Yes / In this technical note, stochastic coupling attenuation is investigated for a class of multi-variable bilinear stochastic systems and a novel output feedback m-block backstepping controller with linear estimator is designed, where gradient descent optimization is used to tune the design parameters of the controller. It has been shown that the trajectories of the closed-loop stochastic systems are bounded in probability sense and the stochastic coupling of the system outputs can be effectively attenuated by the proposed control algorithm. Moreover, the stability of the stochastic systems is analyzed and the effectiveness of the proposed method has been demonstrated using a simulated example.

A novel statistical technique introduced by Besieris is used to study solutions of the nonlinear stochastic complex parabolic equation in the presence of two profiles. Specifically, the randomly modulated linear potential and the randomly perturbed quadratic focusing medium. In the former, a class of solutions is shown to admit an exact statistical description in terms of the moments of the wave function. In the latter, all even-order moments are computed exactly, whereas the odd-order moments are solved asymptotically. Lastly, it is shown that this statistical technique is isomorphic to mappings of nonconstant coefficient partial differential equations to constant coefficient equations. A generalization of this mapping and its inherent restrictions are discussed. / Ph. D. / incomplete_metadata

Call number: LD2668 .T4 1986 D83 / Master of Science / Chemical Engineering

An attempt is made in this thesis to study some stochastic models of both reliability and inventory systems with reference to the following aspects: (i) the confidence limits with the introduction of common-cause failures. (ii) the Erlangian repair time distributions. (iii) the product interactions and demand interactions. (iv) the products are perishable. This thesis contains six chapters. Chaper 1 is introductory in nature and gives a review of the literature and the techniques used in the analysis of reliability systems. Chapter 2 is a study of component common-cause failure systems. Such failures may greatly reduce the reliability indices. Two models of such systems (series and parallel) have been studied in this chapter. The expressions such as, reliability, availability and expected number of repairs have been obtained. The confidence limits for the steady state availability of these two systems have also been obtained. A numerical example illustrates the results. A 100 (1 - a) % confidence limit for the steady state availability of a two unit hot and warm standby system has been studied, when the failure of an online unit is constant and the repair time of a failed unit is Erlangian. The general introduction of various inventory systems and the techniques used in the analysis of such systems have been explained in chapter 4. Chapter 5 provides two models of two component continuous review inventory systems. Here we assume that demand occurs according to a poisson process and that a demand can be satisfied only if both the components are available in inventory. Back-orders are not permitted. The two components are bought from outside suppliers and are replenished according to (s, S) policy. In model 1 we assume that the lead-time of the components follow an exponential distribution. By identifying the inventory level as a Markov process, a system of difference-differential equations at any time and the steady-state for the state of inventory level are obtained. Tn model 2 we assume that the lead-time distribution of one product is arbitrary and the other is exponential. Identifying the underlying process as a semi-regenerative process we find the stationary distribution of the inventory level. For both these models, we find out the performance measures such as the mean stationary rate of the number of lost demands, the demands satisfied and the reorders made. Numerical examples for the two models are also considered. Chaper 6 is devoted to the study of a two perishable product inventory model in which the products are substitutable. The perishable rates of product 1 and product 2 are two different constants. Demand for product 1 and product 2 follow two independent Poisson processes. For replenishment of product 1 (s, S) ordering policy is followed and the associated lead-time is arbitrary. Replenishment of product 2 is instantaneous. A demand for product 1 which occurs during its stock-out period can be substituted by product 2 with some probability. Expressions are derived for the stationary distribution of the inventor}' level by identifying the underlying stochastic process as a semi-regenerative process. An expression for the expected profit rate is obtained. A numerical illustration is provided and an optimal reordering level maximising the profit rate is also studied. To sum up, this thesis is an effort to improve the state the of art of (i) complex reliability systems and their estimation study (ii) muitiproduct inventory systems. The salient features of the thesis are: (i) Analysis of a two-component reliability system with common-cause failures. (ii) Estimation study of a complex system in which the repair time for both hot standby and warm standby systems are assumed to be Eriangian. (iii) A multi-product continuous review inventory system with product interaction, with a (s, S) policy. (iv) Introduction of the concept of substitutability for products. (v) Derivation of expressions for various statistical measures. (vi) Effective use of the regeneration point technique in deriving various measures for both reliability and inventory systems. (vii) Illustration of the various results by extensive numerical work. (vii) Consideration of relevant optimization problems. / Mathematical Sciences / PhD (Statistics)

