A comparative study of type-1 fuzzy logic systems, interval type-2 fuzzy logic systems and generalized type-2 fuzzy logic systems in control problems

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A comparative study of type-1 fuzzy logic systems, interval type-2 fuzzy logic systems and generalized type-2 fuzzy logic systems in control problems Castillo Oscar∗, Amador-Angulo Leticia, Juan R. Castro, Mario Garcia-Valdez

Q1

Tijuana Institute of Technology, 22379 Tijuana, Mexico

a r t i c l e

i n f o

Article history: Received 26 March 2015 Revised 8 March 2016 Accepted 14 March 2016 Available online xxx Keywords: Alpha plane representation Fuzzy controller Generalized type-2 fuzzy logic Footprint uncertainty

a b s t r a c t This paper presents a comparative study of type-2 fuzzy logic systems with respect to interval type-2 and type-1 fuzzy logic systems to show the efficiency and performance of a generalized type-2 fuzzy logic controller (GT2FLC). We used different types of fuzzy logic systems for designing the fuzzy controllers of complex non-linear plants. The theory of alpha planes is used for approximating generalized type-2 fuzzy logic in fuzzy controllers. In the defuzzification process, the Karnik and Mendel Algorithm is used. Simulation results with a type-1 fuzzy logic controller (T1FLC), an interval type-2 fuzzy logic controller (IT2FLC) and with a generalized type-2 fuzzy logic controller (GT2FLC) for benchmark plants are presented. The advantage of using generalized type-2 fuzzy logic in fuzzy controllers is verified with four benchmark problems. We considered different levels of noise, number of alpha planes and four types of membership functions in the simulations for comparison and to analyze the approach of generalized type-2 fuzzy logic systems when applied in fuzzy control. © 2016 Published by Elsevier Inc.

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1. Introduction

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Fuzzy control systems combine information of human experts (natural language) with measurements and mathematical models. Fuzzy systems transform the knowledge base into a mathematical formulation that has proven to be very efficient in many applications [1,4,5,6,14,45,48]. Type-2 fuzzy models have emerged as an interesting generalization of type-1 fuzzy models based upon fuzzy sets. In fact, these models are also called generalized type-2 fuzzy models. There have been a number of claims put forward as to the relevance of type-2 fuzzy sets being regarded as generic building constructs of fuzzy models. Fuzzy controllers have the advantage that they can be adaptive when disturbances in the plant are present. The main advantage of using generalized type-2 fuzzy control (GT2FLC) is that the plants show a higher degree of stability in the simulations. When we consider noise presence for the GT2FLC, the results show that the Generalized Type-2 Fuzzy Logic System has better stability characteristics. This paper considers several experiments in the simulation of four benchmark control problems with a type-1 fuzzy logic controller (T1FLC) and with an interval type-2 fuzzy logic controller (IT2FLC), where the authors conclude that the IT2FLC is

3 4 5 6 7 8 9 10 11 12

∗

Corresponding author. Tel.: +526646236318. E-mail addresses: [email protected], [email protected] (C. Oscar), [email protected] (A.-A. Leticia), [email protected] (J.R. Castro), [email protected] (M. Garcia-Valdez). http://dx.doi.org/10.1016/j.ins.2016.03.026 0020-0255/© 2016 Published by Elsevier Inc.

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better than T1FLC based on the simulation error minimization. We also realized the comparison with the three methods to observe the behavior and the improvement that a GT2FLC can offer with respect to T1FLC and IT2FLC. Few works can be found in the literature related to the implementation of generalized type-2 fuzzy logic systems in the area of fuzzy control. It is because of this fact that we use the concepts of the α − planes representation and generalized type-2 fuzzy sets to improve the design of fuzzy logic systems and allow the analysis of benchmark problems more efficiently with high levels noise in the model. In the last five years, there has been a significant increase of the research on higher order forms of fuzzy logic systems, in particular the use of interval type-2 fuzzy logic [1,2,11,12,33,44,51,52]. In addition with the advancement of IT2FLS, uncertainty could finally be directly incorporated into the Fuzzy Sets. Although the boom of research with IT2FLS is recent, there is still much to be explored, and some current examples of research are shown by, a fuzzy model of computing with words [11], fuzzy operations [12], a simplified IT2FLS [22], type-reduction algorithms [30], the centroid of triangular and Gaussian IT2FSs [38], and enhanced type-reduction [40]. More recently generalized type-2 fuzzy logic systems have been proposed; of course, the idea of going into higher orders or types of fuzzy logic is to construct better models of uncertainty. In this sense, it is theoretically expected that generalized type-2 fuzzy logic will allow better management of uncertainty [49,50]. However, generalized type-2 requires a higher computational overhead and several efforts have been put forward in order to limit the complexity of generalized type-2 fuzzy logic systems; for example, Wagner and Hagras [3],[41–43] have introduced the zSlices-based representation, and Mendel and Liu [21,22], have put forward a representation based on α -planes, which both enable the representation of, and computation with, generalized type-2 fuzzy sets [22,23]. And as such, there are also very few application research works where GT2FS are used, for example, in a GT2FS based on face-space approach to emotion recognition [8], a general type-2 fuzzy inference systems; analysis, design and computational aspects [16], fuzzy c-means for uncertain fuzzy clustering [17], monotone centroid flow algorithm for type-reduction [18], multi-criteria group decision making [19], edge detection for image processing [20], matching GT2FS by comparing the vertical slices [34], generalized type-2 fuzzy systems with interval type-2 and type-1 fuzzy systems applied to fuzzy control [35], the information granule numerical evidence [50], computing with words for discrete GT2FS [51]. In addition in the field of fuzzy control, a comparison of fuzzy controllers for the Water Tank with Type-1 and Type-2 Fuzzy Logic is presented in [1], and the improvement of GT2FLS over IT2FLS and T1FLS for controlling a mobile robot is presented in [35]. In this paper, we use the α -plane representation, which enables the representation of and computation with generalized type-2 fuzzy sets. The main contribution of this paper is a comparative study based on generalized type-2 fuzzy logic for the design and implementation of fuzzy controllers, which allows for better modeling of the uncertainty that exists in achieving control of non-linear plants. Also, the comparative study of T1FLC, IT2FLC and GT2FLC as tools for modeling complex problems in control. Based on the literature and experience in the area of fuzzy control, we expect that GT2FLC can demonstrate that the stabilization and the performance of the simulations are improved, especially when noise levels are considered in the model, and better results are obtained when compared with respect to type-1 FLC and interval type-2 FLC. The rest of the paper is organized as follows. Section 2 describes some basic concepts of fuzzy logic systems; type-1 FLS, interval type-2 FLS, generalized type-2 fuzzy logic systems and the theory of alpha planes. Section 3 shows the benchmark problems that we consider in the simulations. Section 4 shows the simulation results with the implemented generalized type-2 fuzzy logic controller and a comparison with the type-1 and interval type-2 fuzzy logic controllers, so that the advantage of using a generalized type-2 fuzzy logic controller is fully appreciated. Finally, Section 5 offers some conclusions of this work.

