Fuzzy inference modeling method based on T-S fuzzy system

Abstract

In this paper, a kind of fuzzy inference modeling method based on T-S fuzzy system is proposed. Firstly, the T-S fuzzy inference rules “If x is A_i and y is B_j, then $z = c_{ij}^{0} + c_{ij}^{1} x + c_{ij}^{2} y$ ” are transformed into a fuzzy system $z = \sum_{k = i}^{i + 1} \sum_{l = j}^{j + 1} A_{k} (x) B_{l} (y) z_{kl}$ . Secondly, the variants (y, $\dot{y}$ ) and $\ddot{y}$ in the second-order freedom movement systems are considered as input variants and output variant respectively, then we obtain a nonlinear differential equation model with variable coefficients. Thirdly, this method is applied to build the mathematical modeling for time-invariant and time-varying second-order freedom movement systems, and new input-output models and new state-space models are obtained respectively. Finally, we have given the approximation accuracy of the constructed differential equation model and shown the effectiveness and superiority of presented method through simulation experiments and comparing the maximum errors.

Keywords

Fuzzy control fuzzy inference modeling method T-S fuzzy system input-output model state-space model

1 Introduction

It is well known that the theory of automatic control is mainly based on mathematical models of systems which are often represented as differential equations [1]. By using these models, we can design control scheme and analyze system properties such as stability and controllability. Fuzzy control is suitable for the control systems with fuzzy environment being hard to model [1].

The structure of fuzzy system is one of the important problems of fuzzy system. Fuzzy system mainly consists of fuzzification, fuzzy inference and defuzzification [1 –9]. Great progress, both in theory and in application, has been made in the fuzzy system. Li revealed that the commonly used fuzzy controller can be seen as some kinds of interpolation algorithms [10]. In another paper, Li put forward a new kind of modeling method for fuzzy system [11]. The method presented by Li transformed fuzzy inference rule base into nonlinear differential equation with variable coefficients. Furthermore, Li realized time-varying system modeling and also got a mathematical model in the form of differential equations through using this method together with taking into account time factor [12]. As a generalization of method presented by Li, Yuan et al. gave a fuzzy inference modeling method based on fuzzy transformation [13].

In recent years, T-S fuzzy system has drawn broad attention of scholars. As we know, in this system, the rule consequent is represented by linear or nonlinear relationship between input and output, so each rule contains more system information such that few rules are needed to express system information. Moreover, this provides a crisp output that need not to be defuzzified, which makes the technique computationally efficient for describing nonlinear systems [1, 14]. Therefore, T-S system has been recognized as one of the successful tools to handle complex nonlinear systems. T-S fuzzy models enjoy great advantages in stability analysis and controller design for nonlinear systems because of relatively simple and fixed structure, and universal function approximation capability [15 –20].

In this paper, we present a fuzzy inference modeling method based on T-S fuzzy system. We first transform the T-S fuzzy inference rules “If x is A_i and y is B_j, then $z = c_{ij}^{0} + c_{ij}^{1} x + c_{ij}^{2} y$ ” into a fuzzy system $z = \sum_{k = i}^{i + 1} \sum_{l = j}^{j + 1} A_{k} (x) B_{l} (y) z_{kl}$ , then we obtain a nonlinear differential equation model with variable coefficients by considering the variants (y, $\dot{y}$ ) and $\ddot{y}$ in the second-order freedom movement systems as input variants and output variant respectively. We apply this method to time-invariant and time-varying second-order freedom movement systems modeling, and establish new input-output models and new state-space models respectively. Moreover, we give the approximation accuracy of the constructed differential equation model and present some examples to show the effectiveness and superiority of this method through comparative simulation experiments on approximation property.

2 Preliminaries

Takagi and Sugeno first proposed T-S fuzzy system [14], this system is essentially a nonlinear model that can effectively deal with complex multi-variable nonlinear system. Unlike Mamdani system, the rule consequent of the T-S fuzzy system is represented by a linear or nonlinear relationship between input and output so that more information can be contained in each rule. Therefore, the use of fewer rules can achieve the demanded control effect, which means that the controller can be relatively simple [1 , 14–20]. The fuzzy rule base of T-S fuzzy system comprises of a collection of fuzzy if-then rules in the following form:

$\begin{matrix} If x is A_{i} and y is B_{j} then z = c_{ij}^{0} + c_{ij}^{1} x + c_{ij}^{2} y \\ (i = 1, 2, \dots, m; j = 1, 2, \dots n), \end{matrix}$ (1) where A_i (i = 1, ⋯ , m) and B_j (j = 1, ⋯ , n) in (1) are both fuzzy sets with triangular membership functions. The fuzzy rules (1) can be transformed a fuzzy system as follows [1, 14] $z = f (x, y) = \sum_{k = i}^{i + 1} \sum_{l = j}^{j + 1} A_{k} (x) B_{l} (y) z_{kl}$ (2)

The coefficients $c_{ij}^{0}$ , $c_{ij}^{1}$ and $c_{ij}^{2}$ in (1) can be defined as follows:

If g ∈ C²([a₁, b₁] × [a₂, b₂]), then g (x_i, y_j) = z_ij, $c_{ij}^{0} = g (x_{i}, y_{j}) - c_{ij}^{1} x_{i} - c_{ij}^{2} y_{j}$ , $c_{ij}^{1} = {\frac{\partial g}{\partial x} |}_{(x, y) = (x_{i}, y_{j})},$ $c_{ij}^{2} = {\frac{\partial g}{\partial y} |}_{(x, y) = (x_{i}, y_{j})}$ .

3 Fuzzy inference modeling method based on T-S fuzzy system

3.1 Input-output model of time-invariant system

In this section, we will discuss the problem of second-order time-invariant freedom movement system (input u (t) =0) modeling. Let the universes of the input variables respectively be Y = [a₁, b₁], $\dot{Y} = [a_{2}, b_{2}]$ and the output variable $\ddot{Y} = [c, d]$ , then y (t) ∈ Y, $\dot{y} (t) \in \dot{Y}$ , $\ddot{y} (t) \in \ddot{Y}$ .

Let A = {A_i (y)} _1≤i≤m and $B = {B_{j} (\dot{y})}_{1 \leq j \leq n}$ be the fuzzy partitions of the universes Y and $\dot{Y}$ respectively, and y_i and ${\dot{y}}_{j}$ with the conditions $\begin{matrix} a_{1} & = & y_{1} < y_{2} < \dots < y_{m} = b_{1}, a_{2} \\ = & {\dot{y}}_{1} < {\dot{y}}_{2} < \dots < {\dot{y}}_{n} = b_{2}, \end{matrix}$ are respectively the peak points of A_i (y) and $B_{j} (\dot{y})$ which are all triangular membership functions. For fuzzy inference rules $\begin{matrix} If y (t) is A_{i} and \dot{y} (t) is B_{j}, then \\ \ddot{y} (t) = c_{ij}^{0} + c_{ij}^{1} y (t) + c_{ij}^{2} \dot{y} (t) \\ (i = 1, 2, \dots m; j = 1, 2, \dots n) \end{matrix}$ (3) we are ready to prove the following theorem.

