AbstractIn this paper, a new approach being different from the concept of DTC and IFOC for a robust torque control design forinduction motor is addressed. The design is based on the framework of singularly perturbed system theory and linear varyingparameter systems. In these systems, the rotor flux is considered to be a time-varying parameter in order to guarantee a robusttorque control with LPV flux observer with respect to the speed and resistance variations. In fact, this observer is designed toestimate the rotor flux as well as an MRAS observer is introduced to estimate the mechanical speed and rotor resistance. Themain feature of this proposed structure is the enhancement of robustness with flux, speed and rotor resistance variation. Thisimprovement leads to a considerable decrease of the torque ripples and ensures the stability for the entire operating range.The obtained simulations and experimental results are used to validate the effectiveness of the proposed control strategy.
Keywords H∞ LPV controller · Singularly perturbed system · LMI · MRAS observer · Induction motor
Abbreviations
� Rotor speed (rd/s)ωs Stator current frequency (rd/s)ωr Induced rotor current frequency (rd/s)ωc Injected rotor current frequency (rd/s)J In Inertiaf Coefficient of viscousCm Maximal electromagnetic torque∼ Symbol indicating measured valueˆ Symbol indicating the estimated value
3 PowerElectronics and Electrical Drives Laboratory, AalenUniversity, Aalen, Germany
4 University of Picardie Jules Verne, Cuffies, France
∗ Symbol indicating the command valueIM Induction motorMRAS Model reference adaptive systemLPV Linear parameter varyings, r Rotor and stator indicesd, q Direct and quadrate indices for orthogonal compo-
nents¯x Variable complex such as: ¯x� _e [¯x] + j·_m[¯x]
with j2 � − 1¯x Represents either a voltage as ¯u, a current as¯ior a
flux as ¯φ¯x∗ Complex conjugateRs, Rr Stator and rotor resistancesLs, Lr Stator and rotor inductancesτs, τr Stator and rotor time-constants (τsr � Ls, r/Rs, r)σ Leakage flux total coefficient (σ � 1 − M2/LrLs)M Mutual inductanceP Number of pole pairsω Mechanical rotor frequency (rd/s)
1 Introduction
Research interest in induction motor sensor less drives hasgrown significantly over the past years due to their advan-tages such as mechanical robustness, simple construction
and low maintenance. Since the dynamic performances ofsuch machine satisfy the majority of industrial applications,the notion of robustness is then considered as an additionalobjective of themachines control. Thus, field oriented control(FOC) and direct torque control (DTC) are the most well-known control structures used in AC-drives [1, 2]. However,the major disadvantage of the FOC control is the sensitiv-ity to parameters variation, the inaccuracy on the rotor fluxand rotor speeds estimation especially at low speed region.This drawback causes an uncertainty on the knowledge of theangular position of the referential (d, q). Additionally, vectorcontrol generates flux amplitudes less than those attainablewith the given DC bus voltage. Therefore, the peak torquecapacity of the drive is decreasing with the power lossesincrease. For the IM drives undergo gradual torque changesweakened field, the fluxmust be further reduced and the volt-age margin must be increased in order to avoid saturation ofthe current loops. On the other hand, DTC technique allowsa fast and accurate torque response without the complex fluxorientation bloc. In the scientific literature, there are severalDTC-based structures such as direct self-control [3], vectorselection strategy with switching table [4, 5], and deadbeatcontrol strategies [6]. However, their switching frequencyvaries according to the motor speed and the hysteresis bandsof torque and flux comparators. In turn, this results in a largetorque ripple unless a very short sampling time is provided[7]. Also, the most DTC concepts are based on kind of inver-sion of the IM model [8, 9] and therefore they are sensitiveto parameters variation. A comparative analysis of the IFOCand DTC methods, taking into account the effects of param-eters variation in the estimation of the rotor flux [7], as wellas the characteristics of the motor and their performance isprovided [10–12].
In this paper, a new approach is proposed for the torquecontrol of an induction motor where the rotor flux is con-sidered, as a varying parameter working in all the operatingrange. This approach contributes to:
i. A robustness modeling defined by a singularly perturbednonlinear model [13] (decomposition of the systemmodel in two subsystems) of the induction motor, wherethe axis d of the referential (d, q) is collinear at all timeswith the vector linked to the stator current. This dealswith isq � 0 and disq
dt � 0 with isd ≥ 0 or the conditionisd ≥ 0 means that the axis d is oriented in the directionof the stator current vector isd and the choice to orient thereference work frame on the stator current was guided bythe following motivations:First motivationRobustnesswith respect to stator currentsensors and in this case, the closed-loop control of the
working referential orientation is naturally robust sincethe regulation of the component q is realized from amea-surement and not from an estimation, Second motivationRobustness with respect of the inaccuracy on the speedmeasurement where the angular position of the referen-tial (d, q) is known at all times from the measurement ofthe components
(isα, isβ
)it follows that the imprecision
on the measurement of the rotor speed (Ω) does not rep-resent a disturbance for the reference (d, q) orientation.This naturally leads to consider a speed sensor less con-trol. Note that this choice provides a simple and efficientsolution to the main robustness problem of IFOC.
