A Lyapunov Approach for a PI-Controller withAnti-windup in a Permanent Magnet Synchronous

Motor using Chopper ControlPaolo Mercorelli, Member, IEEE,

Abstract—This paper deals with a parameter set up of a PI-regulator to be applied in a system for permanent magnet three-phase synchronous motors to obtain a smooth tracking dynamicseven though a chopper control structure is included in the drive.In particular, an anti-windup control structure is considered toavoid saturation and conditions on all controller parameters arefound which guarantee stability. High performance applicationof permanent magnet synchronous motors (PMSM) is increasing.In particular, application in electrical vehicles is very much used.The technique uses a geometric decoupling procedure and aLyapunov approach to perform a PWM control to be used asa chopper. Chopper control structures are very popular becausethey are very cheap and easy to be realised. Nevertheless, usinga chopper control structure smooth tracking dynamics couldbe difficult to be obtained without increasing the switchingfrequency because of the discontinuity of the control signals.No smooth tracking dynamics lead to a not comfortable traveleffect for the passengers of an electrical vehicle or, more ingeneral, it could be difficult to generate an efficient motionplanning if the tracking dynamics are not smooth. The paperpresents a technique to minimise these undesired effects. Thepresented technique is generally applicable and could be usedfor other types of electrical motors, as well as for other dynamicsystems with nonlinear model structures. Through simulations ofa synchronous motor used in automotive applications, this paperverifies the effectiveness of the proposed method and discussesthe limits of the results.

Index Terms—PD regulators, velocity control, adaptive control

I. INTRODUCTION

AS the high field strength neodymium-iron-boron (NdFeB)magnets become commercially available with affordable

prices. PMSMs are receiving increasing attention due to theirhigh speed, high power density and high efficiency. Thesecharacteristics are very favorable in some special high perfor-mance applications, e.g. robotics, aerospace, and electric shippropulsion systems [1], [2]. Permanent magnet synchronousmotors (PMSMs) as traction motors are common in electricor hybrid road vehicles. For rail vehicles, PMSMs as tractionmotors are not widely used yet. Although the traction PMSMcan bring many advantages, just a few prototypes of vehicleswere built and tested. The next two new prototypes of railvehicles with traction PMSMs were presented on InnoTransfair in Berlin 2008. They were Alstom AGV high speedtrain and Skoda Transportation low floor tram 15T ForCity.

Paolo Mercorelli is with the Institute of Product and Process Innova-tion, Leuphana University of Lueneburg, Volgershall 1, D-21339 Lueneb-urg, Germany Phone: +49-(0)4131-677-5571, Fax: +49-(0)4131-677-5300.mercorelli@uni.leuphana.de.

Advantages of PMSM are well known. The greatest advantageis a low volume of the PMSM in comparison with other typesof motors. It makes a direct drive of wheels possible. Onthe other hand, the traction drive with PMSM has to meetspecial requirements typical for overhead line fed vehicles.The drives and specially their control should be robust towide overhead line voltage tolerance (typically from −30%to +20% ), voltage surges and input filter oscillations. Theseaspects may cause problems during flux weakening operation.There are several reasons to use flux weakening operation of atraction drive. The typical reason is a constant power operationin a wide speed range and reaching nominal power duringlow speed (commonly 1/3 of maximum speed). In the caseof common traction, motors like asynchronous or DC motors,it is possible to reach the constant power region using fluxweakening. This is also possible for traction PMSM, however,a problem with high back electromagnetic force (EMF) rises.To achieve the desired system performance, advanced controlsystems are usually required to provide fast and accurateresponse, quick disturbance recovery, and parameter varia-tions insensitivity [3]. In [4] it is shown how using a fluxweakening control strategy for PMSM a prediction controlstructure improves the dynamic performance of traditionalfeedback control strategies in terms, for instance, of overshootand rising time. To realize an effective prediction control,it is known that an accurate knowledge of the model andits parameters is necessary. In [5] an identification techniqueis shown to detect parameters such as Rs, Ldq and Φ orthe PMSM is shown. In the existing applications choppercontrol structures are very popular because they are verycheap and easy to be realised. Nevertheless, using a choppercontrol structure smooth tracking dynamics could be difficultto be obtained without increasing the switching frequencybecause of the discontinuity of the control signals. No smoothtracking dynamics lead to a not comfortable travel effect forthe passengers of the electrical vehicle. PI regulators are veryoften used in industrial applications because of their simplestructure and in the last years advanced PI controllers havebeen developed to control nonlinear systems, [6], [7]. One ofthe most important problems concerning PI or PID controllersis tuning their parameters. In the scientific literature one ofthe most important approaches is the optimal tuning. Aninteresting contribution in this context is that presented in [8]in which the authors proposes as tuning techniques two setsof evolutionary strategies which are differential evolution andgenetic algorithms. In [8] the optimal PID control parameters

INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8, 2014

ISSN: 1998-0140 44

are applied for a high order system, system with time delayand non-minimum phase system. This paper extends the resultsproposed in [9] and deals with a parameter set up of a PIregulator to be applied in a system for a permanent magnetthree-phase synchronous motor to obtain smooth trackingdynamics even though a chopper control structure is includedin the drive. Moreover, the proposed PI controller uses an anti-windup scheme to avoid saturations. When saturation happens,the feedback loop is effectively broken and if a regulator withan integrator is used, the error will continue to be integrated.The value at the output of the regulator can become verylarge and often degrades the closed-loop performance in theform of large overshoot, long settling time and sometimeseven instability. This is more evident if compared with theexpected linear performance for the systems. The phenomenondescribed is called windup. The windup phenomenon hasattracted interest in academic and in industrial communityalready at the end of the eighties. The problem became to havethe first solution, an overview of the basic schemes is availablein [10]. Other contributions like in [11] analyse the conditionsin order to find invariant subspaces and in [12] a pre-actionis employed to avoid windup. The paper is organized in thefollowing way. In Section II a sketch of the model of thesynchronous motor and its behaviour are given. Section IIIis devoted to derive a decoupling controller which will beused to calculate parameters Kp and Ki of the controller withLyapunov approach. Section V shows simulation results usingreal data. The conclusions close the paper.

II. MODEL AND BEHAVIOR OF A SYNCHRONOUS MOTOR

To aid advanced controller design for PMSM, it is veryimportant to obtain an appropriate model of the motor. Agood model should not only be an accurate representation ofsystem dynamics but also facilitate the application of existingcontrol techniques. Among a variety of models presentedin the literature, since the introduction of PMSM, the twoaxis dq-model obtained using Park’s transformation is themost widely used in variable speed PMSM drive controlapplications, see [3] and [13]. The Park’s dq-transformationis a coordinate transformation that converts the three-phasestationary variables into variables in a rotating coordinatesystem. In dq-transformation, the rotating coordinate is definedrelative to a stationary reference angle as illustrated in Fig. 1.The dq-model is considered in this work.[

ud(t)uq(t)u0(t)

]=[

2 sin(ωelt)3

2 sin(ωel−2π/3)3

2 sin(ωel+2π/3)3

2 cos(ωelt)3

2 cos(ωel−2π/3)3

2 cos(ωel+2π/3)3

13

13

13

] [ua(t)ub(t)uc(t)

], (1)

[id(t)iq(t)i0(t)

]=[

2 cos(ωelt)3

2 cos(ωel−2π/3)3

2 cos(ωel+2π/3)3

−2 sin(ωelt)3

−2 sin(ωel−2π/3)3

−2 sin(ωel+2π/3)3

13

13

13

] [ia(t)ib(t)ic(t)

].

(2)

Fig. 1. Park’s transformation for the motor

The dynamic model of the synchronous motor in d-q-coordinates can be represented as follows:[

did(t)dt

diq(t)dt

]=

[−Rs

Ld

Lq

Ldωel(t)

−Rs

Lq−Ld

Lqωel(t)

][id(t)iq(t)

]+[

1Ld

0

0 1Lq

][ud(t)uq(t)

]−

[0

Φωel(t)Lq

], (3)

and

Mm =3

2p{Φiq(t) + (Ld − Lq)id(t)iq(t)}. (4)

In (3) and (4), id(t), iq(t), ud(t) and uq(t) are the dq-components of the stator currents and voltages in syn-chronously rotating rotor reference frame; ωel(t) is the rotorelectrical angular speed; parameters Rs, Ld, Lq , Φ and pare the stator resistance, d-axis and q-axis inductance, theamplitude of the permanent magnet flux linkage, and p thenumber of couples of permanent magnets, respectively. At theend the motor torque is indicated with Mm. Considering anisotropic motor for that Ld ≃ Lq = Ldq , it follows:[

did(t)dt

diq(t)dt

]=

[− Rs

Ldqωel(t)

− Rs

Ldqωel(t)

] [id(t)iq(t)

]+[

1Ldq

0

0 1Ldq

][ud(t)uq(t)

]−

[0

Φωel(t)Ldq

], (5)

and

Mm =3

2pΦiq(t) (6)

with the following movement equation:

Mm −Mw = Jdωmec(t)

dt, (7)

where pωmech(t) = ωel(t) and Mw is an unknown mechanicalload.

