ANALYTICAL STRUCTURES FOR FUZZY PID CONTROLLERS AND APPLICATIONS

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1 International Journal of Electrical Engineering and Technology (IJEET), ISSN (Print) ISSN (Online), Volume 1 Number 1, May - June (2010), pp IAEME, International Journal of Electrical Engineering and Technology (IJEET), IJEET I A E M E ABSTRACT ANALYTICAL STRUCTURES FOR FUZZY PID CONTROLLERS AND APPLICATIONS VenkataRamesh.Edara Department of EE A.U.C.Eat Visakhapatnam venkatarameshedara@yahoo.com B.Amarendra Reddy Assistant Professor Department of EE,AU College of Engineering Andhra University, bamarendrareddy@yahoo.com Srikanth Monangi Junior manager in Vizag Steel Plant srim250@gmail.com In the present work, analytical structures for fuzzy proportional-integralderivative (PID) controllers are derived via triangular membership functions for inputs; triangular membership functions for output; minimum triangular norm; different combinations of two triangular co-norms (maximum, drastic sum) and five inference methods (such as Mamdani minimum, Larsen product, drastic product, bounded product and standard sequence) and center-of-sum defuzzification method. Computer simulations are included to demonstrate the effectiveness of the fuzzy PID controller over the conventional controller for time-delay and non-linear systems. Index Terms Analytical Structures, Bounded product, Drastic product, Larsen product, Standard Sequence, Triangular co-norm. INTRODUCTION PID controller is the most widely used control structure in industrial plants due to their simple and robust design, low cost and effectiveness for linear systems. Due to their linear structure, conventional PID controllers are usually not effective if the processes 1

2 involved are higher order and time delay systems, nonlinear systems, complex and vague systems without precise mathematical models, and systems with uncertainties [1]. There are researches worldwide aiming to improve the performance of complex processes and, so, leading to several advanced control techniques such as: adaptive control, auto-tuning, optimal control and predictive control. Most advanced control algorithms not only has had a great advance on the control theory but also depends on the mathematical model, where, in hard practical conditions, is not always possible to come up the model. Therefore, there is a necessity for a generalpurpose intelligent controller that can be used easily and effectively to control a wide variety of complex plants, in other words, a fuzzy control. Controllers based on fuzzy logic theory not only try to mimic the behavior of an expert operator but also do not demand the model identification. Fuzzy control systems have been investigated in many technical and industrial applications. The main contribution of the fuzzy control theory is its ability to handle many practical problems that cannot be adequately handled by conventional control methods. Fuzzy controllers are known to be effective in solving practical control problems. They are usually designed and tuned using trial and error method with computer simulations, since basically fuzzy control is model independent. However if model based design is to be carried out, an analytical structure of a fuzzy controller if available will result in a good design. By analytical structure we mean the mathematical expression of a fuzzy controller that represents precisely the fuzzy controller without any approximation. The mathematical structure of a fuzzy controller is determined by its components including input fuzzy sets, output fuzzy sets, fuzzy rules, fuzzy inference, fuzzy logic operators, and defuzzifier. Input-output structure derivation and precise understanding of the resulting structures in relation to conventional controllers is fundamentally important because without such knowledge, systematic analysis and design will be difficult and ineffective at best, so will be effective utilization of the conventional tools. In this work, different classes of fuzzy PID controllers are derived by employing minimum triangular norm, different triangular co-norms (maximum or drastic sum), different inference methods (Mamdani minimum or Larsen product or drastic product or 2

3 bounded product or standard sequence) and center-of-sums method for deffuzification. Fuzzy PID-controller has been employed to meet the desired performances by tuning the controller parameters appropriately. To demonstrate the superiority of fuzzy PID controllers over the conventional PID controller, an attempt has been made to simulate fuzzy PID controller by using triangular membership function as input and output membership functions for non-linear and time delay systems. Figure 1 Structure of fuzzy PID controller STRUCTURE OF FUZZY PID CONTROLLERS The principle structure of a fuzzy PID controller consists of following components and is shown in Figure (1). (i) Fuzzification (ii) Control Rule Base (iii) Inference engine (iv) Defuzzification A. Fuzzification Fuzzification can be defined as a mapping from an observed input space to fuzzy sets in certain input universe of discourse. 3

