling property. Indeed, authors in [4–6,35] have proposed a speed/torque and flux tracking adaptive controller without measurements of the rotor fluxes while adapting to both rotor resistance and unknown load torque. However, to ensure the convergence, the persistent excitation condition should be satisfied. It is stated in [7] that the persistent excitation condition ⇑ Corresponding author. Fax: +966 1 4676990. E-mail address: [email protected] (R. Hedjar). 0142-0615/$ - see front matter  2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.10.007

is not satisfied when the electric torque is absent due to lack of currents. While in [9], authors have proposed an indirect field oriented control (IFOC) which guarantees global exponential speed-flux tracking but under the condition of constant value of load torque and rotor resistance. The load torque estimator was utilized in this control scheme. Also in [10], authors have proved the asymptotic stability of indirect field oriented control of induction motor in the presence of rotor resistance and load torque variations. Moreover, the stator resistance may also vary up to 50% during motor operation. It becomes critical in low speed region [8]. Thus, both of stator and rotor resistances have been estimated online and the convergence is also under persistent excitation condition [7,8,17]. Many other researchers have designed nonlinear controllers for induction motors which are robust to mechanical parameters variations and load torque variations only [11–13]. A discrete-time sliding mode control has designed and applied to induction motor [14]. This control algorithm is robust to unknown load torque variations. However, the robustness of this algorithm to resistances variations has not been reported. A predictive optimal control strategy has been proposed for the control of the flux and speed of an induction machine [15]. In this work, authors use a linearized model and good tracking performances have been obtained. The load torque is estimated and the robustness of this control scheme to resistances variation has not been mentioned. Recently, artificial intelligence has been used to control the speed and rotor flux of induction motor [16]. Here also the robustness of the algorithm to resistances variations has not been reported.

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Consequently, for better performances, many authors have used online parameters estimation schemes, especially for unknown rotor resistance and load torque variations, which are available in the literature, and broadly classified in [18]. The proposed algorithm in this work does not need to estimate unknown mechanical parameters neither resistances variations nor load torque variations. Nonlinear model predictive control (NMPC) of nonlinear systems has received considerable attention in the last years owing to its robustness with respect to parameter variations and allows the explicit consideration of state and input constraints [19]. Thus, the NMPC is evaluated implicitly online by solving a constrained optimization problem. A serious limitation in using NMPC is the presence of fast plant dynamics, which require high sampling frequencies that do not allow to perform the constrained nonlinear optimization problem online [19–21]. For the above reason, this kind of control scheme is usually applied to industrial processes characterized by a slow dynamics such as chemical processes. Consequently, the application of NMPC with online optimization to systems characterized by fast dynamics (like asynchronous machine), sounds like an unusual proposal. To avoid this, several offline nonlinear predictive laws have been developed in [22,23], where the one-step ahead predictive error is obtained by expanding the output signal and reference signal in a r i th order Taylor series, where ri is the relative degree of the ith output. Then, the continuous minimization of the predicted tracking errors is used to drive the control law. This work examines the nonlinear receding-horizon control approach with integral action based on a finite horizon dynamic minimization of the tracking errors, to achieve torque and rotor flux amplitude tracking objectives. An extension to speed control is realized using a cascaded structure. This is a modified version of the Ping’s method [22]. Note that the proposed approach in [22] cannot be applied to induction motor since the derivative of the control signal will appear in the cost function. The advantages of the proposed control law include good tracking performance and good robustness property with respect to mechanical parameters uncertainties, unknown and variable load torque and resistances variations. Moreover, the flux weakening operation has no effect on the speed behavior. This paper aims at the development of: 1. Asymptotically stable output feedback controller with speed-flux-tracking capability in the presence of unknown load torque. 2. Robustness of the closed loop algorithm against stator and rotor resistances variations. 3. Robustness of the closed loop algorithm against mechanical parameters uncertainties. 4. Less power consumption with regards to many proposed algorithms in [9,32–34]. The present contribution is an improvement of the previous work [24] in terms of the robustness of the nonlinear recedinghorizon control algorithm against mechanical parameters uncertainties. Moreover, a global convergence and stability analysis of the closed-loop system have been investigated in this work. The paper is organized as follows. After the mathematical model of the induction motor developed in Section 2, a brief overview of the optimal nonlinear receding-horizon control theory is presented in Section 3. In Section 4, the previous scheme is extended to speed control by a cascaded nonlinear control structure. Stability and robustness of the proposed algorithm are treated in this section. The rotor flux observer is presented in Section 5. Significant simulation results are given in Section 6 for the nominal and mismatched model of the induction motor with bounded

and unknown time-varying load disturbance. The paper ends up with the concluding remarks and suggestions in Section 7. 2. Mathematical model Assuming linear magnetic circuits, the dynamics of induction motor are given by the well-known fifth-order model, see for instance [1] for its derivation and modeling assumptions:

x_ ¼ fðxÞ þ g u

ð1Þ

with x ¼ ½isa isb /ra /rb XT ; u ¼ ½usa usb T where isa, isb: stator currents, /ra, /rb: rotor fluxes, X: speed, usa, usb: stator voltages. Vector function f(x) and constant matrix g are defined as follows:

2

3

c isa þ Tkr /ra þ pXk/rb

7 6 6 c isb þ k /rb  pXk/ra 7 Tr 7 6 7 6 Lm 1 7; i  /  p X / fðxÞ ¼ 6 s a r a rb Tr Tr 7 6 7 6 Lm 1 7 6 i  / þ p X / sb rb r a T T r r 5 4 p LJLm ð/ra isb  /rb isa Þ  ðT L þfJ XÞ r " 1 #T 0 0 0 0 r Ls g ¼ ½ g1 g2  ¼ 0 r 1Ls 0 0 0 All required parameters above have the following meanings:

L2 r¼1 m ; Ls Lr

Lm k¼ ; rLs Lr

1 L2 c¼ Rs þ Rr m2 rLs Lr

!

