Genetic algorithm in parameter estimation of nonlinear dynamic systems

Abstract

We introduce a multi-model parameter estimation method for nonlinear dynamic systems. The method employs a genetic search
with a recursive probability selection mechanism for parameter estimation. The method is applied to nonlinear systems with
known structure and unknown parameters. A new technique is used to determine the selection probabilities. First, a population
of models with random parameter vectors is produced. Second, a probability is recursively assigned to each member of a generation
of models. The probabilities reflect the closeness of each model output to the true system output. The probabilities have
to satisfy an entropy criterion so as to enable the genetic algorithm to avoid poor solutions. This is a new feature that
enhances the performance of the GA on the parameter estimation problem. Finally, the probabilities are used to create a new
generation of models by the genetic algorithm. Numerical simulations are given concerning the parameter estimation of a planar
robotic manipulator.

Full text

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Genetic Algorithm in Parameter
Estimation of Nonlinear Dynamic Systems
E. Paterakis
manos@egnatia.ee.auth.gr
V. Petridis
petridis@vergina.eng.auth.gr
Ath. Kehagias
kehagias@egnatia.ee.auth.gr
http://skiron.control.ee.auth.gr/kehagias/index.htm
Automation and Robotics Lab
http://skiron.control.ee.auth.gr
Department of Electrical and Computer Engineering
Aristotle University -Thessaloniki – 54006
GREECE

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Genetic Algorithm in Parameter Estimation of Nonlinear
Dynamic Systems
E. PaterakisV. PetridisA. Kehagias
Department of Electrical and Computer Engineering
Aristotle University– Thessaloniki - Greece
1.Introduction
• We introduce a parameter estimation method for nonlinear dynamic systems.


• A simple genetic algorithm (GA) is used with a recursive probability selection
mechanism.

• The method is applied to nonlinear systems with known structure and unknown
parameters.

• The nonlinear dynamic system has to be identifiable.

• The selection probabilities have to satisfy an entropy criterion so as to enable the
genetic algorithm to avoid poor solutions. This is a new feature that enhances the
performance of the GA on the parameter estimation problem.

• Numerical simulations are given concerning the parameter estimation of a planar
robotic manipulator with four (4) parameters.

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2. Parameter Estimation Problem
The estimation problem consists in determining the member of the set that best describes the
data according to a given criterion. The purpose of the estimation has to be outlined. If the
final goal is a control strategy for a particular system the accuracy of estimation should be
judged on the basis of the time response. However, if the final goal is to analyze the
properties of a system the accuracy of estimation should be judged on the basis of deviations
in the model parameters. Such problems can be found in engineering problems and also in
biology, economy, medicine etc.
The data are produced by the actual system I when it operates under a given experimental
condition. The output of the system at time t will be denoted by ( )
a sequence of outputs ( )
y 0 , … , ( )
y t will be denoted by yt. The vectors ( )
( )
u t
∈ℜ ), ut denote the input at time t and the sequence of applied inputs, respectively.
Similarly ( )
ϕ t (where ( )
ϕ t
∈ℜ ), ϕt denote the system states at time t and the sequence of
system states, respectively. The vector ϑ (where ϑ ∈ℜd) denotes the parameter vector.
Consider now the following family of discrete time dynamic systems I(
y t (where y t
q
( ) ∈ℜ ), and
u t (where
m
n
)
f g t y, , , , , ,u
ϕϑ0:
( )
t
() ( )
u t
−1,
(
)
ϕϕϑ
ft
=
0
;
, ( )
y t
( ) ( )
t u t
ϕ
,
()
g
=
ϑ
;
0
(1)
In the system of type (1) the functions ( )
initial conditions are also known. The output yt is measurable and only the parameter vector
ϑ0 is unknown. All the possible ϑ that can be applied to I construct a model set which will
be denoted by M . The model set is indexed by the finite dimensional parameter vector ϑ ,
so that a particular model in the model set will be denoted by M ( )
over a set denoted by D
compact. It is also assumed that ϑ0∈D
real world problems, the I changes to:
f ⋅ and ( )
g ⋅ are known. The input utand the r
ϑ . The vector ϑ ranges
M. (where D
M. ⊆ ℜd), which generally will be assumed to be
M.. Taking into account noisy measurements, as in
( )
t
()() ( )
r u t,
()
ϕϕϕϑ
ftt
=−−
1
0
,...,;, ( )( )
y t
( )
~y t
w t
=+
(2)
Now ! yt is the measurable output and ( )
noisy system output
~y t for T time steps, a data record C can be constructed, where
dimC=q T
⋅ and q is the number of outputs. Note that the dynamic system requires that a
sufficient number of output are measured otherwise the system may be unidentifiable. An
exhaustive search of M has to be precluded in realistic situations where M has a vast
number of members. On the other hand GA is a global search scheme taking into account
only a small part of M . The need for a global optimum search scheme arises from the fact
that the fitness function may have local minima, either inherent or due to noisy measurements
(Ljung, ’87), (Söderström and Stoica ’89). Also, GA has not the drawbacks of the standard
estimation techniques in case of discontinuous objective functions. Hence, the use of this
evolutionary scheme is very attractive for parameter estimation of nonlinear dynamical
systems. In the GA, an evolvable population P of potential solutions has to be defined taking
values over the D
ϑ ∈D
w t is white, zero mean output noise. Sampling the
( )
M. Hence as P (where P ⊆D
M . The set P is finite dimensional, namely
M) is denoted a set of distinct values of
{}
P
K12
,,...,
, where K is a fixed
= ϑ ϑ
ϑ

