Middle-East Journal of Scientific Research 24 (3): 571-580, 2016
ISSN 1990-9233
© IDOSI Publications, 2016
DOI: 10.5829/idosi.mejsr.2016.24.03.23054
Multivariable State Feedback Control of
Three-Phase Voltage Source-PWM Current Regulator
1
Ahmed G. Abo-Khalil and 2Mohammad Abdul Baseer
Assiut University, Assiut Egypt (On Leave to Majmaah)
College of Engineering, Majmaah University and PhD Scholar in KBVAU Assam, India
1
2
Abstract: In this paper, a multivariable state feedback current controller is proposed to control a three-phase
PWM converter. Error compensation, dynamic response and high accuracy control are the most important
requirements for current controller which is used in a PWM converter. The multivariable state feedback with
feedforward control for input reference and disturbance is applied to improve the converter input current
waveform, unity power factor control and fast transient response. A feedback gain matrix is derived by utilizing
the pole assignment technique to guarantee sufficient damping and then the feedforward gain matrix is derived
to reduce the transient error. The equivalent circuit and modelling of the power conversion system are derived.
The performance of the proposed current controller is verified by experiment. The results confirm superior
performance of the multivariable state feedback controller to conventional PI controllers.
Key words: PMW Converter Voltage Controllers
Controller Current Controller
INTRODUCTION
Multivariable state
PI Controller
Fuzzy Logic
Existing current controller techniques can be
classified in different ways [2], [3] and [4]. These
techniques can be represented in two main groups, linear
and nonlinear controllers. Several linear control
techniques have been proposed [5], [6] and [7], such as PI
stationary and synchronous, state feedback and deadbeat
controllers.
The main disadvantage of the stationary PI technique
is an inherent amplitude and phase error [1]. To achieve
error compensation, use of additional phase-locked loop
(PLL) circuits [8] or feedforward correction [9] is also
made. In case of synchronous PI controller, the error of
the fundamental component is controlled to zero [9],
but the dynamic properties are still inferior to those
bang-bang controllers [10]. The conventional PI
compensators in the current error compensation part can
be replaced by a state feedback controller working in
stationary [9] or synchronous rotating coordinates [4].
A feedback gain matrix is derived by utilizing the pole
assignment technique to guarantee sufficient damping.
While with integral part the steady state error can be
reduced to zero, the transient error may be unacceptably
large. Therefore, feedforward signals for the reference and
Three-phase voltage source pulse width modulation
VS-PWM converter, shown in Fig. 1, is a very important
component in industrial applications, such as variablespeed drives and ac-dc power supplies for
telecommunications. They are able to supply power in two
directions with unity power factor operation, sinusoidal
source current and low harmonic content. The
performance of the converter system largely depends on
the quality of applied control strategy. Therefore, current
control of VS-PWM is one of the most important subjects
of power electronics circuits. The accuracy of the current
controller can be evaluated with reference to basic
requirements. These requirements are the following [1]:
No phase and amplitude errors over a wide output
frequency range;
To provide high dynamic response of the system;
Limited or constant switching frequency to guarantee
safe operation of converter semiconductor power
devices;
Low harmonic content;
Corresponding Author: Mohammad Abdul Baseer, College of Engineering, Majmaah University and PhD Scholar in KBVAU
Assam, India. Tel: +966530991606.
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Middle-East J. Sci. Res., 24 (3): 571-580, 2016
Fig. 1: Voltage source PWM converter
Fig. 2: Equivalent circuit of VS-PWM
disturbance inputs are added to the feedback control law.
The performance of the state feedback controller is
superior to conventional PI controllers [4].
The deadbeat control algorithm is known to ensure
the best dynamic response [11]. An important advantage
of this technique is that it may not require the voltage
measurements in order to generate the current reference
[12], but inherent delay due to the calculations is indeed
a serious drawback [13]. In addition, the deadbeat
controller doesn’t have an integral control; hence the
steady-state error may exist.
The nonlinear current controller group includes
hysteresis and fuzzy logic controllers (FLC). The
hysteresis band current control is used very often
because of its simplicity of implementation. Also, besides
fast response current loop, the method does not need any
knowledge of load parameters. However, the current
control with a fixed hysteresis band has the disadvantage
that the PWM frequency varies within a band because
peak-to-peak current ripple is required to be controlled at
all points of the fundamental frequency wave [14].
The FLC is used as a substitute for the
conventional PI compensator [15]. The block scheme
of the FL current controller is used instead PI controller.
The design procedure and resulting performance
depend strongly on the knowledge and expertise of the
designer.
In this paper, a multivariable state feedback control is
proposed to obtain a high performance control for threephase VS-PWM. The state feedback control law is
designed by the pole placement technique of multivariable
system and the feedforward control for input reference
and disturbance is incorporated in the control laws for
fast transient response. The simulation and experimental
results are satisfactory both in the transient state and
steady state.
Modeling of PWM Converters: A per-phase equivalent
circuit of VS-PWM is shown in Fig. 2. A system model is
derived from the figure as [4].
es = Ris + L
dis
+ vr
dt
(1)
where es, is and vr are the source voltage, current and
converter input voltage, respectively. And R and L mean
the line resistance and the boost-inductor, respectively.
After (1) is transformed into a synchronous frame, a state
space model can be expressed as.
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Middle-East J. Sci. Res., 24 (3): 571-580, 2016
x = Ax + Bu + Ed
(2)
y = Cx
(3)
. 
 E 0  d 
x   A 0 x  B
=
 .  C 0   p  +  0  u +  0 − I   y 
   

