# Resilient Robust Finite-TimeL2-L∞ Controller Design for Uncertain Neutral System with Mixed Time-Varying Delays

Authors: Xia Chen; Shuping He
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The delay-dependent resilient robust finite-timeL2-L∞ control problem of uncertain neutral time-delayed system is studied. The disturbance input is assumed to be energy bounded and the time delays are time-varying. Based on the Lyapunov function approach and linear matrix
inequalities (LMIs) techniques, a state feedback controller is designed to guarantee that the resulted closed-loop system is finite-time bounded for all uncertainties and to satisfy a given L2-L∞ constraint condition. Simulation results illustrate the validity of the proposed approach.


---

## Body

## 1. Introduction

Dynamical systems with time delays and uncertain parameters have been of considerable interest over the past decades. In fact, time delays are always the important source of system instability and poor performance [1–4]. As a special class of time-delay systems, the neutral type time-delayed system has also received some attention in recent years. This time-delayed system contains time delays both in its state and in the derivative of its states. Moreover, neutral time-delayed systems are frequently encountered in many dynamics, such as automatic control, distributed network system containing lossless transmission line, heat exchangers, and population ecology. Various analysis approaches have been utilized to find stability criteria and control design conditions for asymptotic stability of neutral time delays [5–10].It is now worth pointing out that the control performances mentioned above concern the desired behavior of control dynamics over an infinite-time interval and it always deals with the asymptotic property of system trajectories. For controlling a dynamical system, it can meet the requirements of asymptotic stability, but it will not reflect the transient characteristics. Asymptotic stability is unable to satisfy the transient requirements of industrial production if there exists large amount of overshoot, oscillation change, and nonlinear disturbance within a finite-time interval. To deal with this transient performance of control dynamics, Dorato gave the concept of finite-time stability [11] (or short-time stability) in the early 1960s. Then, the relevant concepts of finite-time bounded (FTB) [12], finite-time stabilization [13], finite-time H∞ control [14], and finite-time L2-L∞ [15] control have been revisited in form of linear matrix inequalities (LMIs) techniques. And this transient performance is widely applied to time-delay systems, uncertain systems, nonlinear systems, stochastic systems, and so forth. However, to the best of our knowledge, very few results in the literature consider the related control problems of neutral time-varying delays in the finite-time interval.On the other hand, theL2-L∞ performance has attracted considerable attention as an important performance evaluation index when it was first proposed in 1989 [16]. In engineering practice, although the study of the impact of noise and delay on the system performance is important, the extremum problem of the controlled output cannot be ignored, because the controlled output should be controlled within a certain range. In control theory and engineering application, the L2-L∞ control has very important significance that lies in its performance index which can control the output value minimization. Unfortunately, up to now, the theme of L2-L∞ control design of uncertain neutral systems with time-varying delays has received little attention.Motivated by the above discussion, this paper focuses on the problem of finite-timeL2-L∞ controller design for a class of neutral systems with mixed time-varying delays and uncertainties. By constructing a suitable Lyapunov function, the sufficient conditions are derived that closed-loop controlled system is FTB and satisfies the given finite-time interval induced L2-L∞ norm of the operator from the unknown disturbance to the output. We also show that the L2-L∞ controller designing problem can be dealt with by solving a set of coupled LMIs. Finally, a numerical example illustrates the effectiveness of the developed techniques.


