T-S fuzzy model identification based on an improved interval type-2 fuzzy c-regression model

Abstract

Fuzzy clustering has been widely applied in T-S fuzzy model identification for nonlinear systems, however, tradition type-1 fuzzy clustering algorithms can’t deal with uncertainties in real world, an improved interval type-2 fuzzy c-regression model (IT2-FCRM) clustering is proposed for T-S fuzzy model identification in this paper. The improved IT2-FCRM adapts a new objective function, which makes the boundary of clustering more clearly and reduces the influence of outliers or noisy data on clustering results. The premise parameters of T-S fuzzy model are upper and lower hyperplanes obtained by improved IT2-FCRM, and the upper and lower hyperplanes are used to build hyper-plane-shaped type-2 Gaussian membership function. Compared with the hyper-sphere-shaped membership function of tradition IT2-FCRM, the hyper-plane-shaped membership function is more coincided with point to plane sample distance described by FCRM clustering. The simulation results of several benchmark problems and a real bed temperature in circulating fluidized bed plant show that the identification algorithm has higher accuracy.

Keywords

Fuzzy identification interval type-2 fuzzy c-regression model fuzzy clustering T-S fuzzy model orthogonal least squares

1 Introduction

For a complex controlled object or uncertain system, it is usually very difficult or even impossible to determine the accurate mathematical model. Fuzzy system can approximate nonlinear dynamic system with arbitrary accuracy, it is widely applied in modeling nonlinear system. The fuzzy identification algorithm can establish fuzzy model of nonlinear system using system’s input and output sample data by certain algorithm. T-S fuzzy model is a local linearization model [1], and its premise parts are generally input variables, output variables and/or state variables of the system, and the consequent parts are linear expressions. It is an important method for fuzzy modeling of complex nonlinear multivariable system.

T-S fuzzy model identification includes structure identification and parameter identification. The main task of structure identification is to select input variables, rules number, membership functions and parameter identification is to define parameters of membership functions and linear expressions for consequent parts. In general, structure identification mainly relies on expert knowledge and experience, which is time consuming and inefficient. Fuzzy clustering can classify sample data into several classes according to their similarity degrees without teacher’s guidance or prior knowledge, and it is widely applied to T-S fuzzy model identification in recent years [2 –4]. The commonly used clustering algorithms in T-S fuzzy model identification mainly includes fuzzy c-means (FCM), Gustafson-Kessel (G-K), Gath-Geva (G-G), subtractive clustering and so on. The above fuzzy clustering algorithms are hyper-sphere-shaped clustering for the sample data distance is Euclidean distance, which indicates that the input-output data in one clustering belongs to the same hypersphere shaped. However, T-S fuzzy model is described by linear functions, which means that the input-output data in one clustering should be sharing the same functional relationship. Then fuzzy c-regression model (FCRM) clustering [5], whose distance is point to plane, is proposed for T-S fuzzy model identification [6 –9]. The premise membership function of T-S fuzzy model identified by FCRM clustering algorithm is bell-shaped, like Gaussian membership function. In order to modeling T-S fuzzy model more accurate, Li proposed a novel hyper-plane-shaped membership function [10, 11], which is more suitable for T-S fuzzy model identification when adapting FCRM clustering algorithm.

For the membership degree of tradition fuzzy sets (type-1 fuzzy sets) is crisp value, and it can’t deal with uncertainties of real applications, like measurement noise of input and output data, different person may define different membership functions, nonlinearity of actuators, sensor drift and so on. With the development of fuzzy sets, type-2 fuzzy sets have been concerned in recent years. The concept of type-2 fuzzy sets was firstly proposed by Zadeh [12]. Type-2 fuzzy sets are defined by primary membership function and secondary membership function. And it contains a type reduction procedure compared with type-1 fuzzy sets. Type reduction is a complex operation and limited the application of type-2 fuzzy sets for a long time. Mendel and his team simplified the secondary membership function of type-2 fuzzy sets to 1, defined interval type-2 fuzz sets [13], which reduced the complexity of type reduction. With the theories development of type-2 fuzzy sets, some efficient type reduction algorithms for general type-2 fuzzy sets have been studied [14]. Based on these efficient type reduction algorithms, type-2 fuzzy logic system has been applied in many fields, such as chaotic time series prediction [15], fuzzy controlling [16], clustering, classification and pattern recognition [17], neuro-fuzzy systems [18], fuzzy model identification [19] and so on. Also, type-2 fuzzy clustering has been applied in T-S fuzzy model identification, such as interval type-2 FCRM (IT2-FCRM) clustering [20, 21], interval type-2 fuzzy c-means [22]. Zou introduced a hyper-plane-shaped membership function in premise parts of T-S fuzzy model when applied IT2-FCRM clustering for T-S fuzzy model identification procedure [23, 24].

In order to overcome the shortcoming of FCM that it didn’t have bounded support or decay rapidly when producing membership functions, Höppner proposed a modified FCM clustering by a new objective function and rewarded more crisp membership degrees [25]. Zhu extended the new objective function and proposed a generalized fuzzy c-means clustering algorithm [26]. In this paper, an improved interval type-2 fuzzy c-regression model clustering is proposed for T-S fuzzy model identification. The main contributions of this paper mainly contains the following twofold. The first contribution is the proposal of an improved interval type-2 fuzzy c-regression model clustering, the improved IT2-FCRM introduces a new objective function in [26]. The new objective function is applied in tradition type-1 fuzzy c-regression model (T1-FCRM) clustering and the converge of this new T1-FCRM will be proofed. Furthermore, this new T1-FCRM will extend to IT2-FCRM by 2 weighting exponents in the membership degree. And the improved IT2-FCRM is applied in T-S fuzzy model primary parameters identification. The secondary contribution is the application of a hyper-plane-shaped membership function, which is more coincidence with the consequent parts of T-S fuzzy model.

