Control and Observability Analysis of Linear Systems
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
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
![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](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)