CHAPTER 5 FUZZY LOGIC CONTROLLER FOR THE CONICAL TANK PROCESS

Transcription

1 99 CHAPTER 5 FUZZY LOGIC CONTROLLER FOR THE CONICAL TANK PROCESS 5.1 INTRODUCTION Fuzzy Logic Controllers are normally preferred for control applications, due to their fault tolerance, knowledge representation, expert knowledge, nonlinearity, uncertainty, real time operation, etc. The Fuzzy Logic Controller basically consists of three elements, viz, fuzzification, fuzzy inference engine, and defuzzification. To control the nonlinear conical tank process, the Mamdani type of Fuzzy Logic Controller is designed. In this work, the Fuzzy Logic Controller is selected as a controller for the nonlinear conical tank process, and its performance is compared with that of the Neuro tuned PI Controller. The simulation results are obtained by the servo, regulatory and servo-regulatory operations, for the above mentioned controllers, for the conical tank level process. 5.2 BLOCK DIAGRAM REPRESENTATION OF THE FUZZY LOGIC CONTROLLER The fuzzy set theory concept was proposed by Lotfi A.Zadeh of the University of California in his paper published in The fuzzy logic representations found in the Fuzzy set theory, try to capture the way humans represent and reason with real world knowledge in the context of uncertainty. The word fuzziness means, vagueness, cloudiness, unclearness and

2 100 indistinctness ; it was used in Washing machines, Refrigerators, Nuclear reactors, Automobiles, and Traffic Light Control, and many such uses were developed by the Japanese. The input variables in the fuzzy control system can be mapped into the set of membership functions which is termed as a Fuzzy set. The set of variables obeys the Boolean logic expression, and has the same characteristics of individual elements called Crisp set. The block diagram representation of a Fuzzy Logic Controller is shown in Figure 5.1. The Fuzzy Logic Controller consists of three basic elements: Fuzzification, Fuzzy Inference Engine, Defuzzification. Knowledge Base Rule Base Reference Input + _ Fuzzification Fuzzy Inference Engine Defuzzification Process Controlled Output Sensor Figure 5.1 Block diagram representation of Fuzzy Logic Controller 1. Fuzzification The Fuzzy Logic Controller requires that each input or output variable, which defines the control surface, can be expressed in fuzzy set notations using Linguistic levels. The linguistic values of each input and output variable divide its universe of discourse into adjacent intervals, to form the membership functions. The membership value denotes the extent to which

3 101 a variable belongs to a particular level. Fuzzification is the process of converting the crisp value into a fuzzy one. The different methods used for fuzzification may be : Ranking order method, Intuition, Inference, Neural networks, Genetic Algorithm, Inductive reasoning method, etc. For this work, the rank ordering method is preferred and used. 2. Knowledge based system The Knowledge based system consists of a data base and a rule base. Data based knowledge is used to form the linguistic variables to match the process with a controlled variable. The physical normalization and scaling factor can be done by this part. 3. Fuzzy Rules The fuzzy rule based system, which is used to represent the human knowledge, is in the form of natural language expression. The behaviour of the control surface relates the input and output variables of the system, and is governed by a set of rules. Normally, the IF-THEN rule form is preferred. A typical rule would be : where If x is A and y is B, then Z = f(x,y) (5.1) A,B = Fuzzy sets in the antecedent Z= f(x,y) = Crisp function consequence to the input variables x and y 4. Fuzzy Inference Engine When a set of input variables are read, a rule that has any degree of truth in its premise, fires and contributes to the forming of the control surface by approximately modifying it. When all the rules are fired, then the resulting

4 102 control surface is expressed as a fuzzy set, to represent the constraint output. The above operation is termed as Fuzzy Inference Engine. Two types of popular methods used for designing the inference engine of the Fuzzy Logic Controller are: Composition based Interface (Through Max-Min Composition operation) and Rule base matrix. For this work, the Mamdani Fuzzy model is used for the conical tank process. The inference used in this work is the max-min method where the min operation is used for the and conjunction, and max is used for the or conjunction. Thus, the membership function of the input is obtained for each rule. The value is the firing value for each rule. 5. Defuzzification Defuzzification is the block which converts the fuzzy quantity into crisp quantity. The different methods available for defuzzification may be : Maximum membership function method, Weighted average method, First in or Last out of the maximum, Centre of the Largest area, etc. (Thimorthy J. Ross). The most prevalent method is the centroid method, which utilizes the following formula: µ ( ) = µ( ) µ( ) (5.2) where µ=membership degree of the output of x 5.3 DEVELOPMENT OF THE FUZZY LOGIC CONTROLLER To design the Fuzzy Logic Controller (FLC), the variable which can represent the dynamic performance of the plant to be controlled, should

