Analysis of Input Delay Systems using Integral Quadratic Constraint
Gabriella SzabóVarga and Gábor Rödönyi
Systems and Control Laboratory, Computer and Automation Research Institute of Hungarian Academy of Sciences,
Budapest, Hungary
Keywords:
Timedelay Systems, LyapunovKrasovskii Functional, Integral Quadratic Constraints, Vehicle Platoon.
Abstract:
The L
2
gain computation of a linear timeinvariant system with state and input delay is discussed. The input
and the state delay are handled separately by using dissipation inequality involving a LyapunovKrasovskii
functional and integral quadratic constraints. A conic combination of IQCs is proposed for characterizing
the input delay, where the coefﬁcients are linear timeinvariant 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
timeinvariant and also in timevarying 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 LyapunovKrasovskii 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 LyapunovKrasovskii functional
and integral quadratic constraints in the timedomain.
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
SzabóVarga, 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 102109
ISBN: 9789897581984
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