Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844
Vol. VI (2011), No. 1 (March), pp. 90-100

Speed Control of a Permanent Magnet Synchronous Machine
Powered by an Inverter Voltage Moment Approach

J. Khedri, M. Chaabane, M. Souissi

Jamel Khedri, Mohamed Chaabane (corresponding author), Mansour Souissi
Research Unit of Industrial Processes Control
National School of Engineers of Sfax
ENIS, Route soukra, km 3.5 - BP.W, 3038, Sfax, Tunisia
E-mail: {khedrijamel,chaabane_uca}@yahoo.fr, mansour.souissi@ipeis.rnu.tn

Abstract: In this paper, the method of moments is presented in order to syn-
thesize controllers for current and speed of a Magnet Permanent Synchronous
Machine (PMSM). The controller’s dynamics are chosen in respect with the
time domain constraints, which have equality relationships between the mo-
ments of the closed-loop system and those of the reference model. The Linear
Matrix Inequality (LMI) formalism is used for the controller synthesis. The
proposed controller is applied for the speed tracking of a PMSM to illustrate
the performances of the method.
Keywords: Times Specifications, method of moments, Reference Model, LMI,
PMSM.

1 Introduction

New industrial applications require variable speed drives with high dynamic performance. In
recent years several techniques of control have been developed, allowing PMSM with variable
speed to achieve these performances. However, vector control, which allows decoupling between
control variables remains the most widely used [1–3]. The main advantages of this configuration
are that the regulatory cascade is a very industrial responded [4, 5]. It gives high dynamic
performance for a wide range of applications. The complexity of the dynamic model of the
permanent magnet synchronous variable speed and the presence of external disturbances and
parametric variations limit the performance of the control law [6]. Thus, our main contribution
in this paper consist in introducing a time constraint with the help of a reference model, simple
to operate and which characterizes the dynamics of the closed loop system. This time constraint
is formulated by the equality of the first time moments of the closed-loop system and those of
the reference model [7–11].

Practically, a first LMI is used to insure closed-loop stability, a second one permits, by
minimizing the norm 2 of time moments cost, to identify time specifications (overshoot, response
time ...).

Without loss of generality, the methodology has been restricted in this paper to the case of
square, invertible, MIMO (multi inputs, multi outputs) systems.

The paper is organised as follows: in section 2, we define the state space representations
of the systems and those of the controller; moreover, the problem of the design of a controller
verifying stability and time performances is stated. In section 3, we propose a solution of this
problem, owing to the demonstration of a specific theorem. Experimental results applied to a
permanent magnet synchronous machine (PMSM) are presented in section 4.

Copyright c⃝ 2006-2011 by CCC Publications



Speed Control of a Permanent Magnet Synchronous Machine Powered by an Inverter Voltage
Moment Approach 91

2 Problem statement

Consider a Linear Time Invariant (LTI) square invertible system H(s) with m inputs and m
outputs: {

ẋsys(t) = Asysxc(t) + Bsysu(t)
y(t) = Csysx(t)

(1)

The controller is modeled in the state space by:{
ẋc(t) = Acxc(t) + Bce(t)
u(t) = Ccxc(t) + Dce(t)

(2)

with
e(t) = r(t)−y(t) (3)

where: xsys(t) ∈ Rl is the state vector of the system,u(t) ∈ Rm is the control input,y(t) ∈ Rm is
the output,r(t) ∈ Rm is the reference of closed-loop system, Figure 1. Matrices Asys, Bsys, Csys
are supposed to be known with appropriate dimensions. The controller model matrices Ac, Bc,
Cc and Dc are to be computed.

Figure 1: Closed-Loop System

The synthesis of controller requires the construction of the augmented model which regroups
the system and the controller model. Consider the augmented state vector z(t) defined as:

z(t) =

[
xsys(t)
xc(t)

]
. (4)

Then, the closed-loop system, Figure 1, can be represented by the following model:{
ż(t) = Abfz(t) + Bbfr(t)

y(t) = Cbfz(t)
(5)

with:

Abf =

[
Asys −BsysDcCsys BsysCc

−BcCsys Ac

]
; Bbf =

[
BsysDc

Bc

]
; Cbf =

[
Csys 0

]
(6)

Let’s note that the closed-loop stability is ensured if and only if there exist a positive definite
matrix P with P = P T , such as the LMI

PAbf + A
T
bfP < 0 (7)



92 J. Khedri, M. Chaabane, M. Souissi

is satisfied.
To facilitate the study, our work is restricted to the case where the controller C(s) is to

be synthesised. The controller design can be considered as a static feedback gain K for the
augmented system.

