Z-number based fuzzy neural network for system identification

Abstract

In this paper, a novel Z-number based Fuzzy Neural Network (Z-FNN) based on the integration of Z-valued fuzzy logic and neural networks is proposed. Z-valued fuzzy rule base is presented and its inference process is described using interpolative approximate reasoning. Accordingly, the structure of the Z-FNN is proposed using a distance measure and interpolative approximate reasoning scheme. Based on presented architecture the learning algorithm of Z-FNN is designed. The updating of the unknown parameters of the network is carried out using Genetic Algorithms (GA). The proposed Z-FNN system is utilized for dynamic plant identification. The effectiveness of Z-FNN has been tested by comparing its performance with the performances of other fuzzy systems available in the literature. The proposed approach has been proven to be a suitable alternative for the identification of nonlinear systems characterized by uncertain and imprecise information.

Keywords

Fuzzy neural networks Z-number fuzzy rule learning

1 Introduction

In industry most of dynamical processes are characterized by unpredictable factors, uncertainty and imprecise information, and therefore deterministic models cannot adequately describe these processes. The fuzzy systems constructed using type-1 or type-2 fuzzy logics are valuable alternatives for describing such kind of processes [1–4]. The kernels of these fuzzy systems are basically fuzzy If-Then rule bases that are generally designed using the knowledge of experienced specialists. However, the systems constructed on the base of type-1 and type-2 fuzzy rule based systems have deficiency related to describing the partial reliabilities and uncertainties of the information used in the knowledge base of the process. Partial reliability of information used in knowledge base are often missing in these fuzzy systems. In the modelling of an uncertain system using a fuzzy rule-based system, the reliabilities of these linguistic values gain importance as they strongly influence the fuzzy system’s performance. Prof. Zadeh has suggested the use of Z-numbers for handling uncertain information with the its reliability degree. In the rule base, Z-numbers provide fuzzy restriction and reliability information for each variable [5]. According to him, “the concept of a Z-number relates to the issue of the reliability of the information. A Z-number, Z, has two components, Z=(A, B). The component, A, is a restriction (constraint) on the values which a real-valued uncertain variable, X, is allowed to take. The next component, B, is a measure of the reliability (certainty) of the first component. Typically, A and B are described in a natural language. Example: (about 45 min, very sure). An important issue relates to computation with Z-numbers. Examples: What is the sum of (about 45 min, very sure) and (about 30 min, sure)? What is the square of (approximately 10, likely)? Computation with Z-numbers falls within the province of Computing with Words (CW or CWW).” Based on the constraint and reliability information, Z-number can better describe uncertain information. It is to be noted that Prof. Lotfi A. Zadeh contributed to literature until the very end of his life when he wrote the referenced work he was almost 90 years of age.

Theoretical aspects and practical applications of Z-number have been described in a number of research studies [5, 6]. In the references [7, 8], multi-criteria decision-making (MCDM) using Z-numbers is considered. The conversion of the Z-number number to fuzzy number is presented in [9]. The reference [10] used Z-number and analytical hierarchy process for the same problem. The referenced research papers use basically the approach proposed in [9]. The paper [11] use fuzzy sets with TOPSIS and Z-numbers for selecting alternatives. In these referenced works, the advantages of the suggested approaches are stated, using the complexity of the algorithms. However, the conversion of Z-numbers to real numbers and solving the stated problems may result in a significant loss of information that can affect the performance of the fuzzy system [12].

Recently, a greater number of research works are reported that are based on the Z-information set, to be able to handle uncertainties, especially for decision-making problems [12, 13]. The reference [12] uses fuzzy probabilities for representing linguistic information and presents a fuzzy Pareto optimality (FPO) for making decision. In [14] using Z-numbers, under incomplete information, a risk-minimized negotiation model is developed for a power purchaser and a transmission company. A fuzzy TOPSIS model, based on Z-numbers is introduced in [15] to handle uncertainty in construction problems. In [16], linguistic Z-number is defined and the Choquet integral is presented for MCDM. [17] used Z-numbers and described a fuzzy MCDM. The designed model is tested on the recruitment and selection of staff. The paper [18] proposed improved Z-number valued MCDM to express and use costumer’s opinions. Based on Z- number customer oriented evaluation is presented to select the best design concept. In the paper [19] linguistic Z-number preference (LZPR) relation is used for the assessment of decision maker’s preferences on creditability. Using the LZPR group decision-making model is developed for digital transformation assessments of small-medium enterprises. The paper [20] presented MCDM model for risk assessment. Z number is integrated with MCDM to rank the risk of debtors. [21] presented the total utility of a Z-number for measuring the total effects. The approach could be utilized for the ordering of Z-numbers in the MCDM model. [22] used linear interpolation and presented a reasoning mechanism using Z-rules. The approach is employed for measuring job satisfaction and for the evaluation of the educational achievements of students. By decomposing Z* macro-parameters into atomic components, the paper [23] presented a model of the endogenous arousal of thoughts.

Some research works are devoted to decision-making problems using the ranking of fuzzy numbers. The reference [24] presented used a ranking approach to transform Z-numbers into probability assignments. The presented model is tested on the medical diagnosis and risk analysis. [25] used Z-numbers for analysis of failure and effects. Using ranking a new risk priority number is determined to prioritize the failure modes. The feasibility of the system is tested on an aircraft turbine. The reference [26] used the weights of centroid points and the fuzziness’ degrees and presented an advanced ranking method of generalized fuzzy numbers. The reference [27] presented the ranking using the optimality degrees of Z-numbers and the adjustment of the obtained degrees by the use of a degree of pessimism. Based on Z numbers, the paper [28] explores outranking relations among alternatives under multiple criteria. Multiply criteria decision-making is employed for carrying outranking.