where Ls, Lr are stator and rotor inductances, Lm is the mutual inductance, Rs, Rr are stator and rotor resistances, Tr = Lr/Rr is the rotor time constant, p is the pole pair number, J is the inertia of the machine, f is the friction coefficient, TL is the load torque considered as an unknown disturbance. Considering the torque and squared rotor flux modulus as outputs of the AC drive, the following equations can be derived, with y1 as the torque and y2 as the rotor flux norm:

(

y1 ¼ h1 ðxÞ ¼ p LLmr ðura isb  urb isa Þ

ð2Þ

y2 ¼ h2 ðxÞ ¼ u2ra þ u2rb ¼ u2r 3. Nonlinear receding-horizon law

In the receding-horizon control strategy, the following control problem is solved at each t > 0 and x(t):

MinuðtÞ JðxðtÞ; t; uðtÞÞ ¼

1 2

Z

tþT

LðsÞds

t

¼ MinuðtÞ

1 2

Z

tþT

½xðsÞT QxðsÞ þ uðsÞT R uðsÞds

t

ð3Þ subject to the Eq. (1) and x(t + T) = 0 for some T > 0, where Q is positive definite matrix and R positive semi-definite matrix. To solve this nonlinear dynamic optimization problem with equality constraints is highly computationally intensive, and in many cases it is impossible to be performed within a reasonable time limit, especially for systems with extremely fast dynamics like induction motor. Furthermore, the global optimization solution cannot be guaranteed in each optimization procedure since; in general, it is a non-convex, constrained nonlinear optimization problem.

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In order to find the current control that improves tracking error along a fixed interval, the output tracking error e(s) is used instead of the state vector x(s) in the above receding-horizon control optimization problem. Thus, the predicted tracking error will be used in the cost function (3) instead of the state vector x(t). The cost function can be rewritten as:

Min Jðe; u; TÞ ¼ Min

Z

tþT

and Pðx; hÞ ¼ j #ðx; hÞ 0    0 j, with #ðx; hÞ ¼ W T ðxÞQ ðKT ð2mhÞ P Pm T T aðt; 2mhÞ þ 2 m1 i¼1 K ð2ihÞaðt; 2ihÞ þ 4 i¼1 K ðð2i  1ÞhÞ aðt; ðð2i  1ÞhÞÞÞ;

aðt; zÞ ¼ eðtÞ þ V y ðx; zÞ  dðt; zÞ;

and bðt; zÞ ¼ KT ðzÞQ KðzÞ:

For the unconstrained case, the minimum is reached at:

UðtÞ ¼ R1 Pðx; hÞ

LðsÞds

ð7Þ

t

where LðsÞ ¼ 12 eT ðsÞQeðsÞ þ 12 uT ðsÞRuðsÞ. The interval [t, t + T] is divided into 2m subintervals of equal T width h ¼ 2m . Using the composite Simpson’s rule for 2m intervals, the objective function can be approximated as: m1 m X X h J¼ LðtÞ þ Lðt þ 2mhÞ þ 2 Lðt þ 2ihÞ þ 4 Lðt þ ð2i  1ÞhÞ 3 i¼1 i¼1

!

Note that for linear constrained case, the optimization problem is convex. Thus, standard numerical procedures are available to solve this quadratic optimization problem. From the receding-horizon principle, only the first control signal u(t) is applied to the process and the whole procedure is repeated at the next time instant. For the application of this nonlinear receding-horizon control to induction motor we took two subintervals (m = 1). Thus the cost function (4) becomes:

ð4Þ Thus, the problem consists in elaborating a control law u(x, t) that improves the tracking accuracy along the interval [t, t + T], such that the output vector yðt þ TÞðyðtÞ 2 Rp1 Þ tracks the desired vector yref(t + T). Note that the desired output trajectory is specified by a smooth vector function yref(t + T) for t 2 bt0, tfc. That is, the tracking error is defined at time jh as:

eðt þ jhÞ ¼ yðt þ jhÞ  yref ðt þ jhÞ

2

ri

ðjhÞ 2 ðjhÞ ri L hi þ . . . . . . þ L hi 2! f ri ! f

ðjhÞri r þ Lg Lf i hi u for i ¼ 1; . . . ; p ri !

1 2

Z

tþT

LðsÞ ds ¼

t

    T T LðtÞ þ 4L t þ þ Lðt þ TÞ 6 2

h ¼ ½LðtÞ þ 4Lðt þ hÞ þ Lðt þ 2hÞ 3 with T = 2h is the prediction horizon. The above performance index can be written using Eq. (4):

J¼

for j = 0–2 m. A simple and effective way of predicting the influence of u(t) on y(t + jh) is to expand it into a ri th order Taylor series expansion, in such a way to obtain, for each component of the vectors:

yi ðt þ jhÞ ¼ hi ðtÞ þ jhLf hi þ

J¼

h T ½e ðtÞQeðtÞ þ uT ðtÞRuðtÞ þ 4eT ðt þ hÞQeðt þ hÞ 3 þ 4uT ðt þ hÞRuðt þ hÞ þ eT ðt þ 2hÞQeðt þ 2hÞ þ uT ðt þ 2hÞRuðt þ 2hÞ

The expansion of the motor outputs y(t + l) in a rth (with r1 = 1 and r2 = 2) order Taylor series in compact form is:

yðt þ lÞ ¼ yðtÞ þ Vy ðx; lÞ þ KðlÞ WðxÞ uðtÞ ð5Þ

For the desired trajectory, Taylor approximation is given by:

ð8Þ

ð9Þ

where          l 0 l  Lf h 1  y1   Lg h1 Lg2 h1   ; KðlÞ ¼  2 ; Vy ðx;TÞ ¼  yðtÞ ¼  ; WðxÞ ¼  1 : 2 2  l l 0 2   l Lf h2 þ Lf h2  Lg 1 Lf h 2 Lg 2 Lf h 2 y2 2

yref ðt þ jhÞ ¼ yref ðtÞ þ dðt; jhÞ ðriÞ

ðjhÞri y

where di ðt; jhÞ ¼ jhy_ refi þ . . . . . . þ r ! refi . i Hence, the dynamic of the ith tracking error can be rewriting as:

yref ðt þ lÞ ¼ yref ðtÞ þ dðt; lÞ

ei ðt þ jhÞ ¼ ei ðtÞ þ V yi ðx; jhÞ  di ðt; jhÞ þ Ki ðjhÞW i ðxÞuðtÞ

eðt þ jhÞ ¼ eðtÞ þ Vy ðx; jhÞ  dðt; jhÞ þ KðjhÞWðxÞuðtÞ 2

ð6Þ

ri

where V yi ðx; jhÞ ¼ ðjhÞLf hi þ ðjhÞ L2f hi þ . . . . . . þ ðjhÞ Lrif hi ; 2! ri !