## 2. Problem Statement

Consider the following neutral time-delayed system with uncertainties:(1)Σ0:{x˙(t)-(C+ΔC(t))x˙(t-τ(t))=(A+ΔA(t))x(t)+(Ad+ΔAd(t))x(t-h(t))+Bu(t)+(D+ΔD(t))w(t)y(t)=(F+ΔF(t))x(t)+Gu(t)x(t0+θ)=ϕ(θ),θ∈[-max⁡{h,τ},0],t0=0,
where x(t)∈Rn is the state, u(t)∈Rm is the controlled input, y(t)∈Rq is the controlled output, and w(t)∈Rp is the disturbance input that belongs to L2[0,+∞) and for a given positive number δ and constant time T, the following form is satisfied:
(2)∫0TwT(t)w(t)dt≤δ,δ≥0.h(t) and τ(t) are time-varying delays and satisfy
(3)0≤h(t)≤h,h˙(t)≤hd,0≤τ(t)≤τ,τ˙(t)≤τd<1,
where h, τ, hd, and τd are constant scalars. ϕ(θ)∈L2[-max⁡{h,τ},0] is the continuous initial function. A, Ad, C, D, and F∈Rn×n are known constant matrices, and ΔA(t), ΔAd(t), ΔC(t), ΔD(t), and ΔF(t) are unknown time-variant matrices representing the norm-bounded parameter uncertainties and satisfy the following form:
(4)[ΔA(t)ΔAd(t)ΔC(t)ΔD(t)]=M1σ(t)[H1H2H3H4],(5)ΔF(t)=M2σ(t)H1,
where M1, M2, H1, H2, H3, and H4 are known real matrices with suitable dimension and σ(t) is an unknown real and possibly time-varying matrix with Lebesgue measurable elements satisfying
(6)σT(t)σ(t)≤I.In this paper, we consider the state feedback controller as follows:(7)u(t)=(K+ΔK(t))x(t),
where K is the unknown controller gain and ΔK(t) is the time-varying controller gain which satisfies
(8)ΔK(t)=Nη(t)S,ηT(t)η(t)≤I.Then, we can get the following closed-loop control system:(9)Σ:{x˙(t)-C-x˙(t-τ(t))=A^x(t)+A-dx(t-h(t))+D-w(t)y(t)=F^x(t)x(t0+θ)=ϕ(θ),θ∈[-max⁡{h,τ},0],t0=0,
where A^=A-+ΔA-(t), A-=A+BK, ΔA-(t)=ΔA(t)+BΔK(t), A-d=Ad+ΔAd(t), C-=C+ΔC(t), D-=D+ΔD(t), F^=F-+ΔF-(t), F-=F+GK, and ΔF-(t)=ΔF(t)+GΔK(t).The main purpose of this paper is to design an appropriate resilient state feedback controller (7), such that the closed-loop control system Σ is finite-time bounded and satisfies the given performance index constraints.Before proceeding with the study, we give the relevant definitions and lemmas first.Definition 1.
For given positive scalarsc1, δ, and T and a symmetrical positive determined matrix R, the closed-loop system Σ is robust finite-time bounded (FTB) with respect to (c1,c2,δ,R,T), if there exists a positive constant c2 with c2>c1, such that, for all the external disturbances w(t) satisfying condition (2), the following formula is satisfied:
(10)ϕT(θ)Rϕ(θ)≤c1⟹xT(t)Rx(t)<c2,∀t∈[0,T].Remark 2.
If the disturbance input is not present in the closed-loop system, that is,w(t)=0, the concept of FTB will reduce into finite-time stability (FTS). It is worth mentioning that Lyapunov stability and finite-time stability are two different concepts. The former is largely known to the control characteristic in infinite-time interval, but the latter concerns the boundedness analysis of the controlled states within a finite-time interval. Obviously, a finite-time stable system may not be Lyapunov stochastically stable and vice versa.Definition 3.
The state feedback controller in the form of (7) is considered as a robust finite-time L2-L∞ controller for the closed-loop system Σ, if the system Σ is FTB with respect to (c1,c2,δ,R,T) and under the zero initial condition, there exist two positive scalars γ and T for all disturbance which satisfy condition (2), such that
(11)∥y(t)∥∞2≤γ2∥w(t)∥22,
where ∥y(t)∥∞2=sup⁡t∈[0,T]|yT(t)y(t)|, ∥w(t)∥22=∫0TwT(t)w(t)dt.Lemma 4 (see [17]).
For any real positive scalarsα, β (where α>β) and a positive definite symmetric matrix S, then the following inequality holds for a vector function ω:[β,α]→Rn which can let the integrals converge:
(12)(∫βαω(σ)dσ)TS(∫βαω(σ)dσ)≤(α-β)(∫βαωT(σ)Sω(σ)dσ).Lemma 5 (see [17]).
For any positive scalarh and positive definite symmetric matrix S, the following inequality is satisfied:
(13)2h2(∫-h0∫t+θtω(σ)dσdθ)TS(∫-h0∫t+θtω(σ)dσdθ)≤∫-h0∫t+θtωT(σ)Sω(σ)dσdθ.Lemma 6 (see [15]).
For any given appropriate dimension matrixH and E, if there exists a matrix W(t) which satisfied WT(t)W(t)≤I and a scalar ε>0, then
(14)HW(t)E+ETWT(t)HT≤ε-1HHT+εETE.


