Structure analysis and universal approximation of IT2 TS FLC using NT algorithm

Abstract

The analytical structure plays an important role in system design and stability analysis of FLC. Structure analysis of traditional IT2 TS FLCs using Zadeh min operator and KM algorithm requires multiple IC dividing, which results in complex calculation and cumbersome parameter adjustment. This article proposes a new IT2 TS FLC by adopting product-type operator and NT algorithm. The proposed controller has such advantages: 1)use product-type operator to skip the partitioning in fuzzy inference process;2) use NT algorithm to avoid determining switching points and sorting rule consequents in type-reduction process. Then, the controller is proved to be universal approximator and sufficient condition is deduced. Finally, we derive the analytical structure of the controller by substituting the parameters, and study the relationship between the uncertainty parameter θ and the analytical structure when the rule consequents are symmetric or asymmetric. Both the computational costs during operation and the computational workload for structural analysis can be reduced significantly by using the new FLC.

Keywords

NT algorithm IT2 TS FLC analytical structure universal approximation

1 Introduction

Fuzzy Logic Controllers (FLCs) have demonstrated strong performance in controlling nonlinear systems with uncertainties [1-3]. However, FLCs have often been treated as black box controllers, resulting in incomplete theories to support system analysis and design. Parameter adjustments for FLCs have traditionally been completed through trial and error. To effectively analyze and design FLCs, it is necessary to obtain their explicit analytical structures. The analytical structure refers to the mathematical expression that precisely describes the FLC’s input-output mapping relationship [4]. Interval Type-2 Fuzzy Logic Controllers (IT2 FLCs) are much more complex than Type-1 Fuzzy Logic Controllers (T1 FLCs), which has led to most existing research on analytical structure being aimed at type-1 T1 FLCs [5-8].

To derive the analytical structure of IT2 FLCs, researchers have used the method of dividing the Input Cases (ICs) according to the input membership function and the Karnik-Mendel (KM) algorithm switching point, and then deducing the analytical structure of each IC separately [9]. By employing the method of partitioning Input Cases (ICs), researchers have successfully obtained the analytical structures of different IT2 Mamdani FLCs and have substantiated the assertion that T2 fuzzy controllers with two input variables and piecewise linear input fuzzy sets can be represented as the sum of two nonlinear PI (or PD) controllers [9-11]. TS FLCs can be viewed as an extension of Mamdani FLCs, making their analysis more challenging. An enhanced method was proposed for analyzing IT2 TS FLCs by dissecting the ICs into three components: input fuzzy sets, rule consequents, and the type reducer [12]. This approach led to the successful derivation of the structures of IT2 TS FLCs with various configurations [12-14]. The aforementioned studies primarily focused on FLCs utilizing Zadeh min operations, while FLCs employing product-type operations are discussed in [15].

Meanwhile, the analytical structure of IT2 FLC is highly correlated with the Footprint Of Uncertainty(FOU) of input membership. When using the traditional triangular input membership function, changing the uncertainty parameter θ will highly affected the analytical structure of the IT2 FLC. Scholars have studied the influence of tuning uncertainty parameter θ of the input membership function on the analytical structure of various types of FLCs, such as IT2 Mamdani FLCs, IT2 TS FLCs, typical IT2 fuzzy PI and PD controllers, and so on [14, 16-18].

As described, an IT2 TS FLC needs to be divided three times, and the dividing results are superimposed to obtain the final ICs, which can reach dozens or hundreds of partitions, and the analytical expression of each IC needs to be deduced independently. Furthermore, changes in the membership function will lead to the re-partitioning of ICs. These two reasons make the structure analysis of IT2 TS FLC very cumbersome and computationally expensive. In response to this challenge, this paper proposes an IT2 TS FLC using product-type operator and Nie-Tan (NT) algorithm [19]. This controller uses the product-type operator so that the input membership function does not need to be partitioned, and NT algorithm is used to avoid sorting the rule consequents and determining the switching point. Therefore, there is no need for partitioning, and the final analytical structure of the controller has only one IC.

The rest of the paper is organized as follows. Section 2 provides the configuration and general structure of the proposed FLC, and advantages are introduced. In Section 3, this FLC is proven to be a universal approximator. Proposed FLC using a typical input membership function and rule consequents is derived, and the influence on analytical structure while changing the uncertainty parameter θ is discussed in Section 4. The last section concludes the paper.

