The fuzzy system based on vague partitions and its application to path tracking control for autonomous vehicles

Abstract

As significant carriers of the application of fuzzy set theories, fuzzy systems have been widely used in many fields. However, selecting fuzzifications, fuzzy reasoning engines, and defuzzifications is subjective for Mamdani fuzzy systems, and the fuzzy rule of Takagi-Sugeno-Kang fuzzy systems is less of a linguistic interpretation. Regarding these shortcomings, this paper proposes a fuzzy system based on vague partitions processing information directly from the fuzzy rule base, in which fuzzy rules have explicit semantics. Firstly, the n-dimensional vague partition of the n-dimensional universe is defined based on 1-dimensional vague partitions and the aggregation function, and its properties are discussed. Based on these, we design the new fuzzy system, and investigate its approximation properties which is the theoretical guarantee for applying the fuzzy system. As an application, we combine the fuzzy system with PID control system to deal with autonomous vehicle path tracking control problems. A series of experiments are constructed, and experimental results indicate that the fuzzy system based on vague partitions makes the fuzzy PID control system strong robustness, and has obvious advantages compared with other traditional fuzzy systems for path tracking control problems.

Keywords

Fuzzy systems vague partitions aggregation functions fuzzy PID control path tracking

1 Introduction

As an intersectional field of robot technology and artificial intelligence technology, the automation of intelligent vehicles has attracted much attention in recent years. In this field, fuzzy set theory introduced by Zadeh [1 plays an important role. Fuzzy systems (FSs) are significant carriers of applying fuzzy set theories in autonomous vehicles 2-6].

Mamdani FS [7] is one of the most widely used FSs. The structure of Mamdani FS consists of fuzzification, defuzzification, fuzzy rule base, and fuzzy reasoning engine, where the fuzzy rule base is the core of the FS. For designing a FS, constructing membership functions on the input and output universe of discourses is a fundamental work. Thus, many scholars focus on the influence of different membership functions on the approximation ability of FSs. Zeng and Singh [8, 9] used triangular and trapezoidal membership functions as fuzzifications and investigated approximation properties of this FS. Chen and Gang [10] proposed the FS with cosine and quadratic polynomial membership functions, as well as discussed its universal approximation property and approximation accuracy. Yuan et al. [11] proposed parameter single fuzzification, and showed first-order and second-order accuracy of FS with this fuzzification. Subsequently, Jiang and Yuan [12] designed a FS with pyramid membership functions constructed on two dimensions to deal with multi-input single-output (MISO) problems. The fuzzy reasoning engine is a decision making logic that outputs the fuzzy set corresponding to the fuzzy input of the FS based on the fuzzy rule base. There have existed many alternative fuzzy reasoning methods available which can be used in FSs, and listed as follows. In 1973, Zadeh [13] proposed the famous compositional rule of inference (CRI) method to solve fuzzy modus ponens problems. However, this method lacks a strict logical basis. To compensate for the deficiency, Wang [14] proposed the triple I(3I) method. Subsequently, scholars researched this method in-depth [15-18]. Nerveless, neither CRI method nor 3I method has unconditional reducibility, where reducibility is a significant index to evaluate the deterioration of fuzzy inference. Therefore, subsethood infer subsethood algorithms [19] were proposed that satisfy the unconditional reducibility. With the continuous development of fuzzy set theory, scholars have improved fuzzy reasoning methods [20-22] from different perspectives. For the fuzzy set output from the fuzzy reasoning engine, defuzzification converts the fuzzy set to a crisp output. Up to now, there are four widely used types of defuzzifications: max-membership, centroid, weighted-average, and mean-max [24]. Although the fuzzifications, fuzzy reasoning methods, and defuzzifications listed proceeding have been successfully applied to various FSs, the selection of fuzzifications, fuzzy reasoning methods, and defuzzifications is still subjective when designing FSs. This leads to less objective results. Therefore, designing a FS that does not rely on fuzzifications, fuzzy reasoning methods, and defuzzifications is one of the motivations of this paper.

Takagi-Sugeno-Kang fuzzy system (TSK-FS) [25, 26] is another widely used FS. For this system, polynomials are used as the rule consequences, and then crisp outputs can be produced directly. Based on this characteristic, TSK-FS designed from data is computationally efficient and numerically accurate. Therefore, many TSK-FSs have been constructed. For example, Kukolj [27] constructed an adaptive fuzzy models generated by the training algorithms, Rezaee and Zarandi [28] designed a first order TSK-FS that can obtain the fuzzy rules without any assumption about the structure, Deng et al. [29] developed a scalable TSK-FS to deal with very large dates, Khater et al. [30] proposed an adaptive interval type-2 TSK-FS based on inforcement learning. Additionally, these TSK-FSs have been successfully applied in several fields such as medicine, engineer, transportation, etc [31-33]. Although improvements have been made in TSK-FSs, its shortcoming that the fuzzy rule of it can not offer linguistic interpretation still exists. Therefore, designing a FS that preserves the advantage of TSK-TSs processing information directly from fuzzy rule base, while also offering linguistic fuzzy rules is another motivation of this paper.

In order to design the desired FS, we first make vague partitions (VPs) for the input and output universes. Until now, different VP methods [34-40] have emerged. We choose the method proposed in [34, 35]. There are three reasons for us to choose this method. One is that Pan and Xu introduced the axiomatic definition of fuzzy sets based on VPs in [34, 35], which is an axiomatic system to govern membership degrees. Then fuzzy phenomena can be understood and described by axiomatic fuzzy sets. Another one is that, based on this VP, the fuzzy rule base is complete naturally. Additionally, if the fuzzy rule based on VPs is consistent, then the fuzzy rule base can be regarded as a mapping between two VPs. Furthermore, the crisp output for correspondingly crisp input can be obtained from the mapping between two VPs. The last one is that Shen et al. [41] introduced five typical VPs that improve the approximation ability of the single-input single-output (SISO) FS. If the input universe is n-dimensional, then aggregation functions [42] are employed to construct a n-dimensional VP on the input universe. Moreover, some properties of the 1-dimensional VP are preserved in the n-dimensional VP if appropriate aggregation functions are chosen.

Based on the above statements, the fuzzy system based on vague partitions (FS-VPs) for MISO model is proposed in this paper. For the proposed FS, it employs a fuzzy rule base to process information directly, and then a concise algorithm calculating the crisp output for the corresponding crisp input is given. Experiments in application to path tracking control for autonomous vehicles are constructed, and the results illustrate that the proposed FS performance better than some widely used FSs.

The main contributions of this paper are stated as below:

As a theoretical basis of the FS, the n-dimensional VP is defined and its properties with respect to aggregation functions are discussed.

The FS-VPs is proposed, which can process information directly from the fuzzy rule base. Compared with Mamdani FS, its structure is more straightforward. Compared with TSK-FS, its fuzzy rules have precise semantics. Furthermore, the algorithm calculating the crisp output for the corresponding crisp input is given.

As an application, the proposed FS is employed as a component in a fuzzy PID control system to deal with path tracking control problems for autonomous vehicles. A series of comparative experiments are conducted. These experimental results illustrate that the proposed FSs with discontinuous aggregation functions make the fuzzy PID control systems more robust than the fuzzy PID control systems consisting of existing FSs, and the proposed FSs with appropriate VPs and aggregation functions make the fuzzy PID control systems perform better than the fuzzy PID control systems consisting of existing FSs to tracking six basic elementary paths and complex paths.

The remaining content of this paper is organized as follows. Some concepts related to this study are briefly reviewed in Section 2. In Section 3, the n-dimensional VP of the n-dimensional universe is defined based on aggregation functions and 1-dimensional VPs, and some properties of n-dimensional are discussed. Section 4 displays the design process of the new FS, and its approximation properties are discussed theoretically. Experimental design and analysis of path tracking control for autonomous vehicles are presented in Section 5. Section 6 summarizes this work and looks forward to future research.

2 Preliminaries

In this section, some essential concepts of VPs and aggregation functions are recalled making the article more fluent, and detailed descriptions can be found in [35 , 47].

2.1 1-dimensional VP

In the sequel, the collection of nonzero natural numbers is denoted by $ℕ^{+}$ . For $n \in ℕ^{+}$ , the set {1, 2, ⋯ , n} is denoted by $\bar{n}$ . Then the definition of VP on real number intervals is as follows, which is called 1-dimensional VP.

Definition 2.1. [35] Let $U = [a, b] \subset ℝ$ . A VP of U is an object of the form $\tilde{U} = {A_{1}, \dots, A_{m}}, m \in ℕ^{+},$ where the membership degree of point x ∈ U that belongs to the class A_i is defined by the function μ_{A
_i} : U → [0, 1] , i = 1, ⋯ , m that satisfy the below conditions:

for arbitrary $i \in \bar{m}$ , μ_{A
_i} (x) is continuous on U;

for arbitrary $i \in \bar{m}$ , there is at least one x₀ ∈ U such that μ_{A
_i} (x₀) =1;

for arbitrary $i \in \bar{m}$ , if μ_{A
_i} (x₀) =1 for x₀ ∈ U, then μ_{A
_i} (x) is not decreasing on [a, x₀], and is not increasing on [x₀, b];

0 < μ_{A
₁} (x) + ⋯ + μ_{A
_m} (x) ≤1 holds for arbitrary x ∈ U.

A one-dimensional VP $\tilde{U}$ is called:

single-kernel if for all $i \in \bar{m}$ , there is only one point x₀ ∈ U such that μ_{A
_i} (x₀) =1;

regular if for arbitrary x ∈ U, μ_{A
₁} (x) + ⋯ + μ_{A
_m} (x) =1.

Shen et al. [41] introduced five typical 1-dimensional VPs which are single-kernel and regular: triangular VP, cosine VP, sine VP, exponential VP, and logarithmic VP. Based on the part (1) of Theorem 3.5 in [41], we have the following corollary without proof.

