I have a nonlinear system and I am doing some simulations in order to reach a situation in which all the states are equal to zero.
I have done a lot of simulations and tried many things, but I couldn't reach the desired state. At this point, I am starting to think that the state I want to reach is not reachable.
So, my question is:
How do I check if a state is reachable when I have a nonlinear system?
For example, consider the equations of the unicycle robot which are the simplest that comes to my mind:
$\dot{x}=vcos\theta$
$\dot{y}=vsin\theta$
$\dot{\theta}=\omega$
In this case I know that the state $(0,0,0)$ is reachable, but how do I show it mathematically?
Just like stable linear systems usually converge to some stable equilibrium, non-linear systems do to. However, non-linear systems can have any amount of equilibria. Undamped systems often get stuck in cycle (for instance the undamped inverted pendulum). This means that every reachable state of the system must be inside this cycle. the cycles shape (doesnt have to be a circle) is undetermined and can be influenced by the initial conditions and the external input. To continue to your system, suppose the velocity $v$ and the angular velocity $\omega$ are constants. Additionally, since you wrote down the difference equations, I suppose your state vector represents $\left[x, y, \theta \right]^T$. From what it appears, I can integrate each of these equations over time directly: $$\theta = \int \omega~~ dt = \omega t + \theta_0$$ $$x = \int vcos\theta~~ dt = \frac{v}{\omega}sin\theta+x_0$$ $$y = \int vsin\theta~~ dt = -\frac{v}{\omega}cos\theta+y_0$$ Now as you might already imagine, these equations describe a circle with radius $\frac{v}{\omega}$ at center $\left[ x_0, y_0 \right]$. This means that only if the constants $v$, $\omega$ and / or the initial positions are chosen correctly, the state $(0,0,0)$ is reachable: $x_0 = 0, y_0 = \frac{v}{\omega} $ Which represents a circle that crosses the origin at $\theta = 0$. This entire story does indeed change if $v$ or $\omega$ are dissipative functions over time, at which the position $x$ and $y$ will always return to zero regardless of $\theta$.
So that concludes your specific example. In an abstract sense, determining reachability of a non-linear system is a very interesting topic that usually comes down to determining the stable equilbria and finding the one it will converge to from the given initial state. I do note that even for dissipative systems, some systems converge to a limit cycle. To make it more interesting, some systems converge to a limit cycle if it is in a certain range or initial states, and to a stable equilibrium point for others.