2 Configurations, general structure and advantages

The controller uses product-type operator, NT method, with linear consequents, and has two inputs and one output. Without loss of generality, we assume that -1≤x₁≤1,-1≤x₂≤1. Each input is composed of N = 2n+1 fuzzy sets. The fuzzy sets of input x₁ are A_-n, …, A₀, …, A_n, and the fuzzy sets of input x₂ are B_-n, …, B₀, …, B_n. Among them, A_-n is defined on [-1,- $\frac{n - 1}{n}$ ], A_n is defined on [ $\frac{n - 1}{n}, 1$ ], and the rest of A_i are defined on [ $\frac{i - 1}{n}$ , $\frac{i + 1}{n}$ ] (-n≤i≤n). B_i is defined similarly. Therefore, when x₁ belongs to a certain fuzzy set A_i, $\frac{i - 1}{n} \leq x_{1} \leq \frac{i + 1}{n}$ .

Fig. 1

Flow chart.

The i^th rule of controller is (1≤i≤N):

If x₁ is A_{p
_i} and x₂ is B_{q
_i}, then u_i = d_ix₁ + e_ix₂ + c_i.

Where d_i, e_i, c_i, p_i, q_i are constants, p_i and q_i are integers between [-n,n].

When FLC using a product-type operator, for the i^th rule, its firing interval is $\begin{matrix} \begin{matrix} f_{i} (x_{1}, x_{2}) = [\underline{f_{i}} (x_{1}, x_{2}), {\bar{f}}_{i} (x_{1}, x_{2})] = \\ [{\underline{μ}}_{A_{p_{i}}} (x_{1}) {\underline{μ}}_{B_{q_{i}}} (x_{2}), {\bar{μ}}_{A_{p_{i}}} (x_{1}) {\bar{μ}}_{B_{q_{i}}} (x_{2})] \end{matrix} \end{matrix}$

Using the NT algorithm, we could replace the firing interval by the average firing level: $\begin{matrix} \begin{matrix} f_{i}^{*} (x_{1}, x_{2}) = \frac{1}{2} (\underline{f_{i}} (x_{1}, x_{2}) + {\bar{f}}_{i} (x_{1}, x_{2})) \\ = \frac{1}{2} ({\underline{μ}}_{A_{p_{i}}} (x_{1}) {\underline{μ}}_{B_{q_{i}}} (x_{2}) + {\bar{μ}}_{A_{p_{i}}} (x_{1}) {\bar{μ}}_{B_{q_{i}}} (x_{2})) \end{matrix} \end{matrix}$ Using the center of sets method, the final output of the TS fuzzy logic controller is:

Compared to the existing IT2 TS FLC, this controller has two distinct advantages: 1. Significantly lower computation cost during operation. 2. Much less computational effort required for structural analysis.

1. Computation cost: When using the Zadeh min operator, it requires 2 N comparison operations, and when using the product operator, it requires 2 N computational operations. According to references [19, 21-24], compared to traditional algorithms like KM algorithm, the NT algorithm has a much faster computation speed. Moreover, as the number of input sets increase, the advantage of the NT algorithm becomes even more pronounced. This is attributed to several characteristics of the NT algorithm: no need for sorting, no need for iteration.

Even with the more efficient EKM algorithm, obtaining u_l and u_r requires (3N+4|[N/2.4-L]|) (3N++4|[N/2.4-R]|) computational operations and (2N+2mN) comparison operations [23]. When using the NT algorithm, the required number of computations is 4N+1 [19]. The comparison is summarized in the Table 1.

Table 1

Computational complexity

FLC	Calculations	Comparisons
Traditional	6N+4\|[N/2.4-L]\|+4\|[N/2.4-R]\|+1	2N
Proposed	6N+1	0

From the table, it can be seen that even without considering the sorting operation required by the traditional FLC, the computational cost of the proposed FLC is still lower than the existing one.

2. Structural analysis: A traditional FLC was divided into 102 partitions during structural analysis [13]. This means that when using methods like constructing Lyapunov function to ensure the stability of the system, each partition needs to be modeled and designed separately. Finally, all the obtained conditions are superimposed. This not only leads to a very large computational burden but also may result in conflicts between conditions. In fact, this is one of the main challenges in designing T2 FLC. On the other hand, when using the proposed FLC, only one partition is obtained. This means that when designing the controller using methods like Lyapunov functions, the required computational cost is only 0.98% of the original, making the analysis and design of type-2 fuzzy controllers possible.

