# Sufficient conditions for global exponential stability of discrete switched time-delay systems with linear fractional perturbations via switching signal design

- Chang-Hua Lien
^{1}Email author, - Ker-Wei Yu
^{1}, - Jenq-Der Chen
^{2}and - Long-Yeu Chung
^{3}

**2013**:39

https://doi.org/10.1186/1687-1847-2013-39

© Lien et al.; licensee Springer 2013

**Received: **7 October 2012

**Accepted: **4 February 2013

**Published: **20 February 2013

## Abstract

The switching signal design for global exponential stability of discrete switched systems with interval time-varying delay and linear fractional perturbations is considered in this paper. Some LMI stability criteria are proposed to design the switching signal and guarantee the global exponential stability for a discrete switched time-delay system. Nonnegative inequalities are introduced to improve the conservativeness of the proposed results. A procedure is provided to guarantee the stability of a switched system and design the switching signal. Finally, some numerical examples are illustrated to show the main results.

## Keywords

## 1 Introduction

A switched system is composed of a family of subsystems and a switching signal that specifies which subsystem is activated to the system trajectories at each instant of time [1]. Switched systems are often encountered in many practical examples such as automated highway systems, automotive engine control system, constrained robotics, robot manufacture, and stepper motors. Many complicate system behaviors, such as multiple limit cycles and chaos, are produced by a switching signal in systems [1–14]. It is also well known that the existence of delay in a system may cause instability or bad performance in closed control systems [15–18]. Time-delay phenomena usually appear in many practical systems such as AIDS epidemic, chemical engineering systems, hydraulic systems, population dynamic model, and rolling mill. Hence stability analysis and stabilization for discrete switched systems with time delay have been studied in recent years [2–4, 6, 7, 9, 10, 12–14].

There are three basic problems in dealing with the stability of discrete switched systems: (1) Find the stability or controller design of switched systems under an arbitrary switching signal [3, 4, 8, 9, 13]; (2) Identify the useful stabilizing switching signal for switched systems [14]; (3) Construct the stabilizing switching signal for switched systems [6, 7]. In this paper, the stability conditions for switching signal design of uncertain discrete switched time-delay systems will be developed. It is interesting to note that the stable property for each subsystem cannot imply that the overall system is also stable under an arbitrary switching signal [4]. Another interesting fact is that the stability of a switched system can be achieved by choosing a switching signal even when each subsystem is unstable [6, 7]. Although many important results have been proposed to design the switching signal of switched time-delay systems, but there are only few reports concerning the switching signal design of discrete switched time-delay systems [6, 7]. Some additional nonnegative inequalities are used to improve the conservativeness for the obtained results [4, 18]. Interval time-varying delay and linear fractional perturbations of a system are also included in our problem under consideration. In this paper, a new scheme for switching signal design is developed to guarantee global exponential stability of a switched system with interval time-varying delay and linear fractional perturbations. By the proposed approach, our results are shown to be less conservative than some recent reports in our demonstrated numerical examples.

The notation used throughout this paper is as follows. For a matrix *A*, we denote the transpose by ${A}^{T}$, symmetric positive (negative) definite by $A>0$ ($A<0$), maximal eigenvalue by ${\lambda}_{max}(A)$, minimal eigenvalue by ${\lambda}_{min}(A)$, $n\times m$ dimension by ${A}_{(n\times m)}$. $A\le B$ means that matrix $B-A$ is symmetric positive semi-definite. *I* denotes the identity matrix. For a vector *x*, we denote the Euclidean norm by $\parallel x\parallel $. Define $\overline{N}=\{1,2,\dots ,N\}$, $A\setminus B=\{x|x\in A\text{and}x\notin B\}$, ${\parallel {x}_{k}\parallel}_{s}={max}_{\theta =-{r}_{M},-{r}_{M}+1,\dots ,0}\parallel x(k+\theta )\parallel $.

