-

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.