MODERN CONTROL THEORY - SJTUMODERN CONTROL THEORY Lecturer：鲍其莲Bao Qilian 1 Chapter 6 Controllability and Observability Objectives: •Definition of controllability •Definition
Post on 19-Mar-2021
11 Views
Preview:
Transcript
MODERN CONTROL THEORYLecturer：鲍其莲 Bao Qilian
1
Chapter 6 Controllability and
Observability
Objectives:
• Definition of controllability
• Definition of observability
• Detective methods for controllability
• Detective methods for observability
• Decomposition
2Chapter 6
6.1 controllability
• Definition:
• The state equation or the pair (A,B) is said to
be controllable if for any initial state x(0)=x0 and any final
state x1, there exists an input that transfer x0 to x1 in a
finite time. Otherwise is said to be uncontrollable.
3
BuAxx
Chapter 6
Example: uncontrollable network.
4
xCxy
uxbuAxx
u, xux cc
10
1
1
21
12
Let 2211
tttt
tttt
t
33
33
eeee
eeee
2
1eA
0
0)0(xIf then ττutx
tτt d)(e
1
1)(
0
)(
21 xx
No matter what the input is, the states are always equivalent.
So the states of the system are uncontrollable.
Chapter 6
• Theorem: The following statements are equivalent.
1. The n-dimensional pair (A,B) is controllable.
2. The following nn matrix is non-singular for any t.
3. The following nnp controllability matrix (Kalman matrix)
has rank n (full row rank).
5
dτeBBedτeBBe(t)W τ)(tAT)A(t-τt
τATAτt
C
TT
00
]1BAB AB C
n-[
Chapter 6
A system is controllable if the following nnp controllability matrix
has rank n (full row rank).
][ BABAABBQ1n2
C
nCQrank
Proof: Caley-Hamilton theorem
1
0
1
110 )()()()(n
i
i
i
n-
n
τ τaτaτaτae AAAIA
τττaxt
i
n
i
i d)()()0(1
0
1
0
uBA
i
ir
i
i
t
i
β
β
β
τττa
2
1
0d)()(
1
u )1,,1,0( ni
Chapter 6
7
Therefore
1
1
0
1 ][)0(
n
n-
β
β
β
BAABBx
n ]rank[rank BABAABBQ1n2
C
The solution of exists according to initial state x(0) only
if following matrix has rank n.
Chapter 6
8
Example: Determine the controllability of following systems
u
9
0
2
10
5
07
xx
u
57
04
10
10
5
07
xx
（1）
（2）
Solution:
(1) Uncontrollable.
(2) Controllable.
Chapter 6
• Example: Determine the controllability of following
systems
• (1)
• (2)
9
u
3
4
0
200
040
014
xx
u
03
00
24
200
040
014
xx
• Solution:
• (1) Controllable.
• (2) Uncontrollable.
Chapter 6
10
Theorem: If all the eigenvalues of a system are different, then the
system can be transformed to the following form using similarity
transformation. The system is controllable if there is no all zero
row in .
uBxx
nλ
λ
λ
0
0
2
1
B
Chapter 6
Other conditions of controllability
11
Theorem: If a system can be transformed to the following Jordan
matrix form using similarity transformation. It is controllable if
each row in matrix corresponding to the last rows of each
Jordan block is not all zero row.B
Chapter 6
uBx
J
J
J
x
k0
0
2
1
i
i
i
i
λ
λ
λ
0
1
01
J
12
Example: Determine the controllability of following systems
using alternative method.
u
9
0
2
10
5
07
xx
u
57
04
10
10
5
07
xx
（1）
（2）
Solution:
(1) Uncontrollable.
(2) Controllable.
Chapter 6
13
Example:Determine the controllability of following systems.
u
3
4
0
200
040
014
xx
u
03
00
24
200
040
014
xx
（1）
（2）
Solution:
（1）Controllable
（2）Uncontrollable
Chapter 6
6.2 Observability
• Definition:
• A linear state system or pair (A,C) is said to be observable
if for any unknown initial state x(0), there exist a finite t1>0
such that the acknowledge of input u and output y over
[0, t1] suffices to determine uniquely the initial state x(0).
