Decentralized Stabilization of Large-Scale Systems
with Prescribed Degree of Exponential Convergence
F.Elmarjany* and N.ELaLami *
*Department Electrical of Engineering ;
*Mohammadia School of Engineering
BP 765 Agdal Rabat
Morocco
Abstract: - In this paper we study the decentralized stabilization problem of linear time invariant, largescale interconnected systems without any assumption of interconnections structures among subsystems .
The overall system can always be stabilized by local state feedback which is optimal for a quadratic
performance index. The decentralized control scheme is based on a stability result that employs the notion
of block diagonal dominance in matrices and considerably improves upon the exiting results for this
problem. Our main result here is the sufficient condition for the decentralized stabilization via eigenvalues
assignment such that by assigning the closed –loop eigenvalues of the isolated subsystem, appropriately
the eigenvalues of the overall closed-loop system are assigned in the desirable range. So, the overall system
can be exponentially stabilized with a prescribed convergence rate. One application of the procedures to the
interconnected system is also presented.
Key-Words: Decentralized control, large scale system, Decentralized stabilization, Diagonal dominance,
eigenvalue bound, continuous Lyapunov equation.
1 Introduction
The decentralized control has been an active field
of research for large-scale systems. Since 1960s,
many authors have considered this problem . A
popular procedure in the design of the
decentralized stabilizing controllers is to begin
with appropriately constructed lyapunov function
for the isolated subsystems such that the overall
system can be proved stable using a lyapunov
function which is a sum of the subsystems
lyapunov functions.
More recently, improvements on the stability
conditions obtained by this approach have been
suggested by many authors, some notable one,
being those of Solheim [9] who proposed a
graphical method based on the Gershgoring
theorem, and of Geromel and Bernusso who gave
a parametric optimization scheme for the choice
of subsystem in Lyapunov function. There is a
number of results who identify classes of
interconnected systems which can always be
stabilized by decentralized feedback [1][2][4][5].
The most popular conditions for decentralized
stabilisability [10] are the matching conditions.
The conditions imply that the interconnections
may be made to enter the subsystem through the
input matrices of the isolated subsystems
introduce an approach for design a decentralized
control by considering the effect of the
interconnections between the subsystem as
uncertainty. Sundareshan in [1][2] design a
constructive procedure based on a stability result
that employs the notion of block diagonal
dominance in matrices. But the obtained design
procedures are generally of a trial and error
nature and if the interconnections do not satisfy
the required conditions, one is obliged to repeat
the process. In addition, the controllers gains
obtained are considerably bigger. In this paper,
we shall present a constructive procedure for
design a decentralized stabilization by utilizing
the interrelations between the block diagonal
dominance matrix and the stability properties of
interconnected system such in
[1] [2]. A
principal contribution of this work is the
algorithm improves upon the existing results in
either enlarging the class of interconnections for
which the design of stabilization scheme can be
yielded lower values of a controller gain for a
given interconnection pattern. We study also the
decentralized stabilization with a prescribed
degree of convergence .
2 Preliminary results
Consider a large-scale system composed of N
linear time invariant subsystems, with the
following state-space equation:
N

x i ( t )  Ai x i ( t )  Bi u i ( t )  Aij x j ( t )

j1

j i

 yi ( t )  Ci x i ( t )
i  1....N
Definition 2.1
(1)
Where xi Rni, ui Rmi, AiRni*ni , BiRmi*ni, Ci
 Rpi*ni
It assumed that all (Ai , Bi) are controllable and
(Ai, Ci) are observable
Where AiiRni*ni, and AijRni*nj, i,j=1,2..N .
If
Aii
are
nonsingular
and

the form:
N
The
 A ij x j
term
is
j1
j i
due
to
the
interconnections of the other subsystems.
The objective in this paper is to design a local
dynamical controller
ui(t) = -Ki xi(t)
(2)
Stabilize the large-scale interconnected system
(1). From (1) and (2) we have
x i ( t )  Fi x i ( t ) 
N

