On Identifiability of Object-Oriented Models

On Identifiability of Object-Oriented Models
On Identifiability of Object-Oriented Models
Markus Gerdin, Torkel Glad
Division of Automatic Control
Department of Electrical Engineering
Linköpings universitet, SE-581 83 Linköping, Sweden
WWW: http://www.control.isy.liu.se
E-mail: [email protected], [email protected]
5th December 2005
OMATIC CONTROL
AU T
CO
MM U
NICATION SYSTE
MS
LINKÖPING
Report no.: LiTH-ISY-R-2710
Submitted to 14th IFAC Symposium on System Identification,
SYSID-2006
Technical reports from the Control & Communication group in Linköping are
available at http://www.control.isy.liu.se/publications.
Abstract
When estimating unknown parameters, it is important that the model is identifiable so that the parameters can be estimated uniquely. For nonlinear differentialalgebraic equation models with polynomial equations, a differential algebra approach to examine identifiability is available. This approach can be slow, so
the present paper describes how this method can be modularized for objectoriented models. A characteristic set of equations is computed for components
in model libraries, and stored together with the components. When an objectoriented model is built using such models, identifiability can be examined using
the stored equations.
Keywords: Identifiability, Nonlinear systems, Algebraic methods, Object
oriented modelling, Modelling, Identification, Descriptor systems
ON IDENTIFIABILITY OF
OBJECT-ORIENTED MODELS 1
Markus Gerdin & Torkel Glad
(gerdin,torkel)@isy.liu.se
Division of Automatic Control, Department of Electrical
Engineering, Linköping University, SE-581 83 Linköping,
Sweden
Abstract: When estimating unknown parameters, it is important that the model
is identifiable so that the parameters can be estimated uniquely. For nonlinear
differential-algebraic equation models with polynomial equations, a differential
algebra approach to examine identifiability is available. This approach can be slow,
so the present paper describes how this method can be modularized for objectoriented models. A characteristic set of equations is computed for components
in model libraries, and stored together with the components. When an objectoriented model is built using such models, identifiability can be examined using
the stored equations.
Keywords: Identifiability, Nonlinear systems, Algebraic methods, Object oriented
modelling, Modelling, Identification, Descriptor systems
1. INTRODUCTION
A model structure is identifiable if it is possible to
estimate its unknown parameters uniquely from
measured input and output data. One important
reason to examine identifiability of a model structure might be that the parameters represent physical properties that are of interest. (If that is not
the case it might be sufficient to estimate certain
combinations of the parameters such as sums,
products or quotients.) Another reason could be
that numerical search methods have difficulties in
computing the parameters if the model structure
is not identifiable.
Identifiability has been studied by many authors,
e.g., Bellman and Åström (1970) and Walter
(1982). Ljung (1999) is a standard reference in
1
This work has been supported by the Swedish Foundation for Strategic Research (SSF) through VISIMOD and
EXCEL and by the Swedish Research Council (VR) which
is gratefully acknowledged.
system identification, including identifiability. In
(Ljung and Glad, 1994), a general method for examining identifiability in linear and nonlinear systems, both state-space systems and differentialalgebraic equations (DAE), is presented. However, this method uses differential algebra which
suffers from high computational complexity, and
can therefore only handle quite small systems.
This contribution discusses how the modularized
structure in object-oriented models can be used
to speed up the computations. Modern modeling tools such as Modelica are based on objectoriented modeling, so this approach can be useful
for models created using such tools.
2. PRELIMINARIES
In this section some preliminaries on objectoriented modeling and identifiability are presented.
2.1 Identifiability
i.e., a linear regression, then there is global identifiability, provided Pi (w, p) 6= 0.
Identifiability of a model structure basically means
that if two parameter values θ1 and θ2 give the
same predictor, then θ1 = θ2 . This property can
be local or global. Identifiability of a model structure actually only means that it is in principle possible to estimate unknown parameters uniquely.
It does not guarantee that identification experiments give good results since the results, among
other things, also depend on input signals and
measurement accuracy. Nevertheless, it of course
important that a model structure is identifiable if
parameters are to be estimated.
