Intelligent Automation & Soft Computing
DOI:10.32604/iasc.2020.012805
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Article

Imperfect Premise Matching Controller Design for Interval Type-2 Fuzzy Systems under Network Environments

Zejian Zhang1, Dawei Wang2,*and Xiao-Zhi Gao3

1Mechanical Engineering College, Beihua University, Jilin, 132013, China
2Mathematic College, Beihua University, Jilin, 132013, China
3School of Computing, University of Eastern Finland, Kuopio, 70211, Finland
*Corresponding Author: Dawei Wang, Email: wdw9211@126.com
Received: 13 July 2020; Accepted: 11 September 2020

Abstract: The interval type-2 fuzzy sets can describe nonlinear plants with uncertain parameters. It exists in nonlinearity. The parameter uncertainties extensively exist in the nonlinear practical Networked Control Systems (NCSs), and it is paramount to investigate the stabilization of the NCSs on account of the section type-2 fuzzy systems. Notice that most of the existing research work is only on account of the convention Parallel Distribution Compensation (PDC). For overcoming the weak point of the PDC and acquire certain guard stability conditions, the state tickling regulator under imperfect premise matching can be constructed to steady the NCSs using the section type-2 indistinct muster, where the fuzzy plant and fuzzy regulator may enjoy together not the same membership functions. By leading into the message of the up and down membership functions of both the fuzzy pattern and fuzzy regulator, we build a new composed term of the section type-2 network systems. The new results we obtained can provide a larger area of stability than the conventional membership unconcerned stability results. Moreover, a novel unmatching state feedback controller design method for the interval type-2 networked systems is explored in our work. The proposed approach can significantly improve the design flexibility of the fuzzy controllers, because their membership functions can be arbitrarily selected. Two numerical examples are further used to demonstrate the less-conservativeness and effectiveness of the novel technique.

Keywords: Network control systems; stability analysis; interval type-2 fuzzy systems; controller design; imperfect premise matching

1  Introduction

As we know, the Takagi-Sugeno can efficacious express non-linear dynamics, and often is called a type-1 system. Unfortunately, the type-1 fuzzy sets have some difficulties in describing the nonlinear plants with uncertain parameters. In order to handle this drawback [1], Zadeh [2] introduced the type-2 fuzzy sets as an extension of the type-1 fuzzy sets. In view of the type-2 fuzzy sets, Mendel et al. [3] proposed the section Type-2 (IT2) fuzzy sets. During the past decades, the IT2 fuzzy model has been widely applied in practice [4,5]. For example, in Lam et al. [6], the stability area and the fuzzy regulator for the IT2 fuzzy model-based system were examined, in which the parameter non-determinacy are captured by both the up and down grades of fuzzy membership. Most of the controller design methods reported in the literature are inspired by the PDC scheme, and the fuzzy pattern and fuzzy regulator are provided with the identical membership functions. Nevertheless, Lam developed the concept of “imperfect premise matching” in Lam et al. [7], i.e., the membership functions of the fuzzy regulator probable be distinct from fuzzy model. The stabilization issue of the type-1 systems was investigated in Zhang et al. [8–11]. Li et al. [12,13] generalized these research results to the IT2 fuzzy model with the imperfect premise matching, thus improving the flexibility in highway design and cutting down the complexity of implementation. More work on the Interval type-2 fuzzy model stability conditions and controller design was presented in Wu et al. [14–16].