ii. The temporal evolution of φrd is directly linked to thevariation of the operating point of the slow subsystemdefined by the torque amplitude (Cm). In this case, thedynamic controllers based on the LPV system theory tak-ing into account the variation of the rotor flux, rotor speedand rotor resistance, respectively, for torque control andflux estimation [16–18] is an alternative to robust con-trol with fixed controller parameters in induction motorcontrol [20].
The LPVmodeling of the electromagnetic torque allowedto have a dynamic controlled only by the slip, it is indepen-dent of the rotation speed (Ω) and it is perturbed by the term(
φ∗2r
φ2rd (t)
).
The major advantage of this method is the power of math-ematical tools computation based on linear matrix inequalitywith convex optimization [14] allowing the on-line computa-tion, the very fast adaptation of the parameters in response tosudden variations of the operating point. Also, the parametersof the controller are modified in open loop without return ofthe system performance in closed loop as proven by severalworks concerning the robust LPV control of the inductionmotor.
In order to get the rotor speed and torque estimation, anMRAS observer algorithm is developed [21–24]. This lateris chosen for its design simplicity and ease implementation.
This paper is organized as follows: The second sectiongives a brief review on singularly perturbed systems theory.The description of singularly perturbed Park model of theinduction model is given in Sect. 3. In Sects. 4 and 5, thereformulation of the electromagnetic torque in the singularlyperturbed system and the synthesis of the LPV controller andLPV flux observer with stability analysis are widely devel-oped, the design of the speed and torque estimator by theMRAS method are explained in details. Finally, simulationsand experimental results of LPV control are discussed inSects. 6 and 7.
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Electrical Engineering (2021) 103:505–518 507
2 Recalls on singular perturbation
Let’s consider the LTI singularly perturbed systems definedby the state representation:
with x ∈ Rn, z ∈ Rm, u ∈ Rr , x is the slow variable,z is thefast variable, uc is the control input and ε is a small parameter.
Note that only the fast mode is controlled by the input uc.These restrictions simplify the study and will allow trans-posing directly the obtained results of system (1) to the Parkmodel.
Generally, the uc command associated with the singularperturbation methodology is decomposed as follows:
uc � uf + us (2)
Or when relation (2) is applied to system (1), uf representsthe fast part control and us the slow part control. These resultsyield to a composite state-feedback control for the originalsystem (1).
According to the Tihonov theorem applied to linear sys-tems, the two-time scale approximation of system (1) resultsin:
˙x �(A11 + A12A
−122 A21
)x − A12A
−122 B2us (3)
where O(ε) represents the approximation error on the slowstates when ε → 0 after a short transitory period defined byη(t f). The states (x, z) are then a good approximation of the
states (x, z).By setting ε � 0, if A22 is stable and invertible,we would obtain the unique solution of the quasi steady staterelating to the fast mode:
z � −A−122 (A21 x + B2us) (4)
By introducing relation (4) in the initial model (1), the slowmodel becomes.
