Robust Stabilization and H∞ Control of UncertainDistributed Delay Systems

Ulrich Münz1, Jochen M. Rieber2 and Frank Allgöwer1

1 Institute for Systems Theory and Automatic Control, University of Stuttgart, Germany,{muenz,allgower}@ist.uni-stuttgart.de

2 Astrium GmbH, Friedrichshafen, Germany, j.rieber@gmx.de

Summary. Linear matrix inequality (LMI) conditions for the robust stabilization and robustH∞ control of uncertain linear systems with distributed delays are presented. All system ma-trices including the delay kernel are uncertain. Yet, the nominal delay kernel is assumed tobe a matrix of rational functions, i.e., it can be written in a linear fractional form. The synthe-sis conditions are derived using a Lyapunov-Krasovskii functional. Techniques from robustcontrol, such as the full-block S-procedure, are used to transform the resulting parametricmatrix inequality into an LMI. As an important feature, the controller synthesis algorithmuses explicitly the information about the continuous delay kernel.

1 Introduction

Time-delay systems (TDS) have attracted an increasing interest over the last years, see forexample [13, 25, 11, 27, 26] and references therein. Most publications deal with linear sys-tems with discrete delays. However, distributed delays play an important role in many fieldsof biology [19, 9, 6] and engineering [28, 20, 21].

The stability of distributed delay systems (DDS) can be analyzed in the frequency domain,see for example [36, 4], in particular if the delay is γ-distributed, e.g., [2, 22]. However, theseapproaches are not suitable for robust analysis and synthesis problems. For this purpose,Lyapunov-based conditions are more suitable. (Robust) stability and stabilization of DDShave been studied in [29, 5, 14, 18, 17] for constant delay kernels, in [10, 12, 8] for piecewiseconstant kernels, and in [16] for continuous kernels. Stabilization of DDS for continuous de-lay kernels has been investigated in [7, 39, 40] using finite dimensional comparison systems.In [41], robust stabilization conditions are presented in terms of parametric matrix inequali-ties. Finally, H∞ control of DDS has been studied in [37] for constant delay kernels, in [8]for piecewise constant delay kernels, and in [25] using generalized Popov theory.

The design conditions presented here expand a recently developed robust stability condi-tion [23] towards robust stabilization and robust H∞ control of uncertain DDS. They requirethat the nominal delay kernel is a matrix of rational functions, i.e., it can be written in alinear fractional form. This assumption is not restrictive as discussed later on. Due to thisassumption, it is possible to reformulate the resulting parametric matrix inequality using thefull-block S-procedure and a convex hull relaxation into linear matrix inequalities (LMI). Thenominal case has been investigated in [24].

J.J. Loiseau et al. (Eds.): Topics in Time Delay Systems, LNCIS 388, pp. 221–231.springerlink.com c© Springer-Verlag Berlin Heidelberg 2009

222 U. Münz, J.M. Rieber, and F. Allgöwer

In contrast to most results from the literature, the new synthesis conditions do not re-quire piecewise constant delay kernels. Compared to [7, 39, 40, 25, 16], the controller designapplies also for uncertain DDS. Unlike [41], the conditions are formulated as LMIs whichconsiderably simplifies the solution and improves the accuracy, see [23]. At the same time,the nominal delay kernel is used explicitly in the LMI conditions, which is a source of conser-vatism for the condition in [37] if the delay kernel is known at least approximately. Finally,this work presents the first H∞ controller design condition for uncertain DDS.

The chapter is structured as follows: We first present the problem statement and somefundamentals on parametric matrix inequalities and uncertain LMIs in Section 2. The maincontribution, the new controller synthesis algorithms, are given in Section 3 before the chapteris concluded in Section 4.

Notation: Our notation is standard. Lα2 [a, b] is the class of all square integrable func-

tions f : [a, b] → Rα. The corresponding L2-norm is ‖f‖2 =√∫ b

afT (t)f(t)dt. We

write

[A B∗ C

]for

[A BBT C

]. A block diagonal matrix with entries M1, . . . ,MN is de-

noted diag(M1, . . . ,MN ). Finally, %,�,≺, and ' indicate positive and negative (semi-)definiteness, respectively.

