linear system by meiling chen1 lesson 6 state transition matrix linear system 1. analysis
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![Page 1: Linear system by Meiling CHEN1 Lesson 6 State transition matrix Linear system 1. Analysis](https://reader035.vdocuments.us/reader035/viewer/2022062308/56649d5f5503460f94a40632/html5/thumbnails/1.jpg)
linear system by Meiling CHEN 1
Lesson 6
State transition matrix
Linear system1. Analysis
![Page 2: Linear system by Meiling CHEN1 Lesson 6 State transition matrix Linear system 1. Analysis](https://reader035.vdocuments.us/reader035/viewer/2022062308/56649d5f5503460f94a40632/html5/thumbnails/2.jpg)
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) :
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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
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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
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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
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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
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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
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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
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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 )(
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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
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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
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linear system by Meiling CHEN 12
Diagonization
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linear system by Meiling CHEN 13
Diagonization
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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
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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
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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
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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
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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
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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
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linear system by Meiling CHEN 20
Method 3:
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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
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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
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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
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linear system by Meiling CHEN 24
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linear system by Meiling CHEN 25
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linear system by Meiling CHEN 26
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linear system by Meiling CHEN 27
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linear system by Meiling CHEN 28
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linear system by Meiling CHEN 29