Adaptive control strategy with flux reference optimizationfor sensorless induction motors

Abderrahim El Fadili n, Fouad Giri, Abdelmounime El Magri, Rachid Lajouad, FatimaZahra ChaouiGREYC Lab, University of Caen Basse-Normandie, 14032 Caen, France

a r t i c l e i n f o

Article history:Received 3 January 2013Accepted 1 December 2013Available online 8 February 2014

Keywords:Output feedback controlInduction machinesAdaptive interconnected observersSpeed and flux regulationMagnetic saturationBackstepping design technique

a b s t r a c t

Avoiding mechanical (speed, torque) sensors in electric motor control entails cost reduction andreliability improvement. Furthermore, sensorless controllers (also referred to output-feedback) areuseful, even in the presence of mechanical sensors, to implement fault tolerant control strategies. Inthis paper, we deal with the problem of output-feedback control for induction motors. The solutionsproposed so far have been developed based on the assumption that the machine magnetic circuitcharacteristic is linear. Ignoring magnetic saturation makes it not possible to meet optimal operationconditions in the presence of wide range speed and load torque variations. Presently, an output-feedbackcontrol strategy is developed on the basis of a motor model that accounts for magnetic saturation. Thecontrol strategy includes an optimal flux reference generator, designed in order to optimize energyconsumption, and an output-feedback designed using the backstepping technique to meet tight speedregulation in the presence of wide range changes in speed reference and load torque. The controllersensorless feature is achieved using an adaptive observer providing the controller with online estimatesof the mechanical variables. Adaptation is resorted to cope with the system parameter uncertainty. Thecontroller performances are theoretically analyzed and illustrated by simulation.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Induction motors offer several features e.g. high power/mass ratioand reduced maintenance cost (as no mechanical commutators areinvolved). The considerable progress in power electronics has led tosophisticated power converters making technically possible speedvariation of AC machines in a flexible way. The problem of inductionmotors control aims at designing controllers ensuring wide rangespeed variation, as well as optimal energetic efficiency, in the presenceof varying and/or uncertain loads. The difficulty in designing suchcontrollers lies in the multivariable and highly nonlinear nature of thecorresponding models. Furthermore, most parameters of these modelsare time-varying even in normal operation conditions. Moreover, mostof the model state variables are not accessible to measurements atleast with cheap and reliable sensors. That is, the difficulty of thecontrol problem comes both from the nature of the control objectivesand the complexity of the model. Therefore, it is not surprising that animportant research activity has been devoted to induction motorcontrol especially over the last two decades. However, most of theproposed controllers necessitate online measurements of the motorspeed and torque using mechanical sensors (Chiasson, 2005; Husson,

2010). The point is that making use of mechanical sensors involvesextra costs and poses additional reliability issues. Therefore, anincreasing interest has recently been paid to the problem of outputfeedback control not resorting to mechanical sensors. Accordingly,mechanical as well as electromagnetic variables are online estimatedusing state observers based only on the measurements of electricalvariables (i.e. stator currents and stator voltages). The output-feedbackcontrollers thus obtained turn out to be composed of a stabilizing(state-feedback) regulator and a state-variable observer. Severaloutput-feedback controllers have been proposed on recent years e.g.Peresada, Tonielli, and Morici (1999), Marino, Peresada, and Tomei(1999), Feemster, Aquino, Dawson, and Behal (2001), Feemster,Dawson, Aquino, and Behal (2000), Aurora and Ferrari (2004),Barambones and Garrido (2004), Khalil and Strangas (2004), Marino,Tomei, and Verrelli (2004), Traoré, Plestan, Glumineau, and DE Leon(2008), Ghanes and Zheng (2009), Ghanes, Barbot, de Leon-Morales,and Glumineau (2010), and Novatnak, Chiason, and Bodson (2002),but they all present a number of limitations e.g.

(i) Most proposed controllers still necessitate, in addition to statorcurrents, the measurement of the speed or position e.g. Peresadaet al. (1999) and Marino et al. (1999), or the measurement of loadtorque e.g. Feemster et al. (2001, 2000), Aurora and Ferrari (2004),Barambones and Garrido (2004), and Marino et al. (2004).

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/conengprac

Control Engineering Practice

0967-0661/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.conengprac.2013.12.005

n Corresponding author.E-mail address: elfadili_abderrahim@yahoo.fr (A. El Fadili).

Control Engineering Practice 26 (2014) 91–106

(ii) The controllers have generally been tested and evaluated at highspeed (Aurora & Ferrari, 2004; Barambones & Garrido, 2004;Feemster et al., 2000, 2001; Khalil and Strangas, 2004; Marinoet al., 1999, 2004; Peresada et al., 1999; Traoré et al., 2008) and,except in (Feemster et al., 2000, 2001; Marino et al., 2004), notheoretical analysis was made to formally prove the stability of theoverall closed-loop system and the global convergence of thetracking errors. Stability analysis at low speed has only been madein Traoré et al. (2008), Ghanes and Zheng (2009), Ghanes et al.(2010), and De León, Glumineau, Traore, and Boisliveau (2013).

One more major limitation of all the previous works is that thecontroller design was based on standard models not accountingfor magnetic saturation of the controlled induction machine. Notethat these models cannot be based solely to perform online fluxoptimization to meet e.g. stator current energy consumption. Infact, quite few works have attempted to account for the non-linearity of the magnetic circuit in induction machine control e.g.Novatnak et al. (2002), Hofmann, Sanders, and Sullivan (2002),and El Fadili, Giri, El Magri, Lajouad, and Chaoui (2012). In thelatter, the control design was done supposing the load torque and/or rotor speed to be accessible to measurements and the obtainedstate-feedback controllers were evaluated only at high speeds.Moreover, the control design in Novatnak et al. (2002), Hofmannet al. (2002), and El Fadili, Giri, El Magri, et al. (2012) wassimplified by supposing that the machine can be consideredseparately and so can be controlled by directly acting on the statorvoltage. The point is that real-life machines are controlled throughinverters (i.e. DC/AC converters) that generate the stator voltages.

In this paper, we address the problem of controlling the overallconverter–machine association including the inverter and theinduction motor. The controlled system is described in Fig. 1 whichshows the interaction between the various elements. The controlobjective is to achieve tight speed reference tracking and rotor fluxnorm optimization (in order to minimize the absorbed statorcurrent). The control problem is dealt with based on a systemmodel that accounts for the nonlinear nature of the machinemagnetic circuit. As the rotor flux and the mechanical variables(speed and load torque) are not supposed to be accessible tomeasurements and some model parameters (including rotorinertia, friction coefficient, stator resistance), an adaptive inter-connected observer is first developed as part of the controllerdesign. A stabilizing state-feedback nonlinear regulator is thendesigned using the backstepping technique. The obtained

nonlinear adaptive output-feedback controller is formally shown,using Lyapunov stability tools, to meet its control performances.

The paper is organized as follows: Section 2 is devoted to themodeling of the controlled inverter–motor association; the adap-tive state observer is designed in Section 3; the development ofthe output-feedback controller is completed in Section 3; theanalysis of the overall closed-loop control system is dealt with inSection 4; the controller performances are illustrated throughnumerical simulations in Section 5.

2. Modeling inverter–motor association

The controlled system, depicted in Fig. 2, is a series combina-tion of an inverter and induction motor. The inverter is an H-bridge converter operating in accordance to the well known PulseWide Modulation (PWM) principle. It consists of six insulated gatebipolar transistors (IGBTs) with anti-parallel diodes for bidirec-tional power flow mode. The achievement of speed regulation andflux optimization, in the presence of wide range load variation, isonly possible if the control design is based on a model that takesinto consideration the nonlinear nature of the machine magneticcircuit. Fortunately, a model presenting this feature has recentlybeen made available (EL Fadili, Giri, & EL Magri, 2013; EL Fadili, ELMagri, Ouadi, & Giri, 2013). This model is presently used letting theinvolved parameters be given the numerical values of a real-life7.5 KW induction motor (Section 5). The magnetic characteristic ofthat machine is shown in Fig. 3 which illustrates well thenonlinearity of the magnetic circuit at high flux values. As themotor is presently considered together with the associated inver-ter, signal averaging is usually resorted to convert the inverter

+

u2

u1

ref

z2

--

+

Speed/Flux

Controller

2r

Inverter

Optimal Flux Generator sI

z1

3/2

Adaptive

Interconnected

Observer

X

is is usu

Induction

motor

is1, is2, is3 us1, us2, us3

PWM

3

h1ref

Fig. 1. Control strategy involving state dependent reference flux.

Fig. 2. Inverter–motor association to be controlled.

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–10692

control inputs into continuous signals (signal averaging is madeover cutting periods). Indeed, the average model (involving aver-age signals) turns out to be more convenient for control design. Inthe αβ-coordinates, this model is defined by the following equa-tions. (EL Fadili, Giri, & EL Magri, 2013; El Fadili, Giri, El Magri, &Besançon, 2013; El Fadili, Giri, El Magri, et al., 2012; EL Fadili,Magri, Ouadi, & Giri, 2013):

_x1 ¼ �a2x1þδ x4þa3px2x5þa3u2vdc ð1Þ

_x2 ¼ � fJx2þ

pJðx4x3�x5x1Þ�

TL

Jð2Þ

_x3 ¼ �a2x3�a3px2x4þδ x5þa3u1vdc ð3Þ

_x4 ¼ a1x1�Lseqδ x4�px2x5 ð4Þ

_x5 ¼ a1x3�Lseqδ x5þpx2x4 ð5Þwhere ðu1;u2Þ represent the average duty ratios in the αβ-coordi-nates, which are obtained through Park's transformation of thethree-phase system ðh1;h2;h3Þ i.e. the switch position functionstaking values in the discrete set {-1,1}. Specifically, one has

hi ¼1 if Hi is ON and H0

i is OFF�1 if Hi is OFF and H0

i is ON

(; i¼ 1;2;3 ð6Þ

In (1)–(5), ðx1; x2; x3; x4; x5Þ are the state variables defined by

x1 ¼ isα; x2 ¼Ω; x3 ¼ isβ;

x4 ¼ ϕrα; x5 ¼ ϕrβ; ð7Þ

where the bar refers to signal averaging over cutting periods. Theremaining quantities in (1)–(7) have the following meaning:

vdc DC link voltageisα,isβ αβ-components of the stator currentΦr amplitude of the instantaneous rotor flux, denoted ϕr;

accordingly one hasΦr ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix42þx52

pϕrα;ϕrβ rotor flux αβ-componentsRs;Rr stator and rotor resistancesf ,J,TL friction coefficient, rotor inertia and load torque,

respectivelyp number of pole pairsLseq equivalent inductance of both stator and rotor leakage

brought to the stator side

δ the only varying parameter depending on the machinemagnetic state as shown in Fig. 4; this dependence hasbeen given a polynomial approximation:

δ¼ ΓðΦrÞ ¼ q0þq1Φrþ⋯þqmΦmr ð9Þ

where the involved coefficients have been identified (based onspline approximation) using the experimental magnetic character-istic of Fig. 3:

a1 ¼ Rr ; a2 ¼ ðRsþRrÞ=Lseq ¼ a3Rsþa1a3; a3 ¼ 1=Lseq

The numerical values of the model parameters are given inTable 1. As already mentioned, the numerical values correspond toan induction motor of 7.5 KW.

