Parallel conjugate gradient-particle swarm optimization and the parameters design based on the polygonal fuzzy neural network

Abstract

Simple binary coded genetic algorithm (GA) and particle swarm optimization (PSO) fall easily into local minimums and fail to find the global optimal solution to the algorithm. Thus, the development of a hybrid algorithm between GA and PSO is urgently demanded. In this paper, a three-layer polygonal fuzzy neural network (PFNN) model and its error function are first given by the arithmetic operations of the polygonal fuzzy numbers. Second, the random sequences are constructed by a chaos random generator, these random sequences are used as the initial population of chaos GA and the optimal individuals for sub-populations gained by chaos search are used as the initial population of PSO, and then an new parallel conjugate gradient-particle swarm optimization (PCG-PSO) is designed. Finally, a case study shows the proposed parallel CG-PS algorithm not only avoids dependence of traditional GA on initial values and overcomes the poor global optimization capability of traditional PSO, but also possesses advantages of rapid convergence and high stability.

Keywords

Polygonal fuzzy number polygonal fuzzy neural network chaos genetic algorithm particle swarm optimization parallel conjugate gradient-particle swarm optimization

1. Introduction

Since Buckler [1, 2] first introduced the concept of fuzzy neural network (FNN) and approximation in 1994, the approximation performance and algorithms of FNN have been widely discussed [2 –6]. In 2002, Liu [7] first introduced n-symmetric polygonal fuzzy number and its extension operations based on trapezoidal fuzzy numbers and constructed a single-input single-output (SISO) polygonal fuzzy neural network (PFNN), and he studied the approximation and learning algorithm of SISO-PFNN in [8]. Given that the connection weights and thresholds of PFNN are all in the polygonal fuzzy numbers, the method avoids complexity of operation based on Zedeh’s extension principle and simplifies the algorithm design significantly. The approximation performance of PFNN relies mainly on the optimization of connection weights and thresholds. However, the key of algorithm design lies in optimizing the adjustment parameters.

Several artificial neural network learning algorithms are available, among which BP and conjugate gradient algorithms (CGA) are the most popular ones. For example, Lang and Liu [9] analyzed the effects of factor of momentum in back propagation (BP) algorithm on network and proposed the improvement method of variable factor of momentum. He and Ye [10] optimized the connection weights of SISO-PFNN by the genetic algorithm. However, GA fails to obtain the optimal solution rapidly and falls into the local optimal solution easily due to the excessive iterations in the optimization process. In 2011, the universal approximation of PFNN was discussed in the sense of K-integral norm [11]. In 2012, a new CG algorithm was designed by combining the Armijo-Goldstein (A-G) linear search criteria [12]. Unfortunately, the A-G criteria often exclude the optimal learning constant from the search interval. In Ref. [13] some influences of disturbance of training mode on the stability of PFNN were discussed. In 2015, a multi input single output - PFNN was proposed and an optimization algorithm was designed by combining Hebb’s criteria and the particle swarm thinking [14]. The algorithm exhibits high stability and rapid convergence but presents certain randomness and parameter diversity. See [15 –17] for some recent related work.

Many training methods for the evolutionary strategy have been proposed. In Ref. [18], a new GA was proposed by combining the principle of survival of the fittest in nature and the search algorithm. This new GA possesses strong global search capability but exhibits certain randomness, initial population dependence, and poor local search capability. Zhang and Xu [19] proposed an artificial ant algorithm by simulating the food hunting process of ant colony, but this algorithm is difficult to solve. In Ref. [20], an artificial neural network learning algorithm based on the particle swarm idea of the predation of birds in nature was proposed; this algorithm can memorize the best particle position and information shared among particles, see [21, 22]. Unfortunately, the above-mentioned algorithms are limited by their dependence on the initial population to a certain extent.

Chaos phenomenon is a universal nonlinear process from nonlinear dynamical system in nature. The process described by chaos phenomenon presents randomness, convergence, and aperiodicity; the process is also extremely sensitively dependent to the initial value, thereby conforming only to the requirements of sequential cipher. In 1989, Robert Matthews proposed the generation function of pseudo-random number sequence based on logistic mapping. Subsequently, randomness is avoided by applying disturbance and introducing chaos state into the algorithms, see [23 –26]. These algorithms possess absolute advantages in solving high-dimensional extreme values but still exhibit shortages, such as low global information sharing. See [27 –29]. In this study, the chaos genetic algorithm and particle swarm optimization are combined on the basis of the extension operation of the polygonal fuzzy numbers. Accordingly, a new PCG-PSO algorithm is proposed by utilizes the strong global search capability of CGA and the rapid convergence of PSO. The algorithm has the characteristics of fast response speed and small overshoot, so it can better overcome the control performance problems in the VAV air conditioning system.

The paper is organized as follows. After the Introduction, the basic concepts of the polygonal fuzzy number and its arithmetic operations are briefly summarized in Section 2. In Section 3, a chaotic random sequence is constructed by logistic and tent mapping. In Section 4, a PFNN is introduced, and its error function is given by the polygonal fuzzy distance. In Section 5, the PCG-PSO is designed by combining the advantages of CGA and PSO. The results show that the new algorithm exhibits smooth response curve, low overshoot, rapid response, and high stability.

2. Polygonal fuzzy number

General fuzzy number cannot simply conduct a linear operation because the arithmetic operation based on Zadeh’s extension principle [30] does not meet the requirement of closure. This brings misfortune to the application of fuzzy sets. Recently, Muhammad et al. introduced the Pythagorean hesitant fuzzy sets and successfully applied them to group decision making. See [31, 32]. However, the proposed polygonal fuzzy number in this paper can approximate a given fuzzy number (set) with arbitrary precision.

In the following part, the definition, extension operation, and metric of polygonal fuzzy number are introduced. Let $ℝ$ be a set of real numbers, $ℝ^{+}$ be a set of nonnegative real numbers, $ℕ$ be a set of natural numbers, and $F_{0} (ℝ)$ be a set of all general fuzzy numbers on $ℝ$ .

