Design of State-Space Controllers with the Help of Signal Flow Graphs
Shown for a Buck Converter
FELIX A. HIMMELSTOSS
Faculty of Electronic Engineering and Entrepreneurship
University of Applied Sciences Technikum Wien
Hoechstaedtplatz 6, 1200 Wien
AUSTRIA
Abstract: - When consulting the text books on control engineering, state space controllers are described and
designed with the help of matrix calculus. This implies good knowledge of linear algebra. Especially in Europe
many Universities of Applied Sciences have extremely reduced mathematics in their curricula. Here a teaching
concept with the help of signal flow graphs can help to explain and design the controller without solving matrix
equations. As example, the bidirectional Buck converter is used. The model of the plant is derived and a simple
state space controller is designed. The plant model is linearized around a working point. Therefore, the simple
state space controller leads to correct results only at this point. Combination with an additional controller or
using the error between the desired value and the actual value as a third state variable improves the quality of
the control. The signal flow graphs for these concepts are given and the controllers designed. With the help of
LTSpice the designs are checked.
Key-Words: - DC/DC converter, Buck converter, modelling, simulation, state space controller, signal flow
graph
Received: May 12, 2022. Revised: December 7, 2022. Accepted: December 24, 2022. Published: December 31, 2022.
1 Introduction
State space controllers lead to a very efficient
control. Perusing the textbooks, the design of these
controllers is described with linear algebra and
matrix calculus. Furthermore, the design is
described for systems of any order. If an example is
given, it uses only simple numbers. Especially in
Europe, many Universities of Applied Sciences
have reduced mathematics extremely in their
curricula. Another way of teaching is therefore
necessary. Here a way to teach using signal-flow
graphs and Mason’s equation [1] is described. The
use of signal flow graphs is very intuitive and helps
the students to a better understanding and the
possibility to design a state space controller. In this
paper the state space controller concept is applied to
a step-down DC/DC converter. An early study of
applying the space controller concept to converters
can be found in [2]. First the model of the plant is
derived and the signal flow graph is drawn. Then the
concept of the state space controller is included into
the graph of the plant and a simple state space
controller designed. The converter is a nonlinear
system and the model changes with the working
point. Therefore, the control has to be improved.
Two concepts, first with an additional PI-controller
and second with the error as additional state variable
are explained and designed with the help of the
signal flow graphs. A comprehensive treatment of
Power Electronics is [5], other valuable textbooks
are e.g. [3-6, 9]. It should be mentioned that the
here-used methodology can be applied to other
DC/DC converters, or also to other plants which are
described by a state-space model.
2 Model of the bidirectional Buck
converter
The Buck converter with synchronous rectification
(an active switch is used instead of the diode)
consists of two active switches S1 and S2, an
inductor L1 and a capacitor C1 (Fig. 1). A pair of
connectors is used for the input voltage and a
second pair serves to apply the load. The two active
switches (S1 & S2) are controlled in push-pull
mode, always one of the switches is turned on (with
a small dead-time between).
In [7] the idealized model for the same Buck
converter is used to design simple P- and PI-
controllers, a compensation controller, and a
feedforward controller, and in [8] the two-loop
control is treated. The idealized model is derived in
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Felix A. Himmelstoss
E-ISSN: 2224-2678
421
Volume 21, 2022
[7], and a model with parasitic resistances is used in
[8]. These models are the starting point for our
design procedures.
Fig. 1. Buck converter with two active switches.
In the CCM (continuous conduction mode) two
modes take place (sometimes also called stages). In
mode M1 the active switch S1 is turned on and
switch S2 is off, and in mode M2 S1 is off and S2 is
on. Including the parasitic resistors of the switches
RS1, RS2, of the inductor RL and of the capacitor RC a
precise model of the converter can be derived
according to
)(
0
1
//
1
u
L
d
u
i
RRCRRC
RRRL
R
L
RRRR
u
i
dt
d
C
L
CC
C
SCL
C
L
.(1)
This model is a nonlinear one. Linearization around
a working point U10, D0, IL0, UC0 leads to the small
signal model (all variables are written as a
combination of the working point value, written as
capital letters with the index 0, and the disturbance
of the variable, written with small letters with a roof
on top)
.