The controller synthesis is ensured by the fact that matrices Ac and Bc are defined a priori
and only the matrices Cc and Dc are to be computed. For this purpose, we consider the following
hypothesis:

The matrix Abf of augmented system can be written as follows:

Abf = Ã + B̃KC̃ (8)

where:

à =

[
Asys 0
−BcCsys Ac

]
; B̃ =

[
Bsys
0

]
; C̃ =

[
−Csys 0

0 I

]
(9)

It is to notice that the gain K is defined as:

K =
[
Dc Cc

]
(10)

The extended system is then characterized by (Ã, B̃, C̃). The objective of this work is to ensure
closed-loop stability and to specify transient performances which are characterized by a reference
model Tref(S) . The Dynamics of the reference model are determined by the use of the time
moments approach. In this case, the time moments of transfer W(S), which are represented in
state space by (AW , BW , CW , DW ) (see [4]), are given by the following relations:

MW,0 = −CW A−1W BW + DW
MW,j = (−1)j+1CW A

−(j+1)
W BW + DW

(11)

(j = 1...∞)
MW,0: is the 0th order moment.

The exposing j refers to the order of moments. For applying the proposed moment approach,
we will define briefly the first three time moments.

The 0th order moment M0 is equal to the static gain of the system and represents the area of
its impulse response. The first order moment M1 characterizes the average time tm , the same
as the average value of a probability density: tm = M1(g)

M0(g)

g(t) is the impulse response of the system, represented by Figure 2.
For a first order system, we have M1 = τM0 where represents the time constant of the system.

More generally, M1 is used to characterize the time response of the considered system.
The second order moment M2, characterizes the variance σ (or dispersion ∆τ) of the impulse

response around its mean time tm.
In practice, the time moments M0, M1, M2 carry sufficient information on the impulse

response: static gain, mean time and dispersion.



Speed Control of a Permanent Magnet Synchronous Machine Powered by an Inverter Voltage
Moment Approach 93

Figure 2: Characterization of an impulse response

The controller has to be designed such that the closed-loop transfer T(S) is equal to the
reference model Tref(s) .

(I + H(s)C(s))−1H(s)C(S) = Tref(s) (12)

or
C(s)(I −Tref(s)) = H−1(s)Tref(s) (13)

It is known that such control law u(t) is in the form: u(t) = Ke(t) Using state space represen-
tation of C(s), we can write:

u(s) = [Cc(sI −Ac)−1Bc + Dc]e(s) (14)

While referring to considered hypothesis given by relation (9), equation (13) can be written as:

K

[
I

(sI −Ac)−1Bc

]
(I −Tref(s)) = H−1(s)Tref(s) (15)

If we set:

F(s) = H−1(s)Tref(s) (16)

W(s) =

[
I

(sI −Ac)−1Bc

]
(I −Tref(s)) (17)

Then, the time constraint (15) can be expressed by:

KW(s) = F(s) (18)

Notice that the last relation (18) is linear, then the formalism LMI can be applied. In fact, an
ideal equality in the form (18) between the closed-loop T(s) and Tref(s) is not possible to obtain.



94 J. Khedri, M. Chaabane, M. Souissi

Therefore, it is recommended to replace an ideal equality by an approximation between T(s) and
Tref(s). In order to achieve this condition, we use the time moments approach, as mentioned
previously.

Therefore, when the objective is to characterize the time response, the equality (18) is trans-
formed to a minimization of the quadratic distance between the first (n + 1) time moments of
the closed-loop system and its reference model, where the minimization criterion is defined as:

(KMW,j −MF,j)T (KMW,j −MF,j) < γI (19)

j = 0, 1, ..., n.
Using this criterion, we can notice that the transient responses of T(s) and Tref(s) become
approximately similar (they have approximately the same overshoot, the same response time,
etc.)

The overall objective is achieved if there exist a symmetric and positive defined matrix P
and a gain K given by relation (10). In such case, the closed-loop stability condition (7) and the
minimization criterion condition (19) are satisfied. In the next section, we present a solution to
this problem, which can be formulated by the theorem 1.

