![Motivation1
[ ]
=
+
−
−
=
2
1
2
1
2
1
01
)(
0
1
10
12
x
x
y
tu
x
x
x
x
1−
s 1−
s 1
1− 2−
u y1x2x
s
x )0(2
s
x )0(1
1 1x2x
1
controllable
uncontrollable](https://image.slidesharecdn.com/k10880deepaknagarcontrolslideshare-160509084812/85/K10880-deepaknagar-control-slide-share-2-320.jpg)
![Motivation2
[ ]
=
+
−
−
=
2
1
2
1
2
1
01
)(
1
3
10
02
x
x
y
tu
x
x
x
x
1−
s 1−
s 1
1− 2−
u y1x2x
s
x )0(2
s
x )0(1
1 1x2x
3
observable
unobservable](https://image.slidesharecdn.com/k10880deepaknagarcontrolslideshare-160509084812/85/K10880-deepaknagar-control-slide-share-3-320.jpg)
![Proof of controllability matrix
[ ]
=−
++++=−
+++++=
++=++=
+=
+=
−+
−+
−
+
−+−++
−−
+
−+−++
−−
+
+++
+++
+
)1(
)2(
1
)1()2(1
21
)1()2(1
21
1
2
12
112
1
)(
nk
nk
k
n
k
n
nk
nknkk
n
k
n
k
n
nk
nknkk
n
k
n
k
n
nk
kkkkkkk
kkk
kkk
u
u
u
BABBAxAx
BuABuBuABuAxAx
BuABuBuABuAxAx
BuABuxABuBuAxAx
BuAxx
BuAxx
Initial condition](https://image.slidesharecdn.com/k10880deepaknagarcontrolslideshare-160509084812/85/K10880-deepaknagar-control-slide-share-5-320.jpg)
![Proof of observability matrix
( )
[ ])1()2()3(11
1
)1()2(1
321
1
111
111
1
)(),2(),1(
)(
)2()(
)1(
−+−+−+++
−
−+−++
−−−
−+
+++
+++
+
−−−−−=
⇒
+++++=
++=++=
+=
+=
+=
nknknkkkkkk
k
n
nknkk
n
k
n
k
n
nk
kkkkkkk
kkk
kkk
kkk
DuCBuCABuDuCBuyDuy
x
CA
CA
C
n
nDuCBuBuCABuCAxCAy
DuCBuCAxDuBuAxCy
DuCxy
DuCxy
BuAxx
Inputs & outputs](https://image.slidesharecdn.com/k10880deepaknagarcontrolslideshare-160509084812/85/K10880-deepaknagar-control-slide-share-6-320.jpg)
![)()()(
)()()(
tDutCxty
tButAxtx
+=
+=
[ ]
=
=
−
−
1
2
12
n
n
CA
CA
CA
C
V
BABAABBU
Controllability matrix
Observability matrix
Then: (1) controllable
(2) observable nVrank
nUrank
=
=
)(
)(](https://image.slidesharecdn.com/k10880deepaknagarcontrolslideshare-160509084812/85/K10880-deepaknagar-control-slide-share-7-320.jpg)
![[ ]
=
+
−
−
=
2
1
2
1
2
1
01
)(
0
1
10
12
x
x
y
tu
x
x
x
x
Example 1
2][
12
01
1][
00
21
][
=
−
=
=
=
−
==
Vrank
CA
C
V
UrankABBU uncontrollable
observable
The rank of a matrix is defined by the number of
linearly independent rows and/or the number of
linearly independent columns](https://image.slidesharecdn.com/k10880deepaknagarcontrolslideshare-160509084812/85/K10880-deepaknagar-control-slide-share-8-320.jpg)
![Example 2
[ ]
=
+
−
−
=
2
1
2
1
2
1
01
)(
1
3
10
02
x
x
y
tu
x
x
x
x
1][
02
01
2][
11
63
][
=
−
=
=
=
−
−
==
Vrank
CA
C
V
UrankABBU controllable
unobservable](https://image.slidesharecdn.com/k10880deepaknagarcontrolslideshare-160509084812/85/K10880-deepaknagar-control-slide-share-9-320.jpg)
![example
[ ] c
cc
xy
uxx
12
1
0
32
10
=
+
−−
=
Controllable canonical form
[ ]
−−
=
=
−
==
12
12
31
10
CA
C
V
ABBU
nVrank
nUrank
≠=
==
1][
2][
[ ] o
oo
xy
uxx
10
1
2
31
20
=
+
−
−
=
Observable canonical form
[ ]
−
=
=
−
−
==
31
10
11
22
CA
C
V
ABBU
nVrank
nUrank
==
≠=
2][
1][
)2)(1(
2
)(
++
+
=
ss
s
sT](https://image.slidesharecdn.com/k10880deepaknagarcontrolslideshare-160509084812/85/K10880-deepaknagar-control-slide-share-11-320.jpg)
![Theorem V
)()()(
)()()(
tDutCxty
tButJxtx
+=
+=
Jordan form
[ ]321
3
2
1
3
2
1
CCCC
B
B
B
B
J
J
J
J
=
=
=
Jordan block
Least row
has no zero
row
First column has no zero column](https://image.slidesharecdn.com/k10880deepaknagarcontrolslideshare-160509084812/85/K10880-deepaknagar-control-slide-share-12-320.jpg)
![Example
[ ]xccy
ub
b
xx
3
1100
020
012
1211
12
11
=
+
=
If 012 =b uncontrollable
