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Page 1: Sliding Mode Observers

Sliding Mode Observers

SOLO HERMELIN

Updated: 15.09.10

1

Page 2: Sliding Mode Observers

Table of Content

SOLO Sliding Mode Observers

2

Sliding Mode Observers

Sliding Mode Observer for a Linear Time Invariant (LTI) System

Generic Observer for a Linear Time Invariant (LTI) System

Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System

Sliding Mode Observers of Target Acceleration

References

Page 3: Sliding Mode Observers

SOLO

Sliding Mode Observers

In most of the Linear and Nonlinear Unknown Input Observers proposed so far, the necessary and sufficient conditions for the construction of Observers is that the Invariant Zeros of the System must lie in the open Left Half Complex Plane, and the transfer Function Matrix between Unknown Inputs and Measurable Outputs satisfies the Observer Matching Condition.

Observers are Dynamical Systems that are used to Estimate the State of a Plant using its Input-Output Measurements; they were first proposed by Luenberger.

David G. LuenbergerProfessor

Management Science and Engineering

Stanford University

In some cases, the inputs of the System are unknown or partially unknown, which led to the development of the so-called Unknown Input Observers (UIO), first for Linear Systems. Motivated by the development of Sliding-Mode Controllers, Sliding Mode UIOs have been developed.

The main advantage of using Sliding-Mode Observer over their Linear counterparts is that while in Sliding, they are Insensitive to the Unknown Inputs and, moreover, they can be used to Reconstruct Unknown Inputs which can be a combination of System Disturbances, Faults or Nonlinearities.

3

Sliding Mode Observers

Page 4: Sliding Mode Observers

SOLO

4

Diagram of the LTI Systemand a Sliding Mode Observer

pxnnxmnxmnxn

pmmn

CBBA

yuux

xCy

uBuBxAx

RRRR

RRRR

,,,

,,,21

21

21

21

2211

Assumptions:

222

222 ,,

mBCrankBrank

pmpCrankmBrank

1

2

3

tu 2

Invariant Zero of the triple (A,B2,C) are in the Open Left-Hand Complex Plane, or equivalently

0Real0 2

2

2

2

smn

C

BAIsrank

pxmpxn

nxmnxnn

Sliding Mode Observer for a Linear Time Invariant (LTI) System

Observer:

2,

ˆˆ

,ˆ,ˆˆˆ 211

mnxp EL

xCy

yyEBuByyLxAx

RR

will stabilize the Observer on the Sliding SurfacenxpL R 0ˆ yyF

xpmF

yyFif

yyFifyyF

yyF

yyE

2,0

0ˆ0

0ˆˆ

ˆ

,ˆ,

RR

Reaching the Sliding Surfaceusing:

0ˆ yyF

2211 uBuBxAx

EBuByyLxAx 211ˆˆˆ

,,ˆ yyE

L

C

C

1u

2ux

y

x y

x

Sliding Mode Observers

LTI System:

Page 5: Sliding Mode Observers

L is obtain by choosing a S.P.D. matrix Q and solving for the S.P.D. matrix P defined by the Lyapunov Algebraic Equation (A.R.E.)

SOLO

5

2211 uBuBxAx

EBuByyLxAx 211ˆˆˆ

,,ˆ yyE

L

C

C

1u

2ux

y

x y

x

pxnnxmnxmnxn

pmmn

CBBA

yuux

xCy

uBuBxAx

RRRR

RRRR

,,,

,,,21

21

21

21

2211

Sliding Mode Observer for a Linear Time Invariant (LTI) System

Observer:

2,

ˆˆ

,ˆ,ˆˆˆ 211

mnxp EL

xCy

yyEBuByyLxAx

RR

will stabilize the Observer on the Sliding SurfacenxpL R 0ˆ yyF

xpmF

yyFif

yyFifyyF

yyF

yyE 2,0

0ˆ0

0ˆˆ

ˆ

,ˆ, RR

Reaching the Sliding Surfaceusing:

