Let $G$ be a second countable, locally compact group which is either compact or abelian, and let $\rho$ be a unitary representation of $G$ on a separable Hilbert space $\mathcal{H}_\rho$. We examine frames of the form $\{ \rho(x) f_j \colon x \in G, j \in I\}$ for families $\{f_j\}_{j \in I}$ in $\mathcal{H}_\rho$. In particular, we give necessary and sufficient conditions for the joint orbit of a family of vectors in $\mathcal{H}_\rho$ to form a continuous frame. We pay special attention to this problem in the setting of shift invariance. In other words, we fix a larger second countable locally compact group $\Gamma \supset G$ containing $G$ as a closed subgroup, and we let $\rho$ be the action of $G$ on $L^2(\Gamma)$ by left translation. In both the compact and the abelian settings, we introduce notions of Zak transforms on $L^2(\Gamma)$ which simplify the analysis of group frames. Meanwhile, we run a parallel program that uses the Zak transform to classify closed subspaces of $L^2(\Gamma)$ which are invariant under left translation by $G$. The two projects give compatible outcomes. This dissertation contains previously published material.

On considère des jeux stochastiques joués sur un graphe fini. La première partie s’intéresse aux jeux stochastiques à deux joueurs et information parfaite. Dans de tels jeux, les joueurs choisissent des actions dans ensemble fini, tour à tour, pour une durée infinie, produisant une histoire infinie. Le but du jeu est donné par une fonction d’utilité qui associe un réel à chaque histoire, la fonction est bornée et Borel-mesurable. Le premier joueur veut maximiser l’utilité espérée, et le deuxième joueur veut la minimiser. On démontre que si la fonction d’utilité est à la fois shift-invariant et submixing alors le jeu est semi-positionnel. C’est-à-dire le premier joueur a une stratégie optimale qui est déterministe et sans mémoire. Les deux joueurs ont information parfaite: ils choisissent leurs actions en ayant une connaissance parfaite de toute l’histoire. Dans la deuxième partie, on étudie des jeux de durée fini où le joueur protagoniste a zéro information. C’est-à-dire qu’il ne reçoit aucune information sur le déroulement du jeu, par conséquent sa stratégie est un mot fini sur l’ensemble des actions. Un automates probabiliste peut être considéré comme un tel jeu qui a un seul joueur. Tout d’abord, on compare deux classes d’automates probabilistes pour lesquelles le problème de valeur 1 est décidable: les automates leaktight et les automates simples. On prouve que la classe des automates simples est un sous-ensemble strict de la classe des automates leaktight. Puis, on considère des jeux semi-aveugles, qui sont des jeux à deux joueurs où le maximiseur a zéro information, et le minimiseur est parfaitement informé. On définit la classe des jeux semi-aveugles leaktight et on montre que le problème d’accessibilité maxmin est décidable sur cette classe. / We consider stochastic games that are played on finite graphs. The subject of the first part are two-player stochastic games with perfect information. In such games the two players take turns choosing actions from a finite set, for an infinite duration, resulting in an infinite play. The objective of the game is given by a Borel-measurable and bounded payoff function that maps infinite plays to real numbers. The first player wants to maximize the expected payoff, and the second player has the opposite objective, that of minimizing the expected payoff. We prove that if the payoff function is both shift-invariant and submixing then the game is half-positional. This means that the first player has an optimal strategy that is at the same time pure and memoryless. Both players have perfect information, so the actions are chosen based on the whole history. In the second part we study finite-duration games where the protagonist player has zero information. That is, he gets no feedback from the game and consequently his strategy is a finite word over the set of actions. Probabilistic finite automata can be seen as an example of such a game that has only a single player. First we compare two classes of probabilistic automata: leaktight automata and simple automata, for which the value 1 problem is known to be decidable. We prove that simple automata are a strict subset of leaktight automata. Then we consider half-blind games, which are two player games where the maximizer has zero information and the minimizer is perfectly informed. We define the class of leaktight half-blind games and prove that it has a decidable maxmin reachability problem.

