Fachgebiet Systemanalyse
Lecture Notes
System Identification
Summer Semester 2014
RCSE course
Prof. Dr.-Ing. habil. Ch. Ament
Date 03/2014
www.tu-ilmenau.de/systemanalyse
TECHNISCHE UNIVERSITÄT
ILMENAU
Lecture „System Identification“
Prof. Dr.-Ing. Ch. Ament
Page 0-1
0. Content
0
Content
Basic methods
1
Introduction
2
Physical model approaches: White box models
3
General model approaches: Black box models
4
Parameter identification
Experimental analysis
5
Design of experiment
6
Identification of signal models
7
Identification of continuous-time systems
8
Identification of discrete-time systems
Online methods
9
Recursive parameter estimation
10
Kalman filtering
Text books

R. Isermann, M. Münchhof: Identification of Dynamic Systems – An Introduction
with Applications, Springer, 2011 (93,80 €)

L. Ljung: System Identification – Theory for the User, 2nd edition, Prentice Hall,
1998 (92 €)

D. Simon: Optimal State Estimation – Kalman, H Infinity, and Nonlinear Approaches, Wiley, 2006 (112 €)
Lecture „System Identification“
Prof. Dr.-Ing. Ch. Ament
Page 1-1
1. Introduction
1 Introduction
1.1 Motivation
For which purposes do we need models of technical processes?





Analysis of a system (stability, time constants, etc.)
Prediction of system behavior (by numerical simulation)
Controller design
Diagnosis and monitoring
Optimization of system design
1.2 Models
Definition
A mathematical model is a description of a system using mathematical concepts and language. […]
Mathematical models are used not only in the natural sciences (such as physics, biology,
earth science, meteorology) and engineering disciplines (e.g. computer science, artificial
intelligence), but also in the social sciences (such as economics, psychology, sociology and
political science); physicists, engineers, statisticians, operations research analysts and
economists use mathematical models most extensively.
A model may help to explain a system and to study the effects of different components,
and to make predictions about behaviour. [Wikipedia]
Classification by the type of mapping
Physical models
Conceptual models
are down-scaled or simplified physical realizations of the real system, e.g. architectural
models, or rapid prototyping components.
exist only in our mind, e.g. describing a
system by mathematical equations or by a
theoretical structure.
Classification by the type of description
White box models
Black box models
describe a system by causally determined
relations. Such an analytical description is
derived from physical laws („first principles“) and is mostly the result of theoretical
modelling.
characterize the relation of inputs and outputs in a descriptive way. General model
approaches or non-parametric models are
used as a result of experimental modeling.
Also mixed models are used; they can be addressed as grey box models.
Lecture „„System Identification“
Prof. Dr.-Ing. Ch. Am
ment
Page 1-2
1. Introduction
Parame
etric models
s
Non--parametric
c models
are built from a strructured model approa
ach
and a finite numbe
er of model parameterrs,
which q
quantify the
e model app
proach. Com
mpared to
o non-parametric mod
dels these
models achieve a more
m
compact represe
entation.
do not have a certain
c
struccture and characc
teriz
ze the syste
em by data that is rece
eived
from
m measurem
ment, e.g. a step respo
onse.
In prrinciple they
y have an i nfinite num
mber of
parameters.
1.3 M
Modelling
Definitio
ons
The pro
ocess of dev
veloping a mathematic
m
cal model is
s termed mathematica
al modelling
g.
[Wikipe
edia].
The field of system
m identification uses sstatistical methods
m
to build mathe
ematical models
m
of
dynamical systems from me
easured datta. System identification also in cludes the optimal
design o
of experime
ents for effficiently gen
nerating infformative data for fittiing such models as
well as model redu
uction. [Wik
kipedia]
The mo
odeling proc
cess
Lecture „„System Identification“
Prof. Dr.-Ing. Ch. Am
ment
Modelin
ng strategie
es
Page 1-3
1. Introduction
Lecture „„System Identification“
Prof. Dr.-Ing. Ch. Am
ment
Page 2-1
2. White bo
ox models
2 Ph
hysical model approac
a
ches: White
W
bo
ox mod
dels
2.1 B
Balance eq
quations
Balancin
ng a conserrved quantiity over tim
me is a very
y universal modelling
m
p
principle, which can
be applied e.g. forr process pla
ants or biollogical syste
ems.
The con
nserved qua
antity m is balanced w
within a spec
cified volum
me :
The balance equattion in integ
gral form is:
t
m(t )  m(0) 
 q
zu ( )
 qab ( )  p( ) d
0
The balance equattion in differrential form
m is:
 (t )  q zu (t )  qab (t )  p(t )
m
Typical conserved quantities are:
a





