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Industry control systems
61
Ya. Dovhopolyi, G. Manko, V. Trishkin, A. Shvachka, 2017
1. Introduction
In modern information technologies an important role
is given to instrumental means and environments for the
development of an automated control system (ACS). Instru-
mental means of development are related to problem-orient-
ed software. A developer can focus on resolving the set task
in order to solve it in the most convenient form. Constituent
parts are the high-level languages [1].
Development of software means for adjustment proceeds
in the direction of broadening the range of supported con-
trollers, the application of artificial intelligence techniques
and methods of diagnosis, development of the user interface
[2]. One of the constituent factors of the complex problem
on the task of control is an automated support of parameters
at the assigned level, using proportional-integral-differential
(PID)-controllers.
During automated adjustment and adaptation the
same methods of identification and calculation of the
controller are used as during manual tuning, but they are
performed in automatic mode. The most effective methods
of automated tuning are those using a computer, tempo-
rarily included in the control ci rcuit. Exploiting the power
of the CPU and due to the absence of constraints on the
software volume, there is the possibility to create a soft-
ware tool with wide service properties and high-quality
mathematical processing [3].
The use of the PID-law leads to the improvement of
regulating quality, but wide acceptance of the algorithm is
limited by the complexity of its tuning. This is explained
by the peculiarity of work of ACS with a PID-controller,
namely, high sensitivity to deviations in the optimum of its
tuning. Thus, the automated tuning and adaptation are the
most urgent tasks when building PID-controllers. Despite
the large number of commercially available products, there
are still many unresolved problems related to quality con-
trol, the influence of nonlinearities in the control object and
external disturbances in the process of identification.
2. Literature review and problem statement
The process of tuning a PID-controller for the objects of
chemical technology based on experimental rules is intuitive
while attempts to set up a controller without preliminary
approximate estimation of coefficients may prove futile.
Historically, there were proposed a great number of methods
for the calculation of parameters of controllers, but most of
them require significant expenditures of time and do not
always yield a satisfactory result, which would ensure the
DEVELOPMENT OF THE
PROGRAM FOR SELF-
TUNING A PROPORTAL-
INTEGRAL-DIFFERENTIAL
CONTROLLER WITH
AN ADDITIONAL
CONTROLLING ACTION
Ya. Dovhopolyi
Рostgraduate student*
E-mail: Xa4ik@3g.ua
G. Manko
PhD, Associate Professor*
E-mail: bsoft@a-teleport.com
V. Trishkin
PhD, Associate Professor*
E-mail: TVYA40@i.ua
A. Shvachka
PhD, Associate Professor*
E-mail: AleksandrShvachka@gmail.com
*Department of Computer-integrated
Technologies and Metrology
Ukrainian State University of Chemical Technology
Gagarina ave., 8, Dnipro, Ukraine, 49005
Розглянуто метод настройки пропорціо-
нально-інтегрально-диференціального регу-
лятору з додатковою керуючою дією дифе-
ренціатора. Встановлено, що запропонований
регулятор забезпечує бажаний вид перехід-
ної характеристики замкнутої системи за
показниками якості: перерегулювання, час
регулювання. Показана можливість створен-
ня графічного інтерфейсу користувача авто-
настройки в середовищі MatLab розробленого
регулятору та ручною підстройкою на основі
досвіду користувача
Ключові слова: пропорціонально-інтегрально-
диференціальний алгоритм, передаточна функ-
ція, якість, інтерфейс, Matlab/Simulink
Рассмотрен метод настройки пропорцио-
нально-интегрально-дифференциального регу-
лятора с дополнительным управляющим воз-
действием дифференциатора. Установлено, что
предложенный регулятор обеспечивает жела-
емый вид переходной характеристики замкну-
той системы по показателям качества: пере-
регулирование, время регулирования. Показана
возможность создания графического интерфей-
са пользователя автонастройки в среде MatLab
разработанного регулятора и ручной подстрой-
ки на основе опыта пользователя
Ключевые слова: пропорционально-инте-
грально-дифференциальный алгоритм, переда-
точная функция, качество, интерфейс, Matlab/
Simulink
UDC 621.03
DOI: 10.15587/1729-4061.2017.114333
Eastern-European Journal of Enterprise Technologies ISSN 1729-3774 6/2 ( 90 ) 2017
62
desired quality of control. Based on the review of existing
methodologies for tuning, one can conclude that the meth-
ods by Ziegler-Nichols, Kopelovich, and CHR, are the most
expedient for the application under industrial conditions.
