This file documents MTT, a set of Model Transformation Tools.

Copyright (C) Peter J. Gawthrop 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003

Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies.

Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided also that the section entitled ``GNU General Public License'' is included exactly as in the original, and provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one.

Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions, except that the section entitled ``GNU General Public License'' may be included in a translation approved by the author instead of in the original English.

MTT
***

MTT is a set of Model Transformation Tools based on bond graphs. MTT implements the theory to be found in the book ``Metamodelling: Bond Graphs and Dynamic Systems'' by Peter Gawthrop and Lorcan Smith published by Prentice Hall in 1996 (ISBN 0-13-489824-9).

It implements two features not discussed in that book:

  * bicausal bond graphs and

  * hierarchical bond graphs.

Introduction
************

MTT is a set of Model Transformation Tools based on bond graphs.  MTT implements the theory to be found in the book ``Metamodelling: Bond Graphs and Dynamic Systems'' by Peter Gawthrop and Lorcan Smith published by Prentice Hall in 1996 (ISBN 0-13-489824-9).

It implements two features not discussed in that book:

  * bicausal bond graphs and

  * hierarchical bond graphs.

In the context of software, it has been said that one good tool is worth many packages. UNIX is a good example of this philosophy: the user can put together applications from a range of ready made tools. This manual describes the application of this philosophy to dynamic system modeling embodied in MTT - a set of Model Transformation Tools each of which implements a single transformation between system representations.

System representations have two attributes.

  * A Form: e.g. acausal bond graph, differential algebraic, linear state-space etc.

  * A Language: e.g. Fig, Matlab, LaTeX, Reduce, postscript etc.

Transformations in MTT are accomplished using appropriate software (e.g. Octave/Matlab, Reduce) encapsulated in UNIX Bourne shell scripts. The relationships between the tools are encoded in a Make File; thus the user can specify a final representation and all the necessary intermediate transformations are automatically generated.

What is a representation?
=========================

Physical systems have many representations. These include

  * a schematic diagram,

  * a block diagram,

  * a bunch of equations,

  * a single differential(-algebraic) equation,

  * simulation code,

  * linearised state-space (or descriptor) equations,

  * transfer function (of the linearised system),

  * frequency response  (of the linearised system),

  * etc...

Each of these representations is related to other representations by an appropriate transformation (see What is a Transformation?. In many cases, a modeler is presented with a physical system and needs to make a model. In particular, a model, in this context, is a representation of the system appropriate to a particular use, for example:

  * simulation,

  * control system design,

  * optimisation

  * etc.

Indeed, for a given physical system, the modeler would need to derive a number of models. This process can be viewed as a series of steps; each involving a transformation between representations (see What is a Transformation?.

In this context, the following considerations are relevant.

  * There is a unique `core' representation of any system. There are many routes  from this core representation, each leading to an appropriate model. There are many possible routes  to this core representation from the physical system: the route chosen is a matter of convenience.

  * Because the core representation is unique, it is easy to expand the tool-box to include additional transformations from the physical system to the core representation and additional transformations from the core representation to the mode.

  * Transformation_1 probably cannot, and certainly should not, be completely automated.  Engineering insight, knowledge and experience is essential to capture the essence (with respect to the particular use) of the physical system whilst discarding irrelevant form.

  * Representation_1 should be `close' in some sense to the Physical system.

  * The core representation, and hence the representations leading to it, must contain enough information to generate all of the required models.

  * Representations must be easily extensible: it must be possible to add extra components or attributes without restructuring the representation.

I happen to believe that Bond graphs (see Bond graphs) provide the most convenient and powerful basis for the core representation.

What is a transformation?
=========================

Each system representation  (see What is a Representation? is related to other representations by an appropriate transformation as follows:

  * Physical system

  * Transformation_1 ---> Representation_1

  * Transformation_2 ---> Representation_2

  * ...

  * Transformation_N ---> Core representation

  * Transformation_N+1 ---> Representation_N+1

  * Transformation_N+2 ---> Representation_N+2

  * ...

  * Transformation_N+M ---> Model

Thus modeling is seen as a sequence of transformations between representations.

What is a bond graph?
=====================

Bond graphs provide a graphical high-level language for describing dynamic systems in a precise and unambiguous fashion. They make a clear distinction between structure (how components are connected together), and behavior (the particular constitutive relationships, or physical laws, describing each component.

They can describe a range of physical systems including:

  * Electrical systems

  * Mechanical systems

  * Hydraulic systems

  * Chemical process systems

More importantly, they can describe systems which contain subsystems drawn from all of these domains in a uniform manner.

Bond graphs are made up of components (see Components) connected by bonds (see Bonds) which define the relationship between variables (see Variables).

Variables
=========

In bond graph terminology there are four sorts of variables:

  * effort variables

  * flow variables

  * integrated effort variables

  * integrated flow variables

Examples of  effort variables are

  * voltage

  * pressure

  * force

  * torque

  * temperature

Examples of  flow variables are

  * current

  * volumetric flow rate

  * velocity

  * angular velocity

  * heat flow

Examples of integrated  flow variables are

  * charge

  * volume

  * momentum

  * angular momentum

  * heat

Bonds
=====

Bonds connect components (see Components) together. Each bond carries two variables:

  * an effort (see Variables) variable and

  * a flow (see Variables) variable.

Each bond has three notations associated with it:

  * a half-arrow,

  * a causal stroke and

  * a causal half-stroke.

The half-arrow indicates two things:

  * the direction of power (or pseudo power) flow and

  * the side of the bond associated with the flow variable.

The causal stroke indicates two things:

  * the effort variable is imposed at the same end as the stroke and

  * the flow variable is imposed at the opposite end to the stroke.

The causal half-stoke indicates one thing:

  * if it is on the effort side of the bond, the effort variable is imposed at the same end as the stroke or

  * if it is on the flow side of the bond, the flow variable is imposed at the opposite end to the stroke.

Components
==========

Components provide the building blocks of a dynamic system when connected by bonds (see bonds). Components have the following attributes:

ports
     provide the connections to other components (see Ports)

constitutive relationships
     define how the port-variables are related (see Constitutive relationship)

Ports
-----

Components have one or more ports. Each port carries two variables, and effort and a flow variable (see Variables). Any pair of ports can be connected by a bond (see Bonds); this connection is equivalent to saying that the effort variables at each port are identical and that the flow variables at each port are identical.

Ports are implemented in MTT using named SS components. (see Named SS components).

The direction of the named SS components. (see Named SS components) is coerced (see Coerced bond direction) to have the same direction as the bons connected to the corresponding port. Thus the direction of the  direction of the named SS components has no significance unless the component is at the top level.

Constitutive relationship
-------------------------

The constitutive relationship of a component defines how the port variables are related. This relationship may be linear or non-linear. This typically contains symbolic parameters (see Symbolic parameters) which may be replaced, for the purposes of numerical analysis by numeric parameters (see Numeric parameters).

Symbolic parameters
-------------------

The constitutive relationship of a system component (see Components) typically contains  symbolic parameters. For example a resistor may have a symbolic resistance r. It is convenient to leave such parameters as symbols when viewing equations or when performing symbolic analysis such as differentiation.

However, MTT allows replacement of symbolic parameters by numeric parameters (see Numeric parameters) when appropriate.

Numeric parameters
------------------

Numerical parameters are needed to give specific values to symbolic parameters (see Symbolic parameters) for the purposes of numeric analysis; for example: simulation, graph plotting or use within a numerical package such as Octave (see Octave).

Algebraic loops
===============

Following Chapter 3 of the book, algebraic loops appear as under-causal components in the bond graph. It is up to the modeler to indicate how these loops are to be resolved by adding appropriate SS elements.

In particular if zero junction is undercausal an SS:loop component (with effort output indicated by a causal stroke) with the following label file entry:

       loop SS unknown,zero

For more information, refer to: ``Metamodelling: Bond Graphs and Dynamic Systems'' by Peter Gawthrop and Lorcan Smith published by Prentice Hall in 1996 (ISBN 0-13-489824-9).

Switched systems
================

Some systems contain switch-like components. For example an electrical system may contain on-off switches and diodes and a hydraulic system may shut-off valves and non-return valves.

Such systems are sometimes called hybrid systems. The modelling an simulation of such systems is the subject of current research. MTT implements a simple pragmatic approach to the modelling and simulation of such systems via two new Bond Graph components:

ISW
     a switched I component

CSW
     a switched C component

These switches are user controlled through the logic representation (see Simulation logic).

User interface
**************

There are two user interfaces to  MTT: a command line interface (see Command line interface) and a menu-driven interface (see Menu-driven interface).

Menu-driven interface
=====================

The Menu-driven interface for MTT is invoked as:

     xmtt

This will bring up a menu which should be self explanatory :-). Various messages will be echoed in the window from whence xMTT was invoked.

Command line interface
======================

The command line interface for MTT is of the form:

     mtt [options] <system_name> <representation> <language>

[options]
     the (optional) option switches  (see Options)

<system_name>
     the name of the system being transformed

<representation>
     the mnemonic for the system representation (see Representation summary)

<language>
     the mnemonic for language for the representation (see Languages)

for example

     mtt rc rep view

creates a view of the report describing system rc and

     mtt rc sm m

creates an m file (suitlable for Octave or Matlab) containing state matrices describing the system rc.

Options
=======

MTT has a number of optional switches to control its operation. These are invoked immediately after `mtt' on the command line; for example:

     mtt -o -ss -cc syst cbg view

invokes the -o, -ss, and -cc options.

If you wish to use an option  all the time, use the alias function appropriate to the shell you are using. For example, using bash:

     alias mtt='mtt -o -ss -cc'

Means that the previous example can be executed using

     mtt syst cbg view

Available options
-----------------

-A
     solve algebraic equations symbolically

-abg
     start at abg.m representation

-ae <method>
     algebraic equation solver: reduce|hybrd|dassl|hooke

-cc
     C++ code generation

-cr
     use cr before resolving equations

-D
     debug -- leave log files etc

-d <dir>
     use directory <dir>

-dc
     maximise derivative (not integral) causality

-dr <dir>
     use files contained in <dir>

-I
     prints more information

-i <method>
     integration method: implicit|euler|rk4|dassl

-o
     ode is same as dae

-oct
     use oct files in place of m files where appropriate

-opt
     optimise code generation (equivalent to -optl)

-optl
     locally optimise code generation (line-by-line)

-ntmpvar <N>
     declare <N> temporary variables (default=$num_tmp_var)

-p
     print environment variables

-partition
     partition hierachical system

-pdf
     generate pdf in place of ps

-q
     quiet mode -- suppress MTT banner

-r
     reset time stamp on representation

-s
     generate sensitivity BG (use mtt -s sSys rep lang)

-ss
     use steady-state info to initialise simulations

-stdin
     read input data from standard input for  simulations

-sub <subsys>
     operate on subsystem <subsys>

-t
     tidy mode (default)

-trace
     just indicate what mtt will do - but do not do it

-u
     untidy mode (leaves files in current dir)

-v
     verbose mode (multiple uses increment the verbosity)

--version
     print version and exit

--versions
     print version of mtt and components and exit

-viewlevel <N>
     view <N> levels of hierachy

Experimental options
....................

There are some experimental options. These have not yet been heavily tested and should be used with caution.

-optg
     globally optimise code generation (full vector)

-make-sort
     use sorted equations (sese, generated by make)

-no-reduce
     try not to use symbolic algebra

-sort
     use sorted equations (sese, generated by seqn)

Deprecated options
..................

C code generation is now deprecated in favour of C++ generation

-c
     c-code generation

Model specific options
----------------------

It is often desirable to keep the options used to build a model associated with the directory in which the model is contained, along with information about compiler options or paths to component libraries. In these cases, it is convenient to create an executable shell script, say call_mtt which sets environment variables and options before calling mtt proper, for example:

     #! /bin/sh
     # call_mtt: sets model specific environment and options

     ## Processor flags

     # use 32 bit Reduce
     export SYMBOLIC="reduce"

     # set compiler debug and optimization options
     export MTT_CXXFLAGS="${MTT_CXXFLAGS}\
             -Wall-Wno-unused -Wuninitialized\
             -O1 -march=pentiumpro -save-temps"

     ## Model flags

     # use directory ../.. as base directory
     export DIR=${DIR:-${PWD}/../..}

     # additional components are in directory ../comp
     export MTT_COMPONENTS="${MTT_COMPONENTS}:${DIR}/comp"

     # additional constitutive relationships are in ../cr
     export MTT_CRS="${MTT_CRS}:${DIR}/cr"

     # C header files in ../cr/h
     export MTT_CXXINCS="${MTT_CXXINCS} -I${DIR}/cr/h"

     # call mtt with model specific options
     exec ${MTTPATH}/mtt -cc -cr -i euler -D -optl -ntmpvar 1200 $*

which may then be used in place of mtt on the command line,

     ./call_mtt sys odeso gnuplot

Utilities
=========

MTT provides some utilities to help you keep track of model building and to keep things clean and tidy. The commands, and there purpose are:

mtt help
     Lists the help/browser commands

mtt copy <system>
     Copies the system (ie directory and enclosed files) to the current working directory.

mtt rename <old_name> <new_name>
     Renames all of the defining representations (see Defining representations) and textually changes each file appropriately.

mtt <system> clean
     Remove all files generated by MTT associated with system `system'.

mtt clean
     Remove all files generated by MTT associated with all systems within the current directory.

mtt system representation vc
     Apply version control to representation `representation' of system `system'.

mtt system vc
     Apply version control to all representations (under version control) system `system'.

These are described in more detail in the following sections.

Help
----

MTT implements a browser to keep track of all the systems, subsystems and constitutive relationships that you, and others may write. It is invoked in the following ways:

            mtt help representations
            mtt help components
            mtt help examples
            mtt help crs
            mtt help representations <match_string>
            mtt help components <match_string>
            mtt help examples  <match_string>
            mtt help crs <match_string>
            mtt help <component_or_example_or_CR_name>

help representations
....................

The command

     mtt help representations

lists all of the representations  (see Representations) available in MTT. These may change as the version number of MTT increases.

The command

     mtt help representations <match_string>

lists those representation which contain the string match_string. This string can be any regular expression  (see standard Linux documentation under awk). For example

     mtt help representations descriptor

gives all representations containing the word descriptor.

help components
...............

The command

     mtt help components

lists all of the components  (see Components) available in MTT. These may change as the version number of MTT increases.

The command

     mtt help components <match_string>

lists those component which contain the string match_string. This string can be any regular expression  (see standard Linux documentation under awk). For example

     mtt help components source

gives all components containing the word component.

help examples
.............

This command provides a good way to get started in MTT. having found an interesting example, copy it to your working directory using

     mtt copy <example_name>

(see Copy)

     mtt help examples

lists all of the examples  available in MTT. This list will change as more examples are added.

The command

     mtt help examples <match_string>

lists those component which contain the string match_string. This string can be any regular expression  (see standard Linux documentation under awk). For example

     mtt help examples pharmokinetic

gives all examples containing the word pharmokinetic.

help crs
........

The command

     mtt help crs

lists all of the constitutive relationships   (see Constitutive relationship) available in MTT. These may change as the version number of MTT increases.

The command

     mtt help crs <match_string>

lists those constitutive relationships which contain the string match_string. This string can be any regular expression  (see standard Linux documentation under awk). For example

     mtt help crs sin

gives all crs containing the word sin.

help <name>
...........

The command

     mtt help <name>

gives a detailed description of the entity called name.

Copy
----

MTT provides a way of copying examples to your working directory:

     mtt copy <example_name>

Use the  command

     mtt help examples

(see help examples) to find something of interest.

Note that components and constitutive relationships are automatically copied when required.

Clean
-----

MTT generates a lot of representations in a number of languages. Some of these you will edit yourself; others can always be recreated by MTT. It makes sense, therefore to have a utility that removes all of these other files when you have finished actively working with a particular system. These are two versions:

 1. mtt system clean

 2. mtt clean

The first removes all files that can be regenerated with MTT associated with system `system'; the second removes all such files associated with all systems in the current working directory.

The files which remain after such a clean are the Defining representations (see Defining representations).

Version control
---------------

When you are working on a modeling project, it is easy to forget what changes you made to a system and why you made them. Sometimes, you may regret some changes and wish to revert to an earlier version: even if you use .old files this may be difficult to achieve safely.

These are very similar problems to those faced by software developers and can be solved in the same way: using version control.MTT provides version control using the standard GNU Revision Control System (RCS). This is hidden from the user, but is fully complementary to direct use of RCS (e.g. via emacs vc commands) to the more experienced user who wishes to do so.

The only files that you should ever change (i.e. the ones never overwritten by MTT) are the Defining representations (see Defining representations).

All of the files, with the exception of system_abg.fig, are initially created by MTT and contain the RCS header for version control.

The MTT version control will automatically expand this part of the text to include all change comments that you give it -- so will direct use of RCS (e.g. via emacs vc commands)

The MTT version commands are as follows:

mtt system representation vc
     Apply version control to representation `representation' of system `system'.

mtt system vc
     Apply version control to all representations (under version control) system `system'.

The first is appropriate after you have made a revision to a single file.  It will prompt you for a change comment; this will be automatically included in the file header. In addition, enough information will be saved to enable any  version to be retrieved via RCS.

The second is appropriate to record the state of the entire model. This assumes that all relevant files have been recorded by the first version of the command.  Once again, old versions of the entire model can be retrieved using the relevant RCS commands.

A subdirectory `RCS' is created to hold this information. You need not bother about the contents, except that you must not delete any files within `RCS'.

Creating Models
***************

MTT helps you to analyse and transform system models -- ultimately the process of capturing the real world in a model is up to you.  This chapter discusses the MTT aspects of creating a model. For convenience, this is divided into creating simple models and creating complex models.

Quick start
===========

It is probably worth a quick skim though MTT to get a flavour of what it can do before plunging into the detail of the rest of this document. Here is a series of commands to do this.

