Solution of a system of nonlinear ODE in the form of a state space with nonzero initial conditions

Question

I have a system of differential equations like this:

$\begin{cases} \frac{dx}{dt} = hpf_1 \cdot \alpha \cdot sin(\omega t) + \frac{d}{dt}(\alpha \cdot sin(\omega t)) \\ \frac{dhpf_1}{dt} + hpf_1 = \frac{d}{dt}(extr(t)) \end{cases}$

where: $extr(t)$ - Any function that has one extreme (minimum or maximum). For example $extr(t) = e^{-(x(t))^2}$

$x$ and $hpf_1$ - variables of the system of differential equations.

I'm trying to get solution, but transforming the system into a state-space form. How to set nonzero initial conditions?

pars = {\[Alpha] = 0.1, h = 1, \[Omega] = 2 Pi 0.5, \[Beta] = 1}

extr = Exp[-(x[t])^2]

AffineStateSpaceModel[{x'[t] == 
   hpf1[t] \[Alpha] Sin[\[Omega] t] + u[t], 
  hpf1'[t] + h hpf1[t] == D[extr, t]}, {x[t], hpf1[t]}, u[t], x[t], t]

Plot[OutputResponse[%, D[\[Alpha] Sin[\[Omega] t], t], {t, 0, 3}] // 
  Evaluate, {t, 0, 3}, PlotRange -> Full]

Answer 1

See this answer for the topic:

https://math.stackexchange.com/questions/3923222/changing-the-quality-of-the-transient-process-in-a-nonlinear-system-part-ii

A notation in the form of an affine state space can be obtained in the form:

extr = Exp[-(x[t])^2]

AffineStateSpaceModel[{x'[t] == 
   hpf1[t] \[Alpha] Sin[\[Omega] t] + u[t], 
  hpf1'[t] + h hpf1[t] == D[extr, t]}, {x[t], hpf1[t]}, u[t], x[t], t]