Theorem 1.For the second-order time-invariant freedom movement system, based on T-S fuzzy system, the obtained input-output model can be expressed by the following nonlinear differential equation with variable coefficients: $\begin{matrix} \ddot{y} (t) = a_{1} (y (t), \dot{y} (t)) y (t) + a_{2} (y (t), \dot{y} (t)) \dot{y} (t) \\ + a_{3} (y (t), \dot{y} (t)) y^{2} (t) + a_{4} (y (t), \dot{y} (t)) y (t) \dot{y} (t) \\ + a_{5} (y (t), \dot{y} (t)) {\dot{y}}^{2} (t) + a_{6} (y (t), \dot{y} (t)) y^{2} (t) \dot{y} (t) \\ + a_{7} (y (t), \dot{y} (t)) y (t) {\dot{y}}^{2} (t) + a_{8} (y (t), \dot{y} (t)) . \end{matrix}$ (4)

Proof. When $(y (t), \dot{y} (t)) \in [y_{i}, y_{i + 1}] \times [{\dot{y}}_{j}, {\dot{y}}_{j + 1}]$ , the local equation on the piece (i, j) can be written as $\begin{matrix} {\ddot{y}}^{ij} (t) & = & A_{i} (y (t)) B_{j} (\dot{y} (t)) {\ddot{y}}_{ij} \\ + A_{i + 1} (y (t)) B_{j} (\dot{y} (t)) {\ddot{y}}_{i + 1 j} \\ + A_{i} (y (t)) B_{j + 1} (\dot{y} (t)) {\ddot{y}}_{ij + 1} \\ + A_{i + 1} (y (t)) B_{j + 1} (\dot{y} (t)) {\ddot{y}}_{i + 1 j + 1} . \end{matrix}$ where the concrete forms of inference antecedents are $A_{i} (y (t)) = \frac{y_{i + 1} - y (t)}{y_{i + 1} - y_{i}}$ , $A_{i + 1} (y (t)) = \frac{y (t) - y_{i}}{y_{i + 1} - y_{i}}$ , $B_{j} (\dot{y} (t)) = \frac{{\dot{y}}_{j + 1} - \dot{y} (t)}{{\dot{y}}_{j + 1} - {\dot{y}}_{j}}, B_{j + 1} (\dot{y} (t)) = \frac{\dot{y} (t) - {\dot{y}}_{j}}{{\dot{y}}_{j + 1} - {\dot{y}}_{j}},$ and those of inference consequents are ${\ddot{y}}_{kl} = c_{kl}^{0} + c_{kl}^{1} y (t) + c_{kl}^{2} \dot{y} (t) (k = i, i + 1; l = j, j + 1),$

After simplification, we obtain $\begin{matrix} {\ddot{y}}^{ij} (t) = a_{1}^{ij} y (t) + a_{2}^{ij} \dot{y} (t) + a_{3}^{ij} y^{2} (t) + a_{4}^{ij} y (t) \dot{y} (t) \\ + a_{5}^{ij} {\dot{y}}^{2} (t) + a_{6}^{ij} y^{2} (t) \dot{y} (t) + a_{7}^{ij} y (t) {\dot{y}}^{2} (t) + a_{8}^{ij} \end{matrix}$

(i) When $(y (t), \dot{y} (t)) \notin [y_{i}, y_{i + 1}] \times [{\dot{y}}_{j}, {\dot{y}}_{j + 1}]$ , $a_{l}^{ij} = 0 (l = 1, 2, \dots, 8) .$

(ii) When $(y (t), \dot{y} (t)) \in [y_{i}, y_{i + 1}] \times [{\dot{y}}_{j}, {\dot{y}}_{j + 1}]$ , we denote $h_{1} = y_{i + 1} - y_{i}, h_{2} = {\dot{y}}_{j + 1} - {\dot{y}}_{j}$ . Then $\begin{matrix} a_{1}^{ij} & = & (- {\dot{y}}_{j + 1} c_{ij}^{0} + y_{i + 1} {\dot{y}}_{j + 1} c_{ij}^{1} + {\dot{y}}_{j + 1} c_{i + 1, j}^{0} \\ - y_{i} {\dot{y}}_{j + 1} c_{i + 1, j}^{1} + {\dot{y}}_{j} c_{i, j + 1}^{0} \\ - y_{i + 1} {\dot{y}}_{j} c_{i, j + 1}^{1} - {\dot{y}}_{j} c_{i + 1, j + 1}^{0} \\ + y_{i} {\dot{y}}_{j} c_{i + 1, j + 1}^{1}) / h_{1} h_{2}, \\ a_{2}^{ij} & = & (- y_{i + 1} c_{ij}^{0} + y_{i + 1} {\dot{y}}_{j + 1} c_{ij}^{2} + y_{i} c_{i + 1, j}^{0} \\ - y_{i} {\dot{y}}_{j + 1} c_{i + 1, j}^{2} + y_{i + 1} c_{i, j + 1}^{0} \\ - y_{i + 1} {\dot{y}}_{j} c_{i, j + 1}^{2} - y_{i} c_{i + 1, j + 1}^{0} \\ + y_{i} {\dot{y}}_{j} c_{i + 1, j + 1}^{2}) / h_{1} h_{2}, \\ a_{3}^{ij} & = & (- {\dot{y}}_{j + 1} c_{ij}^{1} + {\dot{y}}_{j + 1} c_{i + 1, j}^{1} + {\dot{y}}_{j} c_{i, j + 1}^{1} \\ - {\dot{y}}_{j} c_{i + 1, j + 1}^{1}) / h_{1} h_{2}, \end{matrix}$ $\begin{matrix} a_{4}^{ij} = (c_{ij}^{0} - y_{i + 1} c_{ij}^{1} - {\dot{y}}_{j + 1} c_{ij}^{2} - c_{i + 1, j}^{0} + y_{i} c_{i + 1, j}^{1} \\ + {\dot{y}}_{j + 1} c_{i + 1, j}^{2} - c_{i, j + 1}^{0} + y_{i + 1} c_{i, j + 1}^{1} + {\dot{y}}_{j} c_{i, j + 1}^{2} \\ + c_{i + 1, j + 1}^{0} - y_{i} c_{i + 1, j + 1}^{1} - {\dot{y}}_{j} c_{i + 1, j + 1}^{2}) / h_{1} h_{2}, \end{matrix}$ $\begin{matrix} a_{5}^{ij} & = & (- y_{i + 1} c_{ij}^{2} + y_{i} c_{i + 1, j}^{2} + y_{i + 1} c_{i, j + 1}^{2} \\ - y_{i} c_{i + 1, j + 1}^{2}) / h_{1} h_{2}, \\ a_{6}^{ij} & = & (c_{ij}^{1} - c_{i + 1, j}^{1} - c_{i, j + 1}^{1} + c_{i + 1, j + 1}^{1}) / h_{1} h_{2}, \\ a_{7}^{ij} & = & (c_{ij}^{2} - c_{i + 1, j}^{2} - c_{i, j + 1}^{2} + c_{i + 1, j + 1}^{2}) / h_{1} h_{2}, \\ a_{8}^{ij} & = & (y_{i + 1} {\dot{y}}_{j + 1} c_{ij}^{0} - y_{i} {\dot{y}}_{j + 1} c_{i + 1, j}^{0} \\ - y_{i + 1} {\dot{y}}_{j} c_{i, j + 1}^{0} + y_{i} {\dot{y}}_{j} c_{i + 1, j + 1}^{0}) / h_{1} h_{2} \end{matrix}$