00
1
//
1
100
d
u
L
U
L
D
u
i
RRCRRC
RRRL
R
L
RRRR
u
i
dt
d
C
L
CC
C
SCL
C
L
(2)
The output equation can be calculated according to
C
C
LC u
RR
R
iRRu
//
2
. (3)
For simpler writing we use abbreviations for the
matrix elements and the coefficients of the output
equation
(4)
CL uCiCu 12112
. (5)
For getting the matrix description of the plant we
need only the basic equations of electronics:
Kirchhoff’s voltage law (KVL) and Kirchhoff’s
current law (KCL), and the voltage-current
equations for the resistor, the inductor, and the
capacitor. The matrix is only used to make the
description concise. So no knowledge of linear
algebra is necessary. (2, 3) are linear equations and
can be used for small signal calculations. The
complete system can now be drawn as a signal flow
graph in the Laplace domain. The graph can be
found immediately from (4, 5). For a system of
second order we need two integrators 1/s. On the
left side the branches which form the derivative of
the equation are connected. The output variable U2
is achieved according to the output equation with
the branches C11 and C12.
Fig. 3. Signal flow graph of the synchronous Buck
converter.
It should be mentioned that all second order power
converters have the same signal flow graph as
shown in Fig. 3 but different coefficients.
First we calculate the two transfer functions
between the current through the coil and the duty
ratio and between the capacitor voltage and the duty
ratio. We must calculate the forward paths
s
B
FI12
22112
s
AB
FU
(6)
and the loops
s
A
L11
1
,
22112
2s
AA
L
,
s
A
L22
3
22211
31 s
AA
LL
(7)
With the help of Mason’s equation (a short
explanation can be found in the appendix)
31321
1)(
)(
LLLLL
F
sD
sU UC
(8)
one can calculate the transfer functions between the
capacitor voltage, the current through the coil, and
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Felix A. Himmelstoss
E-ISSN: 2224-2678
422
Volume 21, 2022
the output voltage in dependence of the duty cycle,
respectively according to
211222112211
21221
)(
)(
AAAAAAss
BA
sD
sUC
(9)
211222112211
22212
)(
)(
AAAAAAss
AsB
sD
sIL
(10)
211222112211
212212211122)(
)(
AAAAAAss
CAAsCB
sD
sU
(11)
In [7] a simplified model of the Buck converter is
used. The parasitic resistors are omitted. In this case
the plant is a less damped one, so the derived
controller will be slower. The large signal model is
now described by
1
0
11
1
0u
L
d
u
i
CRC
L
u
i
dt
d
C
L
C
L
(12)
and for the linearized model one gets
d
u
L
U
L
D
u
i
CRC
L
u
i
dt
d
C
L
C
L1
100
00
11
1
0
(13)
For a rough design of the controller this plant model
is sufficient. The signal flow graph is now also
simpler and shown in Fig. 4.
Fig. 4. Signal flow graph of the idealized
synchronous Buck converter.
The branches A11 (consists only out of parasitic
resistances) and C11 (RC << R) are now zero and
C12 is equal to one. The transfer functions (9-11)
can now accordingly be simplified. For the design
of the controller these simplified equations can used
for a first design.
3 Basic state space controller
The input variable of the state spaced controlled
Buck converter consists of two P-controllers R1 and
R2 which feedback the state variables IL and UC
and a so-called input filter K which scales the
reference value (the desired output voltage). The
signal flow graph is shown in Fig. 5.
Fig. 5. Simple state space controller for the idealized
plant.
When using the Mason equation, one has first to
find the forward paths (all the series of branches
which are starting at the input (the independent
variable in our case U2ref) and end at the output
(the dependent variable in our case U2) and the
loops. A first-order loop is defined as the product of
branches encountered in a round trip, when one
moves from one node in the direction of the arrows
back to the node where one has started.
Furthermore, we have to find all second order loops
which consist of the product of any first-order loops
which are not touching each other.
