Multivariable Control
Systems
Ali Karimpour
Assistant Professor
Ferdowsi University of Mashhad
Chapter 5
Chapter 5
Controllability, Observability and Realization
Topics to be covered include:
• Controllability of Linear Dynamical Equations
• Observability of Linear Dynamical Equations
• Canonical Decomposition of a Linear Time-invariant
Dynamical Equation
• Realization of Proper Rational Transfer Function Matrices
• Irreducible Realizations
Irreducible realization of proper rational transfer functions
Irreducible Realization of Proper Rational Transfer Function Vectors
Irreducible Realization of Proper Rational Matrices
2
Ali Karimpour Sep 2009
Chapter 5
Controllability and Observability of Linear Dynamical Equations
Definition 5-1
Definition 5-2
3
Ali Karimpour Sep 2009
Chapter 5
Controllability and Observability of Linear Dynamical Equations
Theorem 5-1
4
Ali Karimpour Sep 2009
Chapter 5
Controllability and Observability of Linear Dynamical Equations
Theorem 5-2
5
Ali Karimpour Sep 2009
Chapter 5
Controllability and Observability of Linear Dynamical Equations
Theorem 5-2(continue)
6
Ali Karimpour Sep 2009
Chapter 5
Canonical Decomposition of a Linear Time-invariant
Dynamical Equation
x  Ax  Bu
y  Cx  Eu
x  PAP 1 x  PBu  A x  B u
x  Px
y  CP 1 x  Eu  C x  E u
Theorem 5-3 The controllability and observability of a linear time-invariant
dynamical equation are invariant under any equivalence transformation.
Proof: Let we first consider controllability

S B
AB
 
 PB
A 2 B ..... A n1 B  PB PAP 1 PB PA2 P 1 PB ..... PAn1 P 1 PB


AB A 2 B ..... A n1 B  PS
Similarly we can consider observability
7
Ali Karimpour Sep 2009
Chapter 5
Canonical Decomposition of a Linear Time-invariant
Dynamical Equation
Theorem 5-4
x  Ax  Bu
Consider the n-dimensional linear time –invariant dynamical equation
y  Cx  Eu
If the controllability matrix of the dynamical equation has rank n1 (where n1<n ), then
there exists an equivalence transformation
x  Px
which transform the dynamical equation to
 xc   Ac
   
 xc   0
A12   xc   Bc 
     u
Ac   xc   0 

y  Cc
 xc 
Cc    Eu
 xc 

and the n1-dimensional sub-equation
x c  Ac xc  Bc u
y  Cc xc  Eu
8
is controllable and has the same transfer function matrix as the first system.
Ali Karimpour Sep 2009
Chapter 5
Canonical Decomposition of a Linear Time-invariant
Dynamical Equation
Theorem 5-4 (Continue)
Furthermore P=[q1 q2 … qn1 … qn]-1 where q1, q2, …, qn1 be any n1 linearly
independent column of S (controllability matrix) and the last n-n1 column of P
are entirely arbitrary so long as the matrix [q1 q2 … qn1 … qn] is nonsingular.
Proof: See “Linear system theory and design” Chi-Tsong Chen
x  Ax  Bu
x  Px
xc  Ac xc  Bcu
y  Cx  Eu
y  Cc xc  Eu
n  dimensiona l
n1  n  dimensiona l
G(s)
Hence, we derive the reduced order controllable equation.
9
Ali Karimpour Sep 2009
Chapter 5
Canonical Decomposition of a Linear Time-invariant
Dynamical Equation
Theorem 5-5
x  Ax  Bu
Consider the n-dimensional linear time –invariant dynamical equation
y  Cx  Eu
If the observability matrix of the dynamical equation has rank n2 (where n2<n ), then
there exists an equivalence transformation
x  Px
which transform the dynamical equation to
 xo   A0
   
 xo   A21
0   xo   Bo 
     u
Ao   xo   Bo 

y  C0
 xo 
0    Eu
 xo 

and the n2-dimensional sub-equation
x o  Ao xo  Bo u
y  Co xo  Eu
10
is observable and has the same transfer function matrix as the first system.Ali Karimpour Sep 2009
Chapter 5
Canonical Decomposition of a Linear Time-invariant
Dynamical Equation
Theorem 5-5 (Continue)
Furthermore the first n2 row of P are any n2 linearly independent rows of V
(observability matrix) and the last and the last n-n2 row of P is entirely arbitrary
so long as the matrix P is nonsingular.
Proof: See “Linear system theory and design” Chi-Tsong Chen
x  Ax  Bu
x  Px
y  Cx  Eu
xo  Ao xo  Bou
y  Co xo  Eu
n  dimensiona l
n2  n  dimensiona l
G(s)
Hence, we derive the reduced order observable equation.
11
Ali Karimpour Sep 2009
Chapter 5
Canonical Decomposition of a Linear Time-invariant
Dynamical Equation
x  Ax  Bu
Theorem 5-6 (Canonical decomposition theorem)
Consider the n-dimensional linear time –invariant dynamical equation
y  Cx  Eu
There exists an equivalence transformation
x  Px
which transform the dynamical equation to
 xco   Aco
 x    0
 co  
 xc   0
A12
Aco
0
A13   xco   Bco 

