Abstract
In this paper, mathematical models of Mamdani type simplest fuzzy Proportional Integral (PI)/Proportional Derivative (PD) controllers via Height (Ht) defuzzification are presented. Minimal number of fuzzy sets are chosen for the two inputs and output of the fuzzy controller. L –type and Γ –type membership functions and different Universes of Discourse (UoDs) are considered for the input variables. Membership functions of output variable are chosen in such a way that the sum of all the membership functions at any point is unity. Three linear fuzzy control rules relating all four input fuzzy sets to three output fuzzy sets are chosen. Two triangular norms namely Algebraic Product (AP) and Minimum (Min), and three triangular co –norms (also sometimes called s –norms) such as Bounded Sum (BS), Algebraic Sum (AS) and Maximum (Max) are used. Properties of the fuzzy controller models are studied. Since digital controllers are implemented on the digital processors, finally the computational and memory requirements of these fuzzy controllers and conventional (nonfuzzy) controllers are compared. A rough estimate of the computational time taken by the digital computer while implementing these discrete –time fuzzy controllers is given.
Introduction
Mathematical modeling of fuzzy controllers has been the topic of interest to many researchers. The exact mathematical model showing the relationship between the input and output variables is very useful in analysis, design, and implementation (on a digital computer) of fuzzy controllers. Fuzzy controllers can be realized in many ways. However, control practitioners prefer such a method which introduces less computational time -delay and involves minimal cost of replacement of controllers that are already in use. In this context, implementation of mathematical models of fuzzy controllers on a digital computer is one of the possible choices the practitioner has. Only the existing linear (nonfuzzy) controller expression needs to be replaced by a nonlinear fuzzy controller expression. However, the mathematical expression of a fuzzy controller depends on many factors like membership function, triangular norm (t - norm, in short), triangular co - norm, inference method and defuzzification strategy used. Hence, a fuzzy controller does not have a unique model, i.e., different combinations of these factors lead to different models. Several models of fuzzy controllers were obtained using Center of Sums (CoS) defuzzification, their properties were studied, and their suitability / unsuitability for control was discussed. Mathematical models of fuzzy controllers are useful in the following: establishing stability conditions for feedback systems that contain one of the controller models in the loop, obtaining optimal values of controller parameters using soft computing techniques like Genetic Algorithms (GAs), studying computational and memory requirements in the implementation of controllers on a digital computer, etc. The study on computational and memory requirements helps control practitioners choose the best possible fuzzy controller which produces better performance with less computational delay and reasonable cost of implementation.
Linear PID controllers do not work satisfactorily for nonlinear, time - varyingsystems. Nonlinear controllers such as fuzzy PID controllers will provide better performance for such systems. We now present some important historical developments in the area of mathematical modeling of the simplest fuzzy PI or PD controllers. Mathematical models of the simplest fuzzy PI controller using different inference methods and CoS defuzzification were first obtained by Ying [2]. Using these models, a novel tuning method based on gain margin and phase margin was proposed by Xu et al. [4] to determine the parameters of fuzzy PI controllers. Later models of fuzzy PI controller via different combinations of t - norms, s - norms and inference methods were obtained by Patel and Mohan [5]. Subsequently Ali and Ying [6] developed models by using nonlinear membership functions for the input fuzzy sets. Ying [7] presented a novel method that employs arbitrary trapezoidal input fuzzy sets and Zadeh fuzzy AND operator in the structure derivation of fuzzy PI/PD controllers. Later Ali and Ying [8] studied the analytical structures of two - dimensional and three - dimensional fuzzy controllers by employing nonlinear input fuzzy sets of arbitrary type. Models using algebraic sum s - norm were obtained by Mohan and Sinha [10]. Models of the simplest fuzzy PI or PD controllers with skewed membership functions for input and output fuzzy sets were presented by Mohan and Sinha [11]. The necessity of properly choosing output membership functions while deriving the mathematical models of fuzzy controllers was addressed by Mohan [12]. The influence of output membership function pattern on the nature of control law was shown by Mohan and Naresh [13].
Upon critically examining the above mentioned works, one will notice that the same UoD was considered for both the scaled inputs e
s
(k) and Δe
s
(k) in [2, 13]. different UoDs were considered for the scaled inputs e
s
(k) and Δe
s
(k) in [7]. To fully utilize the nonlinear characteristics of fuzzy controllers, the UoDs of scaled inputs should be different [1]. This point can be clear from the phase portrait, see Fig. 1, of a typical underdamped stable second - order system under control.
The Ht method of defuzzification has low computational complexity while having other desirable properties in the context of control [3]. Properties of the simplest fuzzy PI controllers using Ht defuzzification were studied through computer simulation by Patel [9]. To date, there is no evidence of showing the mathematical models of the simplest fuzzy controllers using Ht defuzzication method.
Therefore, in this paper, we obtain six different mathematical models for the simplest fuzzy PI or PD controller by using Ht defuzzification, different UoDs for the inputs and different combinations of t - norms (AP, Min) and s - norms (BS, AS and Max).
