A mathematical model of the automatic control system with a combined fuzzy PD controller has been developed. The blurring of the P - component of control has been carried out based on three linguistic variables of the normalized error signal. The fuzzy control described by the fuzzy base of rules has been given, membership functions have been chosen, and their parameters have been adjusted, a synthesis and analysis of a hybrid robust system having a simplified rule base with a fuzzy PD-controller for objects with variable parameters has been carried out, a simulator of the system has been designed. For the convenience of adjustment and research of separate blocks and system visualization blocks have been provided. An example of synthesis and modeling of a combined fuzzy system for inertial objects of the third and fourth order has been considered. The settings for adjusting the controller have been defined and adjusted. The results of simulation modeling in the form of transient characteristics for objects of different dimensions that satisfy the specified parameters of quality control and provide better quality indicators than with a traditional controller have been given, and the results of simulation modeling have been presented.
The current state of production, the requirements for the quality of manufactured products put new requirements for the design and development of control systems able to provide a high level of productivity of technological processes and objects with short information concerning the object and its parameters change, depending on the mode of operation and the effects of various perturbations. The main disadvantage of the control systems with traditional controllers is that they do not provide the quality of regulation when changing the object parameters. For example, an increase in the transfer coefficient of an object causes significant fluctuations in the system and it may become unstable, which impairs the quality of regulation and the quality of the finished product. The application of the principles and methods of adaptive control and intelligent management is complex and expensive, which limits their application for simple objects. Therefore, a highly relevant are the tasks of a simulator design and development, synthesis and analysis of a hybrid system with a simplified database having a fuzzy PD controller for objects with variable parameters as well determination and adjustment of parameters and research.
Nowadays there is the development and practical application of fuzzy control systems in various
fields of science and technology. The popularity of fuzzy sets in design is due to the fact that fuzzy
systems are developed faster, hardware implementation is simple, it is even simpler and cheaper than
traditional ones. [
Fundamentals of fuzzy modeling and fuzzy control have been presented in the publications [
To simulate and develop fuzzy systems, Matlab: Simulink software package is used, it is an
important subject in mathematical and engineering education at the universities of Western Europe,
the United States and other countries [23, 24, 25, 26]. However, in the Fuzzy Logic Toolbox library,
the Fuzzy Logic Controllers block does not have a direct access to its individual parts, in particular,
the blocks of normalization and denormalization of output and input signals, which greatly limits the
ability to simulate and study fuzzy control systems, including setting up the fuzzy controller
parameters. Important tasks of normalization and especially denormalization in fuzzy models and
systems have been presented in general [
The main task when designing the control system is the choice of the law (algorithm) of controller
operation, as well as the calculation of the adjustment parameters, which should provide the necessary
quality of regulation, so that the transients and static accuracy in a closed system satisfied the quality
indicators for specified parameters of the regulated object. The existing simplified methods of
calculation allow determining the numerical values of the parameters of debugging the regulators
using formulas or graphs linking these values with object parameters [
The main advantage of traditional proportional P-controllers is the simplicity in implementation
and setting, they create an instant proportional control effect on the object, therefore, relatively
quickly correct the deviations of the adjustable value from the specified value. They are used for
objects of medium capacity and small order, with a slight delay and flood changes of loads [
Deviation of the adjustable value from the set value (error signal)
V( ) = ( ), ( ) = 0( ) − ( ), (1) (2) where yo(t) is the specified value of the regulated quantity, y(t) is the regulated value (output of the system).
To ensure the zero error of regulation, the value of the signal e (t) = 0 is required. However, at zero
value of the error signal according to the expression (1), the control action of the controller V (t) = 0.
Although at zero value of static error, the controller must form a constant value of control on the
object. In order to increase the accuracy of the system with the P controller, it is necessary to increase
the controller's transmission coefficient, however, an increase in the transmission coefficient causes a
fluctuation of the regulated value relatively to the specified value and the system may become
unstable, which is a significant disadvantage that limits the use of the P-controller. To improve the
accuracy of the system I-component of control is introduced [
The second approach to the design of controllers is that the proportional controller additionally introduces the initial value (offset) of the signal, which specifies the neighbourhood (work point) of the controller operation, then the adjusting action on the object is
( ) = ( ) + 0, where U0 is the initial value of the control signal.
In general, the area of the initial control signal depends on the adjustable set value and the load on the object, which depends on the mode of system operation; it can vary widely and in most cases is unknown. Therefore, the initial value of the signal management will be determined on the basis of a predetermined value of the regulated quantity which is known where kc is the desired transmission coefficient which provides the predetermined neighbourhood.