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ISSN: 1998-0140 45

III. STRUCTURE OF THE DECOUPLING CONTROLLER

To achieve a decoupled structure of the system described inequation (5) a matrix F is to be calculated such that:

(A+BF)V ⊆ V, (8)

where u = Fx is a state feedback with u = [ud, uq]T and

x = [id, iq]T ,

A =

[− Rs

Ldqωel(t)

− Rs

Ldqωel(t)

], B =

[ 1Ldq

0

0 1Ldq

], (9)

and V = im([1, 0]T ) is a controlled invariant subspace. Moreexplicitly it follows:

F =

[F11 F12

F21 F22

], and

[ud(t)uq(t)

]= F

[id(t)iq(t)

],

then, according to [14], the decoupling of the dynamics isobtained considering the following relationship:

im

([− Rs

Ldqωel(t)

− Rs

Ldqωel(t)

])+

im

([1

Ldq0

0 1Ldq

] [F11 F12

F21 F22

] [10

])⊆ im

[10

],

(10)

where parameters F11, F12, F21, and F22 are to be calculatedin order to guarantee condition (10) and a suitable dynamicsfor sake of estimation, as it will be explained in the next.Condition (10) is guaranteed if:

F21 = Rs. (11)

After decoupling the first equations of the system representedin (5) becomes as follows:

diq(t)

dt= ωel(t)iq(t) +

uq(t)

Ldq. (12)

IV. A LYAPUNOV APPROACH TO SET THEPI-CONTROLLER PARAMETERS

Considering the following PI-controller:

uc(t) = Kp(ωmecd − ωmec(t))+

Ki

∫ t

0

(ωmecd − ωmec(τ))dτ +Aw(t), (13)

with Aw(t) is the anti-windup signal with

Aw(t) = Kb

(u∗c(t)−Kp(ωmecd − ωmec(t))−

Ki

∫ t

0

(ωmecd − ωmec(τ))dτ), (14)

where

u∗c(t) =

{uc(t) if uc(t) < SatoutSatout if uc(t) ≥ Satout,

(15)

where Satout is the constat of the saturating level of the outputof the controller which corresponds to the saturation level ofthe compressor output, Kb is a constant to be set.

A. PI-Controller parameters conditions in case of no satura-tion

If ωmecd is a constant, it follows that:

∂uc(t)

∂t= −Kp

∂ωmec(t)

∂t+Ki(ωmecd − ωmec(t)), (16)

equation (12) can be written in the following way:

iq(t) =1

ωel(t)

(− diq(t)

dt+

uq(t)

Ldq

). (17)

Combining equations (6) and (17), then the following expres-sion is obtained:

Mm =3

2pΦ

1

ωel(t)

(− diq(t)

dt+

uq(t)

Ldq

). (18)

Considering equation (18) and equation (7), then the followingrelation is obtained:3

2pΦ

1

ωel(t)

(− diq(t)

dt+

uq(t)

Ldq

)−Mw = J

dωmec(t)

dt. (19)

In order to set up parameters Kp and KI of the controller, thefollowing Lyapunov function is chosen:

VL(ωmec(t)) =1

2

(ωmecd(t)− ωmec(t)

)2. (20)

Considering the derivative of (20), then:

∂VL(ωmec(t))

∂t= −(ωmecd(t)− ωmec(t))

∂ωmec(t)

∂t. (21)

From equation(19) it follows that:

∂VL(ωmec(t))

∂t= −

(ωmecd(t)− ωmec(t)

)J

×(32pΦ

1

ωel(t)

(− diq(t)

dt+

uq(t)

Ldq

)−Mw

). (22)

B. Monotonic increasing dynamics

Considering the expression in (16), equation (22) becomesas follows:

∂VL(ωmec(t))

∂t= −

(Kp

Ki

∂ωmec(t)∂t + ∂uc(t)

∂t1Ki

)J

×(32pΦ

1

ωel(t)

(− diq(t)

dt+

uq(t)

Ldq

)−Mw

)≤ 0. (23)

If∂ωmec(t)

∂t≥ 0, (24)

and∂uc(t)

∂t≥ 0, (25)

condition (23) is guaranteed ∀ Kp > 0 and ∀ Ki > 0.

