# H

Authors: Sung Hyun Kim
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper investigates the observer-basedH∞ tracking problem of networked output-feedback control systems with consideration of data transmission delays, data-packet dropouts, and sampling effects. Different from other approaches, this paper offers a single-step procedure to handle nonconvex terms that appear in the
process of designing observer-based output-feedback control, and then establishes a set of linear matrix inequality conditions for the solvability of the tracking problem. Finally, two numerical examples are given to illustrate the effectiveness of our result.


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## Body

## 1. Introduction

Recently, the research on networked control systems (NCSs) has been rapidly growing due to both the fast development of technology of communication networks and the benefits of NCSs that include(1) overcoming the spatial limits of the traditional control system, (2) expanding system setups, (3) increasing flexibility, (4) multitasking, and (5) improving system diagnosis and maintenance (see [1–4]). In particular, more recently, the development of the embedded system that has various communication modules and digital signal processing (DPS) core has confirmed the necessity of further investigations on NCSs. However, it is worth noticing here that the signal transmission over communication channels inevitably gives rise to data transmission delay problem, data-packet dropout problem, and sampling problem (see [3, 5–8]), which may cause instability or serious deterioration in the performance of the resultant control systems. Thus, exploring such problems has been recognized as one of the most important issues in the application of control theory.Over the past several years, numerous researchers have made considerable efforts to propose methods for solving the aforementioned problems, especially based on Lyapunov-Krasovskii functional approach (see [9–11] for stabilization of NCSs (S-NCSs); [12, 13] for stabilization of NOCSs (S-NOCSs); and [5, 14–16] for tracking control of NCSs (T-NCSs), where NOCSs is the abbreviation of networked output-feedback control systems). In addition, [17] investigated the problem of output tracking for NCSs on the basis of the Lyapunov function approach. However, it is worth pointing out here that, regardless of such abundant literature, little progress has been made toward solving the tracking problem of NOCSs (T-NOCSs) in light of the Lyapunov-Krasovskii functional approach. In fact, all states of the controlled plant are not fully measurable in many engineering applications, and thus the tracking problem has emerged as a topic of significant interest in parallel to the stabilization problem. Thus, it is quite meaningful to study the method of designing T-NOCSs, especially by establishing a set of linear matrix inequality (LMI) conditions for the solvability of the tracking problem.Motivated by the above concern, we investigate the problem of designing an observer-based T-NOCS with consideration of data transmission delays, data-packet dropouts, and sampling effects. Specifically, the attention is focused on designing an observer-based NOCS in such a way that the plant state tracks the reference signal in theH∞ sense. The contributions of this paper are mainly threefold.(1)
The problem of designing T-NOCSs is systematically covered with the help of the Lyapunov-Krasovskii functional approach, which helps our results to have more wide applications.(2)
A single-step procedure is proposed to handle nonconvex terms that inherently appear in the process of designing observer-based output-feedback control, which allows the derived sufficient conditions for the solvability of the tracking problem to be established in terms of LMIs.(3)
Through the control synthesis process, this paper shows that the stability criteria derived from the reciprocally convex approach [18] can be clearly applied to the problem of designing T-NOCSs, which offers the possibilities for the extension of the results [19, 20] on the stability analysis toward the design of T-NOCSs. Finally, two numerical examples are given to illustrate the effectiveness of our result.Notation. The Lebesgue space L2+=L2[0,∞) consists of square-integrable functions on [0,∞). Throughout this paper, standard notions will be adopted. The notations X≥Y and X>Y mean that X-Y is positive semidefinite and positive definite, respectively. In symmetric block matrices, (∗) is used as an ellipsis for terms that are induced by symmetry. For a square matrix Q, the notation He(Q) denotes Q+QT, where QT is the transpose of Q. col(q1,q2) is a column vector with entries q1 and q2 and diag(Q1,Q2) is a diagonal matrix with diagonal entries Q1 and Q2. All matrices, if their dimensions are not explicitly stated, are assumed to be compatible with algebraic operation.


## 2. System Description and Preliminaries

Consider a continuous-time plant of the following form:(1)x˙t=Axt+But+Dwt,yt=Cxt,where x(t)∈Rnx, u(t)∈Rnu, and y(t)∈Rny denote the state to be estimated, the control input, and the output, respectively, and w(t)∈Rnw denotes the disturbance input such that w(t)∈L2+. Here, as a way to estimate the immeasurable state variables of (1), we employ the following usual state observer:(2)x^˙t=Ax^t+But-Lyt-y^t,y^t=Cx^t,where x^(t)∈Rnx denotes the estimated state and L∈Rnx×ny is the observer gain to be designed. Further, in parallel to (1) and (2), we incorporate the following dynamic system that generates the reference signal xr(t)∈Rnx:(3)x˙rt=Arxrt+rt,where r(t)∈Rnx denotes the reference input such that r(t)∈L2+ and Ar is constructed to be an asymptotically stable matrix. In this paper, our interest is to design an observer-based networked output-feedback control system (NOCS), based on (1)–(3), such that(1)
the estimated statex^(t) can approach the real state x(t) asymptotically;(2)
the estimated statex^(t) can track a reference signal xr(t) over a communication network; that is, the state x(t) can track xr(t) by (1);(3)
a guaranteedH∞ tracking performance can be achieved.To this end, we first employ the networked control system (NCS) architecture proposed in [3], which contains an observer with time-driven sampler, an event-driven controller, and a packet analyzer with event-driven holder (see Figure 1). For brevity, this paper omits the sophisticated description for the NCS under consideration since it is analogue to that of [3]. However, different from [3], we assume that the initial condition of (2) is given as x^(t)=ϕ(t), for t∈[t0-dM,t0], and the initial condition of (3) is given as xr(t)=φ(t), for t∈[t0-dM,t0], where t0 denotes the initial time.Figure 1
Networked output-feedback control systems (NOCSs) with observer-based controller.Remark 1.
Here, it should be noted that, by the NCS architecture of [3], the communication constraints, such as data transmission delays and packet dropouts, can be represented in terms of piecewise continuous-time-varying delays with the lower and upper bounds.Next, let us consider the following control law, inferred by [3]:(4)ut=Fx^t-dt-xrt-dt,where d(t)∈[dm,dM] corresponds to the piecewise continuous-time-varying delay that occurs from data transmission delays and packet dropouts. Then, by letting e(t)=x(t)-x^(t) and er(t)=x(t)-xr(t), the control law (4) can be rewritten as(5)ut=Fert-dt-et-dt.Further, by setting x~(t)=col(xr(t),er(t),e(t))∈R3nx and w~(t)=col(w(t),r(t))∈Rnw+nx and by combining (1), (2), (3), and (5), the closed-loop system is described as(6)x~˙t=A~x~t+A~dx~t-dt+D~w~t,z~t=C~x~t=Cert,where z~(t)∈Rnx denotes the desired output,(7)A~=Ar00A-ArA000A+LC,A~d=0000BF-BF000,C~T=0CT0,D~=0ID-ID0.Before ending this section, we present the following lemma that will be used in the proof of our main results.Lemma 2 (see [21]).
For real matricesX, Y, and S>0 with appropriate dimensions, it is satisfied that 0≤(X-SY)TS-1(X-SY) and thus the following inequality holds: YTSY≥He(XTY)-XTS-1X. Further if X=μI, then(8)YTSY≥HeμY-μ2S-1,where μ is a scalar. On the other hand, if S<0, then(9)YTSY≤-HeμY-μ2S-1.


