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This dissertation is primarily concerned with studying the invariant subspaces of left-invertible, weighted shifts, with generalizations to left-invertible operators where applicable. The two main problems that are researched can be stated together as When does a weighted shift have the one-dimensional wandering subspace property for all of its closed, invariant subspaces? This can fail either by having a subspace that is not generated by its wandering subspace, or by having a subspace with an index greater than one. For the former we show that every left-invertible, weighted shift is similar to another weighted shift with a residual space, with respect to being generated by the wandering subspace, of dimension $n$, where $n$ is any finite number. For the latter we derive necessary and sufficient conditions for a pure, left-invertible operator with an index of one to have a closed, invariant subspace with an index greater than one. We use these conditions to show that if a closed, invariant subspace for an operator in a class of weighted shifts has a vector in $l^1$, then it must have an index equal to one, and to produce closed, invariant subspaces with an index of two for operators in another class of weighted shifts. / Ph. D.

In this dissertation, we study the partial and generic uniqueness of block term tensor decompositions in signal processing. We present several conditions for generic uniqueness of tensor decompositions of multilinear rank (1, L1, L1), ..., (1, LR, LR) terms. Our proof is based on algebraic geometric methods. Mathematical preliminaries for this dissertation are multilinear algebra, and classical algebraic geometry. In geometric language, we prove that the joins of relevant subspace varieties are not tangentially weakly defective. We also give conditions for partial uniqueness of block term tensor decompositions by proving that the joins of relevant subspace varieties are not defective. The main result is the following. For a tensor Y belong to the tensor product of three complex vector spaces of dimensions I, J, K, we assume that L1, L2, ..., LR is from small to large, K is bigger or equal to J, and J is strictly bigger than LR. If the dimension of ambient space is strictly less than IJK, then for general tensors among those admitting block term tensor decomposition, the block term tensor decomposition is partially unique under the condition that the binomial coefficient indexed by J and LR is bigger or equal to R, and I is bigger or equal to 2; it has infinitely many expressions under the condition IJK is strictly less than the sum from L_1^2 to L_R^2; it is essentially unique under any of the following there conditions: (i) I is bigger or equal to 2, J, K is bigger or equal to the sum from L1 to LR (ii) R is 2, I is bigger or equal to 2 (iii) I is bigger or equal to R, K is bigger or equal to the sum from L1 to LR, J is bigger or equal to 2LR, the binomial coefficient indexed by J and LR is bigger or equal to R.

This dissertation consists of three more or less independent projects. In the first project, we find the microstates free entropy dimension of a large class of L1[0; 1]{ circular operators, in the presence of a generator of the diagonal subalgebra. In the second one, for each sequence {cn}n in l1(N), we de fine an operator A in the hyper finite II1-factor R. We prove that these operators are quasinilpotent and they generate the whole hyper finite II1-factor. We show that they have non-trivial, closed, invariant subspaces affiliated to the von Neumann algebra, and we provide enough evidence to suggest that these operators are interesting for the hyperinvariant subspace problem. We also present some of their properties. In particular, we show that the real and imaginary part of A are equally distributed, and we find a combinatorial formula as well as an analytical way to compute their moments. We present a combinatorial way of computing the moments of A*A. Finally, let fTkg1k =1 be a family of *-free identically distributed operators in a finite von Neumann algebra. In this paper, we prove a multiplicative version of the Free Central Limit Theorem. More precisely, let Bn = T*1T*2...T*nTn...T2T1 then Bn is a positive operator and B1=2n n converges in distribution to an operator A. We completely determine the probability distribution v of A from the distribution u of jTj2. This gives us a natural map G : M M with u G(u) = v. We study how this map behaves with respect to additive and multiplicative free convolution. As an interesting consequence of our results, we illustrate the relation between the probability distribution v and the distribution of the Lyapunov exponents for the sequence fTkg1k=1 introduced by Vladismir Kargin.