0ˆ yyF

On the Sliding Surface 11ˆˆˆ uByLxCLAx

02 QCLAPPCLA T

F is obtain by solving

Sliding Mode Observers

122

TTTT CCCPBFPBCF

Diagram of the LTI Systemand a Sliding Mode Observer

L.T.I. System:

Page 6: Sliding Mode Observers

SOLO

Generic Observer for a Linear Time Invariant (LTI) System

pxmpxnnxmnxn

pmn

DCBA

yux

uDxCy

uBxAx

RRRR

RRR

,,,

,,

Observer

sxpsxmsxq

qxpqxmqxq

sxnsq

RSM

GJF

Lwz

xLwyRuSzMw

yGuJzFz

RRR

RRR

RRR

,,

,,

,,

A Necessary Condition for obtaining an Observer is that (A,C) is Observable.

The Observer will achieve if and only ifxLw

0DRS

LTGCR

DGBTJ

CGTFAT

valueseigenstablehasF

where Tnxn is a Transformation Matrix.

L.T.I. System

uBxAx

uJyGzFz

J

R

u x y

x

C

GM

S

System

Observer

D

s

1

s

1 z

A

F

Sliding Mode Observers

Page 7: Sliding Mode Observers

SOLO

uBxAx

uJyKzFz

J

R

u x y

w

C

KG

S

System

Observer

Generic Observer for a Linear Time Invariant (LTI) System

pxnnxmnxn

pmn

CBA

yux

xCy

uBxAx

RRR

RRR

,,

,,

Observer

sxpsxmsxq

qxpqxmqxq

sxnsq

RSG

KJF

Lwz

xLwyRuSzGw

yKuJzFz

RRR

RRR

RRR

,,

,,

,,

Taking Laplace Transforms we obtain sUBAIsCsYsUBAIssX nn

11

sUBAIsCRSBAIsCJFIsGsW

sUBAIsCJFIssZ

nnq

nq

111

11

L.T.I. System

Sliding Mode Observers

Page 8: Sliding Mode Observers

SOLO

dEuBxAx

uBTyKzFz

BT

u x yC

K

H

System

Observerx

z

dDisturbance

Knowninput

Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System

nxqpxnnxmnxn

qpmn

ECBA

dyux

xCy

dEuBxAx

RRRR

RRRR

,,,

,,,

Observer

DesignedbetoMatricesHKTF

xz

yHzx

yKuBTzFz

nxpnxpnxnnxn

nn

RRRR

RR

,,,

ˆ,

ˆ

dECHIuBCHIxACHIyKxCKuBTzFxCHIze nnn

KKK

n

21

21

The Estimator Error xCHIzxxe n ˆ:

An Unknown Input Observer for an LTI System will derive its State Error

regardless of the unknown input (disturbance)

0ˆ:asymptotic

xxe

td

dECHIuBCHITyHCKACHAKzCKACHAFeCKACHAe nn 1211

eyHzx Substitute in this equation:

L.T.I. System

Sliding Mode Observers

Page 9: Sliding Mode Observers

SOLO

dEuBxAx

uBTyKzFz

BT

u x yC

K

H

System

Observerx

z

dDisturbance

Knowninput

Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System

nxqpxnnxmnxn

qpmn

ECBA

dyux

xCy

dEuBxAx

RRRR

RRRR

,,,

,,,

Observer

An Unknown Input Observer for an LTI System will derive its State Error

regardless of the unknown input (disturbance)

0ˆ:asymptotic

xxe

td

dECHIuBCHITyHCKACHAKzCKACHAFeCKACHAe nn 1211

We can see that if we can make the following relations:

HFK

CKACHAF

CHIT

ECHI

n

n

2

1

0

the State-Estimator Error will be: eFe

We can see that the Observer Error will be zero asymptotically iff all the eigenvalues of F are stable.