We denote $a_{k} (y (t), \dot{y} (t)) = \sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} a_{k}^{ij} (k = 1, 2, \dots, 8) .$

Then we derive the overall nonlinear differential equation when $(y (t), \dot{y} (t)) \in Y \times \dot{Y}$ : $\begin{matrix} \ddot{y} (t) = \sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} {\ddot{y}}^{ij} (t) \\ = \sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} [a_{1}^{ij} y (t) + a_{2}^{ij} \dot{y} (t) + a_{3}^{ij} y^{2} (t) + a_{4}^{ij} y (t) \dot{y} (t) \\ + a_{5}^{ij} {\dot{y}}^{2} (t) + a_{6}^{ij} y^{2} (t) \dot{y} (t) + a_{7}^{ij} y (t) {\dot{y}}^{2} (t) + a_{8}^{ij}] \\ = (\sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} a_{1}^{ij}) y (t) + (\sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} a_{2}^{ij}) \dot{y} (t) \\ + (\sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} a_{3}^{ij}) y^{2} (t) + (\sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} a_{4}^{ij}) y (t) \dot{y} (t) \\ + (\sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} a_{5}^{ij}) {\dot{y}}^{2} (t) + (\sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} a_{6}^{ij}) y^{2} (t) \dot{y} (t) \\ + (\sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} a_{7}^{ij}) y (t) {\dot{y}}^{2} (t) + (\sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} a_{8}^{ij}) \\ = a_{1} (y (t), \dot{y} (t)) y (t) + a_{2} (y (t), \dot{y} (t)) \dot{y} (t) \\ + a_{3} (y (t), \dot{y} (t)) y^{2} (t) + a_{4} (y (t), \dot{y} (t)) y (t) \dot{y} (t) \\ + a_{5} (y (t), \dot{y} (t)) {\dot{y}}^{2} (t) + a_{6} (y (t), \dot{y} (t)) y^{2} (t) \dot{y} (t) \\ + a_{7} (y (t), \dot{y} (t)) y (t) {\dot{y}}^{2} (t) + a_{8} (y (t), \dot{y} (t)) . \end{matrix}$ Example 1: The Van der Pol equation $\ddot{y} (t) - μ (1 - y^{2} (t)) \dot{y} (t) + y (t) = 0$ is simulated, where μ = 1, the initial condition $(y (0), \dot{y} (0)) = (2, 0)$ . Let time T = 20. The universes of Y and $\dot{Y}$ are both divided into seven rules. The state curves of y (t) and $\dot{y} (t)$ are shown in Figs. 1 and 2 where the solid line represents the real curve and the dotted line shows the simulation curve.

More details about the simulation steps can be found in reference [11]. We can see from Figs. 1 and 2, that the simulation curves almost coincide with the original curves, which means that the model established by fuzzy inference modeling method based on T-S fuzzy system has good approximation accuracy.

3.2 State-space models of time-invariant systems

We takes second-order time-invariant freedom movement system as an example to analyze the state-space model in detail. The universes of $x_{i} (t), {\dot{x}}_{i} (t) (i = 1, 2)$ , are respectively denoted by $X_{i} = [a_{i}, b_{i}], {\dot{X}}_{i} = [c_{i}, d_{i}] (i = 1, 2)$ , Let the fuzzy sets A_i and B_j be triangular membership functions, and their peak points be respectively $x_{i}^{(1)}$ , $x_{j}^{(2)}$ , ${\dot{x}}_{ij}^{(1)}$ , ${\dot{x}}_{ij}^{(2)}$ with the conditions $a_{1} = x_{1}^{(1)} < x_{2}^{(1)} < \dots < x_{m}^{(1)} = b_{1}$ and $a_{2} = x_{1}^{(2)} < x_{2}^{(2)} < \dots < x_{m}^{(2)} = b_{2}$ . Then we can get the following fuzzy inference rules: $\begin{matrix} If x_{1} (t) is A_{i}, x_{2} (t) is B_{j}, then \\ {\dot{x}}_{1} (t) = c_{ij}^{0 (1)} + c_{ij}^{1 (1)} x_{1} (t) + c_{ij}^{2 (1)} x_{2} (t) \\ {\dot{x}}_{2} (t) = c_{ij}^{0 (2)} + c_{ij}^{1 (2)} x_{1} (t) + c_{ij}^{2 (2)} x_{2} (t) \\ (i = 1, 2, \dots m; j = 1, 2, \dots n) \end{matrix}$ (5)

The fuzzy rule base (8) can be represented as a piecewise interpolation function of two variables based on T-S fuzzy system and fuzzy logic interpolation mechanism as follows: $({\dot{x}}_{1} (t), {\dot{x}}_{2} (t)) = (S_{1} (x_{1} (t), x_{2} (t)), S_{2} (x_{1} (t), x_{2} (t))),$ (6) where $\begin{matrix} {\dot{x}}_{1} (t) = S_{1} (x_{1} (t), x_{2} (t)) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} A_{i} (x_{1} (t)) B_{j} (x_{2} (t)) {\dot{x}}_{ij}^{(1)}, \\ {\dot{x}}_{2} (t) = S_{2} (x_{1} (t), x_{2} (t)) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} A_{i} (x_{1} (t)) B_{j} (x_{2} (t)) {\dot{x}}_{ij}^{(2)} \end{matrix}$

Thus the fuzzy rule base can be expressed as nonlinear differential equations with variable coefficients, and then we get the following theorem.

Theorem 2.For second-order time-invariant system modeling, based on T-S fuzzy system, ifA_i (x₁ (t)) and B_j (x₂ (t)) are both triangular membership functions, the obtained state-space model can be constructed as the following nonlinear differential equations with variable coefficients: $\begin{matrix} {\dot{x}}_{1} (t) = a_{11} (x_{1} (t), x_{2} (t)) x_{1} (t) + a_{12} (x_{1} (t), x_{2} (t)) x_{2} (t) \\ + a_{13} (x_{1} (t), x_{2} (t)) x_{1}^{2} (t) + a_{14} (x_{1} (t), x_{2} (t)) x_{1} (t) x_{2} (t) \\ + a_{15} (x_{1} (t), x_{2} (t)) x_{2}^{2} (t) + a_{16} (x_{1} (t), x_{2} (t)) x_{1}^{2} (t) x_{2} (t) \\ + a_{17} (x_{1} (t), x_{2} (t)) x_{1} (t) x_{2}^{2} (t) + a_{18} (x_{1} (t), x_{2} (t)) . \end{matrix}$ (7) $\begin{matrix} {\dot{x}}_{2} (t) = a_{21} (x_{1} (t), x_{2} (t)) x_{1} (t) + a_{22} (x_{1} (t), x_{2} (t)) x_{2} (t) \\ + a_{23} (x_{1} (t), x_{2} (t)) x_{1}^{2} (t) + a_{24} (x_{1} (t), x_{2} (t)) x_{1} (t) x_{2} (t) \\ + a_{25} (x_{1} (t), x_{2} (t)) x_{2}^{2} (t) + a_{26} (x_{1} (t), x_{2} (t)) x_{1}^{2} (t) x_{2} (t) \\ + a_{27} (x_{1} (t), x_{2} (t)) x_{1} (t) x_{2}^{2} (t) + a_{28} (x_{1} (t), x_{2} (t)) . \end{matrix}$ (8)