## 3. Main Results

In this section, our main purpose is to solve the design problem of a resilient robust finite-timeL2-L∞ controller for a class of uncertain neutral systems with mixed time-varying delays.Theorem 7.
Given positive scalarsc1, δ, T, and α, positive definite symmetric matrix R, and time-delay parameters h>0, hd>0, τ>0, and τd>0, the closed-loop system Σ is FTB with respect to (c1,c2,δ,R,T), if there exist positive scalars λi,i=1,2,…,6, c2, and symmetric positive definite matrices Pi,i=1,2,…,6, Qi,i=1,2,…,4, and Wi,i=1,2,…,6, such that
(15)Π=[Π1Π2Π3*Π4Π5**Π6]<0,(16)c1[λ2+hλ3+hλ4+τλ5+τλ6]+δ(1-e-αT)<λ1c2e-αT,
where
(17)Π1=[Πij]7×7,Π11=A^TP1+P1A^+P2+P3+P4+P5+W1+W3+W4+W6+W1T+W3T+W4T+W6T-αP1-ατP6-2αQ1-2αQ2-2αQ3-2αQ4,Π12=P1A-d-W1+W2+W1T+W3T+W4T+W6T,Π13=P1C-+W1T+W3T+W4T+W6T,Π14=-W4+W5+W1T+W3T+W4T+W6T+ατP6,Π15=-W2-W3+W1T+W3T+W4T+W6T,Π16=-W5-W6+W1T+W3T+W4T+W6T,Π17=P1D-,Π22=-(1-hd)P2-W1+W2-W1T+W2T,Π23=-W1T+W2T,Π24=-W4+W5-W1T+W2T,Π25=-W2-W3-W1T+W2T,Π26=-W5-W6-W1T+W2T,Π27=0,Π33=-(1-τd)P6,Π34=-W4+W5,Π35=-W2-W3,Π36=-W5-W6,Π37=0,Π44=-(1-τd)P4-W4+W5-W4T+W5T-ατP6,Π45=-W2-W3-W4T+W5T,Π46=-W5-W6-W4T+W5T,Π47=0,Π55=-P3-W2-W3-W2T-W3T,Π56=-W5-W6-W2T-W3T,Π57=0,Π66=-P5-W5-W6-W5T-W6T,Π67=0,Π77=-αI,Π2=[hW1hW2hW3τW4τW5τW6hW1hW2hW3τW4τW5τW6hW1hW2hW3τW4τW5τW6hW1hW2hW3τW4τW5τW6hW1hW2hW3τW4τW5τW6hW1hW2hW3τW4τW5τW6000000],Π3=[A^TP6hA^TQ1hA^TQ2τA^TQ3τA^TQ4A-dTP6hA-dTQ1hA-dTQ2τA-dTQ3τA-dTQ4C-TP6hC-TQ1hC-TQ2τC-TQ3τC-TQ4000000000000000D-TP6hD-TQ1hD-TQ2τD-TQ3τD-TQ4],Π4=diag⁡{-hQ1-hQ1-hQ2-τQ3-τQ3-τQ4},Π5=[0]6×5,Π6=diag⁡{-P6-hQ1-hQ2-τQ3-τQ3}.Proof.
Construct a positive definite Lyapunov function as follows:(18)V(t)=V1(t)+V2(t)+V3(t)+V4(t)+V5(t),
where
(19)V1(t)=xT(t)P1x(t),V2(t)=∫t-h(t)txT(s)P2x(s)ds+∫t-htxT(s)P3x(s)ds,V3(t)=∫t-τ(t)txT(s)P4x(s)ds+∫t-τtxT(s)P5x(s)ds,V4(t)=∫t-τ(t)tx˙T(s)P6x˙(s)ds,V5(t)=∫-h0∫t+θtx˙T(s)(Q1+Q2)x˙(s)dsdθ+∫-τ0∫t+θtx˙T(s)(Q3+Q4)x˙(s)dsdθ.
We take the time derivative of V(t) along the trajectory of system Σ and it yields the following:
(20)V˙1(t)=xT(t)(P1A^+A^TP1)x(t)+xT(t)P1A-dx(t-h(t))+xT(t)P1C-x˙(t-τ(t))+xT(t)P1D-w(t)+xT(t-h(t))A-dTP1x(t)+x˙T(t-τ(t))C-TP1x(t)+ωT(t)D-TP1x(t),V˙2(t)≤xT(t)(P2+P3)x(t)-(1-hd)xT(t-h(t))P2x(t-h(t))-xT(t-h)P3x(t-h),V˙3(t)≤xT(t)(P4+P5)x(t)-(1-τd)xT(t-τ(t))P4x(t-τ(t))-xT(t-τ)P5x(t-τ),V˙4(t)≤x˙T(t)P6x˙(t)-(1-τd)x˙T(t-τ(t))P6x˙(t-τ(t)),V˙5(t)=x˙T(t)(h(Q1+Q2)+τ(Q3+Q4))x˙(t)-∫t-h(t)tx˙T(s)Q1x˙(s)ds-∫t-ht-h(t)x˙T(s)Q1x˙(s)ds-∫t-htx˙T(s)Q2x˙(s)ds-∫t-τ(t)tx˙T(s)Q3x˙(s)ds-∫t-τt-τ(t)x˙T(s)Q3x˙(s)ds-∫t-τtx˙T(s)Q4x˙(s)ds.
For any symmetric positive definite matricesWi,i=1,2,…,6, the following equations are satisfied according to Leibniz-Newton lemma:
(21)2ζT(t)W1[x(t)-x(t-h(t))-∫t-h(t)tx˙(s)ds]=0,2ζT(t)W2[x(t-h(t))-x(t-h)-∫t-ht-h(t)x˙(s)ds]=0,2ζT(t)W3[x(t)-x(t-h)-∫t-htx˙(s)ds]=0,2ζT(t)W4[x(t)-x(t-τ(t))-∫t-τ(t)tx˙(s)ds]=0,2ζT(t)W5[x(t-τ(t))-x(t-τ)-∫t-τt-τ(t)x˙(s)ds]=0,2ζT(t)W6[x(t)-x(t-τ)-∫t-τtx˙(s)ds]=0,
where(22)ζ(t)=[xT(t)xT(t-h(t))x˙T(t-τ(t))xT(t-τ(t))xT(t-h)xT(t-τ)]T,ξ(t)=[ζT(t)wT(t)]T.
According to (20)-(21), we can obtain(23)V˙(t)=V˙1(t)+V˙2(t)+V˙3(t)+V˙4(t)+V˙5(t)≤ξT(t)Ω1ξ(t)-∫t-h(t)tx˙T(s)Q1x˙(s)ds-∫t-ht-h(t)x˙T(s)Q1x˙(s)ds-∫t-htx˙T(s)Q2x˙(s)ds-∫t-τ(t)tx˙T(s)Q3x˙(s)ds-∫t-τt-τ(t)x˙T(s)Q3x˙(s)ds-∫t-τtx˙T(s)Q4x˙(s)ds+2ζT(t)W1[x(t)-x(t-h(t))-∫t-h(t)tx˙(s)ds]+2ζT(t)W2[-∫t-ht-h(t)x˙(s)ds]x(t-h(t))-x(t-h)kkkkkkkkkkkkk-∫t-ht-h(t)x˙(s)ds]+2ζT(t)W3[x(t)-x(t-h)-∫t-htx˙(s)ds]+2ζT(t)W4[x(t)-x(t-τ(t))-∫t-τ(t)tx˙(s)ds]+2ζT(t)W5[∫t-τt-τ(t)x˙(s)dsx(t-τ(t))-x(t-τ)kkkkkkkkkkkkk-∫t-τt-τ(t)x˙(s)ds]+2ζT(t)W6[x(t)-x(t-τ)-∫t-τtx˙(s)ds]=ξT(t)Ω1ξ(t)+ζT(t)Ω2ζ(t)-∫t-h(t)t(ζT(t)W1+x˙T(s)Q1)kkkkkkkkkk×Q1-1(W1Tζ(t)+Q1x˙(s))ds-∫t-ht-h(t)(ζT(t)W2+x˙T(s)Q1)hhhhhhhhhh×Q1-1(W2Tζ(t)+Q1x˙(s))ds-∫t-ht(ζT(t)W3+x˙T(s)Q2)kkkkkkkkk×Q2-1(W3Tζ(t)+Q2x˙(s))ds-∫t-τ(t)t(ζT(t)W4+x˙T(s)Q3)kkkkkkkkkk×Q3-1(W4Tζ(t)+Q3x˙(s))ds-∫t-τt-τ(t)(ζT(t)W5+x˙T(s)Q3)kkkkkkkkkk×Q3-1(W5Tζ(t)+Q3x˙(s))ds-∫t-τt(ζT(t)W6+x˙T(s)Q4)kkkkkkkkk×Q4-1(W6Tζ(t)+Q4x˙(s))ds.
Since Q1, Q2, Q3, and Q4 are positive definite symmetric matrices, we have
(24)V˙(t)≤ξT(t)Ω1ξ(t)+ζT(t)Ω2ζ(t),
where
(25)Ω1=[Ωij]7×7,Ω11=A^TP1+P1A^+P2+P3+P4+P5+A^T(P6+h(Q1+Q2)+τ(Q3+Q4))A^+W1+W3+W4+W6+W1T+W3T+W4T+W6T,Ω12=P1A-d+A^T(P6+h(Q1+Q2)+τ(Q3+Q4))A-d-W1+W2+W1T+W3T+W4T+W6T,Ω13=P1C-+A^T(P6+h(Q1+Q2)+τ(Q3+Q4))C-+W1T+W3T+W4T+W6T,Ω14=-W4+W5+W1T+W3T+W4T+W6T,Ω15=-W2-W3+W1T+W3T+W4T+W6T,Ω16=-W5-W6+W1T+W3T+W4T+W6T,Ω17=P1D-+A^T(P6+h(Q1+Q2)+τ(Q3+Q4))D-,Ω22=-(1-hd)P2+A-dT(P6+h(Q1+Q2)+τ(Q3+Q4))A-d-W1+W2-W1T+W2T,Ω23=A-dT(P6+h(Q1+Q2)+τ(Q3+Q4))C--W1T+W2T,Ω24=-W4+W5-W1T+W2T,Ω25=-W2-W3-W1T+W2T,Ω26=-W5-W6-W1T+W2T,Ω27=A-dT(P6+h(Q1+Q2)+τ(Q3+Q4))D-,Ω33=-(1-τd)P6+C-T(P6+h(Q1+Q2)+τ(Q3+Q4))C-,Ω34=-W4+W5,Ω35=-W2-W3,Ω36=-W5-W6,Ω37=C-T(P6+h(Q1+Q2)+τ(Q3+Q4))D-,Ω44=-(1-τd)P4-W4+W5-W4T+W5T,Ω45=-W2-W3-W4T+W5T,Ω46=-W5-W6-W4T+W5T,Ω47=0,Ω55=-P3-W2-W3-W2T-W3T,Ω56=-W5-W6-W2T-W3T,Ω57=0,Ω66=-P5-W5-W6-W5T-W6T,Ω67=0,Ω77=D-T(P6+h(Q1+Q2)+τ(Q3+Q4))D-,Ω2=hW1Q1-1W1T+hW2Q1-1W3T+hW3Q2-1W3T+τW4Q3-1W4T+τW5Q3-1W5T+τW6Q4-1W6T.
Recalling formula (24) and Lemmas 4 and 5 and using Schur complement, we can get
(26)V˙(t)-αV(t)-αwT(t)w(t)≤ξT(t)Πξ(t)<0;
that is,
(27)V˙(t)<αV(t)+αwT(t)w(t).
Pre- and postmultiplying (27) by e-αt, we have
(28)ddt(e-αtV(t))<αe-αtwT(t)w(t).
Then integrating the aforementioned inequality from 0 to t, where t∈[0,T], it yields
(29)e-αtV(t)-V(0)<α∫0te-ατwT(τ)w(τ)dτ.