KðjhÞ ¼ diag

r2

rp

ðjhÞ ðjhÞ ðjhÞ ; ;...; r1 ! r2 ! rp !

! ;

   L Lr1 h   g f 1     .. WðxÞ ¼  : .     rp  L g L f hp 

Using the approximated tracking error, the cost function (4) can be written under quadratic form:

J¼

h ðJ þ 2 U T Pðx; hÞ þ U T R UÞ 3 0 T

where U ðtÞ ¼ ½ uðtÞ uðt þ hÞ    uðt þ 2mhÞ , J0 represents terms that are independent of U(t), R is a positive definite matrix given by:

R ¼ diagðR þ W T ðbð2mhÞ þ 2

m1 X

bð2ihÞ

i¼1 m 1 X

þ4

ð10Þ

where

or under matrix form:

r1

Similarly, yref(t + T) may be expanded in a same rth order Taylor series:

  y   ref1  yref ðtÞ ¼  ;  yref2 

   l y_ ref1   dðt; TÞ ¼  : 2  l y_ ref þ l y €  2 2 ref2

The tracking error at the next instant (t + l) is then predicted as function of the input u(t) by:

eðt þ lÞ ¼ yðt þ lÞ  yref ðt þ lÞ ¼ eðtÞ þ Vy ðx; lÞ  dðt; lÞ þ KðlÞWðxÞ uðtÞ

By using the predicted tracking error Eq. (11), at the time l = h and l = 2h, the performance index (8) can be written in the conventional quadratic form:

J¼

3 1 J ¼ UT Pðx; hÞU þ UT GðxÞ þ mðe; Q ; hÞ 2h 2

where T

UðtÞ ¼ j uðtÞ uðt þ hÞ uðt þ 2hÞ j ; Pðx;hÞ ¼ diagðR þ WT ðxÞKðx;hÞ WðxÞ;4R;RÞ   GT ðxÞ ¼  WT ðxÞ ðCðhÞ Q eðtÞ þ Zðx;hÞÞ 0 0 ; CðhÞ ¼ 4KðhÞ þ Kð2hÞ; KðQ ;hÞ ¼ 4KðhÞ Q KðhÞ þ Kð2hÞQ Kð2hÞ; Zðx;hÞ ¼ 4KðhÞQ ðVy ðx;hÞ  dðt;hÞÞ þ Kð2hÞQ ðVy ðx;2hÞ  dðt;2hÞÞ;

bð½2i  1hÞÞ; 4R; 2R; . . . ; 4R; RÞ;

i¼1

ð11Þ

m(e, Q, h) are terms that are independent of U(t).

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R. Hedjar et al. / Electrical Power and Energy Systems 46 (2013) 353–365

The minimization of J with respect to U(t), by setting @J=@U ¼ 0, yields to the optimal control:

UðtÞ ¼ P1 ðx; hÞ Gðx; hÞ

ð12Þ

The applied control signal to nonlinear system at time t (receding-horizon principle) is given by:

uðtÞ ¼ j 1 0    0 jUðtÞ

known load torque TL disturb the mechanical equation and induce a steady state error in the rotor speed response. To eliminate this steady state error an integral action is introduced in the external loop by minimizing the predicted rotor position tracking error, instead of rotor speed, with the external control signal. Then from the above equations, the predicted tracking error of the rotor position can be expressed by:

eh ðt þ T v Þ ¼ eh þ T v ev ðtÞ þ V h ðX; T v Þ  dh ðt; T v Þ þ W h ðT v ÞW 1 ðtÞ

or

ð15Þ T

T

1

uðtÞ ¼ ½R þ W ðxÞ Kðx; hÞWðxÞ W ðxÞðCðhÞQe þ Zðx; hÞÞ

ð13Þ

It is noticed that the previous output-tracking control law affects only the torque (y1) and the rotor flux (y2). In the induction machine, the aim is to control speed and flux, thus an extension to speed control is achieved, in the next section, looking at a cascaded nonlinear receding-horizon control structure. 4. Cascade structure of the nonlinear rhc Cascaded control is typically prescribed for linear systems involving time-scale separation assumption. That is, the inner loop is designed to have a faster dynamic than the outer loop. In this paper, the nonlinear continuous receding-horizon control scheme is extended to control speed by using the cascaded structure (Fig. 1). The mechanical equation of the motor is given by:

1 f 1 X_ ðtÞ ¼ y1 ðtÞ  XðtÞ  T L J J J

– For the torque trajectory ðy1 Þ :

yref ðsÞ 1

w1 ðsÞ

yref ðsÞ 2

w2 ðsÞ

– For the velocity trajectory ðXÞ :

T2 _ T 3v € ev ðsÞ ds; ev ðtÞ ¼ XðtÞ  Xref ðtÞ; dh ðt; T v Þ ¼ v X Xref ; ref þ 2 6

     T2 f Tv f T2 Tv f yref 1 1 Xþ v 1 þ w0 V h ðX; T v Þ ¼ v 2 J 3J 2J 3 J

f

Xref ðsÞ w3 ðsÞ

f

The control objective is the tracking of h to a desired reference href and the tracking of y1 and y2 to desired reference signals yref1 and yref2. The performance indexes for the nonlinear system are: Inner loop:

J1 ¼ ¼

1 2

Z

tþT

LðsÞds

t

Z

tþT

eT ðsÞQ eðsÞ þ uT ðsÞR uðsÞ ds

ð16Þ

t

with T = 2 h. External loop:

