MANUEL EDUARDO FLORES MORAN ARTIFICIAL INTELLIGENCE APPLIED TO THE DC MOTOR

Transcription

1 MANUEL EDUARDO FLORES MORAN ARTIFICIAL INTELLIGENCE APPLIED TO THE DC MOTOR A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE DEGREE OF MASTER OF SCIENCE IN AUTOMATION AND CONTROL 2015

2 NEWCASTLE UNIVERSITY SCHOOL OF ELECTRICAL AND ELECTRONIC ENGINEERING Manuel Eduardo Flores Moran, confirm that this report and the work presented in it are my own achievement. I have read and understand the penalties associated with plagiarism. Signed:. Date:.. ii

3 Abstract Technological evolution provides a range of techniques for controlling DC and AC motors. In order to determine the precise controller for speed and position motors, Overshoot percentage, Rise Time and Steady State Error are essentials parameters for analysing the purpose. Proportional Integrative and Derivative (PID) controller offers an acceptable responses, nevertheless it does not represent the optimal one because controller is unsuccessful to prevent the overshoot and reaches the reference value when a torque load disturbance is applied. Fuzzy Logic Controller (FLC) implements the human reasoning for obtaining an optimal response, the representation of this reasoning contributes to achieve an acceptable response however this response presents a significant overshoot percentage and a steady state error. Consequently with these disadvantages, many researchers have developed a controller system that links the benefits for programing the Fuzzy Logic Controller and the easy way for tuning the PID coefficients, defining as supervisor to the FLC to provide the coefficients to the PID, as a result the system offers a response according to the required specifications. This structure provides a response capable to support the noise effects and changing of parameters. In order to find the finest coefficients for the controllers, Genetic Algorithm (GA) is a global optimizer based on natural selection that obtains the controller coefficients to provide an optimal response. The purpose of this examination is to analyse the behaviour of the controllers previously mentioned and determine the efficacy of all of them. This information develops a comparison for all the controller in order to determine the appropriate according to the specifications requested. iii

4 Section 1: Objective 1 Section 2: Academic Background DC motor PID controllers Ziegler Nichols open loop method Ziegler Nichols Closed loop or resonance method Root locus controller design method Fuzzy Logic Controller Fuzzy PD Controller Fuzzy incremental Controller Fuzzy PD+I Controller Fuzzy PID controller Phase plane methodology Genetic Algorithm 12 Section 3: Simulation, discuss and results General simulation settings Classical PID Classical PID controller for Speed Motor Control Ziegler Nichols open loop step response method Ziegler Nichols closed loop or resonance method Root locus controller design method Manual tuning Classical PID controller for Position Motor Control Ziegler Nichols open loop step response method 21 iv

5 Ziegler Nichols closed loop or resonance method Root locus controller design method Manual tuning Fuzzy Logic Controller Fuzzy Logic Controller for Speed Motor Control Fuzzy Logic Controller for Position Motor Control Fuzzy PID Controller Phase plane methodology Effects of the change of parameters For resistance armature (Ra) For coefficient of friction (B) Noise effects Genetic Algorithm 49 Section 4: Conclusions 53 Section 5: References 54 Annexes 57 Annex 1.- Low pass filter. 57 Annex 2.- Genetic Algorithm results 60 v

6 Figure 1. Separately excited DC Motor 1 Figure 2. Separately excited DC Motor Dynamic Model 3 Figure 3. PID controller structure 3 Figure 4. Step response characteristics 4 Figure 5. Response Curve for Ziegler-Nichols Open Loop Method 5 Figure 6. Closed Loop or Resonance Method 6 Figure 7. MATLAB s SISO Design Tool GUI 6 Figure 8. Fuzzy Logic Controller Structure 7 Figure 9. Example of the membership function of e(t) and ce(t) 8 Figure 10. Fuzzy PD Controller 9 Figure 11. Fuzzy Incremental Controller 9 Figure 12. Fuzzy PD+I Controller 9 Figure 13. Self-Tuning Fuzzy PID Controller Structure 10 Figure 14. Step response for a system 11 Figure 15a. Phase Plane for a step response 12 Figure 15b. Relationship between step response and Rule Base 12 Figure 16. Flow chart of the Genetic Algorithm process [1] 13 Figure 17. Ziegler Nichols Open Loop Method Responses for Speed Motor Control 17 Figure 18. Ziegler Nichols Closed Loop Method Responses for Speed Motor Control 18 Figure 19. Root Locus Method Responses for Speed Motor Control 19 Figure 20. Manual Tuning Responses for Speed Motor Control 20 Figure 21. Figure 22. Ziegler Nichols Open Loop Step Method Responses for Position Motor Control 21 Ziegler Nichols Closed Loop Method Responses for Position Motor Control 22 vi

7 Figure 23. Root Locus Method Responses for Position Motor Control 23 Figure 24. Manual Tuning Responses for Position Motor Control 24 Figure 25a. Membership function for the error e(t) 25 Figure 25b. Membership function for the change of error ce(t) 25 Figure 25c. Membership function for output u(t) 26 Figure 26. Figure 27. Output responses for Speed Motor Control applying Fuzzy Logic Controllers 27 Output responses for Position Motor Control applying Fuzzy Logic Controllers 28 Figure 28a. Membership function for the error e(t) for Position Motor Control 30 Figure 28b. Figure 28c. Membership function for the change of error ce(t) for Position Motor Control 30 Membership function for the Proportional Gain Kp for Position Motor Control 30 Figure 28d. Membership function for the Integral Gain Ki for Position Motor Control 30 Figure 29a. Membership function for the error e(t) for Speed Motor Control 31 Figure 29b. Membership function for the change of error ce(t) for Speed Motor 31 Figure 29c. Membership function for the Proportional Gain Kp for Speed Motor 31 Figure 29d. Membership function for the Integral Gain Ki for Position Motor Control 31 Figure 30. Step Response for Position Motor Control 32 Figure 31. Phase Plane of the Step Response for Position Motor Control 33 Figure 33. Figure 33. Representation of the linguistic rules with membership functions of Kp for Position Motor Control 34 Representation of the linguistic rules with membership functions of Ki for Position Motor Control 34 Figure 34. Step Response for Speed Motor Control 35 vii

8 Figure 35. Phase Plane of the Step Response for Speed Motor Control 36 Figure 36a. Output Response for Position Motor Control 38 Figure 36b. Zoom in of the Torque Load disturbance of the Output Response for Position Motor Control 38 Figure 37a. Output Response for Speed Motor Control 39 Figure 37b. Figure 38a. Figure 38b. Figure 38c. Figure 39a. Figure 39b. Figure 39c. Figure 40a. Figure 40b. Figure 40c. Figure 41a. Figure 41b. Zoom in of the Torque Load disturbance of the Output Response for Speed Motor Control 40 PID Manual Tuning Response for Speed Motor Control when resistance of armature increases its value 41 Fuzzy PD+I Response for Speed Motor Control when resistance of armature increases its value 41 Fuzzy PI Response for Speed Motor Control when resistance of armature increases its value 42 PID Manual Tuning Response for Position Motor Control when resistance of armature increases its value 42 Fuzzy PD+I Response for Position Motor Control when resistance of armature increases its value 42 Fuzzy PI Response for Position Motor Control when resistance of armature increases its value 42 PID Manual Tuning Response for Speed Motor Control when coefficient of friction increases its value 44 Fuzzy PD+I Response for Speed Motor Control when coefficient of friction increases its value 44 Fuzzy PI Response for Speed Motor Control when coefficient of friction increases its value 45 PID Manual Tuning Response for Position Motor Control when coefficient of friction increases its value 45 Fuzzy PD+I Response for Control Motor Control when coefficient of friction increases its value 46 viii

