# Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components

Authors: Tiejun Li; Junkang Tian
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper is concerned with delay-dependent stability for continuous systems with two additive time-varying delay components. By constructing a new class of Lyapunov functional and using a new convex polyhedron method, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities. The obtained stability criterion is less conservative than some existing ones. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Robust stability of dynamic interval systems covering interval matrices and interval polynomials has attracted considerable attention over last decades. Reference [1] presents some necessary and sufficient conditions for the quadratic stability and stabilization of dynamic interval systems. It is well known that time delay frequently occurs in many industrial and engineering systems, such as manufacturing systems, telecommunication, and economic systems, and is a major cause of instability and poor performance. Over the past decades, much efforts have been invested in the stability analysis of time-delay systems [2–16]. Reference [2] deals with the problem of quadratic stability analysis and quadratic stabilization for uncertain linear discrete time systems with state delay. Reference [3] deals with the quadratic stability and linear state-feedback and output-feedback stabilization of switched delayed linear dynamic systems. However, almost all the reported results mentioned above on time-delay systems are based on the following basic mathematical model:(1.1)ẋ(t)=Ax(t)+Adx(t-d(t)),
where d(t) is a time delay in the state x(t), which is often assumed to be either constant or time-varying satisfying certain conditions, for example,(1.2)0⩽d(t)⩽d̅<∞,ḋ(t)⩽τ<∞.Sometimes in practical situations, however, signals transmitted from one point to another may experience a few segments of networks, which can possibly induce successive delays with different properties due to the variable network transmission conditions. Thus, in recent papers [15, 16], a new model for time-delay systems with multiple additive time-varying delay components has been proposed: (1.3)ẋ(t)=Ax(t)+Adx(t-∑i=1ndi(t)),(1.4)0⩽di(t)⩽di<∞,ḋi(t)⩽τi<∞.
To make the stability analysis simpler, we proceed by considering the system (1.3) with two additive delay components as follows:
(1.5)ẋ(t)=Ax(t)+Adx(t-d1(t)-d2(t)),x(t)=ϕ(t),t∈[-d,0].
Here, x(t)∈ℝn is the state vector; d1(t) and d2(t) represent the two delay components in the state, and we denote d(t)=d1(t)+d2(t); A, Ad are system matrices with appropriate dimensions. It is assumed that(1.6)0⩽d1(t)⩽d1<∞,ḋ1(t)⩽τ1<∞,0⩽d2(t)⩽d2<∞,ḋ2(t)⩽τ2<∞,
and d=d1+d2, τ=τ1+τ2. ϕ(t) is the initial condition on the segment [-d,0].The purpose of our paper is to derive new stability conditions under which system (1.5) is asymptotically stable for all delays d1(t) and d2(t) satisfying (1.6). One possible approach to check the stability of this system is to simply combine d1(t) and d2(t) into one delay d(t) with(1.7)0⩽d(t)⩽d1+d2<∞,ḋ(t)⩽τ1+τ2<∞.
Then, the system (1.5) becomes
(1.8)ẋ(t)=Ax(t)+Adx(t-d(t)),x(t)=ϕ(t),t∈[-d,0].
By using some existing stability conditions, the stability of system (1.8) can be readily checked. As discussed in [15, 16], however, since this approach does not make full use of the information on d1(t) and d2(t), it would be inevitably conservative for some situations. Recently, some new delay-dependent stability criteria have been obtained for system (1.5) in [15, 16], by making full use of the information on d1(t) and d2(t). However, the stability result is conservative because of overly bounding some integrals appearing in the derivative of the Lyapunov functional. On the one hand, the integral -∫t-d1tẋT(s)Z1ẋ(s)ds in [15] was enlarged as -∫t-d1(t)tẋT(s)Z1ẋ(s)ds, with -∫t-d1t-d1(t)ẋT(s)Z1ẋ(s)ds discarded. On the other hand, some integrals were estimated conservatively. Take -∫t-d1(t)tẋT(s)Z1ẋ(s)ds as an example, by introducing(1.9)0=2ζTS[x(t)-x(t-d1(t))-∫t-d1(t)tẋ(s)ds]
with an appropriate vector ζ(t) and a matrix S, respectively, it was estimated as(1.10)2ζT(t)S[x(t)-x(t-d1(t))]+ζT(t)d1SZ1-1STζ(t)
with d1(t)SZ1-1ST enlarged as d1SZ1-1ST.The problem of delay-dependent stability criterion for continuous systems with two additive time-varying delay components has been considered in this paper. By constructing a new class of Lyapunov functional and using a new convex polyhedron method, a new stability criterion is derived in terms of linear matrix inequalities. The obtained stability criterion is less conservative than some existing ones. Finally, numerical examples are given to indicate less conservatism of the stability results.Definition 1.1.