˙x �(A11 + A12A
−122 A21
)x − A12A
−122 B2us (5)
The slow subsystem can be closed by a state feedback asfollows:
us � −ks x (6)
In the transient state and according to relation (3), the variableis considered as the error between state z and its quasi-statevalue z:
η � z − z (7)
The dynamics of η in the time scale t is written as follows:
εd(η)
dt� A21x + A22(η + z) + B2
(us + u f
) − ε ˙z (8)
By deriving Eq. (4) and considering us � 0, we obtain ε ˙z �0.Then, one can deduce the following relation:
εd(η)
dt� A21x + A22 z + A22(η) + B2us + B2uf (9)
With:
A21x + A22 z + B2us � 0
The term,which cancels out, results from the slow dynam-ics control. The reduced order fast system is then expressedby:
εd(η(t))
dt� A22(η(t)) + B2uf(t) (10)
The above relationship defined in the dilated time scale t f(fast time scale) is still known by(boundary layer) and bytransforming the slow time scale,tf � t−t0
ε0, we can write:
εd(η(t))
dt� A22η(εtf + t0) + B2uf(εtf + t0) (11)
With:
η(0) � z(t0) + z(t0)
In order to reduce the transient state of the fast mode andconsequently accelerate the convergence of z to z, the naturaldynamics of η must be increased by the fast component uf-based feedback such that:
ur � −kfη (12)
Through the feedback looping, the decomposition of system(1) into two distinct time scales is evenmore reinforced sincethe gain kf is important. The feedback loops (6) and (12) areapplied to each subsystem of reduced order such that theinitial system (1) is looped by a composed state feedback:
uc � uf + us (13)
3 Singularly perturbed parkmodelof inductionmotor
If the induction motor is powered by a voltage source, thecomponents Vsd and Vsq are then taken as control variablesand if the components isd and isq of the stator current and
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508 Electrical Engineering (2021) 103:505–518
rotor fluxes φrd and φrq are selected as the state variables, thePark model is expressed by:
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
isd � −γ isd + ωsisq + kτr
φrd + kpΩφrq + 1σ Ls
Vsdisq � −ωsisd − γ isq − kpΩφrd + k
τrφrq + 1
σ LsVsq
φrd � Mτrisd − 1
τrφrd − pΩφrq + ωsφrq
φrq � Mτrisq + pΩφrd − ωs − φrd
1τr
φrq
(14)
Where:
σ � 1 − M2
LsL r, γ � Rs
σ Ls+
RrM2
σ LsL2r, τr � Lr
Rr, k � M
σ LsL r
The electromagnetic torque equation is given by:
Cm � pM
L r
[isqφrd − φrqisd
](15)
Given classical mechanical dynamics, by introducing rela-tion (15), we obtain:
JΩ � pM
L r
[φrdisq − φrqisd
]+ Co + k1Ω + k2Ω
2 (16)
Therefore, by setting:
x � (φrd, φrq,Ω)T slow subsystem
z � (isd, isq)T fast subsystem
u � (vsd, vsq)T control variables
ε � σ � 1 − M2
LsL r.
The scope of this work is restricted to the slow systemcontrol only.
3.1 Slow sub system
Under the standard simplifying hypotheses, the dynamicsingularly perturbed equation with the oriented current com-ponent isq � 0 can be expressed by:
∑
s
�
⎧⎪⎨
⎪⎩
[φrd
φrq
]�
[− 1
τrωs − pΩ
ωs − pΩ − 1τr
]
‖φr‖2 � φ2rd + φ2
rq
[φrd
φrq
]+
[ Mτr0
]isd
(17)
As can be observed that the system output is nonlinear andthe term (ωs − pΩ) occur as the control variable in stateEq. (15). The slow subsystem Σs can be used for controllingboth flux and torque.
Controls
sdi
sln
SlowsubSystem
2rφ
mC
aims
Fig. 1 Slow subsystem control structure
The flux dynamic can be expressed by the slow subsystemΣs as follows:
[φrd, φrq
] [ φrd
φrd
]� [
φrd, φrq][
− 1τr
ωs − pΩ
ωs − pΩ − 1τr
][φrd
φrq
]
+[φrd, φrq
] ( Mτr0
)i∗sd
(18)
Therefore, we can determine the following differential equa-tion:
(19)
φrdφrd + φrqφrq � 1
τrφ2rd + (ω − pΩ)φrdφrq +
M
τrφrdi
∗sd
− 1
τrφ2rq − (ωs − pΩ)φrdφrq
For simplicity of calculation, we note ‖φr‖2 � φ2r .
The differential output equation of system (17) isexpressed by the following equation:
φ2r � 2φrdφrd + 2φrqφrq (20)
Equation (18) becomes:
φ2r � 2
τrφ2r +
2M
τrφrdi
∗sd (21)
As the open loop dynamic of the rotor flux is directly relatedto the rotor time constant, the multiplication of this dynamicby a factor of two is generally acceptable for the regulationof φ∗2
r to obtain a linear closed loop transfer type (22):
φ2r � − 2
τrφ2r +
2
τrφ2∗r (22)
Let’s consider the following linear control law illustrated inFig. 1.
Where:
nsl � ωs − pΩ
and
i∗sd � 1
M
φ2∗r
φrdwith φ2
r � 1
1 + τr/2pφ∗2r (23)
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Electrical Engineering (2021) 103:505–518 509
This control law (23) does not involve the mechanical speedΩ . Through the simplification of coupling terms (18), theperturbation of φr by the term (ωs − pΩ) no longer exists.The regulation of φ2
r by the approach (23) is then totallyindependent of the operating point in the torque-speed plant.
