Analysis of Input Delay Systems using Integral Quadratic Constraint

Gabriella Szabó-Varga and Gábor Rödönyi

Systems and Control Laboratory, Computer and Automation Research Institute of Hungarian Academy of Sciences,

Budapest, Hungary

Keywords:

Time-delay Systems, Lyapunov-Krasovskii Functional, Integral Quadratic Constraints, Vehicle Platoon.

Abstract:

The L

-gain computation of a linear time-invariant system with state and input delay is discussed. The input

and the state delay are handled separately by using dissipation inequality involving a Lyapunov-Krasovskii

functional and integral quadratic constraints. A conic combination of IQCs is proposed for characterizing

the input delay, where the coefﬁcients are linear time-invariant systems. The numerical example (a vehicle

platoon) conﬁrm that using this dissipativity approach a more effective method for L

-gain computation is

obtained.

1 INTRODUCTION

Dynamic systems with both state and input delay

emerge for example in distributed systems and in

large scale systems. The problem of induced L

-gain

computation of systems with input delay can be re-

solved in many special cases.

If only delayed input acts on the system, then it

can be handled as considering this as another input

without delay. Delay on the control input transforms

to state delay when closing the loop (Fridman and

Shaked, 2004). The problem arise when the delayed

and actual disturbance input acts simultaneously on

the system.

In (Cheng et al., 2012), the actual input and the

delayed input were considered as two independent in-

puts, which results in an overestimation of the L

gain, due to disregarding the relation between them.

The other paper, which examined the effects of the

input delay, is (Rödönyi and Varga, 2015). Four dif-

ferent methods were considered to compute the L

gain for state and input delay system. The best of

these methods according to the numerical results in

time-invariant and also in time-varying delay cases is

the augmentation of the system with additional dy-

namics. With this method the input delay is trans-

formed to state delay that can be handled for example

by Lyapunov-Krasovskii functionals (LKFs).

Another method was examined in (Rödönyi and

Varga, 2015), where integral quadratic constrains

(IQCs) was used to describe the input delay in the sys-

tem. A conic combination of two IQCs was used with

constant coefﬁcients.

It is shown in this paper that the upper bound of

the L

-gain can be improved further as compared to

the method of additional dynamics by applying dy-

namic coefﬁcients in the IQC approach.

The structure of the paper is the following: First

the system in consideration is described in Section

2 together with the emerging problem. In Section 3

some preliminary tools are presented together with a

lower bound computation method and additional dy-

namics approach. In Section 4 the new method is

presented to compute the L

-gain in case of input

and state delay using Lyapunov-Krasovskii functional

and integral quadratic constraints in the time-domain.

This method is compared with the other two methods

in Section 5 on an example of vehicle platoon. In Sec-

tion 6 a few conclusion are drawn.

Notations. Matrix inequality M > 0 (M ≥ 0) de-

notes that M is symmetric and positive (semi-) deﬁ-

nite, i.e. all of its eigenvalues are positive (or zero).

Negative (semi-) deﬁniteness is denoted by M < 0

(M ≤ 0). The transpose and conjugate transpose of

a matrix M is denoted by M

and M

∗

, respectively.

σ(M) denotes the maximum singular value of matrix

M. The upper linear fractional transformation is de-

ﬁned by F

(M,∆) = M

+ M

∆(I − M

∆)

−1

where M =





. L

denotes the space

of square integrable signals with norm deﬁned by

kxk



∞

kx(t)k



1/2

, where kx(t)k denotes the

Euclidean norm on R

102

Szabó-Varga, G. and Rödönyi, G.

Analysis of Input Delay Systems using Integral Quadratic Constraint.