 r
 p  
(7)
where x, u, d and y are state, input, disturbance and
output vectors, respectively and A, B, C and E are
coefficient matrices.
In steady state, x 0 and p 0 since d and yr are
assumed constant. Then, steady state solutions xs, ps and
us must satisfy the equation.
where,
 E 0  d 
 A 0   xs   B 
−
0 − I   y  =
   −   us

 r
 C 0   ps   0 
ids 
vdr 
eds 
x =
=
 , u =
,d  
i
v
eqs 
 qs 
 qr 
 R

 1

0
− L

− L
=
A =
, B 

1
R
−
0
− 
− 
L 
L 


1 0 
− B, C =
E=
0 1


Substituting this for the last term in (7) produces.
. 
 x   A 0   x − xs   B 
=
 .  C 0   p − p  + 0  (u − u s )

s  
 p  
and is the source angular frequency. The state variable
is the inductor current and control input is the converter
input voltage and the disturbance is the source voltage
and the output is the source current.
Multivariable State Feedback Control
State Feedback Control: A state space model of a linear
and time-invariant multivariable system is given by (2) and
(3). A target of the control is that as t
, [4], [16].
x
0 and y
yr
(8)
where subscript ”s” denotes steady state values. Now
define new variables as follows, representing the
deviations from this steady state.
. 
 z1   x − xs  .  x 
=
( z  )
z =
=
 
.
 z2   p − ps 
 p 
v = u – us
(10)
(11)
Representing (10) in new variables, a standard for of state
space equation is obtained as
(4)
.
(12)
=
z Aˆ z + Bˆ x
where yr is a reference output.
(9)
where,
Since a state feedback control is basically a type of
 A 0 ˆ  B 
proportional control, the steady-state error may exist due
=
Aˆ =
 , B 0 
C 0 
 
to the model uncertainty. Therefore, to remove this error,
the integral control of the error p is introduced as:
If (12) is a controllable system, a linear state feedback
t
(5)
control can be applied to it. Then,
=
p
( y − y )dt
∫
r
0
v = Kz = K1z1 + K2z2
Assuming yr and d to be constant, differentiating (5)
and using (2) and (2) gives the differential equations.
p = y – yr = Cx – yr
(13)
where K is a feedback gain matrix, with partitioned
matrices K1 and K2, which is derived by pole placement.
From (9), (10) and (13), a control law is obtained as.
(6)
t
∫
Written in matrix form, an augmented state model is
obtained as:
u =K1x + K 2 p =K1x + K 2 ( y − yr )dt
0
573
(14)
Middle-East J. Sci. Res., 24 (3): 571-580, 2016
Feedforward Control: While with integral control static
errors can be made zero, the errors during the transients
may be large. Feedforward control is an important
technique in practice to reduce the effect of disturbance
if these are measurable. The control equations are derived
for feedforward from both reference inputs and
disturbance inputs.
Let the deviation between the reference and the
output be.
where,
=
K ff [ K1
which is a feedforward gain matrix.
The total control law is a function of the state as well
as the disturbance and reference input. When the integral
control in (5) is superimposed on (22), the resultant
control law becomes
as follows:
t
d 
(23)
u =K1x + K 2 ( y − yr )dt + K ff  
yr 