9 Figure 41c. Figure 42a. Figure 42b. Figure 43a. Figure 43b. Fuzzy PI Response for Position Control Motor when coefficient of friction increases its value 46 Fuzzy PD+I Response for Speed Control Motor Control when Noise signal occurs into the feedback loop 47 Fuzzy PI Response for Speed Control Motor Control when Noise signal occurs into the feedback loop 48 Fuzzy PD+I Response for Position Control Motor when Noise signal occurs into the feedback loop 48 Fuzzy PI Response for Position Control Motor when Noise signal occurs into the feedback loop 49 Figure 44a. Simulation block of PID Controller for Speed Motor Control 60 Figure 44b. Response of PID Controller for Speed Motor Control 60 Figure 45a. Simulation block of Fuzzy PD+I and Incremental Controller for Speed Motor Control 61 Figure 45b. Response of Fuzzy PD+I for Speed Motor Control 61 Figure 46a. Simulation block of Fuzzy PI Controller for Speed Motor Control 62 Figure 46b. Response of Fuzzy PI for Speed Motor Control 62 Figure 47a. Simulation block of PID Controller for Position Motor Control 63 Figure 47b. Response of PID Controller for Position Motor Control 63 Figure 48a. Simulation block of Fuzzy PD+I and Incremental Controller for Position Motor 64 Figure 48b. Response of Fuzzy PD+I for Position Motor Control 64 Figure 49a. Simulation block of Fuzzy PI Controller for Position Motor Control 65 Figure 49b. Response of Fuzzy PI for Position Motor Control 65 ix

10 Table I. System interaction due to PID coefficients 4 Table II. Ziegler-Nichols Open Loop Method Table parameters 5 Table III. Closed Loop or Resonance Method Table parameters 6 Table IV. Separately excited DC motor parameters 16 Table V. Table VI. Ziegler Nichols Open Loop Method Table parameters for Speed Motor Control 16 Ziegler Nichols Closed Loop Method Table parameters for Speed Motor Control 17 Table VII. Root Locus Method Table parameters for Speed Motor Control 19 Table VIII. Manual Tuning Method Table parameters for Speed Motor Control 20 Table IX. Table X. Ziegler Nichols Open Loop Method Table parameters for Position Motor Control 21 Ziegler Nichols Closed Loop Method Table parameters for Position Motor Control 22 Table XI. Root Locus Method Table parameters for Position Motor Control 23 Table XII. Manual Tuning Method Table parameters 24 Table XIII. Manual Tuning Method Table parameters 26 Table XIV. Table XV. Parameters of the Fuzzy Logic Controllers for Speed Motor Control 27 Parameters of the Fuzzy Logic Controllers for Position Motor Control 28 Table XVI. Rule Base for Kp of the FLC for Position Motor Control 35 Table XVII. Rule Base for Ki of the FLC for Position Motor Control 35 TABLE XVIII. Rule Base for Kp of the FLC for Speed Motor Control 36 TABLE XIX. Rule Base for Ki of the FLC for Speed Motor Control 37 Table XX. Parameters of the Fuzzy-PI Controllers for Position Motor Control 37 x

11 Table XXI. Parameters of the Fuzzy-PI Controllers for Speed Motor Control 39 Table XXII. Genetic Algorithm Parameters 49 Table XXIII. Table XXIV. Table XXV. Table XXVI. Table XXVII. Table XXVIII. Table XXIX. Table XXX. Comparison between PID Manual Tuning and Genetic Algorithm Parameters for Speed Motor Control 50 Comparison between Fuzzy Incremental and Genetic Algorithm Parameters for Speed Motor Control 50 Comparison between Fuzzy PD+I and Genetic Algorithm Parameters for Speed Motor Control 50 Comparison between Fuzzy Pl and Genetic Algorithm Parameters for Speed Motor Control 51 Comparison between PID Manual Tuning and Genetic Algorithm Parameters for Position Motor Control 51 Comparison between Fuzzy Incremental and Genetic Algorithm Parameters for Position Motor Control 51 Comparison between Fuzzy PD+I and Genetic Algorithm Parameters for Position Motor Control 51 Comparison between Fuzzy Pl and Genetic Algorithm Parameters for Position Motor Control 52 xi

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13 Section 1: Objective To develop a comparative study of the performance and responses of each of the controllers that will be presented during the development of this document, these results are compared according to the specific information in order to obtain an optimal response. Section 2: Academic background 2.1 DC Motor In many industrial processes, DC Motor is the first alternative for this applications because of its multiplicity of benefits. DC Motor is an electro-mechanical equipment that transforms the DC electrical energy to mechanical energy, this equipment is classified into three types such as shunt, series and separately excited. Separately excited DC motors provide different speed values through varying the armature voltage and the field voltage. The main characteristic of this type of motors is to generate high starting torque at low speed [1][15]. For these benefits, separately excited DC motors are most often applied in the industrial processes such as robot arms, train motors and aerospace [1]. Figure 1 shows a graphical representation for the motors previously mentioned. Figure 1. Separately excited DC Motor [15]. From Figure 1, it is possible to observe when an input voltage (Vt) is applied in the armature section, it produces an armature current (Ia) flowing through this section and creates an electro-mechanical force (ea). The magnetic flux (φ) is produced in the separated field section. The mathematical representation of the separately excited DC motor is exposed by the following equations: V t = R a. i a + L a. di a dt + e a (1) T e = K T. i a. φ (2) 1

14 e a = K E. θ. φ (3) T E = T L + Jθ + Bθ (4) Where, Vt = Armature Voltage [V]. Ra = Armature Resistor [Ω]. La = Armature Indcutance [mh]. ea = Back Electromotive Force [V]. TE = Electro mechanic Torque [N.m]. TL = Torque Load [N.m]. J = momentum of inertia [Kg/m2]. B = coefficient of friction [N.m.s]. KT = Torque motor constant [N.m/A]. KE = Back Electro-mechanical Force constant [V.s/rad]. ω = Angular Velocity [rad/s]. θ = Position [rad]. φ = Flux [Wb]. For future purpose, it is necessary to indicate that the armature resistance is influenced by the temperature increasing its values, and the coefficient of friction can be affected due to different factors. Applying Laplace Transforms from the equations 1 and 4, it is possible to determine the state space representation, as shown in Figure 2: V t = R A I A + L A I A S + E a (5) T E = T L + Jω S + Bω (6) 2

15 Figure 2. Separately excited DC Motor Dynamic Model. The state space representation of the DC motor is shown in the next equations: 2.2 PID Controllers θ θ 0 d θ = [ 0 B/J K T /J ] θ + [ 0 ] V dt A (7) i 0 K E /L A R A /L A i 1/L A θ y = [1 1 0] θ (8) i Proportional Integrative Derivative controllers (PID) are widely applied in not less than 90% of the industrial processes due to their robust performance and simplicity to tune their parameters previously mentioned [7]. The Proportional coefficient (Kp) describes the output reaction to the present error (e(t)), the Integrative gain (Ki) defines the responses recognised on the sum of the current errors, and the Derivative coefficient (Kd) defines the relationship established on the percentage at which the error has been varying [1][7]. After tuning coefficients, the control variable (u(t)) provides an action in order to obtain a desired response in function of specific reference. The PID structure is shown in Figure 3 and can be described by the following equations: Figure 3. PID controller structure. 3

16 t 0 u(t) = K p [e(t) + 1 de(t) e(t)dτ + T T d ](9.a) i t u(t) = [K p. e(t) + K p de(t) e(t)dτ + K T p. T d ](9.b) i 0 dt t 0 de(t) u(t) = [K p. e(t) + K i e(t)dτ + K d ](9.c) One of the benefits of these controllers is the facility to eliminate the steady state error of the response, fundamentally executed by the Integral action. Another one is supported by the derivative actions in which is the capacity to anticipate modification in the output response [16]. The step response characteristics are shown in figure 4, Overshoot percentage, Rise time and Steady sate Error are fundamental requirements to determine the appropriate controller of a system. Table I explains the interaction between the variables of the PID controller and the step response of it. dt dt Figure 4. Step response characteristics. Response Rise Time Overshoot Steady State Settling Time Percentage Error Kp Decrease Increase Minor/No Trend Decrease Ki Decrease Increase Increase Eliminate Kd Minor/No Minor/No Decrease Decrease Trend Trend Table I. System interaction due to PID coefficients. 4

