Post on 21-Dec-2015
linear system by Meiling CHEN 1
Lesson 6
State transition matrix
Linear system1. Analysis
linear system by Meiling CHEN 2
)()()(
)()()(
tDutCxty
tButAxtxdt
d
1. Homogeneous solution of x(t) 2. Non-homogeneous solution of x(t)
The behavior of x(t) et y(t) :
linear system by Meiling CHEN 3
Homogeneous solution
)0()()(
)()0()(
)()(
1 xAsIsX
sAXxssX
tAxtx
)0(
)0(])[()( 11
xe
xAsILtxAt
])[()( 11
AsILet At
State transition matrix
)()()()()(
)()0(
)0()(
000)(
0
0
0
00
0
0
txtttxetxeetx
txex
xetx
ttAAtAt
At
At
linear system by Meiling CHEN 4
Properties
)()(.5
)()()(.4
)()()0(.3
)()(.2
)0(.1
020112
1
ktt
tttttt
txtx
tt
I
k
])[()( 11
AsILet At
linear system by Meiling CHEN 5
Non-homogeneous solution
)()()(
)()()(
tDutCxty
tButAxtxdt
d
tdButxttx
sBUAsILxAsILtx
sBUAsIxAsIsX
sBUxsXAsI
sBUsAXxssX
0
1111
11
)()()0()()(
)]()[()0(])[()(
)()()0()()(
)()0()()(
)()()0()(
Convolution
Homogeneous
linear system by Meiling CHEN 6
)()()()()()(
)()()()()(
)()()0()()(
0
0
00
00
0
tDudButCtxttCty
dButtxtttx
dButxttx
t
t
t
t
t
Zero-input response Zero-state response
linear system by Meiling CHEN 7
Example 1
Txlet
tux
x
x
x
00)0(
)(1
0
32
10
2
1
2
1
tttt
tttAt
eeee
eeeeeAsILt
22
21211
222
2])[()(
t
dButxttx0
)()()0()()(
tt
tt
ee
eex
x
2
2
2
1
222
32
2
1
Ans: )]()[( 11 sBUAsIL
linear system by Meiling CHEN 8
Txlet
tux
x
x
x
00)0(
)(1
0
32
10
2
1
2
1
1s 1s1 1
32
u y1x2x
s
x )0(2
s
x )0(1
Using Maison’s gain formula
)()0()0(2
)(
)()0()0()31(
)(
231
1
2
1
1
2
2
2
2
2
1
11
1
21
sUs
xs
xs
sx
sUs
xs
xss
sx
ss
linear system by Meiling CHEN 9
How to find
])[()( 11
AsILet At
State transition matrix
Methode 1: ])[()( 11 AsILt
Methode 3: Cayley-Hamilton Theorem
Methode 2: Atet )(
linear system by Meiling CHEN 10
Methode 1: ])[()( 11 AsILt
3
2
1
2
1
2
1
3
2
1
3
2
1
1
0
0
0
0
1
)(
)(
10
01
00
211
340
010
x
x
x
ty
ty
u
u
x
x
x
x
x
x
ssss
ss
sss
ssss
AsI
AsIadjAsI
414
323
32116
33)2)(4(
1
)()(
2
2
2
1
linear system by Meiling CHEN 11
Methode 2: Atet )(
3
2
1
2
1
2
1
3
2
1
3
2
1
166)(
)(
1
1
1
300
020
001
x
x
x
ty
ty
u
u
x
x
x
x
x
x
t
t
t
At
e
e
e
et3
2
00
00
00
)(
diagonal matrix
linear system by Meiling CHEN 12
Diagonization
linear system by Meiling CHEN 13
Diagonization
linear system by Meiling CHEN 14
Case 1: distincti
)1)(3(43
1
43
10
A
1
3
2
1
3
10
433
13)(
2
1
2
111 v
v
v
vVAI
1
10
33
11)(
2
1
2
122 v
v
v
vVAI
depend
10
03
13
11 121 APPVVP
linear system by Meiling CHEN 15
n 321
In the case of A matrix is phase-variable form and
112
11
2121
111
nn
nn
nnvvvP
Vandermonde matrix
for phase-variable form
4
3
2
1
1
APP
1 PPee tAt
linear system by Meiling CHEN 16
Case 1: distincti
)2)(1)(1(
200
010
101
200
010
101
AIA
0
100
000
100
)(
3
2
1
11
v
v
v
VAI21
depend
0
1
0
000
0
0
1
000
3
2
1
321
3
2
1
321
v
v
v
vvv
v
v
v
vvv
21 VV
linear system by Meiling CHEN 17
0
000
010
101
)(
3
2
1
33
v
v
v
VAI23
1
0
1
00
3
2
1
321
v
v
v
vvv
200
010
001
100
010
1011
321 APPVVVP
linear system by Meiling CHEN 18
Case 3: distincti Jordan form
321
formJordanAPPvvvP 1321
Generalized eigenvectors
231
121
11
)(
)(
0)(
vvAI
vvAI
vAI
1
1
11 1
1ˆ
AAPP
t
tt
tttt
tA
e
tee
etee
e1
11
12
11
2ˆ
linear system by Meiling CHEN 19
Example:
2)2(11
13
11
13
A
1
10
11
11)(
12
11
12
1111 v
v
v
vVAI
0
1
1
1
11
11)(
22
21
22
2121 v
v
v
vVAI
20
12ˆ01
11 121 AAPPVVP
1ˆ
2
22ˆ
PPee
e
teee tAAt
t
tttA
linear system by Meiling CHEN 20
Method 3:
linear system by Meiling CHEN 21
AaAaIaAaAaa
AaAaAaA
IaAaAaA
IaAaAaA
nnn
nn
n
nn
n
nn
n
02
1011
11
02
111
011
1
011
1
)(
0
nn AkAkAkIkAf 2
210)(any
1
0
11
2210)(
n
k
kk
nn
A
AAAIAf
linear system by Meiling CHEN 22
10
21?100 AAExample:
AIAAflet 10100)(
2,1,0)2)(1(20
2121
100210
10022
100110
10011
2)(
1)(
f
f
12
22100
1
1000
10
221
10
21)12(
10
01)22()(
101100100100AAf
linear system by Meiling CHEN 23
02
13? AeAtExample:
2,1,02
1321
2)2(
)1(
102102
10110
t
t
ef
eftt
tt
ee
ee
2
1
20 2
tttt
tttt
ttttAt
eeee
eeee
eeeee
222
2
02
13)(
10
012
22
22
22
linear system by Meiling CHEN 24
linear system by Meiling CHEN 25
linear system by Meiling CHEN 26
linear system by Meiling CHEN 27
linear system by Meiling CHEN 28
linear system by Meiling CHEN 29