Approximation of dynamic systems

Question

I was looking for a formal approach to simplify models of dynamic systems. Say we have a dynamic system given by

$\frac{dx}{dt} = f(t,x,u), ~~~~x(t_0) = x_0$

$y = g(x)$

We know $f$ and $g$ but they are non linear and computationally complex. How does one construct a simplified system model

$\frac{dx}{dt} = \tilde{~f~}(t,x,u), ~~~~x(t_0) = x_0$

$y = g(x)$

I am considering a function approximation approach i.e. to find a function that approximates $f$.

Answer 1

This is a dangerously broad question. Depending on the context and on $f$, a lot of different approximations may be relevant (for instance, if $f(x)=\sin x+ 10^{-6} x^3$, it is obviously very natural to approximate $f$ by $\tilde f(x)=\sin x$).

However one very common approximation is to linearize $f$, especially if you're interested in dynamics near an equilibrium. For instance, if you consider the pendulum equation $\frac{d^2 x}{dt^2}=-\sin x$ and you are interested in the behaviour for $x \simeq 0$, then a reasonnably approximation is $\frac{d^2 x}{dt^2}=-x$ which can be solved explicitly.