LetΦ1,Φ2,…,ΦN:ℛm→ℛn be a given finite number of functions such that they have positive values in an open subset D of ℛm. Then, a reciprocally convex combination of these functions over D is a function of the form
(1.11)1α1Φ1+1α2Φ2+⋯+1αNΦN:D⟶Rn,
where the real numbers αi satisfy αi>0 and ∑iαi=1.The following Lemma1.2 suggests a lower bound for a reciprocally convex combination of scalar positive functions Φi=fi.Lemma 1.2 (See [10]).
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
(1.12)min⁡{αi∣αi>0,∑iαi=1}∑i1αifi(t)=∑ifi(t)+max⁡gi,j(t)∑i≠jgi,j(t)
subject to
(1.13){gi,j:Rm⟶R,gj,i(t)Δ̲̲gi,j(t),[fi(t)gi,j(t)gi,j(t)fj(t)]⩾0}.
In the following, we present a new stability criterion by a convex polyhedron method and Lemma 1.2.


## 2. Main Results

Theorem 2.1.
System (1.5) with delays d1(t) and d2(t) satisfying (1.6) is asymptotically stable if there exist symmetric positive definite matricesP,Q1,Q2,Q3,Q4,Q5,Q6, Z,Z1,Z2 and any matricesS12,N,M,L,S,P1,P2 with appropriate dimensions, such that the following LMIs hold:
(2.1)[ZS12*Z]⩾0,(2.2)E̅13=[E-d1N-d2L*-d1Z10**-d2Z2]<0,(2.3)E̅14=[E-d1N-d2S*-d1Z10**-d2Z2]<0,(2.4)E̅23=[E-d1M-d2L*-d1Z10**-d2Z2]<0,(2.5)E̅24=[E-d1M-d2S*-d1Z10**-d2Z2]<0,
where
(2.6)E=[E11E12S12T0000E18*E22E230000AdTP2T**E3300000***E440000****-Q4000*****E6600******-Q60*******E88]+[N+L00M-N-MS-L-S0]+[N+L00M-N-MS-L-S0]T,E11=Q1+Q2+Q3+Q4+Q5+Q6-Z+P1A+ATP1T,E12=-S12T+Z+P1Ad,E18=P-P1+ATP2T,E22=-(1-τ)Q1-2Z+S12+S12T,E23=-S12T+Z,E33=-Q2-Z,E44=-(1-τ1)Q3,E66=-(1-τ2)Q5,E88=d2Z+d1Z1+d2Z2-P2-P2T.Proof.
Construct a new Lyapunov functional candidate as(2.7)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t)),V1(x(t))=xT(t)Px(t),V2(x(t))=∫t-d(t)txT(s)Q1x(s)ds+∫t-dtxT(s)Q2x(s)ds+∫t-d1(t)txT(s)Q3x(s)ds+∫t-d1txT(s)Q4x(s)ds+∫t-d2(t)txT(s)Q5x(s)ds+∫t-d2txT(s)Q6x(s)ds,V3(x(t))=d∫-d0∫t+θtẋT(s)Zẋ(s)dsdθ,V4(x(t))=∫-d10∫t+θtẋT(s)Z1ẋ(s)dsdθ+∫-d20∫t+θtẋT(s)Z2ẋ(s)dsdθ.Remark 2.2.
Our paper fully uses the information aboutd(t),d1(t), and d2(t), but [15, 16] only use the information about d1(t) and d2(t), when constructing the Lyapunov functional V(x(t)). So the Lyapunov functional in our paper is more general than that in [15, 16], and the stability criteria in our paper may be more applicable.