The proof of this theorem can be completed by the method analogous to that of Theorem 1. In fact, for l = 1,2, we have $a_{lk} (x_{1} (t), x_{2} (t)) = \sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} a_{lk}^{ij} (k = 1, 2, \dots, 8) .$

The local coefficients on the piece (i, j) are defined as: when $(x_{1} (t), x_{2} (t)) \notin [x_{i}^{(1)}, x_{i + 1}^{(1)}] \times [x_{j}^{(2)}, x_{j + 1}^{(2)}]$ , $a_{lp}^{ij} = 0 (p = 1, 2, \dots, 8) .$

When $(x_{1} (t), x_{2} (t)) \in [x_{i}^{(1)}, x_{i + 1}^{(1)}] \times [x_{j}^{(2)}, x_{j + 1}^{(2)}]$ , we let $h_{1} = x_{i + 1}^{(1)} - x_{i}^{(1)}$ , $h_{2} = x_{j + 1}^{(2)} - x_{j}^{(2)}$ , then for l = 1, 2, $\begin{matrix} a_{l 1}^{ij} & = & (- x_{j + 1}^{(2)} c_{ij}^{0 (l)} + x_{i + 1}^{(1)} x_{j + 1}^{(2)} c_{ij}^{1 (l)} + x_{j + 1}^{(2)} \\ c_{i + 1 j}^{0 (l)} - x_{i}^{(1)} x_{j + 1}^{(2)} c_{i + 1 j}^{1 (l)} + x_{j}^{(2)} c_{ij + 1}^{0 (l)} \\ - x_{i + 1}^{(1)} x_{j}^{(2)} c_{ij + 1}^{1 (l)} - x_{j}^{(2)} c_{i + 1 j + 1}^{0 (l)} \\ + x_{i}^{(1)} x_{j}^{(2)} c_{i + 1 j + 1}^{1 (l)}) / h_{1} h_{2}, \\ a_{l 2}^{ij} & = & (- x_{i + 1}^{(1)} c_{ij}^{0 (l)} + x_{i + 1}^{(1)} x_{j + 1}^{(2)} c_{ij}^{2 (l)} \\ + x_{i}^{(1)} c_{i + 1 j}^{0 (l)} - x_{i}^{(1)} x_{j + 1}^{(2)} c_{i + 1 j}^{2 (l)} \\ + x_{i + 1}^{(1)} c_{ij + 1}^{0 (l)} - x_{i + 1}^{(1)} x_{j}^{(2)} c_{ij + 1}^{2 (l)} \\ - x_{i}^{(1)} c_{i + 1 j + 1}^{0 (l)} + x_{i}^{(1)} x_{j}^{(2)} c_{i + 1 j + 1}^{2 (l)}) / h_{1} h_{2}, \\ a_{l 3}^{ij} & = & (- x_{j + 1}^{(2)} c_{ij}^{1 (l)} + x_{j + 1}^{(2)} c_{i + 1 j}^{1 (l)} \\ + x_{j}^{(2)} c_{ij + 1}^{1 (l)} - x_{j}^{(2)} c_{i + 1 j + 1}^{1 (l)}) / h_{1} h_{2}, \\ a_{l 4}^{ij} & = & (c_{ij}^{0 (l)} - x_{i + 1}^{(1)} c_{ij}^{1 (l)} - x_{j + 1}^{(2)} c_{ij}^{2 (l)} - c_{i + 1, j}^{0 (l)} \\ + x_{i}^{(1)} c_{i + 1 j}^{1 (l)} + x_{j + 1}^{(2)} c_{i + 1 j}^{2 (l)} - c_{ij + 1}^{0 (l)} \\ + x_{i + 1}^{(1)} c_{ij + 1}^{1 (l)} + x_{j}^{(2)} c_{ij + 1}^{2 (l)} + c_{i + 1 j + 1}^{0 (l)} \\ - x_{i}^{(1)} c_{i + 1 j + 1}^{1 (l)} - x_{j}^{(2)} c_{i + 1 j + 1}^{2 (l)}) / h_{1} h_{2}, \\ a_{l 5}^{ij} & = & (- x_{i + 1}^{(1)} c_{ij}^{2 (l)} + x_{i}^{(1)} c_{i + 1 j}^{2 (l)} \\ + x_{i + 1}^{(1)} c_{ij + 1}^{2 (l)} - x_{i}^{(1)} c_{i + 1 j + 1}^{2 (l)}) / h_{1} h_{2}, \\ a_{l 6}^{ij} & = & (c_{ij}^{1 (l)} - c_{i + 1 j}^{1 (l)} - c_{ij + 1}^{1 (l)} + c_{i + 1 j + 1}^{1 (l)}) / h_{1} h_{2}, \\ a_{l 7}^{ij} & = & (c_{ij}^{2 (l)} - c_{i + 1 j}^{2 (l)} - c_{ij + 1}^{2 (l)} + c_{i + 1 j + 1}^{2 (l)}) / h_{1} h_{2}, \\ a_{l 8}^{ij} & = & (x_{i + 1}^{(1)} x_{j + 1}^{(2)} c_{ij}^{0 (l)} - x_{i}^{(1)} x_{j + 1}^{(2)} c_{i + 1 j}^{0 (l)} \\ - x_{i + 1}^{(1)} x_{j}^{(2)} c_{ij + 1}^{0 (l)} + x_{i}^{(1)} x_{j}^{(2)} c_{i + 1 j + 1}^{0 (l)}) / h_{1} h_{2} . \end{matrix}$

Example 2: In Example 1, let x₁ (t) = y (t), $x_{2} (t) = \dot{y} (t)$ , then the state curve of x₁ (t) and the simulation curve of x₂ (t) are shown in Figs. 1 and 2, respectively.

3.3 Input-output model of time-varying system

The second-order time-varying freedom movement system (input u (t) =0) is discussed below.

Let time T > 0 be sufficiently large, and the finite time interval [0, T] is equidistantly divided, that is 0 = t₀ < t₁ < ⋯ < t_N = T, t_k = kT/N (k = 0, 1, ⋯ , N - 1).

The time-varying system can be seen as a time-invariant system within each equidistant time interval, so we can use the same modeling method. Note that the universes of the input variables y (t), $\dot{y} (t)$ and the output variable $\ddot{y} (t)$ of time-varying system are affected by the index k, thus for each index k (k = 0, 1, ⋯ , N - 1), they are set to be Y_k = [a_1k, b_1k] and ${\dot{Y}}_{k} = [a_{2 k}, b_{2 k}]$ respectively. Let A_k = {A_ki} _{(1≤i≤m_k)} and B_k = {B_kj} _{(1≤j≤n_k)} be the fuzzy partitions of the universes Y_k and ${\dot{Y}}_{k}$ , where A_ki and B_kj are triangular membership functions. y_ki and ${\dot{y}}_{kj}$ under conditions of a_1k ≤ y_k1 < y_k2 < ⋯ < y_{km
_k} ≤ b_1k and $a_{2 k} \leq {\dot{y}}_{k 1} < {\dot{y}}_{k 2} < \dots < {\dot{y}}_{{kn}_{k}} \leq b_{2 k}$ are the peak points of A_ki and B_kj respectively. For the fuzzy inference rules $\begin{matrix} If y (t) is A_{ki} and \dot{y} (t) is B_{ki}, then \\ \ddot{y} (t) = c_{kij}^{0} + c_{kij}^{1} y (t) + c_{kij}^{2} \dot{y} (t) \\ (i = 1, 2, \dots, m_{k}; j = 1, 2, \dots, n_{k}), \end{matrix}$ (9)

The following theorem is acquired by using fuzzy inference modeling method based on T-S fuzzy system.