Definition 1 [11] Given the fuzzy numbers $A \in F_{0} (ℝ)$ and $n \in ℕ$ . Suppose a closed interval [0,1] on the vertical axis is divided into n subintervals equally, where the points of division are n, i = 0, 1, 2, ·· · , n. If one group of 2n + 2 ordered real numbers ${a_{0}^{1}, a_{1}^{1}, \cdot \cdot \cdot, a_{n}^{1}, a_{n}^{2}, \cdot \cdot \cdot, a_{1}^{2}, a_{0}^{2}} \subset ℝ$ exists, then $a_{0}^{1} \leq a_{1}^{1} \leq \cdot \cdot \cdot \leq a_{n}^{1} \leq$ $a_{n}^{2} \leq \cdot \cdot \cdot \leq a_{1}^{2} \leq a_{0}^{2}$ is satisfied. The straight segments of the membership function A (x) on every M and $ℝ$ , i = 1, 2, ···, n are selected. In other words, for arbitrary $x \in ℝ$ , the membership function A (x) is defined as $\begin{matrix} A (x) = \\ {\begin{matrix} \frac{i - 1}{n} + \frac{x - a_{i - 1}^{1}}{n (a_{i}^{1} - a_{i - 1}^{1})}, x \in [a_{i - 1}^{1}, a_{i}^{1}], i = 1, 2, \cdot \cdot \cdot, n \\ 1, x \in [a_{n}^{1}, a_{n}^{2}] \\ \frac{i - 1}{n} + \frac{a_{i - 1}^{2} - x}{n (a_{i - 1}^{2} - a_{i}^{2})}, x \in [a_{i}^{2}, a_{i - 1}^{2}], i = 1, 2, \cdot \cdot \cdot, n \\ 0, otherwise \end{matrix} \end{matrix}$

Then, A is called an n-polygonal fuzzy number on $ℝ$ and is denoted as $A = (a_{0}^{1}, a_{1}^{1}, \cdot \cdot \cdot, a_{n}^{1}, a_{n}^{2}, \cdot \cdot \cdot, a_{1}^{2}, a_{0}^{2})$ . The ordered real number group $(a_{0}^{1}, a_{1}^{1}, \cdot \cdot \cdot, a_{n}^{1}, a_{n}^{2}, \cdot \cdot \cdot, a_{1}^{2}, a_{0}^{2})$ is an ordered representation of A. See Fig. 1.

Fig. 1

Image of the membership function of n-polygonal fuzzy number A.

Let $F_{0}^{tn} (ℝ)$ be a set of all n-polygonal fuzzy numbers on $ℝ$ . When n = 1, the 1-polygonal fuzzy number A degrades to a triangle or trapezoidal fuzzy number. When n ≥ 2, the superposition of several small trapezoids can be formed by connection lines between adjacent nodes. According to Ref. [11] the extension operation on $F_{0}^{tn} (ℝ)$ is defined as follows:

Definition 2 [11] Given an $n \in ℕ$ , for any $A, B \in F_{0}^{tn} (ℝ)$ and $A = (a_{0}^{1}, a_{1}^{1}, \cdot \cdot \cdot, a_{n}^{1}, a_{n}^{2}, \cdot \cdot \cdot, a_{1}^{2}, a_{0}^{2})$ , $B = (b_{0}^{1}, b_{1}^{1},$ ···, $b_{n}^{1}, b_{n}^{2}, \cdot \cdot \cdot, b_{1}^{2}, b_{0}^{2})$ , then extension operations are defined as

$A + B = (a_{0}^{1} + b_{0}^{1}, a_{1}^{1} + b_{1}^{1}, \cdot \cdot \cdot, a_{n}^{1} + b_{n}^{1}, a_{n}^{2}$ $+ b_{n}^{2}, \cdot \cdot \cdot, a_{1}^{2} + b_{1}^{2}, a_{0}^{2} + b_{0}^{2});$

$A - B = (a_{0}^{1} - b_{0}^{2}, a_{1}^{1} - b_{1}^{2}, \cdot \cdot \cdot, a_{n}^{1} - b_{n}^{2}, a_{n}^{2}$ $- b_{n}^{1}, \cdot \cdot \cdot, a_{1}^{2} - b_{1}^{1}, a_{0}^{2} - b_{0}^{1});$

$A \cdot B = (c_{0}^{1}, c_{1}^{1}, \cdot \cdot \cdot, c_{n}^{1}, c_{n}^{2}, \cdot \cdot \cdot, c_{1}^{2}, c_{0}^{2})$ , where $c_{i}^{1} = a_{i}^{1} b_{i}^{1} \land a_{i}^{2} b_{i}^{1} \land a_{i}^{2} b_{i}^{1} \land a_{i}^{2} b_{i}^{2}$ , $c_{i}^{2} =$ $a_{i}^{1} b_{i}^{1} \lor a_{i}^{2} b_{i}^{1}$ $\lor a_{i}^{2} b_{i}^{1} \lor a_{i}^{2} b_{i}^{2}$ , i = 1, 2, ···, n ;

Let $kA = ({ka}_{0}^{1}, {ka}_{1}^{1}, \cdot \cdot \cdot, {ka}_{n}^{1}, {ka}_{n}^{2}, \cdot \cdot \cdot, {ka}_{1}^{2},$ ${ka}_{0}^{2}),$ when k ≥ 0.

If a transfer function $σ : ℝ \to ℝ$ is a monotone function, then σ can be extended as a n-polygonal fuzzy number through the following form on $F_{0}^{tn} (ℝ)$ , that is

\begin{matrix} σ (A) = {\begin{matrix} (σ (a_{0}^{1}), σ (a_{1}^{1}), \cdot \cdot \cdot, σ (a_{n}^{1}), σ (a_{n}^{2}), \cdot \cdot \cdot, σ (a_{1}^{2}), σ (a_{0}^{2})), σ is increasing \\ (σ (a_{0}^{2}), σ (a_{1}^{2}), \cdot \cdot \cdot, σ (a_{n}^{2}), σ (a_{n}^{1}), \cdot \cdot \cdot, σ (a_{1}^{1}), σ (a_{0}^{1})), σ is descending \end{matrix} . \end{matrix}

Definition 2 shows that the extension operation in $F_{0}^{tn} (ℝ)$ is closed and simple and that the property of linear operation is maintained. This feature is the significance of polygonal fuzzy number and its extension operation.

A fuzzy metric D is introduced to depict the distance between polygonal fuzzy numbers for simplicity. For arbitrary $A, B \in F_{0}^{tn} (ℝ)$ , where $A = (a_{0}^{1}, a_{1}^{1}, \cdot \cdot \cdot, a_{n}^{1}, a_{n}^{2}, \cdot \cdot \cdot, a_{1}^{2}, a_{0}^{2})$ and B = $(b_{0}^{1}, b_{1}^{1}, \cdot \cdot \cdot, b_{n}^{1}, b_{n}^{2}, \cdot \cdot \cdot, b_{1}^{2}, b_{0}^{2})$ . A new fuzzy distance D between A and B is defined as $D (A, B) = {(\sum_{i = 0}^{n} ({(a_{i}^{1} - b_{i}^{1})}^{2} + {(a_{i}^{2} - b_{i}^{2})}^{2}))}^{1 / 2}$ (1)

This polygonal fuzzy distance D plays an important role in defining the error function and designing the hybrid optimization algorithm in the rest of this paper.

