Efficient Approach to Stability Analysis of Discrete-Time Systems with Time-Varying Delay

This paper aims at deriving an efficient criterion for the robust stability analysis of discrete-time systems with time-varying delay. In the derivation, to obtain a larger stability region under the requirement of less computational complexity, this paper proposes a valuable method capable of establishing a less conservative stability criterion without using the free-weighting approach and an extremely augmented state. In parallel, the stabilization problem of systems with time-delayed control input is addressed in connection with the derived stability criterion.


Introduction
Over the last few decades, research on stability analysis of time-delay systems has rapidly accelerated owing to two main reasons.One is that such systems offer suitable mathematical models that can represent practical engineering systems with finite but uncertain signal propagation delays, such as biological systems, network systems, and nuclear reactors [1][2][3].The other is that time-delay can often act as a critical factor that leads to performance degradation and instability of the systems under consideration [4][5][6].Meanwhile, with the growing interest, significant progress has been made toward enlarging the feasible region of a stability criterion (referred to here as "stability region") and reducing its computational complexity.In other words, numerous investigations and research have been carried out to establish further improved stability criteria for time-delay systems by taking one of the following approaches with the use of a more attractive Lyapunov-Krasovskii functional [7][8][9][10][11]: the free-weighting matrix approach [11][12][13], the descriptor system approach [14][15][16], the Jensen inequality approach [17][18][19], or the delaypartitioning approach [20,21].
Recently, the use of the Jensen inequality approach has attracted great attention from the control community since it requires fewer decision variables than other approaches (see [18,22,23] and references therein).In addition, the appearance of the reciprocally convex technique [22] has promoted the use of such an approach as a way to reduce the computational complexity and the conservatism of stability criteria.However, as reported in [24], most of the results of these studies have been confronted with a challenge to establish less conservative stability criteria in terms of performance behavior.Thus, [24] has fully exploited the zero equality terms, introduced by [25], in the process of deriving stability criteria.After that, [10,11] have widely extended the stability criteria in accordance with the structure of Lyapunov-Krasovskii functional containing triple summation terms.However, since the use of such an augmented Lyapunov-Krasovskii function with more terms poses significant computational burdens, there is a need to explore a useful method capable of reducing the computational complexity as well as improving the performance of stability criteria.
Motivated by the above concerns, this paper is focused on deriving an efficient criterion for the robust stability analysis of discrete-time systems with time-varying delay.To be specific, the attention of this paper is paid for simultaneously achieving the following two goals: deceasing the computational complexity and improving the performance of a stability criterion.To accomplish such efficiency, this paper 2 Mathematical Problems in Engineering proposes a valuable method capable of deriving a less conservative stability criterion without using the free-weighting approach and an extremely augmented state, which plays an important role in reducing the computational complexity caused by [10,11].As a result, under the requirement of much less computational complexity, the stability region in the present study is enlarged to the same size as those of [10,11].Moreover, the stabilization problem of systems with time-delay control input is addressed in connection with the derived stability criterion.Finally, three numerical examples are provided to illustrate the efficiency of the proposed stability and stabilization criteria.
This paper is organized as follows.Section 2 provides a system description and useful properties.Section 3 introduces a Lyapunov-Krasovskii functional for deriving the stability criterion of the time-delay system.Section 4 presents the state-feedback controller for the system with delayed control input.Section 5 shows simulation results for validating the proposed results.Finally, Section 6 presents the conclusion along with a summary.
Notation.The notations  ≥  and  >  mean that  −  is positive semidefinite and positive definite, respectively.In symmetric block matrices, ( * ) is used as an ellipsis for terms that are induced by symmetry.For any square matrix Q,

System Description and Useful Properties
Let us consider the following time-delay system: where   ∈ R   and  −() ∈ R   are the state and the delayed state, respectively.Here, the state delay () is assumed to be of an interval time-varying type integer:  ≤ () ≤ , where  and  are known positive integers.To facilitate the derivation of the main result, we set  0 = 0,  1 = ,  2 = (), and  3 =  and use the following notations: where  ∈ {0, 1, 2},  ∈ {1, 2, 3 |  > }, and   denotes any scalar or vector-valued function.

