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?
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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!)
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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.
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Modelica
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Modelica Editor (1)
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Modelica Editor (2)
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Modelica Editor (3)
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Modelica Editor (4)
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Modelica Editor (5)
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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).
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CLAWPACK
CLAWPACK is a software packagedesigned to compute numerical solutionsto hyperbolic partial differential equations
using a wave propagation approach.
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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
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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.
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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.
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VERIFICATION ANDVALIDATION (2)
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.
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VERIFICATION ANDVALIDATION (3)
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.
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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
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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.
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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.
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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.
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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.
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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!
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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.
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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).
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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