# H

Authors: Hanyong Shao; Zhengqiang Zhang; Xunlin Zhu; Guoying Miao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper is concerned withH∞ control for a networked control model of systems with two additive time-varying delays. A new Lyapunov functional is constructed to make full use of the information of the delays, and for the derivative of the Lyapunov functional a novel technique is employed to compute a tighter upper bound, which is dependent on the two time-varying delays instead of the upper bounds of them. Then the convex polyhedron method is proposed to check the upper bound of the derivative of the Lyapunov functional. The resulting stability criteria have fewer matrix variables but less conservatism than some existing ones. The stability criteria are applied to designing a state feedback controller, which guarantees that the closed-loop system is asymptotically stable with a prescribed H∞ disturbance attenuation level. Finally examples are given to show the advantages of the stability criteria and the effectiveness of the proposed control method.


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## Body

## 1. Introduction

For years systems with time delays have received considerable attention since they are often encountered in various practical systems, such as engineering systems, biology, economics, neural networks, networked control systems, and other areas [1–6]. Since time-delay is frequently the main cause of oscillation, divergence, or instability, considerable effort has been devoted to stability for systems with time delays. According to whether stability criteria include the information of the delay, they are divided into two classes: delay-independent stability criteria and delay-dependent ones. It is well known that delay-independent stability criteria tend to be more conservative especially for small size delays. More attention has been paid to delay-dependent stability. For delay-dependent stability results, we refer readers to [7–14]. Among these papers, [11–13] were of systems with interval time-varying delay. Recently these delay-dependent stability results were extended to neutral systems with interval time-varying delay [14]. It should be pointed out that all the stability results mentioned are based on systems with one single delay in the state.On the other hand, networked control systems have been receiving great attention these years due to their advantages in low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability. It is well known that the transmission delay and the data packet dropout are two fundamental issues in networked control systems. The transmission delay generally includes the sensor-to-control delay and the control-to-actuator delay. In most of existing papers the sensor-to-control delay and the control-to-actuator delay were combined into one state delay, while the data packet dropouts were modeled as delays and absorbed by the state delay, thus formulating networked control systems as systems with one state delay [15]. Among recently reported results based on this modeling idea, to mention a few, event-triggered communication and H∞ control codesign problems were addressed for networked control systems in [16], while exponential state estimation problems were considered for Markovian jumping neural networks in [17]. Note that the sensor-to-control delay and the control-to-actuator delay are different in nature because of the network transmission conditions. The transmission delay and the data packet dropout also have different properties. It is not rational to lump them into one state delay. In this paper, to study networked control systems we adopt the model of systems with multiadditive time-varying delay components. For simplicity, the system with two additive time-varying delay components will be employed to address H∞ control problem for networked control systems. Now we write the system as follows:
(1)x˙(t)=Ax(t)+A1x(t-d1(t)-d2(t))+Ew(t)+Bu(t),(2)y(t)=Cx(t)+C1x(t-d1(t)-d2(t))+Fw(t)+Du(t),(3)x(t)=ϕ(t),t∈[-h,0],
where x(t)∈Rn is the state; y(t) is the measurement; u(t) is the control; w(t)∈L2[0,∞] is the disturbance; A, A1, E, B, C, C1, F, and D are known real constant matrices; d1(t) and d2(t) are two time-varying delays satisfying
(4)0≤d1(t)≤h1,0≤d2(t)≤h2,(5)d˙1(t)≤μ1,d˙2(t)≤μ2;
and ϕ(t) is a real-valued initial function on [-h,0] with
(6)h=h1+h2.