A23   xco    Bco u
   
Ac   xc   0 

y 0
Cco
 xco 
Cc  xco   Eu
 
 xc 

and the reduced dimensional sub-equation
x co  Aco xco  Bco u
y  Cco xco  Eu
is observable and controllable and has the same transfer function matrix
12
as the first system.
Ali Karimpour Sep 2009
Chapter 5
Canonical Decomposition of a Linear Time-invariant
Dynamical Equation
Definition 5-3
A linear time-invariant dynamical equation is said to be reducible if and only if there
exist a linear time-invariant dynamical equation of lesser dimension that has the same
transfer function matrix. Otherwise, the equation is irreducible.
Theorem 5-7
A linear time invariant dynamical equation is irreducible if and only if it is controllable
and observable.
Theorem 5-8
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Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Transfer Function Matrices
Dynamical equation
(state-space) description
This transformation
x  Ax  Bu
The input-output description
(transfer function matrix)
is unique
G(s)  C (sI  A) 1 B  E
y  Cx  Eu
The input-output description
(transfer function matrix)
1
G(s)  C (sI  A) B  E
Realization
This transformation
is not unique
Dynamical equation
(state-space) description
x  Ax  Bu
y  Cx  Eu
1. Is it possible at all to obtain the state-space description from the transfer function
matrix of a system?
2. If yes, how do we obtain the state space description from the transfer function matrix?
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Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Transfer Function Matrices
Theorem 5-9
A transfer function matrix G(s) is realizable by a finite dimensional
linear time invariant dynamical equation if and only if G(s) is a proper
rational matrix.
Proof: See “Linear system theory and design” Chi-Tsong Chen
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Ali Karimpour Sep 2009
Chapter 5
Definition 5-4
Irreducible realizations
Theorem 5-10
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Ali Karimpour Sep 2009
Chapter 5
Irreducible realizations
Before considering the general case
(irreducible realization of proper rational matrices)
we start the following parts:
1. Irreducible realization of Proper Rational Transfer Functions
2. Irreducible Realization of Proper Rational Transfer Function Vectors
3. Irreducible Realization of Proper Rational Matrices
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Ali Karimpour Sep 2009
Chapter 5
Irreducible realization of proper rational transfer functions
ˆ0 s n  ˆ1 s n 1  ......  ˆ n
g (s) 
, 0  0
ˆ 0 s n  ˆ1 s n 1  ......  ˆ n
1 s n 1   2 s n  2  ......   n ˆ0
g ( s) 

ˆ 0
s n   1 s n 1  ......   n
1s n 1   2 s n  2  ......   n
ˆ0
ˆ0
y ( s) 
u ( s)  u ( s)  yˆ ( s )  u ( s )  gˆ ( s)u ( s)  eu ( s)
n
n 1
s  1s  ......   n
ˆ 0
ˆ 0
gˆ ( s )
gˆ ( s) 
yˆ ( s)
gˆ ( s) 
u ( s)
g ( s) 
y( s)
u ( s)
yˆ ( s)
u ( s)
x  Ax  bu
yˆ  cx  0u
x  Ax  bu
ˆ0
y  yˆ  u  cx  eu
ˆ 0
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Ali Karimpour Sep 2009
Chapter 5
Irreducible realization of proper rational transfer functions
1 s n 1   2 s n  2  ......   n ˆ0
g ( s) 

n
n 1
ˆ 0
s   1 s  ......   n
yˆ ( s) 1s n 1   2 s n2  ......   n
gˆ ( s) 

u ( s)
s n  1s n1  ......   n
There are different forms of realization
 Observable canonical form realization
 Controllable canonical form realization
 Realization from the Hankel matrix
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Ali Karimpour Sep 2009
Chapter 5
Observable canonical form realization of proper rational transfer functions
1 s n 1   2 s n  2  ......   n ˆ0
g ( s) 

ˆ 0
s n   1 s n 1  ......   n
1s n 1   2 s n 2  ......   n
gˆ ( s) 
s n  1s n 1  ......   n



y ( n )   1 y ( n 1)  ......   n y  1u ( n 1)   2 u ( n  2)  ......   n u

xn (t )  y(t )


(1)
xn1 (t )  y (1) (t )  1 y(t )  1u(t )  xn (t )  1 xn (t )  1u(t )



(1)
xn2 (t )  y ( 2) (t )  1 y (1) (t )  1u (1) (t )   2 y(t )  2u(t )  xn1 (t )   2 xn (t )  2u(t )
.................................



