The LTI state-space equations of a system generally applied in
systems and control theory
dxt
( )
dt=Ax t
( )
+bu t
( )
y t
( )
=cTxt
( )
+dcu t
( )
(1)
Here
u and
y
are the input and output signals of the process,
respectively, and
x
is the state vector. The parameter matrices
of the system are
A,b,c
T
,d
. Since this paper mainly treats
SISO systems, in n-order case, matrix A means a
n×n
( )
square matrix, which is the so-called state matrix,
b
is a
column vector of
n×1
( )
size,
c
T
is a row vector of
1×n
( )
size, and
d
c
is scalar.
The classical model of the dynamic LTI processes, the transfer
function
P s
( )
is defined by the ratio of the
LAPLACE transforms of the output and input signals, which
can be easily derived from the state equation (1)
P s
( )
=
Y s
( )
U s
( )
=c
T
sI−A
( )
−1
b+d
c
=
Bs
( )
As
( )
(2)
where
As
( )
=det sI−A
( )
=s
n
+a
1
s
n−1
+…+a
n
Bs
( )
=b
o
s
m
+b
1
s
m−1
+…+b
m
(3)
The roots of the equation
As
( )
=0
are called poles; the roots
of
Bs
( )
=0
are called zeros. A continuous-time (CT) linear
process is stable, if all roots of the polynomial
As
( )
are
located on the left-hand side of the complex plane. Concerning
the order of the polynomials
As
( )
and
Bs
( )
it should be
noted that the number of the state variables is n,
m
is the
order of the polynomial
Bs
( )
, and the relation
m≤n
exists.
The difference between the order of the numerator and
denominator
p
T
=n−m
is called pole access. If
pT>0
then
P s
( )
is strictly proper, if
pT=0
then the transfer function is
proper. In the practice arbitrary relation
0≤p
T
≤n
might
occur.
Figure 1. Linear regulator with state feedback
Control loops with state feedback
It was shown formerly how processes are represented in state-
space. In many cases this kind of description is available only
and the transfer function of the controlled system is
unavailable. This partly explains why control design
methodology directly based on state-space description has
been evolved. Let us consider the state-space representation of
an LTI process to be controlled such as
dx
dt
=
x=Ax +bu
y=cTx
(4)
which corresponds to (1) for the case of
dc=0
. This does not
violate the generality, because it is very rare for the model to
Evaluation of All Existing Controller Design Methods
LASZLO KEVICZKY, CSILLA BÁNYÁSZ
Institute for Computer Science and Control Budapest, HUNGARY
Abstract — All existing basic regulator design methods are summarized in this paper and compared concerning
their usability and formal algebraic formulations. It is systematically proved that the best usable method is the
YOULA-parameterization based regulator design introduced by the authors.
Keywords — regulator, design, performance, parameterization
Received: June 26, 2021. Revised: March 21, 2022. Accepted: April 23, 2022. Published: May 18, 2022.
1. Introduction
2. Basic Regulator Design Methods
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contain a proportional channel directly affecting the output.
The block scheme of (4) and the classical state-feedback is
shown in Fig. 1, where the thick lines present vector variables
and
r
denotes the reference signal.
In the closed-loop the state vector is fed back with the linear
proportional vector
k
T
according to the expression below
u=k
r
r−k
T
x
(5)
Based on Fig. 1 the state equation of the complete closed
system can be easily written as
dx
dt=A−bk
T
( )
x+k
r
br
y=c
T
x
(6)
i.e., with the state feedback the dynamics represented by the
original system matrix
A
is modified by the dyadic product
bk
T
to
A−bk T
( )
.
The transfer function of the closed-loop control is
c
T
ry
s
( )
=
Y s
( )
R s
( )
=c
T
sI−A+bk
T
( )
−1
bk
r
=
=
c
T
sI−A
( )
−1
bk
r
1+k
T
sI−A
( )
−1
b
=
k
r
1+k
T
sI−A
( )
−1
b
P s
( )
=
=
k
r
Bs
( )
As
( )
+k
T
Ψs
( )
b
(7)
which derives from the comparison of equations valid for the
LAPLACE transforms,
U s
( )
=k
r
R s
( )
−k
T
Xs
( )
(see (6)) and
Y s
( )
=c
T
Xs
( )
(see (4)) using the matrix inversion lemma.