L.T.I. System

Sliding Mode Observers

Page 10: Sliding Mode Observers

SOLO

dEuBxAx

uBTyKzFz

BT

u x yC

K

H

System

Observerx

z

dDisturbance

Knowninput

Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System

Observer

Lemma 1: This Equation is solvable if:

HFK

CKACHAF

CHIT

ECHI

n

n

2

1

0

but:

and a special solution for H is: †1ECEECECECEH TT

nxqspnxp

Proof of Lemma 1:

nxqpxnnxpnxq ECHE nxqpxnnxq ECrankErank

nxqnxqpxnnxqpxn ErankErankCrankECrank ,min

qErankECrank nxqnxqpxn

Necesity

Sufficiency When rank (CE) = rank (E), (CE) is a full column rank matrix, because E is assumed a full column rank matrix, and a left inverse of (CE) exists.

TT ECECECEC1†

and: †ECEH spnxp

q.e.d.

L.T.I. System

qErankECrank nxqnxqpxn Observer Matching Condition

Sliding Mode Observers

Page 11: Sliding Mode Observers

SOLO

dEuBxAx

uBTyKzFz

BT

u x yC

K

H

System

Observerx

z

dDisturbance

Knowninput

Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System

Observer

Lemma 2: Let:

If s1 ϵ C is an unobservable mode of the pair (C1,A) then:

AC

CC1

Then the Detectability for the pair (C,A) is equivalent to that of (C1,A). (A pair (C,A) is Detectable when all the unobservable modes of this pair are Stable).Proof of Lemma 2:

n

AC

C

AIs

rankC

AIsrank

nn

1

1

1

That means that exists a vector α ϵ Cn such that:

nC

AIsrank

C

AIs

AC

C

AIsnn

n

11

1

00

s1 is also an unobservable mode of the pair (C,A).

If s2 ϵ C is an unobservable mode of the pair (C,A) then: nC

AIsrank n

2

That means that always exists a vector β ϵ Cn such that: 02

C

AIs n

s2 is also an unobservable mode of the pair (C1,A).

00

0

0

1

11

222

C

AIs

AC

C

AIs

CssCACC

AIs nn

n

q.e.d.

L.T.I. System

Sliding Mode Observers

Page 12: Sliding Mode Observers

SOLO

dEuBxAx

uBTyKzFz

BT

u x yC

K

H

System

Observerx

z

dDisturbance

Knowninput

Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System

xCy

dEuBxAx

U.I.O. Observer

yHzx

yKuBTzFz

ˆ

Lemma 3: Necessary and Sufficient Conditions to have an U.I.O. Observer for the L.T.I. System are:(1)rank (C E) = rank (E)(2)(C, A1) is a Detectable pair, where ACECECECEAACECEAA TT 1†

1 :

The condition (2) is equivalent to the condition that the Invariant Zeros for the Unknown Input, i.e., of the triplet (A,E,C) must be stable:

qpnsqnC

EAIsrank

pxqpxn

nxqnxnn

C0

Proof of Lemma 3 (Sufficiency):

According to Lemma 1 if rank (C E)= rank (E) exists a solution for H:

†1ECEECECECEH TT

nxqspnxp

and: CKACKACECECECEACKACHAF TTpxnnxnpxn

spnxpnxnnxn nxp 1111

We can see that F may be Stabilized by choosing a proper K1, only if the pair (C, A1) is Detectable.

L.T.I. System

Sliding Mode Observers

Page 13: Sliding Mode Observers

SOLO

dEuBxAx

uBTyKzFz

BT

u x yC

K

H

System

Observerx

z

dDisturbance

Knowninput

Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System

xCy

dEuBxAxL.T.I. System

U.I.O. Observer

yHzx

yKuBTzFz

ˆ

Lemma 3: Necessary and Sufficient Conditions to have an U.I.O. Observer for the L.T.I. System are:(1)rank (C E) = rank (E)(2)(C, A1) is a Detectable pair, where ACECECECEAACECEAA TT 1†

1 :

Proof of Lemma 3 (Necessity):

HFK

CKACHAF

CHIT

ECHI

n

n

2

1

0

A General Solution for H is

†0† ECECIHECEH mnxp

where is an arbitrary matrix and nxmH R0 TT ECECECEC1†

Since the Observer is a U.I.O. Observer for the L.T.I. System we can solve for H, T, K1, F and K2

111

1011011

1

1

CKAAC

CHKA

ACECECI

CHKACECEICKACHAF

C

K

Tm

Tn

Since the Matrix F is Stable the pair is Detectable, therefore the pair (C, A1) is also detectable, according to Lemma 2.