Aij x i ( t )
i  1..N
j1
j i
(3)
where Fi  Ai  Bi  Ki
Let m (.) and M(.) respectively denote the
minimum and the maximum eigenvalues of the
argument matrix A, for any MRn*m let 2norm (ie., the Euclidean norm) is denoted by
M   M (M T M) . Let (.) the matrix measure
induced by some vector or matrix norm and
defined
by
the
formula
( A ) 
I  A  1
lim

0 
Before we derive a sufficient condition of
Decentralized stabilization we need the
following lemmas and definition
Lemma2.1. Let F (.) denote the matrix
measure induced by the vector norm
x  x Fx
T
, where
have
Re ( (A))  max (R ( (A))  inf ( (A) .
1 i  n
i
 A11 A12
 A A
22
A

A
 N1 A N 2
A 1
ii
F N
F
Moreover the matrix measure F(.) is given by
1
1
1
 F (A)   M (FAF 1  A T )   2 (F 2 AF 2 ) . where
2
the Euclidean norm-induced measure is given
by 2(A) = M(A+AT)
1
.... A1N 
.... A 2 N 
....

.... A NN 
(4)
N
  A ,  i  1,., N (5)
ij
j 1
ji
Then A is said to be block diagonal dominant
relative to the partitioning in (4), if strict
inequality holds in (5) then A is strictly block
diagonal dominant.
Lemma 2.2. Let the matrix A Rn*n partitioned
as in (4) satisfy the conditions:
(i)
A = AT, (ii)Aii, i =1, 2,.., N, are positive
definite . (iii)A is strictly block
diagonal dominant .
Then, all eigenvalues of A are release and
positives.
Lemma 2.3
Let consider the state-space
equation (3), let spec (Fi) LHP  i=1, 2 ..., N
and
let Pi Rni*ni, Pi=PiT > 0 be the solution of the
lyapunov equation
FiTPi + PiFi+Qi = 0 (6)
For an arbitrary selected Qi Rni*ni, where Qi is a
positive definite matrix. Then (3) is
asymptotically stable if
 m (Q i )   M ( Pi )   ij 
j1
j i
N
  M (Pj ) A
j1
i j
ji
i  1,..N
F is a positive
definite matrix. For any matrix A,
We
M
Let ARn*n be partitioned in
(7 )
Let v (z) = zT(.) Pz(.) where z (.) =
and
P = diag (P1, P2… PN) as a Lyapunov function
and evaluating its derivative
Along the trajectories of (3), one obtains
Proof.
[z1T(.),z2T(.)……..ZNT(.)] T
v (z)  z T (.)( FT P  PF  H T P  PH)z(.)
 z T (.)( Q  H T P  PH)z(.)
  z T (.) ( W ) z(.)
where Q = diag (Q1, Q2, ...., QN), F = diag (Q1,
Q2, ..., QN) and H = [Aij]i,j=1..N,
From Lemma 2.1, W is positive definite if is
strictly block-diagonal dominant, i.e.
Qi
1 1
  Wij , i  1,2..N
(8)
j1
j i
i  1,....N
and hence
(7) implies (8) 
However the implementation of the decentralized
control by using the Lemma 2.2 is very
complicated because the resolution of (7)
imposes constraining conditions on the
interconnections matrices and led to restricted
classes of the interconnected systems. Also the
obtained gains are very higher. Our contribution
consists in working out a decentralized control
based on the theorem Gershgorin and calling
upon the suitables upper bounds of the trace and
eigenvalues of the Lyapunov equation. As our
result, we will have need for the Lemma and
Theorem following:
Lemma 2.4
Let spec(Fi) LHP and let Pi.
Rni*ni, Pi =PiT > 0 be the solution of the
lyapunov equation (6). For an arbitrary selected
Qi Rni*ni, where Qi is a positive definite matrix.
Let Mi is a positive definite matrix satisfies
Mi(Ai)<0
Then (3) is asymptotically stable if
 M (F) M (F1Q)
 M ( P) 
 2 F (A)
i  1, 2.. N
in
particular
if
 M (Q i )
 M (Pi ) 
 2 2 (A i )
(9)
2(Ai)<0
j 1
j i
j 1
i j
(10)
i 
 M ( M i ) M ( M i Q i )
 2 M i (Fi )
i  1,..N
Proof Let Pi the solution of the Lyapunov
However, it is simple to observe that for i  j,
Wij   M (Pi ) G ij   M (Pj ) Gji
N
1
where
N
N
 m (Qi )   i  ij    j A ji
then
 i  1, .., N
3 An Algorithm for decentralized
stabilization.
In this section we present our main result for
decentralized stabilization
Theorem 3.1 Let spec (Fi)  LHP  i =1,
2…, N and let Mi is a positive definite matrix
satisfies Mi(Ai) < 0. For an arbitrary selected Qi
Rni*ni, where Qi is a positive definite matrix.
Then the system (3) is asymptotically stable if
equation (6), from Lemma 2.2, the system (3) is
stable if the condition (8) is satisfied. From the
theorem 2.1, M(Pi)  i=1, 2…, N has an upper
bound (9). Then the second term of the
inequality (8) satisfies also
N
 M (Pi ) ij    M (Pjj ) A ji 
j 1
i j
N
N
j 1
j i
j 1
i j
 i  ij    j A ji
i  1,2..N
(11)
The systems (3) is asymptotically stable if the
condition
(Q)        A is satisfied 
N
N
i j 1
ji
ij
j 1
i j
j
ji
Remark 3.1 This theorem gives the sufficient
condition of existence of a set of the state
feedback gains stabilizing the whole system by
satisfying the condition (10) which is most robust
than the condition (8) .
To develop the decentralized control with a
prescribed degree of convergence, we need the
following lemmas.
Corollary 3.1 We consider the decoupled
subsystem (3) described by
x i ( t )  Fi x i ( t )
where Fi  Ai  Bi K i i  1...N
Let

