A Lyapunov Approach to Set the Parameters of a PI-Controller tominimise Velocity Oscillations in a Permanent Magnet Synchronous

Motor using Chopper Control for Electrical Vehicles

Paolo Mercorelli

As the high field strength neodymium-iron-boron (NdFeB)magnets become commercially available with affordableprices. PMSMs are receiving increasing attention due totheir high speed, high power density and high efficiency.These characteristics are very favorable in some specialhigh performance applications, e.g. robotics, aerospace, andelectric ship propulsion systems [1], [2]. Permanent magnetsynchronous motors (PMSMs) as traction motors are com-mon in electric or hybrid road vehicles. For rail vehicles,PMSMs as traction motors are not widely used yet. Althoughthe traction PMSM can bring many advantages, just a fewprototypes of vehicles were built and tested. The next twonew prototypes of rail vehicles with traction PMSMs werepresented on InnoTrans fair in Berlin 2008 Alstom AGVhigh speed train and skoda Transportation low floor tram 15TForCity. Advantages of PMSM are well known. The greatestadvantage is low volume of the PMSM in comparison withother types of motors. It makes a direct drive of wheelspossible. On the other hand, the traction drive with PMSMhas to meet special requirements typical for overhead linefed vehicles. The drives and specially their control shouldbe robust to wide overhead line voltage tolerance (typicallyfrom −30% to +20% ), voltage surges and input filteroscillations. These aspects may cause problems during fluxweakening operation. There are several reasons to use fluxweakening operation of a traction drive. The typical reasonis constant power operation in a wide speed range andreaching nominal power during low speed (commonly 1/3of maximum speed). In the case of common traction, motorslike asynchronous or dc motors, it is possible to reach theconstant power region using flux weakening. This is also

possible for traction PMSM, however, problem with highback electromagnetic force (EMF) rises. In [3] it is shownhow using a flux weakening control strategy for PMSM aprediction control structure improves the dynamic perfor-mance of traditional feedback control strategies in terms, forinstance, of overshoot and rising time. To realise an effectiveprediction control, it is known that an accurate knowledgeof the model and its parameters is necessary. To achieve thedesired system performance, advanced control systems areusually required to provide fast and accurate response, quickdisturbance recovery, and parameter variations insensitivity[4]. In [3] it is shown how using a flux weakening controlstrategy for PMSM a prediction control structure improvesthe dynamic performance of traditional feedback controlstrategies in terms, for instance, of overshoot and risingtime. To realise an effective prediction control, it is knownthat an accurate knowledge of the model and its parametersis necessary. In [5] an identification technique is shown todetect parameters such as Rs, Ldq and Φ of the PMSM. Inthe existing applications chopper control structures are verypopular because they are very cheap and easy to realise.Nevertheless, using a chopper control structure smooth track-ing dynamics could be difficult to obtain without increasingthe switching frequency because of the discontinuity of thecontrol signals. No smooth tracking dynamics lead to a notcomfortable travel effect for the passengers of the electricalvehicle. This paper deals with a parameter set up of a PIregulator to be applied in an system for a permanent magnetthree-phase synchronous motors to obtain a smooth trackingdynamics even though a chopper control structure is includedin the drive. The paper is organized in the following way.In Section II a sketch of the model of the synchronous

Proceedings of the 2013 International Conference on Systems, Control and Informatics

1 Introduction

PAOLO MERCORELLI Institute of Product and Process Innovation, Leuphana University of Lueneburg, Volgershall 1, D-21339 Lueneburg, GERMANY

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 dynamics even though a chopper control structure is included in the drive. High performance application of permanent magnet synchronous motors (PMSM) is increasing. In particular, application in electrical vehicles is used very much. The technique uses a geometric decoupling procedure and a Lyapunov approach to perform a PWM control to be used as a chopper. Chopper control structures are very popular because they are very cheap and easy to be realise. Nevertheless, using a chopper control structure smooth tracking dynamics could be difficult to obtain without increasing the switching frequency because of the discontinuity of the control signals. No smooth tracking dynamics lead to a not comfortable travel effect for the passengers of an electrical vehicle or, more in general, it could be difficult to generate an efficient motion planning if the tracking dynamics are not smooth. This paper presents a technique to minimise these undesired effects. The presented technique is generally applicable and could be used for other types of electrical motors, as well as for other dynamic systems with nonlinear model structure. Through simulations of a synchronous motor used in automotive applications, this paper verifies the effectiveness of the proposed method and discusses the limits of the results.

Key-Words: Lyapunov approach; PI-control; velocity control; synchronous motors

WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS DOI: 10.37394/23201.2020.19.13 Paolo Mercorelli

E-ISSN: 2224-266X 111 Volume 19, 2020

motor and its behaviour are given. Section III is devoted toderive a decoupling controller which will be used to calculateparameters Kp and Ki of the controller with Lyapunovapproach. Section V shows simulation results using real 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 Parks transformation is themost widely used in variable speed PMSM drive controlapplications, see [4] and [6]. 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 de-fined relative to a stationary reference angle as illustrated inFig. 1. The dq-model is considered in this work and Park’stransformation is reported in Eqs. (1) and (2) as it can beseen below.

