Delay-dependent robust H control for uncertain systems with ......Information Sciences, vol. 178, no. 8, pp. 3187-3197, August 2009. Delay-dependent robust H∞control for uncertain

Information Sciences, vol. 178, no. 8, pp. 3187-3197, August 2009.

Delay-dependent robust H∞ control for

uncertain systems with time-varying delay

Chen Peng a, Yu-Chu Tian b,1

aSchool of Electrical and Automation Engineering, Nanjing Normal University,

Nanjing, Jiangsu 210042, P. R. China

bSchool of Information Technology, Queensland University of Technology, GPO

Box 2434, Brisbane QLD 4001, Australia

Abstract

This paper proposes a new approach for delay-dependent robust H∞ stability anal-ysis and control synthesis of uncertain systems with time-varying delay. The keyfeatures of the approach include the introduction of a new Lyapunov-Ksrasovskiifunctional, the construction of an augmented matrix with uncorrelated terms, andthe employment of a tighter bounding technique. As a result, significant performanceimprovement is achieved in system analysis and synthesis without using either freeweighting matrices or model transformation. Examples are given to demonstratethe effectiveness of the proposed approach.

Key words: Robust H∞ control; time-delay systems; uncertain systems; boundedreal lemma; delay-dependent stability

1 Introduction

In robust stability analysis of time-delay systems, an early technique for bound-ing cross terms was Park’s inequality. Later, Moon et al. [12] introduced a dif-ferent inequality, which was more general than Park’s one, for conservativenessreduction. Fridman and Shaked [1] proposed a descriptor model transforma-tion of time-delay systems, and used the bounding techniques from both Parkand Moon et al. for H∞ controller design. Following the technique of Moon et

1 Corresponding author. Phone: +61-7-3138 2177, fax: +61-7-3138-2703. Email:[email protected] (Y.-C. Tian).

Preprint submitted to Elsevier Preprint Version on 12 Jan 2009

al. [12], Gao and Wang [3] improved the results of [1]. Several other approacheshave been developed in the last few years to further reduce the conservative-ness of delay-dependent conditions for stability analysis and robust controlsynthesis, e.g., [10,11,21,24].

Recently, much effort has been made in exploring the free weighting matrixtechnique, which was originally proposed by He and colleagues [8,9]. Intro-ducing free variables into the Lyapunov-Krasovskii functional, Lee et al. [11]proposed a delay-dependent robust H∞ control, which was less conservativethan that in [1], for uncertain linear systems with state delay. Xu et al. [21]added null sum terms to Lyapunov functional’s derivative, and obtained lessconservative results than those from previous methods due to the avoidanceof using any bounding technology.

For networked control systems in which data networks induce time-varyingdelay, Peng and Tian [15], Tian et al. [19], and Yue et al. [22,23] introducedsome free matrix parameters to deal with cross terms, and derived stabil-ity criteria which were less conservative than previous ones. Combining thedescriptor model transformation method and the free weighting matrix tech-nique, Parlakçı [13] investigated the delay-dependent stability problem andobtained improved stability conditions. However, system synthesis has been adifficult problem in all these methods [13,15,16,19,22,23] due to the use of freematrix variables.

So far, both the free weighting matrix technique and the model transformationmethod have been popularly used. However, we have realised that althoughintroducing free weighting matrix variables gives a feasible solution to thelinear matrix inequalities (LMIs) in system analysis [17,18], too many freematrix variables will complicate system synthesis and significantly increasethe computational demand. It is also understood that model transformationis a main source of conservativeness [5].

An open question is: is it possible to obtain the same or less conservative resultsusing an approach in which neither model transformation nor free weightingmatrix variables are employed? This paper will develop such an approach thatleads to a positive answer to this question.

The main features of the proposed approach are highlighted as follows:

• Neither free weighting matrices are employed in the Lyapunov-Krasovskiifunctional nor any null sum terms are added to Lyapunov functional’sderivative, leading to decreased computational demand and simplified sys-tem synthesis in comparison with various free weighting matrix methods;• A new Lyapunov–Krasovskii functional is constructed, which gives the ad-

ditional design matrix one more (potential) relaxation in comparison withtraditional methods;

2

• A tighter bounding technology for cross terms is employed to reduce theconservativeness. In comparison with the inequality of Moon et al. [12], thisbounding technology does not apply special structure limitation to matrixvariables for derivation of the controller synthesis conditions in terms ofLMIs; and• Simpler augmented terms are used to derive an improved delay-dependent

bounded real lemma (BRL) for uncertain time-delay systems.

