The University of Sheffield
Department of Automatic Control and Systems
Engineering
Claudia Lizet Navarro Hernández
PhD Student
Supervisor: Professor S.P.Banks
April 2004
Monash University
Australia
April 2004
1.- Iteration Technique for Nonlinear Systems
2.- Design of Observers for Nonlinear Systems
3.- Fault Detection for Nonlinear Systems
4.- Summary and Conclusions
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
1
Having the nonlinear system

x  A( x) x, x(0)  x 0  n .
and introducing the sequence of linear time varying equations:
 [1]
x (t )  A( x 0) x[1] (t ), x[1] (0)  x 0
 [ 2]
x (t )  A( x[1] (t )) x[ 2] (t ), x[ 2] (0)  x0

 [i ]
x (t )  A( x[i 1] (t )) x[ i ] (t ), x[ i ] (0)  x 0
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 A( x)  A( y)   x  y is satisfied.
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
2
x
[i ]

(t )
is a Cauchy sequence and


Limi  x[i ] (t )  x(t )
where
x(t )
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, 89103.
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
3
Solution to Van der Pol oscillator
and for the ith approximation,
     x 2  1 1  x 
 x1    1
 1 

   1
0  x 2 

x
 2
  [i ] 
[i ]
 x1 (t )    ( x1[i 1] (t )) 2  1 1  x1 (t ) 
 [i ] 
  [i ]   
1
0  x 2 (t ) 
 x (t )  
 2

Solution and
Approximations for the
Van der Pol Oscillator
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
4
-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)
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
5
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
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
6
OBSERVER
SYSTEM
Inaccessible
system state x
Reconstructed
state x̂
Fig.1.1 State Reconstruction Process (Open-loop)
Objective
lim x(t )  xˆ (t )  0
t 

Lineal invariant system: x  Ax(t )  Bu (t ) y (t )  Cx(t ), x(t 0)  x 0

Auxiliary dynamical system:
xˆ (t )  Axˆ (t )  Bu (t )  G( y(t )  Cxˆ (t ))

Mismatch

e(t )  xˆ (t )  x(t ).
e(t )  ( A  GC)e(t ), t  t 0

Non-linear system:
x(t )  f ( x(t )), y(t )  h( x(t ))
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
7
PROBLEM:

x(t )  A(t ) x(t )  B(t )u (t )
y (t )  C (t ) x(t ), x(t 0)  x 0
Find state estimator

xˆ (t )  Fxˆ (t )  G(t ) y(t )  H (t )u (t )
Design proposed: “Design of a State Estimator for a Class of TimeVarying Multivariable Systems” (NGUYEN and LEE)
Steps in design:
1.- Canonical transformation of the time-varying system
2.- Construction of a full order dynamical system
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
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EXAMPLE:
e 2t

x(t )   2
 0

0

2
e  3t
0 1
y (t )  
t
0 e
e t 
1

0  x(t )  0u (t )
0
3 
0
x(t )
0.5
1.- Canonical transformation
a) Construction of the observability matrix
b) Check for uniform observability
1
 0
 0
e t

 2
2
N (t )  
t
e t  .5e 3t
 2e
2e  2t  4
4

 3t
t
e  .5e 3t
 3e

0.5 

0

1.5 
2e t 

2e  2t  4.5
0
c) Construction of an (n x n) matrix with rank n by eliminating the
linearly dependent rows
d) Construction of a transformation matrix
e) Transformation of the original system into an equivalent system

x  A x (t )  B u (t )
y (t )  C x (t )
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
9
e 2t  2 1 0 
0 


x (t )   0
0 4e t  x (t )  2u (t )
 5e 3t
0
0 3 

 1 0 0
y (t )   t
x(t )
0 1
e

2.- Construction of asymptotic estimator
a) Choice of n stable eigenvalues for the state estimator
b) Design of matrix G (t ) such that F  A (t )  G (t )C (t ) is a
constant matrix.
c) Construction of the state estimator of form:

xˆ (t )  F (t ) xˆ (t )  G (t ) y(t )  B (t )u(t )
d) Calculation of the estimate using the transformation
.5e 2t 0.5 0
matrix xˆ (t )  P(t ) xˆ (t )

