<p> This thesis studies the multiple description vector quantization with lattice codebooks (MDLVQ).</p> <p> The design of index assignment is crucial to the performance of MDLVQ. However, to our best knowledge, none of previous index assignment algorithms for MDLVQ is optimal. In this thesis, we propose a simple linear-time index assignment algorithm for MDLVQ with any K ≥ 2 balanced descriptions. We prove, under the assumption of high resolution, that the algorithm is optimal for K = 2. The optimality holds for many commonly used good lattices of any dimensions, over the entire range of achievable central distortions given the side entropy rate. The optimality is in terms of minimizing the expected distortion given the side description loss rate and given the side entropy rate. We conjecture it to be optimal for K > 2 in general.</p> <p> We also made progress in the analysis of MDLVQ performance. The first exact closed form expression of the expected distortion was derived for K = 2. For K > 2, we improved the current asymptotic expression of the expected distortion.</p> / Thesis / Master of Applied Science (MASc)

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Techniques for estimating a signal in the presence of noise which contains outliers are currently not well developed. In this thesis, we consider a constant signal superimposed by a family of noise distributions structured as a tunable mixture f(x) = α g(x) + (1 − α) h(x) between finitesupport components of “well-behaved” noise with small variance g(x) and of “impulsive” noise h(x) with a large amplitude and strongly asymmetric character. When α ≈ 1, h(x) can for example model a cosmic ray striking an experimental detector. In the first part of our work, a method for obtaining the expected values of the positive and negative pulses in the first resolution level of a LULU Discrete Pulse Transform (DPT) is established. Subsequent analysis of sequences smoothed by the operators L1U1 or U1L1 of LULU-theory shows that a robust estimator for the location parameter for g is achieved in the sense that the contribution by h to the expected average of the smoothed sequences is suppressed to order (1 − α)2 or higher. In cases where the specific shape of h can be difficult to guess due to the assumed lack of data, it is thus also shown to be of lesser importance. Furthermore, upon smoothing a sequence with L1U1 or U1L1, estimators for the scale parameters of the model distribution become easily available. In the second part of our work, the same problem and data is approached from a Bayesian inference perspective. The Bayesian estimators are found to be optimal in the sense that they make full use of available information in the data. Heuristic comparison shows, however, that Bayes estimators do not always outperform the LULU estimators. Although the Bayesian perspective provides much insight into the logical connections inherent in the problem, its estimators can be difficult to obtain in analytic form and are slow to compute numerically. Suboptimal LULU-estimators are shown to be reasonable practical compromises in practical problems. / AFRIKAANSE OPSOMMING: Tegnieke om ’n sein af te skat in die teenwoordigheid van geraas wat uitskieters bevat is tans nie goed ontwikkel nie. In hierdie tesis aanskou ons ’n konstante sein gesuperponeer met ’n familie van geraasverdelings wat as verstelbare mengsel f(x) = α g(x) + (1 − α) h(x) tussen eindige-uitkomsruimte geraaskomponente g(x) wat “goeie gedrag” en klein variansie toon, plus “impulsiewe” geraas h(x) met groot amplitude en sterk asimmetriese karakter. Wanneer α ≈ 1 kan h(x) byvoorbeeld ’n kosmiese straal wat ’n eksperimentele apparaat tref modelleer. In die eerste gedeelte van ons werk word ’n metode om die verwagtingswaardes van die positiewe en negatiewe pulse in die eerste resolusievlak van ’n LULU Diskrete Pulse Transform (DPT) vasgestel. Die analise van rye verkry deur die inwerking van die gladstrykers L1U1 en U1L1 van die LULU-teorie toon dat hul verwagte gemiddelde waardes as afskatters van die liggingsparameter van g kan dien wat robuus is in die sin dat die bydrae van h tot die gemiddeld van orde grootte (1 − α)2 of hoër is. Die spesifieke vorm van h word dan ook onbelangrik. Daar word verder gewys dat afskatters vir die relevante skaalparameters van die model maklik verkry kan word na gladstryking met die operatore L1U1 of U1L1. In die tweede gedeelte van ons werk word dieselfde probleem en data vanuit ’n Bayesiese inferensie perspektief benader. Die Bayesiese afskatters word as optimaal bevind in die sin dat hulle vol gebruikmaak van die beskikbare inligting in die data. Heuristiese vergelyking wys egter dat Bayesiese afskatters nie altyd beter vaar as die LULU afskatters nie. Alhoewel die Bayesiese sienswyse baie insig in die logiese verbindings van die probleem gee, kan die afskatters moeilik wees om analities af te lei en stadig om numeries te bereken. Suboptimale LULU-beramers word voorgestel as redelike praktiese kompromieë in praktiese probleme.

