Generalized type-2 fuzzy logic in galactic swarm optimization: design of an optimal ball and beam fuzzy controller

Abstract

In this paper we present a modification based on generalized type-2 fuzzy logic to an algorithm that is inspired on the movement of large masses of stars and their attractive force in the universe, known as galactic swarm optimization (GSO). The modification consists on the dynamic adjustment of parameters in GSO using type-1 and type-2 fuzzy logic. The main idea of the proposed approach is the application of fuzzy systems to dynamically adapt the parameters of the GSO algorithm, which is then applied to parameter optimization of the membership functions of the bar and ball fuzzy controller. The experimentation was carried out using the original GSO algorithm, and the type-1 and type-2 fuzzy variants of GSO. In addition a disturbance was added to the bar and ball fuzzy controller plant to be able to validate the effectiveness of the proposed approach in optimizing fuzzy controllers. A formal comparison of results is performed with statistical tests showing that GSO with generalized type-2 fuzzy logic is the best method for optimizing the fuzzy controller.

Keywords

Fuzzy logic galactic swarm optimization fuzzy systems adjustment of parameters fuzzy controller

1 Introduction

Control has become one of the most used and important applications for fuzzy logic given that it provides greater robustness in the design of the controllers, and better performance than traditional control strategies [1].

Fuzzy controllers are often used in complex plants to improve their operation by achieving smooth processes. The advantages offered by the use of fuzzy logic and fuzzy sets in addition to the emergence of type-2 fuzzy sets augment the possibilities of improving the design of fuzzy controllers [2, 3].

The dynamic adjustment of parameters with the use of fuzzy logic has proven recently to be an interesting alternative since several metaheuristic algorithms have been improved in this form. The adjustment is made through the use of fuzzy systems to control the main parameters for the operation of the metaheuristic algorithm and thus obtain an improvement. Since fuzzy systems are designed based on experts’ knowledge, they provide a better adaptation of the algorithm to the problem it is trying to solve [4]. Normally, parameter adaptation in metaheuristic algorithms, during execution, helps to obtain better results than when using fixed parameters [5 –9].

The main advantages of using fuzzy logic in metaheuristics for the optimization of mathematical functions and control problems is their easy implementation and that they even have a good performance in non-linear processes. In addition, their operation is very similar to human reasoning, we do not need to be an expert in area of the problem to be solved, so it is not necessary to know the mathematical model of the problem to be solved, in addition to not depending on complex mathematical expressions.

Regarding fuzzy controllers, in recent decades they have been implemented in a large number of products and processes, such as vehicle systems, robotics, among others. Fuzzy controllers are very intuitive and offer the possibility of using imprecise expressions, it shows a tolerance to noise since its output depends on a set of rules, it will not be affected if a disturbance occurs, as well as being very stable.

In this paper, a method for dynamically adjusting the parameters of the galactic swarm optimization algorithm (GSO) is presented, where the most important parameters are adjusted by means of type-2 fuzzy systems, which are applied to the case of control of the ball and beam that consists in maintaining the balance of a ball on a beam. The main contribution is that in this paper the use of generalized type-2 fuzzy is proposed, unlike previous work that has only used interval type-2 and type-1 fuzzy logic. Generalized type-2 fuzzy logic is an extension of interval type-2 and it can handle higher levels of uncertainty, and this is why we proposed to improve the performance of GSO.

The main idea of the proposed approach is to optimize the fuzzy controller of the ball and beam, which consists of a controller of the Sugeno type [10, 11]. The optimization is carried out by obtaining the necessary parameter values to form the membership functions of the aforementioned controller [12, 13].

Galactic swarm optimization is a metaheuristic algorithm presented by Muthiah-Nakarajan and Noel in 2016 that takes as inspiration the movement of stars and galaxies in the universe. In the GSO algorithm the population is divided into subpopulations where all the individuals of each subpopulation are attracted to the best individual and in this way a balance is maintained between the exploration and exploitation in the algorithm therefore it has a good performance against complex problems [14].

In the current literature we can find different metaheuristic algorithms that have been applied to control problems to find the best design of fuzzy controllers, as in [15 –17]. Also many other authors have worked on hybridizing metaheuristic algorithms, such as particle swarm optimization and gravitational search algorithm, among others [18 –22].

This work is organized as follows: in Section 2 a brief introduction to type-1 and type-2 fuzzy logic is presented. In Section 3 a brief review of the GSO algorithm is given to familiarize the readers with its inspiration and how to perform its processing. In Section 4 the description of the control case of the ball and beam, as well as the fuzzy controller is presented. In Section 5 we show the proposed parameter adjustment approach in the GSO algorithm, as well as the fuzzy controller optimization used to test the proposed method. Section 6 shows the results obtained by the proposed approach applied to the optimization of the ball and beam controller in order to measure its performance and finally Section 7 describes the conclusions.

2 Basic concepts

In the area of artificial intelligence, fuzzy logic proposes allowing a system to deal with information from the real world with validity values between the true and false scale. Fuzzy logic manages to represent vague, incomplete or lack of information concepts to build heuristics capable of interpreting complex to define information [23].