Z Z 1 tþT v 1 tþT v Lh ðsÞds ¼ qh e2h ðsÞ þ r h W 21 ðsÞ ds 2 t 2 t hv ¼ ðLh ðtÞ þ 4 Lh ðt þ hv Þ þ Lh ðt þ 2hv ÞÞ 3

J2 ¼

ð17Þ

Also in this case, the cost function (17) can be expressed in quadratic form:

J2 ¼

  hv 1 mh ðeh ; qh ; hv Þ þ WT1 Gh ðX; hv Þ þ WT1 Ph ðX; hv ÞW1 2 3

where

The external loop is approximated by a second order linear system which can be controlled by any robust linear controller. In this work, the control objective of the external loop is to drive X(t) to the desired reference Xref(t). The parameters uncertainties and un-

NPC2

1 2

with Tv = 2hv.

f 2

xv 2. v xv sþxv

¼ s2 þ2n

and

T 3 w0 : W h ðTÞ ¼ v 6 J

x2

¼ s2 þ2n xf sþx2

1 x0 y ðsÞ  w1 ðsÞ Js þ f 1 ðJs þ f Þ ðs þ x0 Þ

W3

Z

¼ sþxx0 0

Assuming that the torque y1 tracks the reference signal yref1, the global prediction model of the external loop is calculated, including the torque closed loop, in the following manner:

XðsÞ ¼

eh ¼

ð14Þ

The Eq. (14) allows controlling the speed by acting on the torque y1. Thus, the initial system can be decomposed into two subsystems in a cascaded form (Fig. 1). The inner loop incorporates torque-flux model and the external loop is the velocity transfer function deduced from the mechanical equation given above. The desired smoothed reference models, chosen in continuous time, are given by:

– For the flux trajectory ðy2 Þ :

where

W1

U

NPC1

W1 ðtÞ ¼ j W 1 ðtÞ W 1 ðt þ hv Þ W 1 ðt þ 2hv Þ jT ;    0 0 T Gh ðX; hv Þ ¼  8qh W h ðhv Þ ð3eh þ 5hv ev þ K h  dÞ Ph ðX; hv Þ ¼ diagðr h þ 68qh W 2h ðhv Þ; 4r h ; r h Þ;

y1

Inner system

y2

W2

Inner loop External loop Fig. 1. Cascaded control configuration.

External system

Ω (t)

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R. Hedjar et al. / Electrical Power and Energy Systems 46 (2013) 353–365

uðtÞ ¼ WðxÞ1 KðQ ; hÞ1 ðCðhÞ QeðtÞ þ Zðx; hÞÞ

K h ðX; hv Þ ¼ V h ðX; hv Þ þ 2V h ðX; 2hv Þ;  X; hv Þ ¼ dh ðt; hv Þ þ 2dh ðt; 2hv Þ and mh ðeh ; q ; hv Þ dð h

Differentiating the output y1 one time, the output y2 twice and by using the above control equation, it is easy to show that the tracking errors dynamics are:

represents terms that are independent of W 1 . From the minimization of the performance indexes (J1 and J2), and by taking only the control signal applied at time t (recedinghorizon principle), the optimal solutions are:

 For the torque:

e_ 1 ðtÞ þ

– For the external loop:

W 1 ðtÞ ¼ 

 X; hv ÞÞ 4qh W h ðhv Þð3eh þ 5hv ev þ K h ðX; hv Þ  dð

ð21Þ

 For the flux:

r h þ 68qh W 2h ðhv Þ

W 2 ¼ /rnom

3 e1 ðtÞ ¼ 0 2h

€e2 ðtÞ þ

ð18Þ

6 2 e_ 2 ðtÞ þ 2 e2 ðtÞ ¼ 0 5h h

ð22Þ

– For the inner loop: The above dynamics Eqs. (20)–(22) are linear and time invariant. Thus, the proposed tracking controller design technique leads to feedback linearization and the asymptotic stability of the tracking errors dynamics of the overall system can be easily verified.

uðtÞ ¼ ½R þ WT ðxÞ Kðx; hÞWðxÞ1 WT ðxÞðCðhÞQe þ Zðx; hÞÞ ð19Þ 4.1. Tracking performances

4.2. Zero dynamics

For the external loop: the Eq. (18) with rh = 0 gives, by using the second order derivative of X, the following position tracking error dynamics: v

eh þ

27 30 18 €eh þ e ¼0 e_ þ 2 h 3 h 17hv 17hv 17hv

The relative degrees of the outputs are respectively 1 and 2, with:

ð20Þ

8 _ > < y1 ¼ Lf h1 þ Lg h1 u y_ 2 ¼ Lf h2 > :€ y2 ¼ L2f h2 þ Lg Lf h2 u

For the internal loop: it is assumed that W(x) has a full rank (note that this assumption can be satisfied by using flux observer). Let Q = q I2,R = 0 in the controller (19), the applied control signal:

ð23Þ

Rr Variation

5 4.5

(Ω)

4 3.5 3 2.5

0

1

2

3

4

5

6

7

8

9

10

Time (s) T

Tr Variation

0.19 0.18

3

0.17

2.5

0.15

(Nm)

(s)

0.16

0.14 0.13

2 1.5 1

0.12

0.5

0.11 0.1

L

3.5

0

1

2

3

4

5

Time (s)

6

7

8

9

10

0

0

1

2

3

4

5

6

Time (s)

Fig. 2. Rotor resistance variations, induced rotor time constant variations and the load torque variations.

7

8

9

10

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R. Hedjar et al. / Electrical Power and Energy Systems 46 (2013) 353–365

The sum of relative degrees is three, leading to a two order unobservable dynamics. To study the stability of these two zero dynamics, consider a diffeomorphisme change of coordinates as   / follows: z1 = h1(x), z2 = h2(x), z3 = Lfh2, z4 ¼ arctg /rrba and z5 = X.