In addition, the n-polygonal fuzzy number is not only the generalization of triangular or trapezoidal fuzzy numbers, but also it can also approximate a given fuzzy number according to any precision. More importantly, for a given $n \in ℕ$ , each fuzzy number A can determine an n-polygonal fuzzy number Z_n (A (·)) with equidistant subdivision, and Z_n (A (·)) can guarantee the closeness of the arithmetic operations so that some complexities of the operations based on Zadeh’s extension principle are greatly simplified. Normally, a given that polygonal fuzzy numbers can be represented by finite ordered real numbers, they have obvious advantages in fuzzy information processing. That is why introducing the concept of n-polygonal fuzzy number, as shown in Fig. 2, where Z_n (A (x)) = Z_n (A) (x), for any $x \in [a_{0}^{1}, a_{0}^{2}$ ].

Fig. 2

Iimage of Z_n (A) (x) and A (x).

It is not difficult to see from Fig. 2 that an n-polygonal fuzzy number Z_n (A (·)) has properties similar to the triangle or trapezoidal fuzzy number, each Z_n (A) is determined by 2n + 2 orderly real numbers: $a_{0}^{1}, a_{1}^{1}, \cdot \cdot \cdot, a_{n}^{1}, a_{n}^{2}, \cdot \cdot \cdot, a_{1}^{2}, a_{0}^{2}$ . Of course, the greater n is, the more finely the closed interval [0, 1] is subdivided, and the stronger the approximation ability of the obtained membership function Z_n (A (x)) to A (x) is. However, its complexity also increases.

Especially, the connecting lines between two adjacent nodes can constitute the superposition of several small trapezoids (the top triangle is regarded as a special trapezoid). Hence, an n-polygonal fuzzy number Z_n (A (·)) can better describe the fuzzy information of objective things. Next, we take an example to illustrate how to transform a given fuzzy number into a concrete ordered representation.

Example 1. Let n = 3, the membership function of a fuzzy number A is shown as follows: $A (x) = {\begin{matrix} \sqrt{x + 1} - 1, 0 \leq x < 3 \\ 3 - \frac{1}{2} \sqrt{5 x + 1}, 3 \leq x \leq 7 \end{matrix}$

Please solve the ordered representation of 3-polygonal fuzzy number Z₃ (A) according to the Definition 1.

Clearly, the Support set and Kernel of the fuzzy number A are Supp A = [0, 7], KerA = [3, 3] = {3}. From n = 3, it is easy to know that the horizontal lines λ = 1/3 and λ = 2/3. See Fig. 3.

Fig. 3

Image of Z_n (A) (x) and A (x) when n = 3.

Whenever x ∈ [0, 3], let the given membership function $A (x) = \sqrt{x + 1} - 1 = \frac{1}{3}$ or $\frac{2}{3}$ in turn. By solving the equation on, the closed interval [0, 3] we can easily obtain the transverse coordinates $a_{i}^{1}$ of the turning points as $a_{1}^{1} = \frac{7}{9}$ , $a_{2}^{1} = \frac{16}{9}$ .

Analogously, whenever x ∈ [3, 7], let $A (x) = 3 - \frac{1}{2} \sqrt{5 x + 1} = \frac{1}{3}$ or $\frac{2}{3}$ in turn, it is not difficult to obtain the transverse coordinates $a_{i}^{2}$ of the turning points as $a_{1}^{2} = \frac{247}{45}$ , $a_{2}^{2} = \frac{187}{45}$ .

Therefore, the ordered representation of 3-polygonal fuzzy number Z₃ (A) can be expressed as. $Z_{3} (A) = (0, \frac{7}{9}, \frac{16}{9}, 3, 3, \frac{187}{45}, \frac{247}{45}, 7)$

Their geometric interpretation and the equidistance division of the fuzzy number A are shown in Fig. 3.

3. Chaotic sequence generator

A traditional GA is a random search algorithm formed by simulating natural selection and genetic mechanism during the biological evolution process. The random generation of the initial population makes the GA depend on the initial population to some extent. Such dependence is avoided by introducing a chaos generator [33]. In other words, some chaos random sequences (or pseudo-random sequences) are generated by chaos mapping, thereby enabling the sequence to be mapped into the studying space and thus generate the initial population. Chaos variables can be inserted into the GA, and the randomness of the traditional GA can be avoided by applying a certain disturbance. Logistic mapping and tent mapping are common chaos generators. See [34, 35]. They mainly select initial values randomly and then generate chaos random sequences through chaos mapping iterations. In addition, the particle swarm optimization (PSO) algorithm for feedforward neural network and the robust classification of convolutional neural network for multiple fingerprint captures are studied. See [36, 37].

Logistic mapping and tent mapping exhibit a topological conjugate relationship and have identical values. One-dimensional logistic mapping is a type of chaos mapping with simple iterations but is widely applied. The iterative formula is x_k+1 = μx_k (1 – x_k). The said mapping exerts extremely complicated dynamic behavior. Logistic mapping can generate chaos sequences on [0, 1] and [–1, 1] when μ = 4, 8 The probability density for logistic mapping to generate chaos sequence generally conforms to Chebyshev’s distribution. Nevertheless, Chebyshev’s distribution can decrease the global search capability of the algorithm significantly, especially in high-dimensional optimization problems. Usually, the tent mapping is a type of chaos mapping with simple iteration and strong ergodicity, its iterative formula is as $x_{k + 1} = {\begin{matrix} μ x_{k},_{} 0 \leq x_{k} \leq 1 / μ \\ μ (1 - x_{k}), 1 / μ \leq x_{k} \leq 1 \end{matrix} .$

Tent mapping can generally generate chaos sequences on [0, 1] when μ = 2. Experiment has confirmed that the sequence generated by tent mapping must be 0 after several iterations because of the finite precision effect of the computer and piecewise linearity of tent mapping. In practice, the computer often shifts the binary number of the decimal part to the left without a sign. Considering the finite word length of the computer, the sequence approaches 0. Hence, tent mapping falls into fixed point phenomenon easily. When the optimal solution is a marginal solution, the iteration process fails to reach the optimal solution. A new subsection logistic chaos mapping was developed in Ref. [35] to ensure strong randomness of chaos sequence generated by chaos mapping; this proposed mapping is based on the topological conjugate relationship between tent and logistic mapping. The iterative formula is

$x_{n + 1} = {\begin{matrix} 4 μ x_{n} (0.5 - x_{n}) 0 \leq x_{n} < 0.5 \\ 1 - 4 μ (x_{n} - 0.5) (1 - x_{n}), 0.5 \leq x_{n} \leq 1 \end{matrix}$ (2)

The superiority of subsection logistic mapping to logistic mapping or tent mapping is tested. For this purpose, the subsection logistic mapping is generated according to the iterative Formula (2) to verify the mean, variance, first moment, and secondary moment of the chaos random sequence {x_n} generated y in the subsection logistic chaos generator. When μ = 4, then 10,000 random numbers are generated by the three generators (logistic, tent, and subsection logistic mapping). The test results are shown in Table 1.