We start from the input variable D. There is only
one forward path
22112
1s
AB
F
. (14)
The first-order loops are
22112
1s
AA
L
,
s
A
L22
2
,
s
RB
L112
3
,
222112
4s
RAB
L
. (15)
The loops L2 and L3 are not touching each other,
therefore we have also a second-order loop
211222
32 s
RBA
LL
. (16)
All loops touch the forward path. Applying Mason’s
rule, the transfer function between the output
voltage and the duty cycle can be calculated
according to
324321
12 1)(
)(
LLLLLL
F
sD
sU
(17)
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Felix A. Himmelstoss
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423
Volume 21, 2022
Including the pre-filter K, the transfer function
between the output voltage and the reference value
can now be written according to
11222211222112
11222
22112
2
2)(
)(
RBAAARAB
sRBAs
KAB
sU
sU
ref
(18)
This transfer function has to be compared with the
desired transfer function and by comparing the
coefficients the values K, R1 and R2 of the state
space controller can be found.
One can now choose the poles. A typical choice
would be a conjugate complex pole pair
js
2,1
(19)
or a double pole on the negative real axis. The
denominator of the desired transfer functions is
therefore
222 2
ss
jsjsDencomplex
(20)
22
22xsxsxsDenreal
. (21)
To avoid a steady state control error, the numerator
must have the same value as the constant coefficient
of the denominator. The desired transfer functions
are therefore
222
22
2
ss
Gcomplex
(22)
22
2
2xsxs
x
Greal
. (23)
For the idealized converter one gets for the real
double pole
2112
2
AB
x
K
12
22
12
B
Ax
R
2112
211211222
2
2AB
AARBAx
R
(24)
and for the conjugate pole pair
2112
22
AB
K
(25)
2
11222 RBA
12
22
12
B
A
R
(26)
22
11222211222112
RBAAARAB
2112
211211222
22
2AB
AARBA
R
. (27)
For the converter with included losses the
comparison of the coefficients leads for the real pole
pair to
121221
2
CBA
x
K
(28)
12
2211
12
B
AAx
R
(29)
2112
2211211211222
2
2AB
AAAARBAx
R
. (30)
For the conjugate complex pole pair one gets
122112
22
CAB
K
(31)
12
2211
12
B
AA
R
(32)
2112
2211211211222
22
2AB
AAAARBA
R
. (33)
The simulation is done by implementing (Fig. 6) the
model of the converter according the nonlinear
model (12). The integration is done with the help of
the voltage controlled voltage sources E1, E3. The
derivative of the state variables is realized with the
arbitrary voltage sources B1, B2. The arbitrary
voltage source B3 calculates the duty cycle from the
state variables and the reference value. The input
voltage u1 and the reference value Uref are given by
the voltage sources V2 and V1, respectively.
Fig. 6. Simulation of the simple state space
controller.
Fig. 7. Simple state space controller with ideal Buck
with poles at
10001000
2,1 js
: current through
the coil (red); reference value (black), voltage across
the capacitor (green); input voltage (blue), capacitor
voltage (green).
Fig. 7 show the current through the coil, the voltage
across the capacitor, the reference value, and the
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Felix A. Himmelstoss
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424
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input voltage. The controller was designed for the
linearized system; the model of the converter in the
simulation, however, is the nonlinear one.
Therefore, a steady-state error occurs when the input
voltage is changed (at 15 ms).
To achieve a Bode plot (Fig. 9) of the controlled
system one can use a simulation according to Fig. 8.
The desired closed loop transfer function (23) is
calculated with the voltage controlled voltage source
E1.
Fig. 8. Simulation circuit for generating the closed
loop transfer function.
Fig. 9. Closed loop transfer function.
Fig. 10. Circuit simulated Buck converter with
simple state space controller, up to down: current
through the coil (red); duty cycle (black); input
voltage (blue), reference value (brown), output
voltage (green).
With the program LTSpice one can implement the
state space controller into the circuit simulation. The
used simulation circuit is shown in the appendix.
Using a circuit simulation of the Buck converter
gives a very good possibility to prove the controller,
not only around the working point, but in a “real”
surrounding. In Fig. 10 one can see the soft-start by
increasing the reference value and a reference value
step and a step of the input voltage. The current
through the coil and the duty cycle are also depicted.
