10.2478/v10170-010-0041-4
Archives of Control Sciences
Volume 21(LVII), 2011
No. 3, pages 227–242
Pairwise control principle in large-scale systems
ANNA FILASOVÁ and DUŠAN KROKAVEC
The purpose of the paper is present an algorithm of partially decentralized control design
for one type of large-scale linear dynamical system. The pairwise autonomous principle is preferred where design conditions are derived in the bounded real lemma form, and global system
stability is reproven to formulate potential application principle in fault tolerant control. The
validity of the proposed method is demonstrated by the numerical example.
Key words: large-scale systems, decentralized control, state feedback, linear matrix inequality, asymptotic stability
1.
Introduction
Control structure implementation in large-scale systems is computationally pretentious due to high complexity and large dimensionality. Thus, generally all approaches
to large-scale system control (see e.g. [6], [7], [8], [13], [14]) are based on some sort
of model alteration (minimization of interconnection, model reduction through aggregation methods, etc.), and a certain type of decentralization. For well-defined system
inputs in large-scale interconnected dynamical systems where each subsystem input is
decoupled from others, a partially decentralized design approach can be used. This approach provides some advantage in parallel processing and interconnection utilization.
Application of mentioned above ideas for the purpose of decentralization and reduction
of computational requirements in LQ control, and Kalman filtering was subjects of the
papers [12], [2], [3].
A number of problems that arise in the state feedback control can be reduced to a
handful of standard convex and quasi-convex problems that involve matrix inequalities.
It is known that optimal solution can be computed by using interior point method [15]
which converge in polynomial time with respect to the problem size, and efficient interior
point algorithms have recently been developed for and further development of algorithms
of these standard problems is the area of active research. In such approaches, the stability
conditions may be expressed in terms of linear matrix inequalities (LMI), which have a
The Authors are with Technical University of Košice, Faculty of Electrical Engineering and Informatics, Department of Cybernetics and Artificial Intelligence, Košice, Slovak Republic, fax: +421 55 602
2564, e-mail: anna.filasova@tuke.sk, dusan.krokavec@tuke.sk
Received 1.04.2011. Revised 12.09.2011.
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A. FILASOVÁ, D. KROKAVEC
notable practical interest due to the existence of numerical solvers. Some progress review
in this field can be found in [1], [5], [17], and the references therein.
In this paper the pairwise partially decentralized approach to continuous-time largescale linear systems control design is formulated in the bounded real lemma form, and
conditions for the global system stability are reproved. The resulting controller structures
are pairwise autonomous (implying parallel processing) and partially decentralized (implying interconnection utilization). The potential application is in fault tolerant control
systems where such control reconfiguration principle like [10, 11] can be relatively simple introduced in the pairwise partially decentralized structures with respect to sensors
fault in a subsystem.
2. Problem formulation
Considering the system model of the form
q̇q(t) = A q (t) + B u (t)
(1)
y (t) = C q (t) + D u (t)
(2)
reordered in such way that
[
]
[
]
[
]
A = A i,l , C = C i,l , B = diag B i , D = 0
(3)
where i, l = 1, 2, . . . , p, q (t) ∈ ℜ n stands up for the system state, u (t) ∈ ℜ r denotes the
control input, y (t) ∈ ℜ m is the reference output, and the matrices A ∈ ℜ n×n , B ∈ ℜ n×r ,
C ∈ ℜ m×n , and D ∈ ℜ m×r are real finite valued.
Thus, respecting the above give matrix structures it yields
p
Ahl q l (t) + B h u h (t))
q̇qh (t) = A hh q h (t) + ∑ (A
(4)
l=1
l̸=h
p
y h (t) = C hh q h (t) + ∑ C hl q l (t)
(5)
l=1
l̸=h
where q h (t) ∈ ℜ nh , u h (t) ∈ ℜ rh , y h (t) ∈ ℜ mh , A hl ∈ ℜ nh ×nl , B h ∈ ℜ rh ×nh , and C hl ∈
p
p
p
ℜ mh ×nh , respectively, and n = ∑l=1
nl , r = ∑l=1
rl , m = ∑l=1
ml .
Problem of the interest is to design a closed-loop structure based on the pairwise
control in such way that the large-scale system be pairwise asymptotically stable, having
properties of an equivalent state feedback control of the form
K q (t)
u (t) = −K
(6)
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PAIRWISE CONTROL PRINCIPLE IN LARGE-SCALE SYSTEMS
where

K 11 K 12 . . . K 1p

 K 21 K 22 . . . K 2p
K =
..

