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
2
-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
2
-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
2
-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
2
-
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
2
-
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
2
-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
2
-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
T
and M
, respectively.
¯
σ(M) denotes the maximum singular value of matrix
M. The upper linear fractional transformation is de-
fined by F
U
(M,) = M
22
+ M
21
(I M
11
)
1
M
12
,
where M =
M
11
M
12
M
21
M
22
. L
n
2
denotes the space
of square integrable signals with norm defined by
kxk
2
=
R
0
kx(t)k
2
dt
1/2
, where kx(t)k denotes the
Euclidean norm on R
n
.
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
Copyright
c
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
2 PROBLEM FORMULATION
AND SKETCH OF THE
SOLUTION
A linear input and state delayed system denoted by
is described in this paper by the following form
˙x(t) = Ax(t) +A
h
x(t h) + Bd(t) + B
h
d(t h) (1a)
y(t) = Cx(t), (1b)
where x R
n
x
, d R
n
d
and y R
n
y
is the state, distur-
bance and output of the system, respectively. A, A
h
,
B, B
h
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 system is the following
x(t) = φ(t), t [h,0], (2)
where φ : [h,0] R
n
x
is a given continuous func-
tion. Let x
t
(ξ) denote x(t + ξ) for ξ [h,0].
The goal of the paper is to compute the L
2
-gain of
system defined as
kk
= sup
06=dL
n
d
2
,φ=0
kyk
2
kdk
2
. (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
h
(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
h
(d) + d(t) and the sys-
tem is reformulated by the linear fractional transfor-
mation (LFT) form F
U
(G,S
h
), where system G is the
following
˙x(t) = Ax(t) +A
h
x(t h) + (B + B
h
)d(t) + B
h
w(t),
(4a)
v(t)
y(t)
=
0
C
x(t) +
1 0
0 0
d(t)
w(t)
(4b)
and w = S
h
(v). The S
h
perturbation term is described
by integral quadratic constraint. The G plant contains
state delay, the L
2
-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
2
-gain condition the L
2
-gain of the
system can be computed. The exact formulation
of this L
2
-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
2
-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
2
-gain computation of system
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
t
) = x
T
(t)Px(t) + 2x
T
(t)
Z
0
h
Q(ξ)x(t +ξ)dξ
+
Z
0
h
Z
0
h
x
T
(t + ξ)R(ξ,η)x(t + η)dηdξ
+
Z
0
h
x
T
(t + ξ)S(ξ)x(t + ξ)dξ. (5)
The sufficient and necessary conditions for asymp-
totic stability of system are that P = P
T
R
n
x
×n
x
,
for all h ξ, η 0, Q(ξ) R
n
x
×n
x
, R(ξ,η) =
R
T
(ξ,η) R
n
x
×n
x
,S(ξ) = S
T
(ξ) R
n
x
×n
x
and
V (x
t
) εkx(t)k
2
(6)
˙
V (x
t
) εkx(t)k
2
, (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
p
, S
p
,
R
pq
= R
T
qp
, 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
p
+ αQ
p1
S(pl + αl) = (1 α)S
p
+ αS
p1
and
R(pl + αl,ql + βl) =
(1 α)R
pq
+ βR
p1,q1
+ (α β)R
p1,q
α β
(1 β)R
pq
+ αR
p1,q1
+ (β α)R
p,q1
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
2
-gain of system
can be computed exactly in the frequency-domain:
kk
= max
ω
¯
σ
C( jωI A A
h
e
jωh
)
1
×
(B +B
h
e
jωh
)
. (8)
However numerically this maximum can not be calcu-
lated in case of lightly damped modes. The L
2
-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
2
-gain of the 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
d
be the low-pass filter with
kW
d
k
= 1 and
˙x
d
(t) = A
d
x
d
(t) +B
d
d(t),
d
d
(t) = C
d
x
d
(t). (9)
Using this filter the system can be augmented with
the following
˙x(t) = Ax(t) +A
h
x(t h) + Bd(t) + B
h
d
d
(t h)
y(t) = Cx(t). (10)
Then the L
2
-gain of the 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
v
+n
w
)×(n
v
+n
w
)
be a Hermitian-valued func-
tion. Two signals v L
n
v
2
[0,) and w L
n
w
2
[0,)
satisfy the IQC defined by Π if
Z
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 :
L
n
v
2e
[0,) L
n
w
2e
[0,) satisfies the IQC defined by Π,
if (11) holds for all v L
n
v
2
[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
h
the deviation between the delayed and the undelayed
signal as S
h
(v) := v(t h) v(t).