x1 (t )  y ( n 1) (t )  1 y ( n  2) (t )  1u ( n  2 ) (t )  ...   n 1 y (t )   n 1u (t )
 x2 (t )   n 1 xn (t )   n 1u (t )
(1)
 ( n)
 ( n1)
 (1)
( n 1)
x (t )  y (t )  1 y (t )  1u
(t )  ...   n1 y (t )   n1u (1) (t )

  n y(t )   nu (t )   n xn (t )   nu (t )
(1)
1
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Ali Karimpour Sep 2009
Chapter 5
Observable canonical form realization of proper rational transfer functions

x n (t )  y (t )


(1)
x n 1 (t )  y (1) (t )   1 y (t )  1u (t )  x n (t )   1 x n (t )  1u (t )



(1)
x n  2 (t )  y ( 2 ) (t )   1 y (1) (t )  1u (1) (t )   2 y (t )   2 u (t )  x n 1 (t )   2 x n (t )   2 u (t )
.................................



x1 (t )  y ( n 1) (t )   1 y ( n  2 ) (t )  1u ( n  2 ) (t )  ...   n 1 y (t )   n 1u (t )
 x 2 (t )   n 1 x n (t )   n 1u (t )
(1)



(1)
x1 (t )  y ( n ) (t )  1 y ( n1) (t )  1u ( n1) (t )  ...   n1 y (1) (t )   n1u (1) (t )

  n y(t )   nu (t )   n xn (t )   nu (t )
 x1  0
 x  1
 2 
 x3   0
  
 .  .
 xn  0
  n   x1    n 
0 ... 0   n 1   x2    n 1 
1 ... 0   n  2   x3     n  2 u
  

. ... .
.  .   . 
0 ... 1  1   xn   1 
0 ... 0
 x1 
x 
 2
yˆ  0 0 ... 0 1 x3 
 
.
 xn 
21
Ali Karimpour Sep 2009
Chapter 5
Observable canonical form realization of proper rational transfer functions
1s n 1   2 s n 2  ......   n
gˆ ( s) 
s n  1s n 1  ......   n
 x1  0
 x  1
 2 
 x3   0
  
 .  .
 xn  0
  n   x1    n 
0 ... 0   n 1   x2    n 1 
1 ... 0   n  2   x3     n  2 u
  

. ... .
.  .   . 
0 ... 1  1   xn   1 
0 ... 0
 x1 
x 
 2
yˆ  0 0 ... 0 1 x3 
 
.
 xn 
1s n 1   2 s n  2  ......   n ˆ0
g (s) 

n
n 1
s  1s  ......   n
ˆ 0
 x1  0
 x  1
 2 
 x3   0
  
 .  .
 x n  0
  n   x1    n 
0 ... 0   n 1   x2    n 1 
1 ... 0   n  2   x3     n  2 u
  

. ... .
.  .   . 
0 ... 1  1   xn   1 
0 ... 0
 x1 
x 
 2  ˆ
yˆ  0 0 ... 0 1 x3   0 u
  ˆ 0
.
 xn22

Ali Karimpour Sep 2009
Chapter 5
Observable canonical form realization of proper rational transfer functions
1s n 1   2 s n  2  ......   n ˆ0
g (s) 

s n  1s n 1  ......   n
ˆ 0
 x1  0
 x  1
 2 
 x3   0
  
 .  .
 x n  0
  n   x1    n 
0 ... 0   n 1   x2    n 1 
1 ... 0   n  2   x3     n  2 u
  

. ... .
.  .   . 
0 ... 1  1   xn   1 
0 ... 0
The derived dynamical equation is observable.
1s n 1   2 s n 2  ......   n
gˆ ( s) 
s n  1s n 1  ......   n
 x1 
x 
 2  ˆ
yˆ  0 0 ... 0 1 x3   0 u
  ˆ 0
.
 xn 
Exersise 1: Why?
The derived dynamical equation controllable as well if numerator and
denominator of g(s) are coprime.
Exersise 2: Why?
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Ali Karimpour Sep 2009
Chapter 5
Controllable canonical form realization of proper rational transfer functions
1 s n 1   2 s n  2  ......   n ˆ0
g ( s) 

n
n 1
ˆ 0
s   1 s  ......   n
Let us introduce a new variable
N ( s) 1 s n 1   2 s n 2  ......   n

g ( s) 

D( s )
s n   1 s n 1  ......   n
D ( s )v ( s )  u ( s )

y ( s )  N ( s )v ( s )
We may define the state variable as:
Clearly

 x1 (t )  v(t )

 x (t )   (1)
v
(
t
)

 2  

x(t )   ..    ..