The Networked Control Systems (NCSs) have been got a lot of attention owing to their theory and reality purport [17]. Compared with the conventional point-point connections, the network systems reduce the heavy expenses of cables and facility maintenances. The network systems have found numerous applications in the fields of automobile, aerospace, industrial manufacturing, etc. [18–21]. Moreover, It is difficult if not impossible to guarantee the steadiness of the NCSs. what to analyze and devise the stability requirement for the T-S under NCSs has been a popular research topic [22,23]. Such as in Chi et al. [24], the networked images filtration with many output and many transducer nonsynchronous take sample for the T-S fuzzy systems was explored. By drawing into Laypunov-Krasovskii function to analysis the robust stability and the project of the state feedback for the NCSs in Rouamel et al. [25]. Actually, the above issues were received in view of the type-1 fuzzy set theory. NCSs also have parameter uncertainties, and considerable research work concerning the sector type-2 fuzzy pattern has been carried out [26,27]. However, the current efforts on the controller design are mainly inspired by the convention PDC idea. For the sake of overcoming the shortcomings of the PDC and receive a number of less conservative stability conditions, a novel devise way was put forward to manage the unmatching premise of the type-1 systems [28]. It is necessary to further investigate the controller design scheme under the imperfect premise matching for the Interval type-2 T-S fuzzy systems under NCSs.

In this article, the innovations of the paper are as follows:

A less conservative stability conditions are obtained.

The unmatching regulator is designed for the nonlinear NCSs in view of the Interval typ-2 T-S, which makes the regulator design simpler and more flexible.

The proposed method can be generalized to other types of nonlinear control systems.

The rest of our article is arranged as follows. In Section 2, the controller design problem under consideration is represented in details. In Section 3, the synthesis of the state-feedback type-2 fuzzy controller under the imperfect premise matching is presented. In Section 4, for explaining the practicability and validity of the recommend technique, we give two numerical examples. Ultimately, In Section 5, we summarize this paper with some remarks and conclusions

2  Preparatory Knowledge

Here the interval type-2 T-S fuzzy model can be described as follows:

The imagesth rule can be represented as follows:

IF images is images … images is images THEN

images, images l is the number of the fuzzy rules. images are the constant matrices. images is the input vector, images is the state vector.

We have:

where

They content the character of images, and images and images show the down and up grades of the membership. Then we have

where

in which images, and images are the nonlinear functions with images.

In this article, The system in Eq. (1) through IP based network control, system status can be used for feedback, as shown in Fig. 1 [29].

Figure 1: The interval type-2 networking system diagram

Distinct of the famous Parallel Distributed Compensation (PDC) plan means, we adopt a new unmatching fuzzy control law to make the interval type-2 fuzzy systems in Eq. (3) stable. For the NCSs, the transducer is clock-driven and images express the sampling period is constant. The regulator and actuator are eventdriven. Suppose that images express the sampling cycle of the semaphore, and the images express the Δth acquisition time.

Under imperfect premise matching, we have the following fuzzy regulator:

Rule images: IF images is images… images is images THEN

where images are the unknown tickling gains to be pending. On the other hand,

where

For images, we have

Here the images and images express the down and up membership functions. They satisfy the character images. images and images express the down and up grades of the membership functions. We have

Owing to the network slowdown is invariably bounded, The input delay is:

where images is the upper limit of the lag. Putting Eq. (8) into Eq. (6), the following regulator can be shown:

We have the following closed-loop networked system:

Lemma 1 [30] For images , images, supposing that there has images, for images, images, we have:

Lemma 2 [31] images, and images are constant matrices, and images. We have

if and only if images and images set up.

Lemma 3 [32] The images is equivalent to images, images, where images, images, and images is a matrix .

3  Main Results

Theorem 1 When the membership functions satisfy images and images for all images, where images, and for given invariant images and matrix images, and there exist matrices images andimages of such that the following LMIs hold:

where

Here, “*” represents the transposition of symmetric position elements, then the fuzzy type-2 NCSs described by Eq. (10) is asymptotically stable.

Proof. Select the following Lyapunov function:

where

We take the derivative of images:

where

.

where

For any matrices images, we have :

By Lemma 1, we get

With the above equation, we have

Let

where

From Eq. (34), it is obvious that if

we have images. Next, we will consider the following equation

where images. For reducing the conservativeness, the above equations are added to Eqs. (39) and (40), we have

If images and images, imagesand images hold. With images and images, let

The Eqs. (45)–(50) are equivalent to the following inequalities on the basis of Eq. (37), Eq. (38) and Lemma 2:

We can see from Shur complement, the above inequalities Eqs. (51)–(62) are equivalent to Eqs. (13)–(24). Since images, the system described in Eq. (10) is asymptotically stable.