17 In order to find the coefficients of the PID controller, three types of methods have been developed: Ziegler-Nichols Open Loop Method, Closed Loop or Resonance Method and Root Locus provided by Matlab Ziegler Nichols Open Loop Method Ziegler Nichols Open Loop Method is used to obtain the coefficients of the parameters P, PI and PID controllers when a Step Input is applied in the open loop system. The behaviour of the step response is usually of first order system with transportation delay [6][12][16]. Drawing a tangent line at the point of inflection is possible to determine the method variables such as gain steady state (K), Delay Time (L) and Time Constant (T). From this values, the relationship between all of them is represented by the equations A = K.L/T. Figure 5 illustrates a graphic representation of this method and Table II provides the information to obtain the coefficients previously mentioned. Figure 5. Response Curve for Ziegler-Nichols Open Loop Method [16]. Controller Type Kp Ki Kd P T/L PI 0.9T/L L/0.3 PID 1.2T/L 2L 0.5L Table II. Ziegler-Nichols Open Loop Method Table parameters Ziegler Nichols Closed Loop or Resonance Method To determine the critically gain (Ku), it is necessary to set to zero the integrative and derivative coefficients. Proportional gain must be increased from zero to a specific value where the system shows a continuous oscillations [6][12]. Finally, the period time (Tu) could be determined in order to obtain the coefficients for the controllers, as show in Table III. Figure 6 clarifies the controller method explained and its variables. 5

18 Figure 6. Closed Loop or Resonance Method. Controller Kp Ki Kd Type P 0.5 Ku PI 0.4 Ku 0.8 Tu PID 0.6 Ku 0.5 Tu Tu Table III. Closed Loop or Resonance Method Table parameters Root Locus Controller Design Method Root locus controller design method assigns the feasible closed loop poles location of a plant or system as a parameter (Controller Gain) varying from zero to infinity. This method could be explained as a plot of the zeros and poles of a system in the S domain through their real and imaginary components. MATLAB s SISO Design Tool GUI provides a powerful tool in order to obtain the coefficients of the PID controller. Basically, SISO Design Tool is a graphical user interface that provides easily the design of the controller or compensator for a different type of system. Figure 7 illustrates the SISO Design Tool GUI applying a step input for a PID controller. Figure 7. MATLAB s SISO Design Tool GUI. 6

19 In order to find an appropriate controller, it is fundamental to analyse the step response. Overshoot percentage and Steady State Error are the vital variables to decide the coefficient of the PID controller, clicking and dragging the closed loop poles location (red square) to new location. Instantly, this change modifies the gain parameters and updates them. 2.3 Fuzzy Logic Controller In 1965, Lotfi A. Zadeh defined a Fuzzy Logic theory to control non-linear and complex system using the relationship between knowledge and abilities of the programmer applying a linguistic representation, resolving the weakness of PID controller [4][5]. Fuzzy Logic Controller (FLC) offers a defined method for representing, applying and executing an illustration of human knowledge according to the procedures to control the system. The four components of the Fuzzy Logic Controller are: Fuzzifier, Knowledge Base, Inference Mechanism and Defuzzifier. Figure 8 shows its structure, denoting that error (e(t)) and change of error (ce(t)) are the inputs and u(t) is the FLC output. Figure 8. Fuzzy Logic Controller Structure. Fuzzifier: In this sector, a Fuzzy linguistic reference or variable can be obtained due to conversion of a normalized and classified crisp input value. For instance if the crisp input value is 0.25, its linguistic representation could be Positive Small (PS). Due to its symmetry, the most common shape is the triangular. A general idea of this element could be explained observing Figure 9. 7

20 Figure 9. Example of the membership function of e(t) and ce(t) [7]. Knowledge Base: This mechanism offers the database and group of rules for all other sections or components of the Fuzzy Logic Controller. The rules base is a set of linguistic designation to corresponding the input and output. Inference Mechanism: Fundamentally, Inference process sets the output value evaluating the rules provided by the Knowledge Base. The experience or Human Knowledge delimits the rules that provides the output values, for instance: If e(t) is Em and ce(t) is den then f(t) is Cmn. Where f(t) is the output of the controller, Em and den are the proper input functions and Cmn is the output function. Defuzzifier: In this last section the fuzzy output value is converted into numerical value. Bisector, weight average and centre of area are the most applied deffuzifier. For this case, the Fuzzy PD, Fuzzy Incremental and Fuzzy PD+I Controllers will be examined to understand their functionality, benefits and drawbacks Fuzzy PD Controller The proportional and derivative (PD) actions apply the derivation action in order to predict the error as consequence the stability of the closed loop system is improved. This action decreases the overshoot percentage of the step response however the system becomes more sensitive to the noise as well as any sudden change in the reference producing a derivative kick. Figure 10 shows the Fuzzy PD Controller structure [17]. 8

21 Figure 10. Fuzzy PD Controller Fuzzy Incremental Controller This controller keeps the same configuration of the previous one but the integral term is added in the output. The integral action provides a mechanism to avoid the steady state error due to the control variable produce modification to return to zero. However, when the plant or system has limitations, the control variable will be constant but the error will be constantly integrated producing the integrator windup. Another disadvantage is the high experience to set the rules for the integrative action. Figure 11 illustrates the Fuzzy incremental controller [17]. Figure 11. Fuzzy Incremental Controller Fuzzy PD+I Controller The Fuzzy PD+I offers all the benefits of PID controller, however the derivative kick and integrator windup will be present as well as the elaboration of the rules for the integrative action. Figure 12 shows the Fuzzy PD+I Controller structure [17]. Figure 12. Fuzzy PD+I Controller. 9

22 2.4 Fuzzy PID Controller Fuzzy PID or Self-Tuning PID controller defines as supervisor to the FLC meanwhile the PID determines the control variable functioning directly in the system. Fuzzy Logic Controller implements on-line tuning of PID gains in order to obtain a response according to the specifications requested. As mentioned before, Fuzzy Supervisor Controller (FSC) tuning the optimal coefficients to the PID controller to obtain an ideal response. For this purpose, it is necessary to normalize the inputs through scaling factors such as GE and GCE for the error and change of error respectively. Using this values, the fuzzifier provides the particular linguistic value to be processed by the Inference Mechanism which provides the outputs. These outputs are Kp, Ki, and Kd that are normalized in a specific range to be processed for the PID Controller. Kp, Ki and Kd are the inputs to the classical PID controller, and belong to the next ranges [Kpmax, Kpmin], [Kimax, Kimin], [Kdmax, Kdmin]. The values of this range fundamentally is determined by the experience or developing a simulation of the process [10][11]. The next equations indicate the relationship between these variables and a graphical representation of the entire system, figure 13 an graphical representation of this system. Figure 13. Self-Tuning Fuzzy PID Controller Structure. e(t) = r(t) y(t) (10) ce(t) = e(t) e(t 1) (11) K p = K pmin + K p (K pmax K pmin )(12) K i = K imin + K i (K imax K imin ) (13) K d = K dmin + K d (K dmax K dmin )(14) 10

23 Because of the subjectivity of creating rules and the lack of scientific method for developing it, scientists suggest phase plane methodology in order to resolve these inconveniences. Many documents describe a methodology of this technique using the analysis of the closed-loop step response of the system Phase Plane Methodology As previously mentioned, the inference mechanism delimits the rules that provides the output values using the grammar syntax such as: If e(t) is Em and ce(t) is den then f(t) is Cmn. Figure 14 represents a general type of the step response for a closed-loop system. In order to be clear with this methodology is necessary to divide it in the next areas: starting position (a), four regions (A1 A4), the starting position (a), cross-over (b1 b2) and peak valley (c1 c2). Figure 14. Step response for a system [19]. Starting Point (a): To prevent an integral saturation and high overshoot coefficient of the output signal, a BIG Proportional Gain and ZERO Gain for the other coefficients must be provided to the PID controller. Section A1: Since error is greater than zero and change of error is less than zero, the Proportional Gain should be set to ZERO, and the Integrative Gain must be increased as well as Derivative Coefficient in order to prevent the overshoot. Section A2: In this section the error is less than zero as well as change of error, the control action should decrease the overshoot, for this purpose the Proportional Gain is ZERO and Integrative and Derivative coefficients must be BIG. Section A3: In this zone, the error is less than zero and the change of error is greater than zero, the response must return to the reference. The integrative and Derivative Coefficients need to be reduced and Proportional Gain increased, BIG. 11