To determine a neighbourhood let us consider the work of the control system supplied with the presented controller for constant operation mode. After substitution of expression (4) in (1) we got the adjusting action on the object
Hence output of the system is (3) (4) (5) (6) (7) (8) (9) In order that static error equals zero we have to do the equation.
Hence let us determine the desired value of transfer coefficient that provides a set neighborhood. = 1 .
1 − 0 = 0. where k0 is a transfer coefficient of the static control system.
Based on the facts let us determine the dependence of regulated value from the set value for closed system.
= 1 + + 0 .
By analogy let us determine static error of the closed system.
If transfer coefficient kc is determined on the condition (9), then the static error (8) of the proportional controller system equals 0. Hence in a given hybrid system with P-controller in theory it is possible to obtain 0 static regulation error without introducing I-component into the regulation algorithm, which adds additional inertia to the system and worsens its dynamic properties. Since the coefficient kc is in the numerator of the closed system equation (7) and (8), the introduction of the initial control signal channel (4) in the controller will not affect the stability of the closed system, which is an advantage of the system.
To improve the dynamic properties of the proposed hybrid system and provide robustness with in
relation to the variation of the object parameters, we use a simple version of the fuzzy PD controller
which forms a controlling action based on fuzzy logic [
1: 1:
( = ) ℎ ( = ) ℎ 1: ( = ) ℎ
( = ) ( = ) ( = ), (11) (13) where E is the normalized input of control error, Un is normalized control, N, O, P are fuzzy linguistic sets that qualitatively evaluate the control error: N is a negative error, O is a zero error P is a positive error.
Fuzzy linguistic models and sets A, B, C, which are described by the membership functions given
by the type L are left, and P as right are the external ones however the mean triangular membership
function [
= 10 , (12) where y0 is the set rate of the regulated value at the input of the system, which is known and can vary widely, depending on the operation system mode .
The normalized control formed in accordance with fuzzy rules (11) can be within [
= + 1 0 0, where M is the scale factor which is the main parameter of adjustment of the fuzzy controller and is chosen depending on the predefined quality control criterion.
Based on rule (11), selected membership functions, fuzification and defuzification operations, and regulatory actions on object (13), a structural scheme of a hybrid control system model with a simplified version of a PD-controller with a third-order inertial object in a Matlab: Simulink package has been designed (Fig 1).
The fuzzy controller consists of two blocks: a normalized PD control algorithm disguised in the Subsystem block and Fuzyfication and Defuzyfication blocks disguised in the Enabled Subsystem block. The control object model located to the right is provided by the Transfer Fcn blocks. The normalization of the error is carried out by the expressions (2) and (12). An Integer Delay unit was used to determine the first difference in the error signal.
The scheme of the fuzyfication and defuzyfication units disguised in the Enabled Subsystem block is shown in Fig.2
Fuzzy block consists of three blocks of Triangular membership function MF, two of which are
configured for L-left and P-right outer, but the middle of the triangular membership functions that are
activated by a signal X from block Ramp. Defuzzification blurred signal is carried out by Mamdani
method [
Based on the above-mentioned, schemes Fig.1 and Fig. 2 a structural model diagram of a hybrid control system with a simplified version of the fuzzy PD-controller with inertial object of the third order has been made by means of the graphical editor and block libraries in the Simulink work window. Blocks of fuzzy function Triangular MF for the width of the windows a = 1 were adjusted. The time of quantization 0.15 sec. was set in the dialog window Integer Delay block. To study the object k0=10, T1=5, T2=3s, T3=2s, parameters were set. The value of the controlled variable y0=100 was taken.
The quality of the designed system was evaluated with a static error, overshooting and speed of operation. The main parameter setting of the fuzzy PD controller is a scaling factor M. The objective of the modeling was to determine the properties of the designed system at a variation of parameters of the object at the nominal parameters of the object. The controller for 20% and 5% overshooting was adjusted with the coefficient M = 5.2 and 1, 2. The simulation results are presented in Fig. 3 in the form of transients graphs with the step setting y0=100.
Analyzing transient processes, we can conclude that the combination of fuzzy PD-controller in the control system with the inertial object of the third order provides the set of overshooting 20% and 5% and sufficiently small static errors of 0.009 and 0.001%. The time to reach the set rate of adjustable values is 5.4 and 10 seconds and the adjustment time is about 20 seconds. After reaching the equilibrium state at time t=30C it was set a sufficiently large disturbance of z0=20 per the object. The overshooting is 7%, and the static error of the system by the disturbance is 2.5%. Thus a designed hybrid management system with a simple version of the fuzzy PD controller provides a rather small error without the input of I-component in the control algorithm for a given overshooting 20 and 5%. If the quality of the system (overshooting and static error) with the fuzzy PD - controller with three membership functions satisfies the customer, it is possible to recommend the hybrid application of the developed fuzzy controller, as the most simple.
