In chemical industry and biotechnology a
common problem of monitoring and control applications is the general
lack of sensors. In order to overcome this difficulty, model-based
state estimation techniques (e.g. Kalman filters, Luenberger observers)
can be used to estimate state variables, that are not available
from on-line measurements. However, it has to be verified, that
the used sensors are sufficient, namely that the process is observable.
In the case of distributed parameter systems, observability is
affected not only by the choice of sensors, but also by the sensor
location.

In the present contribution we will discuss
the choice of optimal sensor positions for a fixed bed reactor
example. Different observability measures, based on a process
description by ODE's via spatial discretization (Damak *et al*.,
1992; Eising 1984; Kailath, 1980), as well as on the process description
by PDE's (Curtain and Zwart, 1995), will be addressed for locating
optimal sensor positions. Further, investigations will be shown
for the influence of space discretization schemes on observability.

Results will be demonstrated in simulation studies, where state estimation is applied to a fixed bed reactor example.

Curtain, R.F., Zwart, H.J. "An Introduction
to Infinite-Dimensional Linear Systems Theory", Springer-Verlag,
1995.

Damak, T.,Babary, J.P., Nihtila, M.T. "Observer
design and sensor location in distributed parameter bioreactors",
315-320, DYCORD'92, 1992.

Eising, R. "Between controllable and uncontrollable",
Systems and Control Letters, 4, 263-264, 1984.

Kailath, T. "Linear Systems", Prentice-Hall
Information and System Sciences Series, 1980.

The quality control of polymerisation processes is an extremely important but yet unsolved problem. The main purpose is the development of a dynamic structure interconnecting the measured and manipulated variables, so that the process would operate safely, in the most profitable fashion and certain desired processing objectives will be satisfied almost continuously. The fact that polymerisation processes are strongly nonlinear imposes the design and implementation of nonlinear controllers.

Accurate deterministic models, capable of predicting the propagation of all the fundamental state of the process with time have been developed. However, industrial polymerisation processes are subject to stochastic disturbances. Furthermore, it is not always technically feasible (mainly due to lack of robust and reliable sensors) and economically justifiable to measure the controlled variables. Measurement noises, often make the situation more complicated. Another obstacle, especially in batch processes, stem from the fact that most of the model parameters vary significantly with time during the process, due to changes in concentrations, temperature, viscosity, etc. These adverse effects, which sometimes cannot be acceptably overcome, even by the use of an advanced control algorithm, prevent the achievement of the desired control objectives and lead to unsatisfactory system performance and poor product quality.

For all the aforementioned reasons, if one simply solves the dynamic model, starting from some, not completely determined a priori, initial value, it will soon diverge from the real process. Thus, the first problem that must be addressed is the design of an effective non-linear state estimator that allows one to infer the important polymer properties during the reaction from a limited set of noisy measurements, so that necessary feedback control action can be applied. It is proven that the resulting dynamic-stochastic model with feedback information coming from reliable on-line sensors, can achieve excellent on-line tracking of the observable reactor states, providing the control system with its inference to adjust the control effort to erase the effect of the disturbances on product quality.

Several important considerations must be addressed in any state estimation procedure. Firstly, process nonlinearities and continually changing reactor conditions cannot be ignored. Secondly, the state estimator must be formulated in such a way that bias free state estimates can be obtained under all types of unknown model mismatch and process disturbances that might be encountered during implementation. Finally, for the case of a short duration reaction the speed of convergence of the state estimator from initialisation errors is critical.

A well established estimation technique, Extended Kalman Filtering, (EKF) can then be employed to calculate and correct the states. The application of EKF to batch and continuous processes has been studied, especially in the case of process-model mismatch. A modification able to handle multirate sampled data has also been successfully applied. The significance of introducing meaningful stochastic states for the development of a simultaneous state and parameter estimator has been emphasised. Finally, the application of reiterative EKF to batch processes with uncertain initial values has been examined. In all cases, experimental results have been presented, displaying significant improvement.