 
..
..


 
 xn (t ) v ( n 1) (t )

 

x1  x2 , x 2  x3 , ..... , x n1  xn
xn  v ( n ) (t )  u (t )   n v(t )   n1v (1) (t )  ...  1v ( n1) (t )
 u(t )   n x1 (t )   n1 x2 (t )  ...  1 xn (t )
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Ali Karimpour Sep 2009
Chapter 5
Controllable canonical form realization of proper rational transfer functions
g ( s) 
1 s

  2 s  ......   n  0
 
n
n 1
0
s   1 s  ......   n
n 1
n2
N ( s) 1 s n 1   2 s n 2  ......   n

g ( s) 

D( s )
s n   1 s n 1  ......   n
x1  x2 , x 2  x3 , ..... , x n1  xn
xn  v ( n ) (t )  u (t )   n v(t )   n1v (1) (t )  ...  1v ( n1) (t )
 u(t )   n x1 (t )   n1 x2 (t )  ...  1 xn (t )
 x1   0
 x   0
 2 
 x3    .
  
.  0
 xn    n
1
0
...
0
1
...
.
.
...
0
0
...
  n 1
  n  2 ...
0   x1  0
0   x2  0
.   x3   0u
   
1  .   . 
 1   xn  1
y   n
 n 1  n  2
 x1 
x 
 2
... 1  x3 
 
.
 xn 
25
Ali Karimpour Sep 2009
Chapter 5
Controllable canonical form realization of proper rational transfer functions
N ( s) 1 s n 1   2 s n 2  ......   n

g ( s) 

D( s )
s n   1 s n 1  ......   n
 x1   0
 x   0
 2 
 x3    .
  
.  0
 xn    n
1
0
...
0
1
...
.
.
...
0
0
...
  n 1
  n  2 ...
0   x1  0
0   x2  0
.   x3   0u
   
1  .   . 
 1   xn  1
y   n
 n 1  n  2
 x1 
x 
 2
... 1  x3 
 
.
 xn 
1s n 1   2 s n  2  ......   n ˆ0
g (s) 

s n  1s n 1  ......   n
ˆ 0
 x1   0
 x   0
 2 
 x3    .
  
.  0
 xn    n
1
0
...
0
1
...
.
.
...
0
0
...
  n 1
  n  2 ...
0   x1  0
0   x2  0
.   x3   0u
   
1  .   . 
 1   xn  1
y   n
 n 1  n  2
 x1 
x 
 2  ˆ
... 1  x3   0 u
  ˆ 0
.
 xn 
26
Ali Karimpour Sep 2009
Chapter 5
Controllable canonical form realization of proper rational transfer functions
1s n 1   2 s n  2  ......   n ˆ0
g (s) 

n
n 1
s  1s  ......   n
ˆ 0
 x1   0
 x   0
 2 
 x3    .
  
.  0
 xn    n
1
0
...
0
1
...
.
.
...
0
0
...
  n 1
  n  2 ...
0   x1  0
0   x2  0
.   x3   0u
   
1  .   . 
 1   xn  1
The derived dynamical equation is controllable .
N ( s) 1 s n 1   2 s n 2  ......   n

g ( s) 

D( s )
s n   1 s n 1  ......   n
y   n
 n 1  n  2
 x1 
x 
 2  ˆ
... 1  x3   0 u
  ˆ 0
.
 xn 
Exersise 3: Why?
The derived dynamical equation observable as well if numerator and
denominator of g(s) are coprime.
Exersise 4: Why?
27
Ali Karimpour Sep 2009
Chapter 5
Controllable and observable canonical form realization of
proper rational transfer functions
Example 5-2 Derive controllable and observable canonical realization for following system.
2s 3  18s 2  48s  32
g ( s) 
s 3  6s 2  11s  6
2s 3  18s 2  48s  32
6s 2  26s  20
g ( s) 
 3
2
3
2
2
s  6s  11s  6
s  6s  11s  6
Observable canonical form realization is:
 x1  0 0  6   x1  20
 x   1 0  11  x   26u
 2 
 2   
 x 3  0 1  6   x3  6 
 x1 
y  0 0 1 x 2   2u
 x3 
It is not controllable.
Why?
Controllable canonical form realization is:
1
0   x1  0
 x1   0
 x    0
  x   0  u
0
1
 2 
 2   
 x3   6  11  6  x3  1 
 x1 
y  20 26 6 x2   2u
 x3 
It is not observable.
28
Why?
Ali Karimpour Sep 2009
Chapter 5
Irreducible realization of proper rational transfer functions
Example 5-3 Derive irreducible realization for following transfer function.
2s 3  18s 2  48s  32
g ( s) 
s 3  6s 2  11s  6
6 s  20
2s 3  18s 2  48s  32
6s 2  26s  20

2
g ( s) 