5 103 be chosen as the input to the controller. The Fuzzy Logic Controller has two input variables, namely, the error (e) and the rate of change of error (de), which produces the control signal (cs). The input and output variables are converted into Linguistic variables. This case consists of five fuzzy subsets, labeled as: Negatively Big (NB), Negatively Small (NS), Absolutely Zero (AZ), Positively Small (PS) and Positively Big (PB). Each membership function has two parameters, that may be the centre and the width of the triangular functions Selection of the Triangular Membership function of the Fuzzy Logic Controller For the design of the membership function, normally the triangle is selected. The Justification for selecting the triangular membership function in the Fuzzy Logic Controller is as follows: i. The triangular membership function has a left point, centre point and right point, by which we can calculate the slope of the triangle and obtain the best performance. ii. By changing the overlap (Point of cross-over between successive triangles) and sensitivity (Modify the shape of the membership function), the effect on the performance of the controller is modified. iii. The triangular membership function response produces : satisfactory transient response without any oscillations and also produces minimum settling time. minimum steady state error. speed of response is fast, and similarly, the speed of the overshoot is to be exactly equal to zero.

6 104 The minimum error in the set point changes, and a negligible overshoot at sudden load also changes Membership function of the Fuzzy Logic Controller follows: The membership functions of the Fuzzy Logic Controller are as 1. Membership function of the input variable error For the Fuzzy Logic Controller, the membership function of the input variable error, and the degree of membership function are shown in Figure 5.2. Figure 5.2 Membership function for the input variable error representation in FLC 2. Membership function for the input variable of the rate of change of error For the Fuzzy Logic Controller, the membership function of the input variable of the rate of change of error, and the degree of membership function are shown in Figure 5.3.

7 105 Figure 5.3 Membership functions for the input variable of the rate of change of error representation in FLC 3. Membership function for the Output or Controller variable For the Fuzzy Logic Controller, the membership function of the output variable or control signal, and the degree of membership function are shown in Figure 5.4. Figure 5.4 Membership functions for the output variable representation in FLC

8 Rule base tabulation of the Fuzzy Logic Controller The input error (e) and the rate of change of error (de) combination which produces the change in the control signal (cs) for the fuzzy rule base, is shown in Tabulation 5.1. The tabulation can be obtained, based on five fuzzy Linguistic variables as: Negatively Big (NB), Negatively Small (NS), Absolutely Zero (AZ), Positively Small (PS) and Positively Big (PB). Table 5.1 Rule base tabulation for the Fuzzy Logic Controller de e PB PS AZ NS NB PB PB PB PB PS AZ PS PB PS PS AZ NS AZ PB PS AZ NS NB NS PS AZ NS NS NB NB AZ NS NB NB NB 5.4 SIMULATION RESULTS Servo operation with the Fuzzy Logic Controller In the servo operation, the process with load variable needs to be a constant, and the set point value is a variable. The closed loop servo response for the nonlinear conical tank process is obtained, for the variation of the set point value, using the Fuzzy Logic Controller. For the servo operation, the simulation diagram for the Fuzzy Logic Controller with the height of 20 cm from 0 to 500 Seconds and further the height is increased 10 cm from 500 to 1000 Seconds is shown in Figure 5.5.

9 107 e ISE ITSE ITAE IAE Performance Indices 1 Display 1 de Memory1 Fin Lev el e Fuzzy Logic Controller1 Conical Tank 1 Scope 1 Setpoint Level 1 Setpoint Level 2 Figure 5.5 Simulation diagram of the Fuzzy Logic Controller for the conical tank process in the Servo operation The simulated diagram of the Fuzzy Inference System (FIS), Rule viewer representation and Surface viewer representation of the designed Fuzzy Logic Controller are shown in Figures 5.6, 5.7 and Fuzzy Inference System Editor The Fuzzy Inference System (FIS) editor displays the information about a fuzzy inference system. In the fuzzy inference system, the two inputs represented are: the left side of the inputs are error (e) and rate of change of error (de), and the right side of the output is the control signal (cs). The

10 108 simulated diagram of the Fuzzy Inference System (FIS) for the design of the Fuzzy Logic Controller is shown in Figure 5.6. Figure 5.6 Simulated diagram of the Fuzzy Inference System 2. Rule viewer The rule viewer shows how the shape of certain membership functions influences the overall result. The two inputs variables are : error (e) and the rate of change of error (de), which produce the control signal (cs). The simulated diagram of the rule viewer for the design of the Fuzzy Logic Controller is shown in Figure 5.7.

11 109 Figure 5.7 Simulated rule viewer of the Fuzzy Logic Controller 3. Surface Viewer The surface viewer is a three dimensional curve that represents the mapping. The surface viewer represents the 2-inputs for our application X (error), Y (rate of change of error) and the single output Z (control signal). The simulated diagram of the surface viewer for the design of the Fuzzy Logic Controller is shown in Figure 5.8.

