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What are some control theory tools for solving problems of the following form?:

Given a system model, control input constraints $I$, and control output constraints $O$, what is the largest set of states $S$ such that if the system starts in $S$, there exists a control signal $u\in I$ such that the system will stay in $O$ for all time?

To be more concrete, suppose we have a system

$$x’(t) = v(t)$$ $$v’(t) = a(t)$$

and we want to compute the largest set of states $S$ for which there exists a control input $a(t)$ such that if $[x(0), v(0)]\in S$,

$$\forall t, a(t) \geq A$$ $$\forall t, x(t) \leq X$$

for some constants $A$ and $X$. Since this system is simple, it's straightforward to come up with such a set. However, are there tools from control theory to solve these kinds of problems when the system is not as simple? How would a control theorist solve it?

For example, suppose we have the following system

$$x’(t) = v_x(t)$$ $$v_x’(t) = a(t) * \sin{(\theta(t))}$$ $$y’(t) = v_y(t)$$ $$v_y’(t) = a(t) * \cos{(\theta(t))}$$

and we want to compute the largest set of states $S$ for which there exists a control inputs $a(t)$ and $\theta(t)$ such that if $[x(0), v_x(0),y(0),v_y(0)]\in S$,

$$\forall t, a(t) \geq A$$ $$\forall t, \theta_{min} \leq \theta(t) \leq \theta_{max}$$ $$y \leq Y$$ $$-X \leq x \leq X$$

How would a control theorist solve this problem? Are these kinds of problems easier if we don't require the largest initial set for which the constraints can be satisfied, but any non-empty set?

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  • $\begingroup$ You provided a perfectly clear problem description. But just for clarification: What does $u \in > I$ mean? Thank you. $\endgroup$ – Max Herrmann May 13 '15 at 6:51
  • $\begingroup$ @MaxHerrmann Ah sorry, just a typo. It's supposed to be $u \in I$. I fixed the description. Thanks for pointing it out. $\endgroup$ – BmoreDaniel May 13 '15 at 16:00
  • $\begingroup$ I think you will probably need a form of MPC controllers. $\endgroup$ – WG- May 13 '15 at 18:11
  • $\begingroup$ @WG-MPC seems very promising at a high level. However, I see two issues with applying it to our problem. First, I'm not sure how to use the objective function of MPC. It seems like it should be used to find the largest set satisfying the constraints, but I'm not sure how to express that as an arithmetic objective function. Second, the work I've looked at on MPC seems to be limited to finite-horizon, whereas we want to ensure constraints for all time. However, thanks for the suggestion, and I'll keep reading about it. $\endgroup$ – BmoreDaniel May 13 '15 at 21:10
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I think what you're looking for is computation of Regions of Attraction (RoA), and as far as I know there isn't an established general solution for this (depending on the system, I've seen different approaches being used, including empirical exploration).

The most promising computational estimation approach I've seen so far is from Tedrake's lab, using Sum-of-Squares and therefore requiring your model to be a polynomial. Since formulated as an optimization problem, it can easily accept constraints on the control input. The really nice thing is that they put their toolbox on github, with examples as well. Calculating RoAs for limit-cycles (stable orbits) is still in the dev folder, but check out the VanDerPol oscillator in the examples folder, which shows RoA estimation for a stable fixpoint.

Note that this is slightly different from what you asked: the calculation of the RoA finds the set of initial states $S$ that can be brought back to a single point, instead of to the set $O$. The trick is to then iteratively do this for samplings of $S$, and then take the union of these sets.

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