# New Stability Analysis Results for Linear System with Two Additive Time-Varying Delay Components

Authors: Haitao Liu; Fang Liu
Journal: Complexity
 (2020)

---

## Abstract

This paper focuses on the stability problems of continuous linear systems with two additive time-varying delay components. Firstly, an effective and simple Lyapunov–Krasovskii functional (LKF) is established, which not only takes adequate information of delay components and their upper bounds into consideration but also establishes a simpler form to decrease the computational complexity. Secondly, an improved delay-dependent stability criterion together with its corollary is obtained by employing the generalized free-weighting-matrix-based (GFWM-based) inequality and some other techniques to calculate the derivative of the constructed LKF, which will further reduce the introduced estimation error and make the criteria less conservative. Lastly, a numerical example is presented to illustrate the less conservatism and lower computational complexity of the derived results.


---

## Body

## 1. Introduction

It is a phenomenon that time delay often occurs in various dynamic systems such as economics, network transportation, circuit signal system, engineering systems, and so on [1–3]. The existence of the time-varying delay has a negative influence on system performance, which will have an adverse effect on the dynamic performance of the system and even make it instable. Moreover, for the remote control systems or networked control systems, signals transmitted between different points may pass different segments of networks, in which there may exist time delays of different characteristics. For instance, the signal transmitted from the sensor to the controller and the other from the controller to the actuator have different properties for the reason of variable transmission conditions [4]. Therefore, it is necessary to investigate the stability problem of systems with additive time delays and assess the effects from different delays with various properties, which motivates the research about linear system with two additive time delay components in this paper.Since the distribution characteristic of delays mentioned above has been modeled firstly as a system with two additive time-varying delay components in [4], many significant results have been devoted to this issue. As a most general form, linear systems with two additive delay components have drawn lots of attention [5–15]. Moreover, neural networks [16–21], T-S fuzzy systems [22, 23], and Lur’e systems [24], considering two additive delay components in the state, have been also investigated recently. Among these works, two main techniques that contain constructing an appropriate LKF [25, 26] and estimating its derivative [27–32] are usually used to reduce conservatism of the derived criteria. Additionally, reciprocally convex combination lemma [33] and its extensive form [34] also contribute to analyzing the time-delay system stability [15]. Some criteria with less conservatism are obtained in [8] by taking the delay, its upper bound, and the relationship between them into account. A novel LKF [9] is constructed, for which positive definiteness is ensured by the whole LKF rather than each term. In [10], a delay-partitioning-based LKF was established, which greatly reduces the conservatism. In [11], a novel LKF containing delay-product type terms [12, 13] is constructed; meanwhile, both the Wirtinger-based inequality [28] and reciprocally convex combination technique [33] are utilized to estimate the derivative of LKF. Recently, a delay interconnection LKF and its improved form are constructed in [14, 15], respectively; meanwhile, the free-matrix-based inequality [30] is utilized to calculate the single integral terms in their derivatives, which is demonstrated to be effective in reducing the conservatism.For the sake of further study, the relevant effort is to decrease the conservatism and computational complexity of the derived result from the viewpoint of constructing an appropriate LKF and estimating its derivative. Although great development has been devoted to the issue, there still exists room for further study to take both the conservatism and the computational complexity into account. For example, in [14, 15], delay interconnection LKF and its improved form are verified for their effectiveness for reducing the conservatism of the derived results; however, the reduction in conservatism is based on the increase in computational complexity. Therefore, it is a challenging and meaningful problem to ensure the low conservativeness of the stability criteria and reduce the computational complexity of the result as possible. This paper will study this issue from two aspects, one is to construct an effective and simple LKF which not only considers adequate information of delay components and their upper bounds but also establishes a simpler form in comparison with the existing works to decrease the computational complexity. In addition, the GFWM-based inequality [35] reduces the estimation error of the single integral terms appearing in the derivative of the proposed LKF, which has not been utilized for analyzing stability of the system with additive delay components. Hence, improved results can be derived of less conservatism and lower computational complexity by combining two techniques mentioned above, which motivates the following research.This paper analyzes delay-dependent stability of the linear systems with two additive time delay components, mainly aiming at considering the conservatism and computational complexity of the results comprehensively. The main contributions are summarized as follows:(i)
To simplify the analysis process and derive a stability criterion with less computational complexity, an effective and simple LKF containing adequate information about delay components and their upper bounds is constructed, in which the useless terms that contribute nothing to reducing the introduced conservatism are not included.(ii)
GFWM-based inequality together with some other advanced techniques is utilized to estimate the single integral terms appearing in the derivative of the proposed LKF, which further decreases the estimation error and drives less conservative stability criteria. In addition, any vectors in their simplest form are chosen to reduce the computational complexity while using GFWM-based inequality to estimate the single integral terms, which will further reduce the computational complexity.(iii)
With the techniques mentioned above employed in this paper, improved results with less conservatism and lower computational complexity can be presented, whose validity and superiority can be demonstrated with a representative numerical example in comparison with the works in [4, 6–9, 14, 15].This paper is organized as follows. Problem formulation and preliminary are given in Section2. In Section 3, an improved stability criterion and its corollary are developed. A numerical example is used to demonstrate the validity and superiority of the proposed criteria in Section 4. At last, Section 5 presents the conclusion.Notations: the notationℜn denotes the n-dimensional Euclidean space; P>0≥0 means that P is a real symmetric and positive definite (semipositive definite) matrix; I and 0 represent an appropriately dimensioned identify matrix and zero matrix, respectively; ∗ stands for the symmetric term in the symmetric matrix; the transpose and the inverse of a matrix are denoted by the superscripts T and −1, respectively; and SymX=X+XT.


