Fuzzy Control Systems Process of Fuzzy Control

Transcription

1 Fuzzy Control Systems The most widespread use of fuzzy logic today is in fuzzy control applications. Across section of applications that have successfully used fuzzy control includes: Environmental Control Air conditioners Humidifiers Domestic Goods Washing machines/dryers Vacuum cleaners Microwave ovens Refrigerators Consumer Electronics Television Photocopier Video camera auto focus Automotive Systems Automatic Gearbox Four Wheel steering Seat/Mirror control system Process of Fuzzy Control Before the development of fuzzy logic controller ( FLC) systems there were essentially two alternatives to process control: A process was controlled by either a human operator or a computerized direct digital control system (DDC). The function of such a direct digital control system which is shown in figure-27 can be described as follows: The problem consists in dimensioning a control algorithm based on the error vector e = (e 1, e 2,, e p ) that generates an output vector u = (u 1, u 2,, u r ) to the process so that the output vector y = (y 1, y 2,, y p ) of the process is close to or eventually equal to the set point vector r = (r 1, r 2,.. 41

2 ., r p ). In other words, we want to control the process by means of an algorithm of the following general form u[(k + 1)T] = f(u[kt], u[(k - 1)T],, u[0], e(k+ l)t],e[kt],..., e[0]) where k = 1,2,..., and T is the sampling time. Figure -27 DDC control system Figure -28 The Fuzzy controller and its relation to a conventional control loop The structure of a fuzzy logical controller is depicted in figure 28. The process of fuzzy control can roughly be described as shown in figure 29. Fuzzy Rule- Based Real world Fuzzification Fuzzy value Fuzzy Inference Engine Fuzzy value Defuzzification Real world Figure -29 The Fuzzy controller System 42

3 1. Fuzzification The fuzzification is defined as a mapping from a real-word point to a fuzzy set using a specific membership function that is described previously. Thus, the first step is to convert the measured signal x (which might be the error signal in a control system) into a set of fuzzy variables. It is done by giving values (these will be our fuzzy variables) to each of a set of membership functions. The values for each membership function (x) are determined by the original signal x and the shape of the membership. For example, let us say that the fuzzifier splits the signal x into five fuzzy levels as follows (see figure 30): x is large positive: LP x is medium positive: MP x is small: S x is medium negative: MN x is large negative: LN Input Signal x Figure-30 Five level Fuzzifier As the input to the fuzzifier changes in the range -10v to +10v, then the corresponding fuzzy variables will also change. A practical fuzzifier would have a measured signal sensor at its input and would provide at its output the values (fuzzy variables) corresponding to the membership functions. For example, if a sensor signal with an output voltage of 2v is applied to the five level fuzzifier, the resulting set of fuzzy variables is (as shown in figure 31): LN = 0 MN = 0 S = 0.6 MP = 0.4 LP = 0 43

4 Figure-31 Complete set of membership functions for five level fuzzification 2. Fuzzy Rule-Base A fuzzy rule-base consists of a set of fuzzy IF-THEN rules. The control rules are defined as fuzzy conditional statements of this type. As an example. the fuzzy IF-THEN rule can be used to control the speed (SP) of a motor by changing the speed (CS). This can take the following form: "If SP is PB then CS is NB" 3. Fuzzy Inference Engine In fuzzy inference engine, fuzzy logic principles are used to combine the fuzzy IF-THEN rules in the fuzzy rule-base into a mapping from one fuzzy set to another fuzzy set. The min-max compositional rule of inference is then used to derive fuzzy control statements from observed observations of the states of the process. If several rules are combined by "else," this is interpreted as the union operator "max". for example: IF {error S} AND {output_rate LP} THEN {control LN} OR IF {error S} AND {output_rate LN} THEN {control LP} 4. Defuzzification The defuzzification represents the last step in building a fuzzy logic process. Defuzzification can be defined as a mapping from a fuzzy 44

