Selling Robustness Margins: A Framework for Optimizing Reserve Capacities for Uncertain Systems

X. Zhang, M. Kamgarpour, P. J. Goulart and J. Lygeros

in IEEE Conference on Decision and Control, Los Angeles, CA, USA, pp. 6419-6424, December 2014.
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@inproceedings{ZKGL:2014,
  author = {X. Zhang and M. Kamgarpour and P. J. Goulart and J. Lygeros},
  title = {Selling Robustness Margins: A Framework for Optimizing Reserve Capacities for Uncertain Systems},
  booktitle = {IEEE Conference on Decision and Control},
  year = {2014},
  pages = {6419-6424},
  url = {http://dx.doi.org/10.1109/CDC.2014.7040396},
  doi = {10.1109/CDC.2014.7040396}
}

This paper proposes a method for solving robust optimal control problems with modulated uncertainty sets. We consider constrained uncertain linear systems and interpret the uncertainty sets as ?robustness margins? or ?reserve capacities?. In particular, given a certain reward for offering such a reserve capacity, we address the problem of determining the optimal size and shape of the uncertainty set, i.e. how much reserve capacity our system should offer. By assuming polyhedral constraints, restricting the class of the uncertainty sets and using affine decision rules, we formulate a convex program to solve this problem. We discuss several specific families of uncertainty sets, whose respective constraints can be reformulated as linear constraints, second-order cone constraints, or linear matrix inequalities. A numerical example demonstrates our approach.