24 Section A4: In this area, the error is greater than zero as well as change of error. Proportional Coefficient must be Zero, Integrative Coefficient equal to BIG in the same way to Derivative Coefficient. Cross over (b1 b2): The control variable must have the same sign as change of error. Peak valley (c1 c2): Represents to stage, the first one when the change of error is zero and the error is less than zero, and when the change of error remains the same value but the error is greater than zero. Phase plane acts such as a tool for determining the stability of a non-linear system, for the selftuning is the link between the rules base and the system performance [19]. Figure 15a provides a representation of the phase plane for the four sections explained previously and a bridge between the Rule Base and the same response can be shown in Figure 15b. Figure 15a. Phase Plane for a step response [19]. Figure 15b. Relationship between step response and Rule Base [19]. 2.5 Genetic Algorithm Based on Natural Selection of biological individuals, Genetic Algorithm (GA) was introduced by J. H. Holland in GA is an optimizer that starts choosing a precise number of chromosomes from an initial population in which each one describes a solution of the problem, its performance is constantly tested by the fitness function [13]. New individuals, named new generations, are produced by the current population. Owing to the evolutionary process, these new generations are supposed to reach better performances than the previous ones. This process remains running until reaching the optimal or best solution of the system [14]. Figure 16 shows a flow-chart of the Genetic Algorithm process. 12

25 Figure 16. Flow-chart of the Genetic Algorithm process [1]. For instance, the following sections describe each components of the GA process for tuning PID Controller specifically: 1. Encoding: PID coefficients are transformed into a binary string. As example, the representation of this section is X = [Kp, Ki, Kd]. 2. Initialization: In this section, GA creates the first generation randomly. This generation is set in a specific range to ensure the stability of the plant. In order to obtain this information, Ziegler-Nichols method can be applied however in many case the experience is the base for this purpose. Objective function: This function is used to calculate the fitness of each individual value. Depending of the system requirements, the most applied functions are the next ones: Integral Absolute Error (IAE): this function integrates the absolute e(t) over time. It provides a slow response because it does not incorporate weight to any of the error in the response [18]. t IAE = e(t) dt 0 (15) Integral Time Absolute Error (ITAE): Mainly, this function provides a small overshoot instead of the previous one because of the error is multiplied by the time. t ITAE = t e(t) dt 0 (16) 13

26 Integral Square Error (ISE): this function integrates the square of the e(t) over time. As consequence of this operation, ISE will penalise large error instead of smaller ones. Frequently, this function provides a fast response and low oscillations. t ISE = e(t) 2 dt 0 (17) 3. Selection: In the current population, the offspring are produced based on the standard roulette wheel selection. The fitness values determines the probability of each individual, a bigger value of the fitness function is likely to obtain more offspring. When this process finish, all chromosomes are put in the mating pool. 4. Crossover: Basically, this procedure selects two bit strings (parents) from the mating pool in order to contribute with their best characteristics to produce the child. It is necessary to understand that a large population involves a smaller crossover proportion. 5. Mutation: This procedure alters the structure of the string (change a bit of the string randomly). The main purpose of this function is to avoid individuals falling into a local optimum. In many case, mutation can solve a specific problem in GA, however it could spoil of the current population. 14

27 Section 3: Simulation, discuss and results The previous section provides an academic background in order to understand the principle according to the PID classical, Fuzzy Logic Controller as well as Self Tuning PID. This information presents the essential concepts of the functionalities of each controllers and architecture, the techniques to develop them and their limitations in function of the overshoot, time rise, settling time and steady state error. Furthermore, Genetic Algorithm contributes to obtain the optimal values of their variables. The different methodology for PID controller and the architectures of the Fuzzy Logic Controllers are applied to the Separately Excited DC motor in order to examine the results of the simulation for speed and position, these results show the system behaviour for a multi reference input and torque load disturbance. Matlab is the informatics platform to obtain the different simulations and results. First of all, the simulation settings must be defined to obtain the response according the controller applied. The section 3.2 shows the output signals for the motor speed and position applying classical PID controller through of all the methods explained in the previous section. Section 3.3 illustrates the effects of the Fuzzy Logic Controller in the system. Section 3.4 provides the output response for the system applying Fuzzy PI Controller. 3.1 General simulation settings To emphasise the interaction of the classical PID controller, initially the speed multiple reference input is set to 100 rad/sec. After 1 second, this signal increases to 300 rad/sec, finally it decreases to 200 rad/sec at 2 seconds. Basically, for position multiple reference input is in the same period of time however the first values is 1 rad, 3 rad and 2 rad respectively. The torque load values is 0.5 N.m at 1.5 seconds. For Fuzzy Logic and Self-tuning Controllers, the position multiple reference input is set to 1 rad. After 30 seconds, this signal increases to 3 rad, finally it decreases to 2 rad at 60 seconds. For speed multiple input, the first value is 100 rad/sec, 300 rad/sec and finally 200 rad/sec in the same period of time as the previous one. The torque load values is 0.5 N.m at 50 seconds. The table IV provides the separately excited DC motor parameters in order to calculate the mathematical representation of this system. 15

28 Armature Resistor (Ra) 2.45 Ω Armature Inductance (La) H Momentum of inertia (J) [Kg/m2] Coefficient of friction (B) [N.m.s] Torque motor constant (Kt) 1.2 [N.m/A] Back Electro-mechanical Force constant (Ke) 1.2 [V.s/rad] Table IV. Separately excited DC motor parameters. 3.2 Classical PID Controller Classical PID Controller for Speed Motor Control Zeigler Nichols Open Loop Step Response Method A fundamental objective of this methodology is to determine the parameters A and L. Applying a step response to this system, the values of these parameters are and respectively. Once determined these parameters, the value of the coefficients of the PID controller could be obtained as shown in Table V. Controller Type Kp Ki Kd P PI PID Table V. Ziegler Nichols Open Loop Method Table parameters for Speed Motor Control. 16

29 Figure 17. Ziegler Nichols Open Loop Method Responses for Speed Motor Control. Figure 17 illustrates the step response for the P, PI and PID controllers. The first one (green) provides many oscillations and cannot reach the reference signal (blue) due to the lack of the integral action. PI controller (red) reaches the reference value reducing significantly the steady state error however the oscillations remain present. PID controller (turquoise) reduces radically the oscillation, it provides a rejection after a torque load disturbance as well as PI controller, however the overshoot percentage is 12.5%. This methodology does not provide an optimal controller due to the simplicity of it structure Zeigler Nichols Closed Loop or Resonance Method The critical oscillation (Ku) and its period (Tu) are vital to obtain the coefficients for this type of controllers, as a result the values are 75 and 0.5 respectively. The coefficients of these controllers are shown in table VI. Controller Kp Ki Kd Type P PI PID Table VI. Ziegler Nichols Closed Loop Method Table parameters for Speed Motor Control. 17

30 Figure 18. Ziegler Nichols Closed Loop Method Responses for Speed Motor Control. Similarly to the previous methodology, the P controller (green) offers a response with oscillations, with a significant steady state error and does not reject the torque load disturbance. The integral action reduces the steady state error, for this reason the PI controller (red) decrease this variable, and can reject torque load disturbance however the oscillations remain present. The last controller (turquoise) reduces the number of oscillations but the overshoot percentage presents a high value. As well as in the previous methodology, this one does not provide an optimal efficiency due to oscillations and overshoot percentage. All these results are shown in Figure Root Locus Controller Design Method Using the SISO tool provided by Matlab, it is possible to determine the coefficients of each controller. For the P controller (green), the response presents significant oscillations and cannot reject the torque load disturbance. For PI controller (red), the output signal presents an undershoot response and rejects the torque load disturbance. The last controller (turquoise) offers a response without oscillations and an excellent performance to reject torque load disturbance. It is important to understand that root locus method can provides a controller according with the specification requested however the major limitation is that the mathematical representation of the plant must be known. Table VII illustrates the coefficients of this method, and Figure 19 shows the responses for all of controllers. 18