The basic assumption behind EKF is that
the process can be described adequately by a deterministic model.
If this is not the case, Neural Network (NN) techniques can be
used to model the process and to implement a controller design.
The application of NN on estimation has been presented briefly,
along with a comparison between the EKF estimation techniques
and a Neural Network based estimator.

in a Batch Chemical Reactor

This work deals with the problem of temperature
control of batch reactor processes. Generally the control schemes
used in chemical reactors are complex due to the time dependent
variables, presence of different kind of uncertainties in batch
cycles, utility restrictions and to accomplish safety rules. The
most common control mechanisms adopted in industrial processes,
are single or cascade control systems based in standard proportional-integral-derivative
(PID) algorithms.

Although this type of control is satisfactory
in most situations, new control schemes are being developed to
enhance performance and reliability. The objective of this study
is to explore a new strategy to control the reaction temperature
in highly exothermic reactions.

An artificial neural network (ANN) was developed
to control the batch chemical reactor. The database used to generate
the structure of the controller is obtained from the dynamic simulations
of the process model.

The results obtained from the simulations concerning conventional control system and the artificial neural networks control model show a better performance for the artificial neural network based control system. However, our approach should be proved in a pilot scale process and field results will be the final test of our theoretical previsions.

Due to environmental and safety regulations, growing demands on product quality as well as increasingly competitive markets, the continuous improvement of chemical processes and the development of new ones is an important prerequisite for the success of chemical process industries. Tight time and cost constraints force these industries to continuously reduce their experimental effort during process development and to facilitate, even routinize, the application of model-based process technology. Nevertheless, the effort of setting up a detailed mathematical model for a chemical process remains still high because of the large variety of chemical process units and physical phenomena as well as increasing requirements on the sophistication of models. In order to overcome these limitations as systematization of modeling as well as the development of computer-aided modeling environments is required. Established tools and most of the recent explorative prototypical environments developed by other university groups are confined to the specification and representation of model building blocks and lack support for the process of model development. The modeling environment ModKit that is currently being developed provides high-level knowledge on modeling concepts and offers guidance in order to facilitate the development of detailed mathematical models.

The development of any software tool to support an engineering activity requires the conceptualization of the problem domain which finally results in a sound methodology of process modeling. Such a methodology comprises two principal issues: the definition of model building blocks which can be aggregated to form a consistent model and the development of modeling procedures which support both, the derivation of models from scratch, and the reuse and evolutionary modification of an existing model. The formal representation of model building blocks and modeling procedures requires a data model that supports a declarative description of modeling knowledge. At the same time this data model forms the basis for the design of the knowledge-based modeling environment ModKit. This environment is based on Gensym's knowledge-based software environment G2 which provides procedural, object-oriented, and rule-based programming facilities as well as a sophisticated graphical user interface and an interface for inter-process communication.

Canonical modeling objects for the structural and behavioral description of chemical processes have been developed and implemented in ModKit. A graphical model editor is used to build a chemical process model just by selection, modification and aggregation of modeling objects. Experts as well as less experienced users are supported by a predefined but extensible set of modeling steps, guiding the user through the process of modeling. These modeling steps are defined using a Grafcet approach which is similar to Petri nets. The structural description can be decomposed into arbitrary hierarchical levels. Elementary modeling objects contain the behavioral description in an equation-oriented manner. Moreover, alternative models together with comprehensive hypertext documentation can be represented in order to support the multifacetted nature of modeling.

ModKit is an open environment intended to utilize different well-known and state-of-the-art software tools. First of all, Aspen Technology's simulation software SpeedUp as well as gPROMS from Imperial College have been integrated. A code generator automatically maps ModKit's internal model representation into the representation required by a specific simulator. During a simulation run inter-process communication through G2's Standard Interface GSI is used in order to visualize the simulation results within ModKit.

The development and application of computer-aided
design tools has become increasily important in the past fourty
years and the perspectives for the near future are even more promising.
The present work describes the development of an integrated software
package for the simulation and control of high pressure polyethylene
tubular reactors. The overall goal of this software package is
to build powefull, flexible, adaptive simulation tools for the
prediction of the operating conditions and the molecular properties
of various polyethylene products.

Low Density PolyEthylene (LDPE) and its
copolymers are widely used for a large variety of applications
and commonly produced in either autoclave-type vessels or tubular
reactors. A fairly general reaction mechanism is employed to describe
the complex free-radical kinetics of ethylene polymerization.
The mechanism includes, initiator decomposition, chain initiation
and propagation reactions, chain transfer to monomer, solvent
and polymer, intramolecular transfer, and b-scission of sec and
tert radicals and termination and disproportionation reactions.