2
2
3
2
3
2
s  5s  6
s  6s  11s  6
s  6s  11s  6
Observable canonical form realization is:
 x1  0  6  x1  20
 x   1  5  x   6 u
 2   
 2 
 x1 
y  0 1   2u
 x2 
It is controllable too.
Why?
Controllable canonical form realization is:
1   x1  0
 x1   0

 x   6  5  x   1 u
 2   
 2 
 x1 
y  20
6   2u
 x2 
It is observable too.
Why?
29
Ali Karimpour Sep 2009
Chapter 5
Irreducible realization of proper rational transfer functions
Realization from the Hankel matrix
 0 s n  1s n 1  ......   n
g ( s)  n
s  1s n 1  ......   n
g (s)  h(0)  h(1)s 1  h(2)s 2  h(3) s 3  ......
The coefficients h(i) will be called Markov parameters.
h(2)
h(3)
 h(1)
 h(2)
h(3)
h(4)

 h(3)
h(4)
h(5)
H ( ,  )  
.
.
 .
 .
.
.

h( ) h(  1) h(  2)
...
...
...
...
...
...
h(  )

h(   1) 
h(   2) 

.


.

h(    1)
30
Ali Karimpour Sep 2009
Chapter 5
Irreducible realization of proper rational transfer functions
Realization from the Hankel matrix
Theorem 5-11 Consider the proper transfer function g(s) as
 0 s n  1 s n 1   2 s n 2  ......   n
g ( s) 
s n   1 s n 1  ......   n
then g(s) has degree m if and only if
 H (m, m)   H (m  k , m  l )
for every k , l  1, 2 , 3, ....
31
Ali Karimpour Sep 2009
Chapter 5
Irreducible realization of proper rational transfer functions
Realization from the Hankel matrix
Now consider the dynamical equation
g (s)  c(sI  A) 1 b  e  s 1c( I  s 1 A) 1 b  e
x  Ax  bu
y  cx  eu
 e  cbs 1  cAbs 2  cA2bs 3  .....
h(i)  cAi 1b
h(2)
 h(1)
 h(2)

 .
.
H (n  1, n)  
.
 .
 h( n)
h(n  1)

h(n  1) h(n  2)
.....
.....
.....
.....
.....
.....
i  1, 2 , 3 , ......
h( n) 
h(n  1) 

.

.

h(2n  1)

h(2n) 
Let the first σ rows be linearly independent and the (σ+1) th row of H(n+1,n) be
linearly dependent on its previous rows. So
[a1
a2
..... a
1 0 ..... 0] H (n  1, n)  0
32
Ali Karimpour Sep 2009
Chapter 5
Irreducible realization of proper rational transfer functions
Realization from the Hankel matrix
[a1
a2
1 0 ..... 0] H (n  1, n)  0
..... a
We claim that the σ-dimensional dynamical equation
1
0
 0
 0
0
1

 0
0
0

x  . .
.
.
 .
.
.

0
0
 0
 a a a
2
3
 1
y  1 0 0 ..... 0
.....
0
.....
0
.....
0
.....
.
.....
.
.....
0
.....  a 1
0x  h(0)u
0   h(1) 
0   h(2) 


0   h(3) 
 

.  x   . u
.   . 
 

1  h(  1)
 a   h( ) 
(I)
is a controllable and observable (irreducible realization).
Exercise 5: Show that (I) is a controllable and observable (irreducible realization) of
x  Ax  bu
y  cx  eu
33
Ali Karimpour Sep 2009
Chapter 5
Irreducible realization of proper rational transfer functions
Example 5-4 Derive irreducible realization for following transfer function.
2s 3  18s 2  48s  32
g ( s) 
s 3  6s 2  11s  6
g (s)  2  6s 1  10s 2  14s 3  10s 4  34s 5  230s 6  .....
 10 14 
 6
 10 14  10 

H (4,3)  
 14  10  34



10

34
230


We can show that the rank of H(4,3) is 2. So
6
5 1 0H (4,3)  0
Hence an irreducible realization of g(s) is
1
0
 6 
x  
 x   10u

6

5




y  1 0x  2u
34
Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Transfer Function Vectors
Consider the rational function vector
 g1 ( s ) 
 g (s)
 2 
G (s)   . 


.


 g q ( s)



e
g
 1   1 (s) 
e   g ( s ) 
 2  2 
G ( s)   .    . 
  

.
.
  


e q   g q ( s ) 
  

 e1 
e 
 2
1


.
G(s) 
 n
  s   1 s n 1  .....  n
.
e q 
 
 11 s n 1  12 s n  2  ....  1n 


n 1
n2

s


s

....


22
2n 
 21


.


.


  s n 1   s n  2  ....   
q2
qn 
 q1
35
Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Transfer Function Vectors
 e1 
e 
 2
1
G(s)   .   n
  s   1 s n 1  .....  n
.
e q 
 
 0
 0

 .
x  
 .
 0

  n
1
0
...
0
1
...
.
.
...
.
.
...
0
0
...
  n 1
  n  2 ...
0  0 
0  0
 
.  .
 x   u
.  .
1  0 
  
 1  1
 11 s n 1  12 s n  2  ....  1n 


n2
n 1


....

s


s

2n 
22
 21


.