Stability Analysis
Let Δ  =  +1 −   and choose a Lyapunov-Krasovskii functional of the following form: where P,  1 ,  2 , R 1 , and R 2 are taken to be positive definite.
To facilitate later steps, we define an augmented state   as and establish block entry matrices e  such that   = e 0   , Property 3 (see [25]).For symmetric matrices  0 ,  1 , and  2 , the following equalities hold: The following theorem presents the delay-and rangedependent stability criterion for (1).
Remark 3.For the given   , the number of scalar variables (NSVs) used in Theorem 2 is given as 15 2  + 6  .Our approach leads to a significant decrease in the computational burden compared with [10], [11], and [24]  Remark 4. To make up for the weakness in [10,11,24], this paper proposes a valuable method capable of deriving a less conservative stability criterion without using the freeweighting approach and the inclusion of Δ − 1 , Δ − 2 , and Δ − 3 in the augmented state.
Proof.Let us redefine the augmented state   as and establish a block entry matrix e 8 such that   = e 8   .

Control Synthesis
Let us consider a linear system of the following form: where  −() ∈ R   denotes the delayed control input.Then, under the state-feedback control law   =   , the closedloop control system is described as follows: where   =  and  denotes the control gain to be designed.
Proof.First of all, let us consider a nonsingular matrix W of the following form: which satisfies that e  W = e  for all .Further, define In other words, the stabilization condition is given by which becomes (30).Here, (30) implies e 7 (∑

Numerical Examples
Three numerical examples are considered in order to illustrate the effectiveness of the obtained results.
Example 1 (stability analysis).Let us consider a delayed discrete-time system used in [17]: where  ≤ () ≤ .For (40) with various , Table 1 lists the maximum allowable upper bounds (MAUBs) of (), obtained by Theorem 2 and different methods.From Table 1, it can be seen that the stability criteria established in [10,11] and Theorem 2 offer the most improved of the results.In particular, it is noteworthy that Theorem 2 provides the same delay bounds (i.e., stability region) as those of [10,11] under the requirement of much less computational complexity with respect to the number of scalar variables (NSVs), as mentioned in Remark 3.That is, in contrast with [10,11,24], Theorem 2 offers a more efficient approach in terms of both performance and computational complexity.
Example 2 (robust stability analysis).Consider the following uncertain discrete-time system with time-varying delay used in [24]:  2, we can see that the robust stability criterion given in Corollary 6 is much more efficient than the ones in [24] from the viewpoint of both performance and computational complexity.
Example 3 (control synthesis).Consider the following discrete-time system transformed from the continuous-time model of an inverted pendulum (refer to [7]): The goal of this example is to design the control   =   that stabilizes (42) with 1 ≤ () ≤ , such that the closed-loop system is asymptotically stable.Table 3 shows the MAUBs of () and the corresponding control gains, obtained by Theorem 7 ( = 100).From Table 3, we can see that the proposed stabilization condition is significantly valuable in the sense that it requires less computational complexity as well as providing larger MAUB than that of [26].Meanwhile, based on the obtained control gain, Figure 1 shows the state responses of (42) with   = [1 −1]  for  ∈ {−7, . . ., 0} and () = 6 ⋅ |⌈sin(/2)⌉| + 1 ∈ [1,7].Here, it can be found that the state converges to zero as time goes to infinity.

Concluding Remarks
In this paper, the problem of deriving an efficient stability criterion is investigated for discrete-time systems with timevarying delay.The main feature herein is that the conservatism of a stability criterion is reduced in spite of the requirement of less computational complexity.In addition, the stabilization problem of systems with time-delayed control input is addressed in the LMI framework.
For each  ≤ () ≤ , the MAUBs of  such that (23) is robustly asymptotically stable are listed in Table2, obtained by Corollary 6 and different methods.That is, from Table