T The equilibrium point is Z0 ¼ 0 /rnorm 0 z04 z05 . The observable dynamics are given by:



z_ 1 €z2

 ¼

Lf h1

! þ WðxÞ u

L2f h2

ð24Þ

The nonlinear control u of the inner loop stabilizes the nonlinear system and the behaviors of the observable dynamics (z1, z2 and z3) are described by the tracking errors equations given by (21) and (22). The zero dynamics are:

(

Note that the decoupling matrix W(x) in Eq. (24) is only singular at the start-up (detðWðxÞÞ ¼ 2pkRr ðU2ra þ U2rb ÞÞ, this singularity can be avoided by using flux observer with initial condition b r ð0Þ – 0. U 4.3. Robustness For the external loop, the mechanical parameters uncertainties and load torque variations introduce a short steady state error. The velocity is represented by a linear system and it is known in literature that the integral action eliminates the steady state error induced by a low frequency disturbances. For the inner loop, the system is nonlinear and it can be represented by the system (23) in compact form:

Z_ ¼ pðxÞ þ BWðxÞ U where

z_ 4 ¼ p z5 þ Rpzr z1 2

ð25Þ

z_ 5 ¼ 1J z1  fJ z5  1J T L

The stability of angle z4 is guaranteed since it is an angle and it is reset to 0 every 2p. The second internal dynamic z5, which has been made unobservable from the output y1 and y2 (inner loop), is marginally stable (the pole f/J is near the jx axis). Thus, this zero dynamic can cause a stability problem to the inner loop. However, the external loop and especially the controller (18) renders this internal dynamic asymptotically stable.

z2 z3 jT ; z1 ¼ y2 ; z2 ¼ y1 ; z3 ¼ y_ 2 ;  T   pðxÞ ¼  Lf h2 Lf h1 L2f h2  ;    0 1 0 T    and WðxÞ ¼  Lg h1 ðxÞ Lf Lg h2 ðxÞ T : B ¼   0 0 1 Z ¼ j z1

The resistances uncertainties lead to the unmodeled dynamics Dp(x) and DW(x) in the system (26) such that the inner loop is governed by:

Torque: y1 and yref1

10

flux: y2 and yref2

1.4 1.2

8

1

(Wb)

6

(Nm)

ð26Þ

4 2

0.8 0.6 0.4

0

0.2

-2 0

1

2

3

4

5

6

7

8

9

0

10

0

1

2

3

4

Time (s)

5

6

7

8

9

10

Time (s)

ev(t) = Ω - Ω ref

0.8 0.6 0.4

(rad/s)

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2

0

1

2

3

4

5

6

Time (s) Fig. 3. Rotor torque, rotor flux and speed tracking error.

7

8

9

10

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R. Hedjar et al. / Electrical Power and Energy Systems 46 (2013) 353–365

Z_ ¼ pðxÞ þ DpðxÞ þ BðDW þ WðxÞÞ U

Note that the resistance uncertainties considered here are defined as: Rr = RrN + DRr and Rs = RsN + DRs, where RrN and RsN represent the nominal values of the rotor and stator resistances respectively with d ¼ DRRrr , where kdk 6 1. Let e1 ¼ y2  yref 2 ; e2 ¼ y1  yref 1 ; e3 ¼ e_ 1 ; Q ¼ qI2 ; Q i ¼ qi I2 and Ri = 0, the dynamic of the inner loop is given by:

e_ ¼ ð1 þ dÞ KðhÞe þ DpðxÞ þ dWðx; y_ ref 1 ; y€ref 2 Þ

ð28Þ

where

  0   KðhÞ ¼  0   a

    0 ;  a3 

    0     €ref 2 Þ ¼  0 Wðx; y ;   2 y €ref 2  Lf h2 

1 1þd

0 a1 1þd

0 2 q þ hqi =2 a1 ¼ ; hðq þ hqi =3Þ

a2 ¼

q þ hqi =3 2

0:5h ðq þ hqi =5Þ

Since the matrix K(h) is Hurwitz, there exist two positives definite matrices Ph and Qh solution of the Lyapunov equation:

KðhÞT Ph þ Ph KðhÞ ¼ Q h It is noted that the torque Te depends on the square of stator current I2s [25], then L2f h2 can be written as a combination of e1, e2 and e3:

   1 2 2 L2f h2 ¼  c þ e1 þ e3 þ yref 2 þ y_ ref 2 Tr Tr Tr  2 Lm 2 lðe1 þ yref 2 Þ þ 2 ðLm k þ 2Þðe1 þ yref 2 Þ þ2 Tr Tr

and

þ 2Rr ðe2 þ yref 1 ÞX:

q þ hqi =4 a3 ¼ : 0:5hðq þ hqi =5Þ

By using the above equation, the equation below is obtained:

ref ; €ref2 þ KðXÞT e þ bðXÞT y €ref 2  L2f h2 ¼ y y

Assumptions: The unmodeled dynamics in (24) satisfy the following condition: Dp(x) is bounded by a constant N0 > 0, kDp(x)k 6 N0"x 2 X, where X is functional domain of the machine. The reference signals and their derivatives are bounded: $ (N1,  ref k 6 N 1 where y ref ¼ €ref 2 j 6 N 2 and ky N2) > 0, such that jy jyref 2 yref 1 y_ ref 2 jT .

Est. error on φ 0.06

where KðXÞ 2 R31 and bðXÞ 2 R31 . Consider a Lyapunov function candidate V = eTPhe with time derivative along the trajectories (28) is:

V_ ¼ ð1 þ dÞeT Q h e þ 2eT Ph DpðxÞ þ 2deT Ph W or

€ref 2 þ K T e þ bT yref Þ V_ ¼ ð1 þ dÞ eT Q h e þ 2eT P h DpðxÞ þ 2 d eT Pðy Est. error on φ

rα

0.04

0.04

0.02

0.02

0

0

-0.02

-0.02

-0.04

-0.04

-0.06

-0.06

-0.08

-0.08

-0.1

0

1

2

3

rβ

0.06

(Wb)

(Wb)

The rotor speed X is bounded. This is ensured by the external loop, which is stable.

ð27Þ

4

5

6

7

8

9

-0.1

10

0

1

2

3

4

5

6

7

8

9

10

Time (s)

Time (s) Ω and Ω ref

120 100

(rad/s)

80 60 40 20 0 -20

0

1

2

3

4

5

6

7

Time (s) Fig. 4. The estimated error of the rotor fluxes and the speed tracking performance.