4 LPV torque controller design
Consider the torque equation in (d, q) frame given by thefollowing relation:
Cm � M
L r(φ∗
rdi∗sq − φ∗
rqi∗sd) (24)
The equality isq � i∗sq � 0 is satisfied at any moment andleads to the torque equation below:
Cm � −M
L rφrqisd (25)
The substitution of (23) in (25) gives:
Cm � − 1
L r
(φrq
φrd
)φ2∗r (26)
By supposing that the φ2∗r is constant, the torque dynamic
can be obtained by the differential of Eq. (26):
Cm � 1
L r
[φ2∗r
φ2rd
]
(φrqφrd − φrdφrq) (27)
By substituting the expression of φrd and φrq into Eq. (27),one can write:
Cm � 1
L r
[φ2∗r
φ2rd
]((ωs − pΩ)
(φ2rd + φ2
rq
)+M
τrφrqisd
)
(28)
From (27) and (27), the torque dynamic is given by:
Cm � − 1
τr
[φ2∗r
φ2rd
]
Cm +φ2∗r
L r
[φ2∗r
φ2rd
]
(ωs − pΩ) (29)
The system can still be put in the general LPV form:
Cm � Ac(θ )Cm + Bc(θ )(ωs − pΩ) (30)
With:
⎧⎪⎪⎨
⎪⎪⎩
l A(θ (t)) � − 1
τrθ (t),
B(θ (t)) � φ2∗r
Lr(ωs − pΩ)θ (t)
Avec θ (t) �[
φ2∗r
φ2rd(t)
]
4.1 LPV controller synthesis
Consider an open loop LPV system P described by the fol-lowing set of equations:
where y denotes the measured output, z is the controlledoutput, w is the reference and the disturbance inputs and u isthe control inputs. The matrices in (9) are affine functions ofthe parameter vector that varies in polytope Θ with verticesθ1, . . . , θ j that is:
θ (t) ∈ Θ � conv{θ1, . . . , θ j
}�
⎧⎨
⎩
r∑
j�1
αiθ j , α j ≥ 0,r∑
j�1
α j � 1
⎫⎬
⎭
The LPV synthesis problem consists in finding a controllerK (θ ) described by:
K (θ ) :
{xK (t) � AK (θ (t))xK (t) + BK (θ (t))y(t),
u(t) � CK (θ (t))xK(32)
Such that closed-loop system (31) (with input w and outputz) is internally stable and the induced L2 norm of w → z isbounded by a given number γ 0 for all possible parametertrajectories.
Pcl :
[ξ (t)z(t)
]�
[Acl(θ (t)) Bcl(θ (t))Ccl(θ (t)) Dcl(θ (t))
][ξ (t)w(t)
](33)
The characterization of robust stability and performance forclosed-loop system Pcl (31) is proved by the following the-orem:LPV system (31) would have a quadratic stability andgain level if there exists a matrix such that:
⎡
⎣ATcl(θ )X + X Acl(θ ) XBcl(θ ) Ccl(θ )T
Bcl(θ )T X −γ I Dcl(θ )T
Ccl(θ ) Dcl(θ ) −γ I
⎤
⎦ ≺ 0 (34)
This implies for synthesis inequalities (35) that, without lossof generality, we can replace the search over the polytopeΘ by the search over the vertices of this set consequently,condition (34) can be reduced to a finite set of linear matrixinequalities (LMI).
4.2 Computation of self-gain scheduled LPVcontroller
We assume that parameter dependence of the plant P is affineandΘ is polytope with vertices θ j , j � 1, 2, . . . , r .Accord-
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510 Electrical Engineering (2021) 103:505–518
Fig. 2 Diagram of flux variation
ing to the result in [14], one LPV controller K (θ ) can becomputed through the following steps:
• Computing the vertex controllers as follows:
K j�(AK j , BK j ,CK j , 0), (1 ≤ j ≤ r )
• Solving the set of LMIs (13) and (14):
⎡
⎢⎢⎢⎣
X A j + BK j C2 j + ∗ ∗ ∗ ∗ATK j
+ A j A jY + B2 j CK j + ∗ ∗ ∗(XB1 j + BK j D21 j )
T BT1 j −γ I ∗
C1 j C1 j Y + D12 j CK j D11 j −γ I
⎤
⎥⎥⎥⎦
< 0 and
[X II Y
]< 0
(35)
Where (*, *) denotes terms whose expressions follow therequirement that the matrix is self-adjoint. This step gives( AK j , BK j , CK j ) and symmetric matrices X and Y .
• Computing AK j , BK j and CK j by:
AK j � N−1( AK j − X A jY − BK jC2 j Y
− XB2 j CK j )M−T and BK j � N−1 BK j
• CK j � CK j M−T , where N and M are matrices such thatI − XY � NMT .
Finally, the state-spacematrices of theLPVpolytopic con-troller K (θ ) as a convex combination of the vertex controllersare given by.
