## Theoretical and Applied Contributions to Robust Stability Analysis of Complex Systems

View this Special IssueResearch Article | Open Access

# Reachable Set Bounding for Homogeneous Nonlinear Systems with Delay and Disturbance

**Academic Editor:**Raúl Villafuerte-Segura

#### Abstract

Reachable set bounding for homogeneous nonlinear systems with delay and disturbance is studied. By the usage of a new method for stability analysis of positive systems, an explicit necessary and sufficient condition is first derived to guarantee that all the states of positive homogeneous time-delay systems with degree converge asymptotically within a specific ball. Furthermore, the main result is extended to a class of nonlinear time variant systems. A numerical example is given to demonstrate the effectiveness of the obtained results.

#### 1. Introduction

Recent years have witnessed a rapid development of reachable set bounding for linear systems in [1–11], to name a few. In most of existing references, the traditional Lyapunov-Krasovskii function method is most commonly used. However, such a method is usually difficult to derive explicit conditions for reachable set estimation of nonlinear systems with delay and disturbance.

Due to the ubiquitous existence of time delay in practical engineering and its adverse effect on stability [12–15] and oscillation [16–19], it has attracted wide attention in recent years. So far, less attention has been paid to reachable set bounding for nonlinear time-delay systems. Such a problem was discussed in [20, 21] for certain nonlinear perturbed systems with delay, where the involved nonlinear terms satisfy a linear growth condition. Reachable set bounding for continuous-time and discrete-time homogeneous time-delay positive systems of degree one was studied in [22]. The decay rates of homogeneous positive systems of any degree with time-varying delays were given in [23]. Recently, the same problem was considered in [24] for homogeneous positive systems of degree , while time delay was not taken into consideration. The problem of reachable set estimation of switched positive systems with discrete and distributed delays subject to bounded disturbances was investigated in [25].

Positive systems are dynamical systems whose states remain nonnegative whenever the initial states are nonnegative ([26, 27]). In view of the special structure of positive systems, a special method was commonly used for stability analysis of positive systems in [28–33], which is different from the traditional Lyapunov-Krasovskii function method.

Motivated by the work in [23, 24], we study in this paper reachable set bounding for homogeneous nonlinear time-delay systems with bounded disturbance. By developing the methods used in [23, 24], we first establish a necessary and sufficient condition such that all the solutions of positive homogeneous time-delay systems with degree converge asymptotically within a specific ball, which contains those results in [23, 24] in special cases. The main result is also applied to certain nonlinear time variant systems with delay and disturbance.

Throughout this paper, is the set of -dimensional real vectors. Denote by the th coordinate of for . Given , say (or ) if , (or ) if , . Denote . For , denote and . Let =, where is a constant. For given , denote . An -dimensional matrix is called Metzler if all its off-diagonal entries are nonnegative.

#### 2. Preliminaries

In this paper, nonlinear time-delay systems of the formare investigated, where is the state vector, are continuous vector functions satisfying , is a time delay satisfying , is a constant, is the disturbance, and the initial state is continuous. Note that when , system (1) takes the form of the system considered in [24].

The following definitions and lemma in [34] will be required.

*Definition 1. *Assume that is continuous on and continuously differentiable on . The vector function is called cooperative if the Jacobian matrix , , is Metzler.

*Definition 2. *A vector function is called homogeneous of degree if , , .

*Definition 3. *A vector function is called order-preserving on provided that , where , .

Lemma 4. *A cooperative vector function satisfies , where , , , .*

In this paper, we need the following assumptions:(H1) and are continuously differentiable on and homogeneous of degree ;(H2) is cooperative and is order-preserving on ;(H3) for .

Following the proof given in [22], we can easily obtain the following lemma.

