Zero overshoot criterion from the initial point $x_0$ to the final $x_*$, $x_*$ unknown in advance

Suppose a system is described by the following ODE:

$$\dot{x} = f(t,x)+u$$

where

• $$x$$ denotes the state of the system
• $$f(x,t)$$ is an unknown nonlinear function which meets the following condition: $$f(t,x) = 0 \text{ at steady state}$$
• $$u$$ is the control input

Task: bring $$f(x,t)$$ to zero (steady state) without overshoot.

Condition: variable $$x$$ passes from initial state $$x_0$$ to final state $$x_*$$, that do not know in advance.

Is it possible to make a transient in such a system without overshooting, knowing only $$\dot{x}$$,$$x$$ and $$f(t,x)$$ ?

• You need to provide us with more details regarding the $f(t,x)$ function. Your question now seems too general. Maybe tell us any conditions that are met by $f(t,x)$. Apr 21 '21 at 10:13
• @Teo Protoulis There is just black-box with unimodal function, that have input $x$ and there is an output $f$. Nothing more is known according to the conditions of the problem.
– dtn
Apr 21 '21 at 10:20
• So, unimodal means that $f(t,x)$ has either a global minimum or a global maximum meaning it is bounded from below or from above correspondingly. Due to the black box could you perform some system identification in order to come up with a model of the system ? Or you just have to answer this issue about overshooting without anything else ? Apr 21 '21 at 10:26
• @TeoProtoulis It is strictly forbidden to use identification. All that is available is signal $\dot{x}$,$x$ and $f(x,t)$. The problem must be solved in real time.
– dtn
Apr 21 '21 at 10:45
• And what's the order of $x$ and $\dot{x}$ ? I mean is it $x = [x_1 \ x_2 \ \dots \ x_n]$ or is it jut order of 1 ? Apr 21 '21 at 10:48

Im going to step out of the comment section as it is fairly limited. The quickest answer to the question:

Is it possible to somehow form a transient process (with given properties) if the steady state is not known in advance, but it is known that in the steady state f(x,t)=0?

Given the proposed conditions is simple: no. And the explanation boils down to one simple reason: There does not exist a one-size fits all model-free controller (yet). In fact, the field of control engineering revolves around developing a controller that is capable of controlling a system according to requirement (just as you stated). However, in order to guarantee these requirements, a model of the system is required (as these guarantees are usually made before the controller is plugged into the system). For instance, lets look at the definition of Lyapunov stability:

Suppose: $$\dot{x}(t) = f(t, x)$$ Where $$f(t,x)$$ has an equilibrium at $$x^*$$, then this system is considered lyapunov stable if for every $$\epsilon > 0$$ there exists a $$\delta >0$$ such that if $$\|x(0) - x^*\| < \delta$$ then for every $$t\geq 0$$ $$\|x(t) - x^*\| < \epsilon$$. This implies already that $$\epsilon \leq \delta$$. Or in simpler terms: a system is stable around a given equilibrium if the state of the system does not continuously diverge from this equilibrium. In order to ensure the controller you created stabilizes the system, you could use this statement. However, to do so it is required to predict the response of the system with the controller. As you lack a model to predict this response, that is impossible. The guarantees around performance and robustness differ, but they do require a model to ensure this.

To quickly go off-track from classical control, machine learning methods such as reinforcement learning or on--line Data driven control could yield a controller that is both stabilizing and satisfies the performance requirements. However, these types of controllers yield terrible control inputs the first N samples. Internally, they basically develop an internal predictor of the response (practically a model) such that they can optimally decide which input should be applied.

Since I know you have been struggling with this question for over half a year now, I would nearly suggest reverting the statement and find out that it is indeed impossible to do this without a model. If a proper proof can be found somewhere in the literature, you could use that as an argument against however contracts you.

So TL:DR, you need a model to guarantee stability and performance requirements. Otherwise everything just boils down to guessing and keeping your fingers crossed it doesnt explode. If system identification is not permitted, an empirical model can be established using physics first principles.

• I have one hypothesis, but it needs to be tested. Are you familiar with tools like feedback linearization and asymptotic output tracking?
– dtn
Apr 21 '21 at 14:43
• I need some help with this question. could you please take the time. engineering.stackexchange.com/questions/42818/…
– dtn
Apr 27 '21 at 4:34