Fuzzy logic is an extension of classical logic, therefore it can be adapted to everyday expressions that are used in the real world, and for example “it’s very high". This adaptation to natural language is due to the fact that each fuzzy set has a membership function related to each element [24, 25].

A type-1 fuzzy set represented by B is defined with a membership function in the universe X in a range of [0, 1], normally restricted to belonging to a family of contiguous membership functions μ_B:⟶ [0,1]. The membership function of B is expressed by μ_B (x) and is called the type-1 membership function [26]. In this case μ_B (x) is the membership degree of element x ∈ X in set B. In Equation 1 we can find the mathematical representation of a type-1 fuzzy set [27]. $B = {(x, μ_{B} (x)) | x \in X}$ (1)

Taking into account that the extension of classic sets to type-1 fuzzy sets comprises the membership of an element to a set, when this cannot be determined with a value of 0 or 1, then type-1 fuzzy sets are used. Similarly, when it is very difficult to determine the membership degree of an element in a type-1 fuzzy set, then type-2 fuzzy sets are used [28].

An interval type-2 fuzzy set that is expressed by $\tilde{B}$ is a bivariate function called membership function of $\tilde{B}$ in the Cartesian product of X × [0, 1] in the range of [0,1], where X is the universe for the primary variable of $\tilde{B}$ , x which is expressed with $μ_{\tilde{B}} (x, u)$ and is called an interval type-2 membership function. Equation 2 shows the mathematical representation of an interval type-2 fuzzy set, where $μ_{\tilde{B}} (x, u)$ is the secondary function and all memberships are 1. $\tilde{B} = {((x, u), 1) | x \in X, u \in U \equiv [0, 1]}$ (2)

In the case of generalized type-2 fuzzy sets, the same logic is used as in type-1 fuzzy sets adding some complexity to its operations, in generalized type-2 fuzzy sets f_x (u) is used through the vertical axis this due to the complexity shown in generalized type-2 fuzzy sets in comparison to the other fuzzy sets [29]. In this case we have the secondary membership $μ_{\tilde{B}} (x, u)$ defined by another function and it is not constant value of 1. $\tilde{\tilde{B}} = {((x, u), μ_{\tilde{B}} (x, u)) | x \in X, u \in U \equiv [0, 1]}$ (3)

3 Galactic swarm optimization algorithm

Galactic swarm optimization (GSO) is a metaheuristic algorithm presented by Muthiah-Nakarajan and Noel in 2016 that takes as inspiration the movement of stars and galaxies in the universe [14]. In the GSO algorithm the population is divided into subpopulations, where all the individuals of each subpopulation are attracted to the best individual, in this way a balance is maintained between the exploration and exploitation in the algorithm and therefore it has a good performance against complex problems [30].

This metaheuristic algorithm consists of two levels to perform its operation, at the first level in all subpopulations initially formed individuals are attracted to the best individual of each subpopulation, all this is done by calculating the velocity and position of individuals using particle swarm optimization (PSO).

Once the first level is completed, a super swarm is formed based on the best individuals of each of the subpopulations, then level 2 begins where the velocity and position of the individuals are calculated using the PSO algorithm until the best individual is found for the entire population [9].

The particle swarm optimization algorithm is a metaheuristic that uses bird behavior as inspiration, where each individual or particle has a velocity and position with which it moves along the established search space [31, 32]. Individuals in the real world have a small amount of inertia that keeps them where they moved, they also have an acceleration or change of velocity [33].

Level 1:

At this level, the main objective is to explore in all subpopulations or subswarms in parallel to obtain a better exploration of the solutions. Galactic swarm optimization begins its operation through iterations calculating the velocity and position of individuals with the expressions shown below [9]: $v_{j}^{(i)} \leftarrow W_{1} v^{(i)} + c_{1} r_{1} (p_{j}^{(i)} - x_{j}^{(i)}) + c_{2} r_{2} (g^{(i)} - x_{j}^{(i)})$ (4) $x_{j}^{(i)} \leftarrow x_{j}^{(i)} + v_{j}^{(i)}$ (5) where:

v⁽ⁱ⁾ is the current velocity.

$p_{j}^{(i)}$ is the personal best.

g⁽ⁱ⁾ is the global best.

c₁ and c₂ are acceleration constants.

W_i is the inertia weight, r₁ and r₂ are obtained by the following expressions [14]: $W_{i} = 1 - \frac{K}{I_{1} + 1}$ (6) $r_{i} = \cup (- 1, 1)$ (7) where:

K is the current iteration.

r_i are random numbers obtained with a uniform distribution between – 1 and 1.

Level 2:

In the second level of the GSO algorithm the best individuals of each of the subpopulations form a super swarm, just as it was done in the first level the velocity and position of the individuals are calculated, but in this case the equations suffer some modifications as shown below: $v^{(i)} \leftarrow W_{2} v^{(i)} + c_{3} r_{3} (p^{(i)} - y^{(i)}) + c_{4} r_{4} (g - y^{(i)})$ (8) $y^{(i)} \leftarrow y^{(i)} + v^{(i)}$ (9) where:

p⁽ⁱ⁾ is the personal best.

g is the global best.

c₃ and c₄ are the acceleration constants.