3 Universal approximation

In this chapter, we will prove that the IT2 FLC with an analytical structure like Eq.(1) is an universal approximator, meaning it can approximate any two-variable continuous function with any desired accuracy in a compact closed domain. The proof method used in this paper is based on the proof of the universal approximation property of T1 TS fuzzy controllers proposed by Ying Hao [20]. The controller can approximate any continuous polynomial.

First, we prove that F_n (x₁, x₂) can approximate any continuous polynomial P_d (x₁, x₂) defined on [-1,1] with any desired accuracy.

Without loss of generality, we assume that $\begin{matrix} P_{d} (x) = P_{d} (x_{1}, x_{2}) = \sum_{i = 0}^{d} α_{i} x_{1}^{i} + \sum_{i = 0}^{d} β_{i} x_{2}^{i} \end{matrix}$ is a d-order polynomial defined on [-1,1].

Theorem 1: F_n (x₁, x₂) could approximate P_d (x₁, x₂) with arbitrarily high precision.

Proof:

Since P_d ( x ) has been given, we could construct i^th rule as (1≤i≤N).

If x₁ is A_i and x₂ is B_i, then $u_{i} = P_{d} (\frac{p_{i}}{n}, \frac{q_{i}}{n}) - α_{1} \frac{p_{i}}{n} - β_{1} \frac{q_{i}}{n} + α_{1} x_{1} + β_{1} x_{2}$ .

And we could calculate the final output F_n (x₁, x₂) as:

$\begin{matrix} F_{n} (x_{1}, x_{2}) = \frac{\sum_{i = 1}^{N} \frac{1}{2} ({\underline{μ}}_{A_{p_{i}}} (x_{1}) {\underline{μ}}_{B_{q_{i}}} (x_{2}) + {\bar{μ}}_{A_{p_{i}}} (x_{1}) {\bar{μ}}_{B_{q_{i}}} (x_{2})) (P_{d} (\frac{p_{i}}{n}, \frac{q_{i}}{n}) - α_{1} \frac{p_{i}}{n} - β_{1} \frac{q_{i}}{n} + α_{1} x_{1} + β_{1} x_{2})}{\sum_{i = 1}^{n} \frac{1}{2} ({\underline{μ}}_{A_{p_{i}}} (x_{1}) {\underline{μ}}_{B_{q_{i}}} (x_{2}) + {\bar{μ}}_{A_{p_{i}}} (x_{1}) {\bar{μ}}_{B_{q_{i}}} (x_{2}))} \\ = \frac{\sum_{i = 1}^{N} f_{i}^{*} (x_{1}, x_{2}) (P_{d} (\frac{p_{i}}{n}, \frac{q_{i}}{n}) - α_{1} \frac{p_{i}}{n} - β_{1} \frac{q_{i}}{n} + α_{1} x_{1} + β_{1} x_{2})}{\sum_{i = 1}^{n} f_{i}^{*} (x_{1}, x_{2})} \end{matrix}$ (2)

As aforementioned, $\begin{matrix} \begin{matrix} \frac{p_{i} - 1}{n} \leq x_{1} \leq \frac{p_{i} + 1}{n} \\ lim_{n \to \infty} \frac{p_{i} - 1}{n} \leq x_{1} \leq lim_{n \to \infty} \frac{p_{i} + 1}{n} \\ lim_{n \to \infty} \frac{p_{i}}{n} = x_{1} \end{matrix} \end{matrix}$ and similarly, $\begin{matrix} lim_{n \to \infty} \frac{q_{i}}{n} = x_{2} \end{matrix}$ Therefore, $\begin{matrix} \begin{matrix} \lim_{n \to \infty} P_{d} (\frac{p_{i}}{n}, \frac{q_{i}}{n}) = P_{d} (x_{1}, x_{2}) \\ \lim_{n \to \infty} α_{1} \frac{p_{i}}{n} = α_{1} x_{1} \\ \lim_{n \to \infty} β_{1} \frac{q_{i}}{n} = β_{1} x_{2} \end{matrix} \end{matrix}$ Thus, $\begin{matrix} \begin{matrix} \lim_{n \to \infty} F_{n} (x_{1}, x_{2}) \\ = \frac{\sum_{i = 1}^{N} f_{i}^{*} (x_{1}, x_{2}) (P_{d} (\frac{p_{i}}{n}, \frac{q_{i}}{n}) - α_{1} \frac{p_{i}}{n} - β_{1} \frac{q_{i}}{n} + α_{1} x_{1} + β_{1} x_{2})}{\sum_{i = 1}^{n} f_{i}^{*} (x_{1}, x_{2})} \\ = P_{d} (x_{1}, x_{2}) \end{matrix} \end{matrix}$ which means when n→ ∞, F_n (x₁, x₂) could approximate P_d (x₁, x₂).

