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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: [email protected] (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.
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