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The problem of an unmatching observer-based controller design for discrete-time fuzzy systems with time delay is investigated, in which the fuzzy controller shares different membership functions from the fuzzy model. The objective is to design a state observer and unmatching fuzzy controller such that the discrete closed-loop system with time delay is asymptotically stable. A sufficient condition that contains the information of the membership functions of fuzzy model and fuzzy controller for the stabilization via an unmatching observer-based output feedback is presented. The proposed control scheme is well capable of enhancing the design flexibility, and the stability condition is less conservative. Three numerical examples are given to illustrate the effectiveness and advantages of the proposed method.

Time-delay phenomena intensively exist in natural science and engineering and social life, such as nonlinear and time-delay models. For example, they are in numerous dynamical systems including biology systems, mechanics, economics, chemical systems, and network systems. Generally, time delay can often lead to instability and poor performances. Therefore, it is of great significance to investigate the issue of the stability for time-delay systems. A lot of relevant research work has been reported in [

In this paper, we focused on the observer-based stabilization control for discrete-time T-S fuzzy time-delay systems and a novel observer-based output feedback controller is investigated for the discrete-time T-S fuzzy time-delay systems. The main contributions of this paper can be summarized as follows:

As the membership functions of the fuzzy model and the fuzzy controller are all considered in the analysis, some membership-function-dependent stability conditions are derived, which are less conservative than membership-function-independent stability conditions

As the membership functions of the controller are distinctive from those of the fuzzy model, the membership functions can be chosen arbitrarily, which makes a higher degree of design flexibility of the fuzzy controller achieved, and its structure can be further simplified

The remainder of this paper is organized as follows. In Section

In this paper, if not explicitly stated, matrices are assumed to have compatible dimensions. The notation

Let

If

With the fuzzy inference methods, the final outputs of the fuzzy time-delay model can be formalized as follows:

In engineering practices, the state information of system equation (

Observer rule

The overall fuzzy observer is represented by

With the aforementioned fuzzy observer, an unmatching fuzzy observer control law is defined as follows.

If

The overall observer-based fuzzy control law is represented by

Combining the fuzzy controller in equation (

Therefore, the augmented system can be written as

It can be observed from equation (

(see [

For any

For convenience, we use

Especially, if we select

Provided that

Notice that

Defining

The equilibrium of the closed-loop discrete fuzzy time-delay system in equation (

The state feedback gains and observer gains can be constructed as

Select a Lyapunov function as

In this section, we will prove the asymptotic stability of the discrete time-delay system in equation (

If

By Schur complement,

If

It is obvious that equation (

Premultiplying and postmultiplying equation (

For arbitrary

These terms are introduced to equation (

With

On the other hand, for equation (

Let

If

In this section, three examples will be given to illustrate the less conservativeness and the effectiveness of the proposed results.

From the proposed results, we can see the stability conditions are connected with the values of

Rule

It is assumed that the membership functions of the discrete fuzzy model with time delay and fuzzy controller are different and satisfy

Stability region of Theorem

Stability conditions in [

An unmatching observer-based output feedback controller is designed to backing up control of a computer-simulated truck-trailer [

The fuzzy truck-trailer model can be modeled by a two-rule fuzzy model.

Rule

If

The fuzzy state observer for the T-S fuzzy model in equations (

Observer rule

The output of equation (

Here, we set

From Theorem

Rule

The fuzzy controller is defined as

The membership functions are selected as

We assume that

Assuming

With the initial condition

State responses

The same fuzzy state observer model equation (

Comparing with

State responses.

Comparing with the PDC scheme, as the membership functions of the controller are distinctive from those of the fuzzy model in our paper, the membership functions can be chosen arbitrarily. Therefore, the proposed method can enhance the design flexibility of the fuzzy controller as well as retain the robustness property of the T-S fuzzy control systems.

In this paper, an unmatching observer-based stabilization control for discrete-time T-S fuzzy systems with time delay is investigated, in which the discrete fuzzy observer model and fuzzy controller do not share the same membership functions. The information of the membership functions of the fuzzy model and controller is considered in the observer design scheme. Furthermore, the design flexibility can be enhanced by arbitrarily selecting simple membership functions for the observer-based controller. The advantages and effectiveness of the proposed method have been illustrated using simulation examples. According to the obtained results, the proposed method only fits for the systems with constant delay. Therefore, how to extend the abovementioned method to time-varying delay systems will be considered in the future.

The process data used to support the findings of this study have not been made available because they form part of an ongoing study.

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

This work was supported by projects of the National Natural Science Foundation of China (NSFC) (51805008) and project of the Scientific Research of Jilin Provincial Department of Education (JJKH20190644KJ and JJKH20190639KJ).