Stability analysis for this kind of system was conducted in [18], and a delay-dependent stability criterion was obtained. An improved stability criterion was derived in [19] by constructing a Lyapunov functional to employ the information of the marginally delayed state x(t-h). However, another marginally delayed state x(t-h1) was not considered, which caused -∫t-h1t-d1(t)x˙(α)TZ1x˙(α)dα to be discarded when bounding the derivative of the Lyapunov functional. On the other hand, in the process of the bounding, many free weighting matrices were introduced, making the stability result complicated.In this paper we first revisit delay-dependent stability for system (1) and (2). We will construct a new Lyapunov functional to employ the information of the marginally delayed state x(t-h1) as well as x(t-h). Motivated by [13], when bounding the derivative of the Lyapunov functional, we use a novel technique to avoid introducing too many matrix variables and compute a tighter upper bound. Considering that the upper bound depends on the two time-varying delays, we propose the so-called convex polyhedron method to check the negative definiteness for it. The resulting delay-dependent stability criteria turn out to be less conservative with fewer matrix variables. Then we take the advantages of the stability results to investigate the H∞ state feedback control problem, which is to design a state feedback controller u(t)=Kx(t) for the system such that the closed-loop system is asymptotically stable with an H∞ disturbance attenuation level γ>0 satisfying ∥y∥2<γ∥w∥2 for nonzero w(t)∈L2[0,∞] under zero initial condition. A delay-dependent condition will be presented for the state feedback controller such that the closed-loop system is asymptotically stable with a prescribed H∞ disturbance attenuation level. Formulated in LMIs the condition is readily verified, and when it is feasible the controller can be constructed.Notation. Throughout this paper the superscript “T” stands for matrix transposition. I refers to an identity matrix with appropriate dimensions. For real symmetric matrices X and Y, the notation X>Y means that the matrix X-Y is positive definite. The X≥Y follows similarly. The symmetric term in a matrix is denoted by *. Matrices, if not explicitly stated, are assumed to have compatible dimensions.First we go about the stability analysis. To the end, a lemma is given, which will play an important role in deriving our criteria.Lemma 1 (see [20]).
For any symmetric positive definite matrixM>0, scalar γ>0, and vector function ω:[0,γ]→Rn such that the integrations concerned are well defined, the following inequality holds:
(7)(∫0γω(s)ds)TM(∫0γω(s)ds)≤γ(∫0γω(s)TMω(s)ds).


## 2. Stability Analysis

Consider system (1) with w(t)=u(t)=0, namely,
(8)x˙(t)=Ax(t)+A1x(t-d1(t)-d2(t)).
Set
(9)d(t)=d1(t)+d2(t),(10)μ=μ1+μ2.
Taking d1(t)+d2(t) as one delay d(t) we have the following system:
(11)x˙(t)=Ax(t)+A1x(t-d(t)),
with 0≤d(t)≤h, d˙(t)≤μ.For this system there are many delay-dependent stability criteria available, but when used to check the stability for (8), they are more conservative [18]. In the following we present a new stability result for system (8) by considering the two delays separately.Theorem 2.
The system (8) subject to (4) and (5) is asymptotically stable for given h1, h2, μ1, and μ2 if there exist matrices P>0, Qi>0,  i=1,2,3,4, and Zj>0, j=1,2, such that the following LMIs hold:
(12)Φ-e13h1-1Z2e13T-e23h2-1Z2e23T<0,Φ-e13h1-1Z2e13T-e24h2h-2Z2e24T<0,Φ-e24h1h-2Z2e24T-e23h2-1Z2e23T<0,Φ-e24h-1Z2e24T<0,
where e13=[I0-I00]T, e23, e24, and e35 follow similarly, h is defined in (6), and
(13)Φ=[φ1PA1h1-1(Z1+Z2)00*φ2h2-1Z2h-1Z20**φ30h1-1Z1***-Q2-h-1Z20****-Q4-h1-1Z1]+[ATA1T000][h1Z1+hZ2][ATA1T000]T,
with μ given in (10) and
(14)φ1=PA+ATP+∑i=14Qi-h1-1(Z1+Z2),φ2=-(1-μ)Q3-(h2-1+h-1)Z2,φ3=-(1-μ1)Q1-(h2-1+h1-1)Z2-2h1-1Z1.Proof.