.


2

n
1

n
  s   s  ....   
qn 
q2
 q1
 y1   1n
 y  
 2   2n
 .  .
  
.  .
 yq    qn
  
1( n 1)
 2( n 1)
1( n  2) ... 11 
 2( n  2) ...  21 
.
.
.
.
 q ( n 1)
 q ( n2)
 e1 
e 
 2
... .  x   . u
  
... .   . 
...  q1  eq 
This is a controllable form realization of G(s).
We see that the transfer function from
u to yi is equal to
 i1 s n 1   i 2 s n 2  ....   in
ei 
36
n
n 1
s   1 s  .....  AlinKarimpour Sep 2009
Chapter 5
Realization of Proper Rational Transfer Function Vectors
Example 5-5 Derive a realization for following transfer function vector.
s3




G ( s )   ( s  1)( s  2) 
s4




s3
s3


 ( s  1)( s  2)  0
 ( s  3) 2 
1
G (s)  
 


s

4

 1  ( s  1)( s  2)( s  3) ( s  1)( s  2)
s3


 s 2  6 s  9
0 
1
  3
 2

2
1  s  6 s  11s  6  s  3s  2
Hence a minimal dimensional realization of G(s) is given by
1
0  0
0
x   0
0
1  x  0u
 6  11  6 1 
9 6 1 0
y
x   u

2 3 1 1 
37
Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Matrices
There are many approaches to find irreducible realizations for proper
rational matrices.
1. One approach is to first find a reducible realization and then apply
the reduction procedure to reduce it to an irreducible one.
Method I, Method II, Method III and Method IV
2.
In the second approach irreducible realization will yield directly.
38
Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Matrices
Method I: Given a proper rational matrix G(s), if we first find
xij  Aij xij  biju j
an irreducible realization for every element gij(s) of G(s) as
yij  cij xi j  eiju j
y1  y11  y12
y2  y22  y22
 x11   A11
 x   0
 12   
 x 21   0
  
 x 22   0
 y1  c11
y    0
 2 
0   x11  b11 0 
A12 0
0   x12   0 b12  u1 



b21 0  u 2 
0 A21 0  x 21
  

0
0 A22   x 22   0 b22 
 x11 
c12 0
0   x12   e11 e12  u1 

0 c 21 c 22   x 21  e21 e22  u 2 
 
 x 22 
0
0
Clearly this equation is generally not controllable and not observable.
To reduce this realization to irreducible one requires the application of the reduction
procedure twice (theorems 5-4 and 5-5).
39
Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Matrices
 x11   A11
 x   0
 12   
 x 21   0
  
 x 22   0
 y1  c11
y    0
 2 
Proof:
c11 c12
0
0

0
c 21
( sI

0 
c 22  


 A11 ) 1
0   x11  b11 0 
A12 0
0   x12   0 b12  u1 




b21 0  u 2 
0 A21 0 x 21
  

0
0 A22   x 22   0 b22 
 x11 
c12 0
0   x12   e11 e12  u1 

0 c 21 c 22   x 21  e21 e22  u 2 
 
 x 22 
0
0
0
0
0
( sI  A12 ) 1
0
0
0
( sI  A21 ) 1
0
0
0
 b11 0 

 e
0
b
0
12

   11 e12 
 b21 0  e21 e22 
0


( sI  A22 ) 1   0 b22 
0
 c11 ( sI  A11 ) 1 b11  e11 c12 ( sI  A12 ) 1 b12  e12   g11 ( s) g12 ( s ) 

 G(s)


1
1
c 21 ( sI  A21 ) b21  e21 c 22 ( sI  A22 ) b22  e22   g 21 ( s ) g 22 ( s)
40
Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Matrices
Method II: Given a proper rational matrix G(s), if we find the controllable canonicalform realization for the ith column, Gi(s), of G(s) say,
x i  Ai xi  bi u i
 x1   A1
 x   0
 2
....   .
  
 x p   0
y  C1 C 2

Proof:
C
1

C2
0
....
A2
....
.
....
0
....
.... C p
( sI

.... C p 




 C1 ( sI  A1 ) 1 b1  e1
0   x1  b1
0   x 2   0

.   ..   .
  
A p   x p   0
x  e1 e2 ....
 
0  u1 
b2 ... 0  u 2 
. ... .   . 
 
0 ... b p  u p 
ep u
0
...

y i  Ci xi  ei u i
This realization is always
controllable. It is however
generally not observable.
 b1 0 ... 0 


0
b
...
0
0
( sI  A2 ) 1 ...
0
2

  e e .... e
1
2
p

.
. ... . 
.
.
...
.


0
0
... ( sI  A p ) 1   0 0 ... b p 
 g11 ( s ) g12 ( s ) ... g1 p ( s ) 
 g ( s ) g ( s ) ... g ( s )
21
22
2p
1
  G ( s)
.....C p ( sI  A p ) b p  e p  
 .
.
...
. 


