Abstract
This paper studies the problem of reachable set estimation for a class of nonlinear time-varying systems with disturbances. New necessary and (or) sufficient conditions are derived for the existence of a ball such that all the solutions of the system converge asymptotically within it. Explicit estimation on the decay rate is also presented. The method used in this paper is motivated by that for positive systems, which is different from the conventional Lyapunov-Krasovskii functional method. A numerical example is given to illustrate the effectiveness of the obtained result.
1. Introduction
The problem of reachable set estimation (bounding) for linear systems has been addressed in [1–10], to name a few. In all of the aforementioned works, the involved system was mainly aimed at linear time invariant systems. Moreover, the Lyapunov-Krasovskii functional method was most commonly used, which is usually invalid to time-varying systems because they lead to either unsolvable matrix Riccati differential equations or indefinite linear matrix inequalities.
Recently, Hien and Trinh considered the problem of reachable set bounding for linear time-varying systems with delay and bounded disturbances for the first time in [11]. By using a new approach which does not involve the Lyapunov-Krasovskii functional method, an explicit delay-independent condition for state bounding of the system was given in terms of Metzler matrix. The reachable set bounding for a class of nonlinear perturbed time-delay systems was investigated in [12], where the involved nonlinear term satisfies linear growth condition. Recently, state bounding for homogeneous positive systems of degree one with time-varying delay and exogenous input was studied in [13]. In this paper, we will further study the reachable set estimation for a class of nonlinear time-varying systems without satisfying the linear growth condition or the homogeneous condition of degree one. For the particular case when the system is positive, we first establish a necessary and sufficient condition for reachable set bounding. Then, we extend the result to the more general case by using the comparison principle.
Positive systems, whose state trajectories remain nonnegative for all time provided that initial states are nonnegative, have received much attention in recent years (see the books [14, 15] and the references therein). When dealing with stability problems of positive systems, an approach independent of Lyapunov-Krasovskii functional was commonly used in [16–21]. Inspired by this, the paper will apply such a method to the reachable set estimation for nonlinear time-varying systems with disturbances. Necessary and (or) sufficient conditions have been established such that all the solutions of the system converge asymptotically within a ball determined by the upper bound of disturbances. Moreover, the explicit decay rate is also presented.
The paper is organized as follows. In Section 2, we present the notation used through this paper as well as preliminaries for our results. Section 3 then focuses on deriving explicit conditions under which all the solutions of the system converge asymptotically within a ball. Section 4 provides an illustrative example to show the effectiveness of the obtained result. The paper is concluded in Section 5.
2. Preliminaries
Throughout this paper, the following notation will be used. Let and denote the set of -dimensional real vectors and the -dimensional real Euclidean space, respectively. The matrix is said to be Metzler if all its off-diagonal entries are nonnegative. For , we denote by the th coordinate of . For two vectors , we write if , if , if , and if , . Let . For , let and . Given an -dimensional vector , the weighted -norm of the vector is defined by . For , denote a ball by .
Consider the following continuous-time nonlinear time-varying system described by where is the state vector, is the unknown disturbance input, and the vector field is continuous and locally Lipschtiz with respect to , which ensures the existence and uniqueness of solutions of system (1) [22]. We now begin with the following definitions.
Definition 1. System (1) is said to be positive if, for any initial condition , the corresponding state trajectory remains nonnegative for all .
The following proposition gives a sufficient condition guaranteeing the positivity of system (1).
Proposition 2. System (1) is positive if where is the th coordinate of .
Proof. Let be the solution of system (1) with the initial condition . In order to prove that for all , it is sufficient to check that the vector does not point toward the outside of . This is equivalent to verifying that the components of the vector corresponding to the zero components of are nonnegative, which can be derived from condition (2) immediately. The proof of Proposition 2 is complete.
Definition 3 (see [21]). A vector field is said to be homogeneous of degree if for all and all .
Definition 4 (see [23]). A continuous vector field , which is continuously differentiable on , is said to be cooperative if the Jacobian matrix is Metzler for all .
Denote by the th coordinate of . It is well known that a cooperative vector field satisfies the following Kamke condition (see [22, Remark 1.1, pp. 33]).
Proposition 5. Let be a cooperative vector field. For any two vectors satisfying and , one has .
3. Main Results
We first consider the following particular case of system (1), wherewhere the vector field is cooperative and is a constant. By Propositions 2 and 5, we see that system (1) is positive. For this case, we have the following necessary and sufficient condition for the reachable set bounding of system (1).
Theorem 6. Assume that (3) holds, and the vector field is cooperative and homogeneous of degree . Then the following two statements are equivalent.