## 2. Problem Formulation and Preliminary

Consider the following continuous linear system with two additive time-varying delay components:(1)x˙t=Axt+Adxt−d1t−d2t,∀t∈−h,0,xt=ϕt,where xt=x1tx2t⋯xntT∈ℜn is the state vector; A∈ℜn×n and Ad∈ℜn×n are known constant matrices; ϕt is the initial condition; d1t and d2t are continuous and differential functions, which stand for two time delays and satisfy the following conditions:(2)0≤d1t≤h1,d˙1t≤μ1,0≤d2t≤h2,d˙2t≤μ2,where hi,i=1,2 and μi,i=1,2 are constants. Let h=h1+h2, μ=μ1+μ2, dt=d1t+d2t.The necessary lemmas are introduced as follows to develop the results in Section3.Lemma 1.
(GFWM-based inequality [35]). For symmetric positive definite matrix R∈ℜn×n, any matrices L, M, and an arbitrary vector ω:a,b↦ℜn. Then, the following inequality holds:(3)∫abωTsRωsds≥−Symχ¯0TLχ¯1+χ¯0TMχ¯2−b−aχ¯0TLR−1LT+13MR−1MTχ¯0,where χ¯0 is an arbitrary vector and χ¯1=∫abωsds, χ¯2=−χ¯1+2/b−a∫ab∫sbωududs.Lemma 2.
(Jensen’s inequality [27]). For any symmetric positive definite matrix R∈ℜn×n, scalars a and b: a<b, and vector x:a,b↦ℜn, the following inequality holds:(4)b−a22∫ab∫θbxTsRxsdsdθ≥∫ab∫θbxsdsdθTR∫ab∫θbxsdsdθ.Lemma 3.
(Wirtinger-based inequality [28]). For any symmetric positive definite matrix R∈ℜn×n, scalars a and b: a<b, and vector x:a,b↦ℜn, the following inequality holds:(5)∫abx˙TsRx˙sds≥1b−aχ¯1χ¯2TR003Rχ¯1χ¯2,where χ¯1 and χ¯2 are defined in Lemma 1.Lemma 4.
(extended reciprocally convex inequality [34]). For a real scalar α∈0,1, symmetric matrices X1>0 and X2>0, and any matrices S1 and S2, the following matrix inequality holds:(6)1αX10011−αX2≥X1+1−αT11−αS1+αS2∗X2+αT2,where T1=X1−S2X2−1S2T and T2=X2−S1TX1−1S1.


## 3. Results and Discussion

In this section, by constructing a novel LKF and using GFWM-based inequality together with some other advanced techniques to calculate its derivative, a stability criterion and its corollary are derived for system (1). Meanwhile, some discussions are provided to illustrate the superiority of the methods used in this paper.