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2. Basic concepts of fuzzy logic systems

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2.1. Definition of type-1 fuzzy logic systems

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A type-1 fuzzy set in the universe X is characterized by a membership function uA (x) taking values on the interval [0,1] and can be represented as a set of ordered pairs of an element and the membership degree of an element to the set and are defined by the following Eq. (1) [45–48]:

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

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A = {(x, μA (x ))|x ∈ X }

(1)

60

Where μA : X → [0, 1]. In this definition μA (x ) represents the membership degree of the element x ∈ X to the set A. In this work we are going the use the following notation: A(x ) = μA (x ) for all x ∈ X.

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2.2. Definition of Interval type-2 fuzzy logic systems

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Based on Zadeh’s ideas, in 1979 Mizumoto and Tanaka [3] presented the mathematical definition of a type-2 fuzzy set. Since then, several authors have studied these sets, in [25–27] Mendel, John and Mouzouris defined these sets as follows [3].

58 59

63 64

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Fig. 1. Generalized type-2 membership function.

Fig. 2. FOU of the generalized type-2 membership function.

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An Interval Type-2 Fuzzy Set A˜ , denoted by μA(x ) and μ ¯ A˜ (x ) is represented by the lower and upper membership functions of μ (x ). Where x ∈ X. In this case, Eq. (2) shows a sample IT2FS [6,10,24,25,26,27,31]. A

A˜ = {( (x, u ), 1 )|∀x ∈ X, ∀u ∈ Jx ⊆ [0, 1]}

(2)

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Where X is the primary domain, Jx is the secondary domain. All secondary grades (μA˜ (x, u ) ) are equal to 1.

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2.3. Definition of Generalized type-2 fuzzy logic systems

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With GT2FLS the logic is generally the same as for T1FLS and IT2FLS, but their operations are somewhat different, due to the nature of GT2FS [35]. There are several mathematical definitions of a generalized type-2 fuzzy logic system, and we used the representation based on [20,35,36] to define Generalized Type-2 Fuzzy Sets and are defined by the following Eq. (3):

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˜ A = {( (x, u ), μA˜ (x, u ))|∀x ∈ X, ∀u ∈ Jx ⊆ [0, 1]}

(3)

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Where Jx ⊆ [0, 1], x is the partition of the primary membership function, and u is the partition of the secondary membership function. In Fig. 1 we can find a representation of a generalized type-2 membership function, and in Fig. 2, the footprint of uncertainty (FOU) is illustrated, which is associated with the third dimension and allows a better modeling of real world uncertainty. It must be noted that there is a small difference in notation when compared with Type-1 and Interval Type-2, this is, T1FS and IT2FS use the notation μ(x ), but GT2FS uses f x (u ), in the vertical axis, and this is due to the complexity involved in GT2FLS in comparison with the others, as well as how GT2FLS has been described in the literature.

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2.4. α -planes representation

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The α -plane for a generalized T2 FLS, in this case A˜ , is denoted by Ãα , and it is the union of all primary membership functions of Ã, which secondary membership degrees are higher or equal to α (0 ≤ α ≤ 1) [21,23]. The equation of an alpha Please cite this article as: C. Oscar et al., A comparative study of type-1 fuzzy logic systems, interval type-2 fuzzy logic systems and generalized type-2 fuzzy logic systems in control problems, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2016.03.026

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Fig. 3. An example of the associated type-2 fuzzy set for the alpha-plane.

Fig. 4. Graphic representation of the problem of water tank.

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plane is represented by Eq. (4). In Fig. 3 the representation of an alpha plane is illustrated [23,35].

A˜ α = {(x, u ), μA˜ (x, u ) ≥ α|∀x ∈ X, ∀u ∈ JX ⊆ [0, 1]}

(4)

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3. Benchmark problems

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For the evaluation of the generalized type-2 fuzzy logic control approach we used four benchmark problems, which in fuzzy control are mathematical models that allow us to analyze the behavior of the generated fuzzy controllers. We describe each of the benchmark problems and the corresponding implemented fuzzy controllers, in the following subsections.