0
The system is described from (2), (3) and (16) as
follows.
 .   A B   x  E 0  d 
x 
=
  + 
 
 y  C 0  u  0 − I   yr 
 
∫
(16)
The total control block diagram of (23) for feedback
and feedforward components is shown in Fig. 3.
In the steady state, the left hand side of (16) becomes
zero. Thus,
 xs 
ˆ −1 ˆ  d 
  = −G H  
y 
us 
Pole Placement Technique: To derive the feedback and
feedforward gain matrices of (23), a pole placement using
the Generalized Control Conical Form (GCCF) can be used.
The system equation of the open-loop system is
given as.
.
(24)
=
x Ax + Bu
(17)
where,
B ˆ  E
A
=
Gˆ =
 , H 0
0
C



0
− 1
where, x is n x 1, B is n x m, u is m x 1 and the rank of B is
m. Equation (24) is converted to the GCCF through three
transformation described below.
Now, new variables are defined for deviations from
steady state as.
x = x − xs
.
.
( x = x), u= u − u s
.
.
(18)
where,
AG = block digonal ( A 1 A 2 ... A m )
BG = block digonal (b 1 b 2 ...b m )
(19)
and i is control invariant which is equal to the rank of u.
That is.
The control law is of the same form as shown earlier.
(20)
u = K1 x
1
and is given by.
=
u K1x + [− K1
≥
2
m
 xs 
I ] 
u s 
(25)
=
z AG z + BG u
Substituting (18) into (17)
x =
A x + B u , y =
C x
− I ] Gˆ −1Hˆ
= [ K ff 1 K ff 2 ]
(15)
y= y − yr
(22)
d 
=
u K1x + K ff  
 yr 
∑
(21)
i =1
i
=
≥ ... ≥
1+ 2
m
(26)
=
n
+ ... +
m
The matrices A i and b i are of order
1, respectively and have the forms.
Substituting (17) into (21)
574
(27)
=
n
i
×
i
and
i
×
Middle-East J. Sci. Res., 24 (3): 571-580, 2016
Fig. 3: Block diagram of multivariable state feed back control with feedforward control
0 1 0...0 0 
0 0 1...0 0 


.................. 
=
A i =
, b i
0 0 0...1 0 
0 0 0...0 1 


0 0 0...0 0 
0 
0 
 
. 
 
0 
0 
 
1 
(28)
The three transformations which can be applied to
the state equation of (26) in order to achieve the GCCF of
(26) are.
.
(34)
Forming the inverse of (34) yields (35),
. 
. 
 
−1  T 
M c = eij
 
. 
 