Define a Lyapunov functional as follows:(15)V(t)=x(t)TPx(t)+∫t-d1(t)tx(α)TQ1x(α)dα+∫t-htx(α)TQ2x(α)dα+∫t-d(t)tx(α)TQ3x(α)dα+∫t-h1tx(α)TQ4x(α)dα+∫t-h1t∫stx˙(α)TZ1x˙(α)dαds+∫t-ht∫stx˙(α)TZ2x˙(α)dαds,
where d(t) is defined in (9). Then calculating the time derivative of the Lyapunov functional along the trajectory of (8) yields
(16)V˙(xt)≤2x(t)TP(Ax(t)+A1x(t-d(t)))+∑i=14x(t)TQix(t)-x(t-h1)T×Q4x(t-h1)-x(t-h)TQ2x(t-h)-(1-μ)x(t-d(t))TQ3x(t-d(t))-(1-μ1)x(t-d1(t))TQ1x(t-d1(t))+(Ax(t)+A1x(t-d(t)))T(h1Z1+hZ2)×[Ax(t)+A1x(t-d(t))]-∫t-h1tx˙(α)TZ1x˙(α)dα-∫t-htx˙(α)TZ2x˙(α)dα.
Note that
(17)-∫t-h1tx˙(α)TZ1x˙(α)dα-∫t-htx˙(α)TZ2x˙(α)dα=-∫t-d1(t)tx˙(s)TZ1x˙(s)ds-∫t-h1t-d1(t)x˙(s)TZ1x˙(s)ds-∫t-d1(t)tx˙(s)TZ2x˙(s)ds-∫t-d(t)t-d1(t)x˙(s)TZ2x˙(s)ds-∫t-ht-d(t)x˙(s)TZ2x˙(s)ds.
Write α=d1(t)/h1 and β=d2(t)/h2. Then
(18)-∫t-d1(t)tx˙(s)TZ2x˙(s)dα=-h1-1∫t-d1(t)th1x˙(s)TZ2x˙(s)ds=-h1-1∫t-d1(t)td1(t)x˙(s)TZ2x˙(s)ds-h1-1∫t-d1(t)t[h1-d1(t)]x˙(s)TZ2x˙(s)ds.
It follows from (18) that
(19)-∫t-d1(t)tx˙(s)TZ2x˙(s)ds≤-h1-1∫t-d1(t)td1(t)x˙(s)TZ2x˙(s)ds.
Using (19) we have
(20)-h1-1∫t-d1(t)t[h1-d1(t)]x˙(s)TZ2x˙(s)ds=-(1-α)∫t-d1(t)tx˙(s)TZ2x˙(s)ds≤-(1-α)h1-1∫t-d1(t)td1(t)x˙(s)TZ2x˙(s)ds.
By Lemma 1, (18) and (20) imply
(21)-∫t-d1(t)tx˙(s)TZ2x˙(s)ds≤-[x(t)-x(t-d1(t))]Th1-1Z2×[x(t)-x(t-d1(t))]-(1-α)[x(t)-x(t-d1(t))]Th1-1Z2×[x(t)-x(t-d1(t))].
Similarly it can be derived that
(22)-∫t-d(t)t-d1(t)x˙(s)TZ2x˙(s)ds≤-[x(t-d(t))-x(t-d1(t))]Th2-1Z2×[x(t-d(t))-x(t-d1(t))]-(1-β)[x(t-d(t))-x(t-d1(t))]Th2-1Z2×[x(t-d(t))-x(t-d1(t))],-∫t-ht-d(t)x˙(s)TZ2x˙(s)ds≤-[x(t-d(t))-x(t-h)]Th-1Z2×[x(t-d(t))-x(t-h)]-α[x(t-d(t))-x(t-h)]Th1h-2Z2×[x(t-d(t))-x(t-h)]-β[x(t-d(t))-x(t-h)]Th2h-2Z2×[x(t-d(t))-x(t-h)].
