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

ISA Transactions 48 (2009) 336–346

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Passivity-based current controller design for a permanent-magnet synchronous motor A.Y. Achour a,∗ , B. Mendil b , S. Bacha c , I. Munteanu c a

Department of Electrical Engineering, A. Mira University, 06000, Bejaia, Algeria

b

Department of Electronics, A. Mira University, 06000, Bejaia, Algeria

c

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

article

info

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

abstract 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. 2.

3. 4.

5. 6. 7. 8.

Introduction........................................................................................................................................................................................................................336 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 Passivity property of a PMSM in the general dq reference frame ...................................................................................................................................338 Analysis of tracking error convergence using the passivity-based method...................................................................................................................338 4.1. Flux reference computation ..................................................................................................................................................................................338 4.2. Torque reference computation .............................................................................................................................................................................338 Passivity property of a closed-loop system in the general dq reference frame .............................................................................................................339 PBCC structure for a PMSM................................................................................................................................................................................................339 Simulation results ..............................................................................................................................................................................................................339 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 numerous 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: [email protected] (A.Y. Achour), [email protected] (B. Mendil), [email protected] (S. Bacha), [email protected] (I. Munteanu).

used for AC servo drives. It has higher torque/inertia ratio and power density when compared to an induction motor or a woundrotor synchronous motor. This makes it suitable for some applications 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

A.Y. Achour et al. / ISA Transactions 48 (2009) 336–346

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 portcontrolled 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 the αβ model which depends on the rotor position. This αβ model is obtained through the threephase 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 Section 3, the passivity property of the PMSM in the dq reference frame is introduced. Section 4 deals 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 Sections 5 and 6, respectively. Simulation results are presented in

337

Section 7. Section 8 concludes the paper. The proof of the passivity property of the PMSM in the dq frame is given in Appendix A. In Appendix B, the analysis and proof of the exponential stability of the flux tracking error is introduced. Appendix C contains 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): Ldq˙idq + Rdq idq + np ωm =Ldq idq + np ωm =ψf = vdq .

(1)

In this equation the following notations have been employed:



Ldq

ψf =

 



L = d 0

0 ; Lq

  φf 0

idq



;

i = d ; iq

0 == 1



−1 0

;



Rdq

R = S 0



0 ; RS

  v vdq = d . vq

In these equations, Ld and Lq are the stator inductances in the dq frame, RS is the stator winding resistance, the φf are the flux linkages due to the permanent magnets, np is the number of polepairs, ωm is the mechanical speed, vd and vq are the stator voltages in the dq frame, and id and iq are the stator currents in the dq frame. The mechanical equation of the PMSM is given by Jω ˙ m + fVF ωm = τe − τL

(2)

where J is the rotor moment of inertia, fVF is the viscous friction coefficient, and τL is the load torque. The electromagnetic torque τe can be expressed in the dq frame as follows:

τe =

3 2

Ld − Lq id iq + φf iq .




np




(3)

The rotor position θm is given by Eq. (4):

θ˙m = ωm .

(4)

The interdependence between the flux linkage motor ψdq and the current vector idq can be expressed as follow [23]:

  ψd = Ldq idq + ψf ψq

(5)

where ψd and ψq are the flux linkages in the dq frame. Substituting the idq value obtained from (5) in Eqs. (1) and (3) yields

ψ˙ dq + np ωm =ψdq = vdq − Rdq idq 3

τe = − np ψdq =idq .

(6) (7)

2

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 idq is denoted by ∗

idq =

 ∗ id

i∗q

.

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The proportional–integral (PI) current loops, used to force id ∗ T

∗

to track the reference id below:

vd = kdp i∗d − id + kdi

T

, are of the form of the equations




i∗d − id dt ,

kdp , kdi > 0

(8)

i∗q − iq dt ,

kqp , kqi > 0.

