_{1}

The robust stability analysis for discrete large-scale uncertain systems with multiple time delays is addressed in this paper. We establish a method for selecting properly a positive definite matrix Q to derive a very simple upper solution bound of the discrete algebraic Lyapunov equation (DALE). Then, using the Lyapunov equation approach method with this upper bound, several sufficient conditions are presented to guarantee the robust stability of the overall systems. Comparisons between the proposed results with a previous one are also given.

It is known that the system is called an interval system and can be considered as a system with parametric perturbations when matrices of a state equation are interval matrix. In practice, time delay(s) exist(s) in real-life systems and should be integrated into system model. During the past decades, the research of systems with time delay(s) has also become an attractive topic. However, surveying the existing ones, only few works have been devoted to stability analysis and/or stabilization controller design of interval time-delay systems [

Consider the discrete composite interval time-delay system S which is described as an interconnection of N subsystems

, (1)

where _{ij} > 0 denotes the delay, A_{ijI} = A_{iI} for i = j, and A_{iI} and A_{ijI} are interval matrices with appropriate dimensions and have the properties:

Define matrices U_{i}, V_{i}, E_{ij}, and F_{ij}, respectively, as

In fact, system (1) can also be represented as follows.

, (5)

, (6)

, (7)

where A_{i} and A_{ij}, respectively, is defined by

Here, ΔA_{i} and ΔA_{ij} denote the parametric uncertainties with the following properties:

where R_{i} and S_{ij} are defined as

Then, we derive the following criteria.

Theorem 1. For

then the composite uncertain time-delay system (1) or (5) is robustly stable

Proof. The condition (11) infers

which can further implies

Therefore, we obtain _{i} for any given positive definite matrix Q_{i}.

, (14)

Here, we choose Q_{i} as

where

Then, from the Lyapunov Equation (14), we have

Due to the fact that

Then, utilizing these inequalities, we obtain

Substituting this inequality into (16), it is seen that if the condition (12) is satisfied then the right-hand side of (16) is a negative definite matrix. This means that the solution P_{i} of the Lyapunov Equation (16) has the upper bound

Here, we construct a Lyapunov function as follows.

Taking the forward difference for the Lyapunov function (19) results in

where (15) and the following relation are used.

Therefore, obviously the condition (11) can infer ΔV < 0 and hence the composite uncertain time-delay system (5) is robustly stable. Thus, this completes the proof.

Following the same approach that proposed in the proof of Theorem 1, we have the following results.

Theorem 2. The composite uncertain system (6) or (1) is robustly stable if the following condition is satisfied for

Proof. Using the Lyapunov Equation (7) with

Then, let

It is easy to obtain the stability condition (23). Details of the proof are omitted.

Theorem 3. If the following condition holds for

then the composite uncertain system (7) or (1) is robustly stable.

Proof. From (14), if we choose

and

then the condition (26) can assure the stability of the large-scale system (7) or (1). We also omit the remaining proof.

Remark 1. In [

where N_{i} denotes the number of _{i} is defined by

It is obvious that the condition (28) involves an inverse matrix and how to determine the positive constant β_{i} such that (28) is satisfied is an open problem. Furthermore, it is assumed that all eigenvalues of A_{i} are distinct. The conditions (11), (23), and (26) do not involve any inverse matrix and free variable. It is also not necessary to assume all eigenvalues of A_{i} are distinct. Therefore, they are less restrictive than (28). Besides, we have found that the tightness of the obtained results cannot be compared.

A new approach of the analysis of the robust stability for discrete large-scale interval systems with timedelays has been proposed in this paper. By utilizing the Lyapunov equation approach associated with a simple upper solution bound, several concise criteria have been derived to guarantee the robust stability of the aforementioned systems. The feature of these obtained results is that they do not involve any Lyapunov equation although the Lyapunov approach is utilized. Furthermore, comparing to a previous one, all eigenvalues of the system matrix A are not needed to be distinct in this work and the obtained results do not involve any inverse matrix and free variable. Therefore, they are less restrictive and easy to be checked. It is believed that this work is useful for the stabilization problem of discrete large-scale interval systems with timedelays.

The author would like to thank the Ministry of Science and Technology, Taiwan, for financial support of this research under the grant MOST 105-2221-E-230- 003.

Lee, C.-H. (2017) Sufficient Conditions for Robust Stability of Discrete Large-Scale Interval Systems with Multiple Time Delays. Journal of Applied Mathematics and Physics, 5, 759- 765. https://doi.org/10.4236/jamp.2017.54064