Passivity-based current controller design for a permanent-magnet synchronous motor

Passivity property of a PMSM in the general dq reference frame...338 4.. To overcome this drawback a new passivity-based current controller PBCC designed using the dq model of the PMSM i

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Passivity-based current controller design for a permanent-magnet

synchronous motor

A.Y Achoura,∗, B Mendilb, S Bachac, I Munteanuc

aDepartment of Electrical Engineering, A Mira University, 06000, Bejaia, Algeria

bDepartment of Electronics, A Mira University, 06000, Bejaia, Algeria

cGrenoble Electrical Engineering laboratory (G2Elab), CNRS UMR 5269 INPG/UJF, ENSIEG-BP 42, F-38402 Saint-Martin d’Héres Cedex, France

a r t i c l e i n f o

Article history:

Received 24 August 2008

Received in revised form

1 April 2009

Accepted 7 April 2009

Available online 7 May 2009

Keywords:

Permanent-magnet synchronous motor

Passivity-based approach

Rotational speed control

Current control

a b s t r a c t The control of a permanent-magnet synchronous motor is a nontrivial issue in AC drives, because of its nonlinear dynamics and time-varying parameters Within this paper, a new passivity-based controller designed to force the motor to track time-varying speed and torque trajectories is presented Its design

avoids the use of the Euler–Lagrange model and destructuring since it uses a flux-based dq modelling, independent of the rotor angular position This dq model is obtained through the three-phase abc model

of the motor, using a Park transform The proposed control law does not compensate the model’s workless

force terms which appear in the machine’s dq model, as they have no effect on the system’s energy balance

and they do not influence the system’s stability properties Another feature is that the cancellation of the plant’s primary dynamics and nonlinearities is not done by exact zeroing, but by imposing a desired damped transient The effectiveness of the proposed control is illustrated by numerical simulation results

© 2009 ISA Published by Elsevier Ltd All rights reserved

Contents

1 Introduction 336

2 Permanent-magnet synchronous motor model 337

2.1 PMSM model in the general direct-quadrature reference frame 337

2.2 Current-controlled dq model of PMSM 337

3 Passivity property of a PMSM in the general dq reference frame 338

4 Analysis of tracking error convergence using the passivity-based method 338

4.1 Flux reference computation 338

4.2 Torque reference computation 338

5 Passivity property of a closed-loop system in the general dq reference frame 339

6 PBCC structure for a PMSM 339

7 Simulation results 339

8 Conclusion 341

Appendix A Proof of Lemma 1 342

Appendix B Proof of the exponential stability of the flux tracking error 343

Appendix C Proof of Lemma 2 344

References 345

1 Introduction

The permanent-magnet synchronous motor (PMSM) has

nu-merous advantages over other types of machines conventionally

∗Corresponding address: Electrical Engineering Department, University of

Bejaia, Targa Ouzemour, 06000, Algeria Tel.: +213 777 037 698; fax: +213 34 21

51 05.

E-mail addresses:achouryazid@yahoo.fr (A.Y Achour), bmendil@yahoo.fr

(B Mendil), Seddik.Bacha@g2elab.grenoble-inp.fr (S Bacha),

Iulian.Munteanu@g2elab.grenoble-inp.fr (I Munteanu).

used for AC servo drives It has higher torque/inertia ratio and power density when compared to an induction motor or a wound-rotor synchronous motor This makes it suitable for some ap-plications like robotics and aerospace actuators However, it is difficult to control because of its nonlinear dynamical behaviour and its time-varying parameters

In this paper, a control strategy, based on the passivity concept that forces the PMSM to track velocity and electrical torque trajectories, is developed The idea of passivity-based control (PBC) design is to reshape the natural energy of the system and inject the required damping in such a way that the objective is achieved The

0019-0578/$ – see front matter © 2009 ISA Published by Elsevier Ltd All rights reserved.