With these features, the proposed approach can lead to significant perfor-mance improvement in system analysis and synthesis for a large class of delaysystems. Linear systems with time-varying delay and norm-bounded uncer-tainties will be addressed in this paper with regard to delay-dependent robustH∞ control. Two performance indices, which are popularly used to evaluatethe conservativeness of the stability conditions, will be adopted to quantifythe system performance:

• One is the H∞ performance index γ. For a prescribed upper bound τM ofthe delay, the smaller the value of γ is the better the stability conditionsare.• The other is the upper bound τM of the delay. For a prescribed performance

index γ, the larger the value of τM is the less conservative the stabilityconditions are.

Notation: Throughout this paper, N stands for positive integers; Rn denotesthe n-dimensional Euclidean space; Rn×m is the set of n×m real matrices; I isthe identity matrix of appropriate dimensions. The notation X > 0 (or X ≥ 0)for X ∈ Rn×n means that the matrix X is a real symmetric positive definite (orpositive semi-definite). For an arbitrary matrix B and two symmetric matrices

A and C,

A B

∗ C

is a symmetric matrix, where ∗ denotes the entries implied

by symmetry.

2 Problem statement

Consider the following uncertain system with time-varying delay

ẋ(t) = (A + ∆A)x(t) + (Ad + ∆Ad)x(t− τ(t)) + Bu(t) + B̟̟(t)

z(t) = Cx(t) + D̟̟(t) + Cdx(t− τ(t)) + Du(t)

x(t) = φ(t), t ∈ [−τM , 0]

(1)

3

where x(t) ∈ Rn is the state vector; u(t) ∈ Rm is the control input; ̟(t) ∈ Rp

is the disturbance input that belongs to L2[0,∞); z(t) ∈ Rq is the controlled

output. A and B are constant matrices with appropriate dimensions. ∆A and∆B denote the parameter uncertainties satisfying the following conditions:

[∆A, ∆B] = MF (t) [Ea, Eb] , (2)

where M , Ea and Eb are constant matrices with appropriate dimensions; andF (t) is an unknown time-varying matrix, which is Lebesque measurable int and satisfies F T (t)F (t) ≤ I. The time delay, τ(t), is a time-varying andcontinuous-time function satisfying:

0 ≤ τ(t) ≤ τM , |τ̇(t)| ≤ d < 1,∀t ≥ 0 (3)

where τM and d are constants and ϕ(t) is the initial condition of the system.

In the following, we will develop some practically computable criteria for anal-ysis and synthesis of the system governed by (1). The following definition andlemma are useful in deriving the criteria.

Definition 1 A system governed by (1) is said to be robustly asymptoticallystable with an H∞ norm bound γ if the following conditions hold:

1) For the system with w(t) ≡ 0, the trivial solution (equilibrium point) isglobally asymptotically stable if limt→∞ x(t) = 0; and

2) Under the assumption of zero initial condition, the controlled output z(t)satisfies

‖z(t)‖2 ≤ γ ‖̟(t)‖2 (4)

for any nonzero ̟(t) ∈ L2[0,∞).

Lemma 1 [17] For any constant matrices Q11, Q22, Q12 ∈ Rn×n, Q11 > 0,

Q22 > 0,

Q11 Q12

∗ Q22

≥ 0, scalar 0 ≤ τ(t) ≤ τM , and vector function ẋ :

[−τM , 0]→ Rn such that the following integration is well defined, then

−τM

∫ t

t−τM

x(t)

ẋ(t)

T

Q11 Q12

∗ Q22

x(t)

ẋ(t)

dt

≤

x(t)

x(t− τ(t))∫ tt−τ(t) x(t)dt

T

−Q22 Q22 −QT12

∗ −Q22 QT12

∗ ∗ −Q11

x(t)

x(t− τ(t))∫ tt−τ(t) x(t)dt

(5)

4

3 Delay-dependent bounded real lemma

In this section, we will establish a new version of delay-dependent boundedreal lemma (BRL) for time-delay system (1) with u(t) ≡ 0. Setting u(t) ≡ 0in (1), we obtain the following nominal time-delay system