The main functions of the package are built
around a comprehensive mathematical model which describes the
important chemical and physical phenomena that occur in this type
of polyethylene reactor. The mathematical model includes the material,
energy and momentum balances of the tubular reactor supported
by the thermodynamics and transport properties of the reacting
mixture evaluation equations. The method of moments is employed
for the solution of the infinite polymer species balances, while
for the case of multi-component polymerization the pseudo-kinetic
rate constants approach is adopted.

The incorporation of an on-line parameter
estimator provides the capabillity of real time prediction of
the molecular properties of produced polymer, the estimation of
control moves of key process variables as well as the prediction
of the operational and product characteristics of alternative
design options. Using the appropriate reactor measurements, the
on-line parameter estimator adjusts certain key process variables
in order to match the actual reactor performance.

A very powerful and user friendly interface is developed under Windows environment, which operates as a front-end for running the various support programs (reactor model and a number of on-line estimators) that do the actual design. A number of databases is available to the user to load automatically whole sets of input data and several state and output variables can be graphically represented. The whole design process is carried out under the Windows environment, offering maximum flexibility, extensive error checking during data input, using plethora of dialog boxes, buttons and controls that aid the engineers to spend their time in the design process.

The dynamics of fixed bed reactors are described
by partial differential equations (PDE's) derived from mass and
energy balances. Either for (dynamic) simulation or for control
design, the PDE's model is commonly reduced to a set of ordinary
differential equations (ODE's) by using approximation methods
(e.g. finite differences, orthogonal collocation,...) (Georgakis
*et al.*, 1977). The approximation procedure may result in
extensive computation studies before obtaining a satisfactory
model approximation. For orthogonal collocation (which presents
the advantage of substantially reducing the number of required
ODE's), in spite of interesting attempts, e.g. Cho and Joseph
(1983), there exists no systematic procedures for choosing the
reduction parameters (like the number of collocation points, or
the value of the parameters and of the Jacobi polynomial, which
influence the location of the discretization points). The systematic
use of orthogonal collocation requires a better understanding
of its properties and of its influence on the discrtetized process
model dynamics. The objective of this paper is to present some
properties of orthogonal collocation concerning the choice of
the Jacobi polynomials for locating the collocation points and
its interpretation as a possible optimal choice, and the properties
of the matrices which represents the approximation of the space
derivatives with respect to the number of collocation points and
the parameters and of the Jacobi polynomial.

Georgakis, C. Aris, R.; Amundson, R. "Studies in the control of tubular reactors - I. General considerations", Chem. Eng. Sci., 32, 1359-1369, 1977.

Cho, Y.S. and Joseph, B**. "**Reduced-order
steady state and dynamic models. Part I and II", AIChE J.,
29(2), 261-276, 1983.

and Demonstration

Centre for Process Analysis, Chemometrics
and Control, University of Newcastle

Statistical Process Control (SPC) is a tool
for achieving and maintaining product quality. Classical univariate
statistical techniques have focused on the monitoring of one quality
variable at a time and are not appropriate for analysing process
data where variables exhibit collinear behaviour. Minimal information
is derived on the interactions between variables which are so
important in complex manufacturing processes. These limitations
are addressed through the application of multivariate statistical
process control (MSPC) to the process. The basis of MSPC are the
projection techniques of principal components analysis and projection
to latent structures. The philosophy behind these approaches is
to reduce the dimensionality of the problem by forming a new set
of latent variables to obtain an enhanced understanding of the
process behaviour. If the variables are highly correlated, then
the process can be defined in terms of a reduced set of latent
variables, which are a linear combination of the original variables.
This talk presents an overview of multivariate statistical process
control. The power of the methodology is demonstrated by application
to an industrial process.

Principal components analysis was applied
to paper quality data and paper machine process data. Four to
seven principal components were required to explain the majority
of the underlying variability. A specialised procrustes rotation
technique was then used to interpret these principal components
as underlying physical phenomena that determine the interrelationships
among the paper quality variables. To do this, process experience
is used to manipulate the principal components into a recognisable
phenomena. For example an experienced paper maker knows that when
the sheet becomes highly oriented, the CD tear will increase while
the MD tear will decrease. In addition, MD tensile and elongation
will increase while the CD properties decrease.

