An LMI Based Criterion for Global Asymptotic Stability of Discrete-Time State-Delayed Systems with Saturation Nonlinearities

A linear matrix inequality (LMI) based criterion for the global asymptotic stability of discrete-time systems with multiple state-delays employing saturation nonlinearities is presented. Numerical examples highlighting the effectiveness of the proposed criterion are given.

Stability analysis of discrete-time systems in the simultaneous presence of nonlinearities and time delays in their physical models is an important problem. This paper, therefore, deals with the problem of stability analysis of a class of discrete-time state-delayed systems in state-space realization employing saturation overflow arithmetic. The paper is organized as follows. Section 2 introduces the system under consideration. A computationally tractable criterion for the global asymptotic stability of discretetime state-delayed systems employing saturation overflow arithmetic is established in Section 3. It is demonstrated in Section 4 that a previously reported criterion is recovered from the presented approach as a special case. In Section 5, two examples highlighting the effectiveness of the presented approach are given.

Remark 4. Note that condition
Remark 5. Condition (10) provides a limit cycle-free realizability condition for the system with saturation arithmetic.
Remark 6. Stability of the system can be established via Theorem 2 for one combination of the elements of the matrix A, that is, where the elements of first rows of A satisfy (5a) and those of the remaining ( − ) rows satisfy (5b). The stability results for the other possible combinations of the elements of matrix A can easily be worked out.

Comparison
In this section, we will compare the main result of this paper with the result stated in [41].
Remark 9. The present work may be treated as an extension of [41]. Moreover, the present approach leads to generalized and improved result over the result appearing in [41].

Numerical Examples
In this section, two numerical examples are given to demonstrate the usefulness of the present result.
We now apply Theorem 2 in the example under consideration. To check the feasibility of (10), we choose the matrix C in the following form: where 12 > 0 and 12 > 0. With the help of MATLAB LMI toolbox [39,40], it turns out that (10) yields the following solutions for the present system: Therefore, Theorem 2 affirms the global asymptotic stability of the present system. Figure 1 shows the trajectory of the state variable for the present example with The global asymptotic stability of the system under consideration (via Theorem 2) has also been verified for a number of randomly generated initial conditions with the help of trajectories traces of the system.
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Conclusions
An LMI-based sufficient condition (Theorem 2) for the global asymptotic stability of discrete-time systems with multiple state-delays employing saturation nonlinearities has been established. It is shown that Theorem 2 is less stringent than Theorem 7. Two numerical examples highlighting the usefulness of the presented result have been discussed. The potential extensions of the proposed idea to the problems of stability of linear discrete-time systems with intervallike time-varying delay in the state [42,43], stability of fixedpoint state-space digital filters with saturation arithmetic [44], robust stability of discrete-time state-delayed systems using generalized overflow nonlinearities [19], stability of linear systems with input saturation and asymmetric constraints on the control increment or rate [45], and stability of linear two-dimensional systems with multidelays and input saturation [46], to other situations such as [47,48], appear to be appealing problems for future investigation.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.