4 The fuzzy PID controller has 3 inputs: the error signal e (kt) [displacement d (kt)], the first-order time derivative of e (kt) [velocity v (kt)], and the first-order time derivative of v (kt) [acceleration a (kt)]. It has a single output, called the incremental control output u (kt). The input and output membership functions for the fuzzy PID controller are shown in the Figure 2 and Figure 3, respectively, where d N, v N and a N are the inputs and u N is the output. K P, K I and K D are the normalized scaling factors at input and K UPID is the normalized scaling factor at output. L and M are the constants chosen by the designer in the Universe of Discourse. Figure 2 Input membership functions Figure 3 Output membership functions B. Control Rule Base given below. As each input consists of two membership functions, the eight possible rules are R1) If d N is n.d AND v N is n.v AND a N is n.a, then u N is NM. R2) If d N is p.d AND v N is n.v AND a N is n.a, then u N is NS. 4

5 R3) If d N is p.d AND v N is n.v AND a N is p.a, then u N is PS. R4) If d N is n.d AND v N is n.v AND a N is p.a, then u N is NS. R5) If d N is n.d AND v N is p.v AND a N is p.a, then u N is PS. R6) If d N is n.d AND v N is p.v AND a N is n.a, then u N is NS. R7) If d N is p.d AND v N is p.v AND a N is n.a, then u N is PS. R8) If d N is p.d AND v N is p.v AND a N is p.a, then u N is PM. The AND in the antecedent part represents the fuzzy logical AND operation which is considered here as Zadeh s minimum (triangular norm). C. Inference Engine The fuzzy inference engine evaluates the control rules stored in the rule base. Its function is to compute the overall value of the control output variable based on the individual contributions of each rule in the rule base. Each such individual contribution represents the value of the control output variable as computed by a single rule. Based on this degree of match, the clipped fuzzy set representing the value of the control output variable is determined via a particular type of implication. Some fuzzy implication functions, which are often employed in an FLC are (a) Triangular Norms: The triangular norm is a two-place function from [0,1]X[0,1] to [0,1], which includes intersection, algebraic product, bounded product, and drastic product. The operations associated with triangular norms are defined for all x, y є [0,1]: Zadeh s minimum x y = min{ x, y} Algebraic product x y = Bounded product x y = max{ 0, x + y 1} xy (b) Triangular Co-Norms: The triangular co-norms is a two-place function from [0,1]X[0,1] to [0,1], which includes Zadeh OR, Lukasiewicz OR, algebraic sum, drastic sum, and disjoint sum. The operations associated with triangular co-norms are defined for all x, y є [0,1]: Zadeh OR x y = max{ x, y} Lukasiewicz OR x y = min{ 1, x + y} Algebraic sum x y = x + y xy 5

6 Drastic sum D. Defuzzification x x y = y 1 y = 0 x = 0 x, y Defuzzification module converts the set of modified control output values into a single point-wise value. The most commonly used center of sums method is employed > 0 to defuzzify the fuzzy incremental control law. This is expressed as (1) Where A (µ (Ri)) is the area of the inferred fuzzy set corresponding to rule Ri. ANALYTICAL STRUCTURES Analytical Structures can serve as a platform on which analysis can be theoretically and mathematically developed. Deriving the analytical structure of fuzzy controllers is very important as it creates solid information for better understanding, insightful analysis and more effective design of fuzzy control systems. As fuzzy PID controller have three inputs, it is difficult to visualize a point in a 3- D space. A point say (x1, y1, z1), in a 3-D space can always be shown by taking its projection on the xy-, yz- and zx-planes. So, as shown in the Fig.4, eight input combinations are considered in each d N v N -, d N a N - and v N a N -plane so that the state points (d N *, v N *, a N *) can be uniquely located with the 3-D cell represented by the triplet (ni, nii, niii) where ni with i = I, II, III is the input combination number (1 to 8). 6

7 Figure 4 Input combination regions of the fuzzy PID controller A cell (n I, n II, n III ) is said to be valid if and only if the relations between d N and υ N, and d N and a N, produce the relation between υ N and a N. The mathematical description of the input fuzzy set of Figure 2 is µ n.x =, -L x L (2) µ p.x =, -L x L (3) Table1 Outcomes of Minimum operation of premise part of Fuzzy Control Rules R1 R8 Cells (R1) (R2) (R3) (R4) (R5) (R6) (R7) (R8) (1,2,4), (8,2,5) n.d n.a# n.v* n.d n.d n.d n.a p.v (1,3,4), (8,3,5) n.a n.a n.v* n.d# n.d n.a n.a p.v (6,4,5), (7,3,5) n.a n.a n.v n.v# n.d* n.a n.a p.d (6,4,6), (7,3,6) n.v n.v n.v n.v n.d* n.a# n.a p.d (7,1,7), (7,2,6) n.v n.v n.v n.v n.d n.d# n.a* p.a (8,1,7), (8,2,6) n.d n.v# n.v n.d n.d n.d n.a* p.a 7

