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

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]

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]