Remark 1.

(1) The model (1)–(9) is obtained from induction motor modelwhich ignores the nonlinearity of the magnetic characteristicby letting the magnetic inductance Lm be varying with themagnetic state (EL Fadili, EL Magri, Ouadi, & Giri, 2013).

(2) The effectives inverter signals control ðh1;h2;h3Þ are obtainedfrom the control signals ðu1;u2Þ, using the Park transforminverse, which are computed so far. ðh01;h02;h03Þ are the comple-ment, with respect to one, of ðh1;h2;h3Þ, respectively.

(3) In (1)–(5), the DC link voltage vdc is assumed to be constant,preferably equal to the nominal voltage of the inductionmachine. This condition is crucial for the motor to workcorrectly. The regulation of vdc can be dealt with as in ElFadili, Giri, El Magri, Lajouad, and Chaoui (2012) and El Fadili,Giri, El Magri, Dugard, and Chaoui (2012).

3. Interconnected observer design

3.1. System model reorganization

The first step in the observer design consists in separating themodel (1)–(5) in two interconnected state-affine two subsystems.Specifically, one has

Σ1

_X1 ¼ A1ðX2; yÞX1þg1ðu; y;X2ÞþϖðX2ÞρþΔϖρþΔA1ðX2; yÞX1þΔg1ðu; y;X2Þy1 ¼ C1X1

(

ð10Þ

Σ2

_X2 ¼ A2ðX1; ρÞX2þg2ðu; y;X1ÞþΔA2ðX1; ρÞX2þΔg2ðu; y;X1Þy2 ¼ C2X2

(

ð11Þ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9500

1000

1500

2000

2500

3000

3500

4000

Rotor flux norm (Wb)

Del

ta

Fig. 4. Characteristic (δ, Φr): directly computed points (þþ) and polynomialinterpolation (solid). Unities: δðΩH�2Þ, Φr(Wb).0 5 10 15 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Magnetic current (A)

Rot

or fl

ux c

urre

nt (

Wb)

Fig. 3. Magnetic characteristic experimentally built up in EL Fadili, Giri, and ELMagri (2013) for a 7.5 KW induction motor: rotor flux norm Φr(Wb) versusmagnetic current Iμ (A).

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–106 93

with

A1ðX2; yÞ ¼0 a3px5 �a3y1

�px5J

� fJ 0

0 0 0

264

375;

A2ðX1; ρÞ ¼�a3a1 �a3px2 δ

0 �δ Lseq �px20 px2 �δLseq

264

375 ð12Þ

g1ðu; y;X2Þ ¼�a3a1y1þa3vdcu2

pJx4y20

264

375;

g2ðu; y;X1Þ ¼�a3Rsy2þa3vdcu1

a1y1a1y2

264

375 ð13Þ

ϖðX2Þ ¼0 x4 x4Φr x4Φr

2 ⋯ x4Φrm

�1=J 0 0 0 ⋯ 00 0 0 0 ⋯ 0

264

375;

C1 ¼ C2 ¼ 1 0 0� � ð14Þ

Δϖ ¼0 0 0 0 ⋯ 0

�Δð1=JÞ 0 0 0 ⋯ 00 0 0 0 ⋯ 0

264

375; δ¼ q0þq1Φrþ⋯þqmΦ

mr

ð15Þ

ΔA1ð:Þ ¼0 Δa3px5 �Δa3y1

�px5Δ 1J

� ��Δ f

J

� �0

0 0 0

2664

3775;

Δg1ð:Þ ¼�Δða3a1Þy1þΔa3vdcu2

Δ pJ

� �x4y2

0

2664

3775 ð16Þ

ΔA2ð:Þ ¼�Δða3a1Þ �Δa3px2 0

0 �δ ΔðLseqÞ 00 0 �δ ðLseqÞ

264

375;

Δg2ð:Þ ¼�Δa3Rsy2þΔa3vdcu1

Δa1y1Δa1y2

264

375 ð17Þ

with Δðf =JÞ, Δð1=JÞ, Δa1, Δa2, ΔLseq and Δa3 denote the uncer-

tainties on the involved quantities.where X1 ¼ x1 x2 Rs� �T and

X2 ¼ x3 x4 x5� �T are the state vectors of (10) and (11), respec-

tively, ρ¼ TL q0 q1 q2 ⋯ qmh iT

is a parameter vector,

u¼ ½u1 u2 �T and y¼ ½ y1 y2 �T ¼ ½ x1 x3 �T are the input and

output vectors of the induction machine. Obviously, the output yis accessible to measurements.

3.2. Observer design assumptions

Inspired by Traoré et al. (2008), Ghanes et al. (2010), Besançon,de Leon-Morales, and Guevara (2006), and Ghanes, De Leon, andGlumineau (2006), an interconnected observer will now bedesigned for induction machine based on the model (1)–(5) whichwe know accounts for the magnetic saturation. The observerdesign and analysis is a direct application of the more generaltheory of interconnected observers developed in Besançon et al.(2006) and Besançon and Hammouri (1998). There, the followingassumptions are needed:

A.1. The following Lipschitz assumptions are required to ensurethe existence of a unique solution of the differential Eqs. (10) and(11) and their corresponding observer equations (defined later):

A1ðX2; yÞ is globally Lipschitz with respect to X2, and uniformlywith respect to ðyÞ.

A2ðX1; ρÞ is globally Lipschitz with respect to (X1,ρ).g1ðu; y;X2Þ is globally Lipschitz with respect to X2 and

uniformly with respect to ðu; yÞ.g2ðu; y;X1Þ is globally Lipschitz with respect to X1 and

uniformly with respect to ðu; yÞ.

A.2. (Observability condition): A sufficient condition for the sub-system (10) (resp. (11)) to be observable it that the pair ðu; y;X2Þ(resp., ðu; y; X1Þ) is a bounded and regularly persistent input for Σ1

(resp.Σ2) in the sense that there exist αi; βiT i40 and ti0Z0 suchthat, for all initial condition Xi0, one has:

αiIrZ tþTi

tΨ iðu;y;Xi0Þðτ; tÞTCT

i CiΨ iðu;y;Xi0Þðτ; tÞdτrβiI; 8 tZ0 ði¼ 1;2Þ

ð18Þwhere Ψ 1ðu;y;X10Þ (resp., Ψ2ðu;y;X20Þ) denotes the transition matrix forthe system _X1 ¼ A1ðX2; yÞX1; y1 ¼ C1X1 (resp., _X2 ¼ A2ðX1; ρÞX2; y2 ¼ C2X2)

A.3. There exists a time-varying bounded vector KAR3 such that_Λ¼ ðA1�KC1ÞΛ is exponentially stable.

A.4. The solution ΛðtÞ of _Λ¼ ðA1�KC1ÞΛþϖ is persistently excitingso that there areα3; β3 and T3 such that, α3Ir

R tþT3t

ΛðτÞTCT1C1ΛðτÞdτrβ3I, 8 tZt0, I being the identity matrix and

t0Z0

A.5. The load torque TL, the stator resistance and the coefficientsqiði¼ 0;…;mÞ are all unknown but undergo the following equations:_TL ¼ 0; _Rs ¼ 0; _q0 ¼ _q1 ¼⋯¼ _qm ¼ 0 ð19aÞAll remaining machine parameters are knownwith sufficient accuracy.

A.6. All machine states are bounded so that Assumption A.1 aswell as the following bounding holds:‖ΔA1ðX2; yÞ‖rs1; ‖ΔA2ðX1; ρÞ‖rs2; ‖Δg1ðu; y;X2Þ‖rs3;

‖Δg2ðu; y;X1Þ‖rs4; ‖Δϖ‖rs5 ð19bÞfor some real scalars si40 (i¼ 1;…;5).

Remark 2.

(1) In the case of no magnetic saturation, the observability of thesensorless induction machine has been analyzed in Ghaneset al. (2006). It has been formally proved that observability canactually be lost in some operation conditions e.g. in the case ofnull speed. This fact will be shown (see Section 5, Fig. 15) to bealso the case in the presence of magnetic saturation. On theother hand, it was formally shown that (Assumption A.2) is

Table 1Motor characteristics.

Nominal power PN 7.5 KWNominal voltage Usn 380 VNominal flux Φrn 0.56 WbNominal current Isn 15.4 ANominal troque Tem 49 N mStator resistance Rs 0.63 ΩRotor resistance Rr 0.56 ΩInertia moment J 0.22 kg m2

Friction coefficient f 0.001 N m s rd�1

Number of pole pairs p 2Leakage equivalent inductance a Lseq 7 mH

a Equivalent inductance of stator and rotor leakage seen from the stator.

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–10694

a sufficient condition of observability (Besançon et al., 2006;Besançon & Hammouri, 1998). It turns out that, for thatassumption to hold the speed must be nonzero.

(2) The motivation of (Assumption A.5) is twofold:(i) In most applications, the change of load torque TL is

actually infrequent i.e. the load takes a constant valueand keeps it unchanged for a long time before a newchange happens. Also, it is quite often that the motorspeed is required to undergo a step-like variation. In thepresent study, the focus is precisely made on inductionmachine applications involving step-like variation of theload torque and the rotor speed.