DOI: 10.5220/0005987101020109

In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 2, pages 102-109

ISBN: 978-989-758-198-4

2 PROBLEM FORMULATION

AND SKETCH OF THE

SOLUTION

A linear input and state delayed system denoted by Ω

is described in this paper by the following form

˙x(t) = Ax(t) +A

x(t −h) + Bd(t) + B

d(t − h) (1a)

y(t) = Cx(t), (1b)

where x ∈ R

, d ∈ R

and y ∈ R

is the state, distur-

bance and output of the system, respectively. A, A

B, B

and C are constant matrices with appropriate

dimensions. Here only time-invariant delay is consid-

ered, therefore h is constant. For the sake of simpli-

fying the discussion a single delay is considered, but

the method can be generalized to handle multiple de-

lays and different state and input delays. The initial

condition for the Ω system is the following

x(t) = φ(t), t ∈ [−h,0], (2)

where φ : [−h,0] → R

is a given continuous func-

tion. Let x

(ξ) denote x(t + ξ) for ξ ∈ [−h,0].

The goal of the paper is to compute the L

-gain of

system Ω deﬁned as

kΩk

∞

= sup

06=d∈L

,φ=0

kyk

kdk

. (3)

2.1 Sketch of the Solution

To analyse delayed systems different methods exist

like Lyapunov-Krasovskii functionals, Razumikhin

theorem, integral quadratic constraints approach and

frequency-domain methods. The advantage of using

the complete Lyapunov-Krasovskii functional is that

it gives sufﬁcient and necessary condition of stability

in case of constant delays.

The combination of Lyapunov-Krasovskii func-

tional and integral quadratic constraint approach is

used: LKF for the state delay and IQC for the input

delay.

Let S

(d) := d(t − h) − d(t) denote the differ-

ence between the delayed input and the input. Then

d(t − h) is replaced in (1) by S

(d) + d(t) and the sys-

tem is reformulated by the linear fractional transfor-

mation (LFT) form F

(G,S

), where system G is the

following

˙x(t) = Ax(t) +A

x(t −h) + (B + B

)d(t) + B

w(t),

(4a)



v(t)

y(t)







x(t) +



1 0

0 0



d(t)

w(t)



(4b)

and w = S

(v). The S

perturbation term is described

by integral quadratic constraint. The G plant contains

state delay, the L

-gain of such a system can be com-

puted using complete Lyapunov-Krasovskii function-

als. Adding the time-domain IQC to the derivative of

the LKF and the L

-gain condition the L

-gain of the

Ω system can be computed. The exact formulation

of this L

-gain bound computation technique will be

discussed in the following sections.

3 PRELIMINARIES

In this section the complete Lyapunov-Krasovskii

functional is presented for establishing stability of a

time-delay system. Then two L

-gain computation

methods are described for input delayed systems. One

of them is a frequency-domain formula and the other

one is a time-domain method described in (Rödönyi

and Varga, 2015).

The focus of this paper is to give an efﬁcient time-

domain method for L

-gain computation of system Ω

using integral quadratic constraints. An introduction

to IQCs is given in Section 3.4.

3.1 Lyapunov-Krasovskii Functional

One method to establish stability of a time-delay sys-

tem is to use Lyapunov-Krasovskii functionals. In

the literature different LKFs are proposed for time-

invariant delay. Here the so called complete LKF will

be used presented in (Gu et al., 2003):

V (x

) = x

(t)Px(t) + 2x

(t)

−h

Q(ξ)x(t +ξ)dξ

−h

(t + ξ)R(ξ,η)x(t + η)dηdξ

−h

(t + ξ)S(ξ)x(t + ξ)dξ. (5)

The sufﬁcient and necessary conditions for asymp-

totic stability of system Ω are that P = P

∈ R

×n

for all −h ≤ ξ, η ≤ 0, Q(ξ) ∈ R

×n

, R(ξ,η) =

(ξ,η) ∈ R

×n

,S(ξ) = S

(ξ) ∈ R

×n

and

V (x

) ≥ εkx(t)k

(6)

V (x

) ≤ −εkx(t)k

, (7)

for some ε > 0. In this LKF the variables Q,R and

S are matrix functions, which are approximated by

piece-wise linear functions in the analysis.