The time derivative ofV(x(t)) along the trajectory of system (1.5) is given by
(2.8)V̇1(x(t))=2xT(t)Pẋ(t),(2.9)V̇2(x(t))=xT(t)(Q1+Q2+Q3+Q4+Q5+Q6)x(t)-(1-τ)xT(t-d(t))Q1x(t-d(t))-xT(t-d)Q2x(t-d)-(1-τ1)xT(t-d1(t))Q3x(t-d1(t))-xT(t-d1)Q4x(t-d1)-(1-τ2)xT(t-d2(t))Q5x(t-d2(t))-xT(t-d2)Q6x(t-d2),(2.10)V̇3(x(t))=d2ẋT(t)Zẋ(t)-d∫t-dt-d(t)ẋT(s)Zẋ(s)ds-d∫t-d(t)tẋT(s)Zẋ(s)ds,(2.11)V̇4(x(t))=ẋT(t)(d1Z1+d2Z2)ẋ(t)-∫t-d1t-d1(t)ẋT(s)Z1ẋ(s)ds-∫t-d1(t)tẋT(s)Z1ẋ(s)ds-∫t-d2t-d2(t)ẋT(s)Z2ẋ(s)ds-∫t-d2(t)tẋT(s)Z2ẋ(s)ds.
The V̇3(x(t)) is upper bounded by
(2.12)V̇3(x(t))⩽d2ẋT(t)Zẋ(t)-dd-d(t)ζT(t)(e2-e3)Z(e2-e3)Tζ(t)-dd(t)ζT(t)(e1-e2)Z(e1-e2)Tζ(t)(2.13)⩽d2ẋT(t)Zẋ(t)-ζT(t)[e2T-e3Te1T-e2T]T[ZS12S12TZ][e2T-e3Te1T-e2T]ζ(t),
where the inequality in (2.12) comes from the Jensen inequality lemma, and that of (2.13) from Lemma 1.2 as
(2.14)-ζT(t)[βα(e2-e3)T-αβ(e1-e2)T]T[ZS12S12TZ][βα(e2-e3)T-αβ(e1-e2)T]ζ(t)⩽0,
where
(2.15)ζT(t)=[xT(t)xT(t-d(t))xT(t-d)xT(t-d1(t))xT(t-d1)xT(t-d2(t))xT(t-d2)ẋT(t)],e1=(I0000000)T,e2=(0I000000)T,e3=(00I00000)T,α=(d-d(t))/d,  β=d(t)/d. Note that when d(t)=d or d(t)=0, one can obtain ζT(t)(e2-e3)=0 or ζT(t)(e1-e2)=0, respectively. So the relation (2.13) also holds.
By the Jensen inequality lemma, it is easy to obtain(2.16)-∫t-d1(t)tẋT(s)Z1ẋ(s)ds⩽-d1(t)U1TZ1U1,-∫t-d1t-d1(t)ẋT(s)Z1ẋ(s)ds⩽-(d1-d1(t))U2TZ1U2,-∫t-d2(t)tẋT(s)Z2ẋ(s)ds⩽-d2(t)U3TZ2U3,-∫t-d2t-d2(t)ẋT(s)Z2ẋ(s)ds⩽-(d2-d2(t))U4TZ2U4,
where
(2.17)U1=1d1(t)∫t-d1(t)tẋ(s)ds,U2=1d1-d1(t)∫t-d1t-d1(t)ẋ(s)ds,U3=1d2(t)∫t-d2(t)tẋ(s)ds,U4=1d2-d2(t)∫t-d2t-d2(t)ẋ(s)ds,(2.18)lim⁡d1(t)→0⁡1d1(t)∫t-d1(t)tẋ(s)ds=ẋ(t),lim⁡d1(t)→d1⁡1d1-d1(t)∫t-d1t-d1(t)ẋ(s)ds=ẋ(t-d1),lim⁡d2(t)→0⁡1d2(t)∫t-d2(t)tẋ(s)ds=ẋ(t),lim⁡d2(t)→d2⁡1d2-d2(t)∫t-d2t-d2(t)ẋ(s)ds=ẋ(t-d2).
From the Leibniz-Newton formula, the following equations are true for any matricesN,M, L,S,P1,P2 with appropriate dimensions
(2.19)2ζT(t)N[x(t)-x(t-d1(t))-d1(t)U1]=0,2ζT(t)M[x(t-d1(t))-x(t-d1)-(d1-d1(t))U2]=0,2ζT(t)L[x(t)-x(t-d2(t))-d2(t)U3]=0,2ζT(t)S[x(t-d2(t))-x(t-d2)-(d2-d2(t))U4]=0,2[xT(t)P1+ẋT(t)P2ẋT][-ẋ(t)+Ax(t)+Adx(t-d(t))]=0.