Copy an initial set of files describing the bond graph.

     mtt copy rc

Move to it.

     cd rc

View the acausal bond graph (the system is called ``rc'').

     mtt rc abg view

View the causal bond graph of the system.

     mtt rc cbg view

View the corresponding ordinary differential equations (ode).

     mtt rc ode view

View the system (output) step response

     mtt rc sro view

An alternative (but more general) way of achieving the same result is

     mtt -cc rc odeso view

View the system transfer function

     mtt rc tf view

View the log modulus frequency response of the system.

     mtt rc lmfr view

View the log modulus frequency response of the system for 100 logarithmically spaced frequencies in the range 0.1 to 10 radians per second.

     mtt rc lmfr view 'W=logspace(-1,1,100);'

MTT has a report generation ((see Report) facility which can generate a hypertext description of the system.

     mtt rc rep hview

The report contents are specified by the rep representation (see Report), in this case the corresponding file is:

     % %% Outline report file for system rc (rc_rep.txt)

     mtt rc abg tex
     mtt rc struc tex
     mtt rc cbg ps
     mtt rc ode tex
     mtt rc ode dvi
     mtt rc sm tex
     mtt rc tf tex
     mtt rc tf dvi
     mtt rc sro ps
     mtt rc lmfr ps
     mtt rc odes h
     mtt rc numpar txt
     mtt rc input txt
     mtt -cc rc odeso ps
     mtt rc rep txt

A non-hypertext version can be viewed using:

     mtt rc rep view

Now have a go at modifying the bond graph.

     mtt rc abg fig

This brings up the bond graph in Xfig (see Xfig).  Try creating a system with two rs and 2 cs.

More examples can be found using

     mtt help examples

Details of an example can be found using

     mtt help <example_name>

and copied using

     mtt copy <example_name>

Lots of examples are available.

     mtt help examples

lists them and

     mtt copy <name>

gets you an example.

Creating simple models
======================

For then purposes of this section, simple models are those which are built up from bond graphs involving predefined components. In contrast, more complex systems (see Creating complex models) need to be built up hierarchically.

The recommended sequence of steps to create a simple model is:

 1. Decide on a name for the system; let us call it `syst' for the purposes of this discussion.

 2. Invoke the Bond Graph editor to draw the acausal Bond Graph.

            mtt syst abg fig

 3. Draw the Bond Graph  (see Language fig (abg.fig)), including the bonds (see Bonds), the components (see Components) and any artwork (see artwork) to make the Bond Graph more readable. The graphical editor xfig is (see Xfig) is self-explanatory. The icon library is helpful here (see icon library).

 4. Add causal strokes (see strokes) where needed to define causality. As a general rule, use the minimum number of strokes needed to define the problem; this will often be only on the SS components. (see SS components).

     Save the bond graph.

 5. View the corresponding causal bond graph.

            mtt syst cbg view

      1. At this stage, MTT will warn you that the labeled components do not appear in the label file - this can safely be ignored.

      2. MTT will indicate the percentage of components which are causally complete -- ideally this will be 100%. Components which are not causally complete will be listed.

      3. A view of the causal bond graph will be created. The added causal strokes are indicated in blue, undercausal components in green and overcausal components in red.

      4. If the bond graph is causally complete, proceed to the next step, otherwise think hard and return to the first step.

 6. At this stage, no constitutive relationships have been defined. Nevertheless, MTT will proceed in a semi-qualitative fashion by assuming that all constitutive relationships are unity (and therefore linear). It may be useful at this stage to view various derived representations to check the overall model properties before proceeding further. For example:

      1. View the system Differential-algebraic equations

               mtt syst dae view

      2. View the system state matrices

               mtt syst sm view

      3. View the system transfer function

               mtt syst tf view

      4. View the system step response

               mtt syst sro view

 7. As well as creating the causal bond graph, MTT has also generated templates for other text files (see Defining representations) used to further specify the system. These can now be edited using your favorite text editor (see Text editors).

 8. MTT will now generate the representations (see Representation summary)that you desire. For example the system can be simulated by

          mtt syst odeso view

     MTT will complain if a component is named in the bond graph but not in the label file and vice versa. This mainly to catch typing errors.

Creating complex models
=======================

Complex models -- in distinction to simple models (see Creating simple models) -- have a hierarchical structure. In particular, bond graph components can be created by specifying their bond graph. Typically, such components will have more than one port (see Ports); within each component, ports are represented by named SS components (see Named SS components); outwith each component, ports are unambiguously identified by labels (see Port labels) and vector labels (see Vector port labels).

Complex models are thus created by conceptually decomposing the system into simple subsystems, and then creating the corresponding bond graphs. The procedure for simple systems (see Creating simple models) is then followed using the top level system (see Top level); MTT then recursively operates on the lower level systems.

The report representation (see Report) provides a convenient way of viewing a complex system.

An example of such a system can be created as follows:

     mtt copy twolink
     mtt twolink rep hview

Top level
---------

The top level of a complex model contains subsystems but is not, itself, contained by other systems. It has the following special features:

  * its name is used in the mtt command as the system name.

  * all named SS componenents (see Named SS components) are treated as ordinary SS components (see SS components).

Simulation
**********

One purpose of modelling is to simulate the modeled dynamic system. Although this is just another transformation (see What is a Transformation?) and therefore is covered in the appropriate chapter (see Representations), it is important enough to be given its own chapter.

Simulation is typically performed using an appropriate simulation language (which is often inappropriately conflated with modelling tools). MTT provides a number of alternative routes to simulation based on the following representations (see Representations):

cse
     constrained-state differential equation form

ode
     ordinary differential (or state-space) equations

in each case these equations may be linear or nonlinear.

Special cases of numerical simulation, appropriate to linear systems, are:

ir
     impulse response - state

iro
     impulse response - output

sr
     impulse response - state

sro
     impulse response - output

There are a number of languages (see Languages) which can be used to describe these representations for the purposes of numerical simulation:

m
     octave a high-level interactive language for numerical computation.

c
     gcc a c compiler.

cc
     g++ a C++ front-end to gcc.

There are a number solution algorithms available:

  * explicit solution via the matrix exponential

  * backward Euler integration (implicit)

  * forward Euler integration (explicit)

  * Runge Kutta IV integration (explicit, fixed step)

  * Hybrd algebraic solver (MINPACK, Octave fsolve)

However, all combinations of representation, language and solution method are not supported by MTT at the moment. Given a system `system', some recommended commands are:

mtt system iro view
     creates the impulse response of a linear system via the system_sm.m representation using explicit solution via the matrix exponential.

mtt system sro view
     creates the step response of a linear system via the system_sm.m representation using explicit solution via the matrix exponential.

mtt -cc system odeso view
     creates the response of a nonlinear system via the system_ode.cc representation using implicit integration.

mtt -cc -i euler system odeso view
     creates the response of a nonlinear system via the system_ode.cc representation using euler integration.

Simulation parameters are described in the system_simpar.txt file (see Simulation parameters).

The steady-state solution of a system can also be ``simulated''(see Steady-state solutions).

Steady-state solutions
======================

Steady-state solutions (odess)
------------------------------

MTT can compute the steady-state solutions of an ordinary differential equation; this used the octave function `fsolve'. The solution is computed as a function of time using the input specified in the input file. The simulation parameter file (see Simulation parameters) is used to provide the time scales.

For example

     mtt copy rc
     cd rc
     mtt rc odess view

Steady-state solutions (ss)
---------------------------

A rudimentary form of steady-state solution exists in mtt. The steady states and inouts are supplied by the user in the file system_simpar.r and the corresponding output and sate derivative computed by MTT using

     mtt system ss view

For example

     mtt copy rc
     cd rc
     mtt rc sspar view
     mtt rc ss view

Simulation parameters
=====================

Simulation parameters are set in the system_simpar.txt file. At the moment this sets the following variables:

  * LAST the last simulation time

  * DT the incremental time (for plotting)

  * STEPFACTOR the number of integration steps per DT -- thus the integration interval is DT/STEPFACTOR

  * WMIN Minimum frequency = 10^WMIN

  * WMAX Maximum frequency = 10^WMAX

  * WSTEPS Number of Frequency steps.

  * INPUT The input index for frequency response

There are a number of solution algorithms

  * Euler basic Euler integration (see Euler integration). This method is simple, but not recommended for stiff systems.

  * Implicit semi-implicit integration  (see Implicit integration) - uses the smx representation to give stability.

  * Runge Kutta IV fixed step Runge Kutta fourth order integration (see Runge Kutta IV integration).

  * Hybrd numerical algebraic equation solver

Euler integration
-----------------

Euler integration approximates the solution of the Ordinary Differential Equation

     dx/dt = f(x,u)

by

     x := x + f(x,u)*DDT

where

     DDT = DT/STEPFACTOR

If the system is linear, stability is ensured if the integer STEPFACTOR is chosen to be greater than the real number

     (maximum eigenvalue of -A)*DT/2

where A is the nxn matrix appearing in

     f(x,u) = Ax + Bu

If the system is non linear, the linearised system matrix A should act as a guide to the choice of STEPFACTOR.

Implicit integration
--------------------

Implicit integration approximates the solution of the Ordinary Differential Equation

     dx/dt = f(x,u)

by

     (I-A*DT)x := (I-A*DT)x + f(x,u)DT

where A is the linearised system matrix. This implies the solution of N (=number of states) linear equations at each sample interval. The OCTAVE version used the `\' operator to solve the set of linear equations, the C version uses LU decomposition.

If the system is linear, stability is ensured unconditionaly. If the system is non-linear, then the method still works well.

This method is nice in that choice of DT trades of accuracy against computation time without compromising stability. In addition, the correct stready-state values are achieved.

This approach can also be used for constrained state equations of the form:

     E(x) dx/dt = f(x,u)

where E(x) is a state-dependent matrix. The approximate solution is then given by:

     (E(x)-A*DT)x := (E(x)-A*DT)x + f(x,u)DT

which reduces to the ordinary differential equation case when E(x)=I.

The _smx representation includes the E matrix.

Runge Kutta IV integration
--------------------------

Runge Kutta IV approximates the solution of the Ordinary Differential Equation

     dx/dt = f(x,t)

by

     x := x + (DT/6)*(k1 + 2*k2 + 2*k3 + k4)

where

     k1 := f(x,t)
     k2 := f(x+(1/2)*k1,t+(1/2)*DT)
     k3 := f(x+(1/2)*k2,t+(1/2)*DT)
     k4 := f(x+k3,t+DT)

The MTT implementation of Runge-Kutta integration is a fourth order, fixed-step, explicit integration method.

For some systems of equations, the increased accuracy of using a fourth order method can allow larger step-lengths to be used than would allowed by the lower order Euler integration method.

It should be noted that during the interemediate calculations (k1...k4), the input vector u is not advanced w.r.t. time; the system inputs are assumed to be constant over the period of the integration step-length.

Hybrd algebraic solver
----------------------

The hybrd algebraic solver of MINPACK (http://www.netlib.org/minpack/hybrd.f), which is used by Octave in the fsolve routine, may be used in conjunction with one of the other integration methods to solve semi-explicit, index 1, differential algebraic equations; these may be generated in MTT models by use of unknown SS Components see SS component labels.

This method requires that compiled simulation code is used; either -cc or -oct. To perform a simulation based on a model sys,

     mtt -cc -ae hybrd -i euler sys odeso view

MTT will attempt to minimise the residual error at each integration time-step using the hybrd routine.

This method of simulation is particularly well suited to stiff systems where very fast dynamics are of little interest. Care must be taken to ensure that an acceptable level of convergence is achieved by the solver for the system under investigation.

Simulation input
================

This is defined in the system_input.txt file. A default file is created automatically by MTT. This is done explicitly by

     mtt system input txt

If the file already exists, the same command checks that all inputs are defined and that all defined inputs exist in the system and promts the user to correct discrepancies.

Inputs are defined by the full system name appearing in the structure file (see Structure (struc)). They can depend on states (again defined by name), time (defined by t) and parameters

For example:

     system_pump_l_1_u	= 4e5*atm;
     system_pump_r_1_u	= 4e5*(t<10)*atm;
     system_ss_i	        = 0*kg;
     system_ss_o	        = 3e-3*kg;
     system_v_1_u	        = (t>10);
     system_v_ll_1_u         = 1;
     system_v_lr_1_u         = (t<10);
     system_v_ul_1_u         = 0;
     system_v_ur_1_u         = (t>10);

Simulation logic
================

This is defined in the system_logic.txt file. A default file is created automatically by MTT. This is done explicitly by

     mtt system logic txt

If the file already exists, the same command checks that the logic corresponding to all switch states (see Switched systems) are defined and that all defined logic exists in the system and promts the user to correct discrepancies.

Logical inputs are defined by the full system name corresponding to MTT_switch components appearing in the structure file (see Structure (struc)) with `_logic' appended. They can depend on states (again defined by name), time (defined by t) and parameters

For example:

     bounce_ground_1_mtt_switch_logic	= bounce_intf_1_mtt3<0;

Simulation initial state
========================

This is defined in the system_state.txt file. A default file is created automatically by MTT. This is done explicitly by

     mtt system state txt

If the file already exists, the same command checks that all states are defined and that all defined states exist in the system and prompts the user to correct discrepancies.

States are defined by the full system name appearing in the structure file (see Structure (struc)). They can depend on parameters. For example

     system_c_l	= (1e4/k_l)/kg;
     system_c_ll	= (1e4/k_s)/kg;
     system_c_lr	= (1e4/k_s)/kg;
     system_c_u	= (1e4/k_l)/kg;

Simulation code
===============

simulation code can be generated by MTT in the form of the ode2odes transformation. This can be produced in a number of languages, including .m, .oct, C and C++ see Languages.

To generate simulation code in C (deprecated):

     mtt -c [options] sys ode2odes c

Similarly, to generate C++ code:

     mtt -cc [options] sys ode2odes cc

To generate an executable based on the C++ representation:

     mtt -cc [options] sys ode2odes exe

Dynamically linked functions
----------------------------

Some model representations can be compiled into dynamically loaded code (shared objects) which are compiled prior to use in other modelling and simulation environments; in particular, .oct files can be generated for use in GNU Octave (see Creating GNU Octave .oct files) and .mex files can be generated for use in Matlab (see Creating Matlab .mex files) or Simulink (see Embedding MTT models in Simulink).  The use of compiled (and possibly compiler-optimised) code can offer significant processing speed advantages over equivalent interpreted functions (e.g. .m files) for computationally intensive procedures.

The C++ code generated by MTT allows the same code to be generated as standalone code, Octave .oct files or Matlab .mexglx files. Although MTT usually takes care of the compilation options, if it is necessary to compile the code on a machine on which MTT is not installed, the appropriate flag should be passed to the compiler pre-processor:

  * -DCODEGENTARGET=STANDALONE

  * -DCODEGENTARGET=OCTAVEDLD

  * -DCODEGENTARGET=MATLABMEX

Simulation output
=================

The view (see Views) representation provides a graphical representation of the results of a simulation; the postscript language provides the same thing in a form that can be included in a document.

These are two simulation output representations

odes
     ordinary differential equation solution (states)

odeso
     ordinary differential equation solution (output)

Particular output variables can be selected by adding a fourth argument in one of 2 forms

'name1;name2;..;namen'
     plot the variables with names na1 .. namen against time

'name1:name2'
     plot the variable with  name2 against that with name 1

An example of plotting a single variable against time is:

     mtt -o -cc -ss OttoCycle odeso ps 'OttoCycle_cycle_V'

An example of plotting one variable against another is:

     mtt -o -cc -ss OttoCycle odeso ps 'OttoCycle_cycle_V:OttoCycle_cycle_P'

Viewing results with gnuplot
----------------------------

Simulation plots may be conveniently selected, viewed with gnuplot (http://www.gnuplot.org) and saved to file (in PostScript format) using the command

     mtt [options] rc odeso gnuplot

This will cause a menu to be displayed, from which states and outputs may be selected for viewing. Clicking on a parameter name will, by default, cause the time history of the selected parameter to be displayed.

As with xMTT (see Menu-driven interface), the Wish Tcl/Tk interpreter must be installed to make use of this feature.

Exporting results to SciGraphica
--------------------------------

Simulation results can be converted into an XML-format SciGraphica (http://scigraphica.sourceforge.net) (version 0.61) .sg file with the command

     mtt [options] sys odes sg

The SciGraphica file will contain two worksheets, X_sys and Y_sys, containing the state and output time-histories from the simulation.

Sensitivity models
******************

The sensitivity model of a system is a set of equations giving the sensitivity of the system outputs with respect to system parameters. MTT has built in methods for assisting with the development of such models.

This feature is experimental at the moment, but the following example gives an idea of what can be achieved.

     mtt copy rc
     cd rc
     mtt -s src ode view
     mtt -s src odeso view

The sensitivity system src is automatically created from the system rc using the predefined sR and sC components together with vector junctions (see Vector components).  The four outputs are the two system outputs plus the two sensitivity functions.

An alternative route is to create the sensitivity functions by symbolic differentiation. The following sensitivity representations are available:

scse
     sensitivity constrained-state equations

sm
     sensitivity state matrices

scsm
     sensitivity constrained-state matrices

Representations
***************

As discussed in What is a Representation?, a system has many representations. The purpose of MTT is to provide an easy way to generate such representation by applying the appropriate sequence of transformations. The representations supported by MTT are summarised in Representation summary.

There is a two-fold division of representations into those with which the user defines the system and its various attributes, and those which are derived from these. The defining representations are listed in Defining representations.

Each representation is implemented in one or more languages depending on its use. These languages are discussed in Languages and are associated with appropriate tools for modifying or viewing the representations.