To describe the constant time-delay term S
h
three
IQCs were proposed in (Pfifer and Seiler, 2015)
Π
1
=
0 1
1 1
, (12)
Π
2
( jω) =
jω + 1
10 jω + 1
2
|ζ
2
( jω)|
2
0
0 1
, (13)
Π
3
( jω) =
0 ζ
3
( jω)
ζ
3
( jω) 1
, (14)
where
ζ
2
( jω) := 2
( jωh)
2
+ 3.5 jωh + 10
6
( jωh)
2
+ 4.5 jωh + 7.1
, (15)
ζ
3
( jω) :=
2.19( jωh)
2
+ 9.02 jωh + 0.089
( jωh)
2
5.64 jωh 17
. (16)
A combination of IQCs still an IQC, if an operator
satisfies the IQCs Π
i
, i = 1,2,...,M, then it also sat-
isfies the IQC Π
d
( jω) =
M
i=1
λ
i
( jω)Π
i
( jω), where
λ
i
> 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
i
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) =
Ψ
( jω)MΨ( jω), where M = M
T
R
n
z
×n
z
and Ψ
RH
n
z
×(n
v
+n
w
)
. (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
v
w
. 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
0
z(t)
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
0
z(t)
T
Mz(t)dt 0 (18)
for all T 0, v L
n
v
2
[0,), w = (v) and z =
Ψ
v
w
.
Let Π = Π
and partition as
Π
11
Π
21
Π
21
Π
22
. Ac-
cording to (Seiler, 2015, Theorem 4) if Π
11
( jω) > 0
and Π
22
( 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
z
×(n
v
+n
w
)
. 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
h
the plant of the intercon-
nected system reveals the following:
˙
ˆx =
ˆ
A ˆx(t) +
ˆ
A
h
ˆx(t h) +
ˆ
B
w(t)
d(t)
, (19)
z
y
=
ˆ
C ˆx(t) +
ˆ
D
w(t)
d(t)
, (20)
where ˆx :=
x
x
Ψ
are the extended state, x
Ψ
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
2
-gain for system
F
U
(G,S
h
) can be computed.
Theorem 1. Assume F
U
(G,S
h
) is well-posed and
S
h
satisfies the hard IQC defined by (Ψ,M). Then
kF
U
(G,S
h
)k
γ if there exists a λ > 0 and a
bounded quadratic Lyapunov-Krasovskii functional
V ( ˆx
t
) such that for some ε > 0
V ( ˆx
t
) εk ˆx(t)k
2
,
the following inequality holds
λz
T
Mz +
˙
V ( ˆx
t
)γ
2
d
T
d + y
T
y
εkˆx(t)k
2
εkd(t)k
2
. (21)
Proof 1. Integrate the inequality (21) from t = 0 to
t = T with the initial condition φ(t) = 0 t [h,0]
λ
Z
T
0
z(τ)
T
Mz(τ)dτ +V (ˆx
T
) γ
2
Z
T
0
d(τ)
T
d(τ)dτ
+
Z
T
0
y
T
(τ)y(τ)dτ
ε
Z
T
0
(k ˆx(τ)k
2
+ kd(τ)k
2
)dτ 0.
Using that the IQC and the LKF V are non-negative
this inequality is equivalent to kF
U
(G,S
h
)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
i
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
i
coefficients together with IQC Π
i
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
i
Π
i
. Therefore a different factorization is nec-
essary for the λ
i
coefficients. In the next section a
method will be shown to preserve the dynamics of the
λ
i
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:
λ
i
( jω) = Θ
i
( jω)
Λ
i
Θ
i
( jω) (22)
where i = 1,2,...,M. One possible factorization pro-
posed in (Veenman and Scherer, 2014) is the follow-
ing
Θ
i
( jω) =
1
1
jω ρ
i
...
1
( jω ρ
i
)
ν
i
T
, (23)
Analysis of Input Delay Systems using Integral Quadratic Constraint
105
ρ
i
< 0,ν
i
N fixed parameters and 0 < Λ
i
R
(ν
i
+1)×(ν
i
+1)
constant real symmetric matrix. This
Θ
i
is a basis function, using a fixed pole location (ρ
i
constant), then by increasing the dimension of the ba-
sis function (ν
i
) a better approximation of the λ
i
coef-
ficient can be established.
