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A cyclist is a vulnerable road user whose safety is affected by several externalities. The global aim of the research is to investigate the effect of critically identified variables of rider attributes of age, gender, varied environmental condition of lighting, meteorology, and micro-infrastructure variables on the safe usage of the infrastructure for a cyclist. Presently, very few works have attempted to undertake such modelling. A novel methodological framework is developed, consisting of descriptive, statistical, artificial intelligence and mathematical approaches. Accurate prediction models are developed, and in-depth knowledge of how different variables affect cyclist safety are identified, modelled, and quantified. It is found that the variables of age, gender, varied environmental conditions, and micro-infrastructure variable are critical variables affecting the safe usage of infrastructure. These variables, both individually and in combination, impact cyclist safety. Cycling safety is a dynamic variable that varies temporally and spatially. The spatial and environmental variables have a significantly varied effect on safety depending upon the rider personal attribute. As the number of safety variables that the cyclist must conform to grows, so does the risk. The riskiest environmental conditions are exacerbated by the prevailing traffic flow regime, posing a significant safety risk to cyclists. The modelling requirement of a cyclist is significantly different from motorists. A hybrid intelligent modelling paradigm is required, as demonstrated in this research. The study results can significantly impact the route choice, modelling, and planning of infrastructure. A shift in the road safety analysis towards nanoscopic modelling can help achieve zero-vision road traffic fatality. The research reinforces a need for planning and design of infrastructure to move towards a more holistic approach while considering the limitations of this vulnerable road user.

Multi-Agent Systems (MASs) have attracted much popularity, since the previous decade due to their potential wide range of applications. Indeed, connected MASs are deployed in order to achieve more complex objectives that could otherwise not be achievable by a single agent. In distributed schemes, agents must share their information with their neighbours, which are then used for common control and fault detection purposes, and thus do not require any central monitoring unit. This translates into the necessity to develop efficient distributed algorithms in terms of robustness and safety. Indeed, the problem of safety in connected cooperative MASs has arisen as a consequence of their complexity and the nature of their operations and wireless communication exchanges, which renders them vulnerable to not only physical faults, but also to cyber-attacks. The main focus of this thesis is the study of distributed fault and attack detection and isolation in connected MASs. First, a distributed methodology for global detection of actuator faults in a class of linear MASs with unknown disturbances is proposed using a cascade of fixed-time Sliding Mode Observers (SMOs), where each agent having access to their state, and neighbouring information exchanges, can give an exact estimate of the state of the overall MAS. An LMI-based approach is then applied to design distributed global robust residual signals at each agent capable of detecting faults anywhere in the network. This is then extended to agents with nonlinear nonholonomic dynamics where a new distributed robust Fault Detection and Isolation (FDI) scheme is proposed using predefined-time stability techniques to derive adequate distributed SMOs. This enables to reconstruct the global system state in a predefined-time and generate proper residual signals. The case of MASs with higher order integrator dynamics, where only the first state variable is measurable and the topology is switching is investigated, where a new approach to identify faults and deception attacks is introduced. The proposed protocol makes an agent act as a central node monitoring the whole system activities in a distributed fashion whereby a bank of distributed predefined-time SMOs for global state estimation are designed, which are then used to generate residual signals capable of identifying cyber-attacks despite the switching topology. The problem of attack and FDI in connected heterogeneous MASs with directed graphs, is then studied. First, the problem of distributed fault detection for a team of heterogeneous MASs with linear dynamics is investigated, where a new output observer scheme is proposed which is effective for both directed and undirected topologies. The main advantage of this approach is that the design, being dependant only on the input-output relations, renders the computational cost, information exchange and scalability very effective compared to other FDI approaches that employ the whole state estimation of the agents and their neighbours as a basis for their design. A more general model is then studied, where actuator, sensor and communication faults/attacks are considered in the robust detection and isolation process for nonlinear heterogeneous MASs with measurement noise, dynamic disturbances and communication parameter uncertainties, where the topology is not required to be undirected. This is done using a distributed finite-frequency mixed H_/H1 nonlinear UIO-based approach. Simulation examples are given for each of the proposed algorithms to show their effectiveness and robustness.