Representation summary
======================

Some of the the representations available in MTT are (in alphabetical order):

abg
     acausal bond graph

cbg
     causal bond graph

cr
     constitutive relationship for each subsystem

cse
     constrained-state equations

csm
     constrained-state matrices

dae
     differential-algebraic equations

daes
     dae solution - state

daeso
     dae solution - output

def
     definitions - system orders etc.

desc
     Verbal description of system

dm
     descriptor matrices

ese
     elementary system equations

fr
     frequency response

input
     numerical input declaration

ir
     impulse response - state

iro
     impulse response - output

lbl
     label file

lmfr
     loglog modulus frequency response

lpfr
     semilog phase frequency response

nifr
     Nichols style frequency response

numpar
     numerical parameter declaration

nyfr
     Nyquist style frequency response

obs
     observer equations for CGPC

ode
     ordinary differential equations

odes
     ode solution - state

odes
     ODE simulation header file

odeso
     ode solution - output

odess
     ode numerical steady-states - states

odesso
     ode numerical steady-states - outputs

rbg
     raw bond graph

rep
     report

rfe
     robot-form equations

sabg
     stripped acausal bond graph

simp
     simplification information

sm
     state matrices

smx
     state matrices containing explicit states and inputs

sms
     ode

smss
     SM simulation header file

sr
     step response - state

sro
     step response - output

ss
     steady-state equations

sspar
     steady-state definition

struc
     structure - list of inputs, outputs and states

sub
     Executable subsystem list

sub
     LaTeX subsystem list

sympar
     symbolic parameters

tf
     transfer function

A complete list can be found via the help representations command (see help representations).

Many of these representations have more than one language (see Representations) associated with them.

Some of these representations define the system (see Defining representations).

Defining representations
========================

The following representations define the system and therefore must, ultimately, be defined by the user. However, all of these are assigned default values by MTT and may then be subsequently edited (see Text editors) viewed or operated on by the appropriate tools (see Language tools).

system_abg.fig
     the acausal bond graph (see Acausal bond graph (abg))

system_lbl.txt
     the label file (see Labels (lbl))

system_desc.tex
     the description file (see Detailed)

system_simp.r
     algebraic simplifications to make output more readable (see Symbolic parameters for simplification (simp.r))

system_subs.r
     algebraic substitutions to resolve, eq trig. identities (see Symbolic parameters (subs.r))

system_simpar.txt
     simulation parameters (see Simulation parameters)

system_numpar.txt
     numerical parameters (see Numeric parameters (numpar))

system_input.txt
     the system input for simulations (see Simulation input)

system_logic.txt
     the  switching logic for simulations (see Simulation logic)

system_sspar.r
     defines the system steady-state (see Steady-state solutions - symbolic (ss))

Verbal description (desc)
=========================

Systems can be documented in LaTeX using the _desc.tex file. This file is included in the report (see Report) if the abg tex option is included in the rep.txt file.  As usual, MTT provides a default text file to be edited by the user (see Text editors).

Acausal bond graph (abg)
========================

The acausal bond graph is the main input to MTT. It is up to you, as a system modeler, to distill the essential aspects of the system that you wish to model and capture this information in the form of a bond graph.

The inexperienced modeler may wish to look in one of the standard textbooks and copy some bond graphs of  systems to get going.

To create the acausal bond graph of system `sys' in language fig type:

     mtt sys abg fig

To create the acausal bond graph of system `sys' in language m type:

     mtt sys abg m

To view the acausal bond graph of system `sys' type:

     mtt sys abg view

Language fig (abg.fig)
----------------------

A bond graph is made up of:

bonds
     To connect components together.

strokes
     To indicate causality.

components
     Either simple or compound.

artwork
     Irrelevant to the system but useful to the user.

An icon library of bonds, components and other symbols is available within xfig (see icon library).

Icon library
............

A number of predefined iconic symbols are available within xfig.

     Click onto the library icon
     Click onto the library pull-down menu and select BondGraph
     Select iconic symbols from the presented list

Bonds
.....

Bonds are represented by polylines with two segments. They must be the default style (i.e. plain not dashed or dotted). The shortest segment is taken to be the half-arrow. its positioning is significant because:

  * It points in the direction of power flow; thus a bond normally points towards C, I and R components.

  * the corresponding side of the bond indicates flow causality; the other side represents effort causality. This is significant when using causal half-strokes (see strokes). Please adopt the convention of having the half-arrows below horizontal bonds and to the right of vertical bonds.

Strokes
.......

Causal strokes are represented by single-segment polylines. There are two sorts of strokes:

  * Full strokes: these are the usual bond-graph strokes and determine both the effort and flow causality in the usual way. The centre of the stroke should be at about one end of the bond and be at right angles to it.

  * Half strokes: these are an innovation in MTT and allow you to specify the effort and flow causality independently. The end of the stroke should be at about one end of the bond and be at right angles to it. If the causal half-stroke is on the same side as the half-arrow (see bonds) then it determines flow causality; if, on the other hand, it is on the opposite side to the half-arrow (see bonds) then it determines effort causality. Two half strokes on the same, but on opposite sides of the bond are equivalent to a a full stroke at the same end of the bond.

MTT is reasonably forgiving; but a neat diagram will be less ambiguous to you as well as to MTT.

Causality is indicated as follows:

  * Effort is imposed at the same end as the stroke.

  * Flow is imposed at the opposite end as the stroke.

Components
..........

Components are represented by a text string in fig.  The recommended style is: 20pt, Times-Roman and centre justified.

The component text string can be of the following forms:

type
     Just the type of the component is indicated. Components may be either Simple components (see Simple components) or Compound components (see Compound components).  For example:

          R

type:label
     Both the type and the label of the component are given. The type must be a valid name (see Valid names.The name provides a link to more information to be found in see Labels (lbl). For example:

          R:r

type:label:cr
     Not only are the type and the label of the component given, but also the component cr argument. The type must be a valid name (see Valid names.The name provides a link to more information to be found in see Labels (lbl). For example:

          R:r:flow,r

type:label:expression
     Expression is a mathematical expression relating the effort (called mtt_e) to the flow (called mtt_f). For example the following three forms are equivalent

          R:r:mtt_e=r*mtt_f
          R:r:mtt_e-r*mtt_f=0
          R:r:mtt_f=mtt_e/r

     A non-linear example is:

          R:r:mtt_e = sin(mtt_f)

type*n
     The name, together with the number n of repetitions of the component, are given. This repetition only makes sense if the component has an even number of ports (see Port labels); n copies of the component are concatenated with odd Named ports (see Port labels) of the component being connected to the even Named ports of the previous component in the chain in numerical order.  This feature is particularly useful if the component is compound and can be used for, example to give a lumped approximation of a distributed system. For example:

          MySystem*25

type:label*n
     This complete form and is a combination of the simpler forms. For example:

          MySystem:MyLabel*25

Simple components
.................

The following simple components are defined in MTT.

R
     Standard one-port R

C
     Standard one-port C

I
     Standard one-port I

SS
     Source-sensor

TF
     Transformer

GY
     Gyrator

AE
     Effort amplifier

AF
     Flow amplifier

CSW
     Switched one-port C

ISW
     Switched one-port I

SS components
.............

SS components provide input and output variables for a system; Named SS components (see Named SS components) provide this for subsystems.

Simple components - implementation
..................................

Each simple component, with name NAME, is defined by two m files:

NAME_cause.m
     defines the possible causal patterns for the component

NAME_eqn.m
     defines the equations generated

Only the experienced user would normally define simple components - Compound components (see Compound components) are recommended for DIY components.

Compound components
...................

Compound components are systems described by bond graphs and implemented by MTT. They have special SS components, Named SS components (see Named SS components), to indicate connections to the encapsulating system.

Like any other system, they are described by a graphical Bond Graph description (see Language fig (abg.fig)), and a label file (see Labels (lbl)).

By convention, all of the files describing a component live in a directory with the same name as the component.

Named SS components
...................

Named SS components provide the link from the system which defines compound component to the system which uses a compound component see Compound components. A named SS components is of the form SS:[name];

Where `name' is a name consisting of alphanumeric characters and underscore; for example:

     SS:[Mechanical_1]

Each such named SS provides one of the ports (see Ports). The direction of the named SS components. (see Named SS components) is coerced (see Coerced bond direction) to have the same direction as the bond connected to the corresponding port. Thus the direction of the  direction of the named SS components has no significance unless the component is at the top level of a system.

If a named SS component exists at the top level (see Top level) and is treated as an ordinary SS component with the given direction and with the attributes specified in the label file (see Labels (lbl)).

Coerced bond direction
......................

Named SS components (see Named SS components) provide the mechanism for declaring the ports (see Ports) of a component. The corresponding bond has a direction. However, under some circumstances, it may be useful to reverse this direction. MTT provides a coercion mechanism for this: the the direction of the bond attached to the named SS component (see Named SS components) is replaced by the direction of the bond attached to the component port.

Port labels
...........

Most multi-port components have ports see Ports)which display different behaviors; the exception to this is the junction (0 and 1) components. For this reason, MTT provides a method for unambiguously identifying the ports of a multi-port component by port labels.

A port label is indicated by a name within parentheses of the form [name], where `name' is a name consisting of alphanumeric characters and underscore; for example:

     [Mechanical_1]

This provides a label for corresponding to the component to which the nearest bond-end is attached.

The following rules must be be obeyed:

  * If a component has any port labels at all, there must be one for each port of the component.

Port labels may be grouped into vector port labels (see Vector port labels). Components with compatible (ie containing the same number of ports) vector ports may be connected by a single bond (see Bonds); such a bond implies the corresponding number of bonds (one for each element of the vector port label). All such bonds inherit the same direction and any explicit causal strokes (see strokes)

Vector port labels
..................

Port labels (see Port labels) may be grouped into vector port labels of the form [name1,name2,name3].

     [Mechanical_1,Electrical,Hydraulic_5]

Port label defaults
...................

Whether impicitly or explicity, all ports of components (with the exception of 0 and 1 junctions) must have labels  (see Port labels). However, these can be omitted from the bond graph in the following circumstances and default labels are supplied by MTT.

 1. A single unlabled inport defaults to [in]

 2. A single unlabled outport defaults to [out]

These defaults may, in turn be aliases (see Aliases) for port labels (see Port labels) or vector port labels (see Vector port labels).  Combining the default and alias mechanism is a powerful tool for creating uncluttered, yet complex, bond graph models.

Vector Components
.................

Vectors of components can be created in four cases: 0 junctions, 1 junctions, SS components and SS port components.

In each case, the presence of a vector component is indicated by a single port label  (see Port labels) of one of two forms:

 1. containing numerals from 1 to the order of the vector. Thus a vector of 3 components is indicated by a port label of the form [1,2,3].

 2. 1: followed by the order of the vector. Thus a vector of 3 components is indicated by a port label of the form [1:3].

Within the corresponding label file (see Labels (lbl)), the components of a vector port can be accessed using _i where i is the corresponding index. Thus a port SS:[Electrical] appearing near the port label  [1,2,3] could contain the port alias (see Port aliases)

     %ALIAS  in Electrical_1,Electrical_2,Electrical_3

Artwork
.......

You are encouraged to annotate your bond graphs extensively - this makes them an immediately readable document whilst retaining the precise and unambiguous expressive power of the bond graph.

You may add any Fig (see Fig) object to the bond graph as long as it will not be interpreted as part of the bond graph. The reccommended way to acheive this is to put the  Bond Graph at depth 0,10,20 etc (ie depth modulo 10 is zero) and artwork at any other depth.

For compatibility with earlier versions of MTT, the following objects are ignored even at level 0. However, their use is strongly discouraged.

  * Adding text is OK as long as it cannot be confused with components (see components). In particular, you can include invalid component characters such as white space, ", ', ! etc.

  * Adding boxes, arcs etc is always OK.

  * Adding dotted or dashes lines is always OK.

The stripped abg file (sabg) (see Stripped acausal bond graph (sabg)) shows only those parts of the diagram recognised by MTT and is therefore useful for distinguishing artwork.

Valid Names
...........

A valid name is a text string containing alphanumeric characters.  It must NOT contain underscore _, hyphen -, : or *.

The following names should be avoided

     if endif

The following reserved words in reduce should also be avoided (with any case)

     Commands ALGEBRAIC ANTISYMMETRIC ARRAY BYE CLEAR CLEARRULES COMMENT
     CONT DECOMPOSE DEFINE DEPEND DISPLAY ED EDITDEF END EVEN FACTOR FOR
     FORALL FOREACH GO GOTO IF IN INDEX INFIX INPUT INTEGER KORDER LET
     LINEAR LISP LISTARGP LOAD LOAD PACKAGE MASS MATCH MATRIX MSHELL
     NODEPEND NONCOM NONZERO NOSPUR ODD OFF ON OPERATOR ORDER OUT PAUSE
     PRECEDENCE PRINT PRECISION PROCEDURE QUIT REAL REMFAC REMIND RETRY
     RETURN SAVEAS SCALAR SETMOD SHARE SHOWTIME SHUT SPUR SYMBOLIC
     SYMMETRIC VECDIM VECTOR WEIGHT WRITE WTLEVEL

     Boolean Operators EVENP FIXP FREEOF NUMBERP ORDP PRIMEP

     Infix Operators := = >= > <= < => + * / ^ ** . WHERE SETQ OR AND
     MEMBER MEMQ EQUAL NEQ EQ GEQ GREATERP LEQ LESSP PLUS DIFFERENCE MINUS
     TIMES QUOTIENT EXPT CONS Numerical Operators ABS ACOS ACOSH ACOT ACOTH
     ACSC ACSCH ASEC ASECH ASIN ASINH ATAN ATANH ATAN2 COS COSH COT COTH
     CSC CSCH EXP FACTORIAL FIX FLOOR HYPOT LN LOG LOGB LOG10 NEXTPRIME
     ROUND SEC SECH SIN SINH SQRT TAN TANH

     Prefix Operators APPEND ARGLENGTH CEILING COEFF COEFFN COFACTOR CONJ
     DEG DEN DET DF DILOG EI EPS ERF FACTORIZE FIRST GCD G IMPART INT
     INTERPOL LCM LCOF LENGTH LHS LINELENGTH LTERM MAINVAR MAT MATEIGEN MAX
     MIN MKID NULLSPACE NUM PART PF PRECISION RANDOM RANDOM NEW SEED RANK
     REDERR REDUCT REMAINDER REPART REST RESULTANT REVERSE RHS SECOND SET
     SHOWRULES SIGN SOLVE STRUCTR SUB SUM THIRD TP TRACE VARNAME

     Reserved Variables CARD NO E EVAL MODE FORT WIDTH HIGH POW I INFINITY
     K!* LOW POW NIL PI ROOT MULTIPLICITY T

     Switches ADJPREC ALGINT ALLBRANCH ALLFAC BFSPACE COMBINEEXPT
     COMBINELOGS COMP COMPLEX CRAMER CREF DEFN DEMO DIV ECHO ERRCONT
     EVALLHSEQP EXP EXPANDLOGS EZGCD FACTOR FORT FULLROOTS GCD IFACTOR INT
     INTSTR LCM LIST LISTARGS MCD MODULAR MSG MULTIPLICITIES NAT NERO
     NOSPLIT OUTPUT PERIOD PRECISE PRET PRI RAT RATARG RATIONAL RATIONALIZE
     RATPRI REVPRI RLISP88 ROUNDALL ROUNDBF ROUNDED SAVESTRUCTR
     SOLVESINGULAR TIME TRA TRFAC TRIGFORM TRINT

     Other Reserved Ids BEGIN DO EXPR FEXPR INPUT LAMBDA LISP MACRO PRODUCT
     REPEAT SMACRO SUM UNTIL WHEN WHILE WS

Language m (rbg.m)
------------------

The raw bond graph of system `sys' is represented as an m file with heading:

     function [rbonds, rstrokes,rcomponents,rports,n_ports] = sys_rbg

This representation is a half-way house between the fig (see Language fig (abg.fig)) and m (see Language m (abg.m)) representations. It contains the geometric information from the fig file in a form digestible by Octave (see Octave).

The five outputs of this function are:

  * rbonds

  * rstrokes

  * rcomponents

  * rports

  * n_ports

rbonds is  a matrix with

  * one row for each bond (see bonds)

  * columns 1 and 2 containing the x,y coordinates for one end of the bond

  * columns 3 and 4 containing the x,y coordinates for the corner of the bond

  * columns 5 and 6 containing the x,y coordinates for the other end of the bond

rstrokes is  a matrix with (see strokes)

  * one row for each stroke or half-stroke

  * columns 1 and 2 containing the x,y coordinates for one end of the stroke

  * columns 3 and 4 containing the x,y coordinates for the other end of the stroke

rcomponents is  a matrix with (see components)

  * one row for each component

  * columns 1 and 2 containing the x,y coordinates of the component

  * the remaining columns containing fig file information

rports is  a matrix with (see Port labels)

  * one row for each component port that is explicitly labeled

  * columns 1 and 2 containing the x,y coordinates of the port label

  * column 3 contains the port number.

n_ports is the number of ports associated with the system -- i.e. the number of Named SS components (see Named SS components).

Transformation abg2rbg_fig2m
............................

This transformation takes the acausal bond graph as a fig file (see Language fig (abg.fig)) and transforms it into a raw bond graph in m-file format (see Language m (rbg.m)).

This transformation is implemented in GNU awk (gawk). It scans both the fig file (see Language fig (abg.fig)) and the label file (see Labels (lbl)) and generates the rbg (see Language m (rbg.m)) with components sorted according to the label file. It also generates a file sys_fig.fig containing details of the bond graph with the components removed.

Language m (abg.m)
------------------

The acausal bond graph of system `sys' is represented as an m file with heading:

     function [bonds,components,n_ports] = sys_abg

The three outputs of this function are:

  * bonds

  * components

  * n_ports

bonds is  a matrix with

  * one row for each bond

  * the first column contains the arrow-orientated (see Arrow-orientated causality) causality of the effort variable.

  * the second column contains the arrow-orientated (see Arrow-orientated causality) causality of the flow variable.

components is  a matrix with

  * one row for each component

  * one column for each bond impinging on the component. The magnitude of each entry corresponds to the bond number (the appropriate row index of` bonds'); the sign is positive if the bond arrow points into the component and negative otherwise.

n_ports is the number of ports associated with the system -- i.e. the number of Named SS components (see Named SS components).