The Fourier transform (spectrum) of a signal is a complex function and is characterized by the magnitude and phase spectra. Phase retrieval is the reconstruction of the phase spectrum from the measurements of the magnitude spectrum. Such problems are encountered in imaging modalities such as X-ray crystallography, frequency-domain optical coherence tomography (FDOCT), quantitative phase microscopy, digital holography, etc., where only the magnitudes of the wavefront are detected by the sensors. The phase retrieval problem is ill-posed in general, since an in nite number of signals can have the same magnitude spectrum. Typical phase retrieval techniques rely on certain prior knowledge about the signal, such as its support or sparsity, to reconstruct the signal. A classical result in phase retrieval is that minimum-phase signals have log-magnitude and phase spectra that satisfy the Hilbert integral equations, thus facilitating exact phase retrieval. In this thesis, we demonstrate that there exist larger classes of signals beyond minimum-phase signals, for which exact phase retrieval is possible. We generalize Hilbert integral equations to 2-D, and also introduce a variant that we call the composite Hilbert transform in the context of 2-D periodic signals. Our first extension pertains to a particular type of parametric modelling of 2-D signals. While 1-D minimum-phase signals have a parametric representation, in terms of poles and zeros, there exists no such 2-D counterpart. We introduce a new class of parametric 2-D signals that possess the exact phase retrieval property, that is, their magnitude spectrum completely characterizes the signal. Starting from the magnitude spectrum, a sequence of non-linear operations lead us to a sum-of-exponentials signal, from which the parameters are computed employing concepts from high-resolution spectral estimation such as the annihilating filter and algebraically coupled matrix-pencil methods. We demonstrate that, for this new class of signals, our method outperforms existing techniques even in the presence of noise. Our second extension is to continuous-domain signals that lie in a principal shift-invariant space spanned by a known basis. Such signals are characterized by the basis combining coefficients. These signals need not be minimum-phase, but certain conditions on the coefficients lead to exact phase retrieval of the continuous-domain signal. In particular, we introduce the concept of causal, delta dominant (CDD) sequences, and show that such signals are characterized by their magnitude spectra. This condition pertains to the time/spatial-domain description of the signal, in contrast to the minimum-phase condition, which is described in the spectral domain. We show that there exist CDD sequences that are not minimum-phase, and vice versa. However, finite-length CDD sequences are always minimum-phase. Our method reconstructs the signal from the magnitude spectrum up to ma-chine precision. We thus have a class of continuous-domain signals that are neither causal nor minimum phase, and yet allow for exact phase retrieval. The shift-invariant structure is applicable to modelling signals encountered in imaging modalities such as FDOCT. We next present an application of 2-D phase retrieval to continuous-domain CDD signals in the context of quantiative phase microscopy. We develop sufficient conditions on the interfering reference wave for exact phase retrieval from magnitude measurements. In particular, we show that when the reference wave is a plane wave with magnitude greater that the intensity of the object wave, and when the carrier frequency is larger than the band-width of the object wave, we can reconstruct the object wave exactly. We demonstrate high-resolution reconstruction of our method on USAF target images. Our final and perhaps the most unifying contribution is in developing Hilbert integral equations for 2-D first-quadrant signals and in introducing the notion of generalized minimum-phase signals for both 1-D and 2-D signals. For 2-D continuous-domain, first-quadrant signals, we establish partial Hilbert transform relations between the real and imaginary parts of the spectrum. In the context of 2-D discrete-domain signals, we show that the partial Hilbert transform does not suffice and introduce the notion of composite Hilbert transform and establish the integral equations. We then introduce four classes of signals (combinations of 1-D/2-D and continuous/discrete-domain) that we call generalized minimum-phase signals, which satisfy corresponding Hilbert integral equations between log-magnitude and phase spectra, hence facilitating exact phase retrieval. This class of generalized minimum-phase signals subsumes the well known class of minimum-phase signals. We further show that, akin to minimum-phase signals, these signals also have stable inverses, which are also generalized minimum-phase signals.

Closed-loop system identification and fault detection and isolation are the two fundamental building blocks of process monitoring. Efficient and accurate process monitoring increases plant availability and utilisation. This dissertation investigates a subspace system identification and fault detection methodology for the Benfield process, used by Sasol, Synfuels in Secunda, South Africa, to remove CO2 from CO2-rich tail gas. Subspace identification methods originated between system theory, geometry and numerical linear algebra which makes it a computationally efficient tool to estimate system parameters. Subspace identification methods are classified as Black-Box identification techniques, where it does not rely on a-priori process information and estimates the process model structure and order automatically. Typical subspace identification algorithms use non-parsimonious model formulation, with extra terms in the model that appear to be non-causal (stochastic noise components). These extra terms are included to conveniently perform subspace projection, but are the cause for inflated variance in the estimates, and partially responsible for the loss of closed-loop identifiably. The subspace identification methodology proposed in this dissertation incorporates two successive LQ decompositions to remove stochastic components and obtain state-space models of the plant respectively. The stability of the identified plant is further guaranteed by using the shift invariant property of the extended observability matrix by appending the shifted extended observability matrix by a block of zeros. It is shown that the spectral radius of the identified system matrices all lies within a unit boundary, when the system matrices are derived from the newly appended extended observability matrix. The proposed subspace identification methodology is validated and verified by re-identifying the Benfield process operating in closed-loop, with an RMPCT controller, using measured closed-loop process data. Models that have been identified from data measured from the Benfield process operating in closed-loop with an RMPCT controller produced validation data fits of 65% and higher. From residual analysis results, it was concluded that the proposed subspace identification method produce models that are accurate in predicting future outputs and represent a wide variety of process inputs. A parametric fault detection methodology is proposed that monitors the estimated system parameters as identified from the subspace identification methodology. The fault detection methodology is based on the monitoring of parameter discrepancies, where sporadic parameter deviations will be detected as faults. Extended Kalman filter theory is implemented to estimate system parameters, instead of system states, as new process data becomes readily available. The extended Kalman filter needs accurate initial parameter estimates and is thus periodically updated by the subspace identification methodology, as a new set of more accurate parameters have been identified. The proposed fault detection methodology is validated and verified by monitoring process behaviour of the Benfield process. Faults that were monitored for, and detected include foaming, flooding and sensor faults. Initial process parameters as identified from the subspace method can be tracked efficiently by using an extended Kalman filter. This enables the fault detection methodology to identify process parameter deviations, with a process parameter deviation sensitivity of 2% or higher. This means that a 2% parameter deviation will be detected which greatly enhances the fault detection efficiency and sensitivity. / Dissertation (MEng)--University of Pretoria, 2008. / Electrical, Electronic and Computer Engineering / unrestricted