Abstract
This paper studies the relationship between quantified multivariable polynomial inequalities and robust control problems. We show that there is a hierarchy to the difficulty of control problems expressed as polynomial inequalities and a similar hierarchy to the methods used to solve them. At one end, we have quantifier elimination methods which are exact, but doubly exponential in their computational complexity and thus may only be used to solve small size problems. The Branch-and-Bound methods sacrifice the exactness of quantifier elimination to approximately solve a larger class of problems, while Monte Carlo and statistical learning methods solve very large problems, but only probabilistically. We also present novel sequential learning methods to illustrate the power of the statistical methods.
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Abdallah, C.T., Ariola, M., Dorato, P., Koltchinskii, V. (1999). Quantified inequalities and robust control. In: Garulli, A., Tesi, A. (eds) Robustness in identification and control. Lecture Notes in Control and Information Sciences, vol 245. Springer, London. https://doi.org/10.1007/BFb0109881
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DOI: https://doi.org/10.1007/BFb0109881
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