Changing the quality of the transient process in a nonlinear system (in Mathematica)

I urgently need advice and help.

I have a system of differential equations like this:

$begin{cases} frac{dx}{dt} == y[t] cdot alpha cdot sin(omega t) + frac{d}{dt}(alpha cdot sin(omega t))
frac{dy}{dt} + h cdot y(t) == frac{d}{dt}(e^{-(x[t] – 2)^2}) end{cases}$

Parameters: $alpha = 0.3, h = 1, omega = 2 pi 0.5, x(0)=1/4, y(0)=0$

It corresponds to the following structural scheme:

The code that simulates such a system is shown below:

ClearAll["Global`*"]

pars = {[Alpha]1 = 0.3, h1 = 1, [Omega]1 = 2 Pi 0.5}

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

sys = 
 NDSolve[{x'[t] == 
    hpf1[t] [Alpha]1 Sin[[Omega]1 t] + 
     D[[Alpha]1 Sin[[Omega]1 t], t], 
   y'[t] + h1 y[t] == D[extr, t], x[0] == 1/4, y[0] == 0}, 
  x, {t, 0, 500}]

The numerical solution is presented below:

Plot[{Evaluate[x[t] /. sys]}, {t, 0, 150}, PlotRange -> Full, 
 PlotPoints -> 50]

It can be seen that the transition process is a transition from the initial point to the final one with a certain character.

I need to change this character i.e. make the transition from one point to another exponentially.
Like this:

What are the ways to solve this problem?

What to do, add a regulator or manipulate the system of differential equations?

extr = Exp[-(x[t] - 2)^2];
ω1 = 2 Pi 0.5;

Manipulate[
 sys = NDSolve[{x'[t] == 
     y[t] α1 Sin[ω1 t] + 
      D[α1 Sin[ω1 t], t], y'[t] + h1 y[t] == D[extr, t],
     x[0] == 1/4, y[0] == 0}, x, {t, 0, 150}];
 Plot[{Evaluate[x[t] /. sys]}, {t, 0, 150}, 
  PlotRange -> All], {α1, 0, 1}, {h1, 0.5, 1.5}]