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

Authors: Hao Chen
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

This paper focuses on bound of reachable sets for delayed linear systems with polytopic uncertainties. Based on Lyapunov-Krasovskii 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.


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## Body

## 1. 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 [1–5]. 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 [6–18] 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 time-delayed 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 eα(s-t). 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, 8–12, 14]. Numerical examples illustrate the effectiveness and improvement of the obtained results.Notations.The notations are used in this paper except where otherwise specified. Rn is the n-dimension Euclidean space and Rn×m denotes the set of n×m-dimension real matrices; real matrix P>0(≥0) means that P is a symmetric positive definite (positive semidefinite) matrix. Superscript “T” denotes transposition of a vector or a matrix; ⋆ represents the elements below the main diagonal of a symmetric block matrix; I denotes an identity matrix; “—” in tables represents no feasible solution for linear matrix inequality.


## 2. Preliminaries

Consider the following uncertain polytopic time-delayed linear systems with disturbances:(1)z˙t=(A+ΔA)z(t)+(D+ΔD)z(t-τ(t))+(B+ΔB)w(t),z(t)=0,t∈[-τM,0],where z(t)∈Rn is the state vector; w(t)∈Rm is the disturbance. One has A,ΔA∈Rn×n, D,ΔD∈Rn×n, and B,ΔB∈Rn×m. A, D, B are known constant matrices. τ(t) is the time-varying delay. For disturbance w(t), we assume that wT(t)w(t)≤wm2, where wm is a constant.The uncertainties are expressed as a linear convex-hull of known matricesAi, Bi, and Di: (2)ΔA=∑i=1N‍θi(t)Ai,ΔB=∑i=1N‍θi(t)Bi,ΔD=∑i=1N‍θitDi,with θi(t)∈[0,1] and ∑i=1N‍θi(t)=1,∀t>0.In this paper, time-varying delayτ(t) will be considered in two cases:(a)
τm≤τt≤τM,(b)
τm≤τ(t)≤τM, τ˙(t)≤μ.The following lemmas are useful in deriving the criteria.Lemma 1 (see [19]).
The following relation is known as the Leibniz rule:(3)ddt∫atbt‍ft,sds=b˙(t)f[t,b(t)]-a˙(t)f[t,a(t)]+∫a(t)b(t)‍∂∂tf(t,s)ds.Lemma 2 (see [4]).
For any constant matrixP=PT>0 and h2>h1≥0 such that the following integrations are well defined, then(4)-(h2-h1)∫t-h2t-h1‍zT(s)Pz(s)ds≤-∫t-h2t-h1‍zsdsTP(∫t-h2t-h1‍z(s)ds).Lemma 3 (see [5]).
For any constant matrixR>0, scalars h2>h1≥0 such that the following integrations are well defined, then (5)-12(h2-h1)2∫-h2-h1‍∫t+θt-h1‍zT(s)Rz(s)dsdθ≤-∫-h2-h1‍∫t+θt-h1‍zTsdsdθR∫-h2-h1‍∫t+θt-h1‍zsdsdθ.Proof.
By using Lemma2, one can obtain (6)-(-h1-θ)∫t+θt-h1‍zT(s)Rz(s)ds≤-∫t+θt-h1‍zsdsTR∫t+θt-h1‍zsds.
According to Schur complement, the following inequality holds:(7)-∫t+θt-h1‍zT(s)Rz(s)ds∫t+θt-h1‍zsdsT⋆(-h1-θ)R-1≤0.
Integrating both sides of the above inequality from-h2 to -h1, we have(8)-∫-h2-h1‍∫t+θt-h1‍zT(s)Rz(s)dsdθ∫-h2-h1‍∫t+θt-h1‍zsdsdθT⋆∫-h2-h1‍(-h1-θ)dθR-1≤0.
By using Schur complement again, inequality (8) is equivalent to the inequality in Lemma 3. This completes the proof.Lemma 4 (see [1]).
Letf1,f2,…,fN: Rm↦R have positive values in an open subset D of Rm. Then, the reciprocally convex combination of fi over D satisfies (9)min⁡{αi∣αi>0,∑i=1Nαi=1}∑i=1N‍1αifi(t)+max⁡gi,j(t)∑i≠j‍gi,j(t),subject to (10){gi,j(t):Rm⟼R,gj,it=gi,j(t),fi(t)gi,j(t)gi,j(t)fj(t)≥0}.Lemma 5 (see [2]).