ẋ(t) = Ax(t) + Adx(t− τ(t)) + B̟̟(t)

z(t) = Cx(t) + Cdx(t− τ(t)) + D̟̟(t)

x(t) = φ(t), t ∈ [−τM , 0]

(6)

Theorem 1 Given scalars τM > 0, γ > 0 and d > 0, if there exist matricesZ, S, R, Q22 and P11 > 0, Q11 and P22 ≥ 0, and any matrices Q12 and P12with appropriate dimensions such that the following LMIs hold

Ω1 =

Ω11 Ω12 Ω13 Ω14 CT τMA

T Q22 dP12 0

∗ Ω22 Ω23 0 CTd τMA

Td Q22 0 dP22

∗ ∗ Ω33 Ω34 0 0 0 0

∗ ∗ ∗ −γ2I DT̟ τMBT̟Q22 0 0

∗ ∗ ∗ ∗ −I 0 0 0

∗ ∗ ∗ ∗ ∗ −Q22 0 0

∗ ∗ ∗ ∗ ∗ ∗ −dS 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −dZ

< 0 (7)

Q11 Q12

∗ Q22

≥ 0,

P11 P12

∗ P22

≥ 0. (8)

where

Ω11 = R−Q22 + τ2M(Q11 + Q12A + A

T QT12) + P12 + PT12 + P11A + A

T P T11

Ω12 = Q22 + τ2MQ12Ad + P11Ad − P12

Ω13 = AT P12 + P

T22 −Q

T12

Ω14 = P11B̟ + τ2MQ12B̟

Ω22 = dS − (1− d)R−Q22

Ω23 = ATd P12 − P22 + Q

T12

Ω33 = dZ −Q11Ω34 = P

T12B̟

then system (6) is asymptotically stable.

5

PROOF. Construct a Lyapunov functional candidate as

V (xt) = V1(t) + V2(t) + V3(t) (9)

where

V1(t) =∫ t

t−τ(t)xT (s)Rx(s)ds, (10)

V2(t) = τM

∫ 0

−τM

∫ t

t+s

x(θ)

ẋ(θ)

T

Q11 Q12

∗ Q22

x(θ)

ẋ(θ)

dθds, (11)

V3(t) =

x(t)∫ tt−τ(t) x(s)ds

T

P11 P12

∗ P22

x(t)∫ tt−τ(t) x(s)ds

, (12)

and R > 0,

Q11 Q12

∗ Q22

≥ 0,

P11 P12

∗ P22

≥ 0 are to be determined. The time

derivative of V (t) is taken along the state trajectory (6), yielding

V̇1(t) = xT (t)Rx(t)− (1− τ̇(t))xT (t− τ(t))Rx(t− τ(t)) (13)

V̇2(t) = τ2M

x(t)

ẋ(t)

T

Q11 Q12

∗ Q22

x(t)

ẋ(t)

−τM

∫ t

t−τM

x(t)

ẋ(t)

T

Q11 Q12

∗ Q22

x(t)

ẋ(t)

dt (14)

For the first term of (14), it is seen that:

τ 2M

x(t)

ẋ(t)

T

Q11 Q12

∗ Q22

x(t)

ẋ(t)

= τ 2MξT (t)

I AT

0 ATd

0 0

0 BT̟

Q11 Q12

∗ Q22

I AT

0 ATd

0 0

0 BT̟

T

ξ(t) (15)

6

where the augmented matrix ξT (t) =[

xT (t), xT (t− τ(t)), (∫ tt−τ(t) x(t)dt)

T , ̟T (t)]

;it is constructed differently from those in the existing methods.

Furthermore, for the last term of (14), from Lemma 1, we have:

−τM

∫ t

t−τM

x(t)

ẋ(t)

T

Q11 Q12

∗ Q22

x(t)

ẋ(t)

dt ≤ ξT (t)

−Q22 Q22 −QT12 0

∗ −Q22 QT12 0

∗ ∗ −Q11 0

0 0 0 0

ξ(t)

(16)

V̇3(t) = 2

x(t)∫ tt−τ(t) x(s)ds

T

P11 P12

∗ P22

ẋ(t)

x(t)− (1− τ̇(t))x(t− τ(t))

= 2ξT (t)

I 0

0 0

0 I

0 0

P11 P12

∗ P22

A Ad 0 B̟

I −(1− τ̇(t)) 0 0

ξ(t) (17)

For some matrices Z > 0, S > 0, the following two inequities always holdbased on (3).