spec(Ai )  i1 , i2 ,......in

spec(Fi )   ,  ,......
i
1
i
2
i
n

,
(12)
and
where  Ai  Bi  Ki , K i  BiT Pi
The control law ui(t) = -BiTP xi(t)i ,
where Pi Rni*ni; Pi=PiT >0 is the solution of the
matrix Riccati equation:
Pi Ai + AiT Pi – PiBiBiTPi = 0
(13)
The subsystem (12) is asymptotically stable if
Re(i) < 0 ; i2 = i2 ;
Lemma 3.1 since (Ai, Bi) is a controllable pair,
then for a prescribed number i and Ri
is a definite positive, the matrix Ricatti equation
Pi (Ai +i Ii)+ (AiT +i Ii)Pi – PiBi Ri-1BiTPi = 0
(14)
has a solution Pi > 0 ; Pi = PiT such that
-i < Re(i ) < i , (i +i)2 = ( i+i)2 and
Ki = Ri-1BiTPi . The control law ui(t) = -Ki*xi(t)
minimizes the criterion of the form

J i   x iT Q i x i  u iT R i u i dt
0
where Qi = 2 i Pi
Lemma3.5
For any matrix A, we have
M (eAA )  e( 2( A))
T
Theorm3.2 Let a prescribed number i such
that i+Re(i)>0
Then the Lyapunov equation :
Vi ( Ai + i.I) + ( Ai + i I)T Vi = Bi Ri-1BiT
 i =1..N ;
(15)
Where Ri is a symmetric definite positive
matrix, has a solution Vi >0; Vi =ViT such that :
Re(i ) = -(2i +Re(i)) ; Im(i ) = Im(i)
for i = 1, 2….N
and the control law ui(t) = - Ri-1BiTPi xi(t)
where Pi = Vi-1 minimize the criterion of the

form J i   x i Q i x i  u i R i u i dt
T
T
since
M(Qi)Ii  Qi  m(Qi)Ii
then M(Pi)  M(Qi)

