sliding mode observers
TRANSCRIPT
Sliding Mode Observers
SOLO HERMELIN
Updated: 15.09.10
1
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
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
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:
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:
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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