1. Robust and Optimal Control
A Two-port Framework Approach
Some Basic Concepts
1
2. Robust and Optimal Control - A Two-port Framework Approach
Concept
of
Norm
2
3. Robust and Optimal Control - A Two-port Framework Approach
Max(x)
sup(x)
Sup(x) appears in one of max(x) or “boundary”
sup(x) may not be one of max(x)
boundary
Max(x)
sup(x)
Difference between sup(x) and max(x)
t
x
x
t
3
4. Robust and Optimal Control - A Two-port Framework Approach
(1) Vector p-norm (for ):
Definition 2.3: Let be a vector in . The followings are norms on
<Vector norm>
Norm can be loosely understood as a description of size or volume.
x
A vector norm, denoted by of any vector must have the following properties:
(1) unless,
(2) where is any constant and is the absolute value of
(3)
0
x 0 0
cx c x
c | |
c c
x y x y
1
2
n
x
x
x
x
n
c n
c
1 p
1
1/
n
p
p i
i
p
x
x
(2) Vector 1-norm:
1
1
n
i
i
x x
(3) Vector 2-norm:
2
*
2
1
n
i
i
x x x x
(4) Vector - norm:
1
max i
i n
x x
4
5. Robust and Optimal Control - A Two-port Framework Approach
Examples of vector norms
Given the vectors of a、b、c
find the vector norm respectively
( i.e., 1-norm, 2-norm, ∞-norm )
5
1
2
15
a
8
9
10
b
4
-6
12
c
1-norm of a vector
Definition : The 1-norm of any vector x is defined as 1
1
n
i
i
x x
1
1
1 2 15 18
n
i
i
a a
1
1
4 6 12 22
n
i
i
c c
1
1
7
9 0 2
8 1
n
i
i
b b
6. Robust and Optimal Control - A Two-port Framework Approach
6
Definition : The 2-norm of any vector x is defined as
2
*
2
1
n
i
i
x x x x
2-norm of vector
2 2 2 2
*
2
1
1 2 15 230
n
i
i
a a a a
2 2 2 2
*
2
1
4 6 12 196 14
n
i
i
c c c c
2 2 2 2
*
2
1
8 9 10 245
n
i
i
b b b b
∞ -norm of vector
Definition : The ∞-norm of any vector x is defined as
1
max i
i n
x x
5 15
1
a
10 10
b
12 12
c
7. Robust and Optimal Control - A Two-port Framework Approach
Properties of vector norms
7
0 , 0 0
x unless
x x
x y x y
any constant and is the absolute value of
where is
(1)
(2)
(3)
1
2
15
a
2
4
30
d
2
d a
1
18
a
2
230
a
15
a
vector norm of a：
1
36
d
2
2 230
d
30
d
vector norm of d：
2
d a
1 8 9
2 9 11
15 10 25
e a b
1
45
e
2
827
28.7576
e
25
e
vector norm of e：
1 1
45
a b
2 2
245 230
30.6577
a b
25
a b
vector norm of a+b：
Ex:
8. Robust and Optimal Control - A Two-port Framework Approach
In the case of matrices, a matrix norm satisfies
<Matrix norm>
(1) unless
0
A 0
A
(2) where is any constant
A A
(3) A B A B
(4) AB A B
Definition : Let be a matrix in
The following gives a list of different matrix norms ：
11 12 1
21 22 2
1 2
n
n
m m mn
m m m
m m m
M
m m m
m n
c
(1) Matrix 1-norm (column sum):
1
1
max
m
j
i
ij
M m
(2) Matrix 2-norm:
*
ma
2 x
M M M
(3) Matrix -norm (row sum):
1
max
n
j
i
ij
M m
(4) Frobenius norm:
*
F
M trace M M
8
9. Robust and Optimal Control - A Two-port Framework Approach
Examples of 1-norm
9
1
1
max 3 7
4
m
ij
j
i
A a
1
1
max 2 4 6
m
ij
j
i
C c
1
1
max 0 6 6
m
ij
j
i
B b
1-norm of a matrix (column sum)
Definition : The 1-norm of matrix M is defined as 1
1
max
m
ij
j
i
M m
Let
11 12 1
21 22 2
1 2
n
n
m n
m m mn
m m m
m m m
M
m m m
Choose the maximum