A set of research studies dedicated to the solving of optimization problems using Z-numbers. In the paper [29] based on Z-number, the solution of the optimization problem using a multi-objective evolutionary algorithm is considered. The paper [30] presented a portfolio model using Z number and neural networks. The presented model uses the fundamental and market values of assets. Z-number is used to handle uncertainties related to these data and NN is employed for the estimation of the market values.

In the paper [31] the acquisition of fuzzy and probabilistic information is considered for the formation of Z rules. Using Z number, FCM and genetic algorithms the clustering algorithm is proposed. Extending fuzzy if-then rules, the authors of [32] presented the Z-number-valued if-then rules for classification problem. In the paper [33], Z-number is combined fault tree analysis in order to handle the uncertainty and make risk management decision. In the article [34], a Z-number based neural network is used for the solution of regression. Using Z-number, the calculation of the network weights is presented and the stability of the model is discussed. The reference [35] showed that by using some data processing Z-numbers can be transformed into “monotonic” type-2 sets. [36] combined decision-making trail and evaluating laboratory and cognitive mapping with the hesitant fuzzy Z-numbers for blockchain risk evaluation. In the articles [37, 38] Z-rules are developed for the control of omnidirectional mobile robots and for the estimation of food security risk levels correspondingly. In these research works inference engine mechanism using interpolative reasoning is proposed for Z-rules. The paper [39] considered modelling of the uncertain system using Z-number based fuzzy equations. For this purpose, the neural network structure is utilized to find coefficients of the model presented by Z-numbers [40, 41].

In the above papers, Z-number is used to handle uncertainty and solve different scientific problems. But still, the number of papers that deal with fuzzy reasoning under Z information is relatively few [42]. Fuzzy reasoning models are relevant to a wide variety of areas, such as engineering, economics, psychology, sociology, finance, and education. In [1, 5, 7, 43, 44] different fuzzy reasoning techniques are proposed for processing uncertain information. These inference techniques are basically based on compositional rules, interpolation, similarity and the concept of distance. The performances of these techniques, their processing capabilities, speed, and complexity are important issues. It is, therefore, necessary to develop efficient reasoning mechanisms for solving practical problems. [42, 43] proposed an interpolative inference mechanism for sparse fuzzy rules. [41] used interpolative reasoning and developed a fuzzy system for dynamic plant control. The reasoning mechanism is proposed for Z-rules using α- cuts and the fuzzy rule interpolation mechanism.

The development of a rule base is one of the main problems in the construction of fuzzy systems. In the above research papers, the development of Z-rules is accomplished by human experts. In these Z-rules, the reliability of each linguistic value is also determined by experts [6, 7, 37, 38, 41]. The automation of the design process of Z-rules’ is very important. One of the effective techniques for the fuzzy rules’ design is artificial neural networks (ANNs). ANNs are used for the design of rule bases of type-1 [44–46] and type-2 fuzzy systems [3, 47]. In this paper, Using the learning property of neural networks, the design of the Z-rules is considered. Based on the integration of Z-rules and neural networks, the Z number-based Fuzzy Neural Network (Z-FNN) is proposed. The main contributions of the article include:

Z number based fuzzy neural networks (Z-FNN) is proposed using integration of Z valued fuzzy logic and neural networks.

Based on interpolative approximate reasoning and distance measure the architecture of Z-FNN is proposed.

The update of unknown parameters of Z-FNN is carried out using genetic algorithms

The designed algorithm of Z-FNN is presented and used for identification of dynamic plant.

The simulations were carried out in order to demonstrate the efficiency of the presented Z-FNN structure in identification of dynamic plant.

The organization of the paper is as follows. In Section 2 the fuzzy interpolative reasoning is presented. Section 3 presents the architecture and algorithmic design of Z-FNN. The parameter update of the Z-FNN is presented in Section 4. Section 5 describes the application of the Z-FNN system for dynamic plants’ identification. The conclusions are presented in Section 6.

2 Fuzzy rule interpolation

Recently, various fuzzy reasoning methods are designed for fuzzy systems in the literature. Here fuzzy reasoning based on linear rule interpolation is considered, as proposed by Koczy and Hirota [42, 43]. The present work adds onto theirs because it considers fuzzy reasoning using Z-numbers with a fuzzy neural network for adaptability. The presented method needs the fulfilment of several conditions. These are: the considered fuzzy sets need to be normal, convex and continuous with bounded support. This inference mechanism is designed for use with a sparse rule base. The fuzzy interpolative reasoning method uses α- cut and is based on a distance measure.

Suppose we have single-input and single-output (SISO) fuzzy rules. Let the currently observed value of the input variable X be A_* and this input A_* lies between the fuzzy sets A₁ and A₂. Let us now find the output Y of the rule-based system, using the rules that include A₁ and A₂ fuzzy sets. The stated condition can be formulated as $\begin{matrix} It is given that X is A * \\ Rule 1 : & If X is A_{1} Then Y is B_{1} \\ Rule 2 : & \underline{If X is A_{2} Then Y is B_{2}} \\ Find Conclusion Y = B * ? \end{matrix}$ (1)

Here, we need to determine the output fuzzy sets B_* for given input A_*. Let us consider α cut of fuzzy sets A₁ and A₂ and denote them as inf{ A₁^α}, sup{ A₁^α} and inf{ A₂^α}, sup{ A₂^α}. Here inf is infimum, sup is supremum of A₁ and A₂, correspondingly (see Fig. 1). Fuzzy sets A₁ is less than A₂, i.e. A₁ _< A₂, iff

Fig. 1

Infimum inf{A^α} and supremum sup{A^α} of the A.