In galactic swarm optimization the movement of the sub-swarms in the first level is an exploratory phase and the second level is an exploitation phase. In this way the GSO algorithm alternates between an exploratory phase and an exploitation phase. Thus offering greater possibilities of finding optimal solutions.

4 Proposed method (FGSO)

Finding the appropriate parameter values of a metaheuristic algorithm can be a complex task. For this reason in recent years, fuzzy systems have been used to dynamically control the adjustment of the parameters and thus improve the performance of the metaheuristic algorithms. In this case we use galactic swarm optimization with dynamic adjustment of parameter, which is called as fuzzy galactic swarm optimization (FGSO).

The values recommended in the literature [25, 34] for the cognitive and social components used in the GSO algorithm are in the range of 0.5 to 2.5. In some previous papers, the use of the adaptation of parameters during the execution of the algorithms is recommended to improve their performance in the face of different complex optimization problems.

The main advantage of using parameter adjustment instead of fixed parameters is that the parameters are taking different values during the execution of the metaheuristic algorithm, so there is a greater variety and better solutions.

The design of optimal fuzzy controllers for complex plants is a difficult task since knowing all the components or characteristics of a plant is almost impossible to achieve. For this reason the use of fuzzy sets and the principles of fuzzy logic have been receiving increasing attention and their importance has been growing in recent years because they offer greater stability when simulating the processes to be controlled [35].

In the proposed approach we present the dynamic adjustment of parameters in galactic swarm optimization with the use of type-1, interval type-2 and generalized type-2 fuzzy systems with the aim of having a dynamic behavior in the parameters of the algorithm along the execution. In this particular case, it is proposed to use the parameter adjusting in the GSO algorithm and its application to optimize the ball and beam fuzzy controller, as shown in Fig. 1.

Fig. 1

Proposed method.

The proposed approach for optimizing the ball and beam fuzzy controller is to generate the appropriate parameter values to build the membership functions of the fuzzy controller. All individuals in the GSO algorithm contain all the parameters to generate the controller membership functions expecting to obtain better control than the ones obtained with the original controller. Below we can find the flow chart to obtain the best RMSE and fuzzy controller after executing the GSO algorithm. In Fig. 3 we can observe the composition of the individuals in the algorithm [36].

Fig. 2

Flowchart to obtain the best RMSE and fuzzy controller.

Fig. 3

Individual composition.

The fuzzy system used in GSO to optimize the fuzzy controller consists of a fuzzy system with an input variable called “iteration” and two output variables c₃ and c₄ that represent the cognitive and social component respectively. Each of the used variables is partitioned into three triangular membership functions as shown below in Fig. 4.

Fig. 4

FGSO Type-1 Fuzzy system.

For the design of the interval type-2 and generalized type-2 fuzzy systems, an extension of the type-1 fuzzy system shown in Fig. 4 was made using it as the basis for dynamic adaptation of parameters in the GSO algorithm. The resulting type-2 fuzzy logic systems are in Figs. 5 and 6 for interval type-2 and generalized type-2 fuzzy logic, respectively.

Fig. 5

FGSO IT2 interval Type-2 Fuzzy system.

Fig. 6

FGSO GT2 generalized Type-2 Fuzzy system.

As can be noted in the graphical description of the fuzzy systems, they are formed by triangular membership functions, then we show the equations that represent an interval type-2 triangular membership function and a generalized type-2 triangular membership function [37].

An interval type-2 triangular membership function is built with 6 parameters f₁, s₁, t₁, f₂, s₂ and t₂, where f₁ < f₂, s₁ < s₂ and t₁ < t₂ and is shown as follows: $\begin{matrix} μ (x) = itritype 2 (x, [f_{1}, s_{1}, t_{1}, f_{2}, s_{2}, t_{2}]), \\ where f_{1} < f_{2}, s_{1} < s_{2}, t_{1} < t_{2} \\ μ_{1} (x) = \max (\min (\frac{x - f_{1}}{s_{1} - f_{1}}, \frac{t_{1} - x}{t_{1} - s_{1}}), 0) \\ μ_{2} (x) = \max (\min (\frac{x - f_{2}}{s_{2} - f_{2}}, \frac{t_{2} - x}{t_{2} - s_{2}}), 0) \\ \bar{μ} (x) = \max (μ_{1} (x), μ_{2} (x)) \forall_{x} (s_{1}, s_{2}) \\ \bar{μ} (x) = 1 \forall_{x} \in (s_{1}, s_{2}) \\ \underline{μ} (x) = \min (μ_{1} (x), μ_{2} (x)) \end{matrix}$ (10)

In this case, f1, s1 and t1 are the parameters for the upper membership function and f2, s2 and t2 are the parameter for the lower membership function.

