claudia lizet navarro hernández phd student supervisor: professor s.p.banks april 2004 monash...
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Claudia Lizet Navarro Hernández
PhD StudentSupervisor: Professor S.P.Banks
April 2004Monash University Australia April 2004
The University of Sheffield
Department of Automatic Control and Systems Engineering
1.- Iteration Technique for Nonlinear Systems
2.- Design of Observers for Nonlinear Systems
3.- Fault Detection for Nonlinear Systems
4.- Summary and Conclusions
1“Iteration technique for nonlinear systems and its applications to control theory” Monash University
.)0(,)( 0nxxxxAx
0][][]1[
][
0]2[]2[]1[
]2[
0]1[]1[
0
]1[
)0(),())(()(
)0(),())(()(
)0(),()()(
xxtxtxAtx
xxtxtxAtx
xxtxxAtx
iiii
Having the nonlinear system
where i=number of approximations, it can be shown that the solution of this sequence converges to the solution of the original nonlinear system if the Lipschitz condition is satisfied.
and introducing the sequence of linear time varying equations:
2“Iteration technique for nonlinear systems and its applications to control theory” Monash University
yxyAxA )()(
)(][ tx i
3“Iteration technique for nonlinear systems and its applications to control theory” Monash University
where is the solution of the original nonlinear
system.
-Tomas-Rodriguez, M., Banks, S., (2003) Linear approximations to nonlinear dynamical systems with applications to stability and spectral theory, IMA Journal of Mathematical Control and Information, 20, 89-103.
)()(][ txtxLim ii
is a Cauchy sequence and
)(tx
Solution to Van der Pol oscillator
2
12
1
2
1
01
11
x
xx
x
x
)(
)(
01
11))((
)(
)(][
2
][1
2]1[1
][
2
][
1
tx
txtx
tx
txi
ii
i
i
and for the ith approximation,
Solution and Approximations for the Van der Pol Oscillator
4“Iteration technique for nonlinear systems and its applications to control theory” Monash University
5“Iteration technique for nonlinear systems and its applications to control theory” Monash University
-Optimal Control Theory (Banks & Dinesh, 2000)
-Nonlinear delay systems (Banks, 2002)
-Theory of chaos (Banks & McCaffrey, 1998)
-Stability and spectral theory (Tomas-Rodriguez & Banks, 2003)
-Design of Observers (Navarro Hernandez & Banks, 2003)
1.- To represent a nonlinear system by a sequence of linear time-varying approximations
2.- To design an identity observer for linear time-varying systems
3.- To test the performance of the observer for the nonlinear system
6“Iteration technique for nonlinear systems and its applications to control theory” Monash University
)()( tButAxx
))(ˆ)(()()(ˆ)(ˆ txCtyGtButxAtx
).()(ˆ)( txtxte
0),()()( ttteGCAte
Fig.1.1 State Reconstruction Process (Open-loop)
Lineal invariant system:
Auxiliary dynamical system:
Mismatch
Objective 0)(ˆ)(lim
txtxt
Non-linear system: ))(()()),(()( txhtytxftx
Inaccessiblesystem state
x
SYSTEM OBSERVER Reconstructed state
00)(),()( xtxtCxty
7
x̂
“Iteration technique for nonlinear systems and its applications to control theory” Monash University
00)(),()()(
)()()()()(
xtxtxtCty
tutBtxtAtx
PROBLEM:
Find state estimator )()()()()(ˆ)(ˆ tutHtytGtxFtx
Design proposed: “Design of a State Estimator for a Class of Time-Varying Multivariable Systems” (NGUYEN and LEE)
Steps in design:
1.- Canonical transformation of the time-varying system
2.- Construction of a full order dynamical system
8“Iteration technique for nonlinear systems and its applications to control theory” Monash University
EXAMPLE:
)(5.00
010)(
)(
0
0
1
)(
30
022
0
)(3
2
txe
ty
tutx
e
ee
tx
t
t
tt
1.- Canonical transformation
a) Construction of the observability matrix
5.425.3
2442
5.15.2
022
5.00
010
)(
233
2
3
tttt
tt
ttt
t
eeee
ee
eee
e
tN
b) Check for uniform observability
c) Construction of an (n x n) matrix with rank n by eliminating the linearly dependent rows
d) Construction of a transformation matrix
9“Iteration technique for nonlinear systems and its applications to control theory” Monash University
e) Transformation of the original system into an equivalent system
)()(
)()(
txCty
tuBtxAx
)(10
001)(
)(
0
2
0
)(
305
400
012
)(3
2
txe
ty
tutx
e
e
e
tx
t
t
t
t
2.- Construction of asymptotic estimator
a) Choice of n stable eigenvalues for the state estimator
b) Design of matrix such that is a constant matrix.
