Computation of Structural Decomposition for Linear Singular Systems
MINGHUA HE and BEN M. CHEN
Department of Electrical and Computer Engineering
The National University of Singapore
10 Kent Ridge Crescents, Singapore 117576
REPUBLIC OF SINGAPORE
Abstract: We present in this paper computation algorithms for the structural decomposition of general linear
multivariable singular systems. Such kind of decomposition has a distinct feature of capturing and displaying all
the structural properties, such as the finite and infinite zero structures, invertibility structures and redundant
dynamics of the given system. The computation will make it a powerful and convenient tool in solving control
problems for singular systems.
Keywords: Computation; MATLAB programs; singular systems; structure decomposition; structure properties.
1. Introduction
Consider the following linear singular system
Ex  Ax  Bu
: 
 y  Cx  Du
x  n , y   p , u  m ,
(1)
where rank ( E )  n . It is also alternatively called a
descriptor system, an implicit system or a
generalized linear system in literature. Linear
singular system has attracted researchers for more
than three decades since many real systems, such as
electrical power systems, can be naturally modeled
as singular systems. Among the issues discussed for
linear singular system [7], system structure and
equivalence is an essential topic. Campbell [1]
presented an effective structural decomposition
method and got the corresponding equivalent
system, while Verghese et al. [11] defined a strong
system equivalence using a trivial augmentation
and deflation technique. On the other hand,
structural invariants also received much attention in
literature. Further, Misra et al. [10] and Liu et al. [9]
have presented their algorithms to compute the
invariant structural indices of singular systems.
More recently, He and Chen [5] and He et al [6]
have developed a structural decomposition method
for single-input single-output and multivariable
linear singular systems respectively. Such a
structural decomposition can not only give the
invariant structural indices but also explicitly
display the structural properties, such as the finite
and infinite zero dynamics, invertibility structures
and redundant dynamics of the given systems. And
it is expected to be a powerful tool in solving
system and control problems as its counterpart in
nonsingular linear system [3].
This paper focuses on the computation algorithms
of the structural decomposition. First, to make this
paper more self-contained, we briefly describe the
structural decomposition theorem in Section 2. And
in Section 3, the main MATLAB computation
programs are presented while several numerical
examples will be included in Section 4. Finally, a
conclusion will be drawn in Section 5.
2. Structural Decomposition Theorem
and Its Properties
We first summarize the structural decomposition of
general multivariable singular systems in compact
matrix form. And its essential properties will also
be given in brief in this section.
Theorem 2.1 Consider the general multivariable
linear singular system  in (1). Then, there exist
nonsingular state and output transformations
s   nn and o   p p , and a nonsingular
transformation e   nn , as well as an mm input
transformation i (s) , whose inverse has all its
elements being some polynomials of s (i.e., its
inverse contains various differentiation operators),
which together give a structural decomposition of
 and display explicitly its structural properties.
This structural decomposition can be described in
the following equation form,
 xz 
 
 xe 
 y0 
 u0  (2)
x 
 
 
x  s ~
x  s  a  , y  o ~
y   yb  , u  i ( s )u~   uc  ,
 xb 
y 
u 
x 
 d
 d
c
 
x 
 d
and
C 0z 0 C 0a C 0b C 0c C 0d 
~
~ 
~
1
C  o Cs  C v  C k  C dz 0 0
0
0 Cd   Ck ,


0 
C bz 0 0 C b 0
 xd1 
 yd1 
 ud1 






 xd 2 
 yd 2 
 ud 2 
xd  
,
y

,
u

,
  d    d   






 x dm 
 ydm 
 udm 
 d
 d
 d
 I m0
~
~
1
D  o Di ( s)  D v  Dk ( s)   0

 0
xz  0,
xe  Be0u0  Becuc  Bedud ,
xa  Aaa xa  B0a y0  Lad yd  Lab yb ,
xb  Abb xb  B0b y0  Lbd yd , yb  Cb xb ,
where
(3)
B0  0 0 B0a
~
~
~
~
Ak ~
x  Bk ( s) u~  0, Ck ~
x  Dk ( s) u~  0.
and for each i  1, 2, , md ,
x di  Aqi x di  Li 0 y 0  Lid y d  Bqi [u di 
md
M ia xa  M ib x b  M ic x c   M ij xdj ],
y d  C d xd .
The structural decomposition can also be expressed
in the following compact form.
0
0
0
0
0
0
0
0
0 I na
0
0
0
0
I nb
0
0
0
0
I nc
0
0
0
0
0
0