The domain of this matrix functions [−h, 0] (or

[−h,0] × [−h, 0]) are divided into N (or N by N) seg-

ments. Each segment indexed by p or (p, q) can be

described with the help of matrix parameters Q

, S

= R

, p,q = 0, 1, 2, . . . , N so that for 0 ≤ α ≤ 1

Analysis of Input Delay Systems using Integral Quadratic Constraint

103

and 0 ≤ β ≤ 1

Q(−pl + αl) = (1 − α)Q

+ αQ

p−1

S(−pl + αl) = (1 −α)S

+ αS

p−1

and

R(−pl + αl,−ql + βl) =



(1 −α)R

+ βR

p−1,q−1

+ (α− β)R

p−1,q

α ≥ β

(1 −β)R

+ αR

p−1,q−1

+ (β− α)R

p,q−1

α < β

Using this technique the stability conditions can be

described in a linear matrix inequality (LMI) form.

This method is known as discretized complete LKF

and is described in (Gu, 1997).

3.2 A Lower Bound Computation

In case of time-invariant delay the L

-gain of system

Ω can be computed exactly in the frequency-domain:

kΩk

∞

= max



C( jωI − A − A

− jωh

)

−1

(B +B

− jωh

)



. (8)

However numerically this maximum can not be calcu-

lated in case of lightly damped modes. The L

-gain

was computed on a grid of the frequency interval ac-

cording to (8), which gives a lower bound of the gain.

The other methods will give an upper bound on

the L

-gain of the Ω system. Those will be compared

with this method.

3.3 Additional Dynamics

This method was proposed in (Rödönyi and Varga,

2015), where the input delay was transformed to state

delay using a low-pass ﬁlter. Assume, that the input

d(t) is band limited. Let W

be the low-pass ﬁlter with

∞

= 1 and

˙x

(t) = A

(t) +B

d(t),

(t) = C

(t). (9)

Using this ﬁlter the Ω system can be augmented with

the following

˙x(t) = Ax(t) +A

x(t −h) + Bd(t) + B

(t − h)

y(t) = Cx(t). (10)

Then the L

-gain of the Ω system can be computed

using LKFs on system (9) and (10).

Using a low-pass ﬁlter the high-frequency compo-

nents of d are ﬁltered out, therefore the ﬁlter need to

be chosen carefully.

3.4 Integral Quadratic Constraint

In system analysis a very powerful tool to describe the

robustness in the system is to use integral quadratic

constraints.

Deﬁnition 1 ((Megretski and Rantzer, 1997)). Let Π :

jR → C

)×(n

)

be a Hermitian-valued func-

tion. Two signals v ∈ L

[0,∞) and w ∈ L

[0,∞)

satisfy the IQC deﬁned by Π if

∞

−∞



ˆv( jω)

ˆw( jω)



∗

Π( jω)



ˆv( jω)

ˆw( jω)



dω ≥ 0, (11)

where ˆv( jω) and ˆw( jω) are Fourier transforms of v

and w, respectively. A bounded, causal operator ∆ :

[0,∞) → L

[0,∞) satisﬁes the IQC deﬁned by Π,

if (11) holds for all v ∈ L

[0,∞) and w = ∆(v).

The input delay of the system can be described via

IQCs. To this end, the system has to be given by the

interconnection of a plant and a perturbation term, S

the deviation between the delayed and the undelayed

signal as S

(v) := v(t − h) − v(t).

To describe the constant time-delay term S

three

IQCs were proposed in (Pﬁfer and Seiler, 2015)



0 −1

−1 −1



, (12)

( jω) =



jω + 1

10 jω + 1





|ζ

( jω)|

0 −1



, (13)

( jω) =



0 ζ

∗

( jω)

( jω) −1



, (14)

where

( jω) := 2

( jωh)

+ 3.5 jωh + 10

−6

( jωh)

+ 4.5 jωh + 7.1

, (15)

( jω) :=

−2.19( jωh)

+ 9.02 jωh + 0.089

( jωh)

− 5.64 jωh − 17

. (16)

A combination of IQCs still an IQC, if an operator

∆ satisﬁes the IQCs Π

, i = 1,2,...,M, then it also sat-

isﬁes the IQC Π

( jω) =

∑

i=1

( jω)Π

( jω), where

> 0. Usually combined IQC is used in numerical

examples, therefore here these IQCs are studied. The

goal is to preserve the dynamics of the combination

coefﬁcients λ

in the time-domain.

For time-domain system analysis an equivalent

representation of the general IQC (11) is required.