Theorem 3.For second-order time-varying system modeling, based on T-S fuzzy system, ifA_ki (y (t)) and $B_{kj} (\dot{y} (t))$ are both triangular membership functions, the obtained input-output model can be constructed as the following form of nonlinear differential equation with variable coefficients: $\begin{matrix} \ddot{y} (t) = a_{1} (t, y (t), \dot{y} (t)) y (t) + a_{2} (t, y (t), \dot{y} (t)) \dot{y} (t) \\ + a_{3} (t, y (t), \dot{y} (t)) y^{2} (t) + a_{4} (t, y (t), \dot{y} (t)) y (t) \dot{y} (t) \\ + a_{5} (t, y (t), \dot{y} (t)) {\dot{y}}^{2} (t) + a_{6} (t, y (t), \dot{y} (t)) y^{2} (t) \dot{y} (t) \\ + a_{7} (t, y (t), \dot{y} (t)) y (t) {\dot{y}}^{2} (t) + a_{8} (t, y (t), \dot{y} (t)) . \end{matrix}$ (10)

Proof. We divide the universes of input and output into fuzzy subsets. When $(t, y (t), \dot{y} (t)) \in [t_{k}, t_{k + 1}] \times [y_{ki}, y_{ki + 1}] \times [{\dot{y}}_{kj}, {\dot{y}}_{kj + 1}]$ , the local equation on the piece (k, i, j) can be expressed as the followingform: $\begin{matrix} {\ddot{y}}_{k}^{ij} (t) = A_{ki} (y (t)) B_{kj} (\dot{y} (t)) {\ddot{y}}_{kij} \\ + A_{ki + 1} (y (t)) B_{kj} (\dot{y} (t)) {\ddot{y}}_{ki + 1 j} + A_{ki} (y (t)) \\ B_{kj + 1} (\dot{y} (t)) {\ddot{y}}_{kij + 1} + A_{ki + 1} (y (t)) B_{kj + 1} (\dot{y} (t)) {\ddot{y}}_{ki + 1 j + 1} . \end{matrix}$ (11)

The concrete forms of inference antecedents are $\begin{matrix} A_{ki} (y (t)) = \frac{y_{k, i + 1} - y (t)}{y_{ki + 1} - y_{ki}}, A_{ki + 1} (y (t)) = \frac{y (t) - y_{ki}}{y_{ki + 1} - y_{ki}}, \\ B_{kj} (\dot{y} (t)) = \frac{{\dot{y}}_{kj + 1} - \dot{y} (t)}{{\dot{y}}_{kj + 1} - {\dot{y}}_{kj}}, B_{kj + 1} (\dot{y} (t)) = \frac{\dot{y} (t) - {\dot{y}}_{kj}}{{\dot{y}}_{kj + 1} - {\dot{y}}_{kj}}, \end{matrix}$ and the concrete forms of inference consequents are ${\ddot{y}}_{kpq} = c_{kpq}^{0} + c_{kpq}^{1} y (t) + c_{kpq}^{2} \dot{y} (t) (p = i, i + 1; q = j, j + 1),$ (11) can be written as follows by substituting the concrete forms of inference antecedents and consequents into it; after simplification, we have $\begin{matrix} {\ddot{y}}_{k}^{ij} (t) = a_{k 1}^{ij} y (t) + a_{k 2}^{ij} \dot{y} (t) + a_{k 3}^{ij} y^{2} (t) + a_{k 4}^{ij} y (t) \dot{y} (t) \\ + a_{k 5}^{ij} {\dot{y}}^{2} (t) + a_{k 6}^{ij} y^{2} (t) \dot{y} (t) + a_{k 7}^{ij} y (t) {\dot{y}}^{2} (t) + a_{k 8}^{ij} . \end{matrix}$

We define the local coefficients $a_{kl}^{ij} (l = 1, 2,$ ⋯, 8) on the piece (k, i, j) by two cases:

When $(t, y (t), \dot{y} (t)) \notin [t_{k}, t_{k + 1}] \times [y_{ki}, y_{ki + 1}] \times [{\dot{y}}_{kj}, {\dot{y}}_{kj + 1}]$ , the coefficients of local equation are $a_{kl}^{ij} = 0 (l = 1, 2, \dots, 8) .$ when $(t, y (t), \dot{y} (t)) \in [t_{k}, t_{k + 1}] \times [y_{ki}, y_{ki + 1}] \times [{\dot{y}}_{kj}, {\dot{y}}_{kj + 1}]$ , $h_{k 1} = y_{ki + 1} - y_{ki}, h_{k 2} = {\dot{y}}_{kj + 1} - {\dot{y}}_{kj},$ the coefficients of local equation are as follows: $\begin{matrix} a_{k 1}^{ij} = (- {\dot{y}}_{kj + 1} c_{kij}^{0} + y_{k, i + 1} {\dot{y}}_{kj + 1} c_{kij}^{1} + {\dot{y}}_{kj + 1} c_{ki + 1 j}^{0} \\ - y_{ki} {\dot{y}}_{kj + 1} c_{ki + 1 j}^{1} + {\dot{y}}_{kj} c_{kij + 1}^{0} - y_{ki + 1} {\dot{y}}_{kj} c_{kij + 1}^{1} \\ - {\dot{y}}_{kj} c_{ki + 1 j + 1}^{0} + y_{ki} {\dot{y}}_{kj} c_{ki + 1 j + 1}^{1}) / h_{k 1} h_{k 2}, \\ a_{k 2}^{ij} = (- y_{ki + 1} c_{kij}^{0} + y_{ki + 1} {\dot{y}}_{kj + 1} c_{kij}^{2} + y_{ki} c_{ki + 1 j}^{0} \\ - y_{ki} {\dot{y}}_{kj + 1} c_{ki + 1 j}^{2} + y_{ki + 1} c_{kij + 1}^{0} - y_{ki + 1} {\dot{y}}_{kj} c_{kij + 1}^{2} \\ - y_{ki} c_{ki + 1 j + 1}^{0} + y_{ki} {\dot{y}}_{kj} c_{ki + 1 j + 1}^{2}) / h_{k 1} h_{k 2}, \\ a_{k 3}^{ij} = (- {\dot{y}}_{kj + 1} c_{kij}^{1} + {\dot{y}}_{kj + 1} c_{ki + 1 j}^{1} \\ + {\dot{y}}_{kj} c_{kij + 1}^{1} - {\dot{y}}_{kj} c_{ki + 1 j + 1}^{1}) / h_{k 1} h_{k 2}, \\ a_{k 4}^{ij} = (c_{kij}^{0} - y_{ki + 1} c_{kij}^{1} - {\dot{y}}_{kj + 1} c_{kij}^{2} - c_{ki + 1 j}^{0} + y_{ki} c_{ki + 1 j}^{1} \\ + {\dot{y}}_{kj + 1} c_{ki + 1 j}^{2} - c_{kij + 1}^{0} + y_{ki + 1} c_{kij + 1}^{1} + {\dot{y}}_{kj} c_{kij + 1}^{2} \\ + c_{ki + 1 j + 1}^{0} - y_{ki} c_{ki + 1 j + 1}^{1} - {\dot{y}}_{kj} c_{ki + 1 j + 1}^{2}) / h_{k 1} h_{k 2}, \\ a_{k 5}^{ij} = (- y_{ki + 1} c_{kij}^{2} + y_{ki} c_{ki + 1 j}^{2} + y_{ki + 1} c_{kij + 1}^{2} \\ - y_{ki} c_{ki + 1 j + 1}^{2}) / h_{k 1} h_{k 2}, \\ a_{k 6}^{ij} = (c_{kij}^{1} - c_{ki + 1 j}^{1} - c_{kij + 1}^{1} + c_{ki + 1 j + 1}^{1}) / h_{k 1} h_{k 2}, \\ a_{k 7}^{ij} = (c_{kij}^{2} - c_{ki + 1 j}^{2} - c_{kij + 1}^{2} + c_{ki + 1 j + 1}^{2}) / h_{k 1} h_{k 2}, \\ a_{k 8}^{ij} = (y_{ki + 1} {\dot{y}}_{kj + 1} c_{kij}^{0} - y_{ki} {\dot{y}}_{kj + 1} c_{ki + 1 j}^{0} \\ - y_{ki + 1} {\dot{y}}_{kj} c_{kij + 1}^{0} + y_{ki} {\dot{y}}_{kj} c_{ki + 1 j + 1}^{0}) / h_{k 1} h_{k 2} . \end{matrix}$

