Linear Systems Analysis in the Time Domain IV
- Transient Response -
2 2 3 3
2 3
2 3 2 2 2
x( ) x(0) ( ) , x A x ( )
1 1
I A A A (*1)
2! 3!
( ) ( )
1 2! 3!
1 1
( ) A A A A I A A
2 2
A x( ) x(0) A x(0) A x( )
At
At
At
at
At
At
At At
t e
e
e t t t
at at
e at
d e t t t t
dt
e
t d e e t
dt
Matrix exponential
0
2 3
2 3
2 2 3 3 2 2 3 3
2 2
2
( )
( )
( )
exp(A ) A
! A
AB BA AB BA
(A B) (A B) I (A B)
2! 3!
A A B B
I A I B
2! 3! 2! 3!
I (A B) A AB
2!
k k
k At
At
At
A B t At Bt A B t At Bt
A B t
At Bt
t t
k d
dt
if if
t t t
t t t t
t t
t t t
e
e e
e e e
e e e
e e e
2 2 3 3 2 3 2 3 3 3
B B A B AB B
2! 3! 2! 2! 3!
t t t t t
2 2 2 2
2 3
( )
2 2
2! 3!
A B t At Bt
BA AB BA ABA B A BAB A B AB
t t
e
e e
(A B t) At Bt
e
e e
The difference between and vanishes, if A and B commute.
Hence,
1
2
3
1
2
3
2 3
1 1
2 3
2 2
3 3
2 2 3 3
1 1 1
(*1) 0
A
0
1 1
2 3!
1 1
1 2 3!
At
t
t
t
e I At t t
t t t
e
e
e
a) Diagonalized Form
1 1
1
2
1 1 1
1 1
1
1
A 1
2
2 1
A 1
1
1 0 0
A 0 0 , 0 0
0 0 0 0
1
1 0 2
A 0 1 , 0
0 0 0 0
t t
t t
t
t t t
t t
t
t
e te
e e
e
e te t e
e e te
e
b) Jordan Form
1
1
11 -1
x A x B i) ( ) det 0 :
A :
1, , ii) :
. iii) ( I A) 0 :
A , A P P
P AP :
i i
i i i
i i i
n n
n
i
u Q I A
n n
n sol i n
n distinct eigenvalues
p p eigen vector
p p
p p p p
n distint eigenvalues
Characteristic equation.
eigenvalue of A
Diagonalizable if c) General A : diagonalize !!
1
1
1T
1, .
A A
A A 0
A A 0
A
P AP
i i i
T
i i i
T
T T
i i i j i i j i
T T
j j j i j j i j
n n
n
p p
if p p p is a normalized eigenvector Orthogonality of eigenvector
if
p p p p p p
p p p p p p
p p p p
1 1 1
( ) 1
0
1 ( ) 1
0
1 ( ) 1
0
ˆ ˆ
Px x, x = Px x A x B
ˆ ˆ
P x APx B
ˆ ˆ ˆ
x P APx P B x P B
ˆ ˆ
x( ) x(0) P B ( )
Px( ) ˆ P P x(0) P P B ( )
x( ) P P x(0) P P B ( )
t
t t
t
t t
t
t t
let
u u
u u
t e e u d
t e e u d
t e e u d
d) General A (-> Jordan form)
repeated : multiple eigenvalue
i
1
2
21 1 1
2 2 2
2 3
1
2 1 1 1
2
2 2 2
det I A A : 3 3
( I A) 0 ( I A) 0
( I A) 2 ?
AP P 1 A ,
A
i
p p
p p
if rank then p
p such that
p p
p p
Find
2
1 2 3
1,2 1,2 1 2
3
1 0 1 1 0 1
A 0 1 0 ( ) det I A 0 1 0
0 0 2 0 0 2
1 2
: 1, 1, 2
: I A 0
0 0 1 1 0
1, 0 0 0 0 0 , 1
0 0 1 0 0
1 0 1 2, 0 1 0 0 0 0
i i i i i
Q
eigenvalue
eigenvector Ap p p
p choose p p
3 31
0 0
1
p p
ex1)
1 2 3
1-1
1 0 1 1 0 1
P 0 1 0 , P 0 1 0
0 0 1 0 0 1
1 0 0 P AP 0 1 0 0 0 2
p p p
1
1 1
1 A
A 1
x A x
ˆ ˆ ˆ
x P APx x
ˆ ˆ ˆ
x( ) x(0), P x x P x( ) P x(0) x( ) P P x(0) x(0)
0 0 0
P 0 0 P 0 0
t t
t t
t t t
t t t
t e t e
t e e
e e e
e e e
2
1,2 3
1,2 1,2 1
3 3 3
1 1 2 1 1 2
A 0 1 3 ( ) det I A 0 1 3
0 0 2 0 0 2
1 2
: 1, 2
: A I A 0
0 1 2 1
1, 0 0 3 0 0
0 0 1 0
1 1 2 5
2, 0 1 3 0
0 0 0
i i i i i
Q
eigenvalue
eigenvector p p p
p p
p p
3 1
We can find only 1 eigenvector for
1
If not able to find 3 independent vectors, find which transforms A as Jordan form.