Because of the nonlinearity of the converter a
steady-state error occurs especially after the input
voltage and the reference value steps.
Fig. 11. Circuit simulated Buck converter with
simple state space controller, up to down: current
through the coil (red); duty cycle (black), sawteeth
for the pwm-generator (turquoise); input voltage
(blue), reference value (brown), output voltage
(green), pwm-signal (violet).
Fig. 11 shows the steady-state signals: the current
through the coil, the analog duty cycle, and the saw-
teeth of the pwm-modulator. In the picture at the
bottom the input voltage, the reference value, the
output voltage, and the output of the pwm-
modulator are depicted. The analog duty cycle
signal changes with the current of the coil. With a
comparator this signal is compared with a saw-teeth
signal to generate the pulse width signal to control
the two active switches of the converter. When the
analog duty cycle signal is lower than the saw-teeth,
the comparator output signal is high, otherwise it is
low. The steady state error of the output voltage can
be seen clearly.
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4 Improvements of the state space
controller
The state space controller only works precisely,
when the system is linear. When the controller is
designed for a linearized plant (as in the case of the
Buck converter which is a nonlinear plant), it works
precisely only at the working point where the
linearization was done. To improve the control, we
have two possibilities. First we control the state-
spaced controlled plant by an additional controller
with an integral part, or second another state
variable representing the error is included and a
state space controller with increased order is
designed.
4.1 Improved state space controller type 1
The concept is shown with the help of the signal
flow graph (Fig. 12). Instead of the pre-filter, a
simple linear controller is used. To avoid a steady-
state error, we choose a PI-controller.
Fig. 12. Simple state space controller with
additional linear controller.
With the transfer function of the PI-controller in
Bode nominal form
s
sTK
sR )1(
)(
(34)
and choosing the zero of the controller equal to the
real part of the poles of the plant, the open loop
system can be written according to
6232
621001.0
2ees
e
s
s
KGO
. (35)
Now one can draw a Bode plot for the open loop
system with K=1 (Fig. 13).
Fig. 13. Bode plot of the simple state space
controlled system with additional PI-controller.
Fig. 14. State space controlled Buck converter with
additional PI-controller, controller gain 1000, input
voltage (red), reference value (blue), capacitor
voltage (green).
For a phase margin of 60 degrees the gain is -60 dB.
Therefore, the controller gain can be set at 1000.
The start-up, a reference value step, and an input
voltage step are shown in Fig. 14.
The additional PI-controller leads to an additional
ringing at the reference value step, but it
compensates the input voltage step! Reducing the
controller gain (Fig. 15) avoids the ringing and the
control speed is still satisfactory.
Fig. 15. State space controlled Buck converter with
additional PI-controller, controller gain 100, input
voltage (red), reference value (blue), capacitor
voltage (green).
The simulations in Figs. 14, 15 were done with
using integrators to calculate the nonlinear model of
the converter. Fig. 16 shows the results of the
simple state space controlled system with additional
PI-controller. The plant is circuit-simulated as
synchronous rectified Buck converter.
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Fig. 16. State space controlled Buck converter
(modelled as a circuit) with additional PI-controller,
controller gain 100: input voltage (blue), reference
value (brown), output voltage (green).
The improvement of the state space controller by an
additional controller leads to a very satisfying result.
With the state space controller, the poles of the
system to be controlled are committed to a desired
value. This system is subsequently controlled by an
additional normal linear controller. The design can
easily be done with the help of a Bode diagram of
the open loop with a controller gain of one.
Choosing a phase margin and shifting the open loop
diagram, so that at a phase shift of -120o the gain is
1 (0 dB), leads to a fast and stable controller. To
avoid ringing one has to reduce the gain. Optimizing
can be done easily using some simulations. By
including the controllers into a circuit simulation of
the converter one gets the final check of the design.
4.2 Improved state space controller type 2
A second method to improve the state space
controller is to add the integral of the error between
the output variable and the reference value as an
additional state variable. This concept is also called
extended state space controller. Fig. 17 shows the
signal flow graph of this concept.
Fig. 17. Signal flow graph of the extended state
space controller.
We find two forward paths between the output node
U2 and the input node U2ref
211123
1s
CBR
F
,
31221123
2s
CABR
F
.