After the principal components have been
identified, the components can then be predicted using the process
variables through principal components regression.

These techniques are used to trace changes in paper quality due to the underlying causes. This can be used to improve the control and/or the operation of the paper machine.

for Fed-batch Fermentation Processes

Fed-batch fermentation processes are characterized
by their nonlinear dynamics, which is in general poorly understood.

Due to the large number of types of microorganisms
that the industry uses in fermentation processes the effort that
is needed to develop deductive dynamic models for simulation,
fault detection and control design seems too large to be overcome
in the near future.

Existing databases of already running processes
containing process operations and conditions can be used for inductive
modelling instead.

This paper describes the use of such databases
mainly for statistical process control (SPC) and fault diagnosis.
Statistical methods such as principal component analysis (PCA)
and projection to latent structures (PLS) provides new latent
variables based on the original data sets. The latent variables
form a basis for on-line monitoring of the process in a reduced
variable space. This dimensionality reduction simplifies the task
of detecting faults and isolating the variables that contain information
about the origin of the fault.

The methods described can also be used for
prediction purposes. The product concentration is one of the key
variables of the process when the performance is to be evaluated,
but it is not measured on-line due to the expenses of on-line
chemical analysis.

One of the ways to evaluate the process
performance it to predict the final product concentration. This
is done using PLS and is thus not more expensive in terms of computer
time than the fault detection. The accuracy of the prediction
has been found to be approximately equal to the accuracy of the
chemical analysis, but is faster and cheaper.

Aristotle University of Thessaloniki, P.O. Box 472, Greece

Advanced process control of polymerisation
processes is of great importance. Polymer manufacturers face increasing
pressures for production cost reductions and more stringent "polymer
quality" requirements. Significant economic advantages can
be obtained by operating polymerisation reactors in an optimal
way such that product uniformity is maintained. The main goals
in operating a polymer reactor include high yield, consistent
product quality and safe operation. To achieve these goals one
needs well-structure control strategies, efficient process monitoring
and reliable polymer characterisation techniques. A number of
limitations have inhibited the polymer reactor monitoring and
control schemes: the lack of on-line sensors for measuring quality
variables, the poor understanding of the process dynamics, the
highly sensitive and non-linear behaviour of polymer reactors
and the difficulty of developing accurate mechanistic models.

Although important batch process issues
have been extensively studied, such as design, scheduling, simulation,
operation planning, optimisation and control, practically very
important aspects, which have not been considered, are the problems
of estimation of reactive impurities and reactor fouling, which
commonly exist in practice. When there are reactive impurities
and reactor fouling, the calculated optimal control policies and
batch ending times are not appropriate and corrective actions
should be taken to prevent off-specification production.

This paper addresses the issues of reactive
impurities and reactor fouling estimation and present a multivariate
statistical approach, based on the statistical projection methods
of Multiway Principal Components Analysis (MPCA) and Multiway
Projection to Latent Structures (MPLS), for on-line estimation
of reactive impurities and reactor fouling at an early stage of
the polymerisation process. These techniques have been successfully
applied to a pilot-scale methyl-methacrylate (MMA) polymerisation
reactor.

A number of limitations have inhibited the
success of batch process monitoring: the finite and variable duration
of a batch, the presence of significant non-linearities, the lack
of on-line sensors for measuring quality variables, the absence
of steady-state operation, the difficulty of developing accurate
mechanistic models, and process measurements that are autocorrelated
in time as well as being correlated with one another. Recent approaches
to the monitoring of batch behaviour have been based on extensions
of the statistical projection methods of Principal Components
Analysis (PCA) and Projection to Latent Structures (PLS) - multi-way
PCA and multi-way PLS. These techniques form the bases of the
multivariate statistical process control charts for batch process
monitoring. The control limits for detecting when a process is
moving out of control for multivariate SPC charts are based upon
Hotelling's T^{2} statistic. A new approach which allows
the nominal data to dictate the form and shape of the bound, the
M^{2} statistic, is reviewed. Finally, an application
of multivariate SPC and the impact the different confidence bounds
have on process operation is highlighted by application to a batch
methyl methacrylate polymerisation reactor.