2τ̇(t)xT (t)P12x(t− τ(t))

≤ dxT (t)P12S−1P T12x(t) + dx

T (t− τ(t))Sx(t− τ(t)) (18)

2τ̇(t)xT (t− τ(t))P22

∫ t

t−τ(t)x(t)dt ≤ dxT (t− τ(t))P22Z

−1P T22x(t− τ(t))

+d(∫ t

t−τ(t)x(t)dt)T Z

∫ t

t−τ(t)x(t)dt (19)

According to (17), (18) and (19), we have

V̇3(t) ≤ ξT (t)Θξ(t) (20)

where

7

Θ =

Θ11 P11Ad − P12 AT P12 + P

T22 P11B̟

∗ dS + dP22Z−1P T22 A

Td P12 − P22 0

∗ ∗ dZ P T12B̟

∗ ∗ ∗ ∗

Θ11 = P12 + PT12 + P11A + A

T P T11 + dP12S−1P T12

Considering (9), (13), (15), (16) and (20) together, we have

V̇ (xt) ≤ ξT (t)Ωξ(t)− z(t)T z(t) + γ2̟T (t)̟(t) (21)

where

Ω =

Ω11 + dP12S−1P T12 Ω12 Ω13 Ω14

∗ Ω22 + dP22Z−1P T22 Ω23 0

∗ ∗ Ω33 Ω34

∗ ∗ ∗ −γ2I

+

τMAT Q22

τMATd Q22

0

τMBT̟Q22

Q−122

τMAT Q22

τMATd Q22

0

τMBT̟Q22

T

+

CT

CTd

0

DT̟

CT

CTd

0

DT̟

T

Ωij (i, j = 1, . . . , 4) is defined in Theorem 1.

Based on (7) and by Schur complement, (21) implies that

V̇ (xt) ≤ −z(t)T z(t) + γ2̟(t)T̟(t) (22)

Integrating both sides of (22) from t0 to t, we obtain

V (t)− V (t0) ≤ −∫ t

t0

z(s)T z(s)ds +∫ t

t0

γ2̟(s)T ̟(s)ds (23)

Then, letting t → ∞ and under zero initial condition, we obtain from (23)that

∫

∞

t0

z(s)T z(s)ds ≤∫

∞

t0

γ2̟(s)T ̟(s)ds

thus ‖z(t)‖2 ≤ γ ‖̟(t)‖2 is satisfied for any non-zero ̟(t) ∈L2[0,∞).

8

Next, we can prove the asymptotic stability of systems (6). When ̟(t) ≡ 0,we obtain the following result based on (21)

V̇ (xt) ≤ ζT (t)Ω̃ζ(t) (24)

where ζT (t) =[

xT (t), xT (t− τ(t)), (∫ tt−τ(t) x(t)dt)

T]

,

Ω̃ =

Ω11 + dP12S−1P T12 Ω12 Ω13

∗ Ω22 + dP22Z−1P T22 Ω23

∗ ∗ Ω33

+

τMAT Q22

τMATd Q22

0

Q−122

τMAT Q22

τMATd Q22

0

T

Combining (7) and using Schur complement, we have V̇ (xt) < 0, which givesV̇ (xt) < −ρ ‖x(t)‖

2 for a sufficiently small ρ > 0, and ensures the asymptoticstability of system (6) for any delay satisfying (3). Then, by Definition 1, theresult is established. This completes the proof. �

Remark 1 The idea of introducing free weighting matrices has been exten-sively used in analysis and synthesis of time-delay systems [2,9,13,21] andnetworked control systems [15,22,23,16,19]. Although the free weighting matrixtechnique shows the flexibility in solving LMIs and somewhat leads to less con-servativeness than Park’s bounding technology and some model transformationmethods, its defective functions are unavoidable, e.g., the high computationaldemand and the difficulty in controller synthesis due to the superfluous freevariables. It is seen that neither free weighting matrices nor any model trans-formations have been introduced in our proof of Theorem 1; this is achievedbecause only uncorrelated terms are used in the construction of the augmentedmatrix ξT (t) in (15). However, better results can be obtained from our approachas will be shown later in Section 5.