$\inf {A_{1}^{α}} < \inf {A_{2}^{α}} and \sup {A_{1}^{α}} < \sup {A_{2}^{α}}$ (2) here inf{ $A_{1}^{α}$ } and inf{ $A_{2}^{α}$ } are infimums of A₁ and A₂, sup{ $A_{1}^{α}$ } and sup{ $A_{2}^{α}$ } are the supremums of A₁ and A₂.

Using the conditions A₁ _< A*<A₂ and B₁ _< B₂, Koczy and Hirota [41, 42] showed that $\frac{d (A_{*}, A_{1})}{d (A_{*}, A_{2})} = \frac{d (B_{*}, B_{1})}{d (B_{*}, B_{2})}$ (3) where d(*) is the distance between the two fuzzy sets. In literature, different formulas were used to determine distance. These are Euclidian distance, Hamming distance, Hausdorff distance, etc. In this study, the Euclidian distance is utilized.

The papers [42, 43] utilised α-cuts for determining the distance d(*). By using α-cut the lower d_L and upper d_U distances are calculated between the two fuzzy sets. Let’s consider the rule-based fuzzy controller including n fuzzy If-Then rules. Assume that the inputs X_i are A_ij (i = 1, … ,m; j = 1, … ,n). Here m and n are the numbers of inputs and rules, correspondingly. The d_L and d_U distances are calculated using α-cut as follows $\begin{matrix} d_{ij}^{L} = d^{L} (A_{ij}^{α}, X_{i}^{α}) = d (inf {A_{ij}^{α}}, inf {X_{i}^{α}}) \\ = inf {A_{ij}^{α}} - inf {X_{i}^{α}} \\ d_{ij}^{U} = d^{U} (A_{ij}^{α}, X_{i}^{α}) = d (sup {A_{ij}^{α}}, sup {X_{i}^{α}}) \\ = sup {A_{ij}^{α}} - sup {X_{i}^{α}} \\ d_{j}^{L} = d^{L} (B_{j}^{α}, Y_{j}^{α}) = d (inf {B_{j}^{α}}, inf {Y_{j}^{α}}) \\ = inf {B_{j}^{α}} - inf {Y_{j}^{α}} \\ d_{j}^{U} = d^{U} (B_{j}^{α}, Y_{j}^{α}) = d (sup {B_{j}^{α}}, sup {Y_{j}^{α}}) \\ = sup {B_{j}^{α}} - sup {Y_{j}^{α}} \end{matrix}$ (4)

Based on (3) and the concept of distance measure, Koczy and Hirota [42, 43] have proposed an interpolative reasoning mechanism. In this work, as a novel extension of their work, we consider the construction of an adaptive Z-FNN system using interpolative reasoning.

3 Z-number based fuzzy neural networks

3.1 Z-number

Definition 1 [1]. Let X be a nonempty set whose elements are x. A fuzzy set A in X is a set of ordered pairs $A = {(x, μ_{A} (x) | x \in X)}$ . Here μ_A (x) : X → [0, 1] is the grade of membership of x in A and called a membership function. It is real number in the interval [0,1] and associated with each x point in X.

Definition 2 [6]. A triangular fuzzy number A is a fuzzy subset characterized by (a₁, a₂, a₃) on R, and the membership function is determined as $μ_{A} (x) = {\begin{matrix} 0, - \leq x \leq a_{1} \\ \frac{x - a_{1}}{a_{2} - a_{1}}, a_{1} \leq x \leq a_{2}, \\ \frac{a_{2} - x}{a_{3} - a_{2}}, a_{2} \leq x \leq a_{3}, \\ 0, a_{3} \leq x \leq \infty \end{matrix}$ (5)

This membership function is utilized for describing the constraint and reliability parts of the Z-number.

Definition 3 [6, 39]. α – cut of fuzzy set A is $A^{α} = (x \in X; μ_{A} (x) \geq α)$ (6) Here α∈[0,1]. Each α cut defines $A^{α} = (A^{L}, A^{U})$ (7)

Here A^L and A^U lower and upper boundaries. A^α is bounded, as A^L ≤ A^α ≤ A^U.

Fuzzy operations on two fuzzy sets A₁ and A₂ using α – cut and interval arithmetic are given below.

Submission: $A_{1}^{α} + A_{2}^{α} = (A_{1}^{L} + A_{2}^{L}, A_{1}^{U} + A_{2}^{U})$ (8) Subtraction: $A_{1}^{α} - A_{2}^{α} = (A_{1}^{L} - A_{2}^{U}, A_{1}^{U} - A_{2}^{L})$ (9) Multiplication: $\begin{matrix} A_{1}^{α} \cdot A_{2}^{α} = ({min {A_{1}^{L} \cdot A_{2}^{L}, A_{1}^{L} \cdot A_{2}^{U}, A_{1}^{U} \cdot A_{2}^{L}, A_{1}^{U} \cdot A_{2}^{U}}, \\ max {A_{1}^{L} \cdot A_{2}^{L}, A_{1}^{L} \cdot A_{2}^{U}, A_{1}^{U} \cdot A_{2}^{L}, A_{1}^{U} \cdot A_{2}^{U}}) \end{matrix}$ (10) Division: $\begin{matrix} A_{1}^{α} / A_{2}^{α} = ({min {A_{1}^{L} / A_{2}^{L}, A_{1}^{L} / A_{2}^{U}, A_{1}^{U} / A_{2}^{L}, A_{1}^{U} / A_{2}^{U}}, \\ max {A_{1}^{L} / A_{2}^{L}, A_{1}^{L} / A_{2}^{U}, A_{1}^{U} / A_{2}^{L}, A_{1}^{U} / A_{2}^{U}}) \end{matrix}$ (11) Scalar multiplication: $A^{α} = {\begin{matrix} (A^{L}, A^{U}), \geq 0, \\ (A^{U}, A^{L}), < 0 \end{matrix}$ (12)

Definition 4 [5, 6]. A Z-number Z=(A, B) is an ordered pair of fuzzy numbers, where A is a restriction on values of fuzzy variable X, B is reliability defined for A.