12 110 Figure 5.8 Simulated surface viewer of the Fuzzy Logic Controller The simulated output graphs for Level versus Time, Error signal versus Time and Control signal versus Time are shown in Figures 5.9, 5.10 and 5.11.The control signal magnitude is made consistent through gain adjustment. For the servo operation, the performance index values such as ISE, ITSE, IAE, ITAE, and the Time domain specifications are obtained and tabulated in Table 5.2.

13 111 Figure 5.9 Process variables versus Time graph for the Servo operation of the Fuzzy Logic Controller for the conical tank process with the height of 20 cm and further the height is increased 10 cm Figure 5.10 Error signal versus Time graph for the Servo operation of the Fuzzy Logic Controller for the conical tank process

14 112 Figure 5.11 Control signal versus Time graph for the Servo Operation of the Fuzzy Logic Controller for the Conical tank process Regulatory operation with the Fuzzy Logic Controller In the regulatory operation, the set point value needs to be a constant and the process with the load variable is a variable. The closed loop regulatory response for the nonlinear conical tank process, using the Fuzzy Logic Controller is obtained for the variation of the load. For the regulatory operation, the simulation diagram of the Fuzzy Logic Controller for the height of 30 cm from 0 to 1000 Seconds with +10% Load changes after 800 Seconds, is shown in Figure The simulated output graphs for Level versus Time, Error signal versus Time and Control signal versus Time, are shown in Figures 5.13, 5.14 and For the regulatory operation, the performance index values such as ISE, ITSE, IAE, ITAE, and the Time domain specifications are obtained and tabulated in Table 5.2.

15 113 e ISE ITSE ITAE IAE Perfo rma nce Ind ices 1 Display 1 de M em ory1 Fin e Fuzzy Logic Control ler1 D Lev el Conica l T ank 1 Scope 1 Setpoint Level 1 Disturbance 1 Figure 5.12 Simulation diagram of the Fuzzy Logic Controller for the conical tank process in the Regulatory operation Figure 5.13 Process Level versus Time graph for the Regulatory Operation of the Fuzzy Logic Controller for the Conical tank for the height of 30 cm with +10% Load changes after 800 Seconds

16 114 Figure 5.14 Error signal versus Time graph for the Regulatory operation of the Fuzzy Logic Controller for the Conical tank process Figure 5.15 Control signal versus Time graph for the Regulatory Operation of the Fuzzy Logic Controller for the Conical tank process

17 Servo-Regulatory operation with the Fuzzy Logic Controller In the Servo-regulatory operation, the set point is variable and the process with load variable is also variable. For the variation of the set point value with the load variable changes, the closed loop servo-regulatory response for the nonlinear conical tank process with the Fuzzy Logic Controller is obtained. The simulation diagram of the Fuzzy Logic Controller for height of 20 cm from 0 to 500 Seconds and further height is increased 10 cm from 500 to 1000 Seconds with the load changes of +10% after 800 Seconds is shown in Figure The simulated output graphs for Level versus Time, Error signal versus Time and Control signal versus Time, are shown in Figures 5.17, 5.18 and For the Servo-regulatory operation, the performance index values such as ISE, ITSE, IAE, ITAE, and the Time domain specifications are obtained and tabulated in table 5.1. ISE e ITSE ITAE IAE Performance Indices 1 Display 1 de Memory1 Fin e Fuzzy Logic Controller1 D Lev el Conical T ank 1 Scope 1 Setpoint Level 1 Setpoint Level 2 Disturbance 1 Figure 5.16 Simulation diagram of the Fuzzy Logic Controller for the Conical tank process in the Servo-Regulatory operation

18 116 Figure 5.17 Process variable versus Time graph for the Servo- Regulatory operation of the Fuzzy Logic Controller for the conical tank process with the height of 20 cm and further the height is increased 10 cm with +10% Load changes after 800 Seconds Figure 5.18 Error signal versus Time graph for the Servo-Regulatory operation of the Fuzzy Logic Controller for the conical tank process

19 117 Figure 5.19 Control signal versus Time graph for the Servo- Regulatory operation of the Fuzzy Logic Controller for the conical tank process In Table 5.2 of results, the performance index values, and time domain specifications depend on the dead time or delay of the process. In the present case, the delay time is chosen as 15 Seconds, corresponding to the height of 30 cm of the conical tank. The designed Fuzzy Logic Controller is compared with the Neuro tuned PI Controller for the conical tank process, and operated by the servo tracking, regulatory, and servo-regulatory operations. The controller performance is evaluated by the performance index values of error minimization, and Time domain specification criteria.