The multiplier Π in IQC (11) is factorized as Π( jω) =

∗

( jω)MΨ( jω), where M = M

∈ R

×n

and Ψ ∈

×(n

)

∞

. (The factorization method is described

according to (Pﬁfer and Seiler, 2015), where also a

detailed description can be found.) Let z be the out-

put of the system Ψ, namely z := Ψ





. Using

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

104

the Parseval’s theorem the frequency-domain IQC is

equivalent with the following expression in the time-

domain:

∞

z(t)

Mz(t)dt ≥ 0. (17)

For general IQCs the constraint (17) holds only

over inﬁnite time, for hard IQCs a restrictive con-

straint of this holds.

Deﬁnition 2 ((Megretski et al., 2010)). Let Π factor-

ized as Ψ

∗

MΨ with Ψ stable. Then (Ψ,M) is a hard

IQC factorization of Π if for any bounded, causal op-

erator ∆ satisfying the IQC deﬁned by Π the following

inequality holds

z(t)

Mz(t)dt ≥ 0 (18)

for all T ≥ 0, v ∈ L

[0,∞), w = ∆(v) and z =





Let Π = Π

∗

and partition as



∗



. Ac-

cording to (Seiler, 2015, Theorem 4) if Π

( jω) > 0

and Π

( jω) < 0 then Π has J-spectral factoriza-

tion (Ψ,M), which is a hard factorization. A fac-

torization (Ψ,M) is a J-spectral factorization if M =



I 0

0 −I



and Ψ,Ψ

−1

∈ RH

×(n

)

∞

. Here the J-

spectral factorization from (Pﬁfer and Seiler, 2015)

is used, which provide a square, stable and minimum

phase Ψ.

Appending this Ψ to S

the plant of the intercon-

nected system reveals the following:

ˆx =

A ˆx(t) +

ˆx(t −h) +



w(t)

d(t)



, (19)





C ˆx(t) +



w(t)

d(t)



, (20)

where ˆx :=





are the extended state, x

is the

state vector of the Ψ system. The exact formula about

the computation of the constant state matrices is omit-

ted, however it can be derived easily from the descrip-

tion of plant G (4).

Using a slightly modiﬁed version of Theorem 3

from (Pﬁfer and Seiler, 2015) the L

-gain for system

(G,S

) can be computed.

Theorem 1. Assume F

(G,S

) is well-posed and

satisﬁes the hard IQC deﬁned by (Ψ,M). Then

(G,S

∞

≤ γ if there exists a λ > 0 and a

bounded quadratic Lyapunov-Krasovskii functional

V ( ˆx

) such that for some ε > 0

• V ( ˆx

) ≥ εk ˆx(t)k

• the following inequality holds

λz

Mz +

V ( ˆx

)−γ

d + y

y ≤

− εkˆx(t)k

− εkd(t)k

. (21)

Proof 1. Integrate the inequality (21) from t = 0 to

t = T with the initial condition φ(t) = 0 t ∈ [−h,0]

z(τ)

Mz(τ)dτ +V (ˆx

) −γ

d(τ)

d(τ)dτ

(τ)y(τ)dτ ≤

− ε

(k ˆx(τ)k

+ kd(τ)k

)dτ ≤ 0.

Using that the IQC and the LKF V are non-negative

this inequality is equivalent to kF

(G,S

)k ≤ γ.

The inequality (21) gives a linear matrix inequal-

ity (LMI), if the LKF condition for stability can be

formulated as an LMI.

In case of combined IQCs using Theorem 1 the

different factorized IQCs are adding to inequality

(21) with different λ

coefﬁcient. However in the

frequency-domain these coefﬁcients were frequency-

dependent. This dynamics in the time-domain now is

omitted.

Arise the question, that why not factorize the λ

coefﬁcients together with IQC Π

using J-spectral fac-

torization. For J-spectral factorization the exact dy-

namics of the IQC is necessary, which is not the case

with λ

. Therefore a different factorization is nec-

essary for the λ

coefﬁcients. In the next section a

method will be shown to preserve the dynamics of the

coefﬁcients in the time-domain.

4 MAIN RESULTS

4.1 IQC Factorization Preserving the λ

Dynamics

The combined IQC in Section 3.4 omit the dynamics

of the λ coefﬁcient in time-domain, therefore a new

factorization method is presented to preserve this dy-

namics.