Table 1

Parameter test of chaotic sequence generator

Types of chaos generators	Mean value	First moment	Second moment	Variance
Logistic mapping	0.5009	0.4997	0.3751	0.1249
Tent mapping	0.4970	0.4960	0.3312	0.0841
Subsection logistic mapping	0.5017	0.5002	0.3330	0.0833

Table 1 indicates that all parameters in subsection logistic mapping are superior to those in the two other mapping methods. In 2010, Hu and Yang [23] proved that the average number of iterations of logistic mapping is far higher than that of subsection logistic mapping, indicating that the chaos sequence generated by subsection logistic mapping exhibits strong randomness.

4. Polygonal fuzzy neural network

A PFNN refers to a type of network system that makes extension operation in accordance with n-polygonal fuzzy number with connection weight and threshold valued in the $F_{0}^{tn} (ℝ)$ space. PFNN can process fuzzy information through 2n + 2 finite points of polygonal fuzzy number. For simplicity, only SISO-PFNN is examined in this study. The topological structure graph a three-layer forward SISO-PFNN is represented as follows:

Let the input signal of PFNN be X, the connection weights be U_j and V_j, threshold be Θ_j, and all their values are in $F_{0}^{tn} (ℝ)$ . Suppose the transfer function σ is a continuous monotone sigmoidal function, then the analytic expression of the three-layer SISO-PFNN is $Y = F_{n n} (X) = \sum_{j = 1}^{p} V_{j} \cdot σ (U_{j} \cdot X + Θ_{j}) .$ (3)

Usually, human knowledge can be represented by a fuzzy IF-THEN rule, and each fuzzy IF-THEN rule is a conditional statement based on a fuzzy proposition, that is, it can be expressed as follows:

R_l : IF x₁ is X (l) ₁, x₂ is X (l) ₂, ···, x_L is X (l) _L, THEN y is O (l), where x₁, x₂, ···, x_L is the input variable, each X (l) _i is an antecedent fuzzy set, y is the output variable and O (l) is a consequent fuzzy set.

For convenient discussion, let (X (1), O (1)), (X (2), O (2)), ···, (X (L), O (L)) be a pattern pair for training in $F_{0}^{tn} (ℝ)$ , where X (l), $O 1 (l) \in F_{0}^{t 1 n} (ℝ^{+})$ , and X (l) is an actual input of PFNN, O (l) is an expected output, and Y (l) = F_nn (X (l))(l = 1, 2, ···, L) is an actual output. Let ${\begin{cases} X (l) = (x_{0}^{1} (l), x_{1}^{1} (l), \cdot \cdot \cdot, x_{n}^{1} (l), x_{n}^{2} (l), \cdot \cdot \cdot, x_{1}^{2} (l), x_{0}^{2} (l)) \\ Y (l) = (y_{0}^{1} (l), y_{1}^{1} (l), \cdot \cdot \cdot, y_{n}^{1} (l), y_{n}^{2} (l), \cdot \cdot \cdot, y_{1}^{2} (l), y_{0}^{2} (l)) \\ O (l) = (o_{0}^{1} (l), o_{1}^{1} (l), \cdot \cdot \cdot, o_{n}^{1} (l), o_{n}^{2} (l), \cdot \cdot \cdot, o_{1}^{2} (l), o_{0}^{2} (l)) \end{cases}$

Fig. 4

Topological graph of a three-layer forward SISO-PFNN.

From the formula (1) of the above-mentioned polygonal fuzzy distance D, an error function E in SISO-PFNN can be defined as

$\begin{matrix} E = \frac{1}{2} \sum_{l = 1}^{L} D {(O (l), Y (l))}^{2} = \frac{1}{2} \sum_{l = 1}^{L} \\ (\sum_{i = 0}^{n} ((o_{i}^{1} (l) - y_{i}^{1} (l))^{2} + (o_{i}^{2} (l) - y_{i}^{2} (l))^{2})) \end{matrix}$ (4)

Notably, this error function plays a key role in the following design algorithms.

5. Parameters design of PFNN

GA is an algorithm based on the prototype of biological evolutionism. This algorithm exhibits strong global search capability, convergence accuracy and robustness, but presents slow convergence and initial population dependence. A CGA applies chaos variables into the variable population of GA and searches for the optimal solution by fully utilizing the tiny disturbance of chaos variables on subpopulation, thereby overcoming the randomness of GA. PSO was proposed by PhD Ebethart and Kennedy in 1995 when they were studying the law of motion of birds and fish. PSO is an optimization tool based on iterative operation and exhibits high search speed; however, PSO falls into the local optimal solution easily when processing a discrete optimization problem. Therefore, a new PCG-PSO can be designed by combining the advantages of the CGA and PSO.

Suppose CGA searches with N subpopulations and PSO searches within single subpopulation. First, the connection weights U_j, V_j and threshold Θ_j of PFNN are optimized using the formulas (3) and (4). Second, U_j, V_j and Θ_j are adjusted appropriately on the basis of the structural characteristics of PFNN to create the expected output O (l) approximate to the actual output Y (l). Finally, these connection weights or thresholds are updated through design rules, and a new output is acquired after recombining these parameters.

For simplicity, the adjustment parameters $u_{i}^{q} (j)$ , $v_{i}^{q} (j)$ , and $θ_{i}^{q} (j)$ (i = 0, 1, ···, n ; j = 1, 2, ···, p ; q = 1, 2) of PFNN are uniformly denoted as one parameter vector H, where $\begin{matrix} H = (u_{0}^{1} (1), \cdot \cdot \cdot, u_{0}^{2} (1), \cdot \cdot \cdot, u_{0}^{1} (p), \cdot \cdot \cdot, u_{0}^{2} (p), v_{0}^{1} (1), \\ \cdot \cdot \cdot, v_{0}^{2} (1), \cdot \cdot \cdot, v_{0}^{1} (p), \cdot \cdot \cdot, v_{0}^{2} (p), θ_{0}^{1} (1), \cdot \cdot \cdot, \\ θ_{0}^{2} (1), \cdot \cdot \cdot, θ_{0}^{1} (p), \cdot \cdot \cdot, θ_{0}^{2} (p)) 1 ≜ (h_{1}, h_{2}, \cdot \cdot \cdot, h_{d}) \end{matrix}$ (5)

Therefore, the adjustment of U_j, V_j and Θ_j can be changed into the optimization of component parameters h_1pti (i = 1, 2, ···, d). The expression of PFNN can be particularly determined using Equation (3) when H is fixed, and the error function E can be expressed abstractly as E (H).

