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11THE SYNTHESIS METHOD OF THE DIGITAL CONTROLLER IN THE AUTOMATIC CONTROL SYSTEM OF A DYNAMIC OBJECT BASED ON THE FUZZY LOGIC
Kolomiiets Y.,
Postgraduate student of the Aviation Department National Defence University of Ukraine named after Ivan Cherniakhovskyi,
Kyiv, Ukraine ORCID: 0000-0002-9767-0750 Korotin S.,
Deputy Head of the Aviation and Air Defence Institute, PhD in Technical sciences, associate professor
National Defence University of Ukraine named after Ivan Cherniakhovskyi,
Kyiv, Ukraine ORCID: 0000-0003-2123-6103 Blyskun O.,
Postgraduate student of the Aviation Department National Defence University of Ukraine named after Ivan Cherniakhovskyi,
Kyiv, Ukraine ORCID: 0000-0002-7751-8313 Korovin I.,
Associate professor of the Aviation Department, PhD in Technical sciences National Defence University of Ukraine named after Ivan Cherniakhovskyi,
Kyiv, Ukraine ORCID: 0000-0003-4645-9688 Honcharenko Y.
Senior Lecturer of the Aviation Department, PhD in Technical sciences National Defence University of Ukraine named after Ivan Cherniakhovskyi,
Kyiv, Ukraine ORCID: 0000-0001-7654-6083
Abstract
The article proposes a method for improving the quality of functioning of the automatic control system of a dynamic object using a digital fuzzy controller. The authors used dynamic optimization method to tune the digital fuzzy controller. The results of mathematical modelling of the control system with a digital fuzzy controller, forming a control effect on the dynamic object being controlled, were presented. The automatic control systems of a dynamic object based on a proportional-integral-derivative controller and a controller based on fuzzy logic are compared. The analysis of the obtained results simulation showed that in the system with a fuzzy controller the deviation error was reduced by 5.6 times, and the control time was reduced by 1.7 times.
Keywords: automatic control system, dynamic object, fuzzy logic controller, PID controller, mathematical model.
Introduction
It is known that "Automatic Control Systems (ACS)" must provide accurate and rapid response to control effects, despite certain changes in conditions. Therefore, an important requirement in the development of ACS is to ensure high dynamic properties with significant parametric uncertainty [1]. Currently, "Proportional-Integral-Differential Controllers (PIDC)" are often used to implement the set indicators of control quality in different technical systems, the advantages of which are described in [2]. It is known from the theory of automatic control that the application of PIDC does not always allow to obtain optimal dynamic characteristics of nonlinear controlled objects. There are many known methods of optimal adjustment of regulators, which are used in various technological processes [35]. However, the presence in the ACS of numerous internal cross-links, nonlinear and dynamic elements, delayed links, inaccurate knowledge of the structure of the model leads to changes that are impossible to predict. As a rule, the traditional method of proportional-integral-differential regulation cannot provide an acceptable quality of controlling, so various adaptive methods
become relevant [6-8]. The creation of ACS for complex technical objects in conditions of uncertainty and incomplete knowledge about the object has shown the ineffectiveness of the methods of control theory using PIDC. Recently, to solve such problems, methods of intellectual control are used, namely the method of fuzzy logic. This method is implemented in the form of different types of controllers [9-14]. In contrast to the PIDC, the functioning of these controllers is based on the application of linguistic variables of fuzzy logic theory [15, 16]. The "Fuzzy Logic Controllers (FLC)" control system has proven itself in a positive way in controlling complex objects with parameters that vary over a wide range [17, 18].
However, in the considered works [6-18], only one specified type of regulator is used; no comparison of the performance indicators of the ACS "Dynamic Object (DO)" with PIDC and ACS DO with FLC. The control signal at the output of the FLC is given by analytical expressions.
Therefore, the task is to synthesize ACS DO with FLC to identify the typical characteristics, as well as to compare its work with PIDC. In this regard, the purpose
of this article is to study the possibility of improving the performance of the ACS DO by using FLC.
The paper considers one of the possible ways to optimize the process of DO control in the pitch channel by using FLC as the main section of the ACS DO.
The mathematical model of longitudinal motion of DO described in [19, 20] is used in the article.
To study the chosen principle of building an automatic control system, a modeling scheme was developed in the MATLAB application package.