8

9

10

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R. Hedjar et al. / Electrical Power and Energy Systems 46 (2013) 353–365

where P is the last column of the matrix Ph.

It follows that e is uniformly bounded for all t P 0. The solution qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o n ðP h Þ a  . e=kek 6 4 3kkmax min ðP Þ j

converges towards the compact set:

V_ 6 ð1 þ dÞkmin ðQ h Þkek2 þ 2kmax ðP h ÞN0 kek þ 2dkPk N 2 kek þ 2d kP K k kek þ 2dkP b kN1 kek where kmax(), kmin() represent the maximum and minimum eigenvalue respectively. Define j ¼ ð1 þ dÞkmin ðQ h Þ  2dkPKT k > 0 and there exist matrices Q h ; P and K such that kPKT k > kmin ðQ h Þ, then the previous equation gives the upper bound of the rotor resistance uncertainty:

06d<

5. Observer The controller presented above requires full state feedback. This is a problem that needs to be addressed since it is not completely available. The stator currents isa, isb are easily measured and this will be assumed to be the case here. As the primary interest here is the speed control, it is assumed that the rotor speed X is also measured. This ensures the observability of the induction motor [26]. Rotor fluxes are difficult to measure and several papers are devoted to the problem of flux estimation from measurements of angular speed and stator current [26,28]. Since the Kalman filter takes into account the effects of the disturbances noises of any control system. Thus, parameters uncertainties of the machine will be handled as noise [27,28]. The Kalman filter will be used in this paper to estimate the rotor flux and it is derived from the electrical equations of the induction motor model. The electrical dynamics from the first four equations in (1) are given by:

kmin ðQ h Þ ¼ d 6 1: 2kPKk  kmin ðQ h Þ

Thus, the inequality above becomes:

 kek V_ 6 jkek2 þ 2a

ð29Þ

 ¼ kmax ðPh ÞN0 þ dkPkN2 þ dkPbT kN1 . where a The well-known inequality: 2 ab 6 za2 þ b4z for any real positive a, b and z > 0 is used in this part. 2 . The inequality (29) with z ¼ j4 becomes: V_ 6  34 kmaxjðP Þ V þ j4 a h

The solution of the previous differential inequality is given by:

 2 !  2 j 16 a 16 a 3 t e 4kmax ðPh Þ þ Vð0Þ  kmax ðPh Þ kmax ðPh Þ 3 3 j j

(

_ ¼ AðXÞx  þ Bu þ wðtÞ x  ¼ Cx  þ v ðtÞ y

β

α

250

150

150

100

100

50

50

(v)

200

(v)

200

0

0

-50

-50

-100

-100

-150

-150

-200

-200

-250

0

1

2

3

4

5

ð30Þ

Us

Us

250

6

7

8

9

-250

10

0

1

2

3

4

5

6

7

8

9

10

Time (s)

Time (s) Is

6 4 2

(A)

VðtÞ 6

h

Therefore, the parameters variations will induce only a steady state error in the inner loop.

T

2

T

0 -2 -4 -6

0

1

2

3

4

5

6

Time (s) Fig. 5. Stator voltage (usa, usb) and stator current is.

7

8

9

10

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R. Hedjar et al. / Electrical Power and Energy Systems 46 (2013) 353–365

^kþ1=k ¼ Ak x ^k=k þ Bk uk x

where

 ¼ isa x

  1 0 0 0

T ; /ra /rb ; C ¼  0 1 0 0 T 0 k 0   ; ðrLs Þ1 0 k 

isb

  ðrL Þ1  s B¼  0  ¼ ½ i sa y

isb    c    0  AðXÞ ¼  L  m  Tr   0

T

Pkþ1=k ¼ Ak ðPk=k þ T s Q 0 ÞATk þ Q k Kkþ1 ¼ Pkþ1=k CT ðCPkþ1=k CT þ Rk Þ1 ^kþ1=kþ1 ¼ x ^kþ1=k þ Kkþ1 ðy ^kþ1=k Þ ðkÞ  Cx x Pkþ1=kþ1 ¼ Pkþ1=k  Kk C Pkþ1=k

T

u ¼ ½ usa

This Kalman filter (one step ahead predictor) minimizes in a least square sense the covariance:

usb  and  0 pXk   k  c pXk Tr  ; 1 0  Tr pX   Lm pX  T1r  Tr

;

k Tr

n o ^kþ1=k ÞT ðx ^kþ1=k Þ kþ1  x kþ1  x Pkþ1=kþ1 ¼ E ðx

and the discrete model of induction motor (30) is obtained with following third order approximation:

w(t) is the noise vector of state model (system noise), and v(t) is the noise vector of output model (measurement noise). The covariance matrices Qk and Rk of these noises are defined as:

Q k ¼ cov ðwÞ ¼ EfwwT g;

Rk ¼ cov ðv Þ ¼ EfvvT g:

where E{.} denotes the expected value. To implement the Kalman filter algorithm in a DSP for real time application, the induction motor must be discretized. Thus, the recursive form of the Kalman filter may be expressed by the following system equations [28]:

Ak ¼ eAðXÞTs 

i!

Z

Ts

eAðXÞs Bds 

0

3 X Ai ðXÞT iþ1 s

i¼0

ði þ 1Þi

11 10 9

(Ω )

5 4.5 4

8 7

3.5

6

3

5

2.5 0

1

2

3

4

5

4

6

0

1

2

Time (s)

3

4

5

6

4

5

6

Time (s) T

Tr Variation

0.2

L

5 4.5

0.18

4

0.16

3.5

(Nm)

0.14

(s)

Bk ¼

Rs Variation

12

5.5

0.12

3 2.5 2 1.5

0.1

1

0.08 0.06

;

0.5 0

1

2

3

4

5

6

B:

discrete algebraic Riccati equation (DARE) of the Kalman algorithm is a stabilizing solution. This solution converges to the unique positive semi-definite solution [29]. In this work, the cascade structure has been used to separate two subsystems with different dynamics, mechanical part (external loop) and electro-magnetic part (inner loop). Consequently,

6

(Ω )

s

where Ts is the sampling period and Q 0 2 R22 is a positive semidefinite matrix.  1=2 is controllable then the solution of the Note that, Ak ; Q k

6.5

2

3 X Ai ðXÞT i i¼0

Rr Variation

7

ð31Þ

0

0

Time (s)

1

2

3

Time (s) Fig. 6. Electrical parameter variations.