### 3.1. A Stability Criterion with Less Conservatism and Lower Computational Complexity

A novel LKF together with the aforementioned lemmas is used to derive the following criterion.Theorem 1.
For given scalarsh1, h2, μ1, and μ2, system (1) with two additive time-varying delay components satisfying (2) is asymptotically stable if there exist symmetric positive definite matrices P∈ℜ2n×2n, Qi∈ℜn×n, (i = 1, 2, 3, 4), Ri∈ℜn×n, (i = 1, 2, 3), Z,Za∈ℜn×n and any matrices Lk,Mk∈ℜ5n×n, (k = 1, 2, 3, 4, 5, 6), S1,S2∈ℜ2n×2n such that the following matrix inequalities hold:(7)Σ0,0+Ω0,0<0,Σ0,h2+Ω0,h2<0,Σh1,0+Ωh1,0<0,Σh1,h2+Ωh1,h2<0,where(8)Σd1t,d2t=Ξ1d1t,d2t+Ξ2+Ξ31+Ξ41+Ξ42d1t,d2t,Ωd1t,d2t=Ξ32d1t,d2t+Ξ43d1t,d2t,Ξ1d1t,d2t=SymΠ1TPΠ2,Ξ2=e1TQ1+Q2+Q3+Q4e1−e4TQ4e4−1−μ1e5TQ1e5−1−μ2e6TQ2e6−1−μe7TQ3e7,Ξ31=Π3Th1R1+h2R2+hR3Π3+Symeg1TL1Π4+eg1TM1Π5+Symeg1TL2Π6+eg1TM2Π7+Symeg2TL3Π8+eg2TM3Π9+Symeg2TL4Π10+eg2TM4Π11+Symeg3TL5Π12+eg3TM5Π13+Symeg3TL6Π14+eg3TM6Π15,Ξ32d1t,d2t=h1−d1teg1TL1R1−1L1T+13M1R1−1M1Teg1+d1teg1TL2R1−1L2T+13M2R1−1M2Teg1+h2−d2teg2TL3R2−1L3T+13M3R2−1M3Teg2+d2teg2TL4R2−1L4T+13M4R2−1M4Teg2+h−dteg3TL5R3−1L5T+13M5R3−1M5Teg3+dteg3TL6R3−1L6T+13M6R3−1M6Teg3,Ξ41=Π3T12h2Z+h2ZaΠ3−2Π16TZΠ16−2Π17TZΠ17−Π18TZ˜aΠ18−Π19TZ˜aΠ19,Ξ42d1t,d2t=−h−dthΠ18TZ˜+Z˜aΠ18−h−dthSymΠ18TS1Π19−dthΠ19TZ˜aΠ19−dthSymΠ18TS2Π19,Ξ43d1t,d2t=h−dthΠ18TS2Z˜a−1S2TΠ18+dthΠ19TS1TZ˜+Z˜a−1S1Π19,Π1=e1Tdte12T+h−dte13TT,Π2=e1TAT+e7TAdTe1T−e4TT,Π3=Ae1+Ade7,Π4=e5−e2,Π5=e2+e5−2e9,Π6=e1−e5,Π7=e1+e5−2e8,Π8=e6−e3,Π9=e3+e6−2e11,Π10=e1−e6,Π11=e1+e6−2e10,Π12=e7−e4,Π13=e4+e7−2e13,Π14=e1−e7,Π15=e1+e7−2e12,Π16=e7−e13,Π17=e1−e12,Π18=e1T−e7Te1T+e7T−2e12TT,Π19=e7T−e4Te4T+e7T−2e13TT,ei=0n×i−1nIn0n×13−in,eg1=e1Te2Te5Te8Te9TT,eg2=e1Te3Te6Te10Te11TT,eg3=e1Te4Te7Te12Te13TT,Z˜=Z0∗3Z,Z˜a=Za0∗3Za.Proof.
A vectorξt∈ℜ13n is defined as follows:(9)ξt=xTtxTt−h1xTt−h2xTt−hxTt−d1txTt−d2txTt−dt1d1t∫t−d1ttxTsds1h1−d1t∫t−h1t−d1txTsds1d2t∫t−d2ttxTsds1h2−d2t∫t−h2t−d2txTsds1dt∫t−dttxTsds1h−dt∫t−ht−dtxTsdsT.
A novel LKF containing adequate information about delay components and their derivatives with a simper form is constructed as follows:(10)Vxt=V1xt+V2xt+V3xt+V4xt,where(11)V1xt=ςTtPςt,V2xt=∫t−d1ttxTsQ1xsds+∫t−d2ttxTsQ2xsds+∫t−dttxTsQ3xsds+∫t−htxTsQ4xsds,V3xt=∫−h10∫t+θtx˙TsR1x˙sdsdθ+∫−h20∫t+θtx˙TsR2x˙sdsdθ+∫−h0∫t+θtx˙TsR3x˙sdsdθ,V4xt=∫t−ht∫θt∫μtx˙TsZx˙sdsdμdθ+h∫−h0∫t+θtx˙TsZax˙sdsdθ,with(12)ςt=xTt∫t−htxTsdsT.
Firstly, the matrices declared in Theorem1 including P,Qii=1,2,3,4,Rii=1,2,3,Z,Za are positive definite, which insures the positive definitiveness of the proposed LKF. Therefore, Vxt≥ε1xt holds for a sufficiently small scalar ε1>0.
Secondly, the derivative of the proposed LKF with respect to time along the trajectory of system (1) is calculated as follows:(13)V˙xt=V˙1xt+V˙2xt+V˙3xt+V˙4xt.
Calculating the derivatives ofV1xt, V2xt, V3xt, V4xt yields(14)V˙1xt=2ςTtPς˙t=ξTtΠ1TPΠ2+Π2TPΠ1ξt=ξTtSymΠ1TPΠ2ξt=ξTtΞ1d1t,d2tξt,(15)V˙2xt=ξTte1TQ1+Q2+Q3+Q4e1−e4TQ4e4−1−d˙1te5TQ1e5−1−d˙2te6TQ2e6−1−d˙te7TQ3e7ξt≤ξTte1TQ1+Q2+Q3+Q4e1−e4TQ4e4−1−μ1e5TQ1e5−1−μ2e6TQ2e6−1−μe7TQ3e7ξt,=ξTtΞ2ξt,(16)V˙3xt=ξTtΠ3Th1R1+h2R2+hR3Π3ξt−∫t−h1tx˙TsR1x˙sds−∫t−h2tx˙TsR2x˙sds−∫t−htx˙TsR3x˙sds.
Applying Lemma1 to estimate the R1-dependent single integral term in equation (16) yields(17)−∫t−h1tx˙TsR1x˙sds=−∫t−h1t−d1tx˙TsR1x˙sds−∫t−d1ttx˙TsR1x˙sds≤Symη1TtL1χ1+η1TtM1χ2+h1−d1tη1TtL1R1−1L1T+13M1R1−1M1Tη1t+Symη1TtL2χ3+η1TtM2χ4+d1tη1TtL2R1−1L2T+13M2R1−1M2Tη1t,where η1t is selected as(18)η1t=xTtxTt−h1xTt−d1t1d1t∫t−d1ttxTsds1h1−d1t∫t−h1t−d1txTsdsT=e1Te2Te5Te8Te9TTξt=eg1ξt,and the other vectors are derived based on Lemma 1:(19)χ1t=xt−d1t−xt−h1=Π4ξt,χ2t=xt−h1+xt−d1t−2h1−d1t∫t−h1t−d1txsds=Π5ξt,χ3t=xt−xt−d1t=Π6ξt,χ4t=xt+xt−d1t−2d1t∫t−d1ttxsds=Π7ξt,which implies(20)−∫t−h1tx˙TsR1x˙sds≤ξTtSymeg1TL1Π4+eg1TM1Π5+Symeg1TL2Π6+eg1TM2Π7+h1−d1teg1TL1R1−1L1T+13M1R1−1M1Teg1+d1teg1TL2R1−1L2T+13M2R1−1M2Teg1ξt.
Similarly, theR2-dependent single integral term in equation (16) can be estimated via Lemma 1:(21)−∫t−h2tx˙TsR2x˙sds=−∫t−h2t−d2tx˙TsR2x˙sds−∫t−d2ttx˙TsR2x˙sds≤Symη2TtL3χ5+η2TtM3χ6+h2−d2tη2TtL3R2−1L3T+13M3R2−1M3Tη2t+Symη2TtL4χ7+η2TtM4χ8+d2tη2TtL4R2−1L4T+13M4R2−1M4Tη2t,where η2t is selected as(22)η2t=xTtxTt−h2xTt−d2t1d2t∫t−d2ttxTsds1h2−d2t∫t−h2t−d2txTsdsT=e1Te3Te6Te10Te11TTξt=eg2ξt,(23)χ5=xt−d2t−xt−h2=Π8ξt,χ6=xt−h2+xt−d2t−2h2−d2t∫t−h2t−d2txsds=Π9ξt,χ7=xt−xt−d2t=Π10ξt,χ8=xt+xt−d2t−2d2t∫t−d2ttxsds=Π11ξt.
Thus, inequality (21) can be rewritten as(24)−∫t−h2tx˙TsR2x˙sds≤ξTtSymeg2TL3Π8+eg2TM3Π9+Symeg2TL4Π10+eg2TM4Π11+h2−d2teg2TL3R2−1L3T+13M3R2−1M3Teg2+d2teg2TL4R2−1L4T+13M4R2−1M4Teg2ξt.
TheR3-dependent single integral term in equation (16) is also calculated with Lemma 1, which yields(25)−∫t−htx˙TsR3x˙sds=−∫t−ht−dtx˙TsR3x˙sds−∫t−dttx˙TsR3x˙sds≤Symη3TtL5χ9+η3TtM5χ10+h−dtη3TtL5R3−1L5T+13M5R3−1M5Tη3t+Symη3TtL6χ11+η3TtM6χ12+dtη3TtL6R3−1L6T+13M6R3−1M6Tη3t,where η3t is selected as(26)η3t=xTtxTt−hxTt−dt1dt∫t−dttxTsds1h−dt∫t−ht−dtxTsdsT=e1Te4Te7Te12Te13TTξt=eg3ξt,(27)χ9=xt−dt−xt−h=Π12ξt,χ10=xt−h+xt−dt−2h−dt∫t−ht−dtxsds=Π13ξt,χ11=xt−xt−dt=Π14ξt,χ12=xt+xt−dt−2dt∫t−dttxsds=Π15ξt.
Therefore, inequality (25) can be rewritten as(28)−∫t−htx˙TsR3x˙sds≤ξTtSymeg3TL5Π12+eg3TM5Π13+Symeg3TL6Π14+eg3TM6Π15+h−dteg3TL5R3−1L5T+13M5R3−1M5Teg3+dteg3TL6R3−1L6T+13M6R3−1M6Teg3ξt.
Combining inequalities (16), (20), (24), and (28), the estimation of V˙3xt can be derived as follows:(29)V˙3xt≤ξTtΞ31+Ξ32d1t,d2t,where Ξ31 and Ξ32d1t,d2t are defined in equation (8).(30)V˙4xt=12h2x˙TtZx˙t+h2x˙TtZax˙t−∫t−ht∫θtx˙TsZx˙sdsdθ−h∫t−htx˙TsZax˙sds=ξTtΠ3T12h2Z+h2ZaΠ3ξt−∫t−ht−dt∫θt−dtx˙TsZx˙sdsdθ−∫t−dtt∫θtx˙TsZx˙sdsdθ−h−dt∫t−dttx˙TsZx˙s−h∫t−htx˙TsZax˙sds.
The double integral terms in equation (30) can be estimated via Lemma 2, which yields(31)−∫t−ht−dt∫θt−dtx˙TsZx˙sdsdθ−∫t−dtt∫θtx˙TsZx˙sdsdθ≤−2h−dt∫t−ht−dt∫θt−dtx˙sdsdθTZ∫t−ht−dt∫θt−dtx˙sdsdθ−2dt∫t−dtt∫θtx˙sdsdθTZ∫t−dtt∫θtx˙sdsdθ=−2Π16TZΠ16−2Π17TZΠ17.
The single integral terms in equation (30) is estimated via Lemma 3, which yields(32)−h−dt∫t−dttx˙TsZx˙sds−h∫t−htx˙TsZax˙sds=−h−dt∫t−dttx˙TsZx˙sds−h∫t−ht−dtx˙TsZax˙sds−h∫t−dttx˙TsZax˙sds≤−hdt−1ξTtΠ18TZ˜Π18ξt−hh−dtξTtΠ19TZ˜aΠ19ξt−hdtξTtΠ18TZ˜aΠ18ξt=ξTtΠ18TZ˜Π18ξt−hh−dtξTtΠ19TZ˜aΠ19ξt−hdtξTtΠ18TZ˜+Z˜aΠ18ξt,where Z˜ and Z˜a are defined in equation (8), and the dt-dependent terms in equation (32) can be calculated with Lemma 4.(33)−hdtΠ18TZ˜+Z˜aΠ18−hh−dtΠ19TZ˜aΠ19=−Π18Π19ThdtZ˜+Z˜a0∗hh−dtZ˜aΠ18Π19≤−Π18TZ˜+Z˜aΠ18−h−dthΠ18TZ˜+Z˜aΠ18+h−dthΠ18TS2Z˜a−1S2TΠ18−h−dthSymΠ18TS1Π19−dthSymΠ18TS2Π19−Π19TZ˜aΠ19−dthΠ19TZ˜aΠ19+dthΠ19TS1TZ˜+Z˜a−1S1Π19.
Taking equations (30)∼(33) into consideration comprehensively, the upper bound of V˙4xt can be obtained as follows:(34)V˙4xt≤ξTtΞ41+Ξ42d1t,d2t+Ξ43d1t,d2tξt.
Taking equations (13)∼(15), (29), and (34) into account, the upper bound of V˙xt can be obtained as follows:(35)V˙xt≤ξTtΣd1t,d2t+Ωd1t,d2tξt.
Therefore, the derivative of the proposed LKF is negative definite if inequalityΣd1t,d2t+Ωd1t,d2t<0 holds for all d1t∈0,h1 and d2t∈0,h2, which is equivalent to the inequalities in equation (7) based on the convex combination method [3].
From what has been mentioned above, the positive definiteness of the matrices declared in Theorem1 and the inequalities in (7) ensure Vxt≥ε1xt and V˙xt≤−ε2xt, respectively, for any existing sufficient small scalars ε1,ε2>0, which implies system (1) with two time-varying delay components satisfying (2) is asymptotically stable. This completes the proof.