doi:10.1016/j.isatra.2009.04.004

key issue is to identify the workless force terms which appear in

the process model, but which do not have any effect on the energy

balance These terms do not influence the stability properties;

hence, there is no need for their cancellation This leads to simple

control structures and enhances the system robustness

PBC has its roots in classical mechanics [1,2], and it was

introduced in the control theory in [3] This method has been

instrumented as the solution of several robotics [4–7], induction

motors [8–13], and power electronics [14] problems It has also

been combined with other techniques [15–22] A PBC design

with simultaneous energy shaping and damping injection for

an induction motor using the dq model has been presented

in [8] This dq model is obtained through the three-phase abc

model of the motor, using a Park transform [23] The design of

two single-input single-output controllers for induction motors

based on adaptive passivity is presented in [15] Given their

nature, the two controllers work together with a field orientation

block In [16], a cascade passivity-based control scheme for speed

tracking purposes is proposed The scheme is valid for a certain

class of nonlinear system even with unstable zero dynamic,

and it is also useful for regulation and stabilization purposes A

methodology based on energy shaping and passivation principles

has been applied to a PMSM in [17] The interconnection and

damping structures of the system were assigned using a

port-controlled Hamiltonian (PCH) structure The resulting scheme

consists of a steady state feedback to which a nonlinear observer

is added to estimate the unknown load torque The authors

of [18] developed a PMSM speed control law based on a PCH

that achieves stabilization via system passivity In particular, the

PCH interconnection and damping matrices were shaped so that

the physical (Hamiltonian) system structure is preserved at the

closed-loop level The difference between the physical energy of

the system and the energy supplied by the controller forms the

closed-loop energy function A review of the fundamental theory

of the interconnection and damping assignment passivity-based

control (IDA-PBC) technique can be found in [19,20] These papers

showed the role played by the three matrices (i.e interconnection,

damping, kernel of system input) of the PCH model in the IDA-PBC

design

This paper is related to previous work concerning the voltage

control of a PMSM [22] The PBC has been combined with a

variable structure compensator (VSC) in order to deal with a

plant with important parameter uncertainties, without raising the

damping values of the controller The dynamics of the PMSM were

represented as a feedback interconnection of a passive electrical

and mechanical subsystem The PBC is applied only to the electrical

subsystem while the mechanical subsystem is treated as a passive

perturbation

Nevertheless, the passivity-based voltage controller (PBVC)

uses system inversion along the reference trajectory This leads to

singularities and the destruction of the original Lagrangian model

structure [13], because the PBVC uses theαβmodel which depends

on the rotor position Thisαβmodel is obtained through the

three-phase abc model of the PMSM, using the Blondel transform [23] To

overcome this drawback a new passivity-based current controller

(PBCC) designed using the dq model of the PMSM is proposed in

this paper This avoids the model’s structure destruction due to

singularities, since the dq model does not depend explicitly on the

rotor angular position

The paper is organized as follows The PMSM dq model and

the inner current loop design are presented in Section 2 In

Section3, the passivity property of the PMSM in the dq reference

frame is introduced Section4deals with the computation of the

current, flux and torque references The passivity property of the

closed-loop system and the resulting control structure are given in

Sections5and6, respectively Simulation results are presented in

Section7 Section8concludes the paper The proof of the passivity

property of the PMSM in the dq frame is given inAppendix A In Appendix B, the analysis and proof of the exponential stability of the flux tracking error is introduced.Appendix Ccontains the proof

of the passivity property of the closed-loop system

2 Permanent-magnet synchronous motor model

2.1 PMSM model in the general direct-quadrature reference frame

The PMSM uses buried rare earth magnets Its electrical

behaviour is described here by the well known dq model [23], given

by Eq.(1):

L dq˙i dq+R dq i dq+n pωm=L dq i dq+n pωm= ψf = vdq. (1)

In this equation the following notations have been employed:

L dq=



0 L q





i d

i q





0 R S



;

ψf =



φf

0





0 −1

1 0





vd

vq



In these equations, L d and L q are the stator inductances in the

linkages due to the permanent magnets, n pis the number of pole-pairs,ωmis the mechanical speed,vdandvqare the stator voltages

in the dq frame, and i d and i q are the stator currents in the dq frame.