Autonomous Driving Vehicles (ADVs) has received considerable attention in recent years by academia and industry, bringing about a paradigm shift in Intelligent Transportation Systems (ITS), where vehicles operate in close proximity through wireless communication. It is envisioned as a promising technology for realising efficient and intelligent transportation systems, with potential applications for civilian and military purposes. Vehicular network management for ADVs is challenging as it demands mobility, location awareness, high reliability, and low latency data traffic. This research aims to develop and implement vehicular communication in conjunction with a driving algorithm for ADVs feedback control system with a specific focus on the safe displacement of vehicle platoon while sensing the surrounding environment, such as detecting road signs and communicate with other road users such as pedestrian, motorbikes, non-motorised vehicles and infrastructure. However, in order to do so, one must investigate crucial aspects related to the available technology, such as driving behaviour, low latency communication requirement, communication standards, and the reliability of such a mechanism to decrease the number of traffic accidents and casualties significantly. To understand the behaviour of wireless communication compared to the theoretical data rates, throughput, and roaming behaviour in a congested indoor line-of-sight heterogeneous environment, we first carried out an experimental study for IEEE 802.11a, 802.11n and 802.11ac standards in a 5 GHz frequency spectrum. We validated the results with an analytical path loss model as it is essential to understand how the client device roams or decides to roam from one Access Point to another and vice-versa. We observed seamless roaming between the tested protocols irrespective of their operational environment (indoor or outdoor); their throughput efficiency and data rate were also improved by 8-12% when configured with Short Guard Interval (SGI) of 400ns compared to the theoretical specification of the tested protocols. Moreover, we also investigated the Software-Defined Networking (SDN) for vehicular communication and compared it with the traditional network, which is generally incorporated vertically where control and data planes are bundled collectively. The SDN helped gain more flexibility to support multiple core networks for vehicular communication and tackle the potential challenges of network scalability for vehicular applications raised by the ADVs. In particular, we demonstrate that the SDN improves throughput efficiency by 4% compared to the traditional network while ensuring efficient bandwidth and resource management. Finally, we proposed a novel data-driven coordination model which incorporates Vehicle-to-Everything (V2X) communication and Intelligent Driver Model (IDM), together called V2X Enabled Intelligent Driver Model (VX-IDM). Our model incorporates a Car-Following Model (CFM), i.e., IDM, to model a vehicle platoon in an urban and highway traffic scenario while ensuring the vehicle platoon's safety with the integration of IEEE 802.11p Vehicle-to-Infrastructure (V2I) communication scheme. The model integrates the 802.11p V2I communication channel with the IDM in MATLAB using ODE‐45 and utilises the 802.11p simulation toolbox for configuring vehicular channels. To demonstrate model functionality in urban and highway traffic environments, we developed six case studies. We also addressed the heterogeneity issue of wireless networks to improve the overall network reliability and efficiency by estimating the Signal-to-Noise Ratio (SNR) parameters for the platoon vehicle's displacement and location on the road from Road-Side-Units (RSUs). The simulation results showed that inter-vehicle spacing could be steadily maintained at a minimum safe value at all the time. Moreover, the model has a fault-tolerant mechanism that works even when communication with infrastructure is interrupted or unavailable, making the VX-IDM model collision-free.

With the increasing integration of electric vehicles and renewable energy sources in electricity networks, key opportunities in terms of a cleaner environment and a sustainable energy portfolio are unlocked. However, the widespread deployment of these two technologies, can entail significant challenges for the electricity grid and in a larger context for the society, when they are not optimally integrated. In this context, smart charging of electric vehicles and vehicle-to-grid technologies are being proposed as crucial solutions to achieve economic, technical and environmental benefits in future smart grids. The implementation of these technologies involves a number of key stakeholders, namely, the end-electricity user, the electric vehicle owner, the system operators and policy makers. For a wider and efficient implementation of the smart grid vision, these stakeholders must be engaged and their aims must be fulfilled. However, the financial, technical and environmental objectives of these stakeholders are often conflicting, which leads to an intricate paradigm requiring efficient and fair policies. With this focus in mind, the present research work develops multi-objective optimisation algorithms to control the charging and discharging process of electric vehicles. Decentralised, hybrid and real-time optimisation algorithms are proposed, modelled, simulated and validated. End user energy cost, battery degradation, grid interaction and CO2 emissions are optimised in this work and their trade-offs are highlighted. Multi-criteria-decision-making approaches and game theoretical frameworks are developed to conciliate the interests of the involved stakeholders. The results, in the form of optimal electric vehicle charging/discharging schedules, show improvements along all the objectives while complying with the user requirements. The outcome of the present research work serves as a benchmark for informing system operators and policy makers on the necessary measures to ensure an efficient and sustainable implementation of electro-mobility as a fundamental part of current and future smart grids.

