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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:

enter image description here

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] == 
    y[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]

enter image description here

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:

enter image description here

What are the ways to solve this problem? What to do, add a regulator or manipulate the system of differential equations?

Please help me!

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  • $\begingroup$ what is hpf1[t]? $\endgroup$
    – NMech
    Commented Nov 17, 2020 at 10:09
  • $\begingroup$ I'm sorry. It is y(t) $\endgroup$
    – ayr
    Commented Nov 17, 2020 at 10:09
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    $\begingroup$ Well, generally you are trying to control a nonlinear system ? Is this the case or not ? You have a nonlinear system and you want to force certain behaviour to its output, right ? $\endgroup$ Commented Nov 17, 2020 at 12:09
  • $\begingroup$ @Teo Protoulis, Yes, you are absolutely right. It is necessary to introduce something into the system (a link or an additional control signal) that provides a given transient process (in this case, exponential). Those. endow the system with adaptive properties. $\endgroup$
    – ayr
    Commented Nov 17, 2020 at 15:47
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    $\begingroup$ Well, then you have to introduce some design specifications first of all and then choose which control scheme for nonlinear systems is appropriate to design for this specific system. Is the ES Controller block fixed or have you designed it ? $\endgroup$ Commented Nov 17, 2020 at 15:53

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