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For discrete fuzzy descriptor systems with time-delays, the problem of designing fuzzy observers is investigated in this paper. Based on an equivalent transformation, discrete fuzzy descriptor systems with time-delays are converted into standard discrete systems with time-delays. Then, via linear matrix inequality (LMI) approach, both delay-dependent and delay-independent conditions for the existence of fuzzy state observers are obtained. Finally, two numerical examples are provided to illustrate the proposed method.

For many practical engineering systems, increased productivity has led to new operating conditions, which are more challenging. Such conditions would affect system’s performance. To improve efficiency, a number of methods have been proposed, such as fault-tolerant control [

Interconnected systems, which are described as different names in [

Utilizing the approach of linear matrix inequality (LMI), [

For a system, the control is often based on state feedback. However, all states of a system are not always available. At this time, it seems very important to estimate system states. So many scholars begin to focus on the problem of designing observers and filters. For example, a proportional multiple-integral observer was investigated for fuzzy chaotic model with unknown input in [

The paper is organized as follows. Section

In this section, we will briefly describe the problem to be studied. Through this paper, the following discrete-time-delayed descriptor system is considered.

Plant form is as follows.

We make the following assumption.

Consider

Because of the singularity of matrix

Now we try to transform system (

According to equality (

Let

Define

Through the above analysis, singular fuzzy system (

Thus the problem of this paper focuses on finding matrices

This section discusses how to design fuzzy state observers for discrete descriptor systems with time-delays. And for existence of fuzzy observer, two different sufficient conditions are derived. We give the Schur complement Lemma first.

The LMI

Now a sufficient condition about existence of considered observer, which does not rely on time-delay, is presented by the following theorem.

Model (

Define

Take a Lyapunov function as

The inequality

On the other hand, according to Lemma

Obviously, conditions in Theorem

For fuzzy system (

According to proof of Theorem

By Lemma

Taking

Combining with inequalities (

It is easy to see that Theorem

For fuzzy state-space systems, the results of this paper can also be applied. Choosing

At the end, we give an algorithm to design observer for fuzzy descriptor system (

The following steps are introduced to determine observer (

Verify condition (

According to formulation (

Solve LMIs (

Give fuzzy observer (

The steps of designing observer for fuzzy descriptor system (

In this paper, we firstly consider observer design for fuzzy discrete systems with singularity and time-delays simultaneously. Compared with existing work [

In this section, two numerical examples are provided. The first one is to design a fuzzy observer according to Theorem

Consider a discrete fuzzy descriptor system with time-delays as follows.

Coefficient matrices of fuzzy descriptor system are given, and it can be easily verified that matrices

Then fuzzy descriptor system is converted into the following ordinary system:

By using MATLAB LMI Toolbox and solving LMIs in Theorem

Membership functions.

State of the fuzzy descriptor system.

State of the fuzzy observer.

Error system.

Consider the fuzzy system in Example

State trajectories of fuzzy system are the same as those in Figure

State of the fuzzy observer.

Error system.

From these two examples above, it is obvious that error in Figure

This paper has discussed how to design fuzzy observers for discrete descriptor systems with time-delays. According to a simple transformation, the singularity of considered fuzzy systems has been eliminated. Then two sufficient conditions for the existence of fuzzy observer have been derived. Notably, one condition depends on time-delays and the other does not. Finally, different fuzzy observers have been designed for the same discrete-time-delayed descriptor system. By comparison, the fuzzy observer depending on delay condition is better. Additionally, our future work will deal with fuzzy time-delayed descriptor systems by using delay-partitioning method.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was partially supported by NNSF of China (61174141, 61374025), Research Awards Young and Middle-Aged Scientists of Shandong Province (BS2011SF009, BS2011DX019), and Excellent Youth Foundation of Shandong Province (JQ201219).

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