On Positive Solutions and Mann Iterative Schemes of a Third Order Difference Equation

The existence of uncountably many positive solutions and convergence of the Mann iterative schemes for a third order nonlinear neutral delay difference equation are proved. Six examples are given to illustrate the results presented in this paper

Positive Solutions for a Third Order Nonlinear Neutral Delay Difference Equation

The existence, multiplicity, and properties of positive solutions for a third order nonlinear neutral delay difference equation are discussed. Six examples are given to illustrate the results presented in this paper

Strategic flexibility

A flexible system is defined as one that can change the entity\u27s stance, capability or status reacting to a change of the entity\u27s environment. Flexibility has gathered the attention of academic researchers and industry practitioners as an efficient approach to cope with today\u27s volatile environment. As the environments become more unpredictable and volatile, it is imperative for a flexible system to respond quickly to a change in its circumstance. How much flexibility is embedded into the system also has a critical impact on the long-term effectiveness of the flexible system. Moreover, this research focuses on the strategic environment where a decision maker\u27s behavior influences other decision makers\u27 and vice versa. ^ The primary objectives of this dissertation are developing a concrete framework for designing a flexible system by considering the exercise delay as a measure of flexibility and investigating the rational behaviors of decision makers who operate flexible systems under strategic environments. The general approach employed to develop the theoretical models for this dissertation includes optimal control theory, non-linear optimization, stochastic differential equation and game theory. ^ The first part of this research studies the optimal decisions on a flexible system with exercise delay within stochastic environments by postulating two level decisions, operational level and design level decisions. The operational level problem is modeled as a delayed optimal stopping time problem, and this research provides a comprehensive profile of the optimal operational policies according to the parameters representing the market conditions and characteristics of the alternative and designed features of the flexible system. In addition, the profile elucidates the interdependence between the operational level decision and the design level decision separating the entire domain of the design problem into sub-regions. This research effort finds that the design problem is decomposable with well-behaved non-linear optimization problems, and provides illustrative examples to show the usefulness of the developed framework. ^ The second part of this research concentrates on strategic environments which force a decision maker to cope with both exogenous uncertainty and endogenous interactions among decision makers. As the strategic environment, a duopoly market share competition is postulated where the total market profit is regarded as the underlying uncertainty. The player retaining an exclusive patent is regarded as a player competing in the market with a flexible system that does not have exercise delay, and the other competitor is interpreted as a player operating a flexible system with exercise delay. The open loop and closed loop information structures are considered for each model. The results showed that the open loop equilibrium is unique dominant strategy equilibrium. An interesting implication of the open loop equilibria is that the profitability of the flexible option decides the role of its owner in the duopoly market competition. This research finds that the closed loop equilibrium has two distinctive forms. When the asymmetry of exercise delay is large, the closed loop equilibrium is identical to the open loop equilibrium. On the other hand, if the asymmetry provides only a small enough advantage to the player who has a flexible option without exercise delay, the rational behaviors of the players are complicated in the closed loop equilibrium. The first insight from the closed loop equilibrium with large asymmetry is that the closed loop information structure hastens the execution of flexible options, and it results in lower payoffs to both of the players. Second, the role of each player is determined not only by the characteristics of the flexible options but also by the value of stochastic factor. Third, even the player with a competitive disadvantage from the asymmetry has a positive chance to be the leader of the market. ^ This research contributes to the area within industrial engineering and operations research by improving the current theoretical achievement of flexibility. The accomplishments of this work provides insights to various domains those would benefit from enhanced flexibility in the decision making process

Parameter Identification And Fault Detection For Reliable Control Of Permanent Magnet Motors

The objective of this dissertation is to develop new fault detection, identification, estimation and control algorithms that will be used to detect winding stator fault, identify the motor parameters and optimally control machine during faulty condition. Quality or proposed algorithms for Fault detection, parameter identification and control under faulty condition will validated through analytical study (Cramer-Rao bound) and simulation. Simulation will be performed for three most applied control schemes: Proportional-Integral-Derivative (PID), Direct Torque Control (DTC) and Field Oriented Control (FOC) for Permanent Magnet Machines. New detection schemes forfault detection, isolation and machine parameter identification are presented and analyzed. Different control schemes as PID, DTC, FOC for Control of PM machines have different control loops and therefore it is probable that fault detection and isolation will have different sensitivity. It is very probable that one of these control schemes (PID, DTC or FOC) are more convenient for fault detection and isolation which this dissertation will analyze through computer simulation and, if laboratory condition exist, also running a physical models

Conceptual models of the climate : 2001 program of studies in geophysical fluid dynamics

In 2001, the Geophysical Fluid Dynamics Summer Study Program grappled with Conceptual Models of the Climate. Eli Tziperman (Weizman Institute), Paola Cessi (Scripps Institution of Oceanography) and Ray Pierre- Humbert (University of Chicago) provided the principal lectures. This introduction gave us all a glimpse into the complex problem of the climate, both in the present, past and future, and even on other planets. As always, the next weeks of the program were filled with many seminars from the visitors, and culminated in the fellow's reports