For any vectorsz1, z2, constant matrices Ti (i=1,2,3,4), S, and scalars α>0, β>0 satisfying α+β=1, then following inequality holds: (11)-1αz1TT1z1-1βz2TT2z2-βαz1TT3z1-αβz2TT4z2≤-z1z2TT1SSTT2z1z2,subject to (12)T1+T3SSTT2+T4≥0.Lemma 6 (see [13]).
LetV be a Lyapunov function for system (1) with wT(t)w(t)≤wm2. If (13)V˙+αV-αwm2wT(t)w(t)≤0,then V≤1.


## 3. Main Results

In order to study the reachable set bounding of uncertain system (1), firstly, we consider ΔA=0, ΔD=0, ΔB=0 in system (1); that is,(14)z˙t=Azt+Dzt-τt+Bwt,zt=0,t∈-h,0.The reachable set bounding of system (14) with time-varying delay τ(t) for case (a) and case (b) is stated in Theorems 7 and 8, respectively.After that, the reachable set bounding of uncertain system (1) with time-varying delay τ(t) for case (a) and case (b) is stated in Theorems 9 and 10, respectively.Theorem 7.
If there exist matricesP>0, Q1>0, Q2>0, R1>0, R2>0, K1>0, K2>0, K3>0, and K4>0,  S, N with appropriate dimensions, and a scalar α>0, such that the following inequalities hold,(15)R2+K3SSTR2+K4≥0,Φ=Φ11PDΦ130Φ1500Φ18PB⋆Φ22Φ23Φ240Φ26Φ27DTNT0⋆⋆Φ33Φ34Φ35Φ36000⋆⋆⋆Φ4400Φ4700⋆⋆⋆⋆Φ550000⋆⋆⋆⋆⋆Φ66000⋆⋆⋆⋆⋆⋆Φ7700⋆⋆⋆⋆⋆⋆⋆Φ88NB⋆⋆⋆⋆⋆⋆⋆⋆-αwm2I≤0,where (16)Φ11=αP+PA+ATP+Q1-2e-ατmK1-e-ατmR1,Φ13=-e-ατmR1,Φ15=2e-ατmK1,Φ18=ATNT,Φ22=-2e-ατMK3-2e-ατMK4-2e-ατMR2+e-ατM(ST+S),Φ23=e-ατMR2-e-ατMST,Φ24=e-ατMR2-e-ατMS,Φ26=2e-ατMK4,Φ27=2e-ατMK3,Φ33=-e-ατmQ1+e-ατmQ2-e-ατmR1-2e-ατmK2Φ88=-2e-ατMK3-e-ατMR2,Φ34=e-ατMS,Φ35=2e-ατmK2,Φ36=2e-ατMK3,Φ44=-e-ατMQ2-2e-ατMK4-e-ατMR2,Φ47=2e-ατMK4,Φ55=-2e-ατmK1-2e-ατmK2,Φ66=-2e-ατMK3-2e-ατMK4,Φ77=-2e-ατMK3-2e-ατMK4,Φ88=τm2R1+τM-τm2R2+12τm2K1+12τm2K2Φ88=+12τM-τm2K3+12τM-τm2K4-N-NT,then the reachable sets of system (14) are bounded by a ball B(0,r)={z∈Rn∣z≤r} with(17)r=1λmin⁡(P).Proof.
