Block Triangular Decoupling of General Neutral Multi Delay Systems
Fotis N. Koumboulis and Nikolaos D. Kouvakas
Halkis Institute of Technology, Department of Automation, 34400, Psahna Evoias, Greece
Keywords: General Neutral Multi Delay Systems, Dynamic Controllers, Realizable Controllers.
Abstract: The problem of block triangular decoupling is studied for the case of general neutral multi delay systems.
The system is not restricted to be square and invertible. The controller is of the general neutral dynamic type
involving a dynamic feedback and dynamic precompensator. Two different cases of feedback are studied.
The first is the case of measurable output feedback and second is the case of performance output feedback.
The controller is restricted to be realizable. The necessary and sufficient conditions for the problem to be
solvable are established. The general class of the realizable controllers solving the problem is derived. The
closed loop transfer function is proven to have arbitrary characteristic polynomial thus facilitating command
tracking and stability.
1 INTRODUCTION
The block triangular decoupling problem, has
attracted considerable attention (see Commault and
Dion, 1983; Lohmann, 1991; Morse and Wonham,
1970; Otsuka and Inaba, 1992; Otsuka, 1992; Park
2008; Park and Choi, 2011; Sourlas, 2001 and the
references therein). The problem appears to be of
great importance particularly for large scale and
interconnected MIMO plants. For the case of
retarded time delay systems (or more generally for
systems over a ring or a principal ideal domain) the
problem has been studied in Caturiyati (2003), Ito
and Inaba (1997a) and Ito and Inaba (1997b).
The category of general neutral multi delay
systems is more general than the aforementioned
system cases and covers a wide range of applications
(see Koumboulis and Panagiotakis 2008;
Koumboulis, Kouvakas and Paraskevopoulos,
2009a-c and the references therein). In the present
paper the block triangular decoupling problem is
studied for the first time for the category of general
neutral multi delay systems. The controller is of the
measurement output feedback type with a dynamic
feedback matrix and a dynamic precompensator. The
controller is required to be realizable. The controller
type covers the state feedback and the performance
output feedback cases as special cases. The
contribution of the present paper consists in
establishing the necessary and sufficient conditions
for the problem to be solvable and deriving the
general class of the realizable controllers solving the
problem. The closed loop transfer function is proven
not to be restricted by the design requirement except
of its realization index, thus achieving tracking and
BIBO stability. It is important to mention that the
special case of row by row triangular decoupling for
the category of general neutral multi delay systems
has been solved in Koumboulis and Panagiotakis
(2008) using static controllers and in Koumboulis
and Kouvakas (2010) using dynamic controllers.
Also, the problem of diagonal block decoupling for
the same system category has been solved in
Koumboulis and Kouvakas (2011).
2 PRELIMINARIES
Consider the general class of linear neutral multi-
delay differential systems
00
,,
11 11
qq
qq
jjiijjii
ji ji
Extq Axtqττ
== ==
⎛⎞⎛⎞
⎟⎟
⎜⎜
⎟⎟
⎜⎜
−=−+
⎟⎟
⎜⎜
⎟⎟
⎟⎟
⎜⎜
⎝⎠⎝⎠
∑∑∑∑



0
,
11
q
q
jjii
ji
Bu t q τ
==
⎛⎞
⎟
⎜
⎟
⎜
+−
⎟
⎜
⎟
⎟
⎜
⎝⎠
∑∑

(1a)
00
,,
11 11
qq
qq
j jii j jii
ji ji
Cytq Cxtqττ
== ==
⎛⎞⎛⎞
⎟⎟
⎜⎜
⎟⎟
⎜⎜
−=−
⎟⎟
⎜⎜
⎟⎟
⎟⎟
⎜⎜
⎝⎠⎝⎠
∑∑∑∑