Note that the state feedback leaves the zeros of the process
untouched and only the poles of the closed-loop system can be
designed by
k
T
.
The so-called calibration factor
k
r
is introduced in order to
make the gain of
T
ry
equal to unity (
T
ry
0
( )
=1
). The open
loop is obviously not of type one, so it cannot provide zero
error and unity static transfer gain. It can be ensured only if
the condition
kr=−1
cTA−bkT
( )
−1
b
=kTA−1b−1
cTA−1b
(8)
is fulfilled. The above special control loop is called state
feedback.
Pole placement by state feedback
The most natural design method of state feedback is the so-
called pole placement. In this case the feedback vector
k
T
needs to be chosen to make the characteristic equation of the
closed-loop equal to the prescribed, so-called design
polynomial
Rs
( )
, i.e.,
Rs
( )
=s
n
+r
1
s
n−1
+…+r
n−1
s+r
n
=
=det sI−A+bk
T
( )
=As
( )
+k
T
Ψs
( )
b
(9)
The solution always exists if the process is controllable. (It is
reasonable if the order of
R
is equal to that of
A
.) In the
exceptional case when the transfer function of the controlled
system is known, the canonical state equations can be directly
written. Based on the controllable canonical form the system
matrices are
A
c
=
−a
1
−a
2
…−a
n−1
−a
n
1 0 …0 0
0 1 0 0
0 0 0 1 0
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
cc
T=b
1,b2,…,bn
⎡
⎣⎤
⎦
;
bc=1,0,…,0
⎡
⎣⎤
⎦
T
(10)
Considering the special forms of
A
c
and
b
c
, it can be seen
that the design equation (9) results in
kT=kc
T=r
1−a1,r2−a2,…,rn−an
⎡
⎣⎤
⎦
(11)
ensuring the characteristic equation (
Rs
( )
=0
), i.e., the
prescribed poles. The choice of the calibration factor can be
determined by simple calculation
kr=
an+rn−an
( )
bn
=
rn
bn
(12)
Based on equations (7), (8) and (9) it can be seen that in the
case of state feedback pole placement the closed-loop transfer
function results in
Try s
( )
=
krBs
( )
Rs
( )
(13)
The most common case of state feedback is when not the
transfer function but the state-space form of the control system
is given. It has to be observed that all controllable systems can
be described in a controllable canonical form by using the
transformation matrix
Tc=Mc
cMc
( )
−1
. This linear
transformation also refers to the feedback vector
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k
T
=k
c
T
T
c
=k
c
T
M
c
c
M
c
-1
k
T
=b
c
T
M
c
-1
RA
( )
=0,0,…,1
⎡
⎣⎤
⎦M
c
-1
RA
( )
(14)
The design relating to the controllable canonical form (10),
together with the linear transformation relationship
corresponding to the first row of the non-controllable form
(14), is known as the BASS-GURA algorithm. The algorithm in
the second row of (14) is called ACKERMANN method after its
elaborator.
(a)
(b)
(c)
Figure 2. Equivalent schemes to the state feedback design
using transfer functions and polynomials
In the BASS-GURA algorithm, the inverse of the controllability
matrix
Mc
needs to be determined by the general system
matrices
A
and
b
on the one hand and the controllability
matrix
M
c
c
of the controllable canonical form, on the other.
Since this latter term depends only on the coefficients
a
i
in
the denominator of the process transfer function, the
denominator needs to be calculated:
As
( )
=det sI−A
( )
. Since
0,0,…,1
⎡
⎣⎤
⎦M
c
-1
is the last row of the inverse of the
controllability matrix, and
RA
( )
also need to be calculated;
the ACKERMANN method does not need the calculation of
As
( )
.
It is worth mentioning that the state feedback formally
corresponds to a conventional PD control and therefore over-
actuating peaks are expected at the input of the process
because the pole placement tries to make the process faster. In
practice, however, the actuator usually limits the amplitude of
the peaks, which needs to be taken into account during the
design of the poles of the characteristic polynomial
Rs
( )
.