Next we will prove that F_n (x₁, x₂) could approximate P_d (x₁, x₂) uniformly. When an approximation error ɛ is given, the least n required could be calculated as: $\begin{matrix} n > \frac{| α_{1} | + | β_{1} | + \sum_{i = 0}^{d} | α_{i} | i + \sum_{i = 0}^{d} | β_{i} | i}{ɛ} \end{matrix}$

Proof:

$\begin{matrix} | | F_{n} (x_{1}, x_{2}) - P_{d} (x_{1}, x_{2}) | | \\ = max | F_{n} (x_{1}, x_{2}) - P_{d} (x_{1}, x_{2}) | \\ = max | \frac{\sum_{i = 1}^{N} f_{i}^{*} (x_{1}, x_{2}) (P_{d} (\frac{p_{i}}{n}, \frac{q_{i}}{n}) - α_{1} \frac{p_{i}}{n} - β_{1} \frac{q_{i}}{n} + α_{1} x_{1} + β_{1} x_{2})}{\sum_{i = 1}^{n} f_{i}^{*} (x_{1}, x_{2})} - P_{d} (x_{1}, x_{2}) | \\ = max | \frac{\sum_{i = 1}^{N} f_{i}^{*} (x_{1}, x_{2}) (P_{d} (\frac{p_{i}}{n}, \frac{q_{i}}{n}) - α_{1} \frac{p_{i}}{n} - β_{1} \frac{q_{i}}{n} + α_{1} x_{1} + β_{1} x_{2} - P_{d} (x_{1}, x_{2}))}{\sum_{i = 1}^{n} f_{i}^{*} (x_{1}, x_{2})} | \end{matrix}$ (3) In Equation (3) $\begin{matrix} \begin{matrix} | P_{d} (\frac{p_{i}}{n}, \frac{q_{i}}{n}) - α_{1} \frac{p_{i}}{n} - β_{1} \frac{q_{i}}{n} + α_{1} x_{1} \\ + β_{1} x_{2} - P_{d} (x_{1}, x_{2}) | \\ \leq | P_{d} (\frac{p_{i}}{n}, \frac{q_{i}}{n}) - P_{d} (x_{1}, x_{2}) \\ | + | α_{1} (x_{1} - \frac{p_{i}}{n}) | + | β_{1} (x_{2} - \frac{q_{i}}{n}) | \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} | α_{1} (x_{1} - \frac{p_{i}}{n}) | < | \frac{α_{1}}{n} | | β_{1} (x_{2} - \frac{q_{i}}{n}) | < | \frac{β_{1}}{n} | \\ | P_{d} (\frac{p_{i}}{n}, \frac{q_{i}}{n}) - P_{d} (x_{1}, x_{2}) | \\ = | \sum_{i = 0}^{d} α_{i} (x_{1}^{i} - {(\frac{p_{i}}{n})}^{i}) + \sum_{i = 0}^{d} β_{i} (x_{2}^{i} - {(\frac{q_{i}}{n})}^{i}) | \\ \leq \sum_{i = 0}^{d} | α_{i} | | x_{1}^{i} - {(\frac{p_{i}}{n})}^{i} | + \sum_{i = 0}^{d} | β_{i} | | x_{2}^{i} - {(\frac{q_{i}}{n})}^{i} | \end{matrix} \end{matrix}$