(1b)
where
()
n
xt ∈ 
denotes the vector of state
542
N. Koumboulis F. and D. Kouvakas N..
Block Triangular Decoupling of General Neutral Multi Delay Systems.
DOI: 10.5220/0004048205420547
In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 542-547
ISBN: 978-989-8565-21-1
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

variables,
(
)
m
ut ∈ 
the vector of control inputs,
()
p
yt ∈ 
the vector of performance outputs,
i
τ
(
1, ,iq= …
) are positive real numbers denoting
point delays, and
,ji
q
(
0
1, ,jq= …
;
1, ,iq= …
) is a
finite sequence of integers with regard to
i and
j
.
The quantities
q and
0
q
are positive integers.
Clearly, if the quantity
,
1
q
ji i
i
q τ
=
∑
is negative then it
denotes prediction. The real matrices
j
E

,
j
A

,
j
B

have
n
rows while the real matrices
j
C
,
j
C

have
p rows. In general, mp≠ .
The interest is focused on the forced behaviour of
the system, i.e. for zero initial and past conditions
(
(
)
0xt =
,
(
)
0ut =
for
0t <
). Defining
1 q
ττ
⎡⎤
=
⎢⎥
⎣⎦
T 
,
()
()
1
exp exp
q
ssττ
⎡⎤
=− −
⎢⎥
⎣⎦
-sT
e 
the system (1) can be described in the frequency
domain by the following set of equations
()
()
()
()
()
()
sE X s A X s B U s=+
-sT -sT -sT
eee


(2a)
()
()
()
()
CYsC Xs=
-sT -sT
ee

(2b)
where
() ()
{}
Xs xt
−
= L
,
() ()
{}
Us ut
−
= L
,
(
)
Ys=
(
)
{
}
yt
−
L
with
{
}
−
•L
be the Laplace
transform of the argument signal, while
()
()
()
0
,
11
exp
j
q
q
j
ji i
ji
j
E
E
A
sq
A
C
C
τ
==
⎡⎤
⎡⎤
⎢⎥
⎢⎥
⎡
⎤
⎛⎞
⎢⎥
⎟
⎜⎢⎥
⎢
⎥
⎟
⎜=−
⎢⎥
⎟
⎢⎥
⎢
⎥
⎜
⎟
⎟
⎜
⎢⎥
⎝⎠
⎢⎥
⎢
⎥
⎣
⎦
⎢⎥
⎢⎥
⎣⎦
⎢⎥
⎣⎦
∑∑
-sT
-sT
-sT
e
e
e






()
0
,
11
exp
q
q
jjii
ji
BBsqτ
==
⎡⎤
⎛⎞
⎟
⎜
⎢⎥
⎟
⎜=−
⎟
⎢⎥
⎜
⎟
⎟
⎜
⎝⎠
⎢⎥
⎣⎦
∑∑
-sT
e

()
0
,
11
exp
q
q
jjii
ji
CCsqτ
==
⎡⎤
⎛⎞
⎟
⎜
⎢⎥
⎟
⎜=−
⎟
⎢⎥
⎜
⎟
⎟
⎜
⎝⎠
⎢⎥
⎣⎦
∑∑
-sT
e
where
exp e
⎡⎤
⋅
⎢⎥
⎣⎦
⎡⎤
⋅=
⎢⎥
⎣⎦
is the exponential of the
argument quantity. The matrices
()
E
-sT
e

and
()
C
-sT
e
are assumed to be invertible. Hence, the
system of equations in (2) can be expressed in
normal system form as follows
()
()
()
()
()
sX s A X s B U s=+
-sT -sT
ee
(3a)
(
)
(
)
(
)
Ys C Xs=
-sT
e
(3b)
where
() () ()
1
AEA
−
⎡⎤
=
⎢⎥
⎣
⎦
-sT -sT -sT
eee