Corollary 2.2. Let $U = [a, b] \subseteq ℝ$ , $\tilde{U} = {A_{1}, \dots, A_{n}}$ , $n \in ℕ^{+}$ , be a regular single-kernel VP of U. Then for any x ∈ U, there are no more than two fuzzy sets with membership degrees more than 0. In particular, if there is a fuzzy set $A_{i} \in \tilde{U}$ satisfying μ_{A
_i} (x) =1 for x ∈ U, then there is only one fuzzy set with membership degree more than 0.

2.2 Aggregation functions

Definition 2.3. [42] An increasing function Θ : [0, 1] ⁿ → [0, 1] is called an n-ary aggregation function if it satisfies Θ (1, ⋯ , 1) =1 and Θ (0, ⋯ , 0) =0.

An aggregation function is continuous if it is continuous on each component.

Definition 2.4. [43] An n-ary aggregation function T : [0, 1] ⁿ → [0, 1] is a t-norm if it is commutative, associative and satisfies for any $i \in \bar{n}$ , T (1, ⋯ , 1, x_i, 1, ⋯ , 1) = x_i, where x_i is the i-th component of (1, ⋯ , 1, x_i, 1, ⋯ , 1) ∈ [0, 1] ⁿ.

Example 2.5. The following are four basic t-norms which are minimum, product, Łukasiewicz and drastic product, denoted by T_M, T_P, T_L and T_D respectively.

T_M (x₁, x₂) = min {x₁, x₂}.

T_P (x₁, x₂) = x₁ · x₂.

T_L (x₁ . x₂) = max {x₁ + x₂ - 1, 0}.

$T_{D} (x_{1}, x_{2}) = {\begin{matrix} 0, & (x_{1}, x_{2}) \in [0, 1)^{2}, \\ min {x_{1}, x_{2}}, & otherwise . \end{matrix}$

Definition 2.6. [44] An n-ary aggregation operator U : [0, 1] ⁿ → [0, 1] is an uninorm if it is commutative, associative, and satisfies for any $i \in \bar{n}$ and e ∈ [0, 1], U (e, ⋯ , e, x_i, e, ⋯ , e) = x_i, where x_i is the i-th component of (e, ⋯ , e, x_i, e, ⋯ , e) ∈ [0, 1] ⁿ.

If e = 1, then a uninorm is a t-norm.

Example 2.7. The following aggregation functions are uninorm but not t-norm.

$U_{1} (x_{1}, x_{2}) = {\begin{matrix} 2 x_{1} x_{2}, & if (x_{1}, x_{2}) \in [0, 0.5]^{2}, \\ x_{1} + x_{2} - 0.5 - (2 x_{1} - 1) (2 x_{2} - 1), & if (x_{1}, x_{2}) \in [0.5, 1]^{2}, \\ min {x_{1}, x_{2}}, & otherwise . \end{matrix}$

$U_{2} (x_{1}, x_{2}) = {\begin{matrix} 0.5 min {2 x_{1}, 2 x_{1}}, & if (x_{1}, x_{2}) \in [0, 0.5]^{2}, \\ 0.5 + 0.5 max {2 x_{1} - 1, 2 x_{2} - 1}, & if (x_{1}, x_{2}) \in [0.5, 1]^{2}, \\ min {x_{1}, x_{2}}, & otherwise . \end{matrix}$

$U_{3} (x_{1}, x_{2}) = {\begin{matrix} 0.5 max {2 x_{1} + 2 x_{1} - 1, 0}, & if (x_{1}, x_{2}) \in [0, 0.5]^{2}, \\ 0.5 + 0.5 min {2 x_{1} - 2 x_{2} - 2, 1}, & (x_{1}, x_{2}) \in [0.5, 1]^{2}, \\ min {x_{1}, x_{2}}, & otherwise . \end{matrix}$

Definition 2.8. [42] An n-ary aggregation function O : [0, 1] ⁿ → [0, 1] is an overlap function if it commutative, continuous, and satisfies

O (x₁, ⋯ , x_n) =0 if and only if $\overset{n}{\prod_{i = 1}} x_{i} = 0$ ,

O (x₁, ⋯ , x_n) =1 if and only if $\overset{n}{\prod_{i = 1}} x_{i} = 1$ .

Example 2.9. The following aggregation functions are overlap functions.

$O_{1}^{p} (x_{1}, x_{2}) = (x_{1} x_{2})^{p}$ , p > 0.

$O_{2}^{p} (x_{1}, x_{2}) = \frac{11}{12} min {x_{1}, x_{2}}^{p} + \frac{1}{12} (x_{1} x_{2})^{p}$ , p > 0.

$O_{3}^{p} (x_{1}, x_{3}) = sin (\frac{π}{2} (x_{1} x_{2})^{p})$ , p > 0.

3 The n-dimensional VP and its properties

In this section, we define the n-dimensional VP of the n-dimensional universe using the aggregation function to aggregate 1-dimensional VPs, and investigate its properties. Unless mentioned otherwise, we consider the 1-dimensional VP $\tilde{U}$ is regular and single-kernel. For the convenience of the following description, we define the order of the fuzzy sets in $\tilde{U}$ , and number the fuzzy sets in $\tilde{U}$ according to the order. Let $A_{i}, A_{j} \in \tilde{U}$ . Then there exist x_i, x_j ∈ U such that kerA_i = {x_i}, kerA_j = {x_j}. We say A_i < A_j if and only if x_i < x_j if and only if i < j.

Consider the n-dimensional universe D = U₁ × ⋯ × U_n, where $U_{1} = [a_{1}, b_{1}] \subset ℝ$ , ⋯, $U_{n} = [a_{n}, b_{n}] \subset ℝ$ . Then we can construct a VP $\tilde{D}$ on D by using n-ary aggregation functions to aggregate 1-dimensional VPs $\tilde{U_{1}}, \dots, \tilde{U_{n}}$ of U₁, ⋯ , U_n respectively, where $\tilde{U_{l}} = {A_{1}^{l}, \dots A_{m_{l}}^{l}}$ , $m_{l} \in ℕ^{+}$ , $l \in \bar{n}$ , which is called an n-dimensional VP.

Definition 3.1. Let $D = U_{1} \times \dots \times U_{n} \subset ℝ^{n}$ , $\tilde{U_{l}} = {A_{1}^{l}, \dots A_{m_{l}}^{l}}$ be a 1-dimensional VP of U_l, $m_{l} \in ℕ^{+}$ , $l \in \bar{n}$ . Then a n-dimensional VP of D is an object of the form $\tilde{D} = {A_{I} ∣ I = (i_{1}, \dots, i_{n}) \in \bar{m_{1}} \times \dots \times \bar{m_{n}}},$ and the membership degree of point X = (x₁, ⋯ , x_n) ∈ D that belongs to the class A_I is defined as $μ_{A_{I}} (X) = Θ (μ_{A_{i_{1}}^{1}} (x_{1}), \dots, μ_{A_{i_{n}}^{n}} (x_{n})),$ where Θ is an aggregation function.

In what follows, for the convenience of writing, A_I is often written as A_{i₁⋯i_n}. Concretely, we give an example to show how 1-dimensional VPs and the aggregation function determine the 2-dimensional VP.

Example 3.2. Let D = U₁ × U₂, where U₁ = [0, 1], U₂ = [0, 2]. The 1-dimensional VP of U₁ is $\tilde{U_{1}} = {A_{1}^{1}, A_{2}^{1}}$ , where $\begin{matrix} μ_{A_{1}^{1}} (x_{1}) = 1 - x_{1}, x_{1} \in [0, 1], \\ μ_{A_{2}^{1}} (x_{1}) = x_{1}, x_{1} \in [0, 1] . \end{matrix}$ The 1-dimensional VP of U₂ is $\tilde{U_{2}} = {A_{1}^{2}, A_{2}^{2}, A_{3}^{2}}$ , where

$\begin{matrix} μ_{A_{1}^{2}} (x_{2}) = {\begin{matrix} 1 - x_{2}, & x_{2} \in [0, 1), \\ 0, & x_{2} \in [1, 2] . \end{matrix} \\ μ_{A_{2}^{2}} (x_{2}) = {\begin{matrix} x_{2}, & x_{2} \in [0, 1), \\ 2 - x_{2}, & x_{2} \in [1, 2] . \end{matrix} \\ μ_{A_{3}^{2}} (x_{2}) = {\begin{matrix} 0, & x_{2} \in [0, 1), \\ x_{2} - 1, & x_{2} \in [1, 2] . \end{matrix} \end{matrix}$ Aggregation function T_P, which is both a t-norm and an overlap function, is used to aggregate $\tilde{U_{1}}$ and $\tilde{U_{2}}$ . The 2-dimensional VP of D is $\tilde{D} = {A_{11}, A_{12}, A_{13}, A_{21},$