5 values that results from the previous stages into a real-word value. In other word, It combines the fuzzy variables to give corresponding real (crisp or non-fuzzy) signal which can then be used to perform some action. For example a five level defuzzifier block which is shown in figure 32, will have inputs corresponding to the following five actions: LP : Output signal large (positive) MP : Output medium (positive) S : Output signal small MN: Output signal medium (negative) LN : Output signal large (negative) Figure -32 Block diagram of defuzzifier The defuzzifier combines the information in the fuzzy inputs to obtain a single crisp (non -fuzzy) output variable. There are a number of defuzzification methods, such as, center average, maximum defuzzifier, and center of gravity. The simplest and the most used one is the center of gravity. It works as like this: If the fuzzy level LP,, LN have membership values that are labeled 1,..., 5, then the crisp output signal u is defined as: For example, the values of the u i of the membership functions shown in figure 31 are, u 1 = 10V, u 2 = 5V, u 3 = 0V, u 4 = -5V, and u 5 = -10V, and 45

6 corresponding to the central points of the fuzzy classes LP; MP; S; MN; LN at the input to the defuzzifier. Proportional plus derivative fuzzy controller The fuzzy proportional controller can be extended to cover integral and derivative control. Here we outline just the derivative control extension. In this case the fuzzy logic operates on the error signal e(t) and the derivative of the output signal dy(t)/dt and produces an output from its defuzzifier which is the control signal u(t). The fuzzy logic controller bases its actions on the two signals, the error and the rate of change of the output. The output derivative is either available as a direct measurement from the system or by using an observer of the system states. Design principle of fuzzy logic controller To design a fuzzy logic controller that has multi inputs and single output, the following steps must be considered: 1- Determine the inputs and the output. 2- Put the control knowledge into rule-base, which includes. 2.1 Linguistic description, that describes the input and output linguistic variable. 2.2 Quantify the linguistic variable: linguistic variables assume linguistic values, such as: LP, MP, S, MN, LN. next the designer determine the type of membership functions used to fully quantify these fuzzy sets so that the user may automate the control rules specified by expert. 2.3 Specify the set of rules. A convenient way to list all possible rules for the case where there are not many inputs to he fuzzy controller is to use tabular representation. 46

7 3- Matching: Determine which rule to use: this step is done using the inference mechanism, which involves two steps: 3.1 The IF parts of all the rules are compared to the controller inputs to determine which rules apply to the current situation. 3.2 Next, THEN parts (what control action to take) are determined using the rules that have been determined to apply at the current time. However, the steps used by the user to calculate the control action are: a- Calculate error and change in error. According to that define the fuzzy sets. b- Calculate the degree of activation for each rule, this can be achieved by implementing the IF pars of all fuzzy rules (finding the minimum values of membership functions for error and change in error. c- Calculate the control vector (u i ) for each rule. d- Calculate the control action by using the defuzzification operation. The center of gravity (CoG) defuzzification method can be used u N I n n 1 N n 1 Where, I n is the value of the interval, n= 1, 2,, N, and N is the total no. of intervals. Example 22: The following table illustrates the linguistic sets and its membership degree corresponding to each interval, suppose that the number of intervals are 21. The linguistic variables are quantified into five linguistic values (fussy sets). NB stands for (Negative Big), NS n n 47

8 stands for Negative small, Z stands for Zero, PS stands for Positive Small, PB stands for Positive Big. Find the control action if the error is ( - 0.9) and the change in error is (0.1) using the fuzzy rules and the rules sets given: Fuzzy rules are: NB NS Z PS PB NB PB PB PB PS Z NS PB PS PS Z NS Z PB PS Z NS NB PS PS Z NS NS NB PB Z NS NB NB NB The rules set: Interval Measured Scaled PB PS Z NS NB signal signal Solution: Error (E) = scaled E = -9 Then it is NB Change in error (CE) = 0.1 scaled CE = 1 Then it is PS and Z 48

9 Therefore two rules will be fired: IF E is NB AND CE is PS THEN U is PS IF E is NB AND CE is Z THEN U is PB For the first rule E = 1 and CE = 0.3 thus, u = 0.3 For the second rule E = 1 and CE = 0.7 thus, u = 0.7 Now, the control action will be: U = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,.3,.3,.3,.3,.3,.5,.7,.7,.7,.7} Finally, we apply center of gravity (CoG) defuzzification in order to obtain final crisp output: (0* 10) (0* 9) (0* 8)... (.3*1) (.3*2)... (.5*6)... (.7*10) 31.3 U NB NS Z PS PB E=-9 CE=1 NB NS Z PS PB CoG 6.5 Control Action 49