() () ()
1
BEB
−
⎡⎤
=
⎢⎥
⎣
⎦
-sT -sT -sT
eee

() () ()
1
CCC
−
⎡⎤
=
⎢⎥
⎣
⎦
-sT -sT -sT
eee

Consider the open loop transfer matrix
( ) () () ()
1
,
n
Ps C sI A B
−
⎡⎤
=−
⎢⎥
⎣
⎦
-sT -sT -sT -sT
ee ee
(4)
3 SOLUTION OF THE BLOCK
TRIANGULAR DECOUPLING
PROBLEM
Here, the design goal is that of block triangular
decoupling via dynamic state feedback and dynamic
precompensator, namely to derive a closed loop
system in a block lower triangular form. The outputs
of the system are grouped into blocks and each block
of outputs is controlled by the corresponding group
of external inputs and the previous groups of
external inputs. To present the formal definition of
the problem we will first present the form of the
controller.
Let
(
)
sΨ
be the Laplace transform of the vector
()
r
tψ ∈ 
, denoting the measurement output of the
system. It holds that
()
()
()
sL XsΨ=
-sT
e
, where
()
L
-sT
e
is a
rn×
matrix with elements being
rational functions of
1
,,
q
s
s
ee
τ
τ
−
−
…
. The feedback is
proposed to be of the form
()
()
()
()
()
,,Us Ks s Gs s=Ψ+Ω
-sT -sT
ee
(5)
where
()
sΩ
is the
1p×
vector of external inputs.
The elements of the matrices
()
,Ks
-sT
e
and
(
)
,Gs
-sT
e
are rational functions of
s
. The
BlockTriangularDecouplingofGeneralNeutralMultiDelaySystems
543

respective numerator and denominator polynomial
coefficients are multivariable rational functions of
1
,,
q
s
s
ee
τ
τ
−
−
…
. The controller is restricted to be
realizable. This means that the elements of
()
,Ks
-sT
e
and
(
)
,Gs
-sT
e
are restricted to be
realizable, i.e. their realizability index should be
greater than or equal to zero. Substituting controller
(5) to the open loop system (3) the problem of block
decoupling is formulated as follows
() ()
n
CsIA
⎡
−−
⎢
⎣
-sT -sT
ee
()( )() ()( )
1
,,BKsL BGs
−
⎤
−=
⎥
⎦
-sT -sT -sT -sT -sT
eee ee
()
{}
,
1,..., ; 1,..,
block.triang ,
ij
iji
Hs
ν==
=
-sT
e
(6)
where
()
{}
,
1,..., ; 1,..,
block.triang ,
ij
iji
Hs
ν==
=
-sT
e
()
() ()
() () ()
1,1
2,1 2,2
,1 ,2 ,
,0 0
,, 0
,, ,
Hs
Hs Hs
Hs Hs Hs
νν νν
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎢⎥
=
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
-sT
-sT -sT
-sT -sT -sT
e
ee
ee e




and where
()()
,
,,
ij
pp
ij
Hs s
×
∈
-sT -sT
ee
(
{}
,
1, ,
ij
ν
∈
…
) are matrices whose elements are
rational functions of
s
while the respective
numerator and denominator polynomial coefficients
are multivariable rational functions of
1
,,
q
s
s
ee
τ
τ
−
−
…
.
Obviously it holds that
1
i
i
pp
ν
=
=
∑
. The matrices
()
,
,
ii
Hs
-sT
e
(
1, ,i ν= …
) are square and invertible.
From (6) we also observe that
(
)
,Gs
-sT
e
is
constrained to be of full column rank while
()
,Ks
-sT
e
is constrained to preserve the solvability
of the closed loop system, i.e.
()
det
n
sI A
⎡
−−
⎢
⎣
-sT
e
()( )()
,0BKsL
⎤
−≡/
⎥
⎦
-sT -sT -sT
eee(7)
Defining
( ) () () ()
1
,
n
Qs L sI A B
−
⎡⎤
=−
⎢⎥
⎣⎦
-sT -sT -sT -sT
ee ee
and applying elementary computations, relation (6)
can be rewritten as
() ()()
{}
()
1
,,,,
n
Ps sI Ks Qs Gs
−
−=
-sT -sT -sT -sT
eeee
()
{}
,
1,..., ; 1,..,
block.triang ,
ij
iji
Hs
ν==
=
-sT
e
(8)
Define
(
)
(
)
(
)
{}
1
,,,
n
Gs sI Ks Qs
−
=−
-sT -sT -sT
eee