(ii) The nonlinear adaptive control theory is well developedmainly for systems involving constant uncertain para-meters e.g. in Besançon et al. (2006), Besançon &Hammouri (1998), Krstic, Kanellakopoulos, and Kokotovic(1995). Assumption A.5 makes that theory presentlyapplicable

The design strategy consists in separately synthesizing anobserver for each one of the subsystems (10) and (11). Whenfocusing on one subsystem, the state of the other is supposed to beavailable (Fig. 5). The global observer (that applies to the wholesensorless induction machine) is simply obtained by combiningthe separately obtained observers. The design of the individualobservers is performed using the Kalman like technique developedin Besançon and Hammouri (1998 and Zhang, Xu, and Besançon(2003) for state affine systems. Doing so, one obtains the followinginterconnected observers:

_X1 ¼ A1ðX2; yÞX1þg1ðu; y; X2ÞþϖðX2Þ ρþλð ΛS�13 ΛT þS�1

1 ÞCT1ðy1� y1Þ

ð20Þ

_ρ¼ λS�13 ΛTCT

1ðy1� y1Þ ð21Þ

_S1 ¼ �θ1S1�AT1ðX2; yÞS1�S1A1ðX2; yÞþλCT

1C1 ð22Þ

_S3 ¼ �θ3S3þλΛTCT1C1Λ ð23Þ

_Λ¼ ðA1ðX2; yÞ�λS�11 CT

1C1ÞΛþϖðX2Þ ð24Þ

y1 ¼ C1X1 ð25Þ

_X2 ¼ A2ðX1; ρÞX2þg2ðu; y; X1ÞþS�12 CT

2ðy2� y2Þ ð26Þ

_S2 ¼ �θ2S2�AT2ðX1; ρÞS2�S2A2ðX1; ρÞþCT

2C2 ð27Þ

y2 ¼ C2X2 ð28Þwith

A1ðX2; yÞ ¼0 a3px5 �a3y1

�px5J

� fJ 0

0 0 0

264

375 ð29Þ

A2ðX1; ρÞ ¼�a3a1 �a3px2 δ

0 � δ Lesq �px20 px2 � δ Lesq

2664

3775 ð30Þ

g1ðu; y; X2Þ ¼�a3a1y1þa3vdcu2

pJ x4y20

264

375 ð31Þ

g2ðu; y; X1Þ ¼�a3Rsy2þa3vdcu1

a1y1a1y2

2664

3775 ð32Þ

ϖðX2Þ ¼0 x4 x4Φr x4Φr

2⋯ x4Φr

m

�1=J 0 0 0 ⋯ 00 0 0 0 ⋯ 0

264

375 ð33Þ

whereθ1, θ2, θ3 and λ are positive real constants to be selected by the

user (generally using try-error method).

ΛA IR3�mþ2, S1A IR3�3, S2A IR3�3, S3A IRmþ2�mþ2.

X1 ¼ x1 x2 Rs

h iT, ρ¼ TL q0 q1 ⋯ qm

h iTand X2 ¼

x3 x4 x5h iT

denote the estimates of X1, ρ and X2

Φr ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix4

2þ x52

qand δ¼ ΓðΦrÞ ¼ q0þ q1Φrþ⋯þ qmΦ

mr denote

the estimates of Φr and δ

Remark 3. The X1-state observer includes Eqs. (20)–(25), whilethe X2-state observer is composed of Eqs. (26)–(28). Note that Eq.(20) is a copy of the first equation in (10) augmented by acorrection term depending on the output estimation errorðy1� y1Þ. Of course, the modeling error terms in (10) (i.e. thoseinvolving Δϖ, ΔA1 and Δg1) are not copied in the observerequation (20) because they are not known. The correction terminvolves a gain matrix λð ΛS�1

3 ΛT þS�11 ÞCT

1 depends on the varyingmatrix gains Λ; S1; S3 which undergo Eqs. (25)–(28). A similarcomment can be made to motive the form of the observerequation (26) with a copy (of the first equation of (11)) augmentedwith a correction term.□The convergence properties of the interconnected observers

(20)–(28) have been analyzed for general interconnected systemse.g. Besançon and Hammouri (1998), Krstic et al. (1995), andBesançon, Bornard, and Hammouri (1996). They are adapted inProposition 1 to the present particular case, using the followingnotations:

e01 ¼ X1� X1; e2 ¼ X2� X2; e3 ¼ ρ� ρ; e1 ¼ e01�Λ e3 ð34Þ

Proposition 1. Consider the inverter–motor association illu-strated in Fig. 1, analytically represented by the model (1)–(5)or, equivalently, by the interconnected models (10) and (11),subject to Assumptions A.1 to A.6. Suppose that the pair ðu; X2Þ(resp. ðu; X1Þ) is a persistently exciting input for Σ1 (resp. for Σ2)in the sense of (Assumption A.2). Consider the interconnected

X2-Observer

1x 2u

3x 1udcv

,LT

TxxxX 5432 ,,

T

sRxxX ,, 211

dcv

X1-Observer

Fig. 5. Interconnected structure observer.

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–106 95

observer (20)–(28) letting there the initial conditions(S1ð0Þ,S2ð0Þ,S3ð0Þ), of the Lyapunov equations (22), (23), (27), beany positive definite matrices. Then, the observer enjoys thefollowing property, whatever the (finite) initial estimates (X1ð0Þ,ρð0Þ, X2ð0Þ):

If θ1, θ2 and θ3 are large enough then, the solutions (S1, S2, S3) ofEqs. (22), (23), (27) exist and are bounded positive definitematrices i.e. there are positive constants αi and βi (i¼ 1;2;3) suchthat αiIrSiðtÞrβiI, 8 t4t0, where I denotes the identity matrix.□

Remark 4.

(1) The above proposition is a direct application of a more generalresult established in Besançon et al. (2006) and Besançon andHammouri (1998) for interconnected observers.

(2) The boundedness of all motor signals is not an issue becauseinduction motors are BIBO stable (e.g. Popovic, Hiskens, & Hill,1998) and, presently, all motor input signals are bounded.Indeed, it has already been pointed out that, correct inductionmotor operation assumes a constant inverter DC input voltagevdc. Furthermore, the duty ratios ðu1;u2Þ are constructivelyvarying between �1 and 1. Finally, the load torque TL isbounded by assumption (11).

(3) Signal boundedness is required to make Lipschitz the functionsdefined by (12)–(15). The Lipschitz property is required inBesançon et al. (2006) to prove Proposition 1. It guarantees,among others, the existence of solutions of (22), (23), (27). In thisrespect, one may also notice that all equations defining theobserver i.e. (20)–(28). This particularly entails the impossibilityof determining a priori the solutions of (22), (23), (27), since theright side of each equation depends on the solutions of the others.For instance, the right side of (22) involves the state X2 which isthe solution of (26). But, X2 itself depends on X1, which is thesolution of (20), and so on.□

4. Output-feedback adaptive control design

4.1. Optimal flux reference generator

The model (1)–(5) takes into account the magnetic saturationof the induction motor. One can get benefit of this feature todesign a rotor flux reference optimizer. Presently, flux optimiza-tion aims at minimizing the absorbed stator current. This optimi-zation problem has been dealt with in El Fadili, Giri, El Magri, et al.

(2012) based on the machine model (1)–(5) with the numericalvalues of Table 1. An optimal current–flux curve, Φref ¼ FðIsÞ, hasthere been constructed using, in addition to the model (1)–(5), themachine experimental magnetic characteristic (Fig. 3). Theobtained optimal current–flux curve is illustrated in Fig. 6 and,for ease of analytical manipulation, was given a polynomialapproximation (El Fadili, Giri, El Magri, et al., 2012):

Φref ¼ FðIsÞ ¼ f nIns þ f n�1I

n�1s þ⋯þ f 1Isþ f 0 ð35Þ

Accordingly, the provided Φref represents the rotor flux valuethat, for a given load torque Te, leads to minimal stator currentabsorption i.e. if Φr ¼Φref (with Φr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix42þx52

pthe stator current

norm Is is minimal). For convenience, the polynomial (35) will bereferred to flux reference optimizer (FRO).

4.2. Speed and flux controller design and analysis

The adaptive controller developed is expected to ensure thefollowing objectives:

(i) Speed regulation: the machine speed Ω must track, as closelyas possible, any step-like bounded reference signal Ωref .

(ii) Flux optimization: the rotor flux norm Φr must track asaccurately as possible a state-dependent flux referenceΦref ¼ FðIsÞ.

As the model (1)–(5) involves unknown parameters and una-vailable signals, it is natural to base the control design on theobserver (20)–(28) which is rewritten in the following explicitform:

_x1 ¼ �ðRsa3þa1a3Þx1þ δx4þa3px2x5þa3u2vdcþ ~Θ1 ð36Þ

_x2 ¼ � fJx2þ

pJðx4x3� x5x1Þ�

TL

Jþ ~Θ2 ð37Þ

_x3 ¼ �ðRsa3þa1a3Þx3�a3px2x4þ δx5þa3u1vdcþ ~Θ3 ð38Þ

_x4 ¼ a1x1�Lseqδx4�px2x5þ ~Θ4 ð39Þ

_x5 ¼ a1x3�Lseqδx5þpx2x4þ ~Θ5 ð40Þwhere

~Θ1 ¼ 1 0 0� �ðλΛS�1

3 ΛT þλS�11 ÞðCT

1C1e1þCT1C1Λe3Þ ð41Þ

~Θ2 ¼ 0 1 0� �ðλΛS�1

3 ΛT þλS�11 ÞðCT

1C1e1þCT1C1Λe3Þ ð42Þ

~Θ3 ¼ 1 0 0� �

S�12 CT

2C2e2 ð43Þ

~Θ4 ¼ 0 1 0� �

S�12 CT

2C2e2 ð44Þ

~Θ5 ¼ 0 0 1� �

S�12 CT

2C2e2 ð45Þ

Remark 5. It is readily checked using (34) that C1e1þC1Λe3 ¼x1� x1 and C2e2 ¼ x3� x3. This shows that the quantities~Θi(i¼ 1…5) are available.□

The adaptive speed/flux controller design will now be per-formed in two steps using the backstepping technique (Krsticet al., 1995). First, introduce the tracking errors:

z1 ¼Ωref � x2 ð46Þ

z2 ¼Φ2ref �ðx42þ x5

2Þ ð47Þ

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Stator current norm (A)

roro

r flu

x no

rm re

fere

nce

(Wb)

Fig. 6. Optimal current–flux characteristic for the induction machine with numer-ical characteristics of Table 1.