Impulsive Hybrid Discrete-Continuous Delay Differential Equations

This thesis deals with impulsive hybrid discrete-continuous delay differential equations (IHDDEs). This new class of differential equations is highly challenging for two reasons. First, because of a dependency of the right-hand-side function on past states, with time delays that depend on the current state. Second, because both the right-hand-side function and the state itself are discontinuous at implicitly defined time points. The theoretical results and numerical methods presented in this thesis are related to the following subject areas: First, solutions of initial value problems (IVPs) in IHDDEs. Second, derivatives of IVP solutions with respect to parameters (“sensitivities”). Third, estimation of parameters in IHDDE models from experimental data. Amongst others, this thesis thereby makes the following contributions: - The theoretical basis of IHDDE-IVPs is established. This includes the definition of a solution concept, the existence of solutions, the uniqueness of solutions, and the differentiability of solutions with respect to parameters. - A new approach for numerically solving IVPs in differential equations with time delays is introduced. A key aspect is the use of extrapolations beyond past discontinuities. Convergence of continuous Runge-Kutta methods realized in the framework of the new approach is shown, and numerical results are presented that demonstrate the benefit of using extrapolations on a practical example. - A “first discretize, then differentiate” approach and a “first differentiate, then discretize” approach for forward sensitivity computation in IHDDEs are investigated. It is revealed that the presence of time delays destroys commutativity of differentiation and discretization in the case of continuous Runge-Kutta methods. - An extension of the concept of Internal Numerical Differentiation is proposed for differential equations with time delays. The use of the extended concept ensures that numerically computed sensitivities converge to the exact sensitivities, and that the convergence order is identical to the convergence order of the method that is used for solving the nominal IVP. - The first practical forward and adjoint schemes are developed that realize Internal Numerical Differentiation for IHDDEs. Numerical investigations show that the developed schemes are drastically more efficient than classical methods for sensitivity computation. - The new numerical methods for solving IVPs and for computing sensitivites are successfully applied to several challenging test cases, and the properties of the methods are analysed. - Numerical methods are presented for solving nonlinear least-squares parameter estimation problems constrained by IHDDEs. - A new epidemiological IHDDE model is developed. Therein, an impulse accounts for the arrival of an infected population. Further, the zeros of state-dependent switching functions characterize the time points at which new medical treatments become available. - A delay differential equation model is presented for the crosstalk of the signaling pathways of two cytokines. In comparison to an ordinary differential equation model, a better fit to experimental data is obtained with a smaller number of differential states. - A novel model is proposed to describe the voting behavior of the viewers of the TV singing competition “Unser Star für Baku” aired in 2012. Numerical results show that the use of a time delay is crucial for a qualitative correct description of the voting behavior. Furthermore, parameter estimation results yield a good quantitative agreeement with data from the TV show. - The practical implementation of all developed methods in the new software packages Colsol-DDE and ParamEDE is described

Estimation and stability of nonlinear control systems under intermittent information with applications to multi-agent robotics

This dissertation investigates the role of intermittent information in estimation and control problems and applies the obtained results to multi-agent tasks in robotics. First, we develop a stochastic hybrid model of mobile networks able to capture a large variety of heterogeneous multi-agent problems and phenomena. This model is applied to a case study where a heterogeneous mobile sensor network cooperatively detects and tracks mobile targets based on intermittent observations. When these observations form a satisfactory target trajectory, a mobile sensor is switched to the pursuit mode and deployed to capture the target. The cost of operating the sensors is determined from the geometric properties of the network, environment and probability of target detection. The above case study is motivated by the Marco Polo game played by children in swimming pools. Second, we develop adaptive sampling of targets positions in order to minimize energy consumption, while satisfying performance guarantees such as increased probability of detection over time, and no-escape conditions. A parsimonious predictor-corrector tracking filter, that uses geometrical properties of targets\u27 tracks to estimate their positions using imperfect and intermittent measurements, is presented. It is shown that this filter requires substantially less information and processing power than the Unscented Kalman Filter and Sampling Importance Resampling Particle Filter, while providing comparable estimation performance in the presence of intermittent information. Third, we investigate stability of nonlinear control systems under intermittent information. We replace the traditional periodic paradigm, where the up-to-date information is transmitted and control laws are executed in a periodic fashion, with the event-triggered paradigm. Building on the small gain theorem, we develop input-output triggered control algorithms yielding stable closed-loop systems. In other words, based on the currently available (but outdated) measurements of the outputs and external inputs of a plant, a mechanism triggering when to obtain new measurements and update the control inputs is provided. Depending on the noise environment, the developed algorithm yields stable, asymptotically stable, and Lp-stable (with bias) closed-loop systems. Control loops are modeled as interconnections of hybrid systems for which novel results on Lp-stability are presented. Prediction of a triggering event is achieved by employing Lp-gains over a finite horizon in the small gain theorem. By resorting to convex programming, a method to compute Lp-gains over a finite horizon is devised. Next, we investigate optimal intermittent feedback for nonlinear control systems. Using the currently available measurements from a plant, we develop a methodology that outputs when to update the control law with new measurements such that a given cost function is minimized. Our cost function captures trade-offs between the performance and energy consumption of the control system. The optimization problem is formulated as a Dynamic Programming problem, and Approximate Dynamic Programming is employed to solve it. Instead of advocating a particular approximation architecture for Approximate Dynamic Programming, we formulate properties that successful approximation architectures satisfy. In addition, we consider problems with partially observable states, and propose Particle Filtering to deal with partially observable states and intermittent feedback. Finally, we investigate a decentralized output synchronization problem of heterogeneous linear systems. We develop a self-triggered output broadcasting policy for the interconnected systems. Broadcasting time instants adapt to the current communication topology. For a fixed topology, our broadcasting policy yields global exponential output synchronization, and Lp-stable output synchronization in the presence of disturbances. Employing a converse Lyapunov theorem for impulsive systems, we provide an average dwell time condition that yields disturbance-to-state stable output synchronization in case of switching topology. Our approach is applicable to directed and unbalanced communication topologies.\u2