(9)




0 t

Z


vq = kqp i∗q − iq + kqi




0

We assume that by the proper choice of positive gains kdp , kdi , kqp , and kqi , these loops work satisfactory. Then, the reference vector i∗dq can be considered as the control input for the PMSM model. This results in the simplified dynamic dq model of the PMSM given below:

ψ˙ dq + np ωm =ψdq = −Rdq idq

(10)

Jω ˙ m + fVF ωm = τe − τL

(11)

θ˙m = ωm

(12)

∗

3

T τe = − np ψdq =i∗dq .

(13)

2

This simplified form of the PMSM model is further used to design the control input i∗dq using 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.



4. Analysis of tracking error convergence using the passivitybased method The desired value of the flux linkage vector ψdq is ∗ ψdq =

ψd ψd∗

 ∗ (14)

∗ and the difference between ψdq and ψdq , representing the flux tracking error, is

  efd efq

ef =

∗ = ψdq − ψdq .

(15)

Rearranging Eq. (15), ∗ ψdq = ef + ψdq .

(16)

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


∗ ∗ ˙ dq e˙ f + np ωm =ef = −Rdq idq − ψ + np ωm =ψdq . ∗

(17)

The aim is to find the control input i∗dq which ensures the convergence of the error vector ef to zero. The energy function of the closed-loop system is defined as V (ef ) =

1 2

eTf ef .

(18)




Taking the time derivative of V ef along trajectory (17) gives ∗ ∗ ˙ dq V˙ ef = −eTf Rdq i∗dq + ψ np ωm =ψdq .







Note that the term np ω property of the matrix =.

T m ef

where Kf =

(19)

=ef = 0 due to the skew-symmetric

h

kfd 0

0 kfq

i

(20)

with kfd > 0 and kfq > 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 in Appendix B. 4.1. Flux reference computation The computation of the control signal i∗dq requires the desired ∗ flux vector ψdq . If the direct current id 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

ψd∗ = φf

(21)

ψq = Lq iq . ∗

∗

(22)

The torque set-point value τe corresponding to ψdq is given by Eq. (7). Substituting ψd∗ from (21) and i∗q from (22) in (7), it results that ∗

τe∗ =

3 np φf 2 Lq

∗

ψq∗ .

(23)

Therefore the value of the flux reference is deduced as

ψq∗ =

The proof of this lemma is given in Appendix A.



The convergence to zero of the error vector ef is ensured by taking 1 ˙∗ −1 ∗ i∗dq = −R− dq ψdq + np ωm =ψdq + Rdq Kf ef

t

Z




iq

iq

2 Lq 3 np φf

τe∗ .

(24)

4.2. Torque reference computation The desired torque τe∗ is computed from the mechanical dynamic equation (11). Taking the rotor speed ωm equal to its set∗ point value ωm yields ◦ ∗ τe∗ = J ω˙ m + fVF ωm + τˆL .

(25)

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 /fVF . In order to overcome these drawbacks, the following expression for the desired torque has been proposed [13]: ∗ τe∗ = J ω˙ m − z + τˆL .

(26)

where z is the output of the lower filter with speed error input ∗ ωm − ωm satisfying ∗ z˙ = −az + b ωm − ωm ,




a > 0, b > 0.

(27)

∗ With this choice, the convergence rate of the speed error ωm − ωm 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: ∗ τ˙ˆ L = −kL (ωm − ωm ),

kL > 0.

(28)

A.Y. Achour et al. / ISA Transactions 48 (2009) 336–346

339

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 1 ˙∗ ∗ ϑ = −R− dq ψdq + np ωm =ψdq




(29)

and the flux linkage vector ψdq are considered as input and output, respectively.  The proof of this lemma is given in Appendix 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 in Fig. 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, ψdq , 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 in Table 1. The plant and its corresponding control structure of Fig. 1 are implemented using Matlab and Simulink software environments. The PMSM is simulated using Eqs. (1)–(4) whose parameters are given in Table 1. The chosen solver is based on the Runge–Kutta algorithm (ODE4) and it employs an integration time step of 10−4 s. 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 kdp = 95, kdi = 0.85, kqp = 95, and kdi = 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