It can be clearly seen that state feedback formally corresponds
to a serial compensation
R
s
=k
r
As
( )
Rs
( )
(Fig. 2a). The
real operation and effect of the state feedback can be easily
understood by the equivalent block schemes using the transfer
functions shown in Fig. 2. The “regulator”
R
f
s
( )
of the
closed-loop is in the feedback line (Fig. 2b). The transfer
function of the closed-loop is
Try s
( )
=
krBs
( )
Rs
( )
=
krBs
( )
As
( )
+Bs
( )
=
krP s
( )
1+Kks
( )
P s
( )
=
=
krAs
( )
Rs
( )
Bs
( )
As
( )
=krRss
( )
P s
( )
(15)
where
Rf=Kks
( )
=
Ks
( )
Bs
( )
=
Rs
( )
−As
( )
Bs
( )
=
=
kTsI−A
( )
−1
b
cTsI−A
( )
−1
b
(16)
and the calibration factor is
kr=kTA−1b−1
cTA−1b
=
1+Kk0
( )
P0
( )
P0
( )
(17)
Given the block schemes of Fig. 2 it can be stated that the
state feedback also stabilizes the unstable terms, since due to
the effect of the polynomial
Ks
( )
=Rs
( )
−As
( )
there is a pole
placement for any process, so with the stable
Rs
( )
the
stabilization is fulfilled. The feedback polynomial
Ks
( )
formally corresponds to
k
T
. The fact that the numerator
Bs
( )
of the process is present in the denominator of
K
k
s
( )
needs
special consideration. The regulator can be applied only for
minimum phase (inverse stable) processes, where the roots of
Bs
( )
are stable. As a consequence of this special character of
the state feedback, however, here
Bs
( )
is not substituted by
its model
ˆ
Bs
( )
, but the method itself realizes the exact
1Bs
( )
.
Pole placement with pole cancellation
Consider the closed control system shown in Fig. 3, where the
regulator
C=A X
is used to place the poles of the closed
control system according to the characteristic equation
R=0
,
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(
R
is the design polynomial) by the cancellation of the
process poles. To do this,
X
needs to be expressed by the
equation
R=X+B
. The complementary sensitivity function
of the closed-loop is
T=
A
X
B
A
1+A
X
B
A
=AB
AX +AB
=B
X+B
=B
R
(18)
The regulator is
C=A
X
=A
R−B
=
B
R
1−B
R
A
B
=
Rr
1−Rr
P−1
(19)
and actually corresponds to an ideal YOULA regulator with
reference model
R
r
=R
n
=B R
. This regulator places the
poles in
R
and leaves the zeros in
B
untouched, if they are
inverse stable.
Figure 3. Pole canceling regulator
Pole placement with feedback regulator
An other solution when the regulator is put in the feedback is
shown in Fig. 4.
Figure 4. Regulator in the feedback
Now the task is again to place the poles of the closed system
according to the characteristic equation
R=0
(
R
is the
design polynomial). To do this,
K
needs to be determined
from the equation
R=K+A
. The complementary sensitivity
function of the closed system is
T=
B
A
1+K
B
B
A
=B
A+K
=B
R
(20)
and thus this regulator places the poles in
R
and leaves the
zeros in
B
untouched, if they are inverse stable.
The characteristic equation of the closed system has the form
R=0
and it doesnot depend on the unstable property of the
process.
The block diagram in Fig. 5. can be redrawn as Fig. 2c. (The
state feedback methods are discussed in detail in Section 2.1,
and the same control principle is represented in Fig. 2c among
the schemes showing the equivalent transfer function
representations for state feedback.)
Figure 5. The regulator feeds back the internal signal of the
process
Pole placement with characteristic polynomial design
The characteristic polynomial
R
of the closed-loop control
can be directly designed by algebraic methods. In Fig. 6 the
regulator
C=Y X
is the quotient of two polynomials. Under
certain conditions, the DE
AX +BY=R
can be solved for
X
and
Y
. Thus from the characteristic equation
R=0
the
regulator can be directly determined.
Figure 6. Direct control design on the basis of the
characteristic polynomial
The complementary sensitivity function of the closed system
is
T=
Y
X
B
A
1+Y
X
B
A
=BY
AX +BY
=BY
R
(21)
and thus this regulator also places the poles in
R
and leaves
the zeros in
B
untouched, but in the nominator
Y
appears,
which depends on the desired properties and also on DE.
