![Page 1: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/1.jpg)
![Page 2: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/2.jpg)
-
![Page 3: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/3.jpg)
![Page 4: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/4.jpg)
State Observers for Linear SystemsConventional Asymptotic Observers
Observer equation
Any desired spectrum of A+LC can be assignedReduced order observer
![Page 5: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/5.jpg)
Sliding mode State Observer
Mismatch equation
Reduced order Luenberger observer
![Page 6: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/6.jpg)
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
![Page 7: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/7.jpg)
Observers for Time-varying Systems
Block-Observable Form
Ai,i+1, y=yo.
. . . . . . .
01A
![Page 8: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/8.jpg)
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
![Page 9: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/9.jpg)
![Page 10: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/10.jpg)
![Page 11: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/11.jpg)
![Page 12: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/12.jpg)
![Page 13: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/13.jpg)
Observer Design
![Page 14: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/14.jpg)
But matrix Fk-1 is not constant
![Page 15: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/15.jpg)
TheExample
The observer is governed by the equations
Obswerver
![Page 16: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/16.jpg)
Remark
![Page 17: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/17.jpg)
![Page 18: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/18.jpg)
![Page 19: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/19.jpg)
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
![Page 20: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/20.jpg)
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
![Page 21: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/21.jpg)
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
![Page 22: - State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649ec65503460f94bd11a7/html5/thumbnails/22.jpg)
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.