Similar to [12] we have
(23)-∫t-d1(t)tx˙(s)TZ1x˙(s)ds≤-[x(t)-x(t-d1(t))]Th1-1Z1[x(t)-x(t-d1(t))],-∫t-h1t-d1(t)x˙(s)TZ1x˙(s)ds≤-[x(t-d1(t))-x(t-h1)]Th1-1Z1×[x(t-d1(t))-x(t-h1)].
Define
(24)ζ(t)=[x(t)Tx(t-d(t))Tx(t-d1(t))Tx(t-h)Tx(t-h1)T]T.
Combining (16), (17), and (21)–(23) and using (13) yield
(25)V˙(t)≤ζ(t)TΦζ(t)-(1-α)×[x(t)-x(t-d1(t))]Th1-1Z2[x(t)-x(t-d1(t))]-α[x(t-d(t))-x(t-h)]Th1h-2Z2×[x(t-d(t))-x(t-h)]-β[x(t-d(t))-x(t-h)]Th2h-1Z2×[x(t-d(t))-x(t-h)]-(1-β)[x(t-d(t))-x(t-d1(t))]Th2-1Z2×[x(t-d(t))-x(t-d1(t))]=ζ(t)TM(α,β)ζ(t),
where
(26)M(α,β)=Φ-αe24h1h-2Z2e24T-(1-α)e13h1-1Z2e13T-βe24h2h-2Z2e24T-(1-β)e23h2-1Z2e23T=α[Φ-e24h1h-2Z2e24T]+(1-α)[Φ-e13h1-1Z2e13T]-βe24h2h-2Z2e24T-(1-β)e23h2-1Z2e23T=α[Φ-e24h1h-2Z2e24T-βe24h2h-2Z2e24Tmmm-(1-β)e23h2-1Z2e23T]+(1-α)[Φ-e13h1-1Z2e13T-βe24h2h-2Z2e24Tmmmnmmm-(1-β)e23h2-1Z2e23T]=α[β(Φ-e24h-1Z2e24T)mmn+(1-β)(Φ-e24h1h-2Z2e24T-e23h2-1Z2e23T)]+(1-α)[β(Φ-e13h1-1Z2e13T-e24h2h-2Z2e24T)mmmmmmm+(1-β)×(Φ-e13h1-1Z2e13Tmmmmmmmmmmmmmm-e23h2-1Z2e23T)].
By (12) it is derived that M(α,β)<0. Therefore system (8) is asymptotically stable. This ends the proof.Remark 3.
Theorem2 provides a new delay-dependent stability criterion for system (8) with two additive time-varying delay components. In a form of LMIs the criterion can be checked easily.Remark 4.
Note that the corresponding matrixM(α,β) to the upper bound of V˙(xt) is dependent on the two time-varying delays while those in [18, 19] are dependent on the upper bounds of the two time-varying delays. To check the negative definiteness of the function matrix M(α,β), one has to adopt a new method, which is motivated by [13]. The basic idea is that a function matrix is negative definite over a convex polyhedron only if the matrix is negative definite at the vertices. Note that
(27)M(1,1)=Φ-e24h-1Z2e24T,M(1,0)=Φ+e24h1h-2Z2e24T-e23h2-1Z2e23T,M(0,1)=Φ-e13h1-1Z2e13T-e24h2h-2Z2e24T,M(0,0)=Φ-e13h1-1Z2e13T-e23h2-1Z2e23T.
From this it can be seen the negative definiteness of M(α,β) over the rectangle: 0≤α≤1, 0≤β≤1, is determined by that of M(α,β) at the vertices. One calls this approach to the negative definiteness of a function matrix a convex polyhedron method. Apparently the convex polyhedron method can be extended to more than two time-varying delays.Remark 5.