The mechanical equation of the PMSM is given by

where J is the rotor moment of inertia, f VFis the viscous friction coefficient, andτLis the load torque

The electromagnetic torqueτe can be expressed in the dq frame

as follows:

τe= 3

2n p L d−L q

The rotor positionθmis given by Eq.(4):

˙

The interdependence between the flux linkage motorψdq and the current vector i dqcan be expressed as follow [23]:



ψd

ψq



whereψdandψq are the flux linkages in the dq frame.

Substituting the i dqvalue obtained from(5)in Eqs.(1)and(3) yields

˙

τe= −3

2n pψdq=i dq. (7)

2.2 Current-controlled dq model of PMSM

Let us define the state model of the PMSM using the state vector



ψd ψq ωm θm

T

and Eqs.(2),(4),(6)and(7) The reference

value of the current vector i dqis denoted by

i∗dq=

i∗d

i∗



The proportional–integral (PI) current loops, used to forcei d i qT

to track the reference

i∗d i∗qT

, are of the form of the equations below:

vd=k dp i∗d−i d
+k di

Z t

0

i∗d−i d
dt, k dp,k di>0 (8)

vq=k qp i∗q−i q
+k qi

Z t

0

i∗q−i q

dt, k qp,k qi>0. (9)

We assume that by the proper choice of positive gains k dp , k di , k qp,

and k qi, these loops work satisfactory Then, the reference vector

i∗

dqcan be considered as the control input for the PMSM model

This results in the simplified dynamic dq model of the PMSM given

below:

˙

˙

τe= −3

2n pψT

This simplified form of the PMSM model is further used to design

the control input i∗

dqusing the passivity approach

3 Passivity property of a PMSM in the general dq reference

frame

Lemma 1 A PMSM represents a strictly passive system if the

reference vector of the stator currents, i∗dq , and the flux linkage vector,

ψdq , are considered as the input and the output vectors, respectively.



The proof of this lemma is given inAppendix A

4 Analysis of tracking error convergence using the

passivity-based method

The desired value of the flux linkage vectorψdqis

ψ∗

dq=



ψ∗

d

ψ∗

d



(14)

and the difference between ψdq and ψ∗

dq, representing the flux tracking error, is

e f =



e fd

e fq



= ψdq− ψ∗

Rearranging Eq.(15),

ψdq=e f+ ψ∗

Substituting Eq.(16)in Eq.(10)yields

˙

e f+n pωm=e f = −R dq i∗dq− ψ ˙∗

dq+n pωm= ψ∗

dq

The aim is to find the control input i∗

dq which ensures the

convergence of the error vector e f to zero The energy function of

the closed-loop system is defined as

V(e f) =1

2e

T

Taking the time derivative of V e f
along trajectory(17)gives

˙

V e f
= −eT

f R dq i∗dq+ ˙ ψ∗

dq n pωm= ψ∗

dq

Note that the term n pωm eTf=e f = 0 due to the skew-symmetric

property of the matrix=

The convergence to zero of the error vector e f is ensured by taking

i∗dq= −R−dq1 ψ ˙∗

dq+n pωm= ψ∗

dq
+R− 1

where K f =

hk fd 0

0 k fq

i

with k fd>0 and k fq>0

The control input signal, i∗dq, consists of two parts: the term which encloses the reference dynamics and the damping term injected to make the closed-loop system strictly passive

The PBCC ensures the exponential stability of the flux tracking error The corresponding proof is given inAppendix B

4.1 Flux reference computation

The computation of the control signal i∗dqrequires the desired flux vectorψ∗

dq If the direct current i d in the dq frame is maintained

equal to zero, then the PMSM operates under maximum torque Under this condition, and using Eq.(5), it results that