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a

b

250

15

200 10

150 ωm in rad/s

100

5 ia in A

50 0

0

-50 -5

-100 -150

-10

-200 -250

c

0

0.5

1

1.5

2 2.5 3 Time in sec

3.5

4

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-15

5

d

14

80

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2 2.5 3 Time in sec

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2 2.5 3 Time in sec

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Te in Nm

0.5

100

12

8 6 4 2

20 0 -20 -40

0

e

0

-60

-2

-80

-4

-100

0

0.5

1

1.5

2 2.5 3 Time in sec

3.5

4

4.5

5

f

200 150

30 20

100 10 ia in A

Va in V

50 0 -50

0 -10

-100 -20

-150 -200

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

-30

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

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

Table 1 PMSM parameters. Motor parameter

Value

Rated power Rated speed Stator winding resistance Stator winding direct inductance Stator winding quadrate inductance Rotor flux Viscous friction Inertia Pairs pole number Nominal current line Nominal voltage line Machine type: Siemens 1FT6084-8SK71-1TGO

6 kw 3000 rpm 173.77 e-3 Ω 0.8524 e-3 H 0.9515 e-3 H 0.1112 Wb 0.0085 Nm/rad/s 48 e-4 kg m2 4 31 A 310 V

trial-and-error procedure starting from initial values based on the stability condition (20); their final values are kfd = kfq = 650. The gain of the load torque adaptive law is set to kL = 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. 2 shows 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 of τL = 10 N m has been applied at time 0.5 s and has been removed at time 4.5 s (see

A.Y. Achour et al. / ISA Transactions 48 (2009) 336–346

a

b

250 200

341

20 15

150 10

ωm in rad/s

100

5 ia in A

50 0 -50

0 -5

-100 -10

-150

-15

-200 -250

c

0

0.5

1

1.5

2 2.5 3 Time in sec

3.5

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-20

5

d

22

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60 Vd and Vq in V

Te in N.m

e

14 12 10 8

-60 -80

0

-100

2 2.5 3 Time in sec

3.5

4

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5

f

200 150

3.5

4

4.5

5

0

0.5

1

1.5

2 2.5 3 Time in sec

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4

4.5

5

0

2 1.5

2 2.5 3 Time in sec

-20 -40

1

1.5

20

4

0.5

1

40

6

0

0.5

100

20 16

0

30 20

100 10 ia in A

Va in V

50 0 -50

0 -10

-100 -20 -150 -200

2.5 2.505 2.51 2.515 2.52 2.525 2.53 2.535 2.54 2.545 2.55

-30

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

Time in sec

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 in Figs. 3 and 4 show 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, Rs , only slightly affects the dynamic motor response (see Fig. 5). This is due to the fact that the electrical time constant ρf of closed-loop system appearing in Eq. (42) is compensated by the imposed damping gain, Kf , 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 (see Fig. 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 in Fig. 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 (see Fig. 6). This

is due to the fact that the PBCC designed using the procedure in Sections 2 and 4 is 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 currentcontrolled synchronous machine.

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a

b

15

120

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100

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80

-5

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ia in A

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1

1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04 1.045 1.05

Time in sec

Time in sec

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 of Lemma 1 First, multiplying both sides of Eq. (10) by T ∗ ψdq idq = −

T 1 d ψdq ψdq

2Rs

T ψdq

Rs

yields


(30)

dt

T where ψdq is the transpose of vector ψdq . np ωm Rs

T ψdq =ψdq does not appear on the rightT hand side of (30), since ψdq =ψdq = 0 due to skew-symmetric property of the matrix =. Integrating both sides of Eq. (30)

Note that the term

yields t

Z 0


1 T ∗ ψdq idq dt = −

2Rs


1 T ψdq ψdq (t ) +

2Rs


T ψdq ψdq (0).