The rests of this paper are organized as follows: Section 2 is the definition of type-2 T-S fuzzy model. Section 3 describes general IT2-FCRM clustering algorithm. Section 4 proposes the improved IT2-FCRM clustering algorithm and T-S fuzzy model identification procedure. Section 5 presents the simulation results by the proposed identification algorithm and section 6 draws the conclusions and future works.

2 Type-2 T-S fuzzy model

T-S fuzzy model is defined by several if-then fuzzy rules, the i-th rule is shown as follow: $\begin{matrix} R^{i} : if x_{1} is A_{1}^{i} and \dots and x_{M} is A_{M}^{i} \end{matrix}$ then $y_{i} = p_{1}^{i} x_{1} + \dots + p_{M}^{i} x_{M} + p_{0}^{i}$

and $y = \frac{\sum_{i = 1}^{c} w^{i} y_{i}}{\sum_{i = 1}^{c} w^{i}}$ , $w^{i} = \prod_{j = 1}^{M} μ_{j}^{i} (x_{j})$

where i = 1,2, . . . ,c, c is the number of fuzzy rules, x_j(j = 1,2, . . . ,M) are the fuzzy model inputs, y_i is the i-th fuzzy rule output, $A_{j}^{i}$ is the fuzzy subset defined in the discourse of universe for x_j and $p_{j}^{i}$ is the consequent parameter of the i-th fuzzy rule, $μ_{j}^{i} (x_{j})$ is the membership degree of x_j in fuzzy subset $A_{j}^{i}$ , $\prod_{j = 1}^{M}$ is fuzzy operator, usually be min or product operator.

Type-2 T-S model shares the same forms with type-1, it contains 3 conditions:

Premise fuzzy subset ${\tilde{A}}_{j}^{i}$ is type-2 fuzzy set, consequent parameters of linear function are crisp vales, abbreviated as A2-C0: $\begin{matrix} R^{i} : if x_{1} is {\tilde{A}}_{1}^{i} and \dots and x_{M} is {\tilde{A}}_{M}^{i} \end{matrix}$

then $y_{i} = p_{1}^{i} x_{1} + \dots + p_{M}^{i} x_{M} + p_{0}^{i}$

Premise fuzzy subset ${\tilde{A}}_{j}^{i}$ is type-2 fuzzy set, consequent parameters of linear function are type-1 fuzzy sets, abbreviated as A2-C1: $\begin{matrix} R^{i} : if x_{1} is {\tilde{A}}_{1}^{i} and \dots and x_{M} is {\tilde{A}}_{M}^{i} \end{matrix}$

then $y_{i} = {\tilde{p}}_{1}^{i} x_{1} + \dots + {\tilde{p}}_{M}^{i} x_{M} + {\tilde{p}}_{0}^{i}$

where, ${\tilde{p}}_{j}^{i}$ is an interval set of the form as follow: $\begin{matrix} {\tilde{p}}_{j}^{i} = [c_{j}^{i} + s_{j}^{i}, c_{j}^{i} - s_{j}^{i}] \end{matrix}$

Premise fuzzy subset $A_{j}^{i}$ is type-1 fuzzy set, consequent parameters of linear function are type-1 fuzzy sets, abbreviated as A1-C1: $\begin{matrix} R^{i} : if x_{1} is A_{1}^{i} and \dots and x_{M} is A_{M}^{i} \end{matrix}$

then $y_{i} = {\tilde{p}}_{1}^{i} x_{1} + \dots + {\tilde{p}}_{M}^{i} x_{M} + {\tilde{p}}_{0}^{i}$

In this paper, A2-C0 type-2 T-S fuzzy model is applied for nonlinear system identification.

3 Interval type-2 fuzzy c-regression model

For N pairs of sample data (x_k, y_k), FCRM clustering classifies the data into c regression functions and the i-th regression function is shown as (1).

$\begin{matrix} y_{i} = & ω_{1}^{i} x_{1} + . . . + ω_{M}^{i} x_{M} + ω_{0}^{i} \\ = [x^{T}, 1] ω_{i}, i = 1, 2, . . ., c \end{matrix}$ (1)

In (1), x^T = [x₁, x₂, . . . , x_M], 2≤c≤N,c is clustering number, and $ω_{i} = [ω_{1}^{i}, . . ., ω_{M}^{i}, ω_{0}^{i}]^{T}$ .

The task of FCRM is to minimize the objective function as (2).

$J (U, ω) = \sum_{k = 1}^{N} \sum_{i = 1}^{c} u_{ik}^{m} d_{ik}^{2}$ (2)

In (2), m is weighting exponent, usually is 2, u_ik is the degree of input-output data (x_k, y_k) belonging to the i-th class, and satisfies the following constrains.

$\sum_{i = 1}^{c} u_{ik} = 1, 0 < \sum_{k = 1}^{N} u_{ik} < N$ (3)

d_ik in (2) is the distance of (x_k, y_k) to clustering plane ω_i, which is denoted as (4).

$d_{ik} (ω_{i}) = | {\hat{x}}_{k} ω_{i} - y_{k} |$ (4)

In (4), ${\hat{x}}_{k} = [x_{k}^{T}, 1]$ , k = 1,2, . . . ,N.

The FCRM clustering iterative equations are shown as (5) and (6).

