=Paper=
{{Paper
|id=Vol-2608/paper8
|storemode=property
|title=Parametric synthesis of a dynamic object control system with nonlinear characteristics
|pdfUrl=https://ceur-ws.org/Vol-2608/paper8.pdf
|volume=Vol-2608
|authors=Oleg Nikonov,Igor Kyrychenko,Vladyslav Shuliakov,Fastovec Valentyna
|dblpUrl=https://dblp.org/rec/conf/cmis/NikonovKSV20
}}
==Parametric synthesis of a dynamic object control system with nonlinear characteristics==
https://ceur-ws.org/Vol-2608/paper8.pdf
Parametric synthesis of a dynamic object control system
with nonlinear characteristics
Oleg Nikonov[0000-0002-8878-4318], Igor Kyrychenko [0000-0002-2128-3500],
Vladyslav Shuliakov[0000-0002-6654-6749], Fastovec Valentyna[0000-0002-8428-747X]
Kharkiv National Automobile and Highway University,
Str. Yaroslava Mudrogo 25, Kharkiv, 61002, Ukraine
nikonov.oj@gmail.com, igk160450@gmail.com, jason07@ukr.net,
tinafast2013@gmail.com
Abstract. The aim of the research is to solve the problem of using the method
of factor experiment for the problem of parametric synthesis of a closed dy-
namic object control system. The method of factor experiment allows, on the
basis of modeling the behavior of the system in a random external environment,
to select its parameters that satisfy the requirements of the minimum of the re-
gression functional. A quality functional that realize the requirements for a dy-
namic object control system and reduces the computational resource in the pa-
rametric synthesis of the system was proposed. The behavior of a closed control
system was simulated considering the random external perturbations acting on
the control object. The introduction of a nonlinear link with variable amplifica-
tion factor widens the stability area of a closed dynamic object control system.
The parametric synthesis of the closed dynamic object control system consider-
ing nonlinear characteristics on the basis of the factor experiment method was
considered.
Keywords: Car, Parametric Synthesis, Quality Functional, Transient Process,
Dynamic Object.
1 Introduction
Recently, the simulation methods become widespread use, in particular the method of
factor experiment. This method allows, on the basis of modeling the behavior of the
system in a random external environment, to select its parameters that satisfy the re-
quirements of the minimum of the regression functional. Additive quality functional
of the closed system can be used as a regression function. In this case, the behavior of
a closed control system is simulated into account of the random external perturbations
acting on the control object.
Copyright © 2020 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0).
2 Formulation of the problem
The aim of the research is to solve the problem of applying the method of factor ex-
periment to the problem of parametric synthesis of the dynamic object control system.
Due to continuous external disturbance Med t , the position of the dynamic object
is constantly changing. Changing the position of a dynamic object Med t is a ran-
dom function. In this situation, random functions are also functions c t and
c t (Fig. 1). The accuracy of stabilizing a dynamic object is higher, the smaller
the area under the dynamic process curves.
3 Literature review
The methods of classical automatic control theory do not work for high-order mathe-
matical models. If the order of the differential equations that make up the mathemati-
cal model of the control object is higher than five, then the problem of parametric
synthesis can be solved only by the methods of modern control theory, in particular,
the method of state-space, the methods of the theory of analytic design of optimal
regulators [1-3], minimax methods, methods of Liapunov functions, methods of simu-
lation modeling.
Minimax methods and methods of Liapunov functions, which were applied by E.E.
Alexandrov to solve the problem of parametric synthesis of the control system give
satisfactory results [4-6], because they provide for the use of information in the con-
trol algorithm only about those components of the vector of the state of the control
object, the measurement of which is not difficult. But in the process of parametric
synthesis of a digital control system, these methods involve the transition from differ-
ential equations of the mathematical model of the control object to equations, which
introduces a certain error in the synthesis process.
In addition, the high order of the differential equations describing the perturbed
motion of the control object results in considerable computational difficulties associ-
ated with limited computer memory when applying the minimax method and the
Liapunov function method. However, if the objects contain non-analytic nonlineari-
ties, then the above synthesis methods are inactive. Therefore, in recent years, the
simulation methods become widespread use, in particular the method of factor ex-
periment [7, 8].
4 Research Methodology
Therefore, the quantitative accuracy of the stabilization of the dynamic object relative
to the base position can be estimated by the following functional
T 2
Ic M c t dt (1)
0
where M – is the symbol of mathematical expectation.