Since |x_i| ≤ 1, $| \frac{p_{i}}{n} | \leq 1$ , so $| {(x_{i})}^{a} {(\frac{p_{i}}{n})}^{b} | \leq 1$ (a,b are natural number) $\begin{matrix} \begin{matrix} \sum_{i = 0}^{d} | α_{i} | | x_{1}^{i} - {(\frac{p_{i}}{n})}^{i} | \leq \sum_{i = 0}^{d} | α_{i} | | x_{1} - \frac{p_{i}}{n} | \\ (| x_{1}^{i - 1} | + | x_{1}^{i - 2} {(\frac{p_{i}}{n})}^{1} | + \dots + | {(\frac{p_{i}}{n})}^{i - 1} |) \\ \leq \sum_{i = 0}^{d} | α_{i} | \frac{1}{n} | i | \end{matrix} \end{matrix}$ Similarly, $\begin{matrix} \sum_{i = 0}^{d} | β_{i} | | x_{2}^{i} - {(\frac{q_{i}}{n})}^{i} | \leq \sum_{i = 0}^{d} | β_{i} | \frac{i}{n} \end{matrix}$ In order to make $\begin{matrix} max | F_{n} (x_{1}, x_{2}) - P_{d} (x_{1}, x_{2}) | < ɛ \end{matrix}$ Which means $\begin{matrix} ɛ > | \frac{α_{1}}{n} | + | \frac{β_{1}}{n} | + \sum_{i = 0}^{d} | α_{i} | \frac{i}{n} + \sum_{i = 0}^{d} | β_{i} | \frac{i}{n} \end{matrix}$

We need n as

$\begin{matrix} n > \frac{| α_{1} | + | β_{1} | + \sum_{i = 0}^{d} | α_{i} | i + \sum_{i = 0}^{d} | β_{i} | i}{ɛ} \end{matrix}$ (4)

3.1 The controller can approximate any continuous function

Theorem 2: F_n (x₁, x₂) can approximate any continuous function G (x₁, x₂) with any desired accuracy.

Weierstrass approximation theorem [20]: To any continuous function G (x₁, x₂) on a closed interval, given a approximation error bound ɛ > 0, there always exists a polynomial P_d (x₁, x₂) that can approximate G (x₁, x₂) G (x₁, x₂) uniformly with the desired accuracy. Generally, the smaller the ɛ, the higher the polynomial degree.

According to the Weierstrass approximation theorem, polynomial P_d (x₁, x₂) can uniformly approximate G (x₁, x₂) with arbitrary accuracy. Meanwhile, F_n (x₁, x₂) can approximate P_d (x₁, x₂) uniformly with the desired accuracy. That is, F_n (x₁, x₂) can approximate any continuous function G (x₁, x₂) uniformly with the desired accuracy.

Given an approximation error bound ɛ > 0, let ||P_d (x₁, x₂) - G (x₁, x₂)|| < ɛ₁, ||F_n (x₁, x₂) - P_d (x₁, x₂) || < ɛ₂, ɛ = ɛ₁ + ɛ₂. Replace the ɛ in Equation (4) with ɛ₂, then we get the sufficient condition for the universal approximation as

$\begin{matrix} n > \frac{| α_{1} | + | β_{1} | + \sum_{i = 0}^{d} | α_{i} | i + \sum_{i = 0}^{d} | β_{i} | i}{ɛ_{2}} \end{matrix}$ (5)

4 Influence of changing the uncertainty parameter θ on analytical structure

In this section, we will substitute the parameters and configurations to derive the analytical structure of the new controller, and obtain the influence of changing the uncertainty parameter θ on the controller output when the rule consequents are symmetric and asymmetric.

Input x₁ and x₂ are triangular membership functions with the domain of [–1, 1], and the θ₁ and θ₂ represent the uncertainty parameters of two inputs respectively. $\begin{matrix} \begin{matrix} {\bar{μ}}_{A_{1}} (x_{1}) = \frac{θ_{1} - 1}{2} x_{1} + \frac{1 + θ_{1}}{2} \\ {\underline{μ}}_{A_{1}} (x_{1}) = \frac{θ_{1} - 1}{2} x_{1} + \frac{1 - θ_{1}}{2} \\ {\bar{μ}}_{A_{2}} (x_{1}) = \frac{1 - θ_{1}}{2} x_{1} + \frac{1 + θ_{1}}{2} \\ {\underline{μ}}_{A_{2}} (x_{1}) = \frac{1 - θ_{1}}{2} x_{1} + \frac{1 - θ_{1}}{2} \\ {\bar{μ}}_{B_{1}} (x_{2}) = \frac{θ_{2} - 1}{2} x_{2} + \frac{1 + θ_{2}}{2} \\ {\underline{μ}}_{B_{1}} (x_{2}) = \frac{θ_{2} - 1}{2} x_{2} + \frac{1 - θ_{2}}{2} \\ {\bar{μ}}_{B_{2}} (x_{2}) = \frac{1 - θ_{2}}{2} x_{2} + \frac{1 + θ_{2}}{2} \\ {\underline{μ}}_{B_{2}} (x_{2}) = \frac{1 - θ_{2}}{2} x_{2} + \frac{1 - θ_{2}}{2} \end{matrix} \end{matrix}$

Fig. 2

Membership functions of x₁.