As is common when examining identifiability
for nonlinear systems, noise-free models will be
treated in this contribution, so here the predictor
will be the simulated output. For a discussion on
this see, e.g., Ljung and Glad (1994) from which
the basic setting that will be used is taken. This
means that the model is specified by
hi l(t), w(t), θ(t), p = 0
i = 1, 2, . . . , r. (1)
Here w(t) ∈ Rnw is a vector of measured input
and output signals, l(t) ∈ Rnl is a vector of
internal variables, θ ∈ Rnθ is a vector of unknown parameters, and hi (·) ∈ R while p is the
differentiation operator with respect to time, p ·
x(t) = dx(t)
dt .
In the paper by Ljung and Glad (1994), w(t) is
partitioned into inputs and outputs, but this is
not necessary for our purposes.
The following result from Ljung and Glad (1994)
will be used: Assume that a model structure is
specified by (1) where the equations are polynomials and that the unknown parameters are timeinvariant, i.e. the equations
θ̇(t) = 0
(2)
are included among the equations (1). Using Ritt’s
algorithm from differential algebra, (Ritt, 1966;
Glad and Ljung, 1998), it is possible to compute
a new set of equations of the form
A1 (w, p), . . . , AnA (w, p), B1 (w, θ1 , p),
B2 (w, θ1 , θ2 , p), . . . , Bnθ (w, θ1 , θ2 , . . . , θnθ , p),
C1 (w, θ, l, p), . . . , Cnl (w, θ, l, p). (3)
Typically it is possible to prove that the sets of
equations (1) and (3) are equivalent, provided
some conditions of the form
si l(t), w(t), θ(t), p 6= 0
i = 1, 2, . . . , ns (4)
are satisfied. Identifiability is determined by the
polynomials Bi in (3). If the variables θ1 , θ2 ,.. all
occur exclusively in undifferentiated form in the
Bi , then these polynomials give a triangular set
of nonlinear equations for determining the θi . If
the Bi have the form
Bi = Pi (w, p)θi − Qi (w, p),
(5)
While the result above, which is based on differential algebra, gives definite answers on identifiability for a wide class of systems, the computational
complexity is high. Therefore this paper discusses
how the modularized structure of object-oriented
models can be used to reduce the computational
complexity.
2.2 Object-Oriented Modeling
A central idea in object-oriented modeling is to
build models by connecting submodels that represent physical parts of the system. For example
when modeling an electrical circuit, the submodels can be components like voltage sources and
resistors. These submodels are often standardized
modules from model libraries. The modeling is
usually performed through a graphical user interface where the submodels are connected. The most
common language for object-oriented modeling is
Modelica (Fritzson, 2004; Tiller, 2001).
In object-oriented modeling a complete model
thus consists of a number of components, with
equations describing them, and a number of equations describing the connections between the components. The complete model is therefore a system
of differential-algebraic equations (DAE). Since
the components represent different physical parts
of the system, it is natural that they have independent parameters so that will be assumed in the
present paper. For a model with m components,
the equations describing the components are written as
fi li (t), wi (t), θi , p = 0
i = 1, . . . , m. (6)
Here, li (t) ∈ Rnli are internal variables, wi (t) ∈
Rnwi external variables that are used in the connections and θi ∈ Rθi unknown parameters, all
in component i. As before p is the differentiation
operator with respect to time, p · x(t) = dx(t)
dt .
With fi (·) ∈ Rnfi , it is assumed that nfi ≥ nli so
that there are at least as many equations as internal variables for each component. The equations
describing the connections are written
g t, w(t) = 0
(7)
where


w1 (t)


w(t) =  ...  .
(8)
wm (t)
The connection equations are typically simple
equations like a = b or a + b + c = 0, but the
framework also allows more complex connections.
There are also typically equations of the form
wi (t) = y(t), where y is an external signal. To
keep notation simple we assume that all external
functions are represented by the time-variability
of the function g(·). The dimension of g(·) is ng .