Mass or vollume
Energy
Number of cells
Electrical ch
harge
Momentum
m
For each volume  we obtain one first order linear differential equation.
Lecture „System Identification“
Prof. Dr.-Ing. Ch. Ament
Page 2-2
2. White box models
2.2 Electrical systems
Equations of RLC Networks
Kirchhoff voltage law:
 u (t )  0
j
j
Kirchhoff current law:
 i j (t )  0
j
Current voltage relation of network elements:
Element
Current voltage relation
(time domain)
u(t )  R  i(t )
Resistance R
Capacitor C
1
u(t ) 
C
Inductor L
t

Ohm’s law
(frequency domain)
U(s)  R  I(s)
1
 I(s)
sC
i( ) d
U(s) 
di(t )
dt
U(s)  s L  I(s)
0
u(t )  L 
The general Ohm’s law with impedance Z(s) is:
U(s)  Z(s)  I(s)
Mesh current analysis of RCL Networks
1. If applicable, current sources have to be transformed to voltage sources.
2. Plot the network graph.
Select a tree out of this graph.
Definitions:
Nodes correspond to the connecting wires, edges to the devices.
A closed loop within the graph is called a mesh.
A tree connects all nodes of a graph without any meshes.
3. Introduce currents Ii with i=1,...,n in all edges of the graph that do not belong to
the tree.
Lecture „System Identification“
Prof. Dr.-Ing. Ch. Ament
Page 2-3
2. White box models
4. The system of equations is:
 Z11 Z12  Z1n   I1  U1 
    

 Z21 Z22  Z2n    I2   U2 
 


     
    

Z n1 Z n2  Z nn   I n  U n 


  
I
U
Z


Z impedance matrix of the system with the main diagonal elements Zii and all
other elements Zik:
Zik :
Impedance that is passed by Ii and Ik commonly. If the direction of both
currents is similar use positive sign, otherwise negative sign.
Zii :
Sum of all impedances along mesh i
I vector of currents
Ii :

Current of mesh i (with i=1,...,n)
U vector of voltage sources
Ui :
Sum of independent voltage sources in mesh i. If the voltage source
supports the mesh current, use positive sign.
2.3 Modeling analogies
The analysis of systems with lumped elements (devices) can be generalized. For its characterization two complementary variables are used:

Flow q(t): The flow is passing a network device and can be measured within the device. Due to the Kirchhoff current law the sum of all currents in relation to a network
node is zero.

Potential V(t): The potential is assigned to the connectors of a network device; a potential difference can be measured between two connectors. Due to the Kirchhoff
voltage law the sum of potential differences within a network mesh is zero.
A generalized Ohm’s law characterizes the relation of flow and potential difference regarding a lumped network element.
Lecture „„System Identification“
Prof. Dr.-Ing. Ch. Am
ment
Page 2-4
2. White bo
ox models
Applica
ation in diffferent physical dom
mains:
Domain
n
Flow
w q(t)
Potential
P
(difference)
(
V(t)
Resistance
Capacitor
C
(Storage
(
of
flow)
f
Inducto
or
(Storag
ge of
potentia
al diff.)
Electric
cal system
El. current
I(t)) in A
El.
E voltage
U(t)
U
in V
El. res
sistance
R in V/A

El.
E capacitor
C in F = As/V
V
uctor
El. indu
L in H = Vs/A
Mechan
nical
system
(transla
atory)
Forrce
F(t)) in N
Velocity
V
v(t)
v
in m/s
Reciprrocal friction co
onstant
1/k in m/Ns
Mas
M
m in kg
Recipro
ocal
spring constant
c
1/c in m/N
m
Mechan
nical
system
(rotatory)
Torrque
M(tt) in Nm
Angular
A
velo
ocity
(t) in 1/s
Reciprrocal friction co
onstant
1/k in 1/Nms
Moment
M
of
inertia
J in kg m2
Recipro
ocal
spring constant
c
1/c in 1/Nm
1
Pneuma
atic
system
s flow
Mas
 (tt ) in kg/s
m
Pressure
P
Pneum
matic rep(t)
p
in Pa= N
N/m2 sistance
in 1/m
ms
Storage
S
of m
mas
in kg m2/N
–
Therma
al system
Hea
at flow
Temperature
T
e
q(t)) in W=J/s T(t)
T
in K, °C
Thermal
T
capacity
c
in Ws/K
–
Therm
mal
resista
ance
in K/W
W
The pro
oduct of flow
w and poten
ntial (differe
ence) is usually a pow
wer: P(t) = q
q(t) V(t)
Examp
ple: Pneum
matic syste
em
In the following a pneumatic
c system ( including a supply pipe, a valve
e and a pn
neumatic
muscle)) is describe
ed using ele
ectrical sym
mbols:
2.4 B
Block diag
gram
Advanta
ages of bloc
ck diagrams
s:

C
Clear graph
hical representation.

Model imple
ementation can be don
ne step by step.

S
Substructurres may he
elp to make it clearer.

W
Within a project blocks
s can be ex
xchanged ea
asily.

Numerical simulations
s
can be carrried out ba
ased on bloc
ck diagramss
((e.g. by Sim
mulink)
Lecture „System Identification“
Prof. Dr.-Ing. Ch. Ament

Page 2-5
2. White box models
C-Code can be generated from block diagrams.
Classification:
Object orientated modeling:
Nodes are physical components; edges are
physical interactions between these components.
Description is close to physical reality.
Causal modeling:
Nodes are transfer functions; edges are
signals.
Useful for system analysis.
Software: e.g. Matlab/Simulink
Software: e.g. Modelica/Dymola
2.5 State space description
The state of a dynamical system at time t is completely characterized by a set of states
x1(t), … xn(t). These states are summarized as state vector x(t).
System dynamics is defined by the state differential equation. It describes the change of
states in time x (t ) as a function of the current state x(t) and the system’s input u(t). It is a
system of first-order differential equations.
Measurable outputs are summarized as output vector y(t). The output equation defines
y(t) based on the current state x(t) and (in rare cases) the system’s input u(t). The output
equation is an algebraic equation.
System description
State differential equation:
Output equation:
x (t )  f x(t ), u(t )
y(t)  gx(t), u(t )
with the following variables:
State vector
 x1 (t )


x 2 (t )

x(t ) 
  


 x n (t )
with n: Number of states
Input vector
 u1(t) 


u(t )    
u m (t )
with m: Number of inputs
 y 1(t) 


Output vector y(t )    
y p (t )


with p: Number of outputs
Lecture „„System Identification“
Prof. Dr.-Ing. Ch. Am
ment
Page 3-1
3. Black bo
ox models
3 Ge
eneral model
m
approac
a
ches: Bllack box
x mode
els
If no ph
hysical laws
s (first princ
ciples) are available or their deriv
vation is to
oo time-con
nsuming,
we can use genera
al (nonphys
sical) mode
el approach
hes. They describe the
e input outp
put relation of tthe real pro
ocess. In th
he following
g commonly
y used gene
eral model approaches will be
presentted.
The mo
odel parame
eters have no
n physical interpretattion. They are
a unknow
wn at that point and
have to
o be identifie
ed based on
n measurem
ment (see chapter
c
4).
3.1 Lo
ook-up ta
able mode
els
Look-up
p tables deffine a functtional relatiion by sam
mpling the whole
w
range
e of an inpu
ut u and
storing the corresponding va
alues for th
he output ŷ explicitly. This migh
ht be a sim
mple and
useful m
method to map
m
e.g. a calibration curve.
It can b
be considere
ed as a non
n-parametriic model (se
ee chapter 1).
Examp
ple of a loo
ok-up table
e model
Resistan
nce values of a PT100 sensor as a function of
o temperatture [from w
www.pt100
0.de]:
ples for a graphical
g
representa
r
ation of mo
odels:
Examp
1. Pneumatic muscle: Functional rellation of mu
uscle air pressure and muscle con
ntracttion h to the resulting muscle forrce F. Data sheet of the pneumatiic muscle with
w
20 mm diam
meter from
m Festo:
Lecture „„System Identification“
Prof. Dr.-Ing. Ch. Am
ment
Page 3-2
3. Black bo
ox models
2. Performanc
ce diagram of a combu
ustion engin
ne: Function
nal relation
n of motor speed
s
a
and specific
c motor work to motorr consumpttion:
3.2 Polynomia
al models
Approac
ch for a sys
stem with single
s
inputt (m=1) an
nd single ou
utput (SISO
O system) with the
order n of the poly
ynomial app
proach:
ŷ  a0  a1 u  a 2 u 2    a n u n 
n
 ai u i
i 0
uts (m>1), and single output (MIISO system
Approac
ch for a sys
stem with multiple
m
inpu
m):
m
ŷ  a0 