Assume that the dimension of v is 1. Using J-
spectral factorization of the IQC (Ψ
i
,
1 0
0 1
)
and the factorization of λ
i
as Equation (22) the fac-
torized combined IQC reveals
Π
d
( jω) =
N
i=1

Θ
i
( jω) 0
0 Θ
i
( jω)
Ψ
i
( jω)
×
Λ
i
0
0 Λ
i
Θ
i
( jω) 0
0 Θ
i
( jω)
Ψ
i
( jω).
(24)
In case of a frequency independent IQC only the λ
coefficient is factorized
Π
i
( jω) = []
Π
11
Λ
i
Π
21
Λ
i
Π
21
Λ
i
Π
22
Λ
i
×
Θ
i
( jω) 0
0 Θ
i
( jω)
. (25)
4.2 LMI Formulation of L
2
-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
Γ
(t) = A
Γ
x
Γ
(t) +B
Γ1
d(t) + B
Γ2
w(t)
z(t) = C
Γ
x
Γ
(t) +D
Γ1
d(t) + D
Γ2
w(t),
and the dimension of the x
Γ
denoted by n
Γ
. 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
Γ
(t)
= A
c
x(t)
x
Γ
(t)
+ A
ch
x(t h)
x
Γ
(t h)
+
+
B
c1
B
c2
w(t)
d(t)
,
(26)
z(t)
y(t)
=
C
Γ
C
c
x(t)
x
Γ
(t)
+
D
Γ
D
c
w(t)
d(t)
,
(27)
where
A
c
=
A 0
0 A
Γ
, A
ch
=
A
h
0
0 0
,
B
c1
=
B
h
B
Γ2
, B
c2
=
B +B
h
B
Γ1
,
C
Γ
=
0 C
Γ
, C
c
=
C 0
,
D
Γ
=
D
Γ2
D
Γ1
, D
c
=
0 0
.
Before presenting the LMI formulation of the L
2
-
gain computation some notations must be introduced.
¯
Q =
Q
0
Q
1
... Q
N
¯
S =
1
l
diag(S
0
S
1
... S
N
)
¯
R =
R
00
R
T
10
... R
T
N0
R
10
R
11
... R
T
N1
.
.
.
.
.
.
.
.
.
.
.
.
R
N0
R
N1
... R
NN
=
11
Q
T
N
A
T
ch
P S
N
B
T
c
P 0 γ
2
0 0
0 1
C
c
0
D
c
[]
T
C
Γ
0
D
Γ
Λ 0
0 Λ
[]
T
11
= PA
c
A
T
c
P Q
0
Q
T
0
S
0
C
T
c
C
c
S
d
= diag{S
0
S
1
,S
1
S
2
,...,S
N1
S
N
}
R
d
=
R
d11
R
d12
... R
d1N
R
d21
R
d22
... R
d2N
.
.
.
.
.
.
.
.
.
.
.
.
R
dN1
R
dN2
... R
dNN
R
d pq
= l(R
p1,q1
R
pq
)
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106
D
s
=
D
s
01
D
s
02
... D
s
0N
D
s
11
D
s
12
... D
s
1N
D
s
w1
D
s
w2
... D
s
wN
D
s
0p
=
l
2
A
T
c
(Q
p1
+ Q
p
) +
l
2
(R
0,p1
+ R
0p
)
(Q
p1
Q
p
)
D
s
1p
=
l
2
A
T
ch
(Q
p1
+ Q
p
)
l
2
(R
N,p1
+ R
N p
)
D
s
wp
=
l
2
B
T
c
(Q
p1
+ Q
p
)
D
a
=
D
a
01
D
a
02
... D
a
0N
D
a
11
D
a
12
... D
a
1N
D
a
w1
D
a
w2
... D
a
wN
D
a
0p
=
l
2
A
T
c
(Q
p1
Q
p
)
l
2
(R
0,p1
R
0p
)
D
a
1p
=
l
2
A
T
ch
(Q
p1
Q
p
) +
l
2
(R
N,p1
R
N p
)
D
a
wp
=
l
2
B
T
c
(Q
p1
Q
p
)
Theorem 2. Assume F
U
(G,S
h
) is well-posed and
S
h
satisfies the hard IQC defined by (Ψ,M).