In this case, the system input u(t) = 9i(t) - set pitch angle, system output x(t) = &2(t) - pitch angle processed by DO, m(t) - control signal at the output of the controller, the angle of deviation of the rudder is taken as the input signal of DO mi(t) = 8(t), and for an output signal is taken - x(t) = &2(t), the control object (the control object includes an analog steering mechanism and
the DO itself) is described by a common transfer function [19, 20]:
_ ^ +mp + ■>
0 P2(T2P2 + 24TaP +1)
(1)
where K - transmission ratio, T3, T4 i Ta - time constants, £ - damping factor.
The parameters of the transfer function (1) are determined according to [19, 20].
The block diagrams of ACS DO by pitch angle with digital PIDC and FLC were composed and shown in Fig. 1-2.
The mismatch error 6(t), which came on input to the PIDC and FLC, is the difference between the specified pitch angle 9i(t) and the processed pitch angle by
DO &2(t) : 0(t) = &i(t) - &2(t) = u(t) - x(t).
/ f(u)
V
Ramp Fen
-Q+
an object
8(t) m(t) 1 mjt) ö2(t)
0.000025S2+0.0025S+1
PIDC steering gear dynamic object
Fig. 1 The block diagram of ACS DO with digital PIDC
/ f(u) m
w
Ramp Fen
0(t)
an object
m(t) 1 mt(t) 02(t)
0.000025S2+0.0025S+1
FLC steering gear dynamic object
Fig. 2 The block diagram of ACS DO with digital FLC
Because digital integration and differentiation can be performed in a variety of ways, the digital PIDC transfer function (PIDC in Figure 1) can be written in a variety of ways. The PIDC transfer function is described as:
W ( z) = K +
Kih0 z +1 Kj z-1
d
2 z -1 h
(2)
CD-
em
z+1 z-1
where ho - sampling step. This transfer function is derived from the transfer function of analog PIDC W(s) = K + Ki / s+Kd s by approximating the derivative by the first difference and integrating by the trapezoidal method.
The block diagram of digital PIDC is shown in Fig. 3. At small simulation steps, digital PIDC is equivalent to analog.
->Kp Gain
Discrete Transfer Fen
Gainl V
Z-1
z
Discrete Transfer Fcn1
l-K-Gain2
z
Fig. 3 Block diagram of digital PIDC
The adjustment of the controller is carried out in order to obtain the minimum current error of deviation of the adjustable value by the method of dynamic optimization [12].
After adjusting the PIDC with a sampling step
ho=0.001 s the following coefficients of the transfer function were obtained (2): K = - 4.964; K = 39.455; Kd = 19.423.
In the synthesis of digital FLC, the number of controller terms used to estimate the linguistic
variables (input and output parameters of FLC), system error 6, rate of change (first derivative) of error 6, acceleration (second derivative) of error 6' is equal to three. The control effect at the output of the fuzzy controller m is determined by the Mamdani algorithm [21].
The general structure of fuzzy regulators can differ significantly in the methods of fuzzification and de-fuzzification, the algorithm for obtaining input and output variables, and so on. The most widespread is the fuzzification by triangular membership functions and the logical conclusions of Mamdani and Sugeno [12].
The FLC design was performed by a method based on analytical expressions for the control effect at the
£L„ 6L„
Block of calculating the first derivative
G
u
r
Block of normalization of input variables
Block of calculating the second derivative
output of the fuzzy controller with symmetrical triangular membership functions [12], which simplified the design procedure of the fuzzy controller.
The fuzzy controller is built according to the functional scheme shown in Fig. 4. It includes the following elements: blocks for estimating the first and second derivatives of the system error, the block for normalization of input variables, the constraint element, which describes the universal set U = [0, 1], the block for calculating the values of A, B, and C, the comparison block values of A, B, and C and the calculation of uc, the normalization block of the original variable.
he functional diagram of the FLC (FLC block in Fig. 2) can be represented in the form shown in Fig. 4.
Specifying the
membership functions
Block of A, B, and C, values calculating
C
Block of
comparison of uc Normalization
values A, B, C -C block of the
and calculation output variable
uc
,m(t )
Fig. 4 Functional diagram of fuzzy controller
The error 6 at the input of the FLC, its first 6 and second 6 differences come to the input of the normalization block of the input variables.
