In this paper, we study randomized methods for feedback design of uncertain systems. The first contribution is to derive the sample complexity of various constrained control problems. In particular, we show the key role played by the binomial distribution and related tail inequalities, and compute the sample complexity. This contribution significantly improves the existing results by reducing the number of required samples in the randomized algorithm. These results are then applied to the analysis of worst-case performance and design with robust optimization. The second contribution of the paper is to introduce a general class of sequential algorithms, denoted as Sequential Probabilistic Validation (SPV). In these sequential algorithms, at each iteration, a candidate solution is probabilistically validated, and corrected if necessary, to meet the required specifications. The results we derive provide the sample complexity which guarantees that the solutions obtained with SPV algorithms meet some pre-specified probabilistic accuracy and confidence. The performance of these algorithms is illustrated and compared with other existing methods using a numerical example dealing with robust system identification.

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By combining a description of the potential profile at electrodes coated with acid thiol monolayers with a quadratic relationship between activation energy and electrode potential, a rather simple expression for proton transfer voltammograms is derived. Our electrostatic analysis shows that proton transfer can only produce narrow voltammetric peaks when the immobilized acid groups lie close to the metal substrate. Quantitative fits of experimental voltammograms obtained with an Au(111) electrode modified with a 11-mercaptoundecanoic monolayer at pH 8.5 reveal that less than 1% of the carboxylic groups in the monolayer participate in the potential induced proton transfer process and that these groups lay close to the metal surface. A preliminary analysis of the kinetic parameters suggests that the interfacial electric field facilitates an intrinsically slow proton exchange between a proton donor and acceptor pair that are not in close contact with each other at the interface.

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