In the current digital age where automatic controlled systems are used in many fields such as industrial plants, means of transports and domestic electronics applications, the issue of safety and reliability has become of paramount importance. With that came the need to develop techniques to be implemented in control systems that would allow monitoring those systems and detect if some malfunctions or abnormalities are occurring. Fault detection (FD) emerged as one of the most widely used solutions to this issue and the so-called model based fault detection has received a lot of attention. In this approach, a model of the target system is involved to estimate the expected output of the system under healthy condition and then a fault can be detected by comparing the actual measured output to the estimated healthy output. By making use of the state estimation capability of observers, various observer-based fault detection schemes have been proposed to estimate the system output for the purpose of fault detection. However, it is worth noting that, state observers have been designed for state estimation and have found use in being adapted for fault detection. In order to try to answer the increasing requirements on systems performances, this thesis focuses on developing observers to be used specifically in FD schemes and for which a systematic way to be designed is proposed. The first solution based on proportional integral (PI) observer, relies on integrating the systems output to construct an augmented model. This technique has a double effect on achieving better fault detection performances. Indeed, disturbances effect is reduced by using a technique based on replacing part of the model information in the observer design. Besides, it allows having an additional degree of freedom when optimising the observer design to detect faults. The second solution to the FD problem, a new type of observers, referred to as output observer is proposed for fault detection in both linear and nonlinear systems, where unnecessary state estimation in observer-based fault detection can be avoided. First, an input/output system representation, upon which the output observer design is based on, is introduced. Then, a new approach of output observer design, in which only the output variables are estimated, is developed. The convergence of the observer with respect to arbitrary initial conditions is proved and the fault detectability capabilities of the scheme are established. Another benefit of the proposed output observer design is the output injection feature, where the measured output is directly injected in the observer so as to linearise the estimation error dynamics. This feature is fundamental to the solution proposed in this thesis to deal with nonlinearities in the system's model so that the inclusion of those nonlinearities when tuning the observer is avoided. Furthermore, as time delays are ubiquitous in systems, and are one of the most important sources of estimation errors, a solution based on output injection is also proposed to set the condition that ensures the convergence of the observer while maintaining its output estimation performances.

Experimental studies of the flowering of Arabidopsis Thaliana have shown that a large complex gene regulatory network (GRN) is responsible for its regulation. This process has recently been modelled with deterministic differential equations by considering the interactions between gene activators and inhibitors [Valentim et al., 2015, van Mourik et al., 2010]. However, due to the complexity of the models, the properties of the network and the roles of the individual genes cannot be deduced from the numerical solution the published work offers. In this study, deterministic and stochastic dynamic models of Arabidopsis flowering GRN are considered by following the deterministic delayed model introduced in [Valentim et al., 2015]. A stable solution of this model is sought by its linearisation, which contributes to further investigation of the role of the individual genes to the flowering. By decoupling some concentrations, the system has been reduced to emphasise the role played by the transcription factor Suppressor of Overexpression of Constants1 (SOC1) and the important floral meristem identity genes, Leafy (LFY) and Apetala1 (AP1). Two-dimensional motifs, based on the dynamics of LFY and AP1, are obtained from the reduced network and parameter ranges ensuring flowering are determined. Their stability analysis shows that LFY and AP1 are regulating each other for flowering, matching experimental findings (see e.g. [Bl ́azquez et al., 2001, Welch et al., 2004, Yeap et al., 2014]). Moreover, the role of noise is studied by introducing and investigating two types of stochastic elements into the motifs. New suffient conditions of mean square stability and their domain are obtained analytically for the stochastic models using Lyapunov stability theory. Numerical solutions are obtained by using Euler-Maruyama method and Ito stochastic formula. We demonstrate that the stochastic motifs of Arabidopsis flowering time can capture the essential behaviour of the full system and that stochastic effects can change the behaviour of the stability region through a stability switch. Furthermore, the problem of designing an observer and a controller, in which FT is seen as a control input, is considered in the objective of ensuring flowering conditions are met. This study thus contributes to a better understanding of the role of LFY and AP1 in Arabidopsis flowering.

