Reachable Set Bounding for a Class of Nonlinear Time-Varying Systems with Delay

We investigate the problem of reachable set bounding for a class of continuous-time and discrete-time nonlinear time-varying systems with time-varying delay. Unlike some preceding works, the involved disturbance input and time-varying delay are not assumed to be bounded. By employing an approach which does not involve the conventional Lyapunov-Krasovskii functional, new conditions are proposed such that all the state trajectories of the system converge asymptotically within a ball. Two illustrative examples are also given to show the effectiveness of the obtained results.


Introduction
Reachable set bounding plays an important role in ensuring safe operation in practical engineering through synthesizing controllers to avoid undesirable (or unsafe) regions in the state space [1].Therefore, the problem of reachable set bounding with different dynamics has been investigated by many researchers in recent years in [2][3][4][5][6][7], to name a few.
Time delay has received a lot of attention due to its common presence in practical engineering and its detrimental effects on stability [8][9][10][11][12][13][14] and performance of systems such as oscillation [15][16][17][18][19]. Therefore, the problem of the reachable set bounding for systems with delay becomes very important.By using the Lyapunov-Razumikhin approach, an ellipsoid was given to bound the reachable set of linear systems with delay and bounded peak inputs in [20].An improved condition for reachable set bounding for linear systems with delay was proposed in [21] by virtue of a Lyapunov-Krasovskii type functional and the rate of delay.Less conservative estimation results on the reachable set for delayed systems with polytopic uncertainties were established in [22,23] by choosing pointwise maximum Lyapunov functional corresponding to a vertex of the polytope.A delay-partitioning method was applied to study the reachable set bounding problem of delayed systems in [24], which further reduced the conservatism of some existing results.By using the Lyapunov method, LMI conditions for the existence of ellipsoid-based bounds of reachable sets of a linear uncertain discrete system were given in [25].
However, most of the aforementioned works on the reachable set bounding have been mainly focused on linear time-delay systems with constant matrices or a combination of constant matrices (polytopic uncertainties).It seems to us that little has been known about the explicit estimation of reachable set for nonlinear time-varying systems with timevarying delay.Note that it is difficult to apply the usual Lyapunov-Krasovskii functional method to time-varying systems, because it may lead to unsolvable matrix Riccati differential equations or indefinite linear matrix inequalities.
Based on a method developed in positive systems which does not involve the Lyapunov-Krasovskii functional, a delayindependent condition was derived such that all the state trajectories of linear time-varying systems converge exponentially within a ball in [26].Recently, the result in [26] was extended to homogeneous positive systems of degree one with time-varying delays in [27].For a class of nonlinear time-delay systems with bounded disturbances, a new approach to obtain the smallest box which bounds all reachable sets was proposed in [28].For a switched system with nonlinear disturbance which can be bounded by a linear system, global exponential stability criteria were established in [29].

Advances in Mathematical Physics
Inspired by this and motivated by the work in [30,31], the paper will introduce a new approach which is different from the Lyapunov-Krasovskii functional method to derive new explicit conditions such that all the state trajectories of a class of continuous-time and discrete-time nonlinear time-varying systems with delay converge asymptotically within a ball.The main contribution of this paper is threefold: (1) the nonlinear term considered in this paper takes the more general form, which contains the systems studied in [26][27][28][29] as special cases; (2) the involved disturbance input and time-varying delay are not assumed to be bounded; (3) unlike some existing works, we do not need to transform the system to a timeinvariant one, which leads to less conservative conditions for reachable set bounding.
The rest of this paper is briefly outlined 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 state trajectories of the system converge asymptotically within a ball.Section 4 provides two illustrative examples to show the effectiveness of the obtained results.The paper is concluded in Section 5.

Preliminaries
Throughout this paper, the following notation will be used.Let R  and R × denote the set of -dimensional real vectors and the -dimensional real Euclidean space, respectively.Denote N 0 = {0, 1, 2, . ..} and ⟨⟩ = {1, 2, . . ., }.The matrix  ∈ R × is said to be Metzler if all its off-diagonal entries are nonnegative.For  ∈ R  , we denote by   the th coordinate of .For two vectors ,  ∈ R  , we write We first consider the continuous-time nonlinear timevarying system with delay where () ∈ R  is the state vector, the time-varying delay is the continuous vector valued function specifying the initial state of the system, and the vector fields (, ), (, ) : [0, ∞) × R  → R  are continuous and locally Lipschitz with respect to , which ensures the existence and uniqueness of solutions of system (1) [32].We also consider the following discrete-time nonlinear time-varying system with delay described by where () ∈ R  is the state vector, the time-varying delay () : N 0 → N 0 satisfies  − () ≥ − max with  max > 0, the vector fields ,  : N 0 × R  → R  , () : N 0 → R  is the disturbance input, and  : {− max , . . ., 0} → R  is the vector sequence specifying the initial state of the system.
We first extend some definitions given in [33] to the time variant case.Definition 1.A continuous vector field (, ) : [0, ∞) × R  → R  , which is continuously differentiable with respect to  on R  \{0}, is said to be cooperative if the Jacobian matrix , and all real  > 0, (, ) = (, ).

Main Results
We first study the reachable set bounding for the continuoustime system (1).Assume that the vector fields  and  satisfy the following assumption.

Advances in Mathematical Physics
Proof.We first have Assume that That is, Since  and  are homogeneous and order-preserving, we get from Assumption A2 that Therefore, Note that conditions ( 16) and ( 18) imply that This together with ( 17) and ( 24) yields that       ( + 1) By induction, we have that (19) holds.This completes the proof of Theorem 7 Remark 8.In Theorem 7, although conditions ( 16), (17), and (18) arisen from Assumption A2 depend on the time , they may be less conservative for some cases since they do not require that the disturbance and the time-varying delay are bounded.If we further assume that (, V) ⪯ (V), (, V) ⪯ (V), and |  ()| ≤   for  ∈ N 0 and  ∈ ⟨⟩, where (V) and (V) are time-invariant, homogeneous, and order-preserving vector fields, and   are constants, then conditions ( 16) and ( 18) are independent of time and verifiable, and () can be chosen to be   for some 1 >  > 0 in condition (17).

Numerical Examples
We now present two numerical examples to illustrate the main results of this paper.
Generally speaking, the minimal parameter constant  0 can be determined by the following nonlinear programming problem: minimize  0 defined by (31) subject to V ≻ 0,  > 0, and ( 29) and (30).

Conclusion
In this paper, the problem of reachable set bounding for a class of continuous-time and discrete-time nonlinear timevarying systems with delay has been investigated, where the involved disturbance input and time-varying delay may be unbounded.By using an approach which is different from the Lyapnov-Krasovskii functional method, we establish sufficient conditions such that all the state trajectories Noting that  * − ( * ) ∈ [− max ,  * ], we obtain      ( * −  ( * ))     ⪯ ( 0 + K ( * −  ( * ))) V.

Figure 1 :
Figure 1: State and its upper bound of continuous-time system (1).