A generalized type-2 triangular membership function is built with 7 parameters f₁, s₁, t₁, f₂, s₂, t₂ and ρ where f₁ < f₂, s₁ < s₂ and t₁ < t₂ and is shown as follows [37, 38]:

$\begin{matrix} μ (x, u) = trigausstype 2 (x, u, [f_{1}, s_{1}, t_{1}, f_{2}, s_{2}, t_{2}, ρ]), \\ μ (x, u) = \exp [- \frac{1}{2} {(\frac{u - P_{x}}{σ_{u}})}^{2}] where \\ μ_{1} (x) = max (min (\frac{x - f_{1}}{s_{1} - f_{1}}, \frac{t_{1} - x}{t_{1} - s_{1}}), 0) \\ μ_{2} (x) = \max (\min (\frac{x - f_{2}}{s_{2} - f_{2}}, \frac{t_{2} - x}{t_{2} - s_{2}}), 0) \\ \bar{μ} (x) = {\begin{matrix} \max (μ_{1} (x), μ_{2} (x)) \forall_{x} \notin (s_{1}, s_{2}) \\ 1 \forall_{x} \in (s_{1}, s_{2}) \end{matrix} \\ \underline{μ} (x) = min (μ_{1} (x), μ_{2} (x)) \\ P_{x} = \max (\min (\frac{x - f_{x}}{s_{x} - f_{x}}, \frac{t_{x} - x}{t_{x} - s_{x}}), 0), \\ {wheref}_{x} = \frac{f_{1} + f_{2}}{2}, s_{x} = \frac{s_{1} + s_{2}}{2}, t_{x} = \frac{t_{1} + t_{2}}{2} \\ δ = \bar{μ} (x) - \underline{μ} (x) \\ σ_{u} = \bar{\frac{1 + ρ}{2 \sqrt{3}} δ + ɛ} \end{matrix}$ (11)

In this case, f₁, s₁ and t₁ are the parameters for the upper membership function and f₂, s₂ and t₂ are the parameters for the lower membership function. Also ρ represents a fraction of the uncertainty in supporting the secondary membership function.

The expression used to define the “iteration” variable represents a percentage of the current iteration with respect to the total number of iterations established as shown below [39]: $Iteration = \frac{CI}{TI}$ (12) where CI represents the current iteration and TI the total number of iterations used during the execution of the metaheuristic algorithm.

The fuzzy rules of the fuzzy system used for dynamic adjustment of the parameters are the same ones used in the type-1 fuzzy system, which consist in achieving a balance between the exploration and exploitation of the algorithm while the iterations pass [40].

Fuzzy rules of the FGSO, FGSO IT2 and FGSO GT2 fuzzy systems:

If (Iteration is Low) then (c3 is Low) and (c4 is High)

If (Iteration is Medium) then (c3 is Medium) and (c4 is Medium)

If (Iteration is High) then (c3 is High) and (c4 is Low)

5 Case of study (Ball and Beam Controller)

Optimization and control problems usually require a very high computational cost, so given the shortage of time and resources, in recent years, different methods have emerged to deal with these types of problems, which have shown good performance at a relatively low cost.

The case study chosen to test the proposed approach of galactic swarm optimization with parameter adjustment using type-1 and type-2 fuzzy logic is the ball and beam fuzzy controller model, which consists in achieving the balance of a ball in a beam, where we are allowed to roll with a degree of freedom along the beam.

This model consists of an arm attached to the end of a beam and a servo gear attached to the other end. As the gear rotates at an angle of θ then the angle of the lever beam changes to α. Once the angle changes from horizontal position, the force of gravity causes the ball to roll along the beam. A fuzzy controller is used for this model where the ball can be manipulated. Below we can find in Fig. 7 the graphic description of this model [38].

Fig. 7

Ball and Beam System.

The Lagrangian equation of motion for the ball is represented by the following expression: $0 = (\frac{J}{R^{2}} + m) \ddot{r} + mg \sin \propto - mr {\dot{\propto}}^{2}$ (13)

The angle of the beam is α and can be expressed in terms of the θ angle which represents the angle of the gear as shown below: $\propto = \frac{d}{L} θ$ (14) where:

m = Mass of the ball.

J = Moment of inertia of the ball.

R = Radius of the ball

g = Gravitational acceleration.

α = Angle of the bar.

d = Shift of the lever arm.

L = Length of the bar.

r = Position coordinate of the ball.

θ = Gear angle.

The fuzzy system for the ball and beam controller consists of four inputs labeled r, $\frac{dr}{dt}$ , α and $\frac{d α}{dt}$ , which are granulated with two generalized bell membership functions. The outputs are represented by 16 linear equations since the fuzzy system used in this controller is of the Takagi Sugeno type. The structure of the fuzzy system is presented in Fig. 8, where we can find a graphic description of the input and output variables.

Fig. 8

Ball and Beam Fuzzy System.

In Figs. 9, 10, 11 and 12 respectively we can find a graphical description of each of the input variables of the ball and beam fuzzy controller, which are granulated into two generalized bell membership functions.

Fig. 9

r input.

Fig. 10

$\frac{dr}{dt}$ input.

Fig. 11

Alfa input.

Fig. 12

$\frac{d α}{dt}$ input.