If 011 =c unobservable](https://image.slidesharecdn.com/k10880deepaknagarcontrolslideshare-160509084812/85/K10880-deepaknagar-control-slide-share-13-320.jpg)
K10880 deepaknagar control slide share
- 1. CAREER POINT UNIVERSITY MAJOR ASSIGNMENT ON CONTROLABILITY AND OBSERVABILITY SUBMITTED TO MR. SOMESH CHATURVEDY SUBMITTED BY Deepak Nagar KID-K10880 B.Tech. (M.E.) 6th Semester
- 2. Motivation1 [ ] = + − − = 2 1 2 1 2 1 01 )( 0 1 10 12 x x y tu x x x x 1− s 1− s 1 1− 2− u y1x2x s x )0(2 s x )0(1 1 1x2x 1 controllable uncontrollable
- 3. Motivation2 [ ] = + − − = 2 1 2 1 2 1 01 )( 1 3 10 02 x x y tu x x x x 1− s 1− s 1 1− 2− u y1x2x s x )0(2 s x )0(1 1 1x2x 3 observable unobservable
- 4. Definition A linear system is said to be completely controllable if, for all initial times and all initial states , there exists some input function (or sequence for discrete systems) that drives the state vector to any final state at some finite time . 0t )( 0tx )( 1tx 10 tt < A linear system is said to be completely observable if, for all initial times , the state vector can be determined from the output function (or sequence) , defined over a finite time . Definition 0t )( 0tx )( 1ty 10 tt <
- 5. Proof of controllability matrix [ ] =− ++++=− +++++= ++=++= += += −+ −+ − + −+−++ −− + −+−++ −− + +++ +++ + )1( )2( 1 )1()2(1 21 )1()2(1 21 1 2 12 112 1 )( nk nk k n k n nk nknkk n k n k n nk nknkk n k n k n nk kkkkkkk kkk kkk u u u BABBAxAx BuABuBuABuAxAx BuABuBuABuAxAx BuABuxABuBuAxAx BuAxx BuAxx Initial condition
- 6. Proof of observability matrix ( ) [ ])1()2()3(11 1 )1()2(1 321 1 111 111 1 )(),2(),1( )( )2()( )1( −+−+−+++ − −+−++ −−− −+ +++ +++ + −−−−−= ⇒ +++++= ++=++= += += += nknknkkkkkk k n nknkk n k n k n nk kkkkkkk kkk kkk kkk DuCBuCABuDuCBuyDuy x CA CA C n nDuCBuBuCABuCAxCAy DuCBuCAxDuBuAxCy DuCxy DuCxy BuAxx Inputs & outputs
- 7. )()()( )()()( tDutCxty tButAxtx += += [ ] = = − − 1 2 12 n n CA CA CA C V BABAABBU Controllability matrix Observability matrix Then: (1) controllable (2) observable nVrank nUrank = = )( )(
- 8. [ ] = + − − = 2 1 2 1 2 1 01 )( 0 1 10 12 x x y tu x x x x Example 1 2][ 12 01 1][ 00 21 ][ = − = = = − == Vrank CA C V UrankABBU uncontrollable observable The rank of a matrix is defined by the number of linearly independent rows and/or the number of linearly independent columns
- 9. Example 2 [ ] = + − − = 2 1 2 1 2 1 01 )( 1 3 10 02 x x y tu x x x x 1][ 02 01 2][ 11 63 ][ = − = = = − − == Vrank CA C V UrankABBU controllable unobservable
- 10. Theorem III )()()( tuBtxAtx cccc += Controllable canonical form Controllable Theorem IV )()( )()()( txCty tuBtxAtx oo oooo = += Observable canonical form Observable
- 11. example [ ] c cc xy uxx 12 1 0 32 10 = + −− = Controllable canonical form [ ] −− = = − == 12 12 31 10 CA C V ABBU nVrank nUrank ≠= == 1][ 2][ [ ] o oo xy uxx 10 1 2 31 20 = + − − = Observable canonical form [ ] − = = − − == 31 10 11 22 CA C V ABBU nVrank nUrank == ≠= 2][ 1][ )2)(1( 2 )( ++ + = ss s sT
- 12. Theorem V )()()( )()()( tDutCxty tButJxtx += += Jordan form [ ]321 3 2 1 3 2 1 CCCC B B B B J J J J = = = Jordan block Least row has no zero row First column has no zero column
- 13. Example [ ]xccy ub b xx 3 1100 020 012 1211 12 11 = + = If 012 =b uncontrollable If 011 =c unobservable
- 15. { } { } { } { } .... .... 21131211 21131211 ILCILCCC ILbILbbb + + controllable observable In the previous example { } { } { } { } .... .... 21131211 21131211 DLCILCCC ILbILbbb + + controllable unobservable