Figure 2 shows the restriction A and reliability B components of the Z-number. Here A and B components are described using triangular membership functions. In the figure, A represents the value of the fuzzy X variable, and B represents the reliability, degree of truth, probability or possibility measure of A. X is A is referred to as a possibilistic restriction, that is $B (X) : X is A \to Poss (X = u) = μ_{A} (x)$ (13)

Fig. 2

The components of Z-number. (a) restriction A, (b) reliability B.

Here the membership function μ_A(x) of A may describe a constraint defined for B(X), u is a generic value of X. μ_A(x) is the membership degree to which u satisfies.

Z-number can be efficiently used to describe uncertain information. As an example, consider the prediction of the electricity consumption of the city. It is well known that the consumption of electricity is affected by the number of consumers, time period and other factors. For this reason when we express “the next weak electricity consumption will be a little higher” will not be reliable as 100%, and is considered as a probable event. By assigning a reliability degree the event can exactly be described as

“The next weak electricity consumption will be a little higher”, very likely

This example can be exactly presented using Z-number. If we denote the fuzzy variable “electricity consumption” as X, its value can be represented by Z=(A,B). Here the fuzzy X variable is presented by (A, B), where A will be the constraint having the value “little higher”, B will be the truth degree (or reliability) of A having value “very likely”.

Definition 5 [6]. Let=(A₁,B₁) and Z₂=(A₂,B₂) are Z-numbers. The supremum metrics on $F$ are defined as $D (Z_{1}, Z_{2}) = D (A_{1}, A_{2}) + D (R_{1}, R_{2})$ (14)

Using α-cut the distance between Z-numbers Z₁ and Z₂ is computed as $\begin{matrix} D (Z_{1}, Z_{2}^{α}) = | a_{1}^{L} - a_{2}^{L} | + | a_{1}^{U} - a_{2}^{U} | \\ + | r_{1}^{L} - r_{2}^{L} | + | r_{1}^{U} - r_{2}^{U} | \end{matrix}$ (15)

Where $a_{1}^{L}, a_{2}^{L}, r_{1}^{L}, r_{2}^{L}, a_{1}^{U}, a_{2}^{U}, r_{1}^{U}, r_{2}^{U}$ are α-cuts of fuzzy sets that present lower and upper sides of fuzzy triangular numbers A₁, R₁, A₂ and R₂, respectively [32].

3.2 Design of Z-FNN

The design of Z-FNN is a process of the construction of Z fuzzy rules using neural networks’ training capabilities. This is carried out by learning the parameters of the premise and the consequent parts of the fuzzy If-Then rules. Mamdani- and TSK-type fuzzy rules are extensively used for solving different practical problems. Consider the reasoning mechanism of the system based on Mamdani type Z If-Then rules. MISO Z-rules are presented below. $\begin{matrix} If x_{1} is (A_{11}, R_{11}) and \dots and x_{m} is (A_{m 1}, R_{m 1}) Then y is (B_{1}, R_{1}) \\ \dots \\ If x_{1} is (A_{1 n}, R_{1 n}) and \dots and x_{m} is (A_{mn}, R_{mn}) Then y is (B_{n}, R_{n}) \end{matrix}$ (16)

Here m and n are the numbers of input signals and rules, correspondingly. Let A_ij and B_j be the restrictions assigned for the input and output fuzzy sets, R_ij and R_j be the reliabilities of the corresponding fuzzy sets A_ij and B_j. x_i and y are the input and output variables, correspondingly. Here we are considering the case when the reliability parameters are characterised by fuzzy values.

Fuzzy reasoning using Z-numbers is used for decision-making purposes [7, 8], dynamic plant control [41], omnidirectional robot control [36], estimation of the security’s risk level [38]. In the following, using the interpolative reasoning method, we consider the construction of Z-FNN and its parameter update rule. Figure 3 presents the architecture of the proposed Z-FNN. Here the main problem is the development of Z-type If-Then rules using the proposed Z-FNN structure.

Fig. 3

Structure of Z-FNN.

The proposed Z-FNN structure implements the inference process that is presented in [6, 37]. The Z-FNN structure consists of five layers. The input signals x_i, i = 1, … ,m are distributed in layer 1. The next layer 2 is the rule layer, where the rules are represented by R₁, R₂, … , R_n (see Fig. 3). The weights of the connections between layer 1 and layer 2 are denoted by a_ij and ra_ij, between layer 3 and layer 4- by b_j and rb_j, correspondingly. Here i = 1, … ,m; j = 1, … ,n. The restrictions a_ij ∈R, b_ij ∈R and the reliability parameters ra_ij∈[0,1], rb_ij∈[0,1] are characterised by fuzzy numbers of triangle type. In layer 1 of Z-FNN the α-cut is applied in order to find the differences between the input signals x_i and the parameters a_ij, in the premise parts of the rules are calculated. Here α-level of network parameters will be represented as