## 3. Main Results

Choose a Lyapunov-Krasovskii functional of the following form:(10)Vt=V1t+V2t+V3t,V1t=x~TtPx~t,V2t=∫t-dmt‍x~TαQ1x~αdα+∫t-dMt‍x~TαQ2x~αdα,V3t=dm∫-dm0‍∫t+αt‍x~˙TβR1x~˙βdβdα+dI∫-dM-dm‍∫t+αt‍x~˙TβR2x~˙βdβdα,where P, Q1, Q2, R1, and R2∈R3nx×3nx are positive definite matrices and dI=dM-dm. For later convenience, we define an augmented state ζ(t)=col(x~(t),x~(t-dm),x~(t-d(t)),x~(t-dM),w~(t))∈Rnη, nη=13nx+nw, and then establish some block entry matrices ei such that x~(t)=e1ζ(t), x~(t-dm)=e2ζ(t), x~(t-d(t))=e3ζ(t), x~(t-dM)=e4ζ(t), and w~(t)=e5ζ(t). Then the closed-loop system (6) can be rewritten as x~˙(t)=Φtζ(t), where Φt=A~e1+A~de3+D~e5. As a result, the time derivative of Vi(t) along the trajectories of (6) is given by(11)V˙1t=ζTtHee1TPΦtζt,V˙2t=ζTte1TQ1+Q2e1-e2TQ1e2-e4TQ2e4ζt,V˙3t=ζTtΦtTdm2R1+dI2R2Φtζt+O,where(12)O=-dm∫t-dmt‍x~˙TαR1x~˙αdα-dI∫t-dtt-dm‍x~˙TαR2x~˙αdα-dI∫t-dMt-dt‍x~˙TαR2x~˙αdα.By (11), the time derivative of V(t) becomes(13)V˙t=ζTtΠ0ζt+O,where Π0=He(e1TPΦt)+e1T(Q1+Q2)e1-e2TQ1e2-e4TQ2e4+ΦtTdm2R1+dI2R2Φt. To deal with O, we apply the Jensen inequality [22] to O, which results in(14)O≤-∫t-dmt‍x~˙αdαTR1∫t-dmt‍x~˙αdα-1θ1t∫t-dtt-dm‍x~˙αdαTR2∫t-dtt-dm‍x~˙αdα-1θ2t∫t-dMt-dt‍x~˙αdαTR2∫t-dMt-dt‍x~˙αdα=-ζTte1-e2TR1e1-e2ζt-1θ1tζTte2-e3TR2e2-e3ζt-1θ2tζTte3-e4TR2e3-e4ζt,where θ1(t)=(d(t)-dm)/dI≥0, θ2(t)=(dM-d(t))/dI≥0, and θ1(t)+θ2(t)=1; that is, the set of θi(t) is convex. Furthermore, by taking the convexity of θi(t) into account, we can get the following equality:(15)RHSof14=ζTte1-e2TR1e2-e1+e2-e3TR2e3-e2+e3-e4TR2e4-e3ζt-θ2tθ1tζTte2-e3TR2e2-e3ζt-θ1tθ2tζTte3-e4TR2e3-e4ζt=ζTtΠ1ζt-θ2θ1e2-e3ζtθ1θ2e3-e4ζtT×Π2θ2θ1e2-e3ζtθ1θ2e3-e4ζt,where Π1=(e1-e2)TR1(e2-e1)+(e2-e3)TR2(e3-e2)+(e3-e4)TR2(e4-e3)+He((e2-e3)TS(e3-e4)),(16)Π2=R2S∗R2.Hence, we can see that the time derivative of V(t) satisfies that V˙(t)≤ζT(t)(Π0+Π1)ζ(t)-(⋆)TΠ2(⋆), where (⋆)=colθ2/θ1(e2-e3)ζ(t),θ1/θ2(e3-e4)ζ(t). As a result, based on this derivation, the following stability criteria can be established.Lemma 3 (stability criterion).
Forw~(t)=0, the stability criterion is given by(17)0>Π0+Π1,0≤Π2,where Φt=A~e1+A~de3.Proof.
IfΠ2≥0 holds, then V˙(t)≤ζT(t)(Π0+Π1)ζ(t).Lemma 4 (stability criterion in theH∞ sense).
The stability criterion in theH∞ sense is given by(18)0>Π0+Π1+Π3,0≤Π2,where Φt=A~e1+A~de3+D~e5, Π3=e1TC~TC~e1-γ2e5Te5.Proof.
Let us consider theH∞ tracking performance such that supw~(z~2/w~2)<γ. Then, as reported in [19], the H∞ stability criterion can be readily derived by V˙(t)+z~T(t)z~(t)-γ2w~T(t)w~(t)<0, which is assured by (18).Based on Lemma3, the stabilization problem of (6) with w~(t)=0 will be addressed in Section 3.1, and further, based on Lemma 4, the H∞ stabilization problem of (6) with w~(t)≠0 will be investigated in Section 3.2. Here, to derive a set of linear matrix inequalities (LMIs), we first set P=diag(P1,P2,P3) and P¯=P-1=diag(P¯1,P¯2,P¯3), where P¯1=P1-1, P¯2=P2-1, and P¯3=P3-1. Then, from (7), it follows that(19)PA~=P1Ar00P2A-P2ArP2A000P3A+L¯C,PA~d=0000P2BF¯P2-P2BF¯P2000,PD~=0P1P2D-P2P3D0,where L¯=P3L and F¯=FP¯2. Accordingly, the term PΦt becomes(20)PΦt=XA˘Xe1+XA˘dXe3+XD˘e5,where X=diag(I,P2,P2),(21)A˘=P1Ar00A-ArAP¯2000P¯2P3A+L¯CP¯2,A˘d=0000BF¯-BF¯000,D˘=0P1D-IP¯2P3D0.Remark 5.
Inspired by the work of [18], this paper also applied the reciprocally convex approach to reduce the computational complexity and the conservatism of the delay-dependent stability criteria that will be used to derive our main results.


