SYNERGETIC CONTROL FOR GROUP OF DC/DC BUCK CONVERTERS
ANATOLY KOLESNIKOV, GENNADY VESELOV,
ANDREY POPOV, ALEXANDER KOLESNIKOV,
MIKHAIL MEDVEDEV
Department of Automatic Control Systems
Taganrog State University of Radio Engineering
Taganrog, Russia
Abstract: The application of synergetic control theory
to the design of control systems for DC/DC buck
converters is considered. Controllers are designed for
single converters as well as for a group of converters
connected to a common load. The synthesis procedure
allows to suppress the influence of disturbances and to
ensure robustness against variation of the converter’s
internal parameters. Simulation results are presented to
validate the design.
Keywords: Synergetic Control, Buck Converter
Introduction
It is often the case in DC distributed power systems
that several buck converters are connected to a common
load. So the task of controlling a group of converters is
practically important. Here, we consider a system
consisting of several buck converters connected to a
common load. The load consists of an unknown
capacitance, unknown resistance and a constant power
sink.
ROGER A. DOUGAL, IGOR KONDRATIEV
Department of Electrical Engineering
University of South Carolina
Columbia, SC 29208, USA
Consider the task of synthesizing a controller for a
buck converter. The equations of the converter working
on a constant load are [2]
Asymptotic stability to the desired state;

Limitation of currents during transients;

Suppression of external disturbances (load changing)
and robustness to variation of converter’s parameters
We present next the main results of the research, how
the new class of controllers are synthesized and how they
are superior to the existing types of linear PI-regulators.
Synthesis of Scalar Regulators
(1)
dvc1
 c1
P
1
  iL1 
 load  M t ,
dt
Ct
Rext  Ct  c1  Ct
(2)
where i L1 – current,  c1 – voltage at the output of the
converter, Pload – load power, M t   M 0i  const
uncontrolled piecewise constant disturbance. The variable
capacitance C t includes capacitance C1 plus the external
capacitance of the connected loads (fig. 1).
Synergetic Control Theory [1] is applied to the task of
designing the control algorithms for the group of
converters. The method allows analytic synthesis of
effective controllers based on the mathematically
nonlinear model of the converter. The synthesized
algorithms ensure:

diL1
1
d E dE
   c1  ss  c ,
dt
L1
L1
L1
Fig. 1
Using the new variables
z1  C1  c1  Vc ,ref 
(3)
and
z 2  iL1  i01 

P 
C1 
 iL1  c1  load  ,
Ct 
Rext  c1 
(4)
we write equations (1) and (2) in the following form [1]:
z1 t   z2  M t ,
z2 t   

1
1
C
z1 
z 2  1 uc 
L1Ct
Rext Ct
Ct
Plo a dC12
z2
,
2
Ct z1  C1Vc ,ref 
(5)
(6)
where uc 
dc E
L1
– control; Vc ,ref  d ss E
y t   1 z1 ,
z1 t   z 2  y,
– nominal
voltage.
z2 t   
Using the method of analytical design of aggregated
regulators – ADAR -- we write the control law uc z1 , z 2 
for a buck converter (5), (6). At first we assume that
M t   0 . This law should ensure asymptotic stability of
the converter in the whole with respect to the stabilized
state z1s  z2 s  0 , i.e  c1  Vc ,ref , iL1  i01 . According to
the ADAR method we introduce a macrovariable
 1  z1  z2 ,   0
(7)
Let’s substitute  1 (7) into the functional equation
T11   1  0, T1  0 ,
(8)
Then accounting for (5), (6) we get the control law
C1
1
  Rext Ct
uc  u1 
z1 
z2 
Ct
L1Ct
Rext Ct

Plo a dC12
z2
1
 1
2
Ct z1  C1Vc ,ref  T1
(9)
The law (9) will inevitably move the representing
point of the closed-loop system (5), (6), (9) to the
manifold  1  0 (7). Motion along this manifold is
described by the equation
z1 t   
1

z1
(10)
Obviously for   0 , T1  0 the closed-loop nonlinear
system (5), (6), (9) will possess the quality of asymptotic
stability of the motion with respect to the state
z1s  z2 s  0 . Here according to (8) and (10) the top
estimate of the duration of transients is expressed as:
t  3T1    .
So the synthesized control law u1 (9) ensures the
asymptotic stability of the converter (1), (2) with respect
to the desired steady state z1s  z 2 s  0 , i.e.  c1  Vc ,ref ,
iL1  i01 .