Arrow-orientated causality
..........................

The  arrow-orientated causality convention assigns -1, 0 or 1 to both the effort and flow (see Variables) sides of a bond to represent the causal stroke (see strokes) as follows:

0
     if there is no causality set.

1
     if the causal stroke is at the arrow end of the bond.

-1
     if the causal stroke is at the other end of the bond.

see Component-orientated causality.

Component-orientated causality
..............................

The  component-orientated causality convention assigns -1, 0 or 1 to both the effort and flow (see Variables) sides of a bond to represent the causal stroke (see strokes) as follows:

0
     if there is no causality set.

1
     if the causal stroke is at the component end of the bond.

-1
     if the causal stroke is at the other end of the bond.

see Arrow-orientated causality.

Transformation rbg2abg_m
........................

This transformation takes the raw bond graph and, by doing some geometrical computation, determines the topology of the bond graph -- ie what is close to what.

Language tex (abg.tex)
----------------------

For the purpose of producing a report (see Report), MTT generates a LaTeX (see LaTeX) file describing the bond graph and its subsystems. Additional information may be supplied using the description representation (see Detailed).

Stripped acausal bond graph (sabg)
==================================

The stripped acausal bond graph is the acausal bond graph representation (see Acausal bond graph (abg)) without the artwork (see artwork). It is useful to check for mistakes by showing precisely what is recognised by MTT.

Language fig (sabg.fig)
-----------------------

The stripped acausal bond graph can be generated as a fig (see Fig) file using

     mtt syst sabg fig

Stripped acausal bond graph (view)
----------------------------------

This representation has the standard text view (see Views).

Labels (lbl)
============

Bond graph components have optional labels. These provide pointers to further information relating to the component; this avoids clutter on the bond graph.

The label file contains the following non-blank lines (blank lines are ignored)

  * Summary - lines beginning with #SUMMARY

  * Description - lines beginning with #DESCRIPTION

  * Alias - lines beginning with #ALIAS

  * Comments - lines beginning with #

  * Labels - other non-blank lines

Note, for compatability with old versions, % may be used in place of #; but the use of % is deprecated. Each label contains three fields (in the following order) separated by white space and on one line:

 1. The component name see Component names. This must be a valid name  (see Valid names.

 2. The component constitutive relationship see Component constitutive relationship

 3. The component arguments see Component arguments

Not each component see components needs a label, only those which are explicitly labeled on the Bond Graph see Acausal bond graph (abg). MTT checks whether all  components labelled on the bond graph have labels and vice versa.

If no lbl file exists, MTT will create a valid one for you; including a default set of arguments and crs for both simplae and compound components.

If wish to create one to edit yourself, type

     mtt system_name lbl txt

An example lbl file (for the RC system is):

     %% Label file for system RC (RC_lbl.txt)
     %SUMMARY RC
     %DESCRIPTION <Detailed description here>
     % Port aliases
     %ALIAS  in      in
     %ALIAS  out     out

     % Argument aliases
     %ALIAS  $1      c
     %ALIAS  $2      r

     %% Each line should be of one of the following forms:
     %            a comment (ie starting with %)
     %            component-name     cr_name arg1,arg2,..argn
     %            blank

     % ---- Component labels ----

     % Component type C
             c               lin     effort,c

     % Component type R
             r               lin     flow,r

     % Component type SS
             [in]    SS              external,external
             [out]   SS              external,external

The old-style lbl files (see Old-style labels (lbl)) are NO LONGER supported -- you are encouraged to convert them ASAP.

SS component labels
-------------------

In addition to the label there are two information fields, see Labels (lbl). The first must be `SS', the second contains two information fields of the form info_field_1,info_field_2.

These two information fields correspond to the effort and flow variables of the of the SS components as follows

info_field_1
     effort

info_field_2
     flow

Each of these two fields contains one of the following attributes:

external
     indicates that the corresponding variable is a system input or output

internal
     indicates that the variable does not appear as a system output; it is an error to label an input in this way.

a number
     the value of the input; or the value of the (imposed) output

a symbol
     the symbolic value of the input; or the value of the (imposed) output

unknown
     used for the SS method of solving algebraic loops. This indicates that the corresponding system input (SS output) is to be chosen to set the corresponding system output (SS input) to zero.

zero
     used for the SS method of solving algebraic loops. This indicates that the corresponding system output (SS input) is to be set to zero using the variable indicted by the corresponding `unknown' label.

Some examples are:

     %% ss1 is both a source and sensor
     ss1     SS              external,external
     %% ss1 acts as a flow sensor - it imposes zero effort.
     ss2     SS              0,external

Other component labels
----------------------

In addition to the label there are two information fields, see Labels (lbl). They correspond to the constitutive relationship (see Constitutive relationship and arguments of the component as follows

info_field_1
     constitutive relationship

info_field_2
     parameters

Some examples are:

     %Armature resistance
     r_a     lin     effort,r_a

     %Gearbox ratio
     n       lin     effort,n

MTT supports parameter-passing to  (see Parameter passing) subsystems.

Component names
---------------

The component name field must contain a valid name  (see Valid names corresponding to the name (the bit after the :) of each named component (see components) on  the bond graph (see Acausal bond graph (abg)).

Component constitutive relationship
-----------------------------------

The constitutive relationship field contains the name of a constitutive relationship for the component. There are three sorts of constitutive relationship recognised by MTT:

 1. A generic constitutive relationship such as LIN (the generic linear constitutive relationship.

 2. A local constitutive relationship with the same name as the component type

 3. The SS constitutive relationship reserved for SS components. All labels for SS components must contain SS in this field.

Component arguments
-------------------

Parameter declarations
----------------------

It is sometimes useful to use parameters (in addition to those implied by the Component arguments see Component arguments) to compute values in, for example the numpar file. These can be declared in the label file; for examples , the two parameters par1 and par 2 can be declared as:

     #PAR par1
     #PAR par2

On the other hand, some CR arguments (eg foo and bar) may not correspond to parameters. These can be excluded from the sympar list  using the NOTPAR declaration

     #NOTPAR foo
     #NOTPAR bar

For comapability with old code, VAR may be used in place of PAR, but this usage is deprecated.

Units declarations
------------------

The units and domains of ports (see Ports) are declared as:

     #UNITS Port_name domain effort_units flow_units

where "Port_name" is the name of the port, domain is one of:

electrical
     the electrical domain

translational
     the translational mechanical domain

rotational
     the rotational mechanical domain

fluid
     the fluid domain

thermal
     the thermal domain

and effort_units and flow_units are corresponding units for the effort and the flow.

Allowed units are those defined in the units package.

MTT checks that units are

  * defined consistently with the domain

  * the same for connected ports when both ports have defined units.

No checks are done if one or both ends of a bond are not connected to a port with defined units.

The word ``none'' can be specified in place of a unit to prevent MTT from checking the corresponding effort or flow while still checking the units of the other variable, if it is specified. This can be used to force checking of either the effort or flow on signal bonds or in pseudo-bond graphs where the domain of the two variables may not be identical.

Interface Control Definition
----------------------------

It is sometimes useful to be able to automatically generate a set of assignments mapping MTT inputs and outputs to an external interface definition. This can be achieved with use of the #ICD directive.

     #ICD    PressureSensor		PUMP1_PRESSURE_SENSOR,Pa;null,none
     #ICD    Electrical		PUMP1_VOLTAGE,volt;PUMP1_CURRENT,amp

     % Component type De
     	PressureSensor	SS      external

     % Component type SS
     	Electrical	SS	external,external

The ICD directive consists of 3 whitespace delimited fields:

 1. [%|#]ICD

 2. component name

 3. Four comma (,) or semi-colon (;) delimited fields:

      1. name of effort parameter

      2. unit of effort parameter

      3. name of flow parameter

      4. unit of flow parameter

If no parameter name is required, a value of "null" should be used. If the parameter does not have any units, a value of "none" should be used.

ICD parameters may be aliased see Aliases in the same way as normal parameters, thus it is possible to define some or all of the ICD in higher level components.

The command

     mtt sys ICD txt

will generate a text file containing a list of mappings:

     ## Interface Control Definition for System sys
     ## sys_ICD.txt: Generated by MTT Thu Jul 12 21:21:21 CDT 2001

     Input:  PUMP1_VOLTAGE           sys_P1_1_Electrical      Causality: Effort   Units: volt
     Output: PUMP1_CURRENT           sys_P1_1_Electrical      Causality: Flow     Units: amp
     Output: PUMP1_PRESSURE_SENSOR   sys_P1_1_PressureSensor  Causality: Effort   Units: Pa

A set of assignments can be generated with the command

     mtt sys ICD m

resulting in:

     # Interface Control Definition mappings for system sys
     # sys_ICD.m: Generated by MTT Thu Jul 12 21:26:56 CDT 2001

     # Inputs

             mttu(1) = PUMP1_VOLTAGE;

     # Outputs

             PUMP1_CURRENT                  = mtty(1);
             PUMP1_PRESSURE_SENSOR          = mtty(2);

A similar file will be generated by the command

     mtt sys ICD cc

Aliases
-------

Aliases provide a convenient mechanism for relabelling words appearing in the label file (see Labels (lbl)). There are three contexts in which the alias mechanism is used:

 1. renaming ports (see Port aliases),

 2. renaming parameters (see Parameter aliases) and

 3. renaming components (see Component aliases).

All three mechanisms use the same form of statement within the label file

     %ALIAS short_label       real_label

MTT distinguishes between the three forms as follows:

  * Parameter aliases: `short_label' starts with a `$'

  * Component aliases: `real_label' contains the directory separator `/'

  * Port aliases: neither of the above

Port aliases
............

Aliases provide a way of refering to (see Port labels) or vector port labels (see Vector port labels) on the bond graph using a short-hand notation. With in a component label file (see Labels (lbl)) statements of the following forms can occur

     %ALIAS short_label       real_label

When the component is used within another component, the short_lable may be used in place of the real_label. More than one alias per label can be used, for example

     %ALIAS short_label_1       real_label
     %ALIAS short_label_2       real_label
     %ALIAS short_label_3       real_label

The port can then be refered to in four ways: as real_label, short_label_1, short_label_2 or short_label_3. An alternative notation for the ALIAS statement in this case is

     %ALIAS short_label_1|short_label_2|short_label_3       real_label

The alias feature is particularly powerful in conjunction with vector port labels (see Vector port labels) and the port label default (see Port label defaults) mechanisms. For example, a component with 5 ports appearing in the lbl file as:

             [Hydraulic_in]  external        external
             [Hydraulic_out] external        external
             [Power_Shaft]           external        external
             [Thermal_in]    external        external
             [Thermal_out]   external        external

together with the following statements in the label file:

     %ALIAS  in              Thermal_in,Hyydraulic_in
     %ALIAS  out             Thermal_out,Hydraulic_out
     %ALIAS  shaft|power     Power_Shaft

can appear in the bond graph containing that component with one bond labeled either [shaft] or [power] or [Power_Shaft], one unlabeled vector bond pointing in and one unlabeled vector bond pointing out.

Parameter aliases
.................

Parameter aliases are of the form

     %ALIAS $n       actual parameter

where n is an integer (unique within the label file). For example

     %ALIAS  $1              c_v
     %ALIAS  $2              density,ideal_gas,r
     %ALIAS  $3              alpha
     %ALIAS  $4              flow,k_p

Assigns four symbolic parameters to the corresponding strings These four parameters ($1--$4) can then be used for parameter passing(see Parameter passing).

CR aliases
..........

CR aliases are of the form

     %ALIAS $an       actual parameter

where n is an integer (unique within the label file). For example

     %ALIAS  $a1  lin

assigns the symbolic parameter to be lin. This parameter $1 can then be used for passing a diofferent cr to the component (see Parameter passing).

Component aliases
.................

Component aliases are of the form

     %ALIAS Component_name   Component_location

An example appears in the following label file fragment

     ...
     %ALIAS  wPipe   CompressibleFlow/wPipe
     %ALIAS  Poly    CompressibleFlow/Poly
     ....

The two components `wPipe' and `Poly' are both to be found within the library `Compressible flow' and the respective subdirectories. This follows the MTT convention that compound components (see Compound components) live within a directory of the same name.

Parameter passing
-----------------

MTT supports parameter-passing to subsystems within label files (see Labels (lbl)). Within a subsystem, explicit constitutive relationships and parameters (or groups thereof) can be replaced by postitional parameters such as $1, $2 etc.  Although this can be done directly, it is recommended that this is done via the alias mechanism (see Parameter aliases).

In a subsystem $i, is replaced by the ith field of a colon ; separated field in the calling label file. This field may include commas , and the four arithmetic operators +, -, * and /.

For example, consider the following example label file fragment (associated with a component called Pump:

     ...

     %ALIAS  $1              c_v
     %ALIAS  $2              density,ideal_gas,r
     %ALIAS  $3              alpha
     %ALIAS  $4              flow,k_p

     %ALIAS  wPipe   CompressibleFlow/wPipe
     %ALIAS  Poly    CompressibleFlow/Poly

     % Component type wPipe
             pipe    none                    c_v;density,ideal_gas,r

     % Component type Poly
             poly            Poly            alpha

The 4 parameters $1, $2, $3, and $4 can be passed from a higher level component as in the following label file fragment:

     % Component type Pump
             comp            none            c_v;rho,ideal_gas,r;alpha;effort,k_c
             turb            none            c_v;rho,ideal_gas,r;alpha;effort,k_t

Thus in component `comp':

  * $1 is replaced by c_v

  * $2 is replaced by rho,ideal_gas

  * $3 is replaced by alpha

  * $4 is replaced by effort,k_c

whereas in component `turb' the first three parameters are the same but

  * $4 is replaced by effort,k_t

Old-style labels (lbl)
----------------------

Old syle labels (mtt version 2.x) are supported by mtt version 3.x. However, you are advised to use the new form (see Labels (lbl)).

Each line of the _label.txt file is of one of three forms:

 1. Contains three fields (separated by white space) of the form

          label   field_1   field_2

 2. Blank

 3. Preceded by %

Only the first is noticed by MTT; the second and third are for providing helpful commenting.

The role of the two information fields depends on the component with the corresponding label. In particular the classes of components are:

  * SS components, see SS components.

  * Other components,  see components.

Named SS component, see Named SS components never have labels.

SS component labels (old-style)
...............................

In addition to the label there are two information fields, see Labels (lbl). They correspond to the effort and flow of the components as follows

info_field_1
     effort

info_field_2
     flow

Each of these two fields contains one of the following attributes:

     external indicates that the corresponding variable is a system input or output

internal
     indicates that the variable does not appear as a system output; it is an error to label an input in this way.

a number
     the value of the input; or the value of the (imposed) output

a symbol
     the symbolic value of the input; or the value of the (imposed) output

unknown
     used for the SS method of solving algebraic loops. This indicates that the corresponding system input (SS output) is to be chosen to set the corresponding system output (SS input) to zero.

zero
     used for the SS method of solving algebraic loops. This indicates that the corresponding system output (SS input) is to be set to zero using the variable indicted by the corresponding `unknown' label.

Some examples are:

     %Label  field1          field2
     ss1     external        external
     ss2     0               external

Other component labels (old-style)
..................................

In addition to the label there are two information fields, see Labels (lbl). They correspond to the constitutive relationship (see Constitutive relationship and arguments of the component as follows

info_field_1
     constitutive relationship

info_field_2
     parameters

Some examples are:

     %Armature resistance
     r_a     lin     effort,r_a

     %Gearbox ratio
     n       lin     effort,n

MTT supports parameter-passing to  (see Parameter passing (old-style)) subsystems.

Parameter passing (old-style)
.............................

MTT supports parameter-passing to  (see Parameter passing (old-style)) subsystems within label files (see Labels (lbl)). Within a subsystem, explicit constitutive relationships and parameters (or groups thereof) can be replaced by $1, $2, etc.

In a subsystem $i, is replaced by the ith field of a colon ; separated field in the calling label file. This field may include commas ,.

For example subsystem ROD contains the following lines in the label file:

     %DESCRIPTION    Parameter 1:    length from end 1 to mass centre
     %DESCRIPTION    Parameter 2:    length from end 2 to mass centre
     %DESCRIPTION    Parameter 3:    inertia about mass centre
     %DESCRIPTION    Parameter 4:    mass
     %DESCRIPTION    See Section 10.2 of "Metamodelling"

     %Inertias
     J       lin     flow,$3
     m_x     lin     flow,$4
     m_y     lin     flow,$4

     %Integrate angular velocity to get angle
     th

     %Modulated transformers
     s1      lsin    flow,$1
     s2      lsin    flow,$2
     c1      lcos    flow,$1
     c2      lcos    flow,$2

This can be used in a higher-level lbl (see Labels (lbl)) file as:

     %SUMMARY Pendulum example from Section 10.3 of "Metamodelling"

     %Rod parameters
     rod     none    l;l;j;m

Language tex (desc.tex)
-----------------------

This file may contain any LaTeX compatible commands. Any mathematics should conform to the AMSmath package.

Structure (struc)
=================

The causal bond graph implies a set of equations describing the system. The Structure (struc) representation describes the structure of these equations in terms of the input, outputs, states and non-states of the system.

Language txt (struc.txt)
------------------------

This text tile contains a description of the system structure (see Structure (struc) with 5 tab-separated columns containing the following information:

type
     input, output state or nonstate

     index an integer corresponding to the array index

     component name the name of the component corresponding to the variable

system name
     the name of the system containing the component

repetition
     an integer corresponding to the repetition of a repeated subsystem.

An example of such a file (corresponding to rc) (see Quick start) is:

     input           1       e1      rc      1
     output          1       e2      rc      1
     state           1       c       rc      1

Language tex (struc.tex)
------------------------

This LaTeX  (see LaTeX) file contains  a description of the system structure (see Structure (struc) in longtable format. It is a useful item to include in a report(see Report).

Language tex (view)
-------------------

This representation has the standard text view (see Views).