This study investigates seismic response of the Moment-Resisting-Steel Frames (MRSF) with the architectural setbacks. The main objective of the study is to understand the variation of the elastic and inelastic, static and dynamic behavior with changes in the geometric dimensions of the tower portion. A second objective of the study is to determine the adequacy of the analysis procedures of various rigors, specified in current seismic design provision, in predicting those behaviors for MRSF with various size of setback. The analytical study is conducted using a regular and 16 irregular models to capture all possible combinations of configuration of setback in five-story, five-bay MRSFs. An irregular model is developed by gradually changing the horizontal and vertical dimensions of the tower portion of the regular base 2D frame-model. All models were designed for (a) equal global displacement and uniform distribution of inter-story drift under First-Mode (FM) lateral force distribution pattern at first significant yield, and (b) equal period of vibration at the first mode, using Nonlinear Static Seismic analysis procedure. Among the conclusions derived from the research is that the variation of (a) the elastic and inelastic inter-story drift, the ductility demand for the top three stories, and (b) the elastic and inelastic global displacement exhibited a pattern similar to the variation of the FM participation factor at the roof, PF1Φr,1. The square-root-of-sum-of-square (SRSS) distribution provided accurate estimates of elastic story shear and inter-story drift demand as well as the story yield strength and drift.

In the study of linear time-invariant systems, a number of definitions, such as controllability, observability, not having decentralized fixed modes, minimum phase, etc., have been made. These definitions are highly useful in obtaining existence results for solving various types of control problems, but a drawback to these definitions is that they are binary, which simply determines whether a system is, for instance, either controllable or uncontrollable. In practical situations, however, there are many uncertainties in a system’s parameters caused by linearization, modelling errors, discretizations, and other numerical approximations and/or errors. So knowing that a system is controllable can sometimes be misleading if the controllable system is actually "almost" uncontrollable as a result of such uncertainties. Since an "almost" uncontrollable system poses significant difficulty in designing a quality controller, a continuous measure of controllability, called a controllability radius, is more desirable to use and has been widely studied in the past. The main focus of this thesis is to extend the development behind the controllability radius, with an emphasis on real parametric perturbations, to other definitions, replacing the traditional binary 'yes/no' metrics with continuous measures. We study four topics related to this development. First, we generalize the concept of real perturbation values of a matrix to the cases of matrix pairs and matrix triplets. By doing so, we are able to deal with more general perturbation structures and subsequently study, in addition to standard LTI systems, other types of systems such as LTI descriptor and time-delay systems. Second, we introduce the real decentralized fixed mode (DFM) radius, the real transmission zero at s radius, and the real minimum phase radius, which respectively measure how "close" i) a decentralized LTI system is to having a DFM, ii) a centralized system is to having a transmission zero at a particular point s in the complex plane, and iii) a minimum phase system is to being a nonminimum phase system. These radii are defined in terms of real parametric perturbations, and computable formulas for these radii are derived using a characterization based on real perturbation values and the aforementioned generalizations. Third, we present two efficient algorithms to i) solve the general real perturbation value problem, and ii) evaluate the various real LTI robustness radii introduced in this thesis. Finally as the last topic, we study the ability of a LTI system to achieve high performance control, and characterize the difficulty of achieving high performance control using a new continuous measure called the Toughness Index. A number of examples involving the various measures are studied in this thesis.