In a Takagi Sugeno system, the outputs are represented by linear functions, so for the case of the ball and beam controller, the output would be represented by 16 linear functions, which are shown below:

f₁= 1.015r+ 2.234 r′–12.67α–4.046α′+ 0.02624

f₂= 1.161r+ 1.969r′ –9.396α –6.165α′+ 0.474

f₃= 1.506r+ 2.234r′– 12.99α – 1.865α′+ 1.426

f₄= 0.7339r+ 1.969r′ – 9.381α – 4.688α′ – 0.8804

f₅= 0.7343r+ 2.234 r′ – 12.85α – 6.11α′ – 1.034

f₆= 1.413r+ 1.969 r′ – 9.485α – 6.592α′+ 1.159

f₇= 1.225r+ 2.234 r′ – 12.8α – 3.929α′+ 0.3662

f₈= 0.9853r+ 1.969r′ – 9.291α – 5.115α′ – 0.195

f₉= 0.9853r+ 1.969 r′ – 9.292α – 5.115α′+ 0.195

f₁₀= 1.225r+ 2.234 r′ – 12.8α – 3.929α′ – 0.3662

f₁₁= 1.413r+ 1.969 r′ – 9.485α – 6.592α′ – 1.159

f₁₂= 0.7343r+ 2.234 r′– 12.85α – 6.11α′+ 1.034

f₁₃= 0.7339r+ 1.969r′– 9.381α– 4.688α′+ 0.8804

f₁₄= 1.506r+ 2.234 r′ – 12.99α – 1.865α′ – 1.426

f₁₅= 1.161r+ 1.969r′ – 9.396α – 6.165α′ – 0.474

f₁₆= 1.015r+ 2.234 r′–12.67α–4.046α′ – 0.02624

The design of the rules of the fuzzy controller is devised with the objective of presenting a rule for each of the outputs and thus balancing the movement of the ball in the beam as shown below:

If r is Small and $\frac{dr}{dx}$ is Small and α is Small and $\frac{d α}{dt}$ is Small then f = f1

If r is Small and $\frac{dr}{dt}$ is Small and α is Small and $\frac{d α}{dt}$ is Large then f = f2

If r is Small and $\frac{dr}{dt}$ is Small and α is Large and $\frac{d α}{dt}$ is Small then f = f3

If r is Small and $\frac{dr}{dt}$ is Small and α is Large and $\frac{d α}{dt}$ is Large then f = f4

If r is Small and $\frac{dr}{dt}$ is Large and α is Small and $\frac{d α}{dt}$ is Small then f = f5

If r is Small and $\frac{dr}{dt}$ is Large and α is Small and $\frac{d α}{dt}$ is Large then f = f6

If r is Small and $\frac{dr}{dt}$ is Large and α is Large and $\frac{d α}{dt}$ is Small then f = f7

If r is Small and $\frac{dr}{dt}$ is Large and α is Large and $\frac{d α}{dt}$ is Large then f = f8

If r is Large and $\frac{dr}{dt}$ is Small and α is Small and $\frac{d α}{dt}$ is Small then f = f9

If r is Large and $\frac{dr}{dt}$ is Small and α is Small and $\frac{d α}{dt}$ is Large then f = f10

If r is Large and $\frac{dr}{dt}$ is Small and α is Large and $\frac{d α}{dt}$ is Small then f = f11

If r is Large and $\frac{dr}{dt}$ is Small and α is Large and $\frac{d α}{dt}$ is Large then f = f12

If r is Large and $\frac{dr}{dt}$ is Large and α is Small and $\frac{d α}{dt}$ is Small then f = f13

If r is Large and $\frac{dr}{dt}$ is Large and α is Small and $\frac{d α}{dt}$ is Large then f = f14

If r is Large and $\frac{dr}{dt}$ is Large and α is Large and $\frac{d α}{dt}$ is Small then f = f15

If r is Large and $\frac{dr}{dt}$ is Large and α is Large and $\frac{d α}{dt}$ is Large then f = f16

In Fig. 13 we can find the plant for the ball and beam fuzzy controller, where a fuzzy system is used to control the desired position of a ball in a beam.

Fig. 13

Ball and beam plant.

6 Experiments and results

As mentioned in the previous section, the case study used to test the proposed methodology in which type-1 and type-2 fuzzy logic is used to adjust the parameters of the GSO algorithm is the ball and beam system. The proposal is applied to the optimization of the fuzzy controller of the case study, which consists of generating the necessary parameter values to form a fuzzy controller as best as possible. In Table 1 we can find the parameters used to perform the optimization.

Table 1
Parameters for GSO, FGSO, FGSO IT2 and FGSO GT2

Parameter GSO FGSO FGSO IT2 FGSO GT2

Population 10 10 10 10

Subpopulation 5 5 5 5

Iteration 1 50 50 50 50

Iteration 2 100 100 100 100

No epochs 5 5 5 5

C₁ and C₂ 2.5 2.5 2.5 2.5

C₃ and C₄ 2.5 Dynamic Dynamic Dynamic

Parameter	GSO	FGSO	FGSO IT2	FGSO GT2
Population	10	10	10	10
Subpopulation	5	5	5	5
Iteration 1	50	50	50	50
Iteration 2	100	100	100	100
No epochs	5	5	5	5
C₁ and C₂	2.5	2.5	2.5	2.5
C₃ and C₄	2.5	Dynamic	Dynamic	Dynamic

The nomenclature used in the Table is summarized as follows:

GSO: Is the original galactic swarm optimization.

