Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 297324, 6 pages

http://dx.doi.org/10.1155/2014/297324

Research Article

Partitioning Technique for Relaxed Stability Criteria of Discrete-Time Systems with Interval Time-Varying Delay

Kuang-Yow Lian,¹ Wen-Tsung Yang,² and Peter Liu³

¹Department of Electrical Engineering, National Taipei University of Technology, Taipei 10608, Taiwan

²Department of Electrical Engineering, Chung-Yuan Christian University, Chung-Li 32023, Taiwan

³Department of Electrical Engineering, Tamkang University, New Taipei City 25137, Taiwan

Received 23 March 2014; Accepted 10 June 2014; Published 30 June 2014

Academic Editor: Mustafa Kulenović

Copyright © 2014 Kuang-Yow Lian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We demonstrate an improved stability analysis based on a partition oriented technique for discrete-time systems with interval time-varying delay. The partition oriented technique introduces beneficial terms contributing to the negative definiteness of the Lyapunov function difference, meanwhile completely avoiding traditional inequality based approaches. In contrast, nonpartitioning oriented techniques do not put emphasis on further dividing the interval of the summation in the Lyapunov function. Herein, we demonstrate that the advantages of exploiting partitioning techniques manifest the relaxed stability criteria, as well as the flexibility to tune tradeoff between allowable timedelay range performance and computational load. Simulation carried out on a benchmark discrete-time system reveals the significant improvement in terms of maximum allowable upper bound in comparison.

1. Introduction

Stability analysis on continuous and discrete-time systems with interval time-varying delay has been extensively researched in the past decade [1–7]. Traditional techniques focused on introducing inequalities (e.g., Moon et al. [8], Park [9], model transformation [10, 11]) to cope with the double integral problem in the Lyapunov function whereas we term these approaches as nonpartitioning oriented techniques. However, these nonpartitioning oriented techniques increase conservativeness by introducing positive terms to the Lyapunov function difference. This setback led to techniques (e.g., zero equalities [12], averaging technique [13]) which further partition the time-delay interval to introduce beneficial terms contributing to the negative definiteness of the Lyapunov function difference (whereas we term this as partitioning techniques), resulting in relaxed stability criteria. From the above observation, we are motivated to completely avoid nonpartitioning oriented techniques while exploiting partitioning oriented techniques. The advantages are (i) relaxed stability criteria and (ii) flexibility to tune tradeoff between allowable time-delay range performance and computational load. Simulation results on a discrete-time example reveal the best performance in comparison to previous literature.

2. Main Results

Using the partitioning technique, we have the following main results.

Theorem 1. Given nonnegative integer scalars and , the system

is asymptotically stable, where is the state vector; , are constant matrices with appropriate dimensions; is the initial condition of ; for any time varying delay satisfying , there exist real matrices , , , , and any appropriately dimensioned matrices , , , and such that the following two sets of LMIs hold.

Case I (). Consider

where

Case II (). Consider

where

Proof. First, denote , that is, the floor function of , , and . Let

Then, it is clear that

We divide the proof into two cases.

Case I. We choose the Lyapunov function candidate

where

and , , , , , , and are positive definite matrices to be determined. Define which leads to

From (6), the following zero equations

hold for any appropriate dimensions matrices , , , and , where , and semi-positive-definite matrices

Taking the forward difference of and adding all the terms of (11)–(16) to leads to

where and . Thus, LMIs (2) assure that is negative definite.

Case II. We choose the Lyapunov function candidate

where the for (are same as Case I) except for where

which yields

Similar to (11)–(16), (11) and the following zero equations

Taking the forward difference of and adding all the terms of (11) and (22) to allow us to write as

Thus, LMIs (4) assure that is negative definite.

Remark 2. The main goal of the partitioning technique is to divide the summation interval of the Lyapunov function difference to as many subintervals (e.g., , , , , , , , and in the analysis) as possible. Using this technique along with the zero equations (11)–(16) and (22), we introduce free weighting matrices which increase feasibility of the resulting LMIs.

3. Numerical Simulations

In this section, two examples are used to demonstrate the effectiveness of theorem.