Individual quality in nature is generally evaluated by the fitness function. The higher the fitness function, the better the individual. Hence, individuals can be replaced by H as the optimization object. Optimization of H requires minimizing E (H), whereas optimization of individual requires maximizing the fitness. In this study, the fitness function S (H) is defined as S (H) = 1/ - (1 + E (H)), and then the evaluation standard is indirectly changed into individual’s evaluation to the fitness function.

Suppose N populations {G₁, G₂, ···, G_N} exist in nature and each subpopulation $G_{j} = (H_{1}^{j}, H_{2}^{j},$ $\cdot \cdot \cdot, H_{M}^{j})$ covers M individuals, where each individual may be shown as $H_{k}^{j} = (h_{k 1}^{j}, h_{k 2}^{j}, \cdot \cdot \cdot, h_{kd}^{j})$ , j = 1, 2, ···, N; k = 1, 2, ···, M, and the fitness value of each individual is $S_{k}^{j}$ . The optimal individual of each subpopulation G_j is denoted as H^j_best. The fitness value, maximum iteration number, crossover probability, mutation probability, and error are S^j_best, T, p_c,p_m and ɛ, respectively.

Below, some adjusting parametersh₁, h₂, ···, h_1ptd for PFNN can be designed as

Step 1 Select initial values $h 1_{11 i}^{1}$ (i = 1, 2, ···, d) randomly on [0, 1] and generate a d-dimensional chaos variable $(h_{11}^{1}, h_{12}^{1}, \cdot \cdot \cdot, h_{1 d}^{1})$ in accordance with the iterative the formula (2). Then, obtain individuals $H_{1}^{1} [0] =$ $(h_{11}^{1}, h_{12}^{1}, \cdot \cdot \cdot, h_{1 d}^{1})$ of the initial population G₁ using Definition 2 and Equation (5). Similarly, N initial populations can be acquired from N executions of M individuals. The individuals of these initial populations are recorded as $H_{k}^{j} [0]$ .

Step 2 For subpopulation G_j, calculate the error function $E (H_{k}^{j} [0])$ of the individual $H_{k}^{j} [0]$ . If $E (H_{k}^{j} [0]) < ɛ$ , then output $H_{k}^{j} [0]$ . Otherwise,calculate every fitness value $S_{k}^{j}$ , determine the optimal individual $H_{bes t}^{j}$ of G_j, and calculate its fitness value $S 1_{bes t}^{j}$ , where $S_{best}^{j} = max {S (H_{k}^{j}) | k = 1, 2, . . ., M}$

Step 3 Perform chaos crossing of individuals $H_{s}^{j}$ $, H_{t}^{j}$ (s, t = 1, 2, ···, M) depending on the crossover probability p_c. Let ${\begin{matrix} H_{s}^{j_{1}} = α H_{s}^{j} [t_{1}] + (1 - α) H_{t}^{j} [t_{1}] \\ H_{t}^{j_{1}} [t_{1}] = α H_{t}^{j} [t_{1}] + (1 - α) H_{s}^{j} [t_{1}] \end{matrix} .$ (6) where $H_{s}^{j_{1}} [t_{1}], H_{t}^{j_{1}} [t_{1}]$ are individuals after the chaos crossing and α ∈ [0, 1] is a chaos variable that can reflect p_c.

Let $H_{k}^{j_{1}} [t_{1}] = (h_{k 1}^{j_{1}}, h_{k 2}^{j_{1}}, \cdot \cdot \cdot, h_{kd}^{j_{1}})$ and select a mutation equation as $h_{ki}^{j_{2}} = h_{ki}^{j_{1}} + β (h_{k}^{j_{1}} (U) - h_{ki}^{j_{1}})$ , i = 1, 2, ···, d and β ∈ [- p_k, p_k], where $h_{k}^{j_{1}} (U)$ is the maximum values of the components. Mutate every dimension of the component of individuals with the mutation probability p_m and re-calculate the fitness value. Mutated individuals are expressed as $H_{k}^{j_{2}} [t_{1}] = (h_{k 1}^{j_{2}}, h_{k 2}^{j_{2}}, \cdot \cdot \cdot, h_{kd}^{j_{2}}) .$

Step 4 Search the current optimal individual $H_{k}^{j_{2}} [t_{1}]$ on the basis of the mutated individuals. If $H_{k}^{j_{2}} [t_{1}]$ is superior to the original H^j_best, then update the original optimal individuals. Otherwise, replace by H^j_best. Let t₁ = t₁ + 1 and turn to Step 2. Iterate for 30 steps and find the optimal individual w_j of the current subpopulation G_j.

Step 5 Repeat Steps 2-3 for N times to j = 1, 2, . . ., N and determine the optimal individuals w_j of every subpopulation. If w = (w₁, w₂, ···, w_N) and w_j = (w_j1, w_j2, ···, w_jd) is the optimal individual of the subpopulation G_j, then select the initial single subpopulation w of PSO and initialize the flight speed vector V_j [t₂] = (v_j1, v_j2, ···, v_jd) .

Step 6 Calculate the optimal position vector pb_j [t₂](j = 1, 2, ···, N) of individuals and the optimal position vector gb_j [t₂] of subpopulation: ${\begin{matrix} {pb}_{j} [t_{2}] = w_{j} [t^{″}], 1 t^{″} = max_{i^{'}} {i^{'} | E (w_{j} [i^{'} 11]) = min_{0 \leq i \leq t_{2}} E (w_{j} [i])} \\ {gb}_{j} [t_{2}] = {pb}_{j^{″}} [t_{2}], j^{″} = max_{j^{'}} {j^{'} 1 | E ({pb}_{j^{'}} [t_{2}]) = min_{1 \leq j \leq N} E ({pb}_{j} [t_{2}])} \end{matrix} .$

Next, update pb_j [t₂] and V_j [t₂] = (v_j1, v_j2, ···, v_jd) by using the following iterative formula: ${\begin{matrix} V_{j} [t_{2} + 1] = ω [t_{2}] \cdot V_{j} [t_{2}] + c_{1} ξ \cdot ({pb}_{j} [t_{2}] \\ - w_{j} [t_{2}]) + c_{2} η \cdot ({gb}_{j} [t_{2}] - w_{j} [t_{2}]) \\ w_{j} [t_{2} + 1] = w_{j} [t_{2}] + V_{j} [t_{2} + 1] . \end{matrix}$

Step 7 Determine whether 2n + 2 component parameters of n-polygonal fuzzy number in w_j [t₂ + 1] is in an ascending order. If yes, then turn to Step 9 and output; otherwise, rank them in the ascending order and proceed to Step 8.

Step 8 Determine whether t₁ + t₂ > T or j ∈ {1, 2, ···, N} exists such that E (w_j [t₂ + 1) < ɛ.