Considering condition (2), (29) can be simplified as
(30)V(t)<eαt[V(0)+α∫0te-ατwT(τ)w(τ)dτ]<eαT[V(0)+δ(1-e-αT)].
On the other hand,
(31)V(t)≥V1(t)=xT(t)P1x(t)≥λmin⁡(P~1)xT(t)Rx(t),V(0)≤ϕT(θ)P1ϕ(θ)+hϕT(θ)P2ϕ(θ)+hϕT(θ)P3ϕ(θ)+τϕT(θ)P4ϕ(θ)+τϕT(θ)P5ϕ(θ)≤λmax⁡(P~1)ϕT(θ)Rϕ(θ)+hλmax⁡(P~2)ϕT(θ)Rϕ(θ)+hλmax⁡(P~3)ϕT(θ)Rϕ(θ)+τλmax⁡(P~4)ϕT(θ)Rϕ(θ)+τλmax⁡(P~5)ϕT(θ)Rϕ(θ)≤λmax⁡(P~1)c1+hλmax⁡(P~2)c1+hλmax⁡(P~3)c1+τλmax⁡(P~4)c1+τλmax⁡(P~5)c1.
Then, formula (27) can be written as
(32)xT(t)Rx(t)≤c1[λ2+hλ3+hλ4+τλ5+τλ6]+δ(1-e-αT)λ1e-αT,
which can be guaranteed by condition (16). This completes the proof.According to Theorem7, we will obtain the resilient robust finite-time L2-L∞ controller for a class of uncertain neutral system with mixed time-varying delays.Theorem 8.
Given positive scalarsc1, T, δ, and α, positive definite symmetric matrix R, and time-delay parameters h>0, hd>0, τ>0, and τd>0, the closed-loop neutral system Σ is FTB with respect to (c1,c2,δ,R,T) and satisfies the cost function (11) for all admissible disturbance w(t), if there exist positive scalars c2 and β and symmetric positive definite matrices Pi,i=1,2,…,6, Qi,i=1,2,…,4, Wi,i=1,2,…,6, such that conditions (15) and (16) and the following LMI hold:
(33)Ψ=[-P1F^T*-βI]<0.Proof.
Similar to the proof of Theorem7, (29) can be rewritten as
(34)e-αtV(t)<α∫0te-ατwT(τ)w(τ)dτ.
Then, we have
(35)xT(t)P1x(t)≤V(t)<αeαT∫0twT(τ)w(τ)dτ.
From (33), we can obviously get
(36)F^TF^<βP1.
Considering system Σ, we have
(37)yT(t)y(t)=[F^x(t)]T[F^x(t)]=xT(t)F^TF^x(t).
Combining (35)–(37), we can obtain
(38)xT(t)F^TF^x(t)≤βxT(t)P1x(t)≤βαeαT∫0twT(τ)w(τ)dτ;
that is,
(39)yT(t)y(t)≤βαeαT∫0twT(τ)w(τ)dτ.
Lettingγ2=βαeαT, we have ∥y(t)∥∞2<γ2∥w(t)∥22. This completes the proof.Theorem 9.
Given positive scalarsc1, T, δ, and α, positive definite symmetric matrix R, and time-delay parameters h>0, hd>0, τ>0, and τd>0, the closed-loop neutral system Σ is FTB with respect to (c1,c2,δ,R,T), satisfies the cost function (11) for all admissible disturbance w(t), and exists as a state feedback controller in the form of (7) with K=UP1-1, if there exist positive scalars c2, β, εi, i=1,2,…,4, and μi, i=1,2,…,5, and symmetric positive definite matrices Li, i=1,2,…,10, Ti, i=1,2,…,6, Q-i, i=1,2,…,4, Pi, i=2,3,…,5, P-6, and U, such that the following LMIs are feasible:
(40)Π~=[Π~1Π~2Π~3Π~7*Π~4Π~5Π~8**Π~6Π~9***Π~10]<0,(41)Ψ~=[-L1L1FT+UTGTL1H1TL1ST*Ψ~2200**-ε3I0***-ε4I]<0,(42)μ1R-1<L1<R-1,(43)0<P2<μ2R,(44)0<P3<μ3R,(45)0<P4<μ4R,(46)0<P5<μ5R,(47)[c1[h(μ2+μ3)+τ(μ4+μ5)]+δ(1-e-αT)-c2e-αTc1*-μ1]<0,
where(48)
Π
~
1
=
[
Π
~
i
j
]
7
×
7
,
Π
~
11
=
L
1
A
T
+
A
L
1
+
U
T
B
T
+
B
U
+
L
2
+