$u_{ik} = \frac{(1 / d_{ik} (ω_{i}))^{\frac{2}{m - 1}}}{\sum_{j = 1}^{c} (1 / d_{jk} (ω_{j}))^{\frac{2}{m - 1}}}$ (5)

$ω_{i} = [X^{T} W_{i} X]^{- 1} X^{T} W_{i} Y$ (6)

In (6): $\begin{matrix} X = [\begin{matrix} x_{1}^{T}, 1 \\ x_{2}^{T}, 1 \\ ⋮ \\ x_{N}^{T}, 1 \end{matrix}] \in R^{N \times (M + 1)}, Y = [\begin{matrix} y_{1} \\ y_{2} \\ ⋮ \\ y_{N} \end{matrix}] \in R^{N}, \end{matrix}$ $\begin{matrix} W_{i} = [\begin{matrix} u_{i 1} & 0 & \dots & 0 \\ 0 & u_{i 2} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & u_{iN} \end{matrix}] \in R^{N \times N} \end{matrix}$

IT2-FCRM clustering contains 2 weighting exponents in the objective function, m1 and m2, and there exits 2 objective functions to be minimized as (7).

${\begin{matrix} J_{m 1} (U, ω) = \sum_{k = 1}^{N} \sum_{i = 1}^{c} u_{ik}^{m 1} d_{ik}^{2} \\ J_{m 2} (U, ω) = \sum_{k = 1}^{N} \sum_{i = 1}^{c} u_{ik}^{m 2} d_{ik}^{2} \end{matrix}$ (7)

The upper and lower membership degrees can be represented as (8) and (9).

$\begin{matrix} {\bar{u}}_{ik} = & max (\frac{(1 / d_{ik} (ω_{i}))^{\frac{2}{m 1 - 1}}}{\sum_{j = 1}^{c} (1 / d_{jk} (ω_{j}))^{\frac{2}{m 1 - 1}}}, \\ \frac{(1 / d_{ik} (ω_{i}))^{\frac{2}{m 2 - 1}}}{\sum_{j = 1}^{c} (1 / d_{jk} (ω_{j}))^{\frac{2}{m 2 - 1}}}) \end{matrix}$ (8)

$\begin{matrix} {\underline{u}}_{ik} = & min (\frac{(1 / d_{ik} (ω_{i}))^{\frac{2}{m 1 - 1}}}{\sum_{j = 1}^{c} (1 / d_{jk} (ω_{j}))^{\frac{2}{m 1 - 1}}}, \\ \frac{(1 / d_{ik} (ω_{i}))^{\frac{2}{m 2 - 1}}}{\sum_{j = 1}^{c} (1 / d_{jk} (ω_{j}))^{\frac{2}{m 2 - 1}}}) \end{matrix}$ (9)

The average of ${\bar{u}}_{ik}$ and ${\underline{u}}_{ik}$ will be used to update ω by (6), where u_ik in W_i is $({\bar{u}}_{ik} + {\underline{u}}_{ik}) / 2$ .

Gaussian function is applied in premise membership function for T-S fuzzy model, and the upper and lower Gaussian membership functions are defined as (10) and (11).

${\bar{μ}}_{j}^{i} (x_{j}) = exp (- (\frac{x_{j} - {\bar{ν}}_{j}^{i}}{{\bar{σ}}_{j}^{i}})^{2})$ (10)

In (10): $\begin{matrix} {\bar{ν}}_{j}^{i} = \frac{\sum_{k = 1}^{N} {\bar{u}}_{ik} x_{kj}}{\sum_{k = 1}^{N} {\bar{u}}_{ik}}, {\bar{σ}}_{j}^{i} = \sqrt{\frac{2 \sum_{k = 1}^{N} {\bar{u}}_{ik} (x_{kj} - {\bar{ν}}_{j}^{i})^{2}}{\sum_{k = 1}^{N} {\bar{u}}_{ik}}} \end{matrix}$

${\underline{μ}}_{j}^{i} (x_{j}) = exp (- (\frac{x_{j} - {\underline{ν}}_{j}^{i}}{{\underline{σ}}_{j}^{i}})^{2})$ (11)

In (11): $\begin{matrix} {\underline{ν}}_{j}^{i} = \frac{\sum_{k = 1}^{N} {\underline{u}}_{ik} x_{kj}}{\sum_{k = 1}^{N} {\underline{u}}_{ik}}, {\underline{σ}}_{j}^{i} = \sqrt{\frac{2 \sum_{k = 1}^{N} {\underline{u}}_{ik} (x_{kj} - {\underline{ν}}_{j}^{i})^{2}}{\sum_{k = 1}^{N} {\underline{u}}_{ik}}} \end{matrix}$

4 Proposed type-2 T-S fuzzy model identification algorithm

4.1 New objective function

An improved fuzzy partitions by modifying the objective function of FCM was proposed in [25], which is shown as (12).

$J = \sum_{i = 1}^{c} \sum_{k = 1}^{n} u_{ik}^{2} d_{ik}^{2} - \sum_{i = 1}^{c} {\sum_{k = 1}^{n} a_{k} (u_{ik} - \frac{1}{2})}^{2}$ (12)

The first part of (12) is the objective function of FCM and weighting exponent m is 2. More generally, Zhu extended this fuzzy partitions and proposed a general objective function as (13) [26].

$J = \sum_{i = 1}^{c} \sum_{k = 1}^{N} u_{ik}^{m} d_{ik}^{2} + \sum_{k = 1}^{N} a_{k} \sum_{i = 1}^{c} u_{ik} (1 - u_{ik}^{m - 1})$ (13)

The degree u_ik can be updated by (14).