2 Problem Statement and Preliminaries

2.1 Problem Statement

We consider the following class of uncertain distributed delay systems (DDS)

x(t) = A(t)x(t) +

∫ r

0

F (t, θ)x(t− θ)dθ + B1(t)u(t) + B2(t)w(t)

z(t) = C(t)x(t) + D1(t)u(t) + D2(t)w(t)

(1)

x(t) = φ(t) , t ∈ Ω = [−r, 0],where x(t) ∈ Rn, u(t) ∈ Rq1 , w(t) ∈ Rq2 , z(t) ∈ Rp are the state, controlled input,disturbance input, and performance output, respectively. The initial condition is φ and r > 0is the range of the distributed delay. All matrices consist of a known constant part and a time-varying uncertainty, i.e., A(t) = A + ΔA(t), Bi(t) = Bi + ΔBi(t), C(t) = C + ΔC(t),Di(t) = Di + ΔDi(t), and F = F (θ) + ΔF (t, θ), where A ∈ Rn×n, Bi ∈ Rn×qi ,C ∈ Rp×n, Di ∈ Rp×qi , and F : Ω → Rn×n are known matrices and ΔA : R → Rn×n,ΔBi : R → Rn×qi , ΔC : R → Rp×n, ΔDi : R → Rp×qi , and ΔF : R × Ω → Rn×n

are time-varying uncertainties. To shorten the notation, we often drop the time dependenceof the matrices in (1). The presented synthesis conditions can be easily extended to systemswith additional discrete and distributed delays.

The uncertainties are assumed to satisfy the following:

Assumption 1. The admissible uncertainties are bounded and of the form

ΔA(t) = UAΔA0(t)VA, ∀t ∈ R (2)

ΔBi(t) = UBiΔBi,0(t)VBi , ∀t ∈ R (3)

ΔC(t) = UCΔC0(t)VC , ∀t ∈ R (4)

ΔDi(t) = UDiΔDi,0(t)VDi , ∀t ∈ R (5)

ΔF (t, θ) = UFΔF0(t, θ)VF , ∀t ∈ R, θ ∈ Ω, (6)

Robust Control of Distributed Delay Systems 223

where UA, UBi , UC , UDi , UF and VA, VBi , VC , VDi , VF are known constant matrices ofappropriate dimensions. Without loss of generality, we assume moreover that the inducedmatrix 2-norm of ΔA0(t),ΔBi,0(t),ΔC0(t),ΔDi,0(t) and ΔF0(t, θ) satisfy ‖ΔA0(t)‖ ≤1, ‖ΔBi,0(t)‖ ≤ 1, ‖ΔC0(t)‖ ≤ 1, ‖ΔDi,0(t)‖ ≤ 1 for all t ∈ R and ‖ΔF0(t, θ)‖ ≤ 1for all t ∈ R and θ ∈ Ω, respectively.

In addition, we assume that the known part of the kernel F is a matrix of rational func-tions in θ. This is particularly important in order to obtain LMI conditions for the controllersynthesis. More formally, we require the following:

Assumption 2. The matrix function F can be written as a linear fractional representation(LFR)

F (θ) = DF + CF (I − θAF )−1θBF , (7)

with AF ∈ RnF×nF and BF , CF , and DF of appropriate dimensions for some positiveinteger nF .

Note that there exist different LFRs of F (θ). Here, we assume that LFR (7) is minimalin the sense that there are no pole-zero-cancellations. Since F does not have poles on Ω, weknow that det(I − θAF ) �= 0 for all θ ∈ Ω, i.e., the inverse in (7) exists. Assumption 2is not restrictive for F . It is a well-known fact that any continuous function on a closed andbounded interval can be approximated by a polynomial of sufficiently high order. Clearly, theset of polynomial functions is a subset of the set of rational functions. Given a non-rationaldelay kernel F ∗, the best rational approximation is given by the Padé approximation, see e.g.[1]. The error of this approximation can be included in the uncertainty ΔF . Note that it is notassumed that F is piecewise constant nor F (θ) ≥ 0, ∀ θ ∈ Ω, nor ‖F (θ)‖ = 1 as in otherpublications, e.g. [10, 29, 2].

In order to illustrate Assumption 2, we consider∫ r

0

F (θ)x(t− θ)dθ (8)

as a finite convolution between the “input” x(t) ∈ Rn and the truncated multi-input multi-output (MIMO) “impulse response” F : Ω → Rn×n of some linear system. However, thisimpulse response is not, as usual, a sum of exponential and sinusoidal functions but a matrixof rational functions, i.e., F takes the form of a MIMO “transfer function” with real variableθ instead of a complex frequency s. This transfer function can be represented by the LFR (7).It is important to note that AF , BF , CF , and DF have nothing in common with the afore-mentioned linear system with impulse response F . These matrices can be chosen such that(7) is satisfied.