.

K p1 K p2 . . . K pp


p

 , K hh = ∑ K l
h

l=1

(7)
l̸=h
A matrix on the main diagonal of the block matrix K is considered as a summation
of p − 1 matrices, where p is the number of all subsystems. Structurally, K lh represents
a partial gain matrix through the h-th subsystem input is affected by the h-th subsystem
state if the h-th subsystem is paired with the l-th subsystem.
3.
Preliminaries
Proposition 1 (Quadratic performance) Given a stable system (1), (2), then there exists
such γ > 0, γ ∈ ℜ that
∫t
(yyT (r)yy(r) − γuuT (r)uu(r))dr > 0
(8)
0
Proof. Let e
y (s), e
u (s) stand for Laplace transform of m dimensional output vector y (t)
and r dimensional input vector u (t), respectively, which are identically zero for t < 0.
Then
e
y (s) = G (s)e
u (s)
(9)
is the operator relation, where G (s) is noted generally as the transfer function matrix
(short referred as the transfer matrix). Its elements are fraction of scalar polynomials of
s, and its dimension is m × r, and subsequently (9) implies
G(s)∥∥e
∥e
y (s)∥ ¬ ∥G
u (s)∥
(10)
where ∥ · ∥ represents the Euclidean norm for vectors and the spectral norm for matrices.
Since the infinity norm property implies
√
1
G(s)∥∞ ¬ ∥G
G(s)∥ ¬ r∥G
G(s)∥∞
√ ∥G
m
with the notation
G(s)∥∞ =
∥G
(11)
√
γ
(12)
√
where γ is the value of the infinity norm of the transfer matrix G (s), then (11) can be
rewritten as
√
∥e
y (s)∥
1
G(s)∥ ¬ r
¬ ∥G
(13)
0< √ ¬1< √
m
γ∥e
u (s)∥
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A. FILASOVÁ, D. KROKAVEC
Now (13) results in the inequality
√
∥e
y (s)∥ − γ∥e
u (s)∥ > 0
(14)
u(s)∥2 > 0
∥ey(s)∥2 − γ∥e
(15)
respectively, and using the Parceval’s theorem property then (15) gives
∫∞
(yyT (r)yy(r) − γuuT (r)uu(r))dr > 0
(16)
0
which implies that with such defined γ > 0 (16) is satisfied.
Thus, if the system is stable (16) implies (8). This concludes the proof.
Proposition 2 (Bounded real lemma) System (1), (2) is stable with quadratic perfor√
C (sII −A
A)−1 B +D
D∥∞ ¬ γ if there exist a symmetric positive definite matrix
mance ∥C
P ∈ ℜ n×n and a positive scalar γ ∈ ℜ such that
P = P T > 0,



γ>0
(17)

A TP + P A P B
CT

∗
−γII r D T  < 0
∗
∗
−II m
(18)
where I r ∈ ℜ r×r , I m ∈ ℜ m×m are identity matrices, respectively,
Hereafter, ∗ denotes the symmetric item in a symmetric matrix.
Proof. In the sense of the quadratic performance there exists an enough large γ > 0 such
that Lyapunov function of the form
Pq (t) +
v(qq(t)) = q T(t)P
∫t (
)
y T (r)yy(r) − γuuT (r)uu(r) dr > 0
(19)
0
be positive definite, and the time derivative of (19) be negative definite. Thus, evaluating
derivative of v(qq(t)) with respect to t along a trajectory of the system (1), (2) it yields
Pq (t) + q T (t)P
Pq̇q(t) + y T (t)yy(t) − γuuT (t)uu(t) < 0
v̇(qq(t)) = q̇qT (t)P
Aq (t) + B u (t))T P q (t) + q T (t)P
P(A
Aq (t) + B u (t))+
v̇(qq(t)) = (A
C q (t) + D u (t))T (C
C q (t) + D u (t)) < 0
+(C
respectively. Introducing the next notation
[
]
q Tc (t) = q T (t) u T (t)
(20)
(21)
(22)
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PAIRWISE CONTROL PRINCIPLE IN LARGE-SCALE SYSTEMS
then v̇(qq(t)) can be written as follows
Pc q c (t) < 0
v̇(qq(t)) = q Tc(t)P
[
where
Pc =
Since
[
A TP + P A P B u
∗
−γII r
C TC C TD
∗
D TD
]
[
=
]
[
+
CT
DT
]
(23)
C TC C TD
∗
D TD
]
[
C D
]
<0
­0
applying the Schur complement formula to the inequality (25) it can be obtained