The structure of this paper is as follows: Section 2 describes principal components of fuzzy two - term controllers. In Section 3 analytical structures of thesimplest fuzzy two - term controllers are presented using Ht deuzzification. In Section 4 properties of analytical structures are provided. In Section 5 computational aspects of fuzzy two - term controllers are discussed.Section 6 concludes the paper.
Principal components of the simplest fuzzy PI or PD controller
The block diagram of a typical computer controlled system is shown in Fig. 2.
The expression for conventional PI control law in velocity form is given by
Moreover, the expression for conventional PD control law in position form is given by
It can be seen from Equations (1) and (2) that the inputs to PI and PD controllers are simply the scaled versions of e (k) and Δe (k). The output of PI controller is Δu (k) where as the output of PD controller is u (k). Similarly, fuzzy PI and fuzzy PD controllers have the same scaled inputs e s (k) and Δe s (k) (or and ). The scaled outputs of fuzzy PI and fuzzy PD controllers are denoted by Δu s (k) and u s (k) respectively, and they are in general a nonlinear function of scaled inputs. The nature of nonlinear function depends on the choice of components of fuzzy controller. The block diagram of fuzzy two-term controllers is shown in Fig. 3 in which S e , S Δe , S PI and S PD represent the scaling factors of the fuzzy controller. We now present the details of components of the fuzzy controllers considered in this paper.
L - type and Γ - type membership functions, shown in Fig. 4, are chosen for the input variables. Their mathematical description is as follows:
Notice that μ N e (e s (k)) + μ P e (e s (k)) =1 and μ N Δe (Δe s (k)) + μ P Δe (Δe s (k)) =1. The membership functions for the scaled output are shown in Fig. 5. The value of A can be chosen anything in [0, B). When A = 0 the trapezoid μ Z Δu (Δu s (k)) becomes a triangle.
Rule base
Three linear control rules for fuzzy PI control are given below:
R 1 : IF e s (k) is N e AND Δe s (k) is N Δe THEN Δu s (k) is N Δu R 2 : IF (e s (k) is N e AND Δe s (k) is P Δe ) OR (e s (k) is P e AND Δe s (k) is N Δe ) THEN Δu s (k) is Z Δu R 3 : IF e s (k) is P e AND Δe s (k) is P Δe THEN Δu s (k) is P Δu
The justification of this rule base is as follows: Consider Fig. 6 showing the reference (unit step) input, unit step response, error and time rate of change of error in the case of a second - order underdamped system. If we look at rule R 1, condition N e implies y > 1 and N Δe implies > 0. This means that the controller is driving the system output upward and the output is above the reference, as shown by point P 1 in Fig. 6. Hence we set Δu (k) negative in order to turn the system output downward. Similarly, for the first premise part of the rule R 2, the output is above the reference and moving downward (point P 21 in Fig. 6), and therefore we set Δu (k) zero. The second premise part of the rule R 2 and the rule R 3 can be justified in a similar manner at the ponits P 22 and P 3 in Fig. 6. This rule base is also applicable to fuzzy PD controller provided Δu s (k) is replaced with u s (k).The t - norms Algebraic Product and Minimum (to perform AND operation in rule base) are defined as follows:
Algebraic Product: t(μ X (x) , μ Y (y)) = μ X (x) · μ Y (y)
Minimum: t(μ X (x) , μ Y (y)) = min{μ X (x) , μ Y (y)}
The s-norms Bounded Sum, Maximum and Algebraic Sum (to perform OR operation in the rule R 2) are defined in the following manner:
Bounded Sum: s (μ X (x) , μ Y (y)) = min {1, μ X (x) + μ Y (y)}
Maximum: s (μ X (x) , μ Y (y)) = max {μ X (x) , μ Y (y)}
Algebraic Sum (AS):s (μ X (x) , μ Y (y)) = μ X (x) + μ Y (y) - μ X (x) · μ Y (y) All possible combinations of input variables are shown in Fig. 7. As a matter of fact, this figure shows the top view (seen down along μ axis) of the three - dimensional plot with the axes e s (k), Δe s (k) and μ. The control rules R 1 - R 3 are used to obtain appropriate control law in each region. The resultant expressions (μ 1, μ 2 and μ 3) of the antecedent parts of all three rules are given in Table 1 (using AP AND) and Table 2 (using Min AND).
Defuzzification
The crisp value of scaled control output via Height defuzzification is given by
Here, the input - output structural relationships of the simplest fuzzy PI or PD controllers are presented using Ht defuzzification and various combinations of it t - norms and s - norms.
We introduce the following six different classes of controllers: Class 1: Using AP AND, BS OR Class 2: Using AP AND, AS OR Class 3: Using AP AND, Max OR Class 4: Using Min AND, BS OR Class 5: Using Min AND, AS OR Class 6: Using Min AND, Max OR
The following models can be obtained by substituting the expressions of μ 1, μ 2 and μ 3 (defined in Tables 1 and 2) into Equation (3) where μ N e , μ P e , μ N Δe and μ P Δe are defined in Section 2.1.