Construct the following Lyapunov-Krasovskii functional,(18)Vzt=∑i=17‍Vizt,where (19)V1(zt)=zT(t)Pz(t),V2zt=∫t-τmt‍eα(s-t)zT(s)Q1z(s)ds+∫t-τMt-τm‍eα(s-t)zT(s)Q2z(s)ds,V3(zt)=τm∫-τm0‍∫t+θt‍eα(s-t)z˙T(s)R1z˙(s)ds,V4(zt)=(τM-τm)∫-τM-τm‍∫t+θt‍eα(s-t)z˙T(s)R2z˙(s)ds,V5zt=∫-τm0‍∫η0‍∫t+θt‍eα(s-t)z˙T(s)K1z˙(s)dsdθdη+∫-τm0‍∫-τmη‍∫t+θt‍eα(s-t)z˙T(s)K2z˙(s)dsdθdη,V6(zt)=∫-τM-τm‍∫η-τm‍∫t+θt‍eα(s-t)z˙T(s)K3z˙(s)dsdθdη,V7zt=∫-τM-τm‍∫-τMη‍∫t+θt‍eαs-tz˙TsK4z˙sdsdθdη.Taking the time derivative of V(zt) along the trajectory of system (14), we obtain(20)V˙1zt=2zT(t)Pz˙(t)=-αV1(zt)+2zT(t)Pz˙(t)V˙1zt2zT(t)Pz˙(t)=c=+αzT(t)Pz(t)V˙1zt=-αV1(zt)+αzT(t)Pz(t)+2zT(t)V˙1zt=·PAzt+Dzt-τt+Bwt,V˙2zt=-αV2(zt)+zT(t)Q1z(t)V˙1zt=+e-ατmzT(t-τm)Q2z(t-τm)V˙1zt=-e-ατmzT(t-τm)Q1z(t-τm)V˙1zt=-e-ατMzT(t-τM)Q2z(t-τM),V˙3zt=-αV3(zt)+τm2z˙T(t)R1z˙(t)V˙1zt=-τm∫t-τmt‍eα(s-t)z˙T(s)R1z˙(s)dsV˙1zt≤-αV3(zt)+τm2z˙T(t)R1z˙(t)V˙1zt=-e-ατmzt-zt-τmTV˙1zt=·R1zt-zt-τm,V˙4zt=-αV4(zt)+(τM-τm)2z˙T(t)R2z˙(t)V˙1zt=-(τM-τm)∫t-τMt-τm‍eα(s-t)z˙T(s)R2z˙(s)dsV˙1zt≤-αV4(zt)+τM-τm2z˙T(t)R2z˙(t)V˙1zt=-e-ατM(τM-τm)∫t-τ(t)t-τm‍z˙T(s)R2z˙(s)dsV˙1zt=-e-ατM(τM-τm)∫t-τMt-τ(t)‍z˙T(s)R2z˙(s)ds,V˙5zt=-αV5(zt)+12τm2z˙(t)K1z˙(t)+12τm2z˙(t)K2z˙(t)V˙1zt=-∫-τm0‍∫t+θt‍eαs-tz˙TsK1z˙sdsdθV˙1zt=-∫-τm0‍∫t-τmt+θ‍eα(s-t)z˙T(s)K2z˙(s)dsdθV˙1zt≤-αV5(zt)+12τm2z˙(t)K1z˙(t)+12τm2z˙(t)K2z˙(t)V˙1zt=-2e-ατm1τm2∫-τm0‍∫t+θt‍z˙T(s)dsdθV˙1zt=·K1∫-τm0‍∫t+θt‍z˙(s)dsdθV˙1zt=-2e-ατm1τm2∫-τm0‍∫t-τmt+θ‍z˙T(s)dsdθV˙1zt=·K2∫-τm0‍∫t-τmt+θ‍z˙(s)dsdθV˙1zt=-αV5(zt)+12τm2z˙(t)K1z˙(t)+12τm2z˙(t)K2z˙(t)V˙1zt=-2e-ατm(zT(t)-1τm∫t-τmt‍zT(s)ds)V˙1zt=·K1zt-1τm∫t-τmt‍zsdsV˙1zt=-2e-ατm1τm∫t-τmt‍zTsds-zTt-τmV˙1zt=·K21τm∫t-τmt‍zTsds-zTt-τm,V˙6zt=-αV6(zt)+12(τM-τm)2z˙(t)K3z˙(t)V˙1zt=-∫-τM-τm‍∫t+θt-τm‍eα(s-t)z˙T(s)K3z˙(s)dsdθV˙1zt≤-αV6(zt)+12(τM-τm)2z˙(t)