Constitutive relationship (cr)
==============================

The constitutive relationship (see Constitutive relationship) of a simple component (see Simple components is defined in the symbolic algebra language Reduce (see Reduce). The constitutive relationship of a compound components (see Compound components) is implied by the constitutive relationships of its constituent components.

Predefined constitutive relationships
-------------------------------------

Some common cr's are predefined by MTT; these are:

lin
     a linear constitutive relationship

exotherm
     an exothermic reaction

lin
...

The constitutive relationship lin is predefined for the following components.

R
     (one-port) R component

TF
     transformer

GY
     gyrator

MTF
     modulated transformer

MGY
     modulated gyrator

FMR
     flow-modulated resistor

Lin takes two arguments in the form causality,gain

causality
     the causality (effort or flow) of the input to the constitutive relationship

gain
     the gain of the component when the input causality is as specified in the first argument.

For example the arguments

     flow,r

given to an R component corresponds to

     e = rf

if if the input causality is flow or

     f = e/r

if if the input causality is effort.

exotherm
........

DIY constitutive relationships
------------------------------

You can write your own constitutive relationships using Reduce (see Reduce). This requires some understanding as to how MTT represent the elementary system equations (see Elementary system equations). Looking at the predefined constitutive relationships is a good way to get started (see File structure).

Unresolved constitutive relationships
-------------------------------------

Consider the following CR file.

     FOR ALL rho,g,vol,h,topt,bott,flowin,press
     LET tktf2(rho,g,vol,h,topt,bott,effort,2,press,effort,1)
             = tank(rho,g,vol,h,topt,bott,press);

Assuming that `tank' is not defined in a reduce file, MTT will leave it unresolved when generating m or c code.

The resulting function can then be expressed as octave (see Unresolved constitutive relationships - Octave) or c++ code as (see Unresolved constitutive relationships - c++) appropriate.

Unresolved constitutive relationships - Octave
----------------------------------------------

Following the example of the previous section, the unresolved CR `tank' can be expressed as an Octave m-file. For example:

     function p = tank (rho,g,vol,h,topt,bott,press)

       ## usage:  p = tank (vol,h,topt,bott,press)
       ##
       ##

        val = press; zt = topt; zb = bott;
        zval = 0.5*(abs(zb+(zt-zb)*val-h)+(zb+(zt-zb)*val-h));

        p = rho*g*zval + 0.5*(1+tanh((press-0.98)*500))*100000;

     endfunction

This will be automatically loaded into octave.

Unresolved constitutive relationships - c++
-------------------------------------------

Following the example of the previous section, the unresolved CR `tank' can be expressed in c++ code. For example:

     inline double tank(const double rho,
     		   const double g,
     		   const double vol,
     		   const double h,
     		   const double topt,
     		   const double bott,
     		   const double press)

       /*  ## usage:  p = tank (vol,h,topt,bott,press)
         ##
         ##
       */
       double p, val, zval, zt, zb;

       val = press;
       zt = topt;
       zb = bott;
       zval = 0.5 * (abs(zb + (zt - zb) * val - h) + zb + (zt - zb) * val - h);

       p = rho * g * zval + 0.5 * (1 + tanh((press - 0.98) * 500)) * 100000L;

       return p;

To make sure that this is used in system `model', the model_cr.h file must be as follows:

     // CR headers for system model
     #include "tank.c"

Parameters
==========

In general, lbl (see Labels (lbl)) files contain symbolic parameters. MTT provides three ways of substituting for these parameters:

  * symbolic substitution

  * symbolic substitution for simplification of displayed equations

  * numeric

Symbolic parameters (subs.r)
----------------------------

This file contains reduce statements to symbolically change the expressions describing the system. For example, a useful set of trig substitutions is:

     LET cos(~x)*cos(~y) = (cos(x+y)+cos(x-y))/2;
     LET cos(~x)*sin(~y) = (sin(x+y)-sin(x-y))/2;
     LET sin(~x)*sin(~y) = (cos(x-y)-cos(x+y))/2;
     LET cos(~x)^2       = (1+cos(2*x))/2;
     LET sin(~x)^2       = (1-cos(2*x));

Symbolic parameters for simplification (simp.r)
-----------------------------------------------

This file contains reduce statements to symbolically change the expressions describing the system. Unlike the subs.r file (see Symbolic parameters (subs.r)) it does not affect all system transformations; only those converting to LaTeX form.

Numeric parameters (numpar)
---------------------------

When computing time and frequency responses; or when evaluating functions in Octave (see Octave); symbolic parameters need numerical instantiations.

The numpar representation provides the relevant numerical information. It comes in a number of languages:

txt
     a textual description of the parameter values -- this is the defining representation (see Defining representations).

m
     readable by octave a high-level interactive language for numerical computation -- translated by mtt from the txt version.

c
     readable by gcc a c compiler -- translated by mtt from the txt version.

Text form (numpar.txt)
......................

This is the textual form of the numerical parameters representation (see Numeric parameters (numpar)). Lines are either

assignment statements
     variable = value

comments
     lines beginning with #

commented assignment statements
     variable = value # comments

An example file is:

     # Numerical parameter file (rc_numpar.txt)
     # Generated by MTT at Mon Jun 16 15:10:17 BST 1997

     # %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     # %% Version control history
     # %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     # %% $Id$
     # %% $Log$
     # %% Revision 1.27  2005/01/06 14:42:33  geraint
     # %% Fixed explicit/implicit descriptions of forward/backward Euler.
     # %% Replaced -c examples with -cc.
     # %%
     # %% Revision 1.26  2005/01/06 12:28:36  geraint
     # %% Minor typos.
     # %%
     # %% Revision 1.25  2004/08/27 20:12:34  geraint
     # %% Added note about "none" as an option for units.
     # %%
     # %% Revision 1.24  2004/08/13 01:39:24  geraint
     # %% Fixed variable names in diy makefile example (again)
     # %%
     # %% Revision 1.23  2004/08/11 08:24:28  geraint
     # %% Replaced "gnuplot view" with "odeso gnuplot" in examples.
     # %%
     # %% Revision 1.21  2004/07/23 11:05:26  geraint
     # %% Updated Reduce URL: http://www.reduce-algebra.com
     # %%
     # %% Revision 1.20  2003/10/23 18:14:43  geraint
     # %% Added subsection with script for model-specific options.
     # %% Fixed typo in url to mtt.sf.net.
     # %%
     # %% Revision 1.19  2003/10/10 22:22:18  geraint
     # %% typo.
     # %%
     # %% Revision 1.18  2003/09/07 20:41:19  geraint
     # %% *** empty log message ***
     # %%
     # %% Revision 1.17  2003/08/19 14:20:38  gawthrop
     # %% Version 5.0 of MTT
     # %% Remove xref errors (spurious spaces)
     # %%
     # %% Revision 1.16  2003/08/19 14:11:23  gawthrop
     # %% Links to legal stuff
     # %%
     # %% Revision 1.15  2003/08/19 14:01:45  gawthrop
     # %% Added legal appendices
     # %%
     # %% Revision 1.14  2003/08/06 14:50:56  gawthrop
     # %% Describe the alias mechanism for invoking mtt options
     # %%
     # %% Revision 1.13  2002/12/13 10:07:07  gawthrop
     # %% Added example in sh section of DIY reps
     # %%
     # %% Revision 1.12  2002/09/19 08:09:31  gawthrop
     # %% Updated documentation documentation
     # %%
     # %% Revision 1.11  2002/08/20 15:51:17  gawthrop
     # %% Update to work with ident DIY rep
     # %%
     # %% Revision 1.10  2002/07/22 10:45:22  geraint
     # %% Fixed gnuplot rep so that it correctly re-runs the simulation if input files have changed.
     # %%
     # %% Revision 1.9  2002/07/05 13:29:34  geraint
     # %% Added notes about generating dynamically linked functions for Octave and Matlab.
     # %%
     # %% Revision 1.8  2002/07/04 21:34:12  geraint
     # %% Updated gnuplot view description to describe Tcl/Tk interface instead of obsolete txt method.
     # %%
     # %% Revision 1.7  2002/04/23 09:51:54  gawthrop
     # %% Changed incorrect statement about searching for components.
     # %%
     # %% Revision 1.6  2001/10/15 14:29:50  gawthrop
     # %% Added documentaton on  [1:N] style port labels
     # %%
     # %% Revision 1.5  2001/07/23 03:35:29  geraint
     # %% Updated file structure (mtt/bin).
     # %%
     # %% Revision 1.4  2001/07/23 03:25:02  geraint
     # %% Added notes on -ae hybrd, rk4, ode2odes.cc, .oct dependencies.
     # %%
     # %% Revision 1.3  2001/07/13 03:02:38  geraint
     # %% Added notes on #ICD, gnuplot.txt and odes.sg rep.
     # %%
     # %% Revision 1.2  2001/07/03 22:59:10  gawthrop
     # %% Fixed problems with argument passing for CRs
     # %%
     # %% Revision 1.1  2001/06/04 08:18:52  gawthrop
     # %% Putting documentation under CVS
     # %%
     # %% Revision 1.66  2000/12/05 14:20:55  peterg
     # %% Added the c++  anf m CR info.
     # %%
     # %% Revision 1.65  2000/11/27 15:36:15  peterg
     # %% NOPAR --> NOTPAR
     # %%
     # %% Revision 1.64  2000/11/16 14:22:48  peterg
     # %% added UNITS declaration
     # %%
     # %% Revision 1.63  2000/11/03 14:41:08  peterg
     # %% Added PAR and NOTPAR stuff
     # %%
     # %% Revision 1.62  2000/10/17 17:53:34  peterg
     # %% Added some simulation details
     # %%
     # %% Revision 1.61  2000/09/14 17:13:06  peterg
     # %% New options table
     # %%
     # %% Revision 1.60  2000/09/14 17:09:20  peterg
     # %% Tidied up valid name sections
     # %% Tidied up defining represnetations table
     # %% Verion 4.6
     # %%
     # %% Revision 1.59  2000/08/30 13:09:00  peterg
     # %% Updated option table
     # %%
     # %% Revision 1.58  2000/08/01 13:30:19  peterg
     # %% Version 4.4
     # %% updated STEPFACTOR info
     # %% describes octave and OCST interfaces
     # %%
     # %% Revision 1.57  2000/07/20 07:55:44  peterg
     # %% Version 4.3
     # %%
     # %% Revision 1.56  2000/05/19 17:49:17  peterg
     # %% Extended the user defined representation section -- new nppp rep.
     # %%
     # %% Revision 1.55  2000/03/16 13:53:31  peterg
     # %% Correct date
     # %%
     # %% Revision 1.54  2000/03/15 21:22:57  peterg
     # %% Updated to 4.1 -- old style SS no longer supported
     # %%
     # %% Revision 1.53  1999/12/22 05:33:10  peterg
     # %% Updated for 4.0
     # %%
     # %% Revision 1.52  1999/11/23 00:25:11  peterg
     # %% Added the sensitivity reps
     # %%
     # %% Revision 1.51  1999/11/16 04:43:47  peterg
     # %% Added start of sensitivity section
     # %%
     # %% Revision 1.50  1999/11/16 00:30:35  peterg
     # %% Updated simulation section
     # %% Added vector components
     # %%
     # %% Revision 1.49  1999/07/20 23:44:58  peterg
     # %% V 3.8
     # %%
     # %% Revision 1.48  1999/07/19 03:08:33  peterg
     # %% Added documentation for (new) SS lbl fields
     # %%
     # %% Revision 1.47  1999/03/09 01:42:22  peterg
     # %% Rearranged the User interface section
     # %%
     # %% Revision 1.46  1999/03/09 01:18:01  peterg
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     # %%
     # %% Revision 1.45  1999/03/03 02:39:26  peterg
     # %% Minor updates
     # %%
     # %% Revision 1.44  1999/02/17 06:52:14  peterg
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     # %%
     # %% Revision 1.43  1998/11/25 16:49:24  peterg
     # %% Put in subs.r documentation (was called params.r)
     # %%
     # %% Revision 1.42  1998/11/24 12:24:59  peterg
     # %% Added section on simulation output
     # %% Version 3.4
     # %%
     # %% Revision 1.41  1998/09/02 12:04:15  peterg
     # %% Version 3.2
     # %%
     # %% Revision 1.40  1998/08/27 08:36:39  peterg
     # %% Removed in. methods except Euler anf implicit
     # %%
     # %% Revision 1.39  1998/08/18 10:44:28  peterg
     # %% Typo
     # %%
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     # %% Version 3.1
     # %%
     # %% Revision 1.37  1998/08/17 16:14:30  peterg
     # %% Version 3.1 - includes documentation on METHOD=IMPLICIT
     # %%
     # %% Revision 1.36  1998/07/30 17:33:15  peterg
     # %% VERSION 3.0
     # %%
     # %% Revision 1.35  1998/07/22 11:00:53  peterg
     # %% Correct date!
     # %%
     # %% Revision 1.34  1998/07/22 11:00:13  peterg
     # %% Version to BAe
     # %%
     # %% Revision 1.33  1998/07/17 19:32:19  peterg
     # %% Added more about aliases
     # %%
     # %% Revision 1.32  1998/07/05 14:21:56  peterg
     # %% Further additions (Carlisle-Glasgow)
     # %%
     # %% Revision 1.31  1998/07/04 11:35:57  peterg
     # %% Strarted new lbl description
     # %%
     # %% Revision 1.30  1998/07/02 18:39:20  peterg
     # %% Started 3.0
     # %% Added alias and default sections.
     # %%
     # %% Revision 1.29  1998/05/19 19:46:58  peterg
     # %% Added the odess description
     # %%
     # %% Revision 1.28  1998/05/14 09:17:22  peterg
     # %% Added METHOD variable to the simpar file
     # %%
     # %% Revision 1.27  1998/05/13 10:03:09  peterg
     # %% Added unknown/zero SS label documentation.
     # %%
     # %% Revision 1.26  1998/04/29 15:12:46  peterg
     # %% Version 2.9.
     # %%
     # %% Revision 1.25  1998/04/12 17:00:26  peterg
     # %% Added new port features: coerced direction and top-level behaviour.
     # %%
     # %% Revision 1.24  1998/04/05 18:27:20  peterg
     # %% This was the 2.6 version
     # %%
     # Revision 1.23  1997/08/24  11:17:51  peterg
     # This is the released  version 2.5
     #
     # %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     # Parameters
     c =     1.0; # Default value
     r =     1.0; # Default value
     # Initial states
     x(1) =  0.0; # Initial state for rc (c)

As usual, MTT provides a default text file to be edited by the user (see Text editors).

Causal bond graph (cbg)
=======================

The causal bond graph is the causally complete version of the Acausal bond graph (see Acausal bond graph (abg)).

To create the causal bond graph of system `sys' in language fig type:

     mtt sys cbg fig

To create the causal bond graph of system `sys' in language m type:

     mtt sys cbg m

To view the causal bond graph of system `sys' type:

     mtt sys cbg view

Language fig (cbg.fig)
----------------------

The fig file is created by MTT. It is identical to the corresponding acausal representation (see Language fig (abg.fig)) except that

  * the new causal strokes are added (using a double thickness line in blue)

  * components that are undercausal are bold and green

  * components that are overcausal are bold and red

Language m (cbg.m)
------------------

The causal bond graph of system `sys' is represented as an m file with heading:

     function [cbonds,status] = sys_cbg

The two outputs of this function are:

  * cbonds

  * status

cbonds is  a matrix with

  * one row for each bond

  * the first column contains the arrow-orientated (see Arrow-orientated causality) causality of the effort variable.

  * the second column contains the arrow-orientated (see Arrow-orientated causality) causality of the flow variable.

status is  a matrix with

  * one row for each component

  * the first column contains 1 if the component is overcausal; 0 if the component is causally complete and -1 if the component is undercausal.

A successful model would therefore have all zeros in the status matrix.

Transformation abg2cbg_m
........................

This transformation takes the acausal bond graph as an m file (see Language m (abg.m)) and transforms it into a causal bond graph in m-file format (see Language m (cbg.m)).

It is based on the m-function abg2cbg.m which iteratively tries to complete causality whilst recursively searching the bond graph structure. If causality is incomplete, it picks the first acausal dynamic (C or I) component, asserts integral causality, and tries again.

This is essentially the sequential causality assignment procedure of Karnopp and Rosenberg.

The transformation informs the user of the final status in terms of the percentage of causally complete components; a successful model will yield 100% here.

Elementary system equations (ese)
=================================

The elementary system equations are a complete set of assignment statements describing the dynamic system corresponding to the bond graph. They are in the Reduce (see Reduce) language.

Because these are based on a causally complete system, these assignment statements are directly soluble by substitution.

Unlike early versions of MTT, MTT does not sort the equations in order of solution, but rather leaves them sorted by component and subsystem.

These are not supposed to be read by the user, so there is no view facility as such. However, you may read these with your favourite text editor and, to this end,  helpful comment lines have been added.

Wherever components have an explicit constitutive relationship, the corresponding RHS of the equation has a standard form.

     cr(arguments,out_causality,outport,
             input_1, causality_1, port_1,
             ....
             input_i, causality_i, port_i,
             ....
             input_n, causality_n, port_n
             );

where the symbols have the following meaning

arguments
     the constitutive relationship arguments

out_causality
     the causality (effort or flow) of the output variable (see Variables)

outport
     the number (integer) of the output port of the system

input_i
     the ith input to the component

causality_i
     the causality (effort or flow) of the ith input variable (see Variables)

port_i
     the number (integer) of the ith input port of the system

An example for a resistor with linear constitutive relationship is:

     rc_1_bond4_flow := lin(flow,r,flow,1,
             rc_1_bond4_effort,effort,1
             );

Transformation cbg2ese_m2r
..........................

This transformation takes the causal bond graph as an m file (see Language m (cbg.m)) and transforms it into elementary system equations in Reduce (see Reduce) form.

It is based on the m-function cbg2ese.m which iteratively traverses the causal bond graph writing equations as it goes.