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3.1. Water Tank Controller

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The first problem to be considered is known as the Water Tank Controller, which aims at controlling the water level in a tank, therefore, based on the actual water level in the tank the controller has to be able to provide the proper activation of the valve. Fig. 4 shows graphically the way in which the valve opening operates and hence the filling process in the tank, and this has two variables, which are the water level and the speed of opening of the output valve for the tank filling. To calculate the valve opening in a precise way we rely on fuzzy logic, which is implemented as a fuzzy controller that performs the control on the valve that determines how fast the water can enter the tank to maintain the level of water in a better way. The model equation of the Water Tank is represented as a differential equation and the height of water in the tank is given by Eq. (5):

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89 90 91 92 93 94 95

√ d dH Vol = A = bV − a H dt dt 96 97 98 99 100 101 102 103

(5)

Where Vol is the volume of water in the tank, A is the cross-sectional area of the tank, b is a constant related to the flow rate into the tank, and a is a constant related to the flow rate out of the tank. The equation describes the height of water H as a function of time, due to the difference between flow rates into and out of the tank. The implementation of the type-1 and type-2 Fuzzy Logic Controllers are presented to experimentally observe the behavior of each controller. We later present the characteristics of the generalized type-2 fuzzy logic controller, besides the results of the model evaluation. The membership functions are for the two inputs of the fuzzy system: the first is called level, which has three membership functions with linguistic values of high, okay and low. The second input variable is called rate with three membership functions corresponding to the linguistic values of negative, none and positive. The names of the linguistic labels are assigned Please cite this article as: C. Oscar et al., A comparative study of type-1 fuzzy logic systems, interval type-2 fuzzy logic systems and generalized type-2 fuzzy logic systems in control problems, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2016.03.026

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Table 1 Fuzzy rules of the fuzzy controller for the Water Tank Controller. # Rule

Inputs

1 2 3 4 5

Output

Level

Operator

Rate

Valve

Okay low High Okay Okay

—— —— —— And And

—— —— —— positive Negative

Nochange Openfast Closefast Closeslow Openslow

Fig. 5. Representation of a generic fuzzy controller.

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based on the empirical process of the filling behavior of a water tank [1]. The generalized type-2 fuzzy logic controller has an output called valve, which is composed of five triangular membership functions with the following linguistic values: closefast, closeslow, nochange, openslow and openfast. The knowledge about the problem provide us with five rules, which are detailed in Table 1. The combination of rules for this benchmark problem can be found and are described in [1], and these 5 rules are used to visualize the behavior of the generalized type-2 fuzzy logic controller. Finally, at the output of the adder we have the error signal, which is applied to the fuzzy controller together with the change in the error signal over time [1,4,6,9,35,37]. The representation of a generic fuzzy controller is shown in Fig. 5.

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3.2. Temperature Controller

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The second problem to be considered is the Temperature Controller, which aims to establish the temperature of a water regulator. We now present the characteristics of the generalized type-2 fuzzy logic controller, besides the results of the model evaluation. The Membership functions are for the two inputs to the fuzzy system: the first is called temp, which has three membership functions with linguistic values of cold, good and hot. The second input variable is called flow with three membership functions with linguistic values of light, good and hard.. The names of the linguistic labels are assigned based on the empirical process to simulate the Temperature Controller. The generalized type-2 fuzzy logic controller has two outputs called cold, and hot, which are composed of five triangular membership functions with the following linguistic values, respectively: closefast, closeslow, steady, openslow and openfast. The knowledge about the problem provide us with nine rules, which are detailed in Table 2.

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Table 2 Fuzzy rules of the fuzzy controller for the Temperature Controller. # Rule

1 2 3 4 5 6 7 8 9

Inputs

Outputs

Temp

Operator

Flow

Cold

Hot

Cold Cold Cold Good Good Good Hot Hot Hot

And And and And And And And And And

Light Good Hard Light Good Hard Light Good Hard

Openslow Closeslow Closefast Openslow Steady Closeslow Openfast Openslow Closeslow

openfast Openslow Closeslow Openslow Steady Closeslow Openslow Closeslow Closefast

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Fig. 6. Mobile robot model.

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3.3. Mobile robot controller

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The model which is used is of a unicycle mobile robot [35], consisting of two actuated wheels located on the same axis and a front free wheel, and Fig. 6 shows a graphical description of the robot model. The robot model assumes that the motion of the free wheel can be ignored in its dynamics, as shown by Eqs. (6) and (7).

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M (q )v˙ + C (q, q˙ )v + Dv = τ + P (t ) 127

(6)

Where,

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q = (x, y, θ )T is the vector of the configuration coordinates, υ = (v, w)T is the vector of velocities, τ = (τ1 , τ2 ) is the vector of torques applied to the wheels of the robot where τ1 and τ2 denote the torques of the right and left wheel, respectively P ∈ R2 is the uniformly bounded disturbance vector, M (q ) ∈ R2×2 is the positive-definite inertia matrix, C (q, q˙ )ϑ is the vector of centripetal and Coriolis forces, and D ∈ R2×2 is a diagonal positive-definite damping matrix.

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The kinematic system is represented by Eq. (7).

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q˙ =

cos θ 0 sin θ 0 0 1



v w

(7)

    υ J (q )

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Where,

(x, y ) is the position in the X – Y (world) reference frame, θ is the angle between the heading direction and the x-axis, v and w are the linear and angular velocities. Furthermore, Eq. (8) shows the non-holonomic constraint, which this system has, which corresponds to a no-slip wheel condition preventing the robot from moving sideways.

y˙ cos θ − x˙ sin θ = 0

(8)

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The system fails to meet Brockett’s necessary condition for feedback stabilization, which implies that no continuous static state-feedback controller exists that can stabilize the closed-loop system around the equilibrium point. The fuzzy logic system has two inputs: the first is called ev (error in the linear velocity), which has three membership functions with linguistic values of N, Z and P. The second input variable is called ew (error in the angular velocity) with three membership functions with the same linguistic values. The generalized type-2 fuzzy logic controller has two outputs called T1 (Torque 1), and T2 (Torque 2), which are granulated into three triangular membership functions with the following linguistic values, respectively: N, Z, P. The combination of the rules is presented in Table 3 and their description can be found in [35].