In the direct sequence-code division multiple access (DS-CDMA) system, which uses direct sequence spread spectrum (DSSS) technique to perform multiple-access, the major limitation of the system capacity is the capability of interference rejection. In this system, multiuser receivers usually divided into two groups, the first group is called the ¡§centralized receiver,¡¨ because it must know the information of total users, including the spreading sequence of each user, channel response, etc. Due to the complexity of computation, this kind of receivers is suitable for the base station. The second group is called the ¡§decentralized receiver,¡¨ because it only needs to know the information of desired user, therefore, it is very suitable for mobile station. The decentralized receiver can be further separate into two kinds: data-aided and non-data-aided receivers. Usually, the non-data-aided receiver is also called the blind receiver; our proposed interference cancellator belongs to this blind one. This thesis mainly discusses the performance of our proposed interference cancellator in different conditions. There is a novel interference detector which can efficiently detect strong interferers in our proposed interference cancellator. When strong interferers exist, the received signal will be passed through the interference-blocking transformer, which exploits the subspace approach to block strong interference. After interference cancelled, conventional de-spreading technique is used to obtain the desired data. In this thesis, besides the complete mathematical analysis of our proposed interference cancellator, computer simulations are also used to observe its performance behavior in different conditions. The simulation results exhibit that this interference cancellator has good performance in different conditions, and due to have the property of low complexity, our proposed interference cancellator is very suitable for the mobile station. Finally, we make a conclusion for this blind interference cancellator, and expect to realize a mature multiuser receiver based on this technique in the future.

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This thesis focuses on subspace based data-driven modeling and control techniques for batch and continuous processes. Motivated by the increasing amount of process data, data-driven modeling approaches have become more popular. These approaches are better in comparison to first-principles models due to their ability to capture true process dynamics. However, data-driven models rely solely on mathematical correlations and are subject to overfitting. As such, applying first-principles based constraints to the subspace model can lead to better predictions and subsequently better control. This thesis demonstrates that the addition of process gain constraints leads to a more accurate constrained model. In addition, this thesis also shows that using the constrained model in a model predictive control (MPC) algorithm allows the system to reach desired setpoints faster. The novel MPC algorithm described in this thesis is specially designed as a quadratic program to include a feedthrough matrix. This is traditionally ignored in industry however this thesis portrays that its inclusion leads to more accurate process control. Given the importance of accurate process data during model identification, the missing data problem is another area that needs improvement. There are two main scenarios with missing data: infrequent sampling/ sensor errors and quality variables. In the infrequent sampling case, data points are missing in set intervals and so correlating between different batches is not possible as the data is missing in the same place everywhere. The quality variable case is different in that quality measurements require additional expensive test making them unavailable for over 90\% of the observations at the regular sampling frequency. This thesis presents a novel subspace approach using partial least squares and principal component analysis to identify a subspace model. This algorithm is used to solve each case of missing data in both simulation (polymethyl methacrylate) and industrial (bioreactor) processes with improved performance. / Dissertation / Doctor of Philosophy (PhD) / An important consideration of chemical processes is the maximization of production and product quality. To that end developing an accurate controller is necessary to avoid wasting resources and off-spec products. All advance process control approaches rely on the accuracy of the process model, therefore, it is important to identify the best model. This thesis presents two novel subspace based modeling approaches the first using first principles based constraints and the second handling missing data approaches. These models are then applied to a modified state space model with a predictive control strategy to show that the improved models lead to improved control. The approaches in this work are tested on both simulation (polymethyl methacrylate) and industrial (bioreactor) processes.