$\begin{matrix} a_{ij}^{α} = (a_{ij}^{L}, a_{ij}^{U}); {ra}_{ij}^{α} = ({ra}_{ij}^{L}, {ra}_{ij}^{U}); b_{j}^{α} = (b_{j}^{L}, b_{j}^{U}); \\ {rb}_{j}^{α} = ({rb}_{j}^{L}, {rb}_{j}^{U}); i = 1, \dots, m; j = 1, \dots, n \end{matrix}$ (17)

We used α-cuts with the Euclidian distance to calculate the distances for constraint and reliability variables. At first, the distances $({da}_{ji}^{L}, {da}_{ji}^{U})$ between the constraint(restriction) parameter a_ij and the input x_i (i = 1,.,m; j = 1,.,n) are computed, and then the same formulas are adapted for finding the distances $({dr}_{ji}^{L}, {dr}_{ji}^{U})$ between the reliability parameters ra_ij and rx_i. ${da}_{ij}^{α} = {(a_{ij}^{α} - x_{i}^{α})}^{2}$ (18) where i = 1,.,m, j = 1,.,n. We used α={0,1}. Then $({da}_{ji}^{L}, {da}_{ji}^{U})$ . The formula (18) is employed to find the lower ${da}_{ij}^{L}$ and the upper ${da}_{ij}^{U}$ distances between the two fuzzy sets. Then ${Da}_{j}^{α} = ({Da}_{j}^{L}, {Da}_{j}^{U}) = (\sum_{i}^{m} {da}_{ij}^{L}, \sum_{i}^{m} {da}_{ij}^{U})$ (19)

Here Da_j^α=(Da_j^L,Da_j^U) is the sum of the lower and upper distances for restriction parameters a_ij of Z-FNN in j-th node of Layer 2 determined by the α-level. ${dr}_{ij}^{α} = {({ra}_{ij}^{α} - {rx}_{i}^{α})}^{2}$ (20)

The formula is used to compute the lower ${dr}_{ij}^{L}$ and the upper ${dr}_{ij}^{U}$ distances for the reliability parameters. Here ${dr}_{ij}^{α} = {{dr}_{ij}^{L} {dr}_{ij}^{U}}$ , (i=1,…,m;j=1,…,n). ${Dr}_{j}^{α} = ({Dr}_{j}^{L}, {Dr}_{j}^{U}) = (\sum_{i}^{m} {dr}_{ij}^{L}, \sum_{i}^{m} {dr}_{ij}^{U})$ (21)Dr_j^α=(Dr_j^L,Dr_j^U) is the sum of the lower and upper distances. The final total distance is computed by calculating sum of the distances as $D_{j}^{L} = {Da}_{j}^{L} + {Dr}_{j}^{L}; D_{j}^{U} = {Da}_{j}^{U} + {Dr}_{j}^{U}$ (22) The third layer is employed to calculate the inverse of the output signals of the second layer. $Q_{j}^{L} = \frac{1}{D_{j}^{U}}; Q_{j}^{U} = \frac{1}{D_{j}^{L}}$ (23) $Q_{j}^{α} = (Q_{j}^{L}, Q_{j}^{U})$ . The output of the fourth layer will be calculated as $Y^{α} = \sum_{j = 1}^{n} Q_{j}^{α} \cdot b_{j}^{α}; R_{Y}^{α} = \sum_{j = 1}^{n} Q_{j}^{α} \cdot {rb}_{j}^{α}$ (24) $S^{α} = \sum_{j = 1}^{n} Q_{j}^{α}$ (25)

Here $Y^{α} = (Y^{L}, Y^{U}), R_{Y}^{α} = (R_{Y}^{L}, R_{Y}^{U})$ , $S^{α} = (S^{L}, S^{U})$ . The network output is computed in the last layer as $U^{α} = \frac{Y^{α}}{S^{α}} = \frac{Y^{α}}{\sum_{j = 1}^{n} Q_{j}^{α}}; R_{U}^{α} = \frac{R_{Y}^{α}}{S^{α}} = \frac{R_{Y}^{α}}{\sum_{j = 1}^{n} Q_{j}^{α}}$ (26)

Here U^α=(U^L,U^U) and $R_{U}^{α} = (R^{L}, R^{U})$ are the network outputs, defined for constraint and reliability parameters. In the above formulas, all the mathematical operations have been performed using interval arithmetic and α-level procedure.

The output of the Z-FNN system will be Z-number and is determined using (18)-(26). After the determination of the output of Z-FNN, the conversion of the output signal into a crisp number is performed. The conversion depends on the form of membership functions used. In this paper, triangular-type fuzzy sets are used in the “If” and “Then” parts of the fuzzy rule base. The Z-FNN output is calculated using the graded mean of two fuzzy numbers [8, 9]. If we consider α=0 and α=1, then the left U_l, the middle U_m and the right U_r values of the triangle are determined. Here the left (U_l,R_Ul) and (U_r,R_Ur) values correspond to α=0 and the middle (U_m,R_Um) value corresponds to α=1 level, which is the highest value. After finding the left and right values for α=0 level using (11) [32], the highest value is determined [48]. After deriving the output signals, the obtained Z-number is converted to the crisp number. The formula U=((U_l+4×U_m+U_r)/6)× ((R_Ul+4×R_Um+R_Ur)/6) given in [9] is used to determine the output crisp value.

