(1) Consider discrete-time nonlinear time-
varying systems described by the difference equation
$x(k+1)=f(k, x(k)), \quad x(k) \in \mathbb{R}^{n}, k \in \mathbb{Z}$
where $f: \mathbb{Z} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is continous and $x\left(k_{0}\right)=\xi \in \mathbb{R}^{n}$.
My question is why they are saying the system is time-varying, by an example of such? what does it mean by time-varying? Can anyone give me an example of a not-time varying system in this context? Thanks.
(2) If my system becomes $x(k+1)= f(x(k), u(k))$ where $u(k):\mathbb Z\to \mathbb R^n$ is non-constant, is it still a time-varying?
(3) A solution for system described in $(1)$ is a function $\phi: \mathbb Z\to \mathbb R^n$ parametrized by initial state and time i.e $\phi(k_0; k_0,\xi)=\xi$, i.e $\phi(k+1; k_0, \xi)= f(k, \phi(k;k_0,\xi))$ Could any one tell me how to define a solution for the system described in (2)?
Thanks!
Best Answer
It's time varying because the function $ f $ has an explicit dependence on the discrete time $ k $ beyond the implicit dependence it has through the changing value of $ x(k) $. A system that doesn't vary over time would look like $ x(k+1) = f(x(k)) $.