After describing the considered uncertain distributed delay system, we can not turn to thesynthesis problems we are looking at in this contribution. Our goal is to design a static statefeedback controller

u(t) = Kx(t) (9)

that solves the following synthesis problems:

Problem 1 (Stabilization). Given system (1) with w ≡ 0 that satisfies Assumption 1 and 2,find a state-feedback controller (9) that stabilizes the closed loop system for all admissibleuncertainties.

224 U. Münz, J.M. Rieber, and F. Allgöwer

Problem 2 (H∞ Control). Given system (1) that satisfies Assumption 1 and 2, find a stabi-lizing controller (9) that minimizes an upper bound γ > 0 of the L2-gain of the closed loopsystem, i.e.,

γ > sup0<‖w‖2<∞

‖z‖2

‖w‖2, (10)

for all admissible uncertainties and all w ∈ Lq22 [0,∞).

>From [35, 30] we know that (10) is equivalent to system (1) with controller (9) beingstrictly dissipative with respect to the supply function s(w, z) = γ2wTw − zT z, i.e., thereexists a storage function V such that

V (x(t)) +1

γzT (t)z(t) − γwT (t)w(t) < 0 , (11)

for all t and all w ∈ Lq22 [0,∞). Solutions to these controller synthesis problems are given inTheorem 1 and 2, respectively.

2.2 Preliminaries on Parametric Matrix Inequalities

In the proof of the main results, we use the full-block S-procedure in order to transformparametric matrix inequalities into LMIs. It is used intensively for solving robust analysisand synthesis problems, e.g. [31, 33, 15]. In this subsection, we present some basic resultsrelated to this tool.

First, we define a matrix function G(δ) : Δ → Rn1×n2 that is rational in δ ∈ Δ ⊆ R.Hence, G can be written as an LFR

G(δ) = DG +CG(I − δAG)−1δBG , (12)

with AG ∈ RnG×nG and BG, CG, DG of appropriate dimensions. The LFR (12) is well-posed if det (I − δAG) �= 0 for all δ ∈ Δ, cf. [33, 31]. This is obviously fulfilled if G(δ)has no poles for δ ∈ Δ and if (12) is a minimal realization without pole-zero-cancellations.

It is possible to simplify some parametric matrix inequalities related to G using the full-block S-procedure:

Lemma 1 ([33, 32, 15]). Suppose Rp = RTp , Qp = QTp , Sp, G(δ) according to (12), anda compact set Δ ⊆ R are given. Then[

IG(δ)

]T [Qp SpSTp Rp

] [I

G(δ)

]≺ 0, ∀δ ∈ Δ, (13)

if and only if there exist matrices Q,R, S ∈ RnG×nG with Q = QT , R = RT , satisfying[δII

]T [Q SST R

] [δII

]� 0, ∀δ ∈ Δ, and (14)[

I 0AG BG

]T [Q SST R

] [I 0AG BG

]+

[0 ICG DG

]T [Qp SpSTp Rp

] [0 ICG DG

]≺ 0 . (15)

Clearly, we still have a parametric matrix inequality (14). However, using the convex hullrelaxation from [33, 31], it is possible to transform this into a finite set of non-parametricmatrix inequality.

Robust Control of Distributed Delay Systems 225

Lemma 2 ([33, 31]). Suppose that Q ≺ 0 and Δ is the convex hull of two points δ1, δ2 ∈ R

with δ1 < δ2, i.e., Δ = Co({δ1, δ2}) = [δ1, δ2]. Then[δII

]T [Q SST R

] [δII

]� 0, ∀δ ∈ Δ (16)

if and only if [δiII

]T [Q SST R

] [δiII

]� 0, i = 1, 2 . (17)

Summarizing, the parametric LMI (13) can be replaced by (15) and (17). The only intro-duced conservatism is Q ≺ 0.