[AK BK
CK 0
](θ ) �
r∑
j�1
α j
[AK j BK j
CK j 0
](36)
4.3 Synthesis of the LPV torque controller and rotorflux observer
Before the synthesis of the torque controller, we shouldbeforehand to characterize, the variations of the φrd compo-nent as a function of |Cm|. Figure 2 illustrates the evolution
for three remarkable values of Cm by taking the vector φr asa phase reference:
Figure 2Evolution of the componentφrd according to |Cm|for 3 remarkable values with φr vector as a phase reference.In this figure, the stator current vector isd is not visible sinceit coincides with the d axis.
4.4 Polytopic torque controller design
In this section, attention is focused on the design of thetorque feedback controller. Let us consider state Eq. (30).It constitutes an affine parameter-dependent plant if the rotor
flux[
φ∗2r
φrd
]is taken as the varying parameter, and φrd ∈
[0.8 1.11
].
More precisely, let us define the subsystem G with statevector x � Cm having y � Cm as output and u � (ωs− pΩ)as input. The system can be written as follows:
G :
⎧⎨
⎩Cm � (A0 +
[φ2∗r
φ2rd
]A1)Cm + B
[φ2∗r
φ2rd
]u
Cm � CCm
(37)
Alternatively, G admits the following LPV polytopic state-space representation:
G : α(t)S1 + (1 − α(t))S0 (38)
With:
A1 � A0 +
[φ2∗r
φ2rdmax
]
A1, 0 ≤ α(t) �
[φ2∗r
φ2rd
]−
[φ2∗r
φ2rdmin
]
[φ2∗r
φ2rdmax
]−
[φ2∗r
φ2rdmin
] ≤ 1
(39)
The reader may refer to [17] for a general theory and controlof LPV systems.
The polytopic controller K (θ (t)) as it is shown in Fig. 3is a torque LPV feedback controller allowing to track theset point reference where w �.The input of controller e isthe difference between C∗
m and Cm obtained from G andU � ωs − pΩ .
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Electrical Engineering (2021) 103:505–518 511
Fig. 3 Structure of LPV controller with mixed sensitivity method
Fig. 4 Sensitivity structure for LPV flux observer
K (θ ) has to provide satisfactory performance over thewhole motor operating range. The L2 − gain bound y guar-anteeing the closed-loop system performance and stabilityis equal in this case to (γ � 0.039283). The used weightingfunctions are defined by:
Ws � 0.0015 and WT � 5
s + 0.02
4.5 Design of rotor flux and torque observer
The LPV rotor flux observer and torque estimator are acrucial part of the control system design, since they providethe necessary information upon which the controllers actionis based.
The rotor flux observer, as it is shown in Fig. 4, has beenperformed using the induction motor model [17] and stan-dard problem structure where the controller is in fact theobserver and the same optimization mechanism is used toachieve the synthesis. The robustness is improved by takinginto account rotor resistance and mechanical speed varia-tions where θ(t) � (Ω, Rr). The design consists of findingu � Gobs · y to minimize, closed loop H∞ LPV norm fromw to z.
Where w � [Vsα, Vsβ, ηm
]constitutes the exogenous
inputs, z � [eα, eβ
]represent the outputs y � [
isα, isβ]are
the measurements and uT � [φsα, φsβ ] is the control input.The tracking errors of rotor flux components are given by
eα � φrα − φrα and eβ � φrβ − φrβ . The robust quadratic
stability and performance is achieved for γ � 0.0086 usingthe following shaping filter:
W � diag
(0.006
s + 7.3 × 103,
0.006
s + 7.3 × 103
)
4.6 Stator current-basedMRAS torque observer
The MRAS technique based on the stator current uses thelatter as a state variable for speed and torque estimation. Itprovides a reference to the torque controller. This techniqueproposed by [22] allows a good performance not only forspeed and torque estimation but also for sensorless control[21, 22].Themeasured and estimated stator currents are givenby the following equations:
isα � 1
M
(φrα + ωrτrφrβ + τr φrα
)
isβ � 1
M
(φrβ + ωrτrφrα + τr φrβ
)(40)
isα � 1
M
(φrα + ωrτrφrβ + τr φrα
)
isβ � 1
M
(φrβ + ωrτrφrα + τr φrβ
)(41)
The difference between the measured and estimated currentsis given by:
isα − isα � τr
M
(φrβ
(ωr − ωr
))
isβ − isβ � τr
M
(φrα
(ωr − ωr
))(42)
In order to determine the speed error, the currents differencesare multiplied by the two-flux components:
(isα − isα
)φrβ +
(isβ − isβ
)φrα � M
τr
(φ2rα + φ2
rβ
)(ωr − ωr
)
(43)
This operation leads to the following expression:
(ωr − ωr
) �[(
isα − isα)
φrβ +(isβ − isβ
)φrα
]· τr
M
1(φ2rα + φ2
rβ
)
(44)
The estimated torque is directly found by multiplying theestimated stator current by the rotor flux.