A₂₂, A₂₃}, where $\begin{matrix} μ_{A_{11}} (X) = μ_{A_{1}^{1}} (x_{1}) \cdot μ_{A_{1}^{2}} (x_{2}) \\ = {\begin{matrix} (1 - x_{1}) (1 - x_{2}), & (x_{1}, x_{2}) \in [0, 1] \times [0, 1), \\ 0, & (x_{1}, x_{2}) \in [0, 1] \times [1, 2] . \end{matrix} \\ μ_{A_{12}} (X) = μ_{A_{1}^{1}} (x_{1}) \cdot μ_{A_{2}^{2}} (x_{2}) \\ = {\begin{matrix} (1 - x_{1}) x_{2}, & (x_{1}, x_{2}) \in [0, 1] \times [0, 1), \\ (1 - x_{1}) (2 - x_{2}), & (x_{1}, x_{2}) \in [0, 1] \times [1, 2] . \end{matrix} \\ μ_{A_{13}} (X) = μ_{A_{1}^{1}} (x_{1}) \cdot μ_{A_{3}^{2}} (x_{2}) \\ = {\begin{matrix} 0, & (x_{1}, x_{2}) \in [0, 1] \times [0, 1), \\ (1 - x_{1}) (x_{2} - 1), & (x_{1}, x_{2}) \in [0, 1] \times [1, 2] . \end{matrix} \\ μ_{A_{21}} (X) = μ_{A_{2}^{1}} (x_{1}) \cdot μ_{A_{1}^{2}} (x_{2}) \\ = {\begin{matrix} x_{1} (1 - x_{2}), & (x_{1}, x_{2}) \in [0, 1] \times [0, 1), \\ 0, & (x_{1}, x_{2}) \in [0, 1] \times [1, 2] . \end{matrix} \\ μ_{A_{22}} (X) = μ_{A_{2}^{1}} (x_{1}) \cdot μ_{A_{2}^{2}} (x_{2}) \\ = {\begin{matrix} x_{1} x_{2}, & (x_{1}, x_{2}) \in [0, 1] \times [0, 1), \\ x_{1} (2 - x_{2}), & (x_{1}, x_{2}) \in [0, 1] \times [1, 2] . \end{matrix} \end{matrix}$

$\begin{matrix} μ_{A_{23}} (X) = μ_{A_{2}^{1}} (x_{1}) \cdot μ_{A_{3}^{2}} (x_{2}) \\ = {\begin{matrix} 0, & (x_{1}, x_{2}) \in [0, 1] \times [0, 1), \\ x_{1} (x_{2} - 1), & (x_{1}, x_{2}) \in [0, 1] \times [1, 2] . \end{matrix} \end{matrix}$

Next, some properties of the n-dimensional VP of D with respect to aggregation functions are discussed.

Proposition 3.3. Let $D = U_{1} \times \dots \times U_{n} \subset ℝ^{n}$ , $\tilde{D}$ be a n-dimensional VP of D. If the aggregation Θ is strictly increasing, then

there exists at least one $I \in \bar{m_{1}} \times \dots \times \bar{m_{n}}$ such that μ_{A
_I} (X) >0;

for any $I \in \bar{m_{1}} \times \dots \times \bar{m_{n}}$ , there exists only one point X₀ ∈ D such that μ_{A
_I} (X₀) =1;

for X₀ ∈ D, $\overset{m_{n}}{\underset{i_{n} = 1}{Σ}} \dots \overset{m_{1}}{\underset{i_{1} = 1}{Σ}} μ_{A_{I}} (X_{0}) = 1$ holds if there exists $I^{*} \in \bar{m_{1}} \times \dots \times \bar{m_{n}}$ such that μ_{A
_{I
^*}} (X₀) =1.

Proof. (1) From Corollary 2.2, we see for any x₁ ∈ U₁, ⋯ , x_n ∈ U_n, there exists $i_{1} \in \bar{m_{1}}, \dots, i_{n} \in \bar{m_{n}}$ such that $μ_{A_{i_{1}}^{1}} (x_{1}) > 0, \dots, μ_{A_{i_{n}}^{n}} (x_{n}) > 0$ . If the aggregation function Θ is strictly increasing, then there exists at least I = (i₁, ⋯ , i_n) such that $μ_{A_{I}} (X) = Θ (μ_{A_{i_{1}}^{1}} (x_{1}), \dots, μ_{A_{i_{n}}^{n}} (x_{n})) > 0 .$ (2) Since $\tilde{U_{1}}, \dots, \tilde{U_{n}}$ are regular single-kernel 1-dimensional VPs of U₁, ⋯ , U_n respectively, it follows that, for any $i_{1} \in \bar{m_{1}}, \dots, i_{n} \in \bar{m_{n}}$ , there exists only one point x₁₀ ∈ U₁, ⋯ , x_n0 ∈ U_n respectively such that $μ_{A_{i_{1}}^{1}} (x_{10}) = 1, \dots, μ_{A_{i_{n}}^{n}} (x_{n 0}) = 1$ . According to the aggregation function Θ is strictly increasing, then for any $I \in \bar{m_{1}} \times \dots \times \bar{m_{n}}$ , there only one point X₀ = (x₁₀, ⋯ , x_n0) ∈ D such that $μ_{A_{I}} (X_{0}) = Θ (μ_{A_{i_{1}}^{1}} (x_{10}), \dots, μ_{A_{i_{n}}^{n}} (x_{n 0})) = 1 .$ (3) Let $I^{*} = (i_{1}^{*}, \dots, i_{n}^{*}) \in \bar{m_{1}} \times \dots \times \bar{m_{n}}$ . Since the aggregation function Θ is strictly increasing, it follows that $μ_{A_{I^{*}}} (X_{0}) = Θ (μ_{A_{i_{1}^{*}}^{1}} (x_{10}), \dots, μ_{A_{i_{n}^{*}}^{n}} (x_{n 0})) = 1$ if and only if $μ_{A_{i_{1}^{*}}^{1}} (x_{10}) = 1$ , ⋯, $μ_{A_{i_{n}^{*}}^{n}} (x_{n 0}) = 1$ . According to $\tilde{U_{1}}, \dots, \tilde{U_{n}}$ are regular single-kernel 1-dimensional VPs of U₁, ⋯ , U_n respectively, we have $μ_{A_{i_{1}}^{1}} (x_{10}) = 0$ , ⋯, $μ_{A_{i_{n}}^{n}} (x_{n 0}) = 0$ for $i_{1} \neq i_{1}^{*}$ , ⋯, $i_{n} \neq i_{n}^{*}$ . Therefore, μ_{A
_I} (X₀) =0 for I ≠ I^*. As a consequence, $\overset{m_{n}}{\underset{i_{n} = 1}{Σ}} \dots \overset{m_{1}}{\underset{i_{1} = 1}{Σ}} μ_{A_{I}} (X_{0}) = μ_{A_{I^{*}}} (X_{0}) = 1 .$

□

In fact, aggregation functions illustrated in the previous section are all strictly increasing. In the rest of paper, we concern the strictly increasing aggregation functions. Also, the following corollary can be obtained from Corollary 2.2 without proof.

Corollary 3.4. Let $D = U_{1} \times \dots \times U_{n} \subset ℝ^{n}$ , $\tilde{D}$ be a n-dimensional VP of D. Then for any X ∈ D, the number of fuzzy sets with membership degree more than 0 is 2^p, p = 0, 1, ⋯ , n.

Proposition 3.5. Let $D = U_{1} \times \dots \times U_{n} \subset ℝ^{n}$ , $\tilde{D}$ be an n- dimensional VP of D. If the aggregation function Θ is continuous, then for any $I \in \bar{m_{1}} \times \dots \times \bar{m_{n}}$ , μ_{A
_I} (X) is continuous.

Proof. It follows from the part (1) of Definition 2.1 that μ_{A
_I} (X) is continuous for any $I \in \bar{m_{1}} \times \dots \times \bar{m_{n}}$ .

□

Proposition 3.6. Let $D = U_{1} \times \dots \times U_{n} \subseteq ℝ^{n}$ , $\tilde{D}$ be an n-dimensional VP of D. If the aggregation function Θ satisfies the distribution law with respect to addition, then for any X ∈ D and for any $I \in \bar{m_{1}} \times \dots \times \bar{m_{n}}$ , it holds that $\overset{m_{n}}{\underset{i_{n} = 1}{Σ}} \dots \overset{m_{1}}{\underset{i_{1} = 1}{Σ}} μ_{A_{I}} (X) = 1 .$

Proof. According to the assumption, for any X = (x₁, ⋯ , x_n) ∈ D, it holds that $\begin{matrix} \overset{m_{n}}{\underset{i_{n} = 1}{Σ}} \dots \overset{m_{1}}{\underset{i_{1} = 1}{Σ}} μ_{A_{I}} (X) \\ = & \overset{m_{n}}{\underset{i_{n} = 1}{Σ}} \dots \overset{m_{1}}{\underset{i_{1} = 1}{Σ}} Θ (μ_{A_{i_{1}}^{1}} (x_{1}), \dots, μ_{A_{i_{n}}^{n}} (x_{n})) \\ = & Θ (\overset{m_{1}}{\underset{i_{1} = 1}{Σ}} μ_{A_{i_{1}}^{1}} (x_{1}), \dots, \overset{m_{n}}{\underset{i_{n} = 1}{Σ}} μ_{A_{i_{n}}^{n}} (x_{n})) \\ = & Θ (1, \dots, 1) \\ = & 1 . \end{matrix}$ □

4 The FS-VPs and its approximation properties

In this section, we first introduce the fuzzy rule base based on VPs. Furthermore, the new FS is proposed, which can process information directly from its fuzzy rule base without fuzzication, defuzzication and fuzzy reasoning engine. At last, we discuss the approximation properties of the new FS, including universal approximation property and approximation precision.

4.1 Fuzzy rule base

Let $U = [a, b] \subset ℝ$ , $V = [c, d] \subset ℝ$ , $\tilde{U} = {A_{1}, \dots, A_{n}}$ , $n \in ℕ^{+}$ , $\tilde{V} = {B_{1}, \dots, B_{k}}$ , $k \in ℕ^{+}$ be 1-dimensional VPs of U and V respectively. Based on VPs, the fuzzy rule for SISO model is expressed as follows: $R_{i} : IF x is A_{i} THEN y is B_{k_{i}},$ (4.1) in which R_i, $i \in \bar{n}$ denotes the i-th rule, n denotes the number of fuzzy rules, x ∈ U and $A_{i} \in \tilde{U}$ , y ∈ V and $B_{k_{i}} \in \tilde{V}$ , $k_{i} \in \bar{k}$ . From Corollary 2.2, it follows that the fuzzy rule base composed of fuzzy rules formed as (4.1) is complete. In general, we require the fuzzy rule base is consistent. Then the fuzzy rule base for SISO model can be regarded as a mapping $\tilde{f}$ from $\tilde{U}$ to $\tilde{V}$ . Reversely, $\tilde{f}$ can represent a fuzzy rule base.