1 3
2 4
A
2 0
1 6
B
1 2
4 4
C
10. Robust and Optimal Control - A Two-port Framework Approach
Examples of matrix 2-norm
10
max
2
29.8661 5.4650
A A A
max
2
36.5624 6.0467
C C C
max
2
37.1208 6.0927
B B B
Definition : The 2-norm of matrix M is defined as
max
2
M M M
5 11
11 25
A A
1
2
0.1339
29.8661
5 6
6 36
B B
1
2
3.8792
37.1208
17 18
18 20
C C
1
2
0.4376
36.5624
11. Robust and Optimal Control - A Two-port Framework Approach
Examples of matrix ∞ -norm (row sum)
11
1
max 2 4 6
m
ij
i
j
A a
1
max 8
4 4
m
ij
i
j
C c
1
max 1 6 7
m
ij
i
j
B b
Definition : The ∞ -norm of matrix M is defined as
1
max
m
ij
i
j
M m
Let
11 12 1
21 22 2
1 2
n
n
m n
m m mn
m m m
m m m
M
m m m
Choose the maximum
1 3
2 4
A
2 0
1 6
B
1 2
4 4
C
12. Robust and Optimal Control - A Two-port Framework Approach
12
30 5.4772
F
A trace A A
37 6.0828
F
C trace C C
6.4031
41
F
B trace B B
Forbenius-norm of a matrix
Definition : The Forbenius-norm of matrix M is defined as
F
M trace M M
5 11
11 25
A A
5 6
6 36
B B
17 18
18 20
C C
11 12 1
21 22 2
1 2
n
n
m m mn
n n n
n n n
M M
n n n
ii
trace M M M
13. Robust and Optimal Control - A Two-port Framework Approach
1 1
224
A B 2 2
134.5222
A B
147
A B
Matrix norm of AB：
134.8110
F F
A B
Properties of matrix norms
13
0 , A 0
A unless
x x
A B A B
AB A B
any constant
where is
(1)
(2)
(3) Let D AB
(4)
1 3
2 5
A
20 1
-8 -1
B
4 4
0 7
D
1
11
D 2
8.3523
D
8
D
Matrix norms of D：
9
F
D
14. Robust and Optimal Control - A Two-port Framework Approach
<Signal norm>
Let the time signal be measurable.
Definition : The 1-norm of a signal is defined as
y
1
: y dt
y
Definition : The 2-norm of a signal is defined as 2
2
: y dt
y
y
Definition : The -norm of a signal is defined as
y
: sup
y y
<System norm>
Let the system should be linear, time-invariant and causal.
Definition :The 1-norm of a system is defined as
( )
G s 1
1
( ) : ( )
2
G s G j d
Definition : The 2-norm of a system is defined as
( )
G s 2
2
1
( ) : ( )
2
G s G j d
Let the state-space realization of a system be . The norm can be
determined by , where is the observability gramian .
( )
s A B
G s
C D
2
( ) T
G s tr B PB
P
Definition The -norm of a system is defined as
( )
G s ( ) : sup ( )
G s G j
14
15. Robust and Optimal Control - A Two-port Framework Approach
Example : Given a linear system below, please determine its -norm and Bode plot.
0 1 0
-5 2 1
5 1
x x u
y x
15
0 1 0
-5 2 1
5 1
x x u
y x
2
5
(s)
2 5
s
G
s s
Matlab Code:
(s) 1.3214
G
16. Robust and Optimal Control - A Two-port Framework Approach
<Hankel norm>
Definition : Hankel norm is used for determining the residual energy of a
system before . For a stable system described as , the
Hankel norm is defined as
0
t ( ) ( )
y t Hu t
2
0
0
( ,0)
( ) ( )
sup
( ) ( )
T
H T
u L
y t y t
H
u t u t dt
This can be determined by
max
H
H PQ
are the observability gramian and controllability gramian
and
P Q
16
17. Robust and Optimal Control - A Two-port Framework Approach
Example : Given a linear system as below, please determine its norm
and Hankel norm.