Otherwise the system is said to be unobservable.
14Chapter 6
Example: Unobservable network
15
xCxy
ux-
-BuAxx
11
0
1
21
12
tttt
tttt
t
33
33
eeee
eeee
2
1eA
ττtut τtt
t d)(e)0(e)( )(
0
bxx AA
0)( tu
• If then
)0(e)( xxAtt
tt xxty 3
21 e)]0()0([)0(e)( xCA
• The value of output only determined by the difference of
initial states. Therefore, the states are unobservable.
Chapter 6
• Theorem: The following statements are equivalent.
1. The n-dimensional pair (A,C) is observable.
2. The following nn matrix is nonsingular for any t>0.
3. The nqn observability matrix has rank n ( full column
rank)
16
tAτTτA
C dτCeCe(t)WT
0
nnm
n
1CA
CA
C
QO
Chapter 6
A system is observable if the nqn observability matrix has rank n
( full column rank)
nnm
n
1CA
CA
C
QO
nOQrank
Proof: if then0)( tu
)0(e)()( xCCxyAttt
)0(e)( xxAtt
1
0
)(en
i
i
i
τ τa AA )0()()(
1
0
xACy
n
i
i
i τat
)0()()()()(
1
110 x
CA
CA
C
y
n
n tatatat
known
Condition: rank is n
unknown
Chapter 6
• Example: Is the following state equation observable?
18
u
2
1
50
02xx
x10y
Solution:
nrankrank
21
50
10
CA
C
Unobservable.
Chapter 6
Example: Determine the observability of following systems
（1） xx
10
5
07
x540y
（2） xx
10
5
07
x
130
023y
Solution:
（1）Unobservable.
（2）Observable.
Chapter 6
Other conditions of observability
20
Theorem: If all the eigenvalues of a system are different, then the
system can be transformed to the following form using similarity
transformation. The system is observable if there is no all zero
column in .
uBxx
nλ
λ
λ
0
0
2
1
Chapter 6
C
xCy
Other conditions of controllability
21
Theorem: If a system can be transformed to the following Jordan
matrix form using similarity transformation. It is controllable if
each column in matrix corresponding to the first column of
each Jordan block is not all zero column.
Chapter 6
uBx
J
J
J
x
k0
0
2
1
i
i
i
i
λ
λ
λ
0
1
01
J
xCy
C
22
Example: Are the following systems observable?
（1） xx
10
5
07
x540y
（2） xx
10
5
07
x
130
023y
Solution:
(1)unobservable.
(2)Observable
Chapter 6
23
Example: is the state equation observable?
xx
20000
12000
00300
00130
00013
-
-
xy
00110
01111
Using two different ways to determine the observability.
(Observable)
Chapter 6
24Chapter 6
Example : Is the following state equation controllable?
duβββy n x110
u
aaa n
1
0
0
0
100
10
010
110
xx
25Chapter 6
duβββy n x110
Theorem: If a system (A, B) is controllable, it can be
tranformed into canonical controllable form as follows:
u
aaa n
1
0
0
0
100
10
010
110
xx
01
1
1
1
1
2
121
1
n
n
n
a
a
aaa
BAABBP
01
1
1]det[ aλaλaλAλI n
n
n
26
Example: Find the controllable form for the following
system
u
1
1
0
001
010
101
xx x011y
（1）
Solution:
101
111
110
][ 2bAAbbQC
3rank CQ Controllable
（2） 12]det[ 23 λλλ AI
Chapter 6
27
（3）Compute matrix P
121
111
011
001
01
1
][][ 2
21
2
321 a
aa
bAAbbppp
213
112
111
121
111
011
][
1
1
321 pppP
（4） 102
121
111
011
0111
CPC
（5）0 1 0 0
x 0 0 1 x 0
-1 0 2 1
u
2 0 1y x
Chapter 6
28Chapter 6
Example : Is the following state equation observable?