If yes, then turn to Step 9; otherwise, let t₂ = t₂ + 1 and proceed to Steps 5-6. Choose w_j randomly to replace the absent optimal individual in subpopulation in Step 3 and then turn to Steps 3-4.

Step 9 Output the optimal position vector gb_j [t₂ + 1], j ∈ {1, 2, ···, N} of the population.

In testing the reasonability of the above-mentioned algorithms, an individual organization of population in the parallel CG-PS algorithm is proposed and shown in Fig. 5.

Fig. 5

Sub-population individual organization in CGA-PSO.

Note 1 In the chaos crossing in Step 3, the chaos variable α can be updated properly on the basis of α_i = p_cx_i+1 (i = 1, 2, ···, M), where the chaos sequences {x_i+1} meets Equation (2), and x₀ is a given random generation number. This process reflects the ergodicity of chaos variable α_i and overcomes the randomness of the algorithm. In addition, the variable amplitude p_k can be introduced into chaos mutation through the variable probability p_m. Low fitness of individual mutation represents strong iteration chaos disturbance. For example, if m₀ individuals are available for mutation, then choose p_k = p_me^{(m₀-k/-m₀)} for individual k.

Note 2 The parameters ξ, η in Step 6 are random numbers generated by rand (·) between [0, 1], the coefficients c₁ and c₂ generally have a value of 2, and ω [t] is the weight of inertia. When ω [t] is large, the algorithm presents a strong global searching capability. When ω [t] is small, this algorithm exhibits a strong local search capability. ω [t] can be determined by the method in Ref. [10].

In addition, the flowchart of PCG-PSO is presented in Fig. 6.

Fig. 6

Flowchart of the PCG-PSO.

Actually, the PCG-PSO can use an CGA of multiple subpopulations to account for the global searching of the entire algorithm. The optimal individuals of subpopulations form the initial population of PSO to account for local searching, thereby accelerating the algorithm convergence. The hybrid algorithm uses CGA to ensure global search capability and maintain population diversity. PSO is also used for local searching of optimal individuals to accelerate convergence significantly.

6. A simulation example

The optimization effect of the PCG-PSO is verified by applying the hybrid algorithm into the temperature control in air-conditioning system. See [38, 39]. A variable-air-volume (VAV) conditioning system is composed mainly of temperature controller, VAV controller, terminal air valve, air volume sensor, and temperature sensor. This system is considered a neural network system, and the structure flow chart of this air-conditioning system is shown in Fig. 7.

Fig. 7

Room temperature control chart of variable air volume conditioning control.

Normally, the mathematical model of VAV air conditioning system mainly includes control law, actuator, controlled object, measurement transmitter and so on. Most of the actual control objects in air conditioning can be described by higher order differential equations. However, in order to simplify the analysis, as long as the control accuracy can meet the requirements, the low-order model can be used to approximate the dynamic characteristics of the control object. As the control object of the system, room temperature can be modeled according to the relationship between material and energy balance. Under VAV mode, air conditioning room is a model of first order inertia link and pure lag.

According to the Ref. [29], the control model of this network system can be expressed as $G (s) = \frac{{Ke}^{- τ s}}{1 + TS},$ where K is the amplification constant, T is time, τ is the delay parameter, and s is the Laplace operator. After an appropriate selection of parameters, the transfer function can be expressed as $G (s) = \frac{4 (1 - e^{- 224 s})}{472 s + 1}$

To design the controller of the VAV conditioning system, the SISO fuzzy deduction model can be simulated by PFNN. Suppose the inference rule library is composed of a total of L fuzzy rules:

R_l :IF x is X (l), THEN y is O (l), l = 1, 2, ···, L .

Let L = 5, X (l) be the actual input polygonal fuzzy number (setting temperature error) and O (l) be the expected output polygonal fuzzy number (expected temperature), and $X (l), O (l) \in F_{0}^{tn} (ℝ)$ . From the fuzzy inference rule of SISO-PFNN, the fuzzy pattern pairs (X (l), O (l)), l = 1, 2, 3, 4, 5 can be acquired. If let n = 3, for a given the accuracy ɛ = 0.001 > 0. Without losing generality, the ordered representations of these given input-output polygonal fuzzy number are represented as ${\begin{matrix} X (l) = (x_{0}^{1} (l), x_{1}^{1} (l), x_{2}^{1} (l), x_{3}^{1} (l), x_{3}^{2} (l), x_{2}^{2} (l), x_{1}^{2} (l), x_{0}^{2} (l)) \\ O (l) = (o_{0}^{1} (l), o_{1}^{1} (l), o_{2}^{1} (l), o_{3}^{1} (l), o_{3}^{2} (l), o_{2}^{2} (l), o_{1}^{2} (l), o_{0}^{2} (l)) \end{matrix} .$

These given fuzzy pattern pairs (X (1), O (1)), (X (2), O (2)), ···, (X (5), O (5)) of the 3-polygonal fuzzy number to be waited for training are listed in Tables 2-3.

From the two 3-polygonal fuzzy pattern pairs in Tables 2-3, the membership function curve between X (l) and O (l) (l = 1, 2, ···, 5)can be easily obtained. See Figs. 8, 9.

Table 2

Actual inputs of 3-polygonal fuzzy number

Actual input	Ordered representation
X(1)	(0, 0, 0, 0, 0, 0.3, 0.7, 1)
X(2)	(0, 0.3, 0.7, 1, 1, 1.3, 1.7, 2)
X(3)	(1, 1.3, 1.7, 2, 2, 2.3, 2.7, 3)
X(4)	(1, 2.3, 2.7, 3, 3, 3.3, 3.7, 4)
X(5)	(3, 3.3, 3.7, 4, 4, 4, 4, 4)

Table 3

Expected outputs of 3-polygonal fuzzy number

Cxpected output	Ordered representation
O(1)	(0, 0, 0, 0, 0.05, 0.15, 0.45, 0.55)
O(2)	(0.3, 0.4, 0.5, 0.6, 0.8, 0.9, 1.2, 1.3)
O(3)	(1.1, 1.2, 1.45, 1.55, 1.65, 1.75, 2, 2.1)
O(4)	(1.9, 2, 2.2, 2.35, 2.45, 2.55, 2.8, 2.9)
O(5)	(2.7, 2.8, 3.05, 3.15, 3.2, 3.2, 3.2, 3.2)

Fig. 8

Images of the actual inputs 3-fuzzy number.

Fig. 9

Images of the expected outputs 3-fuzzy number.

Several parameters of 3-PFNN are optimized by CGA, PSO and the PCG-PSO. The above-mentioned transfer function G (s) is chosen. The number of hidden layers and populations are set to 14 and N = 30, respectively. Each subpopulation G_j covers M = 16 individuals, and the optimal individuals of each G_j form the optimal position vector with 30 particle numbers. The number of iterations is T = 500, and the operation time is 50. In each time of iteration, CGA and PSO are run for 50 times separately until the error function E (w) meets the given accuracy or it reaches the number of iteration. The optimization results are shown in Table 4.