The search for methods to improve tuning a PID-con-
troller is still under way. Specifically, author of [4], by way of
generalizing the disarranged terminology in the calculation
of controllers, proposed the unified rules for tuning PI and
PID- controllers taking into consideration a zone of insensi-
tivity, though any information on other types of nonlinear-
ities is missing. While developing the subject of non-linear
controllers, authors of ref. [5] obtained a transfer function of
the resonator, which predetermines the non-linear properties
of control object. To conduct the study, the authors con-
structed a simulation model in order to determine stability
of the amplitude and the frequency of auto-oscillations. The
use of the z-transform when applying traditional methods
for the synthesis of analog controllers was described in [6].
Practical implementation, however, turns out to be difficult
and has limitations. Authors of article [7], in order to cal-
culate parameters of tuning a PID-controller, employed the
Matlab programming environment and applied optimization
rules for the criterion of effectiveness (ITAE). The specified
algorithm ensures prompt response of the system, reducing
the impact of disturbances, but it represents only the final
stage of the synthesis of a controller. Author of [8] proposed
a PID-controller design methodology based on the reference
model of reliable tuning (MoReRT). Achieving the robust-
ness and stability reserve of the system is achieved by a set
of rules for tuning, which are characterized by ambiguity for
the objects with the same dynamic properties. In study [9],
author focuses on the issue of using PID-controllers in the
multi-circuit systems taking into consideration the principle
of decentralization and the use of methods from the theory
of phase space for interpreting the results. A search-free
method for estimating the parameters of linear controllers
is given in [10]. It implies building a complex frequency
characteristic of suboptotic controller and bringing it closer
to the appropriate characteristic of a typical controller. The
main unresolved issue is determining the frequencies of
approximation and the smoothing constant. In paper [11],
authors devised a technique for tuning a PI-controller and
a correcting device with a phase advance, which makes it
possible to create an adaptive system. The model does not
imply the estimation of influence of variable coefficients and
examining the transient processes at the same time.
Despite the presence of a totality of sufficiently devel-
oped and inhomogeneous algorithms, there is still a big gap
between theory and practice. In many ways, it is associated
with the use of well-developed reliable algorithms for tuning
and their modifications. The needs of the industry grow each
year, caused by the increasing pace of production, technolo-
gy change, flexibility. The studies conducted into the quality
of work of 100 thousand control circuits at 350 enterprises
have revealed that from 49 to 63 % of circuits operate with
“weak” (close to breaking the circuit) settings [12]. On aver-
age, one-third of circuits run with normal settings, one third
with “weakened”, and one third with practically “weak” set-
tings. This sets the task on improving the methods of control
by employing sophisticated models that allow optimization
of control, but it also gives rise to the problem of search for
the algorithms of their tuning, to study the influence of con-
troller’s coefficients on the transient processes. This requires
multiple launch of the model with modified coefficients and
editing the properties of the model under condition of pro-
viding the user with a convenient interface.
3. The aim and objectives of the study
The aim of present work is to improve operation quality
of proportional-integral-differential controller by introduc-
ing an additional controlling influence of the differentiator,
and to implement its auto-tuning under interactive mode
from the stage of identification of the control object to the
construction of the curve of the transition process.
To achieve the set aim, the following tasks have to be
solved:
– perform an analysis of the impact of components of a
PID-controller on its cumulative effect in order to ensure
stability reserve and system robustness at unsatisfactory
dynamic properties of the control object;
– to assess the information base of a control system (in-
troduction of additional signals) and to devise the models of
structures of PID-controllers with an additional controlling
action of the differentiator and to estimate its settings;
– to create a software application that would enable au-
tomated design of the control circuit of the proposed struc-
tures to the controller with the possibility of manual tuning
based on the results of system’s operation.