0
,
0
0

I nd 
0
0
0
0
0
0
Aaa
0
Bc M ca
Bd M da
0
0
Property 2.1 The given system  in (1) is
stabilizable if and only if  Acon , Bcon  is
stabilizable, and it is detectable if and only if
 Aobs , Bobs 
is
detectable,
where
A
Acon :  aa
0
Lab Cb 
B
, Bcon :  0a

Abb 
 B0b
Lad 
,
Lbd 
(8)
and
 A
Aobs :  aa
 Bc M ca
~
~
A  e1 As  Av  B0 C 0  Ak
0
I ne
(7)
The equation form of this theorem and detail proof
can be found in He, Chen and Lin [6]. Here, we
briefly introduce the essential properties of this
structural decomposition.
(4)
j 1
 I nz
0

0

0
0

0
(6)
and
y0  C0a xa  C0b xb  C0c xc  C0d xd  u0
 J nz
E
 ez
 E az
~
E  e1 Es  E v  
 E bz
 E cz

 E dz

B0b B0c B0d  ,
C 0  0 0 C 0a C 0b C 0c C 0d  ,
xc  Acc xc  B0c y0  Lcd yd  Lcb yb  Bc M ca xa  Bcuc
y di  C qi xi ,
0 0
~
0 0  Dk ( s),

0 0
0 
C
, Bcon :  0a

Acc 
M da
C0c 
M dc 
.
(9)
0
0
Lab C b
0
Abb
0
Lcb C b
Acc
Bd M db Bd M dc
0 
0 

Lad C d 
~
  B0 C 0  Ak ,
Lbd C d 
Lcd C d 

Add 
0
0
 0
B
Bde Bce 
 0e

0 ~
 B0a 0
~
~
1
B  e Bi ( s )  B v  B k ( s )  
  B k ( s ),
0
 B0b 0
 B0c 0 Bc 


 B0d Bd 0 
Property 2.2 The invariant zeros of the given
system  are the eigenvalues of Aaa . The normal
rank of  is equal to m0  md . Here md is the
dimension of u d .
Property 2.3 The given system  has m0 infinite
zero of order 0. And its infinite zero structure (of
order greater than 0) is given by
(10)
S ()  q1 , q2 , , qmd ,


that is, for each i  1, 2, , md ,  has an infinite
zero of order qi , respectively.
(5)
Property 2.4 The given system  is right
invertible if and only if x b and hence y b are non-
existent, is left invertible if and only if xc and
hence uc are non-existent, and is invertible if and
only if both x b and xc are non-existent.
The properties show that our structural
decomposition can explicitly display the structure
properties of the given singular system, and hence
it is expected to be a powerful tool in solving
singular system and control problems.
3. MATLAB Computation Programs
for the Structural Decomposition
singular. The decomposition can be characterized
as the following transformations,
0
 A1 0 
I
PEQ   n1
, PAQ  
,

0
I
n
 0 N
2 

B 
PB   1  , CQ  C1 C2 ,
 B2 
where N is a nilpotent matrix.
(11)
ctr_cf.m
The function transform a matrix pair  A, B into its
control canonical form as follows,
As mentioned before, a detailed constructive
decomposition algorithm will not be given here due
to the limit pages. And in this section, we will give
brief descriptions of the main functions for the
computation.
The
computation
programs
introduced are all in MATLAB codes.
 A Acc 
B 
(12)
T 1 AT   c
, T 1 B   c  ,

0
 0 Ac 
where  Ac , Bc  is completely controllable while
 Ac , 0 is totally uncontrollable.
SD.m
bdc_cf.m
This is the main function, that is, structural
decomposition function for general linear singular
systems. The function transforms the given singular
system ( E , A, B, C , D ) into its structural
This function decomposes a complete controllable
pair  Ac , Bc  into a special block controllability
canonical form [2], in which every submatrix block
corresponds to a distinct input channel. The
decomposition process can be described as follows,
~ ~ ~ ~ ~
decomposition form ( E , A, B , C , D ) , which can
explicitly display all the structural properties, such
as the finite and infinite zero structures, invertibility
structures and even redundant dynamics of the
given system.
sys_hat.m
This function separates two kinds of redundant
states from the original system. One kind of
redundant states x z are static and identical zero all
the time, whereas the other redundant states xe are
linear combination of appropriate order of system
input's derivatives. Such states are associated with
the so called impulse modes, which are introduced
by the derivatives of the system input.
pre_decom.m
This one is to perform a fast-slow decomposition
(see e.g., [4] for details) for the given singular
system. With two constant transform matrices P
and Q , it transforms the given singular system into
two subsystems, one is nonsingular and the other is
J1
0
1
R Ac R  
0