ψ∗

ψ∗

The torque set-point valueτ∗

e corresponding toψ∗

dqis given by

Eq.(7) Substitutingψ∗

dfrom(21)and i∗qfrom(22)in(7), it results that

τ∗

e = 3 2

n pφf

L q ψ∗

Therefore the value of the flux reference is deduced as

ψ∗

q = 2 3

L q

n pφfτ∗

4.2 Torque reference computation

The desired torque τ∗

e is computed from the mechanical dynamic equation(11) Taking the rotor speedωmequal to its set-point valueω∗

myields

τ∗

e =Jω ˙◦

m+f VFω∗

This control structure has two drawbacks [13]:

(i) It is in an open loop and (ii) its convergence rate is limited by

the mechanical time constant J/f VF

In order to overcome these drawbacks, the following expression for the desired torque has been proposed [13]:

τ∗

e =Jω ˙∗

where z is the output of the lower filter with speed error input

ωm− ω∗

msatisfying

˙

m

With this choice, the convergence rate of the speed errorωm− ω∗

m

does not depend only on the natural mechanical damping This

rate can be adjusted by means of the positives gains b and awhich

have the same role as the proportional–derivative (PD) control law

In practical applications, the load torque is unknown; therefore it must be estimated For that purpose, an adaptive law [13] has been used:

˙ˆτL= −k L(ωm− ω∗

Fig 1 The block diagram for the passivity-based current controller.

5 Passivity property of a closed-loop system in the general dq

reference frame

Lemma 2 A closed-loop system represents a strictly passive system

if the desired dynamic output vector given by

ϑ = −R−dq1 ψ ˙∗

dq+n pωm= ψ∗

dq

(29)

The proof of this lemma is given inAppendix C

6 PBCC structure for a PMSM

The design procedure of the passivity-based current controller

for a PMSM leads to the control structure described by the block

diagram inFig 1 It consists of three main parts: the load torque

estimator given by Eq.(28), the desired dynamics expressed by

the relations (21)–(27), and the controller given by Eqs (8),

(9) and (20) In this design the imposed flux vector, ψ∗

dq, is determined from maximum torque operation conditions allowing

the computation of the desired currents i∗

dq Furthermore, the load torque is estimated through speed error, and directly taken into

account in the desired dynamics

The inner loops of the PMSM control are based on well known proportional–integral controllers A Park transform is used

for passing electrical variables between the three-phase and dq

frames

The actuator used in the control application is based on a PWM voltage source inverter Voltage, currents, rotational speed and PMSM angular position are considered measurable variables

7 Simulation results

The parameters of the PMSM used for testing the previously given control structure are given inTable 1

The plant and its corresponding control structure ofFig 1are implemented using Matlab and Simulink software environments The PMSM is simulated using Eqs.(1)–(4)whose parameters are given inTable 1 The chosen solver is based on the Runge–Kutta algorithm (ODE4) and it employs an integration time step of 10− 4s The parameter values of the control system are determined using the procedures detailed in Sections 2 and 4 as follows From the imposed pole locations, the gains of the current PI controller

are computed as k dp = 95, k di = 0.85, k qp = 95, and

k di = 0.8 The gains concerning the desired torque are set at

a = 75 and b = 400 using the pole placement method also The damping parameter values have been obtained by using a

-250 -200 -150 -100 -50 0 50 100 150 200 250

-200 -150 -100 -50 0 50 100 150 200

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time in sec

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time in sec

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time in sec

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time in sec -4

-2 0 2 4 6 8 10 12 14

2.5 2.505 2.51 2.515 2.52 2.525 2.53 2.535 2.54 2.545 2.55

Time in sec

2.5 2.505 2.51 2.515 2.52 2.525 2.53 2.535 2.54 2.545 2.55

Time in sec

-15 -10 -5 0 5 10 15

-100 -80 -60 -40 -20 0 20 40 60 80 100

-30 -20 -10 0 10 20 30

Fig 2 Motor response to a square speed reference signal at zero load torque.

Table 1

PMSM parameters.