$u_{ik} = \frac{(\frac{1}{d_{ik}^{2} - a_{k}})^{\frac{1}{m - 1}}}{\sum_{j = 1}^{c} (\frac{1}{d_{jk}^{2} - a_{k}})^{\frac{1}{m - 1}}}$ (14)

To ensure $d_{ik}^{2} - a_{k} > 0$ , define:

$a_{k} = α \times min d_{* k}^{2} = α \times min {d_{ik}^{2} | i \in {1, . . ., c}}$ (15) where 0≤α<1, so (14) can be rewrote as (16).

$u_{ik} = \frac{(\frac{1}{d_{ik}^{2} - α \times min d_{* k}^{2}})^{\frac{1}{m - 1}}}{\sum_{j = 1}^{c} (\frac{1}{d_{jk}^{2} - α \times min d_{* k}^{2}})^{\frac{1}{m - 1}}}$ (16)

If α= 0, then the new objective function (13) is the same as (2), so the objective function of FCM is a special case of this new objective function.

In this paper, this new objective function is applied in FCRM clustering, and next, the convergence of this new FCRM clustering algorithm will be discussed.

Define, U = [u_ik] _c×N

Firstly, U is a local minimum of (13), if u_ik is calculated by (16).

Proof:

This theorem can be proved by the derivation of u_ik and calculate the H(U), the Hessian of the Lagrangian of J as (17).

$\begin{matrix} J_{L} = & \sum_{i = 1}^{c} \sum_{k = 1}^{N} u_{ik}^{m} d_{ik}^{2} + \sum_{k = 1}^{N} a_{k} \sum_{i = 1}^{c} u_{ik} (1 - u_{ik}^{m - 1}) \\ - \sum_{k = 1}^{N} λ_{k} (\sum_{i = 1}^{c} (u_{ik} - 1)) \end{matrix}$ (17)

According to (17), H(U) can be computed by (18).

$\begin{matrix} h_{ts, ik} (U) = \frac{\partial}{\partial u_{ts}} (\frac{\partial J_{L}}{\partial u_{ik}}) \\ = {\begin{matrix} m (m - 1) u_{ij}^{m - 2} \sum_{k = 1}^{N} (d_{ik}^{2} - a_{k}) if t = i, s = j \\ 0 else \end{matrix} \end{matrix}$ (18)

Since m > 1 and $(d_{ik}^{2} - a_{k}) > 0$ (indicated from (15)), so $H (U) = m (m - 1) u_{ij}^{m - 2} \sum_{k = 1}^{N} (d_{ik}^{2} - a_{k}) > 0$ and [h_ts,ik (U)] is a diagonal matrix, and the Hessian H (U is positive definite and consequently. Thus (16) is a sufficient condition for minimizing function J_L.

Secondary, in order to minimum the function J_L, the regression function parameters ω_i are obtained by using the weighted least square (WLS), so ω is a local minimum of (13) if ω_i is calculated by (6).

Similar to Bezdek’s proof in [27], by the above information:

$J_{L} (U^{t + 1}, ω^{t + 1}) ⩽ J_{L} (U^{t}, ω^{t})$ (19) so J_L is a decreasing function of t,which indicates that the FCRM clustering algorithm using this new objective function is convergence.

According to (16), if type-1 FCRM extends to interval type-2 FCRM, the upper and lower fuzzy membership degrees can be represented as (20) and (21).

$\begin{matrix} {\bar{u}}_{ik} = & max (\frac{(1 / {\hat{d}}_{ik} (ω_{i}))^{\frac{1}{m 1 - 1}}}{\sum_{j = 1}^{c} (1 / {\hat{d}}_{jk} (ω_{j}))^{\frac{1}{m 1 - 1}}}, \\ \frac{(1 / {\hat{d}}_{ik} (ω_{i}))^{\frac{1}{m 2 - 1}}}{\sum_{j = 1}^{c} (1 / {\hat{d}}_{jk} (ω_{j}))^{\frac{1}{m 2 - 1}}}) \end{matrix}$ (20)

$\begin{matrix} {\underline{u}}_{ik} = & min (\frac{(1 / {\hat{d}}_{ik} (ω_{i}))^{\frac{1}{m 1 - 1}}}{\sum_{j = 1}^{c} (1 / {\hat{d}}_{jk} (ω_{j}))^{\frac{1}{m 1 - 1}}}, \\ \frac{(1 / {\hat{d}}_{ik} (ω_{i}))^{\frac{1}{m 2 - 1}}}{\sum_{j = 1}^{c} (1 / {\hat{d}}_{jk} (ω_{j}))^{\frac{1}{m 2 - 1}}}) \end{matrix}$ (21)

In (20) and (21):

${\hat{d}}_{ik} (ω_{i}) = d_{ik}^{2} - α \times min d_{* k}^{2}$ (22)

4.2 Hyper-plane-shaped membership function

Li proposed a hyper-plane-shaped membership function as (23), which is more suitable to describe fuzzy clustering algorithm whose sample distance is point to plane, like FCRM clustering.

$ω_{i} (x_{k}) = exp (- η \frac{d_{ik} (ω_{i})}{max {d_{jk} (ω_{j}), j = 1, . . ., c}})$ (23)

η is a constant that tunes the fuzzy membership grades, and d_ik( ω_i) is defined as (24).

$d_{ik} (ω_{i}) = \frac{| {\hat{x}}_{k} . ω_{i} |}{∥ ω_{i} ∥}$ (24) where, ${\hat{x}}_{k} = [x^{T}, 1]$ .

Based on upper and lower membership degrees, the upper and lower hyperplanes parameters can be calculated by (25) and (26) [23].