A factorization of the dynamical λ is necessary in

similar forms as the IQCs are factorized:

( jω) = Θ

( jω)

∗

( jω) (22)

where i = 1,2,...,M. One possible factorization pro-

posed in (Veenman and Scherer, 2014) is the follow-

ing

( jω) =



jω − ρ

...

( jω − ρ

)



, (23)

Analysis of Input Delay Systems using Integral Quadratic Constraint

105

< 0,ν

∈ N ﬁxed parameters and 0 < Λ

∈

(ν

+1)×(ν

+1)

constant real symmetric matrix. This

is a basis function, using a ﬁxed pole location (ρ

constant), then by increasing the dimension of the ba-

sis function (ν

) a better approximation of the λ

coef-

ﬁcient can be established.

Assume that the dimension of v is 1. Using J-

spectral factorization of the IQC (Ψ



1 0

0 −1



)

and the factorization of λ

as Equation (22) the fac-

torized combined IQC reveals

( jω) =

∑

i=1



( jω) 0

0 Θ

( jω)



( jω)



∗



0 −Λ



( jω) 0

0 Θ

( jω)



( jω).

(24)

In case of a frequency independent IQC only the λ

coefﬁcient is factorized

( jω) = [∗]

∗



∗





( jω) 0

0 Θ

( jω)



. (25)

4.2 LMI Formulation of L

-gain

Computation

Using this new factorized IQC and the discretized

complete LKF the inequality (21) in Theorem 1 can

be formulated as an LMI. For brevity in this section

only one IQC multiplier is considered in form λΠ as

it would be in case of combined multipliers. The LMI

formulation of the problem can be easily extended for

more IQCs.

Let the state-space form of the connected system

Γ :=



Θ 0

0 Θ



Ψ be given as

˙x

(t) = A

(t) +B

Γ1

d(t) + B

Γ2

w(t)

z(t) = C

(t) +D

Γ1

d(t) + D

Γ2

w(t),

and the dimension of the x

denoted by n

. The con-

stant state-space matrices can be computed using the

J-spectral factorization of the Π multiplier and the λ

factorization as in (23).

The state-space form of the connected system of

Γ and G (4) is the following



˙x(t)

˙x

(t)



= A



x(t)

(t)



+ A



x(t −h)

(t − h)









w(t)

d(t)



(26)



z(t)

y(t)







x(t)

(t)







w(t)

d(t)



(27)

where



A 0

0 A



, A



0 0





Γ2



, B



B +B

Γ1





0 C



, C



C 0





Γ2

Γ1



, D



0 0



Before presenting the LMI formulation of the L

gain computation some notations must be introduced.

Q =



... Q



S =

diag(S

... S

)

R =







... R







∆ =







∆

∗ ∗

− A

P S

∗

−B

P 0 γ



0 0

0 1









−









[∗]

−











Λ 0

0 −Λ



[∗]

∆

= −PA

− A

P −Q

− Q

− S

−C

= diag{S

− S

,...,S

N−1

− S

}







d11

d12

... R

d1N

d21

d22

... R

d2N

dN1

dN2

... R

dNN







d pq

= l(R

p−1,q−1

− R

)

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

106





... D





p−1

+ Q

) +

0,p−1

+ R

)

−(Q

p−1

− Q

)

p−1

+ Q

) −

N,p−1

+ R

N p

)

p−1

+ Q

)





... D





= −

p−1

− Q

) −

0,p−1

− R

)

= −

p−1

− Q

) +

N,p−1

− R

N p

)

= −

p−1

− Q

)

Theorem 2. Assume F

(G,S

) is well-posed and

satisﬁes the hard IQC deﬁned by (Ψ,M).

Then kF

(G,S

∞

≤ γ if there exists 0 < Λ ∈

(ν+1)×(ν+1)

and {P = P

= R

} ∈

)×(n

)

, p = 0,1,...,N,q = 0, 1, . . . , N such

that the following holds:



R +



> 0 (28a)





∆ ∗ ∗

−D

+ S

∗

−D

0 3S





> 0. (28b)

The LMI formulation of the L

-gain computation

in case of state delay system using the discretized

complete LKF is described in (Gu et al., 2003, Prop.

