Automatic time scaling

[baggepinnen]

Anecdotally, most issues related to simulation on the ControlSystems repo arise because someone has modeled an electronics circuit in seconds, but the time scales appearing in the model are on nano-micro seconds. The continuous-time solvers really dislike this.

For linear systems, you can always perform a similarity transform on the state-transition matrix without changing the input-output behavior,

à = T⁻¹AT
B̃ = T⁻¹ B
C̃ = CT
D̃ = D

we use this extensively in ControlSystems to keep systems well balanced. T can be any invertible matrix. A similarity transform can never change the eigenvalues of A and can thus not model time scaling, but it can change the typical size of state components during simulation.

For linear systems, scaling of time $t$ changes the Laplace transform $F$ of a function $f$ according to
$f(at) \leftrightarrow \dfrac{1}{a} F\big(\dfrac{s}{a}\big)$
which is also straightforward to apply. We could provide a function that applies this transformation automatically.