Re: Discretization of nonlinear differential equation
Tim Wescott
> Hi.
>
> I'm working on modeling a system which have resulted in a system of
> first order nonlinear differential equations s'(t) = f(t, s(t), u(t))
> with state vector s(t) and control input u(t) and s(t), u(t) \in R^3 The
> measurements are simply given by y(t) = s(t).
>
> The continuous time model is embedded in a digital control loop where
> there output y(n) is a sampled version of y(t) with zero order hold and
> Ts = 0.1 s. Similarly the discrete control inputs u(n) is converted
> through zero order hold to u(t) with the same sample time.
>
> I now want to write this model as: s(n + 1) = g(n, s(n), u(n)). I know
> how to do this is for a linear continous time continous model s'(t) =
> Ax(t) + Bu(t) with the c2d fuction but I am in doubt how to do this for
> the nonlinear model.
>
> Should I linearize the continous time differential equation first or
> what is the best method?
>
> Thanks in advance.
(I'm cross-posting my reply to sci.engr.control - really this time)
The best way to do this is the way that works right for your problem.
Isn't that helpful?
This being a nonlinear problem chances are that any solution is going to
be approximate. If you're lucky then that's not the case, you can do an
exact solution at each time step -- but then, if that were the case you
probably wouldn't be asking.
What you have to fall back onto is your bag of tricks. Any effort to
control a nonlinear system involves having a bag of mathematical tricks.
You try out tricks from your bag, and when you find one that works well
enough you test the heck out of it and you call it good.
In general when I do this sort of thing I figure out the amount of error
that I can tolerate, and I choose the easiest method for solving the
differential equation that meets that error criterion. This may mean
trying to find an exact solution, it may mean finding a way to express the
system so that I can do a one-time linearization and keep the errors in
check through feedback, it may mean that I use simple rectangular
integration at each step (and maybe crank the sample rate way up) -- there
are a number of different methods that one can try, but if there were One
True Method then folks wouldn't bother teaching so much linear systems
theory.
iman way
>
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Jerry Avins
إخفاء النص المقتبس -
>> - عرض النص المقتبس -
>
> I am happy to be a member of this group and I consider you my friends.
No friend would be as disruptive as you. You haven't even the decency to
start your own (off topic) thread. You hijack an unrelated one. Shame!
Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Tim Wescott
> iman way wrote:
>
> إخفاء النص المقتبس -
>
>>> - عرض النص المقتبس -
>>
>> I am happy to be a member of this group and I consider you my friends.
>
> No friend would be as disruptive as you. You haven't even the decency to
> start your own (off topic) thread. You hijack an unrelated one. Shame!
Jerry, I think it's a bot. This stuff gets splattered all over a number
of newsgroups; I doubt that there's any human there to read any replies.
Jerry Avins
I know the origin is gmail and groups.google.com. I gurss they think
it might not be PC to kill the account.