The quality of stabilization processes is estimated not only by the changing of posi-
tion of the dynamic object, but also by the angular velocity of motion of the dynamic
object relative to its base position [9, 10]. The quantitative characteristic of this
movement may be functional
T 2
Iс M с t dt (2)
0
It was an ideal case, when closed dynamic object control system solutions, both func-
tionalities (1) and (2) would be minimized. But this is not possible and the minimums
of functionals (1) and (2) correspond to different values of the parameters of the con-
troller k and k . Therefore, it is advisable to choose the parameters k and k in
condition of a minimum of additive functionality
T
I 12 Iс 22 Iс M 12с2 (t ) 22с2 (t ) dt , (3)
0
where 1 and 2 - are the weighting factors to be selected.
But, many experiments [11] proved that the obtained values of the varied parame-
ters, which give a minimum of functionals (1) and (2) are almost not different (within
5%). Considering this fact, as well as the fact that the output signal from the control
unit is limited and does not allow the dynamic object to pass excessive speed, it is
advisable to use functional (1) or its modifications for this particular system.
In general, a functional that evaluates the quality of the stochastic system looks like
T
I M f e t , x t , y t , t dt , (4)
0
where f – is the function of error, input and output signals, as well as time. Using
different combinations of system and time variables, different quality estimates can be
obtained.
To reduce the contribution of a significant initial error and to account for a future
error, it is more appropriate to use the form functionality
T
I M t с t dt , (5)
t1
t 0 functional (5) is transformed into I t t dt .
T
and when Med с
t1
Fig. 1. Dynamic processes in a closed dynamic object control system
*
Choosing as variables parameters the coefficients k and k , and as the optimization
parameters, the functional (5), we use the factor experiment theory [7, 8] to find the
values of the variable parameters of the controller that give the least functional (5) for
the dynamic object control system.
The points a, b correspond to the minimum of the objective function (5) without
and taking into account WН1 A respectively. Parameters for (5): t1 0.25 s,
Н1
T 5 s. For the case without taking into account W A – I 0.005958 ,
Н1 Н1
k 199.5 , k 16.3 . For the case considered W A – I 8.186339 ,
* *
Н1 Н1
k 323.2 , k 21.0 .
* *
In Fig. 2 the transients processes of a closed dynamic object control system with
* *
the obtained values of the variable parameters of the controller k and k for the case
without taking into account WН1 A are presented. As you can see, transients
Н1
processes are smooth without significant fluctuations. The amount of overshoot was
0.5 %.
a) b)
c) d)
Fig. 2. Transients processes of a closed dynamic object control system when k 199.5 ,
*
k* 16.3 : control voltage U y t (a), dynamic object speed с t (b) and for the value
of moving a dynamic object 100mm с t (c) and the objective function I t (d)
In Fig. 3 the transients processes of a closed dynamic object control system with the
* *
obtained values of the variable parameters of the controller k and k in the case of
the given WН1 A are presented. As you can see, transients processes are smooth
Н1
without significant fluctuations. The amount of overshoot was 0.1 %.
A structural diagram of a dynamic object control system, in addition to a nonlinear
A and characteristic, of the limited (saturation)
link with a transfer function WН1 Н1
zone type also contains a nonlinear link with a transfer function W A and Н2 Н2
characteristic with variable amplification factor.
In this case, this nonlinear link is introduced into the structural diagram in order to
improve the quality of transients processes and increase the reliability of the system as
a whole by extending the stability area.
a) b)
c) d)
Fig. 3. Transients processes of a closed dynamic object control system when k 323.2 ,
*
k* 21.0 : control voltage U y t (a), dynamic object speed c t (b) and for the value
of moving a dynamic object 100mm c t (c) and the objective function I t (d)
From the structural diagram we find the transfer function of the closed circuit with the
nonlinear link with the transfer function WН 2 A Н2
W3НН s
W3 Н s WН 2 AН 2 . (6)
k gs
1 W3 Н s WН 2 AН 2 k
T21s 2 T 2 s 1
s
We find the transfer function of the whole open system
W4 HH s
k W3 Н s WН 2 AН 2
k gs
1 W3 Н s WН 2 AН 2 k
T21s 2 T 2 s 1
s
and a closed system
W4 НН s
W4НН . (7)
1 W4 НН s
We construct the stability domain of a closed dynamic object control system in the
plane of variable parameters of the controller k and k , taking into account a
nonlinear link such as a zone of limitation (saturation) and a nonlinear link with a
variable amplification factor.
Ty A3 T21 s7 Ty A3 T 2 Ty A2 T21 A3 T21 s6
A2 T21 Ty A2 T 2 Ty A3 A3 T 2 Ty A1 T21 s5
Ty T21 A2 T 2 Ty A2 A3 Ty A1 T 2 A1 T21 s4
T21 A1 T 2 Ty A1 Ty T 2 A2 s3 (8)
T 2 A1 Ty k k WН1 AН1 W A k k T s
Н2 Н2 у 2
2
1
2
1 k W A W A k k k T k k s
Н1 Н1 Н2 Н2 у 2 2 2 gs
k k WН1 AН1 W A k k =0.
