УПРАВЛІННЯ У ТЕХНІЧНИХ СИСТЕМАХ УПРАВЛЕНИЕ В ТЕХНИЧЕСКИХ СИСТЕМАХ СО N Т ROL IN TECHNICAL SYSTEMS

Context. The task of a mobile robot control on the base of soft algorithm fuzzy inference has been solved. Objective is the creation of soft algorithm of the fuzzy inference which allows to provide additivity of fuzzy control system. Method. A soft algorithm of fuzzy inference used to control the mobile robot is suggested. Given algorithm allows to compensate errors inherent to the traditional models of fuzzy inference. Errors include: the curse of dimensionality, the absence of additivity and the fuzzy partition. This soft algorithm of fuzzy inference at the expense of rational allocation of premises and conclusions in a matrix of fuzzy relations, reduces the number of operations of the fuzzy inference. Another distinctive feature of the proposed soft algorithm is that in fuzzy inference to find minima and maxima used soft arithmetic operators. The paper shows that during the work of hard formulas for the implementation of these formulas while controlling the mobile robot the situations will appear when the robot loses control. The article points out that the implementation of the possibility in soft algorithm of fuzzy inference option of changes in parameters of sigmoidal membership functions will minimize the error at fuzzy system output. Dynamics of changing RMSE ratio from the varying parameters of sigmoidal membership functions proves it. The additional simulations presented in the article shows that during varying the parameters of sigmoidal membership function, during in increasing of the parameter a , is being observed decrease in the value of RMSE. The effectiveness of the proposed soft algorithm is confirmed by numerical simulation and experiments in the researching of a mobile robot movement along a line. Results. The specialized software for microcontroller Arduino Uno is developed and it realizes the proposed soft algorithm which allows to carry out an experimental study of its properties. Conclusions. The software realizing proposed algorithm has been developed and used in computational experiments investigating the properties of the algorithm. The experiments confirmed the efficiency of the proposed algorithm and software.


NOMENCLATURE
RMSE is a root mean square error; MF is a membership functions; MISO is a multi input single output; FR is a fuzzy rules; FIS is a fuzzy inference system; y is an output parameter; x i is an input parameter; n is a number of input parameters; X is a vector of the input parameters; k is a number of fuzzy rules; m is a number of terms; l is a number input parameters; a,b,c are the parameters of the Gaussian membership function; δ is a softness operator; γ is an operator of the parameterization; M is the number of observations in the learning sample.

INTRODUCTION
Algorithms of the fuzzy-logic inference are successfully applied in modeling of difficult technological processes of the modern control systems [1,2,3]. The algorithms allow increasing accuracy of the control process by balancing outside effects in real-time mode.
However, in practice, implementation of the system based on algorithms of the fuzzy-logic inference has some problems concerning continuous differentiability of MF. They are curse of dimensionality [4,5,6]; non-observance of the condition of the partition of unity condition [7] and non-compliance of the condition of additivity [8].
The main purpose in the article is to design a soft algorithm of fuzzy inference that allows to control robotic systems in real time.

PROBLEM STATEMENT
Fuzzy inference systems, based on the Zadeh rule, are used for modeling modern systems [12,13,14,15]. For example, when MISO-system with n-input and one output parameter is designated, then dependence between these is defined as where y -an output parameter, х i -input parameters, i=1…n, n -number of input parameters.
The vector X of the input parameters is shown on the Cartesian product of the determination of input parameters Х 1 × Х 2 ×…× Х n : Х=[х 1 , х 2 ,…,х n ]. At the same time, the function f shows a condition multiplicity of the output parameter Y in the area of the determination of the input parameters Х According to the Zadeh rule the fuzzy multiplicity in a MISO-system is determined as where ∨ is a symbol, which means the operation of the hard fuzzy maximum; ∧ is a symbol, which means the operation of the hard fuzzy minimum. At the same time, the membership function of the output parameter takes the form The disadvantage of the MF relation, based on the Zade rule using hard formulas is evident. When the equation (2) min(х 1 ; х 2 )=min(0,7; 0)=0, the result at the output is zero. Therefore, a fuzzy system will be non-sensitive to changes of the input parameter х 1 , because its value will depend on the second parameter х 2 =0; if x 1 = 0.2 and the output is 0. Thus, the FIS does not possess additivity. Therefore, values of the RMSE parameter are increasing when modeling. For example, in [16] when modeling a similar system, RMSE (without training) is more than 120.
Another huge disadvantage of ANFIS models, which are created on the Zadeh rule, is existence of empty decisions in conclusions of the fuzzy inference. They increase with the conclusion of input parameters and FR, which are the knowledge base. The growth of the conclusion of FR is so fast, that it causes an error, which is concerned the dimension damnation. In recommendation [17], the conclusion of fuzzy

