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The problems of the admissibility and state feedback stabilization for discrete-time singular systems with interval time-varying delay and norm-bounded uncertainty are studied. The system is equivalently transformed into a new comparison form by decomposition. By taking advantage of the Seuret summation inequality, the reciprocally convex inequality, and some relaxation techniques, a delay-dependent criterion that ensures the admissibility of the concerned systems is established. The result on robust stabilization is also obtained by fixing some parameters. It should be pointed out that the results are less dependent on the parameters so that some conservatism is reduced. A numerical example is included to illustrate the effectiveness and improvement of the proposed methods.

Discrete-time systems, which are the analogues of the continuous-time systems, have more advantages in flexibility, anti-interference, precision, and economy. The discrete-time systems inherit the dynamic behavior of the corresponding continuous-time systems, while there is a lot of difference of the stability analysis and control synthesis between them. Intensive results have been given to study the problems for various discrete-time systems.

Most physical systems and processes in the real world can be represented as singular systems which contain not only differential equations but also nondynamic constraints and even improper parts of the systems. Considerable attention has been paid to singular systems [

In this paper, the problem of a robust controller design for discrete-time singular systems with time delay is investigated. By utilizing the Seuret summation inequality, the reciprocally convex technique, and some transformations, new sufficient condition which contains auxiliary matrices for the admissibility is derived in terms of LMIs. Moreover, the problem of robust admissibilization of uncertain systems is also studied and a new condition is proposed. If the parameter

Notation: throughout the paper, standard notations are used. The superscript

Consider the discrete-time singular system with time delay which can be represented by

For the discrete time-delay system:

(i) The pair

(i) For given integers

Moreover, for the pair

In this paper, the state feedback controller with the following form is used:

Now, we will use some lemmas in the proof which should be introduced first.

Let

Let

Consider the following inequality in the variable

For any matrix

For matrices

In this section, a new condition of admissibility for discrete-time singular systems with time-varying delay is derived and it is extended to design the state feedback controllers. For simplicity, in rest of the section, we will consider the transformation of system (

Now, let

We first consider the stability of system (

It is easy to see that the stability of the singular delay system (

The closed-loop system is

For system (

The discrete-time delay singular system (

By Lemma 1 and equations (

Multiplying (

Next, we will consider the stability of the system. We choose the Lyapunov–Krasovskii functional as follows:

Calculating the forward difference of

By Lemmas 4 and 5, the cross terms in (

Noted that

Combining (

It is known that

Hence, there exists a scalar

Next, we will decompose (

For the special structure of the matrix

Since the matrix

Next, we will give that inequality (

Noted that equation (

By Lemma 3, (

By Schur’s complement, (

Theorem 1 gives a new criterion for discrete-time singular systems with time-varying delay. By Lemmas 2 and 3, the admissible condition (

Next, we will further deal with the state feedback control problem for discrete-time systems (

The closed-loop system (

Replacing

Usually, an additional equation

The auxiliary matrices which are added in the criteria can really increase the computational complexity.

Consider the discrete time-delay singular system (

It is proved in [

Control results of example 1. (a) State response curves for the closed-loop system. (b) Input signals.

In this paper, the state feedback admissibilization for discrete-time singular systems with time-varying delay is under consideration. By using the Seuret summation inequality, the reciprocally convex inequality, and some relaxation techniques, a new admissible condition for the considered system is obtained. And the state feedback controller design problem can be solved by the linear matrix inequalities (LMIs) such that the closed-loop system is admissible. Finally, a simulation is provided to demonstrate the effectiveness of the proposed methods. Furthermore, research topics can include the extension of our results to more complex cases with performance control and performance analysis and to further reduce the conservatism.

Our manuscript involves the theory work and is not related to the data. So, we do not provide the information about the data availability statement.

The authors declare that they have no conflicts of interest.

This work was supported in part by the National Natural Science Foundation of China (61803220, 61503222, and 21606141).

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