### 3.2. Discussion

By constructing a novel LKF and using GFWM-based inequality to estimate its derivative, Theorem1 is derived with less conservatism and lower computational complexity. Moreover, its superiority can be reflected in the following discussion.Firstly, among the conditions of Theorem1, four matrix inequalities included in equation (7) are not linear matrix inequalities (LMIs); hence, LMI Toolbox in MATLAB cannot solve these inequalities directly on account of the existence of the inverse matrices in Ωd1t,d2t. Nevertheless, based on Schur complement, the matrix inequalities in equation (7) can be transformed into LMIs. Therefore, the stability problem of system (1) is equivalent to the feasibility checking problem of LMIs.Secondly, some advanced techniques are applied to derive Theorem1, which will be conducive to reducing its conservativeness and computational complexity.(i)
The LKF constructed in this paper plays an important role to decrease the computational complexity. It not only contains adequate information about delay components and their upper bounds but also has a simpler form in comparison with the existing works. It is obvious that the nonintegral terms, quadratic single integral terms, and quadratic double integral terms in the LKF constructed in this paper have been simplified compared with the latest works [14, 15], and the useless terms that contribute nothing to reducing the introduced conservatism are not included. The simplification of LKF will undoubtedly greatly reduce the computational complexity of Theorem 1; moreover, it ensures the low conservatism of Theorem 1.(ii)
GFWM-based inequality is used to estimate the single integral term inV˙3xt, which further decreases the conservatism and the computational complexity of Theorem 1. The superiority of GFWM-based inequality in comparison with FMBII utilized in [14, 15] is verified theoretically in [35]. Moreover, in the proof process of Theorem 1 in this paper, three arbitrary vectors η1t, η2t, and η3t defined in equations (18), (22), and (26) are chosen to estimate the R1-, R2-, and R3-dependent single integral terms in V˙3xt, respectively, which are verified to be the simplest form to minimize the conservatism introduced in equations (20), (24), and (28). In other words, adding additional terms to η1t, η2t, and η3t will not contribute to the reduction of conservatism. Hence, both the conservatism and the computational complexity will be further reduced with the techniques used in estimating the derivative of the LKF.Last but not the least, both the conservatism and the computational complexity are considered in this paper, while the only objective is to decrease the conservatism in most existing works. The effect of LKF components is fully investigated; thus, an effective and simple LKF constructed in this paper will greatly reduce the decision variables without adversely affecting the conservatism introduced. Moreover, GFWM-based inequality is utilized to deal with three important single integral terms appearing inV˙3xt, which reduces both the introduced conservatism and decision variables. Therefore, the method proposed in this paper contributes to analyzing stability of system (1) with delay components satisfying condition (2).Triple integral term in LKF is verified beneficial to analyzing stability of the systems with additive delays. Therefore, to further illustrate the effect of the GFWM-based inequality used in this paper, the following corollary is derived from Theorem1 directly without considering V4xt in LKF.Corollary 1.
For given scalarsh1, h2, μ1, and μ2, system (1) with two additive time-varying delay components satisfying (2) is asymptotically stable if there exist symmetric positive definite matrices P∈ℜ2n×2n, Qi∈ℜn×n, (i = 1, 2, 3, 4), Ri∈ℜn×n, (i = 1, 2, 3), and any matrices Lk,Mk∈ℜ5n×n, (k = 1, 2, 3, 4, 5, 6), such that the following matrix inequalities hold:(36)Σc0,0+Ξ320,0<0,Σc0,h2+Ξ320,h2<0,Σch1,0+Ξ32h1,0<0,Σch1,h2+Ξ32h1,h2<0,where Σcd1t,d2t=Ξ1d1t,d2t+Ξ2+Ξ31, and the related notations are given in equation (8).


