Re: Discretization of nonlinear differential equation


"pnachtwey" <pnachtwey@xxxxxxxxx> schrieb im Newsbeitrag news:171a1cc5-0723-47f5-9980-4e846f08eb2e@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Nov 17, 11:40 pm, "JCH" <ja...@xxxxxxxxxxxxxxxxxxx> wrote:
"Edward Jensen" <edw...@xxxxxxxxxxxxxx> schrieb im Newsbeitragnews:hdubk2$psm$1@xxxxxxxxxxxxxxxxxxxx





> 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?

EXAMPLE

Linearizing a nonlinear differential equation (DE):

*http://home.arcor.de/janch/janch/_control/20091118-non-linear-ode/

Red points: nonlinear DE
Black points: approximated linear DE

If system is load dependent then optimize the controller (PID) on at least 3
load points and adjust the controller parameters automatically using a
polynomial.

See Note in Math Model: Model Adaptation 2

--
Regards JCH
Stop your nonsense. You have been able to identify any of the systems
I have provided in the past. Your system identification doesn't have
an concept of a ambient value, dead time or know the difference
between type 0 and type 1 systems. It is easy to do what you are
doing if you know the solution and then generate the problem.


What is the bigger problem? Have a data set and find the differential equation(s)? Or vice versa?

It's much easier to solve e.g. 8 known non-linear differential equations, even on higher order:

EXAMPLE FOR 1st ORDER SYSTEM

* http://home.arcor.de/janch/janch/_control/20091120-non-linear-de/

Nothing is better than use the 'real' data values!


--
Regards JCH

.

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