0 0 CT


 ∗ 0 DT  ­ 0
∗ ∗ −II m
(24)
(25)
(26)
and using (26) the LMI condition (24) can be written compactly as (18). This concludes
the proof.
Proposition 3 Autonomous system (1)-(3) is stable if there exists a set of symmetric
matrices, h = 1, 2 . . . , p − 1, k = h + 1, h + 2 . . . , p, h ̸= k
[
]
Pkh Phk
◦
P hk =
>0
(27)
Pkh Phk
such that

[
]
k

P
P

hk
h

qhk (t)+
 q̇qT (t)
p−1 p  hk
P kh P hk
[
]
∑ ∑
k
P
P
h=1 k=h+1 

hk
h
T

q̇qhk (t)

 +qqhk (t) P
P hk
kh
where
q Thk (t) =
[
q Th (t) q Tk (t)











<0
(28)
]
(29)
Proof. Defining Lyapunov function as follows
Pq (i) > 0
v(qq(i)) = q T(i)P
(30)
where P = P T > 0, P ∈ ℜn×n , then the time rate of change of v(qq(i)) along a solution of
the system (1)-(3) is
Pq (t) + q T(t)P
Pq̇q(t) < 0
v̇(qq(t)) = q̇qT(t)P
(31)
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A. FILASOVÁ, D. KROKAVEC
Considering the same form of P with respect to K , i.e.


P 11 P 12 . . . P 1p


p
 P 21 P 22 . . . P 2p 
l

P
P=
..

 , hh = ∑ Ph
l=1
.


(32)
l̸=h
Pp1 Pp2 . . . Ppp
then the next separation is possible
 2
P 1 P 12 0 . . . 0

 P 21 P 12 0 . . . 0

P=
..

.

0
0
0 ...

0


+···+ 

 0
0
Writing
q̇qhk (t) =
[
A hh A hk
A kh A kk






0

P 1p 0

 0 0
+···+ 


Pp1 0
... 0
..
.
0
0
... 0
p
Pp−1
Pp−1,p
p
q hk (t) + ∑
l=1
l̸=h,k
[
A hl
A kl
... 0




 +


Pp1
(33)





Ppp−1
. . . 0 Pp,p−1
]
. . . 0 P 1p
... 0 0
..
.
]
[
q l (t) +
Bh 0
0 Bk
][
u h (t)
u k (t)
]
(34)
then with (33), (34) the inequality (31) implies (28), owing to that for unforced system
u l (t) = 0 , l = 1, . . . p. This concludes the proof.
4.
Pairwise decentralized control
The subsystem interaction implies that the control law u h (t) generally takes form
p
K hh q h (t) − ∑ K hl q l (t)
u h (t) = −K
(35)
l=1
l̸=h
where K hl , h, l = 1, 2, . . . , p are non-zero gain matrices. Considering the structure of K
as is defined in (7) then it yields
[
]
] q (t)
p [
h
u h (t) = − ∑ K hl K hl
=
l=1
q l (t)
[
] l̸=h
[
]
(36)
[
] q (t)
] q (t)
p [
p
h
h
k
l
k
l
= − K h K hk
− ∑ K h K hl
= u h (t) + ∑ u h (t)
l=1
l=1
q k (t)
q l (t)
l̸=h,k
l̸=h,k
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PAIRWISE CONTROL PRINCIPLE IN LARGE-SCALE SYSTEMS
where
u lh (t)
=−
[
]
K hl
[
K hl
q h (t)
q l (t)
]
(37)
i = 1, 2, . . . , p, i ̸= h. Defining
][
]
[
]
[
] [
q h (t)
q
(t)
u kh (t)
K hk K hk
h
◦
= K hk
−
=
u hk (t)
K kh K kh
q k (t)
q k (t)
[
◦
K hk
=
(38)
]
K hk K hk
K kh K kh
(39)
h = 1, 2 . . . , p − 1, k = h + 1, h + 2 . . . , p, and combining (36) for h and k it is obtained