In the study of linear time-invariant systems, a number of definitions, such as controllability, observability, not having decentralized fixed modes, minimum phase, etc., have been made. These definitions are highly useful in obtaining existence results for solving various types of control problems, but a drawback to these definitions is that they are binary, which simply determines whether a system is, for instance, either controllable or uncontrollable. In practical situations, however, there are many uncertainties in a system’s parameters caused by linearization, modelling errors, discretizations, and other numerical approximations and/or errors. So knowing that a system is controllable can sometimes be misleading if the controllable system is actually "almost" uncontrollable as a result of such uncertainties. Since an "almost" uncontrollable system poses significant difficulty in designing a quality controller, a continuous measure of controllability, called a controllability radius, is more desirable to use and has been widely studied in the past. The main focus of this thesis is to extend the development behind the controllability radius, with an emphasis on real parametric perturbations, to other definitions, replacing the traditional binary 'yes/no' metrics with continuous measures. We study four topics related to this development. First, we generalize the concept of real perturbation values of a matrix to the cases of matrix pairs and matrix triplets. By doing so, we are able to deal with more general perturbation structures and subsequently study, in addition to standard LTI systems, other types of systems such as LTI descriptor and time-delay systems. Second, we introduce the real decentralized fixed mode (DFM) radius, the real transmission zero at s radius, and the real minimum phase radius, which respectively measure how "close" i) a decentralized LTI system is to having a DFM, ii) a centralized system is to having a transmission zero at a particular point s in the complex plane, and iii) a minimum phase system is to being a nonminimum phase system. These radii are defined in terms of real parametric perturbations, and computable formulas for these radii are derived using a characterization based on real perturbation values and the aforementioned generalizations. Third, we present two efficient algorithms to i) solve the general real perturbation value problem, and ii) evaluate the various real LTI robustness radii introduced in this thesis. Finally as the last topic, we study the ability of a LTI system to achieve high performance control, and characterize the difficulty of achieving high performance control using a new continuous measure called the Toughness Index. A number of examples involving the various measures are studied in this thesis.

The objective of adaptive control is to design a controller that can adjust its behaviour to tolerate uncertain or time-varying parameters. An adaptive controller typically consists of a linear time-invariant (LTI) compensator together with a tuning mechanism which adjusts the compensator parameters and yields a nonlinear controller. Because of the nonlinearity, the transient closed-loop behaviour is often poor and the control signal may become unduly large. Although the initial objective of adaptive control was to deal with time-varying plant parameters, most classical adaptive controllers cannot handle rapidly changing parameters. Recently, the use of a linear periodic (LP) controller has been proposed as a new approach in the field of model reference adaptive control [1]. In this new approach, instead of estimating plant parameters, the “ideal control signal” (what the control signal would be if the plant parameters and states were measurable) is estimated. The resulting controller has a number of desirable features: (1) it handles rapid changes in the plant parameters, (2) it provides nice transient behaviour of the closed-loop system, (3) it guarantees that the effect of the initial conditions declines to zero exponentially, and (4) it generates control signals which are modest in size. Although the linear periodic controller (LPC) has the above advantages, it has some imperfections. In order to achieve the desirable features, a rapidly varying control signal and a small sampling period are used. The rapidly time-varying control signal requires fast actuators which may not be practical. The second weakness of the LPC [1] is poor noise rejection behaviour. The small sampling period results in large controller gains and correspondingly poor noise sensitivity, since there is a clear trade-off between tracking and noise tolerance. As the last drawback, this controller requires knowledge of the exact plant relative degree. Here we extend this work in several directions: (i) In [1], the infinity-norm is used to measure the signal size. Here we redesign the controller to yield a new version which provides comparable results when the more common 2-norm is used to measure signal size, (ii) A key drawback of the controller of [1] is that the control signal moves rapidly. Here we redesign the control law to significantly alleviate this problem, (iii) The redesigned controller can handle large parameter variation and in the case that the sign of high frequency gain is known, the closed-loop system is remarkably noise-tolerant, (iv) We prove that in an important special case, we can replace the requirement of knowledge of the exact relative degree with that of an upper bound on the relative degree, at least from the point of view of providing stability, and (v) A number of approaches to improve the noise behaviour of the controller are presented. Reference: [1] D. E. Miller, “A New Approach to Model Reference Adaptive Control”, IEEE Transaction on Automatic Control, Vol. 48, No. 5, pages 743-756, May 2003.

