Difference between revisions of "SURF 2019: Geometry of ControlInvariant Sets"
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Revision as of 00:05, 28 November 2018
The concept of a controlinvariant seta 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]. Unfortunately, analytical expressions for controlinvariant 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 closedform expression of the maximal controlinvariant set contained inside the unite hypercube is known [2. eq. (53)], but for higher orders of only numerical approaches are available. The Figure above shows numerical approximations for equal to 2 and equal to 3. The objective of this project is to investigate geometrical properties of these sets via a combination of analytical and numerical techniques. In particular,
 Search for closedform expressions or cheaptoevaluate algorithms that characterize controlinvariant sets of for n larger than 3.
 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/S00051098(99)001132
[2] 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