Fig. 3

Membership functions of x₂.

Rules of controller:

Rule#1: If x₁ is A₁ and x₂ is B₁, then u₁ = d₁x₁ + e₁x₂ + c₁.

Rule#2: If x₁ is A₁ and x₂ is B₂, then u₂ = d₂x₁ + e₂x₂ + c₂.

Rule#3: If x₁ is A₂ and x₂ is B₁, then u₃ = d₃x₁ + e₃x₂ + c₃.

Rule#4: If x₁ is A₂ and x₂ is B₂, then u₄ = d₄x₁ + e₄x₂ + c₄.

Firing level of each rule: $\begin{matrix} \begin{matrix} f_{1}^{*} (x_{1}, x_{2}) = \frac{1}{2} [(\frac{θ_{1} - 1}{2} x_{1} + \frac{1 + θ_{1}}{2}) (\frac{θ_{2} - 1}{2} x_{2} + \frac{1 + θ_{2}}{2}) + (\frac{θ_{1} - 1}{2} x_{1} + \frac{1 - θ_{1}}{2}) (\frac{θ_{2} - 1}{2} x_{2} + \frac{1 - θ_{2}}{2})] \\ = \frac{1}{2} (2 \frac{θ_{1} - 1}{2} \frac{θ_{2} - 1}{2} x_{1} x_{2} + \frac{θ_{1} - 1}{2} x_{1} + \frac{θ_{2} - 1}{2} x_{2} + \frac{1 + θ_{1} θ_{2}}{2}) \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} f_{2}^{*} (x_{1}, x_{2}) = \frac{1}{2} [(\frac{θ_{1} - 1}{2} x_{1} + \frac{1 + θ_{1}}{2}) (\frac{1 - θ_{2}}{2} x_{2} + \frac{1 + θ_{2}}{2}) + (\frac{θ_{1} - 1}{2} x_{1} + \frac{1 - θ_{1}}{2}) (\frac{1 - θ_{2}}{2} x_{2} + \frac{1 - θ_{2}}{2})] \\ = \frac{1}{2} (2 \frac{θ_{1} - 1}{2} \frac{1 - θ_{2}}{2} x_{1} x_{2} + \frac{θ_{1} - 1}{2} x_{1} - \frac{θ_{2} - 1}{2} x_{2} + \frac{1 + θ_{1} θ_{2}}{2}) \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} f_{3}^{*} (x_{1}, x_{2}) = \frac{1}{2} [(\frac{1 - θ_{1}}{2} x_{1} + \frac{1 + θ_{1}}{2}) (\frac{θ_{2} - 1}{2} x_{2} + \frac{1 + θ_{2}}{2}) + (\frac{1 - θ_{1}}{2} x_{1} + \frac{1 - θ_{1}}{2}) (\frac{θ_{2} - 1}{2} x_{2} + \frac{1 - θ_{2}}{2})] \\ = \frac{1}{2} (2 \frac{1 - θ_{1}}{2} \frac{θ_{2} - 1}{2} x_{1} x_{2} - \frac{θ_{1} - 1}{2} x_{1} + \frac{θ_{2} - 1}{2} x_{2} + \frac{1 + θ_{1} θ_{2}}{2}) \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} f_{4}^{*} (x_{1}, x_{2}) = \frac{1}{2} [(\frac{1 - θ_{1}}{2} x_{1} + \frac{1 + θ_{1}}{2}) (\frac{1 - θ_{2}}{2} x_{2} + \frac{1 + θ_{2}}{2}) + (\frac{1 - θ_{1}}{2} x_{1} + \frac{1 - θ_{1}}{2}) (\frac{1 - θ_{2}}{2} x_{2} + \frac{1 - θ_{2}}{2})] \\ = \frac{1}{2} (2 \frac{θ_{1} - 1}{2} \frac{θ_{2} - 1}{2} x_{1} x_{2} - \frac{θ_{1} - 1}{2} x_{1} - \frac{θ_{2} - 1}{2} x_{2} + \frac{1 + θ_{1} θ_{2}}{2}) \end{matrix} \end{matrix}$

