Before the microcomputers appeared, the automatic process control and machine control was based on analog regulators. Digital control algorithms are slower then the analog ones.

 
 Experiment system detail

Flexibility in control realization and modifications are easy using software, simply changing the code. New methods of control are introduced. In many different conditions, these prove to be very efficient. A specific application: neural networks and fuzzy logic use on system control.

Digital Controlling

Digital control offers opportunities for new modalities, that analog regulators were not able to offer (or are difficult to implement): data acquisition, system identification, adaptive control etc. (terms known from the theory of digital controlling). Two important terms are connected with digital systems: discretisation by time (sampling time) and quantisation by level. The sample rate defines successive moments in which sampling of continual values is being performed. As a result of AD conversion, quatisation of continual values appear. Intelligent control application can be understood on the need of performing a regulation of a physical dimension in next circumstances:

Fuzzy Logic

In the classical theory of groups, we say that a certain element belongs or does not belong to the appropriate group.

The experiment - water temperature regulation. The water can be hot or cold. Such approach is simplification of real situation. In this example, the graduations of temperature are neglected. This approach is thus unprecise. In everyday life, we can hear terms like, little bit colder, little bit warmer, cold, hot, very hot, etc. One solution is to introduce terms like in the picture 1, : Very Negative, Little Negative, Around Zero, Little Positive, and Very Positive (VN, LN, AZ, LP, VP). The fuzzy logic theory was introduced for this kind of representation. Lotfi Zadeh was its inventor. A value can belong to different representations, this way its description becomes uncertain, fuzzy. For instance, the fuzzy value F=3.5 is LP with belonging degree n=0.25 and VP with n=0.75.

Expert Systems

Fuzzy logic usually is associated with expert systems. In these, the actions perform in real life by experts, can be used for programming computer systems that later can imitate or perform the job of experts. Manual control involves man's intelligent action of monitoring and controlling  the certain process. With this approach, we can accomplish good control, because the expert knows how to guide the system. In order to write a program for FLC, it is necessary to introduce a group of fuzzy rules. The rules describe the expert's action. The situation in the control system can be described as next: the expert monitors the system (example: the deviation of temperature value from the given, initial, temperature) and decides how much to warm up the water.

Fuzzy rules

In expert systems, the knowledge of the expert is described the next way: IF the mistake has this value THEN heat for that value long time. In most of the programming languages, it can be implemented with IF THEN rule. The next situation shows an example of control: water is heated. At the start it has the room temperature, lets say about 22 C degrees centigrade and has to be heated to 40 C. An  "average" expert will probably react the next way: lets heat the water until it reaches the 40 C, then, we will maintain the temperature to 40 C, on the principle of work of an ON/OFF regulator. Instead of using a classical ON/OFF, we often use the ON/OFF regulator with hysteresis. This means the temperature is maintained in certain intervals, for instance between 39 C and 41 C. The question of quality of this way of control depends on the demands of the buyer, that orders the system (temperature regulator). First, it is useful to notice that the temperature is oscillating between 39 and 41 C. The second, if in the system exists a noticeable delay (the energy of the heater is not instantly transferred) it is logical to expect that in temperature reaching, there will be an overstepping of the value of 41 C (the higher delay, the higher temperature) and the interval (39, 41 C) is expanding.

In case of ON/OFF regulator, in fuzzy logic, the temperature regulation will be implemented the next way :

IF temperature < 39 C THEN heater ON

IF temperature > 41 C THEN heater OFF

This regulator can be easily implemented both in software or hardware environment (digital or analog control respectively) and it is pretty simple. We have to be aware that there also exist groups of rules that describe the temperature regulation more precisely, in this system.

Bart Kosko showed theoretically that the Boolean algebra (1 or 0) is only a special case of fuzzy logic, actually its subgroup. The ON/OFF regulator is only a special case of FLC. FLC should be considered as an algorithm that based on two values (error and error change) calculates the control. It means how much (inside a period T=4 seconds) to heat, to achieve the desired temperature (Picture 5).

The error (given value - instant value) gives the information how much the instant temperature is away from the given one. The error change (instant error - previous error) gives the information about the speed of achieving the given value (Picture 3). These two values are the input in FLC and they have specific values (in C).

Achieving the real value of control

The data processing of the achieved information is performed in next 3 phases:

1 . Scaling

2 . Look-up table  (calculating the fuzzy control)

3 . Inverse scaling

Scaling: The error and the error change are scaled. For instance, if the given value is 40 C, and the instant value is 30 C then the error is e=10 C. If the previous error value was 28 C, then de=2 C. These values are sampled in k*T moments and k is an integer.  The error and error change are recalculated into fuzzy variables ef and def. The fuzzy values are supposed to be in the interval between -4 and 4. To these values an interval of real numbers is attached (e and de). On the basis of given value and starting (initial) temperature value, we determine the emin and emax. For instance, if the given temperature value is 40C and starting (initial) value is 10 C, we calculate the emin = - 30C and emax = 30C. We presume that during 4 sec,  the temperature can maximally rise for 1 C; it comes out that demin = - 1 C and demax = 1 C.