4 Parameter update rule

The a_ij,ra_ij, b_j and rb_j are unknown parameters of Z-FNN architecture (see Fig. 3). The design of Z-FNN is the finding such values of the unknown parameters using them in the network the output desired values will be generated for any input. Z-FNN is constructed using Z rules presented in (14). The input region is partitioned into a set of fuzzy regions and described by the premise part of the fuzzy If-Then rules. The consequent part of these rules is used to describe the system behaviour in these regions. Nowadays, a set of algorithms is utilized for the construction of fuzzy If-Then rules. These are gradient algorithms [44, 46], the least-squares method (LSM), clustering techniques and evolutionary algorithms [44–46]. The parameters of Z-FNN are initialised randomly. We used genetic algorithms for updating the parameters of the Z-FNN system.

Using the output errors, the adjusting of the parameters of Z-FNN is carried out. In the output neuron of Fig. 3, the error cost function E^α = (E^Lα, E^Uα) is determined as $E^{α} = | U_{d}^{α} - U^{α} |; E^{α} = \frac{1}{2} \sum_{k = 1}^{K} (U_{d}^{α k} - U^{α k})^{2}$ (27) where $U_{d}^{α} = (U_{d}^{L}, U_{d}^{U})$ and U^α = (U^L, U^U) are the α-level set of desired and the current outputs of the Z-FNN. For one rule K = 1. We use formula (15) for finding the distance between Z numbers. Using errors calculated for the lower and upper sides, the fitness function is constructed as F = 1/E^α.

In the paper, GA is applied for the training of the Z-FNN controller. Genetic Algorithms (GA) is a random search algorithm used for updating the parameters of different neural and fuzzy neural structures. Real-coded GA is employed for training parameters of the Z-FNN. During learning a set of chromosomes that represent solution parameters of Z-FNN are generated randomly. Each chromosome consists of a set of genes. In modelling, population size denotes the number of generated chromosomes. Selection, crossover and mutation operators are applied for updating the chromosomes. In the paper, we applied tournament selection. Using pairwise comparison the solutions that have high fitness are selected for the next generation. After the selection of the solutions, we applied a multipoint crossover operation. The crossover rate (cr) is utilized to select solutions for the crossover operations. The high value (selected in the interval [0.5, 1]) leads to the quick generation of new solutions. In crossover operation two solutions V=(v1, v2, … ,vn) and W=(w1, w2, … ,w_n) are selected for crossover operation. If denote pair of new solutions by V’=(v’₁,v’₂, … ,v’_n) and W’=(w’₁, w’₂, … ,w’_n) then the crossover operation will be presented by the following formula when F(V)> F(W). $\begin{matrix} v_{i}^{'} = v_{i} + δ (w_{i} - v_{i}) \\ w_{i}^{'} = v_{i} + δ (v_{i} - w_{i}) \end{matrix}$ (28) where the δ is changed in the interval [0, 1].

After the crossover operation, the mutation is applied for updating the selected solutions. For this purpose, a random number is generated for each gene. If the mutation rate (mr) will be more than this random number then the selected gene is updated. $v_{i}^{'} = v_{i} + μ \cdot rand$ (29) where μ is mutation rung, rand is a random number generated in the interval of (0,1). In mutation operation, the small number selected from [0, 1] is added to the selected gene and a new value of solution will be determined. A large mutation rate leads to a pure random search. Usually, the mutation rate is taken from [0, 0.1].

In the paper, for improving learning accuracy we used adaptive formulas for updating the values of δ, μ, cr and mr. We have done a set of simulations and determined minimum g_i and maximum h_i values for the parameters δ, μ, cr and mr as [g_i, h_i], here i = 1,.,4. During learning, the sum of fitness functions is calculated for each iteration and called a total fitness function. In each iteration, we calculated the gradient of the total fitness function (ftot). According to the value of this gradient (decay), the updates of δ, μ and mr values are carried out. The rules used for adaptive change of the parameters is given below

decay(k)=(ftot(k)-ftot(k-1))/ftot(k-1);

if(decay(k)< 0)

δ=δ/a; μ=μ×b; mr = mr+c;

elseif (decay(k)> 0) and (decay(k)< 0.001)

mr = mr+c;

elseif (decay(k)> 0.001)

δ=δ ×a; μ=μ×b;

end

Here a, b and c are small coefficients.

5 Simulations

In industry, many dynamic systems run under uncertain conditions. These uncertainties are reflected in the inputs and outputs of dynamic systems. The use of fuzzy set theory can be a valuable and viable alternative for constructing suitable controllers for these systems. Here, the proposed Z-FNN model is used for dynamic plant identification. The Z-FNN system is tested on a benchmark example and its performance is compared with the performances of other control systems.

Identification is the process of finding the relations between the system’s inputs and outputs. Figure 4 presents the structure of the identification system. Here u(k) is input, y(k) is output of the plant. The problem is finding the association between u(k) and y(k) variables. Here we are applying the Z-FNN system for modelling this relationship. The inputs of the Z-FNN identifier are u(k) input signal, its one-, two-,.., d_i step delays and one-, two-,.., d_o step delays of plant output y(k) signals. The problem is to construct such a Z-FNN system, the difference between the identifier output y_z(k) and plant output y(k) will be the acceptable minimum for all input signals. The stated problem can be solved by learning the unknown parameters of Z-FNN system.

Fig. 4

Identification scheme.

In the simulation, to imitate the plant behaviour, we are using the difference equation describing the model of the plant. We try to identify the plant using the Z If-Then rules. We are applying the same excitation signal to the inputs of the dynamic plant and Z-FNN model, at the same time. The outputs of the model and plant are evaluated. Here, the basic problem is finding such values of the Z-FNN parameters using them in the identification scheme the deviation of the identifier’s output from the plant’s output will be minimal for all input values.

In this example, the use of the Z-FNN system for the nonlinear plant’s identification is presented. The plant is the same as that given in [44].

