On the connection between discrete linear repetitive processes and 2-D discrete linear systems

A direct method is developed that reduces a polynomial system matrix describinga discrete linear repetitive process to a 2-D singular state-space form such that all the relevant properties, including the zero structure of the system matrix, are retained. It is shown that the transformation linking the original polynomial system matrix with its associated 2-D singular form is zero coprime system equivalence. The exact nature of the resulting system matrix in singular form and the transformation involved are established

Model based fault detection for two-dimensional systems

Fault detection and isolation (FDI) are essential in ensuring safe and reliable operations in industrial systems. Extensive research has been carried out on FDI for one dimensional (1-D) systems, where variables vary only with time. The existing FDI strategies are mainly focussed on 1-D systems and can generally be classified as model based and process history data based methods. In many industrial systems, the state variables change with space and time (e.g., sheet forming, fixed bed reactors, and furnaces). These systems are termed as distributed parameter systems (DPS) or two dimensional (2-D) systems. 2-D systems have been commonly represented by the Roesser Model and the F-M model. Fault detection and isolation for 2-D systems represent a great challenge in both theoretical development and applications and only limited research results are available. In this thesis, model based fault detection strategies for 2-D systems have been investigated based on the F-M and the Roesser models. A dead-beat observer based fault detection has been available for the F-M model. In this work, an observer based fault detection strategy is investigated for systems modelled by the Roesser model. Using the 2-D polynomial matrix technique, a dead-beat observer is developed and the state estimate from the observer is then input to a residual generator to monitor occurrence of faults. An enhanced realization technique is combined to achieve efficient fault detection with reduced computations. Simulation results indicate that the proposed method is effective in detecting faults for systems without disturbances as well as those affected by unknown disturbances.The dead-beat observer based fault detection has been shown to be effective for 2-D systems but strict conditions are required in order for an observer and a residual generator to exist. These strict conditions may not be satisfied for some systems. The effect of process noises are also not considered in the observer based fault detection approaches for 2-D systems. To overcome the disadvantages, 2-D Kalman filter based fault detection algorithms are proposed in the thesis. A recursive 2-D Kalman filter is applied to obtain state estimate minimizing the estimation error variances. Based on the state estimate from the Kalman filter, a residual is generated reflecting fault information. A model is formulated for the relation of the residual with faults over a moving evaluation window. Simulations are performed on two F-M models and results indicate that faults can be detected effectively and efficiently using the Kalman filter based fault detection. In the observer based and Kalman filter based fault detection approaches, the residual signals are used to determine whether a fault occurs. For systems with complicated fault information and/or noises, it is necessary to evaluate the residual signals using statistical techniques. Fault detection of 2-D systems is proposed with the residuals evaluated using dynamic principal component analysis (DPCA). Based on historical data, the reference residuals are first generated using either the observer or the Kalman filter based approach. Based on the residual time-lagged data matrices for the reference data, the principal components are calculated and the threshold value obtained. In online applications, the T2 value of the residual signals are compared with the threshold value to determine fault occurrence. Simulation results show that applying DPCA to evaluation of 2-D residuals is effective.Doctoral These

Convolutional Neural Networks as 2-D systems

This paper introduces a novel representation of convolutional Neural Networks (CNNs) in terms of 2-D dynamical systems. To this end, the usual description of convolutional layers with convolution kernels, i.e., the impulse responses of linear filters, is realized in state space as a linear time-invariant 2-D system. The overall convolutional Neural Network composed of convolutional layers and nonlinear activation functions is then viewed as a 2-D version of a Lur'e system, i.e., a linear dynamical system interconnected with static nonlinear components. One benefit of this 2-D Lur'e system perspective on CNNs is that we can use robust control theory much more efficiently for Lipschitz constant estimation than previously possible

Model implementation and analysis of a true three-dimensional display system

To model a true three-dimensional (3D) display system, we introduced the method of voxel molding to obtain the stereoscopic imaging space of the system. For the distribution of each voxel, we proposed a four-dimensional (4D) Givone–Roessor (GR) model for state-space representation—that is, we established a local state-space model with the 3D position and one-dimensional time coordinates to describe the system. First, we extended the original elementary operation approach to a 4D condition and proposed the implementation steps of the realization matrix of the 4D GR model. Then, we described the working process of a true 3D display system, analyzed its real-time performance, introduced the fixed-point quantization model to simplify the system matrix, and derived the conditions for the global asymptotic stability of the system after quantization. Finally, we provided an example to prove the true 3D display system’s feasibility by simulation. The GR-model-representation method and its implementation steps proposed in this paper simplified the system’s mathematical expression and facilitated the microcontroller software implementation. Real-time and stability analyses can be used widely to analyze and design true 3D display systems

Matrix fraction descriptions in convolutional coding

Doutoramento em MatemáticaOs objectos de estudo desta tese são os códigos convolucionais sobre um corpo, constituídos por sequências com suporte compacto à esquerda. Aplicando a abordagem comportamental à teoria dos sistemas, é obtida uma nova definição de código convolucional baseada em propriedades estruturais do próprio código. Os codificadores e os formadores de síndrome de um código convolucional são, respectivamente, as representações de imagem e as representações de núcleo do código. As suas estruturas e propriedades são estudadas, utilizando representações matriciais fraccionárias (RMF's). Seguidamente, são analisados os codificadores e formadores de síndrome minimais de um código convolucional, sendo apresentada uma parametrização simples das suas RMF's. Mostra-se também como obter todos os codificadores minimais de um código convolucional por aplicação de realimentação estática do estado e précompensação. De modo análogo, obtêm-se todos os formadores de síndrome minimais utilizando injecção da saída e pós-compensação. Finalmente, estudam-se os codificadores desacoplados de um código convolucional, que estão directamente ligados à sua decomposição. Apresenta-se um algoritmo para determinação de um codificador desacoplado maximal, que permitirá obter a decomposição máxima do código. Quando se restringe a análise dos codificadores desacoplados aos minimais, obtém-se um codificador canónico desacoplado e parametriza-se, utilizando RMF's, todos os codificadores minimais que apresentam grau máximo de desacoplamento.The objects of study of this thesis are the convolutional codes over a field, constituted by left compact sequences. To define a convolutional code we consider the behavioral approach to systems theory, and present a new definition of convolutional code, taking into account its structural properties. Matrix Fractions Descriptions (MFD’s) are used as a tool for investigating the structure of the encoders and the syndrome formers of a convolutional code, which are, respectively, the image and the kernel representations of the code. Next, we concentrate on the study of the minimal encoders and syndrome formers, and obtain a simple parametrization of their MFD’s. We also show that static feedback and precompensation allow to obtain all minimal encoders of the code. The same is done for the minimal syndrome formers, using output injection and postcompensation. Finally, we analyse the decoupled encoders of a convolutional code, which are associated with code decomposition. We provide an algorithm to determine a maximally decoupled encoder, and, consequently, the finest decomposition of the code. Restricting to minimal decoupled encoders, we first obtain a canonical decoupled one, and parametrize, via MFD’s, all minimal decoupled encoders realizing the finest decomposition of the code