-
State Observers for Linear SystemsConventional Asymptotic Observers
Observer equation
Any desired spectrum of A+LC can be assignedReduced order observer
Sliding mode State Observer
Mismatch equation
Reduced order Luenberger observer
Sliding mode State Observer
Mismatch equation
Reduced order Luenberger observer
Noise intensity
Adaptive Kalman filter
Kalman filter without adaptation
S.M. filter without adaptation
Variance
Observers for Time-varying Systems
Block-Observable Form
Ai,i+1, y=yo.
. . . . . . .
01A
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:
Disturbances
Estimates ofDisturbances
T
Observer Design
But matrix Fk-1 is not constant
TheExample
The observer is governed by the equations
Obswerver
Remark
Parameter estimation
Lyapunov function (t). ),(ˆˆ
t.independenlinearly are )( of components ,)(, ),(
TT
nT
aytay
ttatay
aaVya
aaV
TTT
T
,
2
1
.02 yV
0)(lim
tyt
consttat
)(lim .0)(lim
tat
???
aaVysigna
aaV
TTT
T
,)(
2
1
Sliding mode estimator
.0 yV
, TT aay Taysigny 2)( finite time convergence to 0y
??? ,)[(22
TT
TT
eq
aa
aysign
Sliiding mode estimator with finite timeconvergence of to zero
. ,1,...,1 ),()( ),()( 01 nkttLttL kk
Linear operator
ik ,0det . ,ˆˆ k Tkk
Tk ayay
),...,( , ,)( 10 nTTT
jTi aaQQsignYY
time.finiteafter 0)( 0det and 0
e,convergenc timefinite definite positive is
tayk
1
0
1
0
, ],)([
2
1
n
kk
TTkk
n
k
T
T
yVaaVysigna
aaV
),,...,( 10 nT yyY
Example of operator
t).determinan d(Vandermon 0det
,...,1 ;1,...,0 , ,
operatordelay id
),,...,()( 1
V
ninkeVVe
L
eet
kt
ttT
ii
n
Application: Linear system with unknown parameters
).ˆ( filter. pass low aby obtained becan mode slidingin
, ),( ),(ˆ
),(
AAAxAv
xyssMsignvtfvxAy
tfAxx
eq
X is known, A can be found, if component of X are linearly independent, as components of vector
kie
DIFFERENTIATORSThe first-order system
+
-f(t)x
u
z
Low pass filter
The second-order system
+-
- + f(t)s
xv u
Second-order sliding mode u is continuous, low-pass filter is not needed.