8 By using the minimum triangular norm, the outcome of premise (antecedent) part of each rule is found for some valid cells as shown in the Table I. Consider the point (d N *, υ N *, a N * ) in the cells (6, 4, 5) and (7, 3, 5), rule R1 and minimum triangular norm together lead to the fuzzy set n.a. Therefore, the rule R1 yields the input fuzzy set n.a and the corresponding output fuzzy set NM. Similarly in the cells (6, 4, 5) and (7, 3, 5), rules R2 R8 and minimum triangular norm yield the inputoutput fuzzy set pairs as (n.a, NS) for R2, (n.υ, PS) for R3, (n.υ, NS) for R4, (n.d, PS) for R5, (n.a, NS) for R6, (n.a, PS) for R7 and (p.d, PM) for R8. It has seen from the control rules that the output fuzzy sets NS and PS are fired three times. In such a situation, a fuzzy triangular co-norm (shown in Table II) is used to evaluate combined output fuzzy set corresponding to the rule sets {R2, R4, R6} and {R3, R5, R7}. In Table I, symbol # indicates max-min operation outcome of the premise (antecedent) part of the rule set {R2, R4, R6}. Similarly, symbol * indicates max-min operation outcome corresponding to the rule set {R3, R5, R7}. Table III. Different inference methods along with their definitions are tabulated as shown in Table 2 Triangular co-norms T-conorm Definition Maximum max{µ A (x), µ B (y)} max{µ A (x), µ B (y)} if min{µ A (x), Drastic sum µ B (y)}=0 1 if µ A (x), µ B (y) > 0 Since the fuzzy controller has 3 inputs and the membership values corresponding to the cells in Table I are greater than zero, the drastic sum triangular co-norm operation outcome is unity for the rule sets {R2, R4, R6} and {R3, R5, R7}[4]. In the present work, we have classified fuzzy PID controllers into different classes depending upon the combination of output fuzzy sets, triangular co-norms and inference method used. Considering Mamdani minimum (R MM ) inference method, triangular output fuzzy set, maximum triangular co-norm and center of sums for defuzzification, the derivation of analytical structures for the cells (1, 2, 4) and (8, 2, 5) of related fuzzy controller is 8

9 explained here. The inferred area A (µ (R i )), i = 1,2,,8, corresponding to the eight control rules R1 R8, are obtained using the expression for area (corresponding to R MM ) given in Table III. Substituting the inferred area into equation (1) and simplifying lead to the analytical structure classified as class IV fuzzy PID controller. And also, for each class of controller, we have computed u at some arbitrary points in 3-D input space to see if this controller is possessing the desirable property of generating smooth control effort at these points. We choose the points in 3-D input space to be (d N, υ N, a N ): (0, 0, 0), (0, 0, L), (0, L, 0), (L, 0, 0), (0, L, L), (L, L, 0), (L, 0, L) and (L, L, L) as all these points can be approached from the cells (8,2,5), (8, 3, 5), (7, 3, 5), (7, 3, 6), (7, 2, 6), (8, 2, 6). The computed values of u are shown in Tables IV XII. A. Class I: Triangular co-norm: nil; inference method: drastic product or Larsen product; output fuzzy sets: triangular. Where N u is the normalization factor of u; x, y, z are d N, υ N and a N respectively. The incremental control output for other cells is derived in a similar manner. Drastic Product, R Table 3 Inference methods and their definitions Inference method Definition Area of inferred output fuzzy sets Mamdani minimum, R MM min(, µ( u N )) 2M (2- )/3 Larsen product, R LP.µ( u N ) 2M /3 Case2: µ( u N ), if = 1 2M/3 Case1:, if µ( u N ) = 1 - DP Case3: 0, otherwise 0 Bounded product, R BP max{0, + µ( u N ) 1} 2M 2 / 3 Standard -Sequence, R SS Case1: 1, if ( u N ) (4M/3){1 - } Case2: 0, if > ( u N ) 9

