State constraints for continuous control system

Question

I have the following system:

$$x_1' = x_2$$ $$x_2' = f(x_1, x_2, x_2^{ref})$$

In this system, $x_1$ and $x_2$ are state variables (e.g. position and velocity) while $x_2^{ref}$ is a possibly changing reference value for $x_2$ (e.g. the velocity desired by a pilot). The function $f$ is some control law for driving $x_2$ to $x_2^{ref}$.

I want to impose constraints on $x_2^{ref}$ in order to ensure that $x_1$ stays in some safe region. More precisely, I want some $C$ such that if $x_2^{ref} \in C(x_1, x_2)$, then for all trajectories of the system, $x_1(t) \in S$ for some set $S \subseteq \mathcal{R}$ (assuming necessary initial conditions). That is, C gives a set of safe values for $x_2^{ref}$ (e.g. safe velocities) as a function of $x_1$ and $x_2$.

How can I approach this problem? Are there techniques from control theory? What about if $f$ is linear?

Answer 1

If $f$ is invertible, then you can take a two step process:

1. Compute constraints on $u$ in the following system $$\dot{x_1} = x_2$$ $$\dot{x_2} = u$$ that keep $x_1 \in S$.
2. Invert $f$ to get constraints on $x_2^{ref}$.

Answer 2

The set $C(x_1,x_2)$, taken for a specific pair of $(x_1,x_2)$, is generally referred to as the region of attraction (RoA) or basin of attraction. For this sub-problem, as @IvanAbraham points out, Lyapunov analysis is a good start. I recommend the online textbook Underactuated Robotics (which accompanies an edX mooc), in particular chapter 11, as a reference. A lot of the algorithms there focus on actually designing $f$ in addition to guaranteeing its stability within a specific RoA.

In particular, note that for stable linear systems, it is known that a quadratic Lyapunov candidate function exists: this means you can formulate this as a convex optimisation problem, and if the solver returns "infeasible", it means your system is unstable (see above textbook for details).

If your system isn't too complicated, you can do bifurcation analysis: basically linear stability analysis, where you study how changes of the parameters affect existence and stability of fixpoints. The book Nonlinear Dynamics and Chaos by Strogatz is a great introduction to this.