For a MISO model, let $D = U_{1} \times \dots \times U_{n} \subset ℝ^{n}$ , $V \subset ℝ$ , $\tilde{U_{1}} = {A_{1}^{1}, \dots, A_{m_{1}}^{1}},$ $m_{1} \in ℕ^{+},$ ⋯, $\tilde{U_{n}} = {A_{1}^{n}, \dots, A_{m_{n}}^{n}},$ $m_{n} \in ℕ^{+}$ , $\tilde{V} = {B_{1}, \dots, B_{k}},$ $k \in ℕ^{+}$ , be 1-dimensional VPs of U₁, ⋯, U_n and V respectively. Then the fuzzy rule for MISO model could be expressed as follows:

$R_{i_{1} \dots i_{n}} : IF x_{1} is A_{i_{1}}^{1} and \dots and x_{n} is A_{i_{n}}^{n} THEN y is B_{k_{i}},$ (4.2) in which R_{i₁⋯i_n}, $i_{1} \in \bar{m_{1}}, \dots, i_{n} \in \bar{m_{n}}$ denotes the i₁ ⋯ i_n-th rule, m₁ × ⋯ × m_n denotes the number of fuzzy rules, x₁ ∈ U₁ and $A_{i_{1}}^{1} \in \tilde{U_{1}}$ , ⋯, x_n ∈ U_n and $A_{i_{n}}^{n} \in \tilde{U_{n}}$ , y ∈ V and $B_{k_{i}} \in \tilde{V}$ , $k_{i} \in \bar{k}$ . According to discussion in Section 3, the fuzzy rule formed as (4.2) can be aggregated to a SISO model by an aggregation function. Let $\tilde{D}$ be an n-dimensional VP of D. For X ∈ D, the fuzzy rule for MISO model is simplified as follows: $R_{I} : IF X is A_{I} THEN y is B_{k_{I}},$ (4.3) in which R_I, $I \in \bar{m_{1}} \times \dots \times \bar{m_{n}}$ denotes the I-th rule, m₁ × ⋯ × m_n denotes the number of fuzzy rules, X ∈ D and $A_{I} \in \tilde{D}$ , y ∈ V and $B_{k_{I}} \in \tilde{V}$ , $k_{I} \in \bar{k}$ . From the analysis of Section 3, we see the fuzzy rule base consisting of fuzzy rules in the form of (4.3) is complete. Also, consistency is required for the fuzzy rule base of MISO model. Consequently, the fuzzy rule base for MISO model also can be regarded as a mapping $\tilde{F}$ which is similar to multivariate function from $\tilde{D}$ to $\tilde{V}$ .

4.2 Design of the FS

We first design a SISO FS, which can process information directly from a fuzzy rule base based on VPs. Let $\tilde{U}$ be a VP of input universe U, $\tilde{V}$ be a VP of output universe V. Then a complete and consistent fuzzy rule base can be seen as a mapping $\tilde{f}$ from $\tilde{U}$ to $\tilde{V}$ . Reversely, $\tilde{f}$ can represent a fuzzy rule base. The SISO FS aims to generate the mapping f from U to V. The basic configurations of the new SISO FS are shown schematically in Fig. 1.

Fig. 1

Basic configurations of the SISO FS.

The paper [41] analyzed the properties of $\tilde{f}$ , and designed an algorithm for the SISO FS, of which configurations are shown in Fig. 1, to obtain the mapping f from U to V. The FS with this algorithm is proved that it achieved the quite approximation effect from both theoretical and numerical results. Now, for the convenience of calculation, we make $\tilde{V}$ in [41, Algorithm 1] a triangular VP, and then obtain a simplified algorithm to calculate corresponding output f (x) ∈ V for input x ∈ U for the SISO FS. Let $\tilde{U} = {A_{1}, \dots, A_{m}}$ , $m \in ℕ^{+}$ , $\tilde{V} = {B_{1}, \dots, B_{k}}$ , $k \in ℕ^{+}$ , $\tilde{f} (A_{i}) = B_{k_{i}}$ , $i \in \bar{m}$ , $k_{i} \in \bar{k}$ , kerA_i = {x_i}, x_i ∈ U, $i \in \bar{m}$ , kerB_{k
_i} = {y_{k
_i}}, y_{k
_i} ∈ V, $k_{i} \in \bar{k}$ . Then for the SISO FS, the algorithm (Algorithm 1) to calculate corresponding output f (x) ∈ V for input x ∈ U is shown as follows.

Algorithm 1

Input: A set of fuzzy rules $R_{i} (i \in \bar{m})$ with $\tilde{U}$ and $\tilde{V}$ and a point x ∈ U.

Output: f (x).

1: If x = x_i then

2: f (x) = y_{k
_i};

3: Else

4: $f (x) = μ_{A_{i}} (x) y_{k_{i}} + μ_{A_{i + 1}} (x) y_{k_{i + 1}} .$

5: end if

6: return f (x).

As a consequence, the SISO FS can be represented by $f (x) = {\begin{matrix} y_{k_{i}} & x = x_{i}, \\ μ_{A_{i}} (x) y_{k_{i}} + μ_{A_{i + 1}} (x) y_{k_{i + 1}}, & x \in (x_{i}, x_{i + 1}) . \end{matrix}$ (4.4)

For the MISO FS, we employ the idea that transforms it into the SISO model by an aggregation function. Then the complete and consistent fuzzy rule base is a mapping $\tilde{F}$ from $\tilde{D}$ to $\tilde{V}$ , where $\tilde{D}$ is a VP of input universe $D = U_{1} \times \dots U_{n} \subset ℝ^{n}$ , $\tilde{V}$ is a VP of output universe $V \subset ℝ$ . The configurations of the MISO FS are shown in Fig. 2.

Fig. 2

Basic configurations of the MISO FS.

See back to Eq. (4.4). According to Corollary 2.2 and the regularity of the 1-dimensional VP, Eq. (4.4) is equal to $f (x) = \frac{\underset{i = 1}{\overset{m}{Σ}} μ_{A_{i}} (x) y_{k_{i}}}{\underset{i = 1}{\overset{m}{Σ}} μ_{A_{i}} (x)} .$ Then f (x) can be viewed as the weighted average of all y_{k
_i}, $i \in \bar{m}$ . For each $i \in \bar{m}$ , the weight of y_{k
_i} is the membership degree of x with respect to A_i, that is μ_{A
_i} (x). Based on the idea of weighting, we design an algorithm for the MISO FS to calculate the corresponding output F (X) ∈ V for the input X ∈ D. Let $\tilde{U_{1}} = {A_{1}^{1}, \dots, A_{m_{1}}^{1}}$ , $m_{1} \in ℕ^{+}$ , $\ker A_{i_{1}}^{1} = {x_{1 i_{1}}}$ , x_1i₁ ∈ U₁, $i_{1} \in \bar{m_{1}},$ ⋯, $\tilde{U_{n}} = {A_{1}^{n}, \dots, A_{m_{n}}^{n}}$ , $m_{n} \in ℕ^{+}$ , $\ker A_{i_{1}}^{n} = {x_{{ni}_{n}}}$ , x_{ni
_n} ∈ U_n, $i_{n} \in \bar{m_{n}}$ , $\tilde{V} = {B_{1}, \dots, B_{k}}$ , $k \in ℕ^{+}$ , kerB_j = {y_j}, $j \in \bar{k}$ . Then $\tilde{D} = {A_{I} ∣ I \in \bar{m_{1}} \times \dots \times \bar{m_{n}}}$ , kerA_I = {X_I}, where X_I = (x_1i₁, ⋯ x_{ni
_n}). Assume that $\tilde{F} (A_{I}) = B_{k_{I}}$ . Then for the MISO FS, the algorithm (Algorithm 2) to calculate corresponding output F (X) ∈ V for the input X ∈ D is shown as follows.

Algorithm 2

Input: A set of fuzzy rules $R_{i_{1} \dots i_{n}} (i_{1} \in \bar{m_{1}}, \dots, i_{n} \in \bar{m_{n}})$ with $\tilde{U_{1}}$ , ⋯, $\tilde{U_{n}}$ and $\tilde{V}$ , and a point X = (x₁, ⋯ , x_n) ∈ D.

Output: F (X).

1: Select an aggregation function Θ.

2: Obtain a set of fuzzy rules $R_{I} (I \in \bar{m_{1}} \times \dots \times \bar{m_{n}})$ with $\tilde{D}$ and $\tilde{V}$ .

3: if X = X_I then

4: F (X) = y_{k
_I};

5: else

6: $F (X) = \frac{\overset{m_{n}}{\underset{i_{n} = 1}{Σ}} \dots \overset{m_{1}}{\underset{i_{1} = 1}{Σ}} μ_{A_{I}} (X) y_{k_{I}}}{\overset{m_{n}}{\underset{i_{n} = 1}{Σ}} \dots \overset{m_{1}}{\underset{i_{1} = 1}{Σ}} μ_{A_{I}} (X)} .$

7: end if

8: return F (X).

As a consequence, the MISO FS can be represented by

$F (X) = {\begin{matrix} y_{k_{i}} & X = X_{I}, \\ \frac{\overset{m_{n}}{\underset{i_{n} = 1}{Σ}} \dots \overset{m_{1}}{\underset{i_{1} = 1}{Σ}} μ_{A_{I}} (X) y_{k_{I}}}{\overset{m_{n}}{\underset{i_{n} = 1}{Σ}} \dots \overset{m_{1}}{\underset{i_{1} = 1}{Σ}} μ_{A_{I}} (X)}, & otherwise . \end{matrix}$ (4.5)

Remark 4.1. According to Corollary 3.4, there are no more 2ⁿ terms such that μ_{A
_I} (X) >0 for $X \in D / {X_{I} ∣ I \in \bar{m_{1}} \times \dots \times \bar{m_{n}}}$ . Therefore, for $\overset{m_{n}}{\underset{i_{n} = 1}{Σ}} \dots \overset{m_{1}}{\underset{i_{1} = 1}{Σ}} μ_{A_{I}} (X) y_{k_{I}}$ and $\overset{m_{n}}{\underset{i_{n} = 1}{Σ}} \dots \overset{m_{1}}{\underset{i_{1} = 1}{Σ}} μ_{A_{I}} (X)$ , there are no more 2ⁿ terms need to calculate.