This doctoral thesis is concerned with some theoretical and practical questions related to backward stochastic differential equations (BSDEs) and more specifically their connection with some parabolic partial differential equations (PDEs). The thesis is made of three parts. In the first part, we study the probabilistic representation for a class of multidimensional PDEs with quadratic nonlinearities of a special form. We obtain a representation formula for the PDE solution in terms of the solutions to a Lipschitz BSDE. We then use this representation to obtain an estimate on the gradient of the PDE solutions by probabilistic means. In the course of our analysis, we are led to prove some results for the associated multidimensional quadratic BSDEs, namely an existence result and a partial uniqueness result. In the second part, we study the well-posedness of a very general quadratic reflected BSDE driven by a continuous martingale. We obtain the comparison theorem, the special comparison theorem for reflected BSDEs (which allows to compare the increasing processes of two solutions), the uniqueness and existence of solutions, as well as a stability result. The comparison theorem (from which uniqueness follows) and the special comparison theorem are obtained through natural techniques and minimal assumptions. The existence is based on a perturbative procedure, and holds for a driver whis is Lipschitz, or slightly-superlinear, or monotone with arbitrary growth in y. Finally, we obtain a stability result, which gives in particular a local Lipschitz estimate in BMO for the martingale part of the solution. In the third and last part, we study the time-discretization of BSDEs having nonlinearities that are monotone but with polynomial growth in the primary variable. We show that in that case, the explicit Euler scheme is likely to diverge, while the implicit scheme converges. In fact, by studying the family of θ-schemes, which are mixed explicit-implicit, θ characterizing the degree of implicitness, we find that the scheme converges when the implicit component is dominant (θ ≥ 1/2 ). We then propose a tamed explicit scheme, which converges. We show that the implicit-dominant schemes with θ > 1/2 and our tamed explicit scheme converge with order 1/2 , while the trapezoidal scheme (θ = 1/2) converges with order 7/4.

A substantial segment of the error in numerical weather prediction and climate projections comes from the intrinsic uncertainties of General Circulation Models of the atmosphere. Stochastic physics schemes are one of the preferred methods to represent the model uncertainty in Ensemble Prediction Systems, where different realizations of the same forecast are created to quantify the probabilities of different outcomes in the atmospheric flow. Stochastic physics schemes have been successfully employed in medium-range and seasonal forecasting systems, as they increase the skill of probabilistic forecasts. Similarly it has been demonstrated than these schemes can improve certain aspects of the model's climate. However, it is still not clear whether they are a truthful representation of the model uncertainties they aim to represent. In this thesis, a collection of stochastic physics schemes are evaluated using a seamless approach. It is found that they can improve the representation of the tropical climate and extra-tropical cyclones, but they degrade the individual representation of these processes deteriorating the deterministic skill of the model. Some important features of the model can be degraded by the stochastic physics schemes, like energy and moisture conservation on climate scales. Some closures to the schemes are proposed and successfully tested to remove or reduce some of the problems found. Alternative approaches in the development of stochastic parametrizations are also investigated. Stochastic physics schemes have some benefits but still require further development to produce a realistic representation of model error. It is also recommended that evaluation methodologies must be expanded to include process-based diagnostics to display the realism of its perturbations.

Uncertainties in chemical process performance behaviour continue to cause considerable concern to engineers and other stakeholders. The traditional deterministic uncertainty modelling methods lead to excess overdesign, which is expensive, and have also been shown to give limited insight into the behaviour of complex chemical engineering systems. The present work develops a new framework, termed “Stochastic Process Performance Modelling Framework (SPPMF)”, which combines traditional deterministic process simulation, response surface modelling techniques and advanced structural reliability analysis methods to facilitate efficient performance modelling and optimisation of chemical process systems under uncertainties. Cross application of structural reliability principles to chemical processes presents some challenges; however, means of addressing such issues are proposed and discussed in this thesis. For instance, to facilitate Process Reliability Analysis (PRA), stochastic constraints have been added to the conventional process optimisation formulation. Both first order reliability method and Monte Carlo simulation are then applied to gain a wide range of performance measures. In addition, to allow for automated response surface generation, an interface for linking process simulators and a new stochastic module has been developed; making it possible to obtain samples in the order of thousands, typically in minutes. A number of Structural Reliability Analysis (SRA) concepts have been re-defined to reflect the unique characteristics of chemical processes. For example, while SRA is mainly concerned with the effects of random forces and mechanical properties on structural performance, PRA is focused on random process conditions (e.g. changes in pH, reaction rates, etc) and their effects on both product quantity and quality. Finally, SPPMF has been successfully applied to model stochastic properties of a range of typical process systems. The results show that the new framework can be efficiently implemented in process engineering with significant benefits over the traditional methods. Limitations of SPPMF and directions for future work are also highlighted. This thesis contains commercially confidential information which should not be divulged to any third party without the written consent of the author.

The discovery of genetic regulatory networks was an important advancement in science. Not only do they help understand how organisms behave but the development of synthetic genetic networks has aided in other fields of science and industry. Many genetic networks have been modelled deterministically by using differential equations to provide an insight into the network's behaviour. However, within a biological environment, a certain degree of intrinsic noise should be expected and the robustness of these networks should be tested. Creating and analysing a genetic network in a biological environment can be a time consuming task so applying stochastic methods, such as the Gillespie Algorithm, to a computer model will provide an important, initial insight into the behaviour of the system. One interesting genetic network is the coupled repressilator due to its relatively simplistic design and the broad, multistable dynamics it offers. The inhomogeneous solutions that it can yield are particularly interesting as they may help explain certain biological phenomena, and may be used as a tool to assist with further research into genetic networks. In this thesis, the Gillespie Algorithm will be applied to the coupled repressilator so that its robustness can be tested. Biologically feasible modifications will be made to the system to produce much more stable and predictable dynamics so that the broad range of solutions can exist within a noisy environment. The methods developed will take into account previously made assumptions and potential errors in biological data so that they can be applied to other genetic system. One further objective in this thesis is to explore computational limitations that may occur when modelling large, stochastic networks. Issues such as rounding errors and dealing with very small and very large numbers were encountered and methods to circumvent these without sacrificing computational run-time will be developed and applied.

published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

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