And if we denote that $a_{p} (t, y (t), \dot{y} (t)) = \sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k} - 1} \sum_{j = 1}^{n_{k} - 1} a_{kp}^{ij} (p = 1, 2, 3, 4) .$

Then we get the overall nonlinear differential equation when $(t, y (t), \dot{y} (t)) \in T \times Y \times \dot{Y}$ , $\begin{matrix} \ddot{y} (t) = \sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k} - 1} \sum_{j = 1}^{n_{k} - 1} {\ddot{y}}^{ij} (t) \\ = \sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k} - 1} \sum_{j = 1}^{n_{k} - 1} [a_{k 1}^{ij} y (t) + a_{k 2}^{ij} \dot{y} (t) + a_{k 3}^{ij} y^{2} (t) + a_{k 4}^{ij} y (t) \dot{y} (t) \\ + a_{k 5}^{ij} {\dot{y}}^{2} (t) + a_{k 6}^{ij} y^{2} (t) \dot{y} (t) + a_{k 7}^{ij} y (t) {\dot{y}}^{2} (t) + a_{k 8}^{ij}] \\ = (\sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k} - 1} \sum_{j = 1}^{n_{k} - 1} a_{k 1}^{ij}) y (t) + (\sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k} - 1} \sum_{j = 1}^{n_{k} - 1} a_{k 2}^{ij}) \dot{y} (t) \\ + (\sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k} - 1} \sum_{j = 1}^{n_{k} - 1} a_{k 3}^{ij}) y^{2} (t) + (\sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k} - 1} \sum_{j = 1}^{n_{k} - 1} a_{k 4}^{ij}) y (t) \dot{y} (t) \\ + (\sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k} - 1} \sum_{j = 1}^{n_{k} - 1} a_{k 5}^{ij}) {\dot{y}}^{2} (t) + (\sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k} - 1} \sum_{j = 1}^{n_{k} - 1} a_{k 6}^{ij}) y^{2} (t) \dot{y} (t) \\ + (\sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k} - 1} \sum_{j = 1}^{n_{k} - 1} a_{k 7}^{ij}) y (t) {\dot{y}}^{2} (t) + (\sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k} - 1} \sum_{j = 1}^{n_{k} - 1} a_{k 8}^{ij}) \\ = a_{1} (t, y (t), \dot{y} (t)) y (t) + a_{2} (t, y (t), \dot{y} (t)) \dot{y} (t) \\ + a_{3} (t, y (t), \dot{y} (t)) y^{2} (t) + a_{4} (t, y (t), \dot{y} (t)) y (t) \dot{y} (t) \\ + a_{5} (t, y (t), \dot{y} (t)) {\dot{y}}^{2} (t) + a_{6} (t, y (t), \dot{y} (t)) y^{2} (t) \dot{y} (t) \\ + a_{7} (t, y (t), \dot{y} (t)) y (t) {\dot{y}}^{2} (t) + a_{8} (t, y (t), \dot{y} (t)) . \end{matrix}$

It shows that second-order nonlinear time-varying differential equation can be transformed into piecewise nonlinear time-invariant differential equation, and the coefficients of each piece are all constant.

Example 3: The second-order nonlinear time-varying differential equation $\ddot{y} (t) - \dot{y} (t)^{2} + 12 ty = 0$ is simulated when the initial condition $(y (0), \dot{y} (0)) = (0.6, 0)$ . Let T = 5, N = 8, m_k = 6, n_k = 8, k = 0, 1, ⋯ 7, then the simulation state curves of y (t) and $\dot{y} (t)$ are shown in Figs. 4 and 5, respectively.

3.3 State-space models of time-varying systems

We take second-order time-varying freedom movement system as an example. The finite time interval [0, T] is equidistantly divided, where $0 = t_{0} < t_{1} < \dots < t_{N} = T, t_{k} = \frac{kT}{N} (k = 0, 1, \dots, N - 1)$ .

In each time intervals [t_k, t_k+1] (k = 0, 1, …, N - 1), the universes of x₁ (t), x₂ (t), ${\dot{x}}_{1} (t)$ , ${\dot{x}}_{2} (t)$ are X_1k = [a_1k, b_1k], X_2k = [a_2k, b_2k], ${\dot{X}}_{1 k} = [c_{1 k}, d_{1 k}]$ , ${\dot{X}}_{2 k} = [c_{2 k}, d_{2 k}]$ respectively, and the fuzzy partitions of the universes are respectively A_k = {A_ki} _{(1≤i≤m_k)} and B_k = {B_kj} _{(1≤j≤n_k)}. Let A_ki and B_kj be triangular membership functions and their peak points are respectively $x_{ki}^{(1)}$ and $x_{kj}^{(2)}$ satisfying $a_{k 1} = x_{k 1}^{(1)} < x_{k 2}^{(1)} < \dots x_{{km}_{k}}^{(1)} = b_{k 1}$ and $a_{k 2} = x_{k 1}^{(2)} < x_{k 2}^{(2)} < \dots x_{{kn}_{k}}^{(2)} = b_{k 2}$ .