4.3 Approximation properties of the proposed FS

In this part, approximation properties of the proposed FS are investigated. Before that, we discuss the interpolation property of the proposed FS.

For the proposed FS, we consider the input universe $D = U_{1} \times \dots \times U_{n} \subset ℝ^{n}$ , and the output universe $V \subset ℝ$ . Let H (X) be a function on D and kerA_I = {X_I} for $A_{I} \in \tilde{D}$ , kerB_{k
_I} = {H (X_I)} for $B_{k_{I}} \in \tilde{V}$ . Then, from Eq. (4.5), the proposed FS can be expressed as $F (X) = {\begin{matrix} H (X_{I}) & X = X_{I}, \\ \frac{\overset{m_{n}}{\underset{i_{n} = 1}{Σ}} \dots \overset{m_{1}}{\underset{i_{1} = 1}{Σ}} μ_{A_{I}} (X) H (X_{I})}{\overset{m_{n}}{\underset{i_{n} = 1}{Σ}} \dots \overset{m_{1}}{\underset{i_{1} = 1}{Σ}} μ_{A_{I}} (X)}, & otherwise . \end{matrix}$ Obviously, it holds that F (X_I) = H (X_I). For X ≠ X_I, the output of the proposed FS can be viewed as a weighted average of H (X_I). It implies that F (X) is an interpolation function of H (X) with respect to interpolators $X_{I}, I \in \bar{m_{1}} \times \dots \bar{m_{n}}$ .

Recall that the ∞-norm for a bounded function G (X) on the domain is $‖ G (X) ‖_{\infty} = sup_{X \in D} | G (X) | .$ In the sequel, we prove the proposed FS is a universal approximator under the ∞-norm errors. For this end, a polynomial is introduced since it has continuous differentiable properties on its domain.

Lemma 4.2. (Weierstrass’s Approximation Theorem [45]) Let G (X) be a continuous function on D. Then for arbitrary ɛ > 0, there exists a polynomial $P (X)$ such that $‖ G (X) - P (X) ‖_{\infty} < ɛ .$

In the sequel, we prove the proposed FS is a universal approximator under the ∞-norm errors. For this end, a polynomial is introduced since it has continuous differentiable properties on its domain.

Lemma 4.3. Let $P (X)$ be a polynomial on D. Then for arbitrary ɛ > 0, there is a FS F (X) such that $‖ F (X) - P (X) ‖_{\infty} < ɛ .$

Proof. For arbitrary X = (x₁, ⋯ , x_n) ∈ D, there exist $i_{1} \in \bar{m_{1} - 1}, \dots, i_{n} \in \bar{m_{n} - 1}$ such that x₁ ∈ [x_1i₁, x_1i₁+1] , ⋯ , x_n ∈ [x_{ni
_n}, x_{ni_n+1}]. According to the interpolation properties of F (X), we have $∣ F (X) - P (X) ∣ = ∣ F (X) - F (X_{I}) + P (X_{I}) - P (X) ∣ \leq ∣ F (X) - F (X_{I}) ∣ + ∣ P (X_{I}) - P (X) ∣ .$ It follows from Lagrange’s Mean Theorem that there are Λ = (λ₁, ⋯ , λ_n) ∈ (x_1i₁, x_1i₁+1) × ⋯ × (x_{ni
_n}, x_{ni_n+1}) and Ω = (ω₁, ⋯ , ω_n) ∈ (x_1i₁, x_1i₁+1) × ⋯ × (x_{ni
_n}, x_{ni_n+1}) making that $\begin{matrix} | F (X) - G (X) | = & | \frac{\partial F (X)}{\partial x_{1}} |_{Λ} (x_{1} - x_{1 i_{1}}) + \dots + \frac{\partial F (X)}{\partial x_{n}} |_{Λ} (x_{n} - x_{{ni}_{n}}) | \\ + | \frac{\partial P (X)}{\partial x_{1}} |_{Ω} (x_{1} - x_{1 i_{1}}) + \dots + \frac{\partial P (X)}{\partial x_{n}} |_{Ω} (x_{n} - x_{{ni}_{n}}) | \\ \leq & | \frac{\partial F (X)}{\partial x_{1}} |_{Λ} + \frac{\partial P (X)}{\partial x_{1}} |_{Ω} | h_{1} + \dots + | \frac{\partial F (X)}{\partial x_{n}} |_{Λ} + \frac{\partial P (X)}{\partial x_{n}} |_{Ω} | h_{n}, \end{matrix}$ where $h_{1} = max_{i_{1} \in \bar{m_{1} - 1}} | x_{1 i_{1}} - x_{1 i_{1} + 1} |, \dots, h_{n} = max_{i_{n} \in \bar{m_{n} - 1}} | x_{{ni}_{n}} - x_{{ni}_{n} + 1} |$ . Let h = max {h₁, ⋯ , h_n} and the constant $| \frac{\partial F (X)}{\partial x_{1}} |_{Λ} | + \dots + | \frac{\partial F (X)}{\partial x_{n}} |_{Λ} | + | \frac{\partial P (X)}{\partial x_{1}} |_{Ω} | + \dots + | \frac{\partial P (X)}{\partial x_{n}} |_{Ω} | = M .$ Consequently, $| F (X) - P (X) | \leq Mh$ . If h is small enough, then for arbitrary ɛ > 0, $| F (X) - P (X) | < ɛ$ holds. Furthermore, $‖ F (X) - P (X) ‖_{\infty} = sup_{X \in D} | F (X) - P (X) | < ɛ .$ □

Theorem 4.4. Let G (X) be an arbitrary continuous function on D. Then for arbitrary ɛ > 0, there exists a FS F (X) such that $‖ G (X) - F (X) ‖_{\infty} < ɛ .$

Proof. For arbitrary ɛ > 0, it follows from Lemma 4.2 that there exists a polynomial $P (X)$ on D such that $‖ G (X) - P (X) ‖_{\infty} < \frac{ɛ}{2} .$ For the polynomial $P (X)$ , from Lemma 4.3 we have there exits a proposed FS F (X) such that $‖ F (X) - P (X) ‖_{\infty} < \frac{ɛ}{2} .$ According to the definition of the norm, it holds that $\begin{matrix} ‖ G (X) - F (X) ‖_{\infty} = ‖ G (X) - P (X) + P (X) - F (X) ‖_{\infty} \leq ‖ G (X) - P (X) ‖_{\infty} + ‖ P (X) - F (X) ‖_{\infty} < ɛ . \end{matrix}$ □

Theorem 4.4 illustrates that the proposed FS is an universal approximator. Furthermore, the following theorem show the approximation accuracy of the proposed FS.

Theorem 4.5. Let G (X) be continuous and second-order differentiable on D. If the FS F (X) is second order differentiable on $D ∖ {X_{I} | I \in \bar{m_{1}} \times \dots \bar{m_{n}}}$ , then

$‖ G (X) - F (X) ‖_{\infty} \leq (‖ \frac{\partial G (X)}{\partial x_{1}} - \frac{\partial F (X)}{\partial x_{1}} ‖_{\infty} + \dots + ‖ \frac{\partial G (X)}{\partial x_{n}} - \frac{\partial F (X)}{\partial x_{n}} ‖_{\infty}) h,$ where $h = max {h_{l} | h_{l} = max | x_{{li}_{l}} - x_{{li}_{l} + 1} |, i_{l} \in \bar{m_{l} - 1}, l \in \bar{n}}$ .

$‖ G (X) - F (X) ‖_{\infty} \leq (‖ \frac{\partial G (X)}{\partial x_{1}} - \frac{\partial F (X)}{\partial x_{1}} ‖_{\infty} + \dots + ‖ \frac{\partial G (X)}{\partial x_{n}} - \frac{\partial F (X)}{\partial x_{n}} ‖_{\infty}) \frac{h^{2}}{2},$ where $h = max {h_{l} | h_{l} = max | x_{{li}_{l}} - x_{{li}_{l} + 1} |, i_{l} \in \bar{m_{l} - 1}, l \in \bar{n}}$ .