Gao et al. [19] took advantages of x(t-h) to derive a stability criterion, which improved over that in [18], but another marginally delayed state x(t-h1) was not employed. In this paper one makes use of it to construct the Lyapunov functional V(t) in (15), thus making -∫t-h1t-d1(t)x˙(α)TZ1x˙(α)dα retained in the estimate of V˙(t). On the other hand, when estimating integrals in V˙(xt) one does not introduce any free weighting matrix as [18, 19], but one uses new techniques reported recently in [12, 13]. Take -∫t-h1tx˙(α)TZ1x˙(α)dα as an example. One first divides it into two parts as (17) and then calculates them as (23). As for -∫t-d1(t)tx˙(α)TZ2x˙(α)dα and so forth, one deals with it in such a new way as (18)–(22). Thanks to the new techniques to calculate integrals in V˙(xt) and the convex polyhedron method to check the negative definite for the upper bound of V˙(xt), the resulting Theorem 2 is expected to be less conservative with fewer matrix variables, as shown in the following example.
Whenμ1 and μ2 are unknown, eliminating Q1 and Q3 one can obtain a delay-rate-independent stability criterion from Theorem 2 as follows.Corollary 6.
The system (8) subject to (4) is asymptotically stable for given h1 and h2 if there exist matrices P>0, Q2>0, Q4>0, and  Zj>0, j=1,2, such that the following LMIs hold:
(28)Φ1-e13h1-1Z2e13T-e23h2-1Z2e23T<0,Φ1-e13h1-1Z2e13T-e24h2h-2Z2e24T<0,Φ1-e24h1h-2Z2e24T-e23h2-1Z2e23T<0,Φ1-e24h-1Z2e24T<0,
where
(29)Φ1=[φ~1PA1h1-1(Z1+Z2)00*φ~2h2-1Z2h-1Z20**φ~30h1-1Z1***-Q2-h-1Z20****-Q4-h1-1Z1]+[ATA1T000][h1Z1+hZ2][ATA1T000]T,
with φ~1=PA+ATP+Q2+Q4-h1-1(Z1+Z2), φ~2=-(h2-1+h-1)Z2, and φ~3=-(h2-1+h1-1)Z2-2h1-1Z1.Whend1(t)≡h1, that is, d1(t) is a constant delay, Theorem 2 reduces to the following corollary, which was reported recently in [13].Corollary 7.
The system (8) with d1(t)≡h1 and d2(t) satisfying 0≤d2(t)≤h2 and d˙(t)≤μ2 is asymptotically stable for given h2>0, h1>0, and μ2 if there exist P>0, Qi>0, i=1,2,3, and Zj>0, j=1,2, such that the following LMIs hold:
(30)Φ2-[0I-I0]TZ2[0I-I0]<0,Φ2-[0I0-I]TZ2[0I0-I]<0,
where
(31)Φ2=[φ^1PA10h1-1Z1*φ^2h2-1Z2h2-1Z2**-Q2-h2-1Z20***-Q4-h2-1Z2-h1-1Z1]+[ATA1T00](h1Z1+h2Z2)[ATA1T00]T,
with φ^1=PA+ATP+∑i=13Qi-h1-1Z1 and φ^2=-(1-μ2)Q3-2h2-1Z2.Proof.
Define the Lyapunov functional. Consider the following:(32)V(xt)=x(t)TPx(t)+∫t-d(t)tx(α)TQ3x(α)dα+∫t-h1tx(α)TQ1x(α)dα+∫t-htx(α)TQ2x(α)dα+∫t-h1t∫stx˙(α)TZ1x˙(α)dαds+∫t-ht-h1∫stx˙(α)TZ2x˙(α)dαds.
Along a similar line as in the derivation of Theorem 2 the asymptotic stability can be established, and details are thus omitted.Remark 8.