G s 2
H
1 0 1
0 2 1
1 2
x x u
y x
Observability Gramian:
0
1 1
2 3
1 1
3 4
T
A T A
P e C Ce d
0
1 2
2 3
2
1
3
T
A T A
Q e BB e d
2
1
6
T
G tr B PB
max
1
6
H
G PQ
Hence
Controllability Gramian
17
18. Robust and Optimal Control - A Two-port Framework Approach
Controllability and observability
<Controllability>
Physical meaning：There exists a state feedback gain F such that all eigenvalues of
A+BF can be assigned arbitrarily.
The following statements are equivalent
(1) The n-dimensional pair (A,B) is controllable
(2) The controllability matrix has full row rank.
(3) The matrix has full row rank at every eigenvalue of A
(4) The matrix is nonsingular for any
In addition, if all eigenvalues of A have negative real parts, then the unique solution of
is positive definite. The solution is called the controllability Gramain defined by
1
n
B AB A B
I A B
0
0
T
t
A T A
c
W t e BB e d
0
t
n n
T T
c c
AW W A BB
0
T
A T A
c
W e BB e d
18
19. Robust and Optimal Control - A Two-port Framework Approach
(1) The n-dimensional pair (A,C) is observable
(2) The observability matrix has full column rank.
(3) The matrix has full column rank at every eigenvalue of A
(4) The matrix is nonsingular for any
In addition, if all eigenvalues of A have negative real parts, then the unique solution of
is positive definite. The solution is called the observability Gramain defined by
Controllability and observability
<Observability>
Physical meaning：There exists an observer gain H such that all eigenvalues of A+HC
can be assigned arbitrarily.
The following statements are equivalent
1
n
C
CA
CA
I A
C
0
( ) 0
T
t
A T A
o
W t e C Ce d
0
t
n n
T T
o o
A W W A C C
0
T
A T A
o
W t e C Ce d
19
20. Robust and Optimal Control - A Two-port Framework Approach
Functional spaces
Laplace transform
Inverse transform
2
H
2 0,
L
Laplace transform
Inverse transform
P
P
2 ,
L
2
L jR
Laplace transform
Inverse transform
P
P
2
H
2 ,0
L
The space (for ) consists of all Lebesgue measurable functions
defined in the interval such that
The space consists of all Lebesgue measureable functions such that
p
L 1 p
( )
w t
( , )
1
: ( )
p p
p
w w t dt
L
( )
w t
: sup ( )
t R
w ess w t
Let be a subspace of in which every
function is analytic in , and be a
subspace of in which every function is
analytic and bounded in the open right half
plane.
2
H 2
L
Re( ) 0
s H
L
20
21. Robust and Optimal Control - A Two-port Framework Approach
2
2
( )
( )
G s L
G s H
2
2
[ *( ) ( )]
1
( ) ( )
2
Trace G j G j d
G s G j d
( )
( )
G s L
G s H
( ) sup [ ( )]
sup [ ( )]
G j ess G j
H G j
21
22. Robust and Optimal Control - A Two-port Framework Approach
<System norm>
Let G(s) be linear, time-invariant and causal stable system.