K3z˙(t)V˙1zt=-e-ατM∫-τM-τm‍∫t+θt-τm‍z˙T(s)K3z˙(s)dsdθV˙1zt=-αV6(zt)+12(τM-τm)2z˙(t)K3z˙(t)V˙1zt=-e-ατM∫-τ(t)-τm‍∫t+θt-τm‍z˙T(s)K3z˙(s)dsdθV˙1zt=-e-ατM∫-τM-τt‍∫t+θt-τt‍z˙T(s)K3z˙(s)dsdθV˙1zt=-e-ατM∫-τM-τ(t)‍∫t-τ(t)t-τm‍z˙T(s)K3z˙(s)dsdθV˙7zt=-αV7(zt)+12(τM-τm)2z˙(t)K4z˙(t)V˙1zt=-∫-τM-τm‍∫t-τMt+θ‍eα(s-t)z˙T(s)K4z˙(s)dsdθV˙1zt≤-αV7(zt)+12(τM-τm)2z˙(t)K3z˙(t)V˙1zt=-e-ατM∫-τM-τm‍∫t-τMt+θ‍z˙T(s)K4z˙(s)dsdθV˙1zt=-αV7(zt)+12(τM-τm)2z˙(t)K4z˙(t)V˙1zt=-e-ατM∫-τ(t)-τm‍∫t-τ(t)t+θ‍z˙T(s)K4z˙(s)dsdθV˙1zt=-e-ατM∫-τt-τm‍∫t-τMt-τt‍z˙T(s)K4z˙(s)dsdθV˙1zt=-e-ατM∫-τM-τt‍∫t-τMt+θ‍z˙TsK4z˙sdsdθ.
Using Lemma2,(21)V˙4zt≤-αV4(zt)+(τM-τm)2z˙T(t)R2z˙(t)-e-ατMτM-τmτt-τmzTt-τm-zTt-τt·R2zt-τm-zt-τt-e-ατMτM-τmτM-τtzTt-τt-zTt-τM·R2zt-τt-zt-τM=-αV4(zt)+(τM-τm)2z˙T(t)R2z˙(t)-e-ατM1β1zTt-τm-zTt-τt·R2zt-τm-zt-τt-e-ατM1β2zTt-τt-zTt-τM·R2zt-τt-zt-τM,where β1=(τ(t)-τm)/(τM-τm), β2=(τM-τt)/(τM-τm), and β1+β2=1.
Using Lemma3,(22)V˙6zt≤-αV6(zt)+12(τM-τm)2z˙(t)K3z˙(t)-2e-ατM1τt-τm2∫-τt-τm‍∫t+θt-τm‍z˙Tsdsdθ·K3∫-τt-τm‍∫t+θt-τm‍z˙sdsdθ-2e-ατM1τM-τt2∫-τM-τt‍∫t+θt-τt‍z˙Tsdsdθ·K3∫-τM-τt‍∫t+θt-τt‍z˙sdsdθ-e-ατMτM-τtτt-τm∫t-τtt-τm‍z˙Tsdsdθ·K3∫t-τtt-τm‍z˙Tsdsdθ=-αV6(zt)+12τM-τm2z˙(t)K3z˙(t)-2e-ατMzTt-τm-1τt-τm∫t-τtt-τm‍zTsds·K3(z(t-τm)-1τ(t)-τm∫t-τ(t)t-τm‍z(s)ds)-2e-ατMzt-τt-1τM-τt∫t-τMt-τt‍zsdsT·K3zt-τt-1τM-τt∫t-τMt-τt‍zsds-e-ατMβ2β1zTt-τm-zTt-τt·K3zt-τm-zt-τt,V˙7zt≤-αV7(zt)+12(τM-τm)2z˙(t)K4z˙(t)-2e-ατM1τt-τm2∫-τt-τm‍∫t-τtt+θ‍z˙Tsdsdθ·K4∫-τt-τm‍∫t-τtt+θ‍z˙Tsdsdθ-e-ατMτt-τmτM-τt∫t-τMt-τt‍z˙Tsdsdθ·K4∫t-τMt-τt‍z˙sdsdθ-2e-ατM1τM-τt2∫-τM-τt‍∫t-τMt+θ‍z˙Tsdsdθ·K4∫-τM-τt‍∫t-τMt+θ‍z˙sdsdθ=-αV7(zt)+12(τM-τm)2z˙(t)K4z˙(t)-2e-ατM1τt-τm∫t-τtt-τm‍zTsds-zt-τt·K41τt-τm∫t-τtt-τm‍zsds-zt-τt-e-ατMβ1β2zTt-τt-zTt-τM·K4zt-τt-zt-τM-2e-ατM1τM-τt∫t-τMt-τt‍zTsds-zTt-τM·K41τM-τt∫t-τMt-τt‍zsds-zt-τM.