Proof. (1) For arbitrary X ∈ D, there is $I \in \bar{m_{1}} \times \dots \times \bar{m_{n}}$ such that X = (x₁, ⋯ , x_n) ∈ [x_1i₁, x_1i₁+1] × ⋯ × [x_{ni
_n}, x_{ni_n+1}]. Let R (X) = G (X) - F (X). Since F (X_I) = G (X_I), $I \in \bar{m_{1}} \times \dots \times \bar{m_{n}}$ , it follows that $R (X) = T_{1} (X) (x_{1} - x_{1 i_{1}}) + \dots + T_{n} (X) (x_{n} - x_{{ni}_{n}}),$ where T₁ (X) , ⋯ , T_n (X) are continuous on [x_1i₁, x_1i₁+1] × ⋯ × [x_{1i_n}, x_{1i_n+1}] and differentiable on (x_1i₁, x_1i₁+1) × ⋯ × (x_{1i_n}, x_{1i_n+1}). Fix a point X ∈ D. An auxiliary function is defined as $Γ (S) = G (S) - F (S) - T_{1} (X) (s_{1} - x_{1 i_{1}}) - \dots - T_{n} (X) (s_{n} - x_{{ni}_{n}}),$ S = (s₁, ⋯ , s_n) ∈ (x_1i₁, x_1i₁+1) × ⋯ × (x_{ni
_n}, x_{ni_n+1}). It is evidently that X_I and X are two roots of Γ (S). From Roll’s Theorem, there exists Λ = (λ₁, ⋯ , λ_n) ∈ (x_1i₁, x_1i₁+1) × ⋯ × (x_{ni
_n}, x_{ni_n+1}), where λ₁, ⋯ , λ_n are related to x₁, ⋯ , x_n respectively, such that $\frac{\partial Γ}{\partial x_{1}} |_{Λ} = 0, \dots, \frac{\partial Γ}{\partial x_{n}} |_{Λ} = 0 .$ Therefore, $T_{1} (X) = \frac{\partial G (S)}{\partial s_{1}} |_{Λ} - \frac{\partial F (S)}{\partial s_{1}} |_{Λ}, \dots, T_{n} (X) = \frac{\partial G (S)}{\partial s_{n}} |_{Λ} - \frac{\partial F (S)}{\partial s_{n}} |_{Λ},$ and $R (X) = (\frac{\partial G (S)}{\partial s_{1}} |_{Λ} - \frac{\partial F (S)}{\partial s_{1}} |_{Λ}) (x_{1} - x_{1 i_{1}}) + \dots + (\frac{\partial G (S)}{\partial s_{n}} |_{Λ} - \frac{\partial F (S)}{\partial s_{n}} |_{Λ}) (x_{n} - x_{{ni}_{n}}) .$ From $| R (X) | \leq (| \frac{\partial G (S)}{\partial s_{1}} |_{Λ} - \frac{\partial F (S)}{\partial s_{1}} |_{Λ} | + \dots + | \frac{\partial G (S)}{\partial s_{n}} |_{Λ} - \frac{\partial F (S)}{\partial s_{n}} |_{Λ} |) h$ and arbitrariness of X, it holds that $‖ R (X) ‖_{\infty} \leq (‖ \frac{\partial G (X)}{\partial x_{1}} - \frac{\partial F (X)}{\partial x_{1}} ‖_{2} + \dots + ‖ \frac{\partial G (X)}{\partial x_{n}} - \frac{\partial F (X)}{\partial x_{n}} ‖_{2}) h .$ (2) The proof is similar to (1). □

5 An application to path tracking control

Path tracking control is one of the most important issues in autonomous vehicles technology. It can achieve efficient autonomous driving of vehicles and provides core technology for unmanned driving. There are several methods to realize path tracking control, including Model Predictive Control, Inverse Control, Local Optimal Control, and PID Control. This section combines the proposed FS with a PID control system to deal with path tracking problems. Firstly, the robustness of the fuzzy PID control system is discussed. Moreover, a series of comparative experiments are designed to illustrate the effectiveness and advantages of the proposed FS in a fuzzy PID control system to deal with path tracking problems.

5.1 Description of fuzzy PID control system

Proportional-integral-derivative (PID) [47] control is a type of linear control system. It obtains a control deviation e (t) from the given value r (t) and actual output value y (t). The success of the PID control system depends on selecting the appropriate parameters K_p, K_i, and K_d, which are given by experience usually. In this subsection, we introduce a fuzzy PID control system, which employs the FSs-VPs to output the PID control parameters, and its basic configurations are shown in Fig. 3. In the fuzzy PID control system, there are three proposed FSs. The error between the actual position and predetermined, the differentiation of the error, denoted by e, ed respectively, are two inputs of these three FSs. ΔKp, ΔKi and ΔKd are outputs of these three FSs respectively. Set Kp = 0.4, Ki = 0, and Kd = 1 are the initial values of the fuzzy PID control system. Let the domain of e, ed, ΔKp, ΔKi and ΔKd be U_e = [-3, 3], U_ed = [-3, 3], U_ΔKp = [-0.3, 0.3], U_ΔKi = [-0.06, 0.06] and U_ΔKd = [-3, 3] respectively. Considering the possible vehicle moving states in path tracking problems, e, ed, ΔKp, ΔKi, ΔKd can be divided into 7 categories, negative big (NB), negative middle (NM), negative small (NS). zero (ZO), positive small (PS), positive meddle (PS), positive big (PB). Based on experiences, the fuzzy rules with respect to ΔKp, ΔKi, ΔKd and e, ed are shown in Tables 1-3.

Fig. 3

Basic structure of fuzzy PID control system.

Table 1

Fuzzy rules for ΔKp

e∖ΔKp∖ed
NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	PS	ZO	ZO
NS	PM	PM	PM	PS	ZO	NS	NS
ZO	PM	PM	PS	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NM	NM	NM
PM	ZO	ZO	ZO	NS	NM	NM	NM
PB	ZO	ZO	NM	NM	NM	NB	NB

Table 2

Fuzzy rules for ΔKi

e∖ΔKi∖ed
NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NS	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	NS	ZO
NS	NB	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	NS	ZO	PS	PM	PM
PS	NM	NM	ZO	PS	PM	PM	PB
PM	ZO	PS	PS	PS	PM	PB	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

Table 3

Fuzzy rules for ΔKd

e∖ΔKd∖ed
NB	NM	NS	ZO	PS	PM	PB
NB	PS	NS	NB	NB	NB	NS	PS
NM	PS	NB	NB	NB	NM	NS	ZO
NS	ZO	NS	NM	NM	NM	NS	ZO
ZO	ZO	NS	NS	NS	NS	NS	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PB	NS	PS	PS	PS	NS	PB
PB	PB	PS	PM	PM	PM	PS	PB

5.2 Robustness analysis of the fuzzy PID control system

Robustness refers to the anti-interference ability of a control system to external factors such as parameter changes and disturbances, and it is an essential indicator for evaluating control systems. In this subsection, the controlled object is $G (s) = \frac{523500}{s^{3} + 87.35 s^{2} + 10470 s} .$ In fuzzy PID control, the step response curve refers to the response curve of the system when the input signal undergoes a sudden change. In general, the step response curve is used to evaluate the robustness of the control system. The step response curve typically includes the following indicators:

Settling Time: It refers to the time required for the system to reach the stable state from the initial state. Settling time reflects the response speed and stability of the system.

Rise Time: It refers to the time required for the system to reach the vicinity of the setpoint from the initial state. Rise time reflects the response speed of the system.

Peak Time: It refers to the time at which the system response reaches the maximum deviation. Peak time reflects the oscillation performance of the system.

Overshoot: It refers to the ratio of the maximum deviation (y_max) of the system response from the setpoint (y_set) before reaching the stable state to the set point. It is usually expressed as percentage: (y_max - y_set)/y_set× 100 %. Overshoot reflects the degree of overshoot of the system, i.e., the excessive response of the system to the input signal.

Let VPs of U_e = [-3, 3] and U_ed = [-3, 3] be triangular VPs shown in Fig. 4(a) and 4(b) respectively. We first investigate the influence of aggregation functions of FSs on the robustness of the fuzzy PID control system. Some simulation experiments on MATLAB are conducted, and the step response curves of the fuzzy PID control system with respect to t-norms, uninorms, and overlap functions are obtained, as shown in Fig. 11a, 11b, and 11c respectively. Secondly, we discuss the influence of types of VPs introduced in [41] on the robustness of the fuzzy PID control system. According to Eq. (4.5), it can be seen that the VP type of the consequents has no impact on the output. Therefore, we only need to discuss the impact of the VP type of the antecedent of the proposed FS on the robustness of fuzzy PID control systems. The five types of VPs of U_e and U_ed are shown in Figs. 4-8. This paper only discusses that types of VPs of U_e and U_ed are the same. In the sequel, FS with a certain type of VP refers that the VP type of fuzzy rules antecedents is that type. Set the aggregation function to be T_D. Some simulation experiments are conducted on MATLAB, and the step response curves of the fuzzy PID control system with respect to five types of VPs are shown in Fig. 11d. At last, we compare the influence of the proposed FS with three widely used FSs [24] on the robustness of fuzzy PID control systems. These three FSs have fuzzifications, defizzifications, and fuzzy reasoning engines, and their configurations are shown in Table 4. Some simulation experiments are conducted on MATLAB, and the step response curves of the fuzzy PID control system with respect to three FSs are shown in Fig. 11e.

Fig. 4

VPs of antecedents for System 1₁.

Fig. 5

VPs of antecedents for System 1₂

Fig. 6

VPs of antecedents for System 1₃

Fig. 7

VPs of antecedents for System 1₄

Fig. 8

VPs of antecedents for System 1₅

Observing Fig. 11, we see that settling times, rise times, and peak times of all step response curves are less than 0.5 seconds. Therefore, we do not evaluate the robustness of the fuzzy PID control system by analyzing the settling time, rise time, and peak time of the response curve. Since y_set = 1, the overshoot is equal to (y_max - 1)×100 %. This means that the larger the max deviation of the system response before reaching the stable state, the larger the overshoot, and the poorer the robustness of the fuzzy PID control system. Therefore, we can directly observe the maximum deviation of the response curve to judge the robustness of the fuzzy PID control system.

It can be seen from Fig. 11(a) that the FS with T_D makes a fuzzy PID control system more robust than the fuzzy PID control systems consisting of FSs with T_M, T_P, and T_D. From Fig. 11(b), we see that FSs with different uninorms make no significant difference in the robustness of the fuzzy PID control system. From Fig. 11(c), the same conclusion as the FSs with uninorms is obtained. Comparing Fig. 11(a), (b), and (c), it is evident that FSs with T_D and uninorms make fuzzy PID control systems more robust than the fuzzy PID control systems consisting of FSs with T_M, T_P, T_L, and overlap functions. Note that T_D, U₁, U₂, U₃ are all discontinuous aggregation functions. Therefore, it can be concluded that FSs with discontinuous aggregation functions make fuzzy PID control systems more robust than fuzzy PID control systems consisting of FSs with continuous aggregation functions.

