Regularization of Descriptor Systems by Derivative and Proportional State Feedback

For linear multivariable time-invariant continuous or discrete-time singular systems it is customary to use a proportional feedback control in order to achieve a desired closed loop behaviour. Derivative feedback is rarely considered. This paper examines how derivative feedback in descriptor systems can be used to alter the structure of the system pencil under various controllability conditions. It is shown that derivative and proportional feedback controls can be constructed such that the closed loop system has a given form and is also regular and has index at most 1. This property ensures the solvability of the resulting system of dynamic-algebraic equations. The construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way. The problem of pole placement with derivative feedback alone and in combination with proportional state feedback is also investigated. A computational algorithm for improving the “conditioning” of the regularized closed loop system is derived.

Numerical Methods for Stiff Two-Point Boundary Value Problems

We consider the two-point boundary value problem for stiff systems of ordinary differential equations. For systems that can be transformed to essentially diagonally dominant form with appropriate smoothness conditions, a priori estimates are obtained. Problems with turning points can be treated with this theory, and we discuss this in detail. We give robust difference approximations and present error estimates for these schemes. In particular we give a detailed description of how to transform a general system to essentially diagonally dominant form and then stretch the independent variable so that the system will satisfy the correct smoothness conditions. Numerical examples are presented for both linear and nonlinear problems.

Eigenstructure assignment in descriptor systems

Coordinate free conditions are given for pole assignment by feedback in linear descriptor (singular) systems which guarantee closed-loop regularity. These conditions are shown to be both necessary and sufficient for assignment of the maximum possible number of finite poles. Transformation to special coordinates are not used and the results provide a robust algorithm for the computation of the required feedback.

Modelling and inverting complex-valued Wiener systems

We develop a complex-valued (CV) B-spline neural network approach for efficient identification and inversion of CV Wiener systems. The CV nonlinear static function in the Wiener system is represented using the tensor product of two univariate B-spline neural networks. With the aid of a least squares parameter initialisation, the Gauss-Newton algorithm effectively estimates the model parameters that include the CV linear dynamic model coefficients and B-spline neural network weights. The identification algorithm naturally incorporates the efficient De Boor algorithm with both the B-spline curve and first order derivative recursions. An accurate inverse of the CV Wiener system is then obtained, in which the inverse of the CV nonlinear static function of the Wiener system is calculated efficiently using the Gaussian-Newton algorithm based on the estimated B-spline neural network model, with the aid of the De Boor recursions. The effectiveness of our approach for identification and inversion of CV Wiener systems is demonstrated using the application of digital predistorter design for high power amplifiers with memory

Simulation models in farming systems research : potential and challenges

Integrated simulation models can be useful tools in farming system research. This chapter reviews three commonly used approaches, i.e. linear programming, system dynamics and agent-based models. Applications of each approach are presented and strengths and drawbacks discussed. We argue that, despite some challenges, mainly related to the integration of different approaches, model validation and the representation of human agents, integrated simulation models contribute important insights to the analysis of farming systems. They help unravelling the complex and dynamic interactions and feedbacks among bio-physical, socio-economic, and institutional components across scales and levels in farming systems. In addition, they can provide a platform for integrative research, and can support transdisciplinary research by functioning as learning platforms in participatory processes.

Optimal linearization trajectories for tangent linear models

We examine differential equations where nonlinearity is a result of the advection part of the total derivative or the use of quadratic algebraic constraints between state variables (such as the ideal gas law). We show that these types of nonlinearity can be accounted for in the tangent linear model by a suitable choice of the linearization trajectory. Using this optimal linearization trajectory, we show that the tangent linear model can be used to reproduce the exact nonlinear error growth of perturbations for more than 200 days in a quasi-geostrophic model and more than (the equivalent of) 150 days in the Lorenz 96 model. We introduce an iterative method, purely based on tangent linear integrations, that converges to this optimal linearization trajectory. The main conclusion from this article is that this iterative method can be used to account for nonlinearity in estimation problems without using the nonlinear model. We demonstrate this by performing forecast sensitivity experiments in the Lorenz 96 model and show that we are able to estimate analysis increments that improve the two-day forecast using only four backward integrations with the tangent linear model. Copyright © 2011 Royal Meteorological Society

Femtosecond pulse shaping using differential evolutionary algorithm and wavelet operators