j 1
m
ai u j 
m
 a
j  1 k 1
j, k
u j uk  
Lecture „System Identification“
Prof. Dr.-Ing. Ch. Ament
Page 3-3
3. Black box models
The output ŷ of this approach is a nonlinear function of the input in case n>1. Hence, nonlinear input output relations can be describes with this model approach. On the other hand,
output ŷ is a linear function of the model parameters ai, which will be turn out to be a major advantage in the following chapter 4.
2
2
1
1
0
0
y
y
An unknown function can be characterized by a polynomial approach throughout its whole
input range. But, local aberration can be characterized only insufficiently. Moreover, it cannot be expected to obtain good extrapolation properties as the following example will illustrate [from Nelles: Nonlinear System Identification]. A sin function is approximated by a
polynomial approach with n=5 (left) und n=15 (right).
-1
-1
-2
-2
0
2
4
6
8
-2
-2
0
2
4
6
8
u
u
3.3 Radial basis function models
A radial basis function approximates a given system within a local area of the input space
that is defined by the input u. For the description of the entire input space multiple of these
local descriptions are superimposed.
Suitable radial basis functions  have the following properties:



The functional values are non-negative.
It has a maximum value in the centroid point u0.
With increasing distance to the centroid u0 the functional values are monotonically
decreasing.
The following figure shows two commonly used radial basis functions. In the upper part the
one-dimensional case is shown. The bell-shaped Gaussian function
1
 2
1
ˆ   (u)  e 2
y
with   u  u0 

is continuously differentiable, whereas the triangular function

1 


ˆ   (u)  1 
y

0

1
u  u0  with u0  u  u0  2
2
1
u  u0  with für u0  2  u  u0
2
otherwise
Lecture „„System Identification“
Prof. Dr.-Ing. Ch. Am
ment
Page 3-4
3. Black bo
ox models
is not d
differentiable in the co
orner pointss. In both functions,
f
parameter
p
 defines th
he width
of the b
basis functio
on.
In the lower part of the figu
ure the exttension for the two-dimensional case is plo
otted for
both rad
dial basis fu
unctions.
In orde
er to approx
ximate a fu
unctional re
elation, mu
ultiple radia
al basis fun
nctions  i with
w
i =
1,…,n w
will be supe
erimposed. Each basiss function has
h
a differe
ent centroid
d u0,i and is scaled
with a w
weighing factor ai:
n
ŷ 
a
i
  i (u)
i 1
This app
proach is able to describe a nonliinear function relations
s between tthe input u and the
output ŷ . Similar to the poly
ynomial app
proach in se
ection 3.2 the model o
output ŷ is a linear
function
n of the mo
odel parame
eters.
Lecture „„System Identification“
Prof. Dr.-Ing. Ch. Am
ment
Page 3-5
3. Black bo
ox models
As an e
example, the following figure show
ws the superposition of
o a functio
on (red line) by five
triangullar radial ba
asis functions (dashed blue lines)).
3.4 N
Neural nettworks
Artificia
al neural networks are an abstracction of biolo
ogical neural networkss. They are used to
obtain a parallel and meshed
d approach to the mod
deling of te
echnical pro
ocesses. This architecture can also be
b implemented and e
evaluated in
n parallel, which
w
can b
be advanta
ageous if
subsysttems fail. Then,
T
the model
m
beha
aves more robust suc
ch that the basic func
ction remains a
available in most cases
s, although the overalll performan
nce is reducced.
In the ffollowing, a very comm
mon netwo
ork architec
cture is introduced, in which the neurons
are arra
anged in la
ayers that do not of any feedba
acks. It is called mult
lti layer perrceptron
(MLP) o
or feed forw
ward networrk.
Neuron
ns
The inp
puts u1,…,um at the ne
euron j are weighed with
w
the nettwork weig hts wj,1,…,w
wj,m. Together w
with the bia
as bj they are
a summe
ed up to the
e signal xj, which is th
he input of the
t
activation ffunction f. [see
[
also Ma
atlab, Neurral Network Toolbox, User’s
U
Guide
e]:
The fun
nction f can be a nonlinear function, in man
ny cases fun
nctions are used that model a
thresho
old as
1
.
f (x
x )  tanh( x ) oder f ( x ) 
1  ex
Lecture „„System Identification“
Prof. Dr.-Ing. Ch. Am
ment
Page 3-6
3. Black bo
ox models
Especially in the la
ast layer of a network a linear fun
nction f(x) = x may be
e used. Intrroducing