$y (k) = \frac{y (k - 1) y (k - 2) y (k - 3) u (k - 1) (y (k - 3) - 1) + u (k)}{1 + y {(k - 2)}^{2} + y {(k - 3)}^{2}}$ (30) where y(k-i), (i = 1,.3) are the i-step delayed outputs of the plant, u(k-1) and u(k) are the one-step delayed and current inputs to the plant.

The one-step delayed output and the one-step delayed input (control signal) of the plant are used as inputs for the Z-FNN identifier. The output of the Z-FNN identifier y_n(k) is compared with the output of the plant. Training is done using 900-time steps and 1000 epochs. The signal given below is applied for testing the system. $u (k) = {\begin{matrix} sin (π k / 25), & k < 250 \\ 1.0, & 250 \leq k < 500 \\ - 1.0, & 500 \leq k < 750 \\ 0.3 sin (π k / 25) + 0 . 1 \sin (π k / 32) \\ + 0 . 6 \sin (π k / 10), & 750 \leq k < 1000 \end{matrix}$ (31)

Our goal is to derive the same output signal in the Z-FNN system output. For this purpose, after defuzzification, the output of the Z-FNN model is compared with the output of the plant and as a result, the error is determined. The Z-FNN model output is determined using the structure given in Fig. 3 and formulas given in Section 3. Using the error signal, the A and B parameters of the Z If-Then rules are updated (see Section 4). The update is continued until the output error values for all input signals become acceptable small. Because of minimum and maximum values of B are 0 and 1, in the If-Then rules the training of B is implemented according to this interval. At first, we have defined minimum and maximum values of δ, μ, cr and mr. A set of simulations have been done and as a result the minimum and maximum values of the parameters δ, μ, cr and mr are determined as [0,0.8], [0, 0.2], [0.3, 0.8] and [0.01,0.2], respectively. The initial values of these parameters were set as 0.4, 0.02, 0.6 and 0.1. During training these values are updated.

The identification of the dynamic plant is implemented using different number of rules (or hidden nodes in the second layer). At first, simulation studies are carried out using three fuzzy rules, and then five and teen fuzzy rules are used for the dynamic plant identification, expressed by (30). The simulations have been carried out using Mamdani and TSK types of Z fuzzy rules. At first, Mamdani-type rules were used in controller design. The parameters of the Z-FNN identifier are the a_ij and ra_ij coefficients of the second layer and the b_j and rb_j coefficients of the fourth layer. During simulation, the network’s parameters are initialized as triangle-type fuzzy numbers. Training of the Z-FNN parameters is performed using the above-described real-coded genetic algorithm. In the paper tournament selection, real codded crossover and mutation operations as presented in Section 4 are used for learning the parameters of the Z-FNN identifier. The learning of the identifier is performed using the update rule, the excitation signal (31) and the dynamic plant model (30). During the learning, the sum of square error (SSE) and root-mean-square error (RMSE) indicated in (32) is used for measuring the performance of the Z-FNN identifier.

$\begin{matrix} SSE = \sum_{k = 1}^{K} {(y (k) - y_{n} (k))}^{2}; RMSE \\ = \sqrt{\frac{1}{K} \sum_{k = 1}^{K} {(y (k) - y_{n} (k))}^{2}} \end{matrix}$ (32)

Here K = 1000 is the number of steps, y(k) and y_n(k) are plant and network outputs, respectively. Z-FNN model inputs are y(k-1), y(k-2), y(k-3) outputs and u(k-1) and u(k) inputs to the plant. During simulation the crossover rate is selected as 0.6, mutation rate- 0.02, probability of mutation is set as 0.1and probability of crossover- 0.7. As a result of learning, the fuzzy rules presented by Z-FNN structure are derived and the unknown parameters of the system are determined. The training is performed using three, five and ten Mamdani type Z rules. The simulation was performed for 500 epochs. The population size was 50. During simulation, the SSE and RMSE values are calculated. During GA learning the value of the fitness function is calculated as F = 1/SSE. The values of a and b are defined in the interval [1, 1.01]. c is a small number defined in the interval [0, 0.01].GA operators- selection, crossover and mutation are employed to train Z_FNN. The selection operator (tournament selection) uses pairwise comparison to select the solution that has a high fitness value. After the selection of the solutions, the crossover operation described by the formulas (28) is used for updating the parameters’ values. After, the mutation operation presented by formula (29) is applied to change the parameters’ values. During crossover and mutation operations crossover and mutation rate are used in selecting solutions for update operations. The presented update procedures were used to find unknown parameters of the Z-FNN system. Figure 5 depicts the plots of fitness function for 500 epochs using 5 rules obtained from training. Figure 6 depicts the sum of RMSE (that is $\sum_{i = 1}^{ps} RMS E_{i}$ ) for each generation. Figure 7 depicts the identification results obtained using five Z-rules.

Fig. 5

The plot of the fitness function.

Fig. 6

The plot of the sum of RMSE for all population.

Fig. 7

Identification results, where the plant’s output is the solid line and the Z-FNN output is the dashed line.

When the number of rules used was 5, the average value of RMSE for identification obtained was 0.048132. Table 1 depicts simulation results obtained using 3, 5 and 10 Mamdani type Z rules. The results were obtained using averaged values of twenty simulations.We find that Z-FNN based on Mamdani rules has better performence that other neuro-fuzzy system based on Mamdani rules.