Fig. 1. Park’s transformation for the motor

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

=

2 sin(ωelt)

32 sin(ωelt−2π/3)

32 sin(ωelt+2π/3)

3

−2 cos(ωelt)3

−2 cos(ωelt−2π/3)3

−2 cos(ωelt+2π/3)3

13

13

13

×

ua(t)ub(t)uc(t)

(1)

and id(t)iq(t)i0(t)

=

2 cos(ωelt)

32 cos(ωelt−2π/3)

32 cos(ωelt+2π/3)

3

−2 sin(ωelt)3

−2 sin(ωelt−2π/3)3

−2 sin(ωelt+2π/3)3

13

13

13

×

ia(t)ib(t)ic(t)

. (2)

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)Ldq

], (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. Atthe end with Mm the motor torque is indicated. Consideringa isotropic 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 mechan-ical load.

III. STRUCTURE OF THE DECOUPLING CONTROLLER

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

(A+BF)V ⊆ V, (8)

Proceedings of the 2013 International Conference on Systems, Control and Informatics

2 Model and Behavior of a Synchronous Motor2 Model and Behavior of a Synchronous Motor

3 Structure of the Decoupling Controller

WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS DOI: 10.37394/23201.2020.19.13 Paolo Mercorelli

E-ISSN: 2224-266X 112 Volume 19, 2020

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.Concerning the meaning of relation (A + BF)V ⊆ V ,it is to remark that state feedback matrix F transformsV = im([1, 0]T ), which is a controlled invariant subspace,in an invariant subspace. This practically means that currentid(t) does not influence current iq(t) and thus the system isdecoupled. More explicitly it follows:

F =

[F11 F12

F21 F22

], and

[ud(t)uq(t)

]= F

[id(t)iq(t)

],

then, according to [7], 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 second equations of the system repre-sented in (5) becomes as follows:

diq(t)

dt= ωel(t)iq(t) +

uq(t)

Ldq. (12)

IV. A LYAPUNOV APPROACH TO SET THE PICONTROLLER PARAMETERS

Considering the following PI controller

uc(t) = Kp(ωmecd−ωmec(t))+Ki

∫ t

0

(ωmecd−ωmec(t))dτ,

(13)if ωmecd is a constant, it follows that

∂uc(t)

∂t= −Kp

∂ωmec(t)

∂t+Ki(ωmecd − ωmec(t)), (14)

Eq. (12) can be written in the following way:

iq(t) =1

ωel(t)

(− diq(t)

dt+

uq(t)

Ldq

). (15)

Combining Eqs. (6) and (15), then the following expressionis obtained:

Mm =3

2pΦ

1

ωel(t)

(− diq(t)

dt+

uq(t)

Ldq

). (16)

If Eq. (16) is inserted into Eq. (7), then the following relationis obtained:3

2pΦ

1

ωel(t)

(− diq(t)

dt+

uq(t)

Ldq

)−Mw = J

dωmec(t)

dt. (17)

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

VL(ωmec(t)) =1

2

(ωmecd(t)− ωmec(t)

)2. (18)

Considering the derivative of (18), then it must hold:

∂VL(ωmec(t))

∂t= −(ωmecd(t)− ωmec(t))

∂ωmec(t)

∂t≤ 0.

(19)From Eq.(17) it follows that:

∂VL(ωmec(t))

∂t= −

(ωmecd(t)− ωmec(t)

)J

×(32pΦ

1

ωel(t)

(− diq(t)

dt+

uq(t)

Ldq

)−Mw

). (20)

Considering the expression in (14), Eq. (20) becomes asfollows:

∂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. (21)

If∂ωmec(t)

∂t≥ 0, (22)

and∂uc(t)

∂t≥ 0, (23)

condition (21) is guaranteed ∀ Kp > 0 and ∀ Ki > 0.Remark 1: Condition (22) should be guaranteed with a

suitable choice of the parameters of the controller. Thiscondition states a monotonic dynamics and thus a dynamicsof the motor without oscillations. In automotive field, thiscondition is an ideal one for optimality of the electricaldriver comsuption and comfort of the passengers. �

A. A monotonic dynamics

To show under which conditions the following relation

∂ωmec(t)

∂t≥ 0 (24)

holds, let consider PI controller defined by Eq. (14) whichcan be rewritten in the following way:

∂uc(t)

∂t+Ki(ωmecd − ωmec(t)) = Kp

∂ωmec(t)

∂t≥ 0. (25)

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

∂uc(t)

∂t<< Ki(ωmecd − ωmec(t))−Kp

∂ωmec(t)

∂t. (26)

The following condition must hold:

Ki(ωmecd − ωmec(t))−Kp∂ωmec(t)

∂t≥ 0, (27)

Proceedings of the 2013 International Conference on Systems, Control and Informatics