REVIEW OF THE LITERATURE
Currently, there are a large number FIS used as a decisionmaking system [3ч9]. However, these FIS in the composite rule Zadeh use hard arithmetic operations of finding the minimum and maximum [12,13,15,16]. Typically, such systems are not additive, and are not monotone [8,15]. Also hard fuzzy inference system may cause such error as the curse of dimensionality [4ч6]. This leads to the fact that in some cases FIS cannot learned. In some cases, FIS do not provide the condition of the partition of unity [7]. To resolve this issue we recommend to use parameterized MF for ensuring the partition of unity condition and continuity through the intersection point of the adjustment linguistic terms which equal 0.5. Also, to overcome the above errors in the studies recommended to use soft arithmetic operations [9,10,11,21].

MATERIALS AND METHODS
A soft algorithm of the fuzzy logic inference includes the following steps: Step 1. Fuzzification of the input parameters. The fuzzy MISO-system γ =f(х 1 х 2 ), which has 2 input and 1 output parameters should be considered. If each input parameter has 3 fuzzy terms ( Fig. 1) with a MF then the output parameter has 5 fuzzy terms , where a, b, c are the parameters of the MF; x is a quantitative value of the input parameter. rules k should be the same as the conclusion of terms m. They describe a fuzzy input parameter. For example, if MISOsystem has 2 input parameters and each of them is described by 3 terms, than the amount of FR will be defined as k=m l =3 2 =9. If the fuzzy MISO-system has 12 input parameters and each of them consists of 3 terms, than the amount of FR will be defined as k=531441. The growth of the conclusion of input parameters from 2 to 12 will increase the conclusion of fuzzy rules in 59049 times [18].
To compensate for the above drawbacks is solved the following task: the development of a fuzzy algorithm uses of soft arithmetic operations. A comparison of the proposed method with the traditional models of fuzzy inference is implemented based on the RMSE evaluation.

УПРАВЛІННЯ У ТЕХНІЧНИХ СИСТЕМАХ
Step 2. Determination of values of MF for every implication of the input parameters, based on the information received from sensors of active control systems, for example ( Fig.1 Step 3. Synthesis of the knowledge base, which contains fuzzy rules of the type «If … then» (see Table 1) [19].
Step 4. Creation of a fuzzy ratio matrix. Soft arithmetic operations are used while calculating MIN and MAX [8].
The operation of determination the soft MIN is obtained as follows: The Eq.
Therefore, the soft MISO-system, will give the value different from zero and respond to the changing parameter х 1 , if the second parameter is х 2 =0. In this case the fuzzy system will be additive in the whole range of input parameters.
The equation of parameterized soft maximum capture is obtained as follows where γ is an operator of the parameterization. If =1, then Eq. (6) will apply operations of hard-MAX. If γ =0, then Eq. (6) will apply arithmetic operations.
The Eq. (6) can be applied only with 2 operands. If there are more than 2 operands, then it is necessary to apply the soft-MAX operator 1 2 1
The conclusion of conclusions equals to the conclusion of terms of the output parameters, i.e. 5. In a hard model of the fuzzy inference the conclusion of conclusions will be   Table 2 -Fuzzy ratio matrix equal to 9. Therefore, a rational location of elements in the fuzzy ration matrix will be of the curse dimension.
Step 5. Truncation of output parameter terms, according to the equation .n is the conclusion of the conclusion of the fuzzy logic inference; n is the amount of conclusions of the fuzzy logic inference.
Step 6. Integration of the transformed terms of the output parameters [ ] Step 7. The obtained result is defuzzification by the weighted average method [20,21] Therefore (Eqs. (3)÷(10)) present a soft algorithm of the fuzzy logic inference.