(i) For any initial condition and any disturbance satisfying (3), there exist nonnegative constants , , and such that the solution of system (1) satisfies where depends on , if , is related to and , and is independent of and .
(ii) There exists an -dimensional vector such that .
In addition, if condition (ii) holds, we can choose , , and , where and , , and are defined as follows:
Proof. (ii) (i) Since system (1) is positive, each solution of system (1) with the initial condition satisfies for all . For any , we can conclude from the definition of that Set From (6), we have . By the continuity of at , there exists a real number such that for and . We now prove that for and . Otherwise, there exist a real number and an index such that for and , and . Therefore, By Proposition 5 and the homogeneity of , we obtain From (1) and the definition of , we have By using the basic inequality for and , we get from (10) and (11) that Since , by the definitions of and , we obtain Combining this with (12), we have , which contradicts (8). Consequently, for and . That is, for any , As tends to 1 from the right, we have which implies (4) with , , and defined as in Theorem 6.
(i) (ii) Assume that condition (4) holds for any initial condition and any satisfying (3). In particular, each solution of the system with the nonnegative initial condition satisfies That is, system (1) without disturbances is globally asymptotically stable. Invoking [23, Proposition 4.1], we have that there exists a vector such that . This completes the proof of Theorem 6.
Next, we study the reachable set bounding for system (1) without condition (3). Based on a straightforward computation, we can get where denotes the derivative of from the right and . The right-hand side of the above inequality is interpreted as when .
Similar to Definitions 3 and 4, a vector field is said to be cooperative, if it is continuously differentiable on with respect to , and the Jacobian matrix is Metzler for and . The vector field is said to be homogeneous of degree if for , , and . In the following, assume that the vector field satisfies the following.
(iii) There exists a vector field which is cooperative and homogeneous of degree , such that where is also interpreted as when .
Since is cooperative and homogeneous, we can get from Propositions 2 and 5 that the following system is positive: By using the comparison principle, we have that solution of system (1) with the initial condition satisfies where is the solution of system (19) with the initial condition . Consequently, by using Theorem 6, we have the following sufficient condition for reachable set estimation of system (1).
Theorem 7. Assume that condition (iii) holds. If there exists a vector such that then for any initial condition and any disturbance , the solution of system (1) satisfies where
Proof. Based on the above analysis, it is sufficient to prove that (22) holds for each solution of system (19) with any nonnegative initial condition. Denote by the solution of system (22) with . For any , let Since , , we show that for . Otherwise, there exist a real number and an index such that , , , and . It implies that , Note that is a cooperative and homogeneous vector field of degree . Based on the analysis in Theorem 6, we can eventually get By the definitions of and , we conclude that , which is a contradiction. The remainder of the proof is similar to that of Theorem 6. This completes the proof of Theorem 7.
Remark 8. Compared with Theorem 6, we do not require in Theorem 7 that the initial condition and the disturbance input for . Moreover, the disturbance may be unbounded.
Remark 9. If , then Theorem 7 guarantees that all the solutions of system (1) converge asymptotically within an ellipsoid.
Remark 10. Assume that there exists a vector field such that for and . Suppose also that there exist positive constants , , such that for . Then we get . It can be seen from Theorem 7 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 some nonlinear optimization problem (for details, please see analysis in the following numerical example).
4. Numerical Example
Consider system (1) on with and We see that condition (18) holds for the vector field which is cooperative and homogeneous of degree . If we choose , then (21) is valid. A straightforward computation yields that and . Therefore, all the solutions of system (1) converge asymptotically within the ball by Theorem 7. If we choose the initial condition , then , and hence the solution of the system satisfies . The simulation result is presented in Figure 1.
In the above numerical example, a better estimation of the reachable set can be obtained by choosing appropriate vector . In fact, the minimal bound of the reachable set can be chosen to be subject to , which is a nonlinear optimization problem, where .
5. Conclusion
In this paper, we have studied the problem of reachable set estimation for a class of continuous-time nonlinear time-varying systems with disturbances. When the involved system is a particular positive system, we establish a necessary and sufficient condition such that all the solutions of the system converge asymptotically within a ball. By using the comparison principle, an explicit sufficient condition for reachable set bounding of the general system is also presented. In both cases, the decay rate is estimated precisely. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result. The state bounding for both the continuous-time system with delay and the discrete-time system is an interesting problem, which will be studied in the next paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This paper is supported by the National Natural Science Foundation of China under Grant nos. 61473133, 11671227, and 61374074 and the Natural Science Foundation of Shandong Province under Grant no. JQ201119.