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3.4. Beam and ball controller

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This problem has the goal of stabilizing a ball that is placed on a beam, where it is allowed to roll with one degree of freedom along the length of the beam. A lever arm is attached to the beam at one end and a servo gear at the other. As

143 144 145 146 147 148

152

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Table 3 Fuzzy rules of the fuzzy controller for the Mobile Robot Controller. # Rule

Inputs

1 2 3 4 5 6 7 8 9

Outputs

ev

Operator

ew

T1

T2

N N N Z Z Z P P P

And And And And And And And And And

N Z P N Z P N Z P

N N N Z Z Z P P P

N Z P N Z P N Z P

Fig. 7. Representation of the beam and ball problem.

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the servo gear turns by an angle θ , the lever changes the angle of the beam by α . When the angle is changed from the horizontal position, gravity causes the ball to roll along the beam. A controller is designed for this system so that the ball’s position can be manipulated. The representation of this is shown in Fig. 7. In the system parameters in this problem, we assume that the ball rolls without slipping and friction between the beam and ball is negligible. The constants and variables in this case are defined as follows: (m) mass of the ball 0.11 kg (R) radius of the ball 0.015 m (d) lever arm offset 0.03 m (g) gravitational acceleration 9.8 m/s^2 (L) length of the beam 1.0 m (J) ball’s moment of inertia 9.99e-6 kg.m^2 (r) ball position coordinate (alpha) beam angle coordinate (theta) servo gear angle The fuzzy logic system that stabilizes the beam and ball has four inputs called in1, in2, in3 and in4, respectively. The membership functions used in this case are of the generalized bell form. The number of rules is 16. The fuzzy logic system is of Takagi-Sugeno type [39]. The test criteria are a series of Performance Indices; where the Integral Square Error (ISE), Integral Absolute Error (IAE), Integral Time Squared Error (ITSE), Integral Time Absolute Error (ITAE) and the Root Mean Square Error (RMSE) are used, respectively shown in Eqs. (9)–(13).



ISE =

∞ 0



173

IAE =

∞ 0



174

IT SE = IT AE =

(9)

|e(t )|dt

(10)

∞ 0



175

e2 (t )dt

∞ 0

e2 (t )t dt

(11)

|e(t )|tdt

(12)

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Fig. 8. Proposed architecture of the model for the GT2FLC.



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ε=

2 1 N (Xt − Xˆt ) t=1 N

(13)

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4. Simulations of the Generalized type-2 fuzzy logic controllers

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In this section the proposed model for the implementation and the simulations of the Generalized type-2 fuzzy logic controller (GT2FLC) are presented. In Fig. 8, the block diagram for the simulation of the generalized type-2 fuzzy logic controller is illustrated.

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4.1. Input

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The first step in the whole process is the simulation of the model and in this way obtaining the initial errors of the simulation. In the initial iterations of the model, the Generalized type-2 fuzzy logic controller is evaluated; therefore, the mean square error generated by the model becomes the input of the second iteration.

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4.2. Fuzzification

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The fuzzifier maps crisp inputs into generalized type-2 fuzzy sets to process within the FLC. In this paper, we will focus on the type-2 singleton fuzzifier as it is fast to compute and, thus, suitable for the generalized type-2 FLC real-time operation. Singleton fuzzification maps the crisp input into a fuzzy set, which has a single point of nonzero membership. Hence, the singleton fuzzifier maps the crisp input xp into a type-2 fuzzy singleton, whose MF is μA˜ p (x p ) = 1/1 for x p = xp and

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183

187 188 189 190

μA˜ p (x p ) = 0 for all x p = xp for all p = 1, 2, . . . , P, where P is the number of FLS inputs [3,20].

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4.3. Input linguistic variables

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For the particular case of the water tank, two inputs are defined, in which each one has three Gaussian membership functions with uncertain mean. The Gaussian membership functions for each input are obtained with (20) and (21), and the means of each function are obtained with (15) and (16). For example, for the high membership functions, the first mean is obtained with (15), the second mean is calculated by (16) and the σ value is obtained with (14).

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Fig. 9. Generalized type-2 fuzzy logic controller of the benchmark problem of the water tank.

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The inference system has one output called valve, the linguistic values used for the output are: closefast, closeslow, nochange, openslow and openfast, and we selected the range [−1, 1]. The Gaussian membership functions for the output are obtained with (20)–(24), the means of each function are obtained with (15)–(19) and the σ value with (17). The FOU for the valve output variable, is calculated in a similar way as for the input variables. This is the method that we propose to analyze the four types of membership functions and manually we change the distribution of the parameters. Simulations with 4 types of membership functions in order to observe the behavior of the model are performed. We show the different types of the membership functions with the example of the linguistic variables called good of generalized type-2 form, where the value for the FOU is 0.9 for the inputs and outputs, respectively. Experiments are performed with 4 types of membership functions in order to observe the behavior of the model, which are: Gaussian, Trapezoidal, Triangular and Generalized Bell. The representation of the generalized type-2 fuzzy logic controllers are shown in Fig. 9–12, respectively, for the four benchmark problems studied in this work. The Gaussian membership functions are calculated by Eqs. (20)–(24). This membership function is considered in the GT2FLC and is characterized as having uncertainty in the mean and is calculated by Eqs. (21)–(24). The most detailed representation of the Gaussian membership function in the GT2FLC is presented in the following equations.

