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In this paper, we present a geometric approach for computing the controlled invariant set of a continuous-time control system. While the problem is well studied for in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support function. We compare it with the common algebraic approach for polynomial sublevel sets and show that it is significantly more conservative than our method.

In the analysis of biological samples, control over experimental design and data acquisition procedures alone cannot ensure well-conditioned 1H NMR spectra with maximal information recovery for data analysis. A third major element affects the accuracy and robustness of results: the data pre-processing/ pre-treatment for which not enough attention is usually devoted, in particular in metabolomic studies. The usual approach is to use proprietary software provided by the analytical instruments' manufacturers to conduct the entire pre-processing strategy. This widespread practice has a number of advantages such as a user-friendly interface with graphical facilities, but it involves non-negligible drawbacks: a lack of methodological information and automation, a dependency of subjective human choices, only standard processing possibilities and an absence of objective quality criteria to evaluate pre-processing quality. This paper introduces PepsNMR to meet these needs, an R package dedicated to the whole processing chain prior to multivariate data analysis, including, among other tools, solvent signal suppression, internal calibration, phase, baseline and misalignment corrections, bucketing and normalisation. Methodological aspects are discussed and the package is compared to the gold standard procedure with two metabolomic case studies. The use of PepsNMR on these data shows better information recovery and predictive power based on objective and quantitative quality criteria. Other key assets of the package are workflow processing speed, reproducibility, reporting and flexibility, graphical outputs and documented routines.

In this paper, we develop a method for computing controlled invariant sets using Semidefinite Programming. We apply our method to the controller design problem for switching affine systems with polytopic safe sets. The task is reduced to a semidefinite programming problem by enforcing an invariance relation in the dual space of the geometric problem. The paper ends with an application to safety critical model predictive control.