Example 1. Consider the time-delay system with the following matrices:

Comparing the results of this work to [14], the maximum allowable upper bound (MAUB) of with various is presented in Table 1, which shows that our conditions give significantly better results.

Table 1: Maximum allowable upper bound for various .

(a)

Methods

Shao and Han [14] Proposition 2

16
18
21

Shao and Han [14] Proposition 1

18
20
23

Theorem 1

23

27

(b)

Methods

Shao and Han [14] Proposition 2

25
30
34

Shao and Han [14] Proposition 1

27
31
35

Theorem 1

31
35
38

Example 2. Consider the time-delay system with the following matrices:

This example is discussed in [15], which assumes that and . For this example and by Theorem 1 in this paper, the MAUB .

4. Conclusion

In this work, we have presented an exploitation of the partitioning oriented technique to deal with the stability problem of discrete-time systems with an interval time-varying delay. In the analysis, a new Lyapunov function was defined and some conventional inequalities were avoided resulting in relaxed stability criteria formulated into LMIs. A benchmark numerical example was given and results reflect the criteria which show significant improvement over existing results.

Conflict of Interests

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

References

E. Fridman and U. Shaked, “Stability and guaranteed cost control of uncertain discrete delay systems,” International Journal of Control, vol. 78, no. 4, pp. 235–246, 2005.

H. Gao and T. Chen, “New results on stability of discrete-time systems with time-varying state delay,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 52, no. 2, pp. 328–334, 2007.

Y. He, Q. G. Wang, C. Lin, and M. Wu, “Delay-range-dependent stability for systems with time-varying delay,” Automatica, vol. 43, no. 2, pp. 371–376, 2007.

B. Zhang, S. Xu, and Y. Zou, “Improved stability criterion and its applications in delayed controller design for discrete-time systems,” Automatica, vol. 44, no. 11, pp. 2963–2967, 2008.

T.-S. Chiang and P. Liu, “Robust output tracking control for discrete-time nonlinear systems with time-varying delay: virtual fuzzy model LMI-based approach,” Expert Systems with Applications, vol. 39, no. 9, pp. 8239–8247, 2012.

J. Lin, “Exponential estimates and stabilization of discrete-time singular time-delay systems subject to actuator saturation,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 414373, 27 pages, 2012.

Y. Wang, H. Yu, X. Zhang, and D. Li, “Stability analysis and design of time-varying nonlinear systems based on impulsive fuzzy model,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 192546, 16 pages, 2012.

Y. S. Moon, P. Park, W. H. Kwon, and Y. S. Lee, “Delay-dependent robust stabilization of uncertain state-delayed systems,” International Journal of Control, vol. 74, no. 14, pp. 1447–1455, 2001.

P. Park, “A delay-dependent stability criterion for systems with uncertain time-invariant delays,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 44, no. 4, pp. 876–877, 1999.

H. Shao, “New delay-dependent stability criteria for systems with interval delay,” Automatica, vol. 45, no. 3, pp. 744–749, 2009.

J. Sun, G. P. Liu, J. Chen, and D. Rees, “Improved delay-range-dependent stability criteria for linear systems with time-varying delays,” Automatica, vol. 46, no. 2, pp. 466–470, 2010.

M. Wu, Y. He, and H. J. She, Stability Analysis and Robust Control of Time-Delay Systems, Springer, Berlin, Germany, 2010.

Y. He, G. P. Liu, and D. Rees, “Augmented Lyapunov functional for the calculation of stability interval for time-varying delay,” IET Control Theory & Applications, vol. 1, no. 1, pp. 381–386, 2007.

H. Shao and Q.-L. Han, “New stability criteria for linear discrete-time systems with interval-like time-varying delays,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 56, no. 3, pp. 619–625, 2011.

X. G. Liu, R. R. Martin, M. Wu, and M. L. Tang, “Delay-dependent robust stabilisation of discrete-time systems with time-varying delay,” IEE Proceedings: Control Theory and Applications, vol. 153, no. 6, pp. 689–702, 2006.