11, AC

q.e.d.

Sliding Mode Observers

Page 14: Sliding Mode Observers

SOLO

dEuBxAx

uBTyKzFz

BT

u x yC

K

H

System

Observerx

z

dDisturbance

Knowninput

Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System

xCy

dEuBxAxL.T.I. System

U.I.O. Observer

yHzx

yKuBTzFz

ˆ

Lemma 3: Necessary and Sufficient Conditions to have an U.I.O. Observer for the L.T.I. System are:(1)rank (C E) = rank (E)(2)(C, A1) is a Detectable pair, where ACECECECEAACECEAA TT 1†

1 :

Proof of Lemma 3 (Necessity) (continue):

q.e.d.

Csqn

C

EAIs

ECEsCECE

I

ECEsCECEI

rankC

EAIsrank

pxqpxn

nxqnxnn

p

n

pxqpxn

nxqnxnn

00

0††

††

CsErank

CAECE

C

AIs

rank

ECAECE

C

ACECEAIs

rankC

EAIsrank

q

n

nn

pxqpxn

nxqnxnnnxn

1

0

0

0

The condition that the pair (C, A1) is detectable, is equivalent totherefore equivalent to the Invariant Zeros of the triplet (A,E,C) being stable

Csn

C

AIsrank nxnn 1

The Condition that the Invariant Zeros of the triplet (A,E,C) are stable is:

Sliding Mode Observers

Page 15: Sliding Mode Observers

SOLO

dEuBxAx

uBTyKzFz

BT

u x yC

K

H

System

Observerx

z

dDisturbance

Knowninput

Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System

xCy

dEuBxAxL.T.I. System

U.I.O. ObserverDesign Procedure

yHzx

yKuBTzFz

ˆ

1 Check if rank (CE)=rank(E). If rank (CE)≠rank(E) go to . 10

2 Compute: ATACHTECECECEH pxnnxpnxnTT

nxqspnxp

1

1,,

3 If (C1, A) Observable a U.I.O. exists, and K1 can be computed using Pole Placements or any other Method. Go to 9

Tn

T pp1

,,1

4 Construct a Transformation Matrix P by choosing n1=rank (WO) (whereWO=[C, CA1,…,CA1

n-1]) row vectors , together othe n-n1 row vectors to construct the nonsingular

Tn

Tn pp ,,11

Tn

Tn

Tn

T ppppP ,,11 11

5 Perform 0*0 1

2221

1111 CPC

AA

APAP

6 Check Detectability of (C,A1). If one of eigenvalues of A22 is unstable, a U.I.O. doesn’t exist and go to

10

7 Select n1 eigenvalues and assign them to using Pole Placement. *111 CKA p

8 Compute where is any (n-n1)xn matrix. TT

p

T

pp KKPKPK 21111 2

pK

9 Compute HFKKKKCKAF 12111 ,

10 Stop

Sliding Mode Observers

Page 16: Sliding Mode Observers

16

Sliding Mode Observers of Target Acceleration

Kinematics:

tataRRtd

dMT 11

We want to Observe (Estimate) the Unknown Target Acceleration Component:

taT 1

Define:0:1_ vtaestAt

EstT

mEstEst AvRztd

d 00

The Differential Equation of the Observer will be a copy of the kinematics:

mM Ata

:1

Define the Observer Error: EstEstO Rz 0:

Define the Sliding Mode Observers that must drive σO→0:

