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MathGroup Archive 2005

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Re: Re: Representation and Simulation of Dynamic Systems

  • To: mathgroup at smc.vnet.net
  • Subject: [mg56989] Re: [mg56968] Re: [mg56928] Representation and Simulation of Dynamic Systems
  • From: Chris Chiasson <chris.chiasson at gmail.com>
  • Date: Thu, 12 May 2005 02:32:23 -0400 (EDT)
  • References: <200505110924.FAA24079@smc.vnet.net>
  • Reply-to: Chris Chiasson <chris.chiasson at gmail.com>
  • Sender: owner-wri-mathgroup at wolfram.com

Dr. Caffa Vittorio,
Do you have a good link (http://....) that describes covariance
simulation and/or the method of adjoints?
Regards,

On 5/11/05, Caffa Vittorio Dr. <Caffa at iabg.de> wrote:
> 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: [mg56989] [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
> 
> 


-- 
Chris Chiasson
http://chrischiasson.com/
1 (810) 265-3161


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