Note that whend1(t) is a constant delay h1, system (8) can be regarded as a system in the form of (11) with interval time-varying delay: h1≤d(t)≤h, 0≤d˙(t)≤μ2. The system can serve as a model for networked control systems with both network-induced delay and data dropout phenomenon [15, 16]. In the form of LMIs Corollary 7 can provide a delay-dependent stability criterion for the model. Derived by the convex polyhedron method Corollary 7 is less conservative than those recently reported in [11]; see [13].In the following, we take the example in [19] to show that our stability criteria, though having much fewer matrix variables, are less conservative.Example 9.
Consider the system (8) with
(33)A=[-200-0.9],A1=[-10-1-1],d˙1(t)≤0.1,d˙2(t)≤0.8.
For given upper bound h1 of d1(t), we intend to find the admissible upper bound h2 of d2(t), which guarantees the asymptotic stability of (8).
Whenh2 is given, the admissible h1 can be seen from Table 2.
As seen in Tables1 and 2, Theorem 2 is less conservative than those in [15, 16]. It is worth noting that with fewer matrix variables involved, Theorem 2 needs less computational requirements.
Whend1(t) is a constant delay h1, the system can be looked upon as those with interval time-varying delay. As indicated in Remark 8, the stability result in Corollary 7 as well as that in [11] can be turned to for computing the admissible upper bound h of d(t), which are shown in Table 3.
Even as a delay-dependent criterion for systems with interval time-varying delay, Corollary7 has advantages over [11] in the sense that the computed admissible upper bound of the time-varying delay is larger.Table 1
Admissible upper boundh2 for various h1.
Method
h
1
1
1.2
1.5
[18]
h
2
0.415
0.376
0.248
[19]
h
2
0.512
0.406
0.283
Theorem2
h
2
0.5955
0.4632
0.3129Table 2
Admissible upper boundh1 for various h2.
Method
h
2
0.1
0.2
0.3
[18]
h
1
2.263
1.696
1.324
[19]
h
1
2.300
1.779
1.453
Theorem2
h
1
2.3400
1.8337
1.5318Table 3
Admissible upper boundh for various h1.
Method
h
1
1
2
3
4
[11]
h
1.7423
2.4328
3.2234
4.0644
Corollary7
h
1.8737
2.5049
3.2592
4.0745


## 3. State Feedback Control

Without a free weighting matrix introduced, Theorem2 only involves the matrices in the Lyapunov functional employed. It can be expected as a useful tool for the H∞ state feedback control problem formulated above. We first present an H∞ performance analysis result in the following.Theorem 10.
System (1) and (2) with u(t)=0 and delays subject to (4) and (5) is asymptotically stable with an H∞ disturbance attenuation level γ for given h1, h2, μ1, and μ2, if there exist matrices P>0, Qi>0, i=1,2,3,4, and  Zj>0, j=1,2 such that the following LMIs hold:
(34)Φ--e-13h1-1Z2e-13T-e-23h2-1Z2e-23T<0,Φ--e-13h1-1Z2e-13T-e-24h2h-2Z2e-24T<0,Φ--e-24h1h-2Z2e-24T-e-23h2-1Z2e-23T<0,Φ--e-24h-1Z2e-24T<0,
where e-13=[e13T0]T, e-23,   e-24, and   e-35  follow similarly and
(35)Φ-=[φ1PA1h1-1(Z1+Z2)00PE*φ2h2-1Z2h-1Z200**φ30h1-1Z10***-Q2-h-1Z200****-Q4-h1-1Z10*****-γ2I]+[ATA1T000ET][h1Z1+hZ2][ATA1T000ET]T+[CTC1T000FT][CTC1T000FT]T,
with φ1, φ2, φ3, and h given in Theorem 2.Proof.