4. Study into properties of the control system
The initial stage in the estimation of controller’s tuning
is the experimental study of the control object and construc-
tion of the acceleration curve. In order to obtain it, we give
a step influence to the input of the object. We determine the
driver that affects the controlled magnitude. We change by
a jump the incoming action by a few percent of the motion
of the controlling mechanism and observe a change in the
controlled magnitude followed by processing of the resulting
curve [3].
A control object can be improperly designed (dependent
control circuits, a large delay, high order of the object) and be
nonlinear. Sensors may have bad contact with the object, the
noise in the measured channel can be large, the resolution of
the sensor may not be high enough, the source of the incom-
ing influence on the object may have a large inertia. Before
starting the tuning, it is required to make sure there are no
such problems. Despite the diversity and complexity of actu-
al control objects, the PID-controllers typically employ only
two structures of mathematical models: the first-order model
with a delay, and the second-order model with a delay. Much
less frequently used are the models of higher orders, though
they better correspond to an object.
Calculation of parameters of the controller based on
formulae cannot enable optimal tuning because results
obtained analytically are based on extremely simplified
models of the object. Specifically, they do not take into
consideration the nonlinearity of the type of “constraint” for
the controlling influence. In addition, the models employ pa-
rameters that are identified with a certain error. That is why,
after calculating parameters of the controller, it is advisable
to fine-tune it, based on practice, theoretical analysis, and
experiment:
– a growth of the proportional coefficient of amplifica-
tion (P-controller) ensures greater accuracy and operational
Industry control systems
63
performance of the system, but can lead to the loss of sta-
bility. In this case, it is necessary to take into consideration
that changes in the load must be small so that the static error
remains in the permissible limits;
– a reduction in the integral component (I-controller)
provides a more intensive decrease in the control error, but a
low speed of performance. Such a controller is not applicable
for use in systems without self-adjustment;
– an increase in the differential component (D-control-
ler) increases stability reserve and operational performance.
Differential control suffers worse from noise than other
types of control since it amplifies this noise.
A shortcoming of all experimental techniques is the in-
completeness of information about system’s stability reserve
and robustness, which determine operational reliability of
the controller. The examined factors require analysis on a
change in the dynamics of the control object; in this case, it
is better to obtain a transition characteristic of the control
object, to simulate it taking into consideration a range of
change in the parameters and receive settings that are close
to optimal.
A shortcoming of all experimental techniques is the in-
completeness of information about system’s stability reserve
and robustness, which determine operational reliability of the
controller. The examined factors require analysis on a change
in the dynamics of the control object; in this case, it is better
to obtain a transition characteristic of the control object, to
simulate it taking into consideration a range of change in the
parameters and receive settings that are close to optimal.
5. Development of structures of the PID-controller with
an additional controlling action of the differentiator
Proportional control corresponds to the use of “in-
stantaneous” information about the system, the integral
control ‒ of the “past” of the system. An element that is
responsible for utilizing “predictive” information about
the system is the differentiator.
We propose the structures of PID-controllers [13, 14, 15]
with an additional controlling action of the differentiator
(PID-CACD). The structure of the controllers includes a
differentiation unit whose input receives the output signal,
proportional to the sum of the output signals of one (two,
three) components of a standard PID-controller. The output
signal of the additional unit is added to the output signal of
the PID-controller as an additional controlling action.
The block diagram of PID-PD-controller controller is
shown in Fig. 1. The input of additional differentiator 7
receives the output signal of adder 5, which summarizes the
output signals of proportional conversion unit 2 and differ-
entiation unit 4 of a standard PID-controller. Additional
controlling action of differentiator 7 is added to the basic
signal of PID-controller in adder 6.
A transfer function of the devised structure of the con-
troller takes the form:
( )
( )
= + + + ⋅ ⋅+ ⋅ ⋅
⋅
2
1,
p d pd dd
i
Ws K T K K s T K s
Ts
(1)
where Kp is the proportional amplification coefficient; Ti is
the time constant of integration; Td is the time constant of
differentiation; Kd is the amplification coefficient of addi-
tional differentiator.