## 3.1. A Stability Criterion with Less Conservatism and Lower Computational Complexity

A novel LKF together with the aforementioned lemmas is used to derive the following criterion.Theorem 1.
For given scalarsh1, h2, μ1, and μ2, system (1) with two additive time-varying delay components satisfying (2) is asymptotically stable if there exist symmetric positive definite matrices P∈ℜ2n×2n, Qi∈ℜn×n, (i = 1, 2, 3, 4), Ri∈ℜn×n, (i = 1, 2, 3), Z,Za∈ℜn×n and any matrices Lk,Mk∈ℜ5n×n, (k = 1, 2, 3, 4, 5, 6), S1,S2∈ℜ2n×2n such that the following matrix inequalities hold:(7)Σ0,0+Ω0,0<0,Σ0,h2+Ω0,h2<0,Σh1,0+Ωh1,0<0,Σh1,h2+Ωh1,h2<0,where(8)Σd1t,d2t=Ξ1d1t,d2t+Ξ2+Ξ31+Ξ41+Ξ42d1t,d2t,Ωd1t,d2t=Ξ32d1t,d2t+Ξ43d1t,d2t,Ξ1d1t,d2t=SymΠ1TPΠ2,Ξ2=e1TQ1+Q2+Q3+Q4e1−e4TQ4e4−1−μ1e5TQ1e5−1−μ2e6TQ2e6−1−μe7TQ3e7,Ξ31=Π3Th1R1+h2R2+hR3Π3+Symeg1TL1Π4+eg1TM1Π5+Symeg1TL2Π6+eg1TM2Π7+Symeg2TL3Π8+eg2TM3Π9+Symeg2TL4Π10+eg2TM4Π11+Symeg3TL5Π12+eg3TM5Π13+Symeg3TL6Π14+eg3TM6Π15,Ξ32d1t,d2t=h1−d1teg1TL1R1−1L1T+13M1R1−1M1Teg1+d1teg1TL2R1−1L2T+13M2R1−1M2Teg1+h2−d2teg2TL3R2−1L3T+13M3R2−1M3Teg2+d2teg2TL4R2−1L4T+13M4R2−1M4Teg2+h−dteg3TL5R3−1L5T+13M5R3−1M5Teg3+dteg3TL6R3−1L6T+13M6R3−1M6Teg3,Ξ41=Π3T12h2Z+h2ZaΠ3−2Π16TZΠ16−2Π17TZΠ17−Π18TZ˜aΠ18−Π19TZ˜aΠ19,Ξ42d1t,d2t=−h−dthΠ18TZ˜+Z˜aΠ18−h−dthSymΠ18TS1Π19−dthΠ19TZ˜aΠ19−dthSymΠ18TS2Π19,Ξ43d1t,d2t=h−dthΠ18TS2Z˜a−1S2TΠ18+dthΠ19TS1TZ˜+Z˜a−1S1Π19,Π1=e1Tdte12T+h−dte13TT,Π2=e1TAT+e7TAdTe1T−e4TT,Π3=Ae1+Ade7,Π4=e5−e2,Π5=e2+e5−2e9,Π6=e1−e5,Π7=e1+e5−2e8,Π8=e6−e3,Π9=e3+e6−2e11,Π10=e1−e6,Π11=e1+e6−2e10,Π12=e7−e4,Π13=e4+e7−2e13,Π14=e1−e7,Π15=e1+e7−2e12,Π16=e7−e13,Π17=e1−e12,Π18=e1T−e7Te1T+e7T−2e12TT,Π19=e7T−e4Te4T+e7T−2e13TT,ei=0n×i−1nIn0n×13−in,eg1=e1Te2Te5Te8Te9TT,eg2=e1Te3Te6Te10Te11TT,eg3=e1Te4Te7Te12Te13TT,Z˜=Z0∗3Z,Z˜a=Za0∗3Za.Proof.
A vectorξt∈ℜ13n is defined as follows:(9)ξt=xTtxTt−h1xTt−h2xTt−hxTt−d1txTt−d2txTt−dt1d1t∫t−d1ttxTsds1h1−d1t∫t−h1t−d1txTsds1d2t∫t−d2ttxTsds1h2−d2t∫t−h2t−d2txTsds1dt∫t−dttxTsds1h−dt∫t−ht−dtxTsdsT.
A novel LKF containing adequate information about delay components and their derivatives with a simper form is constructed as follows:(10)Vxt=V1xt+V2xt+V3xt+V4xt,where(11)V1xt=ςTtPςt,V2xt=∫t−d1ttxTsQ1xsds+∫t−d2ttxTsQ2xsds+∫t−dttxTsQ3xsds+∫t−htxTsQ4xsds,V3xt=∫−h10∫t+θtx˙TsR1x˙sdsdθ+∫−h20∫t+θtx˙TsR2x˙sdsdθ+∫−h0∫t+θtx˙TsR3x˙sdsdθ,V4xt=∫t−ht∫θt∫μtx˙TsZx˙sdsdμdθ+h∫−h0∫t+θtx˙TsZax˙sdsdθ,with(12)ςt=xTt∫t−htxTsdsT.
Firstly, the matrices declared in Theorem1 including P,Qii=1,2,3,4,Rii=1,2,3,Z,Za are positive definite, which insures the positive definitiveness of the proposed LKF. Therefore, Vxt≥ε1xt holds for a sufficiently small scalar ε1>0.
Secondly, the derivative of the proposed LKF with respect to time along the trajectory of system (1) is calculated as follows:(13)V˙xt=V˙1xt+V˙2xt+V˙3xt+V˙4xt.
Calculating the derivatives ofV1xt, V2xt, V3xt, V4xt yields(14)V˙1xt=2ςTtPς˙t=ξTtΠ1TPΠ2+Π2TPΠ1ξt=ξTtSymΠ1TPΠ2ξt=ξTtΞ1d1t,d2tξt,(15)V˙2xt=ξTte1TQ1+Q2+Q3+Q4e1−e4TQ4e4−1−d˙1te5TQ1e5−1−d˙2te6TQ2e6−1−d˙te7TQ3e7ξt≤ξTte1TQ1+Q2+Q3+Q4e1−e4TQ4e4−1−μ1e5TQ1e5−1−μ2e6TQ2e6−1−μe7TQ3e7ξt,=ξTtΞ2ξt,(16)V˙3xt=ξTtΠ3Th1R1+h2R2+hR3Π3ξt−∫t−h1tx˙TsR1x˙sds−∫t−h2tx˙TsR2x˙sds−∫t−htx˙TsR3x˙sds.
Applying Lemma1 to estimate the R1-dependent single integral term in equation (16) yields(17)−∫t−h1tx˙TsR1x˙sds=−∫t−h1t−d1tx˙TsR1x˙sds−∫t−d1ttx˙TsR1x˙sds≤Symη1TtL1χ1+η1TtM1χ2+h1−d1tη1TtL1R1−1L1T+13M1R1−1M1Tη1t+Symη1TtL2χ3+η1TtM2χ4+d1tη1TtL2R1−1L2T+13M2R1−1M2Tη1t,where η1t is selected as(18)η1t=xTtxTt−h1xTt−d1t1d1t∫t−d1ttxTsds1h1−d1t∫t−h1t−d1txTsdsT=e1Te2Te5Te8Te9TTξt=eg1ξt,and the other vectors are derived based on Lemma 1:(19)χ1t=xt−d1t−xt−h1=Π4ξt,χ2t=xt−h1+xt−d1t−2h1−d1t∫t−h1t−d1txsds=Π5ξt,χ3t=xt−xt−d1t=Π6ξt,χ4t=xt+xt−d1t−2d1t∫t−d1ttxsds=Π7ξt,which implies(20)−∫t−h1tx˙TsR1x˙sds≤ξTtSymeg1TL1Π4+eg1TM1Π5+Symeg1TL2Π6+eg1TM2Π7+h1−d1teg1TL1R1−1L1T+13M1R1−1M1Teg1+d1teg1TL2R1−1L2T+13M2R1−1M2Teg1ξt.