FGSO: Is the galactic swarm optimization using type-1 fuzzy logic.

FGSO IT2: Is the galactic swarm optimization using interval type-2 fuzzy logic.

FGSO GT2: Is the galactic swarm optimization using generalized type-2 fuzzy logic.

The results of the parameter optimization of the fuzzy controller are obtained by the fitness value of each individual, which is defined by the calculation of the root mean square error (RMSE) metric. The RMSE help us measure the behavior of the controller obtained by means of the GSO algorithm and the use of type-1 and type-2 fuzzy logic for the adjustment of parameters in the GSO algorithm. The expression of the mean square error is as shown below [35]: $RMSE = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(X_{i} - Y_{i})}^{2}}$ (15) where X_i is the reference value and Y_i is the value produced by the system.

In order to add more complexity to the plant used to test the optimized controllers, a percentage of noise (uniform random numbers) is added to the inputs of the plant system. The equation that expresses the noise that was applied is presented below: $y_{i} = y_{i} + (percent * NA)$ (16) where “percent” is the percentage of noise and NA is a random number within fixed interval. In Table 2 we can find the root mean square error (RMSE) values ordered from smallest to highest for 30 executions of the original GSO algorithm after controller optimization.

Table 2

Results of the optimization controller using the GSO algorithm

Experiment	RMSE	Experiment	RMSE
1	1.488728	16	1.497685
2	1.489140	17	1.498768
3	1.490186	18	1.498855
4	1.491930	19	1.499110
5	1.492200	20	1.499790
6	1.49254	21	1.499868
7	1.492553	22	1.499936
8	1.493226	23	1.500721
9	1.494038	24	1.500766
10	1.494744	25	1.500850
11	1.496545	26	1.501097
12	1.496571	27	1.502056
13	1.496817	28	1.502232
14	1.496951	29	1.502935
15	1.497637	30	1.502975

In Table 3 we can find the root mean square error (RMSE) values ordered from smallest to highest for 30 executions of the variant FGSO after controller optimization.

Table 3

Results of the optimization controller using the FGSO variant

Experiment	RMSE	Experiment	RMSE
1	1.491239	16	1.501709
2	1.493238	17	1.503027
3	1.493526	18	1.503230
4	1.493870	19	1.504874
5	1.496763	20	1.504874
6	1.496763	21	1.505332
7	1.497514	22	1.505332
8	1.497714	23	1.507396
9	1.499293	24	1.507473
10	1.499293	25	1.507475
11	1.499645	26	1.510487
12	1.499645	27	1.512520
13	1.499789	28	1.513396
14	1.499978	29	1.513396
15	1.501410	30	1.513737

In Table 4 we can find the root mean square error (RMSE) values ordered from smallest to highest for 30 executions of the FGSO IT2 variant after controller optimization.

Table 4

Results of the optimization controller using the FGSO IT2 variant

Experiment	RMSE	Experiment	RMSE
1	1.488961	16	1.495451
2	1.489864	17	1.496858
3	1.489934	18	1.498979
4	1.491158	19	1.499093
5	1.491464	20	1.499280
6	1.491993	21	1.499926
7	1.492482	22	1.500185
8	1.492519	23	1.500296
9	1.492580	24	1.500371
10	1.492864	25	1.500692
11	1.492924	26	1.501003
12	1.493258	27	1.501596
13	1.494117	28	1.501720
14	1.494612	29	1.502427
15	1.495399	30	1.502509

In Table 5 we can find the root mean square error (RMSE) values ordered from smallest to highest for 30 executions of the FGSO GT2 variant after controller optimization.

Table 5

Results of the optimization controller using the FGSO GT2 variant

Experiment	RMSE	Experiment	RMSE
1	1.486496	16	1.495226
2	1.487782	17	1.495365
3	1.489330	18	1.495885
4	1.489477	19	1.496299
5	1.489562	20	1.496493
6	1.489611	21	1.497046
7	1.490832	22	1.497261
8	1.491071	23	1.498049
9	1.491297	24	1.498096
10	1.492394	25	1.498737
11	1.492589	26	1.499958
12	1.492706	27	1.500372
13	1.494067	28	1.500832
14	1.494362	29	1.501185
15	1.494439	30	1.501302

In Table 6 we can find the best, worst, average and standard deviation for 30 executions of the original GSO algorithm and variants FGSO, FGSO IT2 and FGSO GT2 after controller optimization.

Table 6

Summary of results GSO, FGSO, FGSO IT2 and FGSO GT2

RMSE	GSO	FGSO	FGSO IT2	FGSO GT2
Best	1.492640	1.499334	1.488961	1.486496
Worst	1.540816	1.512965	1.502509	1.501302
Average	1.510623	1.506700	1.496150	1.494604
Standard deviation	0.013568	0.003541	0.004346	0.004232

The averages of the results for each of the proposed methods are illustrated in Fig. 14.

Fig. 14

Results for the ball and beam plant without noise.

In Table 7 we can find the mean square error (RMSE) values ordered from smallest to highest for 30 executions of the original GSO algorithm after optimization of the controller by applying 5% noise to the plant of the ball and beam system.