From Lemma5 and inequality (15), one can obtain(23)-e-ατM1β1(zT(t-τm)-zT(t-τ(t)))·R2(z(t-τm)-z(t-τ(t)))-e-ατM1β2(zT(t-τ(t))-zT(t-τM))·R2(z(t-τ(t))-z(t-τM))-e-ατMβ2β1(zT(t-τm)-zT(t-τ(t)))·K3(z(t-τm)-z(t-τ(t)))-e-ατMβ1β2(zT(t-τ(t))-zT(t-τM))·K4(z(t-τ(t))-z(t-τM))≤-e-ατMzt-τm-zt-τtzt-τt-zt-τMTR2SSTR2·z(t-τm)-z(t-τ(t))z(t-τ(t))-z(t-τM).Obviously, the following equation holds:(24)2z˙T(t)N(-z˙(t)+Az(t)+Dz(t-τ(t))+Bw(t))=0.
Combining (20)–(24), one gets(25)V˙(zt)+αV(zt)-αwm2wT(t)w(t)≤ξT(t)Φξ(t),where (26)ξTt=zTt,zTt-τt,zTt-τm,zT(t-τM),1τm∫t-τmt‍zT(s)ds,1τ(t)-τm∫t-τ(t)t-τm‍zT(s)ds,1τM-τt∫t-τMt-τtzTsds,z˙Tt,wTt‍.Since (15) hold, we can conclude that V˙+αV-(α/wm2)wT(t)w)(t)≤0.
Therefore, one can obtainV(zt)≤1 by Lemma 6.
Using the spectral properties of symmetric positive definite matrixP, the following inequality holds:(27)λmin⁡(P)z(t)2≤V(zt).
This further implies thatz(t)≤r=1/λmin⁡(P) due to (27). This completes the proof.If the derivative of time delayμ is known, that is, the case τm≤τ(t)≤τM,τ˙(t)≤μ, bounding for reachable set of system (14) is described in the following.Theorem 8.
If there exist matricesP>0, Q1>0, Q2>0, R1>0, R2>0, K1>0, K2>0, K3>0, K4>0, and M>0, S, N with appropriate dimensions, and a scalar α>0, such that the following inequalities hold,(28)R2+K3SSTR2+K4≥0,Φ=Φ11PDΦ130Φ1500Φ18PB⋆Φ22Φ23Φ240Φ26Φ27DTNT0⋆⋆Φ33Φ34Φ35Φ36000⋆⋆⋆Φ4400Φ4700⋆⋆⋆⋆Φ550000⋆⋆⋆⋆⋆Φ66000⋆⋆⋆⋆⋆⋆Φ7700⋆⋆⋆⋆⋆⋆⋆Φ88NB⋆⋆⋆⋆⋆⋆⋆⋆-αwm2I≤0,where(29)Φ11=αP+PA+ATP+Q1-2e-ατmK1T-e-ατmR1+M,Φ13=-e-ατmR1,Φ18Φ15=2e-ατmK1,Φ18=ATNT,Φ22=-2e-ατMK3-2e-ατMK4-2e-ατMR2Φ22=+e-ατM(ST+S)-(1-μ)e-ατMM,Φ23=e-ατMR2-e-ατMST,Φ24=e-ατMR2-e-ατMS,Φ26=-2e-ατMK4,Φ27=-2e-ατMK3,Φ33=-e-ατmQ1+e-ατmQ2-e-ατmR1Φ22=-2e-ατmK2-2e-ατMK3-e-ατMR2,Φ34=e-ατMS,Φ18Φ35=2e-ατmK2,Φ36=2e-ατMK3,Φ44=-e-ατMQ2-2e-ατMK4-e-ατMR2,Φ47=2e-ατMK4,Φ55=-2e-ατmK1-2e-ατmK2,Φ66=-2e-ατMK3-2e-ατMK4,Φ77=-2e-ατMK3-2e-ατMK4,Φ88=τm2R1+(τM-τm)2R2+12τm2K1+12τm2K2Φ22=+12τM-τm2K3+12τM-τm2K4-N-NT,then the reachable sets of system (14) are bounded by a ball B(0,r)={z∈Rn|∥z∥≤r} with(30)r=1λmin⁡(P).Proof.