u
β
β
β
a
a
a
nn
1
1
0
1
1
0
100
0
10
01
0
xx
x100 y
29Chapter 6
Theorem: If a system(A,C) is observable, it can be tranformed
into canonical observable form as follows:
u
β
β
β
a
a
a
nn
1
1
0
1
1
0
100
0
10
01
0
xx
x100 y
11
2
121
01
1
1
1
nn
n
a
a
aaa
CA
CA
C
P
01
1
1]det[ aλaλaλAλI n
n
n
30
u
1
1
0
001
010
101
xx
x011y
Chapter 6
Example: Find the observable form for the following
system
6.3 Canonical decomposition
System decomposition
• (1) controllable subsystem and uncontrollable subsystem
• (2) observable subsystem and unobservable subsystem
• (3) Transfer function
31Chapter 6
If not all the states of a linear system are controllable, then
the system can be decomposed into two subsystem, one is
controllable, and another one is un-controllable.
Assume:
32
Cxy
BuAxx
12
0 0
[ ]
CC C C
CC C
C
C C
C
xx A A Bu
xx A
xy C C
x
12
1
C C C CC
C C
x A x A x B u
y C x
Transform
Controllable states
Transfer function
xPx C
nnBAABBQ 1
2
C ]rank[rank
Then
Transfer function of the system is the same as controllable subsystem
Chapter 6
11 1 1 1 1
1 1
C
=
C C
G(s) C sI A B CP sI A P B C sP IP P AP B
C sI A B C sI A B
33
11 1 1 1 1
1
-1
12
12
C
=
0 0
1
( )( )
010
0
C C
C C
C
C C C C
C C
C
C C CCC
G(s) C sI A B CP sI A P B C sP IP P AP B
C sI A B
sI A A BC C
sI A
A
sI A sI A sI A BC C
sI A
B
C C C sI AsI A
1
CB
Transfer function
Transfer function of the system is the same as controllable subsystem
34
C
C
CC
C
C
C
C
C
C
C
x
x]CC[y
uB
x
x
A
AA
x
x
1
12
00
controllable
uncontrollable
Chapter 6
• Theorem: ( Chosen of transformation matrix)
• Consider the n-dimensional state equation with
• We form the nn matrix
• Where the first n1 columns are any n1 independent
columns of QC and remaining columns can arbitrarily be
chosen as long as P is nonsingular.
• Then the system will be decomposed by the following
equivalence transformation .
35
nnBAABBQ 1
2
C ]rank[rank
][: nn qqqP 11
1
Pxx
Chapter 6
36
Example: decompose the following system:
u
1
1
21
12xx x10y
Solution
(1)
(2) Chose independent column vector
2111
11rankrankrank
nAbbQC
1
11p
1
02p
11
01
11
011
1
21 ppPC
1 CC APPA BPB C1 CCPC
C
C
C
C
C
C
x
xy
ux
x
x
x
11
0
1
30
11
Chapter 6
37
uB
B
x
x
AA
A
x
x
O
O
O
O
O
O
O
O
21
0
O
O
Ox
xCy ]0[
uBxAx OOOO
OO xCy
Transfer functionOOO BAICBAICG
11 )()()( sss
If not all the states of a linear system are observable, then the
system can be decomposed into two subsystem by equivalence
transformation, one is observable, and another one is
unobservable. The transfer function will be the same as the
controllable subsystem.
Assume:2
O
C
rankQ rank CA n
CA
Cxy
BuAxx Transform
xPx o
Observable subsystem
Chapter 6
• Theorem: ( Chosen of transformation matrix)
• Consider the n-dimensional
• state equation with
• We form the nn matrix
• Where the first n2 rows are any n2 independent rows of QO
and remaining rows can arbitrarily be chosen as long as P
is nonsingular.
• Then the system will be decomposed by the following
equivalence transformation .