xˆ (t )   1
 0

0
0

0 xˆ (t )
2
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
10

Given a nonlinear system
x(t )  A( x) x(t )  B( x)u (t ), x(0)  x 0
y (t )  C ( x) x(t )
1. Reduction to a sequence of linear time varying approximations
 [i ]
x (t )  A( x[ i 1] (t )) x[i ] (t )  B( x[ i 1] (t ))u [ i ] , x[ i ] (0)  x 0
y [i ] (t )  C ( x[ i 1] (t )) x[ i ] (t )
2. Design of observer at each time varying approximation
 [i ]
xˆ (t )  Fxˆ [i ] (t )  G( x[i 1] ) y[i ] (t )  B( x[i 1] )u[i ] , xˆ [i ] (0)  x0
xˆ[i ] (t )  P( x[i 1] (t )) xˆ [i ] (t ), x[i ] (0)  x0
3. Test of observer at final approximation
Proof
lim x[i ] (t )  lim xˆ [i ] (t )  0
i 
i 
as t  
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
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EXAMPLES
a)
1
0
 0
1


x 0
0
1  x(t )  0u (t )
 x1  11  x 2  6
0
6 0 1 
y (t )  
 x(t )
2
2
0



Fig. 2 State X2 and Estimate
Fig. 1 State X1 and Estimate
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
12
Fig. 4 Error of Estimates
Fig. 3 State X3 and Estimate
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
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b)
.5 x 3  2
 0

x   .2 x1
0
  x1  7  x 2 x 3
7 x 2 1 
y (t )  
 x(t )
4 3 0

.5 
1

1  x(t )  0u (t )
0
 9
Fig. 6 State X2 and Estimate
Fig. 5 State X1 and Estimate
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
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Fig. 7 State X3 and Estimate
Fig. 8 Error of Estimates
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
15
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.
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
16
f
u
d
Process
Measurements
y
Unknown Input
Observer
r

Lineal invariant system: x  Ax  Bu  E1d1  K1 f1
f1  0
Objective
and
f1  0 or
y (t )  Cx  E2 d 2  K 2 f 2
f2  0  r  0
f2  0  r  0

Auxiliary dynamical system:
xˆ  Axˆ  Bu  L( y  Cxˆ )
r  H ( y  Cxˆ )

Non-linear system:
x(t )  f ( x(t )), y(t )  h( x(t ))
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
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PROBLEM:x (t )  A(t ) x(t )  B (t )u (t )  B (t )w(t )  F (t )  (t )  F (t )  (t )
u
w
1
1
2
2
y (t )  C (t ) x(t )  v(t ), x(t 0)  x 0
Where
u control input
v noise
y measurement
 i faults
w process noise
Fi faults directions
Find linear observer
and residual
( 1 = target fault)

xˆ (t )  A(t ) xˆ (t )  Bu (t )u (t )  L(t )( y (t )  C (t ) xˆ (t ))
r (t )  H (t )( y (t )  C (t ) xˆ (t ))
Such that r (t ) is primarly affected by the target fault and minimally by noises
and nuissance faults
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
18
Given a nonlinear system

x(t )  A( x) x(t )  B( x)u (t )  F1 ( x) 1 (t )  F2 ( x)  2 (t ), x(0)  x 0
y (t )  C ( x) x(t )
1. Reduction to a sequence of linear time varying approximations
 [i ]
x (t )  A( x[ i 1] (t )) x[ i ] (t )  B( x[i 1] (t ))u [i ] (t )  F1 ( x[i 1] (t )) 1 (t )  F2 ( x[i 1] (t ))  2 (t ), x[i ] (0)  x 0
[i ]
[i ]
y [i ] (t )  C ( x[i 1] (t )) x[i ] (t )
2. Design of observer at each time varying approximation
 [i ]
xˆ (t )  A( x[i 1] (t )) xˆ[i ] (t )  Bu ( x[i 1] (t ))u[i ] (t )  L( x[i 1] (t ))( y[i ] (t )  C ( x[i 1] (t )) xˆ[i ] (t ))
r [i ] (t )  H ( x[i 1] (t ))( y[i ] (t )  C ( x[i 1] (t )) xˆ [i ] (t )), xˆ [i ] (0)  xˆ0
3. Test of observer at final approximation in the presence of
different target and nuissance faults
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
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- 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
“Iteration technique for nonlinear systems and its applications to control theory”
Monash University
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