2
2
1
( ) : ( )
2
G s G j d
( ) : sup ( )
G s G j
22
23. Robust and Optimal Control - A Two-port Framework Approach
RL
GH
BH
RH
Strictly
proper
2
RH
Stable
A
B C
E
F G
H
( 3)( 4) ( 1)
: :
( 1)( 2) ( 4)
( 7) ( 20)
: :
( 5) ( 3)( 5)
( 4)( 5) ( 1)
: :
( 6)( 7) ( 2)( 4)
( 20) ( 1)( 2)
: :
( 3)( 5) ( 3)(
s s s
A B
s s s
s s
C D
s s s
s s s
E F
s s s s
s s s
G H
s s s s
4)( 5)
s
Example
D
23
24. Robust and Optimal Control - A Two-port Framework Approach
RL
GH
BH
RH
Strictly
proper
RH
2
RL
2
RH
2
RH
Stable
Anti-Stable
A
B C
E
F G
H
I
J
( 3)( 4) ( 1)
: :
( 1)( 2) ( 4)
( 7) ( 20)
: :
( 5) ( 3)( 5)
( 4)( 5) ( 1)
: :
( 6)( 7) ( 2)( 4)
( 20) ( 1)( 2)
: :
( 3)( 5) ( 3)(
s s s
A B
s s s
s s
C D
s s s
s s s
E F
s s s s
s s s
G H
s s s s
4)( 5)
( 3) ( 4)( 6)
: :
( 1)( 2) ( 3)( 5)
s
s s s
I J
s s s s
Example
D
24
25. Robust and Optimal Control - A Two-port Framework Approach
Example : Determine the following systems and evaluate which function space is
located.
(1) (2) (3)
2
2
1 3
( ) ( )
1 ( 1)( 2
(
1 )
)
s s
G s G s
s
s
G s
s s s
(1) It is clear that is stable and for
. Hence . By decomposition of , one has
1( )
G s 2
1 sup sup
sup ( ) 1
1 1
j
j
G j
0
1
G H
1
G
1
1
( ) 1
1 1
s
G s
s s
2
1 2
1
( )
2
1 1 1
(1 )(1 )
2 1 1
1 1 1 1
(1 )
2 1 1 ( 1)( 1)
G G j d
d
j j
d
j j j j
1 2 1
( ) , ( )
G s H G s H
25
26. Robust and Optimal Control - A Two-port Framework Approach
Example : Determine the following each system and evaluate which function space
is located.
(1) (2) (3)
1 3
2
2
( ) ( )
1 ( 1)
( )
)
1 ( 2
s
G s
s
s s
G s G s
s s s
(2) By definition, because of ;
, because of ;
, because of .
2 2 2 2
( ) , ( ) , ( )
G s L G s H G s H
2 ( )
G s L
2
2
( )
sup
s
1
up ( )
j
j
G j
2 2
( )
G s H
2
( )
G j d
2 ( )
G s H
2
Re( ) 0
sup
1
s
s
s
26
27. Robust and Optimal Control - A Two-port Framework Approach
Example : Determine the following systems and evaluate which function space is
located.
(1) (2) (3)
2
1 2 3
( ) ( ) ( )
( 1)(
1 1 2)
s s
G s G s
s s
s
G s
s s
(3) , because is not analytic at ;
, because is analytic on axis and satisfies
, because is not analytic at .
3 ( )
G s H
3 ( )
G s 1
s
3 ( )
G s L
3 ( )
G s j sup ( )
G j
3 2
( )
G s H
3 ( )
G s 1
s
3 3 2 3
( ) , ( ) , ( )
G s L G s H G s H
27
28. Robust and Optimal Control - A Two-port Framework Approach
A complex square matrix is called unitary if the inverse is equal to
the complex conjugate transpose: * *
A A AA I
A square matrix is called orthogonal if it is real and satisfies
( )
T T
i
diag
A A AA
A square matrix is called normal if . A normal matrix
has the decomposition of where and is a
diagonal matrix.
* *
ZZ Z Z
Z
*
Z U U
*
UU I
Some definitions of matrix
28
A square matrix is called orthonormal if it is real and satisfies
and 1
T T
A A AA I A
30. Robust and Optimal Control - A Two-port Framework Approach
Concept
of
positive real
30
31. Robust and Optimal Control - A Two-port Framework Approach
31
Positive semi-definite
A n× n matrix M is called to be positive semi-definite if
Example:
*
0
z Mz
2 1 0
1 2 1
0 1 2
M
2 2 2
2 2
2 2
2 2 2
2 2 2 2 2
0
T T
a
z Mz z M z a b a b c b c b
c
a ab b bc c
a a b b c c
a
Let z b
c
M is positive definite
32. Robust and Optimal Control - A Two-port Framework Approach
32
Definition of positive real:
A p×p proper rational transfer function matrix G(s) is positive real
if
poles of all elements of G(s) are in Re[s]≤0 ;
for all real ω in which jω is not a pole of any element of G(s),
G(jω)+GT (-jω)≥0 (positive semi-definite);
any pure imaginary pole jω of any element of G(s) is a simple pole
and (positive semi-definite)
, where
2
lim det 0
p q T
G j G j
T
q rank G G
33. Robust and Optimal Control - A Two-port Framework Approach
33
Definition of strictly positive real:
A p×p proper rational transfer function matrix G(s) is strictly positive real
if
Poles of all elements of G(s) are in Re[s]<0 .