Fig. 1. PID-PD-controller: 1 (5, 6) – adder, 2 – proportional
component, 3 – integral component, 4 – differential
component 7 – additional differentiator, Zc(t) – controlling
action (a setpoint), Z(t) – measuring signal of the controlled
parameter, DZ(t) – error signal
The circuit of a PID-ID controller is shown in Fig. 3.
The structure of the controller differs in that the output of
proportional conversion unit 2 is connected to the input of
second adder 6, while the outputs of integration unit 3 and
of the first differentiation unit 4 are, accordingly, connected
to the inputs of first adder 5, the output of second one is
connected to the input of second differentiation unit 7. The
output of adder 6 is the controlling action of this controller.
The circuit of a PID-ID controller is shown in Fig. 2. The
output of proportional conversion unit 2 is connected to the
input of second adder 6. The outputs of integration unit 3
and of first differentiation unit 4 are connected to the inputs
of first adder 5. The output of adder 5 is connected to the
input of second differentiation unit 7. The output of adder 6
is the controlling action of this controller.
Fig. 2. PID-ID-controller
The calculation of a transfer function of the given struc-
ture of the controller showed:
( )
= + + +⋅+⋅ ⋅
⋅
2
1.
d
p d dd
ii
K
Ws K TsTKs
T Ts
(2)
PID-3D-controller performs additional differentiation
of all three output constituents of a standard PID-controller
(Fig. 3).
Fig. 3. PID-3D-controller
Eastern-European Journal of Enterprise Technologies ISSN 1729-3774 6/2 ( 90 ) 2017
64
Transfer function based on structural diagram (Fig. 4):
( )
= + +⋅+ + +⋅⋅ ⋅
⋅⋅
11
.
pdpdd
ii
Ws K Ts K Ts Ks
Ts Ts (3)
To configure a PID-CACD, it is necessary to determine
the settings: Kp, Ti, Td, Kd.
6. Calculation of settings of PID-CACD
The calculation of PID-CACD settings was conducted
using the express-method [16]. Dependences of settings on
the dynamic characteristics of an object are estimated based
on the experimental-statistical data in the form of regression
equations.
A control object is represented by the sequential connection
of two aperiodic links of the first order with a link of delay:
( )
−t⋅
=⋅⋅
+⋅ +⋅
12
1
.
11
s
o
o
K
Ws e
Ts Ts (4)
The object’s amplification coefficient (KО) varies from
0.4 % to 4 %, time constants (Т1(2)) from 1 to 35 units of
time. Clean delay (t) is calculated in the course of a statisti-
cal experiment using a relative delay:
t =t
/,
ro
T
(5)
where To is the general time constant of the object;
tr»(0,05–1,0).
We shall consider in more detail the calculation of PID-
CACD settings.
To perform the task, we used a classic one-factor exper-
iment, that is, we changed in the parameters of the model of
dynamics of the examined object only one parameter leaving
all the others unchanged. First, we determined the impact of
change in the ratio of time of full delay (τ) to the total time
constant of the object (To). By changing the time of clean
delay, we changed ratio (τ/To) in the range from 0.1 to 0.8
and, through multiple solutions to the system of differential
equations, we searched for optimal parameters of settings of
a multi-parametric controller. During further investigation,
it was found that the optimal settings of the controller are
significantly affected not only by ratio (τ/To), but also by the
very magnitude of time constant.
The second series of experiments was carried out when
changing the overall time constant of objects (To) in a range
from 4 min. to 50 min. Note that in this series of experi-
ments we accurately maintained ratio T1/T2=0.4 by a change
in the time constants in the range from T
1/T2=1/2.5 to
T1/T2=10/25. When processing the statistical data obtained,
in order to achieve more exact calculations, we applied a
method of piecewise-nonlinear approximation, that is, the
observed objects were divided into two groups according
to ratio (τ/To<=0.3) and (τ/To>0.3); we obtained specific
regression equations for each group.