[
] 
] q (t)
p [
h

[
]
[
][
]  ∑ K hl K hl
 l=1

q l (t)
u h (t)
K hk K kh
q h (t)
 l̸=h,k
[
] 
=−
−
 (40)

] q (t)
 p [ l

u k (t)
K hk K kh
q k (t)
k

 ∑ K K kl
k
l=1
ql (t)
l̸=h,k
[
uh (t)
uk (t)
]
[
=
]
ukh (t)
uhk (t)
[
p
+
∑
l=1
l̸=h,k
ulh (t)
ulk (t)
]
(41)
respectively. Then substituting (41) in (34)
[[
=
] [
A hh A hk
Bh 0
−
A kh A kk
0 Bk
Using the next notations
[
Ahh
◦
A hkc =
Akh
p
ω ◦hk (t) = ∑
l=1
l̸=h,k
[
Ahk
Akk
][
]
q̇qhk (t) =
]]
[
]
p
Ahl q l (t)
B h u lh (t)+A
(42)
K hk K hk
q hk (t)+ ∑
h
l=1
K kh K k
Akl q l (t)
B k u lk (t)+A
l̸=h,k
[
−
B h u lh (t) + A hl q l (t)
B k u lk (t) + A kl q l (t)
=
][
Bh 0
0 Bk
]
(
p
= ∑
l=1
l̸=h,k
◦
B hk
ω hk (t) +
K hk K hk
K kh K kh
p
∑
l=1
l̸=h,k
◦
B hk
[
u lh (t)
u lk (t)
]
◦
◦
= A ◦hk − B hk
K hk
] [
+
A hl
A kl
]
(43)
)
q l (t) =
(44)
A l◦
hk q l (t)
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A. FILASOVÁ, D. KROKAVEC
[
where
ωhk (t) =
p
∑
l=1
l̸=h,k
]
u lh (t)
u lk (t)
[
,
A◦l
hk
=
A hl
]
(45)
A kl
(42) can be written as
q̇qhk (t) = A◦C
hk q hk (t) +
p
∑ Al◦hk ql (t) + Bhk◦ ωhk (t)
(46)
l=1
l̸=h,k
On the other hand, if
[
p
C hh = ∑ C hl ,
◦
C hk
=
l=1
l̸=h
then
p−1
y(t) =
p
∑ ∑
h=1 k=h+1
C hk C hk
C kh C kh
]
[
,
l◦
C hk
=
C hl
C kl
]
p
)
(
◦
C hk
qhk (t) + ∑ C hl ql (t)
(47)
(48)
l=1
l̸=h
p
◦
y hk (t) = C hk
q hk (t) + ∑ C hl q l (t) + 0 ω hk (t)
(49)
l=1
l̸=h
Now, taking (46), (49) considered pair of controlled subsystems is fully described.
5. Controller parameter design
Theorem 4 Subsystem pair (34) in system (1), (3), controlled by control law
C ◦hk (sII − A ◦hkc )−1 B ◦hk ∥2∞ ¬ γhk ,
(41) is stable with the quadratic performances ∥C
l◦
◦ −1 l◦ 2
C hk (sII − A hkc ) B hk ∥∞ ¬ εhkl , h = 1, 2 . . . , p − 1, k = h + 1, h + 2 . . . , p,
and ∥C
l = 1, 2 . . . , p, l ̸= h, k, if there exist a symmetric positive definite matrix Y ◦hk ∈
ℜ(nh +nk )×(nh +nk ) , a matrix Z ◦hk ∈ ℜ(rh +rk )×(nh +nk ) , and positive scalars γhk , εhkl ∈ ℜ such
that
Y ◦hk = Y ◦T
εhkl > 0, γhk > 0, h, l = 1, . . . , p, l ̸= h, k, h < k ¬ p
hk > 0,


p◦
◦
◦ ◦T
Φ ◦hk
A 1◦
···
A hk
B hk
Y hk
C hk
hk


1◦T
 ∗ −εhk1 I n · · ·

0
0
C hk
1



 .
..
..
..
..
 ..

.
.
.
.