LOOK UP TABLE: (calculating the fuzzy control). When we want to determine the ef and def, on the basis of the Look_up table we determine the uf (fuzzy control).

Look-up table

Picture 6. 3D projection of Look-up table 

Descaling: Control in real time, demands the translation of fuzzy control into real value. As the period is T=4 sec, the control determines in which area of the interval the heater is ON. (Picture 5). The control has the shape of PWM signal, that serves to turn ON and OFF the heater. In the previous example, the steps of scaling and descaling are pretty simple. Now we are going to explain the step of creating the LOOK_UP table, that in fact is the base of the control algorithm itself.

Creating the Look-up table

Before we start to elaborate the creation the Look-up table, lets imagine we already have one created. Analyzing the Look-up table, we can notice that control can only be positive. That means, the system can only be heated. (it cools by turning OFF the heater). The values of uf are 1,2,3 or 4 that means in the period of 4 sec the heater is ON for 1,2,3 or 4 sec. We can notice that fuzzy control is much more precise then the ON/OFF control. How more precise it is, depends, on the structure of FLC, on the software. The procedure of achieving the values (0,1,2,3 and 4) that are described in the Look-up table can be understood on the basis of a help table usually called Fuzzy Associative Memory-Picture 7.

1.    IF e LN                                   THEN u AZ

2.    IF e AZ                                  THEN u AZ

3.    IF e LP      AND de VN       THEN u LP

4.    IF e LP      AND de LN        THEN u LP

5.    IF e LP      AND de AZ        THEN u VP

6.    IF e VP                                  THEN u VP

A comment on the rules written above: 1. and 2. do not allow the temperature to be higher then the given one. The rules 3. and 4. restrict not to overheat the water if the error is not too big and there exists a tendency to achieve the given value of temperature. Rule 5. says, if the error is not too big, but there does not exist a tendency to achieve the given value, then we have to heat more intensely (longer). Rule 6. says that for a big error we have to heat with much greater intensity. In a certain moment, there exists one fuzzy error (ef) and one fuzzy error change (def). On the basis of picture 7, we determine the degree of belonging of ef and def for the first rule. Later we determine the minimum of these two values, and this way we achieve the dashed surface. The procedure has to be repeated for all the other rules. The surfaces we get from the calculations have to be summed, and by one of the methods, we determine the fuzzy controlling.

Determining the control

For calculating the fuzzy control, it is necessary to determine the center of gravity, using the shapes determined in the previous steps. The simplified calculation is based on the next formula: uf=(F1*n1+...+Fk*k)/(n1+...+nk), k=1 : 6. The analogy is certain with the calculation of center of gravity in mechanics. For instance, if ef = 0.5 and def = - 1 then we can see from Picture 7  that uf=(0*1+2*0.25+4*0.25)/(1+0.25+0.25)=1. Each pair of values of e and de can be  from the domain (-4,4)X(-4,4). This way, uf can be calculated.  For forming the Look-up table, we need integers-positive and negative (-4,-3,...,3,4). We enter the so created Look-up table into the program that is performed by the microcontroller. There,  it is used for real time control. After we described the procedure of creating the Look-up table, it is useful to try to find out the possibilities for adjusting the control algorithm, to determine the parameters upon which the FLC depends:

· Setting up the resolution of fuzzy variables (for fixed limits of real values). The resolution that we introduced in our example goes from - 4 to 4. There are no limits    for the values to be (-2,2) or (-10,10). In the first case, less precise control is chosen, while in the second, we chose a more precise control.

· The number of labels is 5 (very negative,...,very positive). This number can be 3 or 7 or 9 etc. The smaller number of labels supposes smaller precision, while greater number of labels supposes more precise control.

· The function of belonging can be of different dimensions and shapes: trapezoidal, Gaussian, triangular etc. In our example, we used the function of belonging in a shape of triangle.

· The number of IF...THEN rules is not fixed; it is determined by the expert.

From the enumerated objections, we can conclude where we can find ways to make adjustments. The understanding and mastering of the process is of great importance for good FLC adjustment. Before a practical use of FLC, it is desirable to perform a simulation of control on the computer. This way we are able to test the possible behavior of the process in the procedure of control. The most commonly used program package for this purpose is MATLAB.

The description of a real system

The central part of FLC is a board with Motorola's  MC68HC11F1 microcontroller working on 8 MHz. On the board, behind the controller there are included: 32 Kb of memory, power regulator 7805, power monitor with reset circuit TL7705, and all the necessary passive components for work. For indicating the temperature, a 7-segment display is chosen, with 4 digits. driven by Motorola's MC14499 circuit. 

Picture of experimental system

The heater is connected to the mains, via a tyristor, that is activated by an optocupler with zero-crossing circuit MOC 3043. The optocupler is connected to the zero line on port G ( PG0 ), and activates on logical ‘0’. The heater is of 150W. It serves to convert the digital signal of PWM into the action of water heating. Next to the heater, in the box, we have an NTC resistor, termomether and mixer. The mixing is of extreme importance, because it reduces the delay in the system, and guarantees a linear temperature distribution. In this case, the temperature difference between the top and the bottom of the vessel equals 0.2 C. We can consider it as a very good result. The NTC resistor serves as a measuring instrument, whose purpose is to give the information about the instant temperature values.