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R. Hedjar et al. / Electrical Power and Energy Systems 46 (2013) 353–365

the linear system given by Eq. (30) can be seen as a linear time invariant system [26]. Indeed, the dynamic of the rotor speed is very low (constant within many samples) with regards to the dynamic of the electro-magnetic part. Thus, with this valid consideration, the separation principle holds and this proves the stability of the overall system (induction motor in cascade structure with Kalman filter). 6. Simulation results Computer simulations have been performed to check the behavior of the proposed controller. The plant under control is a 1.1 kW induction machine used in [30] with the following parameters:

Rr ¼ 3:6 X;

Rs ¼ 8 X; Lr ¼ 0:47 H; 2

Ls ¼ 0:47 H; Lm ¼ 0:452 H; J ¼ 0:015 kgm ; p ¼ 2;

f ¼ 0:005;

The nominal values of this machine are:

T nom ¼ 7 N; Xnom ¼ 73:3 rad=s; kIsnom k ¼ 4:6 A;

kU nom k ¼ 180 V; / ¼ 1:22 Wb:

Stator current: maximum value: kIsk 6 12A and Stator voltage: maximum value: kUsk 6 300V. To guarantee a cascade structure and the stability of the overall system a suitable selection of the reference trajectories is required. Therefore, after several trials, the optimal parameters values of the three reference models are:

nf ¼ 1; nv ¼ 1;

x0 ¼ 40 rad=s for the torque trajectory: The parameters of the observer are [28]:

T s ¼ 104 s; Q k ¼ 104 I4 ; Rk ¼ 10 I4 ; Pð0Þ ¼ 101 I4

1

flux: y and y

1.2

6

1

4

0.8

2

0.4

-2

0.2 1

2

3

4

5

0

6

ref2

0.6

0

0

2

1.4

(Wb)

(Nm)

ref1

8

-4

and Q 0 ¼ I4 :

In the benchmark experiment [30], it assumed that the signals that are measurable are the stator currents (isa,isb) and the rotor speed X. The load torque is unknown and is variable (no vanishing load torque), and all other parameters are constant and known except for the rotor resistance which will change during the operation and only the nominal value is known. We have to note that the proposed controller will use only the nominal values. To examine the rotor flux and the rotor speed tracking performances for the proposed algorithm, it has been requested [30] that the speed starts at the low speed X = 0.1 Xnom till t = 1 s. After it must reach the value X = Xnom in the interval of time 1–2 s; X = 1.5 Xnom in the interval 2–3 s; X = Xnom in the interval 3–6s; X = 0.25 Xnom in the interval 6–8s and X = Xnom for t > 8 s. The flux must reach the nominal value /r = /rnom Wb in the interval of time 0.1–2 s. The flux reference needs to be reduced from the nominal value as the speed reference is increased above the rated speed in order to keep the required field voltages within the limits [31]. Thus, the flux is reduced to 0.5 /rnom Wb in the interval 2–3 s. To test the disturbance rejection, a 0.5 Tnom Nm unknown load torque is applied between t = 0.2 s and t = 4 s. Afterwards it is decreased to 0.25 Tnom. The rotor resistance starts with the nominal value up to t = 5 s. During the interval 5–7 s, it will increase to 1.3 Rrnom; R = 0.7

Torque: y and y 10

xf ¼ 25 rad=s for the flux trajectory xv ¼ 10 rad=s for the speed trajectory

0

1

2

3

4

5

6

Time (s)

Time (s)

e (t) = Ω - Ω v ref

1.5 1

(rad/s)

0.5 0 -0.5 -1 -1.5 -2

0

1

2

3

4

Time (s) Fig. 7. Rotor torque, rotor flux and speed tracking performance in the mismatched case.

5

6

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R. Hedjar et al. / Electrical Power and Energy Systems 46 (2013) 353–365

Rrnom during the interval 7–9 s and it will increase to its nominal value for t > 9 s. All initial conditions of the motor are set to zero except for the flux observer: /r(0) = 0.02 Wb. Fig. 2 shows the variations of the rotor resistance, the induced rotor time constant and the load torque profile which is requested by the benchmark experiment [30]. Through a series of trial and error, the optimal control parameters obtained are:

Q ¼ 104 I2 ; 4

qh ¼ 10 ;

R ¼ 102 I2 ;

h ¼ 0:002;

hv ¼ 3h; r h ¼ 0:0001:

Fig. 3 shows the behavior of the produced torque, rotor flux and the rotor speed. It appears that the produced torque is inside the saturation limits and the rotor flux is very close to the desired reference flux. It also appears that the proposed controller exhibits a very good speed tracking capabilities and robustness to both rotor resistance variations and load torque variations. The applied load torque has no effect on the flux and its effect on the speed is rapidly compensated. Note that in this figure the depicted speed tracking error is with regards to the smoothed speed reference trajectory. Fig. 4 depicts the estimated tracking error of the Kalman filter and the rotor speed tracking performance. As it is clearly reported in Fig. 4, the rotor resistance variations has a slight effect on the observer tracking error since the dynamic of the rotor flux depends on rotor time constant.