. 
(35)
1 −1
b1 ... bm Abm ... A
bm ]
where j = 1,2,..., i and i =1, 2, …, m.The bottom row of
each partitioned submatrix Mc 1 is used to form the
transformation matrix T-1, using the format .
A change of basis in state variable, that is, x=Tz,
where det T 0. Equation (25) then assumes the form.
z=
T −1 AT z + T −1BT u =
AG z + BG u
m −1
M c = [ b1 Ab1 ... A
(29)
T −1 = [ e1T 1 e1T 1 A... e1T 1 A
1 −1
T
T
... em
eTm m A... em
A
m
m
A change of basis in the control variables, that is,
u = Fw
where F is m x m and det F
becomes
m −1
]T
(36)
T is found by taking the inverse of (36). F is derived from
(30) and then H is from (32).
(30)
0. When (30) is applied, (29)
.
z=
T −1 AT z + T −1BF w =
AG z + BG w
Applying the state feedback control,
v= z
(31)
The introduction of state feedback
w =−
v H z =−
v H T −1x
then,
.
.
(38)
z= ( AG + BG Γ ) z= Ad z
(32)
where Ad is the desired closed-loop system matrix which
has the form
where, H is an m x m matrix and v is an m x 1 input vector.
Substituting w from (32) into (31) yields.
z=
[T −1 AT − T −1BFH ] z + [T −1BF ] z
(37)
Ad = block digonal ( Ad1 Ad 2 ... Adm )
(33)
A formal expression for
=[ AG − BG H ]z + BG v = AG z + BG v
Above transformation require T, T-1, F and H matrices.
The controllability matrix of multivariable system is given.
T
BG
BG = I m
575
(39)
can be obtained since.
(40)
Middle-East J. Sci. Res., 24 (3): 571-580, 2016
Fig. 4: Control block diagram of PWM VSC
Thus, this leads to,
T
=
Γ BG
[ Ad − AG ]
components are used with the feed forward gain
matrix (23) to produce the feedforward component of the
PWM voltage reference. The dq voltage reference
components are then transformed and modulated using
SVPWM.
(41)
Returning to the original state space,
Experimental Results: To demonstrate the performance
of the multivariable state control, the experiment was
carried out. Figure 5 shows the hardware configuration of
the experimental system. The actual operation of the
IGBT PWM converter was tested on a small prototype.
The converter rating is stated in the Appendix. Highperformance DSP chip TMS320C33 was used as a main
controller, which operates at 33.3-MHz clock and is
capable of 32-b floating-point operation.
The sampling rate of the nonlinear control loop is
double the PWM frequency, that is, the sampling period
is 100µs. The space-vector modulation with symmetrical
switching patterns was employed as a PWM strategy.
Figure 6 shows the voltage transient responses for the
step change of the dc voltage reference. For the same size
of the DC capacitor, the multivariable state feedback
control gives fast rising time and excellent decoupling
characteristics of d–q current control.
u = Fw
= F [G − H ]T
−1
x
= Kx
(42)
where,
K = F[G – H]T
1
The total feedback and feed forward controller
components are depicted in Fig. 4. The output DC link
voltage is compared with the reference and the error is fed
to the voltage controller to reduce the produce the current
reference in dq frame axis. The three phase line currents
and source voltages are transformed to dq frame axis.
The latter quantities are used together with the state
feedback matrix (43) to produce the feedback PWM
voltage reference components. In the same way, the dq
voltage source components and the reference dq current
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Middle-East J. Sci. Res., 24 (3): 571-580, 2016
Fig. 5: System hardware configurations
Fig. 6: The proposed controller transient response for a step change in the dc reference voltage
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Middle-East J. Sci. Res., 24 (3): 571-580, 2016
Fig. 7: PI controller transient response for a step change
in the dc reference voltage
Fig. 9: PI controller transient response for a step change
in the load
Fig. 10: Measured waveforms of the input voltage and
current
The transient response for the dc reference step
change in case of PI controller is shown in Fig. 7. It is
noticeable that the current overshoots due to voltage
change are higher than the multivariable state feedback
control. The decoupling characteristics of dq current
control is worst compared with multivariable state
feedback control.
Fig. 8: The proposed controller transient response for a
step change in the load
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Middle-East J. Sci. Res., 24 (3): 571-580, 2016
Appendix
Table 1: Parameters of SV-PWM
Parameters
Value
Rated power
Rated voltage
Main frequency
Rated DC link voltage
DC link capacitor
Input interface inductance
Switching frequency
3 [kW]
220 [V]
60 [Hz]
340[V]
1950[µF]
0.0033[H]
5 kHz
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Fig. 11: Spectra for input voltage and current
Figures 8 and 9 show the transient responses for the
step changes of the load for multivariable state feedback
control and PI controller, respectively. The dc-voltage and
dq currents responses are more satisfactory and
acceptable in case of the multivariable state feedback
control. It can be observed that the multivariable state
feedback control performance is satisfactory in transient
and steady state.
Fig. 10 shows that the source power factor is
controlled at unity, as usual. Lagging and leading
power-factor control can be achieved if necessary.
The spectrum shown in Fig. 11 indicates that the
dominant frequency for the supply voltage and current is
equal to the fundamental frequency 60 Hz.
CONCLUSIONS
In this paper, an overall multivariable state feedback
controller for a three phase PWM converter is proposed.
The controller is designed by pole placement technique of
multivariable system regulation theory. The PWM
controller consists of outer DC link voltage and inner
current controllers. The external DC link voltage controller
incorporating the integral control regulates the DC link
voltage with high dynamics and the zero steady state
error. The internal multivariable state current controller
ensures the true unity power factor operation, fast
transient response and excellent performance. The control
strategy is able to give higher performance in transient
states such as the step change of the voltage reference
and the load since the feedforward control for the input
reference and disturbance is superposed on feedback
control. Experimental results verified the validity of the
proposed control scheme.
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Middle-East J. Sci. Res., 24 (3): 571-580, 2016
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