Cm � −p · ML r
· isd · φrq (45)
The stability of this algorithm is studied by the hyper-stabilityPopov criterion. For more details the reader can consult [22].
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512 Electrical Engineering (2021) 103:505–518
Fig. 5 General block diagram ofthe suggested IM controlscheme
5 Stability analysis
In order to check the stability of the control structure, weshall use a Lyapunov function that depends on the variantparameter as follows:
For such Lyapunov function, the stability conditiondV (x,p)
dt < 0 is equivalent to:
Q(θ )AT (θ ) + A(θ )Q(θ ) − dQ
dt< 0 (47)
• θ (t) �[
φ2∗r
φ2rd
]∈ V × T for the torque controller:
Q(θ)A(θ)T + A(θ )Q(θ)T − Q(τ ) − Q0 (48)
• For θ ∈ V and i � 1, . . . , n:
Ai Qi + Qi Ai + ATi Q(θ) + Q(θ)Ai+
A(θ)T Qi + Qi A(θ) − Q(τ ) − Q0 + Mi ≥ 0 (49)
• Q(θ) > I for all Mi S0θ ∈ V .
TheMATLAB function pdlstabmakes it possible to checkthe feasibility conditions when tmin < 0.
The system composed of LPV torque controller and LPVflux observer is stable since the observer is quadraticallystable and then the system is stable in specified range fortmin � −0.0035.
0 5 10 15 200
2
4
6
8
10
Time (s)
Load
torq
ue (N
.m)
Fig. 6 Evolution of the load torque variation
6 Simulation and experimental results
6.1 Simulation results of the new control structure
Figure 5 gives the general control structure of the inductionmotor where the mechanical speed and the electromagnetictorque are given by the MRAS observer providing refer-ence to the torque controller. The flux given by the LPVflux observer is controlled by a linear controller (23).
The simulation objective is to test the dynamic perfor-mance of the torque loop. The dynamic behavior of themagnitudes φr,Cm and isd is tested under the following sim-ulation conditions: a load torque of 10Nm is applied at 5 s inFig. 6. The rotor speed response follows the specified refer-ence as shown in Figs. 7, 8 and 9. At 12.5 s a reversible speedtest is carried out from 100 rd/s to − 100 rd/s at rated motorload. Figures 10, 11, 12 and 13 confirm the good dynamicbehavior of the cited quantities. One can note that torque isnot affected by the flux variation and the adaptation of theparameters can be very rapid in response to sudden variationsof the operating point, which confirms the choice of the LPVregulator.
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Electrical Engineering (2021) 103:505–518 513
0 5 10 15 20
-100
-50
0
50
100
Time (s)
Rot
or s
peed
(rad
/s) Reference
Estimated
Fig. 7 Evolution of the reference and the estimated speeds of inductionmotor
0 5 10 15 20-0.1
-0.05
0
0.05
0.1
Time (s)
Estim
atio
n er
ror (
rad/
s)
Fig. 8 Speed estimation error
0 5 10 15 20-10
-5
0
5
10
Time (s)
Trac
king
erro
r (ra
d/s)
Fig. 9 Speed tracking error
0 5 10 15 20
0
1
2
3
Time (s)
Stat
or c
urre
nt (A
)
Axis-dAxis-q
Fig. 10 Isd and Isq stator currents
To test the impact of rotor time constant variation at lowspeed, the proposed control structure is simulated where aload torque variation of 10Nm is applied at t � 5 s and at t �7 s a rotor time constant variation is introduced as illustratedin Fig. 14. It is clear that the speed is close to its referenceand the tracking error is small and converges quickly to zerowithout influence of the load torque and rotor time constantvariation as shown in Figs. 15 and 16. Moreover, one can
0 5 10 15 20-5
0
5
10
15
Time (s)
Elec
trom
agne
tic to
rque
(mN
)
ReferenceEstimated
Fig. 11 Electromagnetic torque evolution
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Time (s)R
otor
flux
(Wb)
ReferenceEstimated
Zoom
Fig. 12 Rotor flux
0 0.05 0.1 0.15 0.2
0
0.2
0.4
0.6
0.8
1
Time (s)
ReferenceEstimated
Fig. 13 Zoom of Rotor flux
0 5 10 15 200
5
10
15
Time (s)
Para
met
ers
varia
tion
Load torque (Nm)Rotor time constant (s)
Fig. 14 Load torque and rotor time constant variations
observe a perfect superposition of electromagnetic torqueand its reference as depicted in Fig. 17. All these argumentsprove the efficiency of the proposed control structure.