31 Controller Type Kp Ki Kd P PI PID e-3 Table VII. Root Locus Method Table parameters for Speed Motor Control. Figure 19. Root Locus Method Responses for Speed Motor Control Manual tuning As shown in the previous controllers, the major disadvantages of the performance are the oscillations and the overshoot percentage. For controllers with only proportional action (green), the response presents oscillations and never reaches the steady state error. However the performances for PI (red) and PID (turquoise) do not present any oscillations and overshoot as well as the rejection of torque load disturbance. Manual Tuning could provide an optimal controller according to the specification due to behaviour of this system and the parameter of the motor. Table VIII indicates the coefficients of the controller and Figure 20 illustrates the responses. 19

32 Controller Type Kp Ki Kd P 12.5 PI PID Table VIII. Manual Tuning Method Table parameters for Speed Motor Control. Figure 20. Manual Tuning Responses for Speed Motor Control. The first two methodologies provide the academic background to find the variables of each controller however the response for this system presents oscillations and steady state error. Root Locus methodology improves system response reducing the drawbacks detailed before however the response offers a significant undershoot. Using the information before explained, manual tuning develops an optimal controller that reduces the overshoot percentage. Additionally if the torque load disturbance increases its magnitude, the controller rejects it reaching the reference again Classical PID Controller for Position Motor Control Zeigler Nichols Open Loop Step Response Method First of all is fundamental to understand that the behaviour of a scheme of position is based on a system of third order. As shown in the previous section, the main purpose is to calculate the 20

33 parameters A and L. As a result of applying a step input, the results are and , Table IX provides the coefficients of each type of controller. For the first one, the response presents oscillations and steady state error that increases when the torque load is executed. For PI controller, oscillations are maintained in its magnitude however the frequency of this response increases slightly. For the last controller, the amount of oscillations decreases considerably but the overshoot percentage is greater than 50%. The oscillations and overshoot percentage are the main drawbacks of this methodology. Figure 21 shows the response for all controllers explained before. Controller Type Kp Ki Kd P PI PID Table IX. Ziegler Nichols Open Loop Method Table parameters for Position Motor Control. Figure 21. Ziegler Nichols Open Loop Step Method Responses for Position Motor Control Zeigler Nichols Closed loop or Resonance Method To obtain the coefficients of the controllers is necessary to find the critical gain and its frequency, the results of this exercise are 84.2 and respectively. P controller (green) provides a response with oscillations and does not reach the reference, when the torque load disturbance occurs, the 21

34 steady state error increases due to the lack of the integral action. The response for the second controller (green) presents oscillations with a significant percentage of overshoot. For PID controller (turquoise), the response presents a high value of overshoot percentage however the oscillations are reduced due to the characteristic of this controller. Similarly to previous methodology, the analysis of this system determines that the response of the system is not the optimal due to the information requested as an overshoot percentage and steady state error. Table X shows the coefficients for the controllers and the responses for these systems are shown in Figure 22. Controller Kp Ki Kd Type P 42.1 PI PID Table X. Ziegler Nichols Closed Loop Method Table parameters for Position Motor Control. Figure 22. Ziegler Nichols Closed Loop Method Responses for Position Motor Control Root Locus Controller Design Method In this methodology, The P controller (green) offers a response with a slight overshoot however when the torque load disturbance acts, the steady state error increases and cannot reach again the 22

35 reference signal. For the second one (red), the overshoot percentage increases because of the relationship between proportional and integral action. Finally, PID controller (turquoise) provides a response with a small overshoot percentage, it is important to indicate that when a torque load is applied, any controller previously mentioned can reject it. Table XI indicates the parameters of the controller and Figure 23 illustrates the responses for these controllers. Controller Type Kp Ki Kd P PI PID Table XI. Root Locus Method Table parameters for Position Motor Control. Figure 23. Root Locus Method Responses for Position Motor Control Manual Tuning The P controller (green) provides an output signal with a small overshoot; the response of PI controller (red) shows a signal with any overshoot however when the signal changes, this controller cannot reach the reference. The final Controller (turquoise) provides a controller without overshoot however for all of these controllers when a torque load is applied, the response cannot 23

36 reach the reference providing a significant steady state error. Table XII indicates the coefficients of the controllers and figure 24 shows each response of them. Controller Type Kp Ki Kd P 15 PI PID Table XII. Manual Tuning Method Table parameters. Figure 24. Manual Tuning Responses for Position Motor Control. Similarity to the Speed Motor Control, Ziegler Nichols Open Loop and Resonance Method presents oscillations and overshoot. These factors are reduced in the Root Locus method however this methodology is not the optimal due to fundamentally by the overshoot percentage. Manual tuning reduces these factors but the output response does not reach the reference when a torque load disturbance occurs. If this disturbance increases, the difference is considerable. 3.3 Fuzzy Logic Controller As mentioned in the section 2.3, Fuzzy Logic Controller represents the human reasoning to obtain a response according the specification. FLC are formed by four components: Fuzzifier, Inference mechanism, knowledge Base and Defuzzifier. Basically, the error signal (e(t)) and its derivative 24

37 terms (ce(t)) are converted into a fuzzy value in the first section. After that, these values are processed to determine the output signal by the inference knowledge. Finally, the result is converted to the crisp value to act as output control (u(t)). The main purpose of the Fuzzifier is convert the input values into fuzzy value to be processed in the next stage. Additionally, it must provide a tool to reduce the noise effect for the system and simplify computational methods. For this cause, Triangular membership functions are more suitable and available to analyse the interception with the other ones. Figure 25a illustrates the membership functions for the error. Figure 25b shows the membership function for change of error, and Figure 25c displays the membership for the output. Knowledge base provides the fuzzy rules to the inference mechanism in order to determine the corresponding fuzzy value. These fuzzy rules symbolise the skills and the ability of a human programmer to obtain a response without oscillations, overshoot and steady state error. According to the experience, the error and its derivative signal can be divided into five subset as BIG NEGATIVE (NB), NEGATIVE SMALL (NS), ZERO (Z), POSITIVE SMALL (PS) and POSITIVE BIG (PB), also the output signal can be fractionated in the same subregions. Figure 25a. Membership function for the error e(t). Figure 25b. Membership function for the change of error ce(t). 25

38 Figure 25c. Membership function for output u(t). According with the sub regions of each variables, the fuzzy base are elaborated by the linguistic representation: If e(t) is PB and ce(t) is NB then u(t) is Z. It is critical to remember that the development of the rule is based on the ability of the operator, using this experimentation as a guide and the step response is possible to elaborate them. Table XIII provides the 25 rules for the FLC. ERROR NB NS Z PS PB PB Z PS PB PB PB CHANGE OF ERROR PS NS Z PS PB PB Z NB NS Z PS PB NS NB NS NS Z PS NB NB NB NB NS Z Table XIII. Manual Tuning Method Table parameters. The Defuzzifier adopts the instrument to transform the fuzzy results from the interference mechanism into a crisp value. As well as fuzzifier, another objective of this section is to reduce the computational methods, for this purpose Bisector can approach the previous specification Fuzzy Logic controller for Speed Motor Control According to the section before mentioned, Fuzzy Logic Controller can be set in different type of architectures, these architectures represent the same behaviour of the variable of the PID Controllers. Applying a multiple reference input described in the general simulation settings, it is possible to analyse the performance of each controller. For PD Controller (green), the response is not able to achieve the reference input in the first period of time [0 30] seconds, as well as in the 26

39 other reference inputs and torque load disturbance denoting that the steady state error is significantly high due to the lack of integral term in this architecture. Fuzzy Incremental Controller (red) provides an output response capable of reaching all the references however the main drawbacks of this architecture are that response is slow and presents ripples. Finally, The Fuzzy PD+I (turquoise) provides a response without overshoot that is able to reach the reference for all the entire period as well as the rejection of the torque load disturbance, it is important to denotes that this response is faster than the previous one. Table XIV indicates the parameters for all the Fuzzy Controllers and figure 26 shows the response for all the architectures described before for Speed Motor Control. Controller GE GCE GU GIE Type Fuzzy PD e Fuzzy Incremental Fuzzy PD+I Table XIV. Parameters of the Fuzzy Logic Controllers for Speed Motor Control. Figure 26. Output responses for Speed Motor Control applying Fuzzy Logic Controllers. 27