Thus the characteristic equation of the closed system has the
form
R=0
and it does not depend on the unstable character
of the process.
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Regulators based on YOULA parameterization
The YOULA parameter, as a matter of fact, is a stable (by
definition), regular transfer function
Q s
( )
=C s
( )
1+C s
( )
P s
( )
or shortly
Q=C
1+CP
(22)
where
C s
( )
is a stabilizing regulator, and
P s
( )
is the transfer
function of the stable process.
It follows from the definition of the YOULA parameter that the
structure of the realizable and stabilizing regulator in the
YOULA-parameterized (sometimes called
Q
-parameterized)
control loop is fixed:
C s
( )
=Q s
( )
1−Q s
( )
P s
( )
or shortly
C=Q
1−QP
(23)
The sensitivity and complementary sensitivity functions linear
in
Q
of the closed control systems were defined by (25). It is
interesting to observe that the YP regulator of (23) can be
realized by a simple control loop with positive feedback as
shown in Fig. 7.
Figure 7. Realization of a YP regulator
A YOULA-parameterized (YP) closed-loop is shown in Fig. 8.
Figure 8. YOULA-parametrized closed-loop
The All-Realizable-Stabilizing (ARS) regulator has the form of
(23).
The closed-loop transfer function or Complementary
Sensitivity Function (CFS)
T=CP
1+CP
=QP
(24)
which is linear in the YOULA parameter
Q
. It is well known
that the YP regulator corresponds to the classical IMC
(Internal Model Control) structure.
The relationships between the most important signals of the
closed system can be obtained with simple calculations
u=Qr −Qy
n
e=1−QP
( )
r−1−QP
( )
y
n
=Sr −Sy
n
y=QPr +1−QP
( )
y
n
=Tr +Sy
n
(25)
The effect of
r
and
yn
on
u
and
e
is completely
symmetrical (not considering the sign). Thus the input of the
process depends only on the external signals and
Q s
( )
.
From the equation (24) it can be seen that the
YOULA parameterization has the transfer function
QPr
concerning the reference signal tracking. If the
KB parameterization is introduced on the figure of Fig. 8, then
the YOULA parameterization can be simply extended for
TDOF control systems. To do this, let us simply apply a
parameter
Qr
for the design of the tracking properties, and
connect it in serial to the KB-parameterized loop, so the block
diagram of Fig. 9 is obtained.
Figure 9. Two-degree-of-freedom version of the YP control
loop
The overall transfer characteristics for this system are
u=Q
r
y
r
−Qy
n
e=1−Q
r
P
( )
y
r
−1−QP
( )
y
n
=1−T
r
( )
y
r
−Sy
n
(26)
y=Q
r
Py
r
+1−QP
( )
y
n
=T
r
y
r
+1−T
( )
y
n
=T
r
y
r
+Sy
n
where the tracking properties can be designed by choosing
Qr
in
T
r
=Q
r
P
, and the noise rejection properties by choosing
Q
in
T=QP
. These two properties can be handled
separately. The reference signal of the whole system is
denoted by
y
r
. The conditions for
Qr
are the same as for
Q
.
The meaning of
T
r
is analogous to the meaning of the
complementary sensitivity function
T
of the one-degree-of-
freedom control loop for tracking.
Control loops with state feedback
3. Comparision of the
Previously Discussed Design Methods
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The most important advantage of the state feedback regulator,
is that the calculation of the feedback vector is very simple.
The most important disadvantage is that the internal state
variables, necessary for the feed-back are usually not available
in the practical tasks. This is why the observer topology is
generally necessary to this method. Unfortunately this
topology is not so simply to compute. Another important
disadvantage is that this regulator assigns the pole of the
closed-loop system, unfortunately it leaves the numerator of
the process untouched in
T
. It is important to know that from
the methods discussed in this paper this is the only method
which is applicable for unstable processes.
Pole placement with pole cancellation
The most important advantage of this method that it is very
simple to calculate the regulator. The disadvantage is that this
regulator assigns the pole of the closed-loop system,
unfortunately it also leaves the numerator of the process
untouched in
T
.