σi = highi /2

(14)

m1 = highi

(15)

213

m2 = m1 + (m1 ∗ K ), where K is in (0, 1 )

(16)

214

σi = m 2 / 5

(17)

215

m 1 = σi

(18)

216

m2 = m1 + (m1 ∗ K ), where K is in (0, 1 )

(19)

μ˜ (x, u) = gausmgausstype2 (x, u, [σx , m1 , m2 ])

(20)

217 218

Where “gausmgausstype2” stands for a Gaussian generalized type-2 membership function with uncertain mean

μ˜ (x ) = [μ(x ), μ¯ (x )] = igausmtype2(x, [σx , m1 , m2 ] ) 219

(21)

Where “igausmtype2” stands for a Gaussian interval type-2 membership function with uncertain mean

mx =

m1 + m2 2

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Fig. 10. Generalized type-2 fuzzy logic controller of the benchmark problem the Temperature Controller.

Fig. 11. Generalized type-2 fuzzy logic controller of the mobile robot.

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Fig. 12. Generalized type-2 fuzzy logic controller of the beam and ball.

δ = μ(x ) − μ¯ (x ) δ σu = √ + ε

(22)

2 6

  

1 x − mx 2 2 σx   2

220

px = gaussmf(x, [σx , mx ] ) = exp − 221

μ˜ (x, u) = gaussmf(u, [σu , px ]) = exp −

1 2

x − px

σu

(23)

(24)

222

4.4. Inference

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Once the input and output variables are defined, with their respective membership functions, the inference process is performed in the system, and for this the following steps are needed: Define the fuzzy rules: The structure of the rules in the generalized Type-2 FLS is the standard Mamdani-type FLS rule structure used in the Type-1 FLS and an interval Type-2 FLS, but in the paper, we assume that the antecedents and the consequents sets are represented by generalized type-2 fuzzy sets. So for a type-2 FLS with p inputs x1  X1 , . . . , xp ∈ XP and one output y ∈ Y, which is a Multiple Input Single Output (MISO) system, and if we assume there are M rules, the kth rule in the generalized type-2 FLS can be written as follows [7,24,25]:

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Rk : IF x1 is F˜1k and . . . and x p is F˜pk , 230 231 232 233

234

T HEN y is G˜ k

(25)

For modeling the process with the fuzzy system, we consider rules that help describe the existing relationship between the behaviors of the controllers in the real world; these rules are designed to be experimentally observable for the problem solved by each controller. The inference of a GT2FLS can be simplified into two main operations, meet and join, as shown in Eqs. (26) and (27).

μA˜ (x, u ) μB˜ (x, w ) = {(v, fx (u )∗˜ fx (w ))|v ∈ u ∨ w, u ∈ Jxu ⊆ [0, 1], w ∈ Jxw ⊆ [0, 1]}

(26)

μA˜ (x, u )  μB˜ (x, w ) = {(v, fx (u )∗˜ fx (w ))|v ∈ uw , u ∈ Jxu ⊆ [0, 1], w ∈ Jxw ⊆ [0, 1]}

(27)

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4.5. Type reduction

236

As for the type-reducer for the GT2FLS, one of the techniques uses the centroid, shown in Eq. (28) and this is the definition of the centroid C ˜˜ for type reduction of a GT2FS. Where θi is a combination associated with the secondary degree A fx1 (θ1 )∗˜ . . . ∗˜ fxN (θN ).

237 238



CA˜˜ = 239 240 241 242

(z, μ(z )) |gotl z ∈

θ μ(z ) ∈ fx1 (θ1 ) × . . . × fxN (θN ), θi ∈ Jx1 × . . . × JxN θ

i=1 xi i , N i=1 i



(28)

To perform the defuzzification process, the Karnik and Mendel method is used. (1) Karnik and Mendel algorithm: Type reduction is performed by applying the type reduction of the Karnik and Mendel algorithm [12,13,24,26,27,28,30], and this reduction is given by the following Eqs. (29) and (30).

yl∝ (x ) = 243

yr∝ (x ) = 244

N

L

¯ k∝ x y¯ i k=1 l L k x ¯ ∝ k=1

( ) + ( )+

R

k  ∝ x k=1 R i ∝ i=1

M

j=L+1 M j=L+1

∝j (x )y¯ lj ∝j (x )

(29)



¯k  k ( )y¯ kr + M k=R+1 ∝ (x )y¯ r  ¯i  ( x ) + M i=R+1 ∝ (x )

(30)

(1) Alpha plane integration: the results of the alpha planes are integrated by the following Eqs. (31) and (32) [21,23,29,51].

N yˆlj 245

 

i ∝∝ ylj x i N i=1 ∝i

(31)

i ∝∝ yrj (x ) i N i=1 ∝i

(32)

i=1

( x ) =

N

i=1

yˆrj (x ) = 246

4.6. Defuzzification

247

After realizing the type reduction and integrating the results of all the alpha planes, the defuzzification is performed by using the average of yl and yr , to obtain the defuzzified output of a generalized type-2 non-singleton FLS [15,28,30,32].

248

yˆ j (x ) =

yˆlj (x ) + yˆrj (x ) 2

(33)

249

5. Simulation results

250

254

We used four benchmark problems in control, and we consider fuzzy logic systems with the predetermined structure described in Section 3. We change the types of membership functions of the inputs and outputs. Each benchmark problem has the configuration of the parameters in the specific simulation. In the first case of filling the water tank, we considered changing the type of the membership functions, using a perturbation with a level of 1.5, the value of the rho is 0.2 and 5 alpha planes are used.