40 3.3.2 Fuzzy Logic Controller for Position Motor Control First of all, the multiple reference input is set to 1 rad. After 30 seconds, this signal increases to 3 rad, finally it decreases to 2 rad at 60 seconds. PD controller (green) cannot reach the reference for all the entire period of time including after the torque load disturbance. For Fuzzy Incremental controller (red), the response of the system reaches a consistent development according to the steady state error and overshoot percentage, it can also reject the torque load disturbance however this controller provides some ripples due to the system behaviour. For the last controller, the main disadvantage is that when torque load disturbance occurs it is not able to reject it. Fuzzy PD+I controller (turquoise) reaches the reference signal in the entire period of time without oscillations and overshoot percentage. Table XV indicates the parameters for all the Fuzzy Controllers and figure 27 displays the responses for all the controllers for a multiple reference input. Controller GE GCE GU GIE Type Fuzzy PD Fuzzy Incremental Fuzzy PD+I Table XV. Parameters of the Fuzzy Logic Controllers for Position Motor Control. Figure 27. Output responses for Position Motor Control applying Fuzzy Logic Controllers. 28

41 Many factors as overshoot percentage, time rise and fundamentally the steady state error provide the measurements to determine the optimal controller that could supply the technical information requested. In the case of speed control, Fuzzy PD+I offers an optimal response for a multiple reference input, if this input reference increases the range explained before, this controller reaches the reference without overshoot and oscillations. Furthermore, its performance offers a reasonable output signal still when the torque load disturbance increases its magnitude. For the case of position control, it is necessary to remember that plant behaviour is consistent with a third order system, when a torque load disturbance increased its magnitude the Fuzzy Incremental controller reject this effect however ripples are maintained; Fuzzy PD+I cannot reject the torque load disturbance presenting a significant steady state error. 3.4 Fuzzy PID Controller Previously mentioned, Fuzzy Logic Control replaces the mathematical calculations by linguistic representation instead of conventional controllers. The Linguistic representations are processed through the rule base, and they are based on the practical experience. This system takes advantage of the simplicity of the FLC controllers with the easy parameterization of the PI controller defining as Supervisor to the first one, in other words FLC tunes each variables of the PI controller that acts directly to the plant. Similarly to the previous FLC controller is necessary to define each one of its components. Its inputs are the error (e(t)) and change of error (ce(t)); on the other hand, its outputs are Kp and Ki. The first section of the FLC, Fuzzifier, transforms the input value into the corresponding fuzzy value, considering factors as the simplicity of the calculation and noise effect, the triangular membership offers a symmetrical and simple way to apply this concept. Seven subsets or sub regions have been selected to describe each input variable: BIG NEGATIVE (NB), MEDIUM NEGATIVE (MN), NEGATIVE SMALL (NS), ZERO (Z), POSITIVE SMALL (PS), POSITIVE MEDIUM (PM) and POSITIVE BIG (PB). For the Position Motor Control, Figure 28a shows the membership functions for the error, Figure 28b displays the change of error, the membership function for the output Proportional Gain is shown in Figure 28c, and Figure 28d illustrates the membership function for the Integral Action. 29

42 Figure 28a. Membership function for the error e(t) for Position Motor Control. Figure 28b. Membership function for the change of error ce(t) for Position Motor Control. Figure 28c. Membership function for the Proportional Gain Kp for Position Motor Control. Figure 28d. Membership function for the Integral Gain Ki for Position Motor Control. 30

43 For the Speed Motor Control, Figure 29a displays the membership functions for the error, Figure 29b shows the change of error, the membership function for the output Proportional Gain is illustrated in Figure 29c, and Figure 29d shows the membership function for the Integral Action. Figure 29a. Membership function for the error e(t) for Speed Motor Control. Figure 29b. Membership function for the change of error ce(t) for Speed Motor Control. Figure 29c. Membership function for the Proportional Gain Kp for Speed Motor Control. 31

44 Figure 29d. Membership function for the Integral Gain Ki for Position Motor Control. In many cases, the experience or the ability of the programmer to define the rules of the FLC are subjective and represent the lack of the standard methodology to develop it In order to resolve this weakness, Phase Plane method can provide a relationship between the system behaviour and the rule base of the Fuzzy Logic Controller, providing a procedure more objective and universal Phase Plane Methodology In order to avoid subjectivity for creating the rules of the FLC, the phase plane methodology offers a relationship between the system behaviour and the rules base. The first step is to obtain the step response of the system and analyse it according to the information requested. For the position control, figure 30 shows the step response for the system. This response has been divided in two points a and b1 representing the starting point and the final point respectively. Figure 30. Step Response for Position Motor Control. This methodology demands a deep study of this signal in order to understand the relationship with the rules. It is possible determine the following requirements: 32

45 Starting Point (a): In order to obtain a fast time response, The FLC must provide a BIG Proportional and Integrative Coefficients. Section between a and b1: Since error is greater than zero and change of error is less than zero, the Proportional Gain should decrease to ZERO as well as Integrative in order to prevent the overshoot and saturation. Final Point (b1): The control variable must have the same sign as change of error due to the response try to reach the reference and avoid the Steady State Error. An equivalent diagram of the step response can be represented by the phase plane, figure 31 shows this plot including the points previously mentioned. According with this diagram, the point a is located in the middle-right side of the phase plane. Previously mentioned, the proportional gain must be high as well as the integral action. During the trajectory of the response (section between a and b1), Proportional and Integrative Coefficients must decrease the magnitude to prevent the overshoot and finally (b1) the purpose is reach the reference to avoid the steady state error. Figure 31. Phase Plane of the Step Response for Position Motor Control. Using this graph and the linguistic interpretation, Figure 32 can clarify this relationship, observing all the interaction of the response with the sub regions. It is important to denote that this figure is applied to study the Kp coefficient and the linguistic rules contents the information requested avoiding oscillation and overshoot. It is possible to observe that the previous controller cannot achieve the reference after a torque load disturbance, for this reason is fundamental considering this effects to elaborate the rules. 33

46 Figure 32. Representation of the linguistic rules with membership functions of Kp for Position Motor Control. In the same way, it is possible to examine the analysis for Ki coefficient. Figure 33 illustrates the result of this bridge between the behaviour of the response and the linguistic rules. The determination of the rules is obtained applying the previous concepts considering the torque load disturbance. Figure 33. Representation of the linguistic rules with membership functions of Ki for Position Motor Control. Finally, table XVI and XVII illustrate the linguistic rules for Kp and Ki respectively. Applying these linguistic rules, the inference mechanism will determine the fuzzy value that will be analysed by the defuzzifier. 34

47 Table XVI. Rule Base for Kp of the FLC for Position Motor Control. Table XVII. Rule Base for Ki of the FLC for Position Motor Control. For the speed control, the first step is to obtain the step response for the system and study its behaviour in order to develop the linguistic rules as shown in Figure 34. Figure 34. Step Response for Speed Motor Control. 35

48 According with the respective section and applying the academic background, the step response offers an objective procedures to find the rules of the FLC. Examining this response, it is vital to denote that the signal presents a significant overshoot and does not reach the reference signal. The representation in the phase plane is shown in Figure 35. Figure 35. Phase Plane of the Step Response for Speed Motor Control. In order to determine the rules, it is necessary that the signal needs to reach the reference, for that reason the Proportional and Integral Gain must be high value however the sign of the Proportional coefficient should be opposite sign to the Integral factor. As well as Position control, to avoid the overshoot and reach the reference, the FLC should decrease the magnitude of Proportional and Integral gains. Finally, to avoid any steady state error the signal must share the same sign of the change of error. Table XVIII and XIX illustrate the linguistic rules for Kp and Ki respectively Table XVIII. Rule Base for Kp of the FLC for Speed Motor Control. 36