22

11

1232

201

2/1

0121

1

3/2

10

1.1

5.1

2

vz

vz

vzsigntv

zvzsignvztv

zsigntv OO

Missile Command Acceleration

estAtdRsignRRsignRRNa

EstEstRSM

EstEstEstEstEstEstEstEstEstEstC _'

2

3/1

2

2/1

1

t1, t2, t3 are Design Parameters

Observer 4: Variation of 1

22

11

122

201

2/1

012/1

1

1

3/23/10

1.1

5.1

2

vz

vz

vzsignLv

zvzsignvzLv

zsignLv OO

Observer 1:

02

11

3/1

21

1

2/1

10

z

vz

signv

zsignv

OOO

OOO

02

11

21

1

2/1

10

z

vz

signv

zsignv

OO

OOO

L is a Design Parameter are Design Parameters21, OO are Design Parameters21, OO

Observer 2: Observer 3:

Sliding Mode Observers

Page 17: Sliding Mode Observers

17

Sliding Mode Observer of Target Acceleration: MATLAB Listing

% Nonlinear Sliding Mode Target Acceleration ObserversAt_est=0;v0=0;z0=x1;z1=0;z2=0;Observer=1;%First Observer ParameterL=10;%Second Observer ParametersalphaO1=30;alphaO2=1;%Third Observer Parametersrho1=20;rho2=3;%Fourth Observer Parameterst1=10;t2=3;t3=1;

%Second Order Sliding Mode SigmaSM=Range_est*Lamdadot_est; y2 = alpha1*sign(SigmaSM)*abs(SigmaSM)^0.5+x2; x2_dot =alpha2*sign(SigmaSM)*abs(SigmaSM)^(1/3); %Nonlinear Sliding Mode Target Acceleration Observers z0_dot=v0-Rdot_est*Lamdadot_est-Am; SigmaO=z0-SigmaSM; if(Observer==1) v0=-2*L^(1/3)*abs(SigmaO)^(2/3)*sign(SigmaO)+z1; v1=-1.5*L^(1/2)*abs(z1-v0)^(1/2)*sign(z1-v0)+z2; v2=1.1*L*sign(z2-v1); z1_dot=v1; z2_dot=v2; v0_dot=0; At_est=v0; end if(Observer==2) v0=-alphaO1*abs(SigmaO)^(1/2)*sign(SigmaO)+z1; v1=-alphaO2*abs(SigmaO)^(1/3)*sign(SigmaO); z1_dot=v1; z2_dot=0; v0_dot=0; At_est=v0; end if (Observer==3) v0=-rho1*abs(SigmaO)^(1/2)*sign(SigmaO)+z1; v1=-rho2*sign(SigmaO); z1_dot=v1; z2_dot=0; v0_dot=0; At_est=v0; end if(Observer==4) v0=-2*t1*abs(SigmaO)^(2/3)*sign(SigmaO)+z1; v1=-1.5*t2*abs(z1-v0)^0.5*sign(z1-v0)+z2; v2=1.1*t3*sign(z2-v1); z1_dot=v1; z2_dot=v2; v0_dot=0; At_est=v0; end %Missile Acceleration Command and Autopilot Ac=-N*Rdot_est*Lamdadot_est+y2+At_est;

N = 3;alpha1 =10;alpha2 = 1;

%Nonlinear Sliding Mode Target Acceleration % Observer State Integration z0=z0+z0_dot* delta_time; z1=z1+z1_dot* delta_time; z2=z2+z2_dot* delta_time; v0=v0+v0_dot* delta_time;

