Nonlinear Control Strategies of Three-Phase Boost-type PWM Rectifiers [602110]

Nonlinear Control Strategies of Three-Phase Boost-type PWM Rectifiers
Jiuhe Wang
School of Automation
Beijing Information Science and
Technology University
Beijing, China
[anonimizat] Xia
School of Automation
Beijing Information Science and
Technology University
Beijing, China
[anonimizat] Zhang
School of Automation
Beijing Information Science and
Technology University
Beijing, China
[anonimizat]
Abstract üIn order to improve the properties of
three-phase boost-type PWM rectifiers, scholars in
different countries have began to apply nonlinear
control theory to study PWM rectifiers. For this
purpose, this paper introduces the nonlinear control
strategies of three-phase boost-type PWM rectifiers,
which includes feedback linearization, passivity
control based on Euler-Lagrange( EL) model ˈ
active disturbance rejection control(ADRC) strategy.
Meanwhile this paper reviews nonlinear control
strategies above and points to further research.
Keywords -PWM rectifiers; feedback linearization;
passivity control; ADRC
ĉ. Introduction
Three-phase boost-type PWM rectifier has many
advantages, such as sinusoidal current control and unity
power factor on the AC side, and DC output voltage
control on the DC side, and reversible power flow
between both sides etc.. Therefore, “green
transformation” of electric energy is realized by PWM
rectifier. The nonlinear control strategy of PWM rectifier
has advantages such as fast response and good stable
performances, high power factor and low input current
total harmonic disturbance (THD) etc., which is
interested by scholars in different countries[1]. In order
to promote above strategies, the paper introduces the
control strategies, which includes feedback
linearization[2-3], passivity control[4-6] and Active
disturbance rejection control [7-8].
The paper is organized as follow. In Section Ċ basic
math model is given. Section ċ introduces feedback
linearization control strategy and Section Čdoes
passivity control strategy of one. Section čdoes active
disturbance rejection control strategy of one. Finally,
Section Ď analyzes above control strategies. Section ď
states our conclusions and points to further research.
Ċ. Math model of PWM rectifier The power circuit of three-phase boost type PWM
rectifier is shown in Fig.1. uu,uv and uw are the phase
voltages of three phase balanced voltage source and iu, iv,
and iw are phase current in Fig.1. Su,Sv, and Sw are the
switching function of PWM rectifier. When Sj˄̆˙uǃ
vǃw˅closed, Sj=1, jSclosed, Sj=0. uDC is the DC output
voltage, R and Lmean resistance and inductance of filter
reactor, respectively, C is the DC side capacitance, R˨ is
the DC side load, uru,urv and urw are the input voltages of
rectifier, and iL is load current.
uu
vu
wuR
R
RC
1T3T5T
6T4T2TLR
rv
rw
oN



DCi
Li
DCuvi
wiruu
uuL
LL
LuiuS
uSvSwSwSvS
1D3D5D
6D4D2D
Fig.1 Power circuit of Three-phase boost type PWM rectifier
From Fig.1, the math model of three-phase boost type
PWM rectifier can be written as:
u
uu u D C
v
vv v D C
w
ww w D C
DC
uu vv ww LdiLu R i m udt
diLu R i m udt
diLu R i m udt
duCm i m i S i idt­  °
°
°  °°®
°  °
°
° °¯ (1)
Where
2
3uvw
uSSSm ,2
3vwu
vSSSm ,2
3wuv
wSSSm .
ċ.Feedback linearization control strategy
A. State-feedback linearization
2009 IITA International Conference on Control, Automation and Systems Engineering
978-0-7695-3728-3/09 $25.00 © 2009 IEEE
DOI 10.1109/CASE.2009.102299

