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

Question

My question is a continuation of the topic.

https://math.stackexchange.com/questions/3910814/changing-the-quality-of-the-transient-process-in-a-nonlinear-system-in-mathemat

Unfortunately, last time I didn’t get any help, so I decided to reformulate the question.

I have a system of differential equations like this:

$\begin{cases} \frac{dx}{dt} = y \cdot \alpha \cdot sin(\omega t) + \frac{d}{dt}(\alpha \cdot sin(\omega t)) \\ \frac{dy}{dt} + y = \frac{d}{dt}(f(t)) \end{cases}$

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

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

$x_{*}$ - the point at which the maximum or minimum of the function $f(t)$ is reached.

Let's pay special attention to the variable $x$.

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

The solution is to go from the starting point $x(0)$ to the point $x_{*}$ at which the minimum or maximum of the function $f(t)$ is reached.

For my system with such parameters, the solution will look like this:

It can be seen that the transition process is a transition from the initial point to the final one with a certain character, which is added to the additional sinusoidal signal $\alpha \cdot sin(\omega t)$.

It is necessary to change the nature of the transition from one point to another, making it exponential (naturally retaining an additional sinusoidal signal $\alpha \cdot sin(\omega t)$), i.e. it must be described by law:

$x(t)=(x(0)−x_{*}) \cdot exp(-t) + x_{*} + \alpha \cdot sin(\omega t)$,

Like this:

The problem is complicated by the fact that the function $f(t)$ and the point at which its minimum/maximum $x_{*}$ is located, generally speaking, are not known to us.

How to get out of the situation using the structure of the differential equation? Add additional input signal? Include an additional differential equation in the system? Use adaptive techniques?

Help! Please give at least some idea. I am desperate and my hands give up.

Answer 1

The problem is complicated by the fact that the function f(t) and the point at which its minimum/maximum x∗ is located, generally speaking, are not known to us.

So if I understand it correctly: $f(t)$ is a unknown function, of which your controlled system should approach either the maximum or the minimum value $f(t)$ has over $t$. If I look at your system of equations, this appears to be rather easy. Even though $f(t)$ is not known, we do know its derivative: $$\frac{dy}{dt} + y = \frac{d}{dt}f(t)$$ As commonly known, if a functions derivative equals 0, the function can be in 3 situations: Either a local minimum, local maximum or a saddle point (think of $x^3$). Given is that $f(t)$ only contains one extreme, the best bet is to say that if the derivative reaches $0$, the optimum point $x^*$ is reached. This means that the control goal actually becomes the following: $$\frac{dy}{dt} + y = 0$$ But this is the furthest I can help. Simply because I have no insight in what the physical meaning of the states $x$ and $y$ are, therefore I cannot give an advice on where to apply an external input (which is required to accelerate your problem). Additionally, your definition of $f(t)$ sounds poorly. if $f(t)$ is just a function of time, it implies that $y$ responses independently to $x$ and that $f(t)$ is not affected by either $y$ or $x$. your example showed that $f$ is a function of $x(t)$ instead, which does imply the link.