Remark 2 It is worth mentioning that Q12 in (11) and P12 in (12) play animportant role in reducing conservativeness in our approach. The cross termsof 2xT (t)P12

∫ tt−τ(t) x(s)ds, 2

∫ 0−τM

∫ tt+s ẋ

T (t)Q12ẋ(t)dtds, and 2xT (t)P12x(t −

τ(t)) are introduced into the Lyapunov-Krasovskii functional in which Q12and P12 are selected in accordance with (8). Therefore, these two additionaldesign matrices, i.e., Q12 and P12, give a potential relaxation [13,18], andconsequently less conservative results can be expected. The improved effects ofthese additional terms will also be shown in Section 5.

9

Remark 3 In comparison with the inequality of Moon et al. [12], Lemma 1 isa more general and tighter bounding technology to deal with cross terms. There-fore, the BRL derived here is expected to be less conservative. Furthermore,Lemma 1 and Proposition 1 in [7] are respectively special cases of Lemma 1and Theorem 1 of this paper when we set Q11 = Q12 = 0 and P12 = P22 = 0in system (1). But when Q11, Q12, P12 and P22 6= 0, the results derived in thispaper are less conservative than those in [7] due to the novel construction ofthe Lyapunov functional components in (11) and (12).

For time-delay system (1) with uncertainties satisfying (2) and u(t) ≡ 0, fromTheorem 1, we can obtain the following delay-dependent BRL through using aroutine method [6] handling norm-bounded uncertainties (the proof is omittedhere).

Theorem 2 Given scalars τM , γ and d > 0, if there exist matrices Z, S, R,Q22 and P11 > 0, Q11 and P22 ≥ 0, scalar ε > 0, and any matrices Q12 and P12with appropriate dimensions such that the following matrix inequalities hold

Ω2 =

Ω11 Ω12 Ω13 Ω14 Ω15 CT τMA

T Q22 dP12 0 εETa

∗ Ω22 Ω23 0 0 CTd τMA

Td Q22 0 dP22 εE

Tb

∗ ∗ Ω33 Ω34 Ω35 0 0 0 0 0

∗ ∗ ∗ −γ2I 0 DT̟ τMBT̟Q22 0 0 0

∗ ∗ ∗ ∗ −εI 0 τMMT Q22 0 0 0

∗ ∗ ∗ ∗ ∗ −I 0 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ −Q22 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −dS 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −dZ 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −εI

< 0 (25)

Q11 Q12

∗ Q22

≥ 0,

P11 P12

∗ P22

≥ 0. (26)

where

Ω15 = P11M + τ2MQ12M , Ω35 = P

T12M

Ωij (i, j = 1, . . . , 4) is defined in Theorem 1, then system (1) is asymptoticallystable.

Remark 4 Both Theorems 1 and 2 provide an improved delay-dependent BRLfor systems with time-varying delay. Theorem 1 applies to systems without

10

uncertainties, while Theorem 2 is for uncertain systems. For a given τM , theminimum γ that satisfies (7) for Theorem 1 or (25) for Theorem 2 can beobtained by solving a quasi-convex optimisation problem.

4 Robust H∞ control for uncertain systems

Using the BRL derived in the last section, we now design a feedback controllergain K to make system (1) robustly asymptotically stable with the normbound γ.

4.1 Main Results

Theorem 3 Given scalars τM , γ and d > 0, if there exist matrices Z, S, R,Hi (i = 1, 2, 3), P11 and Q22 > 0, Q11 and P22 ≥ 0, a scalar ε > 0, and anymatrices Q12, P12 with appropriate dimensions such that the following matrixinequalities hold

Σ =

Σ11 Σ12 Σ13

∗ Σ22 0

∗ ∗ Σ33

< 0, (27)

Q11 Q12

∗ Q22

≥ 0,

P11 P12

∗ P22

≥ 0, (28)

where

Σ11 =

Ω11 Ω12 Ω13 Ω14 Ω15

∗ Ω22 Ω23 0 0

∗ ∗ Ω33 Ω34 Ω35

∗ ∗ ∗ −γ2I 0

∗ ∗ ∗ ∗ −εI

,

11

Σ12 =

(C + DK)T τTM(A + BK)T dP12 0 εE

Ta

CTd τMATd 0 dP22 εE

Tb

0 0 0 0 0

DT̟ τMBT̟ 0 0 0

0 τMMT 0 0 0

,

Σ13 =

τMQ12 P11 0 τMKT BT KT BT KT BT

0 0 0 0 0 0

0 0 P T12 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

,

Σ22 = diag{−I,−Q−122 ,−dS,−dZ,−εI},

Σ33 = diag{−H1,−H2,−H3,−H−11 ,−H

−12 ,−H

−13 },

Ωij (i, j = 1, . . . , 4) is defined in Theorem 1, then system (1) is robustly asymp-totically stable with the memoryless feedback controller gain K and H∞ normbound γ.