ComparingΦ- with Φ in (13), we can conclude that (34) implies (12). Therefore system (1) and (2) with u(t)=0 is asymptotically stable. Now using the same Lyapunov functional as V(t) in (15) and calculating V˙(t) similar to the derivation of Theorem 2 along the solution of system (1) and (2) with u(t)=0, we have
(36)y(t)Ty(t)-γ2w(t)Tw(t)+V˙(t)≤ζ-(t)TM-(α,β)ζ-(t),
where
(37)M-(α,β)=Φ--αe-24h1h-2Z2e-24T-(1-α)e-13h1-1Z2e-13T-βe-24h2h-2Z2e-24T-(1-β)e-23h2-1Z2e-23T,
with α and β defined in the proof of Theorem 2 and ζ-(t)=[ζ(t)Tw(t)T]T with ζ(t) in (24). On the one hand, using the convex polyhedron method we can prove that M-(α,β)<0 by (34). On the other hand, under the zero condition we have V(0)=0 and V(∞)≥0. Integrating both sides of (36) leads to ∥y∥2<γ∥w∥2 for all nonzero w(t)∈L2[0,∞]. This ends the proof.Now we are in a position to resolve theH∞ state feedback control problem aforementioned.Theorem 11.
Consider system (1) and (2) with delays subject to (4) and (5). Given h1, h2, μ1, and μ2, there exists a state feedback controller u(t)=Kx(t) ensuring that the closed-loop system is asymptotically stable with an H∞ disturbance attenuation level γ, if there exist matrices P->0, Q-i>0, i=1,2,3,4, and Z-j>0, j=1,2, K- such that the following LMIs hold:
(38)[ΩiΓΓTΛ]<0,i=1,2,3,4,
where
(39)Ω1=Ψ-e-13h1-1Z-2e-13T-e-23h2-1Z-2e-23T,Ω2=Ψ-e-13h1-1Z-2e-13T-e-24h2h-2Z-2e-24T,Ω3=Ψ-e-24h1h-2Z-2e-24T-e-23h2-1Z-2e-23T,Ω4=Ψ-e-24h-1Z-2e-24T,Λ=diag⁡{h1-1(Z-1-2P-),h-1(Z-2-2P-),-I},Γ=[P-AT+K-TBTP-AT+K-TBTP-CT+K-TDTP-A1TP-A1TP-C1T000000000ETETFT],
with
(40)Ψ=[φ-1A1P-h1-1(Z-1+Z-2)00E*φ-2h2-1Z-2h-1Z-200**φ-30h1-1Z-10***-Q-2-h-1Z-200****-Q-4-h1-1Z-10*****-γ2I],φ-1=AP-+BK-+(AP-+BK-)T+∑i=14Q-i-h1-1(Z-1+Z-2),φ-2=-(1-μ)Q-3-(h2-1+h-1)Z-2,φ-3=-(1-μ1)Q-1-(h2-1+h1-1)Z-2-2h1-1Z-1,
and e-13,   e-23,   e-24,   e-35, h, and μ are given in Theorem 10. Moreover, if the foregoing condition holds, a desired controller gain matrix is given by
(41)K=K-P--1.Proof.
Apply the controlleru(t)=Kx(t) to system (1) and (2) and then the closed-loop system is formulated as follows:
(42)x˙(t)=(A+BK)x(t)+A1x(t-d1(t)-d2(t))+Ew(t),y(t)=(C+DK)x(t)+C1x(t-d1(t)-d2(t))+Fw(t).
By Theorem 10 this system is asymptotically stable with an H∞ disturbance attenuation level γ, if there exist matrices P>0, Qi>0,  i=1,2,3,4, and Zj>0, j=1,2, such that
(43)Φ-c-e-13h1-1Z2e-13T-e-23h2-1Z2e-23T<0,Φ-c-e-13h1-1Z2e-13T-e-24h2h-2Z2e-24T<0,Φ-c-e-24h1h-2Z2e-24T-e-23h2-1Z2e-23T<0,Φ-c-e-24h-1Z2e-24T<0,
where e-13,e-23,e-24,e-35, and h are the same as those in Theorem 10 and(44)Φ-c=[φc1PA1h1-1(Z1+Z2)00PE*φ2h2-1Z2h-1Z200**φ30h1-1Z10***-Q2-h-1Z200****-Q4-h1-1Z10*****-γ2I]+[(A+BK)TA1T000ET][h1Z1+hZ2][(A+BK)TA1T000ET]T+[(C+DK)TC1T000FT][(C+DK)TC1T000FT]T,with φ2 and φ3 defined in Theorem 10 and φc1=P(A+BK)+(A+BK)TP+∑i=14Qi-h1-1(Z1+Z2).