Table 1

Results of simulation

Number of Rules	SSE (Mean)	RMSE (Mean)
3	6.44279	0.0802679
5	2.316689	0.048132
10	1.271279	0.035655

In the second simulation, we test the model using TSK type of Z fuzzy rules. In the TSK-type rule the linear functions characterized by the fuzzy coefficients b_ij, and rb_ij are used in the consequent part. Simulations have been done using three, five and ten TSK type Z rules. Table 2 demonstrates the performance of Z-FNN using different rules’ number. For 5 fuzzy rules, the SSE and RMSE values were 1.330658 and 0.036478, respectively. The results were obtained using averaged values of ten simulations. As shown in Table 2, the increasing number of rules decreased the RMSE and improved the accuracy of the system. However, the increase in the number of rules causes the complexity of the network to increase. We simulated the identification of dynamic plant using the neuro-fuzzy network (NFN) also. The simulation results of Z-FNN based identification system is compared with the results of other identification systems. Table 3 depicts the comparison of different identification models.

Table 2

Results of simulation

Number of Rules	SSE	RMSE
3	1.665202	0.040807
5	1.330658	0.036478
10	0.869306	0.029484

Table 3

Comparisons of simulation results

Models	SSE	RMSE
RSONFIN [46]	6.084	0.0780
FFNFN [44]	2.71441	0.0521
TRFN-S [45]	1.19716	0.0346
FWNN [44]	0.90751	0.030125
NFN	2.71441	0.0521
Z-FNN (Mamdani type)	1.271279	0.035655
Z-FNN (TSK type)	0.869306	0.029484

To test the generalization ability and robustness of the Z-FNN system we changed the input signal and tested the identifier. The following signal is applied to the input of the Z-FNN identifier $u (k) = {\begin{matrix} - 0.7 + \frac{mod (k, 60)}{40}, k < 250 \\ 1, 250 \leq k \leq 500 \\ - 1.05 cos (\frac{π k}{45}), 500 \leq k \leq 750 \\ 0.7 - \frac{mod (k, 180)}{120}, k \geq 750 \end{matrix}$ (33)

Figure 8 depicts the plots of the outputs of the Z-FNN model and the dynamic plant. Here, the solid line is the output of the plant, the dashed line is the output of the Z-FNN system. As seen from the experiment, we trained the Z-FNN using the input signal (31), but tested it with the anther signal given above. We simulated the Z-FNN system using 5 and 10 rules. The values of SSE and RMSE for the five rules were obtained as 1.3522 and 0.03677, for ten rules- 0.88124 and 0.2968, respectively. As shown, the results obtained demonstrate the robustness of the Z-FNN identifier.

Fig. 8

The plots of the outputs of the Z-FNN model and the dynamic plant using input signal (33). Here the solid line is the plant output, dashed line is the Z-FNN output.

The complexity depends on the number of neurons and the number of layers used in the Z-FNN (see Fig. 3). If we denote the number of input neurons in the first layer N1, hidden layers N2 (these are layer 2 and layer 3), then from the Z-FNN structure, the number of calculations in terms of O(n) notation will be determined. Here O(n) denotes the longest running time defined by a maximum number of steps, that is an upper bound of running time. It gives us a guarantee that the algorithm will never take any longer. From the structure of the Z-FNN system, we can see that in the second layer (N1·N2), in the third - (N2), in the fourth - (2·N2), in the fifth- (2) calculations are required to determine the output of the network. Here · is the multiplication operation. If we take into account the number of parameters (which is two- fuzzy value and its reliability) and alpha levels (two alpha levels, 0 and 1) then for one variable we will have 6 parameters. If we get a sum of these calculations, then the running time will be determined as T(n)=6· (N1·N2)+(6·N2)+6·(N2·2)+12. Assume that N1 = N2. Then the running time will be described as T(n)=6n² ⁺ 18n+12. The T(n) will denote the approximate running time for real-time operations of Z-FNN (without learning). For simplicity, we did not consider the cost of each operation. If we use O notation, then the upper bound of the running time of Z-FNN for real-time operation will be presented as O(n³).

The Z-FNN system is utilized for the identification of the nonlinear plant, where the association between the input and output signals of the plant is nonlinear. The kernel of Z-FNN system is Z-valued fuzzy rules. In these fuzzy rules input and output variables are represented by Z-fuzzy numbers that use two fuzzy values - restriction on the values of fuzzy variable and the reliability degree of the fuzzy value. The input-output relationship of the controller is described by Z fuzzy rules that allow a more adequate description of the considered problem. Therefore, the system constructed using Z-rules can approximate the nonlinear plants with high accuracy. The comparison of the results of different approaches used for modelling the same plant demonstrated that Z-valued rules can be superior for modelling these kinds of objects. The reason for this is a more accurate description of the situations using the fuzzy values and their reliability degree in the rule base. The advantage of the proposed approach is proved by comparing the developed system with the other existing systems. The obtained results satisfy the efficiency of using the Z-FNN system in the identification of dynamic systems characterized with nonlinearity and uncertainties.

6 Conclusions

A novel Z-FNN system based on interpolative reasoning is designed and used for dynamic plant identification. The kernel of Z-FNN is the Z rules that are characterized by the reliability degrees of the fuzzy values. The architecture of the Z-FNN system is developed. Based on the proposed structure, the GAs is used for adjusting the parameters of Z-FNN. Adaptive update rules are used for improving the learning performance of the system. The designed Z-FNN structure with its learning algorithm is implemented in the Matlab package for dynamic plant identification. The simulation results of Z-FNN identifier are compared with the simulation results of other approaches taken from the literature. It is seen that the proposed approach is a viable and serious alternative for practical applications that are characterized by uncertain and imprecise information. The future studies is based to the design of a Z valued rule-based system using other membership functions such as trapezoidal and Gaussian membership functions and the design of the Z-FNN system using other inference procedures.

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