 M (
0
T
eFio Qi eFio )dt
applying the lemma3.5 to the right side of the
above inequality, we have
 M (Pi )  
 M (Q i )
2(Fi 0 )
(17)
from the lemma 2.2 and et the upper bound of Pi
in the equation (17) we satisfy the condition
(16). 
Remark 3.2 - the inequality (16) provides a
decentralized stabilization condition for the
system(1) with
a prescribed
degree of
convergence which must satisfies the following
conditions : - (Fi + i Ii) <0 and
i + Re(i) >0.
4 Illustrative example
To illustrate the design procedure described
above, we consider the following linear
interconnected system which was treated by
Sundareshan and Elbanna [1]. The system is
composed by two subsystems
0
avec Qi = 2 i Vi-1
BiRi-1BiTPi
Proof : we have Fi = Ai - BiKi = Ai Let Pi = Vi-1, then Fi is described by :
Fi = iIi - Pi-1( AiT + iIi)Pi = -Pi-1( AiT + 2iIi)Pi
Then i = -(2i +i) ;
so Re(i ) = -(2i +Re(i)) ; Im(i ) = Im(i)
 i =1, 2…., N

Remark 1: this result makes it possible to move
all the poles of the system on the real axis with
the same quantity . So the resulting the closedloop dynamic is becoming faster in the optimal
way than the open-loop dynamic :  Re(i ) > 
Re(i)
Theorem 3.3 Let Fio=Fi + i Ii
if Spec(Fi0 )  LHP for i = 1, 2…. N.
Then for any arbitrary selected Q Rni*ni ; Qi =
QiT >0
the interconnected system (1) is
asymptotically stable if
 m (Q i ) 
 M (Q j )
N
  M (Q i ) N
 A ij  
A ji
j 12( F   I )
2(Fi  Ii ) jj1I
i
i i
j i
i  1,2,....N
(16)
Proof It is well known that the solution Pi f of
the following Lyapunov equation :
Fi0TPi + PiFi0 + Qi = 0

Satisfies
T
Pi   e Fio Qi e Fio dt
0
1
0
0
0
0
1
x 1 ( t )  
0
0
0
 2  3  4
0.005
 1
 2
1

 1.3 0.4
 0.6  1.5
0
x 2 ( t )   0
 4
1
  2
 4
0
0 

0 
0
 x1 ( t )    u1 ( t )
1 
0 
 6
1
0.003
0.001
 x (t)
0.01  2
3 
0
0 

0
1  x 2 ( t )  0 u 2 ( t )
1
 6  8
1
 0.001 0.0001
 0.5 0.002
0.001  x1 ( t )
2.5
0.25
0.003 
0.1
The required controller gains K1 and K2 to be
used in the decentralized stabilization scheme are
then : K1= [ 2.4407, 10.0168, 14.5852, 37.4113 ]
and K2=[ 6.3947 0.0932 840229 ]
For comparison, the state feedback gains
determined in [1] are K1= [94132 , 13447, 979,
36] and K2 = [646 234 9] which have
considerably larger values. All the states of the
subsystems are plotted in figures. From the
simulation results, it can be seen that each
subsystems are asymptotically stable.
states of subsystem1
3
2.5
2
1.5
1
0.5
0
-0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
states of subsystem2
6
5
4
3
2
1
0
-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
5 Conclusion
In this note, we have studied the decentralized
stabilization of large scale interconnected
systems via eigenvalues assignment such that by
assigning the closed loop eigenvalues of the
isolated subsystem, appropriately
the
eigenvalues of the overall closed-loop system are
assigned in the desirable range. So, the overall
system can be exponentially stabilized with a
prescribed convergence rate. This algorithm
offers a procedure to move about the poles of
the system on the real axis with the same
quantity , so the closed-loop dynamic is
becoming faster in the optimal way than the
open-loop dynamic. It is shown that the gains
obtained are considerably smaller values than
those found in [1].
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