The operation of the system with fuzzy control has been investigated in course of object's parameters changing, that is the increase of the transmission coefficient of the object k0=30 and 40. In the first case the controller was adjusted to ensure the transition process with a 5% overshooting in the initial system to the scale factor М=5 and the initial displacement U0=2,85. Simulation results for the coefficients of transfer of the object k0=10, 30, 40 shown in Fig. 4.
With the increase of the transmission coefficient of the object overshooting is 13.4% and 23%, and there are damped oscillations in the system. Static error is 0.0032; 2.63; 1.91%. To compare a system with a classic PID controller with a threefold increase in the transmission coefficient of objects of the third and fourth order are unstable.
In the second case the controller was adjusted under the condition to provide a process with 20% overshooting in the initial system. The controller was adjusted at the scale factor М=7 and the initial displacement U0=3,70. Simulation results are represented in Fig.5.
When transmission coefficient of the object is k0=10, the overshooting is 20%. The increase of the transmission coefficient up to 30 and 40 causes an overshooting of 27% and 36%. The static error is 0.0017; 1.27; 3.68%. Increasing the transmission ratio by 4 times degrades the quality of regulation. However, the system is stable and workable. Therefore, the set value of overshooting significantly affects the quality of regulation, which must be taken into account when designing a fuzzy controller and its adjustment.
The influence of the time constant of the objects on the quality of regulation with a fivefold increase and reduced time constants, that is Т1 = 25, Т2 = 15 seconds and with constant transmission coefficients of the object k0 = 10 has been investigated. The simulation results are shown in Fig. 6.
Changing the time constant of the object of regulation has little effect on the quality of regulation. Overshooting is 5% and 25%. Instead, the time of regulation varies widely, which is a natural phenomenon due to the change in inertial properties of the object. Based on the results of simulation and built transition characteristics of the system when changing the parameters of the object in wide limits, we conclude that the proposed hybrid fuzzy PD-controller with a simplified version provides a stable operation of the system and the quality of regulation, has robust properties, is simpler than the traditional adaptive systems. If the system quality (overshooting and static error) at the variation of the parameters of the object within the specified limits with the fuzzy PD-controller with three membership functions satisfies the customer, then the application of the developed hybrid fuzzy controller can be recommended as the simplest, which has robust properties and is simpler than the traditional adaptive systems.
.
Nowadays, there are no generally accepted reasonable methods of synthesis and determination of parameters for setting fuzzy regulators, which makes it impossible to optimize them, complicates their development and implementation. The problem of synthesis of the automatic control system with fuzzy regulators is complex and multifaceted, it contains a significant number of partial problems and possible ways to solve them
In the study a new hybrid fuzzy PD-controller with a simplified version and three membership functions was proposed, a structural diagram of the system model and a simulator for its analysis have been designed and developed.
It is proposed to determine the parameters of setting the regulator on the basis of identification of objects by transitional characteristics, provided that the transfer coefficient of the object is equal to one that ensures the choice of parameters of the regulator, regardless of the transfer factor of the object.
It is proposed to determine the parameters of setting the regulator on the basis of identification of objects by transitional characteristics, provided that the transfer coefficient of the object is equal to one that ensures the choice of parameters of the regulator, regardless of the transfer factor of the object. A structural diagram of the fuzzy system model in the MatLab: Simulink package has been developed, which makes it possible to calculate and build transitional characteristics of the system to analyze its properties and interactively test the optimal adjustment parameters of the regulator.
The results of simulation are transient characteristics of the system and it was established that the proposed PD-controller provides a stable operation of the system and good quality of regulation when changing the parameters of the object in wide limits, it has robust properties.
Based on the results of simulation and built transition characteristics of the system when changing the parameters of the object in wide limits, we conclude that the proposed hybrid fuzzy PD-controller with a simplified version provides a stable operation of the system and the quality of regulation, has robust properties, is simpler than the traditional adaptive systems. If the system quality (overshooting and static error) at the variation of the parameters of the object within the specified limits with the fuzzy PD-controller with three membership functions satisfies the customer, then the application of the developed hybrid fuzzy controller can be recommended as the simplest, which has robust properties and is simpler than the traditional adaptive systems.
The system with a hybrid PD-controller is simpler than traditional adaptive systems, which extends their application to manage simple objects with variable parameters..
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