3 Mains results

The objective of this section is to give solution of the previous problem.

3.1 Theorem 1.

Consider a LTI system (Asys, Bsys, Csys) (1) and the corresponding augmented system (Ã, B̃, C̃)
defined in (9). Consider a static state feedback Ke stabilizing the pair (Ã, B̃).

The stability of the closed-loop system and the time constraints imposed by the reference
model Tref(s) are guaranteed, if the optimisation problem:
Minimum γ
(L, G, P) with the constraints
P > 0 [

Sym{P(Ã + B̃Ke)} (LC̃ −GKe)T + PB̃
(LC̃ −GKe) + B̃T P −(G + GT )

]
< 0 (20)

and [
−I (LMW,j −GMF,j)

(LMW,j −GMF,j)T −γI

]
< 0j = 0, 1, ..., n. (21)

has a solution, then the gain of dynamic controller C(s) is given by :

K = G−1optLopt (22)

Notice that Lopt, Gopt correspond to the minimal value of γ denoted γopt. Before demonstration
of Theorem 1, let us recall the following result [6, 7].

3.2 Lemma 1.

Consider the augmented system (5), where the state matrix Abf is defined by (8). The
following statements are equivalent:



Speed Control of a Permanent Magnet Synchronous Machine Powered by an Inverter Voltage
Moment Approach 95

1) There exist a symmetric and positive defined matrix P = P T > 0 and two matrices Ke and
K such that:

P(Ã + B̃Ke) + (Ã + B̃Ke)
T P < 0 (23)

P(Ã + B̃KC̃) + (Ã + B̃KC̃)T P < 0 (24)

2) There exist a symmetric and positive defined matrix P = P T > 0 , a non singular matrix G
and two matrices Ke and K such that:[

Sym{(Ã + B̃Ke)T P} PB̃
B̃T P 0

]
+ sym{

[
0
I

]
G[(KC̃ −Ke)− I]} (25)

3.3 Proof of lemma 1

First of all, notice that the orthogonal complements of the second term of (25) are:

V =
[
I 0

]
, the orthogonal complement of V , such that V V ⊥ = 0, is V ⊥ =

[
0
I

]
;

and U =
[
I (KC̃ −Ke)T

]
, the orthogonal complement of U, such that UU⊥ = 0, is:

U⊥ =

[
(KC̃ −Ke)T

−I

]
Applications of elimination lemma [9] lead to left and right multiplication of (25) by V and V ⊥

respectively, which gives inequality (23):

[
I 0

]
sym{

[
0
I

]
G
[
(KC̃ −Ke)− I

]
}
[
I
0

]
= P(Ã + B̃Ke) + (Ã + B̃Ke)

T P < 0

Even, if (25) is left and right multiplied respectively by U and U⊥, we get inequality (24).

3.4 Proof of Theorem 1

By using Shur complement, the condition (21) can be expressed as:

(LMW,j −GMF,j)(LMW,j −GMF,j)T < γI (26)

j = 0, 1, ..., n.
Besides, if one multiplies equation (18) by G:

LW(s) = GF(s) (27)

equation (19) becomes:

(LMW,j −GMF,j)(LMW,j −GMF,j)T < γI (28)

j = 0, 1, ..., n.
One demonstrates that (26) and then the condition (21) correspond to the minimization of the 2
norm of the error between the (n+1) first moments of the closed-loop and those of the reference
model. If the minimization of γopt is satisfied, then the two transfers T(s) and Tref(s) are very
close in low frequencies, which implies the desired transients for the closed-loop system. The
stability is also insured due to the fact that condition (20) implies condition (24), equivalent to
condition (7) which expresses closed-loop stability.