The objective of adaptive control is to design a controller that can adjust its behaviour to tolerate uncertain or time-varying parameters. An adaptive controller typically consists of a linear time-invariant (LTI) compensator together with a tuning mechanism which adjusts the compensator parameters and yields a nonlinear controller. Because of the nonlinearity, the transient closed-loop behaviour is often poor and the control signal may become unduly large. Although the initial objective of adaptive control was to deal with time-varying plant parameters, most classical adaptive controllers cannot handle rapidly changing parameters. Recently, the use of a linear periodic (LP) controller has been proposed as a new approach in the field of model reference adaptive control [1]. In this new approach, instead of estimating plant parameters, the “ideal control signal” (what the control signal would be if the plant parameters and states were measurable) is estimated. The resulting controller has a number of desirable features: (1) it handles rapid changes in the plant parameters, (2) it provides nice transient behaviour of the closed-loop system, (3) it guarantees that the effect of the initial conditions declines to zero exponentially, and (4) it generates control signals which are modest in size. Although the linear periodic controller (LPC) has the above advantages, it has some imperfections. In order to achieve the desirable features, a rapidly varying control signal and a small sampling period are used. The rapidly time-varying control signal requires fast actuators which may not be practical. The second weakness of the LPC [1] is poor noise rejection behaviour. The small sampling period results in large controller gains and correspondingly poor noise sensitivity, since there is a clear trade-off between tracking and noise tolerance. As the last drawback, this controller requires knowledge of the exact plant relative degree. Here we extend this work in several directions: (i) In [1], the infinity-norm is used to measure the signal size. Here we redesign the controller to yield a new version which provides comparable results when the more common 2-norm is used to measure signal size, (ii) A key drawback of the controller of [1] is that the control signal moves rapidly. Here we redesign the control law to significantly alleviate this problem, (iii) The redesigned controller can handle large parameter variation and in the case that the sign of high frequency gain is known, the closed-loop system is remarkably noise-tolerant, (iv) We prove that in an important special case, we can replace the requirement of knowledge of the exact relative degree with that of an upper bound on the relative degree, at least from the point of view of providing stability, and (v) A number of approaches to improve the noise behaviour of the controller are presented. Reference: [1] D. E. Miller, “A New Approach to Model Reference Adaptive Control”, IEEE Transaction on Automatic Control, Vol. 48, No. 5, pages 743-756, May 2003.

In this paper, we study the parameter estimation of the multiple linear time series regression model with long memory stochastic regressors and innovations. Robinson and Hidalgo (1997) and Hidalgo and Robinson (2002) proposed a class of frequency-domain weighted least squares estimates. Their estimates are shown to achieve the Gauss-Markov bound with standard convergence rate. In this study, we proposed a time-domain generalized LSE approach, in which the inverse autocovariance matrix of the innovations is estimated via autoregressive coefficients. Simulation studies are performed to compare the proposed estimates with Robinson and Hidalgo (1997) and Hidalgo and Robinson (2002). The results show the time-domain generalized LSE is comparable to Robinson and Hidalgo (1997) and Hidalgo and Robinson (2002) and attains higher efficiencies when the autoregressive or moving average coefficients of the FARIMA models have larger values. A variance reduction estimator, called TF estimator, based on linear combination of the proposed estimator and Hidalgo and Robinson (2002)'s estimator is further proposed to improve the efficiency. Bootstrap method is applied to estimate the weights of the linear combination. Simulation results show the TF estimator outperforms the frequency-domain as well as the time-domain approaches.

Novel methods of system identification are developed in this dissertation. First set of methods are designed to realize time varying linear dynamical system models from input-output experimental data. The preliminary results obtained in a recent paper by the author are extended to establish a new algorithm called the Time Varying Eigensystem Realization Algorithm (TVERA). The central aim of this algorithm is to obtain a linear, time varying, discrete time model sequence of the dynamic system directly from the input-output data. Important results relating to concepts concerning coordinate systems for linear time varying systems are developed (discrete time theory) and an intuitive understanding of equivalent realizations is provided. A procedure to develop first few time step models is detailed, providing a unified solution to the time varying identification problem. The practical problem of identifying the time varying generalized Markov parameters required for TVERA is presented as the next result. In the process, we generalize the classical time invariant input output AutoRegressive model with an eXogenous input (ARX) models to the time varying case and realize an asymptotically stable observer as a byproduct of the calculations. It is further found that the choice of the generalized time varying ARX model (GTV-ARX) can be set to realize a time varying dead beat observer. Methods to use the developed algorithm(s) in this research are then considered for application to the identification of system models that are bilinear in nature. The fact that bilinear plant models become linear for constant inputs is used in the development of an algorithm that generalizes the classical developments of Juang. An intercept problem is considered as a candidate for application of the time varying identification scheme, where departure motion dynamics model sequence is calculated about a nominal trajectory with suboptimal performance owing to the presence of unstructured perturbations. Control application is subsequently demonstrated. The dynamics of a particle in a rotating tube is considered next for identification using the time varying eigensystem realization algorithm. Continuous time bilinear system identification method is demonstrated using the particle example and the identification of an automobile brake model.

In this paper, the gap metric is introduced to study the robustness of the stability of feedback systems. A relation between the gap metric and coprime fractions is also investigated. It is shown that the stability radius of the controller in the gap metric is equal to the stability margin of the controller. In the loop-shaping design procedure in the £h-gap metric, it is practically hard to formulate an ideal controller. Finally, this paper studied the conservatism of the gap metric, and proposed some properties that can help for control design and analysis.