Re: Representation and Simulation of Dynamic Systems

*To*: mathgroup at smc.vnet.net*Subject*: [mg56984] Re: Representation and Simulation of Dynamic Systems*From*: "Jens-Peer Kuska" <kuska at informatik.uni-leipzig.de>*Date*: Thu, 12 May 2005 02:32:17 -0400 (EDT)*Organization*: Uni Leipzig*References*: <d5s9bh$lj9$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Hi, and you can not select your depend output variables (like y[t]) in your example and just insert the solution that NDSolve[] gives for your independent variables ? Strange ... Regards Jens "Caffa Vittorio Dr." <Caffa at iabg.de> schrieb im Newsbeitrag news:d5s9bh$lj9$1 at smc.vnet.net... > 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 > -------------------------------------------- >