Analytical models in the regions 1–8
All the six classes of controllers have the same mathematical models given as
Analytical models in the regions 9–12
Expression for the control law of the simplest fuzzy PI or PD controller is given by
Properties of the simplest fuzzy PI/PD controllers
Their minimum control effort is -0.5 (A + 2B), occuring at (e
s
(k) , Δe
s
(k)) = (- h
e
, - h
Δe
). Their control effort is zero at (e
s
(k) , Δe
s
(k)) = (0, 0). Their maximum control effort is 0.5 (A + 2B), occuring at (e
s
(k) , Δe
s
(k)) = (h
e
, h
Δe
). Their control effort is zero on the line Δe
s
(k) . Surface and contour plots of control output are shown in Figs. 8 and 9. In surface plots Δu
s
(k) versus e
s
(k), Δe
s
(k) is shown. Since all six classes of fuzzy controllers are different from one another, their surface plots also should look different. However, this feature is not distinctly apparent as can be seen in Figs. 8 and 9. To show that these controllers are distinctly different, the value of Δu
s
(k) on Δu
s
(k) axis is frozen to some value and then the so called contour plots are drawn. In other words, the contour (2D) plots are nothing but sectional views of surface (3D) plots. Looking at the contour plots one can quickly conclude that all six classes of fuzzy controllers are actually six different controllers. The control effort is continuous, and symmetric about the line Δe
s
(k) . Moreover, magnitude of control effort increases monotonically from the minimum value to the maximum value. The arrows in the contour plots represent gradients at those points. They are variable structure controllers as their structures are different in different regions shown in Fig. 7. All classes of controllers except class 1 controller are variable gain controllers as gain γ of each controller is a function of inputs e
s
(k) and Δe
s
(k). All classes of controllers except class 1 controller are nonlinear controllers. Class 1 controller is a linear controller. The maximum value of gain γ
1,γ
2,...,or γ
6 is , and this occurs at (e
s
(k) , Δe
s
(k)) = (h
e
, h
Δe
). This information will be useful in the stability analysis of fuzzy feedback control systems.
Computational aspects of the simplest fuzzy PI/PD controllers
Linear PID controllers are still widely used in industries because they provide control output quickly. The computational delay introduced by them is almost insignificant. Infact, it is the smallest when compared with any other control scheme. It can be seen from Equations (1) and (2) that the PI and PD controllers require only one mathematical operation, that is addition. The mathematical operations that are common (like computing Δe (k) from e (k) and e (k - 1); e s (k) and Δe s (k) from e (k) and Δe (k); u (k) from u (k - 1) and Δu (k) etc.) to linear PI / PD controllers and fuzzy PI / PD controllers are ignored here as our objective here is to find the additional computational effort required by the fuzzy controllers. As the mathematical models of different classes of the simplest fuzzy PI / PD controllers have been obtained, we can find the number of mathematical operations and memory locations required in their implementation. We consider the class 2 controller for explanation purpose.
The class 2 controller model in regions 9–12 is given by
h = h e h Δe , and
C = 15h 2.
Now C requires two multiplications and one memory location, h requires one multiplication and one memory location, and N requires four operations (three multiplications and one addition) and one memory location. The expression on the right hand side of Δu s (k) requires fourteen operations (nine multilpications, three additions, one division and one subtraction). We call these operations on - line operations because these fourteen operations are required to be carried out for every k th sampling instant. This is not the case with computation of N, h and C because, once N, h and C are computed they can be stored in memory locations so that these stored values can be used at every k th instant. For this reason, we call these operations off - line operations. Hence, the total number of mathematical operations and memory locations required in the implementation of class 2 fuzzy controller to compute Δu s (k) at k th instant are 21 and 3, respectively.
Similarly, the mathematical operations and memory locations required in the implementation of the other five classes of controllers in regions 9–12 are obtained and presented in Table 4. It can be observed from the mathematical models of class 2 fuzzy controller in different regions of Fig. 7 that the computational and memory burden on digital controller in regions 9–12 is the highest when compared to that of region 1,2,...,7, or 8. This is true for all the 6 classes of fuzzy controllers. It has been observed that the maximum time taken by the digital computer with 3.2 GHz processor to compute Δu s (k) from e s (k) and Δe s (k) is of the order of 10-4 sec for the models obtained in this paper. The computational time delay should be less than the sampling time period T. We believe that this quantitative analysis helps the control practitioners in implementing fuzzy controllers.
Conclusions
Mathematical models of six different classes of the simplest fuzzy PI/PD controllers have been obtained via L - type, Γ - type, and trapezoidal/triangular input -output membership functions, AP / Min AND operation, BS / AS / Max OR operation, and Ht defuzzication. The models derived in this paper are simple and computationally attractive. We believe that this work helps in replacing conventional PI / PD controllers with fuzzy controllers (class 2,...class 6) for better performance.