49 Table XIX. Rule Base for Ki of the FLC for Speed Motor Control. The Defuzzifier converts the fuzzy value into a crisp value. As well as fuzzifier, another objective of this section is to reduce the computational methods, for this purpose Bisector can approach the previous specification. As it was defined in the academic background according to this system, Fuzzy Logic Controller provides the values of Kp and Ki of the classical PID, these values belong to the range [Kpmax, Kpmin] and [Kimax, Kimin]. To determine the values of these elements, Ziegler Nichols Methods could offer an alternative to do it, however these coefficients are determined by the experience or developing a simulation of the process. Table XX illustrates Parameters of the Fuzzy-PI Controllers for Position Motor Control and figure 36a shows the output response for this system. Controller Type FUZZY PI Error 0.65 Change of error Kp 2.5 Ki 7.2 Kpmin 0 Kimin 0 Table XX. Parameters of the Fuzzy-PI Controllers for Position Motor Control. 37

50 Figure 36a. Output Response for Position Motor Control. Figure 36b. Zoom in of the Torque Load disturbance of the Output Response for Position Motor Control. The controller established indicates a response without overshoot for all of the periods of the multiple reference input, observing a rise time appropriate to determine the performance of this controller. When the torque load is applied, the system provides a control variable to reduce substantially the steady state error compared with the previous controllers. Figure 36a illustrates the output response of the system, and figure 36b is a zoom in on the response for a torque load disturbance. In addition, another advantage for this controller is that the output response can reach the reference if it increases its magnitude for the range detailed before. 38

51 For the Speed Motor Control, the controller offers a fast time response without overshoot for all of the period of the input reference. When the torque load disturbance occurs the controller provides the control variable to reach the reference providing a signal with small overshoot. This system provides a response that is able to reach with an input with different input value described in the specific range. If the torque load increases, this system provides a control variable to reject it and reach the steady state error. Table XXI illustrates Parameters of the Fuzzy-PI Controllers for Speed Motor Control, figure 37a shows the output response for this system and figure 37b is a zoom in on the response for a torque load disturbance Controller FUZZY Type PI Error 0.05 Change of error Kp 0.5 Ki 7.2 Kpmin 0.5 Kimin 5.2 Table XXI. Parameters of the Fuzzy-PI Controllers for Speed Motor Control. Figure 37a. Output Response for Speed Motor Control. 39

52 Figure 37b. Zoom in of the Torque Load disturbance of the Output Response for Speed Motor Control. 3.5 Effects of the change of parameters. This section provides the results according to the effects of the modification of the motor parameters. Basically, the parameters that could be affected are the armature resistance and coefficient of friction For resistance armature (ra) In order to emphasis this behaviour, the original value of the armature resistor is 2.45 ohmios, for this demostration this value is modified to ohmios.for the Speed Motor Control, figure 38a represents the performance of this modification applied by PID Manual Tuning for Speed Motor Control. The blue line ilustatres the response using the original parameters, meanwhile the green one shows the response when armature resistor increases by 50%, this repsonse displays a signal with considerable overshoot for each value of the step refence inputs. Figure 38b shows the response of the system controlled by Fuzzy PD+I, this response practically does not change in all of the period of the multiple input reference. In the same way, Fuzzy PI provides a control variable to attend this effect for all of the period, figure 38c illustrates this response. 40

53 Figure 38a. PID Manual Tuning Response for Speed Motor Control when resistance of armature increases its value. Figure 38b. Fuzzy PD+I Response for Speed Motor Control when resistance of armature increases its value. 41

54 Figure 38c. Fuzzy PI Response for Speed Motor Control when resistance of armature increases its value. For the Position Motor Control, PID Manual Tuning cannot reject the change of parameters providing a response with overshoot for each reference input as shown in Figure 39a. For Fuzzy PD+I, the output variable offers a control variable to supply these effects as well as Fuzzy PI. The responses for each controller can be observed by figure 39b and 39c respectively, when torque load disturbance occurs, the steady state error increases its value because of the armature resistance affects directly over armature current. Figure 39a. PID Manual Tuning Response for Position Motor Control when resistance of armature increases its value. 42

55 Figure 39b. Fuzzy PD+I Response for Position Motor Control when resistance of armature increases its value. Figure 39c. Fuzzy PI Response for Position Motor Control when resistance of armature increases its value Coefficient of friction (b) Initially the coefficient of friction is set to [N.m.s]. In order to study the effects of the change of this parameter, the new value is 0.1 [N.m.s]. For the Speed Motor Control, PID manual tuning response presents a slight difference in the rise time comparing with the original one for all of the period of the multiple reference input as shown in figure 40a. For Fuzzy PD+I controller, 43

56 this difference is reduced however it cannot reject completely and affects the specifications required, figure 40b shows the response. Finally, Fuzzy PI provides a control variable to reduce considerably this difference as shown in Figure 40c. Figure 40a. PID Manual Tuning Response for Speed Motor Control when coefficient of friction increases its value. Figure 40b. Fuzzy PD+I Response for Speed Motor Control when coefficient of friction increases its value. 44

57 Figure 40c. Fuzzy PI Response for Speed Motor Control when coefficient of friction increases its value. For Position Motor Control, PID Manual Tuning Controller presents a difference in the rising time as shown in figure 41a. For Fuzzy PD+I, this difference is reduced considerably by the control variable however Fuzzy PI offers a response equally to the original one. Figure 41b and 41c illustrate the response for Fuzzy PD+I and Fuzzy PI respectively. Figure 41a. PID Manual Tuning Response for Position Motor Control when coefficient of friction increases its value. 45

58 Figure 41b. Fuzzy PD+I Response for Control Motor Control when coefficient of friction increases its value. Figure 41c. Fuzzy PI Response for Position Control Motor when coefficient of friction increases its value. 3.6 Noise effects Another aspect to analyse the performance of the controller for Speed and Position Motor Control is the noise effects. As previously mentioned, the analysis to design a filter is according to the specification of the Bode response. Bode Graph provides the specification to elaborate a Low Pass 46

59 Filter, this filter is included next to the derivative parameter. The noise source is Band-Limited White Noise and its power is , this source is added into feedback loop. Annex I provides the academic background to obtain the Low Pass filter, the next equation describes this element: T(s) = 10 4 s s The response without noise effects is represented by the green line, the response with noise effects is illustrated by the red line and the noise source is displayed by the blue line. For the Speed Motor Control, figure 42a illustrates the response controlled by Fuzzy PD+I, the Low Pass Filter reduces considerably the magnitude of the noise. Fuzzy PI, the frequency and magnitude of the noise are reduced, figure 42b shows the response for this system. Figure 42a. Fuzzy PD+I Response for Speed Control Motor Control when Noise signal occurs into the feedback loop. 47

60 Figure 42b. Fuzzy PI Response for Speed Control Motor Control when Noise signal occurs into the feedback loop. For Position Motor Control, the action of the Low Pass Filter reduces the magnitude of the noise source for a system controlled by Fuzzy PD+I and Fuzzy PI Controller, the response for this system can be shown by the Figure 43a and 43b respectively. Figure 43a. Fuzzy PD+I Response for Position Control Motor when Noise signal occurs into the feedback loop. 48

61 Figure 43b. Fuzzy PI Response for Position Control Motor when Noise signal occurs into the feedback loop. 3.7 Genetic Algorithm As explained in section 2.5, Genetic Algorithm is an optimizer that represents the biological process to achieve an optimal solution finding the appropriate controller coefficients. In order to obtain rapidly the different coefficients, it is necessary to set the lower and upper bounds. Normally, the analysis for Classical PID does not request a considerable time, however for Fuzzy and Fuzzy PI this time could be hours. Table XXII illustrates the Genetic Algorithm parameters for setting the Matlab application for each process. Controller Type Kp Population size 100 Generation 35 Stochastic Selection method uniform Crossover Constrain Method depend Mutation Constrain Method depend Table XXII. Genetic Algorithm Parameters. 49