To summarize, a complete object oriented model
consists of the equations for the components and
for the connections,
fi li (t), wi (t), θi , p = 0
i = 1, . . . , m. (9a)
g t, w(t) = 0.
(9b)
This model can be analyzed using the method
described in Section 2.1. Our main idea is however
to separate the identifiability analysis into two
stages. The first stage is to rewrite the model for
a single component using the technique given by
(3). We thus assume that the model
fi li (t), wi (t), θi , p = 0
(10)
can be rewritten in the equivalent form
Bi,1 (wi , θi,1 , p), Bi,2 (w, θi,1 , θi,2 , p),
Bi,nθi (wi , θi,1 , θi,2 , . . . , θi,nθi , p),
Ci,1 (wi , θi , li , p), . . . , Ci,nli (wi , θi , li , p). (11)
An important part of the model for the analysis
below is the set of Ai,j . These relations between
the connecting variables are independent of the
choice of the parameters.
Example 1. Consider a capacitor described by the
voltage drop w1 , current w2 and capacitance θ1 .
It is then described by (10) with
θ1 ẇ1 − w2
.
(12)
f1 =
θ̇1
If we consider only situations where ẇ1 6= 0 we
get the following series of equivalences.
θ1 ẇ1 − w2 = 0,
θ̇1 = 0, ẇ1 6= 0
⇔
θ1 ẇ1 − w2 = 0, θ1 ẅ1 − ẇ2 = 0, ẇ1 6= 0
⇔
θ1 ẇ1 − w2 = 0, θ1 ẇ1 ẅ1 − ẇ1 ẇ2 = 0, ẇ1 6= 0
⇔
θ1 ẇ1 − w2 = 0, w2 ẅ1 − ẇ1 ẇ2 = 0, ẇ1 6= 0
(13)
With the notation (11) we thus have
and the function s1 of (4) is ẇ1 .
(14a)
(14b)
Example 2. Next consider an inductor where w2 is
the current, w1 the voltage and θ1 the inductance.
It is described by
θ1 ẇ2 = w1 ,
θ̇1 = 0
θ1 ẇ2 = w1 ,
ẅ2 w1 = ẇ2 ẇ1
provided ẇ2 6= 0.
(15)
(16)
As discussed earlier, the transformation to (11)
can always be performed for polynomial DAE. To
show that calculations of this type in some cases
also can be done for non-polynomial models, we
consider a nonlinear resistor where the voltage
drop is given by an arbitrary function.
Example 3. Consider a nonlinear resistor with the
equation
(17)
w1 = R(w2 , θ1 )
where it is assumed that the parameter θ1 can be
uniquely solved from (17) if the voltage w1 and
the current w2 are known, so that
θ1 = φ(w1 , w2 ).
Ai,1 (wi , p), . . . , Ai,nAi (wi , p),
A1,1 = w2 ẅ1 − ẇ1 ẇ2
B1,1 = θ1 ẇ1 − w2
Calculations similar to those of the previous example show that this is equivalent to
(18)
Differentiating (17) once with respect to time and
inserting (18) gives
(19)
ẇ1 = Rw2 w2 , φ(w1 , w2 ) ẇ2
which is a relation between the external variables
w1 and w2 . We use the notation Rx for the partial
derivative of R with respect to the variables x.
In the special case with a linear resistor, where
R = θ1 · w2 , this reduces to
w1
ẇ2
(20a)
ẇ1 =
w2
⇔ w2 ẇ1 = w1 ẇ2
(20b)
(assuming w2 6= 0).
3. MAIN RESULTS
The main results of this paper concern how the
modularized structure of object-oriented models
can be used to examine identifiability in an efficient way.
Assume that all components are identifiable if
the external variables wi of each component are
measured. This means, that given measurements
of
wi
i = 1, . . . , m
(21)
the unknown parameters θ can be computed
uniquely from the B polynomials. When examining identifiability of the connected system it is
not a big restriction to assume that the individual
components are identifiable since information is
removed when not all wi are measured. (Remember that all components have unique parameters.)