${\bar{ω}}_{i} = [X^{T} {\bar{W}}_{i} X]^{- 1} X^{T} {\bar{W}}_{i} Y$ (25)

In (25): $\begin{matrix} {\bar{W}}_{i} = [\begin{matrix} {\bar{u}}_{i 1} & 0 & \dots & 0 \\ 0 & {\bar{u}}_{i 2} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & {\bar{u}}_{iN} \end{matrix}] \end{matrix}$

${\underline{ω}}_{i} = [X^{T} {\underline{W}}_{i} X]^{- 1} X^{T} {\underline{W}}_{i} Y$ (26)

In (26): $\begin{matrix} {\underline{W}}_{i} = [\begin{matrix} {\underline{u}}_{i 1} & 0 & \dots & 0 \\ 0 & {\underline{u}}_{i 2} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & {\underline{u}}_{i N} \end{matrix}] \end{matrix}$

So the corresponding upper and lower hyper-plane-shaped membership functions can be defined as (27) and (28).

${\bar{w}}_{i} (x_{k}) = exp (- η \frac{d_{ik} ({\bar{ω}}_{i})}{max {d_{jk} ({\bar{ω}}_{j}), j = 1, . . ., c}})$ (27)

In (27): $\begin{matrix} d_{ik} ({\bar{ω}}_{i}) = \frac{| {\hat{x}}_{k} . {\bar{ω}}_{i} |}{∥ {\bar{ω}}_{i} ∥} \end{matrix}$

$\underline{w} (x_{k}) = exp (- η \frac{d_{ik} ({\underline{ω}}_{i})}{max {d_{jk} ({\underline{ω}}_{j}), j = 1, . . ., c}})$ (28)

In (28): $\begin{matrix} d_{ik} ({\underline{ω}}_{i}) = \frac{| {\hat{x}}_{k} . {\underline{ω}}_{i} |}{∥ {\underline{ω}}_{i} ∥} \end{matrix}$

In consequent parameters identification procedure, the membership degree of sample data (x_k, y_k) to i-th fuzzy rule is the average of ${\bar{w}}_{i} (x_{k})$ and ${\underline{ω}}_{i} (x_{k})$ as (29).

$w_{i} (x_{k}) = \frac{{\bar{w}}_{i} (x_{k}) + {\underline{w}}_{i} (x_{k})}{2}$ (29)

4.3 Procedure of T-S fuzzy model identification

Giving N pairs of input-output sample data (x_k, y_k), T-S fuzzy model identification procedure based on improved IT2-FCRM clustering can be summarized as follows:

Set identification parameters: number of clustering c, 2 weighting exponents m1 and m2, adjustment coefficients η and α, maximum iterations r_max and threshold ɛ. Initialize ω⁽⁰⁾ and set r = 1.

Update the upper and lower fuzzy degrees ${\bar{u}}_{ik}^{(r)}$ and ${\underline{u}}_{ik}^{(r)}$ by (20) and (21).

Update the upper and lower hyperplanes parameters ${\bar{ω}}^{(r)}$ and $ω_{-}^{(r)}$ by (25) and (26).

Update $ω^{(r)} = ({\bar{ω}}^{(r)} + {\underline{ω}}^{(r)}) / 2$ , if $∥ {\bar{ω}}^{(r)} - {\bar{ω}}^{(r - 1)} ∥ ⩽ ɛ$ or r = r_max, go to Step 5), otherwise, set ω ^(r - 1) = ω^(r), r = r + 1 and go to Step 2).

For each training data (x_k, y_k), calculate the membership degree ${\bar{w}}_{i} (x_{k})$ and ${\underline{w}}_{i} (x_{k})$ by (27) and (28), and then get the i-th rule membership degree w_i (x_k)by (29),define:

$f_{i} (x_{k}) = \frac{w_{i} (x_{k})}{\sum_{i = 1}^{c} w_{i} (x_{k})}$ (30)

All N pairs of training data can be represented as (31):

$Y = \hat{X} . P$ (31)

Where: $\begin{matrix} Y = [y_{1}, y_{2}, \dots, y_{N}]^{T} \end{matrix}$ $\begin{matrix} \hat{X} = [\begin{matrix} f_{1} (x_{1}) . x_{11} & \dots & f_{1} (x_{1}) . x_{1 M} & f_{1} (x_{1}) & \dots & f_{c} (x_{1}) . x_{11} & \dots & f_{c} (x_{1}) . x_{1 M} & f_{c} (x_{1}) \\ ⋮ \\ f_{1} (x_{k}) . x_{k 1} & \dots & f_{1} (x_{k}) . x_{kM} & f_{1} (x_{k}) & \dots & f_{c} (x_{k}) . x_{k 1} & \dots & f_{c} (x_{k}) . x_{kM} & f_{c} (x_{k}) \\ ⋮ \\ f_{1} (x_{N}) . x_{N 1} & \dots & f_{1} (x_{N}) . x_{NM} & f_{1} (x_{N}) & \dots & f_{c} (x_{N}) . x_{N 1} & \dots & f_{c} (x_{N}) . x_{NM} & f_{c} (x_{N}) \end{matrix}] \end{matrix}$ $\begin{matrix} P = [p_{1}^{1} \dots p_{M}^{1} p_{0}^{1} \dots p_{1}^{c} \dots p_{M}^{c} p_{0}^{c}]^{T} \end{matrix}$

The optimal estimation of P can be obtained by least square (LS) or orthogonal least squares (OLS), in this paper, OLS is applied.