## 2 Problem statement and preliminaries

*σ*is a switching signal in the finite set $\{1,2,\dots ,N\}$ and will be chosen to preserve the stability of the system, $\varphi (k)\in {\mathrm{\Re}}^{n}$ is an initial state function, time-varying delay $r(k)$ is a function from $\{0,1,2,3,\dots \}$ to $\{0,1,2,3,\dots \}$ and $1\le {r}_{m}\le r(k)\le {r}_{M}$, ${r}_{m}$ and ${r}_{M}$ are two given positive integers. Matrices ${A}_{i},{B}_{i}\in {\mathrm{\Re}}^{n\times n}$, $i=1,2,\dots ,N$, are constant. $\mathrm{\Delta}{A}_{i}(k)$ and $\mathrm{\Delta}{B}_{i}(k)$ are two perturbed matrices satisfying the following condition:

**Definition 1**System (1a)-(1e) is said to be globally exponentially stable with a convergence rate

*α*if there are two positive constants $0<\alpha <1$ and Ψ such that

Now the main results are provided in the following theorem.

**Theorem 1**

*If for some constants*$0<\alpha <1$, $0\le {\alpha}_{i}\le 1$, $i\in \overline{N}$,

*and*${\sum}_{i=1}^{N}{\alpha}_{i}=1$,

*there exist some*$n\times n$

*matrices*$P>0$, $Q>0$, ${R}_{1}>0$, ${R}_{2}>0$, ${R}_{3}>0$, $S>0$, $T>0$, ${U}_{i}>0$, $i\in \overline{N}$, ${V}_{l11}\in {\mathrm{\Re}}^{4n\times 4n}$, ${V}_{l12}\in {\mathrm{\Re}}^{4n\times n}$, ${V}_{l22}\in {\mathrm{\Re}}^{n\times n}$, $l=1,2,3,4$,

*and constants*${\epsilon}_{i}>0$, $i\in \overline{N}$,

*such that the following LMI conditions hold for all*$j=1,2,\dots ,N$:

*Then system*(1a)-(1e)

*is globally exponentially stable with the convergence rate*$0<\alpha <1$

*by the switching signal designed by*

*where* ${\overline{\mathrm{\Omega}}}_{i}$ *is defined in* (2a), (2b).

*Proof*

By Definition 1, system (1a)-(1e) is globally exponentially stable with the convergence rate $0<\alpha <1$ with the switching signal in (3e). This completes this proof. □

**Remark 1**

In order to achieve the exponential stability of a discrete switched system, the condition in Lemma 1 will be a reasonable choice and a feasible setting.

**Remark 2** The matrix uncertainties in (1c)-(1e) are usually called linear fractional perturbations [16]. The parametric perturbations in [4, 9, 10] are the special conditions of the considered perturbations with ${\mathrm{\Xi}}_{i}=0$, $i\in \underline{N}$.

**Remark 3**Under the same switching signal defined in (3e), the switching domains of [6, 7] are selected as:

where matrices $P>0$ and $U>0$. It is noted that above selections are similar to switching signal design in continuous switched systems [8]. Hence the proposed switching domain design approach is the discrete version of [6, 7] and shown to be useful from numerical simulations.

*M*, ${N}_{A}$, and ${N}_{B}$, and Ξ are some given constant matrices with appropriate dimensions. $\mathrm{\Gamma}(k)$ is an unknown matrix representing the perturbation which satisfies

The following sufficient conditions for the stability of system (8a)-(8e) can be obtained in a similar way to Theorem 1.

**Theorem 2**

*If a given constant*$0<\alpha <1$,

*there exist some*$n\times n$

*matrices*$P>0$, $Q>0$, ${R}_{1}>0$, ${R}_{2}>0$, ${R}_{3}>0$, $S>0$, $T>0$, ${V}_{l11}\in {\mathrm{\Re}}^{4n\times 4n}$, ${V}_{l12}\in {\mathrm{\Re}}^{4n\times n}$, ${V}_{l22}\in {\mathrm{\Re}}^{n\times n}$, $l=1,2,3,4$,

*and constants*$\epsilon >0$

*such that*(3a), (3b),

*and the following LMI conditions are satisfied*:

*Then system* (8a)-(8e) *is globally exponentially stable with the convergence rate* $0<\alpha <1$.