Н2 Н2 у 2
From the transfer function of the closed system (7) we write the characteristic equa-
tion (8). In characteristic equation (8) we make a substitution s j , select the real
and imaginary parts and equal them to zero. Obtained algebraic equations we solve
referring to the parameters k and k :
k
T T A T T A A T A T A T
y 1
2
2 2 y 2 3 y 1 2 1
2
1
4
k W A W A k k T 1
2 2
Н1 Н1 Н2 Н2 у 2 1
(9)
T A T T A T A T T A T ;
y 3 2 y 2
2
1 3
2
1
6
2 1 y
2
k W A W A k k T 1
Н1 Н1 Н2 Н2 у 2
2 2
1
k
T A T T A T T A T A T
1
2
1 2 y 1 y 2 2
2
y 3
2
1
6
k W A W A k k k
Н1 Н1 Н2 Н2 у 2 gs
A T T A T T A A T T A T
2
2
1 y 2 2 y 3 3 2 y 1
2
1
4
(10)
k W A W A k k k Н1 Н1 Н2 Н2 у 2 gs
k k WН1 AН1 W A k k T 1 . Н2 Н2 у 2 2
k W A W A k k k
Н1 Н1 Н2 Н2 у 2 gs
In Fig. 4 and 5 are shown the limits of stability area of the closed dynamic object
control systems in the plane of variable parameters of the controller k and k taking
into account a nonlinear variable with amplification factor, constructed using relations
(9) and (10), where a, b, c – are the points of minimum of the objective function (5)
without and with considering WН1 A , W A respectively.
Н1 Н2 Н2
Fig. 4. Stability area of a closed dynamic object control system considering the nonlinear link
with variable amplification factor at AН
2
bН 2 2 : 1 – k Н 2 k Н 3 ; 2 – kН2 0.8 kН3 ;
3 – kН
2
0.7 kН3 ; 4 – kН2 0.6 kН3 ; 5 – kН2 0.5 kН3
For the case with WН1 A and W A based on the method of factor ex-
Н1 Н2 Н2
periment we obtain: I 8.190428 , k 607.1 , k 49.9 . Parameters for (5):
* *
t1 0.25 s, T 5 s.
Fig. 5. Stability area of a closed dynamic object control system considering the nonlinear link
with variable amplification factor at kН2 0.5 kН3 : 1 – AН 2 bН 2 100 ; 2 –
AН2 bН2 10 ; 3 – AН2 bН2 5 ; 4 – AН2 bН2 3 ; 5 – AН 2 bН 2 2
As you can see from Fig. 4, the introduction of a nonlinear link [12 - 14] with vari-
able amplification factor widens the stability area of a closed dynamic object control
system (with k Н 2 k Н 3 (curve 1), the stability area coincides completely with the
stability of the linear system).
The optimal point c in this case moves to the area of high coefficients k and k .
The above helps to improve the reliability and accuracy of a closed dynamic object
control system.
As you can see from Fig. 5, when AН 2 bН 2 100 the equation
WН2 AН 2 kН3 , and the stability area approaches to the linear.
In Fig. 6 the transients processes of a closed dynamic object control system with
the obtained values of the variable parameters of the controller k* and k* consider-
ing WН1 A and W A are presented. The amount of overshoot was 0.1%.
Н1 Н2 Н2
a) b)
c) d)
Fig. 6. Transients processes of a closed dynamic object control system when k 607.1 ,
*
k* 49.9 : control voltage U y t (a), dynamic object speed c t (b) and for the value
of moving a dynamic object 100mm c t (c) and the objective function I t (d)
For a more positive result when using a variable structures need to use a more com-
plicated law of the changing the structure of the system, but it can reduce the reliabil-
ity of the system as a whole, due to the additional elements and connections.
5 Conclusions
The aim of the research to solve the problem of using the method of factor experi-
ment for the problem of parametric synthesis of a closed dynamic object control sys-
tem was considered. Quality functional that realize the requirements of a dynamic
object control system and reduce the calculating resource of parametric system syn-
thesis was proposed. Parametric synthesis of a closed dynamic object control system
with nonlinear characteristics on the basis of the factor experiment method was made
by using three practical examples.
It was proved that the method of factor experiment allows, on the basis of model-
ing the behavior of the system in a random external environment, to select its parame-
ters that satisfy the requirements of the minimum of the regression functional. The
behavior of a closed control system was simulated considering the random external
perturbations acting on the control object. It was defined that the introduction of a
nonlinear link with variable amplification factor widens the stability area of a closed
dynamic object control system.
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