Evolutionary meta-algorithms for pulse shaping of broadband femtosecond duration laser pulses are proposed. The genetic algorithm searching the evolutionary landscape for desired pulse shapes consists of a population of waveforms (genes), each made from two concatenated vectors, specifying phases and magnitudes, respectively, over a range of frequencies. Frequency domain operators such as mutation, two-point crossover average crossover, polynomial phase mutation, creep and three-point smoothing as well as a time-domain crossover are combined to produce fitter offsprings at each iteration step. The algorithm applies roulette wheel selection; elitists and linear fitness scaling to the gene population. A differential evolution (DE) operator that provides a source of directed mutation and new wavelet operators are proposed. Using properly tuned parameters for DE, the meta-algorithm is used to solve a waveform matching problem. Tuning allows either a greedy directed search near the best known solution or a robust search across the entire parameter space.

Analysis of the variability of auditory brainstem response components through linear regression

(ABR) is of fundamental importance to the investiga- tion of the auditory system behavior, though its in- terpretation has a subjective nature because of the manual process employed in its study and the clinical experience required for its analysis. When analyzing the ABR, clinicians are often interested in the identi- fication of ABR signal components referred to as Jewett waves. In particular, the detection and study of the time when these waves occur (i.e., the wave la- tency) is a practical tool for the diagnosis of disorders affecting the auditory system. In this context, the aim of this research is to compare ABR manual/visual analysis provided by different examiners. Methods: The ABR data were collected from 10 normal-hearing subjects (5 men and 5 women, from 20 to 52 years). A total of 160 data samples were analyzed and a pair- wise comparison between four distinct examiners was executed. We carried out a statistical study aiming to identify significant differences between assessments provided by the examiners. For this, we used Linear Regression in conjunction with Bootstrap, as a me- thod for evaluating the relation between the responses given by the examiners. Results: The analysis sug- gests agreement among examiners however reveals differences between assessments of the variability of the waves. We quantified the magnitude of the ob- tained wave latency differences and 18% of the inves- tigated waves presented substantial differences (large and moderate) and of these 3.79% were considered not acceptable for the clinical practice. Conclusions: Our results characterize the variability of the manual analysis of ABR data and the necessity of establishing unified standards and protocols for the analysis of these data. These results may also contribute to the validation and development of automatic systems that are employed in the early diagnosis of hearing loss.

A dynamical systems framework for intermittent data assimilation

We consider the problem of discrete time filtering (intermittent data assimilation) for differential equation models and discuss methods for its numerical approximation. The focus is on methods based on ensemble/particle techniques and on the ensemble Kalman filter technique in particular. We summarize as well as extend recent work on continuous ensemble Kalman filter formulations, which provide a concise dynamical systems formulation of the combined dynamics-assimilation problem. Possible extensions to fully nonlinear ensemble/particle based filters are also outlined using the framework of optimal transportation theory.

Participatory research approaches in the development of improved management practices in indigenous chickens production systems with smallholder farmers in Kenya

This thesis is concerned with development of improved management practices in indigenous chicken production systems in a research process that includes participatory approaches with smallholder farmers and other stakeholders in Kenya. The research process involved a wide range of activities that included on-station experiments, field surveys, stakeholder consultations in workshops, seminars and visits, and on-farm farmer participatory research to evaluate the effect of some improved management interventions on production performance of indigenous chickens. The participatory research was greatly informed from collective experiences and lessons of the previous activities. The on-station studies focused on hatching, growth and nutritional characteristics of the indigenous chickens. Four research publications from these studies are included in this thesis. Quantitative statistical analyses were applied and they involved use of growth models estimated with non-linear regressions for the growth characteristics, chi-square determinations to investigate differences among different reciprocal crosses of indigenous chickens and general linear models and covariance determination for the nutrition study. The on-station studies brought greater understanding of performance and production characteristics of indigenous chickens and the influence of management practices on these characteristics. The field surveys and stakeholder consultations helped in understanding the overarching issues affecting the productivity of the indigenous chickens systems and their place in the livelihoods of smallholder farmers. These activities created strong networking opportunities with stakeholders from a wide spectrum. The on-farm farmer participatory research involved selection of 200 farmers in five regions followed by training and introduction of interventions on improved management practices which included housing, vaccination, deworming and feed supplementation. Implementation and monitoring was mainly done by individual farmers continuously for close to one and half years. Six quarterly visits to the farms were made by the research team to monitor and provide support for on-going project activities. The data collected has been analysed for 5 consecutive 3-monthly periods. Descriptive and inferential statistics were applied to analyse the data collected involving treatment applications, production characteristics and flock demography characteristics. Out of the 200 farmers initially selected, 173 had records on treatment applications and flock demography characteristics while 127 farmers had records on production characteristics. The demographic analysis with a dissimilarity index of flock size produced 7 distinct farm groups from among the 173 farms. Two of these farm groups were represented in similar numbers in each of the five regions. The research process also involved a number of dissemination and communication strategies that have brought the process and project outcomes into the domain of accessibility by wider readership locally and globally. These include workshops, seminars, field visits and consultations, local and international conferences, electronic conferencing, publications and personal communication via emailing and conventional posting. A number of research and development proposals were also developed based on the knowledge and experiences gained from the research process. The thesis captures the research process activities and outcomes in 8 chapters which include in ascending order – introduction, theoretical concepts underpinning FPR, research methodology and process, on-station research output, FPR descriptive statistical analysis, FPR inferential statistical analysis on production characteristics, FPR demographic analysis and conclusions. Various research approaches both quantitative and qualitative have been applied in the research process indicating the possibilities and importance of combining both systems for greater understanding of issues being studied. In our case, participatory studies of the improved management of indigenous chickens indicates their potential importance as livelihood assets for poor people.