**Remark 4** In Theorems 1 and 2, the global asymptotic stability of switched system (1a)-(1e) and non-switched system (8a)-(8e) can be achieved by setting $\alpha =1$. The nonnegative inequalities in (6a)-(6d) are used to improve the conservativeness of the obtained results.

**Remark 5** If we wish to select a switching signal to guarantee the stability of switched system (1a)-(1e), the following procedures are proposed.

Step 1 Test the exponential stability of each subsystem of switched system (1a)-(1e) by Theorem 2 with $A={A}_{i}$, $B={B}_{i}$, $M={M}_{i}$, ${N}_{A}={N}_{Ai}$, ${N}_{B}={N}_{Bi}$, $\mathrm{\Xi}={\mathrm{\Xi}}_{i}$, $i=1,2,\dots ,N$. If the sufficient conditions in Theorem 2 have a feasible solution for some $i\in \underline{N}$, the switching signal is selected by $\sigma =i$ and the stability of the switched system in (1a)-(1e) can be guaranteed.

Step 2 If the obtained stability results for each subsystem of switched time-delay system (1a)-(1e) do not satisfy the requirement in Step 1, we can use Theorem 1 to design the switching signal to guarantee the global exponential stability of switched time-delay system (1a)-(1e).

From the results in Steps 1-2, we can propose a less conservative stability result of the system.

## 3 Illustrative examples

**Example 1**Consider system (1a)-(1e) with no perturbations and the following parameters (Example 3.1 of [6]):

**The obtained results for our proposed results in this paper**

The delay upper bound and stability switching domains for switched system (1a)-(1e) with (9) | |
---|---|

[6] | ${r}_{m}=1$, ${r}_{M}=2$ ( |

${\overline{\mathrm{\Omega}}}_{1}=\{{[{x}_{1}\phantom{\rule{0.25em}{0ex}}{x}_{2}]}^{T}\in {R}^{2}:-12.0213{x}_{1}^{2}-38.8254{x}_{1}{x}_{2}-31.6356{x}_{2}^{2}<0\}$, ${\overline{\mathrm{\Omega}}}_{2}={\mathrm{\Re}}^{2}\setminus {\overline{\mathrm{\Omega}}}_{1}$ | |

Our results | ${r}_{m}=1$, ${r}_{M}=178$ ( |

${r}_{m}=1$, ${r}_{M}=392$ ( | |

${r}_{m}=1$, ${r}_{M}=186$ ( | |

${\overline{\mathrm{\Omega}}}_{1}=\{{[{x}_{1}\phantom{\rule{0.25em}{0ex}}{x}_{2}]}^{T}\in {R}^{2}:0.2018{x}_{1}^{2}+0.3982{x}_{1}{x}_{2}-0.2247{x}_{2}^{2}<0\}$, ${\overline{\mathrm{\Omega}}}_{2}={\mathrm{\Re}}^{2}\setminus {\overline{\mathrm{\Omega}}}_{1}$ | |

${r}_{m}=1$, ${r}_{M}=20$ ( | |

${\overline{\mathrm{\Omega}}}_{1}=\{{[{x}_{1}\phantom{\rule{0.25em}{0ex}}{x}_{2}]}^{T}\in {R}^{2}:0.6762{x}_{1}^{2}+1.8317{x}_{1}{x}_{2}+0.2681{x}_{2}^{2}<0\}$, ${\overline{\mathrm{\Omega}}}_{2}={\mathrm{\Re}}^{2}\setminus {\overline{\mathrm{\Omega}}}_{1}$ |

**Example 2**Consider system (1a)-(1e) with the following parameters (Example 1 of [10]):

**Some obtained results for our proposed results in this paper**

The delay upper bound and stability switching domains for switched system (1a)-(1e) with (10) | |
---|---|