<0


p◦T

 ∗
∗
· · · −εhkp I n p
0
C hk




∗
···
∗
−γhk I(rh+rk )
0

 ∗
∗
∗
···
∗
∗
−II(mh+mk )
(50)
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PAIRWISE CONTROL PRINCIPLE IN LARGE-SCALE SYSTEMS
(51)
◦
l◦
◦
where A ◦hk , B hk
, A l◦
hk , C hk , C hk are defined in (43), (45), (47), respectively,
◦ ◦T
◦
◦ ◦
◦T ◦T
Φ ◦hk = Y hk
A hk + A ◦hkY hk
− B hk
Z hk − Z hk
B hk
(52)
k◦
k◦T
h◦
and where A h◦T
hk , A hk , as well as C hk , C hk are not included into the structure of (51).
◦
Then K hk is given as
◦
◦ ◦−1
K hk
= Z hk
Y hk
(53)
Proof. Considering ω ◦hk (t) given in (44) as an generalized input into the subsystem pair
(46), (49) then using (18), (23) it can be written
[
]
p−1 p [
]
q
(t)
hk
•
<0
(54)
∑ ∑ q Thk (t) ω ◦T
hk (t) Phk
◦ (t)
ω
h=1 k=h+1
hk
where

Φ •hk

•
Phk
=
 ∗
∗
Ψ •hk
◦T
C hk
ϒ◦hk
−ϒ
l◦T
D hk
∗
−II mh +mk


<0

(55)
◦
◦ ◦
Φ •hk = A ◦T
hkc Phk + Phk A hkc
[ { }p
]
l◦
◦
•
◦
◦ l◦
B
A
Ψ hk = Phk
= Phk
B hk
hk
hk
ϒ ◦hk = diag
[
l=1, l̸=h,k
{εhkl I nl } pl=1,l̸=h,k γhk I (rh +rk )
[ { }
]
T p
T (t)
ω l◦T
=
q
ω
hk
l l=1, l̸=h,k
hk
[
]
l◦
D hk
= {C
C hk } pl=1, l̸=h,k 0
Defining the congruence transform matrix
[
]
◦−1
◦
T hk = diag Phk
I hk
(56)
(57)
]
(58)
(59)
(60)
(61)
◦
where I hk
is the identity matrix of appropriate dimension, and multiplying left-hand as
well as right-hand side of (55) by (61) results in


◦−1 ◦T
Φ◦hk Ψ◦hk Phk
C hk


◦
 ∗ −ϒ
<0
l◦T
(62)
ϒ
D


hk
hk
∗
∗
−II (mh +mk )
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A. FILASOVÁ, D. KROKAVEC
◦T
◦
◦−1
◦−1 ◦T ◦T
◦
◦ ◦−1
Φ ⋄hk = P ◦−1
hk A hk + A hk Phk − B hk K hk Phk − Phk K hk B hk
[ { }p
]
◦l
◦
◦
B hk
A hk
Ψ hk =
l=1, l̸=h,k
(63)
(64)
Thus, with the substitutions
◦−1
◦
Phk
,
= Y hk
◦
◦ ◦−1
◦ ◦
= K hk
Phk = K hk
Y hk
Z hk
(65)
(62) implies (51).
Note, this formulation includes in every inequality (51) all state variables of the
large-scale systems (reordering in a prescribed form) and solving these in a symmetrical
structure conditioned by LMIs structure definitions.
Theorem 5 (Control expansion) If a system comprising p−1 interconnected pairwise
controlled linear subsystems is stable then adding the pairwise control of the p-th subsystem the resulting structure be stable.
Proof. Providing the base of mathematical induction principle the number of subsystem
is chosen as j = 2. Thus,
[
]
P 12 P 12
◦
◦T
P12 = P12 =
>0
(66)
P 21 P 21
and this condition is satisfied solving it in the sense of Proposition 3. Moreover, Schur
complement property implies
P 21 > 0,
P21 )−1 P 12 > 0
P 12 − P 21 (P
(67)
Since the statement holds true for at least one value, it is assumed that it will hold true
for an arbitrary fixed value j−1, i.e. P j−1 = P Tj−1 > 0 be a positive definite Lyapunov
weighting matrix of a stable system comprising j−1 interconnected pairwise controlled
linear subsystems.
To prove that the induction hypothesis holds trues for all values of p let the j-th
subsystem is connected into control in such way that
[
]
Phj Ph j
◦
Ph j =
>0
(68)
P jh P jh
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PAIRWISE CONTROL PRINCIPLE IN LARGE-SCALE SYSTEMS
for h = 1, 2, . . . , j − 1. Considering (33) a matrix P j = P Tj associated with the system
comprising j interconnected pairwise controlled linear subsystems takes the form


 
P1 j


 

 P2 j  
[
]
⋄


 
P j−1 + P j−1
..


