Improved Results on Reachable Set Bounding for Linear Delayed Systems with Polytopic Uncertainties

This paper focuses on bound of reachable sets for delayed linear systems with polytopic uncertainties. Based on LyapunovKrasovskii functional theory, delay decomposition technique, and reciprocally convex method, some new results expressed in the form of linear matrix inequalities are derived. It should be noted that triple integral functionals are first to be introduced for reachable set analysis. Consequently, a tighter bound of the reachable set is obtained. Four numerical examples are given to illustrate the effectiveness and advantage of the proposed results comparing with the existing criteria.


Introduction
In real world, many phenomena can be described by time delay systems, such as communication networks, biology, and physical process.It is well known that the presence of time delay may lead to complicated behaviors for dynamic system, including instability, oscillations, and robustness [1][2][3][4][5].In addition to stability and robustness of the state, the property of input-to-state for dynamical systems is also concerned.For a dynamic system, reachable set is the set of all the states in the Euclidean space that are reachable from the origin, in finite time, by inputs with peak value that is bounded by some given positive scalar [6].It was first considered in the late 1960s and it has a wide range of applications, such as peak-to-peak gain minimization problem and control systems with actuator saturation.Thus, the problem of reachable set bounding for time delay systems has received considerable attention in recent years; for instance, see [6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein.
There are already some relevant results about the problem of reachable set bounding for linear systems.An LMI condition for an ellipsoid that bounded the reachable set of linear systems without time delay was given by Boyd in [13].In [6], Fridman and Shaked firstly derived LMIs criteria of an ellipsoid that bounded the reachable set of uncertain systems with time-varying delays and bounded peak input based on the Razumikhin theory.In [11], Kim proposed an improved condition by using the modified Lyapunov-Razumikhin functionals.Nam and Pathirana obtained a smaller reachable set bound by the delay decomposition technique [10].The maximal Lyapunov functionals, combined with the Razumikhin method, were employed to give a nonellipsoidal description of the reachable set in [15].More recently, the authors derived the ellipsoid bounds of reachable sets of linear uncertain linear discrete-time systems based on the idea to minimize the projection distances of the ellipsoids on each axis with different exponential convergence rates [17].Based on property of Metzler matrix, a new approach which did not involve the Lyapunov-Krasovskii functional method was used to get the state bounding for linear timedelayed systems [18].The delays considered in [6-9, 11, 12, 15, 16, 18] are from 0 to an upper bound.However, delays may vary in an interval for which lower bound of delays is not necessary to be 0, such as [10].On the other hand, the authors considered nondifferentiable time-varying delays in [10,18], and differentiable time-varying delays were considered in [6-9, 11, 12, 15, 16].Paper [11] assumed the derivative of delay to be less than 1.As is well known, large value of derivative of delay may yield bigger reachable set bounding.These constraints on the delays are strong and may be relaxed.
In this paper, we study the reachable set bounding for linear delayed systems with polytopic uncertainties.Constraints for delay are relaxed.Time delays vary in an interval for which lower bound of delays is not necessarily 0, and value of derivative of delay is not necessarily less than 1.Inspired by the Lyapunov functionals in [2], we construct Lyapunov-Krasovskii functionals, combining with the delay decomposition technique and reciprocally convex method to derive a more accurate description of the reachable set bound.Different from the Lyapunov functionals in [2], the integral terms of Lyapunov functionals in this paper contain  (−) .Moreover, to the best of our knowledge, it is first time to introduce triple integral functionals for reachable set analysis.We will show that the reachable set bound is tighter than that of [6,[8][9][10][11][12]14].Numerical examples illustrate the effectiveness and improvement of the obtained results.
Notations.The notations are used in this paper except where otherwise specified.  is the -dimension Euclidean space and  × denotes the set of  × -dimension real matrices; real matrix  > 0(≥ 0) means that  is a symmetric positive definite (positive semidefinite) matrix.Superscript "" denotes transposition of a vector or a matrix; ⋆ represents the elements below the main diagonal of a symmetric block matrix;  denotes an identity matrix; "-" in tables represents no feasible solution for linear matrix inequality.
Lemma 1 (see [19]).The following relation is known as the Leibniz rule: Lemma 2 (see [4]).For any constant matrix  =   > 0 and ℎ 2 > ℎ 1 ≥ 0 such that the following integrations are well defined, then Lemma 3 (see [5]).For any constant matrix  > 0, scalars ℎ 2 > ℎ 1 ≥ 0 such that the following integrations are well defined, then Proof.By using Lemma 2, one can obtain According to Schur complement, the following inequality holds: Integrating both sides of the above inequality from −ℎ 2 to −ℎ 1 , we have By using Schur complement again, inequality ( 8) is equivalent to the inequality in Lemma 3.This completes the proof.

Main Results
In order to study the reachable set bounding of uncertain system (1), firstly, we consider Δ = 0, Δ = 0, Δ = 0 in system (1); that is, The reachable set bounding of system (14) with timevarying delay () for case (a) and case (b) is stated in Theorems 7 and 8, respectively.
Using the spectral properties of symmetric positive definite matrix , the following inequality holds: This further implies that ‖()‖ ≤  = 1/√ min () due to (27).This completes the proof.
Remark 12.In this paper, delay decomposition technique and reciprocally convex method are used to construct Lyapunov functionals, and triple integral terms are introduced in Lyapunov functionals for the first time to investigate bounds of reachable set for systems with uncertainties, which may lead to tighter bounding for reachable set.
Remark 14.The reachable set of system (1) can be minimized by solving the following optimization problem for a scalar  > 0:

Examples
In this section, four numerical examples will be presented to show the validity of the main results derived in this paper.
Example 1.Consider the following uncertain time-delayed system with parameters: By solving optimization problems (42), computed 's for the case   ≤ () ≤   with different values of   are listed in Table 1.Computed 's for different values of  with   = 0.7 and   = 0.75 for the case   ≤ () ≤   , τ () ≤  are obtained in Tables 2 and 3, respectively.It is clear to see that the proposed method in this paper yields tighter bounds than literatures [6,8,11].
Computed 's for the case   ≤ () ≤   , τ () ≤  with   = 0,   = 0.1 are listed in Table 4 to compare with the ones in [6,8,11].It should be noted that there is no feasible solution employing the approaches in [6,11], and the derived method in this paper yields much tighter bounding than [8].Hence, the proposed method leads to a wider application range.
From Theorem 9, computed radiuses 's for case (1) and case (2) are listed in Table 5.These results are compared to the ones in [6,8,10,11].It is clear to see that our results decrease radiuses of the ellipsoid.
By employing the method of Theorem 8 in this paper, 's for different values of () with  = 0 are listed in Table 6.It is easy to see that bounds obtained in this paper are better than the ones of literatures [8][9][10][11][12]14].

Example 4 .
Consider the following uncertain time-delayed system with parameters: 2 , . . .,   :    →  have positive values in an open subset  of   .Then, the reciprocally convex combination of   over  satisfies
[3]is clear to see that radius  is smallest if  = min Hence, we can use MATLAB's Toolbox to solve the matrix inequalities in Theorems 7-10.Remark 17.The approach is likely to help further work in this area.It may be used to improve estimate partial state bounding for neural networks with time-varying delays, such as[3].

Table 6 :
Computed 's of Example 4 for different values of  with  = 0. contain triple integral terms, which lead to tighter bounding than previous literatures.Numerical examples have been given to illustrate the effectiveness and improvement of the proposed methods.These results are likely to help further work in this area.One future work is to extend the results in this technical note to linear neutral systems and linear mixed delay systems.