A hybrid model reduction for use with the automated multilevel substructuring (AMLS) method is presented for frequency response analysis of complex structures. Structure responses to harmonic excitations and quasi-static responses to dominant damping forces are included in a reduced approximation subspace. Both types of responses greatly increase the efficiency of the subspace for solving the frequency response problem (FRP) for systems with high modal density and structural damping, and provide a good preparation for future frequency-dependent problems. A distilled subspace assumed to provide accurate frequency responses is generated from the finite element (FE) models by using the AMLS method. Then, the hybrid model reduction method is used to reduce the distilled subspace into a small new subspace. Three types of vectors are used to construct this subspace. The first type is distilled subspace dynamic response vectors (DRVs), which are exact solutions in the distilled subspace at certain chosen frequencies, called the DRV frequencies. The second type is modal DRVs, which are inexpensive approximate solutions calculated in an eigenspace. The third type is damping deformation vectors (DDVs), which provide information about response of the structure to damping effects. As exact responses, the distilled subspace DRVs eliminate frequency response errors at the DRV frequencies, and improve the accuracy at nearby frequencies as well. A small number of DRV frequencies are chosen carefully to offer maximum benefit with minimal computational cost. The modal DRVs are approximated very inexpensively from a suitable eigenspace. Only the diagonal entries in the modal coefficient matrices are used, along with low-rank updates that improve the accuracy of the modal DRVs and are applied using the Sherman-Morrison-Woodbury formula. Because of their low cost, a large number of modal DRVs constitute the major part of the reduced subspace. A small number of DDVs represent response to provide damping with minimal computational cost. The dimension of the final subspace is minimized by removing any redundancy through a special implementation of the QR factorization. This method results in a much smaller new subspace than the one from traditional modal truncation while achieving the same FRP accuracy. Such an efficiency also establishes a good foundation for future application in frequency-dependent problems. / text

In most of state-of-the-art speech recognition systems, Gaussian mixture models (GMMs) are used to model the density of the emitting states in the hidden Markov models (HMMs). In a conventional system, the model parameters of each GMM are estimated directly and independently given the alignment. This results a large number of model parameters to be estimated, and consequently, a large amount of training data is required to fit the model. In addition, different sources of acoustic variability that impact the accuracy of a recogniser such as pronunciation variation, accent, speaker factor and environmental noise are only weakly modelled and factorized by adaptation techniques such as maximum likelihood linear regression (MLLR), maximum a posteriori adaptation (MAP) and vocal tract length normalisation (VTLN). In this thesis, we will discuss an alternative acoustic modelling approach — the subspace Gaussian mixture model (SGMM), which is expected to deal with these two issues better. In an SGMM, the model parameters are derived from low-dimensional model and speaker subspaces that can capture phonetic and speaker correlations. Given these subspaces, only a small number of state-dependent parameters are required to derive the corresponding GMMs. Hence, the total number of model parameters can be reduced, which allows acoustic modelling with a limited amount of training data. In addition, the SGMM-based acoustic model factorizes the phonetic and speaker factors and within this framework, other source of acoustic variability may also be explored. In this thesis, we propose a regularised model estimation for SGMMs, which avoids overtraining in case that the training data is sparse. We will also take advantage of the structure of SGMMs to explore cross-lingual acoustic modelling for low-resource speech recognition. Here, the model subspace is estimated from out-domain data and ported to the target language system. In this case, only the state-dependent parameters need to be estimated which relaxes the requirement of the amount of training data. To improve the robustness of SGMMs against environmental noise, we propose to apply the joint uncertainty decoding (JUD) technique that is shown to be efficient and effective. We will report experimental results on the Wall Street Journal (WSJ) database and GlobalPhone corpora to evaluate the regularisation and cross-lingual modelling of SGMMs. Noise compensation using JUD for SGMM acoustic models is evaluated on the Aurora 4 database.

<p>A model is developed for the Quanser torsion system available at Control Systems Research Laboratory at Chulalongkorn University. The torsion system is a laboratory equipment that is designed for the study of position control. It consists of a DC motor that drives three inertial loads that are coupled in series with the motor, and where all components are coupled to each other through torsional springs.</p><p>Several nonlinearities are observed and the most significant one is an offset in the input signal, which is compensated for. Experiments are carried out under feedback as the system is marginally stable. Different input signals are tested and used for system identification. Linear black-box state-space models are then identified using PEM, N4SID and a subspace method made for closed-loop identification, where the last two are the most successful ones. PEM is used in a second step and successfully enhances the parameter estimates from the other algorithms.</p>