Multivariate process monitoring is founded
on the technique of principal components analysis (PCA). For non-linear
systems, linear PCA may give misleading or uninformative distinctions
between different operating conditions and mask the occurrence
of process malfunctions. The application of non-linear PCA, in
particular, accumulated non-linear PCA scores has provided a significant
improvement in the separation of different operating conditions/faults.
The methodologies are applied to a polymerisation reactor and
a large industrial metals manufacturing process.

The concept of extracting features from
highly non-linear data has been discussed by a number of researchers,
with most techniques being based upon artificial neural networks.
Of particular interest is the ability of the autoassociative neural
network topology to provide a transformation which has been referred
to as non-linear principal component analysis, NLPCA. The architecture
of this network comprises five layers: input layer, mapping layer,
bottleneck layer, demapping layer and output layer. Kramer proposed
that the outputs of the bottleneck layer were non-linear principal
components. This has been questioned by a number of researchers,
since substituting the sigmoids by linear functions does not result
in the extraction of linear principal components. A more appropriate
terminology is that of non-linear feature analysis. The use of
non-linear features has been shown to successfully describe the
underlying structure of non-linear data. The generation of non-linear
principal components requires the use of the statistical technique,
principal curves. Non-linear Principal Components Analysis can
be used in a similar way to PCA; the identification and removal
of correlations amongst process variables as an aid to dimesionality
reduction, data visualisation and data exploration in non-linearly
correlated data.

The application of non-linear PCA, in particular,
accumulated non-linear PCA scores has provided a significant improvement
in the separation of different operating conditions/faults. The
methodologies are applied to a polymerisation reactor.

It is known that many of the traditional
regression techniques display potential instability so that the
regression results are often sensitive to the selection of the
calibration samples. Some techniques are under investigation in
Newcastle to counteract such problems. These include stabilising
these procedures through bagging regression, stacked regression
and other data augmentation techniques.

The techniques of Principal Components Regression
(PCR) and Projection to Latent Structures (PLS) have been shown
to be able to handle collinear and high dimensional data for a
range of problems. However, of interest to industry is whether
by applying these techniques, acceptable calibration models can
be developed from a minimal number of samples. This paper investigates
the problem of sample sparsity through noise augmentation and
bootstrapping techniques.

One approach for the generation of robust
models is based upon the standard bootstrap. A series of new data
sets are created through the statistical technique of the standard
bootstrapping, i.e. sampling with replacement. A calibration model
is then developed for each data set and the corresponding outputs
are averaged to give the final model.

The data augmentation technique is similar
in concept but it is built upon the addition of noise to the data.
The methodology originated in the field of computerised vision.
More recently, it has also received attention as a method for
improving neural network training. In this report the data set
is augmented with Gaussian noise and twenty new data sets, each
the same size as the original are created. The level of noise
added ranged from 0 to 20% of the variability for each individual
wavelength. A PLS model, in the form of a regression vector is
then calculated for each new data set and the resulting twenty
PLS models are then averaged to produce the final calibration
model. Two alternative approaches were examined, the augmentation
of both the spectra (process) and reference (quality) variables
and the augmentation of the spectra alone.

The ability of the data augmentation techniques and bootstrap regression to produce robust calibration models for specific properties of an industrial data set are investigated on an industrial data-set.

Email: D.Rossiter@Lboro.ac.uk, tel: +1509 222507, fax: +1509 223923.

Control structure selection is an important
step in the design of control schemes for chemical processes.
How to efficiently select manipulable inputs and measureable outputs
for multi-input multi-output processes is what is of interest.

This work presents systematic approaches
for considering how using a theoretical basis the process (and/or
control) engineer can make informed choices for which of the available
inputs should be selected to have the most effect on the outputs.
A measure, the Single Input Effectiveness SIE [1,2], for each
candidate input has been derived and used to rank the inputs.
Other work on the selection of inputs that achieve the best disturbance
rejection structure is also covered [3]. A new indicator the Input-Disturbance
Alignment measure (IDA) is presented. Case studies will be used
to validate the design tools.

*Acknowledgement*

This work funded by the Process Engineering Committee of EPSRC Grant No. GR/K74807.

*References*

1. Y. Cao (1996), PhD Thesis, University
of Exeter, UK.

2. Y. Cao and D. Rossiter (1996),
'An input pre-screening technique for control structure selection',
To appear Computers and Chemical Engineering.

3. Y. Cao and D. Rossiter (1996),
'Input screening techniques for disturbance rejection', Proceedings
of CONTROL'96, Exeter, UK, September 2-5.