(31)

Consider that i∗dq is the input vector and ψdq is the output vector. Then, with the positive definite function

A.Y. Achour et al. / ISA Transactions 48 (2009) 336–346

a

b

140

15

120

10

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-5

40

-10

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ωm in rad/s

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343

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Torque in N.m

8 6

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4 2 0

e

2 0

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f

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250 200

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150 100 Va in A

Vq in V

40 30 20

50 0 50 -100 -150

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-200 0

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time in sec

2

-250

1

1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04 1.045 1.05

Time in sec

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.

Vf =

1

T ψdq ψdq

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

Z


1 1 T ∗ ψdq idq dt = − Vf (t ) + Vf (0). Rs

0

Rs

(32)

The square of the standard Euclidian norm of the vector ef is given as

2

ef = e2 + e2 = eT ef , fd fq f (33)

This means that the PMSM is a strictly passive system [13]. Thus, 1 T the term np ωm R− dq ψdq =ψdq has 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∗dq from (20) in (19) yields

2  V˙ ef = −eTf Kf ef ≤ −λmin Kf ef (t ) , ∀t ≥ 0 (34)  where λmin Kf > 0 is the minimum eigenvalue of the matrix Kf and k.k is the standard Euclidian vector norm.

(35)

which, combined with (18), gives V (ef ) =

1 2

2 eTf ef ≤ ef ,



∀t ≥ 0.

(36)

Multiplying both sides of (36) by (−λmin Kf ) leads to





2 −λmin Kf V (ef ) ≥ −λmin Kf ef ,

∀t ≥ 0,

(37)

which, combined with (34), gives V˙ ef ≤ −λmin Kf V (ef ),






∀t ≥ 0.

(38)




Integrating both sides of the inequality (38) yields V (ef ) ≤ V (0)e−ρf t ,

∀t ≥ 0,

(39)

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

where ρf = λmin Kf > 0. Considering the relation (36) at t = 0, and multiplying it by e−ρf t , gives



V (0)e

−ρf t

2 ≤ ef (0) e−ρf t , ∀t ≥ 0.

T ψdq ϑ =−

(41)

The inequalities (36) and (41) give that



ρ

ef (t ) = ef (0) e− 2f t .

(42)

Eq. (42) shows that the flux tracking error ef is exponentially decreasing with a rate of convergence of ρf /2. 

(43)

2Rs

Rs

,


T − ψdq Kf e f .

dt

(44)

n ω

ψ ϑ =−

Substituting the control input vector i∗dq from (20) in Eq. (10) gives

T 1 d ψdq ψdq

T ψdq

T T The term pR m ψdq =ψdq disappears from (44), since ψdq =ψdq = 0 s due to the skew-symmetric property of the matrix =. According to (42), the flux tracking error ef is exponentially decreasing. Thus, T the term ψdq Kf ef becomes insignificant, and Eq. (44) can be written as

T dq

Appendix C. Proof of Lemma 2

ψ˙ dq + np ωm =ψdq = −Rdq ϑ − Kf ef ,

Multiplying both sides of Eq. (43) by

(40)

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

2 V (ef ) ≤ ef (0) e−ρf t ,

where ϑ is given by (29).

T 1 d ψdq ψdq

2Rs

dt


.

(45)

Integrating both sides of Eq. (45) yields t

Z 0


1 T ψdq ϑ dt = −

2Rs


1 T ψdq ψdq (t ) +

2Rs


T ψdq ψdq (0).

(46)

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

Let us consider the positive definite function Vf from (16). The energy balance (46) of the closed-loop system becomes t

Z


1 1 T ψdq ϑ dt = − Vf (t ) + Vf (0).

0

Rs

Rs

(47)

This equation shows that the closed-loop system is strictly n ω T passive [13]. Thus, the term pR m ψdq =ψdq has no influence on s 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 manipulators 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-E20TO/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.

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