EXPERIMENTS
The task of estimation of the created soft algorithm of the fuzzy logic inference assumes finding an optimal decision where RMSE is MIN [22,23] ( ) Experimental data y= f (х 1 , х 2 , … х n ) is given in Table 3. Calculation of RMSE in the hard fuzzy inference system.
The data obtained from the defuzzification of the output result. Table 4 shows applying the hard algorithm of the fuzzy logic interference.  Calculation of the RMSE parameter with the use of the experimental data (see Table 3) and the data, obtained from the modeling of the hard fuzzy inference model (see Table 4) is designated as RMSE hf =247,54.
Calculation of RMSE in the soft fuzzy inference system. RMSE of with the appliance of soft arithmetic operations is calculated in (Eqs. (3)÷(10)). The obtained data is given in Table 5.
As shown in Table 5 there are no non-sensitive zones. Thus, soft fuzzy model reacts to all changes of the input parameters and becomes additive. It helps to minimize the value of the RMSE parameter.
Calculation of the RMSE parameter with the use of the experimental data (see Table 3) and data, obtaned from the modeling of the soft fuzzy inference model (see Table 5) is defined as RMSE sf =16, 19.
It is shown that the soft model of the fuzzy logic inference is in 15,3 times more corresponded to the experimental selection in Table 3.
A graphic version of the modeling is shown in Fig. 2.
Several numerical experiments are necessary to perform to study the soft algorithm of the fuzzy inference system.
Additional imitation modeling of RMSE calculation. Some additional experiments, which allow proving effectiveness of the introduced soft algorithm, are to be performed with the random conclusions from 430 to 470. It is necessary to fill values of the output parameter in Table 3 and to calculate RMSE for hard and soft algorithms of the FIS. It is necessary to change only the values of the coefficients a and b of the output MF y if using fuzzy logic modeling Eq. (3). The value of the parameter c is not changed (see Table 6).
MIN RMSE for the soft fuzzy logic inference is obtained in the 3 rd experiment: RMSE=11.14. MIN RMSE of the hard fuzzy logic inference is obtained in the 4 th experiment: RMSE=244.88, then a=40, then b=2 (see Eq. (3)).
A graphic version of the best variants for the RMSE parameter is illustrated in Fig. 3.
The data, which show dynamics of RMSE depend on the parameter a (see Table 7).
Interpretation of the data (see Table 7) is given in Fig. 4.