4.1 Rule consequents are asymmetric

When the rule consequents are asymmetric, in other words, u₂ = - u₃, u₁ = - u₄, the final output of FLC is

$\begin{matrix} \begin{matrix} F_{n} (x_{1}, x_{2}) = \frac{\begin{matrix} (θ_{1} - 1) (d_{1} + d_{2}) {x_{1}}^{2} + (θ_{2} - 1) (e_{1} - e_{2}) {x_{2}}^{2} \\ + [(θ_{1} - 1) (e_{1} + e_{2}) + (θ_{2} - 1) (d_{1} - d_{2})] x_{1} x_{2} \\ + (θ_{1} - 1) (c_{1} + c_{2}) x_{1} + (θ_{2} - 1) (c_{1} - c_{2}) x_{2} \end{matrix}}{2 (1 + θ_{1} θ_{2})} \\ = \frac{(θ_{1} - 1)}{2 (1 + θ_{1} θ_{2})} [(d_{1} + d_{2}) {x_{1}}^{2} + (e_{1} + e_{2}) x_{1} x_{2} + (c_{1} + c_{2}) x_{1}] \\ + \frac{(θ_{2} - 1)}{2 (1 + θ_{1} θ_{2})} [(e_{1} - e_{2}) {x_{2}}^{2} + (d_{1} - d_{2}) x_{1} x_{2} + (c_{1} - c_{2}) x_{2}] \end{matrix} \end{matrix}$

According to the above equation, it can be seen that only when d₁ = d₂ = e₁ = e₂ = 0, which means the controller is Mamdani type, the output of the controller is a first-order controller. When any of d₁, d₂, e₁, e₂ is not equal to 0, the analytical structure of the controller contains second-order polynomials of x₁ and x₂.

7.4 cm 0.5pt

Assume $\begin{matrix} \frac{(θ_{1} - 1)}{2 (1 + θ_{1} θ_{2})} = K_{1}, \frac{(θ_{2} - 1)}{2 (1 + θ_{1} θ_{2})} = K_{2} \end{matrix}$

When θ₂ is constant, $\begin{matrix} \frac{d K_{1}}{d θ_{1}} = \frac{1 + θ_{2}}{2 {(1 + θ_{1} θ_{2})}^{2}} \end{matrix}$

When θ₁, θ₂ ∈ [0, 1], $\begin{matrix} \frac{1 + θ_{2}}{2 {(1 + θ_{1} θ_{2})}^{2}} > 0 \end{matrix}$

Therefore, K₁ is monotonically increasing, with a minimum value of $- \frac{1}{2}$ at θ₁=0 and a maximum value of 0 at θ₁ =1. Similarly, K₂ is monotonically increasing, with a minimum value of $- \frac{1}{2}$ at θ₂=0 and a maximum value of 0 at θ₂=1.

This indicates that as uncertainty θ increases, the absolute values of the coefficients in the controller output will continuously decrease, eventually leading to a controller output of 0.

4.2 Rule consequents are symmetric

When the rule consequents are symmetric, in other words, u₂ = u₃, u₁ = u₄, the final output of FLC is $\begin{matrix} \begin{matrix} F_{n} (x_{1}, x_{2}) = \frac{\begin{matrix} (θ_{1} - 1) (θ_{2} - 1) [\begin{matrix} (d_{1} + d_{2}) {x_{1}}^{2} (x_{1} x_{2}) \\ + (e_{1} + e_{2}) x_{1} {x_{2}}^{2} + (c_{1} + c_{2}) x_{1} x_{2} \end{matrix}] \\ + (1 + θ_{1} θ_{2}) [(d_{1} + d_{2}) x_{1} + (e_{1} + e_{2}) x_{2} + (c_{1} + c_{2})] \end{matrix}}{2 (1 + θ_{1} θ_{2})} \\ = \frac{(θ_{1} - 1) (θ_{2} - 1)}{2 (1 + θ_{1} θ_{2})} [(d_{1} + d_{2}) {x_{1}}^{2} (x_{1} x_{2}) + (e_{1} + e_{2}) x_{1} {x_{2}}^{2} + (c_{1} + c_{2}) x_{1} x_{2}] \\ + \frac{1}{2} [(d_{1} + d_{2}) x_{1} + (e_{1} + e_{2}) x_{2} + (c_{1} + c_{2})] \end{matrix} \end{matrix}$