an input vector u  u1  u m T and a vector of weights
w
w j  w j ,1  w j , m
T
this rela-
tion can
n be written
n as a comp
pact equatio
on:
T
y j  f (w j u  b j )
Layers
A layer is built up out of n neurons in pa
arallel. With
h the bias vector b  b1  bn T and the

T
matrix of network
k weights W  w1
 w nT
 the functional
f
relation
r
forr the vectorr of outT
e written as:
puts y  y1  y n T can be
y  f (W u  b)
The following figurres show th
he block dia
agram of on
ne layer – on
o the rightt hand side
e using n
single n
neurons, on the right hand
h
side ussing vectorr signals:
Multi L
Layer Perce
eptron (ML
LP)
A typica
al MLP cons
sists of two layers: Forr the first la
ayer a nonlinear activa tion functio
on f (1),
for the second laye
er a linear activation
a
fu
unction f (2) is used:
Lecture „„System Identification“
Prof. Dr.-Ing. Ch. Am
ment
Page 3-7
3. Black bo
ox models
The corrresponding
g functional relation is:
y f
(2)
W
(2)
f
(1)
(W
(1)
(1)
ub
(2)
)b

Parame
eters of this
s model approach are the netwo
ork weights W(.) and t he bias vallues b(.).
They ca
an be obtaiined from a nonlinearr parameter optimizattion that wiill be introd
duced in
the nex
xt chapter 5.
5
3.5 Ex
xtension to dynam
mical mod
dels
The mo
odel approaches presented in thiss section arre static approaches. IIn order to characterize d
dynamical systems,
s
ex
xternal dyna
amics (as a “memory”) has to b
be added. For
F timediscrete
e systems this
t
can be
e realized b
by storing passed
p
valu
ues of the input u(k) and the
output y
y(k) in a de
elay line:
Hence, the static model
m
approaches can
n be used again. But, the
t
length o
of the delay
y lines n
and m a
ave to be defined
as well as the
t
sample
e time T ha
d
in an
a appropriiate way, such that
the dyn
namical prop
perties are included an
nd the num
mber of (n+m
m+1) inputts is not too
o high.
Lecture „„System Identification“
Prof. Dr.-Ing. Ch. Am
ment
Page 4-1
4. Parrameter iden
ntification
4 Pa
aramete
er identtificatio
on
4.1 Problem definition
The rea
al system is defined by
y a set of pa
arameter, which
w
is sum
mmarized in
n a parame
eter vector x p . For the ide
entification of this set o
of parameter, a mode
el is built in parallel to the real
a estimatiion x̂ p of tthese param
meters. The
e measured
d values y(tt) of the
system that uses an
ˆ(t , x
ˆ p ) of the model and
real sys
stem is co
ompared with the outtput y
d the output error
ˆ p ) is determ
ey (t , x
mined.
ˆ p ) will assess th
An obje
ect function
n J( x
he output error
e
within
n a time in
nterval. Usu
ually the
object ffunction is defined suc
ch that its value will increase, iff the outpu
ut error rise
es. Now,
ˆ p ) as a
optimiza
ation methods are req
quired in orrder to min
nimize the objective fu
unction J(x
x
function
n of the mo
odel parameters x̂ p an d to adapt the model behavior tto the real system.
Hence, the problem
m of param
meter identiffication is converted in
nto an optim
mization pro
oblem.
Block d
diagram:
In princ
ciple the ob
bjective function can b
be defined independen
i
ntly in differrent ways. In most
cases th
he squared output erro
or is integra
ated over a time interv
val T:
T
ˆ p ) :  e 2 (t , x
ˆ p ) dt
J(x
0
Lecture „System Identification“
Prof. Dr.-Ing. Ch. Ament
Page 4-2
4. Parameter identification
4.2 Direct optimization
Linear parameter estimation
It is assumed that the output y is a linear function of the unknown parameter vector x p :
T
y(t , x p )  c (t )  x p
The model is defined in the same way:
ˆ(t , x
ˆ p )  c T (t )  x
ˆp
y
Hence, the output error between the model ŷ and the real measurement y is:
T
ˆ p )  y(t )  y
ˆ(t , x
ˆ p )  y(t )  c (t )  x
ˆp
ey (t , x
A set of measurements with k = 1,...,N is obtained and summarized as vectors:
 e(t1 , x
ˆp )
 y ( t1 ) 




