Manufacturing Systems Modeling Using Neural Networks
George A. Rovithakis, Vassilis I. Gaganis, Stelios E. Perrakis
and Manolis A. Christodoulou
Department of Electrical & Computer Engineering Technical University
of Crete
73100 Chania, Crete
GREECE
Abstract: In this paper a neural network approach to the factory
dynamics modelling problem is discussed. A recurrent high-order neural
network structure (RHONN) is employed to identify the manufacturing cell
dynamics, which is supposed to be unknown. The model is constructed in
such a way that enables the design of a controller which will force the
model and thus the original cell to display the required behaviour. Buffer
states as well as control input signals are transformed into continuous
ones so as to be conformant with the RHONN assumptions. A case study demonstrates
the approximation capabilities of the proposed architecture.
George A. Rovithakis was born in Chania, Crete, Greece in 1967.
He received the diploma in Electrical Engineering from the Aristotelian
University of Thessaloniki, Thessaloniki, Greece in 1990 and the MSc. and
Ph.D degrees in Electronic and Computer Engineering from the Technical
University of Crete, in 1994 and 1995 respectively. He served as a Teaching
and Research Assistant at the Laboratory of Automation, Department of Electronic
and Computer Engineering, Technical University of Crete. Currently he is
there as a visiting Assistant Professor. His research interests include
Factory Automation, Nonlinear Systems, Robust Adaptive Control, Intelligent
Control and Control of Unknown Systems using Neural Networks, where he
has authored or co-authored over 40 publications in scientific journals,
refereed conference proceedings and book chapters. He is a member of the
IEEE and of the Technical Chamber of Greece.
Vassilis I. Gaganis was born in Chania, Greece, in 1972. He received
his diploma in Mechanical Engineering from the University of Patras, Greece,
in 1995. Currently he works on obtaining the MSc. degree in Electronics
and Computer Science at the Technical University of Crete, Greece. His
current research interests are in dynamic neural networks, adaptive non-linear
control and real-time manufacturing systems scheduling.
Stelios E. Perrakis was born in Chania, Greece, in 1972. He received
his BSc. degree in Computer Science from the University of Crete, Greece,
in 1995. Currently he works on obtaining the M.Sc. degree in Electronics
and Computer Science at the Technical University of Crete. His current
research interests are in dynamic neural networks and computer implementations
for real-time manufacturing systems scheduling.
Manolis A. Christodoulou was born in Kifissia, Greece in 1955. He
received the diploma from National Technical University of Athens, Greece,
the MSc. degree from University of Maryland, the Degree of Engineer from
USC and the Ph. D from Democritus University of Thrace, Greece. After being
at University of Patras, he joined the Technical University of Crete, Greece
in 1988, where he is currently a Professor of Control and Director of the
Laboratory of Automation. He spent two sabbatical years (1987-1989) at
Syracuse University, Syracuse, NY. Professor Christodoulou has authored
and co-authored over 120 journal articles, book chapters and conference
publications in the areas of control theory and applications in Robotics,
Factory Automation and Computer Integrated Manufacturing in Engineering
and recently in neural networks for dynamic system identification and control.
He is the organiser of various conferences and sessions of IEEE and IFAC
and guest editor of various special issues of international journals. Professor
Christodoulou is managing and co-operating on various research projects
in Greece, within the European Union and in collaboration with the USA.
He bears membership to IEEE and the Technical Chamber of Greece. He has
been active in the IEEE Control Systems Society as the Chairman of the
IEEE CS Greek Chapter and as the founder of the IEEE Mediterranean Symposium
on new directions in Control & Automation.
1. Introduction
The manufacturing cell dynamics modeling problem can be stated as follows
:
Factory Dynamics Modeling Problem : Find a mathematical model that
will describe the unknown factory dynamics with a prescribed degree of
accuracy; in other words the error between the actual dynamics and that
of the proposed model should lie within a ``small'' neighborhood of zero.
The dimension of the above mentioned neighborhood is a design requirement
which will strongly affect the behavior of a future developed control law.
