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MathGroup Archive 2005

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Re: Representation and Simulation of Dynamic Systems

  • To: mathgroup at smc.vnet.net
  • Subject: [mg56968] Re: [mg56928] Representation and Simulation of Dynamic Systems
  • From: "Caffa Vittorio Dr." <Caffa at iabg.de>
  • Date: Wed, 11 May 2005 05:24:48 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Chris,

I have used as a toy example a dynamic system which happens to be linear
with constant coefficients. Actually, I wanted to focus the discussion
on non-linear dynamic systems. 

(As a special case, linear systems, possibly with non-constant
coefficient, could be also analyzed with an extension of the method I am
suggesting. Given the system description, one could for instance compute
symbolically the matrices A(t), B(t), C(t), D(t). Then using the method
of adjoints or a covariance simulation one could compute the system for
example the system response to a stochastic input.)

Actually the method I am suggesting is a surrogate of an interface into
which one could place a block diagram. If such an interface were
available I would probably do all my simulation work with Mathematica
instead of using that-simulation-link-addition-which-must-not-be-named
on that-mathematics-laboratory-program-which-must-not-be-named.

Cheers, Vittorio


>-----Original Message-----
>From: Chris Chiasson [mailto:chris.chiasson at gmail.com]
To: mathgroup at smc.vnet.net
>Sent: Tuesday, May 10, 2005 4:22 PM
>To: Caffa Vittorio Dr.
>Cc: mathgroup at smc.vnet.net
>Subject: [mg56968] Re: [mg56928] Representation and Simulation of Dynamic Systems
>
>It seems like you are solving systems with constant coefficients.
>
>The trickiest parts would be (a) designing an interface into which one
>could place a block diagram and (b) having Mathematica interpret that
>diagram.
>
>If you could do that, control system professional can probably do
>whatever else you might need ...
>
>http://library.wolfram.com/examples/CSPExtending/
>
>Does anyone know if there is a good bond graph or block diagram
>"package" available for Mathematica?
>
>I would really like to see the equivalent of
>that-simulation-link-addition-which-must-not-be-named on
>that-mathematics-laboratory-program-which-must-not-be-named for
>Mathematica.
>
>Regards,
>
>On 5/10/05, Caffa Vittorio Dr. <Caffa at iabg.de> wrote:
>> The behavior of (time-continuous, non-linear) dynamic systems can be
>> numerically investigated with NDSolve. One can first sketch a block
>> diagram of the system and then convert it into equations. Here is a
toy
>> example after the conversion:
>>
>>    pos'[t] = vel[t]
>>    vel'[t] = -k pos[t] + force[t] / m
>>
>> This works fine if the variables are all states, as in the example
>> above. But often, in order to describe a given dynamic system you
want
>> or you have to introduce some auxiliary variables (i.e. variables
which
>> are not states). This is in fact the case if you want to describe a
>> generic dynamic system. Here are the standard equations:
>>
>>    x'[t] = f[x[t], u[t], t]    (state equations)
>>    y[t] = g[x[t], u[t], t]     (output equations)
>>
>> where: x = state vector, u = input vector, y = output vector, t =
time.
>> In this case the components of the output vector are the "auxiliary"
>> variables.
>>
>> I'm considering here a scheme for representing dynamic systems
(possibly
>> using a block diagram as a starting point) which allows the usage of
>> auxiliary variables. This representation can be transformed into
>> equations for NDSolve automatically. After having solved the
equations
>> it is possible to inspect not only the state variables but also the
>> auxiliary variables.
>>
>> Comments or alternative solutions to the problem I'm considering are
>> welcome!
>>
>> Procedure
>>
>> o) Sketch the system on a piece of paper. Here is a toy example:
>>
>>                           ---------- [ -k ] ---------
>>                          |                           |
>>                          V                           |
>> force[t] --> [ 1/m ] --> + --> [ 1/s ] ---> [ 1/s ] --->  pos[t]
>>                             |           |
>>                             |            -------------->  vel[t]
>>                             |
>>                             --------------------------->  acc[t]
>>
>> Note: [ 1/s ] is an integrator block
>>       [  k  ] is a gain block
>>
>> o) Convert the sketch into a system description:
>>
>> In[1]:= sys = {pos'[t] -> vel[t],
>>         vel'[t] -> acc[t],
>>         acc[t] -> -k pos[t] + force[t] / m};
>>
>> Note: the arrow points to the source of the signal.
>>
>> o) Make a list of the state variables:
>>
>> In[2]:= states = {pos[t], vel[t]};
>>
>> o) Form the differential equations (the following steps could be
>> performed by a function):
>>
>> In[3]:= lhs = D[states, t]
>>
>> Out[3]= {pos'[t], vel'[t]}
>>
>> In[4]:= rhs = D[states, t] //. sys
>>
>>                  force[t]
>> Out[4]= {vel[t], -------- - k pos[t]}
>>                     m
>>
>> In[5]:= eqns = Join[Thread[lhs == rhs], {pos[0] == pos0, vel[0] ==
>> vel0}]
>>
>>                                        force[t]
>> Out[5]= {pos'[t] == vel[t], vel'[t] == -------- - k pos[t], pos[0] ==
>> pos0,
>>                                           m
>> vel[0] == vel0}
>>
>> o) Specify the parameters:
>>
>> In[6]:= params = {m -> 10, k -> 2, pos0 -> 0, vel0 -> 0, force[t] ->
>> Sin[t]};
>>
>> o) Solve the differential equations:
>>
>> In[7]:= sol = First[NDSolve[eqns /. params, states, {t, 0, 10}]]
>>
>> Out[7]= {pos[t] -> InterpolatingFunction[{{0., 10.}}, <>][t],
>>
>>          vel[t] -> InterpolatingFunction[{{0., 10.}}, <>][t]}
>>
>> o) Inspect the results (including auxiliary variables)
>>
>> In[8]:= Plot[pos[t] /. sol, {t, 0, 10}]
>>
>> Out[8]= -Graphics-
>>
>> In[9]:= Plot[acc[t] //. sys /. params /. sol, {t, 0, 10}]
>>
>> Out[9]= -Graphics-
>>
>> Cheers, Vittorio
>>
>> --------------------------------------------
>> Dr.-Ing. Vittorio G. Caffa
>> IABG mbH
>> Abt. VG 32
>> Einsteinstr. 20
>> 85521 Ottobrunn / Germany
>>
>> Tel. (089) 6088 2054
>> Fax: (089) 6088 3990
>> E-mail: caffa at iabg.de
>> Website : www.iabg.de
>> --------------------------------------------
>>
>>
>
>
>--
>Chris Chiasson
>http://chrischiasson.com/
>1 (810) 265-3161


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