0
 B1
0
R 1 Bc  
0

0
0  0
J2  0 
,
0  

0 0 Jk 
B12  B1k B1l 
B2  B2 k B2 l 

0  
 

0
0 Bk Bkl 
(13)
where J i , i  1, 2, , k are Jordan blocks with
zero eigenvalue and
 0
 0
Bi    ,

 
1
kronecker.m


Bij    .

 
 0
(14)
The function transform the given system's system
matrix P (s ) to its Kronecker Canonical Form
with two constant transform matrices M and N ,
P~ ( s )  M P ( s ) N
 A  sE B 
M
N
D 
 C
 diag sI  J f I  sJ 

L1
 L p
LT1
 LTq

(15)
Here every block of the diagonal entries in P~ ( s )
1
0

E  0

0
0
0
0
0
0
0
0
0
0
0
1
0
1

0, A 

0
0
0
1
1
0
1
1 
 0
 
B  1 , C  2
 
 0
0
0
1
0

0

0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0

0

0
1
 2 1  1, D  0,
(16)
is associated the distinct structure index.
SCB.m
This is the function of structural decomposition for
linear nonsingular system. The function was
developed by Lin and Chen [8], and it decomposes
a given linear system  A, B, C, D and explicitly
displays its structural properties. The function
SD.m is its natural extension to linear singular
systems.
r_jordan.m
This function transforms a real matrix H to its
Jordan canonical form.
The functions introduced here are only some main
procedures, and there are still many other functions
needed in our computation. But due to the limit of
page, we can not introduce every function here.
However, this omission will not affect our
illustration of computation process in the following
section.
4. Some Numerical Examples
To illustrate the computation of our structural
decomposition algorithm, we present in this section
two numerical examples, one is of single-input and
single-out linear singular system while the other is
of multi-input and multi-output case.
Let us first look at the following single-input and
single-output system,
the statement
Ge , Gs , Gi , Go , E v , Av , Bv , Cv , Dv , nz , ne , na , nb , nc , nd 
 SD E , A, B, C , D ,
returns
0
0

G e  0

1
0
0
0

Gs  0

1
0
Go  1 ,
1  0.3333  0.7071
0 0.6667
0
1
0
0
0
0
0
0  0.6667  0.7071
0  0.3333  0.7071
1
0
0
0  0.6667  0.7071
0
0
0
0 0.6667
0
Gi ( s ) 
0
1
0 ,

0
0
0
0
0

0
1
,
1 ,
s2
(17)
(18)
(19)
and
0 0
 0
 1
0 0

Ev    6
0 1

4
.
2426
0 0

 5
0 0
0
1 0
0 1
0

Av  0 0
1

0
0 0
0 0 0.6667
0 0
0 0
0 0 , Cv   1

1 0
0 1
0
0 
0
0 
0
 3  , Bv 

0 1.4142
0
2 
0 0 0 1,
(20)
0
 1
 
 0 ,
 
0
 1 
Dv  0, nz  1, ne  1, na  2 , nb  0 , nc  0 , nd  1.
And finally this decomposition result can be
verified by the following operation. The statement
Pv , M , N   Kronecker E, A, B, C, D  returns
 1  s 0
 0
s

 0
0
Pv  
0
 0
 0
0

0
 0
0
0
0
0
0
0
1 s 0
0 1 0
0
0
1
0
0
0
0
0 
0
.
0
 s

1 
(21)
From Pv , the Kronecker Canonical Form of the
given system, we can see clearly that the structure
indices are the same as our computation results.
Now we look at the following multi-input multioutput linear singular system,
1 0
0 0