(11)
Plo a dC12
z2
,
2
Ct z1  C1Vc ,ref 

where y  1 z1dt .
The essence of the extended model (11) including
disturbance M t  is as follows. During the transients of
the system (11) the derivative of the additional coordinate
y t   0 . In the steady state when z1s  0 the derivative
y t   0 . This means that the additional coordinate
yt   1 z1dt

is an integrator that will generate a
constant signal in the system (11) that is modeling the
disturbance M t  . In its turn the control law u z1 , z 2 , y 
should form another signal y that compensates the action
of the disturbance M t  . Naturally the control law
u z1 , z 2 , y  should ensure that the system (11) is
asymptotically stable with respect to the reference state
z1s  z 2 s  0 .
Let’s introduce the following macrovariable:
 2  z1  z2  y
(12)
Substituting  2 (12) into the functional equation
T2 2   2  0.
and accounting for the equations (11) we find the control
law
 1

C1
1
uc  u 2 
z1  
   z 2 
Ct
L1Ct
 Rext Ct

(13)
2
PloadC1
z2
1

 y  1 z1   2
2
Ct z1  C1Vc ,ref 
T2
The law (13) moves the plant (11) from the arbitrary
initial conditions to the manifold (12). Motion along this
manifold is described by the decomposed equations:
y t   1 z1 ,
z1 t   z 2  y .
Suppression of Disturbances
Assume now that the converter is influenced by an
uncontrolled
piecewise-constant
disturbance
M t   M 0i  const . Then according to the ADAR
method we extend the system (1), (2) and write it in the
following form:
1
1
C
z1 
z 2  1 uc 
L1Ct
Rext Ct
Ct
(14)
Substituting the relation z2  z1  y into (14 )
from  2
 0 (12) yields
y t   1 z1 ,
z1 t   z1  1    y ,
Let’s write the system (15) as a single equation
(15)
z1 t   z1 t     11 z1  0 .
(16)
Obviously for
  0;   11  0
(17)
where M t   M 0
disturbance.
Introducing the new variables [1]
z1  C1  C 2  c1  Vc1ref  ,
equation (16) is asymptotically stable with respect to the
state z1  z1s  0 and therefore z 2 s  0 . This means that
if the inequalities (17) and T2  0 are satisfied, the
synthesized closed-loop system will also possess the
quality of asymptotic stability in the whole with respect to
the reference state z1s  z2 s  0 .
The equations (15) can also be written in the following
form:
z1 t   z1  1     z1 dt .
C1  C2
Ct
z2 
2  0
PI control law for the coordinate
z1 . An analogous PI law
To account for the limitation 0  d c  1 , the control laws
(9) and (13) can be formed as the following expressions:
ui sup  0.5tanh ui  tanh ui ,
or
u
u
ui sup  0.51  e i  1  e i ,   0, i  1,2


So application of the ADAR method [2] allows
synthesis of effective control laws that ensure asymptotic
stability of the buck converter.
Synthesis of Vector Regulator for Two
Parallel Converters Connected to a
Common Load
According to Fig. 1 the equations describing the shunt
connection of buck converters have the following form
[1]:
d E d E
diL1
1
   c1  ss  c1 ,
dt
L1
L1
L1
(19)
d E d E
diL 2
1
   c1  ss  c 2 ,
dt
L2
L2
L2
(20)
d c1
1
iL1  iL 2    c1  Pload  M t  , (21)

dt
Ct
Rext Ct  c1Ct


p 
 iL1  iL 2  c  load  ,

Rext  c1 

(23)
z 3  iL 2  iL 2 ,
(24)
y1 t   2 z1 ,
(25)
z1 t   z2  y1 ,
(26)
z2 t   a1 z1   uc1  uc 2   a2 z 2 
(12) there is a
is formed for the coordinate z2 . Generally this means
that, according to (16) and (18), the synthesized control
law u2 (13) besides giving the asymptotic stability, also
suppresses the uncontrolled disturbance M(t)=const
influencing the converter.
(22)
using the ADAR method we write Eqs. (19)-(21) in the
following extended form
(18)
This means that on the manifold
– uncontrolled piecewise-constant
 a3
z2
z1  C1  C2 Vc ,ref
,
z3 t   b1 z1  uc1  uc 2 ,
where a1 
a3 
C
1
1
Ct
1 1
   ;
 L1 L2 