### 3.1. Control Design forw~(t)=0

Lemma 6.
Letμ1>0, μ2>0, and ϵ>0 be prescribed. Suppose that there exist matrices F¯∈Rnu×nx, L¯∈Rnx×ny, and S~∈R3nx×3nx and symmetric matrices 0<P1, P¯2,P3∈Rnx×nx, 0<Q~1, Q~2∈R3nx×3nx, 0<R~1, R~2∈R3nx×3nx such that(22)0>1,10dmA¯0dmA˘d0ϵE102,2dIA¯0dIA˘d0ϵE2∗∗3,3R~1A˘d0ϵE300∗4,4R~2+S~-S~0∗∗∗∗5,5R~2+S~0000∗∗6,60∗∗∗0007,7,(23)0≤R~2S~∗R~2,where(24)1,1=μ12R~1+diag-2μ1P1,-2μ1P¯2,-2ϵP¯2,2,2=μ22R~2+diag-2μ2P1,-2μ2P¯2,-2ϵP¯2,3,3=A¯¯+Q~1+Q~2-R~1,4,4=-Q~1-R~1-R~2,5,5=-2R~2-HeS~,6,6=-Q~2-R~2,7,7=-2μ1P30dmP3A+L¯C0-2μ2P3dIP3A+L¯C∗∗HeP3A+L¯C,A˘d=0000BF¯-BF¯000,A¯=P1Ar00A-ArAP¯20000,A̿=HeP1ArA-ArT0∗HeAP¯2000-2ϵP¯2,E1=000000I00,E2=0000000I0,E3=00000000I.Then the closed-loop system (6) is asymptotically stable in the absence of w~(t) for any time-varying delay d(t) satisfying dm≤d(t)≤dM. Moreover, the control and observer gain matrices can be reconstructed as follows:(25)F=F-P-2-1,L=P3-1L-.Proof.
From Lemma3, the stabilization condition is given as follows: (i) 0≤Π2 and (ii) 0>Π0+Π1=He(e1TPΦt)+e1T(Q1+Q2)e1-e2TQ1e2-e4TQ2e4+ΦtTdm2R1+dI2R2Φt + (e1-e2)TR1(e2 − e1) + (e2-e3)TR2(e3-e2)+(e3-e4)TR2(e4-e3)+He((e2-e3)TS(e3-e4)), where Φt=A~e1+A~de3 and thus PΦt=XA˘Xe1+XA˘dXe3. Let us consider the condition given in (ii). Then, by letting R¯1=XP¯R1P¯X and R¯2=XP¯R2P¯X, we can rewrite the condition, 0>Π0+Π1, as follows:(26)0>Hee1TXA˘Xe1+e1TXA˘dXe3+e1TQ1+Q2e1-e2TQ1e2-e4TQ2e4+XA˘Xe1+XA˘dXe3T×X¯dm2R¯1+dI2R¯2X¯XA˘Xe1+XA˘dXe3+e1-e2TPX¯R¯1X¯Pe2-e1+e2-e3TPX¯R¯2X¯Pe3-e2+e3-e4TPX¯R¯2X¯Pe4-e3+Hee2-e3TSe3-e4,where X¯=X-1. Further, since diag(X¯,X¯,X¯,X¯)eiT=eiTX¯ (i=1,2,3,4), pre- and postmultiplying both sides of (26) by diag(X¯,X¯,X¯,X¯) and its transpose yield(27)0>Q+A˘e1+A˘de3Tdm2R¯1+dI2R¯2A˘e1+A˘de3,where Q=He(e1TA˘e1+e1TA˘de3)+e1T(Q~1+Q~2)e1-e2TQ~1e2-e4TQ~2e4+(e1-e2)TR~1(e2-e1) + (e2-e3)TR~2(e3-e2)+(e3-e4)TR~2(e4-e3)+He((e2-e3)TS~(e3-e4)) in which Q~1=X¯Q1X¯, Q~2=X¯Q2X¯, R~1=X~R¯1X~, R~2=X~R¯2X~, S~=X¯SX¯, and X~=X¯PX¯=diag(P1,P¯2,P¯2P3P¯2)>0. That is, by applying the Schur complement to (27), we can get(28)0>-R¯1-10dmA˘e1+A˘de30-R¯2-1dIA˘e1+A˘de3∗∗Q.Here, since R¯1-1=X~R~1-1X~ and R¯2-1=X~R~2-1X~, it follows from Lemma 2 that R¯1-1≥2μ1X~-μ12R~1 and R¯2-1≥2μ2X~-μ22R~2. In this sense, it is clear that (28) holds if(29)0>1,1′01,3′0dmA˘d002,2′2,3′0dIA˘d0∗∗3,3′R~1A˘d000∗4,4R~2+S~-S~∗∗∗∗5,5R~2+S~000∗∗6,6,where(30)1,1′=μ12R~1+diag-2μ1P1,-2μ1P¯2,P¯2-2μ1P3P¯2,2,2′=μ22R~2+diag-2μ2P1,-2μ2P¯2,P¯2-2μ2P3P¯2,3,3′=HeA˘+Q~1+Q~2-R~1,1,3′=dmA˘,2,3′=dIA˘.However, as shown in (29), there exist some nonconvex terms in (1,1)′, (1,3)′, (2,2)′, (2,3)′, and (3,3)′ as follows:(31)1,1′=⋆⋆⋆⋆⋆⋆⋆⋆⋆+P¯2-2μ1P3P¯2,2,2′=⋆⋆⋆⋆⋆⋆⋆⋆⋆+P¯2-2μ2P3P¯2,1,3′=⋆00⋆⋆000P¯2dmP3A+L¯CP¯2,2,3′=⋆00⋆⋆000P¯2dIP3A+L¯CP¯2,3,3′=⋆⋆⋆⋆⋆⋆⋆⋆⋆+P¯2HeP3A+L¯CP¯2.Here, note that all terms associated with P¯2(⋆)P¯2 in(32)1,1′01,3′02,2′2,3′∗∗3,3′can be separated as follows:(33)E1E2E3P¯2000P¯2000P¯2-2μ1P30dmP3A+L¯C0-2μ2P3dIP3A+L¯C∗∗HeP3A+L¯CP¯2000P¯2000P¯2︸ΨE1E2E3T.Furthermore, from Lemma 2, it follows that(34)Ψ≤-2ϵP¯2000P¯2000P¯2-ϵ2-2μ1P30dmP3A+L¯C0-2μ2P3dIP3A+L¯C∗∗HeP3A+L¯C-1,which allows that (22) implies (29), based on the Schur complement. Next, we need to convert the given condition in (i), that is, 0≤Π2, into an LMI. To this end, let us pre- and postmultiply both sides of 0≤Π2 by diag(X¯,X¯) and its transpose. Then we can get(35)0≤X¯R2X¯S~∗X¯R2X¯,which becomes (23) due to R~2=X~R¯2X~=(X¯PX¯)(XP¯R2R¯X)(X¯PX¯)=X¯R2X¯.