Then kF
U
(G,S
h
)k
γ if there exists 0 < Λ
R
(ν+1)×(ν+1)
and {P = P
T
,Q
p
,S
p
,R
pq
= R
T
qp
}
R
(n
x
+n
Γ
)×(n
x
+n
Γ
)
, p = 0,1,...,N,q = 0, 1, . . . , N such
that the following holds:
P
¯
Q
¯
Q
T
¯
R +
¯
S
> 0 (28a)
D
sT
R
d
+ S
d
D
aT
0 3S
d
> 0. (28b)
The LMI formulation of the L
2
-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
2
-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
i
(t) = v
i
(t), (29a)
˙v
i
(t) = q
i
(t) +d
i
(t), (29b)
˙q
i
(t) =
1
τ
i
q
i
(t) +
g
i
τ
i
u
i
(t), (29c)
where p
i
,v
i
denote position and velocity, d
i
is a
disturbance representing both outer effects and mod-
elling error, q
i
is an internal state such that the ac-
celeration of the vehicle is a
i
(t) = q
i
(t) + d
i
(t). L
2
-
gain calculations will be carried out on a homoge-
neous platoon (the vehicles parameter are the same)
with parameters τ
i
= 0.7 and g
i
= 1 i = 0,1,...,n. u
0
is the signal generated by the pedal signal of the first
vehicle driver, u
i
,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
u
1
(t) = k
1
δ
1
(t) k
2
e
1
(t) +a
0
(t h)
u
i
(t) = k
1β
δ
i
(t) k
2β
e
i
(t) +k
a0
a
0
(t h)
+ k
a1
a
i1
(t h) k
1α
(v
i
(t h) v
0
(t h))
k
2α
(p
i
(t h) p
0
(t h)), i = 2,...,n
where δ
i
, v
i
v
i1
and e
i
, p
i
p
i1
+ L
i
are the relative speed and spacing error, respec-
tively. The prescribed spacing L
i
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
1
=
0.7,k
2
= 0.1127,k
1α
= 0.4642,k
2α
= 0.0564,k
1β
=
0.2358,k
2β
= 0.0564, k
a1
= 0.0449, k
a0
= 0.9551.
In the analysis only SISO systems are considered,
namely d
0
7→ e
1
(the effect of the lead vehicle dis-
turbance on the first spacing error) and d
1
7→ e
2
(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 system with state vector
x = [e
1
,δ
1
,q
1
]
T
are the following:
A =
0 1 0
0 0 1
0.16 1 1.43
, B =
0
1
0
,
Analysis of Input Delay Systems using Integral Quadratic Constraint
107
B
h
=
0
0
1.43
, A
h
= 0, C =
1 0 0
.
The system d
1
7→ e
2
has both state and input de-
lay, the state-space matrices using the state vector
x = [e
1
,δ
1
,q
1
,e
2
,δ
2
,q
2
]
T
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
,
A
h
=
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0.08 0.66 0.06 0.08 0.66 0
,
B = [0,1,0, 0, 1,0]
T
, B
h
= [0, 0, 0, 0, 0, 0.06]
T
,
C = [0,0,0,1,0,0].
5.2 L
2
-gain of a Vehicle Platoon
A linear combination of the multipliers will be used
as
Π
d
( jω) = λ
1
( jω)Π
1
+ λ
2
( jω)Π
i
( jω), (31)
where Π
i
can be Π
2
or Π
3
, and λ
1
,λ
2
: jR R that
satisfies λ
i
( jω) > 0,ω.
Before the numerical results notations are neces-
sary for the different L
2
-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
d
=
1
τ
d
s+1
. The τ
d
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
d
= 0.05 is
used.
TD the time-domain IQC method with constant λ
i
us-
ing IQCs Π
2
or Π
3
in the combined Π
d
.
TDλ the time-domain IQC method with dynamic λ
i
us-
ing IQCs Π
2
or Π
3
in the combined Π
d
. At the
factorization of λ
i
the ρ
i
= 1 parameter is used
and the ν
i
is increased from 0 until a tight bound
with the lower bound is established. In the tables
the results using ν
i
= 2 are presented.
In Table 1 the numerical results are shown in case
of system d
0
7→ e
1
for the different L
2
-gain compu-
tation methods. This system does not contain state
delay, therefore a simple quadratic storage function
V = x
T
Px is used instead of an LKF. By the time-
domain IQC method with constant λ coefficient (TD-
2,3) a highly overestimated L
2
-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
2
-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
2
-gain are computed.
Two different IQCs are considered Π
2
and Π
3
. Better
numerical results were established using Π
2
than Π
3
.
In an earlier paper (Rödönyi and Varga, 2015) dif-
ferent methods were suggested for L
2
-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
2
-gain of system (d
0
7→ e
1
) 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
d
1
7→ e
2
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
2
-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
2
-gain of system (d
1
7→ e
2
) 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
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108
6 CONCLUSION
A time-domain method for computing upper bound
of the L
2
-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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