Stator winding resistance 173.77 e-3Ω

Stator winding direct inductance 0.8524 e-3 H

Stator winding quadrate inductance 0.9515 e-3 H

Machine type: Siemens 1FT6084-8SK71-1TGO

trial-and-error procedure starting from initial values based on the

stability condition(20); their final values are k fd = k fq = 650

The gain of the load torque adaptive law is set to k L = 6, a

value which ensures the best asymptotic convergence of the speed

error

In all tests performed in this study, the following signals have been considered as representative for performance analysis: rotational speed (Fig 2(a)), line current (Fig 2(b)), electromagnetic torque (Fig 2(c)), the stator voltages in the dq frame (Fig 2(d)), zoom of voltage at the output of the inverter (Fig 2(e)), and zoom

of line current (Fig 2(f)).Fig 2shows the motor response to a square speed reference signal with magnitude±150 rad/s, without load torque As can be seen, the rotor speed and line current quickly track their references without overshoot and all other signals are well shaped The peaks visible on the electromagnetic torque evolution are due to the high gradients imposed to the rotational speed In practice, these peaks can be easily reduced

by limiting the speed reference changing rate and by limiting the

value of the imposed current i∗

q However, such a situation has been chosen for a better presentation of the control law capabilities and performances

The second aspect of this study concerns the robustness test of the designed control system against disturbances and parameter changes To this end, a load torque step ofτL = 10 N m has been applied at time 0.5 s and has been removed at time 4.5 s (see

-250 -200 -150 -100 -50 0 50 100 150 200 250

-200 -150 -100 -50 0 50 100 150 200

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time in sec

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time in sec

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time in sec

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time in sec 0

2 4 6 8 10 12 14 16 18 20 22

2.5 2.505 2.51 2.515 2.52 2.525 2.53 2.535 2.54 2.545 2.55

Time in sec

2.5 2.505 2.51 2.515 2.52 2.525 2.53 2.535 2.54 2.545 2.55

Time in sec

-20 -15 -10 -5 0 5 10

20

15

-100 -80 -60 -40 -20 0 20 40 60 80 100

-30 -20 -10 0 10 20 30

Fig 3 Motor response to a square speed reference signal with a load torque step of 10 Nm from t=0.5 s to t=4.5 s.

Fig 3) The results inFigs 3and4show that the response of the

rotor speed to the disturbance is quite fast and the electromagnetic

torque, τe, has been increased to a value corresponding to the

load applied The rotational speed and line current tracks the

reference quickly, without overshoot, and all other signals are well

shaped

Three tests of robustness to parameter changes have been

performed The first shows that a change of+50% of the stator

winding resistance, R s, only slightly affects the dynamic motor

response (see Fig 5) This is due to the fact that the electrical

time constantρf of closed-loop system appearing in Eq.(42)is

compensated by the imposed damping gain, K f, from Eq.(20)

However, a change of+100% of the inertia moment J increases

the mechanical time constant and hence the rotor speed settling

time (seeFig 6) The designed PBCC is based only on the electrical

part of the PMSM and has no direct compensation effect on the

mechanical part

As presented inFig 7, a simultaneous change of+50% of the

stator winding resistance and +100% of the moment inertia J

induces a similar behaviour as in the previous case (seeFig 6) This

is due to the fact that the PBCC designed using the procedure in Sections2and4is based only on the electrical part of the PMSM and has no direct compensation effect on the mechanical part

8 Conclusion

A new passivity-based speed control law for a PMSM has been developed in this paper The proposed control law does not compensate the model’s workless force terms as they have no effect on the system energy balance Therefore, the identification of these terms is a key issue in the associated control design Another feature is that the cancellation of the plant primary dynamics is not done by exact zeroing but by imposing a desired damped transient

The design avoids the use of the Euler–Lagrange model and destructuring (singularities effect) since it uses a flux-based

dq modelling, independent of the rotor angular position The

inner current control loops which have been built using classical

PI controllers preserve the passivity property of the current-controlled synchronous machine

0 20 40 60 80 100 120 140 160

-200 -150 -100 -50 0 50 100 150 200

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 2 4 6 8 10 12

Time in sec

-20 -15 -10 -5 0 5 10

20

15

0 10 20 30 40 50 60

-30 -20 -10 0 10 20 30

1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04 1.045 1.05

Time in sec

1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04 1.045 1.05

Fig 4 Motor response to a step speed reference with a load torque step of 10 Nm from t=0.3 s to t=1.3 s.