For all real ω for which jω is not a pole of any element of G(s),
G(jω)+GT (-jω)>0 (positive definite)
Any pure imaginary pole jω of any element of G(s) is a simple pole
and (positive definite)
, where
2
lim det 0
T
p q
G j G j
T
rank
q G G
34. Robust and Optimal Control - A Two-port Framework Approach
34
Example 1:
1
, 0
G s a and a
s a
1. -
pole a G s is Hurwitz
2
2. 0, for all
1 1 2
where q= 0
=
T
T
G j G j
j j
rank G G
a
a a a
2 2
2
2
3. lim det
2
lim 2 , as
0 p=1, q=0
p q T
G j G j
a
a
a
Sol:
G is strictly positive real
35. Robust and Optimal Control - A Two-port Framework Approach
1
sC
R
( )
i
V s
sL
( )
o
V s
( )
I s
35
Example 2:
Sol:
Consider a RLC circuit as shown in Fig.1. Please verify
that whether the system is positive real when
(a)
(b)
Fig.1
2 0.5 2
L C R
0 0.5 2
L C R
( ) ( ) ( ) ( )
( ) ( )
1
1
i
o
V s R I s I s I s
V s I s
sC
s
sL
C
2
( ) 1
( )
( ) 1
o
i
V s
G s
V s LCs RCs
According to KVL rule,
36. Robust and Optimal Control - A Two-port Framework Approach
1
sC
R
( )
i
V s
sL
( )
o
V s
( )
I s
36
Sol:
Fig.1
(a)
2 2
( ) 1 1
1. ( )
( ) 1 1
o
i
V s
G s
V s LCs RCs s s
1
3
2
pole j
2
2
2 2 2
2
2
2
2 2
1 1 2(1 )
(
2.
( ) 1 ( ) 1 1
2
1,
1
1 )
0
T
G j G j
j j j j
if then
Thus, G is not positive real
0.5
2 2
C R
L
Example 2-1:
37. Robust and Optimal Control - A Two-port Framework Approach
1
sC
R
( )
i
V s
sL
( )
o
V s
( )
I s
37
Example 2-2:
Sol:
Fig.1
0.5
0 2
C R
L
(a)
2
( ) 1 1
1. ( )
( ) 1 1
o
i
V s
G s
V s LCs RCs s
1
pole
2 2
1 1 (1- ) (1+ ) 2
2. for all real
1 1- 1
>
1
0
T j j
G j G j
j j
2
2
2
3. lim det
2
2 0
lim , as p=1, q=0
1
p q T
G j G j
Thus, G is strictly positive real
38. Robust and Optimal Control - A Two-port Framework Approach
38
Example 3:
2 1
1 2
1 2
2 1
s
s s
G s
s s
1. 1, 2
pole G s is Hurwitz
2
2 2
2 2
2 1 2 1 2 2 2
1 2 1 2 1 4
2.
1 2 1 2 2 4
2 1 2 1 4 1
T
j j j
j j j j
G j G j
j
j j j j
Sol:
G is strictly positive real
2
2 2
2 2
2 2 2
1 4
2 4
4 1
j
a
a b
b
j
2 2
2 2
2 2 2 2 2 2 2 2
2
2 2
2 2
2 2 2 2
2 2 4 2 2 4
1 4 4 1 1 4 4 1
2 2 4
1 1
0
a
j j j j
a b a b a ab ab b
b
a b
2
3. lim det , s
0 a 2, 1
p q T
G j G j p q