20 118 Table 5.2 Comparison of the Performance index values and Time domain specifications of the Fuzzy Logic Controller with the Neuro tuned PI Controller for the conical tank process in the Servo, Regulatory and Servo-Regulatory operations Performance index values and Time domain specifications Regulatory operation Fuzzy Logic Controller Servo Operation Regulatory Operation Servo- Servo- Regulatory operation Neuro tuned PI Controller Servo Operation Regulatory Operation ISE ITSE ITAE IAE Peak overshoot (M p ) Settling time (T s ) 115Sec 110Sec 152Sec 188.5Sec 185 Sec 195.8Sec Steady state error (e ss ) For the servo operation, the Fuzzy Logic Controller and Neuro tuned PI Controller are operated at the height of 20 cm from 0 to 500 Seconds and further the height is increased 10 cm from 500 to 1000 Seconds. In servo operation, the above controller s performance index values such as ISE, ITSE, IAE, ITAE, and the Time domain specifications such as peak over shoot (M p ), settling time (T s ) and steady state error (e ss ) are obtained and tabulated in Table 5.2. From the table, it is observed that the Fuzzy Logic Controller produces better performance values and Time domain specifications, when compared with the Neuro tuned PI Controller. From this, it is concluded that the Fuzzy Logic Controller can produce faster settling response, when compared with the Neuro tuned PI Controller in the servo operation. In the regulatory operation, the Fuzzy Logic Controller and Neuro tuned PI Controller are operated at the height of 30 cm from 0 to 1000 Seconds with +10 % load changes after 800 Seconds. In the regulatory operation, the Fuzzy Logic Controller s performance index values such as

21 119 ISE, ITSE, IAE, ITAE, and the Time domain specifications, such as peak over shoot (M p ), settling time (T s ) and steady state error (e ss ) are obtained and tabulated in Table 5.2. From the table, it is observed that the Fuzzy Logic Controller produces better performance values and Time domain specifications, when compared with the Neuro tuned PI Controller. From this, it is concluded that the Fuzzy Logic Controller can produce faster settling response when compared with the Neuro tuned PI Controller in regulatory operation. Similarly, in the servo-regulatory operation, the Fuzzy Logic Controller and Neuro tuned PI Controller are operated at the height of 20 cm from 0 to 500 Seconds and further the height is increased 10 cm from 500 Seconds to 1000 Seconds with +10% Load changes after 800 Seconds. In the servo-regulatory operation, the above controller s performance index values such as ISE, ITSE, IAE, ITAE, and the Time domain specifications, such as peak over shoot (M p ), settling time (T s ) and steady state error (e ss ) are obtained, and tabulated in Table 5.2. From the table, it is observed that the Fuzzy Logic Controller produces better performance values and Time domain specifications when compared with the Neuro tuned PI Controller. From this, it is concluded that the Fuzzy Logic Controller can produce faster settling response, when compared with the Neuro tuned PI Controller in the servoregulatory operation. From the servo, regulatory and servo-regulatory operations, it is seen that the Fuzzy Logic Controller is able to produce faster response with minimum error, and is also preferable for the conical tank process, when compared with the Neuro tuned PI Controller. Further, Ganesh Ram et al. (2012) developed the Adaptive tuned Fuzzy PI Controller for conical tank process. But in this work, the Fuzzy Logic Controller for Conical tank process is developed, in which this

22 120 controller response produces a settling time and an error less than the one reported in Ganesh Ram et al. (2012). This shows that the developed Fuzzy Logic Controller performs better than the adaptive tuned Fuzzy PI controller of Ganesh Ram et al. 5.5 SUMMARY AND CONCLUSION This chapter presented the design of the Fuzzy Logic Controller for the conical tank level process. The simulation results are obtained for this controller by adjusting the set point changes, load changes and change in set point with load changes. It shows that the response oscillations are reduced for the set point variations, load variations and change in set point with load variations for the Fuzzy Logic Controller as against the Neuro tuned PI Controller. From the graph, it is seen that the Fuzzy Logic Controller produces the minimum peak over shoot and faster settling time. For the above changes in the set point, load changes and change in setpoint with load changes, the Fuzzy Logic Controller is able to produce a faster response when compared to the Neuro tuned PI Controller, Genetic Algorithm tuned PI Controller and Conventional PI Controller. The Fuzzy Logic Controller produces less error and better performance index values, viz., ISE, ITSE, IAE, ITAE. The Fuzzy Logic Controller of the conical tank process also produces better time domain specifications, when compared to the Neuro tuned PI Controller, Genetic Algorithm tuned PI Controller and Conventional PI Controller. The designed controller s performance is also tested in another non-linear process, called the ph process, and the same is discussed in the next Chapter.