Hence, according to (2.8)–(2.19), we can obtain
(2.20)V̇(x(t))⩽ξT(t)E̅ξ(t),
where
(2.21)ξT(t)=[ζT(t)U1TU2TU3TU4T],E̅=[E-d1(t)N-(d1-d1(t))M-d2(t)L-(d2-d2(t))S*-d1(t)Z1000**-(d1-d1(t))Z100***-d2(t)Z20****-(d2-d2(t))Z2].
If E̅<0, then there exists a scalar ε>0, such that
(2.22)V̇(x(t))⩽ξT(t)E̅ξ(t)⩽-εξT(t)ξ(t)⩽-εxT(t)x(t)<0,∀x(t)≠0.
The E̅<0 leads for d1(t)→d1 to E1<0 and leads for d1(t)→0 to E2<0, where
(2.23)E1=[E-d1N-d2(t)L-(d2-d2(t))S*-d1Z100**-d2(t)Z20***-(d2-d2(t))Z2]<0,(2.24)E2=[E-d1M-d2(t)L-(d2-d2(t))S*-d1Z100**-d2(t)Z20***-(d2-d2(t))Z2]<0.
It is easy to see that E1 results from E̅|d1(t)=d1, where we deleted the zero row and the zero column. Define
(2.25)ξ1T(t)=[ζT(t)U1TU3TU4T],ξ2T(t)=[ζT(t)U2TU3TU4T],
The LMI (2.23) and (2.24) imply (2.22) because
(2.26)d1(t)d1ζ1T(t)E1ζ1(t)+d1-d1(t)d1ζ2T(t)E2ζ2(t)=ξT(t)E̅ξ(t)⩽-εxT(t)x(t)
and E̅ is convex in d1(t)∈[0,d1].
LMI (2.23) leads for d2(t)→d2 to LMI (2.2) and for d2(t)→0 to LMI (2.3). It is easy to see that E̅13 results from E1|d2(t)=d2, where we deleted the zero row and the zero column. The LMI (2.2) and (2.3) imply (2.23) because
(2.27)d2(t)d2ζ13T(t)E̅13ζ13(t)+d2-d2(t)d2ζ14T(t)E̅14ζ14(t)=ξ1T(t)E1ξ1(t)<0
and E1is convex in d2(t)∈[0,d2], where
(2.28)ξ13T(t)=[ζT(t)U1TU3T],ξ14T(t)=[ζT(t)U1TU4T]E̅13and E̅14 are defined in Theorem 2.1.
Similarly, the LMI (2.4) and (2.5) imply (2.24) because
(2.29)d2(t)d2ζ23T(t)E̅23ζ23(t)+d2-d2(t)d2ζ24T(t)E̅24ζ24(t)=ξ2T(t)E2ξ2(t)<0
and E2is convex in d2(t)∈[0,d2], where
(2.30)ξ23T(t)=[ζT(t)U2TU3T],ξ24T(t)=[ζT(t)U2TU4T].E̅23and E̅24are defined in Theorem 2.1. According to the above analysis, one can conclude that the system (1.5) with delays d1(t) and d2(t) satisfying (1.6) is asymptotically stable if the LMIs (2.1)–(2.5) hold.
From the proof of Theorem2.1, one can obtain that E̅ is negative definite in the rectangle 0⩽d1(t)⩽d1, 0⩽d2(t)⩽d2, only if it is negative definite at all vertices. We call this method as the convex polyhedron method.Remark 2.3.
To avoid the emergence of the reciprocally convex combination in (2.12), similar to [9], the integral terms in (2.10) can be upper bounded by
(2.31)-d∫t-dtẋT(s)Zẋ(s)ds⩽-[x(t-τ(t))-x(t-d)]TZ[x(t-τ(t))-x(t-d)]-[x(t)-x(t-τ(t))]TZ[x(t)-z(t-τ(t))]-(1-γ)[x(t-τ(t))-x(t-d)]TZ[x(t-τ(t))-x(t-d)]-γ[x(t)-x(t-τ(t))]TZ[x(t)-x(t-τ(t))]
which results in a convex combination on γ. However, Theorem 2.1 directly handles the inversely weighted convex combination of quadratic terms of integral quantities by utilizing the result of Lemma 1.2, which achieves performance behavior identical to the approaches based on the integral inequality lemma but with much less decision variables, comparable to those based on the Jensen inequality lemma.Remark 2.4.