1) Affine nonlinear model ˖Under R is neglected,
math model in synchronous rotating dq coordinate
according to (2), is given as follows:
dd d D C
q
qq q D C
d
DCd d q q D C
Ldi u m uidt L L
di u m uidt L L
d u mi mi uCdt C R CZ
Z­ °
°
°  ®
°
°  °¯ (2)
where Z is angular frequency of AC voltage.
Consider uDC constant, two input two output affine
nonlinear model of PWM rectifier is given from (2) as
11 2 2
11
22() () ()
()
()f gu gu
yh
yh  ­
° ®
° ¯xx x x
x
x (3)
where   12TT
dq xxi i x ,  12TT
dq uu m m u ,
11 () d yh i x , 22 () q yh i x ,
2
1()d
quxLfuxL§·¨¸
¨¸
¨¸¨¸©¹Z
Zx , 1()
0DCu
g L§·¨¸ ¨¸¨¸©¹x , 20
() DC g u
L§·
¨¸ ¨¸¨¸©¹x .
2) Nonlinear controller ˖PWM rectifiers can be
linearized based on involutivity of smooth vector
field12 1 2() () () () ff gga d ga d gxx x x .
The law of state-feedback linearization is expressed as
12
11
22
21d
d
qq DCukx xum k v L L
um k v u ukx xLZ
Z§·§·  ¨¸¨¸§·§· § ·¨¸¨¸ ¨¸¨¸ ¨ ¸¨¸¨¸ ©¹©¹ © ¹ ¨¸¨¸©¹©¹u (4)
where 12Tvv v are the new control inputs which can
be obtained from LQR[2].
B. input-output linearization
1) Affine nonlinear model ˖Math model in
synchronous rotating dq coordinate according to (1), is
given as follows
d
qd d d D C
q
dq q q D CdiLL i R i u S udt
diLL i R i u S udtZ
Z­ °°®
° °¯ (5)
For fast voltage control, the input power should supply
instantaneously the sum of load power and charging rate
of the capacitor energy. By the power balance between
the AC input and the DC output
3( ) / 2DC
dd qq D C D CLdup ui ui C u u idt   (6)
From(5) and (6), affine nonlinear model of PWM
rectifiers in the following form: 11 2 2
11
22() () ()
()
()f gu gu
yh
yh  ­
° ®
° ¯xx x x
x
x (7)
where 1
2dd D C
qq D Cuu S u
uu S u §·§ · ¨¸¨ ¸ ©¹© ¹,1d
2 DCyi
yu§·§ · ¨¸¨ ¸©¹© ¹,1
2
3d
q
DCx i
x i
xu§·§ ·
¨¸¨ ¸ ¨¸¨ ¸¨¸¨ ¸©¹© ¹x ,
12 21 13()2T
L
m
DCR Rifx x x x U xLL C u CZZ§·     ¨¸©¹x ,
 1() 1 / 0 0TgL x , 2() 0 1 / 0TgL x , mUis the
amplitude of the AC phase voltage.
2) Nonlinear controller ˖From (7),PWM rectifiers can
be linearized based on[3]. The nonlinear control law of
input-output feedback linearization is expressed as
111
22() ()uvEAuv§· §· §·  ¨¸ ¨¸ ¨¸©¹ ©¹ ©¹xx (8)
where
21
2211 21
23 2 33333()39 331
242 2mm L m LmRxxLARUx U x U ix Ux d i
LC x C x C x C x C dtZ
Z§·¨¸
¨¸ ¨¸ ¨¸©¹x ,
310
()3102mLEU
LC x§·
¨¸
¨¸
¨¸¨¸©¹x .
For tracking control and eliminating this tracking
error, the new control inputs are given by
11 1 12 11
221 2 22 2 23 2dref
DCRik e k e d t v
v u k ek eke d t§· §·¨¸ ¨¸¨¸©¹   ©¹³
³
  (9)
where 1 dd r e f eii  ,2 DCR DC eu u  .
From (9), we obtain error dynamics as
11 1 11 2 1
22 1 22 2 22 3 20
0ek ek e
ek ek e k e ­
® ¯ 
   (10)
The gain parameters ijk are determined from assigned
pole locations.
Č.Passivity control strategy
A. EL model of PWM rectifiers
EL model of PWM rectifiers can be expressed [4]as
follow:
00 / 0
00 / 0
002/ 3 / 0
00
00
002 / 3 0dd d
qq q
DCd q D C
dd
qq
LD CLd i d t L S i
Ld i d t L S i
Cd u d t S S u
Ri u
Ri u
RuZ
Z §· § · § · § ·
¨¸ ¨ ¸ ¨ ¸ ¨ ¸¨¸ ¨ ¸ ¨ ¸ ¨ ¸¨¸ ¨ ¸ ¨ ¸ ¨ ¸ ©¹ © ¹ © ¹ © ¹
§· § · § ·
¨¸ ¨ ¸ ¨ ¸ ¨¸ ¨ ¸ ¨ ¸¨¸ ¨ ¸ ¨ ¸©¹ © ¹ © ¹ (11)

300

From(11), EL model of rectifier is obtained as follow:
 Mx Jx x u ƒ (12)
With positive definite diagonal matrix M, a skew
symmetric matrix J, i.e. T J J, and a symmetric
positive semi-definite matrix tƒ . The matrix are
expressed by
00
00
002/ 3L
L
C§·
¨¸ ¨¸¨¸©¹M ,
0d
qu
u§·
¨¸ ¨¸¨¸©¹u ,1
2
3d
q
DCx i
x i
xu§·§ ·
¨¸¨ ¸ ¨¸¨ ¸¨¸¨ ¸©¹© ¹x ,
0
0
0d
q
dqLS
LS
SSZ
Z§·
¨¸ ¨¸¨¸©¹J ,00
00
002 / 3 LR
R
R§·
¨¸ ¨¸¨¸©¹ƒ .
B. Controller Desig Q
1) Controller Desig Qĉ˖ Determining the equilibrium
point x*see[4]. Controller design is as follow.
For the error state vector*e xx x , its dynamic
equation based on[5] is then expressed as:
*ee e