DTU, 2800 Lyngby, Denmark, Email: SBJ@ktuserver.kt.dtu.dk

This contribution concerns a case study
on control structure selection for an almost binary destillation
column. The column is energy integrated with a heat pump to transfer
heat from the condenser to the reboiler.

This integrated configuration renders the
possible control structure somewhat different from what is usually
seen. Further the heat pump enables disturbances to propagate
faster through the system. The plant has six possible actuators
of which three must be used to stabilize the system. Hereby three
actuators are left for product purity control and/or pressure
control. A MILP sceening method based on a linear state space
model is used to determine an economically optimal set of controlled
and manipulated variables. The algorithm searches all possible
structures rejects the ones with forces any output beyond a specified
maximum deviation, when the system is subjected to worst case
disturbances. The generated set of inputs and outputs are analysed
with freqency dependent RGA and singular values to determine the
best pairring of the variables in terms disturbance rejection.
The paring and controller design are implemented and evaluated
through nonlinear simulation.

The suggested control structure is also compared to a control structure applied experimentally.

Recycle streams are very common in chemical plants, where integration of units is favoured for economic and enviromental reasons. The design of the control system for process with recycle presents some specific difficulties, because all the units linked by recycle streams must be accounted for simultaneously. Neglecting the effect of the recycle leads to unsatisfactory performance and in some cases instabilities may appear in the closed loop response.

Studies about the effect of the recycle parameters on the overall dynamics have been the object of intense research in the last years [1, 2, 3,]. In [4] alternative process designs are compared, for a typical reactor/stripper configuration, in order to optimize economics and controllability of the plant. As a rule to avoid amplification of disturbances (snow-ball effect) is proposed to fix the flow rate of the recycle loop. In [5] a plant including three distillation columns and two recycle streams is proposed as benchmark.

Detrimental effect of recycle can be eliminated by acting at the stage of design of process equipments, or at the stage of design of the control system by a careful selection of the control configuration and by a proper design of the controller.

In our reserch activity [6,7] the possibility of eliminating the problem at the stage of design of the control system has been faced by analysing the performance of different control systems with and without recycle compensators [8] for SISO and MIMO processes. It is shown that for SISO systems the recycle compensators becomes necessary for a fast soppression of disturbances. For MIMO processes the adoption of a compensator in many cases largely simplifies the design of the control system; in fact the elimination of the effect of the recycle, which is the main cause of interaction, allows satisfactory results even with simple diagonal PI controllers. The robustness of control structures based on the compensator has been investigated for different dynamics of the forward and of the recycle process in the presence of uncertainty and variation of parameters caused by changes of operating conditions.

In the case of complex plants there is not
a general methodology to assist the choice of alternative control
schemes; therefore a technique to analize all the possible combinations
of controlled and manipulated variables (on the basis of a defined
objective function) has been developed and applied on the proposed
benchmark.

References

[1] Luyben, W. L. (1993a). Dynamics
and control of recycle systems. 1. Simple open-loop and closed-loop
systems. Ind. Eng. Chem. Res.; 32, pp.466-475.

[2] Luyben, W. L. (1993b). Dynamics
and control of recycle systems. 2. Comparison of alternative process
design. Ind. Eng. Chem. Res.; f 32, pp.476-486.

[3] Morud J. and S. Skogestad (1994).
Effects of recycle on dynamics and control of chemical processing
plants. Comp. Chem. Engng.; 18, pp.S529-S534.

[4] Luyben, W. L. (1994). Snowball
effects in reactor/separator processes with recycle. Ind. Eng.
Chem. Res.; 33, pg.299-305.

[5] Luyben L. L. and Luyben W. L.
(1995) Plantwide control for a process with three distillation
columns and two recycle streams. Proc. Dycord+95; pp.269-274.

[6] Scali, C. and R. Antonelli (1995).
Performance of different regulators for plants with recycle. Comp.
Chem. Engng.; 19, pp.S409-S414.

[7] Scali, C. and R. Antonelli (1996).
Issues in the control of systems with recycle; proc IFAC'96 World
Congress, S. Francisco, CA (USA); M, pp.91-96. [8] Taiwo, O. (1986).
The design of robust control systems for plants with recycle.
Int. J. Control.; 43, pp.671-678.