Models for factory dynamics usually fall under one of the categories below
-
Discrete Event Dynamical Systems (DEDS)
-
Discrete Time Dynamical Systems (DTDS)
The basic property of the DEDS model is that the events occur at discrete
but unknown times. The above which is very useful in describing some processes,
cannot take into consideration the so called ``fast dynamics'', which is
the system dynamics when viewed microscopically, and cannot be excluded
since it affects directly the quality of the output product or sometimes
leads to non-optimal energy consumption, just to name a few. On the other
hand, DTDS [1] evolve at fixed discrete times with sampling period T. The
``fast dynamics'' can be more easily introduced, and the connection to
the existing control theory becomes more obvious. However, such approaches
suffer from the ``dimensionality'' drawback, that is a huge number of state
variables is required even for a small manufacturing cell. From the above
discussion it becomes apparent that a more generalized model (GM) is needed
in order to better analyze different kinds of processes that exist in the
real world.
On line learning is the main property these GM's should exhibit in order
to become more efficient and to provide high degree of autonomy, where
what we mean by autonomy is ``the power and ability for self governance
in the performance of control functions'' [2]. The above considerations
lead to more complicated models which are more difficult to analyze, but
they are neat and uniform and possibly describe the factory dynamics with
minimal error.
Since we are concerned with the problem of regulating manufacturing cells
e.g forcing the output of such a dynamical system to reach some desired
constant value [3], and assume that we have no {\em a priori} information
about the dynamics governing the cell, all existing techniques do not apply.
Hence, the objective will be to approximate the unknown nonlinear dynamical
system by neural networks [4][5][6][7] and once a model is obtained, to
use it, to develop adaptive laws for on line adjustment of the weights
of these networks, such that the stability of the overall system is guaranteed
and furthermore, the control objective is asymptotically achieved. Therefore,
in this way the problem is transformed into a nonlinear adaptive control
problem, where the uncertainty in the system is due to some unknown parameters
and to the existence of a modeling error term, which always appears in
such identification procedures.
In order that a neural network architecture is able to approximate in some
sense the behavior of a dynamical system such as a manufacturing cell,
it is clear that it should contain some form of dynamics, or put differently,
feedback connections [8]. In the neural network literature, such networks
are known as recurrent neural networks [9], originally designed for pattern
recognition applications. A special category of recurrent neural networks,
namely Recurrent High Order Neural Networks (RHONNs), possess a linear
in the weights property, thus making the issues of proving stability and
convergence feasible and their incorporation into a control loop promising.
A mathematical analysis of the approximation capabilities of a generalized
type of RHONNs is presented in [10].
This paper focusses on the state-space approach to continuous-time recurrent
(dynamic) neural networks for the purpose of nonlinear control and identification.
The inherent dynamics of recurrent networks seems perfectly suited for
control problems, although involving fairly complex analysis [11].
The remaining of the paper is organized as follows: continuous state and
control input signals are defined in Section 2, while the proposed model
is presented in Section 3. In Section 4 adaptive laws are derived, and
stability and convergence properties are explored and established. Finally,
in Section 5 an implementation example is given and results are discussed.
2.Continuous Signals Definitions
RHONNs structure requires that time evolves
continuously. Additionally, state and control input signals are continuous
functions of time. However manufacturing cell dynamics is commonly described
by models which are based on the assumption that time evolves in steps
(DTDS or DEDS), causing any event such as a machine start or an object
production be brought at integer numbers of time steps. Machine commands
are usually encoded as series of ones and zeros (where one may be interpreted
as a command to start operation), and moreover, buffer states are a subset
of the integers. In order to have a successful relation between these two
different strategies, we assume that an equivalent machine-operation frequency
is used as control input to the system. This equivalent frequency is defined
as the inverse of the time between two successive machine-starts. As shown
in Figure 1, we can translate the absolute times in a piece-wise constant
function of the frequency, and obtain an u(t) diagram.
Obviously, using this definition the frequency ranges between zero and
a fixed finite value. The lower bound is equal to zero, which corresponds
to an infinite period of time e.g the time until the machine operates again
is infinite and thus it will never work again. By forcing the controller
to send a zero-value frequency input to a machine, we can model a machine
breakdown, or express the fact that a machine has reached the production
requirements, and thus completed the prespecified work.