[10] | ${r}_{m}=1$, ${r}_{M}=5$ ( |

[4] | ${r}_{m}=1$, ${r}_{M}=6$ ( |

[6] | Results fail (switching signal cannot be selected even when no perturbations) |

Our result | ${r}_{m}=1$, ${r}_{M}=7$ ( |

${r}_{m}=1$, ${r}_{M}=19$ ( | |

${r}_{m}=1$, ${r}_{M}=10$ ( | |

Switching signal in (3e) with ${\overline{\mathrm{\Omega}}}_{1}=\{{[{x}_{1}\phantom{\rule{0.25em}{0ex}}{x}_{2}]}^{T}\in {R}^{2}:5.8081{x}_{1}^{2}+7.1275{x}_{1}{x}_{2}+1.7783{x}_{2}^{2}<0\}$, ${\overline{\mathrm{\Omega}}}_{2}={\mathrm{\Re}}^{2}\setminus {\overline{\mathrm{\Omega}}}_{1}$ |

**Example 3**Consider system (1a)-(1e) with the following parameters:

**The obtained results for our proposed results in this paper**

The delay upper bound and stability switching domains for switched system (1a)-(1e) with (11) | |
---|---|

Results fail (under arbitrary switching signal condition) | |

Results fail (switching signal cannot be selected even when no perturbations) | |

Our results | Fail to guarantee the stability of each subsystem in Theorem 2 |

${r}_{m}=1$, ${r}_{M}=7$ ( | |

Switching signal in (3e) with ${\overline{\mathrm{\Omega}}}_{1}=\{{[{x}_{1}\phantom{\rule{0.25em}{0ex}}{x}_{2}]}^{T}\in {\mathrm{\Re}}^{2}:27.9396{x}_{1}^{2}+3.3182{x}_{1}{x}_{2}-18.2592{x}_{2}^{2}<0\}$, ${\overline{\mathrm{\Omega}}}_{2}={\mathrm{\Re}}^{2}\setminus {\overline{\mathrm{\Omega}}}_{1}$ |

Note that the matrices ${A}_{1}$ and ${A}_{2}$ in this example are not Hurwitz, the results in [4, 6, 7], and [10] cannot find any feasible solution to guarantee the stability of a switched system for arbitrary and designed switching signals. The proposed results in [4] and [10] are subjected to an arbitrary switching signal condition, we illustrate the comparisons here only for showing the advantage for switching signal design of the proposed results in this paper.

**Example 4**Consider system (1a)-(1e) with no perturbations and the following parameters (Example 1 of [7]):

**The obtained results for our proposed results in this paper**

The delay upper bound and stability switching domains for switched system (1a)-(1e) with (13) | |
---|---|

Results fail (under arbitrary switching signal condition) | |

[7] | ${r}_{m}=1$, ${r}_{M}=4$ ( |

${\overline{\mathrm{\Omega}}}_{1}=\{{[{x}_{1}\phantom{\rule{0.25em}{0ex}}{x}_{2}]}^{T}\in {R}^{2}:-0.9918{x}_{1}^{2}-0.035{x}_{1}{x}_{2}-1.4672{x}_{2}^{2}<0\}$, ${\overline{\mathrm{\Omega}}}_{2}={\mathrm{\Re}}^{2}\setminus {\overline{\mathrm{\Omega}}}_{1}$ | |

Our results | ${r}_{m}=1$, ${r}_{M}=7$ ( |

Note that the eigenvalues of ${A}_{2}$ are 0.9804 and 2.0196, Theorem 1 and Theorem 2 with the second subsystem cannot use to guarantee the stability of system (1a)-(1e) with (13).

