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The influence of sleep on human life, including physiological, psychological, and mental aspects, is remarkable. Therefore, it is essential to apply appropriate therapy in the case of sleep disorders. For this, however, the irregularities must first be recognised, preferably conveniently for the person concerned. This dissertation, structured as a composition of research articles, presents the development of mathematically based algorithmic principles for a sleep analysis system. The particular focus is on the classification of sleep stages with a minimal set of physiological parameters. In addition, the aspects of using the sleep analysis system as part of the more complex healthcare systems are explored. Design of hardware for non-obtrusive measurement of relevant physiological parameters and the use of such systems to detect other sleep disorders, such as sleep apnoea, are also referred to. Multinomial logistic regression was selected as the basis for development resulting from the investigations carried out. By following a methodical procedure, the number of physiological parameters necessary for the classification of sleep stages was successively reduced to two: Respiratory and Movement signals. These signals might be measured in a contactless way. A prototype implementation of the developed algorithms was performed to validate the proposed method, and the evaluation of 19324 sleep epochs was carried out. The results, with the achieved accuracy of 73% in the classification of Wake/NREM/REM stages and Cohen's kappa of 0.44, outperform the state of the art and demonstrate the appropriateness of the selected approach. In the future, this method could enable convenient, cost-effective, and accurate sleep analysis, leading to the detection of sleep disorders at an early stage so that therapy can be initiated as soon as possible, thus improving the general population's health status and quality of life.