Remark 1: Condition (24) should be guaranteed with asuitable choice of the parameters of the controller. Thiscondition states monotonic dynamics and thus dynamics ofthe motor without oscillations for the monotonic increasingdynamics. In automotive field, this condition is an idealone for optimality of the electrical driver consumption and

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ISSN: 1998-0140 46

comfort of the passengers. �

To show under which conditions the following relation

∂ωmec(t)

∂t≤ 0 (26)

holds, let consider PI controller defined by equation (16) whichcan be rewritten in the following way:

∂uc(t)

∂t+Ki(ωmecd − ωmec(t)) = Kp

∂ωmec(t)

∂t≥ 0. (27)

Choosing Kp and Ki big enough, it is possible to consider

∂uc(t)

∂t<< Ki(ωmecd − ωmec(t))−Kp

∂ωmec(t)

∂t. (28)

The following condition must hold:

Ki(ωmecd − ωmec(t))−Kp∂ωmec(t)

∂t≥ 0, (29)

which is equivalent to prove that:

Kp∂ωmec(t)

∂t+Kiωmec(t) ≥ Kiωmecd . (30)

Considering the solution of the following differential equation,then:

Kp∂ωmec(t)

∂t+Kiωmec(t) = 0, (31)

then it must be:

ωmec(t) = ωmec(0)e− Ki

Kpt ≥ Kiωmecd , (32)

then the following final general condition is obtained:

Ki ≤ωmec(0)

ωmecd

Kp. (33)

Concerning assumption (25):

∂uc(t)

∂t≥ 0, (34)

let us consider equation (16) in which ωmecd is a constant,then:

∂uc(t)

∂t−Ki − ωmec(t) = Kp

∂ωmec(t)

∂t≥ 0. (35)

Considering that it should be guaranteed ∂uc(t)∂t ≥ 0, then:

−Kp∂ωmec(t)

∂t+Ki(ωmecd − ωmec(t)) ≥ 0,

which it is equivalent to proof that:

Kp∂ωmec(t)

∂t+Kiωmec(t) ≤ Kiωmecd , (36)

Considering the solution of

Kp∂ωmec(t)

∂t+Kiωmec(t) = 0, (37)

then it must be:∂ωmec(t)

∂t= ωmec(0)e

− KiKp

t ≤ ωmecd

KpKi. (38)

From equation (69) a boundary condition on Ki is obtained:

Ki ≥ωmec(0)

ωmecd

Kp. (39)

Combining condition (33) with (71), the final sufficientcondition on parameters Kp and Ki is obtained:

Ki =ωmec(0)

ωmecd

Kp. (40)

C. Anti-windup and PI-controller parameters conditions incase of saturation and increasing dynamics

Let us consider equation (16) in which ωmecd is a constant,then:∂uc(t)

∂t= [1−Kb]

(−Kp

ωmec(t)

∂t+Ki(ωmecd −ωmec(t))

).

(41)Considering that it should be guaranteed ∂uc(t)

∂t ≥ 0, then:

0 ≤ Kb ≤ 1, (42)

andKp

∂ωmec(t)

∂t−Ki(ωmecd − ωmec(t)) ≤ 0,

which it is equivalent to proof that:

KbKp∂ωmec(t)

∂t+KbKiωmec(t) ≥ KbKiωmecd . (43)

Considering the solution of

Kp∂ωmec(t)

∂t+Kiωmec(t) = 0, (44)

then it must be:

ωmec(t) = ωmecie− Ki

Kpt ≥ ωmecd

KpKi, (45)

where ωmeci = ωmec(0). From equation (69) a boundarycondition on Ki is obtained:

Ki ≤ωmec(0)

ωmecd

Kp. (46)

D. Monotonic decreasing dynamics

Considering the expression in (16), equation (22) becomesas follows:

∂VL(ωmec(t))

∂t= −

(Kp

Ki

∂ωmec(t)∂t + ∂uc(t)

∂t1Ki

)J

×(32pΦ

1

ωel(t)

(− diq(t)

dt+

uq(t)

Ldq

)−Mw

)≤ 0. (47)

If∂ωmec(t)

∂t≤ 0, (48)

and∂uc(t)

∂t≤ 0, (49)

condition (47) is guaranteed ∀ Kp > 0 and ∀ Ki > 0.