## 4 Conclusion

In this paper, the switching signal design to guarantee the global exponential stability for uncertain discrete switched systems with interval time-varying delay and linear fractional perturbations has been considered. Some nonnegative inequalities and LMI approach are used to improve the conservativeness of the proposed results. A procedure has been proposed to test the stability of the switched system and design the switching signal. The obtained results are shown to be less conservative and useful via numerical examples. In the future, switching signal designs for robust stabilization and performance (guaranteed cost control, ${H}_{\mathrm{\infty}}$ control, nonfragile control, passivity analysis and passive control) can be investigated and developed [10, 19, 20].

## Appendix

**Lemma 1**

*If there exist some constants*$0<\alpha <1$, $0\le {\alpha}_{i}\le 1$, $i\in \underline{N}$, ${\sum}_{i=1}^{N}{\alpha}_{i}=1$,

*some matrices*$P>0$, ${U}_{i}>0$

*such that*

*we have*

*where* Φ *is an empty set of* ${\mathrm{\Re}}^{n}$ *and* ${\overline{\mathrm{\Omega}}}_{i}$ *is defined in* (2a), (2b).

*Proof* This lemma can be proved in a similar way to [6–8]. □

**Lemma 2** [21]

*For a given matrix*$S=\left[\begin{array}{cc}{S}_{11}& {S}_{12}\\ 2\ast & {S}_{22}\end{array}\right]$

*with*${S}_{11}={S}_{11}^{T}$, ${S}_{22}={S}_{22}^{T}$,

*the following conditions are equivalent*:

- (1)
$S<0$,

- (2)
${S}_{22}<0$, ${S}_{11}-{S}_{12}{S}_{22}^{-1}{S}_{12}^{T}<0$.

**Lemma 3** [16]

*Suppose that*${\mathrm{\Delta}}_{i}(k)$

*is defined in*(1d)

*and satisfies*(1e),

*then for real matrices*${V}_{i}$, ${W}_{i}$,

*and*${X}_{i}$

*with*${X}_{i}={X}_{i}^{T}$,

*the following statements are equivalent*:

- (I)
*The inequality is satisfied*${X}_{i}+{V}_{i}{\mathrm{\Delta}}_{i}(k){W}_{i}+{W}_{i}^{T}{\mathrm{\Delta}}_{i}^{T}(k){V}_{i}^{T}<0;$ - (II)
*There exists a scalar*${\epsilon}_{i}>0$*such that*$\left[\begin{array}{ccc}{X}_{i}& {V}_{i}& {\epsilon}_{i}\cdot {W}_{i}^{T}\\ \ast & -{\epsilon}_{i}\cdot I& {\epsilon}_{i}\cdot {\mathrm{\Xi}}_{i}^{T}\\ \ast & \ast & -{\epsilon}_{i}\cdot I\end{array}\right]<0,$

*where the matrix* ${\mathrm{\Xi}}_{i}$ *is defined in* (1d).

## Declarations

### Acknowledgements

The research reported here was supported by the National Science Council of Taiwan, ROC under Grant No. NSC 101-2221-E-022-009. The authors would like to thank the editor and anonymous reviewers for their helpful comments.

