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The object of this thesis is (1) to demonstrate the value of an electronic analog computer for the solution of non-linear ordinary differential equations particularly when a large family of solutions is required; and (2) to obtain as a by-product results of practical applicability to servomechanism selection and analysis.

The electronic analog computer is widely used to simulate most types of automatic control systems, Only a limited amount of work has been reported, however, involving analog computer simulation of sampled-data systems, This is to be expected since such systems are essentially digital in nature. The purpose of the work described in this thesis was to develop methods of simulating various phenomena associated with sampled-data system, The techniques which have been developed are described ad evaluated in the report. In addition, experimental data is presented to illustrate the performance of the various simulation circuits. As an illustration of the techniques which have been developed, the simulation of a representative sampled-data system is described. Data obtained from this simulation is included in the report. / Master of Science

This thesis is focused on analyzing a particular notion of complexity of real valued functions through the lens of analog computers. This report features design changes to Pour-El’s notion of an analog computer that reflect this question of complexity in a concrete way. Additionally, these changes to the analog computer allow an extension of Pour-El’s work in which the complexity of a function can be identified with the order of a differentiably algebraic equation that the function satisfies.

Il existe de nombreux modèles de calcul sur les réels. Ces différents modèles<br />calculent diverses fonctions, certains sont plus puissants que d'autres,<br />certains sont deux à deux incomparables. Le calcul sur les réels est donc de<br />ce point de vue bien différent du calcul sur les entiers qui est unifié par la<br />thèse de Church-Turing qui affirme que tous les modèles raisonnables calculent<br />les mêmes fonctions.<br /><br />Les résultats de cette thèse sont de deux sortes. Premièrement, nous<br />montrons des équivalences entre les fonctions récursivement calculables et une<br />certaine classe de fonctions R-récursives et entre les fonctions<br />GPAC-calculables et les fonctions récursivement calculables. Ces deux<br />résultats ne sont cependant valables que si les fonctions présentent quelques<br />caractéristiques : elles doivent être définies sur un compact et dans le<br />premier cas être de classe C^2. Deuxièmement, nous montrons également une<br />hiérarchie de classes de fonctions R-récursives qui caractérisent les<br />fonctions élémentairement calculables, les fonctions<br />En-calculables pour n?3 (où les En sont les<br />fonctions de la hiérarchie de Grzegorczyk), et des fonctions récursivement<br />calculables. Ce résultat utilise un opérateur de limite dont nous avons prouvé<br />la généralité en montrant qu'il transfère une inclusion sur la partie discrète<br />des fonctions en une inclusion sur les fonctions sur les réels elles-mêmes.<br /><br />Ces résultats constituent donc une avancée vers une éventuelle<br />unification des modèles de calcul sur les réels.

Il existe de nombreux modèles de calcul sur les réels. Ces différents modèles calculent diverses fonctions, certains sont plus puissants que d'autres, certains sont deux à deux incomparables. Le calcul sur les réels est donc de ce point de vue bien différent du calcul sur les entiers qui est unifié par la thèse de Church-Turing affirmant que tous les modèles raisonnables calculent les mêmes fonctions. Nous montrons des équivalences entre les fonctions récursivement calculables et une certaine classe de fonctions R-récursives et entre les fonctions GPAC-calculables et les fonctions récursivement calculables. Nous montrons également une hiérarchie de classes de fonctions R-récursives qui caractérisent les fonctions élémentairement calculables, les fonctions de la hiérarchie de Grzegorczyk et les fonctions récursivement calculables à l'aide d'un opérateur de limite. Ces résultats constituent donc une avancée vers une éventuelle unification des modèles de calcul sur les réels / Computation on the real numbers can be modelised in several different ways. There indeed exist a lot of different computation models on the reals. However, there are few results for comparing those models, and most of these results are incomparability results. The case of computation over the reals hence is quite different from the classical case where Church thesis claims that all reasonable models compute exactly the same functions. We show that recursively computable functions (in the sense of computable analysis) can be shown equivalent to some adequately defined class of R-recursive functions, and also to GPAC-computable functions. More than an analog characterization of recursively enumerable functions, we show that the limit operator we defined can be used to provide an analog characterization of elementarily computable functions and functions from Grzegorczyk's hierarchy. Those results can be seen as a first step toward a unification of computable functions over the reals

Indiana University-Purdue University Indianapolis (IUPUI) / Neural signal processing to decode neural activity has been an active research area in the last few decades. The next generation of advanced multi-electrode neuroprosthetic devices aim to detect a multiplicity of channels from multiple electrodes, making the relatively time-critical processing problem massively parallel and pushing the computational demands beyond the limits of current embedded digital signal processing (DSP) techniques. To overcome these limitations, a new hybrid computational technique was explored, the Extended Analog Computer (EAC). The EAC is a digitally confgurable analog computer that takes advantage of the intrinsic ability of manifolds to solve partial diferential equations (PDEs). They are extremely fast, require little power, and have great potential for mobile computing applications. In this thesis, the EAC architecture and the mechanism of the formation of potential/current manifolds was derived and analyzed to capture its theoretical mode of operation. A new mode of operation, resistance mode, was developed and a method was devised to sample temporal data and allow their use on the EAC. The method was validated by demonstration of the device solving linear diferential equations and linear functions, and implementing arbitrary finite impulse response (FIR) and infinite impulse response (IIR) linear flters. These results were compared to conventional DSP results. A practical application to the neural computing task was further demonstrated by implementing a matched filter with the EAC simulator and the physical prototype to detect single fiber action potential from multiunit data streams derived from recorded raw electroneurograms. Exclusion error (type 1 error) and inclusion error (type 2 error) were calculated to evaluate the detection rate of the matched filter implemented on the EAC. The detection rates were found to be statistically equivalent to that from DSP simulations with exclusion and inclusion errors at 0% and 1%, respectively.