-
State Observers for Linear Systems
Conventional Asymptotic Observers
Observer equation
Any desired spectrum of
Reduced order observer
A+LC can be assigned
Sliding mode State Observer
Mismatch equation
Reduced order Luenberger
observer
Sliding mode State Observer
Mismatch equation
Reduced order Luenberger
observer
Variance
Kalman filter without adaptation
S.M. filter without adaptation
Adaptive Kalman filter
Noise intensity
Observers for Time-varying Systems
Block-Observable Form
A 01
Ai,i+1, y=yo.
. . . . . . .
Time-varying Systems with disturbances
The last equation with respect to yr depends on disturbance
vector
f(t), then vr,eq is equal to the disturbance.
Simulation results:
T
Disturba
nces
Estimates of
Disturbances
Observer Design
But matrix Fk-1 is not constant
Example
The
Obswerver
The observer is governed by the equations
Remark
Parameter estimation
y  a  ( t ),
a ,  ( t )   , components
T
yˆ  aˆ  ( t ),
y  a  (t).
T
n
of  ( t ) are linearly
independen
T
Lyapunov function
V 
1
T
a a
2
T
T
a    y  ,
T
V  a a
2

V   y  0.
a ( t )  const  lim a ( t )  0 .   
lim y ( t )  0 lim
t 
t 
???
t 
Sliding mode estimator
V 
1
V    y  0 .
T
a a
2
T
T
a    ( sign y ) ,
T
T
y  a   a  ,
T
V  a a
y    ( sign y ) 
2
T
 a  finite time convergence to y  0
T
T

a 
a
 T
T  
  [( sign y ) eq  
,
a

2
2


???
t.
Sliiding mode estimator with finite time
convergence of to zero
Linear operator
L  ( t )   1 ( t ),
1
k
   ik  yˆ  aˆ T  ,
k
k
det   0 ,
V 
L  ( t )   k ( t ), k  1,..., n  1,    0 .
yk  a  k .
T
T
a a
2
n 1
a
T
    [sign ( y k ) ],
T
k
k 0
Y
T
n 1
V  a a , V     y k
T
k 0
 ( y 0 ,..., y n 1 ),

T
Y     ( signY )  Q ,    i 
 is positive
definite
j
,
Q
T
 finite time
T
T
 ( a  0 ,..., a  n 1 )
convergenc
e,
y k  0 and det   0  a ( t )  0 after finite time.

Example of operator
 (t )  ( e
T
1 t
,..., e
nt
),
L id delay operator


it
i ikk

 e
V , V  ee
, k  0 ,..., n  1; i  1,..., n
det V  0 (Vandermon
d determinan
t).
Application: Linear system with unknown parameters
x  Ax  f ( t ),
y  Aˆ x  v  f ( t ), v   Msign ( s ), s  y  x ,
in sliding
mode
v eq  A x can be obtained
by a low pass filter. ( A  Aˆ  A ).
X is known, A can be found, if component of X are linearly
independent, as components of vector 
DIFFERENTIATORS
The first-order
system
+
f(t)
z
x
u
-
Low pass
filter
The second-order
system
v+
u
-
x -
+
s
f(t)
Second-order sliding
mode
u is continuous, low-pass filter is not needed.