2.3 Preliminaries on Uncertain LMIs

In order to deal with the model uncertainties, we use the following result:

Lemma 3 ([34]). Let U, V,W, and Z be real matrices of appropriate dimensions with Zsatisfying ‖Z‖ ≤ 1, where ‖ · ‖ is the induced matrix 2-norm. Then, we have the following:

(1) For any real number ε > 0, UZV + (UZV )T ' εUUT + ε−1V TV .(2) For any matrix P % 0 and scalar ε > 0 such that εI − V PV T % 0, we have

(W + UZV )P (W + UZV )T ' WPW T

+ WPV T (εI − V PV T )−1V PW T + εUUT .

(3) For any matrix P % 0 and scalar ε > 0 such that P − εUUT % 0, we have

(W + UZV )TP−1(W + UZV ) ' W T (P − εUUT )−1W + ε−1V TV.

3 Robust Controller Design

Now, we are ready to present the main results of this contribution, namely the solutions toProblem 1 and 2 in Subsection 3.1 and 3.2, respectively.

3.1 Robust Stabilization of Uncertain Distributed Delay Systems

Problem 1 can be solved using the LMIs in the next theorem.

Theorem 1. Consider system (1) with w ≡ 0 where the uncertainties satisfy Assump-tion 1 and F satisfies Assumption 2. The static feedback controller (9) solves Problem 1for all admissible uncertainties if there exist real ε1 > 0, ε2 > 0, ε3 > 0 and matri-ces K,Q1, P 2, P 3, R,Q, S where Q1, P 2, P 3, R, Q are symmetric and Q1 % 0, P 2 %0, R � 0 and Q ≺ 0 such that

226 U. Münz, J.M. Rieber, and F. Allgöwer

⎡⎣Ψ1 Q1VTA K

TV TB1

∗ −ε1I 0∗ ∗ −ε2I

⎤⎦ ≺ 0 (18)

r2Q + r(S + ST ) + R � 0 (19)⎡⎢⎢⎣Ψ2 BFQ1 SCTF +AFRC

TF 0

∗ −P 2 Q1DTF Q1V

TF

∗ ∗ P 2 − P 3 + ε3UFUTF + CFRC

TF 0

∗ ∗ ∗ −ε3I

⎤⎥⎥⎦ ≺ 0 , (20)

is feasible, where Ψ1 = AQ1+Q1AT +B1K+K

TBT

1 +rP 3+ε1UAUTA +ε2UB1U

TB1 and

Ψ2 = Q + AFST + SATF + AFRA

TF . The corresponding controller gain is K = KQ−1

1 .

Proof. Consider the following Lyapunov-Krasovskii functional candidate, cf. [11], withP1 % 0, P2 % 0:

V (xt) = xTP1x+

r∫0

t∫t−θ

xT (ξ)P2x(ξ)dξdθ . (21)

The derivative of V along solutions of (1) with controller (9) is

V (xt) = xT (t)((A+ B1K)TP1 + P1(A + B1K) + rP3)x(t)

+

∫ r

0

[x(t)

x(t− θ)

]T [P2 − P3 P1F (θ)

F T (θ)P1 −P2

]︸ ︷︷ ︸

M(θ)

[x(t)

x(t− θ)

]dθ, (22)

where P3 = P T3 ∈ Rn×n. Clearly, V < 0, i.e., the controller (9) stabilizes the system, if

(A+ B1K)TP1 + P1(A+ B1K) + rP3 ≺ 0, (23)

M(θ) ≺ 0 , ∀θ ∈ Ω . (24)

The remainder of the proof shows that LMI (18) to (20) imply (23) and (24) for all admissibleuncertainties.

First, we show that (18) implies (23). We apply the Schur lemma [3] to (18) and pre-and post-multiply P1 = Q−1

1 . With K = KQ1 and P 3 = Q1P3Q1, we obtain Ψ3 ≺ 0where Ψ3 = (AT + KTBT

1 )P1 + P1(A + B1K) + rP3 + ε1P1UAUTAP1 + ε−1

1 V TA VA +

ε2P1UB1UTB1P1 + ε−1

2 KTV TB1VB1K. Now, we apply Lemma 3 to (23) and see that

(AT + KT BT1 )P1 + P1(A+ B1K) + rP3 ' Ψ3 ≺ 0,

for any ε1 > 0 and ε2 > 0, i.e., (18) implies (23).Next, we show that ⎡⎣−P 2 Q1F (θ)T Q1V