We modify the Lyapunov-Krasovskii functional(31)V′(zt)=∑i=17‍Vi(zt)+V8(zt),where (32)V8(zt)=∫t-τ(t)t‍eα(s-t)zT(s)Mz(s)ds.Taking the time derivative of V8(zt) along the trajectory of system (14),(33)V˙8zt≤-αV8(zt)+zT(t)Mz(t)-(1-μ)e-ατMzT(t-τ(t))Mz(t-τ(t)).By the same way as in proof Theorem 7, one can obtain the result easily. The proof is completed.Next, consider the linear system with polytopic uncertainties (1). Bounding for reachable set of system (1) with the case τm≤τ(t)≤τM is obtained in Theorem 9.Theorem 9.
If there exist matricesP>0, Q1>0, Q2>0, R1>0, R2>0, K1>0, K2>0, K3>0, and K4>0, S, N with appropriate dimensions, and a scalar α>0, satisfying the following inequalities for all i=1,2,…,N,(34)R2+K3SSTR2+K4≥0,(35)Φi=Φ11iP(D+Di)Φ13i0Φ15i00Φ18iP(B+Bi)⋆Φ22iΦ23iΦ24i0Φ26iΦ27iD+DiTNT0⋆⋆Φ33iΦ34iΦ35iΦ36i000⋆⋆⋆Φ44i00Φ47i00⋆⋆⋆⋆Φ55i0000⋆⋆⋆⋆⋆Φ66i000⋆⋆⋆⋆⋆⋆Φ77i00⋆⋆⋆⋆⋆⋆⋆Φ88iN(B+Bi)⋆⋆⋆⋆⋆⋆⋆⋆-αwm2I≤0,where (36)Φ11i=αP+P(A+Ai)+A+AiTP+Q1Φ88i=-2e-ατmK1-e-ατmR1,Φ13i=-e-ατmR1,Φ15=2e-ατmK1,Φ18i=(A+Ai)TNT,Φ22i=-2e-ατMK3-2e-ατMK4-2e-ατMR2+e-ατM(ST+S),Φ23i=e-ατMR2-e-ατMST,Φ24i=e-ατMR2-e-ατMS,Φ26i=-2e-ατMK4,Φ27i=-2e-ατMK3,Φ33i=-e-ατmQ1+e-ατmQ2-e-ατmR1Φ88i=-2e-ατmK2-2e-ατMK3-e-ατMR2,Φ34i=e-ατMS,Φ35i=2e-ατmK2,Φ36i=2e-ατMK3,Φ44i=-e-ατMQ2-2e-ατMK4-e-ατMR2,Φ47i=2e-ατMK4,Φ55i=-2e-ατmK1-2e-ατmK2,Φ66i=-2e-ατMK3-2e-ατMK4,Φ77i=-2e-ατMK3-2e-ατMK4,Φ88i=τm2R1+τM-τm2R2+12τm2K1+12τm2K2Φ88i=+12τM-τm2K3+12τM-τm2K4-N-NT,then the reachable sets of system (1) are bounded by a ball B(0,r)={z∈Rn∣z≤r} with(37)r=1λmin⁡(P).Proof.
ReplacingA,B,D with ∑i=1N‍θi(t)(A+Ai), ∑i=1N‍θi(t)(B+Bi), and ∑i=1N‍θi(t)(D+Di) in proof of Theorem 7, respectively, we easily get the conclusion.If the derivative of time delayμ is known, that is, the case τm≤τ(t)≤τM,τ˙(t)≤μ, bounding for reachable set of uncertain system (1) is obtained in Theorem 10.Theorem 10.