38
n
n
p
p
p
P
2
1
1 :
Pxx
nn
n
2
1
rankrank
CA
CA
C
QO
39
O
O
Ox
xCy ]0[
uB
B
x
x
AA
A
x
x
O
O
O
O
O
O
O
O
21
0
observable
unobservable
Chapter 6
40
Example:
u
1
0
0
342
100
010
xx x011y
Solution32
242
110
011
CA
CA
C
Q2
O
nrankrankrank
100
110
011
OP
100
110
0111-
OP
1 OO APPA
BPB O1 OCPC
u
1
1
0
122
022
010
O
O
O
O
x
x
x
x
OC
O
O
x
xy ]001[
}
100
110
011
OP
An independent row
Chapter 6
• For both uncontrollable and unobservable system, it can
be decomposed into
• Transfer function
• The same as the controllable and observable subsystem.
41
uB
B
x
x
x
x
AA
A
AAAA
AA
x
x
x
x
OC
CO
OC
OC
OC
CO
OC
OC
OC
CO
OC
OC
OC
CO
0
0
00
000
00
43
242321
13
OC
OC
OC
CO
OCCO
x
x
x
x
CC 00y
COCOCO BAICBAICG11 )()()( sss
leunobservab and ableuncontroll :
observablebut abeluncontroll:
leunobservabbut lecontrollab:
observable and elcontrollab:
OC
OC
OC
CO
x
x
x
x
Chapter 6
现代控制理论 42Chapter 6
43
Kalman decomposition
Chapter 6
6.4 Discrete-time state equation
• The state model of discrete-time systems
• or
44
)k(Cx)k(y
)k(Bu)k(Ax)k(x
1
)k(Du)k(Cx)k(y
)k(Bu)k(Ax)k(x
1
Chapter 6
1.Controllability of discrete-time system
• Definition:
• The state equation or the pair (A,B)
is said to be controllable if for any initial state x(0)=x0 and
any final state x1, there exists an input sequence of finite
length that transfer x0 to x1. Otherwise is said to be
uncontrollable.
45
)k(Bu)k(Ax)k(x 1
Chapter 6
• Theorem: The following statements are equivalent.
1. The n-dimensional pair (A,B) is controllable.
2. The following nn matrix is non-singular.
3. The following nnp controllability matrix has rank n (full
row rank).
46
1
0
1n
m
mTTm
dc )(ABBA)n(W
]1BAB AB C
n-[
Chapter 6
47
现代控制理论 47
Example: Is the state equation controllable?
)(
1
0
1
)(
011
220
001
)1( kukk
xx
Sol.
nrankBAABBrankrank
3
111
620
111
Q 2
C
Controllable
Chapter 6
2.Observability of discrete-time system
• Definition:
• A discrete time state equation is said to be observable if
for any unknown initial state x(0), there exist a finite integer
k1>0 such that the acknowledge of input sequence u(k)
and output y(k) from k=0 to k1 suffices to determine
uniquely the initial state x(0). Otherwise the system is said
to be unobservable.
48Chapter 6
• Theorem: The following statements are equivalent.
1. The n-dimensional pair (A,C) is observable.
2. The following nn matrix is nonsingular.
3. The nqn observability matrix has rank n ( full column
rank)
49
1
0
1n
m
mTmT
do ACC)A()(nW
nnm
n
1CA
CA
C
QO
Chapter 6
现代控制理论
Example : Is the following state equation observable?
)(
1
0
1
)(
011
220
001
)1( kukk
xx )(111)( kky x
nrank
CA
CA
C
rankrankQO
3
642
230
111
2
Solution:
Observable
Chapter 6
MATLAB
Homework:
• Learn and practice functions related to :
1. Controllability matrix
2. Observability matrix
51
Summary
• Concepts of controllability and observability
• Rules to determine the controllability and observability of linear
system
• Decomposition of LTI system
• Controllability and observability of discrete-time linear system
52Chapter 6
top related