41
 g q1 ( s ) g q 2 ( s ) ... g qp ( s ) 
 A1 ) 1
0
...
0



Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Matrices
Method III: Let a proper rational matrix G(s), where

G ( s )  G ( s )  G ( )
consider
the monic least common denominator of G(s) as  ( s)  s m   1 s m1   2 s m2  ...   m
Then we can write G(s) as
G( s) 


1
R1s m1  R2 s m2  ...  Rm  G()
 ( s)
Then the following dynamic equation is a realization of G(s).
Ip
0p
...
0p 
 0p
0 p 
 0

0 
0
I
...
0
p
p
p
p


 p
.
.
...
.  x   .. u
x   .


 
0
0
0
...
I
p
p
p
p


0 p 
  m I p   m 1 I p   m  2 I p ...   1 I p 
I p 


 
y  Rm Rm 1 Rm  2 ... R1 x  G ()u
Exercise 6: Show that the above dynamical equation is a controllable realization42of G(s)
Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Matrices
Method IV: It is possible to obtain observable realization of a proper G(s). Let
G(s)  H (0)  H (1)s 1  H (2)s 2  ...........
Consider the monic least common denominator of G(s) as
 ( s)  s m   1 s m1   2 s m2  ...   m
Then after deriving H(i) one can simply show
H (m  i)  1H (m  i  1)   2 H (m  i  2)  ...   m H (i)
i 1
(I )
Exercise 7: Proof equation (I)
Let {A, B, C and E} be a realization of G(s) then we have
G(s)  E  C (sI  A) 1 B  E  CBs 1  CABs 2  CA2 Bs 3  ...........
43
Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Matrices
Then {A, B, C and D} be a realization of G(s) if and only if
E  H (0)
and
H (i  1)  CAi B
i  0 , 1, 2 ,....
Now we claim that the following dynamical equation is a realization of G(s).
Iq
0q
 0q
 0
0q
Iq
q

.
.
x   .

0q
0q
 0q
  m I q   m 1 I q   m  2 I q

y  I q 0 0 ... 0 x  H (0)u


...
...
...
...
...
0q 
 H (1) 
 H (2) 
0 q 


. x  
u
..



Iq 
H
(
m

1
)


 H (m) 
  1 I q 
We can readily verify that
 H (2) 
 H (3) 
 H (i  1) 
 H (3) 
 H (4) 
 H (i  2) 
2
i
, A B
 , ... , A B  

AB  
 ... 



...
... 






 H (m  1)
 H (m  2)
 H (m  i )
CAi B  H (i  1)
E  H (0) 44
Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Matrices
Now we shall discuss in the following a method which will yield directly irreducible
realizations. This method is based on the Hankel matrices.
q p
Let G(s) be
G(s)  H (0)  H (1)s 1  H (2)s 2  ...........
Consider the monic least common denominator of G(s) as
 ( s)  s m   1 s m1   2 s m2  ...   m
Define
0 p
I
 p
N  0 p

 .
0 p

0p
...
0p
...
Ip
...
.
...
0p
...
mI p 
0 p   m 1 I p 
..0 p .   m  2 I p 

...
.

Ip
  1 I p 
0p
 0q
 0
q

M  .

 0q
  m I q

Iq
0q
...
0q
Iq
...
.
.
...
0q
0q
...
  m 1 I q
  m2 I q
...
0q 
0 q 
. 

Iq 
  1 I q 
45
Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Matrices
0 p
I
 p
N  0 p

 .
0 p

0p
0p
Ip
 ( s)  s m   1 s m1   2 s m2  ...   m
Iq
 0q
... 0 p
mI p 
 0
0q
... 0 p   m 1 I p 
q

.
M  .
... ..0 p .   m  2 I p 
.
...
...
.
0p
...
Ip
 1 I p





 0q
  m I q

0q
...
Iq
...
.
...
0q
0q
...
  m 1 I q
  m2 I q
...
0q 
0 q 
. 

Iq 
  1 I q 
We also define the two following Hankel matrices
H (2)
 H (1)
 H (2)
H (3)
T 
 .
.

 H (m) H (m  1)
H ( m) 
H (m  1) 

.

H (2m  1)
H (3)
 H (2)

H (4)
~  H (3)
T 

.
.

 H (m  1) H (m  2)
H (m  1) 
H (m  2)

.