L
3
+
L
4
+
L
5
+
T
1
+
T
3
+
T
4
+
T
6
Π
~
1
=
[
Π
~
i
j
]
7
×
7
,
Π
~
11
=
+
T
1
T
+
T
3
T
+
T
4
T
+
T
6
T
-
α
L
1
-
α
τ
L
6
-
2
α
L
7
-
2
α
L
8
-
2
α
L
9
-
2
α
L
10
,
Π
~
12
=
A
d
L
1
-
T
1
+
T
2
+
T
1
T
+
T
3
T
+
T
4
T
+
T
6
T
,
Π
~
13
=
C
L
1
+
T
1
T
+
T
3
T
+
T
4
T
+
T
6
T
,
Π
~
14
=
-
T
4
+
T
5
+
T
1
T
+
T
3
T
+
T
4
T
+
T
6
T
+
α
τ
L
6
,
Π
~
15
=
-
T
2
-
T
3
+
T
1
T
+
T
3
T
+
T
4
T
+
T
6
T
,
Π
~
16
=
-
T
5
-
T
6
+
T
1
T
+
T
3
T
+
T
4
T
+
T
6
T
,
Π
~
17
=
D
,
Π
~
22
=
-
(
1
-
h
d
)
L
2
-
T
1
+
T
2
-
T
1
T
+
T
2
T
,
Π
~
23
=
-
T
1
T
+
T
2
T
,
Π
~
24
=
-
T
4
+
T
5
-
T
1
T
+
T
2
T
,
Π
~
25
=
-
T
2
-
T
3
-
T
1
T
+
T
2
T
,
Π
~
26
=
-
T
5
-
T
6
-
T
1
T
+
T
2
T
,
Π
~
27
=
0
,
Π
~
33
=
-
(
1
-
τ
d
)
L
6
,
Π
~
34
=
-
T
4
+
T
5
,
Π
~
35
=
-
T
2
-
T
3
,
Π
~
36
=
-
T
5
-
T
6
,
Π
~
37
=
0
,
Π
~
44
=
-
(
1
-
τ
d
)
L
4
-
T
4
+
T
5
-
T
4
T
+
T
5
T
-
α
τ
L
6
,
Π
~
45
=
-
T
2
-
T
3
-
T
4
T
+
T
5
T
,
Π
~
46
=
-
T
5
-
T
6
-
T
4
T
+
T
5
T
,
Π
~
47
=
0
,
Π
~
55
=
-
L
3
-
T
2
-
T
3
-
T
2
T
-
T
3
T
,
Π
~
56
=
-
T
5
-
T
6
-
T
2
T
-
T
3
T
,
Π
~
57
=
0
,
Π
~
66
=
-
L
5
-
T
5
-
T
6
-
T
5
T
-
T
6
T
,
Π
~
67
=
0
,
Π
~
77
=
-
α
I
,
Π
~
2
=
[
h
T
1
h
T
2
h
T
3
τ
T
4
τ
T
5
τ
T
6
h
T
1
h
T
2
h
T
3
τ
T
4
τ
T
5
τ
T
6
h
T
1
h
T
2
h
T
3
τ
T
4
τ
T
5
τ
T
6
h
T
1
h
T
2
h
T
3
τ
T
4
τ
T
5
τ
T
6
h
T
1
h
T
2
h
T
3
τ
T
4
τ
T
5
τ
T
6
h
T
1
h
T
2
h
T
3
τ
T
4
τ
T
5
τ
T
6
0
0
0
0
0
0
]
,
Π
~
3
=
[
L
1
A
T
+
U
T
B
T
h
L
1
A
T
+
h
U
T
B
T
h
L
1
A
T
+
h
U
T
B
T
τ
L
1
A
T
+
τ
U
T
B
T
τ
L
1
A
T
+
τ
U
T
B
T
L
1
A
d
T
h
L
1
A
d
T
h
L
1
A
d
T
τ
L
1
A
d
T
τ
L
1
A
d
T
L
1
C
T
h
C
T
Q
1
h
C
T
Q
2
τ
C
T
Q
3
τ
C
T
Q
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
D
T
h
D
T
h
D
T
τ
D
T
τ
D
T
]
,
Π
~
4
=
diag
⁡
{
-
h
L
7
-
h
L
7
-
h
L
8
-
τ
L
9
-
τ
L
9
-
τ
L
10
}
,
Π
~
5
=
[
0
]
6
×
5
,
Π
~
6
=
diag
⁡
{
-
P
-
6
-
h
Q
-
1
-
h
Q
-
2
-
τ
Q
-
3
-
τ
Q
-
3
}
,
Π
~
7
=
[
L
1
H
1
T
L
1
S
T
ε
1
M
1
ε
2
B
N
L
1
H
2
T
0
0
0
L
1
H
3
T
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
L
1
H
4
T
0
0
0
]
,
Π
~
8
=
[
0
]
6
×
4
,
Π
~
10
=
diag
⁡
{
-
ε
1
I
-
ε
1
I
-
ε
2
I
-
ε
2
I
}
,
Ψ
~
22
=
-
β
I
+
ε
3
M
T
M
2
T
+
ε
4
G
N
T
N
G
T
.Proof.
ReplacingA^, A-d, C-, and D- in (15) with A^=A-+ΔA-(t), A-=A+BK, ΔA-(t)=ΔA(t)+BΔK(t), A-d=Ad+ΔAd(t), C-=C+ΔC(t), and D-=D+ΔD(t), respectively, we have
(49)Π=Π-+ΔΠ-<0,