Similarly, theR2-dependent single integral term in equation (16) can be estimated via Lemma 1:(21)−∫t−h2tx˙TsR2x˙sds=−∫t−h2t−d2tx˙TsR2x˙sds−∫t−d2ttx˙TsR2x˙sds≤Symη2TtL3χ5+η2TtM3χ6+h2−d2tη2TtL3R2−1L3T+13M3R2−1M3Tη2t+Symη2TtL4χ7+η2TtM4χ8+d2tη2TtL4R2−1L4T+13M4R2−1M4Tη2t,where η2t is selected as(22)η2t=xTtxTt−h2xTt−d2t1d2t∫t−d2ttxTsds1h2−d2t∫t−h2t−d2txTsdsT=e1Te3Te6Te10Te11TTξt=eg2ξt,(23)χ5=xt−d2t−xt−h2=Π8ξt,χ6=xt−h2+xt−d2t−2h2−d2t∫t−h2t−d2txsds=Π9ξt,χ7=xt−xt−d2t=Π10ξt,χ8=xt+xt−d2t−2d2t∫t−d2ttxsds=Π11ξt.
Thus, inequality (21) can be rewritten as(24)−∫t−h2tx˙TsR2x˙sds≤ξTtSymeg2TL3Π8+eg2TM3Π9+Symeg2TL4Π10+eg2TM4Π11+h2−d2teg2TL3R2−1L3T+13M3R2−1M3Teg2+d2teg2TL4R2−1L4T+13M4R2−1M4Teg2ξt.
TheR3-dependent single integral term in equation (16) is also calculated with Lemma 1, which yields(25)−∫t−htx˙TsR3x˙sds=−∫t−ht−dtx˙TsR3x˙sds−∫t−dttx˙TsR3x˙sds≤Symη3TtL5χ9+η3TtM5χ10+h−dtη3TtL5R3−1L5T+13M5R3−1M5Tη3t+Symη3TtL6χ11+η3TtM6χ12+dtη3TtL6R3−1L6T+13M6R3−1M6Tη3t,where η3t is selected as(26)η3t=xTtxTt−hxTt−dt1dt∫t−dttxTsds1h−dt∫t−ht−dtxTsdsT=e1Te4Te7Te12Te13TTξt=eg3ξt,(27)χ9=xt−dt−xt−h=Π12ξt,χ10=xt−h+xt−dt−2h−dt∫t−ht−dtxsds=Π13ξt,χ11=xt−xt−dt=Π14ξt,χ12=xt+xt−dt−2dt∫t−dttxsds=Π15ξt.
Therefore, inequality (25) can be rewritten as(28)−∫t−htx˙TsR3x˙sds≤ξTtSymeg3TL5Π12+eg3TM5Π13+Symeg3TL6Π14+eg3TM6Π15+h−dteg3TL5R3−1L5T+13M5R3−1M5Teg3+dteg3TL6R3−1L6T+13M6R3−1M6Teg3ξt.
Combining inequalities (16), (20), (24), and (28), the estimation of V˙3xt can be derived as follows:(29)V˙3xt≤ξTtΞ31+Ξ32d1t,d2t,where Ξ31 and Ξ32d1t,d2t are defined in equation (8).(30)V˙4xt=12h2x˙TtZx˙t+h2x˙TtZax˙t−∫t−ht∫θtx˙TsZx˙sdsdθ−h∫t−htx˙TsZax˙sds=ξTtΠ3T12h2Z+h2ZaΠ3ξt−∫t−ht−dt∫θt−dtx˙TsZx˙sdsdθ−∫t−dtt∫θtx˙TsZx˙sdsdθ−h−dt∫t−dttx˙TsZx˙s−h∫t−htx˙TsZax˙sds.
The double integral terms in equation (30) can be estimated via Lemma 2, which yields(31)−∫t−ht−dt∫θt−dtx˙TsZx˙sdsdθ−∫t−dtt∫θtx˙TsZx˙sdsdθ≤−2h−dt∫t−ht−dt∫θt−dtx˙sdsdθTZ∫t−ht−dt∫θt−dtx˙sdsdθ−2dt∫t−dtt∫θtx˙sdsdθTZ∫t−dtt∫θtx˙sdsdθ=−2Π16TZΠ16−2Π17TZΠ17.
The single integral terms in equation (30) is estimated via Lemma 3, which yields(32)−h−dt∫t−dttx˙TsZx˙sds−h∫t−htx˙TsZax˙sds=−h−dt∫t−dttx˙TsZx˙sds−h∫t−ht−dtx˙TsZax˙sds−h∫t−dttx˙TsZax˙sds≤−hdt−1ξTtΠ18TZ˜Π18ξt−hh−dtξTtΠ19TZ˜aΠ19ξt−hdtξTtΠ18TZ˜aΠ18ξt=ξTtΠ18TZ˜Π18ξt−hh−dtξTtΠ19TZ˜aΠ19ξt−hdtξTtΠ18TZ˜+Z˜aΠ18ξt,where Z˜ and Z˜a are defined in equation (8), and the dt-dependent terms in equation (32) can be calculated with Lemma 4.(33)−hdtΠ18TZ˜+Z˜aΠ18−hh−dtΠ19TZ˜aΠ19=−Π18Π19ThdtZ˜+Z˜a0∗hh−dtZ˜aΠ18Π19≤−Π18TZ˜+Z˜aΠ18−h−dthΠ18TZ˜+Z˜aΠ18+h−dthΠ18TS2Z˜a−1S2TΠ18−h−dthSymΠ18TS1Π19−dthSymΠ18TS2Π19−Π19TZ˜aΠ19−dthΠ19TZ˜aΠ19+dthΠ19TS1TZ˜+Z˜a−1S1Π19.
Taking equations (30)∼(33) into consideration comprehensively, the upper bound of V˙4xt can be obtained as follows:(34)V˙4xt≤ξTtΞ41+Ξ42d1t,d2t+Ξ43d1t,d2tξt.
Taking equations (13)∼(15), (29), and (34) into account, the upper bound of V˙xt can be obtained as follows:(35)V˙xt≤ξTtΣd1t,d2t+Ωd1t,d2tξt.
Therefore, the derivative of the proposed LKF is negative definite if inequalityΣd1t,d2t+Ωd1t,d2t<0 holds for all d1t∈0,h1 and d2t∈0,h2, which is equivalent to the inequalities in equation (7) based on the convex combination method [3].
From what has been mentioned above, the positive definiteness of the matrices declared in Theorem1 and the inequalities in (7) ensure Vxt≥ε1xt and V˙xt≤−ε2xt, respectively, for any existing sufficient small scalars ε1,ε2>0, which implies system (1) with two time-varying delay components satisfying (2) is asymptotically stable. This completes the proof.