8.5). Only the ∆ matrix has to be modiﬁed to get the

LMI formulation of (21).

5 NUMERICAL EXAMPLE:

VEHICLE PLATOON

A vehicle platoon model is used as a numerical exam-

ple. Due to imperfect inter-vehicle communication

the system contains input and state delays. Comput-

ing the L

-gain of this system not only stability can

established, but also the effects of the communication

caused delays are illustrated.

5.1 Vehicle Platoon Model

The ﬁrst vehicle in the platoon is driven by a human

driver (indexed by 0), and the followers motion deter-

mined by on-board controller (indexed by 1, 2 . . . , n).

The longitudinal dynamics of the ith vehicle is the fol-

lowing:

˙p

(t) = v

(t), (29a)

˙v

(t) = q

(t) +d

(t), (29b)

˙q

(t) = −

(t) +

(t), (29c)

where p

denote position and velocity, d

is a

disturbance representing both outer effects and mod-

elling error, q

is an internal state such that the ac-

celeration of the vehicle is a

(t) = q

(t) + d

(t). L

gain calculations will be carried out on a homoge-

neous platoon (the vehicles parameter are the same)

with parameters τ

= 0.7 and g

= 1 i = 0,1,...,n. u

is the signal generated by the pedal signal of the ﬁrst

vehicle driver, u

,i = 1,2,...,n is the acceleration de-

mand generated by the controllers.

A leader and predecessor follower control archi-

tecture is used with constant spacing policy proposed

in (Swaroop and Hedrick, 1999). This means that

the controller uses information about the predeces-

sor and also about the leader vehicle, therefore inter-

vehicle communication is necessary. This communi-

cation can be imperfect causing delays in the system

description. Taking this inter-vehicle communication

delays into account, the controllers can be described

by the following equations

(t) = −k

(t) −k

(t) +a

(t − h)

(t) = −k

1β

(t) −k

2β

(t) +k

(t − h)

+ k

i−1

(t − h) − k

1α

(t − h) − v

(t − h))

− k

2α

(t − h) − p

(t − h)), i = 2,...,n

where δ

, v

− v

i−1

and e

, p

− p

i−1

+ L

are the relative speed and spacing error, respec-

tively. The prescribed spacing L

can be set

to zero in the analysis without loss of general-

ity. The k

∗

are constant parameters of the con-

trollers. The aim of the paper is analysis, there-

fore the numerical values of these parameters, which

will be used from (Rödönyi et al., 2012) are k

0.7,k

= 0.1127,k

1α

= 0.4642,k

2α

= 0.0564,k

1β

0.2358,k

2β

= 0.0564, k

= 0.0449, k

= 0.9551.

In the analysis only SISO systems are considered,

namely d

7→ e

(the effect of the lead vehicle dis-

turbance on the ﬁrst spacing error) and d

7→ e

(the

effect of the ﬁrst vehicle disturbance on the second

spacing error).

The ﬁrst system contains only input delay and the

state-space matrices as in Ω system with state vector

x = [e

,δ

]

are the following:

A =





0 1 0

0 0 1

−0.16 −1 −1.43





, B =





−1





Analysis of Input Delay Systems using Integral Quadratic Constraint

107





1.43





, A

= 0, C =



1 0 0



The system d

7→ e

has both state and input de-

lay, the state-space matrices using the state vector

x = [e

,δ

]

are the following:

A =







0 1 0 0 0 0

0 0 1 0 0 0

−0.16 −1 −1.43 0 0 0

0 0 0 0 1 0

0 0 −1 0 0 1

0 0 0 −0.08 −0.34 −1.43













0 0 0 0 0 0

−0.08 −0.66 0.06 −0.08 −0.66 0







B = [0,1,0, 0, −1,0]

, B

= [0, 0, 0, 0, 0, 0.06]

C = [0,0,0,1,0,0].

5.2 L

-gain of a Vehicle Platoon

A linear combination of the multipliers will be used

( jω) = λ

( jω)Π

+ λ

( jω)Π

( jω), (31)

where Π

can be Π

or Π

, and λ

,λ

: jR → R that

satisﬁes λ

( jω) > 0,∀ω.