In the third series of experiments, we changed ration
(T1/T2); in this case, the basic value was accepted from ra-
tios: (T1/T2=0.4) and (τ/To=0.3). The obtained regression
equations express the dependence of corrective coefficients
on a change in the ratio (T1/T2). In this series of experiments
the models of objects were also divided into two groups
according to ratio (T1/T2<=0.4) and (T1/T2>0.4), while the
total change in this ratio was in the range from 0.1 to 0.9.
For objects according to ratio τ/Тo<=0.3, regression
equations:
= − ⋅t − ⋅t
1(0.298 72.0967 ) / (1.0 18.088 ),
p rr
K
(6)
(7)
(8)
(9)
For ratio τ/Тo>0.3:
( )
= − ⋅t + ⋅t2
19.9878 21.2 12.982 ,
rpr
K
(10)
(11)
(12)
(13)
These settings were subsequently refined using the fol-
lowing formulae:
– for ratio T1/T2<=0.4:
=1/ ,
ppo
KK K (14)
( )
( )
= +⋅
−⋅
1 12
2
12
0.764 1.6298 /( –
2.5973 / ),
i i
o
TT TT
T KT (15 )
( )
( )
= −⋅ +
+⋅
1 12
2
12
0.7962 0.1333 /
0.9416 / /
(
),
d
o
d
T T TT
TT K (16 )
( )
( )
=+
1 12
0.8304 0.4262 ./
d d
KK TT
(17 )
– for ratio T1/T2>0.4:
=1/ ,
p p o
KK K (18)
( )
( )
= ⋅+
+⋅
1 12
2
12
(
),
1.4235–1.7065 /
1.615 /
ii
o
TT TT
KTT (19)
( )
( )
= ⋅+
+⋅
1 12
2
12
1.2068–0.8698 /
0.877 / /
(
),
d d
o
T T TT
TT K (20)
( )
( )
= +⋅
− 1 12
0.8333 0.4167 / .
dd
K TTK
(21)
The scope of industrial application of the proposed con-
troller is very wide. It could be used in chemical, oil-refining
and metallurgical industries, in the production of building
materials, as well as in many areas of food production and in
other sectors where there are control objects with a significant
time delay and where standard PID-controllers are used.
7. Building a software tool for tuning a PID-controller
with additional information signals
A full cycle of tuning a PID-CACD controller in inter-
active mode is enabled by GUI developed in the MatLab
programming environment [17]. The program automates the
( )
= + ⋅t + ⋅ t2
10.598 1.477 12 (.857 ), · /18.5
roir
TT
( )
( )
= − ⋅t + ⋅ t2
126.2 72.485 105.714 / .5 ,· 18
d
ro
r
ТT
( ) ( )
( )
( )
= − ⋅t − ⋅t
1
1.6443 16.636 / 1 12.6592 / .· 15
rod r
ТK
( )
( )
= − ⋅t + ⋅t2
10.583 1.12 20.5714 · /1 ,8.5
r r oi
TT
( )
( )
= − ⋅t − ⋅t2
115.267 1.5929 8.2143 · /1 ,8.5
r r od
TT
( )
( )
= − ⋅t + ⋅t2
13.4798 6.2025 2.9296 · /1 .8.5
r r od
KT
Industry control systems
65
stages of development: identification of an object, calculation
of the settings of the controller, construction of ACS transi-
tion process, determining quality indicators (Fig. 4).
The approximation of the object was carried out by a
model in the form of (4) and implemented by the minimi-
zation of square of the difference of values of the output
magnitudes of the model and the object. We use the MatLab
function: fmincon optimset. The approximation is launched
by pressing the button “Identification”. As a result, a graph
of the output magnitude of the model is constructed; the
numerator and denominator of the transfer function are en-
tered into fields “Nominator” and “Denominator”. Next, one
can then choose the type of controller. Pressing the button
“Design” results in calculating the parameters of the con-
troller tuning and in the construction of the ACS transition
process graph.
As a result, parameters of tuning are calculated, which
are displayed in the fields above the corresponding sliders,
with a transition process chart being built. The bottom
of the window displays quality estimated: Overshoot –
over-regulation, Peak – maximum deviation, Setting Time ‒
time of control.