0 0

E  0 0
0  1

0 0
0  1
1 1
 1 1

1 0

B0 0
 1 0

 1 2
  1 0
0 0
0  1

0 0

0 0  , A  I7
0 0 1 1  1

1 0 1 0  1
0 0 1 1  1
0
1

0
 0 1 0

,
1 , D  
0 0 0

1

1
1
1 0 0 0 0 0 1 
C
.
0  1 1 0 2 1  2 
0
1
0
0
0
0
0
0
0
1
0
1
(22)
Ge , Gs , Gi , Go , E v , Av , Bv , Cv , Dv , nz , ne , na , nb , nc , nd 
 SD E , A, B, C , D .
0
0

1

G e  0
0

1
1

1 1
0
1
1
0
0
0
0.7746
0
1  0.2582  0.3162
0
0
0
0  0.5164 0.3162
0 0.2582
0.3162
2  0.2582  0.3162
0 0.2582
0.3162
0
0
0
1
0
0
0
0
0.6
0

0.4 ,
0.4

0.6
0.4
0 0
1
0
0
0
1
0
0
0
1
0
0
0
0.7746
0
0
0
 0.2582  0.3162
 0.2582  0.3162
 0.5164 0.3162
0.2582
0.3162
 0.5164 0.3162
0
0
0
0
1
0
1
0
0 
0.6

0.6
0.4

0.4
0.4
(23)

s 1
 s2
 s  1


-1
Gi ( s )  
s2
1
1  ,
0.8944 s  0.4472 0.8944 1.3416


1 0
Go  
.
0 1
(24)
And
Then its structural decomposition form is in the
results of the following statement,
And the results is
0
0

1

Gs  0
0

1
1

0
0

0

E v  0
0

0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0

0

Av  0
0

0
0

0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0

0 ,
0

0
1
0
0
0
0 
0
0
0
0 
0
0
0
0 

1
0
0
1.4142  ,
1.5215
0.3333 1.8648 0.7172 

 0.7071
0
0
1 
0.7071  0.3873 0.8333  0.3333
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0 
0
 0  0.2 0.8944


 1
0
0 


Bv   0
0
0 ,
0
0
1.0541


0
0 
1
0
1
0 

0 0 0 
Dv  
,
0 0 0 
0 0 0 0 0 1 0 
Cv  
.
0 0 0 0 0 0 1 
(25)
And the corresponding structure indices are
nz  1, ne  2 , na  1, nb  0 , nc  1, nd  2 .
(26)
Again, this result can be verified by the following
computation,
Pv , M , N   Kronecker E, A, B, C, D  ,
and its computation result is:
1  s 0
 0
s

0
 0

0
0

Pv   0
0

0
 0
 0
0

0
 0
 0
0
0
1
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
1 s
0 1
0 0
0 0
0 0
0
0
0
0
0
0
1
0
0
1 1  1 0  2  1 2
0 2
0 0 1 1 1

1 1  2 1  4  1 3

0 0 0 1 0
0 1
M  0 2
0 0 0 1 1

0 0 0
0
0
0 0
0 0
0 0 1 0
1

1 0 1
0 1
0 0
0  1 0 0 1
1 1
0
0
0
1
0
0
0
0
0
0
0
1
1  2 2

1
0
0

0
 2 0
1 1 0
N 
1  3 2
1 1 0

 0 1 0
0
1
0

1
0

1

0 0
0 0
1 s
0 1
0 0
0 0
0 0
0 0
0 0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0

0

0
0 ,

0
0

0
1
1
0

1

0 (27)
0 ,

1
0

0
0
0
0
0 0 0 0
1  1 2 0 0 0

0
0
0 1 0 0

1
0
1 0 0 0
1
0
1 0 0 0
 .
1  1 3 1 0 0
1
0
1 1 0 0

0
0
0 0 1 0
0
0  1 0 0 1

 1 1  1 0 1 0
Thus, with these two numerical examples, we
illustrate the computation process of our structural
decomposition algorithm. It can be seen that the
MATLAB functions are effective in computing the
structural decomposition.
5. Conclusions
We have presented in this paper MATLAB
computation functions for the structural
decomposition technique for general linear singular
systems. The structural decomposition has a
distinct feature of explicitly capturing and
displaying the structure properties, which make it a
powerful tool in solving system and control
problems as its counterpart in nonsingular systems.
The numerical examples showed that our
computation programs are effective in giving a
singular system’s structural decomposition form.
They thus enhance the structural decomposition’s
role as a powerful tool in solving practical
problems.
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