Compared to some existing ones, the estimation ofV̇(x(t)) in the proof of Theorem 2.1 is less conservative due to the convex polyhedron method is employed. More specifically, -∫t-d1t-d1(t)ẋT(s)Z1ẋ(s)ds is retained, while -∫t-d1tẋT(s)Z1ẋ(s)ds is divided into -∫t-d1(t)tẋT(s)Z1ẋ(s)ds and -∫t-d1t-d1(t)ẋT(s)Z1ẋ(s)ds. When the two integrals together with others are handled by using free weighting matrix method, instead of enlarging some term d1(t)SZ1-1ST as d1SZ1-1ST. The convex polyhedron method is employed to verify the negative definiteness of V̇(x(t)). Therefore, Theorem 2.1 is expected to be less conservative than some results in the literature.Remark 2.5.
The case in which only two additive time-varying delay components appear in the state has been considered, and the idea in this paper can be easily extended to the system (1.3) with multiple additive delay components satisfying (1.4). Choose the Lyapunov functional as
(2.32)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t)),V1(x(t))=xT(t)Px(t),V2(x(t))=∫t-d(t)txT(s)Q1x(s)ds+∫t-dtxT(s)Q2x(s)ds+∑i=1n∫t-di(t)txT(s)Q3ix(s)ds+∑i=1n∫t-ditxT(s)Q4ix(s)ds,V3(x(t))=d∫-d0∫t+θtẋT(s)Zẋ(s)dsdθ,V4(x(t))=∑i=1n∫-di0∫t+θtẋT(s)Ziẋ(s)dsdθ.
Then, the corresponding stability result can be easily derived similar to the proof of Theorem 2.1. The result is omitted due to complicated notation.Remark 2.6.
The stability condition presented in Theorem2.1 is for the nominal system. However, it is easy to further extend Theorem 2.1 to uncertain systems, where the system matrices A and Ad contain parameter uncertainties either in norm-bounded or polytopic uncertain forms. The reason why we consider the simplest case is to make our idea more lucid and to avoid complicated notations.


## 3. Illustrative Example

Example 3.1.
Consider system (1.5) with the following parameters:
(3.1)A=[-200-0.9],Ad=[-10-1-1],assumingḋ1(t)⩽0.1,ḋ2(t)⩽0.8.
Our purpose is to calculate the upper boundd1 of delay d1(t), or d2 of delay d2(t), when the other is known, below which the system is asymptotically stable. By combining the two delay components together, some existing stability results can be applied to this system. The calculation results obtained by Theorem 2.1, in this paper, Theorem  1 in [6, 12, 15, 16], [14, Theorem  2] for different cases are listed in Table 1. It can be seen from the Table 1 that Theorem 2.1, in this paper, yields the least conservative stability test than other results.Table 1
Calculated delay bounds for different cases.
Stability conditionsDelay boundd2 for given d1Delay boundd1 for given d2d1=1d1=1.2d1=1.5d2=0.3d2=0.4d2=0.5[6, 12, 14]0.1800.080Infeasible0.8800.7800.680[15]0.4150.3760.2481.3241.0390.806[16]0.5120.4060.2831.4531.2141.021Theorem2.10.8730.6730.3731.5731.4731.373Example 3.2.
Consider system (1.5) with the following parameters:
(3.2)A=[0.01.0-1.0-2.0],Ad=[0.00.0-1.01.0].
We assume condition 1: ḋ1(t)⩽0.2, ḋ2(t)⩽0.5; condition 2: ḋ1(t)⩽0.2, ḋ2(t)⩽0.3, and under the two cases above, respectively. Table 2 lists the corresponding upper bounds of d2 for given d1. This numerical illustrates the effectiveness of the derived results.Table 2
Allowable upper bound ofd2 for various d1.
d10.30.50.70.9Condition 1d20.7670.5670.3670.067Condition 2d20.9680.7680.5680.368


## 4. Conclusions

This paper has investigated the stability problem for continuous systems with two additive time-varying delay components. By constructing a new class of Lyapunov functional and using a new convex polyhedron method, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities. The obtained stability criterion is less conservative than some existing ones. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.



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