62 In order to select the appropriate fitness function, the controller must provide a response without overshoot, a minimum steady state error, fast rising time and small settling time. IAE represents the cumulative error function, this function idealizes the difference between the reference and the response. ITAE provides a function that ponders the present error, meanwhile ITSE considers large error instead of small by the time. For the specification explained before, ITAE provides a small overshoot with a minimum steady state error. The following tables illustrate the parameters obtained by Genetic Algorithm, the response can be examined in the Annex II Controller Type Kp Ki Kd ITAE PID Manual Tuning GA PID Manual Tuning e Table XXIII. Comparison between PID Manual Tuning and Genetic Algorithm Parameters for Speed Motor Control. Controller Type GE GCE GU ITAE Fuzzy Incremental GA Fuzzy Incremental Table XXIV. Comparison between Fuzzy Incremental and Genetic Algorithm Parameters for Speed Motor Control. Controller Type GE GCE GU GIE ITAE Fuzzy PD+I GA Fuzzy PD+I Table XXV. Comparison between Fuzzy PD+I and Genetic Algorithm Parameters for Speed Motor Control. 50

63 Controller Type E CE Kp Kpmin Ki Kimin ITAE Fuzzy PI GA Fuzzy PI e Table XXVI. Comparison between Fuzzy Pl and Genetic Algorithm Parameters for Speed Motor Control. Controller Type Kp Ki Kd ITAE PID Manual Tuning GA PID Manual Tuning Table XXVII. Comparison between PID Manual Tuning and Genetic Algorithm Parameters for Position Motor Control. Controller Type GE GCE GU ITAE Fuzzy Incremental GA Fuzzy Incremental Table XXVIII. Comparison between Fuzzy Incremental and Genetic Algorithm Parameters for Position Motor Control. Controller Type GE GCE GU GIE ITAE Fuzzy PD+I GA Fuzzy PD+I e Table XXVIX. Comparison between Fuzzy PD+I and Genetic Algorithm Parameters for Position Motor Control. 51

64 Controller Type E CE Kp Ki ITAE Fuzzy PI GA Fuzzy PI Table XXX. Comparison between Fuzzy Pl and Genetic Algorithm Parameters for Position Motor Control. 52

65 Section 4: Conclusions Based on the results obtained, in the case for Speed Motor Control, PID Manual Tuning Controller provides a response without overshoot and minimum steady state error for different inputs reference and its control variable is capable to reject the torque load disturbance, nevertheless this controller cannot support the change of parameters, this is fundamental in industrial environments where the temperature affects the resistance For Position Motor Control, PID manual Tuning does not provide a control variable to reject the torque load disturbance producing a significant steady state error. As well as Speed Motor Control, this controller does not support the change of parameters. For Speed Motor Control, because of a slow response and some ripples, Fuzzy Incremental Controller present many drawbacks to attempt these difficulties. On the other hand, Fuzzy PD+I provides a fast response and without oscillations and ripples. It is necessary to indicate that the two controllers can reject the Torque Load disturbance. However, for Position Motor Control, the major difficulty is to reject these effects. Fuzzy incremental presents some ripples and oscillations, meanwhile Fuzzy PD+I offers a stable response with a reduced steady state error comparing with the previous one. An advantage of the architecture of the Fuzzy Logic Controller is that it can attend to change of parameters, but this controller presents problems when the input reference changes suddenly and noise effects occurs because of this phenomenon causes that the rules are not assigned to these values. Fuzzy PI provides an optimal controller according to the specifications explained previously. This controller can support change of parameters unlikely to classical PID, which presents a response with overshoot. Analysing the results in the previous section, it is possible to observe that the difference between the original and the modified signal is minimum. Another benefit of this architecture is that it can achieve different suddenly reference input instead of Fuzzy Logic Controllers which presents a disadvantage in these aspects. For Position Motor Control, according to the results obtained in the GA section and the simulations response, this controller presents the smallest steady state error. For the Speed Motor Control, the response is according to the specifications without overshoot and a good rejection for Torque Load disturbance. It is important to denote that the noise effects in this system can be resolve by the filter and the controller. In conclusion, comparing all controllers, Fuzzy PI provides a response with the major optimal performance. 53

66 Section 5: References [1] Elsrogy, W.M. Speed Control of DC Motor Using PID Controller Based on Artificial Intelligence Techniques, IEEE International Conference on Control, Decision and Information Technologies (CODIT), pp 1-6, [2] Yang Huafen. Study on Fuzzy PID Control in Double Close-Loop Speed Regulation System, IEEE Third International Conference on Measuring Technology and Mechatronics Automation (ICMTMA), Vol. 3 pp , [3] Yunfei Lv, Hui Luo and Yong, Cai. Research on Tuning Method for Fuzzy PID, IEEE Fourth International Workshop on Advanced Computational Intelligence (IWACI), pp , [4] T. Ravichandran and F. Karray, Knowledge Based Approach for Online Self-Tuning of PID-Control, Proceedings of the American Control Conference, Vol. 4 pp , [5] M. S. Abou Omar, T. Y. Khedr and B. A. Abou Zalam. Particle Swarm Optimization of Fuzzy Supervisory Controller for Nonlinear Position Control System, IEEE 8th International Conference on Computer Engineering & Systems (ICCES), pp , [6] J. G. Ziegler and N. B. Nichols. Optimum settings for automatic controllers, Trans. ASME, Vol. 64 pp , [7] Ahmed Z. Alassar, Iyad M. Abuhadrous and Hatem A. Elaydi. Modeling and Control of 5 DOF Robot Arm Using Supervisory Control, IEEE The 2nd International Conference on Computer and Automation Engineering (ICCAE), Vol. 3 pp , [8] Robert P. Copeland and Kuldip S. Rattan. A Fuzzy Logic Supervisor for PID Control of Unknown Systems, IEEE International Symposium on Intelligence Control, pp , [9] Murat Akgul and Gmer Morgul. Fuzzy Controller Design for Parametric Controllers, IEEE International Symposium on Intelligent Control, pp ,

67 [10] Zhen Yu Zhao. Fuzzy Gain Scheduling of PID Controllers, IEEE First Conference on Control Application, Vol. 2 pp , [11] S.-Z. He, S. Tan and F.-L. Xu. Fuzzy self-tuning of PID controllers, IEEE Second International Conference on Fuzzy System, Vol. 2 pp , [12] Payam Solatian, Seyed Hamidreza Abbasi and Fereidoon Shabaninia. Simulation Study of Flow Control Based On PID ANFIS Controller for Non-Linear Process Plants, American Journal of Intelligent Systems, Vol. 2.5, pp , [13] Kamal M.M, Mathew Lini and Chatterji. Speed control of brushless DC motor using intelligent controllers, IEEE Students Conference on Engineering and Systems (SCES), pp. 1-5, [14] Liu Fan and Er Meng Joo. Design for Auto-tuning PID Controller Based on Genetic Algorithms, IEEE 4th Conference on Industrial Electronics and Applications (ICIEA), pp , [15] V. Tipsuwanpom, A. Numsomran, N. Klinsmitth, S. Gulphanich. Separately Excited DC Motor Drive with Fuzzy Self-Organizing, International Conference on Control, Automation and Systems, 2007 ICCAS '07, pp , [16] Meshram, P.M. and Kanojiya, R.G. Tuning of PID controller using Ziegler-Nichols method for speed control of DC motor, IEEE- International Conference On Advances In Engineering, Science And Management (ICAESM), pp , [17] Jan Jantzen. Tuning of fuzzy PID controllers Denmark. Tech. Report no 98- H 871(fpid), pp. 1-22, 30 Sep [18] Deepyaman Maiti, Ayan Acharya, Mithun Chakraborty, Amit Konar and Ramadoss Janarthanan. Tuning PID and PIλ Dδ Controllers using the Integral Time Absolute Error Criterion, IEEE 4th International Conference on Information and Automation for Sustainability (ICIAFS), pp ,

68 [19] Han-Xiong Li and H. B. Gatland. A New Methodology for Designing a Fuzzy Logic Controller. IEEE Transactions on Systems, Man, and cybernetics, pp

69 Annexes Annex 1: Low Pass Filter 57

70 58

71 59