## Authors’ Affiliations

## References

- Sun Z, Ge SS:
*Switched Linear Systems Control and Design*. Springer, London; 2005.View ArticleGoogle Scholar - Han Y, Tang H:Robust ${H}_{\mathrm{\infty}}$ control for a class of discrete switched systems with uncertainties and delays.
*Proc. 26th Chinese Control Conf.*2007, 681-684.Google Scholar - Ibrir S: Stability and robust stabilization of discrete-time switched systems with time-delays: LMI approach.
*Appl. Math. Comput.*2008, 206: 570-578. 10.1016/j.amc.2008.05.149MathSciNetView ArticleGoogle Scholar - Lien CH, Yu KW, Chung YJ, Chang HC, Chung LY, Chen JD:Exponential stability and robust ${H}_{\mathrm{\infty}}$ control for uncertain discrete switched systems with interval time-varying delay.
*IMA J. Math. Control Inf.*2011, 7: 433-444.MathSciNetGoogle Scholar - Liu J, Liu X, Xie WC: Delay-dependent robust control for uncertain switched systems with time-delay.
*Nonlinear Anal. Hybrid Syst.*2008, 2: 81-95. 10.1016/j.nahs.2007.04.001MathSciNetView ArticleGoogle Scholar - Phat VN, Ratchagit K: Stability and stabilization of switched linear discrete-time systems with interval time-varying delay.
*Nonlinear Anal. Hybrid Syst.*2011, 5: 605-612. 10.1016/j.nahs.2011.05.006MathSciNetView ArticleGoogle Scholar - Sangapate P: New sufficient conditions for the asymptotic stability of discrete time-delay systems.
*Adv. Differ. Equ.*2012., 2012: Article ID 28Google Scholar - Sun XM, Wang W, Liu GP, Zhao J: Stability analysis for linear switched systems with time-varying delay.
*IEEE Trans. Syst. Man Cybern., Part B, Cybern.*2008, 38: 528-533.Google Scholar - Sun YG, Wang L, Xie G: Delay-dependent robust stability and stabilization for discrete-time switched systems with mode-dependent time-varying delays.
*Appl. Math. Comput.*2006, 180: 428-435. 10.1016/j.amc.2005.12.028MathSciNetView ArticleGoogle Scholar - Sun YG, Wang L, Xie G:Delay-dependent robust stability and ${H}_{\mathrm{\infty}}$ control for uncertain discrete-time switched systems with mode-dependent time delays.
*Appl. Math. Comput.*2007, 187: 1228-1237. 10.1016/j.amc.2006.09.053MathSciNetView ArticleGoogle Scholar - Xie D, Xu N, Chen X: Stabilisability and observer-based switched control design for switched linear systems.
*IET Control Theory Appl.*2008, 2: 192-199. 10.1049/iet-cta:20060502MathSciNetView ArticleGoogle Scholar - Zhai G, Liu D, Imae J, Kobayashi T: Lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems.
*IEEE Trans. Circuits Syst.*2006, 53: 152-156.View ArticleGoogle Scholar - Zhang L, Shi P, Basin M: Robust stability and stabilisation of uncertain switched linear discrete time-delay systems.
*IET Control Theory Appl.*2008, 2: 606-614. 10.1049/iet-cta:20070327MathSciNetView ArticleGoogle Scholar - Zhang WA, Yu L: Stability analysis for discrete-time switched time-delay systems.
*Automatica*2009, 45: 2265-2271. 10.1016/j.automatica.2009.05.027View ArticleGoogle Scholar - Gau RS, Lien CH, Hsieh JG: Novel stability conditions for interval delayed neural networks with multiple time-varying delays.
*Int. J. Innov. Comput. Inf. Control*2011, 7: 433-444.Google Scholar - Li T, Guo L, Sun C: Robust stability for neural networks with time-varying delays and linear fractional uncertainties.
*Neurocomputing*2007, 71: 421-427. 10.1016/j.neucom.2007.08.012View ArticleGoogle Scholar - Gu K, Kharitonov VL, Chen J:
*Stability of Time-Delay Systems*. Birkhäuser, Boston; 2003.View ArticleGoogle Scholar - Yu KW: Further results on new stability analysis for uncertain neutral systems with time-varying delay.
*Int. J. Innov. Comput. Inf. Control*2010, 6: 1133-1140.Google Scholar - Dong Y, Wei J: Output feedback stabilization of nonlinear discrete-time systems with time-delay.
*Adv. Differ. Equ.*2012., 2012: Article ID 73Google Scholar - Zhang Z, Mou S, Lam J, Gao H: New passivity criteria for neural networks with time-varying delay.
*Neural Netw.*2009, 22: 864-868. 10.1016/j.neunet.2009.05.012View ArticleGoogle Scholar - Boyd SP, El Ghaoui L, Feron E, Balakrishnan V:
*Linear Matrix Inequalities in System and Control Theory*. SIAM, Philadelphia; 1994.View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.