THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
IDENTIFICATION OF DYNAMIC SYSTEMS
J.H BALBI - N. BALBI - G. GIROLAMI - P. ORENGA - G. SIMONNOT
Université de Corse - Laboratoire d'Hélioénergétique
Vignola - 20000 AJACCIO - France
Abstract. In this paper, the dominant approach to the modelling of physical
systems is described : it uses local laws and powerful numerical tools. For linear
problems, it leads to eigenvalues and eigenvectors, in a suitable functional
space, from which it is possible to construct the response to any excitation
using the Green's resolvant. This approach has led to important progress in
engineering physics.
Nevertheless, a systemic approach is useful in physics and irreplaceable for living
systems. This second way uses global laws of the phenomenon in addition with
a dynamical identification. of the system using some adequate experiments.
We illustrate this method on the modelling of a solar plant, which is correctly
represented by a simple ordinary differential equation.
INTRODUCTION
the 19" century gave macroscopic physics its laws. The second half of the
20 th century gave it the tools necessary to apply these laws : power-ful computers,
efficient algorithms and a rigorous mathematical basis.
Today's dominant approach to the modelling of physics systems, the local-numerical
approach, uses these laws and tools, which has led to spectacular progress in
all the fiels of engineering physics.
There is nevertheless a second and quite different approach; which could be
called a systemic approach, it presents many advantages when applied to physics
systems and is almost irreplaceable for the study of living systems (ecology,
biology, economics, etc.).
This article presents these two approaches and proposes an example of the use
of the systemic method applied to a solar energy system.
THE LOCAL-NUMERICAL MODELLING OF PHYSICS SYSTEMS
For all the systems of classical physics, a general method of problem solving,
is used most of the time. It consists in writing the local equations describing
the phenomenon, then in solving them numerically in the suitable space-time
domain. This method is also applied to certain problems in biophysics such as
Circulation of the blood, Cherruault (1977) and marine ecosystems ,Nihoul (1982).
An outline of the method follows.
Definition of the system
First, it is usually necessary to describe the geometry of the structure, (domain
and boundary), the phenomena which occur in it, the external actions, the state
variables, and the internal parameters.
Local Equations
Physical phenomena are often described by partial differential equations with
respect to state and space variables. For example, in structure mechanics, the
problem can be represented by :
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THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
domain: M [y (x, t)] + B Ly (x, t)) + K Ly (x, t)] = F (x, 0)
boundary : Ily (s, t)]= f(, 0, a)
where M, B, K, | are differential space operators, F and f express external actions
and y expresses generalized displacements.
Rough Numerical Solution
As this point, most people using this method integrate these equations numerically
by discretisation of space and time variables (finite differences, for example).
This procedure has become a basic tool in physics and sometimes even replaces
actual experimentations.
Eigen-model
For certain difficult and essentially linear problems, the approach can be quite
different. An eigen-problem can be associated with the original one ; for example,
in structure mechanics we can associate the free undamped system (F = 0,
f= 0, B = 0) for wich harmonic vibrations are sought :
y (x, t) ='u(x) exp (jut)
K fu(x)}- w? M [u(x)]= 0 (2)
Ilu( x)]= 0
The set of the solutions [w,., u,.(x)] constitutes the spectrum and the modal base
of the system. From them, it ¥s possible to obtain the response to any excitation
(F, £), using Green's resolvent. These elements form a model of the system :
the eigen- model.
Theoretical Resolution : Variational Formulation
Every time it is possible, it is very important to be able to give a theoretical
solution. Which is a very good guide for a numerical solution. This can be done
by replacing the local formulation (2) by a global one called variational :
alu, v)- w? b (u, v) = 0 ¥veV (3)
in wich V is a Hilbert's functionnal space and a and: b are bilinear forms. In
certain circumstances it is possible to establish the existence and the properties
of the spectrum [uy] and of the modal base [uy], V being a space of infinite
dimension, Doutroy (1984).
Numerical determination of the eigen-model
Basic functions associated to a descretisation of the space variables are chosen
to facilitate numerical computations (triangle functions in the case of finite
elements, Ciarlet (1978)). .
These functions define a finite dimension sub-space for V ; the problem (3)
is then reduced to finding the eigen-values of a matricial problem.
An 4, 7,7 By Uy =o (4)
wich leads to a spectrum [up,y] and a modal base [up, vy), v = , which constitutes
an approximation of the base of the system described in equation (3).
Controlling the Model
Although these methods are very efficient it is nevertheless necessary to control
the validity of the model. By changing certain parameters, it is possible to adjust
the theoretical model to the reality of the experiments. This is in general
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
essential because either consciously or not many simplifications have been made
to find a more or less simple model. In addition to this, the numerical computations
can also bring about errors.
In conclusion, we can say that this general method, justified by rigorous theory
and poweful algorithms, is a very good tool and has given many examples of
success in technology, such as the design of planes using the finite elements
method.
A SYSTEMIC APPROACH : A GLOBAL-EXPERIMENTAL MODELLING
Considering all this, it might seem that there is nothing to” add. However, this
approach has weak points and, thus _ limits. First of all, it cannot generally
be applied to ecosystems, since most of these systems (economics, - biology,
ecology, etc) do not have universal local laws, but only global and empirical
laws. And even in physics systems, the local-numerical method is not necessarily
the only or the best way to tackle a problem. As soon as a complex structure
is dealt with, the method becomes awkward and costly, and can only be used
if it is profitable.
Therefore we propose a very different modelling which can be applied to a large
class of systems in various fields : physics, chemistry, biology, economics, etc.
Principle of the Method
The method concerns systems which are essentially linear within their normal
limits.
It determines a reduced model experimentally and at the same time, takes into
account the global laws of the phenomenon.