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514 Electrical Engineering (2021) 103:505–518
0 5 10 15 20-10
0
10
20
30
Time (s)
Rot
or s
peed
(rad
/s) Reference
ActualEstimated
Fig. 15 Evolution of reference, actual and estimated speeds of inductionmotor
6.9 6.95 7 7.05 7.1 7.15 7.219.995
20
20.005
20.01
Time (s)
Rot
or s
peed
(rad
/s)
ReferenceActualEstimated
Fig. 16 Zoom of reference, actual and estimated speeds
0 5 10 15 20-5
0
5
10
15
Time (s)
Elec
trom
agne
tic to
rque
(Nm
)
ReferenceEstimated
Fig. 17 Electromagnetic torque
6.2 Comparative study of new approach with IFOCand DTC control strategies
In order to confirm the performances of the new control struc-ture, a comparison between the results obtained by DTCwith 12 sectors, IFOC and the new approach was fulfilledunder MATLAB/Simulink. In the three cases, the speed iscontrolled in a closed loop by a simple PI, using the samesimulation conditions. A load torque is applied to the motorat t � 5 s. It can be clearly observed from Figs. 18 and 19that the new structure can provide a quick response, a betterrejection of disturbance (load torque) compared to the DTCand the IFOC. It confirms the insensitivity to rotor resistanceof the speed response compared to the IFOC control. Fig-ures 20 and 21 show clearly that the torque response underthe proposed control system is similar to the one obtainedunder DTC; but faster than that achieved under IFOC. Fur-
0 5 10 15 20-5
0
5
10
15
20
25
Time (s)
Rot
ors
peed
(rd/
s)
ReferenceDTCIFOCNew structureZoom
Fig. 18 Rotor speed response: new approach, IFOC, DTC
5 5.5 6 6.5 70
5
10
15
20
Time (s)
Rot
or s
peed
(rd/
s)
ReferenceDTCIFOCNew structure
Fig. 19 Zoom of the rotor speed response: new approach, IFOC, DTC
0 5 10 15 20-5
0
5
10
15
Time (s)
Elec
trom
agne
tic to
rque
(Nm
)
ReferenceIFOC DTC New structure
With Loadand
parametersvariation
Without Loadand
parametersvariation
Zoom
Fig. 20 Electromagnetic torque response: new approach, IFOC, DTC
thermore, it shows that the torque ripples under the proposedcontrol are less than those obtained by DTC. In summary,it is evident that new approach can provide a better robust-ness with respect to resistance variation then that achievedby IFOC.
6.3 Comparative study of new approach with Slidingmode order 1 control strategy
In order to evaluate further the control system, a closed-loopspeed and torque control with simple PI (proportional inte-gral) and sliding mode controller, respectively, for speed andtorque are applied to the motor drive with all two controlmethods (LPV and SM) to be able to examine the transient
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Electrical Engineering (2021) 103:505–518 515
4.9 4.95 5 5.05 5.1 5.15 5.2
0
5
10
Time (s)
Elec
trom
agne
tic to
rque
(Nm
)
ReferenceIFOC DTC New structure
Fig. 21 Zoom of the electromagnetic torque response: new approach,IFOC, DTC
0 5 10 15 20
0
10
20
30
Time (s)
Rot
or s
peed
(rd/
s) ReferenceSM-Order 1New structure
Zoom
Fig. 22 Rotor speed response: new approach, SM-Order 1
state and the steady state performances of the motor. Thesame loading is used for all the control systems. After themotor speeds up to the commanded speed, a nominal load isapplied to the motor at t � 5 s. Figures 22 and 23 show themotor speed under the two control systems. It is seen that theproposed control system is capable of controlling the motorspeed as well as sliding mode. Figures 24 and 25 show themotor developed torque under the two control systems. It isseen that the two torque curves are quite similar. Figure 26confirms that the torque response under the proposed con-trol system is slightly slower than that obtained under slidingmode; it is evident that the torque ripples under the proposedcontrol are less than those under sliding mode. In fact, theproposed method provides a better robustness in comparisonwith that sliding mode.
As a final result, the achieve simulation results presentedabove demonstrate that the performance and robustnessunder the new control structure are to a large extent betterthan those obtained by DTC, IFOC and SM order 1.
6.4 Description of the laboratory setup
Experimental tests are conducted by using dSPACE DS1104to implement the new control structure. Figure 27 shows thebasic structure of the laboratory setup with all parts. The IMstator is fed by a converter controlled directly by the DS1104board. The encoder is used to measure the mechanical speed.