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–10696

Step 1: It follows from (36), (39), and (40) that the errors z1 andz2 undergo the differential equations:

_z1 ¼ _Ωref �pJðx4x3� x5x1Þþ

TL

Jþ fJx2� ~Θ2 ð48Þ

_z2 ¼ 2Φref _Φref �2x4 _x4�2x5 _x5¼ 2Φref _Φref �2x4ða1x1�Lseqδ x4�px2x5Þ�2x5ða1x3�Lseqδ x5þpx2x4Þ�2x4 ~Θ4�2x5 ~Θ5

¼ 2Φref _Φref �2a1ðx1x4þ x3x5Þþ2Lseqδ ðx42þ x52Þ�2x4 ~Θ4�2x5 ~Θ5

_z2 ¼ 2Φref _Φref �2a1ðx1x4þ x3x5Þþ2Lseqδ ðΦ2ref �z2Þ�2x4 ~Θ4�2x5 ~Θ5

ð49ÞIn (48) and (49), the quantities pðx4x3� x5x1Þ and

2a1ðx1x4þ x3x5Þ stand up as virtual control signals. Let us tem-porarily suppose these to be the actual control signals andconsider the Lyapunov function candidate:

V1 ¼12ðz21þz22Þ ð50Þ

It can be easily checked that, in the case of null errors~Θ i ¼ 0 ði¼ 1;2;3Þ, the time derivative of (50) can be made anegative definite function of (z1; z2Þ i.e._V1 ¼ �c1z21�c2z22� ~Θ2z1�2x4 ~Θ4z2�2x5 ~Θ5z2 ð51Þ

by letting pðx4x3� x5x1Þ ¼ μ1 and 2a1ðx4x1þ x3x5Þ ¼ ν1 with

μ1 ¼def

Jðc1z1þ _Ωref ÞþTLþ f ðΩref �z1Þ and ν1 ¼def c2z2þ2Φref _Φref

þ2LseqδðΦ2ref �z2Þ ð52Þ

where c1 and c2 are any positive design parameters. Since J and fare unknown, the first equation in (52) is replaced by itscertainty equivalence form, yielding the following adaptivecontrol laws:

μ1 ¼def

Jðc1z1þ _Ωref Þþ TLþ f ðΩref �z1Þ ð53Þ

ν1 ¼def c2z2þ2Φref _Φref þ2LseqδðΦ2ref �z2Þ ð54Þ

where J and f are estimates (yet to be determined) of J and f ,respectively. As the quantities pðx4x3� x5x1Þand 2a1ðx1x4þ x3x5Þare not the actual control signals, they cannot let be equal to μ1and ν1, respectively. Nevertheless, we retain the expressions ofμ1 and ν1 as the first stabilizing functions and introduce the newerrors:

z3 ¼ μ1�pðx4x3� x5x1Þ ð55Þ

z4 ¼ ν1�2a1ðx1x4þ x3x5Þ ð56ÞThen, using the notations (53)–(56), the dynamics of the errors

z1 and z2, initially described by (48) and (49), are now describedby

_z1 ¼ _Ωref �1Jðμ1�z3Þþ

TL

Jþ fJx2� ~Θ2

_z1 ¼ _Ωref �1J½ Jðc1z1þ _Ωref Þþ TLþ f x2�z3�þ

TL

Jþ fJx2� ~Θ2

_z1 ¼ �c1z1þ~JJðc1z1þ _Ωref Þþ

~fJx2þ

1Jz3� ~Θ2 ð57Þ

_z2 ¼ �c2z2þz4�2x4 ~Θ4�2x5 ~Θ5 ð58Þwhere

~J ¼ J� J and ~f ¼ f � f ð59Þ

Similarly, the time-derivative of V1 can be expressed in func-tion of the new errors as follows:

_V1 ¼ �c1z21�c2z22þz1~JJðc1z1þ _Ωref Þþ

~fJx2þ

1Jz3� ~Θ2

" #

þz2ðz4�2x4 ~Θ4�2x5 ~Θ5Þ ð60Þ

Step 2: The second design step consists in choosing the actualcontrol signals, u1 and u2, so that all errors (z1; z2; z3; z4) convergeto zero. To this end, it must be made clear how these errorsdepend on the actual control signals (u1, u2). First, focusing on z3,it follows from (55) that

_z3 ¼ _μ1�pð _x4x3þ x4 _x3� _x5x1� x5 _x1Þ ð61ÞUsing (37)–(40), (63) and (59), one gets from (61):

_z3 ¼ Jðc1 _z1þ €Ωref Þ� _~J ðc1z1þ _Ωref Þþ _TL� _~f x2þ f _x2h i�pð _x4x3þ x4 _x3� _x5x1� x5 _x1Þ

¼ ðc1 J� f Þ �c1z1þ~JJðc1z1þ _Ωref Þþ

~fJx2þ

1Jz3� ~Θ2

" #

þ J €Ωref þ f _Ωref þ _TL�½_~J ðc1z1þ _Ωref Þþ _~f x2��pðða1x1�Lseqδx4�px2x5Þx3þ x3 ~Θ4

�ða1x3�Lseqδx5þpx2x4Þx1� x1 ~Θ5Þ�px4ð�ðRsa3þa1a3Þx3�a3px2x4þ δx5þa3u1vdcÞ�px4 ~Θ3

þpx5ð�ðRsa3þa1a3Þx1þ δx4þa3px2x5þa3u2vdcÞþpx5 ~Θ1

ð62ÞFor convenience, the above equation is given the following

compact form:

_z3 ¼ μ2þc1 J� f

Jz3þpa3vdcðx5u2� x4u1Þ

þðc1 J� f Þ~JJðc1z1þ _Ωref Þþ

~fJx2

" #� _~Jðc1z1þ _Ωref Þþ _~f x2h i

ð63Þ

with

μ2 ¼ �c1z1ðc1 J� f Þþ J €Ωref þ f _Ωref þ _TLþpx5ð�ðRsa3þa1a3Þx1þ δx4þa3px2x5Þ�pðða1x1�Lseqδx4�px2x5Þx3�ða1x3�Lseqδx5þpx2x4Þx1Þ�px4ð�ðRsa3þa1a3Þx3�a3px2x4þ δx5Þ�pðx3 ~Θ4� x1 ~Θ5Þ�pðx4 ~Θ3� x5 ~Θ1Þ�ðc1 J� f Þ ~Θ2 ð64Þ

Note that the derivatives _TL and _δ are given by the observerequation (21). Similarly, it follows from (56) that, z4 undergoes thefollowing differential equation:

_z4 ¼ _ν1�2a1ð _x4x1þ x4 _x1þ _x5x3þ x5 _x3Þ ð65ÞUsing (37)–(40) and (54), it follows from (65):

_z4 ¼ c2ð�c2z2þz4Þþ2Φref €Φref þ2 _Φ2ref þ2Lseqδð2Φref _Φref

�ð�c2z2þz4ÞÞþ2Lseq_δðΦ2

ref �z2Þ�2a1ðða1x1�Lseqδx4�px2x5Þx1þ x1 ~Θ4Þ�2a1ðx4ð�ðRsa3þa1a3Þx1þ δx4þa3px2x5þa3u2vdcÞþ x4 ~Θ1Þ�2a1ðða1x3�Lseqδx5þpx2x4Þx3þ x3 ~Θ5Þ�2ðc2�2LseqδÞðx4 ~Θ4þ x5 ~Θ5Þ�2a1ðx5ð�ðRsa3þa1a3Þx3�a3px2x4þ δx5þa3u1vdcÞþ x5 ~Θ3Þ

ð66Þ

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–106 97

For convenience, Eq. (66) is given the following compact form:

_z4 ¼ ν2�2a1a3vdcðx4u2þ x5u1Þ ð67Þwith

ν2 ¼ c2ð�c2z2þz4Þþ2Φref €Φref þ2 _Φ2ref

þ2Lseqδð2Φref _Φref �ð�c2z2þz4ÞÞ�2a1ðða1x1�Lseqδx4�px2x5Þx2þ x4ð�ðRsa3þa1a3Þx1þ δx4þa3px2x5ÞÞ�2a1ðða1x3�Lseqδx5þpx2x4Þx3þ x5ð�ðRsa3þa1a3Þx3�a3px2x4þ δx5ÞÞþ2Lseq

_δðΦ2ref �z2Þ�2a1ðx3 ~Θ5þ x5 ~Θ3

þ x4 ~Θ1þ x1 ~Θ4Þ�2ðc2�2LseqδÞðx4 ~Θ4þ x5 ~Θ5Þ¼ ðc2�2LseqδÞð�c2z2þz4Þþ2Φref €Φref þ2 _Φ

2ref

þ4LseqδΦref _Φref þ2Lseq_δðΦ2

ref �z2Þþ2a1x3pðx5x1� x3x4Þ�2ða1Þ2ðx12þ x3

2Þþ2a1ðLseqδþ Rsa3þa1a3Þðx4x1þ x3x5Þ�2a1δðΦ2

ref �z2Þ�2a1ðx5 ~Θ3þ x5 ~Θ3þ x4 ~Θ1

þ x1 ~Θ4Þ�2ðc2�2LseqδÞðx4 ~Θ4þ x5 ~Θ5Þ ð68ÞTo stabilize the error system (z1; z2; z3; z4), Eqs. (57), (58), (63),

and (67) suggest the following control law:

u2

u1

" #¼

λ0 λ1λ2 λ3

" #�1UA

UB

" #ð69Þ

and the following parameter adaptive laws:

_~J ¼ �λJ ;_~f ¼ �λf ð70Þ

where

λJ ¼ �c2z23þz1ðc1z1þ _Ωref Þþz3ðc1 J� f Þðc1z1þ _Ωref Þ ð71Þ

λf ¼ z1ðΩref �z1Þþz2ðΩref �z1Þðc1 J� f Þþz23 ð72Þ

UA ¼ �μ2�ðc3þc1Þz3�ðλJðc1z1þ _Ωref Þþλf ðΩref �z1ÞÞ ð73Þ

UB ¼ �z2�c4z4�ν2 ð74Þ

λ0 ¼ pa3x5vdc; λ1 ¼ �pa3x4vdc; λ2 ¼ �2a1a3x4vdc;λ3 ¼ �2a1a3x5vdc ð75Þ

The output-feedback adaptive controller thus establishedincludes the adaptive observer (20)–(28) and the adaptive regu-lator (69) and (70). Its performances are described hereafter inTheorem 1. But, let us first make a number of computationremarks.

Remark 6.

(1) In view of (64), it turns out that the control (69) requires thespeed reference Ωref to be differentiable with respect to timeup to second order and its (first and second) derivative mustbe available. This requirement is not an issue as it can alwaysbe met by filtering the (eventually nondifferentiable) initialreference through second-order linear filter and taking thefiltered version as the reference to be matched.

(2) The computation of _Φref and €Φref , needed in (68), is dealt withusing the available equations and estimates. First, recall thebasic relations:

Φref ¼ FðIsÞ ¼ f 0þ f 1Isþ f 2I2s þ⋯þ f nI

ns with Is ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix21þx23

q

It is readily checked that

_Φref ¼dFðIsÞdIs

_Is ¼ dFðIsÞdIs

dIsdx1

_x1þdIsdx3

_x3

� �ð76Þ

The derivatives _x1 and _x3 are then replaced by the right sidesof Eqs. (2) and (3). In the latter, the unavailable states arereplaced by their estimates which we know they converge totheir true values (Proposition 1). The computation of €Φref isdealt with similarly.

(3) The estimates J and f are computed using Eqs. (70)–(72).Accordingly, one gets

_J ¼ �c1z23þz1ðc1z1þ _Ωref Þþz3ðc1 J� f Þðc1z1þ _Ωref Þ ð77Þ

_f ¼ z1ðΩref �z1Þþz3ðΩref �z1Þðc1 J� f Þþz23 □ ð78Þ

Theorem 1. Consider the overall control system composed of theinverter–motor association, described by the model (1)–(5), and theoutput-feedback adaptive controller including:

(i) the interconnected adaptive observer (20)–(28)(ii) the optimal flux generator (35)(iii) and the adaptive regulator (69) and (70).