It also writes out the system structure as the file sys_def.r.

Differential-Algebraic Equations (dae)
======================================

The system differential algebraic equations describe the system dynamics together together with any algebraic constraints.

They are generated in language lang for system sys by:

     mtt sys dae lang

Valid languages are:

r
     reduce (see Reduce).

m
     m  (see m).

view
     reduce (see Views).

There are five sets of variables describing the system:

x
     the system states (corresponding to C and I components with integral causality.

z
     the system nonstates (corresponding to C and I components with derivative causality.

u
     the system inputs  (corresponding to SS components with external attribute).

ui
     the internal system inputs (corresponding to SS components with internal attribute) used to solve algebraic loops (see Algebraic loops).

y
     the  system outputs (corresponding to SS components with external attribute).

In general there are four sets of equations. The right-hand side of each is a function of x, dz/dt,  u and ui and the left hand sides are:

 1. the derivative of x (dx/dt)

 2. z

 3. w=0 (the algebraic equations)

 4. y

Language reduce (dae.r)
-----------------------

The system DAEs (see Differential-Algebraic Equations) are represented in the reduce  (see Reduce) language as arrays containing the algebraic expressions for the right hand sides of each set of equations. The arrays are:

MTTx
     x -- the system states (corresponding to C and I components with integral causality.

MTTz
     z -- the system nonstates (corresponding to C and I components with derivative causality.

MTTu
     u -- the system inputs  (corresponding to SS components with external attribute).

mttv
     ui -- the internal system inputs (corresponding to SS components with internal attribute) used to solve algebraic loops (see Algebraic loops).

MTTy
     y -- the  system outputs (corresponding to SS components with external attribute).

Transformation ese2dae_r
........................

This transformation (see What is a Transformation?) uses Reduce (see Reduce) to combine the elementary system equations (see Elementary system equations) with the constitutive relationships (see Constitutive relationship) and simplify the result.

Language m (dae.m)
------------------

The system DAEs (see Differential-Algebraic Equations) are represented in the m (see m) language as two m-functions of the form:

     function resid = sys_dae(dx,x,t)
     function y  = sys_dae(dx,x,t)

Where x is the dae descriptor vector and dx its time derivative; t is the time. The first function is of a form suitable for solution by DASSL; the second function can then be used to find the coresponding system output.

Transformation dae_r2m
......................

This transformation (see What is a Transformation?) uses Reduce (see Reduce) to rewrite the elementary system equations (see Elementary system equations) in m-file format  (see m) . Numerical parameters are declared as global.

Constrained-state Equations (cse)
=================================

The system constrained-state equations describe the system dynamics for a special class of systems (see the book for details). The resuting equations are of the form:

     E(x) dx/dt = f(x,u)
     y = g(x,u)

They typically occure where two or more states are constrained to be equal, or proportional, to each other. For example, two capacitors in parallel or two inertias connected by a stiff shaft.

They are generated in language lang for system sys by:

     mtt sys cse lang

Valid languages are:

r
     reduce (see Reduce).

m
     m  (see m).

view
     reduce (see Views).

There are three sets of variables describing the system:

x
     the system states (corresponding to C and I components with integral causality.

u
     the system inputs  (corresponding to SS components with external attribute).

y
     the  system outputs (corresponding to SS components with external attribute).

In general there are two sets of equations. The right-hand side of each is a function of x and  u  and the left hand sides are:

 1. the derivative of x (dx/dt) y

Language reduce (cse.r)
-----------------------

The system CSEs (see Constrained-state Equations) are represented in the reduce  (see Reduce) language as arrays containing the algebraic expressions for the right hand sides of each set of equations. The arrays are:

MTTx
     x -- the system states (corresponding to C and I components with integral causality.

MTTu
     u -- the system inputs  (corresponding to SS components with external attribute).

MTTy
     y -- the  system outputs (corresponding to SS components with external attribute).

together with the array containing the elements of the E matrix.

Transformation dae2cse_r
........................

This transformation (see What is a Transformation?) Reduce (see Reduce) to find various Jacobians which are combined to find the E matrix and the constrained-state equations (see Constrained-state Equations).

Language m (view)
-----------------

This representation has the standard text view (see Views).

Ordinary Differential Equations
===============================

The system ordinary differential equations describe the system dynamics.

They are generated in language lang for system sys by:

     mtt sys ode lang

Valid languages are:

r
     reduce (see Reduce).

m
     m  (see m).

view
     reduce (see Views).

There are three sets of variables describing the system:

x
     the system states (corresponding to C and I components with integral causality.

u
     the system inputs  (corresponding to SS components with external attribute).

y
     the  system outputs (corresponding to SS components with external attribute).

In general there are two sets of equations. The right-hand side of each is a function of x and  u  and the left hand sides are:

 1. the derivative of x (dx/dt) y

Language reduce (ode.r)
-----------------------

The system ODEs (see Ordinary Differential Equations) are represented in the reduce  (see Reduce) language as arrays containing the algebraic expressions for the right hand sides of each set of equations. The arrays are:

MTTx
     x -- the system states (corresponding to C and I components with integral causality.

MTTu
     u -- the system inputs  (corresponding to SS components with external attribute).

MTTy
     y -- the  system outputs (corresponding to SS components with external attribute).

Transformation cse2ode_r
........................

This transformation (see What is a Transformation?) uses Reduce (see Reduce) to invert the E matrix of the constrained-state equations (see Constrained-state Equations) and simplify the result.

Language m (ode.m)
------------------

The system ODEs (see Ordinary Differential Equations) are represented in the m (see m) language as two m-functions of the form:

     function dx = sys_ODE(x,t)
     function y  = sys_ODE(dx,x,t)

Where x is the ODE state vector and dx its time derivative; t is the time. The first function is of a form suitable for solution by odesol; the second function can then be used to find the corresponding system output.

Transformation ode_r2m
......................

This transformation (see What is a Transformation?) uses Reduce (see Reduce) to rewrite the ordinary differential equations (see Ordinary Differential Equations) in m-file format  (see m) . Numerical parameters are declared as global.

Language m (view)
-----------------

This representation has the standard text view (see Views).

Descriptor matrices (dm)
========================

The system descriptor matrices A, B, C, D and E describe the linearised system dynamics in the form

     E dx/dt = Ax + Bu
     y = Cx + Du

They are generated in language lang for system sys by:

     mtt sys dm lang

Valid languages are:

r
     reduce (see Reduce).

m
     m  (see m).

view
     reduce (see Views).

Language reduce (dm.r)
----------------------

The system  descriptor matrices (see Descriptor matrices) are represented in the reduce  (see Reduce) language as arrays containing the four matrices. The arrays are:

MTTA
     A

MTTB
     B

MTTA
     C

MTTD
     D

MTTE
     E

Language m (dm.m)
-----------------

The system descriptor matrices (see Descriptor matrices) are represented in the m (see m) language as an m-function of the form:

     function [A,B,C,D,E] = sys_dm

System numeric parameters (see Numeric parameters) are passed via global variables defined in the _numpar.m file. Thus the system descriptor matrices are typically generated in Octave (see Octave) as follows:

     sys_numpar
     [A,B,C,D,E] = sys_dm

Parameters can be changed from their default values by entering their values directly into Octave (see Octave)  and then invoking sys_dm; for example

     sys_numpar
     par_1 = 25
     par_2 = par_1 + 3
     [A,B,C,D,E] = sys_dm

Report (rep)
============

MTT has a report-generator feature. The user specifies the report contents in a text file (see Report (text)) using an appropriate text editor (see Text editors).

For example, the report can be viewed by typing

     mtt system rep view

Language text (rep.txt)
-----------------------

The user specifies the report contents in a text file (see Report (text)) using an appropriate text editor (see Text editors). The text file contains lines which are either comments (indicated by %) or valid MTT commands. The report will then contain appropriate sections. The following languages are supported by the report generator:

m
     octave a high-level interactive language for numerical computation.

r
     reduce  a high-level interactive language for symbolic computation.

tex
     latex a text processor.

ps
     ghostview another document viewer.

c
     gcc a c compiler.

For example:

     mtt rc abg tex
     mtt rc cbg ps
     mtt rc struc tex
     mtt rc ode tex
     mtt rc sro ps
     mtt rc tf tex
     mtt rc lmfr ps

The acausal bond graph (abg) (see Acausal bond graph (abg)) with the tex language is handled in a special way: the acausal Bond Graph  in fig format (see Language fig (abg.fig)), the label file (see Labels (lbl)) the description file (see Detailed), together with corresponding subsystems are included in the report. It is recommended that the first (non-comment line) in the file should be:

     mtt <system> abg tex

where <system> is the name of the (top-level) system.

As usual, MTT provides a default text file to be edited by the user (see Text editors).

In the special case that the first argument to mtt (normally the system) is a directory, a default text file is provided which generates a report for all systems to be found in that directory tree.

Language view
-------------

This representation has the standard text view (see Views).

Extending MTT
*************

MTT has a number of built-in mechanisms for the user to extend its capabilities. As MTT is based on `Make' it is unsurprising that some of these  involve the creation of `make files'.

Makefiles
=========

If a file called `Makefile' exists in the current directory, MTT executes it using make before doing anything else. This is useful if one of the .txt files contains a reference to, for example, an octave function of which MTT unaware. Such a function can be created using the makefile. An example `Makefile' is

     # Makefile for the Two link GMV example

     all: msdP_tf.m TwoLinkP_obs.m TwoLinkP_sm.m twolinkp_sm.m TwoLinkGMV_numpar.m

     msdP_tf.m: msdP_abg.fig
             mtt -q msdP tf m

     TwoLinkP_obs.m: TwoLinkP_abg.fig TwoLinkP_lbl.txt
             mtt -q TwoLinkP obs m

     TwoLinkP_sm.m: TwoLinkP_abg.fig TwoLinkP_lbl.txt
             mtt -q TwoLinkP sm m

     twolinkp_sm.m: TwoLinkP_sm.m
             cp -v TwoLinkP_sm.m twolinkp_sm.m

     TwoLinkGMV_numpar.m: TwoLinkGMV_numpar.txt
             mtt -q TwoLinkGMV numpar m

All of the files in the line stating `all:' are created when MTT is executed (if they don't already exist).

New (DIY) representations
=========================

It may be convenient to create new representations for MTT; in particular, it is nice to be able to include the result of some numerical or symbolic computations within an MTT report (see Report). Therefore MTT provides a mechanism for doing this.

Future extensions of MTT will use such representations stored in $MTT_REP.

There are three parts to creating a DIY representation called myrep

 1. Creating a make file in Make format called myrep_rep.make

 2. Optionally creating a shell script called myrep_rep.sh

 3. Optionally creating a documentation file in LaTeX format called myrep_rep.tex

Makefile
--------

To create a new representation `myrep' in a language `mylang', create a file with the name

     myrep_rep.make

This file must contain text in `make' syntax. It is executed by MTT and the two arguments `SYS' (the system name) and `LANG' (the language) are passed to it by MTT. Note that MTT cannot know of any prerequisites, but these can be explicitly included in the makefile (which may include execution of MTT itself.

The following example declares the new representation `ident' which is created in conjunction with the shell-script ident_rep.sh (see Shell-script (DIY representations)).

# -*-makefile-*-

#SUMMARY 	Identification
#DESCRIPTION 	Partially know system identification using
#DESCRIPTION    using bond graphs

# Makefile for representation ident
# File ident_rep.make

#Copyright (C) 2000,2001,2002 by Peter J. Gawthrop

## Model targets
model_reps =  ${MTT_SYS}_sympar.m ${MTT_SYS}_simpar.m ${MTT_SYS}_state.m
model_reps += ${MTT_SYS}_numpar.m ${MTT_SYS}_input.m ${MTT_SYS}_ode2odes.m
model_reps += ${MTT_SYS}_def.m

## Prepend s to get the sensitivity targets
sensitivity_reps = ${model_reps:%=s%}

## Model prerequisites
model_pre =  ${MTT_SYS}_abg.fig ${MTT_SYS}_lbl.txt
model_pre += ${MTT_SYS}_rdae.r ${MTT_SYS}_numpar.txt

## Prepend s to get the sensitivity targets
sensitivity_pre = ${model_pre:%=s%}

## Simulation targets
sims = ${MTT_SYS}_sim.m s${MTT_SYS}_ssim.m

## m-files needed for ident
ident_m = ${MTT_SYS}_ident.m ${MTT_SYS}_ident_numpar.m

## Targets for the ident simulation
ident_reps = ${ident_m} ${sims} ${model_reps} ${sensitivity_reps}

## ps output files etc
psfiles = ${MTT_SYS}_ident.ps ${MTT_SYS}_ident.comparison.ps
figfiles = ${psfiles:%.ps=%.fig}
gdatfiles = ${psfiles:%.ps=%.gdat}
datfiles = ${psfiles:%.ps=%.dat2}

## LaTeX files etc
latexfiles = ${MTT_SYS}_ident_par.tex

all: ${MTT_SYS}_ident.${MTT_LANG}

echo:
	echo "sims: ${sims}"
	echo "model_reps: ${model_reps}"
	echo "sensitivity_reps: ${sensitivity_reps}"
	echo "ident_reps: ${ident_reps}"

${MTT_SYS}_ident.view: ${psfiles}
	ident_rep.sh ${MTT_SYS} view

${psfiles}: ${figfiles}
	ident_rep.sh ${MTT_SYS} ps

${figfiles}: ${gdatfiles}
	ident_rep.sh ${MTT_SYS} fig

${gdatfiles}: ${datfiles}
	ident_rep.sh ${MTT_SYS} gdat

${datfiles} ${latexfiles}: ${ident_reps}
	ident_rep.sh ${MTT_SYS} dat2

${MTT_SYS}_ident.m:
	ident_rep.sh ${MTT_SYS} m

${MTT_SYS}_ident_numpar.m:
	ident_rep.sh ${MTT_SYS} numpar.m

## System model reps
## Generic txt files
${MTT_SYS}_%.txt:
	mtt ${MTT_OPTS} -q -stdin ${MTT_SYS} $* txt

## Specific m files
${MTT_SYS}_ode2odes.m: ${model_pre}
	mtt -q -stdin ${MTT_OPTS} ${MTT_SYS} ode2odes m

${MTT_SYS}_sim.m: ${MTT_SYS}_ode2odes.m
	mtt ${MTT_OPTS} -q -stdin ${MTT_SYS} sim m

## Numpar files
${MTT_SYS}_numpar.m:
	mtt ${MTT_SYS} numpar m

## Sympar files
${MTT_SYS}_sympar.m:
	mtt ${MTT_SYS} sympar m

## Generic txt to m
${MTT_SYS}_%.m: ${MTT_SYS}_%.txt
	mtt ${MTT_OPTS} -q -stdin ${MTT_SYS} $* m

## r files
${MTT_SYS}_def.r: ${MTT_SYS}_abg.fig
	mtt ${MTT_OPTS} -q -stdin ${MTT_SYS} def r

${MTT_SYS}_rdae.r:
	mtt ${MTT_OPTS} -q -stdin ${MTT_SYS} rdae r

## Sensitivity model reps
## Generic txt files
s${MTT_SYS}_%.txt:
	mtt ${MTT_OPTS} -q -stdin -s s${MTT_SYS} $* txt

## Specific m files
## Numpar files
s${MTT_SYS}_numpar.m:
	mtt -s s${MTT_SYS} numpar m

## Sympar files
s${MTT_SYS}_sympar.m:
	mtt -s s${MTT_SYS} sympar m

s${MTT_SYS}_ode2odes.m: ${sensitivity_pre}
	mtt -q -stdin ${MTT_OPTS} -s s${MTT_SYS} ode2odes m

s${MTT_SYS}_ssim.m:
	mtt -q -stdin ${MTT_OPTS} -s s${MTT_SYS} ssim m

s${MTT_SYS}_def.m:
	mtt -q -stdin ${MTT_OPTS} -s s${MTT_SYS} def m

## Generic txt to m
s${MTT_SYS}_%.m: s${MTT_SYS}_%.txt
	mtt ${MTT_OPTS} -q -stdin s${MTT_SYS} $* m

## r files
s${MTT_SYS}_rdae.r:
	mtt ${MTT_OPTS} -q -stdin -s s${MTT_SYS} rdae r

Shell-script
------------

For more complex DIY representations, it is convenient to define new commands to be used by the Makefile (see Makefile (DIY representations)).

The following example shows this in the context of the DIY representation `ident' used as an example in the previous section (see Makefile (DIY representations)).

