IEEE Systems, Man and Cybernetics Magazine - July 2022 - 46

Special case: We can also extend to a nonlinear
FOMAS (1) under impulsive control (2) without the T-S
fuzzy model.
The tracking error system is defined as
(
D ==
b =+ -
t pp p
pp
() (),
()
kk kk
C
t ss ss() (( (),) (( ), )), ! k
s
DBtt gt tg tt tt
tb et tt
,
(30)
Table 1. The physical meaning of symbols.
Phase a, b stator current
ii ,as bs
ii ,ss
ab
ii,ds qs
vv ,ss
ab
vv,ds
vdc
qlr
qm
dir
~ r
TT Tab c
,,
qs
Stationary a - b axis stator current
Synchronously rotating d-q axis stator current
Stationary a - b axis stator voltage
Synchronously rotating d-q axis stator voltage
DC-bus voltage
Rotor flux angle
Mechanical angle
Rotor direction
Rotor speed
Phase a-b-c duty cycle ratio of PWM signal
PWM: pulsewidth modulation.
Proof
The proof of Theorem 3 is similar to that of Theorem 1.
Numerical Simulations
In this section, we provide numerical examples to
expla in the benefits of the results proposed in
this ar t icle.
Example 1
We consider the impulsive control of a FOMAS PMSM
chaot ic model. Some manufactur ing industr ies,
including paper textile mills, are looking at PMSMs
that can be modeled as MASs [3]). In this example, we
where BRnn
#
!
is defined as a constant matrix.
Theorem 3
Under Assumption (),AH
12 ,
,, ,
a =
ln{t21 ln
lnht
-- +
+- +
(( )(() ))
ET
mm U
b((mB
f
maxmax
b
+
and (( )) (( ))
bbk
T
for some positive scalars
tt{ h and U , system (30) can achieve finite-time
consensus if the ADT satisfies
() ()
where mmax and maxmf
of (
B
BB)
T
respectively.
xxa2 ) TE Tb
ln 11 B
max U))
b
,
(31)
are the largest eigenva lues
LW IL WI
++ ++ ,NNk
iqs
∗
PI
ids
∗
PI
qlr
iqs
ids
Park
ibs
ias
Clarke
ibs
ias
ADCIN1
Ileg2_
Bus
Driver
ADCIN2
ADCIN3
vqs
∗
PI
vds
∗
Inverter
Park
vbs
∗
Space
Vector
Gen.
vas
∗
Ta
Tb
Tc
PWM
Driver
PWM1
PWM2
PWM3
PWM4
PWM5
PWM6
Voltage
Source
Inverter
Encoder
Wr
SMOSPD
Speed
Estimation
qlr
qm
dir
Ramp
Gen.
QEP
Driver
QEP_A
QEP_B
QEP_inc
PMSM
Figure 3. Real-time model-based design of a PMSM. PI: proportional integral; PWM: pulsewidth modulation;
SMOSPD: sliding mode rotor speed estimator; QEP: quadrature encoder pulse; ADC: analog-to-digital converter.
46 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE July 2022

IEEE Systems, Man and Cybernetics Magazine - July 2022

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