Fig. 9

VPs of consequents for all systems

Fig. 10

Fuzzification

Fig. 11

Step response curves

It can be seen from Fig. 11(d) that FSs with different types of VPs have almost no impact on the robustness of the fuzzy PID control system.

Comparing Fig. 11(a), (11b), (11c), (e), it can be concluded that the fuzzy PID control systems consisting of the FSs with discontinuous aggregation functions are more robust than fuzzy PID control systems consisting of three usually used FSs respectively.

5.3 Comparative experiment for the proposed fuzzy systems with different VPs

Based on the previous subsections, we employ the fuzzy PID control system to deal with path tracking problems. The following are six paths: six types of basic elementary functions.

h₁ (t) =1;

h₂ (t) =0.1 cos(πt);

h₃ (t) =2^t;

h₄ (t) = t²;

h₅ (t) =0.05 ln(0.1t);

h₆ (t) = arctan t.

Table 4
Configurations for different fuzzy systems

Systems∖Configuration Fuzzification Fuzzy reasoning engine Defuzzification

And method Aggregation Implication

System 2 Fig. 10 ‘min’ ‘max’ ‘prod’ ‘centriod’

System 3 Fig. 10 ‘prod’ ‘max’ ‘min’ ‘mom’

System 4 Fig. 10 ‘prod’ ‘max’ ‘min’ ‘bisector’

Systems∖Configuration	Fuzzification	Fuzzy reasoning engine	Defuzzification
System 2	Fig. 10	‘min’	‘max’	‘prod’	‘centriod’
System 3	Fig. 10	‘prod’	‘max’	‘min’	‘mom’
System 4	Fig. 10	‘prod’	‘max’	‘min’	‘bisector’

In fact, most paths in real life can be abstracted into six types of basic elementary functions or compositions of basic elementary functions. In this subsection, regarding the above six paths, we investigate the influence of the proposed FSs with the above five types of VPs on the tracking effect of fuzzy PID control system. It has known that the VP type of the consequents has no impact on the output. Therefore, we only need to discuss the impact of the VP type of the antecedent on the path tracking control problems. In the sequel, the five types of FSs are compared, and their configurations are shown in Table 5.

Table 5

Configurations for the five types of FSs

Configuration Systems	System 1₁	System 1₂	System 1₃	System 1₄	System 1₅
VPs of antecedents	Fig. 4	Fig. 5	Fig. 6	Fig. 7	Fig. 8
VPs of consequents	Fig. 9	Fig. 9	Fig. 9	Fig. 9	Fig. 9
Aggregation function	T _M	T _M	T _M	T _M	T _M

The standard deviation $δ = \sqrt{\frac{\overset{n}{\underset{i = 1}{Σ}} (\hat{h} (t_{i}) - h (t_{i}))^{2}}{n}}$ (6) are used to evaluate the quality of systems. We conducted simulation experiments on MATLAB using the five types of systems mentioned above, and obtain tracking errors for the six functions as shown in Table 6, where the blue font represents the minimum error of five systems regarding the tracked function.

Table 6

Errors of FSs with different VPs for tracked functions

Tracked functions
Errors
Systems
	h₁ (t)	h₂ (t)	h₃ (t)	h₄ (t)	h₅ (t)	h₆ (t)
System 1₁	0.0909	0.0170	0.0979	0.0469	0.0338	0.0170
System 1₂	0.0916	0.0176	0.1216	0.0974	0.0341	0.0236
System 1₃	0.0912	0.0168	0.0989	0.0383	0.0329	0.0153
System 1₄	0.0910	0.0173	0.1003	0.0545	0.0347	0.0182
System 1₅	0.0911	0.0170	0.1275	0.0373	0.0334	0.0151

From the above results, we conclude that the VP type indeed impacts the quality of the systems, and errors between predetermined paths and tracking paths, called tracking errors, can be reduced by choosing appropriate VPs. What is more, we see that, for tracked functions h₁ (t), h₂ (t), h₃ (t), h₄ (t), h₅ (t), and h₆ (t), System 1₁, System 1₃, System 1₁, System 1₅, System 1₃, System 1₅ performance best respectively.

5.4 Comparative experiment of proposed FSs with different aggregation functions

In this subsection, regarding the above six paths, we investigate the influence of the proposed FSs with different aggregation functions on the tracking effect of fuzzy PID control system. For tracked functions h₁ (t), h₂ (t), h₃ (t), h₄ (t), h₅ (t), and h₆ (t), FSs with triangular VPs, sinusoial VPs, triangular VPs, logarithmic VPs, sinusoial VPs, and logarithmic VPs are used respectively, which perform best in the previous subsection. We also employ Eq. (5.1) to evaluate the quality of FSs with different aggregation functions. For some t-norms, uninorms, and overlap functions, we obtain tracking errors for the six tracked functions as shown in Table 7, where the blue font represents the minimum error of systems regarding the tracked function.

Table 7
Errors of different aggregation functions for tracked functions

Tracked functions

Errors

Aggregation functions

h₁ (t) h₂ (t) h₃ (t) h₄ (t) h₅ (t) h₆ (t)

T _M 0.0909 0.0168 0.0979 0.0373 0.0329 0.0151

T _P 0.0909 0.0168 0.0979 0.0373 0.0330 0.0151

T _L 0.0909 0.0168 0.0979 0.0373 0.0327 0.0151

T _D 0.0997 0.0162 0.1660 0.1694 0.0315 0.0240

U ₁ 0.0993 0.0163 0.1651 0.1677 0.0316 0.0240

U ₂ 0.0986 0.0163 0.1653 0.1695 0.0316 0.0240

U ₃ 0.1011 0.0162 0.1666 0.1694 0.0341 0.0240

$O_{1}^{5}$ 0.0904 0.0164 0.1560 0.1360 0.0323 0.0241

$O_{1}^{0.9}$ 0.0910 0.0169 0.0964 0.0331 0.0342 0.0182

$O_{2}^{0.9}$ 0.0910 0.0169 0.0964 0.0331 0.0333 0.0139

$O_{3}^{3}$ 0.0905 0.0164 0.1388 0.1023 0.0328 0.0471

$O_{3}^{1.05}$ 0.0904 0.0169 0.0968 0.0382 0.0342 0.0189

Tracked functions
T _M	0.0909	0.0168	0.0979	0.0373	0.0329	0.0151
T _P	0.0909	0.0168	0.0979	0.0373	0.0330	0.0151
T _L	0.0909	0.0168	0.0979	0.0373	0.0327	0.0151
T _D	0.0997	0.0162	0.1660	0.1694	0.0315	0.0240
U ₁	0.0993	0.0163	0.1651	0.1677	0.0316	0.0240
U ₂	0.0986	0.0163	0.1653	0.1695	0.0316	0.0240
U ₃	0.1011	0.0162	0.1666	0.1694	0.0341	0.0240
$O_{1}^{5}$	0.0904	0.0164	0.1560	0.1360	0.0323	0.0241
$O_{1}^{0.9}$	0.0910	0.0169	0.0964	0.0331	0.0342	0.0182
$O_{2}^{0.9}$	0.0910	0.0169	0.0964	0.0331	0.0333	0.0139
$O_{3}^{3}$	0.0905	0.0164	0.1388	0.1023	0.0328	0.0471
$O_{3}^{1.05}$	0.0904	0.0169	0.0968	0.0382	0.0342	0.0189

It can be seen from Table 7 that, for tracked functions h₁ (t), h₂ (t), h₃ (t), h₄ (t), h₅ (t), and h₆ (t), the best performance FSs are System 1₁ with $O_{1}^{5}$ and System 1₁ with $0_{3}^{1.05}$ , System 1₃ with T_D and System 1₃ U₃, System 1₁ with $O_{2}^{0.9}$ and System 1₁ with $O_{1}^{0.9}$ , System 1₅ with $O_{2}^{0.9}$ and System 1₅ with $O_{1}^{0.9}$ , System 1₃ with T_D, System 1₅ with $O_{2}^{0.9}$ respectively. It indicates that the aggregation functions that make the FSs perform best are different for different tracked functions. Therefore, selecting an aggregation function needs concrete analysis according to actual problems. Fortunately, tracking errors can certainly be further reduced by choosing appropriate aggregation functions. Furthermore, the tracking effect of these FSs are displayed in Figs. 12 and 13 show the difference between predetermined positions and tracking positions for these FSs. Figure 12 vividly demonstrates that the proposed FSs can be effectively applied to fuzzy PID control systems to handle path tracking problems.

Fig. 12

Path tracking graphs for six functions.

Fig. 13

Path tracking deference graphs for six functions

Note that, for aggregation functions shown in Table 7, T_M, T_P, T_L, $O_{1}^{5}$ , $O_{1}^{0.9}$ $O_{2}^{0.9}$ , $O_{3}^{3}$ , $O_{3}^{1.05}$ , are continuous, and T_D, U₁, U₂, U₃ are discontinuous. We use Eq. (5.2) to calculate the average tracking errors of fuzzy PID control systems consisting of FSs with continuous (discontinuous) aggregation functions, as shown in Table 8. $AVE (δ) = \frac{\overset{N}{\underset{i = 1}{Σ}} δ_{i}}{N},$ (7) where δ_i is the tracking error of the fuzzy PID control system consisting of FS with the continuous (discontinuous) aggregation function, N is the number of fuzzy PID control systems consisting of FSs with continuous (discontinuous) aggregation functions.