)(tG )()()( tCtGtAF
c) Construction of the state estimator of form:
)()()()()(ˆ)()(ˆ tutBtytGtxtFtx
d) Calculation of the estimate using the transformation matrix )(ˆ)()(ˆ txtPtx
)(ˆ
200
001
05.05.
)(ˆ
2
tx
e
tx
t
10“Iteration technique for nonlinear systems and its applications to control theory” Monash University
)())(()(
)0(,))(()())(()(][]1[][
0][][]1[][]1[
][
txtxCty
xxutxBtxtxAtxiii
iiiiii
0][][]1[][]1[][
][
)0(ˆ,)()()()(ˆ)(ˆ xxuxBtyxGtxFtx iiiiiii
0][][]1[][ )0(),(ˆ))(()(ˆ xxtxtxPtx iiii
)()()(
)0(),()()()()( 0
txxCty
xxtuxBtxxAtx
Given a nonlinear system
1. Reduction to a sequence of linear time varying approximations
2. Design of observer at each time varying approximation
3. Test of observer at final approximation
Proof as 0)(ˆlim)(lim ][][
txtx i
i
i
it
11“Iteration technique for nonlinear systems and its applications to control theory” Monash University
EXAMPLES
)(022
106)(
)(
0
0
1
)(
62111
100
010
txty
tutx
xx
x
Fig. 1 State X1 and Estimate
a)
12“Iteration technique for nonlinear systems and its applications to control theory” Monash University
Fig. 2 State X2 and Estimate
Fig. 4 Error of Estimates
Fig. 3 State X3 and Estimate
13“Iteration technique for nonlinear systems and its applications to control theory” Monash University
)(034
17)(
)(
0
0
1
)(
971
102.
5.25.0
2
32
1
3
txx
ty
tutx
xxx
x
x
x
b)
Fig. 5 State X1 and Estimate
Fig. 6 State X2 and Estimate
14“Iteration technique for nonlinear systems and its applications to control theory” Monash University
Fig. 7 State X3 and Estimate
Fig. 8 Error of Estimates
15“Iteration technique for nonlinear systems and its applications to control theory” Monash University
1.- To represent a nonlinear system by a sequence of linear time-varying approximations
2.- To design an unknown input observer for linear time-varying systems
3.- To apply the iteration technique to solve the nonlinear problem and test performance.
16“Iteration technique for nonlinear systems and its applications to control theory” Monash University
1111 fKdEBuAxx
Lineal invariant system:
Auxiliary dynamical system:
Non-linear system: ))(()()),(()( txhtytxftx
2222)( fKdECxty
17“Iteration technique for nonlinear systems and its applications to control theory” Monash University
Process
Unknown Input Observer
Measurements y
u
f d
r
Objective 01 f 002 rfand01 f or 002 rf
)ˆ(ˆˆ xCyLBuxAx
)ˆ( xCyHr
18“Iteration technique for nonlinear systems and its applications to control theory” Monash University
00
2211
)(),()()()(
)()()()()()()()()()()(
xtxtvtxtCty
ttFttFtwtBtutBtxtAtx wu
PROBLEM:
Find linear observer ))(ˆ)()()(()()()(ˆ)()(ˆ txtCtytLtutBtxtAtx u
and residual ))(ˆ)()()(()( txtCtytHtr
Where u control input v noise
y measurement faults ( = target fault)
w process noise faults directions
i
iF
Such that is primarly affected by the target fault and minimally by noises
and nuissance faults
1
)(tr
)())(()(
)0(),())(()())(()())(()())(()(][]1[][
0][][
2]1[
2][
1]1[
1][]1[][]1[
][
txtxCty
xxttxFttxFtutxBtxtxAtxiii
iiiiiiiiii
)()()(
)0(),()()()()()()()()( 02211
txxCty
xxtxFtxFtuxBtxxAtx
Given a nonlinear system
1. Reduction to a sequence of linear time varying approximations
2. Design of observer at each time varying approximation
3. Test of observer at final approximation in the presence of different target and nuissance faults
19“Iteration technique for nonlinear systems and its applications to control theory” Monash University
))(ˆ))(()())((()())(()(ˆ))(()(ˆ ][]1[][]1[][]1[][]1[][
txtxCtytxLtutxBtxtxAtx iiiiiiu
iii
0
][][]1[][]1[][ ˆ)0(ˆ)),(ˆ))(()())((()( xxtxtxCtytxHtr iiiiii
20“Iteration technique for nonlinear systems and its applications to control theory” Monash University
- New method to study nonlinear systems by using known linear techniques
- Nonlinear system replaced by a sequence of linear time-varying problems
- The linear time-varying problem must have a solution