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) satisfiesfor 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.