5 Simulations

In this section, some nonlinear systems are tested to verify the identification effects of proposed algorithm. The mean square error (MSE) and root mean square error (RMSE) will be used as the performance indexes when compared with other methods, which are defined as: $\begin{matrix} J_{MSE} = \frac{\sum_{k = 1}^{N} (y_{k} - {\hat{y}}_{k})^{2}}{N} \end{matrix}$ $\begin{matrix} J_{RMSE} = \sqrt{\frac{\sum_{k = 1}^{N} (y_{k} - {\hat{y}}_{k})^{2}}{N}} \end{matrix}$ where, y_k is the original output and ${\hat{y}}_{k}$ is the T-S fuzzy model output, N is the number of training or testing data.

5.1 One-dimensional (1-D) sin function

In this example, a 1-D sin function is defined as follow: $\begin{matrix} y = \frac{sin x}{x}, x \in [- 40, 0) \cup (0, 40] \end{matrix}$

The input of fuzzy model is x, and the fuzzy model is $\hat{y} = f (x)$ .

The sample data points are uniformly sampled between [–40, 0)∪ (0,40], and the identification parameters are c = 4, m1 = 2 m2 = 7, η= 50, α= 0.99. Table 1 lists the performance comparison with other methods and Table 2. shows the premise and consequent identification parameters for one-dimensional sin function.

Table 1
Performance comparison of different identification methods (One-dimensional sin function)

Method No. of rules MSE

Zarandi [20] 4 0.02385

Zou [23] 4 0.0077

Dam [28] 4 0.022

Proposed Method 4 0.0041

Method	No. of rules	MSE
Zarandi [20]	4	0.02385
Zou [23]	4	0.0077
Dam [28]	4	0.022
Proposed Method	4	0.0041

Table 2

Identification parameters of One-dimensional sin function

Rules	Parameters	x	1
Rule 1	${\bar{ω}}_{1}$	–0.0000461	–0.0276
	${\underline{ω}}_{1}$	–0.0000447	–0.0274
	p ₁	4.4841	100.4865
Rule 2	${\bar{ω}}_{2}$	–0.005337	–0.1706
	${\underline{ω}}_{2}$	–0.0053 38	–0.1705
	p ₂	–0.0012	–0.0318
Rule 3	${\bar{ω}}_{3}$	–0.0006131	0.0570
	${\underline{ω}}_{3}$	–0.0006067	0.0567
	p ₃	–0.0001	–0.0031
Rule 4	${\bar{ω}}_{4}$	–0.000023	0.7054
	${\underline{ω}}_{4}$	–0.000000	0.7037
	p ₄	1.8405	–95.6399

5.2 Double input-Single output static function

In this example, a double inputs and single output (DISO) nonlinear function is tested, which is shown as follow: $\begin{matrix} y = (1 + x_{1}^{- 2} + x_{2}^{- 1.5})^{2}, 1 ⩽ x_{1}, x_{2} ⩽ 5 \end{matrix}$

The input of fuzzy model is x₁ and x₂, and the fuzzy model is $\hat{y} = f (x_{1}, x_{2})$ .

The 50 training data are collected from [29], and the identification parameters are c = 6, m1 = 2, m2 = 7, η= 20, α= 0.99. Figure 1 shows the output curves of T-S fuzzy model and original system and Fig. 2 shows the errors curve.

Fig. 1

Fuzzy model and original output curves (DISO function).

Fig. 2

Fuzzy model and original output errors curve (DISO function).

Table 3 lists the performance comparison with other methods and Table 4 shows the premise and consequent identification parameters for DISO function.

Table 3

Performance comparison of different identification methods (DISO function)

Method	No. of rules	MSE
Li [6]	6	0.0085
Li [10]	6	0.0018
Zarandi [20]	4	0.0715
Zou [23]	4	0.0019
Li [30]	6	0.0050
Cheung [31]	6	0.0064
Lin [32]	3	0.0519
Tsai [33]	4	0.0024
Qiao [34]	4	0.0053
Tsai [35]	4	0.0027
Li [36]	6	0.0071
Singh [37]	6	0.0020
Tsai [38]	4	0.0015
Proposed Method	4/6	0.0012/5.27×10^-4

Table 4

Identification parameters of DISO function (c = 6)

Rules	Parameters	x ₁	x ₂	1
Rule 1	${\bar{ω}}_{1}$	–0.8174	–0.9022	7.5767
	${\underline{ω}}_{1}$	–0.8172	–0.9025	7.5771
	p ₁	–0.3057	–0.0887	2.8844
Rule 2	${\bar{ω}}_{2}$	–0.5267	–0.3727	5.3048
	${\underline{ω}}_{2}$	–0.5267	–0.3729	5.3063
	p ₂	251.9757	–138.7129	–1110.8
Rule 3	${\bar{ω}}_{3}$	–0.2585	–0.3882	3.8923
	${\underline{ω}}_{3}$	–0.2584	–0.3886	3.8930
	p ₃	–176.3847	240.0815	–819.1094
Rule 4	${\bar{ω}}_{4}$	–0.5334	–0.5309	6.2476
	${\underline{ω}}_{4}$	–0.5332	–0.5312	6.2483
	p ₄	–138.9578	–227.8815	2036.9
Rule 5	${\bar{ω}}_{5}$	–0.6986	–0.6400	6.8574
	${\underline{ω}}_{5}$	–0.6987	–0.6401	6.8585
	p ₅	–3.4633	–7.5119	52.9808
Rule 6	${\bar{ω}}_{6}$	–0.6797	–0.5930	5.6803
	${\underline{ω}}_{6}$	–0.6801	–0.5935	5.6826
	p ₆	0.1898	–0.1975	1.4944

5.3 Mackey-Glass chaotic system

In this example, the Mackey-Glass (M-G) chaotic time series are tested. The chaos equation is shown as follow: $\begin{matrix} x (t + 1) = 0.9 x (t) + \frac{0.2 x (t - 17)}{1 + x^{10} (t - 17)} \end{matrix}$

The initial value is set to 1.2 from x(0) to x(17). The 1000 pairs chaotic data are selected between t = 124 and t = 1123. The fuzzy model can be expressed as: $\begin{matrix} \hat{x} (t + 1) = f (x (t - 1), x (t - 2), x (t - 3), x (t - 4)) \end{matrix}$

The first 800 pairs of data are used to build T-S fuzzy model, the left 200 pairs data are used to test the fuzzy system. And the identification parameters are c = 2, m1 = 1.5, m2 = 7, η= 20, α= 0.99. Figure 3 shows the output curves of T-S fuzzy model and original system and Fig. 4 shows the errors curve.

