Lecture 10-3. Observability and state Estimation
- State variable feedback plays a key role in the state space approach to the linear feedback control system
- However, often we do not have direct access to all the state variables
- They must be computed from inputs and measurable outputs
The solution
( 1) ( ) ( ), (0)
0(*1)
( ) ( ) ( )
n, ( )
m, ( )
px k Gx k Hu k x x
y k Cx k x k R u k R y k R
+ = + =
= ∈ ∈ =
Consider the discrete-time system
free response forced-response
1 1 0
0
( ) ( )
k
k k j
j
x k G x G Hu j
− − −
=
= + ∑
The question is how and when the state can be determined from the output measurements.
If x0 is found, we can determine x(k) based on the sol of the state equation
1 1 0
0
( ) ( ) ( )
k
k k j
j
y k Cx k CG x C G Hu j
− − −
=
= = + ∑
( ) 0,..., 1 [0, 1] :
( ) :
u j j k k known inputs
y j measured outputs
= − −
Definition (observability)
: A system described byis said to be observable if every initial state x0 can be exactly determined from the output measurements, y(j), in finite time steps, 0 ≤ j ≤ k
( 1) ( ) ( ( ) ), (0)
0( ) ( )
x k Gx k Hu k x x
y k Cx k
+ = + =
=
Once x0 is determined, x(k) can be computed
(0) (0)
(0)
( 1) ( 2)
k k
y u
O x T
y k u k
= +
− −
1 02 0
2
1 2
0
(0) (0)
(1) (1) (0)
(2) (2) (0) (1)
( 1) (0) ( )
k
k k j
j
y Cx
y Cx CG x CHu
y Cx CG x CGHu CHu
y k CG x C G Hu j
− − − −
=
=
= = +
= = + +
− = + ∑
1 2 3
0 0
0 0
(0) (0)
(0)
( 1) ( 1)
k k
k k k
k k
C
CG CH
O T
CG CG H CG H CH
y u
O x T
y k u k
− − −
= =
= −
− −
RHS is known, x(0) is to be determined x(0) can be uniquely determined
if and only if
if
by Cayley-Hamilton theorem
is linear combination of (for any k)
Hence, k ≥ n , where observability matrix
( ) {0}
( )
( )
k
k n
k
rank O n N O
R O R
=
=
=
(0) ( k) 0, ( )j 0, 0,1, , 1
x =N O and u= y = k = k −
0 1
, , n G G −
G
k( ) ( )
( ) ( )
k k
rank O rank O N O N O
=
= 1
n
n
C O O CG
CG −
= =
: unobservable subspace System is called observable if i.e.
( ) N O
( ) {0}
N O =
( )
Rank O = n
Theorem (observability)
:The discrete time system is observable if and only if
( ) rank O = n
Where, is the observability matrix defined by
O
1 n
C O CG
CG
−
=
Proof
For a given arbitrary initial state, x0 and u(i)=0
0 0
1
(0) (1)
( 1)
ny C y CG
y x Ox
y n CG
−
= = =
−
If rank O = n, then there exist n independent row vectors mr1,mr2,…,mrn of O and from the above equation we can have
1 1
0 0
r r
r
m rm
y m
x O x
y m
= =
Or
Or is nonsigular and Or-1exists. Thus x0 can be uniquely determined from yr1,…,yrm
Continuous-time Observability
Continuous-time system(without sensor noise and disturbance)
,
x = Ax + Bu y = Cx + Bu
The question : How and when the state can be determine from the output measurements?
Let’s look at derivatives of y :
2
( 1) 1 2 3 ( 2) ( 1)
2 3
( 1) 1
...
0 ...
, 0
...
n n n n n n
n n
n n
y Cx Du
y Cx Du CAx CBu Du y CA x CABu CBu Du
y CA x CA Bu CA Bu CBu Du
y C D
y CA CB D
hence x
CA B CA B CBD
y CA
− − − − − −
− −
− −
= +
= + = + +
= + + +
= + + + + +
= +
( 1)
( 1) ( 1)
n
n n
u u
u y Ox Tu
−
− −
= +
Hence, if , we can determine x(t) from derivatives of u(t) and y(t) up to order n-1 . i.e., if , i.e.,
then the system is observable
{ }
( k) 0 N O = Rewriting
known
(n 1) (n 1)
Ox = y
−− Tu
−{ }
( k) 0
N O = rank O( )= n
(
( 1) ( 1))
:
n n
x F y Tu
F left inverse of O
− −
= −
Definition
(Observability) C-T systems :
If every x0 can be determined by measuring output y in a “finite” time, the system is “completely observable” or “observable”
( ) rank O = n
Theorem : Continuous system is observable if and only if
Observability-Controllability duality
Let be dual of system
i.e.,
Original system
Dual system : transpose all matrices Swap inputs and outputs
Reverse directions of signal flow arrows
( , , , A B C D ) ( , , , A B C D )
,
x = Ax + Bu y = Cx + Du
Observability-Controllability duality
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
, , ,
T T
T T
T T T T
z t A z t C v t Az t Bu t w t B z t D v t Cz t Dv t
A A B C C B D D
= + = +
= + = +
= = = =
Controllability matrix of dual system
Similary,
Thus, system is observable (controllable) if and only if dual system is controllable (observable)
In fact,
i.e., unobservable subspace is orthogonal complement of controllable subspace of dual system
Observability-Controllability duality
1
1
[ ]
[ ( ) ]
n
T T T T n T
T
C B AB A B
C A C A C
O
−
−
=
=
=
C
O = C
T( ) (
T) ( )
N O = range O
ㅗ= range C
ㅗEnd of 10-3