(1) The closed-loop system, expressed in the error coordinates (z1,z2, z3, z4; e1; e2; e3), undergoes the following equations:

_z1 ¼ �c1z1þ1Jz3þ

~JJðc1z1þ _Ωref Þþ

~fJx2� ~Θ2 ð79Þ

_z2 ¼ �c2z2þz4�2x4 ~Θ4�2x5 ~Θ5 ð80Þ

_z3 ¼ � c1þc3�c1 J� f

J

! !z3

þðc1 J� f Þ~JJðc1z1þ _Ωref Þþ

~fJðΩref �z1Þ

" #ð81Þ

_z4 ¼ �c4z4�z2 ð82Þ

_e1 ¼ bA1ðX2; yÞ�λS�11 CT

1C1ce1þg1ðu; y;X2ÞþΔg1ðu; y;X2Þ�g1ðu; y; X2ÞþbA1ðX2; yÞþΔA1ðX2; yÞ�A1ðX2; yÞcX1þΔϖρþ ~ϖρ

ð83Þ

_e2 ¼ bA2ðX1; ρÞ�S�12 CT

2C2ce2þbA2ðX1; ρÞþΔA2ðX1; ρÞ�A2ðX1; ρÞcX2

þg2ðu; y;X1ÞþΔg2ðu; y;X1Þ�g2ðu; y; X1Þ ð84Þ

_e3 ¼ �½λS�13 ΛTCT

1C1Λ�e3�½λS�13 ΛTCT

1C1�e1 ð85Þ

(2) The system error (e1; e2; e3, z1, z2, z3, z4), described byEqs. (79)–(85), is asymptotically convergent to a neighborhoodof the origin that can be made arbitrarily small by letting thedesign parameters, θ1; θ2; θ3,c1; c2; c3; c4, be sufficiently largeand satisfy the following conditions:

γ1 ¼def

c1�12υ4�

12J

40; γ2 ¼def

c2�2Φr maxυ5�2υ6

� �40; ð86Þ

γ3 ¼def

c3þfJ� 12J

� �40; γ4 ¼

defc440; ð87Þ

γ5 ¼def ðθ1� ~μ13�υ1 ~μ�υ2 ~μ16Þ40;

γ6 ¼def

θ2� ~μ14�1υ1

~μ�υ3 ~μ11

� �40; ð88Þ

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–10698

γ7 ¼def

θ3� ~μ15�1υ2

~μ16�1υ3

~μ11

� �40 ð89Þ

with

~μ1 ¼μ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

λminðS1Þp ; ~μ2 ¼

μ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλminðS2Þ

p ;

~μ ¼def μ3þμ4þμ5þμ9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλminðS1Þ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλminðS2Þ

p ; ~μ7 ¼def μ7þμ8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

λminðS2Þp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

λminðS3Þp ð90Þ

~μ13 ¼def μ213

2υ4λminðS1Þ; ~μ14 ¼

defΦr maxðμ216þμ217Þυ5λminðS2Þ

;

~μ15 ¼def μ213

2υ4λminðS3Þ; ð91Þ

~μ16 ¼def

μ6þ 12υ4

μ213

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλminðS1Þ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλminðS3Þ

p ; ~μ17 ¼def

υ61υ7þ υ7

2 ðμ416þμ417Þ� �

λminðS1Þð Þ2ð92Þ

μ1 ¼ k1ðk11s1þs3Þþk1k8s5; μ2 ¼ k2ðk12s2þs4Þ; μ3 ¼ k1k11

k15þk2k8k9; μ4 ¼ k2k12k16 μ5 ¼ k2k10; μ6 ¼ λk19;μ7 ¼ k2k12k18; μ8 ¼ k2k14; μ9 ¼ k1k8 ð93Þ

whereλminðS1Þ,λminðS2Þ,λminðS3Þ denote the minimal eigenvalues of S1, S2, S3,respectively

k1–k19 are positive real constants defined in the proofυiði¼ 1;…;7Þ and (μ13; μ14; μ16, μ17) are arbitrary positive

constants satisfying:

υiA �0 1 ði¼ 1;…;7Þ½ ð94Þ

~Θ1rμ14ð‖e1‖þ‖e3‖Þ; ~Θ2rμ13ð‖e1‖þ‖e3‖Þ; ~Θ4rμ16‖e2‖;~Θ5rμ17‖e2‖ ð95Þ

Proof. See Appendix.

Remark 7. .

It is worth noting that Theorem 1 does not stipulate whetherthe parameter estimates J and f converge to their true values. As amatter of fact, the convergence of the unknown parameterestimates to their true values is not necessary when the parameteradaptive law is derived from a stabilizing Lypunov function, whichis actually the case in the present paper. The excitation persistentcondition of Assumption A.2, which guarantees the observerconvergence, is useless for the parameter estimation, becausethe unknown parameters are not estimated by the observer.□

5. Simulation

The adaptive output-feedback controller, with flux-reference opti-mizer, described in Theorem 1, will now be evaluated throughsimulation. The experimental setup has been simulated, withinMatlab/Simulink environment. The calculation step is given the value5 ms. This is motivated by the fact that the inverter frequencycommutation 10 kHz. The experimental protocol is illustrated inFig. 1. The controlled machine is simulated by model (1)–(5) usingthe electromechanical characteristics of a real-life machine that existsat GIPSA Lab, in Grenoble-France (Table 1). These correspond to a real-life 7.5 kW induction machine whose magnetic characteristic is that ofFig. 3. The controller performances will be illustrated by consideringquite tough operation conditions described in Figs. 7 and 8. Accord-ingly, the applied load torque and reference speed are profiled so thatthe machine operates successively in high and low speedmodes whilefacing large load torque changes.

Specifically, the machine operates at high speed (Ωref ¼151 rd=s) over the interval 0; 6 s½ � and at low speed(Ωref ¼ 20 rd=s) on 6 s; 8 s½ � (see Fig. 8).

The design parameters of the observer and regulator are giventhe following values selected by try-error: θ1 ¼ 100, θ2 ¼ 200.θ3 ¼ 200, λ¼ 10, c1 ¼ 500, c2 ¼ 800 c3 ¼ 500, c4 ¼ 10 000. In all

0 1 2 3 4 5 6 7 8-5

0

5

10

15

20

25

30

35

40

45

Time (s)

Load

torq

ue (

Nm

) TL mes

TL est

Fig. 7. Load torque TL (solid, estimated; dotted, applied torque).

0 1 2 3 4 5 6 7 8-20

0

20

40

60

80

100

120

140

160

Time (s)

Spee

d (r

d/s)

ref

mes

est

Fig. 8. Rotor speed (rd/s) (solid, speed reference trajectory; dotted, measuredspeed; dashed, estimated speed).

0 0.5 1 1.5-10

0

10

20

Time (s)

Trac

king

err

or (r

d/s)

ref-

mes

0 0.5 1 1.5-15

-10

-5

0

5

Time (s)

Estim

atio

n er

ror (

rd/s

)

mes-

est

Fig. 9. Zoom on the speed (rd/s), over the interval [0 s,1.5 s]. Top: tracking error.Bottom: estimation error.

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–106 99

experiments, the initial conditions of the observed variables aredifferent from those of the true variables. Specifically, we chooseΩð0Þ ¼ 20 rd=s, Ωð0Þ ¼ 0 rd=s, φrαð0Þ ¼ 0:1 wb, φrβð0Þ ¼ �0:1 wb,φrαð0Þ ¼ 0:4 wb, φrβð0Þ ¼ � 0:4 wb.

5.1. Control performances obtained with the controller includingflux-reference optimizer

In this subsection, all machine parameters are supposed to beconstant and known to the designer (and so used in the controldesign) except for those which are online estimated (i.e. f ; J;Rs; TL).The load torque is made time-varying between zero and the

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

Time(s)

Trak

ing

erro

r (w

b)

ref- r

mes

0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

Time (s)

Est

imat

ion

erro

r (w

b)

mes- r

est

Fig. 11. Zoom on the rotor flux norm (Wb), over the interval [0 s, 1 s]. Top, trackingerror; Bottom, estimation error.

0 1 2 3 4 5 6 7 80

100

200

300

400

500

600

700

800

900

1000

1100

Time (s)

para

met

er

( H

-2)

est

Fig. 12. δðΩH�2Þ (dotted, measured; solid, estimated).

0 1 2 3 4 5 6 7 80.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Estim

ated

sta

toriq

ue re

sist

ance

(O

hm)

Fig. 13. Estimated stator resistance (Ω).

5 5.02 5.04 5.06 5.08 5.1-2

-1

0

1

2

Time (s)

cont

rol s

igna

l u 2

4 4.02 4.04 4.06 4.08 4.1-2

-1

0

1

2

Time (s)

cont

rol s

igna

l u 1

Fig. 14. Zoom on the control signal u1 and u2.

0 2 4 6 8 10-60

-40

-20

0

20

40

60

80

100

Time (s)

roto

r spe

ed (r

d/s)

mes

est

Fig. 15. Observer performances at zero speed (dotted, measured; solid, estimated).

0 1 2 3 4 5 6 7 8-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Rot

or fl

ux n

orm

(w

b)

ref

rmes

rest

Fig. 10. Rotor flux norm (Wb) (solid, flux reference; dotted, estimated flux; dashed,measured flux).

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–106100

machine nominal torque which presently equals 49 N m. Fig. 7shows the load torque applied to the machine as well as itsestimate provided online by the observer (20)–(28). It is readilyseen that the estimation error vanishes after a short transientperiod. Rotor speed control and observation are illustrated inFig. 8. It is seen that the speed estimate (provided by the observer(20)–(28)) matches well the measured speed and both track wellthe speed reference trajectory. This is better emphasized in Fig. 9which shows that both speed tracking error and speed estimationerror converge to zero after short periods following load torquechanges. The rotor flux norm control and observation perfor-mances are illustrated in Fig. 10. It is seen that the estimated flux,provided by the observer (20–28), matches well the true flux andboth track well their reference trajectory, provided by the fluxreference optimizer (35). Fig. 11 shows that the corresponding fluxtracking error and flux estimation error vanish after less than 0.2 s,following each load torque change. Notice that the flux referencetrajectory is actually changing with the load torque so that theabsorbed stator current is minimized. Fig. 12 emphasizes the goodestimation accuracy of the parameter δ. It also shows that thevalue of δ is actually changing with the flux, which in turn varieswith the load torque. Nevertheless, δ converges rapidly after eachload torque change. Fig. 13 illustrates the quality of estimation ofthe stator resistance. It is seen that the estimate converges to its

nominal value (0.63 Ω) shortly after each change of load torque orspeed reference. The control signals are shown in Fig. 14, it isparticularly checked that these remain bounded.

