Re: NDSolve Repeated convergence test failure

• To: mathgroup at smc.vnet.net
• Subject: [mg47122] Re: NDSolve Repeated convergence test failure
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Fri, 26 Mar 2004 03:56:26 -0500 (EST)
• Organization: The University of Western Australia
• References: <c3uem8\$9ub\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <c3uem8\$9ub\$1 at smc.vnet.net>,
Steven Rey Ortiz <sro7516 at cs.tamu.edu> wrote:

> Howdy!  I at trying to use NDSolve to evaluate a set of three ODEs (two
> second order and one first order).  However, by construction the third
> function value will eventually eliminate the second order terms of the
> other two equations, and when it does this, Mathematica returns a
> "Repeated convergence test failure" and stops evaluating.  Is there a way
> to continue the integration process beyond this point?  The differential
> equations are still meaningful.  Following is a link to the notebook with
> the specific equations.  The last line shows the error.
>
> http://students.cs.tamu.edu/sro7516/double-pendulum.nb

Having a look at your Notebook, one sees that theta2 stablilizes before
t=2. That is its first and second derivatives are effectively zero (the
first and second derivatives are less than 10^(-6) in absolute value by
t = 1.92).

From this time onwards, the equation for theta2 is satisfied and the
equation for theta1 effectively reduces to an exactly solvable trivial
first order equation:

3 (Pi (Cos[t] + Sin[t]) + 8 theta1'[t]) == 0

Substituting the solution to this equation into the one for mtilde2 one
finds that it is also satisfied. Hence you have the solution, in closed
form, after the singularity.

> Problem Background: I am experimenting with an adaptive controller for a
> double pendulum system.  The notebook first defines the equations of
> motion for the system.  The controller assumes an initial value for the
> mass of the second link, but adjusts this value based on feedback from the
> system.  When it has the correct value for the mass, the control law will
> cancel the secend order terms from the equations of motion (and correctly
> track the desired trajectory).

The question, then, is how to get NDSolve to do this automatically.
Perhaps Rob Knapp from WRI will be able to offer some help here.

Cheers,
Paul

--
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
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AUSTRALIA                            http://physics.uwa.edu.au/~paul

```

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