Table 4

Comparison of parameter optimization results of the CGA, PSO and PCG-PSO

Performance testing	CGA	PSO	PCG-PSO
Average iteration times	432	303	224
Convergence times	565	392	430
Error prediction	0.035	0.026	0.011

Table 4 indicates that the average number of iterations in the PCG-PSO is significantly lower than those of CGA and PSO but takes longer time in convergence than PSO. The PCG-PSO can obtain the fastest optimization speed, followed by CGA and PSO. Of course, PSO exhibits rapid convergence but poor global searching capability, and this algorithm easily causes premature convergence. Thus, the PCG-PSO can increase convergence rate without weakening global convergence capability.

As shown in Fig. 10, the temperature controlled by the PCG-PSO is stable with the adjustment time of about 1,240 s. The maximum overshoot is 0.285°C with the adjustment time of 900 s. The maximum overshoot of conventional PID controller is 5°C with the adjustment time of about 2,420 s. Temperatures controlled by CGA and PSO fluctuate violently. When the temperature decreases from 30°C to 20°C, the maximum overshoot of PID controller is 2.27°C with the adjustment time of about 1,650 s. Therefore, the VAV air-conditioning system based on the PCG-PSO presents smooth response curve, low overshoot, rapid response, and high stability. Thus, the optimized parameters by CGA, PSO and the PCG-PSO are applied in the simulation of temperature controller in the VAV air-conditioning system. The results are compared with the PID controller and fuzzy PID control in [38, 39].

Fig. 10

Comparison of three algorithms in temperature control of VAV air-conditioning system.

Actually, Fig. 10 is a comparison of the response curves of PCG-PSO, PSO and CGA under step input. From the stability performance, the PID control of CGA has the largest overshoot and the longest adjustment time, and its stability is poor; while the stability of PCG-PSO is the best. It not only improves the response speed, but also has short transition time, no oscillation, and the overshoot decreases obviously. The stability of PSO is in the middle of the two. However, the proposed PCG-PSO can directly improve the stability of the PID control system without changing the fuzzy rules or new fuzzy reasoning. In fact, in the operation of VAV air conditioning system, the change of supply air temperature and quantity will cause the change of control object and many parameters. It is a typical time-varying system. From the stability point of view, the conventional PID control needs a period of fluctuation to achieve steady-state, there is about 5% steady-state error. Especially when the parameters change, the transition time of conventional PID control is the longest, the overshoot can increase to about 40%, the oscillation is aggravated, and the adjustment time is obviously longer. Therefore, when the parameters of air-volume air-conditioning system change, the PID control of PCG-PSO obviously reduces the overshoot in the process of rising, basically without oscillation or insensitivity, and then after a short period of adjustment, it can quickly enter a stable working state, and the steady-state error state is close to zero. Therefore, PCG-PSO has a strong stability in air volume air conditioning system.

In addition, using single loop or series PID control for VAV air conditioning system can achieve better results, especially through the first and second disturbance. The simulation is shown in Figs. 11, 12.

Fig. 11

Simulation of temperature controller in VAV air-conditioning system under first disturbance.

Fig. 12

Simulation of temperature controller in VAV air-conditioning system under second disturbance.

By comparing the above simulation results, it can be seen that the control performance of VAV air conditioning system increases significantly after the secondary disturbance, especially the series PID control is better than the single loop PID control. That is to say, the series PID control can more effectively restrain the error caused by the secondary disturbance. This is because the series PID control system has fast response speed and small overshoot, so it can better solve the contradiction between dynamic and stability in the network system. Especially when the model structure and parameters of the controlled object change, the series PID control is insensitive to the changes of network parameters, which can enhance the adaptability and robustness of the network to uncertain factors. See [39 –41].

In fact, if the air valve is directly adjusted according to the room temperature by PID, this is the single loop control. Figs. 11 (1)-(2) and Figs. 12 (1)-(2) are the output responses of single-loop and series-pole control systems under second disturbances. The proportion, integral and differential parameters of inner-loop PID control of series-pole control can be determined by attenuation curve method and experiential trial-and-error method. The maximum short-term deviation of single-loop control system is 2.0, while that of series-pole control is 0.49. There is no residual between the two control modes. The maximum dynamic deviation of series-pole control for secondary disturbance can be reduced by about two times. As a result, the ability to suppress secondary disturbances is significantly enhanced.

7. Conclusions

The proposed PCG-PSO is a new optimization algorithm based on the advantages of CGA and PSO and relies on PFNN as its basic network. This hybrid algorithm can perform global search to finite (N) populations by CGA and thus find the optimal individuals of each subpopulation. These N optimal individuals are used as the initial population of PSO. the PCG-PSO not only maintains satisfying global searching capability and eliminates initial value dependency, but also accelerates the convergence and overcomes the poor global optimization capability of PSO. The hybrid algorithm also presents good stability. These advantages are proven by the simulation calculation. In fact, the response curve overshoot of the conventional PID control is large and the adjustment time is long, which results in the unsatisfactory control effect. The PCG-PSO proposed in this paper will show their advantages when applied to the fuzzy PID controller, especially in the VAV air conditioning system, which has the characteristics of fast response speed and small overshoot, so it can better overcome the control performance problems in the VAV air conditioning system. Hence, the proposed PCG-PSOalgorithm has some originality and advantages. However, the CGA is controlled by many parameters, such as crossover and mutation rates. Selection of these parameters can influence other performance or efficiencies of the PCG-PSO. Thus, this issue will be addressed in future studies.

References

J.J.

Buckley and

Hayashi , Fuzzy neural networks: A survey, Fuzzy Sets and Systems 66(1) 1994, 1–13.

J.J.

Buckley and

Hayashi , Can fuzzy neural nets approximate continuous fuzzy function, Fuzzy Sets and Systems 61(1) 1994, 43–51.

and

Aliev , Genetic algorithm-based learning of fuzzy neural network, International Journal of Approximate Reasoning 118 2001 351–358.

R.A.

Aliev ,

B.G.

Guirimov and

R.R.

Aliev , et al., Evolutionary algorithm-based learning of fuzzy neural networks, Part 2: Recurrent fuzzy neural networks, Fuzzy Sets and Systems 160(17) 2009, 2553–2566.

J.R.

Castro ,

Castillo and

Melin , et al., A hybrid learning algo-rithm for a class of interval type-2 fuzzy neural networks, Information Sciences 179(13) 2009, 2175–2193.

Dahala ,

Almejallib and

M.A.

Hossainc , GA-based learning for rule identification in fuzzy neural networks, Applied Soft Computing 35 2015, 605–617.