Before the numerical results notations are neces-

sary for the different L

-gain computation method:

LB the lower bound computation method described in

Section 3.2.

AD the additional dynamics method described in Sec-

tion 3.3 with ﬁlter W

s+1

. The τ

time-

constant of the ﬁlter has to be chosen carefully,

because higher time constant can cause increased

gain, smaller time-constant can cause numerical

problems by solving the LMI. Here τ

= 0.05 is

used.

TD the time-domain IQC method with constant λ

us-

ing IQCs Π

or Π

in the combined Π

TDλ the time-domain IQC method with dynamic λ

us-

ing IQCs Π

or Π

in the combined Π

. At the

factorization of λ

the ρ

= −1 parameter is used

and the ν

is increased from 0 until a tight bound

with the lower bound is established. In the tables

the results using ν

= 2 are presented.

In Table 1 the numerical results are shown in case

of system d

7→ e

for the different L

-gain compu-

tation methods. This system does not contain state

delay, therefore a simple quadratic storage function

V = x

Px is used instead of an LKF. By the time-

domain IQC method with constant λ coefﬁcient (TD-

2,3) a highly overestimated L

-gain is computed com-

pared to the lower bound (LB).

The suggestion is that preserving the dynamics of

the λ coefﬁcient a lower upper bound of the L

-gain

can be calculated. Using the factorization method de-

scribed in Section 4 (TDλ-2,3), nearly the same re-

sults are received as the lower bound (LB), meanly

a good approximation of the L

-gain are computed.

Two different IQCs are considered Π

and Π

. Better

numerical results were established using Π

than Π

In an earlier paper (Rödönyi and Varga, 2015) dif-

ferent methods were suggested for L

-gain compu-

tation, and the additional dynamics method (Section

3.3) gave the best numerical results. However here

with time-domain IQC preserving the λ dynamics less

conservative norms are established.

Table 1: L

-gain of system (d

7→ e

) with time-invariant

delay (only input delay).

h 0.05 0.1 0.25 0.5 0.8

LB 1.27 1.35 1.6 2.02 2.51

AD 1.35 1.44 1.69 2.1 2.6

TD-2 3.55 3.56 3.56 3.58 3.63

TDλ-2 1.27 1.35 1.6 2.02 2.51

TD-3 14.06 14.06 14.06 14.06 14.07

TDλ-3 1.28 1.36 1.63 2.07 2.6

In Table 2 the numerical results in case of system

7→ e

are shown. This system contains also state

delay, therefore the discretized complete LKF is used

from Section 3.1. The different methods gave similar

results as in Table 1: the time-domain IQC method

using constant λ coefﬁcients (TD-2,3) overestimates

the L

-gain. However if the λ dynamics are preserved

(TDλ-2,3) nearly the same gain can be computed as

the lower bound (LB) by every delay value.

Table 2: L

-gain of system (d

7→ e

) with time-invariant

delay (state and input delay).

h 0.05 0.1 0.25 0.5 0.8

LB 4.44 4.44 4.44 4.44 4.44

AD 4.44 4.44 4.44 4.44 4.44

TD-2 4.52 4.52 4.52 4.52 4.52

TDλ-2 4.44 4.44 4.44 4.44 4.44

TD-3 4.52 4.52 4.52 4.52 4.52

TDλ-3 4.44 4.44 4.44 4.44 4.44

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

108

6 CONCLUSION

A time-domain method for computing upper bound

of the L

-gain of state and input delay systems is

presented using a dissipation inequality involving

Lyapunov-Krasovskii functionals and conic combina-

tion of integral quadratic constraints. The coefﬁcients

of the combination of IQCs are proposed to be dy-

namic systems.

It was shown by a numerical example that the up-

per bound is very tight, and nearly coincides with the

lower bound. As a numerical example a vehicle pla-

toon was examined with leader and predecessor fol-

lowing control architecture and constant spacing pol-

icy.

Future works involves the construction of con-

troller synthesis based on this time-domain method.

Further extension of this time-domain method will be

to consider also uncertainties in the system.

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