Table 7

Results of the optimization controller using the GSO algorithm applying noise

Experiment	RMSE	Experiment	RMSE
1	1.499333	16	1.506648
2	1.501145	17	1.506984
3	1.502126	18	1.508259
4	1.502954	19	1.508323
5	1.502956	20	1.508478
6	1.503058	21	1.508666
7	1.503905	22	1.509163
8	1.503938	23	1.509310
9	1.503939	24	1.509559
10	1.504174	25	1.510375
11	1.504803	26	1.510541
12	1.505516	27	1.511305
13	1.506044	28	1.511886
14	1.506078	29	1.512354
15	1.506197	30	1.512964

In Table 8 we can find the mean square error (RMSE) values ordered from smallest to highest for 30 executions of the FGSO variant after optimization of the controller by applying 5% noise to the plant of the ball and beam system.

Table 8

Results of the optimization controller using the FGSO variant applying noise

Experiment	RMSE	Experiment	RMSE
1	1.499375	16	1.511887
2	1.501592	17	1.512203
3	1.501995	18	1.512889
4	1.502339	19	1.513184
5	1.502356	20	1.515658
6	1.503250	21	1.516784
7	1.503758	22	1.518970
8	1.504192	23	1.519050
9	1.504790	24	1.523933
10	1.505247	25	1.524629
11	1.505895	26	1.524654
12	1.508597	27	1.526679
13	1.508630	28	1.527121
14	1.509495	29	1.527305
15	1.511438	30	1.527610

In Table 9 we can find the RMSE values ordered from smallest to highest for 30 executions of the FGSO IT2 variant after optimization of the controller by applying 5% noise to the plant of the ball and beam system. In Table 10 we can find the RMSE values ordered from smallest to highest for 30 executions of the FGSO GT2 variant after optimization of the controller by applying 5% noise to the plant.

Table 9

Results of the optimization controller using the FGSO IT2 variant applying noise

Experiment	RMSE	Experiment	RMSE
1	1.498652	16	1.507271
2	1.499590	17	1.508681
3	1.501467	18	1.509373
4	1.502245	19	1.509677
5	1.502312	20	1.510409
6	1.502404	21	1.511090
7	1.502776	22	1.511424
8	1.503149	23	1.511447
9	1.504355	24	1.511898
10	1.504566	25	1.512450
11	1.504630	26	1.512995
12	1.505508	27	1.513124
13	1.506436	28	1.513569
14	1.506691	29	1.513713
15	1.507220	30	1.513954

Table 10

Results of the optimization controller using the FGSO GT2 variant applying noise

Experiment	RMSE	Experiment	RMSE
1	1.498622	16	1.504715
2	1.499891	17	1.504795
3	1.500107	18	1.505009
4	1.500547	19	1.505209
5	1.501245	20	1.505438
6	1.501362	21	1.505462
7	1.501457	22	1.505960
8	1.501523	23	1.506500
9	1.501647	24	1.506903
10	1.502320	25	1.507337
11	1.502821	26	1.507761
12	1.502978	27	1.507934
13	1.503538	28	1.508284
14	1.503625	29	1.508808
15	1.504312	30	1.508962

In Table 11 we can find the best, worst, average and standard deviation for 30 executions of the original GSO algorithm and the FGSO, FGSO IT2 and FGSO GT2 variants after controller optimization by applying 5% noise to the plant of the ball and beam system. The averages of the results for each of the proposed methods are illustrated in Fig. 15.

Table 11

Summary of results GSO, FGSO, FGSO IT2 and FGSO GT2 applying noise

RMSE	GSO	FGSO	FGSO IT2	FGSO GT2
Best	1.53406	1.49937	1.49865	1.498622
Worst	1.55061	1.52761	1.51395	1.508962
Average	1.54178	1.51251	1.50743	1.504169
Standard deviation	0.00501	0.00916	0.00461	0.002898

Fig. 15

Results for the ball and beam plant by applying 5% noise.

6.1 Statistical comparison for optimization of the ball and beam fuzzy controller

In order to provide a deeper analysis of the results obtained after the optimization of the parameters of fuzzy controller with the different variants of the proposed approach a statistical test was performed. The test is the statistical Z test and the parameters used in this case are those shown in Table 12:

Table 12
Statistical Z test parameters

Parameter Value

H ₀ μ₁ ⩾ μ₂

H _a μ₁ < μ₂ (Claim)

Level of significance 95%

A 0.05

Critical Value – 1.645

Parameter	Value
H ₀	μ₁ ⩾ μ₂
H _a	μ₁ < μ₂ (Claim)
Level of significance	95%
A	0.05
Critical Value	– 1.645

The null hypothesis (H₀) states that the average of the GSO algorithm and the FGSO, FGSO IT2 and FGSO GT2 variants (μ₁) with 5% noise is greater than or equal to the average of the GSO algorithm and the FGSO, FGSO IT2 and FGSO GT2 variants (μ₂) without applying noise to the plant. The alternative hypothesis (H_a) states that the average of the GSO algorithm and the FGSO, FGSO IT2 and FGSO GT2 variants (μ₁) with 5% noise is less than the average of the FGSO, FGSO IT2 and FGSO GT2 variants (μ₂) without applying noise to the plant of the ball and beam system.