TF

∗ P 2 − P 3 + ε3UFUTF 0

∗ ∗ −ε3I

⎤⎦ ≺ 0. (25)

implies (24) for all admissible uncertainties. Therefore, we apply the Schur lemma first on−ε3I and then on −P 2+ε−1

3 Q1VTF VFQ1 and obtain Ψ4(θ) ≺ 0 where Ψ4(θ) = P 2−P 3+

Robust Control of Distributed Delay Systems 227

ε3UFUTF + F (θ)Q1(P 2 − ε−1

3 Q1VTF VFQ1)

−1Q1FT (θ). Then, we pre- and post-multiply

M with diag(Q1, Q1) and apply the Schur lemma. With Lemma 3, we have

P 2 − P 3 + F (θ)Q1P−12 Q1F

T (θ) ' Ψ4(θ) ≺ 0,

for any ε3 > 0, where P 2 = Q1P2Q1, i.e., (25) implies (24).Finally, we transform the parametric matrix inequality (25) into an LMI using the full-

block S procedure. Therefore, (25) is formulated as follows⎡⎢⎢⎣I 0 00 I 00 0 I0 F T (θ) 0

⎤⎥⎥⎦T ⎡⎢⎢⎣

−P 2 0 Q1VTF Q1

0 P 2 − P 3 + ε3UFUTF 0 0

VFQ1 0 −ε3I 0Q1 0 0 0

⎤⎥⎥⎦︸ ︷︷ ︸

M

⎡⎢⎢⎣I 0 00 I 00 0 I0 F T (θ) 0

⎤⎥⎥⎦ ≺ 0 .

With Assumption 2 and the full-block S-procedure (Lemma 1), we see that this inequality istrue for all θ ∈ Ω if and only if there exist Q = QT , R = RT and S satisfying θ2Q+ θ(S+ST ) + R � 0 for all θ ∈ Ω and

[I 0 0 0ATF 0 CTF 0

]T [Q SST R

] [I 0 0 0ATF 0 CTF 0

]+

⎡⎢⎢⎣0 I 0 00 0 I 00 0 0 IBTF 0 DT

F 0

⎤⎥⎥⎦T

M

⎡⎢⎢⎣0 I 0 00 0 I 00 0 0 IBTF 0 DT

F 0

⎤⎥⎥⎦ ≺ 0.

We can simplify the last LMI and obtain (20). Moreover, we apply the convex hull relaxation(Lemma 2) to θ2Q + θ(S + ST ) + R � 0 with δ1 = 0 and δ2 = rI . The result is R � 0and (19). �

3.2 Robust H∞ Control of Uncertain Distributed Delay Systems

Now, we turn to Problem 2 that can be solved using the LMIs in the following theorem.

Theorem 2. Consider system (1) where the uncertainties satisfy Assumption 1 and F sat-isfies Assumption 2. The static feedback controller (9) solves Problem 2 for all admissibleuncertainties if there exist real ε1 > 0, ε2 > 0 and matrices K,Q1, P 2, P 3, R,Q, S whereQ1, P 2, P 3, R, Q are symmetric and Q1 % 0, P 2 % 0, R � 0, Q ≺ 0 such that theminimization problem

min γ (26)

subject to⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ψ5 B2 Q1VTA K

TV TB1 0 Q1C

T +KTDT

1 Q1VTC K

TV TD1 0

BT2 −γI 0 0 V T

B2 DT2 0 0 V TD2

∗ ∗ −ε1I 0 0 0 0 0 0∗ ∗ 0 −ε1I 0 0 0 0 0∗ ∗ 0 0 −ε1I 0 0 0 0

∗ ∗ ∗ ∗ ∗ −γI + ε2ΣUCD 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ −ε2I 0 0∗ ∗ ∗ ∗ ∗ ∗ 0 −ε2I 0∗ ∗ ∗ ∗ ∗ ∗ 0 0 −ε2I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦≺ 0, (27)

228 U. Münz, J.M. Rieber, and F. Allgöwer

as well as (19) and (20) is feasible, where Ψ5 = Q1AT +K

TBT

1 +AQ1 +B1K + rP 3 +ε1ΣUAB with ΣUAB = UAU

TA +UB1U

TB1 +UB2U

TB2 and ΣUCD = UCU

TC +UD1U

TD1 +

UD2UTD2 . The corresponding controller gain is K = KQ−1

1 .