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number. The set Pj includes the potential solutions at the generation j. The output
( )
y t
i
corresponds to the parameter vector ϑi
∈
and j is the generation index. The model validation is achieved through the discrepancy
between the noisy measured output ( )
given by
( )
E t
i
, where ⋅ denotes Euclidean norm:
( ) ( )
E ty t
ii
=
jjj
P where 1≤ ≤
i K is the index of the model
~y t and the model output
( )
y t
i
j
at time step t and it is
j
( )
y t
jj
−~
2
(3)
The sum of
(MSE) function
( )
( )
E t
i
j
for every time step t, where 0≤ ≤
for the particular model with parameter vector ϑi
T
=
0
is positive so it can stand as a fitness function. The estimate of ϑ0 at the
j*:
( )
ϑis
sK
, ,...
=1 2
Therefore the parameter estimation problem can be stated as a static optimization problem of
finding the value ϑ* that minimizes the fitness function ( )
tT, defines the mean square error
w T
i
j
j at the j generation:
( )( )
w T
i
E t
i
jj
t
=∑
(4)
The function
j-th generation isϑi
( )
w T
i
j
jj
w T
*
argmin
=
(5)
w T .

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3. Production of Selection Probabilities
Based on the fitness function of (4), a probability is assigned to each member of Pj. This
probability expresses the similarity of the model output to system output and it is relative to
the performance of each other member of Pj. Several production schemes of selection
probabilities have been proposed in literature. One of them is the following:
( )
( )
wT
m
m
=
∑
1
1
The trial solutions with parameter vectors ϑi
over the whole measurement data record C in order to calculate these selection probabilities.
Another selection probability scheme is the following:
( )
p T
e
s
=
∑
1
Substituting
( )
w T
i
from (4) we have the Boltzmann selection probability mechanism
where σ is the temperature parameter, which is usually constant.
( )
e
i
Et
∑
=
∑
1
Here, we propose a new method of production of the selection probabilities. The method is
based on the selection scheme (8), although it has two differences. First, the probabilities are
calculated recursively and second, there is no need to take into account all the measurement
record C in order to evaluate the trial solutions. The probability update rule for the time step t
is defined as:
(
( )
p t
i
Et
j
s
−⋅
=
∑
1
1
Where
( )
pKj
s
1
=∀
and sK
=1... is the prior probability distribution.
The main point that makes this method different from the others is that the evaluation of
each model selection probability is performed for a total of T
Tj time steps are defined by the entropy criterion, which is explained in Section 1. The role
of the entropy criterion is twofold. First, it prevents premature convergence of the algorithm
retaining thus the required population diversity. Second, it makes the algorithm able to use
fewer output measurements than it would otherwise be necessary (
times tttTj
===
01, ,..., finally one can obtain:
( )
( )
p T
i
Et
i
⋅
=
∑
0
1
( )
p T
i
w T
i
j
j
j
K
=
1
(6)
jj
P
∈
where 1≤ ≤
iK have to be evaluated
( )
( )
T
e
i
j
wT
w
K
i
j
sj
=
−
−
(7)
j
( )
( )
σ
p T
e
j
Et
s
K
i
σ
j
t
T
sj
t
T
=
∑
=
−
−
=
1
1
(8)
( )
()
)
()
()
( )
p t
i
e
p t
s
e
j
j
Et
j
j
K
i
σ
j
sj
σ
=
−⋅
−
−
−
−
1
1
1
(9)
j0
T
j≤
time steps. The number of
)
TT
j≤
. Repeating for
()
( )
0
( )
( )
( )
j
pe
pe
jj
i
j
E
σ
t
j
s
j
s
K
i
j
t
Ti
∑
=
j
t
Ti
∑
=
=
⋅
−
−
0
0
σ
(10)