For temperature measuring, an NTC resistor of 47K, from Simens is used with a tolerance of 10%. The label  NTC (Negative Temperature Coefficient) describes the nature of the resistor, that in dependence of temperature change, changes its resistance. The change can be positive-the resistance increases with temperature increase and negative-the resistance decreases, with temperature increase. In this case on 0 C  resistance is 165.642K.  On 25 C (that is the referent one) 47K. The label  47K describes the value of the resistor, on the referent temperature.

The resistance dependence can be also mathematically described the next way: R=RO exp(b/T). RO is a constant that depends upon the material and dimensions of the resistor, b is a coefficient of temperature sensitivity and T is the temperature in (K). With temperature change, the resistance of NTC changes, and the voltage with it  that is recorded on the AD converter. We used an internal 8 bit AD converter with precision of   1LSB. For temperature displaying, it is necessary to recalculate the voltage of NTC into temperature. There are two possibilities. One is - using the formula, we directly calculate the temperature. The second one is, to experimentally determine the table that linearise the real characteristic. In our example, we used the second method. The labels PE0 and PG0 represent the ports of the microcontroller. For instance, PE0 is the E port (the input of the AD converter), the zero line. On the basis of the used components, the maximal accuracy (precision) that can be achieved is 1 C, that is usual in temperature measuring. Using higher quality parts and AD converters, and higher resolution, better results can be achieved. On the other side, the costs of the entire project increase.

Picture 10. Program execution diagram

The procedure of control is the next: the referent value is given, and in periodical time intervals (sampling rate is 4 sec) a digitalized information of the instant value of temperature is acquired. After that, we calculate the error and error change. Using these (previously scaled) from the Look-up table, we read the scaled value of control (from 0 to 4) that after this are descaled and as such, represent the interval during which the heater is turned ON. The job of the microcontroller is to insure that the heater will be ON as long as it is calculated. On Picture 10,  a block diagram of the described control system is shown.

Listing 1. Program fazi1.c

Listing 2. Program fazi2.c

Listing 3. Program fazi3.c

The control algorithm is placed in the interrupt routine, and activates each 16.384 ms. To adjust the sampling period of 4 s, we need around 244 RTI interrupts. When the working time of the heater is calculated (0<TIME_ON<4) it is necessary to ensure turning OFF of the heater, after the elapsing of the time interval. This way generating a PWM signal with a 4 s period, that represents the control. On listing 1, variables used in the program are declared, and the initialization of the Look-up table is done.  Listing 2 represents RTI ( Real Time Interrupt) the interrupt routine, in which the results of the AD conversion are processed and the times of turning ON of the heater are calculated. In the interval of 4 seconds, the program enters 244 times the interrupt routine, and each time the value from the AD converter is red. Their total sum (PRIV) after 4 s elapse, is devided by 244, this way very regular temperature readings are accomplished. Calculating the medium values, we can achieve more precise results and we can reduce the influence of noise. Listing 3 is the main program. In the first part, we initialize the variables that appear in the program, SPI communication and RTI interrupt are declared. In the closed-loop, there are only two functions. The first one performs the conversion of temperature from binary into decimal values (necessary for the display) and the second function sends the decimal ciphers of the instant temperature values trough SPI communications.

Comments on Results

Before the implementation of the control algorithm on a system, we performed a simulation. The simulation is performed on the bases of the law of energy transformation. The transformation of electrical energy into heat can be described with next equation: P*dt=m*c*dT. P=150W, heater power, dt is the heater working time, m=0.4 kg mass of water, c=4186J/Kg*K, specific heat capacity of water and dT is the change of water temperature as a result of heating. After the simulation, the control algorithm is tested on a real system. Each 4 sec the values red from the AD converter are stored in the external RAM and after that, this value is transferred by serial communication to the computer, where the data are processed. 

Picture 11.simulated results of the system with Look-up table

Picture 12.real results of the system

Comparing the simulated and achieved real results of the system, we can notice a large degree of similarity. The difference between the simulated and real system, appears as a result of noise in the system, and as the impossibility to simulate the entire real system completely. Instead of conclusion, we offer the answer to the question why fuzzy controllers are convenient for automation engineers? They describe the solution without complex mathematical model of the system.

· Optimizes the already existing solutions, with the purpose to achieve a more simple and efficient end product

· Reduces the price of the end product on the bases of simplified procedure of modeling

· Clarifies the system. System is understandable, easier for maintenance and upgrading

· Higher resistance to errors and changes of the system environment

· Increases the system robustness without decreasing its sensitivity

On the bases of the performed experiments, we can conclude that fuzzy logic controllers are applicable in the problems of temperature regulation. We can not give a final mark about the possibilities and quality, because the system was not tested in the conditions of disturbances, nor compared with other control algorithms (from which the PID regulator is the typical representative).

autors: Stevan Tošić dipl. eng. & Nebojša Matić

translated by: Simeon Todoran

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