PROOF. Assume that a proportional feedback controller is employed, i.e.,u(t) = Kx(t). A and Du(t) in (1) are replaced by A + BK and DKx(t),respectively.

For matrices Hi (i = 1, 2, 3) > 0, the following inequalities always hold

τ 2M(Q12BK + (BK)T QT12) ≤ τ

2M(Q12H

−11 Q

T12 + K

T BT H1BK) (29)

P11BK + (BK)T P T11 ≤ P11H

−12 P

T11 + K

T BT H2BK (30)

2xT (t)(BK)T P12

∫ t

t−τ(t)x(s)ds ≤ xT (t)KT BT H3BKx(t)

+[∫ t

t−τ(t)x(s)ds]T P T12H

−13 P12

∫ t

t−τ(t)x(s)ds (31)

Replacing A with A + BK in (25) and also considering (29), (30) and (31),we have

Π < Ω2 + Σ13Σ−133 Σ

T13 (32)

where Ω2 is given in (25), and Σ13 and Σ33 are defined in Theorem 3.

12

By Schur complements, Σ < 0 in (27) reveals that Ω2 + Σ13Σ−133 Σ

T13 < 0 in

(32). Then, following the process of the proof of Theorem 1, we can obtainthe results of Theorem 3. This completes the proof. �

4.2 Iterative Algorithm

The stability criteria established in Theorem 3 involve non-linear terms, e.g.,Q−122 in Σ22. Therefore, they are no longer LMIs and thus cannot be solveddirectly by using the LMI technique.

However, similar to the idea of [4,18], an iterative algorithm is developedbelow for obtaining a feasible solution set of the stability criteria in Theorem3 through solving a non-linear optimisation problem involving LMI conditions.

Assume that there exist matrix variables M = MT > 0 and Ni = NTi > 0

satisfying

Q−122 ≥M, H−1i ≥ Ni (33)

By Schur complements, (33) is equivalent to

M−1 I

I Q−122

≥ 0,

N−1i I

I H−1i

≥ 0 (34)

Let Z1 = M−1, Z2 = Q

−122 , Si = N

−1i , and Wi = H

−1i . Then, conditions in (34)

become

Z1 I

I Z2

≥ 0,

Si I

I Wi

≥ 0 (35)

Therefore, with these matrix variable transformations, the non-linear terms−Q−122 and H

−1i in (27) are replaced by linear terms M and Ni, respectively,

while the original conditions in (27) are still kept unchanged.

From these discussions, using the idea of the cone complementarity [4], thefollowing non-linear minimisation algorithm is presented to solve the originalnon-convex minimisation problem:

13

Min tr(Z1M + Z2Q22 +3

∑

i=1

(SiNi + WiHi))

S.t.: (27)♯, (28)♯ and (35)

M I

∗ Z1

≥ 0,

Q22 I

∗ Z2

≥ 0,

Ni I

∗ Si

≥ 0,

Wi I

∗ Hi

≥ 0. (36)

where ♯ indicates that Q−122 and H−1i in (27) and (28) are replaced by M and

Ni, respectively; and the same operation applies in (37) below.

Although (36) gives only a suboptimal solution to the original problem (27)and (28), it is much easier to solve than the original non-convex minimisationproblem. To get a feasible solution, the following algorithm is proposed.

Algorithm 1 Finding τM (the upper bound of the delay) and K (the feedbackcontroller gain)

(1) Choose a sufficiently small initial τM > 0 such that there exists a feasibleset (Zi,M,X,G,N, F )

0 satisfying LMIs in (36). Set k = 1.

(2) Solve the following LMI problem, where←−−→MkZ1 means M

kZ1 + Zk1 M

Min tr(←−−→MkZ1 +

←−−→Qk22Z2 +

3∑

i=1

(←−→Ski Ni +

←−−→W ki Hi))

S.t.: LMIs in (36)

Set (Z1, Z2,M,Q22, Si, Ni,Wi, Hi)k+1 = (Z1, Z2,M,Q22, Si, Ni,Wi, Hi)

(37)

(3) Substitute the obtained matrix variables Q22, H i, etc., into (27) and (28).If the conditions in (27) and (28) are satisfied, then set k = k + 1 andgo back to Step 2 after increasing τM to some extent. Otherwise, if theseconditions are not satisfied within a specified number of iterations for thecurrent value of τM , then exit with the last value of τM being the upperbound of the delay and the corresponding value of K being the feedbackgain.

