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

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.

• You have been given an answer and a couple of useful comments. Nov 26, 2020 at 7:09
– dtn
Nov 26, 2020 at 7:10
• Answer has a vote, so someone else considers it appropriate - just because you don’t like it... Nov 26, 2020 at 8:51
• The fact remains that this solution does not fit the problem I reformulated and posed again.
– dtn
Nov 26, 2020 at 8:55
• Input/Output linearization and then a tracking controller. However, it would be helpful to provide is with some background information, like what does your system of ODE's represent, such that we which information is available and which variables can be manipulated. Nov 29, 2020 at 22:13

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.

• I tried to apply your approach in Matlab. This thought also visited me. The problem here is that for some functions, the derivative is zero not only at the maximum / minimum point, but also tends to zero outside it. Take a look at this link please: wolframalpha.com/input/… Do you want to work together on this task?
– dtn
Nov 27, 2020 at 3:36
• The given function approaches zero if $x\rightarrow\pm\infty$, but never reaches it. This means that it is not an equilibrium. $f$ does however have an equilibrium at $x=0$. Through the state equation it can be determined if this equilibrium is also a stable one (think of the pendulum / inverted pendulum case). If it is stable, the system will naturally converge to this optimum, else a controller should make it stable. If you can provide more detailled information of the system I can help you further, but in the end its you who has to solve it :). Nov 27, 2020 at 11:29
• The structure of $f (t)$ is not known to us, but this signal is available for measurement. $f(t)$ - black-box. The block diagram of the control system is as follows. https://ibb.co/fDxJqBD Block G - estimation of the gradient of the function, i.e. $\frac{df}{dt}$.
– dtn
Nov 28, 2020 at 4:21
• i.e. $\frac{df}{dx}$, excuse me
– dtn
Nov 28, 2020 at 4:37
• This block diagram shows nothing about the differential equations you gave. Additionally, it shows the presence of a $u$, but I cannot figure out where this fits in the equations (also because $y$ can not be found in the given block diagram). Can I now assume $f$ is a function of $x$? What do these differential equations describe? the physics of a system, or some controller behaviour? Also, do note that if a system is unknown, controlling it optimally (or even with stability guarantees) is impossible. However, identifying these dynamics with System Identification can solve this. Nov 28, 2020 at 12:37