255

5.1. Simulation results for the Water Tank Controller

256

The results of the simulations for the first benchmark problem are presented in Table 4. In Table 4 it can be noted that when we increase the value of the perturbation, the error decreases in a minimum amount. The GT2FLC finds a smaller error of simulation than both the T1FLC and IT2TLC, .i.e., we realized a comparison with the results based on IAE; in this metric when the type of the membership functions in the inputs is Gaussian, the error for the T1FLC is of 118.1053, for IT2FLC is about 112.261, and for GT2FLC the best result is about 112.305, and these experiments are shown with a perturbation of 1.5. It is very important to say that the errors in the IT2FLC and GT2GLC are similar in the minimization of the errors and in most of the experiments the IT2FLC is better than the GT2FLC. This is because the analyzed problem does not present sufficient need for robustness and the GT2FLC does not improve the performance. For the metric of the root mean square error, for the first three membership functions, it is clear that the GT2FLC obtains better results. The number of alpha planes was also under variation for analyzing its effect on the results. The number of alpha planes indicate the amount of secondary membership function that are evaluated. We changed the number of alpha planes from 5 to 15 to analyze and find the optimal number of planes necessary in the evaluation in the model with GT2FLC. Rho is a

251 252 253

257 258 259 260 261 262 263 264 265 266 267 268

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Table 4 Simulation results for the water tank when adding Pulse Generador noise Perturbations. Type of the membership function

Triangular

Gaussian

Trapezoidal

Generalized bell

Performance index

ITAE ITSE IAE ISE RMSE ITAE ITSE IAE ISE RMSE ITAE ITSE IAE ISE RMSE ITAE ITSE IAE ISE RMSE

VALVE T1FLC

IT2FLC

GT2FLC

2456.6318 6171.7424 118.1053 324.4240 0.15767742 2336.2659 5673.2466 113.6483 303.1137 0.07904827 2519.1318 6551.2959 120.2294 336.9320 0.23091238 2582.30864 6538.35411 121.471442 336.401794 0.19558786

2292.57479 5350.38023 112.610978 295.958673 0.02566181 2292.57479 5350.38023 112.610978 295.958673 0.02566181 2292.57479 5350.38023 112.610978 295.958673 0.02566181 2292.57479 5350.38023 112.610978 295.958673 0.02566181

2318.8949 5466.9907 113.4572 300.1413 0.01201354 2254.0702 5394.5037 112.3051 299.279427 0.00827921 2336.0610 5505.7092 114.0442 302.0869 0.01467692 2601.96672 6617.44945 122.152073 339.206444 0.16977793

5

10

15

2292.7283 5387.3627 112.748225 297.586403 0.02746781 2258.83853 5346.38538 112.134083 296.920303 0.01987987 2273.71797 5364.78746 112.501449 297.63211 0.01666568 2464.70028 6307.41519 118.869528 330.3839 0.15634243

2222.6025 5310.38699 111.281491 295.821877 0.02805744 2278.9592 5371.20269 112.672649 298.242749 0.02014303 2307.26145 5388.41733 113.104372 297.693609 0.01787246 2457.31403 6286.67183 118.666982 329.716817 0.14390615

2211.97854 5295.64 111.024056 295.300678 0.02826925 2267.76852 5352.69995 112.305879 296.997438 0.02014703 2289.46125 5368.71603 112.727711 297.188128 0.01819174 2525.91311 6419.05127 120.486606 334.983859 0.14206562

Table 5 GT2FLC with variations in Alpha Planes of the Water Tank. Type of the membership function

Performance index

VALVE GT2FLC

Triangular

Gaussian

Trapezoidal

Generalized Bell

269 270 271 272 273 274 275 276 277 278 279 280

ITAE ITSE IAE ISE RMSE ITAE ITSE IAE ISE RMSE ITAE ITSE IAE ISE RMSE ITAE ITSE IAE ISE RMSE

parameter used to define the volume (support) in the membership functions, and we made variations of this value of 0.2 and 0.5. In Table 5 we can find the results of the experiments with variation in the number of alpha planes, the value of rho in 0.5 and a level of noise of 0.5. In Table 5 it can be noted that when we increase the number of alpha planes to 5, 10 and 15 that all GT2FLCs find the same errors. For example, when we used trapezoidal membership functions in the inputs and with the same model, the Root Mean Square Error (MSE) generated for the GT2FLC with 5 alpha planes is of 0.016665, with 10 is of 0.017872 and with 15 is of 0.018191. Experimentation was also performed with various scenarios of external perturbations, and different noise generators were used: like pulse generated noise, where the height of the Power Spectral Density of the pulse generated noise for the amplitude, period, phase delay set to 1,10, 0, respectivetively and the pulse width (%) was set to 0.1, 0.3, 0.5 and 1.5. Different noise levels were also applied as a disturbance in the plant to evaluate the generalized type-2 fuzzy logic controller and to visualize the results of the model. Table 6 shows all experiments; which were made with 5 alpha planes and the metric was the Integral Absolute Error (IAE). Please cite this article as: C. Oscar et al., A comparative study of type-1 fuzzy logic systems, interval type-2 fuzzy logic systems and generalized type-2 fuzzy logic systems in control problems, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2016.03.026

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C. Oscar et al. / Information Sciences xxx (2016) xxx–xxx Table 6 GT2FLC with variations in perturbation of Water Tank Controller. Simulations of Problem Benchmark Known as The Water Tank Experiment

Type of membership function in Inputs

GT2FLC Noise levels

1 2 3 4

Gaussian Triangular Trapezoidal Generalized Bell

0.1

0.3

0.5

1.5

111.397396 112.089488 112.797974 120.700749

112.820884 111.84549 112.054702 119.548436

112.305879 111.024056 112.727711 120.486606

112.623036 112.623036 112.623036 112.623036

Fig. 13. Comparative results of the T1FLC, IT2FLC and GT2FLC with a noise level of 0.5.

288

In Table 6 we can note that increasing the noise levels makes the simulation errors to be lower, and this is because the generalized type-2 fuzzy logic controller handles better the uncertainty of the control process. The error when T1FLC is used for noise levels 0.1, 0.3, 0.5 and 1.5, in the triangular membership functions, is of 117.063, 116.988, 118.295 and 118.105 respectively. Then the GT2FLC is better with the level of noise. Fig. 13 shows the comparison in the simulations with a noise level of 0.5. In Fig. 13 we can observe that when we use the model of GT2FLC the fuzzy controller presents more stability. The blue line represents the behavior with the Type -1 Fuzzy Logic System, the green line indicates the Interval Type-2 Fuzzy Logic System and the red line represents the behavior of the Generalized Type-2 Fuzzy Logic System.