### 3.2.  Control Design forw~(t)≠0

Theorem 7.
Letμ1>0, μ2>0, ϵp>0 be prescribed. Suppose that there exist scalars ϵq>0, γ>0; matrices F¯∈Rnu×nx, L¯∈Rnx×ny, S~∈R3nx×3nx; and symmetric matrices 0<P1, P¯2,P3∈Rnx×nx, 0<Q~1,Q~2∈R3nx×3nx, 0<R~1,R~2∈R3nx×3nx such that(36)0>1,10dmA¯0dmA˘d0dmD¯ϵpE1002,2dIA¯0dIA˘d0dID¯ϵpE20∗∗3,3R~1A˘d0D¯ϵpE3X¯C~T00∗4,4R~2+S~-S~000∗∗∗∗5,5R~2+S~000000∗∗6,6000∗∗∗0007,7ϵqE40∗∗∗000∗8,8000∗00000-I,(37)0≤R~2S~∗R~2,where(38)1,1=μ12R~1+diag-2μ1P1,-2μ1P¯2,-2ϵpP¯2,2,2=μ22R~2+diag-2μ2P1,-2μ2P¯2,-2ϵpP¯2,3,3=A̿+Q~1+Q~2-R~1,4,4=-Q~1-R~1-R~2,5,5=-2R~2-HeS~,6,6=-Q~2-R~2,7,7=diag-2ϵqI,-γ2I,8,8=-2μ1P30dmP3A+L¯CdmP3D0-2μ2P3dIP3A+L¯CdIP3D∗∗HeP3A+L¯CP3D∗∗∗-γ2I,A˘d=0000BF¯-BF¯000,A¯=P1Ar00A-ArAP¯20000,A̿=HeP1ArA-ArT0∗HeAP¯2000-2ϵpP¯2,D¯=0P1D-I00,C~T=0CT0,E1=00000000I000,E2=000000000I00,E3=0000000000I0,E4=000I0000.Then the closed-loop system (6) is asymptotically stable and satisfies z~2<γw~2   for all nonzero w~(t)∈L2+ and for any time-varying delay d(t) satisfying dm≤d(t)≤dM. Moreover, the control and observer gain matrices can be reconstructed as follows:(39)F=F¯P¯2-1,L=P3-1L¯.Proof.
From Lemma4, the H∞ stabilization condition is given as follows: (i) 0≤Π2 and (ii) 0>Π0+Π1+Π3=He(e1TPΦt)+e1T(Q1+Q2)e1-e2TQ1e2-e4TQ2e4+ΦtTdm2R1+dI2R2Φt + (e1-e2)TR1(e2-e1)+(e2-e3)TR2(e3-e2)+(e3-e4)TR2(e4-e3)+He((e2-e3)TS(e3-e4))+e1TC~TC~e1 − γ2e5Te5, where Φt=A~e1+A~de3+D~e5. As in the proof of Lemma 6, we first consider the condition given in (ii) by letting R¯1=XP¯R1P¯X and R¯2=XP¯R2P¯X. Then the condition, 0>Π0+Π1+Π3, can be converted by (20) into(40)0>Hee1TXA˘Xe1+e1TXA˘dXe3+e1TXD˘e5+e1TQ1+Q2e1-e2TQ1e2-e4TQ2e4+XA˘Xe1+XA˘dXe3+XD˘e5TX¯dm2R¯1+dI2R¯2×X¯XA˘Xe1+XA˘dXe3+XD˘e5+e1-e2TPX¯R¯1X¯Pe2-e1+e2-e3TPX¯R¯2X¯Pe3-e2+e1TC~TC~e1+e3-e4TPX¯R¯2X¯Pe4-e3+Hee2-e3TSe3-e4-γ2e5Te5,where X¯=X-1. Further, since diag(X¯,X¯,X¯,X¯,I)eiT=eiTX¯, for i=1,2,3,4, and diag(X¯,X¯,X¯,X¯,I)e5T=e5TI, pre- and postmultiplying both sides of (40) by diag(X¯,X¯,X¯,X¯,I) and its transpose yield(41)0>Q+A˘e1+A˘de3+D˘e5T×dm2R¯1+dI2R¯2A˘e1+A˘de3+D˘e5,where Q=He(e1TA˘e1+e1TA˘de3+e1TD˘e5)+e1T(Q~1+Q~2)e1-e2TQ~1e2-e4TQ~2e4+(e1-e2)TR~1(e2-e1) + (e2-e3)TR~2(e3-e2) + (e3-e4)TR~2(e4-e3)+He((e2-e3)TS~(e3-e4))+e1TX¯C~TC~X¯e1-γ2e5Te5, in which Q~1=X¯Q1X¯, Q~2=X¯Q2X¯, R~1=X~R¯1X~, R~2=X~R¯2X~, S~=X¯SX¯, X~=X¯PX¯=diag(P1,P¯2,P¯2P3P¯2)>0. That is, by applying the Schur complement to (41), we can get(42)0>-R¯1-10dmA˘e1+A˘de3+D˘e50-R¯2-1dIA˘e1+A˘de3+D˘e5∗∗Q.Here, since R¯1-1=X~R~1-1X~ and R¯2-1=X~R~2-1X~, it follows from Lemma 2 that R¯1-1≥2μ1X~-μ12R~1 and R¯2-1≥2μ2X~-μ22R~2. In this sense, it is clear that (42) holds if(43)0>1,1′01,3′0dmA˘d01,7′02,2′2,3′0dIA˘d02,7′∗∗3,3′′R~1A˘d03,7′00∗4,4R~2+S~-S~0∗∗∗∗5,5R~2+S~0000∗∗6,60∗∗∗0007,7′,where(44)1,1′=μ12R~1+diag-2μ1P1,-2μ1P¯2,P¯2-2μ1P3P¯2,2,2′=μ22R~2+diag-2μ2P1,-2μ2P¯2,P¯2-2μ2P3P¯2,3,3′′=HeA˘+Q~1+Q~2-R~1+X¯C~TC~X¯,1,3′=dmA˘,1,7′=dmD˘,2,3′=dIA˘,2,7′=dID˘,3,7′=D˘,7,7′=diag-γ2I,-γ2I.However, as shown in (43), there exist some nonconvex terms in (1,1)′, (1,3)′, (1,7)′, (2,2)′, (2,3)′, (2,7)′, (3,3)′′, (3,7)′, and (7,7)′ as follows: (1,1)′, (1,3)′, (2,2)′, and (2,3)′ are the same as those defined in the proof of Lemma 6, and(45)1,7′=0⋆⋆⋆P¯2dmP3DI0,2,7′=0⋆⋆⋆P¯2dIP3DI0,3,7′=0⋆⋆⋆P¯2P3DI0,7,7′=-γ2I00⋆,3,3′′=⋆⋆⋆⋆⋆⋆⋆⋆⋆+P-2HeP3A+L-CP¯2+X-C~TC~X-,where I denotes the identity matrix. As in the proof of Lemma 6, to deal with the nonconvex terms, we apply Lemma 2 to (43), which boils down to(46)0>1,10dmA-0dmA˘d0dmD¯E102,2dIA-0dIA˘d0dID¯E2∗∗3,3′R~1A˘d0D¯E300∗4,4R~2+S~-S~00∗∗∗∗5,5R~2+S~00000∗∗6,600∗∗∗0007,7E4∗∗∗000∗8,8,where (3,3)′=A̿+Q~1+Q~2-R~1+X¯C~TC~X¯. As a result, by applying the Schur complement to X¯C~TC~X¯ in (3,3)′, we can obtain (36). The next step is to convert the given condition in (i), that is, 0≤Π2, into an LMI. To this end, let us pre- and postmultiply both sides of 0≤Π2 by diag(X¯,X¯) and its transpose. Then we can get(47)0≤X-R2X-S~∗X-R2X-,which becomes (23) due to R~2=X¯R2X¯.