96 J. Khedri, M. Chaabane, M. Souissi

4 Application to the Permanent Magnet Synchronous Machine
(PMSM)

Our Permanent Magnet Synchronous Machine (PMSM) is powered by a two levels voltage
inverter. To obtain non reciprocating quantities, our model is done in the Park landmark [10].
Thus we have the following equations in the synchronously d-q reference frame [11, 12]:



İd =
−Rs
Ld

Id +
Lq
Ld

wrIq +
1
Ld

vd

İq =
−Rs
Lq

Iq − LdLq wrId −
ϕ
Lq

wr +
1
Lq

vq

ẇr = p
(ce−cr)

J
− fc

J
wr

ce = p[(Ld −Lq)Id + ϕ]Iq −
fc
J
wr − fcJ wr

ẇr =
p2

J
[(Ld −Lq)Id + ϕ]Iq −

fc
J
wr − pJ cr

(29)

where vd and vq are the stator voltages, Id and Iq are the stator currents, Ldand Lq are the
inductances, Rs is the stator winding resistance, ϕ is the flux linkage of the permanent magnets,
wr is the angular velocity of the motor shaft, fc is the friction coefficient relating to the rotor
speed; J is the moment of inertia of the rotor , ce is the electromagnetic torque, cr is the load
torque and p is the number of pole. In an equivalent manner, these equations can be written in
the form of a general transfer function H(s) who contains the open-loop transfer of variables to
control namely Hid(s) and Hwr(s).

H(s) =

[
Hid(s) 0

0 Hwr(s)

]
(30)

where:

Hid(s) =
1
Rs

1 + τid(s)
, Hwr(s) =

HbfIq
p2ϕ
fc

1 + τ(s)
(31)

where τid =
Ld
Rs

; τ = J
fc

Let HbfIq(s),is the closed-loop transfer of the current Iq: HbfIq = 11+τbfiq(s) τbfiq is chosen equal
to 7.0175 10-4 s in order to perform a time response of the closed-loop system ten times faster
than open-loop.

Notice that: τiq =
Lq
Rs

; Hiq(s) =
1
Rs

1+τiq(s)

The objective is to reduce the time response of the system, to keep the overshoot of the closed-
loop inferior to 5% while decoupling the two outputs. These specifications can be carried out by
a second order reference model with static unity gain for the speed, and a first order one for the
current Id:

Tref(s) =

[ 1
1+τidref s)

0

0 1
1+

2ζ
w
(s)+ s

2

w2

]
(32)

The design choices correspond to τidref = 0.1τid in order to perform a time response of the
closed-loop current ten times faster than the open-loop.
ζ = 7.6205, w = 93.906rds−1 in order to obtain a closed-loop speed 3.3 times faster than the
open-loop one. Let’s note that Hwr has two constants time:
τ = J

fc
= 0.5333s, τ1 = 7.047210−4s−1 The two constants time of the closed-loop speed are

τbf = 0.1618s, τ1bf = 7.017510−4s Notice that the decoupling specification is insured by a di-
agonal reference model. The d-axis current Id is regulated to follow a zero setpoint. The static



Speed Control of a Permanent Magnet Synchronous Machine Powered by an Inverter Voltage
Moment Approach 97

state feedback Ke stabilizing the pair (Ã, B̃) such as (23) feasible has been computed:

Ke =

[
−79.1667 0 0 722 0

0 0 −2.1433 0 0.0939

]
(33)

Solving LMIs (20) and (21) by LMI optimization algorithm, the matrices P , G and L can be
obtained:

G = 108
[
0.0065 0

0 4.5881

]
; γopt = 6.609610

−14 (34)

The obtained static gain is:

K =

[
5.7 0 722 0
0 0.5006 0 0.09386

]
(35)

5 Experimental results and interpretations

The implementation of our model was performed on a test benchmark consisting of two
Permanent Magnet Synchronous Machines (figure 3) fitted by the same encoder and powered by
a voltage inverter; one was used as a motor and the other one as a load. The proposed control
algorithm was executed by the use of Matlab/ Simulink software. Then it was compiled and
implemented on Dspace 1104. The digital sampling period was taken equal to 0.1ms.

Figure 3: Photo of the test bench.



98 J. Khedri, M. Chaabane, M. Souissi

Figures (4)-(6) illustrate the experimental results of the dynamic behaviour of PMSM for
speed operating without charge.