## 3.2. Discussion

By constructing a novel LKF and using GFWM-based inequality to estimate its derivative, Theorem1 is derived with less conservatism and lower computational complexity. Moreover, its superiority can be reflected in the following discussion.Firstly, among the conditions of Theorem1, four matrix inequalities included in equation (7) are not linear matrix inequalities (LMIs); hence, LMI Toolbox in MATLAB cannot solve these inequalities directly on account of the existence of the inverse matrices in Ωd1t,d2t. Nevertheless, based on Schur complement, the matrix inequalities in equation (7) can be transformed into LMIs. Therefore, the stability problem of system (1) is equivalent to the feasibility checking problem of LMIs.Secondly, some advanced techniques are applied to derive Theorem1, which will be conducive to reducing its conservativeness and computational complexity.(i)
The LKF constructed in this paper plays an important role to decrease the computational complexity. It not only contains adequate information about delay components and their upper bounds but also has a simpler form in comparison with the existing works. It is obvious that the nonintegral terms, quadratic single integral terms, and quadratic double integral terms in the LKF constructed in this paper have been simplified compared with the latest works [14, 15], and the useless terms that contribute nothing to reducing the introduced conservatism are not included. The simplification of LKF will undoubtedly greatly reduce the computational complexity of Theorem 1; moreover, it ensures the low conservatism of Theorem 1.(ii)
GFWM-based inequality is used to estimate the single integral term inV˙3xt, which further decreases the conservatism and the computational complexity of Theorem 1. The superiority of GFWM-based inequality in comparison with FMBII utilized in [14, 15] is verified theoretically in [35]. Moreover, in the proof process of Theorem 1 in this paper, three arbitrary vectors η1t, η2t, and η3t defined in equations (18), (22), and (26) are chosen to estimate the R1-, R2-, and R3-dependent single integral terms in V˙3xt, respectively, which are verified to be the simplest form to minimize the conservatism introduced in equations (20), (24), and (28). In other words, adding additional terms to η1t, η2t, and η3t will not contribute to the reduction of conservatism. Hence, both the conservatism and the computational complexity will be further reduced with the techniques used in estimating the derivative of the LKF.Last but not the least, both the conservatism and the computational complexity are considered in this paper, while the only objective is to decrease the conservatism in most existing works. The effect of LKF components is fully investigated; thus, an effective and simple LKF constructed in this paper will greatly reduce the decision variables without adversely affecting the conservatism introduced. Moreover, GFWM-based inequality is utilized to deal with three important single integral terms appearing inV˙3xt, which reduces both the introduced conservatism and decision variables. Therefore, the method proposed in this paper contributes to analyzing stability of system (1) with delay components satisfying condition (2).Triple integral term in LKF is verified beneficial to analyzing stability of the systems with additive delays. Therefore, to further illustrate the effect of the GFWM-based inequality used in this paper, the following corollary is derived from Theorem1 directly without considering V4xt in LKF.Corollary 1.
For given scalarsh1, h2, μ1, and μ2, system (1) with two additive time-varying delay components satisfying (2) is asymptotically stable if there exist symmetric positive definite matrices P∈ℜ2n×2n, Qi∈ℜn×n, (i = 1, 2, 3, 4), Ri∈ℜn×n, (i = 1, 2, 3), and any matrices Lk,Mk∈ℜ5n×n, (k = 1, 2, 3, 4, 5, 6), such that the following matrix inequalities hold:(36)Σc0,0+Ξ320,0<0,Σc0,h2+Ξ320,h2<0,Σch1,0+Ξ32h1,0<0,Σch1,h2+Ξ32h1,h2<0,where Σcd1t,d2t=Ξ1d1t,d2t+Ξ2+Ξ31, and the related notations are given in equation (8).