Complex-valued B-spline neural networks for modeling and inverting Hammerstein systems

Many communication signal processing applications involve modelling and inverting complex-valued (CV) Hammerstein systems. We develops a new CV B-spline neural network approach for efficient identification of the CV Hammerstein system and effective inversion of the estimated CV Hammerstein model. Specifically, the CV nonlinear static function in the Hammerstein system is represented using the tensor product from two univariate B-spline neural networks. An efficient alternating least squares estimation method is adopted for identifying the CV linear dynamic model’s coefficients and the CV B-spline neural network’s weights, which yields the closed-form solutions for both the linear dynamic model’s coefficients and the B-spline neural network’s weights, and this estimation process is guaranteed to converge very fast to a unique minimum solution. Furthermore, an accurate inversion of the CV Hammerstein system can readily be obtained using the estimated model. In particular, the inversion of the CV nonlinear static function in the Hammerstein system can be calculated effectively using a Gaussian-Newton algorithm, which naturally incorporates the efficient De Boor algorithm with both the B-spline curve and first order derivative recursions. The effectiveness of our approach is demonstrated using the application to equalisation of Hammerstein channels.

Single-carrier frequency domain equalisation for hammerstein communication systems using complex-valued neural networks

Single-carrier (SC) block transmission with frequency-domain equalisation (FDE) offers a viable transmission technology for combating the adverse effects of long dispersive channels encountered in high-rate broadband wireless communication systems. However, for high bandwidthefficiency and high power-efficiency systems, the channel can generally be modelled by the Hammerstein system that includes the nonlinear distortion effects of the high power amplifier (HPA) at transmitter. For such nonlinear Hammerstein channels, the standard SC-FDE scheme no longer works. This paper advocates a complex-valued (CV) B-spline neural network based nonlinear SC-FDE scheme for Hammerstein channels. Specifically, We model the nonlinear HPA, which represents the CV static nonlinearity of the Hammerstein channel, by a CV B-spline neural network, and we develop two efficient alternating least squares schemes for estimating the parameters of the Hammerstein channel, including both the channel impulse response coefficients and the parameters of the CV B-spline model. We also use another CV B-spline neural network to model the inversion of the nonlinear HPA, and the parameters of this inverting B-spline model can easily be estimated using the standard least squares algorithm based on the pseudo training data obtained as a natural byproduct of the Hammerstein channel identification. Equalisation of the SC Hammerstein channel can then be accomplished by the usual one-tap linear equalisation in frequency domain as well as the inverse B-spline neural network model obtained in time domain. Extensive simulation results are included to demonstrate the effectiveness of our nonlinear SC-FDE scheme for Hammerstein channels.

A study of anticipatory non-autonomous systems

Rhythms are manifested ubiquitously in dynamical biological processes. These fundamental processes which are necessary for the survival of living organisms include metabolism, breathing, heart beat, and, above all, the circadian rhythm coupled to the diurnal cycle. Thus, in mathematical biology, biological processes are often represented as linear or nonlinear oscillators. In the framework of nonlinear and dissipative systems (ie. the flow of energy, substances, or sensory information), they generate stable internal oscillations as a response to environmental input and, in turn, utilise such output as a means of coupling with the environment.