Re: NDSolve and hybrid dynamics (Differential Algebraic Equation DAE)

*To*: mathgroup at smc.vnet.net*Subject*: [mg113569] Re: NDSolve and hybrid dynamics (Differential Algebraic Equation DAE)*From*: schochet123 <schochet123 at gmail.com>*Date*: Wed, 3 Nov 2010 02:53:52 -0500 (EST)*References*: <iaongc$jf2$1@smc.vnet.net>

Even when a system has binary states it is common to approximate those states numerically by a smoothed function. Consider, for example the simple continuous approximation cpw[x_]=Piecewise[{{1, x < 1/10 - eps}, {1 + (x - (1/10 - eps))/(2 eps), 1/10 - eps <= x <= 1/10 + eps}}, 2] Taking, for example, eps=1/1000; substituting this into the original DAE, i.e., solving the problem appreqns = {x'[t] == y[t] + Sin[t], y[t] == cpw[x[t]], x[-Pi] == 1/2}; apprsol = {x[t], y[t]} /. NDSolve[appreqns, {x[t], y[t]}, {t, -5, 5}] yields a solution which differs from that obtained by using the true binary state by no more than about 10^(-4). If that is not good enough, then take a smaller eps and/or a smoother approximation. One reason for using the smoothed function is that it is not always easy/possible to solve the algebraic equation for the slaved variable. Consider for example alteqns = {x'[t] == y[t] + Sin[t], y[t]^3 - (2 x[t] + 100) y[t] == cpw[x[t]], x[-Pi] == 1/2, y[-Pi] == Chop[N[y /. ysubs[[1]], 50]]}; altsol = {x[t], y[t]} /. NDSolve[alteqns2, {x[t], y[t]}, {t, -5, 5}] where ysubs = Solve[y^3 - 101 y == 2]; In this example it is possible to omit the initial condition for y and then NDSolve picks one of the three possible values. Steve On Nov 2, 12:00 pm, Ravi Balasubramanian <ravi.balasubraman... at yale.edu> wrote: > Hello Oliver, > > Thanks for your suggestion. > > Unfortunately, a smoothing function would not be useful, since the > system has binary states. > > I figured a way to do this is to include the f[x] function into the > x'[t] equation such as > > x'[t] == f[x[t]] + Sin[t] > > (f[x] has only two states and no smoothing) > > and use NDSolve to solve for x[t]. This works. And then I can evalu= ate > f[x[t]] using the NDSolve solution. > > Just to summarize this discussion: NDSolve does not support binary > variables as part of the state? > > Ravi > > Oliver Ruebenkoenig wrote: > > > Ravi, > > > would a smoothing function be useful? > > > f[x_] := Which[ > > x <= -1, 0, > > -1 < x < 1, 1/2 + x (15/16 - x^2 (5/8 - 3/16 x^2)), > > x >= 1, 1] + 1 > > > Plot[f[(x - 0.1)/0.001], {x, -0.2, 0.2}] > > > eqns = {x'[t] == y[t] + Sin[t], y[t] == f[t], x[-\[Pi]] == = 1/2}; > > sol = NDSolve[eqns, {x[t], y[t]}, {t, -5, 5}] > > > Hope this helps, > > > Oliver > > > On Mon, 1 Nov 2010, Ravi Balasubramanian wrote: > > >> Friends, > > >> I am trying to solve a hybrid dynamics problem using NDSolve. > > >> I am aware that I could use the event locator methods in NDSolve and > >> manually take care of the transitions (set initial conditions etc). = But > >> my system is complicated, and I would have to do a lot of book-keeping=