## 3.1. Control Design forw~(t)=0

Lemma 6.
Letμ1>0, μ2>0, and ϵ>0 be prescribed. Suppose that there exist matrices F¯∈Rnu×nx, L¯∈Rnx×ny, and S~∈R3nx×3nx and symmetric matrices 0<P1, P¯2,P3∈Rnx×nx, 0<Q~1, Q~2∈R3nx×3nx, 0<R~1, R~2∈R3nx×3nx such that(22)0>1,10dmA¯0dmA˘d0ϵE102,2dIA¯0dIA˘d0ϵE2∗∗3,3R~1A˘d0ϵE300∗4,4R~2+S~-S~0∗∗∗∗5,5R~2+S~0000∗∗6,60∗∗∗0007,7,(23)0≤R~2S~∗R~2,where(24)1,1=μ12R~1+diag-2μ1P1,-2μ1P¯2,-2ϵP¯2,2,2=μ22R~2+diag-2μ2P1,-2μ2P¯2,-2ϵP¯2,3,3=A¯¯+Q~1+Q~2-R~1,4,4=-Q~1-R~1-R~2,5,5=-2R~2-HeS~,6,6=-Q~2-R~2,7,7=-2μ1P30dmP3A+L¯C0-2μ2P3dIP3A+L¯C∗∗HeP3A+L¯C,A˘d=0000BF¯-BF¯000,A¯=P1Ar00A-ArAP¯20000,A̿=HeP1ArA-ArT0∗HeAP¯2000-2ϵP¯2,E1=000000I00,E2=0000000I0,E3=00000000I.Then the closed-loop system (6) is asymptotically stable in the absence of w~(t) for any time-varying delay d(t) satisfying dm≤d(t)≤dM. Moreover, the control and observer gain matrices can be reconstructed as follows:(25)F=F-P-2-1,L=P3-1L-.Proof.
From Lemma3, the stabilization condition is given as follows: (i) 0≤Π2 and (ii) 0>Π0+Π1=He(e1TPΦt)+e1T(Q1+Q2)e1-e2TQ1e2-e4TQ2e4+ΦtTdm2R1+dI2R2Φt + (e1-e2)TR1(e2 − e1) + (e2-e3)TR2(e3-e2)+(e3-e4)TR2(e4-e3)+He((e2-e3)TS(e3-e4)), where Φt=A~e1+A~de3 and thus PΦt=XA˘Xe1+XA˘dXe3. Let us consider the condition given in (ii). Then, by letting R¯1=XP¯R1P¯X and R¯2=XP¯R2P¯X, we can rewrite the condition, 0>Π0+Π1, as follows:(26)0>Hee1TXA˘Xe1+e1TXA˘dXe3+e1TQ1+Q2e1-e2TQ1e2-e4TQ2e4+XA˘Xe1+XA˘dXe3T×X¯dm2R¯1+dI2R¯2X¯XA˘Xe1+XA˘dXe3+e1-e2TPX¯R¯1X¯Pe2-e1+e2-e3TPX¯R¯2X¯Pe3-e2+e3-e4TPX¯R¯2X¯Pe4-e3+Hee2-e3TSe3-e4,where X¯=X-1. Further, since diag(X¯,X¯,X¯,X¯)eiT=eiTX¯ (i=1,2,3,4), pre- and postmultiplying both sides of (26) by diag(X¯,X¯,X¯,X¯) and its transpose yield(27)0>Q+A˘e1+A˘de3Tdm2R¯1+dI2R¯2A˘e1+A˘de3,where Q=He(e1TA˘e1+e1TA˘de3)+e1T(Q~1+Q~2)e1-e2TQ~1e2-e4TQ~2e4+(e1-e2)TR~1(e2-e1) + (e2-e3)TR~2(e3-e2)+(e3-e4)TR~2(e4-e3)+He((e2-e3)TS~(e3-e4)) in which Q~1=X¯Q1X¯, Q~2=X¯Q2X¯, R~1=X~R¯1X~, R~2=X~R¯2X~, S~=X¯SX¯, and X~=X¯PX¯=diag(P1,P¯2,P¯2P3P¯2)>0. That is, by applying the Schur complement to (27), we can get(28)0>-R¯1-10dmA˘e1+A˘de30-R¯2-1dIA˘e1+A˘de3∗∗Q.Here, since R¯1-1=X~R~1-1X~ and R¯2-1=X~R~2-1X~, it follows from Lemma 2 that R¯1-1≥2μ1X~-μ12R~1 and R¯2-1≥2μ2X~-μ22R~2. In this sense, it is clear that (28) holds if(29)0>1,1′01,3′0dmA˘d002,2′2,3′0dIA˘d0∗∗3,3′R~1A˘d000∗4,4R~2+S~-S~∗∗∗∗5,5R~2+S~000∗∗6,6,where(30)1,1′=μ12R~1+diag-2μ1P1,-2μ1P¯2,P¯2-2μ1P3P¯2,2,2′=μ22R~2+diag-2μ2P1,-2μ2P¯2,P¯2-2μ2P3P¯2,3,3′=HeA˘+Q~1+Q~2-R~1,1,3′=dmA˘,2,3′=dIA˘.However, as shown in (29), there exist some nonconvex terms in (1,1)′, (1,3)′, (2,2)′, (2,3)′, and (3,3)′ as follows:(31)1,1′=⋆⋆⋆⋆⋆⋆⋆⋆⋆+P¯2-2μ1P3P¯2,2,2′=⋆⋆⋆⋆⋆⋆⋆⋆⋆+P¯2-2μ2P3P¯2,1,3′=⋆00⋆⋆000P¯2dmP3A+L¯CP¯2,2,3′=⋆00⋆⋆000P¯2dIP3A+L¯CP¯2,3,3′=⋆⋆⋆⋆⋆⋆⋆⋆⋆+P¯2HeP3A+L¯CP¯2.Here, note that all terms associated with P¯2(⋆)P¯2 in(32)1,1′01,3′02,2′2,3′∗∗3,3′can be separated as follows:(33)E1E2E3P¯2000P¯2000P¯2-2μ1P30dmP3A+L¯C0-2μ2P3dIP3A+L¯C∗∗HeP3A+L¯CP¯2000P¯2000P¯2︸ΨE1E2E3T.Furthermore, from Lemma 2, it follows that(34)Ψ≤-2ϵP¯2000P¯2000P¯2-ϵ2-2μ1P30dmP3A+L¯C0-2μ2P3dIP3A+L¯C∗∗HeP3A+L¯C-1,which allows that (22) implies (29), based on the Schur complement. Next, we need to convert the given condition in (i), that is, 0≤Π2, into an LMI. To this end, let us pre- and postmultiply both sides of 0≤Π2 by diag(X¯,X¯) and its transpose. Then we can get(35)0≤X¯R2X¯S~∗X¯R2X¯,which becomes (23) due to R~2=X~R¯2X~=(X¯PX¯)(XP¯R2R¯X)(X¯PX¯)=X¯R2X¯.