0 1 2 3 4 5 6 7 8-10

-5

0

5

10

Time (s)

Trac

king

err

or (r

d/s)

ref-

mes

0 1 2 3 4 5 6 7 8-10

-5

0

5

10

Time (s)

Estim

atio

n er

ror (

rd/s

)

mes-

est

Fig. 16. Robustness of control laws with respect to noise measurement.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time(s)

Rot

or fl

ux n

orm

(w

b)

rref

rmes

rest

Fig. 17. Rotor flux norm (Wb) (CFR) (solid, reference; dotted, estimated; dashed,measured).

0 1 2 3 4 5 6 7 8

-0.1-0.05

00.05

0.1

Time(s)

Trak

ing

erro

r (w

b)

rref

- rmes

0 1 2 3 4 5 6 7 8-0.1

0

0.1

0.2

0.3

Time (s)

Est

imat

ion

erro

r (w

b)

rmes

- rest

Fig. 18. Zoom on the rotor flux norm (Wb) (CFR). Top, tracking error; Bottom,estimation error.

0 1 2 3 4 5 6 7 8-5

0

5

10

Time (s)

Trac

king

err

or (

rd/s

)

ref-

mes

0 1 2 3 4 5 6 7 8

-5

0

5

Time (s)

Est

imat

ion

erro

r (r

d/s)

mes-

est

Fig. 19. Zoom on the speed (rd/s) (CFR). Top, tracking error; Bottom,estimation error.

1 2 3 4 5 6 7 80

5

10

15

20

Time (s)

stat

or c

urre

nt (A

)

1 2 3 4 5 6 7 80

5

10

15

20

Time (s)

stat

or c

urre

nt (A

)

OFG

CFR

Fig. 20. Stator current norm (A). Top, CFR controller; Bottom, FRO controller.

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–106 101

To illustrate the observability issue, another test is performedwhere the machine speed is enforced to vanish. The simulationprotocol is such that the load torque is constant and the speedreference switches from a high value and very low value. The restof the simulation conditions remain unchanged and the sameobserver is used. The resulting observation performances areillustrated in Fig. 15. This clearly shows that, the performancesdeteriorate whenever the machine speed is close to zero, just as inthe case where the magnetic saturation is ignored (Ghanes et al.,2006).

Finally to checking the robustness of proposed control law withrespect to noise input, a final test has been added. Fig. 16 showsthe speed estimation and tracking errors when the white noise isadded to stator currents. It is clearly seen that the proposed outputcontrol strategy is robust to measurement noise. In fact, thisrobustness is justified by the fact that the control law (69) doesnot depend on any terms relating to the derivative of themeasured signals

5.2. Control performances obtained with constant flux

The same experimental protocol and conditions (as in Section5.1) are presently considered. The only change here concerns the

rotor flux reference: it is presently kept constant equal to thenominal flux Φref ¼ 0:56 wb (while it was online generated inSection 5.1). For convenience, the resulting adaptive output-feedback controller is referred to constant-flux-reference (CFR).

0 1 2 3 4 5-20

-10

0

10

20

Time (s)

Trac

king

err

or (r

d/s)

0 1 2 3 4 5-20

-10

0

10

20

Time (s)

Estim

atio

n er

ror (

rd/s

)

ref-

mes

mes-

est

Fig. 21. The þ20% of J, Zoom on the speed (rd/s): Top, tracking error; Bottom,estimation error.

0 1 2 3 4 5 6-0.2

-0.1

0

0.1

0.2

Time (s)

Trac

king

err

or (

wb)

0 1 2 3 4 5 6-0.2

-0.1

0

0.1

0.2

Time (s)

est

imat

ion

erro

r (w

b)

ref-

rmes

mes-

rest

Fig. 22. The þ20% of J, Zoom on the rotor flux norm (Wb). Top, tracking error;Bottom, estimation error.

0 1 2 3 4 5 6 7 8-0.2

-0.1

0

0.1

0.2

Time (s)

Trac

king

err

or (w

b)

0 1 2 3 4 5 6 7 8-0.2

-0.1

0

0.1

0.2

Time (s)

Estim

atio

n er

ror (

wb)

ref-

rmes

mes-

rest

Fig. 23. The þ20% of f, Zoom on the rotor flux norm (Wb) Top, tracking error;Bottom, estimation error.

0 1 2 3 4 5 6 7 8-10

-5

0

5

10

Time (s)

Trac

king

err

or (r

d/s)

0 1 2 3 4 5 6 7 8-10

-5

0

5

10

Time (s)

Est

imat

ion

erro

r (rd

/s)

ref-

mes

mes-

est

Fig. 24. The þ20% of f; Zoom on the speed (rd/s). Top, tracking error; Bottom,estimation error.

0 1 2 3 4 5 6-0.2

-0.1

0

0.1

0.2

Time (s)

Trac

king

err

or (w

b)

0 1 2 3 4 5 6-0.2

-0.1

0

0.1

0.2

Time (s)

Estim

atio

n e

rror

(wb)

ref-

rmes

rmes-

est

Fig. 25. The þ20% of Rr. Zoom on the rotor flux norm (Wb): Top, tracking error;Bottom, estimation error.

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–106102

The corresponding control performances are partly illustratedfocusing on the flux. Fig. 17 shows that the CFR controller regulateswell the rotor flux norm (Fig. 18). A similar phenomenon isobserved in Fig. 19 concerning rotor speed estimation and track-ing. To complete the comparison between the FRO-based con-troller of Section 5.1 and the present CFR controller, the absorbedstator currents are plotted in Fig. 20. This clearly shows that FRO-based controller requires a smaller current, and so offers a betterenergetic efficiency, than the CFR controller.

5.3. Robustness with respect to model parameter uncertainty

In this subsection, the experimental conditions are made morecomplex as some machine parameters are subject to variations.Specifically, it is supposed that the true rotor resistance Rr is 20%larger than its supposed nominal value (given in Table 1), the truemachine inertia moment and viscous friction coefficient are also20% larger compared to their nominal values. Meanwhile, thedesigner is not supposed to be aware of these model parametererrors and so keeps using the nominal values of Table 1 in controldesign. Except for this modeling error, the experimental protocoland conditions are identical to Section 5.1. Figs. 21–26 show thatthe proposed FRO-based adaptive output-feedback controller isrobust to the above modeling errors. Indeed, its asymptotictracking quality is preserved, only transient performances havebeen slightly affected but they still are quite acceptable.

6. Conclusion

In this paper, we have developed an output-feedback adaptivenonlinear control design approach for induction motors. The twooperational control objectives are tight speed regulation and fluxreference optimization (in the sense of energetic efficiency), in thepresence of wide range variations of the speed reference trajectory andthe load torque. The developed adaptive controller includes anadaptive interconnected state observer, a flux reference optimizerand an adaptive speed/flux regulator. Adaptation is resorted to dealwith parameter uncertainty affecting the machine inertia moment,stator resistance, friction coefficient and load torque. The proposedadaptive controller is formally shown (Theorem 1) to meet its controlobjectives and, besides, presents the following appealing features:

(i) the control design is performed for the overall inverter–motorassociation (Fig. 1), while most previous works considered the

motor as a separate system directly controlled with the statorvoltage which is not the case in real-life

(ii) no mechanical sensors are needed reducing sensor imple-mentation and maintenance costs

(iii) the controller shows interesting robustness capability.

Appendix. Proof of Theorem 1

Proof of Part 1: It follows from (10), (11), (34) and (20)–(28),that e01; e2; e3 undergo the following equations:

_e01 ¼ A1ðX2;yÞ�λΛS�13 ΛTCT

1C1�λS�11 CT

1C1

j ke01þϖðu; y; X2Þe3

�g1ðu; y; X2Þþ A1ðX2;yÞþΔA1ðX2;yÞ�A1ðX2;yÞh i

X1

þg1ðu; y;X2ÞþΔg1ðu; y;X2ÞþΔϖρþ ~ϖρ ð96Þ

_e2 ¼ A2ðX1; ρÞ�S�12 CT

2C2

j ke2þ A2ðX1; ρÞþΔA2ðX1; ρÞ�A2ðX1; ρÞ

j kX2

þg2ðu; y;X1ÞþΔg2ðu; y;X1Þ�g2ðu; y; X1Þ ð97Þ

_e3 ¼ �λ S�13 ΛTCT

1C1e01 ð98aÞwhere

~ϖ ¼ϖðX2Þ�ϖðX2Þ ð98bÞ

Using (34) one readily gets

_e1 ¼ _e01� _Λ e3�Λ _e3 ð99Þ

_e1 ¼ A1ðX2Þ�λS�11 CT

1C1

j ke1þg1ðu; y;X2Þ�g1ðu; y; X2ÞþΔg1ðu; y;X2Þ

þ A1ðX2ÞþΔA1ðX2;yÞ�A1ðX2Þj k

X1þΔϖ ρþ ~ϖ ρ ð100Þ

_e2 ¼ A2ðX1; ρÞ�S�12 CT

2C2

j ke2þ A2ðX1; ρÞþΔA2ðX1; ρÞ�A2ðX1; ρÞ

j kX2

þg2ðu; y;X1ÞþΔg2ðu; y;X1Þ�g2ðu; y; X1Þ ð101Þ

_e3 ¼ � λS�13 ΛTCT

1C1Λj k

e3� λS�13 ΛTCT

1C1

j ke1 ð102Þ

Eqs. (79) and (80) are immediately obtained from (57) and (58).Eq. (81) is obtained substituting the control law (69) to (u1; u2) onthe right side of (63). Eq. (82) is obtained substituting the controllaw (69) to (u1; u2) on the right side of (67). This proves Part 1.

Proof of Part 2: Consider the following augmented Lyapunovfunction candidate:

V6 ¼ V2þV3þV4þV5 ð103Þwith

V2 ¼ eT1S1e1; V3 ¼ eT2S2e2; V4 ¼ eT3S3e3;

V5 ¼12z21þ

12z22þ

12z23þ

12z24þ

12

~J2

Jþ12

~f2

Jð104Þ

From (79) to (82), its time-derivative along the trajectory of thestate vector (z3,z4,z5,z6) is

_V5 ¼ z1 _z1þz2 _z2þz3 _z3þz4 _z4þ~J _~JJþ~f _~fJ

ð105Þ

_V5 ¼ z1 �c1z1þ~JJðc1z1þ _Ωref Þþ

~fJðΩref �z1Þþ

1Jz3� ~Θ2

!