Fig. 3

Fuzzy model and original output curves (M-G chaos data).

Fig. 4

Fuzzy model and original output errors curve (M-G chaos data).

Table 5 lists the performance comparison with other methods and Table 6 shows the premise and consequent identification parameters for M-G chaos data.

Table 5

Performance comparison of different identification methods (M-G chaos data)

Method	No. of rules	RMSE
Zou [23]	10	0.0224
Eyoh [39]	16	0.0040
Zhan [40]	6	0.067
Chao [41]	14	0.0037
Wei [42]	6	0.0038
Lv [43]	2	0.0065
Tsai [44]	10	0.0129
Proposed Method	2	0.0030

Table 6

Identification parameters of M-G chaos data

Rules	Parameters	x(k-1)	x(k-2)	x(k-3)	x(k-4)	1
Rule 1	${\bar{ω}}_{1}$	9.1208	–17.8875	13.3928	–3.6394	0.0121
	${\underline{ω}}_{1}$	9.1738	–18.0125	13.4877	–3.6622	0.0121
	p ₁	–125.0418	259.1002	–174.9520	34.2881	0.6008
Rule 2	${\bar{ω}}_{2}$	8.0126	–15.6187	12.1529	–3.5732	0.0251
	${\underline{ω}}_{2}$	8.0137	–15.6656	12.2468	–3.6224	0.0262
	p ₂	–58.7738	79.9102	350.5781	–312.8170	10.8219

5.4 Box-Jenkins gas-furnace process

In this example, a gas-furnace process contains 296 Box-Jenkins data sets are tested. The input of this actual system is the gas flow rate and the output of this system is carbon dioxide concentration.

The fuzzy model can be expressed as: $\begin{matrix} \hat{y} (t) = & f (u (t), u (t - 1), u (t - 2), y (t - 1), \\ y (t - 2), y (t - 3)) \end{matrix}$

The first 148 pairs of data are used to build T-S fuzzy model, the left 148 pairs of data are used to test the fuzzy system. And the identification parameters are c = 2, m1 = 2, m2 = 7, η= 5, α= 0.99. Figure 5 shows the output curves of T-S fuzzy model and original system and Fig. 6 shows the errors curve.

Fig. 5

Fuzzy model and original output curves (Box-Jenkins data).

Fig. 6

Fuzzy model and original output errors curve (Box-Jenkins data).

Table 7 lists the performance comparison with other methods when the total 296 pairs of data are used for training and Table 8 shows the premise and consequent identification parameters for Box-Jenkins data.

Table 7

Performance comparison of different identification methods (Box-Jenkins data: 296 training data)

Method	No. of rules	MSE
Li [6]	4	0.0498
Li [7]	3	0.056
Zhao [8]	3/4	0.0519/0.0483
Li [30]	3	0.0534
Lv [43]	2	0.0545
Dan [45]	3	0.0748
Yan [46]	3	0.0568
Ren [47]	4	0.0431
Proposed Method	2/3/4	0.0513/0.0465/0.0425

Table 8

Identification parameters of Box-Jenkins data(c = 4)

Rules	Parameters	u(k)	u(k-1)	u(k-2)	y(k-1)	y(k-2)	y(k-3)	1
Rule 1	${\bar{ω}}_{1}$	0.1521	–0.2808	–0.1425	1.9741	–1.2560	0.2319	2.6965
	${\underline{ω}}_{1}$	0.1540	–0.2848	–0.1408	1.9752	–1.2573	0.2321	2.6946
	p ₁	81883.37	45217.69	–85262.68	475648.36	–345303.49	73443.38	–858447.88
Rule 2	${\bar{ω}}_{2}$	0.1846	0.0153	–0.7447	1.7903	–1.5101	0.5581	8.6611
	${\underline{ω}}_{2}$	0.1853	0.0151	–0.7459	1.7901	–1.5110	0.5589	8.6785
	p ₂	9623.93	–15210.20	4325.84	23948.63	–21926.96	7470.97	–3046.37
Rule 3	${\bar{ω}}_{3}$	–0.7626	1.2812	–1.2895	1.1012	–0.2880	–0.0585	12.9583
	${\underline{ω}}_{3}$	–0.7630	1.2805	–1.2901	1.0999	–0.2869	–0.0586	12.9788
	p ₃	9353.21	–22593.11	16878.71	–4642.52	–2055.02	2375.47	–79224.94
Rule 4	${\bar{ω}}_{4}$	–0.8366	2.7369	–2.4110	1.3033	–0.6265	0.1133	11.181
	${\underline{ω}}_{4}$	–0.8382	2.7401	–2.4125	1.3017	–0.6245	0.1128	11.1832
	p ₄	–3306.57	1330.118	430.29	–13219.30	10076.13	–2299.82	33018.06

Table 9 lists the performance comparison with other methods when the 148 pairs of data are used for training and the left 148 pairs of data are used for testing.