If there exist matricesP>0, Q1>0, Q2>0, R1>0, R2>0, K1>0, K2>0, K3>0, K4>0, and M>0, S, N with appropriate dimensions, and a scalar α>0, satisfying the following inequalities for all i=1,2,…,N,(38)R2+K3SSTR2+K4≥0,(39)Φi=Φ11iP(D+Di)Φ13i0Φ15i00Φ18iP(B+Bi)⋆Φ22iΦ23iΦ24i0Φ26iΦ27i(D+Di)TNT0⋆⋆Φ33iΦ34iΦ35iΦ36i000⋆⋆⋆Φ44i00Φ47i00⋆⋆⋆⋆Φ55i0000⋆⋆⋆⋆⋆Φ66i000⋆⋆⋆⋆⋆⋆Φ77i00⋆⋆⋆⋆⋆⋆⋆Φ88iN(B+Bi)⋆⋆⋆⋆⋆⋆⋆⋆-αwm2I≤0,where (40)Φ11=αP+P(A+Ai)+A+AiTP+Q1-2e-ατmK1Φ11=-e-ατmR1+M,Φ13=-e-ατmR1,Φ15=2e-ατmK1,Φ18=(A+Ai)TNT,Φ22=-2e-ατMK3-2e-ατMK4-2e-ατMR2Φ88=+e-ατM(ST+S)-(1-μ)e-ατMM,Φ23=e-ατMR2-e-ατMST,Φ24=e-ατMR2-e-ατMS,Φ26=-2e-ατMK4,Φ27=-2e-ατMK3,Φ33=-e-ατmQ1+e-ατmQ2-e-ατmR1-2e-ατmK2Φ88=-2e-ατMK3-e-ατMR2,Φ34=e-ατMS,Φ35=2e-ατmK2,Φ36=2e-ατMK3,Φ44=-e-ατMQ2-2e-ατMK4-e-ατMR2,Φ47=2e-ατMK4,Φ55=-2e-ατmK1-2e-ατmK2,Φ66=-2e-ατMK3-2e-ατMK4,Φ77=-2e-ατMK3-2e-ατMK4,Φ88=τm2R1+τM-τm2R2+12τm2K1+12τm2K2Φ88=+12τM-τm2K3+12τM-τm2K4-N-NT,then the reachable sets of system (1) are bounded by a ball B(0,r)={z∈Rn∣z≤r} with(41)r=1λmin⁡(P).Proof.
ReplacingA, B, D with ∑i=1N‍θi(t)(A+Ai), ∑i=1N‍θi(t)(B+Bi), and ∑i=1N‍θi(t)(D+Di) in Theorem 8, respectively, one can easily obtain the conclusion.Remark 11.
In [7, 8, 12], the constraint of τ(t) was 0≤τ(t)≤τ. τm≤τ(t)≤τM is considered in this paper; it is more general than [7, 8, 12].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 13.
In order to guarantee negative definite,μ is required that μ<1 in [11]. It should be noted that the value of derivative of time delay is not necessarily less than 1 in Theorems 8 and 10 since the term Φ22=-e-ατM(K3T+K3  −  e-ατM(K4T+K4)-e-ατM(R2T+R2)+e-ατM(ST+S)-(1-μ)e-ατMM can be negative definite by choosing appropriate K3,K4,R2,S,M when μ>1. Obviously, the results in this paper have more scope of application than the one in [11].Remark 14.
The reachable set of system (1) can be minimized by solving the following optimization problem for a scalar δ>0:(42)min⁡δ¯(δ¯=1δ)s.t.(a)P≥δI,b34-35or38-39.Remark 15.
It is clear to see that radiusr is smallest if δ=min⁡i=1,2,…,N is largest in inequalities (35) and (39). Therefore, one can consider Pi (i=1,2,…,N) as decision variable with the additional requirement (42).Remark 16.
It should be noted that the matrix inequalities in Theorems7–10 cannot be simplified to LMIs. However, when α is fixed, then the matrix inequalities reduce to LMIs. 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].


## 4. 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:(43)A+A1=-200-0.7,A+A2=-200-1.1,D+D1=-10-1-0.9,D+D2=-10-1-1.1,B+B1=-0.51=B+B2,wT(t)w(t)≤1.By solving optimization problems (42), computed δ¯’s for the case τm≤τ(t)≤τM with different values of τM are listed in Table 1. Computed δ¯'s for different values of μ with τM=0.7 and τM=0.75 for the case τm≤τ(t)≤τM,τ˙(t)≤μ 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].Table 1
Computedδ-’s of Example 1 for the case τm≤τ(t)≤τM with τm=0.