## 3.2.  Control Design forw~(t)≠0

Theorem 7.
Letμ1>0, μ2>0, ϵp>0 be prescribed. Suppose that there exist scalars ϵq>0, γ>0; matrices F¯∈Rnu×nx, L¯∈Rnx×ny, S~∈R3nx×3nx; and symmetric matrices 0<P1, P¯2,P3∈Rnx×nx, 0<Q~1,Q~2∈R3nx×3nx, 0<R~1,R~2∈R3nx×3nx such that(36)0>1,10dmA¯0dmA˘d0dmD¯ϵpE1002,2dIA¯0dIA˘d0dID¯ϵpE20∗∗3,3R~1A˘d0D¯ϵpE3X¯C~T00∗4,4R~2+S~-S~000∗∗∗∗5,5R~2+S~000000∗∗6,6000∗∗∗0007,7ϵqE40∗∗∗000∗8,8000∗00000-I,(37)0≤R~2S~∗R~2,where(38)1,1=μ12R~1+diag-2μ1P1,-2μ1P¯2,-2ϵpP¯2,2,2=μ22R~2+diag-2μ2P1,-2μ2P¯2,-2ϵpP¯2,3,3=A̿+Q~1+Q~2-R~1,4,4=-Q~1-R~1-R~2,5,5=-2R~2-HeS~,6,6=-Q~2-R~2,7,7=diag-2ϵqI,-γ2I,8,8=-2μ1P30dmP3A+L¯CdmP3D0-2μ2P3dIP3A+L¯CdIP3D∗∗HeP3A+L¯CP3D∗∗∗-γ2I,A˘d=0000BF¯-BF¯000,A¯=P1Ar00A-ArAP¯20000,A̿=HeP1ArA-ArT0∗HeAP¯2000-2ϵpP¯2,D¯=0P1D-I00,C~T=0CT0,E1=00000000I000,E2=000000000I00,E3=0000000000I0,E4=000I0000.Then the closed-loop system (6) is asymptotically stable and satisfies z~2<γw~2   for all nonzero w~(t)∈L2+ and for any time-varying delay d(t) satisfying dm≤d(t)≤dM. Moreover, the control and observer gain matrices can be reconstructed as follows:(39)F=F¯P¯2-1,L=P3-1L¯.Proof.
From Lemma4, the H∞ stabilization condition is given as follows: (i) 0≤Π2 and (ii) 0>Π0+Π1+Π3=He(e1TPΦt)+e1T(Q1+Q2)e1-e2TQ1e2-e4TQ2e4+ΦtTdm2R1+dI2R2Φt + (e1-e2)TR1(e2-e1)+(e2-e3)TR2(e3-e2)+(e3-e4)TR2(e4-e3)+He((e2-e3)TS(e3-e4))+e1TC~TC~e1 − γ2e5Te5, where Φt=A~e1+A~de3+D~e5. As in the proof of Lemma 6, we first consider the condition given in (ii) by letting R¯1=XP¯R1P¯X and R¯2=XP¯R2P¯X. Then the condition, 0>Π0+Π1+Π3, can be converted by (20) into(40)0>Hee1TXA˘Xe1+e1TXA˘dXe3+e1TXD˘e5+e1TQ1+Q2e1-e2TQ1e2-e4TQ2e4+XA˘Xe1+XA˘dXe3+XD˘e5TX¯dm2R¯1+dI2R¯2×X¯XA˘Xe1+XA˘dXe3+XD˘e5+e1-e2TPX¯R¯1X¯Pe2-e1+e2-e3TPX¯R¯2X¯Pe3-e2+e1TC~TC~e1+e3-e4TPX¯R¯2X¯Pe4-e3+Hee2-e3TSe3-e4-γ2e5Te5,where X¯=X-1. Further, since diag(X¯,X¯,X¯,X¯,I)eiT=eiTX¯, for i=1,2,3,4, and diag(X¯,X¯,X¯,X¯,I)e5T=e5TI, pre- and postmultiplying both sides of (40) by diag(X¯,X¯,X¯,X¯,I) and its transpose yield(41)0>Q+A˘e1+A˘de3+D˘e5T×dm2R¯1+dI2R¯2A˘e1+A˘de3+D˘e5,where Q=He(e1TA˘e1+e1TA˘de3+e1TD˘e5)+e1T(Q~1+Q~2)e1-e2TQ~1e2-e4TQ~2e4+(e1-e2)TR~1(e2-e1) + (e2-e3)TR~2(e3-e2) + (e3-e4)TR~2(e4-e3)+He((e2-e3)TS~(e3-e4))+e1TX¯C~TC~X¯e1-γ2e5Te5, in