5 Numerical Examples

This section aims to demonstrate the effectiveness of the proposed approach.For comparisons with existing methods [2,3,21], we have chosen the same sys-

14

tem models as in these references. Example 1 is governed by (6), and Examples2 and 3 have the form of (1). Our simulation results are derived from Theorems1 to 3 using the Matlab LMI Toolbox.

5.1 Example 1

Consider time-delay system (6) with the following parameters:

A =

−0.6238, −1.0132

2.0116, −0.2106

, Ad =

−0.5011, 0.7871

−0.3002, −0.5231

,

C =

0.2134, −0.0191

0.1119, −0.1665

, B̟ =

−0.4326, 0.1253

1.6656, 0.2877

,

Cd =

0.0816, 0.1290

0.0712, 0.0669

, D =

0, 0

0, 0

. (38)

In order to compare our results with those in Fridman and Shaked [2], Gaoand Wang [3], Lee et al. [11] and Xu et al. [21], we assume a constant timedelay, e.g., d = 0. The results are shown in Tables 1 and 2 in terms of the tworespective performance indices: the maximum allowable delay bound τM for aprescribed γ, and the minimum allowable γ for a prescribed τM .

Table 1Example 1 - the maximum allowable delay bound τM for a given γ (NoV: numberof variables).

γ =2.0 γ =3.0 γ =4.0 NoV

Fridam and Shaked [2] 0.4057 0.5047 0.5515 10

Gao et al. [3], Lee et al. [11] 0.4057 0.5046 0.5515 11

Xu et al. [21] 0.4203 0.5146 0.5589 5

This work (Q12 = 0, P12 = 0) 0.4734 0.5545 0.5904 5

This work (Q12 = 0, P12 6= 0) 0.4734 0.5545 0.5904 6

This work (Q12 6= 0, P12 = 0) 0.8289 0.9241 0.9609 6

This work (Q12 6= 0, P12 6= 0) 0.9290 0.9689 0.9885 7

Tables 1 and 2 show that the results obtained from this work outperform thosederived from [2,3,11,21]. They also depict the contribution of the additional

15

Table 2Example 1 - the minimum allowable γ for a given delay bound τM .

τM =0.1 τM =0.3 τM =0.5

Fridam and Shaked [2] 1.0714 1.5067 2.9281

Gao et al. [3], Lee et al. [11] 1.0714 1.5067 2.9281

Xu et al. [21] 1.0577 1.4515 2.7757

This work (Q12 = 0, P12 = 0) 0.9949 1.2286 2.2297

This work (Q12 = 0, P12 6= 0) 0.9949 1.2886 2.2297

This work (Q12 6= 0, P12 = 0) 0.9825 1.1071 1.2100

This work (Q12 6= 0, P12 6= 0) 0.9318 0.9421 0.9623

design matrices Q12 and P12 in (9) to the performance improvement. In gen-eral, the more additional design matrices are considered, the less conservativeperformance can be achieved.

Now, let us have a brief discussion on the computational demand of vari-ous methods. Roughly speaking, to the same upper delay bound, the numberof variables used in the computation is an indication of the computationaldemand: the fewer the variables are used, the less computational power isrequired. Following this idea, some comparisons are given below:

• Compared with [2] and [11] which used 10 and 11 variables, respectively,this work needs only 5 to 7 variables (depending on how many additionaldesign matrices are used) to derive much improved results (the improvementis over 16%);• Xu et al. [21] employed 5 variables to obtain better results than those in

[2,3,11]. Compared with Xu et al. [21], this work also employs 5 variables(for Q12 = 0, P12 = 0) but gives less conservative results;• When additional design matrices Q12 and P12 are considered separately or

in combination, this work uses 6 or 7 variables to give further improvedresults than those when these matrices are not considered.