5 5.5 6 6.5 710
15
20
25
Time (s)
Rot
or s
peed
(rd/
s)
ReferenceSM-Order 1New structure
Fig. 23 Zoom of the rotor speed response: new approach, SM-Order 1
0 5 10 15 20-5
0
5
10
15
20
Time (s)
Elec
trom
agne
tic to
rque
(Nm
)
ReferenceSM-Order 1New structure
With Loadand
parametersvariation
Without Loadand
parametersvariation
Zoom
Fig. 24 Electromagnetic torque response: new approach, SM-Order 1
5 5.5 6 6.5 7
0
5
10
15
Time (s)
Elec
trom
agne
tic to
rque
(Nm
)
ReferenceSM-Order 1New structure
Fig. 25 Zoom of the electromagnetic torque response: new approach,SM-Order 1
4.9 4.95 5 5.05 5.1 5.15 5.2
0
5
10
15
Time (s)
Elec
trom
agne
tic to
rque
(Nm
)
ReferenceSM-Order 1New structure
Fig. 26 Zoom of the electromagnetic torque response: new approach,SM-Order 1
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516 Electrical Engineering (2021) 103:505–518
Fig. 27 Structure of thelaboratory setup
Connector Panel CP1104
DS1104 dSPACEGrid
IMLaod
Con
vert
er
Stat
or v
olta
ge &
cu
rren
t sen
sors
( LEM
)
PWM
0 2 4 6 8 10 12 14 16 18 20
0
10
20
Time (s)
Spe
ed (r
d/s)
ReferenceActualEstimated
Fig. 28 Rotor speed response
The sensors used for the currents and voltages measure are,respectively, LA-55NP and LV-25P. Furthermore, all param-eters of themotor are given in the appendix, and the switchingfrequency is 11 kHz.
6.5 Experimental results
Experimental investigation was focused on the torque con-troller using the proposed approach. Figure 28 shows that thereference, measured (actual), and estimated speed and alsothe speed reference of the proposed sensorless control. It isobserved that the measured and estimated speed is close toeach other and converge to the speed reference. Figure 29shows the estimated and reference torque, we can see that is,the torque has no upper ripple and steady state torque reaches10Nmexactly, also the tracking errors are small and convergequickly to zero. Experimental motor currents are shown inFig. 30 It is observed that is, some current ripples appear atlow speed this is due to stator flux variation. However, statorcurrent orientation is well maintained (isq � 0) It can beseen in Fig. 31 that the rotor flux magnitude is constant, theestimated and the reference rotor flux are close to each otherand the trajectory tracking is satisfactory.
200 2 4 6 8 10 12 14 16 18
0
4
8
12
Reference
EstimatedCem
(Nm
)
Time (s)
Fig. 29 Electromagnetic torque response
0 5 10 15 20-2
0
2
4
6
Time (s)
Sta
tor c
urre
nts
(A) d axis
q axis
Fig. 30 Stator current
It is observed that both simulation and experimental resultsare same. The experimental results confirm the effectivenessof our approach.
7 Conclusion
Robust LPV torque controller performance under flux varia-tions has been achieved for induction motor. An LMI-basedapproach with polytopic model in the singularly perturbedsystem theory frame work has been proposed to design an
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Electrical Engineering (2021) 103:505–518 517
0 5 10 1 20
0.5
1
1.5
2
Time (s)
Rot
or fl
ux (W
eb)
ReferenceEstimated
Fig. 31 Rotor flux response
LPV torque controller to track the electromagnetic torque. Itis clearly turned out thatwith the use of theLPVcontroller therobustness and stability of thewhole drivewas demonstrated.The main advantage of using singularly perturbed systems isthat they provide a systematic way of reducing the inductionmotor model and in the same way the controller. The simu-lation and experimental results reveal and demonstrate highperformances of the inductionmotor control according to theprofile defined above.
Funding Open Access funding provided by Projekt DEAL.
Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing, adap-tation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indi-cate if changes were made. The images or other third party materialin this article are included in the article’s Creative Commons licence,unless indicated otherwise in a credit line to the material. If materialis not included in the article’s Creative Commons licence and yourintended use is not permitted by statutory regulation or exceeds thepermitted use, youwill need to obtain permission directly from the copy-right holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Appendix
Machine parameters:Rr � 5.1498 �; Rs � 12.75�; M � 0.4331H; Ls �
0.4991H; Lr � 0.4331H; J � 0.0035 kg m2;f � 0.001 Nm s/rd;Rated values:P � 0.9 kW, n � 1400 rpm, cosϕ: 0.84; p � 2; f � 50 Hz;
Load torque � 10 Nm.
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