Remark 1 We consider the relationship between the membership and the fuzzy regulator in Theorem 1, which is more relaxed than the stability condition that does not consider the information of the membership (Corollary 1). When images, Theorem 1 can be reduced to Corollary 1 below, i.e., the membership function independent stability condition.

Corollary 1. The scalars images and the matrix images are given, the Eq. (10) is asymptotically stable, if there exist matrices images and images make the under non-equality set up:

Based on Theorem 1, The following theorem 2 will give the controller design method.

Theorem 2. The scalars images are given, the Eq. (10) is asymptotically stable with feedback gains images, if there satisfy satisfy images and images for all images, where images, and the following matrices images andimages exist, and make the following LMIs set up:

where

Proof. Pre- and post-multiply the diag images and its transpose to both sides of Eqs. (13)–(24). Let images, and the new variables are as follows: images,images,images, images, images, images, images, images, images, images, images. With images, the inequalities Eqs. (75)–(86) can be obtained.

4  Numerical Examples

Example 1 [33]. In order to compare the results of Theorem 1 and Corollary 1, stability regions are shown in Figs. 2(a) and 2(b).

Rule images: images is images,

where

We choose the following function as the membership functions of Eq. (87)

Distinct of the above membership, the down and up membership functions of controller are

Here, we select images, and images, and the feedback gains of images take the value when the characteristic root is images, and we obtain the stability area which is shown in Figs. 2(a) and 2(b) based on Theorem 1 and Corollary 1. we can see larger stability area can be obtained from Theorem 1 than that from Corollary 1(based on the PDC method), which means that the stability condition under the incomplete premise matching is less conservative.

Figure 2: (a) Stable area of Theorem 1 with images, images, images (denoted by “o”); (b) Stable area of Corollary 1 with images (denoted by “×”)

Example 2 Consider the following mechanical system [27]

Let images, images, and images. images, images. The mass-spring-damper system in Eq. (88) is described as:

where

In this example, the lower and upper bounds of the membership functions of the plant in Eq. (89) can be selected as the same as those in Example 1. However, according to Theorem 2, the membership functions of the controller cannot be chosen as images. For simplification, they are set to be images, which satisfy images and images. Using Theorem 2 with images and images, we can obtain the maximum value of delay images, and the controller gains are:

With the controller gains and membership functions images, the controller in Eq. (6) is applied to control the mass-spring-damper mechanical system in Eq. (89). The system responses and control inputs are shown in Figs. 3 and 4, respectively under the initial condition of images, which demonstrate that the proposed controller design method is indeed effective and efficient.

Remark 2 Compared with conventional controller design techniques proposed in Lam et al. [6,14–16,26,27] where the fuzzy regulator is bear on the up and down membership functions of the interval type-2 model, the membership functions of our networked fuzzy controller are much simpler, thus significantly enhancing the controller design flexibility.

Figure 3: Answers of the control system in Eq. (10) with images

Figure 4: System control input

5  Conclusions

In this article, we investigate the stability and synthesis for the interval type-2 fuzzy systems. A less conservative stability criterion is got. The stability condition can offer larger stability regions than those obtained from the traditional PDC scheme. Furthermore, a new design technique under the imperfect premise matching is developed in order to stabilize the control systems. Distinct of the PDC controller design method, some simple and certain functions as the membership functions of the fuzzy controller can be selected. Two numerical examples are used to demonstrate the less conservativeness of the above methods. They can significantly improve the design flexibility as well as reduce the implementation complexity of the fuzzy controller.

Funding Statement: The author(s) received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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