þz2ð�c2z2þz4�2x4 ~Θ4�2x5 ~Θ5Þþz3ðμ2þpa3vdcðx5u2� x4u1ÞÞ

þz3ðc1 J� f Þ~JJðc1z1þ _Ωref Þþ

~fJðΩref �z1Þ

" #

0 1 2 3 4 5 6-10

-5

0

5

10

Time (s)

Trac

king

err

or (r

d/s)

0 1 2 3 4 5 6-10

-5

0

5

10

Time (s)

Estim

atio

n er

ror (

rd/s

)

ref-

mes

mes-

est

Fig. 26. The þ20% of Rr. Zoom on the speed (rd/s): Top, tracking error; Bottom,estimation error.

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–106 103

�z3_~Jðc1z1þ _Ωref Þþ _~f x1h i

þc1z23�c1~JJz23�

fJz23þ

~fJz23

þz4ðν2�2a1a3vdcðx4u2þ x5u1ÞÞ ð106ÞAdding c3z23�c3z23þc4z24�c4z24 to the right side of (106) and

rearranging terms, yields

_V5 ¼ �c1z21�c2z22�c3z23�c4z24�z1 ~Θ2�2x4 ~Θ4z2

�2x5 ~Θ5z2þ1Jz1z3�

fJz23þz3ðμ2þðc1þc3Þz3

þpa3vdcðx5u2� x4u1ÞÞþz4ðν2þz2þc4z4

�2a1a3vdcðx4u2þ x5u1ÞÞ�z3_~J ðc1z1þ _Ωref Þþ _~f x1h i

þ~JJ_~Jþðc1z1þ _Ωref Þðz1þz5ðc1 J� f ÞÞ�c1z23h i

þ~fJ_~f þz23þðΩref �z1Þðz1þz3ðc1 J� f ÞÞh i

ð107Þ

where c3 and c4 are new arbitrary positive real design parameters.Substituting the control law (69) to (u2; u1), and adaptation laws(70) on the right side of (107) we find

_V5 ¼ �c1z21�c2z22�c3z23�c4z24�z1 ~Θ2

�2x4 ~Θ4z2�2x5 ~Θ5z2þ1Jz1z3�

fJz23 ð108Þ

replacing x4 ¼ x4� ~x4 and x5 ¼ x5� ~x5in Eq. (108) becomes

_V5 ¼ �c1z21�c2z22�c3z23�c4z24�z1 ~Θ2�2x4 ~Θ4z2

þ2 ~x4 ~Θ4z2�2x5 ~Θ5z2þ2 ~x5 ~Θ5z2þ1Jz1z3�

fJz23 ð109Þ

where ~x4 and ~x5 are estimation errors of x4 andx5, respectively.From Assumptions A.1–A.3 one gets, using the fact that the

machine is powered by an DC/AC inverter which physicallyprovides a finite voltage supply:

‖S1‖rk1; ‖S2‖rk2; ‖S3‖rk3; ‖S�11 ‖rk4;

‖S�12 ‖rk5; ‖S�1

3 ‖rk6; ‖X1‖rk11; ‖X2‖rk12; ð110Þ

‖g1ðu; y;X2Þ�g1ðu; y; X2Þ‖rk7‖e2‖;‖A1ðX2; yÞ�A1ðX2; yÞ‖rk15‖e2‖; ‖ρ‖rk8; ð111Þ

‖g2ðu; y;X1Þ�g2ðu; y; X1Þ‖rk10‖e1‖þk14‖e3‖; ‖ ~ϖ‖rk9‖e2‖ð112Þ

‖A2ðX1; ρÞ�A2ðX1; ρÞ‖rk16‖e1‖þk18‖e3‖; ‖ΛTCT1C1‖rk19 ð113Þ

Due to magnetic saturation characteristic, x4rΦr max andx5rΦr max are bounded and this implies that using Eqs. (110),(41), (44) and (45), there are constant positive, μ13; μ14; μ16; μ17,satisfying the following inequalities:

~Θ2rμ14ð‖e1‖þ‖e3‖Þ; ~Θ1rμ13ð‖e1‖þ‖e3‖Þ;~Θ4rμ16‖e2‖; ~Θ5rμ17‖e2‖ ð114Þwhich hold

j z1 ~Θ2 jr12υ4z12þ

12υ4

μ213ð‖e1‖2þ‖e3‖2þ2‖e1‖‖e3‖Þ ð115Þ

2jx4z2 ~Θ4 jrΦr maxυ5z22þ1υ5μ216Φr max‖e2‖2 ð116Þ

2jx5z2 ~Θ5 jrΦr maxυ5z22þ1υ5μ217Φr max‖e2‖2; ð117Þ

2j ~x4z2 ~Θ4 jrz22υ6

þ υ62υ7

‖ ~x4‖4þυ6υ72

μ416‖ ~Θ4‖4

rz22υ6

þυ612υ7

þυ72μ416

� �‖e2‖4 ð118Þ

2j ~x5z2 ~Θ5 jrz22υ6

þυ612υ7

þυ72μ417

� �‖e2‖4 ð119Þ

8υiA �0;1 ; ði¼ 4;5;6;7Þ½ ð120Þ

Substituting (115)–(120) in (109), one gets

_V5r�c1z21�c2z22�c3z23�c4z24þ12υ4z12þ

1Jz1z3�

fJz23

þ 12υ4

μ213ð‖e1‖2þ‖e3‖2þ2‖e1‖‖e3‖Þ

þΦr maxυ5z22þ1υ5μ216Φr max‖e2‖2þΦr maxυ5z22þ

1υ5μ217Φr max‖e2‖2

þ2z22υ6

þυ612υ7

þυ72μ416

� �‖e2‖4þυ6

12υ7

þυ72μ417

� �‖e2‖4 ð121Þ

On others hand, from (100)–(102), (23) and (27) one gets

_V2þ _V3þ _V4 ¼ eT1f�θ1S1�λCT1C1ge1þ2eT1S1ðA1ðX2; yÞ

þΔA1ðX2; yÞ�A1ðX2; yÞÞX1

þ2eT1S1fg1ðu; y;X2ÞþΔg1ðu; y;X2Þ�g1ðu; y; X2ÞgþeT3f�θ3S3�λΛTCT

1C1Λg e3þeT2f�θ2S2�CT

2C2ge2þ2eT2S2ðA2ðX1; ρÞþΔA2ðX1; ρÞ�A2ðX1; ρÞÞX2

þ2eT2S2fg2ðu; y;X1ÞþΔg2ðu; y;X1Þ�g2ðu; y; X1Þg�2eT3ðλΛTCT1C1Þe1

þ2eT1S1Δϖρþ2eT1S1 ~ϖρ ð122ÞBounding the right side terms with norms, the previous

expression becomes:

_V2þ _V3þ _V4r�θ1eT1S1e1þ2‖e1‖‖S1‖‖A1ðX2; yÞ�A1ðX2; yÞ‖‖X1‖þ2‖e1‖‖S1‖‖ΔA1ðX2; yÞ‖‖X1‖þ2‖e1‖‖S1‖‖g1ðu; y;X2Þ�g1ðu; y; X2Þ‖þ2‖e1‖‖S1‖‖Δg1ðu; y;X2Þ‖�θ3eT3S3e3

�θ2eT2S2e2þ2‖e2‖‖S2‖‖A2ðX1; ρÞ�A2ðX1; ρÞ‖‖X2‖þ2‖e2‖‖S2‖‖ΔA2ðX1; ρÞ‖‖X2‖þ2‖e2‖‖S2‖‖g2ðu; y;X1Þ�g2ðu; y; X1Þ‖þ2‖e2‖‖S2‖‖Δg2ðu; y;X1Þ‖þ2λ‖e3‖‖ΛTCT

1C1‖‖e1‖þ2‖e1‖‖S1‖‖Δϖ‖‖ρ‖þ2‖e1‖‖S1‖‖ ~ϖ‖‖ρ‖ ð123ÞGathering compatible terms in (123) on gets, using the defini-

tions (86)–(89):

_V2þ _V3þ _V4r�θ1eT1S1e1�θ2eT2S2e2�θ3eT3S3e3þμ1‖e1‖þμ2‖e2‖þ2ðμ3þμ4þμ5þμ9Þ‖e1‖‖e2‖þ2μ6‖e1‖‖e3‖þ2ðμ7þμ8Þ‖e2‖‖e3‖ ð124ÞThe time derivative V6 along the trajectory of the state vector

(e1; e2; e3z1; z2; z3; z4) is obtained from (103), using (124) and(121):

_V6r�θ1eT1S1e1�θ2eT2S2e2�θ3eT3S3e3þ2ðμ3þμ4þμ5þμ9Þ‖e1‖‖e2‖þ2μ6‖e1‖‖e3‖þ2ðμ7þμ8Þ‖e2‖‖e3‖

þμ1‖e1‖þμ2‖e2‖� c1�12υ4�

12J

� �z21

� c2�2Φr maxυ5�21υ6

� �z22� c3þ

fJ� 12J

� �z23�c4z24

þ 12υ4

μ213‖e1‖2þ 1

2υ4μ213‖e3‖

2þ 1υ4μ213‖e1‖‖e3‖

þ 1υ5Φr maxðμ216þμ217Þ‖e2‖2þυ6

1υ7

þυ72ðμ416þμ417Þ

� �‖e2‖4 ð125Þ

This rewritten as

_V6r�θ1V2�θ2V3�θ3V4þμ1‖e1‖þμ2‖e2‖

þ 12υ4

μ213‖e1‖2þ 1

υ5Φr maxðμ216þμ217Þ‖e2‖2

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–106104

þ 12υ4

μ213‖e3‖2þ2ðμ3þμ4þμ5þμ9Þ‖e1‖‖e2‖

þ2 μ6þ12υ4

μ213

� �‖e1‖‖e3‖þ2ðμ7þμ8Þ‖e2‖‖e3‖

� c1�12υ4�

12J

� �z21� c2�2Φr maxυ5�

2υ6

� �z22

� c3þfJ� 12J

� �z23�c4z24

þυ61υ7

þυ72ðμ416þμ417Þ

� �‖e2‖4 ð126Þ

Using the following definition:

γ1 ¼def

c1�12υ4�

12J; γ2 ¼

defc2�2Φr maxυ5�

2υ6

� �; ð127Þ

γ3 ¼def

c3þfJ� 12J

� �; γ4 ¼

defc4; ð128Þ

γ5 ¼def ðθ1� ~μ13�υ1 ~μ�υ2 ~μ16Þ; γ6 ¼

defθ2� ~μ14�

1υ1

~μ�υ3 ~μ11

� �; ð129Þ

γ7 ¼def

θ3� ~μ15�1υ2

~μ16�1υ3

~μ11

� �ð130Þ

with:

~μ1 ¼μ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

λminðS1Þp ; ~μ2 ¼

μ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλminðS2Þ

p ; ~μ ¼def μ3þμ4þμ5þμ9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλminðS1Þ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλminðS2Þ

p ;

ð131Þ

~μ7 ¼def μ7þμ8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

λminðS2Þp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

λminðS3Þp ; ~μ13 ¼

def μ2132υ4λminðS1Þ

;