When the components have been connected, the
only knowledge available about the wi is the
A polynomials and the equation g t, w(t) =
0. (Remember that any known external signals
are included in the time-variability of g.) The
connected system is thus identifiable if the wi can
be computed from
(
i = 1, . . . , m
Aij wi (t), p = 0
(22a)
j = 1, . . . , nAi
g t, w(t) = 0.
(22b)
3.2 Local Identifiability
Note that this means that all w(t) are algebraic
variables (not differential), so that no initial conditions can be specified for any component of w(t).
If, on the other hand, there are several solutions
to the equations (22) then these different solutions can be inserted into the B polynomials, so
there are also several possible parameter values.
In this case the connected system is therefore not
identifiable. Note again that measured inputs and
outputs lead to equations of the form wi (t) = u(t),
where the function u is included in the timevariability of g.
Theorem 5. Consider an object-oriented model
where the components are locally identifiable and
thus can be locally described in the form (23).
A sufficient condition for the total model to be
locally identifiable is that (22) can be solved locally uniquely for the wi . If all the functions ψi
of (23) are locally injective then this condition is
also necessary.
The result is formalized in the following theorems.
Note that the distinction between global and local
identifiability was not discussed above, but this
will be done below.
3.1 Global Identifiability
Global identifiability means that there is a unique
solution to the identification problem, given that
the measurements are informative enough. For a
subsystem (10) that can be rewritten in the form
(11) global identifiability means that the Bi,j can
be solved uniquely to give the θi,j . In other words
there exist functions ψ, that can in principle be
calculated from the Bi,j , such that
θi = ψi (wi , p).
(23)
We then have the following formal result on identifiability.
Theorem 4. Consider an object-oriented model
where the components are globally identifiable
and thus can be described in the form (23). A sufficient condition for the total model to be globally
identifiable is that (22) can be solved uniquely for
the wi . If all the functions ψi of (23) are injective
then this condition is also necessary.
Proof. If (22) gives a global solution for w(t),
then this solution can be inserted into the B
polynomials to give a global solution for θ since
the components are globally identifiable. The connected system is thus globally identifiable. If there
are several solutions for wi and the functions ψi
of (23) are injective, then there are also several
solutions for θ, so the system is not globally identifiable since the identification problem has more
than one solution. 2
Local identifiability of a model structure means
that locally there is a unique solutions to the
identification problem, but globally there may
be more than one solution. This means that the
description (23) is valid only locally. We get the
following result on local identifiability.
Proof. If (22) gives a locally unique solution for
w(t), then this solution can be inserted into the
B polynomials to give a local solution for θ
since the components are locally identifiable. The
connected system is thus locally identifiable. If
there locally are several solutions for wi and the
functions ψi of (23) are injective, then there are
also several local solutions for θ, so the system
is not locally identifiable since the identification
problem locally has more than one solution. 2
4. APPLYING THE RESULTS
The techniques discussed above are intended to
be used when examining identifiability for objectoriented models. Since each component must be
transformed into the form
Ai,1 (wi , p), . . . , Ai,nAi (wi , p),
Bi,1 (wi , θi,1 , p), Bi,2 (w, θi,1 , θi,2 , p),
Bi,nθi (wi , θi,1 , θi,2 , . . . , θi,nθi , p),
Ci,1 (wi , θi , li , p), . . . , Ci,nli (wi , θi , li , p), (24)
the first step is to perform these transformations using, e.g., differential algebra (Ljung and
Glad, 1994). The transformed version of the components can then be stored together with the
original model equations in model libraries. As the
transformation is calculated once and for all, it
should also be possible to use other methods than
differential algebra to make the transformation
into the form (24). As mentioned above, this could
make it possible to handle systems described by
non-polynomial differential-algebraic equations.
When an object-oriented model has been composed of components for which the transformation
into the form (24) is known, identifiability of the
complete model,
fi li (t), wi (t), θi , p = 0
i = 1, . . . , m. (25a)
g w(t) = 0,
(25b)
can be checked by examining the solutions of the
differential-algebraic equation
(
i = 1, . . . , m
Aij wi (t), p = 0
(26a)
j = 1, . . . , nAi
g w(t) = 0.