Est. error on φ 0.08

Fig. 5 depicts the variations of the admissible stator voltage (usa, usb) and the stator current is that is also admissible. These variables are close to the nominal values. Therefore, we don’t need to introduce constraints in the optimization problem since the voltages (usa, usb) and the stator current is are inside the nominal values (Isnom = 4.6 A) [30]. From the analysis of the simulation results of the proposed algorithms [9,32–34] to the benchmark experiment given in [30], it is possible to note that these controllers have practically the same performance with regards to the proposed control scheme in this paper, in terms of speed response and field-weakening speed range. However, as it is reported in Fig. 5, both during transients and during steady state the current modulus and the active power consumption of the proposed control algorithm are significantly smaller than those proposed in [9,32–34]. Therefore, with the proposed controller, the machine consumes less power. In the second experiment, we have simulated a more complex mismatched case. Indeed, as it is stated in the introduction, the variation of parameters during operation degrades the tracking performances. The rotor resistance changes during the operation due to ohmic heating. However, the stator resistance value will also vary due also to ohmic heating. To overcome the tarcking performance problem induced by the resistances uncertainties, many authors have added an online adaptive mechanism to identify these two resistances. The convergence of the identification mechanism is ensured only if the persistent excitation condition is satisfied. To check the robustness of the proposed algorithm without the adaptive mechanism, the mismatched case consists of:

Est. error on φ

rα

0.06

0.06

0.04

0.04

0.02 0 -0.02

0

(Wb)

(Wb)

0.02

-0.02 -0.04

-0.04 -0.06 -0.08

-0.06

-0.1

-0.08

-0.12

-0.1

rβ

0

1

2

3

4

5

-0.14

6

0

1

2

3

4

5

6

Time (s)

Time (s)

Ω and Ω ref

140 120

(rad/s)

100 80 60 40 20 0 -20

0

1

2

3

4

Time (s) Fig. 8. Estimated rotor fluxes dynamic and speed tracking performance.

5

6

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R. Hedjar et al. / Electrical Power and Energy Systems 46 (2013) 353–365

- Variation of resistance parameters (Rr, Rs) as it is shown in Fig. 6 which covers the more realistic situation, - The no vanishing load torque (Fig. 6), - Mechanical parameters (uncertainties) used by the controller are: Jc = 2J and fc = 2f. The profile of the desired rotor flux and speed are shown in Figs. 7 and 8 respectively. The simulation results, illustrated in Fig. 7, show good tracking performances of the speed and rotor flux that have been achieved in spite of disturbances in the load torque and resistances (rotor and stator). Therefore, the above results demonstrate that the proposed controller has strong robustness properties in the presence of load disturbance and parameter variations. The stator currents and voltages are close to the nominal values. These results are very interesting in comparison with the solutions using resistances estimation and/or load torque observer under persistent excitation condition of input signals. So, with this control algorithm we don’t need to estimate neither the resistances variations nor load torque variations. Consequently, the use of the proposed feedback nonlinear receding-horizon scheme under cascade structure can solve the control problem of induction machines in the presence of uncertainties like mechanical parameters, load torque and resistances variations. It is noted that when a decoupling control algorithm is used, the variation in load torque and rotor resistance Rr causes the loss of input–output decoupling property and this can deteriorate the transient response. This calls for adaptive version of many algorithms where the convergence of the estimator is under persistency excitation of the induction machine. Thus, the main advantage of the proposed algorithm is that it does not need to estimate neither load torque nor stator and rotor resistances variations. Also, good tracking performances have been achieved despite of mechanical parameters uncertainties. Moreover, in order to keep the required field voltages within the limits, the flux reference was reduced from its nominal value as the speed reference was increased above the rated speed. Operation in this flux weakening regime will excite the coupling between flux and speed, causing undesirable speed fluctuation and perhaps instability [31]. However, with the proposed simpler control strategy, the flux weakening operation has no effect on the speed behavior. In this paper we have proved theoretically the stability of the overall system, convergence of the Kalman filter and the robustness of the controller with respect to load torque variations and resistances variations. These theoretical results have been confirmed by different simulations results. Finally, it is noted that the drawback of the proposed cascaded nonlinear predictive control scheme is how to find the adequate tuning parameters (h, hv, Q, R, qh, and rh) that meet the desired tracking performance of different reference trajectories.

7. Conclusions This paper has presented an approximate nonlinear recedinghorizon controller for induction motors in cascade structure. It was assumed that only the stator currents and the rotor speed were available for measurements. One of the main advantages of this nonlinear receding-horizon control approach is that one does not need to perform an online optimization, and asymptotic tracking of the smooth reference signals is guaranteed. To increase the robustness of the controller with respect to time-varying load torque and mechanical parameters uncertainties, an integral action is introduced in the external loop. Robustness, stability of the closedloop system and convergence of the rotor flux observer have been investigated in this work theoretically and proved by simulations.

Indeed, based on simulation results, it is demonstrated that the proposed control law achieves speed and flux amplitude tracking objectives even with unknown load torque disturbance, and presents sufficient robustness in case of electrical and mechanical parameters uncertainties. Further, with the proposed algorithm we do not need to estimate the rotor and stator resistances neither the unknown load torque. In addition, the proposed nonlinear algorithm ensures asymptotic decoupling of the speed and flux subsystems. Additional research should be oriented first towards a nonlinear sensorless control scheme to reduce anymore the cost, secondly to discrete time-implementation of the proposed nonlinear recedinghorizon controller. Analysis of the influence of sampling rate, truncation errors, measurement noise and saturations are all worthy of further investigation. Acknowledgement This work was supported by the Research Center of College of Computer and Information Sciences, King Saud University. Appendix A Lie-derivatives of the output are:  

1 ðx2 x3  x1 x4 Þ þ pk x23 þ x24 x5 þ pðx1 x3 þ x2 x4 Þx5 Tr

Lm 2 2 x þ x24 ;  Lf h2 ðxÞ ¼ 2 ðx1 x3 þ x2 x4 Þ  Tr 3 Tr    2

2

Lm 1 Lm 2 ðx1 x3 þ x2 x4 Þ þ 2  L2f h2 ðxÞ ¼ 2 cþ x1 þ x22 þ 2 ðLm k þ 2Þ x23 þ x24 Tr Tr Tr Tr  Lf h1 ðxÞ ¼ p

Lm Lr



cþ

Lm þ 2p ðx2 x3  x1 x4 Þx5 Tr  Lg h1 ðxÞ ¼ ½ pkx4 pkx3 ; h i  Lg h2 ðxÞ ¼ r2LLsmT r x3 r2LLsmT r x4 :

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