The upper bound corresponds to a minimum period of time. This can be measured
by assuming that the machine works continuously. In such a case the period
of time is equal to the machine production time, and the frequency is equal
to its inverse. However, since there is always some idle time between two
consecutive part productions, the frequency input is never expected to
reach this upper bound.
If we define the states to represent buffer levels, they will be discontinuous
functions expected to vary by one unit each time a new product is accumulated
or taken out respectively, from the corresponding buffer. However, if we
consider any part as a fluid which is to be processed by a machine at a
constant rate, then the corresponding input (output) buffers will decrease
(increase) linearly, as shown in Figure 2.
The slope of this decrease is such that the buffer reaches its next integer
level at a time interval equal to the one required by the machine to complete
the specific operation. Such an assumption is implementable, since machine-starts
can be on line detected and operation times are supposed to be known. Although,
using the above definitions time discontinuities seem to smooth out, control
input is still a discontinuous function of time, since frequencies are
piecewise constant. It is clear that such discontinuities render both input
and state signals highly nonlinear, and thus very difficult for the neural
network to learn. This problem can be overcome by applying some implementable
smoothing functions, provided that there is an 1-1 mapping between the
original signal and the smoothed one.
Since the control signal is piecewise continuous, changing from value xa
to a value xb , a smart way to smooth values is to use
two consecutive parabolas, one ranging between xa
and \frac{x_a + a _b} {2} $, while the other ranges between $ \frac{x_a
+ a _b} {2} $ and $ x_b $. Parabolas derivatives should be zero at $x_a$
and $x_b$ respectively, while at their intersection they share a common
derivative value. The unknown coefficients can easily be calculated and
produce the required shape . Although such a smoothing function produces
a delay so as to be implementable (the smoothed value is based on current
or previous values only), it can be adjusted to be ``narrow'' enough so
that the deviation from the original signal is neglectable. A sigmoid function
ranging between $x_a$ and $x_b$, with a steep slope could be used instead.
However, such a technique suffers from derivative discontinuities at common
points between the original signal and the smoothing sigmoidal function.
Both situations are shown in Figure 3.
3.The Manufacturing Cell Dynamic Model
Results
In the following Figure, the feasible but randomly selected machine-command
schedule is presented. Each line represents the schedule for the corresponding
machine, where the raised values denote a machine-start command. The corresponding
frequencies have been calculated and presented in Figure 7. Observe the
last frequency values, which are not equal to zero, since the present schedule
is only a small part of the original one used for the network training.
Buffers states evolution as well as the identifier states with the error
term multiplied by a factor of {500}}, are presented in Figure 8. These
states were calculated by our own simulator, and correspond to the control
input schedule defined above. Observe that products are accumulated in
buffers 3 and 6, thus making the signal to be learned more complex. It
is obvious that buffers state identification error is bounded by a constant
value envelope, from both top and below. This value in any case did not
exceed {\bf{0.0005}}, although only 8 weights per equation were used.
The error magnitude is not expected to reach zero, even if a large-time
schedule is used as training function. This is due to the capabilities
of this certain high order connection selected. Using the derived adaptive
laws, the error reaches the minimum value it can get. In \mbox{order} to
increase the accuracy of the model, higher order terms should be introduced
in the dynamical equations. However, such an extension is not always desired,
since it always leads to an error reduction, but also excites an oscillatory
behaviour when used at a very large scale
Conclusions
A neural network identifier structure for identifying manufacturing cell
dynamics is presented. The proposed equations are non-linear, but still
possess a linear-in-the-weights property. Hence stable adaptive laws can
be derived, which guarantee convergence and stability properties. The emerging
non-linearities do not seem to be a problem since the controller, which
is to be designed later, does not require that the model is linear. Instead,
a non-linear controller is expected to be more flexible and capable of
absorbing any possible disturbances.
The results in Figures 8 and 9 show that the proposed model seems to be
satisfactory enough to describe a complex dynamic system, such as manufacturing
cells. Finally, the current model presents two important properties. First,
it can always be improved by adding new high order terms, in order to increase
accuracy. Second, it is self adjusted in environmental changes, that is,
not only does it need no training whenever a specific machine is no more
capable of producing new parts, but also in case the production requirements
are altered. These two properties are of great importance, since the whole
idea is based on the assumption that the control policy will be decided
on-line.