Lemma 5. *System (1) is positive under assumptions (H2) and (H3).*

#### 3. Main Results

Theorem 6. *Suppose that (H1)-(H3) are valid. Then, we have the following equivalent statements:*(i)*There is an -dimensional vector satisfying .*(ii)*The solution of system (1) satisfies**for any , any initial state , any disturbance , and any bounded delay , where , , and are appropriate nonnegative constants dependent on , , and the initial state , and if .**In addition, if condition (i) holds, , , and can be chosen as follows:where , satisfies , and satisfies the following equation:*

*Proof. *(i)(ii) Given the initial state , from Lemma 5 we have , . Based on definitions of and , we haveSetThen (6) and (7) yield , , . Next, we show that for and . If it is not true, there is a constant and an index guaranteeing for , , and . Therefore,Using Lemma 4 and the homogeneity of , we get from (9) and (10) thatFor the case when , it holds thatConsidering is homogeneous and order-preserving, we concludeNote thatWe further get from (13) and (14) thatFor the case when , it holds that ; i.e.,It thus follows thatNext, we can conclude from (1) and (7) thatConsequently, (11), (15), (17), and (18) imply thatOn the other hand, the definitions of and yield that and Combining this with (19), we have , which contradicts (8). Therefore, , , ; i.e., From the well-known inequality for and , we further get It implies (2).

(ii)(i) For the particular case when and , system (1) reduces toGiven the initial condition , each solution of system (24) satisfies That is, system (24) is asymptotically stable. Based on Proposition 4.1 in [35], there is a vector such that . The proof is complete.

*Remark 7. *It can be seen from Theorem 6 that the bound of the reachable set is determined by the bound of disturbances, the choice of , and the value of . When the bound of disturbances and the value of are given, an appropriate vector can be chosen to guarantee a minimal bound of the reachable set by solving the following nonlinear optimization problem: subject to , where is defined as in Theorem 6.

*Remark 8. *If for , then Theorem 6 reduces to the main result given in [23]. If for , then Theorem 6 reduces to the main result given in [24].

Finally, consider the following nonlinear time-varying systemwhere , , , and are the same as in (1), and are vector functions satisfying .

Suppose that and satisfy the following assumption:

(H4) and are continuous on , continuously differentiable with respect to on , and there are vector functions and satisfying (H1) and (H2), and for ,

Without the restriction on the disturbance that for , we can get the following reachable set bounding criterion for system (26).

Theorem 9. *Suppose that (H4) is valid. If there is an -dimensional vector such that , the solution of system (26) satisfies (2), where constants , , and are defined by (3).*

*Proof. *SetBased on definitions of and , it holds that , , . For the case when , , notice thatHere denotes the left derivative. Similar to the analysis in Theorem 6, it is not difficult to conclude that , , . Consequently, (2) holds. The proof is complete.

#### 4. Numerical Example

Consider system (1) withIt is easy to verify that assumptions (H1)-(H2) hold. Let . Then . By a direct calculation, it yields , , , and .

We conclude from Theorem 6 that there is a ball such that all the states of system (1) converge asymptotically within it. Given the initial state , , noting that and , solution (1) satisfies Figure 1 presents the simulation.

#### 5. Conclusion

This paper has been concerned with reachable set bounding for homogeneous nonlinear time-delay systems with disturbance. We not only derive explicit reachable set bounding criterion independent of delay, but also estimate the decay rate. It will be interesting to extend our work to the case of unbounded delays and discrete-time systems.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

We declare that there are no conflicts of interest regarding this paper.

#### Acknowledgments

This work is supported by the National Natural Science Foundation of China under grant 61873110.