 

.

 
Pj = 
(69)



P j−1, j




 [

]
j
Pj
P j1 P j2 · · · P j, j−1
where
⋄
P j−1
= diag
[
j
j
j
P 1 P 2 . . . P j−1
]
>0
(70)
j
j
P j = ∑ P jl > 0
(71)
l=1
l̸= j
⋄
, P j are positive definite, with respect to Schur complement property then P j
Since P j−1
be positive definite if


P 1p


[
]
 P 2p 
j
⋄
−1 

P j−1 +P
P j−1 ) 
P j − P j1 P j2 · · · P j, j−1 (P
(72)
..

.


j
P j−1, j
be positive definite.
Using Shermann–Morrison–Woodbury equality it yields
⋄
⋄
⋄
⋄
⋄
P j−1 + P j−1
P j−1
P j−1
P j−1 )−1 + (P
P j−1
P j−1
(P
)−1 = (P
)−1 − (P
)−1 ((P
)−1 )−1 (P
)−1
(73)
and (72) can be rewritten as

P jj −
[
P j1 P j2 · · · P j, j−1
]


⋄
−1 
P
(P j−1 ) 

P 1p
P 2p
..
.



 + P#


(74)
P j−1, j
Since
•
⋄
P j−1
P j−1 )−1 > 0
P j−1
= (P
)−1 + (P
(75)
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238
A. FILASOVÁ, D. KROKAVEC
P # is positive definite, i.e.

P# =
[
P j1 P j2 · · · P j, j−1
]


⋄
•
⋄
−1
P
P
P
(P j−1 j−1 j−1 ) 


P 1p
P 2p
..
.



>0


(76)
P j−1, j
It is evident that (74) can now be written as follows
j−1 (
∑
)
Phj )−1 Ph j + P #
P jh − P jh (P
(77)
h=1
Since Schur complement of (68) implies, h = 1, 2, . . . , j − 1
Phj )−1 Ph j > 0,
P jh − P jh (P
j
Ph > 0
(78)
such positive definiteness of (77) implies positive definiteness of P j .
Remark 1 Inequality (69) implies that if faulty sensors occur in a subsystem sensor
structure for q h , the rest pairwise autonomous systems stay stable if loops with q h are
blocked, and reconfiguration is necessary only with respect to the h-th subsystem sensors.
6.
Illustrative example
To demonstrate properties of the proposed approach a simple system with four-inputs
and four-outputs is used in the example. The parameters of this system are




3
1
2 −1
3
1 2 1




 −1
 0
2
0
1 
6 1 0 




A=
 , C =  2 −1 3 0 
1
−1
1
3




1 −2 −2
2
0
0 1 3
[
]
[
]
1 0
B = diag 1 1 1 1 , B hk =
, h = 1, 2, 3, k = 2, 3, 4, h < k
0 1
[
]
[ ]
[
]
[
]
[ ]
[ ]
3
1
2
−1
1
1
2
1
C 3◦
C 4◦
C ◦12 =
A ◦12 =
, A 3◦
, A 4◦
,C
,C
,C
12 =
12 =
12 =
12 =
−1 2
0
1
0 2
1
0
[
]
[
]
[
]
[
]
[
]
[ ]
3 2
1
−1
1 2
1
1
A◦13 =
C ◦13 =
C 2◦
C 4◦
, A2◦
, A4◦
,C
,C
,C
13 =
13 =
13 =
13 =
1 1
−1
3
2 1
−1
0
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PAIRWISE CONTROL PRINCIPLE IN LARGE-SCALE SYSTEMS
[
A ◦14 =
[
A ◦23 =
[
A ◦24 =
[
A ◦34 =
3 −1
1
2
2 0
−1 1
2 1
−2 2
1 3
−2 2
]
[
, A 2◦
14 =
]
[
, A 1◦
23 =
]
[
, A 1◦
24 =
]
[
, A 1◦
34 =
]
[
]
[ ]
[
2
1 1
1
2
◦
2◦
3◦
C 14 =
C 14 =
C 14 =
,C
,C
,C
−2
0 1
0
1
]
[ ]
[
]
[ ]
[
−1
1
2 1
0
0
4◦
◦
1◦
4◦
C 23 =
C 23 =
C 23 =
, A 23 =
,C
,C
,C
1
3
−1 1
2
0
]
[
]
[
]
[ ]
[
−1
0
2 0
0
1
3◦
◦
1◦
3◦
C 24 =
C 24 =
C 24 =
, A 24 =
,C
,C
,C
1
−2
0 1
0
1
]
[
]
[
]
[ ]
[
1
−1
1 0
2
−1
2◦
◦
1◦
2◦
C 34 =
C 34 =
C 34 =
, A 34 =
,C
,C
,C
1
−2
1 1
0
0
1
−2
]
[
Considering e.g. h = 2, k = 3 then (51) implies