8. Discussion of the research results
We examine influence of the structure and method of
tuning a proportional-integral-differential controller with
an additional controlling action of the differentiator on
control quality. It is important because the introduction
of an additional controlling action is limited by difficult
tuning (non-linearity of the object, external disturbanc-
es). But there are objective difficulties associated with
a large number of methods for estimating parameters of
the controllers; however, most of them require significant
expenditures of time and do not always produce a satis-
factory result, which would ensure the desired quality of
control.
In the framework of the research described in the
present article, we determined the structures of PID-con-
trollers with an additional controlling action of the differ-
entiator, with settings calculated by the
express-method. This makes it possible
to substantiate the approach to deter-
mining the settings of the controller with
an additional controlling action and to
obtain certain effects from the imple-
mentation into production. Specifically,
it is possible to ensure the desired form
of ACS transitional characteristic for
the indicators of over-regulation, time
of control.
Typically, when tuning controllers,
the experimental techniques are applied,
characterized by incomplete information
about stability reserve of the systems
and robustness. However, the use of such
methods is justified only for extremely
simplified object models. When trying
to overcome these constraints in order
to improve the quality of control, there
are objective difficulties associated with
the need to fine-tune a controller, based
on experience, theoretical analysis, and
experiment. In the framework of the re-
search described in the present paper,
we proposed a procedure for overcoming
these difficulties. It is based on the fact that the output signal
of additional differentiation unit is added to the output con-
trolling signal of a standard PID-controller and is used as an
additional controlling action. This technique has allowed us
to implement control that does not allow significant displace-
ments of the controlling element. It increases stability reserve
and robustness of the system. This is essential for objects
with a limited reserve of the controlling load. This means
that the obtained scientific result in the form of analytical
dependences for calculating the settings of a controller with
an additional controlling action of the differentiator to ensure
robustness depending on the dynamic properties of an object
is interesting from a theoretical point of view.
From a practical point of view, using the Guide environ-
ment of the system MatLab, we demonstrated the possibility
of creating a graphic interface of auto-tuning a controller
that does not require from the user knowledge in the field
of programming and in-depth knowledge of the theory of
automated control, as well as controllers with an additional
control action of the differentiator, and makes it possible to
automate all stages of the development from identification of
an object to assessing the quality of a controller.
The study, which started with a series of papers [13−16],
is planned to continue to improve the structure of PID-
CACD and extend the scope of application.
9. Conclusions
1. It was established that a shortcoming of all experimen-
tal methods for tuning the controllers is the incompleteness
of information on the system’s stability reserve and robust-
ness. The choice of the optimal model of the object should be
based on sufficiency criteria of control quality at minimal
complexity of the model. The calculation of parameters of
Fig. 4. Interface of PID-CACD auto-tuning software
Eastern-European Journal of Enterprise Technologies ISSN 1729-3774 6/2 ( 90 ) 2017
66
tuning a controller based on formulae cannot ensure optimal
tuning of the controller since the results obtained analytical-
ly are based on extremely simplified object models.
2. We proposed a structure of the PID-controller with
an additional controlling action of the differentiator whose
input signal is the output signal of a standard PID-controller
proportional to two or three of its components.
The experiment employed a classic one-factor experi-
ment on changing one of the dynamic characteristics of the
object. By multiple solutions to the system of differential
equations we searched for optimal parameters of settings
of a multi-parametric controller. When processing the sta-
tistical data obtained, in order to achieve more accurate
calculation, we applied the method of piecewise-nonlinear
approximation. Dependences of setting parameters on the
dynamic characteristics of the object are calculated based
on experimental-statistical data taking into consideration
quality indicators: over-regulation, time of control.
3. A software application for auto-tuning PID-CACD
was developed, which displays graphs of the transition pro-
cess and calculates parameters of the controller, using sever-
al methods for tuning. Before starting the system, the user is
given a graphic menu for entering a priori information about
control object: the range of change in the input and output
signals of the object, structure of the controller, initial ap-
proximations of the settings. Identification is performed by
means of an analysis of response to the incoming jump in an
open circuit.
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