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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