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ISSN: 1998-0140 47

Remark 2: Condition (48) should be guaranteed with asuitable choice of the parameters of the controller. Thiscondition states monotonic dynamics and thus dynamicsof the motor without oscillations also for the monotonicdecreasing dinamics. In automotive field, this condition is anideal one for optimality of the electrical driver consumptionand comfort of the passengers. �

To show under which conditions the following relation

∂ωmec(t)

∂t≤ 0 (50)

holds, let consider PI controller defined by equation (16) whichcan be rewritten in the following way:

∂uc(t)

∂t+Ki(ωmecd − ωmec(t)) = Kp

∂ωmec(t)

∂t≤ 0. (51)

Choosing Kp and Ki big enough, it is possible to consider

∂uc(t)

∂t<< Ki(ωmecd − ωmec(t))−Kp

∂ωmec(t)

∂t. (52)

The following condition must hold:

Ki(ωmecd − ωmec(t))−Kp∂ωmec(t)

∂t≤ 0, (53)

which is equivalent to prove that:

Kp∂ωmec(t)

∂t+Kiωmec(t) ≥ Kiωmecd . (54)

Considering the solution of the following differential equation,then:

Kp∂ωmec(t)

∂t+Kiωmec(t) = 0, (55)

then it must be:

ωmec(t) = ωmec(0)e− Ki

Kpt ≥ Kiωmecd , (56)

then the following final general condition is obtained:

Ki ≥ωmec(0)

ωmecd

Kp. (57)

Concerning assumption (25):

∂uc(t)

∂t≤ 0, (58)

let us consider equation (16) in which ωmecd is a constant,then:

∂uc(t)

∂t−Ki − ωmec(t) = Kp

∂ωmec(t)

∂t≤ 0. (59)

Considering that it should be guaranteed ∂uc(t)∂t ≥ 0, then:

−Kp∂ωmec(t)

∂t+Ki(ωmecd − ωmec(t)) ≥ 0,

which is equivalent to proof that:

Kp∂ωmec(t)

∂t+Kiωmec(t) ≥ Kiωmecd , (60)

Considering the solution of

Kp∂ωmec(t)

∂t+Kiωmec(t) = 0, (61)

then it must be:

∂ωmec(t)

∂t= ωmec(0)e

− KiKp

t ≤ ωmecd

KpKi. (62)

From equation (69) a boundary condition on Ki is obtained:

Ki ≤ωmec(0)

ωmecd

Kp. (63)

Combining condition (57) with (63), the final sufficientcondition on parameters Kp and Ki is obtained:

Ki =ωmec(0)

ωmecd

Kp. (64)

E. Anti-windup and PI-controller parameters conditions incase of saturation and decreasing dynamics

Let us consider equation (16) in which pd is a constant,then:

∂uc(t)

∂t= [1−Kb]

(−Kp

ωmec(t)

∂t+Ki(ωmecd −ωmec(t))

).

(65)Considering that it should be guaranteed ∂uc(t)

∂t ≤ 0, then

0 ≤ Kb ≤ 1, (66)

and

−Kp∂ωmec(t)

∂t+Ki(ωmecd − ωmec(t)) ≤ 0,

which it is equivalent to proof that:

Kp∂ωmec(t)

∂t+Kiωmec(t) ≥ Kiωmecd . (67)

Considering the solution of

Kp∂ωmec(t)

∂t+Kiωmec(t) = 0, (68)

then it must be

ωmec(t) = ωmecie− Ki

Kpt ≤ ωmecd

KpKi, (69)

where ωmeci = ωmec(0). From equation (69) a boundarycondition on Ki is obtained:

Ki ≥ωmec(0)

ωmecd

Kp. (70)

To conclude, the intersection of all these conditions is thetwo following constraints:

Ki =ωmec(0)

ωmecd

Kp, (71)

and0 ≤ Kb ≤ 1. (72)

INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8, 2014

ISSN: 1998-0140 48

Fig. 2. Simulink structure of the whole control system

Uc(t)