Unlike the majority of the nonlinear control methods used in

the PMSM field, this control loop compensates the nonlinearities

by means of a damped transient Its computation aims at imposing

the current’s set-points based on the flux references in the

dq frame These latter variables are computed based on the

load torque estimation by imposing maximum torque operation

conditions

The speed control law contains a damping term ensuring

the system’s stability and the adjustment of the tracking error

convergence speed The obtained closed-loop system allows

exponential zeroing of the speed error, also preserving the

passivity property

Simulation studies show the feasibility and the efficiency of

the proposed controller This controller can be easily included

into control structures developed for current-fed induction motors

commonly used in industrial applications Its relatively simple

structure should not involve significant hardware and software

implementation constraints

Appendix A Proof ofLemma 1

First, multiplying both sides of Eq.(10)byψ T

dq

R s yields

ψT

dq i∗dq= − 1

2R s

d ψT

dqψdq

whereψT

dqis the transpose of vectorψdq Note that the term n pωm

R s ψT

dq= ψdqdoes not appear on the right-hand side of(30), since ψT

dq= ψdq = 0 due to skew-symmetric property of the matrix = Integrating both sides of Eq (30) yields

Z t

0

ψT

dq i∗dq
dt= − 1

2R s ψT

dqψdq

(t) + 1

2R s ψT

dqψdq

(0). (31)

Consider that i∗dqis the input vector andψdqis the output vector Then, with the positive definite function

0 20 40 60 80 100 120 140 160

0 10 20 30 40 50 60

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 2 4 6 8 10 12

-20 -15 -10 -5 0 5 10

20

15

0 2 4 6 8 10 12 14 16

-250 -200 -150 -100 50 0 50 100 150 200 250

Time in sec

1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04 1.045 1.05

Fig 5 Motor response to a step reference with a change of+50% of the stator winding resistance, Rs, with a load torque step of 10 Nm from t=0.3 s to t=1.3 s.

2ψT

the energy balance Eq.(31)of the PMSM becomes

Z t

0

ψT

dq i∗dq

dt= −1

R s V f(t) + 1

This means that the PMSM is a strictly passive system [13] Thus,

the term n pωm R−dq1ψT

dq= ψdqhas no influence on the energy balance and on the asymptotic stability of the PMSM also; it is identified as

a workless force term 

Appendix B Proof of the exponential stability of the flux

tracking error

Consider the quadratic function(18)and its time derivative in

Eq.(19) Substituting i∗

dqfrom(20)in(19)yields

˙

V e f
= −eT

f K f e f ≤ − λmin



whereλmin



K f

>0 is the minimum eigenvalue of the matrix K f

andk kis the standard Euclidian vector norm

The square of the standard Euclidian norm of the vector e f is given as

which, combined with(18), gives

V(e f) =1

2e

T

Multiplying both sides of(36)by(−λmin



K f )leads to

− λmin



K f

V(e f) ≥ −λmin



which, combined with(34), gives

˙

V e f
≤ − λmin



K f

Integrating both sides of the inequality(38)yields

0 20 40 60 80 100 120 140 160

0 10 20 30 40 50 60

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 2 4 6 8 10 12

-20 -15 -10 -5 0 5 10

20

15

0 2 4 6 8 10 12 14 16 18 20

-250 -200 -150 -100 50 0 50 100 150 200 250

Time in sec

1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04 1.045 1.05

Fig 6 Motor response to a step reference with a change of+100% of the inertia moment J.

whereρf = λmin



K f >0 Considering the relation(36)at t=0, and multiplying it by e− ρf t, gives

which, combined with(39), leads to the following inequality:

The inequalities(36)and(41)give that

ρf

Eq (42) shows that the flux tracking error e f is exponentially

decreasing with a rate of convergence ofρf/2 

Appendix C Proof ofLemma 2

Substituting the control input vector i∗dqfrom(20)in Eq.(10)

gives

˙

whereϑis given by(29) Multiplying both sides of Eq.(43)byψ T

dq

R s ,

ψT

dqϑ = − 1

2R s

d ψT

dqψdq

dt − ψT

The termn pωm

R s ψT

dq= ψdqdisappears from(44), sinceψT

dq= ψdq = 0 due to the skew-symmetric property of the matrix= According

to(42), the flux tracking error e fis exponentially decreasing Thus, the termψT

dq K f e fbecomes insignificant, and Eq.(44)can be written as

ψT

dqϑ = − 1

2R s

d ψT

dqψdq

Integrating both sides of Eq.(45)yields

Z t

ψT

dqϑ
dt= − 1

2R ψT

dqψdq

(t) + 1

2R ψT

dqψdq

(0). (46)

0 20 40 60 80 100 120 140 160

0 10 20 30 40 50 60

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time in sec

0 2 4 6 8 10 12

-20 -15 -10 -5 0 5 10

20

15

0 5 10 15 20 25

-250 -200 -150 -100 50 0 50 100 150 200 250

Time in sec

1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04 1.045 1.05

Fig 7 Motor response to a step reference with a change of+50% of the stator winding resistance Rsand a change of+100% of the inertia moment J.

Let us consider the positive definite function V f from (16) The

energy balance(46)of the closed-loop system becomes

Z t

0

ψT

dqϑ
dt= −1

R s V f(t) + 1

This equation shows that the closed-loop system is strictly

passive [13] Thus, the term n pωm

R s ψT

dq= ψdq has no influence on the energy balance and the asymptotic stability of the closed-loop

system; it is identified as a workless force term 

References

[1] Goldstein H Classical mechanics New York: Addison-Wesley; 1980.

[2] Arnold VI Mathematical methods of classical mechanics New York: Springer;

1989.

[3] Takegaki M, Arimoto S A new feedback for dynamic control of manipulators.

ASME Journal of Dynamic Systems Measurements Control 1981;102:119–25.

[4] Ortega R, Spong M Adaptive motion control of rigid robots: A tutorial.

Automatica 1989;25(6):877–88.

[5] Berghis H, Nijmeijer H A passivity approach to controller–observer design for

robots IEEE Transaction on robotic and automatic 1993;9(6):740–54.

[6] Lanari L, Wen JT Adaptive PD controller for manipulators System Control Literature 1992;19:119–29.

[7] Ailon A, Ortega R An observer-based set-point controller for robot manipula-tors with flexible joints System Control Literature 1993;21(4):329–35 [8] Ortega R, Espinoza-Pérez G Passivity-based control with simultaneous energy-shaping and damping injection: The induction motor case study In: Proceedings of 16th IFAC world congress Proceeding in CD, Track.We-E20-TO/3 2005 p 6.

[9] Ortega R, Nicklasson PJ, Espinoza-Pérez G On speed control of induction motors Automatica 1996;3(3):455–66.

[10] Ortega R, Nicklasson PJ, Espinoza-Pérez G Passivity-based controller of a class of Blondel–Park transformable electric machines IEEE Transactions on Automatic Control 1997;42(5):629–47.

[11] Gökder LU, Simaan MA A passivity-based control method for induction motor control IEEE Transactions on Industrial Electrical 1997;44(5):688–95 [12] Kim KC, Ortega R, Charara A, Vilain JP Theoretical and experimental Comparison of two nonlinear controllers for current-fed induction motors IEEE Transactions on Control System Techniques 1997;5(5):338–48 [13] Ortega R, Loria A, Nicklasson PJ Passivity-based control of Euler–Lagrange systems New York: Springer; 1998.

[14] Sira-Ramirez H, Ortega R, Espinoza-Pérez G, Garcia M Passivity-based controllers for the stabilization of DC-to-DC power converters In: Proceedings

of 34th IEEE conference on decision and control 1995 p 3471–6.

[15] Travieso-Torres JC, Duarte Mermoud MA Two simple and novel SISO controllers for induction motors based on adaptive passivity ISA Transactions 2008;47:60–79.