#! /bin/sh

## ident_rep.sh
## DIY representation "ident" for mtt
# Copyright (C) 2002 by Peter J. Gawthrop

ps=ps

sys=$1
rep=ident
lang=$2
mtt_parameters=$3
rep_parameters=$4

## Some names
target=${sys}_${rep}.${lang}
def_file=${sys}_def.r
dat2_file=${sys}_ident.dat2
dat2s_file=${sys}_idents.dat2
ident_numpar_file=${sys}_ident_numpar.m
option_file=${sys}_ident_mtt_options.txt

## Get system information
if [ -f "${def_file}" ]; then
 echo Using ${def_file}
else
  mtt -q ${sys} def r
fi

ny=`mtt_getsize $1 y`
nu=`mtt_getsize $1 u`

check_new_options() {
    if [ -f "${option_file}" ]; then
	old_options=`cat ${option_file}`
        if [ "${mtt_options}" != "${old_options}" ]; then
	   echo ${mtt_options} > ${option_file}
	fi
    else
	echo ${mtt_options} > ${option_file}
    fi
}

## Make the _ident.m file
make_ident() {
filename=${sys}_${rep}.m
date=`date`
echo Creating ${filename}

cat > ${filename} <<EOF
function [epar,Y] = ${sys}_ident (y,u,t,par_names,Q,extras)

  ## usage:  [epar,Y] = ${sys}_ident (y,u,t,par_names,Q,extras)
  ##
  ## last      last time in run
  ## ppp_names Column vector of names of ppp params
  ## par_names Column vector of names of estimated params
  ## extras    Structure containing additional info
  ##
  ## Created by MTT on ${date}

  ## Sensitivity system name
  system_name = "s${sys}"

  ##Sanity check
  if nargin<3
    printf("Usage: [y,u,t] = ${sys}_ident(y,u,t,par_names,Q,extras);");
    return
  endif

  if nargin<6
    ## Set up optional parameters
    extras.criterion = 1e-3;
    extras.emulate_timing = 0;
    extras.max_iterations = 10;
    extras.simulate = 2;
    extras.v =  1e-2;
    extras.verbose = 1;
    extras.visual = 1;
  endif

  ## System info
  [n_x,n_y,n_u,n_z,n_yz] = ${sys}_def;
  sympar  = ${sys}_sympar;
  simpar  = ${sys}_simpar;
  sympars  = s${sys}_sympar;
  simpars  = s${sys}_simpar;

  ## Parameter indices
  i_par = ppp_indices (par_names,sympar,sympars);

  ## Initial model state
  x_0 = zeros(2*n_x,1);

  ## Initial model parameters
  par_0 = s${sys}_numpar;

  ## Reset simulation parameters
  [n_data,m_data] = size(y);
  dt = t(2)-t(1);
  simpars.last = (n_data-1)*dt;
  simpars.dt = dt;

  ## Identification
  [epar,Par,Error,Y,iterations,x] = ppp_optimise(system_name,x_0,par_0,simpars,u,y,i_par,Q,extras);

  ## Do some plots
  figure(1);
  title("Comparison of data");
  xlabel("t");
  ylabel("y");
  [N,M] = size(Y);
  plot(t,Y(:,M-n_y+1:M),"1;Estimated;", t,y,"3;Actual;");
  figfig("${sys}_ident_comparison");

  ## Create a table of the parameters
  [n_par,m_par] = size(i_par);
  fid = fopen("${sys}_ident_par.tex", "w");
  fprintf(fid,"\\\\begin{table}[htbp]\\n");
  fprintf(fid," \\\\centering\\n");
  fprintf(fid," \\\\begin{tabular}{|l|l|}\\n");
  fprintf(fid,"  \\\\hline\\n");
  fprintf(fid,"  Name & Value \\\\\\\\ \\n");
  fprintf(fid,"  \\\\hline\\n");
  for i = 1:n_par
    fprintf(fid,"$%s$ & %4.2f \\\\\\\\ \\n", par_names(i,:), epar(i_par(i,1)));
  endfor
  fprintf(fid,"  \\\\hline\\n");
  fprintf(fid,"\\\\end{tabular}\\n");
  fprintf(fid,"\\\\caption{Estimated Parameters}\\n");
  fprintf(fid,"\\\\end{table}\\n");
  fclose(fid);

endfunction
EOF
}

make_ident_numpar() {
echo Creating ${ident_numpar_file}
cat > ${sys}_ident_numpar.m <<EOF
function [y,u,t,par_names,Q,extras] = ${sys}_ident_numpar;

  ## usage: [y,u,t,par_names,Q,extras] = ${sys}_ident_numpar;
  ## Edit for your own requirements
  ## Created by MTT on ${date}

  ## This section sets up the data source
  ## simulate = 0  Real data (you supply ${sys}_ident_data.dat)
  ## simulate = 1  Real data input, simulated output
  ## simulate = 2  Unit step input, simulated output
  simulate = 2;

  ## System info
  [n_x,n_y,n_u,n_z,n_yz] = ${sys}_def;
  simpars = s${sys}_simpar;

  ## Access or create data
  if (simulate<2)		# Get the real data
    if (exist("${sys}_ident_data.dat")==2)
      printf("Loading ${sys}_ident_data.dat\n");
      load ${sys}_ident_data.dat
    else
      printf("Please create a loadable file ${sys}_ident_data.dat containing y,u and t\n");
      return
    endif
  else
    switch simulate
      case 2			# Step simulation
	t = [0:simpars.dt:simpars.last]';
	u = ones(size(t));
      otherwise
	error(sprintf("simulate = %i not implemented", simulate));
    endswitch
  endif

  if (simulate>0)
    par = ${sys}_numpar();
    x_0 = ${sys}_state(par);
    dt = t(2)-t(1);
    simpars.dt = dt;
    simpars.last = t(length(t));
    y =  ${sys}_sim(zeros(n_x,1), par, simpars, u);
  endif

  ## Default parameter names - Put in your own here
  sympar = ${sys}_sympar;	# Symbolic params as structure
  par_names = struct_elements (sympar);	# Symbolic params as strings
  [n,m] = size(par_names);	# Size the string list

  ## Sort by index
  for [i,name] = sympar
    par_names(i,:) = sprintf("%s%s",name, blanks(m-length(name)));
  endfor

  ## Output weighting vector
  Q = ones(n_y,1);

  ## Extra parameters
  extras.criterion = 1e-5;
  extras.emulate_timing = 0;
  extras.max_iterations = 10;
  extras.simulate = simulate;
  extras.v =  1e-2;
  extras.verbose = 1;
  extras.visual = 1;

endfunction
EOF
}

make_dat2() {

## Inform user
echo Creating ${dat2_file}

## Use octave to generate the data
octave -q <<EOF
  [y,u,t,par_names,Q,extras] = ${sys}_ident_numpar;
  [epar,Y] = ${sys}_ident (y,u,t,par_names,Q,extras);
  [N,M] = size(Y);
  y_est = Y(:,M);
  data = [t,y_est,u];
  save -ascii ${dat2_file} data
EOF

## Tidy up the latex stuff - convert foo_123 to foo_{123}
cat ${sys}_ident_par.tex > mtt_junk
sed  -e "s/_\([a-z0-9,]*\)/_{\1}/g" < mtt_junk >${sys}_ident_par.tex
rm mtt_junk
}

case ${lang} in
    numpar.m)
        ## Make the numpar stuff
        make_ident_numpar;
	;;
    m)
        ## Make the code
        make_ident;
	;;
    dat2)
        ## The dat2 language (output data) & fig file
	make_dat2;
	;;
    gdat)
        cp ${dat2_file} ${dat2s_file}
	dat22dat ${sys} ${rep}
        dat2gdat ${sys} ${rep}
	;;
    fig)
        gdat2fig ${sys}_${rep}
	;;
    ps)
        figs=`ls ${sys}_ident*.fig | sed -e 's/\.fig//'`
	for fig in ${figs}; do
            fig2dev -Leps ${fig}.fig > ${fig}.ps
	done
	texs=`ls ${sys}_ident*.tex | sed -e 's/\.tex//'`
	for tex in ${texs}; do
          makedoc "" "${sys}" "ident_par" "tex" "" "" "$ps"
          doc2$ps ${sys}_ident_par "$documenttype"
	done
	;;
    view)
	pss=`ls ${sys}_ident*.ps`
        echo Viewing ${pss}
        for ps in ${pss}; do
          gv ${ps}&
	done
	;;
    *)
	echo Language ${lang} not supported by ${rep} representation
        exit 3
esac

Documentation
-------------

Component library
=================

If MTT does not recognise a component (eg named MyComponent) as a simple component (see Simple components) or as already existing, it searches the library search path $MTT_COMPONENTS (see $MTT_COMPONENTS) for a directory called MyComponent containing MyComponent_lbl.txt. It then copies the entire directory into the current working directory. Thus, for example, the directory could contain MyComponent_desc.tex MyComponent_abg.fig MyComponent_lbl.txt and MyComponent_cr.r in addition to MyComponent_lbl.txt.

Documentation
*************

Manual
======

MTT is documented in this manual. The manual can be invoked in various ways:

mtt manual
     Brings up a pdf version of the manual

mtt info
     Brings up an xterm containing an info version of the manual

mtt hinfo
     Brings up an html browser containing the manual

emacs
     type ^h^i followed by mmtt in the command window

browser
     point browser to http://mtt.sf.net

On-line documentation
=====================

MTT components, constitutive relations, examples and representations in libraries (see Component library) are documented in two ways:

 1. brief

 2. verbose

Brief on-line documentation
---------------------------

Documentation of DIY components, examples, constitutive relationships and representations is provides by the programmer by inserting code of the form

     #SUMMARY     One line summary
     #DESCRIPTION Multi-line
     #DESCRIPTION More detailed description

within the appropriate file (usually at or near the top):

components
     _lbl.txt (see Labels (lbl))

examples
     _lbl.txt (see Labels (lbl))

constitutive relations
     _cr.r (see DIY constitutive relationships)

representations
     _rep.make (see Makefile (DIY representations))

This documentation is accessed by the user in various ways

mtt help name
     prints basic information on the screen

mtt system lbl view
     gives formatted information about the component or example

Including mtt system abg tex in the _rep.txt file
     gives formatted information about the component or example within the report

Detailed on-line documentation
------------------------------

DIY components, examples, constitutive relationships can be described textually in LaTeX (.tex) description file; this is the only language for this representation. This representation is used by the LaTeX language version (see Language tex (abg.tex)) of the acausal bond graph representation (see Acausal bond graph (abg)).

The file may contain any  LaTeX commands valis for the ``article'' document type but must not contain:

  * documentclass commands

  * document environments

Languages
*********

These are a number of languages used by MTT to implement the various representations. Each has associated Language tools (see Language tools) to manipulate and/or view the representation.

fig
     Fig a graphical description language.

m
     octave a high-level interactive language for numerical computation.

r
     reduce  a high-level interactive language for symbolic computation.

tex
     latex a text processor.

dvi
     xdvi a document viewer.

ps
     ghostview another document viewer.

gdat
     gnuplot a data viewer.

c
     gcc a c compiler.

sg
     scigraphica a plotting package.

These tools are automatically invoked as appropriate by MTT; but for more advanced use, these tools can be used directly on files (with the appropriate suffix) generated by MTT.

Fig
===

Please see xfig documentation.

m
=

Please see Octave documentation

Reduce
======

Please see the reduce documentation.

c
=

Please see the gcc documentation.

Language tools
**************

Views
=====

A number of representations (see Representations) have a language representation which is particularly useful for viewing by the user. These views are invoked, where appropriate by the command:

     mtt sys rep view

where sys is the system name and rep a corresponding representation.

Xfig
====

Text editors
============

All representations live in text files and thus may be edited using your favourite text editor; however, the Fig (see Fig) representation is pretty meaningless in this form and so you should use Xfig (see Xfig) for representation in this language.

Its up to you which text editor to use. I recommend emacs, but simpler (and less powerful) editors such as xedit, textedit and vi are also ok.

I usually run MTT out of an emacs shell window and keep the rest of the files in emacs buffers.

Octave
======

Octave is a numerical matrix-based language see Octave.  It is similar to Matlab in many ways. In most cases, m-files generated by MTT can be understood by both Matlab and Octave (and no doubt other Matlab lookalikes).

MTT provides the octave function mtt. The octave command

     help mtt

gives the following information:

      usage:  mtt (system[,representation,language])

      Invokes mtt from octave to generate system_representation.language
      Ie equivalent to "mtt system representation language" at the shell
      Representation and language defualt to "sm" and "m" respectively

Thus for example, if octave is in the directory containing the system rc the following session generates the state matrices of the system "rc" with the defaut capacitance but resitance r=0.1.

     octave> mtt("rc");
     Creating rc_rbg.m
     Creating rc_cmp.m
     Creating rc_fig.fig
     Creating rc_sabg.fig
     Creating rc_alias.txt
     Creating rc_alias.m
     Creating rc_sub.sh
     Creating rc_abg.m
     Creating rc_cbg.m (maximise integral causality)
     Creating rc_type.sh
     Creating rc_ese.r
     Creating rc_def.r
     Creating rc_struc.txt
     Creating rc_rdae.r
     Creating rc_subs.r
     Creating rc_cr.txt
     Creating rc_cr.r
     Copying CR SS to here from
     Copying CR lin to here from
     Creating rc_dae.r
     Creating rc_sympar.txt
     Creating rc_sympar.r
     Creating rc_cse.r
     Creating rc_sspar.r
     Creating rc_csm.r
     Creating rc_ode.r
     Creating rc_ss.r
     Creating rc_sm.r
     Creating rc_switch.txt
     0 switches found
     Creating rc_sympars.txt
     Creating rc_sm.m
     Copying rc_sm.m
     octave> mtt("rc","numpar");
     Creating rc_numpar.txt
     Creating rc_numpar.m
     Copying rc_numpar.m
     octave> mtt("rc","sympar");
     Creating rc_sympar.m
     Copying rc_sympar.m
     octave> par = rc_numpar
     par =

       1
       1

     octave> sym = rc_sympar;

     octave> par(sym.r) = 0.1;
     octave> [A,B,C,D] = rc_sm(par)
     A = -10

     B = 10

     C = 1

     D = 0

     octave>

generates the data structure rc corresponding the the bond graph of the system called `rc'. The following octave commands then generate the step reponse and bode diagram respectively:

     step(rc);
     bode(rc);

Octave control system toolbox (OCST)
------------------------------------

MTT provides an interface to the Octave control system toolbox (OCST) using the mfile mtt2sys. the octave command

     help mtt2sys

gives the following information.

      usage:  sys = mtt2sys (Name[,par])

      Creates a sys structure for the Octave Control Systems Toolbox
      from an MTT system with name "Name"
      Optional second argument is system parameter list
      Assumes that Name_sm.m, Name_struc.m and Name_numpar.m exist

Thus for example, if octave is in the directory containing the system rc:

     rc = mtt2sys("rc");

generates the data structure rc corresponding the the bond graph of the system called `rc'. The following octave commands then generate the step reponse and bode diagram respectively:

     step(rc);
     bode(rc);

Creating GNU Octave .oct files
------------------------------

GNU Octave dynamically loaded functions (.oct files) can be created by instructing MTT to create the ``oct'' representation:

       mtt [options] sys ode oct

This will cause MTT to create the C++ representation of the system (sys_ode.cc) and to then compile it as a shared object suitable for use within Octave. The resultant file may be used in an identical manner to the equivalent, but generally slower, interpreted .m file.

Usage information for the function may be obtained within Octave in the usual manner:

       octave:1> help rc_ode

       rc_ode is the dynamically-linked function from the file
       /home/mttuser/rc/rc_ode.oct

       Usage: [mttdx] = rc_ode(mttx,mttu,mttt,mttpar)
       Octave ode representation of system rc
       Generated by MTT on Fri Jul  5 11:23:08 BST 2002

Note that the first line of output from Octave identifies whether the compiled or interpreted function is being used.

Alternatively, standard representations may be generated using the Octave DLDs by use of the ``-oct'' switch:

       mtt -oct rc odeso view

In order to successfully generate .oct files, Octave must be correctly configured prior to compilation and certain headers and libraries must be correctly installed on the system (see .oct file dependencies).

Creating Matlab .mex files
--------------------------

On GNU/Linux systems, Matlab dynamically linked executables (.mexglx files) can created by instructing MTT to create the ``mexglx'' representation:

       mtt [options] sys ode mexglx

This will cause MTT to create the C++ representation of the system (sys_ode.cc) and to then compile it as a shared object suitable for use within Matlab.

If it is necessary to compile mex files for another platform, then the usual C++ representation (generated with the -cc flag) can be created and the resultant file compiled with the -DCODEGENTARGET=MATLABMEX flag on the target platform.

       mtt_machine:
       mtt -cc rc ode cc

       matlab_machine:
       matlab> mex -DCODEGENTARGET=MATLABMEX rc_ode.cc

Embedding MTT models in Simulink
--------------------------------

It is possible to embed MTT functions or entire MTT models within Simulink simulations as Sfun blocks. If the zip package is installed on the system, the command

       mtt sys sfun zip

will create a compressed archive containing sys.mdl, which may be embedded into a larger Simulink model. Also contained within the archive will be four sys_sfun*.c files,

  * sys_sfun.c model state and output equations

  * sys_sfun_ae.c model algebraic equations

  * sys_sfun_input.c model inputs

  * sys_sfun_interface.c interface between MTT model and Simulink

The last of these files must be edited to correctly map the inputs and outputs between the MTT and Simulink models. The two sections to edit are clearly marked with

       /* Start EDIT */
       ....
       /* End EDIT */

These four files should then be compiled with the Matlab ``mex'' compiler as described in the README file in the archive.

If it is desired to compile the .mex files directly from within MTT on a machine which has the Matlab header files installed, this may be done with the command

       mtt sys sfun mexglx

which will generated the four .mex files and the .mdl file. In this case, the user must ensure that sys_sfun_interface.c has been correctly edited prior to compilation.

Note that solution of algebraic equations within Simulink is not possible unless the Matlab Optimisation Toolbox is installed.

LaTeX
=====

LaTeX is a powerful text processor which MTT uses to provide visual output.

Administration
**************

Software components
===================

MTT is built from a set of readily-available  software tools. These are:

  * General purpose software tools.

  * Octave (see Octave setup)

  * REDUCE  (see REDUCE setup)

The General purpose tools are (these will all be available with a standard Linux distribution):

sh
     Bourne shell

gmake
     Gnu make

gawk
     Gnu awk

sed
     Gnu sed

grep
     Gnu grep

comm
     Gnu Compare sorted files by line

xfig
     Figure editor, version 3 or greater.

fig2dev
     Fig file conversion, version 3 or greater.

ghostview
     postscript viewer

xdvi
     dvi viewer

dvips
     dvi to postscript conversion

latex
     the text processor (LaTeX2e needed)

latex2html
     converts latex to html

perl
     needed for latex2html

gnuplot
     a graph plotting program

gnuscape
     or other web/html browser such as netscape, Red Baron etc.

gcc
     GNU c compiler

REDUCE setup
============

Symbolic algebra is performed by REDUCE, which although not free software is the the result of international collaboration. The version I use is obtained from:

     ZIB ( http://www.zib.de )

Octave setup
============

Octave is available at various web sites including: http://www.octave.org

.octaverc
---------

The .octaverc file should contain the following lines:

     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     %% Startup file for Octave for use with MTT
     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     implicit_str_to_num_ok = 1;
     empty_list_elements_ok = 1;

.oct file dependencies
----------------------

Successful compilation of .oct code requires that Octave has been configured to use dynamically linked libraries and that the Octave libraries liboctave, libcruft and liboctinterp are available on the system.