where(50)Π-=[Π-1Π2Π-3*Π4Π5**Π6]<0,Π-1=[Π-ij]7×7,Π-1=[Π-11Π-12Π-13Π14Π15Π16P1D*Π22Π23Π24Π25Π26Π27**Π33Π34Π35Π36Π37***Π44Π45Π46Π47****Π55Π56Π57*****Π66Π67******Π77],Π-11=A-TP1+P1A-+P2+P3+P4+P5+W1+W3+W4+W6+W1T+W3T+W4T+W6T-αP1-ατP6-2αQ1-2αQ2-2αQ3-2αQ4,Π-12=P1Ad-W1+W2+W1T+W3T+W4T+W6T,Π-13=P1C+W1T+W3T+W4T+W6T,Π-3=[A-TP6hA-TQ1hA-TQ2τA-TQ3τA-TQ4AdTP6hAdTQ1hAdTQ2τAdTQ3τAdTQ4CTP6hCTQ1hCTQ2τCTQ3τCTQ4000000000000000DTP6hDTQ1hDTQ2τDTQ3τDTQ4],ΔΠ-=[ΔΠ-10ΔΠ-3*00**0]<0,ΔΠ-1=[ΔA-T(t)P1+P1ΔA-(t)P1ΔA-d(t)P1ΔC-(t)000P1ΔD-(t)*000000**00000***0000****000*****00******0],ΔΠ-3=[ΔA-T(t)P6hΔA-T(t)Q1hΔA-T(t)Q2τΔA-T(t)Q3τΔA-T(t)Q4ΔAdT(t)P6hΔAdT(t)Q1hΔAdT(t)Q2τΔAdT(t)Q3τΔAdT(t)Q4ΔCT(t)P6hΔCT(t)Q1hΔCT(t)Q2τΔCT(t)Q3τΔCT(t)Q4000000000000000ΔDT(t)P6hΔDT(t)Q1hΔDT(t)Q2τΔDT(t)Q3τΔDT(t)Q4],bring formulas (4) and (8) into ΔΠ-, and according to Lemma 6, we have
(51)Γd1σ(t)Γe1+(Γd1σ(t)Γe1)T≤ε1Γd1Γd1T+ε1-1Γe1TΓe1,Γd2η(t)Γe2+(Γd2η(t)Γe2)T≤ε2Γd2Γd2T+ε2-1Γe2TΓe2,
where(52)Γd1=[M1TP1000000000000M1TP6hM1TQ1hM1TQ2τM1TQ3τM1TQ4]T,Γe1=[H1H2H3000H400000000000],Γd2=[NTBTP1000000000000NTBTP6hNTBTQ1hNTBTQ2τNTBTQ3τNTBTQ4]T,Γe2=[S00000000000000000].Considering
(53)ΔΠ-=Γd1σ(t)Γe1+(Γd1σ(t)Γe1)T+Γd2η(t)Γe2+(Γd2η(t)Γe2)T≤ε1Γd1Γd1T+ε1-1Γe1TΓe1+ε2Γd2Γd2T+ε2-1Γe2TΓe2,
(49) can be guaranteed by
(54)Π-+ε1Γd1Γd1T+ε1-1Γe1TΓe1+ε2Γd2Γd2T+ε2-1Γe2TΓe2<0.
Using Schur complement, equality (54) can be rewritten as
(55)Π^=[Π-1Π2Π-3Π^7*Π4Π-5Π^8**Π-6Π^9***Π^10]<0,
where
(56)Π^7=[S1TNTε1P1M1ε2P1BNS2T000S3T000000000000000S4T000],Π^9=[00ε1P6M1ε2P6BN00hε1Q1M1hε2Q1BN00hε1Q2M1hε2Q2BN00τε1Q3M1τε2Q3BN00τε1Q4M1τε2Q4BN],Π^8=[0]6×4,Π^10=diag⁡{-ε1I-ε2I-ε1I-ε2I}.
LettingL1=P1-1, L2=L1P2L1, L3=L1P3L1, L4=L1P4L1, L5=L1P5L1, L6=L1P6L1, L7=L1Q1L1, L8=L1Q2L1, L9=L1Q3L1, L10=L1Q4L1, T1=L1W1L1, T2=L1W2L1, T3=L1W3L1, T3=L1W3L1, T4=L1W4L1, T5=L1W5L1, T6=L1W6L1, P-6=P6-1, Q-1=Q1-1, Q-2=Q2-1, Q-3=Q.3-1, Q-4=Q4-1, and U=KL1, we can obtain condition (40) by pre- and postmultiplying inequality (55) by block-diagonal matrix(57)diag⁡{P1-1P1-1P1-1P1-1P1-1P1-1IP1-1P1-1P1-1P1-1P1-1P1-1P6-1Q1-1Q2-1Q3-1Q4-1IIII}.
Next, we will prove that condition (33) is equivalent to (41).
Considering(58)Ψ=Ψ-+ΔΨ-<0,
where
(59)Ψ-=[-P1F-T*-βI],ΔΨ-=[0ΔF-T(t)*0],
combining with formulas (5) and (8), and using Schur complement, we have
(60)ΔΨ-=Γd3σ(t)Γe3+(Γd3σ(t)Γe3)T+Γd4η(t)Γe4+(Γd4η(t)Γe4)T≤ε3Γd3Γd3T+ε3-1Γe3TΓe3+ε4Γd4Γd4T+ε4-1Γe4TΓe4,
where Γd3=[0M2T]T, Γe3=[H10], Γd4=[0NTGT]T, and Γe4=[S0].
Then, we can get the following inequality which ensures (58):
(61)Ψ-+ε3Γd3Γd3T+ε3-1Γe3TΓe3+ε4Γd4Γd4T+ε4-1Γe4TΓe4<0,
Using the Schur complement, equality (61) can be rewritten as
(62)Ψ^=[-P1FT+KTGTH1TST*Ψ~2200**-ε3I0***-ε4I]<0.
Then, we can obtain condition (40) by pre- and postmultiplying inequality (62) by block-diagonal matrix diag⁡{P1-1III}.
DenotingL~1=R1/2L1R1/2, P~2=R-1/2P2R-1/2, P~3=R-1/2P3R-1/2, P~4=R-1/2P4R-1/2, and P~5=R-1/2P5R-1/2, we know that condition (16) is equivalent to (47) according to conditions (42)–(46). This completes the proof.