(36)
We find seven first order loops
s
A
L11
1
,
22112
2s
AA
L
,
s
A
L22
3
,
s
RB
L112
4
222112
5s
RAB
L
,
211123
6s
CBR
L
,
31221123
7s
CABR
L
. (37)
The loops L3 and L4, the loops L1 and L3 and the
loops L3 and L6 do not touch each other, therefore
we have also three second-order loops
22211
31 s
AA
LL
,
211222
43 s
RBA
LL
33111222
63 s
RCBA
LL
. (38)
The loop L3 does not touch the forward path F1, but
all loops touch the second forward path F2. The
transfer function can now be calculated according to
634331
7654321
231
21
1
)(
)(
LLLLLL
LLLLLLL
FLF
sD
sU
. (39)
1112221221123
112112211123
221122112
2
1122211
31221112211312
2)(
)(
CBACABR
s
RBAACBR
RABAA
sRBAAs
CACAsCRB
sD
sU
. (40)
Now one has to choose the poles of the closed loop
system. This can be done arbitrarily, but one has to
have in mind that a too fast control leads to high
signal values which may destroy the circuit. It is
ingenious to choose a triple pole on the real axis, or
one real pole and a conjugate complex pole pair. A
triple pole at minus x leads to a desired denominator
of
32233 33)(3 xsxsxsxsDen
. (41)
We can also use one real pole and a conjugate
complex pole-pair. This leads to the denominator
xsxsxs
xsssDencomplex
222223
222
22
23
. (42)
The desired transfer function with zero steady state
error is therefore
3223
3
333
)( xsxsxs
x
sG
(43)
xsxsxs
x
Gcomp 222223
22
322
. (44)
4.2.1 Ideal plant with triple pole at minus x
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Felix A. Himmelstoss
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To achieve the controller values, the desired transfer
function (43) and the transfer function of the system
(40) must be compared. A11 and C11 are very small,
so they can be omitted (A11=0, C11=0). The transfer
function can be reduced to (45)
3121221
11222
221122112
2
11222
3
3121221
2)(
)(
RCBA
s
RBA
RABAA
sRBAs
RCBA
sD
sU
Comparison of the coefficients leads to
xRBA 3
11222
12
22
13
B
Ax
R
(46)
2
11222221122112 3xRBARABAA
2112
112222112
2
23
AB
RBAAAx
R
(47)
3
3121221 xRCBA
121221
3
3CBA
x
R
. (48)
Fig. 18. Simulation circuit: idealized plant of the
synchronous Buck converter with extended state
space controller with triple pole at -1000.
Fig. 18 shows the simulation program. The
converter is modelled with its state space
description.
Fig. 19. State space controlled Buck converter
(modelled as a non-linear system) with extended
state space controller with triple pole at -1000: input
voltage (blue), reference value (brown), output
voltage (green).
In Fig. 19 the start-up, a reference value step, and an
input voltage step for the Buck with extended state
space control is shown. The system is slower and
the reaction to the input voltage change is much
more pronounced.
4.2.2 Ideal controller with a complex pole pair
and a real pole
The complex pole pair is chosen at
j
and the
real pole at –x. To get the controller parameters the
transfer functions (44) and (45) have to be
compared. This leads to
xRBA
2
11222
12
22
12
B
Ax
R
(49)
xRBARABAA
2
22
11222221122112
2112
112222112
22
22
AB
RBAAAx
R
(50)
22
3121221
RCBA
121221
22
3CBA
R
. (51)
The second concept (4.2) is equivalent to (4.1) when
a simple I-controller is taken. The I-controller leads
to a slower system. Therefore, it is better to use
concept (4.1).
5 Conclusion
The design of state controllers for a second order
system without using linear algebra and matrix
operation was shown for a step-down DC/DC
converter. Now only some knowledge about the
signal flow graph and the equation of Mason is
necessary. The system has to be described in state-
space form which is very common and does not
need knowledge about matrix calculus. The design
equations for the controller follow immediately
from the signal flow graph. The Buck converter
used as an example is a nonlinear system, therefore
the simple state space controller works only
correctly for the chosen working point. Two
methods to overcome this limitation are shown. The
better way is to use an additional linear controller
e.g. a PI-controller. The second concept to include
the error as a third state variable is also possible and
is similar, when in the first concept an I-controller is
used instead of a PI-controller. The I-controller
leads however to slower control and is therefore not
so useful. The free simulation tool LTSpice helps in
the design and makes possible to check the system
with a circuit simulation of the plant.