For the fuzzy inference rules: $\begin{matrix} If x_{1} (t) is A_{ki}, x_{2} (t) is B_{kj}, then \\ {\dot{x}}_{1} (t) = c_{kij}^{0 (1)} + c_{kij}^{1 (1)} x_{1} (t) + c_{kij}^{2 (1)} x_{2} (t) \\ {\dot{x}}_{2} (t) = c_{kij}^{0 (2)} + c_{kij}^{1 (2)} x_{1} (t) + c_{kij}^{2 (2)} x_{2} (t) \\ (i = 1, 2, \dots, m_{k}, j = 1, 2, \dots, n_{k}) \end{matrix}$ (12) they can be expressed as piecewise interpolation function of two variables based on T-S fuzzy system as follows: $({\dot{x}}_{1} (t), {\dot{x}}_{2} (t)) = (S_{1} (t, x_{1} (t), x_{2} (t)), S_{2} (t, x_{1} (t), x_{2} (t))),$ where $\begin{matrix} {\dot{x}}_{1} (t) = S_{1} (t, x_{1} (t), x_{2} (t)) \\ = \sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k}} \sum_{j = 1}^{n_{k}} A_{ki} (x_{1} (t)) B_{kj}^{*} (x_{2} (t)) {\dot{x}}_{kij}^{(1)}, \\ {\dot{x}}_{2} (t) = S_{2} (t, x_{1} (t), x_{2} (t)) \\ = \sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k}} \sum_{j = 1}^{n_{k}} A_{ki} (x_{1} (t)) B_{kj}^{*} (x_{2} (t)) {\dot{x}}_{kij}^{(2)} . \end{matrix}$

We learn that the fuzzy rule base can be expressed as nonlinear differential equation with variable coefficients, so we reach the following conclusion.

Theorem 4.For the second-order time-varying system, based on T-S fuzzy system, ifA_ki (x₁ (t)) andB_kj (x₂ (t)) are both triangular membership functions, the obtained state-space model can be expressed as the following form of nonlinear differential equation with variable coefficients:

$\begin{matrix} {\dot{x}}_{1} (t) = a_{11} (t, x_{1} (t), x_{2} (t)) x_{1} (t) + a_{12} (t, x_{1} (t), x_{2} (t)) x_{2} (t) \\ + a_{13} (t, x_{1} (t), x_{2} (t)) x_{1}^{2} (t) + a_{14} (t, x_{1} (t), x_{2} (t)) x_{1} (t) x_{2} (t) \\ + a_{15} (t, x_{1} (t), x_{2} (t)) x_{2}^{2} (t) + a_{16} (t, x_{1} (t), x_{2} (t)) x_{1}^{2} (t) x_{2} (t) \\ + a_{17} (t, x_{1} (t), x_{2} (t)) x_{1} (t) x_{2}^{2} (t) + a_{18} (t, x_{1} (t), x_{2} (t)), \end{matrix}$ (13)

$\begin{matrix} {\dot{x}}_{2} (t) = a_{21} (t, x_{1} (t), x_{2} (t)) x_{1} (t) + a_{22} (t, x_{1} (t), x_{2} (t)) x_{2} (t) \\ + a_{23} (t, x_{1} (t), x_{2} (t)) x_{1}^{2} (t) + a_{24} (t, x_{1} (t), x_{2} (t)) x_{1} (t) x_{2} (t) \\ + a_{25} (t, x_{1} (t), x_{2} (t)) x_{2}^{2} (t) + a_{26} (t, x_{1} (t), x_{2} (t)) x_{1}^{2} (t) x_{2} (t) \\ + a_{27} (t, x_{1} (t), x_{2} (t)) x_{1} (t) x_{2}^{2} (t) + a_{28} (t, x_{1} (t), x_{2} (t)) . \end{matrix}$ (14) The proof of this theorem is similar to that of Theorem 3. In fact, for l = 1,2, we have $\begin{matrix} a_{lp} (t, y (t), \dot{y} (t)) \\ = \sum_{k = 0}^{N - 1} \sum_{i = 1}^{m_{k} - 1} \sum_{j = 1}^{n_{k} - 1} a_{klp}^{ij} (p = 1, 2, \dots, 8) . \end{matrix}$

When $(t, x_{1} (t), x_{2} (t)) \notin [t_{k}, t_{k + 1}] \times [x_{ki}^{(1)}, x_{ki + 1}^{(1)}] \times [x_{kj}^{(2)}, x_{kj + 1}^{(2)}]$ , $a_{klp}^{ij} = 0 (p = 1, 2, \dots, 8) .$

When $(t, x_{1} (t), x_{2} (t)) \in [t_{k}, t_{k + 1}] \times [x_{ki}^{(1)}, x_{ki + 1}^{(1)}] \times [x_{kj}^{(2)}, x_{kj + 1}^{(2)}]$ , let $h_{k 1} = x_{ki + 1}^{(1)} - x_{ki}^{(1)}$ , $h_{k 2} = x_{kj + 1}^{(2)} - x_{kj}^{(2)}$ , then $\begin{matrix} a_{k, l 1}^{ij} = (- x_{kj + 1}^{(2)} c_{kij}^{0 (l)} + x_{ki + 1}^{(1)} x_{kj + 1}^{(2)} c_{kij}^{1 (l)} + x_{kj + 1}^{(2)} c_{ki + 1 j}^{0 (l)} \\ - x_{ki}^{(1)} x_{kj + 1}^{(2)} c_{ki + 1 j}^{1 (l)} + x_{kj}^{(2)} c_{kij + 1}^{0 (l)} - x_{ki + 1}^{(1)} x_{kj}^{(2)} c_{kij + 1}^{1 (l)} \\ - x_{kj}^{(2)} c_{ki + 1 j + 1}^{0 (l)} + x_{ki}^{(1)} x_{kj}^{(2)} c_{ki + 1 j + 1}^{1 (l)}) / h_{k 1} h_{k 2}, \\ a_{k, l 2}^{ij} = (- x_{ki + 1}^{(1)} c_{kij}^{0 (l)} + x_{ki + 1}^{(1)} x_{kj + 1}^{(2)} c_{kij}^{2 (l)} + x_{ki}^{(1)} c_{ki + 1 j}^{0 (l)} \\ - x_{ki}^{(1)} x_{kj + 1}^{(2)} c_{ki + 1 j}^{2 (l)} + x_{ki + 1}^{(1)} c_{kij + 1}^{0 (l)} - x_{ki + 1}^{(1)} x_{kj}^{(2)} c_{kij + 1}^{2 (l)} \\ - x_{ki}^{(1)} c_{ki + 1 j + 1}^{0 (l)} + x_{ki}^{(1)} x_{kj}^{(2)} c_{ki + 1 j + 1}^{2 (l)}) / h_{k 1} h_{k 2}, \\ a_{k, l 3}^{ij} = (- x_{k, j + 1}^{(2)} c_{kij}^{1 (l)} + x_{kj + 1}^{(2)} c_{ki + 1 j}^{1 (l)} \\ + x_{kj}^{(2)} c_{kij + 1}^{1 (l)} - x_{kj}^{(2)} c_{ki + 1 j + 1}^{1 (l)}) / h_{k 1} h_{k 2}, \\ a_{k, l 4}^{ij} = (c_{kij}^{0 (l)} - x_{ki + 1}^{(1)} c_{kij}^{1 (l)} - x_{kj + 1}^{(2)} c_{kij}^{2 (l)} - c_{ki + 1 j}^{0 (l)} \\ + x_{ki}^{(1)} c_{ki + 1 j}^{1 (l)} + x_{kj + 1}^{(2)} c_{ki + 1 j}^{2 (l)} - c_{kij + 1}^{0 (l)} \\ + x_{ki + 1}^{(1)} c_{kij + 1}^{1 (l)} + x_{kj}^{(2)} c_{kij + 1}^{2 (l)} + c_{ki + 1 j + 1}^{0 (l)} \\ - x_{ki}^{(1)} c_{ki + 1 j + 1}^{1 (l)} - x_{kj}^{(2)} c_{ki + 1 j + 1}^{2 (l)}) / h_{k 1} h_{k 2}, \\ a_{k, l 5}^{ij} = (- x_{ki + 1}^{(1)} c_{kij}^{2 (l)} + x_{ki}^{(1)} c_{ki + 1 j}^{2 (l)} \\ + x_{ki + 1}^{(1)} c_{kij + 1}^{2 (l)} - x_{ki}^{(1)} c_{ki + 1 j + 1}^{2 (l)}) / h_{k 1} h_{k 2}, \\ a_{k, l 6}^{ij} = (c_{kij}^{1 (l)} - c_{ki + 1 j}^{1 (l)} - c_{kij + 1}^{1 (l)} + c_{ki + 1 j + 1}^{1 (l)}) / h_{k 1} h_{k 2}, \\ a_{k, l 7}^{ij} = (c_{kij}^{2 (l)} - c_{ki + 1 j}^{2 (l)} - c_{kij + 1}^{2 (l)} + c_{ki + 1 j + 1}^{2 (l)}) / h_{k 1} h_{k 2}, \end{matrix}$ $\begin{matrix} a_{k, l 8}^{ij} = (x_{ki + 1}^{(1)} x_{kj + 1}^{(2)} c_{kij}^{0 (l)} - x_{ki}^{(1)} x_{kj + 1}^{(2)} c_{ki + 1 j}^{0 (l)} \\ - x_{ki + 1}^{(1)} x_{kj}^{(2)} c_{kij + 1}^{0 (l)} + x_{ki}^{(1)} x_{kj}^{(2)} c_{ki + 1 j + 1}^{0 (l)}) / h_{k 1} h_{k 2} . \end{matrix}$