10 Table 4 u for Class I Controller (d N, υ N,a N ) (8,2,5) (8,3,5) (7,3,5) (7,3,6) (7,2,6) (8,2,6) (0,0,L) ¼ ¼ (0,L,0) ¼ ¼ (L,0,0) 1 0 ¼ ¼ 0 1 (0,L,L) 1/3 3/ /2 1/3 (L,L,0) 3/2 1/3 1/3 3/2 2 2 (L,0,L) 2 2 3/2 1/3 1/3 3/2 (L,L,L) B. Class II: Triangular co-norm: maximum; inference method: drastic product or Larsen product; output fuzzy sets: triangular. C. Class III: Table 5 u for Class II Controller (d N,υ N,a N ) (8,2,5) (8,3,5) (7,3,5) (7,3,6) (7,2,6) (8,2,6) (0,0,L) 1/ /3 ½ ½ (0,L,0) ½ ½ 1/ /3 (L,0,0) 1 1/3 ½ ½ 1/3 1 (0,L,L) 1 5/ /3 1 (L,L,0) 5/ /3 2 2 (L,0,L) 2 2 5/ /3 (L,L,L) Triangular co-norm: drastic sum; inference method: drastic product or Larsen product; output fuzzy sets: triangular. 10

11 D. Class IV: Table 6 u for Class III Controller (dn, υn, an) (8,2,5) (8,3,5) (7,3,5) (7,3,6) (7,2,6) (8,2,6) (0,0,L) 0 1/5 1/5 0 1/7 1/7 (0,L,0) 1/7 1/7 0 1/5 1/5 0 (L,0,0) 1/5 0 1/7 1/7 0 1/5 (0,L,L) 1/7 1/3 1/5 1/5 1/3 1/7 (L,L,0) 1/3 1/7 1/7 1/3 1/5 1/5 (L,0,L) 1/5 1/5 1/3 1/7 1/7 1/3 (L,L,L) 1/3 1/3 1/3 1/3 1/3 1/3 Triangular co-norm: maximum; inference method: Mamdani minimum; fuzzy sets: triangular. output E. Class V: Table 7 u for Class IV Controller (dn, υn, an) (8,2,5) (8,3,5) (7,3,5) (7,3,6) (7,2,6) (8,2,6) (0,0,L) 1/ /3 0 0 (0,L,0) 0 0 1/ /3 (L,0,0) 1 1/ /3 1 (0,L,L) 3/7 9/ /7 3/7 (L,L,0) 9/7 3/7 3/7 9/7 2 2 (L,0,L) 2 2 9/7 3/7 3/7 9/7 (L,L,L) Triangular co-norm: drastic sum; inference method: Mamdani minimum; output fuzzy sets: triangular. 11

12 Table 8 u for Class V Controller (dn, υn, an) (8,2,5) (8,3,5) (7,3,5) (7,3,6) (7,2,6) (8,2,6) (0,0,L) 0 3/11 3/11 0 1/15 1/15 (0,L,0) 1/15 1/15 0 3/11 3/11 0 (L,0,0) 3/11 0 1/15 1/15 0 3/11 (0,L,L) 1/15 1/3 3/11 3/11 1/3 1/15 (L,L,0) 1/3 1/15 1/15 1/3 3/11 3/11 (L,0,L) 3/11 3/11 1/3 1/15 1/15 1/3 (L,L,L) 1/3 1/3 1/3 1/3 1/3 1/3 F. Class VI: Triangular co-norm: drastic sum; inference method: Bounded product; output fuzzy sets: triangular. G. Class VII: Table 9 u for Class VI Controller (dn, υn, an) (8,2,5) (8,3,5) (7,3,5) (7,3,6) (7,2,6) (8,2,6) (0,0,L) 0 1/9 1/9 0 3/13 3/13 (0,L,0) 3/13 3/13 0 1/9 1/9 0 (L,0,0) 1/9 0 3/13 3/13 0 1/9 (0,L,L) 3/13 1/3 1/9 1/9 1/3 3/13 (L,L,0) 1/3 3/13 3/13 1/3 1/9 1/9 (L,0,L) 1/9 1/9 1/3 3/13 3/13 1/3 (L,L,L) 1/3 1/3 1/3 1/3 1/3 1/3 Triangular co-norm: drastic sum; inference method: Standard sequence; fuzzy sets: triangular. output 12