which Q~1=X¯Q1X¯, Q~2=X¯Q2X¯, R~1=X~R¯1X~, R~2=X~R¯2X~, S~=X¯SX¯, X~=X¯PX¯=diag(P1,P¯2,P¯2P3P¯2)>0. That is, by applying the Schur complement to (41), we can get(42)0>-R¯1-10dmA˘e1+A˘de3+D˘e50-R¯2-1dIA˘e1+A˘de3+D˘e5∗∗Q.Here, since R¯1-1=X~R~1-1X~ and R¯2-1=X~R~2-1X~, it follows from Lemma 2 that R¯1-1≥2μ1X~-μ12R~1 and R¯2-1≥2μ2X~-μ22R~2. In this sense, it is clear that (42) holds if(43)0>1,1′01,3′0dmA˘d01,7′02,2′2,3′0dIA˘d02,7′∗∗3,3′′R~1A˘d03,7′00∗4,4R~2+S~-S~0∗∗∗∗5,5R~2+S~0000∗∗6,60∗∗∗0007,7′,where(44)1,1′=μ12R~1+diag-2μ1P1,-2μ1P¯2,P¯2-2μ1P3P¯2,2,2′=μ22R~2+diag-2μ2P1,-2μ2P¯2,P¯2-2μ2P3P¯2,3,3′′=HeA˘+Q~1+Q~2-R~1+X¯C~TC~X¯,1,3′=dmA˘,1,7′=dmD˘,2,3′=dIA˘,2,7′=dID˘,3,7′=D˘,7,7′=diag-γ2I,-γ2I.However, as shown in (43), there exist some nonconvex terms in (1,1)′, (1,3)′, (1,7)′, (2,2)′, (2,3)′, (2,7)′, (3,3)′′, (3,7)′, and (7,7)′ as follows: (1,1)′, (1,3)′, (2,2)′, and (2,3)′ are the same as those defined in the proof of Lemma 6, and(45)1,7′=0⋆⋆⋆P¯2dmP3DI0,2,7′=0⋆⋆⋆P¯2dIP3DI0,3,7′=0⋆⋆⋆P¯2P3DI0,7,7′=-γ2I00⋆,3,3′′=⋆⋆⋆⋆⋆⋆⋆⋆⋆+P-2HeP3A+L-CP¯2+X-C~TC~X-,where I denotes the identity matrix. As in the proof of Lemma 6, to deal with the nonconvex terms, we apply Lemma 2 to (43), which boils down to(46)0>1,10dmA-0dmA˘d0dmD¯E102,2dIA-0dIA˘d0dID¯E2∗∗3,3′R~1A˘d0D¯E300∗4,4R~2+S~-S~00∗∗∗∗5,5R~2+S~00000∗∗6,600∗∗∗0007,7E4∗∗∗000∗8,8,where (3,3)′=A̿+Q~1+Q~2-R~1+X¯C~TC~X¯. As a result, by applying the Schur complement to X¯C~TC~X¯ in (3,3)′, we can obtain (36). The next step is to convert the given condition in (i), that is, 0≤Π2, into an LMI. To this end, let us pre- and postmultiply both sides of 0≤Π2 by diag(X¯,X¯) and its transpose. Then we can get(47)0≤X-R2X-S~∗X-R2X-,which becomes (23) due to R~2=X¯R2X¯.


## 4. Numerical Example

We provide two examples to verify the effectiveness of the proposed methods in Lemma6 and Theorem 7. For the networked output-feedback control system (NOCS), we assume that the sampling period h=0.01 and the data transmission delay bounds are given by τm=0.005 and τM=0.01. As a result, from [3], it follows that dm=τm=0.005[s] and dM=(o¯+1)h+τM=0.01o¯+0.02[s], where o¯ denotes the maximum number of data-packet dropouts.