#### References

- E. Fridman and U. Shaked, “On reachable sets for linear systems with delay and bounded peak inputs,”
*Automatica*, vol. 39, no. 11, pp. 2005–2010, 2003. View at: Publisher Site | Google Scholar | MathSciNet - J.-H. Kim, “Improved ellipsoidal bound of reachable sets for time-delayed linear systems with disturbances,”
*Automatica*, vol. 44, no. 11, pp. 2940–2943, 2008. View at: Publisher Site | Google Scholar - Z. Q. Zuo, D. W. C. Ho, and Y. J. Wang, “Reachable set bounding for delayed systems with polytopic uncertainties: the maximal Lyapunov-Krasovskii functional approach,”
*Automatica*, vol. 46, no. 5, pp. 949–952, 2010. View at: Publisher Site | Google Scholar | MathSciNet - O. M. Kwon, S. M. Lee, and J. H. Park, “On the reachable set bounding of uncertain dynamic systems with time-varying delays and disturbances,”
*Information Sciences*, vol. 181, no. 17, pp. 3735–3748, 2011. View at: Publisher Site | Google Scholar | MathSciNet - P. T. Nam and P. N. Pathirana, “Further result on reachable set bounding for linear uncertain polytopic systems with interval time-varying delays,”
*Automatica*, vol. 47, no. 8, pp. 1838–1841, 2011. View at: Publisher Site | Google Scholar | MathSciNet - P. T. Nam, P. N. Pathirana, and H. Trinh, “Exponential convergence of time-delay systems in the presence of bounded disturbances,”
*Journal of Optimization Theory and Applications*, vol. 157, no. 3, pp. 843–852, 2013. View at: Publisher Site | Google Scholar | MathSciNet - N. D. That, P. T. Nam, and Q. P. Ha, “Reachable set bounding for linear discrete-time systems with delays and bounded disturbances,”
*Journal of Optimization Theory and Applications*, vol. 157, no. 1, pp. 96–107, 2013. View at: Publisher Site | Google Scholar | MathSciNet - Z. Zuo, Y. Chen, Y. Wang, D. W. Ho, M. Z. Chen, and H. Li, “A note on reachable set bounding for delayed systems with polytopic uncertainties,”
*Journal of The Franklin Institute*, vol. 350, no. 7, pp. 1827–1835, 2013. View at: Publisher Site | Google Scholar | MathSciNet - J. Lam, B. Zhang, Y. Chen, and S. Xu, “Reachable set estimation for discrete-time linear systems with time delays,”
*International Journal of Robust and Nonlinear Control*, 2013. View at: Publisher Site | Google Scholar - B. Zhang, J. Lam, and S. Xu, “Reachable set estimation and controller design for distributed delay systems with bounded disturbances,”
*Journal of The Franklin Institute*, vol. 351, no. 6, pp. 3068–3088, 2014. View at: Publisher Site | Google Scholar | MathSciNet - L. V. Hien and H. M. Trinh, “A new approach to state bounding for linear time-varying systems with delay and bounded disturbances,”
*Automatica*, vol. 50, no. 6, pp. 1735–1738, 2014. View at: Publisher Site | Google Scholar | MathSciNet - J. Qi and Y. Sun, “Global exponential stability of certain switched systems with time-varying delays,”
*Applied Mathematics Letters*, vol. 26, no. 7, pp. 760–765, 2013. View at: Publisher Site | Google Scholar | MathSciNet - G. Zong, H. Ren, and L. Hou, “Finite-time stability of interconnected impulsive switched systems,”
*IET Control Theory & Applications*, vol. 10, no. 6, pp. 648–654, 2016. View at: Publisher Site | Google Scholar | MathSciNet - Y. Li, Y. Sun, and F. Meng, “New criteria for exponential stability of switched time-varying systems with delays and nonlinear disturbances,”
*Nonlinear Analysis: Hybrid Systems*, vol. 26, pp. 284–291, 2017. View at: Publisher Site | Google Scholar | MathSciNet - Y. Li, Y. Sun, F. Meng, and Y. Tian, “Exponential stabilization of switched time-varying systems with delays and disturbances,”
*Applied Mathematics and Computation*, vol. 324, pp. 131–140, 2018. View at: Publisher Site | Google Scholar | MathSciNet - H. Liu, F. Meng, and P. Liu, “Oscillation and asymptotic analysis on a new generalized Emden-Fowler equation,”
*Applied Mathematics and Computation*, vol. 219, no. 5, pp. 2739–2748, 2012. View at: Publisher Site | Google Scholar | MathSciNet - J. Shao, Z. Zheng, and F. Meng, “Oscillation criteria for fractional differential equations with mixed nonlinearities,”
*Advances in Difference Equations*, p. 323, 2013. View at: Publisher Site | Google Scholar | MathSciNet - J. Shao, F. Meng, and Z. Zheng, “Oscillation theorems for linear matrix Hamiltonian systems,”