p
2
1 1
3
1
1 2 3 1 2 3 1
3
1 2 3 1 1 1 1 2 3 3
1 1 1 1 1 1
1 1 1 2 1 2 1 2
1 0 AP = PJ P 0 0
0 0
1 0
A 0 0
0 0
A A A
A I A 0
A A I
p p p p p p
p p p p p p p
p p p p
p p p p p p
So that,
1
2 2 22
1
-1
0 1 2 1
A I = 0 0 3 0 , ,
0 0 1 0
0
2 1, 3 0, 0 1, 1
0
1 0 5 1 0 5
P 0 1 3 , P 0 1 3
0 0 1 0 0 1
1 0 5 1 1 2 P AP 0 1 3 0 1 3 0 0 1 0 0 2
a
p p let p b
c
b c c c b choose p
1 0 5 1 1 0 0 1 3 0 1 0 0 0 1 0 0 2
Jordan form
A
1 1 2
x 0 1 3 x Ax , x( ) x(0) ? 0 0 2
if t e
t
1
J J
2
ˆ ˆ
Px x, x = Px
1 1 0
ˆ ˆ ˆ ˆ
x( ) P APx J x 0 1 0 x 0 0 2 0
ˆ ˆ
0 0 , x( ) x(0)
0 0
t t
t t t
t
let
t
e te
e e t e
e
A J 1
2
2 2 2
2
2 2
1 0 5 0 1 0 5
P P 0 1 3 0 0 0 1 3
0 0 1 0 0 0 0 1
5 1 0 5
0 3 0 1 3
0 0 0 0 1
5 3 5
0 3 3
0 0
t t
t t t
t
t t t
t t
t
t t t t t
t t t
t
e te
e e e
e
e te e
e e
e
e te e te e
e e e
e
1
2
3
1 1
1
2
1 1 1
1 1
1
1 2
3
1 1
2
2 1
1
0 A
0
1 0 0
A 0 0 0 0 ,
0 0 0 0
1
1 0 2
A 0 1 0
0 0 0 0
t
t At
t
t t
t At
t
t t t
t t
At
t
e
e e
e
e te
e e
e
e te t e
e e te
e
2. Jordan Form 1. Diagonal Form
-1
1 1 1
( ) 1
1 ( ) 1
1 A
x A x B A , P AP
x = Px ˆ
ˆ ˆ ˆ
x P APx P B x P B
ˆ ˆ
x( ) x(0) P B ( )
Px( ) ˆ x( ) P P x(0) P P B ( )
P P
i i i
t t
t t
t t
u p p
let
u u
t e e u d
t t e e u d
e e
3. General A
A A( ) 0
0
1 1
0 0
A -1 1
1 1
2 3
2 3
x A x B , ( ) B ( )
X( ) A X( ) B ( ) , X( ) ( I A) ( I A) B ( ) ( I A)
1 1
( I A) I A
1 1 1 1
I A A A
1 1 1 1
t t
t
u x t e x e u d
s s x s u s s s x s u s
e s
s s s
s s s s
L
Laplace Transformation
2 1-1 1 -1 2 3
2 3 4
2 2 3 3
A 2 2 3 3
1 1 1
, , ( ) ( 1) ( )
( 1)!
1 1 1 1
( I A) I A A A
1 1
I A A A
2! 3!
1 1
I A A A
2! 3!
n
n n n
n n
t
t t t f t d F s
s n s ds
s s s s s
t t t
e t t t
L L L
L L
Laplace transformation table
Method 1. Diagonalization
Method 2. Laplace Transformation
Method 3. Sylvester's Interpolation Formula