(26b)
The number of solutions to this differentialalgebraic equation then determines if the system
is identifiable, as discussed in Theorem 4 and 5.
Note that the number of solutions could vary with
t, so that the system is identifiable at some time
instances and not at other.
The number of solutions of the differentialalgebraic equation (26) could be checked in different ways, and some are listed below.
5. EXAMPLES
In this section the techniques described in the
paper are exemplified on a minimal model library
consisting of a resistor model, an inductor model,
and a capacitor model. Note that these components have corresponding components for example
within mechanics and fluid systems (c.f., bond
graphs). In this small example, all variables are
external.
The transformation into the form (3) was performed in Section 2.2, so we shall here examine
the identifiability of different connections of the
components.
u
w1
+ w2
w3
+
4.1 Differential Algebra
If the system equations are polynomial, then one
obvious way to check the number of solutions is
to use differential algebra in a similar way as was
done to achieve the form (24). This method can
be slow in some cases, but it always gives definite
answers. However, it some cases this approach
should be faster than to derive the transformation
to the form (24) for the complete object-oriented
model.
Differential algebra can be used to examine both
local and global identifiability, but requires that
the equations are polynomial.
Fig. 1. A resistor and an inductor connected in
series.
Example 6. Consider a nonlinear resistor and an
inductor connected in series where the current
w2 = f and total voltage u are measured as shown
in Fig. 1. Denote the voltage over the resistor with
w1 and the voltage over the inductor with w3 .
Using Examples 2 and 3 we get the equations
ẇ1 = Rw2 w2 , φ(w1 , w2 ) ẇ2
(27a)
ẅ2 w3 = ẇ2 ẇ3
(27b)
for the components and the equation
w1 + w3 = u
4.2 Kunkel & Mehrmann’s Test
Kunkel and Mehrmann (2001) describe a method
to examine the properties of nonlinear differentialalgebraic equations through certain rank tests.
Among other things, it is possible to determine the
number of variables that are differential variables,
and the number of variables that are algebraic
variables. The algebraic variables can be calculated from the equations at each time instant, so
(26) is locally uniquely solvable if all w(t) are
algebraic variables, so that no initial conditions
can be specified.
for the connections. Differentiating the last equation once gives
ẇ1 + ẇ3 = u̇.
For smaller models it may be possible to examine
the solvability of (26) by inspection of the equations and manual calculations. This can of course
not be developed into a general procedure, but
may still be a good approach in some cases.
Manual inspection can be used to check both local
and global identifiability.
(27d)
The system of equations (27) (with w1 , ẇ1 , w3 ,
and ẇ3 as unknowns) has the Jacobian


−Rw2 ,w1 ẇ2 1 0 0

0
0 ẅ2 −ẇ2 


(28)

1
0 1 0 
0
1 0 1
where
Rw2 ,w1 =
4.3 Manual Inspection
(27c)
∂
Rw2 w2 , φ(w1 , w2 ) .
∂w1
(29)
The Jacobian has the determinant −Rw2 ,w1 · ẇ22 +
ẅ2 , so the system of equations is solvable for most
values of the external variables. This means that
the system is identifiable.
Example 7. Now consider two capacitors connected in series where the current w2 = f and
total voltage u are measured as shown in Fig. 2.
Denote the voltages over the capacitors with w1
u
w1
w3
+ w2
+
Fig. 2. Two capacitors connected in series.
and w3 respectively. Using Example 1 we get the
equations
w2 ẅ1 = ẇ1 ẇ2
w2 ẅ3 = ẇ3 ẇ2
(30a)
(30b)
for the components. The connection is described
by the equation
w1 + w3 = u.
(31)
These equations directly give that if
w1 (t) = φ1 (t)
w3 (t) = φ3 (t)
(32a)
(32b)
is a solution, then so are all functions of the form
w1 (t) = (1 + λ)φ1 (t)
w3 (t) = φ3 (t) − λφ1 (t)
(33a)
(33b)
for scalar λ. Since (14b) implies that the capacitance is an injective function of the derivative
of the voltage, this shows that the system is not
identifiable.