289

5.2. Simulation results for the Temperature Controllers

290

The same methodology is used to evaluate the second case study, known as the Temperature Controller. We change the type of the membership functions in the order described in Table 7. The results are presented without a level of noise, with 5 alpha planes and the rho is of 0.2. In Table 7 it can be noted that the GT2FLC finds a smaller error of simulation than the T1FLC and IT2TLC., i.e., we realized a comparison with the results of IAE; in this metric when the types of the membership functions is Trapezoidal – Triangular – Trapezoidal for each input, the error for the T1FLC is of 65.88579, for IT2FLC is about 65.879, and for the GT2FLC the best result is about 5.8009. It is very important to say the this problem has more complexity, and therefore the GT2FLC is obtaining good results. Table 8 shows the results for 5 and 10 alpha planes with a 0.1 noise level. These results are based on the first fuzzy logic system with trapezoidal –triangular- trapezoidal membership functions in the inputs and outputs. Table 8 shows that the best results are found with 5 alpha planes. For example; for the GT2FLC with 5 alpha planes using the IAE metric, the error is of 5.8534, with 10 planes is 6.1016 and with 15 is 5.60762 for the output called Cold, and we have errors of 123.1628, 122.4505, 120.3524 for 5,10 and 15 alpha planes respectively for the output called Hot. Table 9 summarizes the errors when we used levels of noise in the model. The design in this FLC is triangular, Trapezoidal and Triangular in each input and triangular en each output. The number of alpha planes is 10.

281 282 283 284 285 286 287

291 292 293 294 295 296 297 298 299 300 301 302 303 304

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Table 7 Simulation results of the Temperature Controller. Type of the Membership Function

Trapezoidal– Triangular- Trapezoidal

Gaussian-Traingular-Gaussian

Generalized BellTriangular- Generalized Bell

Triangular-Triangular-Triangular

Performance Index

Cold

Hot

T1FLC

IT2FLC

GT2FLC

T1FLC

IT2FLC

GT2FLC

ITAE ITSE IAE ISE ITAE ITSE IAE ISE

1663.553 2437.612 65.88579 95.03003 1741.460 2659.436 69.02725 103.7495

1682.536 2462.568 65.879 94.377 1532.269 2090.826 60.57609 81.314

144.4108 21.7304 5.8009 0.9009 287.6101 68.6303 11.6299 2.7980

55734.8903 2604529.04 2197.54 100335.09 55240.2699 2563154.49 2183.04 99057.74

55144.00 2509809.49 2181.53895 97486.9185 57587.8629 2772980.33 2270.10574 106853.884

3491.17351 11958.9318 123.495238 410.408878 1962.60174 3974.46279 78.51589 158.492475

ITAE ITSE IAE ISE ITAE ITSE IAE ISE

1749.005 2675.297 69.23830 104.2121 1779.681 2765.599 70.71690 108.4903

1711.718 2526.223 67.068 97.004 1503.432 2021.935 59.472 78.674

39.7505 6.8328 1.6923 0.2896 178.0814 36.8415 8.3716 1.8201

55033.8407 2544739.23 2172.36 98167.37 56148.9912 2616602.20 2222.73 101739.22

60807.9911 3054325.55 2354.96485 114210.612 54922.9763 2485794.81 2172.57093 96517.6793

3031.47632 9172.3689 119.66858 349.291104 2071.09178 5898.19815 71.9036781 194.119393

Table 8 GT2FLC with variations in the Alpha Planes of the Temperature Controller. Type of the membership function

Performance index

GT2FLC Cold

Trapezoidal– Triangular - Trapezoidal

ITAE ITSE IAE ISE

Hot

5

10

15

5

10

15

144.8882 22.9323 5.8534 0.9609

148.3088 30.4921 6.1016 1.3297

137.3279 21.520 5.60762 0.91585

3472.0214 11884.3544 123.16228 409.135221

3490.03109 11846.0381 122.84505 402.77024

3384.56 11366.56 120.3524 391.01352

Table 9 GT2FLC with variations in the perturbations of the Temperature Controller. Type of the membership function

Performance index

GT2FLC Cold

Trapezoidal– Triangular - Trapezoidal

ITAE ITSE IAE ISE

Hot

0.1

0.3

1.5

0.1

0.3

1.5

50.64094 14.63237 2.20356 0.64670

42.78478 6.97924 1.812506 0.29564

57.85523 21.34890 2.55378 0.95089

0.18162 3028.5748 9348.7351 119.67266

0.48844 3003.7234 9206.4946 119.50530

0.48844 3041.167 9484.03 120.1691

307

In Table 9 we can observe that in the Cold output, the error with noise levels is minimized. In Table 8 for the Cold variable the IAE error is of 5.8534 and considering 5 alpha planes and in Table 9 with 10 alpha planes the result using a 0.3 level of noise is of 1.812506, because GT2FLC manages the uncertainty in a better way.

308

5.3. Simulation results for the Mobile Robot Controller

309

The experiments for the third benchmark problem, which is the mobile robot, were performed with various simulation scenarios; the type of the membership functions are Trapezoidal and Triangular, 5 alpha planes, and different noise generators were used, like the pulse generated noise. In this case, the amplitude, period, pulse width (%) and phase delay were set to 10, 1.5 and 0, respectively. Table 10 shows the results of these simulations. The improvement in GT2FLC is shown with the results presented in Table 10. For example, in the metric of the ITAE for the output ev (error in velocity) using the first fuzzy logic system the errors are of 37.98262, 8.106911 and 6.51420 for T1FLC, T2FLC and GT2FLC, respectively. In this benchmark problem, the errors without noise are of 64.2730, 8.6085, 12.5365 for the T1FLC, T2FLC and GT2FLC, respectively. Fig. 14 shows the behavior of the T1FLC, IT2FLC and GT2FLC applied to the benchmark problem of the mobile robot without noise for the first FLS.