Assume $\begin{matrix} \frac{(θ_{1} - 1) (θ_{2} - 1)}{2 (1 + θ_{1} θ_{2})} = K_{s} \end{matrix}$ and $\begin{matrix} \begin{matrix} \frac{d K_{s}}{d θ_{1}} = \frac{(θ_{2} - 1) (1 + θ_{2})}{2 {(1 + θ_{1} θ_{2})}^{2}} \\ \frac{d K_{s}}{d θ_{1}} = \frac{(θ_{1} - 1) (1 + θ_{1})}{2 {(1 + θ_{1} θ_{2})}^{2}} \end{matrix} \end{matrix}$

When θ₁, θ₂ ∈ [0, 1], the value of K_s will decrease continuously as θ₁ and θ₂ increase, with the maximum value of $\frac{1}{2}$ when both θ₁ and θ₂ are 0, and the minimum value of 0 when both θ₁ and θ₂ are 1.

As the uncertainty in the input functions increases, the coefficients of the higher-order terms in the controller output will decrease. When the uncertainty reaches its maximum value, the controller output will only contain the linear term and the constant term. This means that the controller will turn to a first-order controller, which is less complex but may also have less control authority.

4.3 Simulation

This section will use the proposed controller to control a typical second-order system. The system under consideration is a spring-damper system with m = 1, k = 25, and c = 5. The transfer function of the system is $\begin{matrix} G (s) = \frac{1}{s^{2} + 5 s + 25} \end{matrix}$

Rule consequents of FLC are symmetric, while θ₁=θ₁=1. d₁=d₂=100, e₁=e₂=40, c₁=c₂=0.

Therefore, $\begin{matrix} \begin{matrix} \frac{(θ_{1} - 1) (θ_{2} - 1)}{2 (1 + θ_{1} θ_{2})} = 0 \\ F_{n} (x_{1}, x_{2}) = 100 x_{1} + 40 x_{2} \end{matrix} \end{matrix}$

While x₁ is error, x₂ is error integral. The FLC is a PI controller.

The sampling time is 0.01 second. The simulation result is shown in Fig. 4.

Fig. 4

Simulation result.

5 Conclusion

This article presents an IT2 TS FLC using product-type operator and NT algorithm. The universal approximation property of the controller is proven, and sufficient conditions under given approximation error are provided. The analytical structure of the controller is derived by substituting parameters, and the influence of uncertainty parameter θ on the analytical structure is discussed for output results in symmetric and antisymmetric cases.

Compared to an IT2 TS FLC that typically uses the Zadeh “and” operator and the KM algorithm, the advantages of this controller are: 1) the application of a product-type operator eliminates the need for partitioning during the inference process; 2) the application of the NT algorithm eliminates the need to find switching points and sort rule consequents. The final analytical structure has only one IC, making it easy for analyzing and designing. Furthermore, it is proven that this controller is a universal approximator, and the sufficient condition is given. Finally, the influence of uncertainty θ on the analytical structure can serve as a reference in the design of the controller and input membership function parameters.

The FLC proposed in this article, compared to existing FLC, significantly simplifies the computational burden for performance analysis and parameter designing. Additionally, the proposed FLC has lower computational requirements, leading to improved real-time performance. Furthermore, there is a significant reduction in the computational cost for deriving controller parameters from the controlled object, which have positive implications for the research of fuzzy control theory.

This article has some limitations. For instance, it does not discuss controllers under multiple fuzzy sets and complex rules. It only addresses the sufficient conditions for universal approximation and does not discuss the necessary conditions. The proposed structure could be extended to generalized type-2 fuzzy controllers. These are the issues that need to be addressed in future research.

Funding

This work was supported in part by the Natural Science Foundation of Hunan Province of China under Grant 2022JJ30104, Research Foundation of Education Bureau of Hunan Province of China under Grant 21A0439 (both hosted by Zuqiang Long, the First author of paper).

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