## 4. A Numerical Example

In this section, a typical numerical example is provided to illustrate both the less conservativeness and the lower computational complexity of the derived criteria in comparison with the existing works.Example 1.
Consider system (1) with the following parameters:(37)A=−200−0.9,Ad=−10−1−1.Suppose thatd˙1t≤0.1 and d˙2t≤0.8, which means μ1=0.1 and μ2=0.8.This is a representative number example to analyze the stability of system (1) satisfying condition (2), which is also used in [4, 6–9, 14, 15] to illustrate the conservatism of the derived criteria. As known, the stability criterion obtained with Lyapunov stability theory is usually a sufficient condition. Hence, the calculated upper bounds of delays based on different criteria are less than their analytical value, which represent the conservatism of criteria. In other words, the criterion with bigger calculated values of delays is less conservative.To verify the validity of the method presented in this paper, Tables1 and 2 are listed on behalf of two cases. For one case, while the upper bound of d2t is known, i.e., h2 = 0.3, 0.4, 0.5, the calculated upper bound of d1t obtained in this paper and other existing works is listed in Table 1. For another, while the upper bound of d1t is given, i.e., h1 = 1, 1.1, 1.2, 1.5, the calculated upper bound of d2t obtained by Theorem 1 and Corollary 1 in this paper and other existing works is listed in Table 2. Moreover, the number of decision variables (NoVs) of the stability criteria is also presented in tables to demonstrate the superiority of the proposed method.Based on the comparison of the calculated upper bounds and NoVs for different cases, it is obvious that the method used in this paper contributes to the stability analysis of system (1) on account of not only the further decrease of the conservatism but also the significant reduction of the decision variables. Compared with the existing works, for instance, Theorem 1 in [15], the derived criteria in this paper have greatly reduced the computational complexity while being less conservative, which is also the main contribution of this paper.Moreover, based on the comparison of the calculated upper bounds and NOVs from Theorem1 and Corollary 1, the triple integral term constructed in V4xt is verified beneficial to decrease the conservativeness. Compared with the existing works, Corollary 1 is less conservative without the triple integral terms introduced in LKF, which further illustrate the effect of the method used in this paper.Table 1
The calculated upper bounds ofd1(t) for different h2.
Methods
h
2
NoVs
0.3
0.4
0.5
Theorem 1 [4]
1.324
1.039
0.806
12.5n2 + 4.5n
Theorem 1 [6]
1.453
1.214
1.021
19.5n2 + 3.5n
Theorem 1 [7]
1.572
1.472
1.372
32.5n2 + 8.5n
Proposition 2 [8]
1.573
1.473
1.373
7n2 + 5n
Theorem 1 [9]
1.707
1.637
1.557
8.5n2 + 3.5n
Theorem 1 [14]
1.8804
1.7798
1.6759
189n2 + 30n
Theorem 1 [15]
1.9137
1.8137
1.7136
195.5n2 + 30.5n
Theorem 1
1.9511
1.8511
1.7511
74.5n2 + 5.5n
Corollary 1
1.9432
1.8432
1.7432
65.5n2 + 4.5nTable 2
The calculated upper bounds ofd2(t) for different h1.
Methods
h
1
NoVs
1
1.1
1.2
1.5
Theorem 1 [4]
0.415
0.376
0.340
0.248
12.5n2 + 4.5n
Theorem 1 [6]
0.512
0.457
0.406
0.283
19.5n2 + 3.5n
Theorem 1 [7]
0.872
0.772
0.672
0.371
32.5n2 + 8.5n
Proposition 2 [8]
0.873
0.773
0.673
0.373
7n2 + 5n
Theorem 1 [9]
0.989
0.914
0.836
0.564
8.5n2 + 3.5n
Theorem 1 [14]
0.9999
1.0770
0.9725
0.6807
189n2 + 30n
Theorem 1 [15]
1.2136
1.1136
1.0137
0.7137
195.5n2 + 30.5n
Theorem 1
1.2511
1.1511
1.0511
0.7511
74.5n2 + 5.5n
Corollary 1
1.2432
1.1432
1.0432
0.7432
65.5n2 + 4.5n


## 5. Conclusions

This paper investigates the stability of linear system with two additive time-varying delays. The main contribution of this paper is that the derived results have greatly reduced the computational complexity while being less conservative with comparison with the existing works. To achieve the goal, an effective and simple LKF is established in this paper, which is verified to be a great contribution to reduce the computational complexity. Additionally, GFWM-based inequality together with some other advanced techniques is utilized in this paper to calculate the derivative of the proposed LKF, which is verified to be significant to further decrease the conservatism of the derived criteria. A representative numerical example is presented to illustrate the validity and superiority of the method used in this paper.



---

*Source: 1745758-2020-04-04.xml*