This can be acheived by compiling Octave from the source code, configured with the options --enable-shared and --enable-dl.

A number of additional libraries and headers are also required to be installed on a system. These include,

  * ncurses and readline terminal control routines

  * blas or altas basic linear algebra subprograms, usually optimised for the specific processor

  * fftw fast Fourier transform routines

  * g2c GNU Fortran to C conversion routines

  * kpathsea TeX path search routines

Note that on many GNU/Linux distributions, the necessary headers are contained in development packages which must be installed in addition to the standard library package.

Further information on configuring and installing Octave to handle dynamic libraries (DLDs) can be found in the Octave documentation (http://www.octave.org/docs.html).

Paths
=====

There are a number of paths that must be set correctely for MTT to work. These are normally set up by sourcing the file mttrc that lives in the MTT home directory.

$MTTPATH
--------

The environment variable $MTTPATH points to the mtt home directory. This is usually /usr/local/lib/mtt.

$MTT_COMPONENTS
---------------

The environment variable $MTT_COMPONENTS is a colon-separated path pointing to directories containing components and subsystems. By default

     MTT_COMPONENTS=.:$MTT_LIB/lib/comp/

but you may wish to add your own component libraries:

     MTT_COMPONENTS=my_library_path:$MTT_COMPONENTS

$MTT_CRS
--------

The environment variable $MTT_CRS is a colon-separated path pointing to directories containing constitutive relationships. By default

     MTT_CRS=$MTTPATH/lib/cr

but you may wish to add your own component libraries:

     MTT_CRS=my_cr_path:$MTT_CRS

$MTT_EXAMPLES
-------------

The environment variable $MTT_EXAMPLES is a colon-separated path pointing to directories containing EXAMPLES and subsystems. By default

     MTT_EXAMPLES=$MTTPATH/lib/examples

but you may wish to add your own component libraries:

     MTT_EXAMPLES=my_examples_path:$MTT_EXAMPLES

$OCTAVE_PATH
------------

The $OCTAVE_PATH path must include the relevant paths for mtt to work properly. In particular, it must include:

     $MTTPATH/trans/m
     $MTTPATH/lib/comp/simple
     $MTTPATH/lib/comp/compound

File structure
==============

The recommended installation of MTT uses the following directory structure with corresponding contents. Normally, each of the listed directories is a subdirectory of /usr/local. The directory mtt is pointed to by $MTTPATH (see $MTTPATH).

mtt/bin
     This is the home directory for MTT. MTT  itself lives here along with mttrc.

mtt/bin/trans
     The transformations executed by MTT.

mtt/bin/trans/m
     The  m-files associated with the transformations.

mtt/bin/trans/awk
     The  awk scripts associated with the transformations.

mtt/lib
     The place for components, examples and CRs which will be updated.

mtt/lib/comp/simple
     The  m-files defining the simple components.

mtt/lib/comp/compound
     The  m-files defining the compound components.

mtt/lib/cr/r
     constitutive relationship definitions

mtt/lib/examples
     Some examples.

mtt/examples/metamodelling
     Examples from the book.

mtt/doc
     The  documentation files for MTT.

mtt/doc/Examples
     Examples used in the documentation.

Legal stuff
***********

GNU Free Documentation License
==============================

Version 1.2, November 2002

     Copyright © 2000,2001,2002 Free Software Foundation, Inc.
     59 Temple Place, Suite 330, Boston, MA  02111-1307, USA

     Everyone is permitted to copy and distribute verbatim copies
     of this license document, but changing it is not allowed.

 0. PREAMBLE

     The purpose of this License is to make a manual, textbook, or other functional and useful document free in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others.

     This License is a kind of ``copyleft'', which means that derivative works of the document must themselves be free in the same sense.  It complements the GNU General Public License, which is a copyleft license designed for free software.

     We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does.  But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book.  We recommend this License principally for works whose purpose is instruction or reference.

 1. APPLICABILITY AND DEFINITIONS

     This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License.  Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein.  The ``Document'', below, refers to any such manual or work.  Any member of the public is a licensee, and is addressed as ``you''.  You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law.

     A ``Modified Version'' of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language.

     A ``Secondary Section'' is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document's overall subject (or to related matters) and contains nothing that could fall directly within that overall subject.  (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.)  The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them.

     The ``Invariant Sections'' are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License.  If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant.  The Document may contain zero Invariant Sections.  If the Document does not identify any Invariant Sections then there are none.

     The ``Cover Texts'' are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License.  A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words.

     A ``Transparent'' copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters.  A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text.  A copy that is not ``Transparent'' is called ``Opaque''.

     Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML, PostScript or PDF designed for human modification.  Examples of transparent image formats include PNG, XCF and JPG.  Opaque formats include proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML, PostScript or PDF produced by some word processors for output purposes only.

     The ``Title Page'' means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page.  For works in formats which do not have any title page as such, ``Title Page'' means the text near the most prominent appearance of the work's title, preceding the beginning of the body of the text.

     A section ``Entitled XYZ'' means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language.  (Here XYZ stands for a specific section name mentioned below, such as ``Acknowledgements'', ``Dedications'', ``Endorsements'', or ``History''.)  To ``Preserve the Title'' of such a section when you modify the Document means that it remains a section ``Entitled XYZ'' according to this definition.

     The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document.  These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License.

 2. VERBATIM COPYING

     You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License.  You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute.  However, you may accept compensation in exchange for copies.  If you distribute a large enough number of copies you must also follow the conditions in section 3.

     You may also lend copies, under the same conditions stated above, and you may publicly display copies.

 3. COPYING IN QUANTITY

     If you publish printed copies (or copies in media that commonly have printed covers) of the Document, numbering more than 100, and the Document's license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover.  Both covers must also clearly and legibly identify you as the publisher of these copies.  The front cover must present the full title with all words of the title equally prominent and visible.  You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects.

     If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages.

     If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using public-standard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public.

     It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document.

 4. MODIFICATIONS

     You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it.  In addition, you must do these things in the Modified Version:

      A. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document).  You may use the same title as a previous version if the original publisher of that version gives permission.

      B. List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement.

      C. State on the Title page the name of the publisher of the Modified Version, as the publisher.

      D. Preserve all the copyright notices of the Document.

      E. Add an appropriate copyright notice for your modifications adjacent to the other copyright notices.

      F. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below.

      G. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document's license notice.

      H. Include an unaltered copy of this License.

      I. Preserve the section Entitled ``History'', Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page.  If there is no section Entitled ``History'' in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence.

      J. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on.  These may be placed in the ``History'' section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission.

      K. For any section Entitled ``Acknowledgements'' or ``Dedications'', Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein.

      L. Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles.  Section numbers or the equivalent are not considered part of the section titles.

      M. Delete any section Entitled ``Endorsements''.  Such a section may not be included in the Modified Version.

      N. Do not retitle any existing section to be Entitled ``Endorsements'' or to conflict in title with any Invariant Section.

      O. Preserve any Warranty Disclaimers.

     If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant.  To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles.

     You may add a section Entitled ``Endorsements'', provided it contains nothing but endorsements of your Modified Version by various parties---for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard.

     You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version.  Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity.  If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.

     The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.

 5. COMBINING DOCUMENTS

     You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers.

     The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy.  If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work.

     In the combination, you must combine any sections Entitled ``History'' in the various original documents, forming one section Entitled ``History''; likewise combine any sections Entitled ``Acknowledgements'', and any sections Entitled ``Dedications''.  You must delete all sections Entitled ``Endorsements.''

 6. COLLECTIONS OF DOCUMENTS

     You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects.

     You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.

 7. AGGREGATION WITH INDEPENDENT WORKS

     A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an ``aggregate'' if the copyright resulting from the compilation is not used to limit the legal rights of the compilation's users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document.

     If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document's Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.

 8. TRANSLATION

     Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections.  You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers.  In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail.

     If a section in the Document is Entitled ``Acknowledgements'', ``Dedications'', or ``History'', the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title.

 9. TERMINATION

     You may not copy, modify, sublicense, or distribute the Document except as expressly provided for under this License.  Any other attempt to copy, modify, sublicense or distribute the Document is void, and will automatically terminate your rights under this License.  However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.

 10. FUTURE REVISIONS OF THIS LICENSE

     The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time.  Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.  See http://www.gnu.org/copyleft/.

     Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License ``or any later version'' applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation.  If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation.

ADDENDUM: How to use this License for your documents
----------------------------------------------------

To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:

       Copyright (C)  YEAR  YOUR NAME.
       Permission is granted to copy, distribute and/or modify this document
       under the terms of the GNU Free Documentation License, Version 1.2
       or any later version published by the Free Software Foundation;
       with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
       Texts.  A copy of the license is included in the section entitled ``GNU
       Free Documentation License''.

If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the ``with...Texts.'' line with this:

         with the Invariant Sections being LIST THEIR TITLES, with
         the Front-Cover Texts being LIST, and with the Back-Cover Texts
         being LIST.

If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation.

If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.

GNU GENERAL PUBLIC LICENSE
==========================

Version 2, June 1991

     Copyright © 1989, 1991 Free Software Foundation, Inc.
     59 Temple Place - Suite 330, Boston, MA  02111-1307, USA

     Everyone is permitted to copy and distribute verbatim copies
     of this license document, but changing it is not allowed.

Preamble
--------

The licenses for most software are designed to take away your freedom to share and change it.  By contrast, the GNU General Public License is intended to guarantee your freedom to share and change free software---to make sure the software is free for all its users.  This General Public License applies to most of the Free Software Foundation's software and to any other program whose authors commit to using it.  (Some other Free Software Foundation software is covered by the GNU Library General Public License instead.)  You can apply it to your programs, too.

When we speak of free software, we are referring to freedom, not price.  Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for this service if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs; and that you know you can do these things.

To protect your rights, we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights. These restrictions translate to certain responsibilities for you if you distribute copies of the software, or if you modify it.

For example, if you distribute copies of such a program, whether gratis or for a fee, you must give the recipients all the rights that you have.  You must make sure that they, too, receive or can get the source code.  And you must show them these terms so they know their rights.

We protect your rights with two steps: (1) copyright the software, and (2) offer you this license which gives you legal permission to copy, distribute and/or modify the software.

Also, for each author's protection and ours, we want to make certain that everyone understands that there is no warranty for this free software.  If the software is modified by someone else and passed on, we want its recipients to know that what they have is not the original, so that any problems introduced by others will not reflect on the original authors' reputations.

Finally, any free program is threatened constantly by software patents.  We wish to avoid the danger that redistributors of a free program will individually obtain patent licenses, in effect making the program proprietary.  To prevent this, we have made it clear that any patent must be licensed for everyone's free use or not licensed at all.

The precise terms and conditions for copying, distribution and modification follow.

TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION

 0. This License applies to any program or other work which contains a notice placed by the copyright holder saying it may be distributed under the terms of this General Public License.  The ``Program'', below, refers to any such program or work, and a ``work based on the Program'' means either the Program or any derivative work under copyright law: that is to say, a work containing the Program or a portion of it, either verbatim or with modifications and/or translated into another language.  (Hereinafter, translation is included without limitation in the term ``modification''.)  Each licensee is addressed as ``you''.

     Activities other than copying, distribution and modification are not covered by this License; they are outside its scope.  The act of running the Program is not restricted, and the output from the Program is covered only if its contents constitute a work based on the Program (independent of having been made by running the Program). Whether that is true depends on what the Program does.

 1. You may copy and distribute verbatim copies of the Program's source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty; keep intact all the notices that refer to this License and to the absence of any warranty; and give any other recipients of the Program a copy of this License along with the Program.

     You may charge a fee for the physical act of transferring a copy, and you may at your option offer warranty protection in exchange for a fee.

 2. You may modify your copy or copies of the Program or any portion of it, thus forming a work based on the Program, and copy and distribute such modifications or work under the terms of Section 1 above, provided that you also meet all of these conditions:

      a. You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change.

      b. You must cause any work that you distribute or publish, that in whole or in part contains or is derived from the Program or any part thereof, to be licensed as a whole at no charge to all third parties under the terms of this License.

      c. If the modified program normally reads commands interactively when run, you must cause it, when started running for such interactive use in the most ordinary way, to print or display an announcement including an appropriate copyright notice and a notice that there is no warranty (or else, saying that you provide a warranty) and that users may redistribute the program under these conditions, and telling the user how to view a copy of this License.  (Exception: if the Program itself is interactive but does not normally print such an announcement, your work based on the Program is not required to print an announcement.)

     These requirements apply to the modified work as a whole.  If identifiable sections of that work are not derived from the Program, and can be reasonably considered independent and separate works in themselves, then this License, and its terms, do not apply to those sections when you distribute them as separate works.  But when you distribute the same sections as part of a whole which is a work based on the Program, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it.

     Thus, it is not the intent of this section to claim rights or contest your rights to work written entirely by you; rather, the intent is to exercise the right to control the distribution of derivative or collective works based on the Program.

     In addition, mere aggregation of another work not based on the Program with the Program (or with a work based on the Program) on a volume of a storage or distribution medium does not bring the other work under the scope of this License.

 3. You may copy and distribute the Program (or a work based on it, under Section 2) in object code or executable form under the terms of Sections 1 and 2 above provided that you also do one of the following:

      a. Accompany it with the complete corresponding machine-readable source code, which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,

      b. Accompany it with a written offer, valid for at least three years, to give any third party, for a charge no more than your cost of physically performing source distribution, a complete machine-readable copy of the corresponding source code, to be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,

      c. Accompany it with the information you received as to the offer to distribute corresponding source code.  (This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer, in accord with Subsection b above.)

     The source code for a work means the preferred form of the work for making modifications to it.  For an executable work, complete source code means all the source code for all modules it contains, plus any associated interface definition files, plus the scripts used to control compilation and installation of the executable.  However, as a special exception, the source code distributed need not include anything that is normally distributed (in either source or binary form) with the major components (compiler, kernel, and so on) of the operating system on which the executable runs, unless that component itself accompanies the executable.

     If distribution of executable or object code is made by offering access to copy from a designated place, then offering equivalent access to copy the source code from the same place counts as distribution of the source code, even though third parties are not compelled to copy the source along with the object code.

 4. You may not copy, modify, sublicense, or distribute the Program except as expressly provided under this License.  Any attempt otherwise to copy, modify, sublicense or distribute the Program is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.

 5. You are not required to accept this License, since you have not signed it.  However, nothing else grants you permission to modify or distribute the Program or its derivative works.  These actions are prohibited by law if you do not accept this License.  Therefore, by modifying or distributing the Program (or any work based on the Program), you indicate your acceptance of this License to do so, and all its terms and conditions for copying, distributing or modifying the Program or works based on it.

 6. Each time you redistribute the Program (or any work based on the Program), the recipient automatically receives a license from the original licensor to copy, distribute or modify the Program subject to these terms and conditions.  You may not impose any further restrictions on the recipients' exercise of the rights granted herein. You are not responsible for enforcing compliance by third parties to this License.

 7. If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License.  If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the Program at all.  For example, if a patent license would not permit royalty-free redistribution of the Program by all those who receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program.

     If any portion of this section is held invalid or unenforceable under any particular circumstance, the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances.

     It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system, which is implemented by public license practices.  Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice.

     This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License.

 8. If the distribution and/or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces, the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded.  In such case, this License incorporates the limitation as if written in the body of this License.

 9. The Free Software Foundation may publish revised and/or new versions of the General Public License from time to time.  Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.

     Each version is given a distinguishing version number.  If the Program specifies a version number of this License which applies to it and ``any later version'', you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation.  If the Program does not specify a version number of this License, you may choose any version ever published by the Free Software Foundation.

 10. If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different, write to the author to ask for permission.  For software which is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this.  Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally.

     NO WARRANTY

 11. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW.  EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM ``AS IS'' WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.  THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU.  SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

 12. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

END OF TERMS AND CONDITIONS

Appendix: How to Apply These Terms to Your New Programs
-------------------------------------------------------

If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.

To do so, attach the following notices to the program.  It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty; and each file should have at least the ``copyright'' line and a pointer to where the full notice is found.

     ONE LINE TO GIVE THE PROGRAM'S NAME AND A BRIEF IDEA OF WHAT IT DOES.
     Copyright (C) YYYY  NAME OF AUTHOR

     This program is free software; you can redistribute it and/or modify
     it under the terms of the GNU General Public License as published by
     the Free Software Foundation; either version 2 of the License, or
     (at your option) any later version.

     This program is distributed in the hope that it will be useful,
     but WITHOUT ANY WARRANTY; without even the implied warranty of
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
     GNU General Public License for more details.

     You should have received a copy of the GNU General Public License
     along with this program; if not, write to the Free Software
     Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA.

Also add information on how to contact you by electronic and paper mail.

If the program is interactive, make it output a short notice like this when it starts in an interactive mode:

     Gnomovision version 69, Copyright (C) 19YY NAME OF AUTHOR
     Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
     This is free software, and you are welcome to redistribute it
     under certain conditions; type `show c' for details.

The hypothetical commands show w and show c should show the appropriate parts of the General Public License.  Of course, the commands you use may be called something other than show w and show c; they could even be mouse-clicks or menu items---whatever suits your program.

You should also get your employer (if you work as a programmer) or your school, if any, to sign a ``copyright disclaimer'' for the program, if necessary.  Here is a sample; alter the names:

     Yoyodyne, Inc., hereby disclaims all copyright interest in the program
     `Gnomovision' (which makes passes at compilers) written by James Hacker.

     SIGNATURE OF TY COON, 1 April 1989
     Ty Coon, President of Vice

This General Public License does not permit incorporating your program into proprietary programs.  If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library.  If this is what you want to do, use the GNU Library General Public License instead of this License.