Table 8

Average errors of different aggregation functions for tracked functions

Tracked functions
Errors
Aggregation functions
	h₁ (t)	h₂ (t)	h₃ (t)	h₄ (t)	h₅ (t)	h₆ (t)
Continuous	0.0906	0.0167	0.1098	0.0563	0.0332	0.0209
Discontinuous	0.0992	0.0163	0.1658	0.1689	0.0328	0.0240

It can be seen from Table 8, we see that the FSs with continuous aggregation functions overall perform better than FSs with discontinuous aggregation functions in fuzzy PID control systems to track paths h₁ (t), h₃ (t), h₄ (t), and h₄ (t), and the average errors of FSs with continuous aggregation functions is significantly smaller than the average errors of FSs with discontinuous aggregation functions. When paths h₂ (t), h₅ (t) are tracked, the FSs with discontinuous aggregation function overall perform better than FSs with continuous aggregation function in fuzzy PID control systems. However, the difference between the average errors of FSs with discontinuous aggregation functions and the average errors of FSs with continuous aggregation functions is not obvious. In terms of tracking accuracy, the FSs with continuous aggregation functions perform better than FSs with discontinuous aggregation functions in fuzzy PID control systems to track six basic paths.

5.5 Comparative experiment with other systems

In this subsection, we compare the tracking effect of fuzzy PID control systems consisting of the proposed FS and traditional FSs whose configurations are shown in Table 4.

It is worthy noting that the “Aggregation” in Table 4 is different from aggregation functions mentioned in this paper. In fact, aggregation functions in this paper play a role same as “And method" in Table 4.

Eq. (5.1) is employed again to evaluate the quality of systems. Some simulation experiments on MATLAB are constructed. If paths are those six types of basic elementary functions, then tracking errors with respect to those three systems are displayed in Table 9.

Comparing Tables 6 and 9, we found that when tracking paths h₃ (t), h₄ (t), and h₆ (t), the proposed FS can reduce the tracking error of the fuzzy PID control system by changing the type of VP, compared to the traditional FSs. In fact, by observing Tables 7 and 9, we found that the proposed FS significantly reduces the tracking error of the fuzzy PID control system for the six paths, compared to the traditional FSs, achieved by changing the type of aggregation function, most of which are implemented through overlap functions and t-norm T_D. Consequently, it is obtained that if appropriate VPs and aggregation functions are chosen for proposed FSs, then the proposed FS performs better than traditional FSs in fuzzy PID control systems to track the six paths.

Table 9
Errors of different systems for tracked functions

Tracked functions

Errors

Aggregation functions

h₁ (t) h₂ (t) h₃ (t) h₄ (t) h₅ (t) h₆ (t)

System 2 0.0943 0.0162 0.1054 0.0624 0.0326 0.0188

System 3 0.0904 0.0164 0.1599 0.1697 0.0324 0.0241

System 4 0.0928 0.0163 0.1119 0.0710 0.0325 0.0241

Tracked functions
System 2	0.0943	0.0162	0.1054	0.0624	0.0326	0.0188
System 3	0.0904	0.0164	0.1599	0.1697	0.0324	0.0241
System 4	0.0928	0.0163	0.1119	0.0710	0.0325	0.0241

Next, in the case of more complex paths, we compare the tracking effect of fuzzy PID control systems consisting of proposed FSs and three traditional FSs. The follows are preset paths.

h₇ (t) = (0.2 cos(πt)) ³;

h₈ (t) = t^1.75 + 0.1 (0.1) ^t;

h₉ (t) = arctan t + 0.2 cos(πt);

h₁₀ (t) = t^1.75 + t.

For h₇ (t), it is evident that h₇ (t) is composed of a trigonometric function and a power function. Based on the previous experiments, we know System 1₅ with

O_{2}^{0.9}

and System 1₅ with

O_{1}^{0.9}

make the fuzzy PID control system perform best to track the power function path h₄ (t) = t², System 1₃ with T_D and System 1₃ with U₃ make the fuzzy PID control system perform best to track the trigonometric function path h₂ (t) =0.1 cos(πt). Then we employ the fuzzy PID control systems consisting of these three FSs respectively to track the path h₇ (t), and compare the tracking effect with System 2, System 3, and System 4. Some simulation experiments on MATLAB are constructed, and tracking errors of these systems for h₇ (t) are displayed in Table 10, where the blue font represents the minimum error of systems regarding h₇ (t). It is worthy to note that System

1_{5} (O_{2}^{0.9})

refers to System 1₅ with

O_{2}^{0.9}

, and the rest can be deduced from this. For h₈ (t), it is evident that h₈ (t) is a linear combination of a power function and an exponential function. Based on the previous experiments, we know System 1₅ with

O_{2}^{0.9}

and System 1₅ with

O_{1}^{0.9}

make the fuzzy PID control system perform best to track the power function path h₄ (t) = t², System 1₁ with

O_{2}^{0.9}

and System 1₁ with

O_{1}^{0.9}

make the fuzzy PID control system perform best to track the exponential function path h₃ (t) =2^t. Then we employ the fuzzy PID control systems consisting of these three FSs respectively to track the path h₈ (t), and compare the tracking effect with System 2, System 3, System 4. Some simulation experiments on MATLAB are constructed, and tracking errors of these systems for h₈ (t) are displayed in Table 11, where the blue font represents the minimum error of systems regarding h₈ (t). Similarly, tracking errors of proposed systems and traditional systems for h₉ (t) and h₁₀ (t) are displayed in Tables 12 and 13 respectively.

Table 10

Errors of different systems for h₇ (t)

Systems	System $1_{5} (O_{2}^{0.9})$	System $1_{5} (O_{1}^{0.9})$	System 1₃ (T_D)	System 1₃ (U₃)	System 2	System 3	System 4
Errors	0.0056	0.0056	0.0054	0.0054	0.0055	0.0055	0.0055

Table 11

Errors of different systems for h₈ (t)

Systems	System $1_{5} (O_{2}^{0.9})$	System $1_{5} (O_{1}^{0.9})$	System $1_{1} (O_{2}^{0.9})$	System $1_{1} (O_{1}^{0.9})$	System 2	System 3	System 4
Errors	0.0285	0.0285	0.0349	0.0349	0.0516	0.1238	0.0603

Table 12

Errors of different systems for h₉ (t)

Systems	System $1_{5} (O_{2}^{0.9})$	System 1₃ (T_D)	System 1₃ (U₃)	System 2	System 3	System 4
Errors	0.0247	0.0353	0.0353	0.0289	0.0356	0.0356

Table 13

Errors of different systems for h₁₀ (t)

Systems	System $1_{5} (O_{2}^{0.9})$	System $1_{5} (O_{1}^{0.9})$	System 2	System 3	System 4
Errors	0.0283	0.0283	0.0525	0.1284	0.0614

From Tables 10-13, it also can be concluded that if the appropriate VP and the aggregation functions are chosen for proposed FSs, then the proposed FS performs also better than traditional FSs in fuzzy PID control systems to track complex paths. In what follows, steps how to choose appropriate VPs and aggregation functions for proposed FSs are given.

Observe the complex path function is composed of which basic elementary functions compose.

List the FSs that make the fuzzy PID control system most effective in tracking these elementary functions. VPs and aggregation functions of these FSs are given in section 5.4.

Through experimental testing FSs listed in Step 2, the optimal FS in the fuzzy PID control system to track the complex path is obtained.

Based on the above steps, the proposed FS can be extended to deal with complicated cases.

In conclusion, whether it is tracking six basic paths or tracking complex paths, the proposed FS improves the tracking performance of the fuzzy PID control system composed of traditional FSs. Therefore, the proposed FS is effective in handling path tracking problems. Additionally, the proposed fuzzy system has a simple structure and does not require fuzzification, fuzzy inference, and defuzzification, simplifying the algorithm steps of the FS. However, it still enhances the tracking performance of the fuzzy PID control system composed of traditional FSs, making the proposed FS efficient.

6 Conclusion

This paper presents the MISO FS-VPs processing information directly from the fuzzy rule base, addressing a gap in the literature relating to the inherent subjectivity and linguistic interpretation issues associated with existing FS. We define a n-dimensional VP of the n-dimensional universe utilizing a 1-dimensional vague partition and aggregation functions, subsequently analyzing its properties. Based on this, the algorithm of FS to calculate the output for corresponding input is given, and the proposed FS is proved a universal approximator, providing the theoretical support necessary for its practical application. A practical application of the proposed FS is demonstrated via its combination with a PID control system to resolve autonomous vehicle path tracking control problems. Experimental results illustrate that the proposed FS is flexible and suitable for various scenarios. Suppose there is a high demand for the robustness of the control system. In that case, we can choose the fuzzy PID control system consisting of the proposed FS with a discontinuous aggregation function. Suppose there is a high demand for the tracking accuracy of the control system. In that case, we can choose the fuzzy PID control system consisting of the proposed FS with a continuous aggregation function. What is more, experimental results support the notion that the system is effective and highly efficient compared to traditional fuzzy systems for path tracking control problems. Additionally, FSs also have been successfully applied to financial forecasting [48, 49], decision analysis [50, 51], medical systems [52, 53]. In future work, we intend to apply the FS-VPs to financial market forecasts. Various factors, such as economic data, political events, market sentiment, etc., influence financial market trends. These factors often need more specific and easier to quantify and predict. Through FSs, these fuzzy factors can be transformed into actionable rules. The proposed FS can establish VPs to predict future market trends by analyzing historical data and related factors. Also, trying to apply FSs-VPs to decision analysis and medical systems is our research direction.

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The fuzzy system based on vague partitions and its application to path tracking control for autonomous vehicles

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 1-dimensional VP

2.2 Aggregation functions

3 The n-dimensional VP and its properties

4 The FS-VPs and its approximation properties

4.1 Fuzzy rule base

5 An application to path tracking control

5.1 Description of fuzzy PID control system

Table 9 Errors of different systems for tracked functions Tracked functions Errors Aggregation functions h1 (t) h2 (t) h3 (t) h4 (t) h5 (t) h6 (t) System 2 0.0943 0.0162 0.1054 0.0624 0.0326 0.0188 System 3 0.0904 0.0164 0.1599 0.1697 0.0324 0.0241 System 4 0.0928 0.0163 0.1119 0.0710 0.0325 0.0241

References