*Applied Mathematics and Computation*, vol. 253, pp. 402–409, 2015. View at: Publisher Site | Google Scholar | MathSciNet - H. Liu and F. Meng, “Interval oscillation criteria for second-order nonlinear forced differential equations involving variable exponent,”
*Advances in Difference Equations*, Paper No. 291, 14 pages, 2016. View at: Publisher Site | Google Scholar | MathSciNet - P. T. Nam, P. N. Pathirana, and H. Trinh, “Reachable set bounding for nonlinear perturbed time-delay systems: the smallest bound,”
*Applied Mathematics Letters*, vol. 43, pp. 68–71, 2015. View at: Publisher Site | Google Scholar | MathSciNet - X. Zhu, Y. Sun, and X.-J. Xie, “State bounding for nonlinear time-varying systems with delay and disturbance,”
*Journal of The Franklin Institute*, vol. 355, no. 16, pp. 8213–8224, 2018. View at: Publisher Site | Google Scholar | MathSciNet - N. Zhang, Y. Sun, and P. Zhao, “State bounding for homogeneous positive systems of degree one with time-varying delay and exogenous input,”
*Journal of The Franklin Institute*, vol. 354, no. 7, pp. 2893–2904, 2017. View at: Publisher Site | Google Scholar - J. G. Dong, “On the decay rates of homogeneous positive systems of any degree with time-varying delays,”
*Institute of Electrical and Electronics Engineers Transactions on Automatic Control*, vol. 60, no. 11, pp. 2983–2988, 2015. View at: Publisher Site | Google Scholar | MathSciNet - Yuangong Sun and Fanwei Meng, “Reachable Set Estimation for a Class of Nonlinear Time-Varying Systems,”
*Complexity*, vol. 2017, Article ID 5876371, 6 pages, 2017. View at: Publisher Site | Google Scholar - D. Tian, S. Liu, Y. Sun, and J. Xia, “Reachable set estimation for switched positive systems with mixed time-varying delays and bounded disturbances,”
*IET Control Theory & Applications*, vol. 12, no. 15, pp. 2003–2009, 2018. View at: Publisher Site | Google Scholar | MathSciNet - L. Farina and S. Rinaldi,
*Positive Linear Systems: Theory and Applications*, Wiley, New York, NY, USA, 2000. View at: Publisher Site | MathSciNet - T. Kaczorek,
*Positive 1D and 2D Systems*, Springer-Verlag, New York, NY, USA, 2002. - X. Liu, W. Yu, and L. Wang, “Stability analysis of positive systems with bounded time-varying delays,”
*IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 56, no. 7, pp. 600–604, 2009. View at: Publisher Site | Google Scholar - X. Liu, W. Yu, and L. Wang, “Stability analysis for continuous-time positive systems with time-varying delays,”
*IEEE Transactions on Automatic Control*, vol. 55, no. 4, pp. 1024–1028, 2010. View at: Publisher Site | Google Scholar | MathSciNet - H. R. Feyzmahdavian, T. Charalambous, and M. Johansson, “Exponential stability of homogeneous positive systems of degree one with time-varying delays,”
*IEEE Transactions on Automatic Control*, vol. 59, no. 6, pp. 1594–1599, 2014. View at: Publisher Site | Google Scholar - Y. Sun, Y. Tian, and X.-J. Xie, “Stabilization of positive switched linear systems and its application in consensus of multiagent systems,”
*IEEE Transactions on Automatic Control*, vol. 62, no. 12, pp. 6608–6613, 2017. View at: Publisher Site | Google Scholar | MathSciNet - Yuangong Sun, Zhaorong Wu, and Fanwei Meng, “Common Weak Linear Copositive Lyapunov Functions for Positive Switched Linear Systems,”
*Complexity*, vol. 2018, Article ID 1365960, 7 pages, 2018. View at: Publisher Site | Google Scholar - N. Zhang, Y. Sun, and F. Meng, “State bounding for switched homogeneous positive nonlinear systems with exogenous input,”
*Nonlinear Analysis: Hybrid Systems*, vol. 29, pp. 363–372, 2018. View at: Publisher Site | Google Scholar | MathSciNet - H. L. Smith,
*Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, ameircan mathematical society*, 1995. - V. S. Bokharaie, O. Mason, and F. Wirth, “Stability and positivity of equilibria for subhomogeneous cooperative systems,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 74, no. 17, pp. 6416–6426, 2011. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2019 Xingao Zhu and Yuangong Sun. 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.