~μ14 ¼defΦr maxðμ216þμ217Þ

υ5λminðS2Þ; ð132Þ

~μ15 ¼def μ213

2υ4λminðS3Þ; ~μ16 ¼

defμ6þ 1

2υ4μ213

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλminðS1Þ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλminðS3Þ

p ;

~μ17 ¼def

υ61υ7þ υ7

2 ðμ416þμ417Þ� �

ðλminðS1ÞÞ2ð133Þ

μ1 ¼ 2k1ðk11s1þs3Þþ2k1k8s5; μ2 ¼ 2k2ðk12s2þs4Þ;μ3 ¼ k1k11k15þk2k8k9; μ4 ¼ k2k12k16

μ5 ¼ k2k10; μ6 ¼ λk19; μ7 ¼ k2k12k18; μ8 ¼ k2k14; μ9 ¼ k1k8 ð134ÞUsing (103), (127)–(130), (131)–(134), the inequality (126)

implies

_V6r�θ1V2�θ2V3�θ3V4�γ1z21�γ2z

22�γ3z

23�γ4z

24

þ ~μ13V2þ ~μ14V3þ ~μ15V4þ ~μ1

ffiffiffiffiffiffiV2

pþ ~μ2

ffiffiffiffiffiffiV3

pþ2 ~μ

ffiffiffiffiffiffiV2

p ffiffiffiffiffiffiV3

pþ2 ~μ16

ffiffiffiffiffiffiV2

p ffiffiffiffiffiffiV4

pþ2 ~μ11

ffiffiffiffiffiffiV3

p ffiffiffiffiffiffiV4

pþ ~μ17V3

2

ð135ÞRecall the following inequalities which hold, whatever

υiA �0 1½, ði¼ 1;2;3Þ:

2ffiffiffiffiffiffiV2

p ffiffiffiffiffiffiV3

prυ1V2þ

1υ1V3 ð136Þ

2ffiffiffiffiffiffiV2

p ffiffiffiffiffiffiV4

prυ2V2þ

1υ2V4 ð137Þ

2ffiffiffiffiffiffiV3

p ffiffiffiffiffiffiV4

prυ3V3þ

1υ3V4 ð138Þ

Using (136) and (137), and (127) and (130), one gets from Eq.(135):

_V6r�γ5V2�γ6V3�γ7V4�γ1z21�γ2z

22

�γ3z23�γ4z

24þ ~μ1

ffiffiffiffiffiffiV2

pþ ~μ2

ffiffiffiffiffiffiV3

pþ ~μ17V3

2 ð139ÞChoosing θ1; θ2; θ3, c1; c2; c3; c4 sufficiently large and satisfying

the following conditions:

γ1 ¼def

c1�12υ4�

12J

40;

γ2 ¼def

c2�2Φr maxυ5�2υ6

� �40;

γ3 ¼def

c3þfJ� 12J

� �40;

γ4 ¼def

c440

γ5 ¼def ðθ1� ~μ13�υ1 ~μ�υ2 ~μ16Þ40;

γ7 ¼def

θ3� ~μ15�1υ2

~μ16�1υ3

~μ11

� �40;

γ6 ¼def

θ2� ~μ14�1υ1

~μ�υ3 ~μ11

� �40 ð140Þ

then Eq. (139) can be rewritten as follows:

_V6r�γ5V2�γ6V3þ ~μ1

ffiffiffiffiffiffiV2

pþ ~μ2

ffiffiffiffiffiffiV3

pþ ~μ17V3

2 ð141ÞTherefore

_V6r�γminV23þ ~μ12ϑffiffiffiffiffiffiffiffiV23

pþ ~μ17V23

2 ð142Þ

with _V6r� γmin V23þμ ϑffiffiffiffiffiffiffiffiV23

p, ~μ12 ¼ maxð ~μ1; ~μ2Þ and ϑ40

such that ϑffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2þV3

p4

ffiffiffiffiffiffiV2

pþ

ffiffiffiffiffiffiV3

pand V23 ¼ V2þV3.

Choosing υ6 close to zero and γ5; γ6 be sufficiently large so thatEq. (142) can be rewritten as follows:

_V6r�γminV23þ ~μ12ϑffiffiffiffiffiffiffiffiV23

pð143Þ

Then, it follows from (143) that _V6 is negative definite when-ever the initial conditions are such that V234 ð ~μ12ϑ=γminÞ2. That is,the error system (e1; e2; e3,z1,z2,z3,z5), defined by (79)–(85) isasymptotically stable. It is readily seen from (127) to (130) and(133) that, the quantity γmin= ~μ12ϑ can be made arbitrarily large byletting the θi's and the ci's be sufficiently large and letting the(arbitrary) parameter υ6 be sufficiently close to zero. This provesPart 2 and completes the proof of Theorem 1.

References

Aurora, C., Ferrari, A. (2004). Speed regulation of induction motors: An adaptivesensorless sliding mode control scheme. In American control conference,Boston, Massachusetts, USA (pp. 2586–2591).

Barambones, O., & Garrido, A. J. (2004). A sensorless variable structure control ofinduction motor drives. Electric Power Systems Research, 72(1), 21–32.

Besançon, G., & Hammouri, H. (1998). On observer design for interconnectedsystems. Journal of Mathematical Systems, Estimation and Control, 8(4), 1–25.

Besançon, G., Bornard, G., & Hammouri, H. (1996). Observer synthesis for a class ofnonlinear control systems. European Journal of Control, 2(3), 176–192.

Besançon, G., de Leon-Morales, J., & Guevara, O. H. (2006). On adaptive observersfor state affine systems. Intrenational Journal of Control, 79(6), 581–591.

Chiasson, J. N. (2005). Modeling and high-performance control of electric machines.Hoboken, NJ, USA: John Wiley & Sons, Inc.

De León, J., Glumineau, A., Traore, D., & Boisliveau, R. (2013). Experimentalevaluation of nonlinear control design techniques for sensorless inductionmotor. In: F. Giri (Ed.), AC electric motors control: Advanced design techniques andapplications. Oxford, UK: John Wiley & Sons Ltdhttp://dx.doi.org/10.1002/9781118574263.ch11.

EL Fadili, A., Giri, F., & EL Magri, A. (2013). Control models for induction motors. In:F. Giri (Ed.), AC electric motors control: Advanced design techniques and applica-tions. Oxford, UK: John Wiley & Sons Ltdhttp://dx.doi.org/10.1002/9781118574263.ch2.

EL Fadili, A., EL Magri, A., Ouadi, H., & Giri, F. (2013). Nonlinear control for speedregulation of induction motor with optimal energetic efficiency. In: F. Giri (Ed.),AC electric motors control: Advanced design techniques and applications. Oxford,UK: John Wiley & Sons Ltdhttp://dx.doi.org/10.1002/9781118574263.ch10.

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–106 105

El Fadili, A., Giri, F., El Magri, A., Lajouad, R., & Chaoui, F. Z. (2012). Towards a globalcontrol strategy for induction motor: Speed regulation, flux optimization andpower factor correction. International Journal Electrical Power and EnergySystems, 43, 230–244.

El Fadili, A., Giri, F., El Magri, A., Dugard, L., & Chaoui, F. Z. (2012). Adaptivenonlinear control of induction motors through AC/DC/AC converters. AsianJournal of Control, 14, 1470–1483, http://dx.doi.org/10.1002/asjc.478.

El Fadili, A., Giri, F., El Magri, A., & Besançon, G. (2013). Sensorless inductionmachine observation with nonlinear magnetic characteristic. InternationalJournal of Adaptive Control and Signal Processing., http://dx.doi.org/10.1002/acs.2398.

Feemster, M., Dawson, D. M., Aquino, P., & Behal, A. (2000). A global exponentialoutput-feedback controller for induction motors. Asian Journal of Control, 2(4),230–242.

Feemster, M., Aquino, P., Dawson, D. M., & Behal, A. (2001). Sensorless rotor velocitytracking control for induction motors. IEEE Transactions on Control SystemsTechnology, 9(4), 645–653.

Ghanes, M., & Zheng, G. (2009). On sensorless induction motor drives: Sliding-mode observer and output feedback controller. IEEE Transactions on IndustrialElectronic, 56(9), 3404–3413.

Ghanes, M., De Leon, J., Glumineau, A. (2006). Observability study and observer-based interconnected form for sensorless induction motor. In 45th IEEEconference on decision & control (CDC), San Diego, USA, December 13–15(pp. 1240–1245).

Ghanes, M., Barbot, J. P., de Leon-Morales, J., & Glumineau, A. (2010). A robustsensorless output feedback controller of the induction motor drives: Newdesign and experimental validation. International Journal of Control, 83(3),484–497.

Hofmann, H., Sanders, S. R., & Sullivan, C. R. (2002). Stator-flux-based vector controlof induction machines in magnetic saturation. IEEE Transactions on IndustryApplications, 33(4), 935–942.

Husson, R. (2010). Control methods for electrical machines drives. Hoboken, NJ, USA:John Wiley & Sons.

Khalil, H. K., Strangas, E. (2004). Sensorless speed control of induction motors. InAmerican control conference, Boston, Massachussets, USA (pp. 1127–1132).

Krstic, M., Kanellakopoulos, I., & Kokotovic, P. (1995). Nonlinear and adaptive controldesign. New York, NY: John Wiley and Sons, Inc.

Marino, R., Peresada, S., & Tomei, P. (1999). Global adaptive output feedback controlof induction motors with uncertain motor resistance. IEEE Transactions onAutomatic Control, 44(5), 967–983.

Marino, R., Tomei, P., & Verrelli, C. M. (2004). A global tracking control for speedsensorless induction motors. Automatica, 40(6), 1071–1077.

Novatnak, R., Chiason, J., & Bodson, M. (2002). High performance motion control ofan induction motor with magnetic saturation. IEEE Transactions on ControlSystems Technology, 7(3), 2145–2150.

Peresada, S., Tonielli, A., & Morici, R. (1999). High performance indirect fieldoriented output feedback control of induction motors. Automatica, 35(6),1033–1047.

Popovic, D. H., Hiskens, I. A., & Hill, D. H. (1998). Stability analysis of inductionmotor networks. Electrical Power & Energy Systems, 20(7), 475–487.

Traoré, D., Plestan, F., Glumineau, A., & DE Leon, J. (2008). Sensorless inductionmotor: High order sliding mode controller and adaptive interconnectedobserver. IEEE Transactions on Industrial Electronics, 55(11), 3818–3827.

Zhang, Q., Xu, A., Besançon, G. (2003). An efficient nonlinear adaptive observer withglobal convergence. In IFAC symposium on system identification (SYSID),Rotterdam (pp. 1737–1742).

A. El Fadili et al. / Control Engineering Practice 26 (2014) 91–106106