P.Y.

Liu , A new fuzzy neural network and its approximation performance, Science China (Series E) 32(1) 2002, 76–86.

P.Y.

Liu , Fuzzy Neural Network Theory and Applications [D], Beijing: Beijing Normal University, 2002.

D.L.

Lang and

Z.M.

Liu , The misadjustment analysis of BP algorithm and improved algorithm, Acta Electronica Sinica 23(1) 1995, 117–120.

10.

C.M.

He and

Y.P.

Ye , Evolution computation based learning algorithms of polygonal fuzzy neural networks, International Journal of Intelligent Systems 26(4) 2011, 340–352.

11.

G.J.

Wang and

X.P.

Li , Universal approximation of polygonal fuzzy neural networks in sense of K-integral norms, Science China, Information Science 54(11) 2011, 2307–2323.

12.

He and

G.J.

Wang , The conjugate gradient algorithm of the polygonal fuzzy neural networks, Acta Electronica Sinica 40(10) 2012, 2079–2084.

13.

X.L.

Sui and

G.J.

Wang , Influence of perturbations of training pattern pairs on stability of polygonal fuzzy neural network, Pattern Recognition and Artificial Intelligence 26(6) 2012, 928–936.

14.

G.J.

Wang ,

He and

X.P.

Li , Optimization algorithms for MISO polygonal fuzzy neural networks, Science China Information Science 45(5) 2015, 650–667.

15.

G.J.

Wang and

X.P.

Li , Construction of the polygonal fuzzy neural network and its approximation based on K-integral norm, Neural Network World 24(4) 2014, 357–376.

16.

X.P.

Li and

Li , The structure and realization of a polygonal fuzzy neural network, International Journal of Machine Learning and Cybernetics 7(3) 2016, 375–389.

17.

G.J.

Wang and

C.F.

Suo , The isolation layered optimization algorithm of MIMO polygonal fuzzy neural network,Neural Computing and Applications 29(10) 2018, 721–731.

18.

X.H.

Zhang ,

G.Z.

Dai and

N.P.

Xu , Genetic algorithms-a new optimization and search algorithms, Control Theory and Application 12(3) 1995, 265–273.

19.

J.H.

Zhang and

X.H.

Xu , A new evolutionary algorithm-ant colony algorithm, Systems Engineering Theory and Practice 19(3) 1999, 84–87.

20.

H.B.

Gao and

Gao , Particle swarm optimization based algorithm for neural network learning, Acta Electronica Sinica 32(9) 2004, 1572–1574.

21.

Y.L.

Lee ,

A.A.

Ei-saleh and

Iamail , Gravity-based particle swarm optimization with hybrid cooperative swarm approach for global optimization, Journal of Intelligent & Fuzzy Systems 26(1) 2014, 465–481.

22.

Melo and

Watada , Gaussian-PSO with fuzzy reasoning based on structural learning for training a neural network, Neurocomputing 172 2016, 405–412.

23.

Hu ,

J.S.

Yang and

L.B.

Wang , Improved chaos genetic algorithm and its application, Computer Engineering 36(5) 2010, 170–172.

24.

X.H.

Yuan ,

Y.B.

Yuan and

Wang , et al., A novel self-adaptive chaotic genetic algorithm, Acta Electronica Sinica 34(4) 2006, 708–712.

25.

T.B.

Wu ,

Cheng and

T.Y.

Zhou , et al., Optimization control of PID based on chaos genetic algorithm, Journal of computer simulation 26(5) 2009, 202–204.

26.

Wang ,

Y.S.

Dai and

S.S.

Wang , Modified chaos-genetic algorithm, Computer Engineering and Applications 46(6) 2010, 29–32.

27.

S.H.

Xu and

X.G.

He , A training algorithm of process neural networks based on CGA combined with PSO, Control and Decision 28(9) 2013, 1393–1398.

28.

Z.W.

Zhao ,

G.Q.

Ni and

Y.Y.

Shen , et al., Multiple multidimensional fuzzy reasoning algorithm based on neural network, Journal of Intelligent & Fuzzy Systems 35(4) 2018, 4121–4129.

29.

Y.Q.

Yang ,

G.J.

Wang and

Yang , Parameters optimization of polygonal fuzzy neural networks based on GA-BP hybrid algorithm, International Journal of Machine Learning and Cybernetics 5(5) 2014, 815–822.

30.

L.A.

Zadeh , Fuzzy sets, Information and Control 8 1965, 338–353.

31.

M.S.A.

Khan ,

Abdullah and

Ali , et al., Pythagorean hesitant fuzzy sets and their application to group decision making with incomplete weight information, Journal of Intelligent & Fuzzy Systems 33(6) 2017, 3971–3985.

32.

M.S.A.

Khan ,

Ali and

Abdullah , et al., New extension of TOPSIS method based on Pythagorean hesitant fuzzy sets with incomplete weight information, Journal of Intelligent & Fuzzy Systems 35(5) 2018, 5435–5448.

33.

Q.B.

Luo and

Zhang , A new approach to generate chaotic pseudo-random sequence, Journal of Electronics and Information Technology 28(7) 2006, 1262–1264.

34.

Z.G.

Cheng ,

L.Q.

Zhang and

X.L.

Li , Chaotic hybrid particle swarm optimization algorithm based on tent map, Systems Engineering and Electronics 29(1) 2007, 103–106.

35.

J.L.

Fan and

X.F.

Zhang , Piecewise logistic chaotic map and its performance analysis, Acta Electronica Sinica 37(4) 2009, 720–725.

36.

Han and

J.S.

Zhu , Improved particle swarm optimization combined with backpropagation for feedforward neural networks, International Journal of Intelligent Systems 28(3) (2013), 271–288.

37.

Peralta ,

Triguero and

García , et al., On the use of convolutional neural netwoks for robust classification of multiple fingerprint captures, International Journal of Intelligent Systems 33(1) 2018, 213–230.

38.

Y.H.

Duan , Research of predictive fuzzy-PID controller in temperature of central air-conditioning system, Journal of System Simulation 20(3) 2008, 600–606.

39.

J.Q.

Cao and

Xu , Variable universe self-adaptive fuzzy PID control applied in the VAV air-conditioning systems, Journal of Computer Simulation 28(5) 2011, 197–200.

40.

D.G.

Wang ,

C.L.P.

Chen ,

W.Y.

Song and

H.X.

Li , Error compensated marginal linearization method for modeling a fuzzy system, IEEE Transactions on Fuzzy Systems 23(1) 2015, 215–222.

41.

D.G.

Wang ,

W.Y.

Song and

H.X.

Li , Approximation properties of ELM-fuzzy systems for smooth functions and their derivatives, Neurocomputing 149 2015, 265–274.