Table 13 shows the averages, standard deviations and values of z for each of the comparisons made, where the GSO algorithm and the FGSO, FGSO IT2 and FGSO GT2 variants (μ₁) with 5% noise obtains significant evidence against the FGSO, FGSO IT2 and FGSO GT2 variants (μ₂) without applying noise to the plant. As a consequence H₀ is rejected and H_a is accepted with a 95% of the level of significance and a rejection zone for values lower than – 1.645.

Table 13

Results of Z-test for the ball and beam fuzzy controller

Method	Average	STD	Z value
GSO Noise	1.510623	0.013568	– 11.797208
GSO	1.541782	0.005019
FGSO Noise	1.506700	0.003541	– 3.242148
FGSO	1.512517	0.009167
FGSO IT2 Noise	1.496150	0.004346	– 9.756925
FGSO IT2	1.507436	0.004610
FGSO GT2 Noise	1.494604	0.004232	– 10.214093
FGSO GT2	1.504169	0.002898

After performing the statistical tests among the variants of the proposed approach and observing that there is a good performance, a statistical test is carried out between our variants and the variants presented in [38] that also deal with the ball and beam fuzzy controller. The parameters shown in Table 12 are the same used to perform the statistical test.

Table 14 shows the average, standard deviation and values of z for the statistical test performed between our proposed approach (μ₁) and the Harmony search (HS) algorithm variants (μ₂), where in five of the six cases it is possible to obtain significant evidence, with a 95% level of significance and a rejection zone for values less than – 1,645, thus showing that our methods are better.

Table 14

Results of Z-test for our proposal and the FHS method

Method without noise
Method	Average	STD	Z value
FGSO	1.512517	0.009167	– 19.491404
FHS	1.570000	0.013300
FGSO IT2	1.507436	0.004610	– 18.804380
IT2FHS	1.560000	0.014600
FGSO GT2	1.504169	0.002898	– 7.035448
GT2FHS	1.530000	0.019900
Method with noise
FGSO Noise	1.506700	0.003541	– 17.476893
FHS	1.550000	0.013100
FGSO IT2 Noise	1.496150	0.004346	– 4.032735
IT2FHS	1.520000	0.032100
FGSO GT2 Noise	1.494604	0.004232	2.727522
GT2FHS	1.490000	0.008220

7 Conclusions

In this paper, we proposed an approach for the dynamic adjustment of parameters in galactic swarm optimization, where type-1, interval type-2 and generalized type-2 fuzzy systems are used, and this approach was tested with the optimization of the ball and beam fuzzy controller.

To validate the proposed method, a statistical test was carried out among the variants presented in Section 5, where they were tested by simulating the original plant of the controller and adding a level of disturbance. A statistical test was also carried out with respect to the Harmony Search Algorithm with the objective of observing the performance of our proposal against other metaheuristics that have been applied to the same case study of the ball and beam fuzzy controller.

After analyzing the results, we can conclude that the proposed method for augmenting the GSO algorithm with parameter adjustment using type-1 and type-2 fuzzy logic shows good performance in control problems, specifically when applied to the optimization of the ball and beam fuzzy controller.

We can conclude that the proposed approach was good because in the comparison made among the variants, the approach obtains good results either when a noise level was applied or without applying noise to the plant. In addition, in the comparison made against the harmony search algorithm, significant improvements are achieved in 5 of six cases that were compared. This is based on the results presented in Tables 13 and 14.

Galactic swarm optimization has proven to be a metaheuristic that can offer good performance in the face of control problems. As future work we envision testing the approach with different control problems expecting to also have encouraging results.

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Generalized type-2 fuzzy logic in galactic swarm optimization: design of an optimal ball and beam fuzzy controller

Abstract

Keywords

1 Introduction

2 Basic concepts

Table 1 Parameters for GSO, FGSO, FGSO IT2 and FGSO GT2 Parameter GSO FGSO FGSO IT2 FGSO GT2 Population 10 10 10 10 Subpopulation 5 5 5 5 Iteration 1 50 50 50 50 Iteration 2 100 100 100 100 No epochs 5 5 5 5 C1 and C2 2.5 2.5 2.5 2.5 C3 and C4 2.5 Dynamic Dynamic Dynamic

Table 12 Statistical Z test parameters Parameter Value H 0 μ1 ⩾ μ2 H a μ1 < μ2 (Claim) Level of significance 95% A 0.05 Critical Value – 1.645

References

Table 1
Parameters for GSO, FGSO, FGSO IT2 and FGSO GT2

Parameter GSO FGSO FGSO IT2 FGSO GT2

Population 10 10 10 10

Subpopulation 5 5 5 5

Iteration 1 50 50 50 50

Iteration 2 100 100 100 100

No epochs 5 5 5 5

C₁ and C₂ 2.5 2.5 2.5 2.5

C₃ and C₄ 2.5 Dynamic Dynamic Dynamic

Table 12
Statistical Z test parameters

Parameter Value

H ₀ μ₁ ⩾ μ₂

H _a μ₁ < μ₂ (Claim)

Level of significance 95%

A 0.05

Critical Value – 1.645