1

Uc*(t)

Saturation

Kp

P-Factor

Kb

Kb-Factor

1s

Integrator

Ki

I-Factor

Add 3

Add 2Add 1

1

Velocity error

Fig. 3. Simulink structure of PI controller including anti-windup scheme

V. SIMULATION RESULTS

Simulations have been carried out using a special stand witha 58 kW traction PMSM. The stand consists of PMSM, tramwheel and a continuous rail. The PMSM is a prototype forlow floor trams. PMSM parameters: nominal power 58 kW,nominal torque 852 Nm, nominal phase current 122 A and

Fig. 4. PWM-Simulink-Block

number of poles 44. Model parameters: R = 0.08723 Ohm,Ldq = Ld = Lq = 0.8 mH, Φ = 0.167 Wb. Surfacemounted NdBFe magnets are used in PMSM. Advantage ofthese magnets is inductance up to 1.2 T, but theirs disadvantageis corrosion. The PMSM was designed to meet B curverequirements. In Fig. 2 the complete control scheme is shown.In this Simulink block diagram the transformed dq-observer isindicated together with Park’s and inverse Park’s transforma-tion. PWM frequency equals 100 kHz and the structure of thesimulink PWM block is shown in Fig. 4. Figure 6 shows theobtained and desired motor velocity profiles. Figure 3 showsPI controller with the anti-windup proposed scheme. Figure7 shows the obtained and desired motor acceleration profiles.From these two results it is possible to remark that the effectof the chopper control is visible which does not allow thetracking to be precise. In particular, according to the theoreticalcondition ∂ωmec(t)

∂t the result should not present oscillation.Because of the realisation of the controller using a chopperwhich consists of discontinuous signals this is structurallynot possible. Figure 8 shows PWM signal sequence with themaximal chopper switching frequency equals 2.5 kHz. Fig. 9shows the chopper effect on the input of the motor.

VI. CONCLUSIONS AND FUTURE WORK

This paper deals with a PI-controller for a three-phasesynchronous motor. The technique uses a decoupling proce-dure. A Lyapunov approach is used to calculate parametersKp and Ki to obtain soft velocity variation. Moreover, ananti-windup control structure is considered to avoid saturationand conditions on all controller parameters are found whichguarantee stability. The resulting control structure is generallyapplicable for other dynamic systems with similar nonlinearmodel structure. Through simulations of a synchronous motorused in automotive applications, this paper verifies the effec-tiveness of the proposed method, the found theoretical and

INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8, 2014

ISSN: 1998-0140 49

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

500

1000

1500

2000

2500

Time (sec.)

Mot

or v

eloc

ity (

roun

ds/m

in.)

Obtained velocityDesired velocity

Fig. 5. Profile of the obtained and desired motor velocity using a maximalswitching chopper frequency equals 2.5 kHz in case of positive desiredvelocity

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02−2500

−2000

−1500

−1000

−500

0

Time (sec.)

Mot

or v

eloc

ity (

roun

ds/m

in.)

Obtained velocityDesired velocity

Fig. 6. Profile of the obtained and desired motor velocity using a maximalswitching chopper frequency equals 2.5 kHz in case of negative desiredvelocity

simulation results. Future work includes the calculation of theoptimal value of parameter Kp and Kb.

REFERENCES

[1] M.A. Rahman and P. Zhou. Analysis of brushless permanent magnetsynchronous motors. IEEE Transactions on Industrial Electronics,43(2):256–267, 1996.

[2] M. Ooshima, A. Chiba, A. Rahman, and T. Fukao. An improved controlmethod of buried-type ipm bearingless motors considering magneticsaturation and magnetic pull variation. IEEE Transactions on EnergyConversion, 19(3):569–575, 2004.

[3] M.A. Rahman, D.M. Vilathgamuwa, M.N. Uddin, and T. King-Jet.Nonlinear control of interior permanent magnet synchronous motor.IEEE Transactions on Industry Applications, 39(2):408–416, 2003.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02−2

−1

0

1

2

3

4

5

6

7

8x 10

5

Time (sec.)

Mot

or a

ccel

erat

ion

(rou

nds/

min

.2 )

Obtained accelerationDesired acceleration

Fig. 7. Profile of the obtained and desired motor acceleration using a maximalswitching chopper frequency equals 2.5 kHz in case of positive desiredvelocity

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

0.5

1

1.5

Time (sec.)