Pole placement with feedback regulator
This method practically can be evaluated on the similar way as
the previous method. Unfortunately the most important
disadvantage is that in a practical task it is very rare that the
regulator is in the feedback line.
Pole placement with characteristic polynomial design
This method is a little bit more complex than the pole
cancellation method, because the calculation of the regulator
needs the solution of a DE. The disadvantage is that this
regulator assigns the pole of the closed-loop system,
unfortunately it also leaves the numerator of the process
untouched in
T
and puts another polynomial in the numerator
of
T
. This polynomial comes from the solution of the DE, so
it is not easy to design.
Regulators based on YOULA parameterization
This method is the simplest, because it needs only basic
polynomial operations to calculate the regulator. A further
advantage is that the result of the design is the best reachable
T
even for invariant process zeros, too.
Except the state feedback regulator the other methods are
applicable only for stable processes.
Let us assume the transfer function of the process in the
following factorized form
P s
( )
=P
+
s
( )
P s
( )
−
=P
+
s
( )
P
−
s
( )
e
−sTd
(27)
or shortly
P=P
+P
−=P
+P
−e−sTd
(28)
where
P
+
is stable, and its inverse is also stable (Inverse
Stable: IS) and realizable (ISR). The inverse of
P
−
is unstable
(Inverse Unstable: IU) and not realizable (Non Realizable:
NR), i.e., (IUNR).
P
−
is inverse unstable (IU). Here, in
general, the inverse of the dead-time part
e−sTd
is not
realizable, because it would be an ideal predictor.
In polynomial form a delay free process is given by
P s
( )
=
Bs
( )
As
( )
=
B
+
s
( )
B
−
s
( )
As
( )
(29)
where
B
+
s
( )
and
B
−
s
( )
contain the inverse stable and
inverse unstable zeros, respectively.
If the reference model, formulating our design goal is
R
n
s
( )
=
B
n
s
( )
A
n
s
( )
(30)
then the optimal YOULA parameter is
Q s
( )
=Rns
( )
B+
−1s
( )
(31)
Using this parameterization the optimal YOULA regulator can
be calculated as
C s
( )
=Q s
( )
1−Q s
( )
P s
( )
=R
n
s
( )
B
+
−1
s
( )
1−R
n
s
( )
B
+
−1
s
( )
B
+
s
( )
B
−
s
( )
=
=
B
n
s
( )
As
( )
B
+
s
( )
A
n
s
( )
As
( )
−B
n
s
( )
B
−
s
( )
⎡
⎣⎤
⎦
(32)
The transfer function of the closed-loop system is
T s
( )
=R
n
s
( )
B
−
s
( )
=
B
n
s
( )
A
n
s
( )
B
−
s
( )
(33)
which is the best reachable result for the case of inverse
unstable zeros. This result explains the name: “uncancellable”
for the inverse unstable factors of the numerator of the
process.
For the two-degree-of-freedom version of the YOULA
regulator (see Fig. 9) an additional reference model
R
r
s
( )
=
B
r
s
( )
A
r
s
( )
(34)
must also be calculated.
4. Computation of the
Optimal Youla Regulator
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It can be well seen in this section that the computation of the
YOULA regulator requires only very simple polynomial
operations (additions and multiplications).
Example 4.1. Let the CT process be given by a non-minimum
phase transfer function
P s
( )
=
1+sτ
1
( )
1−sτ
2
( )
1+sT
1
( )
1+sT
2
( )
1+sT
3
( )
(35)
where
T
1
=10sec
;
T
2
=5sec
;
T
3
=2sec
;
τ
1
=6sec
and
τ
2
=4sec
, where
B
+
=1+sτ
1
( )
and
B
−
=1−sτ
2
( )
.
The selected reference model is
Rns
( )
=
Bns
( )
Ans
( )
=
1+sτn1
1+sTn1
=1
1+sTn1
(36)
where
Tn1 =5sec
;
τ
n1
=0
.