  Mx Jx x u Mx Jx x ƒ ƒ (13)
Let V denotes the error storage function in the
following form:
/2Tee V xM x (14)
The desired asymptotic behavior of the error dynamics
(13) can be achieved by performing a damping injection.
To carry out this job, consider the following desired error
dissipation matrix:
() de a e ƒƒ xR x (15)
where aRis positive definite diagonal matrix.
After a straightforward calculation using (13) and (15),
the error dynamics with desired damping becomes
** *() ee e a e   Mx Jx x u Mx Jx x R x ƒ ƒ (16)
From(16), in order to 0 ee e Mx Jx x ƒ uis given by
** *ae   uM x J x x R x ƒ (17)
then () 0Taee V    xR xƒ .
In view of * 0Mx , passivity controller obtained
from(17) is written as:
11
2() ma d m a
d
DCR
qa q m
q
DCRUR iI RRSu
uR i L ISuZ  ­ °°®° °¯ (18)
where mIis the amplitude of the AC phase
current, DCRuis the reference of DCu.
2)Controller Desig QĊ ˖ In order to improve the
properties, a new method designing controller is
introduced below. (13) can be transformed as
** *() ) ed e e a e   Mx x u Mx J(x x x R xƒ ƒ (19)
From(19), uis given by
**ae   uM x J x x R x ƒ (20) Based on * 0Mx , passivity controller obtained
from(20)[6] is written as:
1
2() dq m a d m
d
DC
da q
q
DCuL i R I R i ISu
Li R iSuZ
Z  ­ °°®° °¯ (21)

č. Active disturbance rejection control strategy
A. Active disturbance rejection controller
Active disturbance rejection controller (ADRC)[7] is
not depend on math model of object. ADRC is consist of
nonlinear tracking differentiator(TD), extended state
observer(ESO) and nonlinear state error feedback
( NLSEF).First, un-exceed fast tracking and good
differential of input are realized by TD. Second,
disturbances in self model are seen as inner disturbances,
inner and outside disturbances are defined as total one.
Total disturbance is observed by using ESO, and is
compensated by disturbance component from NLSEF.
System controlled by ADCR has strong robustness.
ADRC structure is shown in Fig.2.
Fig.2 ADRC structure
In Fig.2, each derivatives ( vl,",vn)of input v is
obtained from n-order TD. Each state variable(1,,n zz" )
and total disturbance 1nzora(t) of object are observed
by ESO. The errors between outputs of TD and ESO are
utilized to generate control uby NLSEF. Total
disturbance a(t) is used in the feedforward compensation.
B. Active disturbance rejection controller for PWM
rectifiers
ADRC is used in direct voltage control of direct power
control system for three-phase boost-type PWM rectifiers
[8].Direct voltage equation is written as:
/( )D d du dt t bi D  (22)
where 2DDC uu ,() 2 / DL tu R CD  , 6/ m bU C .
From (22), ()tDincludes LR, and change of LRcan be
observed by ESO. ()tD is used to compensate the change
of LR.ADRC of PWM rectifiers is shown in Fig.3.
301

DCRu
1DCu0e0di*
di
1z2zDCu
ESOPWM TD NLC
b 1/b()wt
Fig.3 ADRC of PWM rectifiers
Ď. Analysis of control strategies
A. Feedback linearization control strategy
Advantages of feedback linearization control strategy
are that nonlinear system is changed into linear one and
PWM rectifiers have fast respond. Shortcomings of one
are complicated controller and singularity( x3=0 in
equation 8).
B. passivity control strategy
1) Controller Desig Qĉ˖ Substituting (18) into
(11),then (11) is not decoupling. When damp aR is
injected the system is a couple one. aRonly accelerates
the rate of convergence. When system operates in ideal
stable state, i.e. dmiI , 0 qi and DCD C Ruu , equations in
(11) hold. In practice dmiIz and 0 qizcaused by various
perturbation influence the dynamic and static state
performances of PWM rectifiers.
2) Controller Desig QĊ˖ In order to analyze the
performances of PWM rectifier using controller design
Ċwesubstitute (19) into(11) and (11) can be written
as:
1
2
2
2120
33 3 3()22 2 2d
dm
a
q
q
a
DCDCD C md md a d d m aq
LLd iiIRRd t
Ld iiRRd t
du uCu U i RI i R i i I R idt R­ °°
° ®°
°      °¯ (23)
When 1aRand 2aRare chosen big, difast steady at
mIand qi at 0, then (23) is changed as
1
2
2
20
33
22d
dm
a
q
q
a
DCDCDC m mm
LLd iiIRRd t
Ld iiRRd t
du uCu U I RIdt R­ °°
° ®°
°   °¯ (24)
The third equation in (24) satisfies power conservation,
and DCu steady at DCRu. (24) shows that PWM rectifiers
can be decoupled by control law (18) and control law (18)
can improve the dynamic and static state performances
of PWM rectifiers.
C. Active disturbance rejection control strategy Advantages of Active disturbance rejection control
strategy are strong robust and fast respond. When order
is greater than 3, the design of ADRC is difficult.
ď.Conclusions
According to treatise and Analysis from above, the
controller (18) can transform (11) into decoupling and
PWM rectifier has most dynamic and static state
performances compared with other control strategies.
Passivity control strategy has good dynamic and static
state performances. So combination of passivity control
strategy and ADRC have application perspective and are
researched further.
Acknowledgement
This work is supported by major science and technology
development project of beijing municipal commission of
education/ major nature science foundation of beijing
(KZ200710772015).
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