()
,Gs×
-sT
e
(9)
After dividing
(
)
,Ps
-sT
e
into row blocks as follows
()
()
()
1
,
,
,
Ps
Ps
Ps
ν
⎡
⎤
⎢
⎥
⎢
⎥
=
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
-sT
-sT
-sT
e
e
e

()
()
,
,
i
pm
i
Ps
s
×
⎡
⎤
∈
⎢
⎥
⎣
⎦
-sT
-sT
e
e
the equation (8) takes on the form
(
)
(
)
() ()
{}
1
,
1,..., ; 1,..,
,
,block.triang,
,
ij
iji
Ps
Gs H s
Ps
ν
ν
==
⎡
⎤
⎢⎥
⎢⎥
=
⎢⎥
⎢⎥
⎢⎥
⎣
⎦
-sT
-sT -sT
-sT
e
ee
e


(10)
or equivalently the form
()()
,,
i
Ps Gs =
-sT -sT
ee

() ()
,1 , 1
,,
iii
Hs H s
−
⎡
⎢
⎣
-sT -sT
ee
()
,
, 0 0 ; 1,...,
ii
Hs i ν
⎤
=
⎥
⎦
-sT
e  (11)
From (11) the following necessary condition is
derived
()
()
1
12
,
Rank
,
Ps
pp p
Ps
ν
ν
⎧⎫
⎡
⎤
⎪⎪
⎪⎪
⎢⎥
⎪⎪
⎪⎪
⎢⎥
⎪⎪
=+++
⎨⎬
⎢⎥
⎪⎪
⎢⎥
⎪⎪
⎪⎪
⎢⎥
⎪⎪
⎣
⎦
⎪⎪
⎩⎭
-sT
-sT
e
e


(12)
Condition (12) implies that
mp≥
. Before
presenting the necessary and sufficient conditions
and the general solutions of the controller matrices,
three definitions will be presented. If
mp= define
()()
ˆ
,,Ps Ps=
-sT -sT
ee
. If
mp>
define
()
()
()
()
,
ˆˆ
,;det,0
,
Ps
Ps Ps
Ps
⎡
⎤
⎢⎥
=≡/
⎢⎥
⎢⎥
⎣
⎦
-sT
-sT -sT
-sT
e
ee
e

(13)
ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics
544

Clearly, the choice of
(
)
,Ps
-sT
e

is not unique. Here,
the following choice is proposed
()
1
1
,
mp
T
TT
mp
Ps ke k e
υυ
−
−
⎡⎤
=
⎢⎥
⎣⎦
-sT
e