22

11

1232

201

2/1

0121

1

3/2

10

1.1

5.1

2

vz

vz

vzsigntv

zvzsignvztv

zsigntv OO

22

11

122

201

2/1

012/1

1

1

3/23/10

1.1

5.1

2

vz

vz

vzsignLv

zvzsignvzLv

zsignLv OO

02

11

3/1

21

1

2/1

10

z

vz

signv

zsignv

OOO

OOO

02

11

21

1

2/1

10

z

vz

signv

zsignv

OO

OOO

Sliding Mode Observers

Page 18: Sliding Mode Observers

18

22

11

1232

201

2/1

0121

1

3/2

10

1.1

5.1

2

vz

vz

vzsigntv

zvzsignvztv

zsigntv OO

t1, t2, t3 are Design Parameters

Observer 4: Variation of 1

22

11

122

201

2/1

012/1

1

1

3/23/10

1.1

5.1

2

vz

vz

vzsignLv

zvzsignvzLv

zsignLv OO

Observer 1:

02

11

3/1

21

1

2/1

10

z

vz

signv

zsignv

OOO

OOO

02

11

21

1

2/1

10

z

vz

signv

zsignv

OO

OOO

L is a Design Parameter are Design Parameters21, OO are Design Parameters21, OO

Observer 2: Observer 3:

Scenario: R0=10000 m, Rdot=-1000 m/s, alpha1=10, alpha2=1, N=3, Ldot0=0.05 rad/sA step pulse Target acceleration At=100 m/s2 starting at t=3 s and finishing at t=7sWith Not Noise

10L 1,30 21 OO 3,20 21 OO 1,3,10 321 ttt

0 1 2 3 4 5 6 7 8 9 10-200

0

200

Atest

0 1 2 3 4 5 6 7 8 9 10-1000

0

1000

z0

0 1 2 3 4 5 6 7 8 9 10-100

0

100

z1

0 1 2 3 4 5 6 7 8 9 10-10

0

10

SigmaO

0 1 2 3 4 5 6 7 8 9 10-200

0

200

Atest

0 1 2 3 4 5 6 7 8 9 10-1000

0

1000

z0

0 1 2 3 4 5 6 7 8 9 10-20

0

20

z1

0 1 2 3 4 5 6 7 8 9 10-50

0

50

SigmaO

0 1 2 3 4 5 6 7 8 9 10-200

0

200

Atest

0 1 2 3 4 5 6 7 8 9 10-1000

0

1000

z0

0 1 2 3 4 5 6 7 8 9 10-10

0

10

z1

0 1 2 3 4 5 6 7 8 9 10-20

0

20

SigmaO

0 1 2 3 4 5 6 7 8 9 10-100

0

100

Atest

0 1 2 3 4 5 6 7 8 9 10-1000

0

1000

z0

0 1 2 3 4 5 6 7 8 9 10-200

0

200

z1

0 1 2 3 4 5 6 7 8 9 10-200

0

200

SigmaO

Sliding Mode Observer of Target acceleration - MATLAB Results

Sliding Mode Observers

Page 19: Sliding Mode Observers

19

22

11

1232

201

2/1

0121

1

3/2

10

1.1

5.1

2

vz

vz

vzsigntv

zvzsignvztv

zsigntv OO

t1, t2, t3 are Design Parameters

Observer 4: Variation of 1

22

11

122

201

2/1

012/1

1

1

3/23/10

1.1

5.1

2

vz

vz

vzsignLv

zvzsignvzLv

zsignLv OO

Observer 1:

02

11

3/1

21

1

2/1

10

z

vz

signv

zsignv

OOO

OOO

02

11

21

1

2/1

10

z

vz

signv

zsignv

OO

OOO

L is a Design Parameter are Design Parameters21, OO are Design Parameters21, OO

Observer 2: Observer 3:

0 1 2 3 4 5 6 7 8 9 10-200

0

200

Atest

0 1 2 3 4 5 6 7 8 9 10-1000

0

1000

z0

0 1 2 3 4 5 6 7 8 9 10-100

0

100

z1

0 1 2 3 4 5 6 7 8 9 10-100

0

100

SigmaO

0 1 2 3 4 5 6 7 8 9 10-200

0

200

Atest

0 1 2 3 4 5 6 7 8 9 10-1000

0

1000

z0

0 1 2 3 4 5 6 7 8 9 10-10

0

10

z1

0 1 2 3 4 5 6 7 8 9 10-20

0

20

SigmaO

0 1 2 3 4 5 6 7 8 9 10-200

0

200

Atest

0 1 2 3 4 5 6 7 8 9 10-1000

0

1000

z0

0 1 2 3 4 5 6 7 8 9 10-20

0

20

z1

0 1 2 3 4 5 6 7 8 9 10-50

0

50

SigmaO

0 1 2 3 4 5 6 7 8 9 10-200

0

200

Atest

0 1 2 3 4 5 6 7 8 9 10-1000

0

1000

z0

0 1 2 3 4 5 6 7 8 9 10-100

0

100

z1

0 1 2 3 4 5 6 7 8 9 10-20

0

20

SigmaO

Scenario: R0=10000 m, Rdot=-1000 m/s, alpha1=10, alpha2=1, N=3, Ldot0=0.05 rad/sA step pulse Target acceleration At=100 m/s2 starting at t=3 s and finishing at t=7sWith Lamda_dot Noise Filtered with Time Constant of 200msec

10L 1,30 21 OO 3,20 21 OO 1,3,10 321 ttt

Sliding Mode Observer of Target acceleration - MATLAB Results

Sliding Mode Observers

Page 20: Sliding Mode Observers

20

Sliding Mode Observer of Target acceleration - MATLAB Results

Scenario: R0=1000 m, Rdot=-1000 m/s, alpha1=30, alpha2=1, N=3, Ldot0=0.05 rad/sNo Target acceleration , No Measurement Noises

Observer Output0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-505

Atest

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1000

100

z0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-101

z1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.10

0.1

SigmaO

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-50

0

50

100

X1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-400

-200

0

200

X1 dot

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

X2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-500

0

500

Am

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

Lamdadot

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.5

0

0.5

Lamdadot2

Sliding Mode Observers

Page 21: Sliding Mode Observers

21

Sliding Mode Observer of Target acceleration - MATLAB ResultsScenario: R0=1000 m, Rdot=-1000 m/s, alpha1=30, alpha2=1, N=3, Ldot0=0.05 rad/sA step pulse Target acceleration At=100 m/s2 starting at t=0.3 s and finishing at t=0.6sWithout Noise

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-50

0

50

100

X1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-400

-200

0

200

X1 dot

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

X2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-500

0

500

Am

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

Lamda

dot

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

0

5

Lamda

dot2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1000

100

Atest

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1000

100

z0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20020

z1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10010

SigmaO

Observer Output

Sliding Mode Observers

Page 22: Sliding Mode Observers

22Return to Table of Content

Sliding Mode Observer of Target acceleration - MATLAB Results

Scenario: R0=1000 m, Rdot=-1000 m/s, alpha1=30, alpha2=1, N=3, Ldot0=0.05 rad/sA step pulse Target acceleration At=100 m/s2 starting at t=0.3 s and finishing at t=0.6sWith Lamda_dot Noise Filtered with Time Constant of 20msec

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-50

0

50

100

X1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-400

-200

0

200

X1 dot

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

X2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-500

0

500

Am

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

Lamda dot

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

0

5

Lamda dot2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2000

200

Atest

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1000

100

z0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20020

z1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10010

SigmaO

ObserverOutput 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.1

0

0.1

Lamdadot

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

0

5

Lamdadot2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.05

0

0.05NoiseLamdadot

Lamda_dotNoise

Sliding Mode Observers

Page 23: Sliding Mode Observers

References

SOLO

O’Reilly, J., “Observers for Linear Systems”, Academic Press, 1983

23

Chen, J., Patton, R., J., “Robust Model-Based Fault Diagnosis for Dynamic Systems”, Kluwer Academic Publishers, 1999

Sliding Mode Observers

Page 24: Sliding Mode Observers

24

SOLO

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 –2013

Stanford University1983 – 1986 PhD AA