13 F. Class VIII: Table 10 u for Class VII Controller (dn, υn, an) (8,2,5) (8,3,5) (7,3,5) (7,3,6) (7,2,6) (8,2,6) (0,0,L) 0 1/3 1/ (0,L,0) /3 1/3 0 (L,0,0) 1/ /3 (0,L,L) 1 1 1/3 1/3 1 1 (L,L,0) /3 1/3 (L,0,L) 1/3 1/ (L,L,L) Triangular co-norm: maximum; inference method: Bounded product; fuzzy sets: triangular. output Table 11 u for Class VIII Controller (dn, υn, an) (8,2,5) (8,3,5) (7,3,5) (7,3,6) (7,2,6) (8,2,6) (0,0,L) 1/ /3 4/3 4/3 (0,L,0) 4/3 4/3 1/ /3 (L,0,0) 1 1/3 4/3 4/3 1/3 1 (0,L,L) 9/5 11/ /5 9/5 (L,L,0) 11/5 9/5 9/5 11/5 2 2 (L,0,L) /5 9/5 9/5 11/5 (L,L,L) I. Class IX: Triangular co-norm: maximum; inference method: Standard sequence; fuzzy sets: triangular. output 13

14 SIMULATION RESULTS Table 12 u for Class IX Controller The control performance of the fuzzy PID controller subjected to a step input is compared with conventional PID controller for linear, non-linear and time-delay systems. These responses are shown in Figure 5 Figure 8. A) FOR TIME DELAY SYSTEMS Example 1: (dn, υn, an) (8,2,5) (8,3,5) (7,3,5) (7,3,6) (7,2,6) (8,2,6) (0,0,L) 1/5 3/5 3/5 1/5 ¾ ¾ (0,L,0) ¾ ¾ 1/5 3/5 3/5 1/5 (L,0,0) 3/5 1/5 ¾ ¾ 1/5 3/5 (0,L,L) 4/5 6/5 2/3 2/3 6/5 4/5 (L,L,0) 6/5 4/5 4/5 6/5 2/3 2/3 (L,0,L) 2/3 2/3 6/5 4/5 4/5 6/5 (L,L,L) 7/6 7/6 7/6 7/6 7/6 7/6 Consider a first order system with time delay described by G(s) = e -0.75s /(s+1) Figure 5Unit Step Response for first order Time delay system 14

15 Example 2: Consider a second order system with time delay described by G(s) = e -s / s(s+3 Figure 6 Unit Step Response for second order Time delay system FOR NON-LINEAR SYSTEMS Example 1: For a non-linear system Figure 7 Unit Step Response for non-linear system 15

16 Example 2: For a non-linear system Figure 8 Unit Step Response for modulus nonlinearity system CONCLUSION In this paper analytical structures for different classes are derived triangular output fuzzy sets, different combinations of (Maximum, Drastic Sum) and by using two triangular co-norms five inference methods (Mamdani Minimum, Larsen Product, Drastic product, Bounded Product and Standard sequence) and center-of-sum defuzzification method. For each class of controller, the controller output u is computed at some arbitrary points and is tabulated. Computer simulations for non-linear and time delay systems shows that fuzzy PID controllers are giving better response over the conventional PID controllers. REFERENCES [1]. Mohan.B.M and Sinha.A, The simplest fuzzy PID controllers: mathematical models and stability analysis, Soft Computing, vol.10, pp , [2]. G.Chen, Conventional and fuzzy PID controllers: An overview, International Journal of Intelligent Control and Systems, 1 (1996), [3]. Patel.A.V., Mohan.B.M.,(2002), Analytical Structures and analysis of the simplest fuzzy PI controllers, Automatica, 38:

17 [4]. Mohan.B.M and Sinha.A, Analytical Structures for fuzzy PID controllers? IEEE Transactions on Fuzzy Systems, Vol. 16, No. 1, February 2008 [5]. Ying, H. The simplest fuzzy controllers using different inference methods and different nonlinear proportional-integral controllers with variable gains. Automatica, 29(6), [6] Astrom, K. J., & Haagglund, T. (2001). The future of PID control Control Engineering Practice, 9, [7] s. Galichet and L. Foulloy, Fuzzy controllers: Synthesis and alences, IEEE Trans. Fuzzy Syst., vol. 3, pp , May [8] B. S. Moon, Equivalence between fuzzy logic controllers and PI controllers for single input systems, Fuzzy Sets Syst., vol. 69, pp ,