Calculations in the block of normalization of input variables are performed according to the equations:
h1=-(ö-6U)/(26U) »2=-(6>-6U)/(26U) «3 =-(ë-ëmm)i( 2¿u)
(3)
Signals from the output of the normalization block of the input variables u, i = 1, 2, 3, are fed to the constraint elements, which describes the universal set U = [0, 1].
fiT(u)
From the constraint element, the signals are fed to the block of A, B, and C, values calculating, where after specifying the membership functions, they are calculated by equations:
A = min[^! U), n (w2), n (u)] (4)
B = min[^2(M1),^2(M2),^2(M3)] (5)
C = min[^3(Mi),^3(M2),^3(M3)] (6)
Whatever the values of the variables ui, u2, and u3 on the universal set U = [0, 1] depending on the ratios of A, B, and C, the "resulting figure" shown in Fig. 5, can take only three configurations [12].
Fig. 5 The resulting figure with triangular membership functions
u
u
u
2
u
3
m
m
m
In the logical block of comparison of values A, B, C and calculation uc, the calculation of the abscissa "the center of gravity of the resulting figure" Uc is carried out, according to the configuration of the "resulting figure" Fig. 5.
For the first configuration, when A < C < B, Uc is determined by the equation:
B/2 + (A3 -4B3 + 3C3)/24
B + (A2 -2B2 + C2)/4
(7)
certain value [12]:
if A < B, then the value of C is defined as C = 2 A ; if A > B, then the value of C is defined as C = 2 B. The obtained values of Uc are then calculated in the normalization block of the output variable in the value of the control effect on the control object. With symmetrical ranges of change of output signals (mm<x = -mmin) the value of the control effect is calculated by the equation:
m = mmin(1-2uc )
(10)
For the second configuration, when A > C > B, Uc is determined by the equation:
A /2 - (2A2 - B2 - C2 ) / 4 + (4A3 - B3 - 3C3 ) / 24
A - (2 A2 - B2 - C 2)/4
(8)
f A < B < C
For the third configuration, when <! , Uc
[B < A < C
is determined by the equation:
C/2 + (B2 - C 2)/4 + (A3 - B3) / 24 _
uc =-„ . ., . .- (9)
C + (A2 + B2 -2C2)/4
For fixed values of A and B the value of C has a u A
The signal from the normalization block of the output variable goes further to the input of the controlled object.
The values of the ranges Am = Qmax = -Omin, Bm = Omax = -Omin, Cm = Omax = -Omin when configuring the FLC were obtained by solving the optimization problem [12].
After solving the optimization problem with the selected quality criterion of ACS DO, the following ranges of changes of input and output variables FLC are obtained: Omax = -Omin = 1.0496, Omax = -Omin = 0.01, Omax Omin = 1.7931 , mmax mmin = 19.377.
Simulation results
The authors simulated ACS DO with FLC in accordance with equations (1-10). The obtained results are shown in Fig. 6-9, which presents the processes in control systems with digital PIDC and FLC.
PIDC
Fig. 6 Transition process in ACS DO with PIDC and FLC
Fig. 7 Error 0(t) in ACS DO with PIDC and FLC
m
V rv A / -PIDC — FLC <
¿k \J / 3 V / 4.^ r: S
-1
-3 -4
Fig. 9 The control effect m(t) in ACS DO with PIDC and FLC
Analyzing the dependences shown in Fig. 6-9, it is established that in the ACS with PIDC, the control time is 0.6 s (Fig. 6), and the maximum current error is 0.02387 (Fig. 7), in the system with FLC the control time is 0.34 s (Fig. 6), and the maximum current error (excluding the initial release when capturing the signal) is 0.0043 (Fig. 7).
According to Fig. 6-8, tracking the specified pitch angle in both ACS DO occurs with small deviation errors. The control signal m(t) coming to the steering mechanism in the system with FLC changes gradually, and in the system, with PIDC this signal is periodic (Fig. 9).
Conclusions
The synthesis of ACS DO with digital FLC makes it possible to identify the characteristic features of the fuzzy controller, as well as to compare its operation with PIDC.
The use of FLC as the main link of the ACS DO allows to obtain high quality of this system.
The authors found that both PIDC and FLC in ACS DO provide stable tracking of a given pitch angle with small deviation errors. But in the system with FLC the deviation error is 5.6 times less than the deviation error in the system with PIDC, and the adjustment time is reduced by 1.7 times. In addition, the control signal received by the steering mechanism in the system with FLC changes gradually, and in the system with PIDC, this signal is periodic, which can lead to sharp fluctuations in the rudders of a dynamic object. Therefore, the system with FLC provides higher accuracy and dynamic control performance, higher functional control reliability, compared to PIDC.
This technique can also be used to calculate the control effects at the output of the fuzzy controller for membership functions of another type.
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