305 306

310 311 312 313 314 315 316 317

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C. Oscar et al. / Information Sciences xxx (2016) xxx–xxx Table 10 GT2FLC results of the Mobile Robot Controller with Level of Noise. Type of the membership function

Trapezoidal– Triangular - Trapezoidal

Triangular-Traingular- Traingular

Performance index

ITAE ITSE IAE ISE ITAE ITSE IAE ISE

Ev

ew

T1FLC

IT2FLC

GT2FLC

T1FLC

IT2FLC

GT2FLC

914.8946 1476.911 37.98262 61.2967 1058.569 3198.932 64.39595 215.1603

191.2268 39.48574 8.106911 0.4602692 156.02476 26.90503 7.89937 4.53556

40.42045 62.30691 6.51420 9.0 0 0 0 25.795155 127.63917 17.091682 113.43763

775.64721 1558.0866 30.18 52.70 979.17493 5406.4796 42.10 249.21

246.81732 69.57077 10.69438 3.0509699 183.84692 35.46252 7.361293 1.4066426

22.176083 13.934322 4.78927 3.197610 4.84641 2.643047 2.8110388 5.085763

Fig. 14. Behavior of the results of the T1FLC, IT2FLC and GT2FLC without level of noise.

318

5.4. Simulation results for the beam and ball controller

319

324

For the last benchmark problem the configuration is set as follows: the Gaussian and generalized bell functions are used for the inputs. This FLS is of Takagi-Sugeno-Kang type with outputs of linear type. Tables 11 and 12 show the simulation results without a level of noise and with a 1.5 level of noise respectively. In Tables 11 and 12 we use the IAE metric error without levels of noise, when we applied T1FLC the error is of 444.618176, with IT2FLC the error is of 113.4594 and with GT2FLC the error is of 68.0607. Therefore, the GT2FLC shows the best errors in the simulations.

325

6. Conclusions

326

In this paper, a comparative study for fuzzy control based on generalized type-2 fuzzy logic has been presented. As it can be noted in Tables 4 and 12, when comparing the simulation results using the generalized type-2 fuzzy inference systems, the errors of the simulations indicate that the GT2FLC shows smaller errors, and this method achieved better

320 321 322 323

327 328

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Table 11 Simulation results for the beam and ball without Pulse Generated Noise Perturbations. Type of the membership function

Gaussian

Generalized Bell

Performance index

ITAE ITSE IAE ISE RMSE ITAE ITSE IAE ISE RMSE

Output T1FLC

IT2FLC

GT2FLC

44478.6894 129326.074 444.618176 1289.91 2.5590909 44520.5674 129642.107 445.006653 1292.87063 1.6009485

2863.5753 8384.43 113.4594 327.32184 1.052616 2850.5442 8316.0535 113.180892 326.5277 1.070965

1020.6964 2970.81133 68.060700 194.835924 0.582874 1019.7583 2971.051 68.02087 195.242088 0.57345323

Table 12 Simulation results for the beam and ball when adding Pulse Generated Noise Perturbations. Type of the membership function

Gaussian

Generalized Bell

Performance index

ITAE ITSE IAE ISE RMSE ITAE ITSE IAE ISE RMSE

Output T1FLC

IT2FLC

GT2FLC

44635.9439 129956.8524 445.887896 1.60711043 2.582039 44699.33 130393.77 446.427124 1297.23573 1.611134

2872.65 8456.79 113.94 330.65 1.05 2856.70 8372.55 113.44 327.91 1.08

2867.522 8389.5888 113.65559 328.284 1.07344 2856.98 8365.43 113.38889 328.1340 1.073449

344

results in the simulations without noise in the model. Changing the levels of noise in the model made the Footprint of Uncertainty to be evaluated in an exhaustive way, and the theory supports that in these cases the generalized type-2 fuzzy logic controller should reflect better results. In the results presented in Figs. 13 and 14, the behavior of the generalized type-2 fuzzy logic controller is better, because the GT2FLC finds the lower simulation errors in the models (for the problems studied in this work, which are of minimization, with respect to the simulation errors in the model), when the proposed method is applied, better results are obtained, and the main reason is that the uncertainty in the membership functions is modeled more closely with generalized type 2 fuzzy sets. This work leads to the conclusion that the use of generalized type2 fuzzy systems can be a good choice when there is a high level of uncertainty in the problem. In other words, generalized type-2 fuzzy logic allows for better modeling of uncertainty, because it gives more degrees of freedom in comparison to interval type-2 and type-1 fuzzy logic. The complex nature of the uncertainty encountered in the real world problems indicates that generalized type-2 fuzzy logic is needed in real-world devices and applications, in particular in problems of control that is the case study in this paper. In the future, we envision using optimization techniques or bio-inspired algorithms that would help finding the optimal parameter values of the membership functions and the optimal number of alpha planes for the automatic implementation of the proposed method. Finally, we look forward to improving the generalized type-2 fuzzy logic algorithms to consider other areas of application.

345

Acknowledgment

346 Q2 347

We thank the program of the Division of Graduate Studies and Research of Tijuana Institute of Technology and the financial support provided by a CONACYT contract grant number: 261565.

348

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329 330 331 332 333 334 335 336 337 338 339 340 341 342 343

349 350 351 352 353 354 355 356 357 358 359

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Please cite this article as: C. Oscar et al., A comparative study of type-1 fuzzy logic systems, interval type-2 fuzzy logic systems and generalized type-2 fuzzy logic systems in control problems, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2016.03.026