WriteJ=diag⁡{P-1,P-1,P-1,P-1,P-1,I}, P-=P-1, Z-i=P-1ZiP-1, i=1,2, Q-j=P-1QjP-1, and j=1,2,3,4. Performing a congruence transformation to (43) by J yields
(45)Ψ+ΓΛ^-1ΓT-e-13h1-1Z-2e-13T-e-23h2-1Z-2e-23T<0,Ψ+ΓΛ^-1ΓT-e-13h1-1Z-2e-13T-e-24h2h-2Z-2e-24T<0,Ψ+ΓΛ^-1ΓT-e-24h1h-2Z-2e-24T-e-23h2-1Z-2e-23T<0,Ψ+ΓΛ^-1ΓT-e-24h-1Z-2e-24T<0,
where
(46)Λ^=diag⁡{h1-1Z1-1,h-1Z2-1,I}.
By Schur complements we have
(47)[ΩiΓΓT-Λ^]<0,i=1,2,3,4.
Note that (47) is not linear in P-, K-, Q-i, and Z-j due to Zi-1=P-Z-i-1P-. However, noting that (P--Z-i)Z-i-1(P--Z-i)≥0, we have P-Z-i-1P-≥-Z-i+2P-. Therefore, Zi-1≥-Z-i+2P-. It follows immediately that -Λ^≤Λ, which means that (38) implies (47). This completes the proof.Due to the fact thatZi-1≥-Z-i+2P-, condition (38) is more conservative than (47). However, based on (38) one can obtain an LMI approach to the H∞ state feedback control problem for systems with two additive time-varying delays. The existence of the state feedback controller is guaranteed by the feasibility of LMIs (38). When LMIs (38) are solvable, the controller can be constructed according to (41). Based on (47), one can obtain a less conservative controller at the cost of more complexity by employing CCL method [21].To illustrate the effectiveness of this method we provide an example.Example 12.
Consider system (1) and (2) with parameters given as follows:
(48)A=[0.1100-0.9],A1=[-20-11.1],E=[0.560.61],B=[0.2-2.5],C=[0.11.8],C1=[0.7-1],F=0.1,D=0.4.
Given h1=0.1, h2=0.2, μ1=0.1, μ2=0.2, and γ=0.6 we can find that LMIs in (38) are feasible with
(49)P-=[2.6423-0.2090-0.20900.7465],K-=[-0.52410.8779].
By Theorem 11, there exists a state feedback controller
(50)u(t)=K-P--1x(t)=[-0.10781.1458]x(t)
such that the closed-loop system is asymptotically stable for 0≤d1(t)≤0.1,  0≤d2(t)≤0.2 with an H∞ disturbance attenuation level γ=0.6.


## 4. Conclusion

This paper is concerned withH∞ control for a networked control model of systems with two additive time-varying delays. For one thing a new delay-dependent stability criterion was derived, which improves over existing ones in that it has less conservatism with fewer matrix variables. A delay-rate-independent criterion was also obtained as a byproduct. When one of the delays is constant, a new stability criterion was given for systems with interval time-varying delay. Then examples were provided to illustrate the reduced conservatism of the criteria. Finally the H∞ state feedback control problem was solved via an LMI approach, which was demonstrated to be effective using another example.



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*Source: 923436-2014-10-19.xml*