RESULTS
The process of mobile robot control, which is moving along the line, is illustrated in Fig. 5. The decision of this task is made for 2 lengths: 2 meters and 2.5 meters.
There are a lot of ways to solve the task [27,28]. A robot based on the two-wheeled platform mini Q, is applied in the experiment.
Scheme and the principle of the mobile robot operation.
Mobile robot includes the following elements: A microcontroller, based on the Arduino Uno scheme, made on ATmega328p processor with 16 MHz CPU speed with 32 Kb memory; A two-wheeled platform mini Q with a disc, 2 wheels 42Ч19 mm and 2 micro motors with 3-9 V; A Motor Shield card for Arduino Uno, based on L298P driver, which allows controlling two micro motors with 5-24 V voltage; Troyka Shield card connects necessary sensors to the mobile robot; Two digital line sensors. A structural scheme of the mobile robot is shown in Fig. 6.
The line sensors, which determine the colour of the surface under the mobile robot, should be used for the robot moving along the line. The output signal from the sensor is a binary signal: logical 0 or 1 depends on the colour under the mobile robot. 1 means "black", 0 means "white".
The use of the Troyka Shield card allows connecting to the Motor Shield. Three different signals are transmitted through the three-wire loopback: G means "ground"; V means "voltage"; S means "signal". The Motor Shiels is  used to control two wheel rotation. Its work is similar to the scheme H-bridge, for example l293d. Two pins are applied to indicate the direction of the wheel rotation. For example, if there is a signal 1:1, then the mobile robot moves straight ahead.
If the signal 1:0, then the mobile robot rotates around its axis. If the signal 0:0, then the mobile robot moves back. The other two pins are used to supply the voltage to the micro motors ranging from 0 to 5 V. The process of control is implemented with the help of pulse-wide modulation. If the pin is under 5V, then the mobile robot moves at the maximum speed. If V=0, then the mobile robot does not move. The Motor Shield card is put on the Arduino Uno.
Fuzzy system of the mobile robot control. Input parameters for the mobile robot control are linguistic variables: power supply voltage -u (V) and robot's weight -m (gr). It is possible to use the range of values of linguistic variables. If the power voltage is within the range from 6 to 9 V (see Fig. 7 Fig. 7, b). The MF is designated as Eq. (3).
The MF of the output linguistic variable Speed, depends on the ADC precision. Therefore its range is occurred as [0 … 255] (see Fig.7, c). This variable is also defined by the sigmoidal function Eq. Hard and soft algorithms of the fuzzy logic inference should be used for calculation the variable Speed. A matrix should be created for the robot with a FIS applying the (Eqs. (5)÷ (7)) outlined in Table 8.
Defuzzification uses a method of the center of gravity to determinate the output variable Speed outlined in Table 9 and Table 10.
Calculation of the RMSE by the methods presented in Section 4 shows that RMSE sf =11.93 in modeling of the soft model of the fuzzy inference and RMSE hf =35.59 in modeling of the hard model of FIS. The area, which is grey (see Table  10), is an area of non-sensitivity of the mobile robot with a FIS. The output of the MISO system will be the same if the robot's weight is in the range from 180 to 190 grams. It will also influence its work. Therefore, the robot will not move if the variable Speed equals 0. The results are illustrated in Fig. 8 and in Table 9     The moving time of the mobile robot along the line t (see Fig. 5) and the amount of successful robot's passes are estimated withing the experiment. The experimental data of the mobile robot when it moves along the first track (see Fig. 5, a) are given in Table 11. The experimental data of the mobile robot when it moves along the second track are given in Fig. 5, b (see Table 12).
The analysis of the experimental results (see Table 11) shows that the mobile robot has not done a move along the track in the experiment no. 1, when using the hard FIS. However, the mobile robot has successfully done all the movements along the track in the all experiments (see Table 11), when using the soft FIS. The result of the robot moving along the track no.2 are similar to the above mentioned result. Thus, the experiments show that the process of the mobile robot control is unstable.
Analyzing the results given in Table 12 one can conclude that if the power voltage supply of the mobile robot increases then the time of its passing the track decreases (see Fig. 9).

DISCUSSION
Different methods are possible to minimize RMSE. For example, ANFIS technology of finding the optimal conclusion of fuzzy rules is described in paper [24,25,26]. Nevertheless, the present research proves that it is possible to decrease RMSE at 30 % off if change the parameters a and b of the sigmoidal function Eq.  Table 3), if the parameters a and b of the sigmoidal function are changed (see Eq. (3)). Other parameters (see Table 3) are constant.
According to the results of the experiments, it is possible to conclude that: RMSE is 20,7 times less when using a soft fuzzy algorithm in comparison with a hard one. It is possible to minimize RMSE in 1,46 times if the parameters of the sigmoidal MF are changed. The mobile robot does not always accomplish a task if the hard FIS is used. The decision time for a track is minimized if the power voltage supply is increased. Therefore, a soft fuzzy system of decision-making is recommended to apply for a mobile robot control.

CONCLUSION
The problem of building a soft algorithm of fuzzy inference used to control a mobile robot is solved in the paper.
Scientific novelty of the results obtained in the article is that for the first time proposed soft algorithm, which in fuzzy inference to find minima and maxima uses soft arithmetic operators that provides additive of fuzzy control systems.
The practical significance of these results is that for the movement of the mobile robot along a line the specialized software based on the soft fuzzy algorithm is developed.