5.2 Example 2

Consider time-delay system (1) with uncertainties and the following parame-ters:

16

A =

−3.0242, 2.7527

0.8104, −4.3988

, Ad =

2.8409, −1.2355

−9.8952, −0.1443

,

B̟ =

−0.9043, 0.4325

−0.7774, 0.1846

, C =

−0.9647, −1.6555

0.8245, −0.8378

,

Cd =

1.2723, 0.2718

0.4810, −0.2368

, D̟ =

0.1352, −1.0236

−0.0125, 0.3368

,

M = [1.5, 0.8]T , Ea = [0.2, 0.3] , Eb = [0.1, 0.2] . (39)

Tables 3 and 4 tabulate the maximum allowable upper delay bound τM fora prescribed γ, and the minimum allowable γ for a prescribed delay boundτM , respectively. They show that the results obtained from this work withconsideration of both Q12 and P12 are less conservative than those computedfrom [3,11,21].

Table 3Example 2 - the maximum allowable delay bound τM for a given γ.

γ =3 γ =4 γ =6

Gao et al. [3], Lee et al. [11] 0.1524 0.2202 0.2791

Xu et al. [21] 0.1677 0.2406 0.3024

This work (Q12 = 0, P12 = 0) 0.1405 0.2202 0.2872

This work (Q12 6= 0, P12 = 0) 0.2176 0.2711 0.3141

This work (Q12 = 0, P12 6= 0) 0.1405 0.2202 0.2872

This work (Q12 6= 0, P12 6= 0) 0.2243 0.2711 0.3141

Table 4Example 2 - the minimum allowable γ for a given delay bound τM .

τM =0.1 τM =0.2 τM =0.3

Gao et al. [3], Lee et al. [11] 2.5766 3.6174 7.3958

Xu et al. [21] 2.5209 3.3495 5.8767

This work (Q12 = 0, P12 = 0) 2.7203 3.6619 6.6663

This work (Q12 6= 0, P12 = 0) 2.5332 2.8628 5.1596

This work (Q12 = 0, P12 6= 0) 2.7203 3.6619 6.6663

This work (Q12 6= 0, P12 6= 0) 2.4699 2.7626 5.1596

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5.3 Example 3

Consider the injection velocity control of a thermoplastic injection mouldingprocess [20]. The objective is to control the injection velocity to guarantee thegiven performance index γ defined in (4). Similar to [14], this work convertsthe original second-order plus delay process model described in [20] into thefollowing state space model:

A =

−0.2449, −0.0165

1.000, 0

, Ad =

0.1, 0

0, 0.1

,

B = [1, 0]T , B̟ = [1, 1]T

, C = [0, 0.0142]T . (40)

Then, the uncertainties in the system identification are described in the formof (2) with the following parameters:

M =

0.01, 0

0, 0.01

, Ea =

0.1, 0

0, 0

, Eb =

0, 0.1

0, 0.1

. (41)

Assume that the initial conditions of the states are x1(t) = 0.5et+1 and x2(t) =

−0.5et+1 for t ∈ [−τM , 0].

Set the performance index γ = 2. Applying Algorithm 1, we can find that themaximum allowable delay bound that guarantees the stability of system (1) isτM = 2.9ms and the corresponding feedback gain K = [3.2931,0.49296]×10

−3.

Given the injection velocity profile as shown in the solid line in Fig. 1, the re-sponse of the set-point tracking of the injection velocity based on the designedfeedback controller is shown in the dotted line in the same figure. It is seenfrom Fig. 1 that the set-point tracking performance is acceptable.

0 50 100 150 200 250 300 350−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Simulation Time(Ms)

Inje

ctio

n ve

loci

ty s

et p

oint

s

Fig. 1. Tracking of the injection velocity setpoint.

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6 Conclusion

An approach has been proposed for delay-dependent robust H∞ stability anal-ysis and control synthesis of uncertain systems with time-varying delay. It in-tegrates a new delay-dependent BRL, a new Lyapunov-Krasovskii functional,and a tighter bounding technology for cross terms, leading to less conservativeresults than the existing ones. The effectiveness of the proposed approach hasbeen demonstrated through numerical examples.

Acknowledgment

Author C. Peng would like to thank the Natural Science Foundation of China(NSFC) for its support under grant number 60704024, and the Natural Sci-ence Foundation of Jiangsu for its support under grant number BK2006573.Author Y.-C. Tian is grateful to the Australian Government’s Department ofEducation, Science and Training (DEST) for its support under InternationalScience Linkages (ISL) grant number CH070083.

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