where
j
e
is an 1 m× unity row vector having the
unity in its
j −
th position and
i
k
are appropriate
different than zero reals. The integers
1
,,
mp
υυ
−
…
are chosen in a way that the set of integers
{}
{}
1
1, , , ,
mp
m υυ
−
−……
corresponds to the
indices of the linear independent columns of
(
)
,Ps
-sT
e
. The parameters
i
k
may be used to
adjust the characteristics of
(
)
1
ˆ
,Ps
−
⎡⎤
⎢⎥
⎣⎦
-sT
e
(e.g. to
adjust its norm, to achieve stability). Divide
(
)
1
ˆ
,Ps
−
⎡⎤
⎢⎥
⎣⎦
-sT
e
in column blocks as follows
() ()()
††
1
,,,Ps Ps P s
ν
⊥
⎡⎤
=
⎢⎥
⎣⎦
-sT -sT -sT
eee
()
1
ˆ
,Ps
−
⎡⎤
=
⎢⎥
⎣⎦
-sT
e
where
()
()
†
,
,
i
mp
i
Ps
s
×
⎡
⎤
∈
⎢
⎥
⎣
⎦
-sT -sT
e
e
()
()
()
,
,
mmp
Ps
s
×−
⊥
⎡⎤
∈
⎢⎥
⎣⎦
-sT -sT
e
e
In the following theorem, the solvability
conditions and the general class of all realizable
controllers solving the problem are presented.
Theorem 1: The necessary and sufficient
condition for the solvability of the Block Triangular
Decoupling problem via a dynamic measurement
output feedback controller of the form (5) is
condition (12). The general class of the realizable
controller matrices solving the problem is
() ()()
{}
,,,
m
Gs I Ks Qs=×−
-sT -sT -sT
eee
()()
†
,1
1
,,Ps H s
ν
κκ
κ=
⎧
⎡
⎪
⎪
⎢
×
⎨
⎢
⎪
⎪
⎣
⎩
∑
-sT -sT
ee
()()
†
,
,,
i
i
Ps H s
ν
κκ
κ=
∑
-sT -sT
ee…
(
)
(
)
†
,
,,Ps H s
ννν
⎤
+
⎥
⎦
-sT -sT
ee
()
() ()
}
1
,, ,Ps s s
ν
⊥
⎡⎤
+Λ Λ
⎢⎥
⎣⎦
-sT
-sT -sT
ee e
(14a)
()
,:Ks
-sT
e
arbitrary, realizable and preserve the
closed loop solvability (14b)
where
()()
()
,,
i
mp p
i
ss
−×
Λ∈
-sT -sT
ee
(1,,)i ν= …
()()
,
,,
ij
pp
ij
Hs s
×
∈
-sT -sT
ee
(
{}
1, ,
i
ν
∈
…
,
{}
1, ,
j
i
∈
…
) are arbitrary matrices being enough
realizable to satisfy the realizability of (14a). The
matrices
()
,
,
ii
Hs
-sT
e
(
1, ,i ν= …
) are square and
invertible.
Proof: Using the definitions before Theorem 1, we
observe that the general solution of (10) is
() ()()
†
,1
1
,,,Gs P s H s
ν
κκ
κ=
⎡
⎢
=
⎢
⎣
∑
-sT -sT -sT
eee


()()
†
,
,,
i
i
Ps H s
ν
κκ
κ=
∑
-sT -sT
ee…
(
)
(
)
†
,
,,Ps H s
ννν
⎤
+
⎥
⎦
-sT -sT
ee
()() ()
1
,, ,Ps s s
ν
⊥
⎡
⎤
+Λ Λ
⎢
⎥
⎣
⎦
-sT
-sT -sT
ee e
(15)
where
()()
()
,,
i
mp p
i
ss
−×
Λ∈
-sT -sT
ee
(
1, ,i ν= …
)
are arbitrary matrices. Substituting (15) to (9), the
relation (14a) is derived. Clearly,
()
,Ks
-sT
e
is
arbitrary but it should also be chosen to be realizable
and to satisfy the constraint (7). For
(
)
,Gs
-sT
e
to be
left invertible (of full column rank) and realizable, it
is necessary for
()
,
,
ij
Hs
-sT
e
and
(
)
,
i
sΛ
-sT
e
(
{}
1, ,
i
ν
∈
…
,
{}
1, ,
j
i
∈
…
) to be sufficiently
realizable and
()
,
,
ii
Hs
-sT
e
to be invertible. For
example the index of realizability of
(
)
,
i
sΛ
-sT
e
should be greater than or equal to the minus of the
index of realizability of
()()
{}
()
,,,
m
PsIKs Qs
⊥
−
-sT-sT -sT
eee
and the
index of realizability of
()
,
,
ij
Hs
-sT
e
should be
greater than or equal to the minus of the realizability
index of
()()
{}
()
†
,
,,
im
Ps
IKs Qs
−
-sT
-sT -sT
e
ee
. 
The expressions of the general class of the
controller matrices proposed in Theorem 1 are
implicit. In the following corollary explicit and
analytic expressions of a class of controllers solving
BlockTriangularDecouplingofGeneralNeutralMultiDelaySystems
545