6. CONCLUSIONS
This paper has shown how the structure of objectoriented models can be used to simplify examination of identifiability. For components in model
libraries, the transformation to the form (24)
is computed once and for all and stored with
the component. This makes it possible to only
consider a smaller number of equations when
examining identifiability for an object-oriented
model composed of such components. Although
the method described in this paper may suffer
from high computational complexity (depending,
among other things, on the method selected for
deciding the number of solutions for (26)), it can
make the situation much better than when trying
to use to use the differential-algebra approach
described in (Ljung and Glad, 1994) on a complete
model.
The technique could be included in tools for
object-oriented modeling such as Dymola and
Openmodelica. Preferably, this could be part of
a complete set of system identification routines
linked to the modeling software. The identification
routines could either be included directly in the
modeling software, or as external software that
interacts with the modeling software.
Future work could include to examine if it is possible to make the method fully automatic, so that it
can be included in modeling tools and to examine
if other system analysis or design methods can
benefit from the modularized structure in objectoriented models. It could also be interesting to
examine the case when several components share
the same parameter. This could occur for example
if the different parts of the system are affected
by environmental parameters such as temperature
and fluid constants.
REFERENCES
Bellman, R. and K. J. Åström (1970). On
structural identifiability. Mathematical Biosciences 7(3–4), 329–339.
Fritzson, P. (2004). Principles of Object-Oriented
Modeling and Simulation with Modelica 2.1.
Wiley-IEEE. New York.
Glad, S. T. and L. Ljung (1998). Identifiability with constraints. In: NOLCOS 1998. Enschede, the Netherlands. pp. 455–458.
Kunkel, P. and V. Mehrmann (2001). Analysis of over- and underdetermined nonlinear
differential-algebraic systems with application to nonlinear control problems. Mathematics of Control, Signals, and Systems
14(3), 233–256.
Ljung, L. (1999). System Identification - Theory
for the User. Information and System Sciences Series. Second ed. Prentice Hall PTR.
Upper Saddle River, N.J.
Ljung, L. and T. Glad (1994). On global identifiability for arbitrary model parametrizations.
Automatica 30(2), 265–276.
Ritt, J. F. (1966). Differential Algebra. Dover.
New York.
Tiller, M. (2001). Introduction to Physical Modeling with Modelica. Kluwer. Boston, Mass.
Walter, E. (1982). Identifiability of State Space
Models with Applications to Transformation
Systems. Vol. 46 of Lecture Notes in Biomathematics. Springer-Verlag. Berlin, Heidelberg,
New York.
Avdelning, Institution
Division, Department
Datum
Date
Division of Automatic Control
Department of Electrical Engineering
2005-12-05
Språk
Language
Rapporttyp
Report category
ISBN
Svenska/Swedish
Licentiatavhandling
ISRN
Engelska/English
Examensarbete
C-uppsats
D-uppsats
—
—
Serietitel och serienummer
Title of series, numbering
Övrig rapport
ISSN
1400-3902
URL för elektronisk version
LiTH-ISY-R-2710
http://www.control.isy.liu.se
Titel
Title
On Identifiability of Object-Oriented Models
Författare
Author
Markus Gerdin, Torkel Glad
Sammanfattning
Abstract
When estimating unknown parameters, it is important that the model is identifiable so
that the parameters can be estimated uniquely. For nonlinear differential-algebraic equation
models with polynomial equations, a differential algebra approach to examine identifiability
is available. This approach can be slow, so the present paper describes how this method can
be modularized for object-oriented models. A characteristic set of equations is computed for
components in model libraries, and stored together with the components. When an objectoriented model is built using such models, identifiability can be examined using the stored
equations.
Nyckelord
Keywords
Identifiability, Nonlinear systems, Algebraic methods, Object oriented modelling, Modelling,
Identification, Descriptor systems
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Related manuals

Download PDF

advertisement