A twomass fuzzy control system is considered. For fuzzification process, classical both linear and nonlinear membership functions are used. To find optimal values of membership function’s parameters, genetic algorithm is used. To take into account values of both output and intermediate parameters of the system, a penalty function is considered. Research is conducted for the case of speed control system and displacement control system. Obtained results are compared with the case of the system with classical, crisp controller.
In the optimal controls synthesis, various approaches are used. Among them are: the method of analytical design of controllers [
Nonlinear control theory, including feedback linearization [
Application of fuzzy logic at synthesis of optimal control is frequently used. In particular, in [
Many studies are devoted to selecting the type of membership function (e.g., [
One approach to solving this problem is the synthesis of control actions that ensure a minimum level of quality desired integral performance index. By means of genetic algorithms, it is possible to determine the unknown parameters of the membership functions.
Research was conducted for the case of twomass dynamic system. In Subsection
Variables that are described in (
In the work [
At synthesis of these regulators, we arrive to a structure that can be represented as
Taking into account that technical systems which can operate at different points in state space, which are characterized by different constraints and different requirements to quality control, are imposed, traditionally one uses a compromise setting and shape control action based on the following criterion:
Optimal control
At the application of fuzzy sets theory, we do not form some sort of trajectory that is optimal for all subsystems of the family but implement an optimal transition from one trajectory to another, specified for a particular subsystem. This approach makes it possible to improve the quality characteristics of the system.
That is, the criterion will be formed as follows:
Each subsystem can generate different types of transitions at different speeds. It is possible to form different transition paths to a given point in output signal’s space by switching between corresponding control actions.
Obviously, at the beginning of system operation (when the error is large), controller which ensures faster transients should be applied to the system, when approaching the area of the setpoint switching to a subsystem that provides a smooth behavior of the system must be done.
As an example, a study for the functional described in [
To investigate the influence of the values of the membership functions parameters on the behavior of the system, classic performance indexes values (see, e.g., [
At synthesis of the controller in real electrical facilities one should comply with the physical feasibility of the processes occurring in them. To take into account constraints on intermediate coordinates value of the following penalty function is calculated
Hence, fitness function has the following form:
The case of a compromise setting of the system which consists of two subsystems
Note that, with this approach, the values of the coefficients do not depend on the state of the system at any current moment of time.
When using fuzzy logic, both linear and nonlinear membership functions have been applied. In this case, the system transient will depend on the parameters of these functions
There are a number of membership functions. However, in the case of two terms, it is possible and appropriate to use limited number of such functions. Conducted studies suggest the possibility of applying the proposed approach to both linear and nonlinear membership functions.
In the case of a linear membership function, study was carried out on the example of the functions of
Membership functions. (a)
In the case of such function, parameters
We believe that, among nonlinear functions, the most common is sigmoid function (
Research was conducted for the case of the systems (
For the comparison of the obtained in this paper results with a classic case a system with crisp controller which is tuned only for one linear standard form (
Hereinafter,
Research is carried out for the case of linear (
Comparison of the characteristics of dynamic system (






max  

But  0.1198  0.1571  0.0451  0.78  0.4718  2.185  1.081 
Bin  0.09375  0.125  0.0394  0.7518  1.404  1.742  1 
Equ  0.06621  0.1093  0.0246  0.5698  0.7215  1.654  1.008 
Fuz  0.05904  0.1026  0.0232  0.4805  0.5805  1.766  1.019 
Comparison of the characteristics of dynamic system (






 

But  0.1198  0.1571  0.0451  0.78  0.4718  2.185  1.081 
Bin  0.09375  0.125  0.0394  0.7518  1.404  1.742  1 
Equ  0.06621  0.1093  0.0246  0.5698  0.7215  1.654  1.008 
Fuz  0.0561  0.09807  0.0215  0.4748  0.5308  1.484  1.038 
In the given tables the following notations are used: But corresponds to the system with controller (
The corresponding transients are shown in Figures
Output signal of the system (
Time dependence of
Similar studies were conducted for the case of the fourth order system (
Comparison of the characteristics of dynamic system (






 

But  0.2672  0.2949  0.0114  0.8808  0.547  2.853  1.111 
Bin  0.183  0.1934  0.0712  0.9305  1.633  1.99  1 
Equ  0.1469  0.1862  0.0557  0.696  0.84  1.826  1.011 
Fuz  0.1455  0.1813  0.0549  0.6458  0.815  1.96  1.009 
Comparison of the characteristics of dynamic system (






 

But  0.2672  0.2949  0.0114  0.8808  0.547  2.853  1.111 
Bin  0.183  0.1934  0.0712  0.9305  1.633  1.99  1 
Equ  0.1469  0.1862  0.0557  0.696  0.84  1.826  1.011 
Fuz  0.1445  0.1802  0.0547  0.7348  0.547  1.906  1.016 
It should be noted that the proposed approach can be generalized to the case of arbitrary linear or nonlinear functions.
The corresponding transients are shown in Figures
Output signal of the system (
Time dependence of
The synthesis system based on functional with variable parameters is proposed in this paper. These parameters are implemented by using fuzzy control and on the basis of genetic algorithms.
This approach can be used as a variant of the optimal control synthesis for both linear and nonlinear systems, which are described as the set of dynamic subsystems. This approach provides a gain compared with systems synthesized based on Pareto optimal solutions.
The proposed approach can be applied to any system, any integral quality indexes, and any membership functions without considerable modifications.
The authors declare that there is no conflict of interests regarding the publication of this paper.