m s a math model of systems

8/3/2019 M S a Math Model of Systems

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Modeling and Simulation ofSystems

Richard Zobel

What is a system?

What is a model?



General sequence from problem specificationto successful system design: (1)

1. Identify problem system, obtain ordevelop specification.

2. Describe system as a series ofmathematical equations.

3. Use a simulation package such asModelica (free!) or CLAWPACK (Conservation Laws Package) (alsofree!)



Modelica ®

Modelica ® is a non-proprietary, object-oriented, equation based language toconveniently model complex physical

systems containing, e.g., mechanical,electrical, electronic, hydraulic, thermal,control, electric power or process-oriented

subcomponents.

https://www.modelica.org/Different-views-of-Modelica



Modelica



Modelica Editor (1)

http://www.et.web.mek.dtu.dk/FME/preview.html



Modelica Editor (2)




Modelica Editor (3)




Modelica Editor (4)




Modelica Editor (5)




OPENMODELICA

OPENMODELICA is an open-sourceModelica-based modeling and simulationenvironment intended for industrial and

academic usage. Its long-termdevelopment is supported by a non-profitorganization – the Open Source Modelica

Consortium (OSMC).

http://www.openmodelica.org/index.php/home/consortium


http://www.openmodelica.org/images/M_images/om.jpg





CLAWPACK

CLAWPACK is a software packagedesigned to compute numerical solutionsto hyperbolic partial differential equations

using a wave propagation approach.

http://www.amath.washington.edu/~claw

http://www.amath.washington.edu/~claw



Various extensions of CLAWPACK

CLAWMAN -- for solving problems on curvedmanifolds

TsunamiClaw -- for solving shallow water wavepropagation and inundation problems

ChomboClaw -- The functionality of CLAWPACK inthe adaptive mesh refinement framework ofCHOMBO.

WENOCLAW -- High order accurate wave-propagation algorithms.

sphereCLAW -- Multi-block code for cubed spheregrids

MHDCLAW -- Constrained transport algorithm forMHD

http://www.amath.washington.edu/~claw/extensions.html

http://www.amath.washington.edu/~claw/clawman.html

http://www.amath.washington.edu/~claw/tsunamiclaw.html

http://www.amath.washington.edu/~calhoun/demos/ChomboClaw/

http://seesar.lbl.gov/anag/chombo/index.html

http://www.amath.washington.edu/~claw/extensions/wenoclaw/www/index.html

http://www.math.wisc.edu/~rossmani/sphereCLAW/

http://www.math.wisc.edu/~rossmani/MHDCLAW/

http://www.math.wisc.edu/~rossmani/MHDCLAW/

http://www.math.wisc.edu/~rossmani/sphereCLAW/

http://www.amath.washington.edu/~claw/extensions/wenoclaw/www/index.html

http://seesar.lbl.gov/anag/chombo/index.html

http://www.amath.washington.edu/~calhoun/demos/ChomboClaw/

http://www.amath.washington.edu/~claw/tsunamiclaw.html

http://www.amath.washington.edu/~claw/clawman.html

http://www.amath.washington.edu/~claw/extensions.html



General sequence from problem specificationto successful system design: (2)

4. Input mathematical model to

simulation package.

(Strong note! No Programming required! )

5. Input required constants, functions,tables, etc.

6. Provide Inputs (forcing functions)and Initial Conditions.

7. Run some test simulations.



VERIFICATION ANDVALIDATION (1)

1. Verify that the test simulationsbehave as predicted or expected.

2. If not, revise model and try again!

3. If OK, this completes theVERIFICATION phase.

4. The next phase is VALIDATION.




T h i s c o n c e r n s c o m p a r i n g t h esimulation results with what actuallyhappens in the real-world system.

If the real system performance closelyagrees with the simulation then thesimulation can be used (with care! ) to

predict general system behavior within

limits. Beware of interpolation andextrapolation.




Common sense and experience isneeded!

If simulation predictions do not agreesufficiently with the real systembehavior, then it is necessary to revisethe simulation model until agreement is

reached.

If they agree sufficiently well,simulation studies of system behavior

may begin.



EXAMPLE SYSTEMBasic Control of a RAMJET Missile

The problem is that the step response must not have an overshoot

Simple 2nd

order Step Response Second Order x 1st

Order Exponential



DYNAMICS EQUATIONS 1

A simple 2nd order system can be expressed asa force equation:

a.d2x/dt2 +bdx/dt + cx = f(x),

where a is a mass. b is a friction element and c is a retarding force, as in a simple mass,damper, spring system as found in any vehicle

which has at least one or one on each wheel! f(x) is an input or command to the system.

This might be for instance,

a command to change course towards the target,

this might be a step of acceleration of 15g left.



DYNAMICS EQUATIONS 2

A missile has a mass (may be variabledue to fuel and/or weaponry use), air actsas a retarding force and also air friction

acts as damper.

As this is a 3D system, there are 3 suchequations, one for each of the 3 axes.

In addition, there is rotary motion aroundthe 3 axes.

Consequently, this is a 6 degree of

freedom system.



DYNAMICS EQUATIONS 2(Cont.)

Just to complicate it even more, it isessential to add the Coriolis effect, where a body has simultaneous translational and

rotary velocities.

Any high speed biker will tell you!

Most real systems are very complex.

Now you know why we use a simulationpackage and/or an actual simulator suchas a flight simulator.



Flame-out Problems (1)

The missile is asked to pull a 15g turn.

Initially the missile will then be pointingaway from its forward direction.

This is because of the inertia of the missile

For a ramjet this can be disastrous.

This is because there is a major loss of aircompression to the ramjet whichextinguishes the engine and the missilefalls out of the sky.



Flame-out Problems (2)

By adding an additional term to theequation, the overshoot does not happen,so the missile achieves maximum

performance.

It is important to be aware that a missilehas 4 fins and 4 wings – all small.

It is highly maneuverable at high speedsand accelerations.

There is no human on board!



SYSTEMS TRIALS (1)

All such models require verification ,and subsequent validation againstactual system trials.

Safety Critical Systems also requireformal proof of correctness ofoperation under all conditions.

Completeness is also important here.

Formal Methods is now a major part ofComputer Science.



SYSTEMS TRIALS (2)

It now also has to interface to the realhardware as well.

Operational training systems are alsoessential.

Safety critical systems include allforms of transport, all machinery, tools,any system with moving parts(including you).



SYSTEMS TRIALS (3)

Examples can be ladders and ropes,motorbikes!, cranes, trucks, planes,your friends! yourself (your worst

enemy).

The list is endless!

Thank you

m s a math model of systems

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