Table 9

Performance comparison of different identification methods (Box-Jenkins data: 148 training data)

Method	No. of rules	Training MSE	Testing MSE
Li [7]	3	0.0159	0.1255
Li [11]	3	0.0149	0.1324
Li [30]	3	0.0150	0.1470
Cheung [31]	3	0.059	0.152
Lv [43]	2/3	0.0152/0.0150	0.1707/0.1567
Dam [45]	3	0.0291	0.138
Yan [46]	3	0.0168	0.1402
Luo [48]	2	0.0254	0.1243
Feng [49]	–	0.0211	0.1210
Han [50]	–	0.1484	0.3307
Zhang [51]	–	0.0196	0.1369
Proposed Method	2/3	0.0161/0.0149	0.1186/0.1098

5.5 CFB boiler bed temperature system

In this example, the bed temperature data of 2×50 MW circulating fluidized bed (CFB) plant was used. The bed temperature of CFB plant is mainly effected by coal feed quantity, primary air flow and secondary air flow. In this case, 1800 pairs of operation data are collected and the sample time is 1 s. The fuzzy model can be expressed as: $\begin{matrix} \hat{y} (k) = & f (y (k - 1), y (k - 2), u_{c} (k - 5), \\ u_{c} (k - 6), u_{p} (k - 1), u_{s} (k - 1)) \end{matrix}$ where y is bed temperature, u_c is coal feed quantity, u_p is primary air flow and u_s is secondary air flow. Figures 7 8 show the bed temperature, coal feed quantity, primary air flow and secondary air flow collected from 2×50 MW CFB plant.

Fig. 7

Bed temperature and coal feed quantity.

Fig. 8

Primary air flow and secondary air flow.

And the identification parameters are c = 2, m1 = 2, m2 = 7, η= 20, α= 0.99. Figure 9 shows the output curves of T-S fuzzy model and original system and Fig. 10 shows the errors curve.

Fig. 9

Fuzzy model and original output curves (CFB data).

Fig. 10

Fuzzy model and original output errors curve (CFB data).

Table 10 lists the performance comparison with other methods when the total 1400 pairs of data are used for training and Table 11 shows the premise and consequent identification parameters for CFB data.

Table 10

Performance comparison of different identification methods (CFB data: 1400 training data)

Method	No. of rules	Training MSE	Testing MSE
Zhang [52]	–	0.1286	0.0483
Shi [53]	2	0.0240	0.0293
Proposed Method	2	0.0242	0.0276

Table 11

Identification parameters of CFB data

Rules	Parameters	y(k-1)	y(k-2)	u_c(k-5)	u_c(k-6)	u_p(k-1)	u_s(k-1)	1
Rule 1	${\bar{ω}}_{1}$	1.0446	–0.0477	–0.0293	0.0816	–0.0000237	0.000141	–14.3680
	${\underline{ω}}_{1}$	1.0436	–0.0467	–0.0276	0.0806	–0.0000235	0.000142	–14.5261
	p ₁	24.8142	37.2456	50.4573	50.0354	0.006017	0.04426	3395.5451
Rule 2	${\bar{ω}}_{2}$	0.6327	0.3615	0.0423	–0.0479	0.0001591	–0.000124	16.3312
	${\underline{ω}}_{2}$	0.6304	0.3638	0.0404	–0.0465	0.0001588	–0.000126	16.5256
	p ₂	4.6417	6.8546	5.2026	5.6051	0.0008667	0.006206	65.9385

6 Conclusions

For type-2 fuzzy logic system can handling uncertainties more robust than type-1, in this paper, an improved IT2-FCRM clustering is proposed for T-S fuzzy model identification. The improved IT2-FCRM clustering algorithm introduced a novel membership constraint function and utilized a new objective function, which makes the IT2-FCRM less sensitive to outliers or noisy data and can classify sample data more accurate. As the distance of FCRM is point to plane, compared with the tradition hyper-sphere-shaped membership function, the proposed IT2-FCRM applies a hyper-plane-shaped membership function, which is more suitable to describe consequent linear expressions of T-S fuzzy model.

The premise parameters of T-S fuzzy model are upper and lower hyperplanes executed by proposed IT2-FCRM clustering algorithm. Then the upper and lower hyperplanes will define hyper-plane-shaped type-2 Gaussian membership functions and the consequent parameters of T-S fuzzy model are determined by OLS algorithm. The simulation results show the effectiveness of proposed T-S fuzzy model identification algorithm.

The future works mainly focus the following 2 aspects:

There are 4 parameters to determine the identification accuracy of T-S fuzzy model, that are m1, m2, η and α. In this paper, these parameters are trialed, and the next step is to optimize these 4 parameters by some intelligent optimization algorithms.

As the development of theories and applications of general type-2 fuzzy sets, another research will focus on general type-2 fuzzy c-regression model clustering applied in T-S fuzzy model identification.

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T-S fuzzy model identification based on an improved interval type-2 fuzzy c-regression model

Abstract

Keywords

1 Introduction

2 Type-2 T-S fuzzy model

3 Interval type-2 fuzzy c-regression model

4.1 New objective function

5.1 One-dimensional (1-D) sin function

Table 1 Performance comparison of different identification methods (One-dimensional sin function) Method No. of rules MSE Zarandi [20] 4 0.02385 Zou [23] 4 0.0077 Dam [28] 4 0.022 Proposed Method 4 0.0041

References

Table 1
Performance comparison of different identification methods (One-dimensional sin function)

Method No. of rules MSE

Zarandi [20] 4 0.02385

Zou [23] 4 0.0077

Dam [28] 4 0.022

Proposed Method 4 0.0041