Φ ◦23

 ∗


 ∗


 ∗
∗
A 1◦
23
A 4◦
23
◦
B 23
−ε231
0
0
∗
−ε232
0
∗
∗
∗
∗
−γ23 I 2
∗
◦ ◦T
Y 23
C 23


◦1T 
C 23

◦4T 
<0
C 23



0
−II 2
◦
◦ ◦T
◦ ◦
◦T ◦T
Y 23
B23
Y 23
Φ ◦23 = A ◦23Y 23
+Y
A 23 −B
Z 23 −Y
B 23
and solving this inequality with respect to the LMI matrix variables Y ◦23 , Z ◦23 , ε231 , ε232 ,
and γ23 the problem was feasible giving the next solutions
[
]
[
]
0.6109 0.0196
5.4325 −0.1854
◦
◦
Y 23 =
, Z 23 =
0.0196 0.8891
−0.1896
5.7929
ε231 = 8.9610,
ε234 = 6.1122,
[
which results in
◦
K 23
=
γ23 = 5.6881
8.9063 −0.4047
−0.5197
6.5267
]
By the same way the gain matrix set was computed as follows
[
]
[
]
7.2425
2.5305
8.1138
4.3414
◦
◦
K 12
=
, K 13
=
2.3707 10.5833
4.1602 9.0512
[
◦
K 14
=
]
, A 3◦
14 =
8.1794 2.0529
1.1264 5.3293
]
[
◦
, K 23
=
8.9063 −0.4047
−0.5197
6.5267
]
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]
]
]
240
A. FILASOVÁ, D. KROKAVEC
[
◦
K 24
=
7.3799 −0.5361
−0.7325
4.3683
]
[
,
◦
K 34
=
5.7969 2.3816
3.7088 6.0249
]
Note, the control laws are realized in the partly-autonomous structure (38), (39), where
every subsystem pair is stable, and the large-scale system be stable, too.
To compare the results, an equivalent gain matrix (7) to centralized control can be
constructed


23.5358
2.5305
4.3414
2.0529


 2.3707 26.8694 −0.4047 −0.5361 


K =

4.1602
−0.5197
21.3748
2.3816


1.1264 −0.7325
3.7088 15.7225
where the closed-loop eigenvalue spectrum is
{
}
A −B
BK ) = −22.7536 −26.2085 −15.2702 ± 2.5280 i
ρ(A
Matrix K structure implies evidently that the control is diagonally dominant.
It is possible to verify - routinely - that using (18) directly to compute K gives


8.7328 −0.6035 5.3992 1.5464


 −0.4476 12.6053 0.7168 0.1421 


K ce = 

4.1167
0.6944
6.3277
0.8524


1.2344
0.2404 1.5967 7.8333
and the resulting closed-loop eigenvalue spectrum be
{
}
A −B
BK ce ) = −8.8985 −10.2202 −4.1902 ± 1.9046 i
ρ(A
However, the purpose of this example is only to illustrate this method and does not
address issues of numerical stability.
7. Concluding remarks
The pairwise autonomous partially decentralized approach to the large-scale linear
systems control design is considered in this paper, where the design conditions are derived as the set of inequalities in the bounded real lemma form. Including subsystem
interconnections the obtained inequalities are separable. Afterwards the resulting controller structures are pairwise autonomous (implying parallel processing) and partially
decentralized (implying interconnection utilization).
Reproving the global system stability condition more informal approach to system
design is outlined. If the interconnected subsystem outputs are assumed to be completely
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PAIRWISE CONTROL PRINCIPLE IN LARGE-SCALE SYSTEMS
241
measurable but through certain corrupted sensors in one subsystem only, the presented
principle seems to be suitable for control reconfiguration in the frame of the partly decentralized fault tolerant control structure.
References
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242
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[12] A.P. L EROS: LSS linear regulator problem. A partially decentralized approach.
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