The optimal YOULA regulator can be calculated as
C s
( )
=
B
n
s
( )
As
( )
B
+
s
( )
A
n
s
( )
As
( )
−B
n
s
( )
B
−
s
( )
⎡
⎣⎤
⎦
=(37)
1+sT
1
( )
1+sT
2
( )
1+sT
3
( )
1+sτ
1
( )
1+sT
n1
( )
1+sT
1
( )
1+sT
2
( )
1+sT
3
( )
−1−sτ
2
( )
⎡
⎣⎤
⎦
Using the numerical values the regulator is
C s
( )
=1+17s+80s
2
+100s
3
s1+6s
( )
1+2.273s+4.454s
2
( )
(38)
The overall transfer function of the closed-loop system is
T s
( )
=R
n
s
( )
B
−
s
( )
=
1−sτ
2
1+sT
n1
=1−4s
1+5s
(39)
Because usually the reference model has unity gain, i.e.
B
n
0
( )
=A
n
0
( )
(40)
it follows, that
T0
( )
=1
has also unity gain.
The usual normalization of the process polynomial means that
A0
( )
=1
and
B
−
0
( )
=1
(while
B
+
0
( )
≠1
) it can be easily
checked that the YOULA regulator is always an integrating
regulator for (40).
Example 4.2. Investigate now a discrete-time (DT) case, when
the pulse transfer function of the process is a second order
system
P z
( )
=
−0.32 z−1.25
( )
z−0.8
( )
z−0.6
( )
(41)
and the reference model is
Rnz
( )
=0.6
z−0.4
(42)
The optimal YOULA regulator can be computed now as
C z
( )
=
−0.624 1 −2.917 z+2.083z
2
( )
1+0.027z−1.248z
2
+0.693z
3
(43)
It was shown that the YOULA regulator design is a very simple
procedure, which is applicable for all kind of (minimum or
non minimum phase) CT and DT processes. The computation
of the regulator is very simple, requires only polynomial
operations.
For reasonable design goal this design results in an integrating
regulator.
This regulator ensures the theoretical best reachable closed-
loop property of the control system.
[1] Athens, M. and F.L.Falb (1966). Optimal control; an
introduction to the theory and its applications. McGraw-
Hill.
[2] Åström, K.J. and B. Wittenmark (1984). Computer
Controlled Systems. Prentice-Hall, p. 430.
[3] Goodwin, G.C., Graebe S.F. and Salgado M.E. (2001).
Control System Design. Prentice-Hall, p. 908.
[4] Horowitz, I.M. (1963). Synthesis of Feedback Systems,
Academic Press, New York.
[5] Keviczky, L. (1995). Combined identification and control:
another way. (Invited plenary paper.) 5th IFAC Symp. on
Adaptive Control and Signal Processing, ACASP'95, 13-
30, Budapest, Hungary.
[6] Keviczky, L. and Cs. Bányász (1999). Optimality of two-
degree of freedom controllers in
H
2
- and
H
∞
-norm
space, their robustness and minimal sensitivity. 14th IFAC
World Congress, F, 331-336, Beijing, PRC.
[7] Keviczky, L. and Cs. Bányász (2015). Two-Degree-of-
Freedom Control Systems (The Youla Parameterization
Approach), Elsevier, Academic Press, p. 512.
[8] Keviczky, L., R. Bars, J. Hetthéssy and Cs. Bányász
(2018). Control Engineering. Springer.
[9] Keviczky, L., R. Bars, J. Hetthéssy and Cs. Bányász
(2018). Control Engineering: MATLAB Exercises,
Springer.
[10]Maciejowski, J.M. (1989). Multivariable Feedback
Design, Addison Wesley, p. 424.
[11]Youla, D.C., Bongiorno, J.J. and C. N. Lu (1974). Single-
5. Conclusions
References
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.11
Laszlo Keviczky, Csilla Bányász
E-ISSN: 2224-2678
102
Volume 21, 2022
loop feedback stabilization of linear multivariable
dynamical plants, Automatica, Vol. 10, 2, pp. 159-173.
[12]Youla, D.C. and J.J. Bongiorno, Jr. (1985). A feedback
theory of two-degree-of-freedom optimal Wiener-Hopf
design," IEEE Trans. Auto. Control, vol. AC-30, No. 7,
pp. 652-665.
P.S.: Professor Youla died two weeks ago. He was 95 years
old. So our paper can be considered a good memorial of the
two famous scientests: Athens and Youla.
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
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.11
Laszlo Keviczky, Csilla Bányász
E-ISSN: 2224-2678
103
Volume 21, 2022