Method
τ
M
=
0.7
τ
M
=
0.75
[6]
19.71
65.42
[8]
5.38
13.44
Theorem9
3.91
6.58Table 2
Computedδ-’s  of Example 1 for the case τm≤τ(t)≤τM,τ˙(t)≤μ with τm=0,τM=0.7.
μ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.9
2
[11]
2.97
3.30
3.85
4.85
6.93
12.84
53.86
—
—
[8]
1.89
1.94
2.00
2.08
2.19
2.35
2.60
3.51
5.30
Theorem10
1.79
1.84
1.90
1.95
1.96
1.98
2.06
3.09
3.23Table 3
Computedδ-’s  of Example 1 for the case τm≤τ(t)≤τM,τ˙(t)≤μ with τm=0,τM=0.75.
μ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.9
2
[11]
3.34
3.79
4.53
5.88
8.85
18.36
127.70
—
—
[8]
2.28
2.35
2.45
2.57
2.68
2.85
4.62
5.57
13.39
Theorem10
2.03
2.33
2.34
2.45
2.60
2.72
4.37
4.92
8.73Example 2.
Consider the following uncertain time-delayed system with parameters(44)A+A1=0-0.09361-0.685,A+A2=0-0.14641-0.245,D+D1=-0.1-0.3500.3=D+D2,B+B1=-11=B+B2,μ=0,τm=0,τM=0.1,wTtwt≤1.
Computedδ¯’s for the case τm≤τ(t)≤τM, τ˙(t)≤μ with τm=0, τM=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.Table 4
Computedδ-’s of Example 2 for the case τm≤τ(t)≤τM,τ˙(t)≤μ with τm=0,τM=0.1.
Method
[6]
[11]
[8]
Theorem10
δ
-
—
—
2.8686
×
1
0
4
11.96Example 3.
Consider the following uncertain time-delayed system with parameters(45)A+A1=-200-0.7,A+A2=-200-1.1,D+D1=-10-1-0.9,D+D2=-10-1-1.1,B+B1=-0.51=B+B2,wT(t)w(t)≤1,and time-varying delay τ(t) is considered in two cases:(1)
τ(t) is nondifferentiable, 0≤τ(t)≤0.7;(2)
τ(t) is nondifferentiable, 0.1≤τ(t)≤0.7.From Theorem9, computed radiuses r’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.Table 5
Computed radiusesr’s of Example 3.
Method
[6]
[11]
[8]
[10]
Theorem9
Case (1)
19.71
—
5.3
1.2
1.17
Case (2)
19.71
—
5.3
1.1
0.87Example 4.
Consider the following uncertain time-delayed system with parameters:(46)z˙t=-200-0.7z(t)+-10-1-0.9z(t-τ(t))+-0.51w(t),and wT(t)w(t)≤1.By employing the method of Theorem8 in this paper, r’s for different values of τ(t) with μ=0 are listed in Table 6. It is easy to see that bounds obtained in this paper are better than the ones of literatures [8–12, 14].Table 6
Computedr’s of Example 4 for different values of τ with μ=0.
τ
0.1
0.3
0.5
0.7
0.9
[11]
0.83
1.28
1.94
2.90
4.46
[10]
0.74
0.92
1.36
2.30
3.51
[12]
0.68
0.80
0.97
1.64
3.22
[8]
0.66
0.75
0.94
1.61
3.14
[9]
0.66
0.75
0.94
1.61
3.14
[14]
0.66
0.75
0.94
1.61
3.14
Theorem8
0.41
0.71
0.92
1.25
2.74


## 5. Conclusions

In this paper, sufficient conditions for the existence of a ball that binds the reachable set of linear uncertain polytopic systems with disturbance and time-varying delays have been proposed in terms of Lyapunov-Krasovskii functional theory, delay decomposition technique, and reciprocally convex method. The proposed Lyapunov-Krasovskii functionals 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.



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