(27)
(28)
C1  C2
1
; a2 
;
Rext Ct
Ct
2
 1  1 1 
 C 2  Plo a d
   .
; b1  
Ct
 C1  C2  L2 L1 
Let’s denote u3  uc1  uc 2 , u4  uc1  uc 2 and state the
vector regulator synthesis task: it is necessary to
synthesize control laws u3 z1 , z2 , z3 , y1  and u4 z1 , z2 , z3 , y1 
that 1) ensure asymptotic stability of the system (25)–(28)
with respect to the desired state z1s  z2 s  z3s  0 2)
suppress the uncontrolled disturbance M t   const . To
solve this task we introduce the following macrovariables:
 3  z1   2 z2  3 z3   4 y1 ,
(29)
 4  z1   2 z2  3 z3   4 y1
(30)
T3 3 t    3  0, T4 4 t    4  0,
T3  0, T4  0
.
(31)
Substituting the macrovariables  3 (29) and  4 (30)
into the Eq. 31 and solving them jointly with the
converter’s equations we find the control laws:












z

 3 4

3 4
2 1
1
    z  y    
u3 
3
3
2
1

3 2  2 3 
3
(32)
 3

 T 3  T 4

4
 3

 a1 z1  a2 z 2 
u4  b1 z1 
a3 z 2
z1  C1  C2 Vc ,ref


2
(33)

where y1  2 z1dt ,  3  2   2  3 ,  3 4   3  4 ,  2   2 ,
 3   3 , u3  uc1  uc 2 , u 4  uc1  uc 2 .
The obtained control laws u 3 (32) and u 4 (33) ensure
that the representing point gets to the intersection of the
manifolds  3  0 (29) and  4  0 (30). Motion of the
system (25)–(28), (32), (33) along this manifold is
described by the following decomposed equations:
(34)
z1 t   z 2  y1 t .
Substituting into (34) the coordinate z 2 the joint
solution of the equations (29), (30) yields
y1 t    2 z1 ,
z1 t  
   34
 (35)
3 3
z1   4 3
 1 y1
32 23
 32 23

or
z1 t  
3  3
z1 t  
32  2 3
    43 
  3 4
 12 z1  0
 32  2 3 
3  3
z1 
32  2 3

1

32  2 3
y1 t    2 z1 ,
z1 t  
    34 
2 z1 dt,
  4 3

 32  2 3 
,
2 z1   2   2 z 2  y1    ,


   2 3  2 4

 T  T

4
 3

These laws also suppress the uncontrolled piecewiseconstant disturbance M t   const . This means from the
equations (35) that can be written in the following form:
.
this means introduction of a PI control law at the
intersection of the manifolds  3  0 (29) and  4  0
(30). Note that according to (31), (35) the duration of
transients in the synthesized system (25-28), (32), (33) is
determined by the parameters T3 , T4 ,  2 ,  i ,  i selected
at the controller design time
So the vector controller synthesized using the ADAR
method has two control channels u 3 (32) and u 4 (34) it
ensures asymptotic stability of the two buck-converters
connected to a common load and suppresses the constant
disturbance.
Modeling results
Modeling is performed using the Virtual Test Bed
software for system simulation. The closed-loop system
contains a buck-converter and the synergetic regulator (9).
Following are the system parameters: l  1.35 mHenry,
Ct  2.7  10 6 Farad, Rext  50 Ohm, Vc ,ref  600 V,
T  27 106 sec,   500 ,   3 ,   10 , e  850 V,
C1  2.6  10 6 Farad, Plo a d  5  10 4 W. Modeling of
designed control system was performed both for the exact
(switching) buck-converter model as well as for the
switching-averaged buck-converter model. Note that the
synergetic regulator was designed by considering the
averaged model. The system schematic with constant
disturbance attenuation at VTB 1.5 is presented in fig. 2,
which shows the buck-converter Buck_Av, DC voltage
source VS0, regulator Reg, inductance L1, internal and
external capacitances, DC current source I, load resistance
R4, resistor R2 as current sensor, and inductive current
sensor.
The stability conditions for these equations with
respect to z1s  0 are
3  3
 0,
3 2   2 3
 3 4   4 3 