Proof. Consider the Lyapunov-Krasovskii functional candidate (21) as a storage function.Then, this storage function has to satisfy (11), i.e.,

V (xt) +1

γzT (t)z(t) − γwT (t)w(t) =

∫ r

0

[x(t)

x(t− θ)

]TM(θ)

[x(t)

x(t− θ)

]dθ

+

[x(t)w(t)

]T [Ψ6 P1B2 + 1γ

(CT + KT DT

1

)D2

∗ −γI + 1γDT

2 D2

]︸ ︷︷ ︸

N

[x(t)w(t)

]< 0, (28)

with M given in (22) and Ψ6 = P1(A + B1K) + (AT + KT BT1 )P1 + rP3 + 1

γ(CT +

KT DT1 )(C + D1K). Obviously, (11) is fulfilled if N ≺ 0 and M(θ) ≺ 0 for all θ ∈ Ω.

M(θ) ≺ 0 can be transformed into (19) and (20) as in the proof of Theorem 1. It remains toshow that (27) implies N ≺ 0.

First, we pre- and post-multiply (27) with diag(P1, I, I, I, I, I, I, I, I), where P1 = Q−11 ,

K = KQ1 and P 3 = Q1P3Q1. Then, we apply the Schur lemma on the upper-left blockand obtain Ψ8 + Ψ9 ≺ 0 with

Ψ8 =

[Ψ7 P1B2

∗ −γI]

+ ε1

[P1ΣUABP1 0

0 0

]+

1

ε1

[V TA KTV T

B1 00 0 V T

B2

] ⎡⎣ VA 0VB1K 0

0 VB2

⎤⎦

Ψ9 =

[CTV TC KTV T

D1 0DT

2 0 0 V TD2

] [(γI − ε2ΣUCD )−1 0

0 ε−12 I

] ⎡⎢⎢⎣C D2

VC 0VD1K 0

0 VD2

⎤⎥⎥⎦ ,where Ψ7 = (A + B1K)TP1 + P1(A + B1K) + rP3 and C = C + D1K.

Then, we separate N as N = N1 +N2 with

N1 =

[Ψ10 P1B2

∗ −γI], N2 =

1

γ

[CT +KT DT

1

DT2

] [C + D1K D2

],

with Ψ10 = P1(A+ B1K) + (AT + KT BT1 )P1 + rP3. Next, we rewrite N1 as

N1 =

[Ψ7 P1B2

∗ −γI]

+ P1

[UA UB1 UB2

0 0 0

] ⎡⎣ΔA0 0 00 ΔB1,0 00 0 ΔB2,0

⎤⎦⎡⎣ VA 0VB1K 0

0 VB2

⎤⎦+

⎛⎝P1

[UA UB1 UB2

0 0 0

] ⎡⎣ΔA0 0 00 ΔB1,0 00 0 ΔB2,0

⎤⎦⎡⎣ VA 0VB1K 0

0 VB2

⎤⎦⎞⎠T

.

We apply Lemma 3 and see that N1 ' Ψ8 for any ε1 > 0. Furthermore, note that the firstpart of N2 can be written as

Robust Control of Distributed Delay Systems 229

[CT + KT DT

1

DT2

]=

[CT

DT2

]+

[V TC KTV TD1 00 0 V T

D2

] ⎡⎣ΔCT0 0 00 ΔD1,0 00 0 ΔDT

2,0

⎤⎦⎡⎣ UTCUTD1

UTD2

⎤⎦ .Hence, we can apply Lemma 3 also to N2 and obtain N2 ' Ψ9. Summarizing, we see thatN1 + N2 ' Ψ8 + Ψ9 ≺ 0, i.e., (27) implies N ≺ 0. �

Note that the applied Lyapunov-Krasovskii functional is very simple. Nonetheless, therobust stability condition in [23] is less conservative than results from the literature. Here, weuse similar techniques as in [23] but for the controller design. Hence, we expect also quiteaccurate results. The conservatism of the synthesis conditions could be further decreasedby using more advanced Lyapunov-Krasovskii functional candidates, as those presented forexample in [18, 5, 14].

4 Conclusions

We presented two LMI conditions for the robust stabilization and robust H∞ controller de-sign for uncertain systems with distributed delays. Both conditions are delay-dependent be-cause they explicitly depend on the range r of the distributed delay as well as on the shape ofthe delay kernel, given by the linear fractional representation of the nominal delay kernel Fin Assumption 2.

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