Description of the Method
- Definition of the System
By observing reality, the experimenter specifies the limits and the variable
necessary to define the system unambiguously.
The choice of the variables is one of the major difficulties encountered : how
many are necessary ? How many sub-systems must be chosen ? Generally the
answers to these questions can only be obtained by the analysis of the dynamic
behavior.
For physics systems with feedback, it will be necessary to modelize the passive
system obtained by doing away with the feedback, whenever possible. (This can
also do away with major non-linearity).
- Structuring the System
Systems can be structured by a fundamental experiment. The parameters must
be kept constant to make the system invariant. It is then excited, appropriately :
for oscillating structures a harmonic excitation is widely used whereas for dissipative
systems, a step is often chosen. Invariant linear sub-systems, their eigenvalues
and therefore the number of necessary state variables, as well as the relations
among the sub-systems, can be determined by the spectral analysis of the response,
combined withs the knowledge of the phenomena.
- Identification of the System
This Same experiment is performed again with a number of captors equal to
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88 — THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
the number of state variables determined previously in order to measure the
eigenvectors of the sub-systems. Then the experiment is repeated again a certain
number of times in order to determine the relations between the coefficients
used in the model and the physical parameters of the system.
Ordinary differential equations for each sub-system and functional relations
among these sub-systems can then be derived.
% = Alg).x, +B, o es
£, Os Xp oore
where x are the state variables
q, are the parameters of the system
f, are the functional relations and
4, are the external excitations.
For every sub-system, the eigen problem is then
% = AG) x,
Conclusion
The proposed method, like the local numerical method, leads to a reduced model,
but much more directly ; it is also inexpensive and easy to use. In fact, its
slimplicity is an important point in.its favor for its widespread use in various
APPLICATIONS.
Structure Mechanics
This method is widely used in structure mechanics where it is in competition
with the finite element method, Fillod (1985). It is easily applied to this field,
because of advanced phenomenological knowledge, and also because it is possible,
using a sweep in the frequency of the excitation, to the precise eigenvalues
of the reduced model find even in presence of some non-linearity in the phenomenon.
Eco-Systems
Applications to biology are beginning to appear under the name of "compartment
models" Atkins (1969). They are less frequent in economics in spite of some
recent uses, Aracil (1984), Pupion (1980).
Thermal System
This procedure has been adapted to a thermal system in the field of solar energy,
Balbi (1986). This application is intersesting because of the contrast between
the complexity of the system, and the simplicity of the model. The system is
composed of an. array of solar collectors, using 1200 m? of mirrors which focalize
the sun on eight twenty meter long pipes. (fig. 1)
Let us now apply the above method to this particular system.
90 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY, SEVILLA, OCTOBER, 1986
~ Definition of the System
Like most artificial systems, this system is clearly defined. The extenal action
is produced by the solar flux, the out put temperature can be used ‘as a state
variable. Other important parameters are :
the rate of flow of the fluid in the pipes (q)
the input temperature Te of the same fluid,
the ambient temperature, Ta.
The output temperature can be maintained constant by the action of a feedback
loop on the rate of flow q, no matter what the value of the solar flux is.
Due to the propagation delays of the fluid, this kind of command is not very
well suited to the system and must be replaced by a previsional command based
‘on the system model.
Following ‘what was said above the feedback loop was disconnected to maintain
the rate of flow constant during the experiment. T, and T, were also constant.
~ Structuring the Model
A solar flux step can be obtained by simultaneously defocalising :the eight
lines of collectors. The response of the array to this excitation is given in Fig. 2.
The analysis of this curve shows that it is necessary and sufficient to consider
two identical invariant linear subsystems. The output temperature of the array
can then be defined by computing the half sum of the output temperatures
of the two subsystems, each temperature being delayed differently, (tj, t ). Each
subsystem is caracterised by two eigenvalues a and & Here, it is not necessary
to use the notion of eigenvectors,. and because: the sub-systems are identical,
it is possible to describe the system by just one differential equation: of second
order and not two. It is then necessary to include the delays t%, 12 in an equivalent
solar flux 9”.
The final modal can be written
T+ +8) T+oB(T-W=ag"
= 1
PH W= % O-wWs o- a)
Where T is the output temperature,
Tz is this temperature when the array is not focalised,
a is an equivalent optical coefficient.
~ Identification
The dependence of the coefficients (a, B, Ty a,
parameters (q, T., T.) is determined by perfor:
\ ty) with respect to the physical
with different set$ of parameters.
ling’ several simrilar experiments
The model can then provide a correct response even if the parameters are time-
dependent.
The efficiency of the model is’ shown in Fig. 3, where the correspondance. betweeri
the experimental aiid theoretical responses to a variable flux can be seen.
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 ch)
(7-14)
soo 1200 1800
805]
7685]
Solar flue. (w/e?)
735)
645}
605)
Fig 3 - Theoretical and experimental responses 2 a vatlable.solar fluxs
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
= Command
Due to the simplicity of the model, a real time previsional command can be
considered. If it is necessary to maintain the output temperature cosntant :
T = T,. The equation a B(T, ad Ty) =a $* (t) must hold, that is, if Te and To
are cosntant, the parameter q must be of the form :
q ='W (t),which is obtained by solving the implicit equation above ; this can
be done in real time with the help of a microcomputer, thanks to the algebraic
form of this equation. >
CONCLUSION
The local-numerical method has many advantages, but the systemic approach
can nevertheless be very useful for complex andi inexpensive systems, particularly
if automatic control is needed. All’ scientists should be familiar with it. Future
research should attempt to extend it to essentially non-linear systems.
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CHERRUAULT, Y., LORIDON, P., (1977). Modélisation et méthodes mathématiques
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CIARLET, P.G., (1978). The finite element method for elliptic problems.
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