### 4.1. Example 1

Consider a continuous-time system of the following form:(48)x˙t=01-1αxt+01ut+01wt,yt=10xt,where α>0 is a variable element. First of all, to show the applicability of the proposed method in Lemma 6, we search the maximum allowable upper bounds (MAUBs) for (48) with w~(t)=col(w(t),r(t))=0∈Rnx+nw. To this end, let us set μ1=μ2=0.1, ϵ=10, and Ar=diag(1,1). Then, from Lemma 6, we can obtain the MAUBs for α=1,2,3, which are tabulated in Table 1. Now, let us analyze the behavior of the tracking response for x(t) and xr(t) of the NOCS in the case where w~(t)≠0 by using the derived condition in Theorem 7. For this purpose, we set α=1, o¯=10, μ1=μ2=0.1, ϵ=10, and Ar=diag(1,1). Then, from Theorem 7, we can obtain the following control and observer gain matrices: F=[-5.5501-4.9075], L=[-31.1609-179.7242]T. In addition, the disturbance attenuation is given by γ=1.1679. Here we assume that w(t)=e-0.5tsin(2πt), x(0)=col(±0.5,±0.5), xr(t)=x^(t)=0, for t∈[-dM,0], and r(t)=col(0.2sin0.2πt,0.04πcos0.2πt), for t≥0, where the initial time t0 is set to zero. Figure 2(a) shows the x1-x2 trajectories for four different initial conditions x(0), which form a specific ellipse, made by the given reference input r(t), as the time t increases. Further, the behavior of the estimation error e1(t)=y(t)-y^(t) is depicted in Figure 2(d), from which we can see that the estimation error goes to zero as the time t increases. Figures 2(b) and 2(c) show the behavior of the state x(t) of (49) for initial condition x(0)=(0.5,-0.5), where the network-induced delay d(t) is generated as shown in Figure 2(e) such that the data transmission delay τ(t)∈[0.005,0.01] and the data-packet dropouts o¯=10. From Figures 2(b) and 2(c), we can see that the state x(t) tracks the reference signal xr(t) well; that is, the tracking response of the NOCS with (2), (5), and (48) is in a good shape with respect to our control goal.Table 1
Maximum allowable upper bounds (MAUBs) forα of Example 1.
α
1
2
3
τ
M
0.32 (o-=30)
0.19 (o-=17)
0.09 (o-=7)Figure 2
(a) Thex1-x2 trajectory for each different initial condition x(0); (b) the tracking response of x1(t) for xr,1(t); (c) the tracking response of x2(t) for xr,2(t); (d) the estimation error e1(t)=y(t)-y^(t); and (e) the network-induced delay d(t). Here, xi, x^i, ei, and xr,i denote the ith element of x, x^, e, and xr, respectively.


### 4.2.  Example 2

Consider the following satellite system, modified from [5]: Ar=diag(-1,-1,-1,-1),(49)A=0.0000.0001.0000.0000.0000.0000.0001.000-0.3000.300-0.0040.0040.300-0.3000.004-0.004,B=0010,CT=0100,D=0001.Through this example, we will achieve the H∞ performance for (49) based on Theorem 7 to design an observer-based NOCS in such a way that the state x(t) of (49) tracks the reference signal xr(t) in the H∞ sense. The obtained H∞ performance for each upper bound dM is tabulated in Table 2, where μ1=μ2=0.1 and ϵp=10 are assumed. From Table 2, we can see that the H∞ performance is improved as dM decreases from 0.1 to 0.02, which is reasonable.Table 2
H
∞ performance for each upper bound dM.
d
M
0.02
0.03
0.05
0.1
γ
0.8265
0.9072
1.1337
2.3290


## 4.1. Example 1

Consider a continuous-time system of the following form:(48)x˙t=01-1αxt+01ut+01wt,yt=10xt,where α>0 is a variable element. First of all, to show the applicability of the proposed method in Lemma 6, we search the maximum allowable upper bounds (MAUBs) for (48) with w~(t)=col(w(t),r(t))=0∈Rnx+nw. To this end, let us set μ1=μ2=0.1, ϵ=10, and Ar=diag(1,1). Then, from Lemma 6, we can obtain the MAUBs for α=1,2,3, which are tabulated in Table 1. Now, let us analyze the behavior of the tracking response for x(t) and xr(t) of the NOCS in the case where w~(t)≠0 by using the derived condition in Theorem 7. For this purpose, we set α=1, o¯=10, μ1=μ2=0.1, ϵ=10, and Ar=diag(1,1). Then, from Theorem 7, we can obtain the following control and observer gain matrices: F=[-5.5501-4.9075], L=[-31.1609-179.7242]T. In addition, the disturbance attenuation is given by γ=1.1679. Here we assume that w(t)=e-0.5tsin(2πt), x(0)=col(±0.5,±0.5), xr(t)=x^(t)=0, for t∈[-dM,0], and r(t)=col(0.2sin0.2πt,0.04πcos0.2πt), for t≥0, where the initial time t0 is set to zero. Figure 2(a) shows the x1-x2 trajectories for four different initial conditions x(0), which form a specific ellipse, made by the given reference input r(t), as the time t increases. Further, the behavior of the estimation error e1(t)=y(t)-y^(t) is depicted in Figure 2(d), from which we can see that the estimation error goes to zero as the time t increases. Figures 2(b) and 2(c) show the behavior of the state x(t) of (49) for initial condition x(0)=(0.5,-0.5), where the network-induced delay d(t) is generated as shown in Figure 2(e) such that the data transmission delay τ(t)∈[0.005,0.01] and the data-packet dropouts o¯=10. From Figures 2(b) and 2(c), we can see that the state x(t) tracks the reference signal xr(t) well; that is, the tracking response of the NOCS with (2), (5), and (48) is in a good shape with respect to our control goal.Table 1
Maximum allowable upper bounds (MAUBs) forα of Example 1.
α
1
2
3
τ
M
0.32 (o-=30)
0.19 (o-=17)
0.09 (o-=7)Figure 2
(a) Thex1-x2 trajectory for each different initial condition x(0); (b) the tracking response of x1(t) for xr,1(t); (c) the tracking response of x2(t) for xr,2(t); (d) the estimation error e1(t)=y(t)-y^(t); and (e) the network-induced delay d(t). Here, xi, x^i, ei, and xr,i denote the ith element of x, x^, e, and xr, respectively.


## 4.2.  Example 2

Consider the following satellite system, modified from [5]: Ar=diag(-1,-1,-1,-1),(49)A=0.0000.0001.0000.0000.0000.0000.0001.000-0.3000.300-0.0040.0040.300-0.3000.004-0.004,B=0010,CT=0100,D=0001.Through this example, we will achieve the H∞ performance for (49) based on Theorem 7 to design an observer-based NOCS in such a way that the state x(t) of (49) tracks the reference signal xr(t) in the H∞ sense. The obtained H∞ performance for each upper bound dM is tabulated in Table 2, where μ1=μ2=0.1 and ϵp=10 are assumed. From Table 2, we can see that the H∞ performance is improved as dM decreases from 0.1 to 0.02, which is reasonable.Table 2
H
∞ performance for each upper bound dM.
d
M
0.02
0.03
0.05
0.1
γ
0.8265
0.9072
1.1337
2.3290


## 5. Concluding Remarks

This paper has addressed the observer-basedH∞ tracking problem of NOCSs with network-induced delays. In the derivation, a single-step procedure is proposed to handle nonconvex terms that appear in the process of designing observer-based output-feedback control, and then a set of linear matrix inequality conditions are established for the solvability of the tracking problem.



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*Source: 724389-2015-07-29.xml*