ˆp )  
e y (x

, y    ,
e(t N , x
 y (t N )
ˆ p )

 c T (t 1 ) 


  
c T (t )
N 

Then, we can write the output error for all N measurements as:
ˆp )  y    x
ˆp
e y (x
As objective function J the sum of the squared errors
T
ˆ p )  e y (x
ˆ p )  e y (x
ˆp )
J(x
has to be minimized. It is also called “least square error” (Carl Friedrich Gauß: „Methode
der kleinsten Quadrate“).
The solution for the optimal parameter vector x̂ p is:

ˆ p  T  
x

T

1
T
 y

The system will be called identifiable, if    is regular. If this is not the case, it might be
that not enough measurements or only very similar measurements are available.
Lecture „„System Identification“
Prof. Dr.-Ing. Ch. Am
ment
Page 4-3
4. Parrameter iden
ntification
Examp
ple: Parameter identtification o
of a vehicle
e with constant acce
eleration
A vehicle with an unknown mass
m
m will be accelerrated with a constant force F0 sta
arting at
t = 0:
F (t )  F0   (t ) with
h F0  4
e vehicle position y(tk) is measure
ed:
At the ttimes tk the
k
1
2
3
4
5
tk
5,0
6,0
7,0
8,0
9,0
y(tk)
0,2
1,2
3,0
4,8
7,1
For mod
deling Newton’s law F (t )  m  y((t ) can be applied. Initially the v
vehicle is lo
ocated in
y(0)  0 . The unkn
nown mass
s m and the
e unknown initial vehic
cle velocity y (0)  v 0 sh
hould be
determiined by parrameter identification.
In the ffigure below
w the mode
el output ŷ (as solid line) and the measured
d values (a
as crossmarks) are plotted
d.
Lecture „System Identification“
Prof. Dr.-Ing. Ch. Ament
Page 4-4
4. Parameter identification
4.3 Recursive Solution
In order to obtain the direct solution for x̂ p as shown in section 4.2, the measurement of
all samples in x̂ p has to be completed. This is no problem, if parameter optimization can
ˆ p (k ) each time,
be done offline. But, if we need an update of the identified parameter x
when a new measurement y(k) is available a recursive solution would be more appropriate
– which should be also called an online parameter identification.
Recursive algorithm
Initial step in k=0: Define
ˆ p (0 ), P (0 )
x
ˆ p (0 ) , matrix P (0 ) should
If we have no information about the initial parameter vector x
be chosen as a “large” positive definite matrix, e.g. P (0 )  10 10  I .
Recursive step from k to k+1:
P (k ) c(k  1)
1. Step:
K (k  1) 
2. Step:
ˆ p (k  1)  x
ˆ p (k )  K (k  1)  y(k  1)  c T (k  1) x
ˆ p (k )
x
3. Step:
P (k  1)  I  K (k  1)c (k  1)  P (k )
T
1  c (k  1)P (k ) c(k  1)




T
T
ˆ p )  e y (x
ˆ p )  e y (x
ˆ p ) (as in section 4.2).
This will minimize the objective function J(x
Recursive algorithm with fading memory
In case of time-varying processes, it is appropriate to introduce time-depending weighing
factors. A weighing factor with exponential forgetting will discount the weights for past
measurements. Introducing a forgetting factor 0    1 (e.g.   0,99 ) the object function
can be modified:
T
ˆ p )  e y (x
ˆ p )  Q  e y (x
ˆp )
J(x
Q  diag(N 1 ,  2 , , 1)
with
Modified recursive step from k to k+1:
P (k ) c(k  1)
1. Step:
K (k  1) 
2. Step:
ˆ p (k  1)  x
ˆ p (k )  K (k  1)  y(k  1)  c T (k  1) x
ˆ p (k )
x
3. Step:
P(k  1) 
T
  c (k  1)P (k ) c(k  1)

1

I  K(k  1)c
T


(k  1)  P(k )