0 5 10 15
−8

−6

−4

−2

0

2

4

6

time in s

id
 a

n
d

 id
 r

e
fe

re
n

ce
 in

 A

id measured
id reference

Figure 4: d-axis current and its reference

0 5 10 15
−8

−6

−4

−2

0

2

4

6

time in s

iq
 a

n
d

 iq
 r

e
fe

re
n

ce
 in

 A

iq measured
iq reference

Figure 5: q-axis current and its reference

0 5 10 15
−250

−200

−150

−100

−50

0

50

100

150

200

250

time in s

S
p

e
e

d
 a

n
d

 s
p

e
e

d
 r

e
fe

re
n

ce
 in

 r
d

/s

wref
w meas

Figure 6: Speed and its reference

Whose data are listed in the Appendix. The results show that the closed loop system with



Speed Control of a Permanent Magnet Synchronous Machine Powered by an Inverter Voltage
Moment Approach 99

the synthesized controller have a good behaviour: indeed, the measured speed and dq-axis cur-
rents track well the trajectory of reference one with good accuracy over the whole speed range,
moreover, the time constraint imposed with the help of time moments has permitted to keep the
closed loop outputs very close to those of the reference model.

6 Conclusion

In this paper, a controller design, for current and speed tracking of a PMSM has been
developed by using the method of moments. The controller’s dynamics are chosen in respect with
the time domain constraints. These constraints have equality relationships between the moments
of the closed-loop system and those of the reference model. The proposed controller has been
achieved in order to ensure the closed-loop stability and to specify transient performances in
regard with a reference model. In this study, the LMI formalism has been used for the controller
synthesis. Finally, we have applied the proposed controller for the speed tracking of a PMSM
which has given satisfactory results.

7 Acknowledgements

The authors wish to thank all the team of the Laboratory of Automatic and Industrial
Informatics of the University of Poitiers FRANCE and special thanks to Pr. Gerard Champenois
for the aid he afforded us for the realization of our experimental work validation.

8 Appendix : Motor parameters

Motor rated power 1 KW
Rated current 6.5 A

Pole pair number (p) 2
d-axis inductance Ld 4.5mH

Stator resistance 0.56 Ω
Motor inertia J 2.08.10−3Kg.m2

Friction coefficient fc 3.9.10−3Nm.s.rad−1

Magnet flux constant ϕ 0.064 wb

Bibliography

[1] Vas, P., Sensorless vector and direct torque control, Oxford, U. K, Oxford Univ. Press, 1998.

[2] Bose, B. K., Power electronics and AC drives, Englewood Cliffs, NJ, Prentice-Hall,1986.

[3] Bose, B. K., Power electronics and variable frequency Drives, Technology and Appliqua-
tion,IEEE press 1997.

[4] Flaus, J. M, Industrial control, Hermes, Paris 1994.

[5] Maret, L., automatic control, Polytechnic Press romandes, 1987.



100 J. Khedri, M. Chaabane, M. Souissi

[6] Lin, F-J, Real time position controller design with torque feedforward control for a PM
synchronous motor , IEEE Trans. On Industrial Electronics, Vol.44, N◦.3, June 1997, pp.
398-407.

[7] Bentayeb, A., N. Maamri and J-C Trigeassou, The Moments in Control: a tool for Analy-
sis, Reduction and Design,International Journal of Computers, Communications & Control
(IJCCC). Vol. 2, num. 1, pp. 82-103, 2007.

[8] Maamri, N. and Trigeassou, J. C, PID design for time delayed systems by the method of
moments ,European Control Conference, Groningen Holland (1993).

[9] Maamri, N. and Trigeassou, J. C, Controllers design by moments placement ,European Control
Conference, Rome Italy (1995).

[10] Maamri, N., Bentayeb, A. and Trigeassou, J. C, Design and Iterative optimization of Re-
duced Robust Controllers with Equality Constraints , RECOND-Milan (2003).

[11] Trigeassou, J. C., Method of moments in automatic . Conférence Internationale Francophone
d’Automatique, CIFA, Lille (2000).

[12] Mehdi, D., Control theory some insights . Report Automatic Control UCA-ENIS Sfax
Tunisia Avril (2004).

[13] Peaucelle, D. and Arzelier, D., An efficient numerical solution for H2 static output feedback
synthesis . European Control Conference, 2001, pp.3800-3805.

[14] Skelton, R. E., Iwasaki, T. and Grigoriadis, K., a unified algebraic approach to linear control
design , Taylor & Francis, (1998).

[15] Bose, B. K., Power electronics and AC drive . New york, Prince Hall, first Edition, (1986),
chapter 2, page 95.

[16] Sturtzer, G. and Sanigiel, E., Modeling and Control of three-phase motors .

[17] Hassaine, S., Applications of new technology command to permanent magnet synchronous
machine. Thesis, ESIP, Poitiers, March 2008.