# SURF 2019: Geometry of Control-Invariant Sets

The concept of a *control-invariant* set---a set with the property that a controller can make the state of a system remain inside of for all positive ---is closely connected with safety and reliability of engineered systems [1]. Invariant sets can be used to construct controllers that make systems such as quadrotors avoid crashes [2]. Unfortunately, analytical expressions for control-invariant sets are not known for many important systems, and are also hard to compute numerically in high dimensions.

Some of the most fundamental dynamical systems are linear -order integrators:

For = 2 a closed-form expression of the maximal control-invariant set contained inside the unit hypercube is known [3. eq. (53)], but for higher orders of only numerical approaches are currently available. The figures above show numerical approximations for equal to 2 and equal to 3. The objective of this project is to investigate geometrical properties of control-invariant sets via a combination of analytical and numerical techniques. In particular,

- Search for closed-form expressions or cheap-to-evaluate algorithms that characterize control-invariant sets for n > 2.
- Investigate incremental algorithms, i.e., if the set for n is known, can we characterize the set for n+1?
- Stretch goal: generalize the incremental ideas to differential extensions of general systems such as the quadrotor dynamics on SE(3).

Familiarity with the following topics is desirable:

- Advanced knowledge of linear ordinary differential equations.
- Optimization.
- Programming in Matlab and/or Python (the figures above were created with code from https://github.com/pettni/pcis).

**References**

[1] Blanchini, F. (1999). Set invariance in control. Automatica, 35(11), 1747–1767. https://doi.org/10.1016/S0005-1098(99)00113-2

[2] https://www.youtube.com/watch?v=rK9oyqccMJw

[3] Ames, A. D., Xu, X., Grizzle, J. W., & Tabuada, P. (2017). Control Barrier Function Based Quadratic Programs for Safety Critical Systems. IEEE Transactions on Automatic Control, 62(8), 3861–3876. https://doi.org/10.1109/TAC.2016.2638961