4 A Lyapunov Approach to set the PI Controller Parameters

WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS DOI: 10.37394/23201.2020.19.13 Paolo Mercorelli

E-ISSN: 2224-266X 113 Volume 19, 2020

which it is equivalent to prove that:

Kp∂ωmec(t)

∂t+Kiωmec(t) ≥ Kiωmecd . (28)

Considering the solution of the following differential equa-tion, then:

Kp∂ωmec(t)

∂t+Kiωmec(t) = 0, (29)

then it must be:

ωmec(t) = ωmec(0)e− Ki

Kpt ≥ Kiωmecd , (30)

then the following final general condition is obtained:

Ki ≤ωmec(0)

ωmecd

Kp. (31)

Concerning assumption (23):

∂uc(t)

∂t≥ 0, (32)

let us consider Eq. (14) in which ωmecd is a constant, then:

∂uc(t)

∂t−Ki − ωmec(t) = Kp

∂ωmec(t)

∂t≥ 0. (33)

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 , (34)

Considering the solution of

Kp∂ωmec(t)

∂t+Kiωmec(t) = 0, (35)

then it must be:∂ωmec(t)

∂t= ωmec(0)e

− KiKp

t ≤ ωmecd

KpKi. (36)

From Eq. (36) a boundary condition on Ki is obtained:

Ki ≥ωmec(0)

ωmecd

Kp. (37)

Combining condition (31) with (37), the final sufficientcondition on parameters Kp and Ki is obtained:

Ki =ωmec(0)

ωmecd

Kp. (38)

V. SIMULATION RESULTS

Simulations have been carried out using a special standwith a 58 kW traction PMSM. The stand consists of PMSM,tram wheel and continuous rail. The PMSM is a prototypefor low floor trams. PMSM parameters: nominal power58 kW, nominal torque 852 Nm, nominal speed 650 rpm,nominal phase current 122 A and number of poles 44. Modelparameters: R = 0.08723 Ohm, Ldq = Ld = Lq = 0.8mH, Φ = 0.167 Wb. Surface mounted NdBFe magnets areused in PMSM. Advantage of these magnets is inductanceup to 1.2 T, but theirs disadvantage is corrosion. The PMSMwas designed to meet B curve requirements. The stand was

Fig. 2. Simulink structure of the whole control system

Fig. 3. PWM-Simulink-Block

loaded by an asynchronous motor. The engine has parametersas follows: nominal power 55 kW, nominal voltage 380V and nominal speed 589 rpm. In Fig. 2 the completecontrol scheme is shown. In this Simulink block diagramthe transformed dq-observer is indicated together with Parkand inverse Park transformation. PWM frequency equals100kHz and the structure of the simulink PWM block isshown in Fig. 3. Figure 4 shows the obtained and desiredmotor velocity profiles. Figure 5 shows the obtained anddesired motor acceleration profiles. From these two resultsit is possible to remark that the effect of the chopper controlis visible which does not allow the tracking to be precise.In particular, according to the theoretical condition ∂ωmec(t)

∂tthe result should not present oscillation. Because of therealisation of the controller using a chopper which consists ofdiscontinuous signals this is structurally not possible. Figure

Proceedings of the 2013 International Conference on Systems, Control and Informatics

5 Simulation Results

WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS DOI: 10.37394/23201.2020.19.13 Paolo Mercorelli

E-ISSN: 2224-266X 114 Volume 19, 2020

Fig. 4. Profile of the obtained and desired motor velocity using a maximalswitching chopper frequency equals 2.5kHz

Fig. 5. Profile of the obtained and desired motor acceleration using amaximal switching chopper frequency equals 2.5kHz

6 shows PWM signal sequence with the maximal chopperswitching frequency equals 2.5kHz. Fig. 7 shows the choppereffect on the input of the motor.

VI. CONCLUSIONS AND FUTURE WORK

This paper deals with a PI-controller set up for a three-phase synchronous motors. The technique uses a decou-pling procedure. A Lyapunov approach is used to calculateparameters Kp and Ki to obtain soft velocity variation.It is generally applicable for other dynamic systems withsimilar nonlinear model structure. Through simulations ofa synchronous motor used in automotive applications, thispaper verifies the effectiveness of the proposed method, thefound theoretical and the simulation results. Future workincludes the calculation of the optimal value of parameterKp.

Fig. 6. PWM signal used as a chopper with a maximal switching frequencyequals 2.5kHz

Fig. 7. Three-phase control signals after the chopper controller using aPWM singnal with a maximal switching frequency equals 2.5kHz

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

[4] 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.

[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] D.A. Khaburi and M. Shahnazari. Parameters identification of perma-nent magnet synchronous machine in vector control. In Proceedings ofthe 10th European Conference on Power Electronics and Applications,EPE 2003, Toulouse, 2003.

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Proceedings of the 2013 International Conference on Systems, Control and Informatics

6 Conclusions and Future Works

References

WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS DOI: 10.37394/23201.2020.19.13 Paolo Mercorelli

E-ISSN: 2224-266X 115 Volume 19, 2020

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