the problem at hand are proposed. The derivation of
this class is based on the observation that
(
)
,Qs
-sT
e
and
(
)
,Ps
-sT
e
are strictly proper with respect to
s
.
Corollary 1: If
(
)
,Qs
-sT
e
is realizable, a class of the
realizable controller matrices solving the problem is
given by (14a) and (14b) with
i)
()
,Ks
-sT
e
proper with respect to
s
and
realizable while the minimum index of
realizability of its elements is restricted to be
greater than or equal to zero
ii) the minimum index of realizability of the
elements of
(
)
,
,
ij
Hs
-sT
e
is restricted to be
greater than or equal to the minus of the
minimum realizability index of the elements of
()
†
,
i
Ps
-sT
e
iii) the minimum index of realizability of the
elements of
()
,
i
sΛ
-sT
e
is restricted to be greater
than or equal to the minus of the minimum
realizability index of the elements of
()
,Ps
⊥ -sT
e
. 
Corollary2: If
()()
,,Qs Ps=
-sT -sT
ee
(the case of
performance output feedback) and
()
,Ps
-sT
e
is
realizable, a class of the realizable controller
matrices solving the problem is given by
() ()()
†
,1
1
,,,Gs P s H s
ν
κκ
κ=
⎧
⎡
⎪
⎪
⎢
=
⎨
⎢
⎪
⎪
⎣
⎩
∑
-sT -sT -sT
eee
()()
†
,
,,
i
i
Ps H s
ν
κκ
κ=
∑
-sT -sT
ee…
(
)
(
)
†
,
,,Ps H s
ννν
⎤
+
⎥
⎦
-sT -sT
ee
()
() ()
}
1
,, ,Ps s s
ν
⊥
⎡⎤
+
+Λ Λ
⎢⎥
⎣⎦
-sT
-sT -sT
ee e
()
()
{}
{}
,
1,..., ; 1,...,
block.triang
,
,
ij
iji
Hs
Ks
ν==
−
-sT
-sT
e
e
(16)
where
()
,Ks
-sT
e
is restricted only to be proper with
respect to
s
and realizable while the minimum index
of realizability of its elements is restricted to be
greater than or equal to zero and the minimum index
of realizability of the elements of
(
)
,
,
ij
Hs
-sT
e
is
restricted to be greater than or equal to the minus of
the minimum realizability index of the elements of
()
†
,
i
Ps
-sT
e
. The minimum index of realizability of
the elements of
(
)
,
i
sΛ
-sT
e
is restricted to be greater
than or equal to the minus of the minimum
realizability index of the elements of
()
,Ps
⊥ -sT
e
. 
Remark 1: In Theorem 1 and Corollaries 1 and 2
the blocks of the closed loop transfer matrix are
restricted only to have enough large realizability
index. This way regional BIBO stability of the
closed loop system can be achieved, while tracking
and command following are satisfied as fast as is
allowed by the realizability indices of the elements
of the closed loop transfer matrix.
Remark 2: The condition (12) is the same with the
necessary and sufficient condition for the solvability
of the diagonal block decoupling (Koumboulis and
Kouvakas, 2011). As was expected, the class of
controller, solving the problem at hand, is much
wider.
4 CONCLUSIONS
In the present paper the problem of block triangular
decoupling has been studied for the first time for the
category of general neutral multi delay systems. The
controller has been selected of the general neutral
dynamic type involving a dynamic feedback and
dynamic precompensator. The controller has been
restricted to be realizable. The necessary and
sufficient condition for the problem to be solvable
has been established and the general class of the
realizable controllers solving the problem has been
derived. The realizability indices of the elements of
the closed loop transfer matrix are restricted to be
large enough. Except this restriction, the closed loop
transfer matrix has been proven to have arbitrary
characteristic polynomial thus offering itself for
command tracking and regional BIBO stability.
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