References:
[1] G. Gonzalez, Microwave Transistor Amplifiers, Prentice-
Hall, 1984.
[2] F. A. Himmelstoss, and F. C. Zach, State Space Control
for Switched Mode Power Supplies, Proceedings of the
International Power Electronics Conference IPEC90-93,
Tokyo, April 2-6, 1990, pp.1157-1164.
[3] F. Zach, in German: Leistungselektronik, Springer, 6th
ed., 2022.
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Felix A. Himmelstoss
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Volume 21, 2022
[4] N. Mohan, T. Undeland and W. Robbins, Power
Electronics, Converters, Applications and Design, 3nd ed.
New York: W. P. John Wiley & Sons, 2003
[5] R.W. Erickson & D. Maksimovic, Fundamentals on
Power Electronics, Springer 2020.
[6] Y. Rozanov, S. Ryvkin, E. Chaplygin, P. Voronin, Power
Electronics Basics, CRC Press, 2016.
[7] F. A. Himmelstoss, Controller design of a Buck converter
with the help of LTSpice, International Asian Congress on
Contemporary Sciences-VI, 2022, pp.195-202, ISBN-978-
625-8323-27-6.
[8] F. A. Himmelstoss, Cascaded control of a Buck converter
designed and simulated with the help of LTSpice, 6th
International European Congress on Interdisciplinary
Scientific Research, Bucharest, pp. 1086-1096, ISBN:
978-625-8213-38-6.
[9] R. H. Bishop (editor), The Mechatronics Handbook, CRC
Press, 2008.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
Appendix
A1. Signal flow graphs - a short summery
The signals are nodes and over the branches which
connect the nodes the transfer coefficient or
function are written. We distinguish between
forward paths and loops. To get the value of a
forward path (between an input node and an output
node), one has to multiply the values of the
connecting branches. A loop starts at a node and
ends at the same node. The value of the loop is the
multiplication of the values of all branches which
form the loop. Second-order loops are formed by the
product of two loops which are not touching each
other. Mason’s equation enables us to find transfer
functions. The denominator is 1 minus the sum of
all first-order loops plus the sum of all second-order
loops minus etc. The numerator is the sum of all
forward paths multiplied by the denominator, where
all loops which are touching the forward path are
deleted.
A2. Circuit simulation
Fig. A shows the used simulation circuit for the
simple state space controller. The synchronous
rectified Buck converter is modelled with the active
switches (MOSFETs) S1 and S2, the inductor L1,
and the capacitor C1. The input voltage is built with
the voltage source V1 and the load by the resistor
R1. The electronic switches are controlled by the
voltage controlled voltage sources E1 and E2. E1 is
necessary, because S1 is a high-side switch, S2 is a
low-side switch, therefore no floating driver is
necessary, but E2 is used to realize the same
propagation delay. The dead-time of the switches is
realized with the capacitors C2, C3, the resistors R2
and R3, the diodes D2 and D3, and the AND-gates
A1 and A2 (which can also be used for connecting
an enable signal). S1 is controlled by the pwm
output of the comparator U1, and the second switch
S2 by the inverted output pwm_q. The pwm-
modulation is achieved by the saw-teeth generator
V2 and the output of the arbitrary voltage source
B1, which calculates the simple state space
controller. With the voltage source V5 the desired
value Uref is produced. The comparator U1 is
double side supplied with the voltage sources V3
and V4. Only 5 V is allowed for this device. The
voltage source V6 with the value zero in series to
the inductor L1, shows the current measurement
device. This current controls the current-controlled
voltage source H1 which is used in the state space
controller B1.
Fig. A. Circuit orientated simulation of the synchronous Buck converter with simple state space controller.
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.46
Felix A. Himmelstoss
E-ISSN: 2224-2678
429
Volume 21, 2022