From this theorem we learn that the solutions of (13) and (14) can be solved piece by piece.

Example 4: In Example 3, let x₁ (t) = y (t) and $x_{2} (t) = \dot{y} (t)$ , then we derive the simulation state curves of x₁ (t) and x₂ (t) as shown in Figs. 4 and 5, respectively.

4 Comparison of different fuzzy inference modeling methods

In this section, we let Model I represent the model using fuzzy inference modeling method proposed by Li [11], Model II represent the model using fuzzy inference modeling method based on fuzzy transformation [13] and Model III represents the model using fuzzy inference modeling method based on T-S fuzzy system.

From the references [1 , 13], we have known that Model I is based on Mamdani fuzzy system f and it has approximation formula for original system g as ${∥ g - f ∥}_{\infty} \leq \frac{1}{8} h^{2} [{∥ \frac{\partial^{2} g}{\partial x^{2}} ∥}_{\infty} + {∥ \frac{\partial^{2} g}{\partial y^{2}} ∥}_{\infty}]$ (15)

Model II is based on the fuzzy system f of transformation and it has approximation formula for original system g as $\begin{matrix} {∥ g - f ∥}_{\infty} \leq \frac{1}{8} h^{2} [{∥ \frac{\partial^{2} g}{\partial x^{2}} ∥}_{\infty} + {∥ \frac{\partial^{2} g}{\partial y^{2}} ∥}_{\infty}] \\ + \frac{2}{3} h [{∥ \frac{\partial g}{\partial x} ∥}_{\infty} + {∥ \frac{\partial g}{\partial y} ∥}_{\infty}] \end{matrix}$ (16)

Model III is based on T-S fuzzy system f it has approximation formula for original system g as ${∥ g - f ∥}_{\infty} \leq \frac{1}{2} h^{2} [{∥ \frac{\partial^{2} g}{\partial x^{2}} ∥}_{\infty} + 2 {∥ \frac{\partial^{2} g}{\partial x \partial y} ∥}_{\infty} + {∥ \frac{\partial^{2} g}{\partial y^{2}} ∥}_{\infty}]$ (17)

We have seen that Model I and Model III are of the second order approximation accuracy and Model II has only the first order approximation accuracy. However, we cannot show which is better from formulas (15) and (17). In order to compare the three models, two simulation experiments under conditions of time-invariant and time-varying be given. By comparing the maximum approximation errors of the input-output models and state-space models, it turns out that the proposed method is effective and superior in approximating the equations.

4.1 Comparison of time-invariant system modeling methods

In order to evaluate approximation capability of the proposed method, different fuzzy inference modeling methods are used to simulate the Var der Pol equation. The set of initial values and the number of rules are the same as those of Example 1. We make careful comparison and overall summary of approximate effect of these models. For simplicity, the following notations are needed:

The simulation curves of the three models of the Var der Pol equation are similar to Figs. 1 and 2. In order to directly distinguish which model has more powerful ability to approach the Var der Pol equation, the maximum approximation errors of the input-output models and the state-space models are given in the Tables 1 and 2.

We can be seen in Tables 1 and 2 that the max errors of the model using fuzzy inference modeling method based on T-S fuzzy system is smaller than those of the model using fuzzy inference modeling method presented by Li and the model using fuzzy inference modeling method based on fuzzy transformation. The approximation effect is better than the other two methods, so fuzzy modeling method based on T-S fuzzy system has the superiority to some extent.

4.2 Comparison of time-varying system modeling

We simulate the second-order nonlinear time-varying differential equation $\ddot{y} (t) - \dot{y} (t)^{2} + 12 ty = 0$ by applying the models obtained from different fuzzy inference modeling methods. The set of initial values and numbers of rules are the same as Example 3. For convenience, some notations are stated as:

Model I^′ represents the model using fuzzy inference modeling method proposed by Li [12].

Model II^′ represents the model using fuzzy inference modeling method based on fuzzy transformation [13].

Model III^′ represents the model using fuzzy inference modeling method based on T-S fuzzy system.

The simulation curves of the three models of the nonlinear time-varying differential equation are similar to Figs. 4 and 5. The Tables 3 and 4 gives the maximum approximation errors of the input-output models and the state-space models to directly discern which model can achieve a better approximation effect of the original differential equation.

Tables 3 and 4 show us that the maximum errors of the model using fuzzy inference modeling method based on T-S fuzzy system is smaller than those of the model using fuzzy inference modeling method presented by Li and the model using fuzzy inference modeling method based on fuzzy transformation. The proposed method can approach the original differential equation in a better way. Hence, fuzzy modeling method based on T-S fuzzy system is advantageous to some degree.

5 Conclusion

This paper presented a fuzzy inference modeling method based on T-S fuzzy system. We established the time-invarying and time-variant second-order freedom movement systems models and simulated the Var der Pol equation and nonlinear time-varying differential equation by the obtained differential equation model. Moreover, we compared the simulation results of this model with another two models using the fuzzy inference modeling method presented by Li and the fuzzy inference modeling method based on fuzzy transformation. It is shown that the proposed method can better approximate the original equation.

Footnotes

Acknowledgments

This work was supported by the National Science Foundation of China(No.61473327).

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