PW

M c

ontr

ol s

igna

l

Fig. 8. PWM signal used as a chopper with a maximal switching frequencyequals 2.5 kHz in case of positive desired velocity

[4] R. Dolecek, J. Novak, and O. Cerny. Traction permanent magnet syn-chronous motor torque control with flux weakening. Radioengineering,18(4):601–605, 2009.

[5] P. Mercorelli. A decoupling dynamic estimator for online parametersindentification of permanent magnet three-phase synchronous motors.In Proceedings of the 16th IFAC Symposium on System Identification,SYSID 2012, pages 757–762, Brussels, 2012.

[6] S. Koprda, Z. Balogh, and M. Turcani. Modeling and comparisonof fuzzy pid controller with psd regulation in the discrete systems.INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNALPROCESSING, 5(5):496–504, 2011.

[7] P. Mercorelli. A switching cascade sliding pid-pid controllers combinedwith a feedforward and an mpc for an actuator in camless internal com-bustion engines. INTERNATIONAL JOURNAL OF MATHEMATICALMODELS AND METHODS IN APPLIED SCIENCES, 7(9):793–801,2013.

[8] Intan Z. M. Darus Mohd S. Saad, Hishamuddin Jamaluddin. Pid con-troller tuning using evolutionary algorithms. WSEAS TRANSACTIONS

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0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02−300

−200

−100

0

100

200

300

400

Time (sec.)

Thr

ee p

hase

vol

tage

inpu

ts

Phase onePhase twoPhase three

Fig. 9. Three-phase control signals after the chopper controller using a PWMsingnal with a maximal switching frequency equals 2.5 kHz in case of positivedesired velocity

on SYSTEMS and CONTROL, 7(4):139–149, 2012.[9] P. Mercorelli, B. Paden, and D. Prattichizzo. Perfect anti-windup

in output tracking scheme with preaction. In Proceedings ofthe 2013 International Conference on Systems, Control andInformatics, WSEAS-Europment, pages 151–155, Venice, Italy,http://www.europment.org/library/2013/venice/bypaper/SCI/SCI-22.pdf,2013.

[10] M.V. Kothare, P.J. Campo, M. Morari, and C.N. Nett. An uni-fied framework for the study of anti-windup designs. Automatica,30(12):18691883, 1994.

[11] N. Kapoor, A. R. Teel, and P. Daoutidis. An invariant subspace techniquefor anti-windup syntesis. In In Proceedings of the 1997 AmericanControl Conference, ACC97, volume 5, pages 3083–3087, Albuquerque,New Messico (U.S.A.), 1997.

[12] P. Mercorelli, B. Paden, and D. Prattichizzo. Perfect anti-windup inoutput tracking scheme with preaction. In Proceedings of the 1998 IEEEInternational Conference on Control Applications, CCA97, volume 1,pages 134–138, Trieste, Italy, 1998.

[13] D.A. Khaburi and M. Shahnazari. Parameters identification of permanentmagnet synchronous machine in vector control. In Proceedings of the10th European Conference on Power Electronics and Applications, EPE2003, Toulouse, 2003.

[14] G. Basile and G. Marro. Controlled and conditioned invariants in linearsystem theory. Prentice Hall, New Jersey-USA, 1992.

Paolo Mercorelli received his Master’s degree inElectronic Engineering from the University of Flo-rence, Italy, in 1992, and Ph.D. degree in SystemsEngineering from the University of Bologna, Italy,in 1998. In 1997, he was a Visiting Researcher forone year in the Department of Mechanical and En-vironmental Engineering, University of California,Santa Barbara, USA. From 1998 to 2001, he held apostdoctoral position at Asea Brown Boveri, Heidel-berg, Germany. Subsequently, from 2002 to 2005, hewas a Senior Researcher and Leader of the Control

Group, Institute of Automation and Informatics, Wernigerode, Germany, andfrom 2005 to 2011, he was an Associate Professor of Process Informaticswith Ostfalia University of Applied Sciences, Wolfsburg, Germany. In 2011he was a Visiting Professor at Villanova University, Philadelphia, USA. Since2012 he has been a Full Professor (Chair) of Control and Drive Systemsat the Institute of Product and Process Innovation, Leuphana University ofLueneburg, Germany. His current research interests include mechatronics,automatic control, and signal processing.

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ISSN: 1998-0140 51