       12  0. (36)
2 3
 3 2

This means that is the inequalities (36) are satisfied
and T3  0, T4  0 the synthesized vector control laws u 3
(32) and u 4 (33) will ensure asymptotic stability (in the
whole) of the closed-loop system (25)-(28), (32), (33)
with respect to the state z1s  z 2 s  z3s  0 .
Fig. 2
Fig. 3.
Simulation results are presented in fig. 3, including the
voltage drop across the capacitance C1 and the current
through resistor Rext . The system starts from zero initial
condition. The moment at which the regulator starts
working is clearly seen as the break point in the current
and voltage waveforms. The load voltage approaches the
setpoint even with unknown value of the constant load.
The simulation results show that the synergetic control
system does provides the desired properties of the closedloop system: asymptotic stability, monotonic transients,
and small error of about 1 % in steady-state.
The schematic for a system containing two buckconverters is shown in fig. 4.
Fig. 5.
Fig 5 shows the transients of the capacitor voltage,
inductor currents, and the control. One can see the system
startup, then the onset of working of the synergetic
regulator (9), (10). It is clear that the designed vector
synergetic control system ensures high performance of the
control system.
To confirm robustness of the designed control system
we present in fig. 6 and fig. 7 simulation results for the
cases of variation of the capacitance, resistance, and load
current. The parameters of load are changed within the
limits of 100% of nominal values.
Fig. 6. Transients of current, voltage and control for
Rext  10 Ohm and I  100 А
Fig. 4
The system contains two Buck-converters Buck_Av,
DC voltage source VS0, regulator, inductances L0, L1,
internal and external capacitances, DC current I, load
resistance R0, and current sensors R1, R2.
Simulation results are shown in fig. 5. The parameters
of the regulator, converters, and load are: l1  1.35
mHenry,
C1  2.6  10
l2  1.25
5
Ct  8.7  10 5
mHenry,
Farad, C2  2.6  10
5
Farad,
Farad, Rext  5 Ohm,
Vc ,ref  700 V, T3  T4  8.7  10 5 sec,  2  5 ,  3  2 ,
 4  3 ,  2  0.2 ,  3  0.3 ,  4  0.4 ,   1 , e  850 V,
Plo a d  25 10 4 Vt.
These results confirm the robust character of the
designed synergetic control system, and confirm that
– synergetic regulators ensure asymptotically stable
performance of the closed-loop system;
– synergetic regulators are robust with regard to
variation of load parameters;
– synergetic regulators ensure attenuation of
unmeasured constant disturbances that may acting on the
converter from the load.
Fig. 7. Transients of voltage, and control for
Rext  10 Оhm and I  50 А, Ct  7.5  10 5 F
Conclusion
Scalar and vector regulators for DC/DC buck
converters connected to a common load were designed
based on Synergetic Control Theory. The design
procedure ensures asymptotic stability in the whole of the
power distribution system and provides a robust control
system.
Acknowledgments
This research was supported by the US Office of
Naval Research under grant N00014-00-1-0131.
References
[1] A.A. Kolesnikov, G.E. Veselov, Al.A. Kolesnikov,
et al. Modern Applied Control Theory: Synergetic
Approach in Control Theory. (Taganrog: TSURE press,
2000).
[2] V.Joseph Thottuvelil, George C. Verghese,
Analysis and Control Design of Paralleled DC/DC
Converters with current Sharing, IEEE Transaction on
Power Electronics, Vol. 13, NO. 4, July 1998.