H ( 2 m) 
It can be readily verified that
~
T  MT  TN
M i T  TN i
i  0 , 1, 2 ,.....
46
Ali Karimpour Sep 2009
Chapter 5
Realization of Proper Rational Matrices
It can be readily verified that
~
T  MT  TN
Let I k ,l
k  l (l  k )
M i T  TN i
i  0 , 1, 2 ,.....
be as the form
I k ,l  I k
0
Note that the left-upper-corner of M iT = TN i is H(i+1) so:
H (i  1)  I q,qm M iTI Tp, pm  I q,qmTN i I Tp, pm
i  0 , 1, 2 , ....
It can be readily verified that
But we want Irreducible Realization of Proper Rational Matrices
47
Ali Karimpour Sep 2009
Chapter 5
Irreducible Realization of Proper Rational Matrices
48
Ali Karimpour Sep 2009
Chapter 5
Irreducible Realization of Proper Rational Matrices
Example 5-6 Derive an irreducible realization for the following proper rational function.
  2s 2  3s  2

2
(
s

1
)
G( s)  
4s  5


s 1
Least common denominator of G(s), is
1 

s 
 3s  5 
s  1 
 ( s)  s( s  1) 2
 2 0  1 1  1  2 0 2 3 0  3  4 0 4 5 0  5  6 0 6
G( s)  
  1  2 s    1 2 s  1  2 s    1 2 s  .1  2 s    1 2 s .....
4

3

 











1 2 0
3
0
1
 1  2 1 2

1

2

 H (1) H (2) H (3)  
 2 0
3
0 4 0 
T   H (2) H (3) H (4)  


1
2
1

2

1
2

 H (3) H (4) H (5)  
3
0 4 0
5
0


1

2

1
2
1

2


Non-zero singular values of T
are 10.23, 5.79, 0.90 and 0.23.
So, r = 4.
49
Ali Karimpour Sep 2009
Chapter 5
Irreducible Realization of Proper Rational Matrices
- 0.3413 0.2545 - 0.8902 - 0.1621 
- 0.2357 - 0.5238 - 0.0581 - 0.0071 


 0.5127 - 0.2078 - 0.1054 - 0.8264 
Yr  

0.2357
05238
0
.
0581
0
.
0071


- 0.6738 0.2627
0.4316  0.5392


- 0.2357 - 0.5238  0.0581  0.0071
Yˆ  Yr S 1 / 2
 - 0.4003 - 0.0196 0.4905 - 0.6574
 0.1049

0.5872
0.6022
0.5306


 0.5496 - 0.1057 - 0.0978 0.1026 
Ur  

0.1382
0.5432

0
.
3875
0.1888


 - 0.6989 0.2311 - 0.2949 0.4522 


0.5432
0.3875
0.1888 
 0.1382









H
Uˆ  S 1 / 2U r
- 1.0915 0.6121 - 0.8443 - 0.0770 
- 0.7539 - 1.2598 - 0.0551 - 0.0034
1.6398 - 0.4999 - 0.1000 - 0.3923 

0.7539 1.2598 0.0551 0.0034 
- 2.1553 0.6317 0.4093 - 0.2560 

- 0.7539 - 1.2598 - 0.0551 - 0.0034
 - 1.2803 0.3355 1.7579 - 0.4421 - 2.2356
 - 0.0471 1.4124 - 0.2543 - 1.3066 0.5557

 0.4652 - 0.5711 - 0.0927 - 0.3675 - 0.2797

 - 0.3121 - 0.2519 0.0487 - 0.0896 0.2147
0.4421 
1.3066 
0.3675

0.0896 
50
Ali Karimpour Sep 2009
Chapter 5
Irreducible Realization of Proper Rational Matrices
 - 0.1067 - 0.0737
 0.1058 - 0.2178
H
†

1
/
2
Yˆ  S Yr  
 - 0.9386 - 0.0613

 - 0.3415 - 0.0149
 - 0.1251
 0.0328

 0.1718
Uˆ †  U r S 1 / 2  
 - 0.0432
 - 0.2185

 0.0432
0.1603 0.0737 - 0.2107 - 0.0737
- 0.0864 0.2178 0.1092 - 0.2178 
- 0.1112 0.0613 0.4551 - 0.0613 

- 1.7406 0.0149 - 1.1356 - 0.0149
- 0.0081 0.5171 - 1.3847 
0.2441 - 0.6349 - 1.1176 
- 0.0440 - 0.1031 0.2161 

- 0.2259 - 0.4086 - 0.3976
0.0961 - 0.3109 0.9525 

0.2259 0.4086 0.3976 
 - 1.2497 0.0369 0.2155 - 0.1904 
 0.1588 - 1.0139 - 0.1604 0.0772 
†~ ˆ†
ˆ

A  Y TU  
 - 0.2227 - 0.1800 - 0.2888 0.8076


 0.1246 - 0.1181 0.1354 - 0.4476 
 - 1.0915 0.6121 - 0.8443 - 0.0770 
ˆ
C  I q ,qmY  

 - 0.7539 - 1.2598 - 0.0551 - 0.0034
 - 1.2803 0.3355 
 - 0.0471 1.4124 

B  UˆI Tp , pm  
 0.4652 - 0.5711 


 - 0.3121 - 0.2519
 2 0 
E  H (0)  

 4  3
51
Ali Karimpour Sep 2009