## 4. Simulation Example

In this part, we consider a class of neutral time-varying delayed systems with parameters described as(63)A=[1.50.22.10.9],Ad=[-1.1-0.2-0.1-1.1],B=[1.00.8],C=[-0.200.2-0.1],R=[1001],D=[0.10.2-0.20.1],F=[1.51.70.20.9],G=[2-1.5],M1=[1.1-0.7],M2=[0.8-0.4],H1=[1.40.8],H2=[0.41.1],H3=[0.70.2],H4=[0.51.3],N=[0.2],S=[0.20.6].In this note, we choose the initial values forc1=1, T=5, α=0.3, and δ=1.0 and the upper bounds on the delays are τ=0.8, h=0.5, hd=0.9, and τd=0.9. By using the LMI toolbox in MATLAB to solve LMIs (40)–(47), we can get the finite-time L2-L∞ controller gain as follows:
(64)L1=[0.6515-0.1789-0.17890.3827],U=[-0.3115-0.0343],K=UL1-1=[-0.5768-0.3593],
with constraint conditions β=14.7085, γ=0.9923, and c2=124.6975.Selectingh(t)=0.9/(1+t2), τ(t)=0.11/(3+t2), σ(t)=(0.9/(1+t2))I, η(t)=(1.5/(1+t2))I, and h(t)=0.9/(1+t2), t∈[0,20], and setting the initial states x0=[-0.50.8]T and w0=[0.040.08]T, we have the open-loop controlled system state simulation graph and the trajectories of closed-loop controlled system state and output as shown in Figures 1, 2, and 3, respectively. Figure 4 shows the evolution of function xT(t)Rx(t)(t∈[0,20]) of the uncertain neutral time-delayed system Σ0. Based on comparison between result in Figure 1 and result in Figure 2, we noted that the design finite-time L2-L∞ controller can make the closed-loop controlled system achieve FTB.Figure 1
The trajectories of open-loop controlled system statex(t).Figure 2
The trajectories of closed-loop controlled system statex(t).Figure 3
The trajectories of closed-loop controlled system outputy(t).Figure 4
The graph ofxT(t)Rx(t)  (t∈[0,T]) of closed-loop controlled system.


## 5. Conclusions

This paper studied the delay-dependent resilient robust finite-timeL2-L∞ control problem for a class of uncertain neutral time-delayed system with mixed time-varying delays. A state feedback controller is designed by using LMI technique and free weighting matrices, such that the closed-loop controlled system is FTB and satisfies the input-output L2-L∞ performance matrices. The simulation results verify the effectiveness of the design method. We will consider the finite-time observer for neutral time-delayed system in the future.



---

*Source: 304824-2014-05-13.xml*