In the past few decades, there has been a growing demand for bandwidth in wireless communication systems due to an increase in the number of mobile users, computer network applications and high speed internet access and services. Visible light communication (VLC) is a newly emerging technology in optical wireless communication (OWC) systems to provide both illumination and high-speed data communication. In addition, it can also be used for indoor positioning or localization with wide applications that has merited a number of attractive research studies. However, security in wireless communications has become a matter of concern due to the possibility of unauthorised access to transmitted data. This thesis investigates received optical power (ROP) distribution to achieve illumination requirements for both optimal and sub-optimal Lambertian order in single and four cell configurations. Furthermore, it proposes VLC-based indoor positioning by using the received optical power levels from the emitting LEDs. Both scenarios of line-of-sight (LOS) and line-of-sight with non-line-of-sight (LOSNLOS) positioning are considered. The performance of the proposed system is evaluated under both noisy and noiseless channel cases and the impact of different location codes on the positioning error is also investigated. An analytical model of the system with noise and the corresponding numerical evaluation for a range of signal to noise ratios (SNRs) are presented. The results show that an accuracy of 8 cm in average is achievable at an SNR of 15 dB in the LOSNLOS scenario. We also introduce a novel approach for recovering 4×4 MIMO–VLC data with partial channel state information (CSI) knowledge both in noisy and noiseless cases. In addition, the proposed indoor localization system utilizing two visible light emitting diodes for different environments offering less complexity for both LOS and LOSNLOS scenarios is offered. The results of the scheme is compared with existing trilateration techniques and the performance of the proposed system is evaluated with a reported accuracy of less than 20 cm for SNR values of greater than 13 dB. This thesis also proposes a new secure single input single output (SISO) VLC system and investigates the bit error rate (BER) performance with and without encryption in the Media Access Control (MAC) layer and physical layer for both the LOS and LOSNLOS scenarios. For a BER of 10-3, the results show that there is 1 to ~ 3.5 dB power penalties for the secured VLC system for data rates of up to 10 Mbps. The key length impact on the error propagation is also investigated. Finally, this thesis proposes a secure MIMO VLC system that relies on the position of the user by incorporating a new modified version of the Rivest-Shamir-Adleman (RSA) technique for encrypting the transmitted data in the MAC layer. Furthermore, the ability of the proposed system to control the size of the encrypted cell depending on the application environment is demonstrated.

Recent advances in machine intelligence, particularly Artificial Neural Networks (ANNs) and Particle Swarm Optimisation (PSO), have introduced conceptually advanced technologies that can be utilised for financial market share trading analysis. The primary goal of the present research is to model short-term daily trading in Financial Times Stock Exchange 100 Index (FTSE 100) shares to make forecasts with certain levels of confidence and associated risk. The hypothesis to be tested is that financial shares time series contain significant non-linearity and that ANN, either separately or in conjunction with PSO, could be utilised effectively. Validation of the proposed model shows that nonlinear models are likely to be better choices than traditional linear regression for short-term trading. Some periodicity and trend lines were apparent in short- and long-term trading. Experiments showed that a model using an ANN with the Discrete Fourier Transform (DFT) and Discrete Wavelet Transform (DWT) model features performed significantly better than analysis in the time domain. Mathematical analysis of the PSO algorithm from a systemic point of view along with stability analysis was performed to determine the choice of parameters, and a possible proportional, integral and derivative (PID) algorithm extension was recommended. The proposed extension was found to perform better than traditional PSO. Furthermore, a chaotic local search operator and exponentially varying inertia weight factor algorithm considering constraints were proposed that gave better ability to converge to a high quality solution without oscillations. A hybrid example combining an ANN with the PSO forecasting regression model significantly outperformed the original ANN and PSO approaches in accuracy and computational complexity. The evaluation of statistical confidence for the models gave good results, which is encouraging for further experimentation considering model cross-validation for generalisation to show how accurately the predictive models perform in practice.

Among the number of optical sources, vertical-cavity surface-emitting lasers (VCSELs) are relatively recent type of semiconductor laser devices, which are attractive for a number of applications particularly for free space optical (FSO) communication systems. In such systems reliable optical devices with lower power consumption and low cost are among the key requirements. VCSELs typically operate with unstable output polarization modes, and there is a need to improve their output power regarding to the polarization instability, particularly when introducing the optical feedback (OF). This thesis investigates a number of key properties of VCSEL including the polarization instability, hysteresis loop (HL), relative intensity noise (RIN) and how to control the polarization switching (PS). The investigations are based on the analytical studies and extensive experimental work. PS properties of VCSEL are investigated by introducing variable polarization optical feedback (VPOF) with the modulation frequency and modulation depth. The dependency conditions for the HL, RIN and PS are determined with VPOF. Under OF, the threshold current (Ith) of VCSEL is reduced by 11.5% and the PS, which is demonstrated theoretically and experimentally, is completely suppressed. The PS positions are depending on the polarization angle of OF, OF levels and the bias current. The PS disappeared with the modulation depth of 78.66%, whereas it is entirely vanished with the modulation frequency of 200 MHz. The hysteresis width of the VCSEL polarization modes is reduced by increasing the feedback level. The minimum RIN value of -156 dB/Hz is achieved at a zero degree of the polarization angle for the dominant polarization mode of VCSEL under VPOF. For the first time, a novel technique based on employing orthogonal polarization OF is proposed to supress the nonlinearity associated with the modulated VCSEL, where the second, third, and fourth harmonics are completely suppressed to the noise floor. Finally, optimal operating conditions for a high-quality polarization-resolved chaos synchronization of the polarization modes of VCSEL with VPOF are experimentally and theoretically studied. A perfect value of 99% of the correlation dynamics for the chaotic synchronization of the polarization modes of VCSEL is found with a zero time delay over a wide range of polarization angle. Finally, Simulink and Origin software version 6.1 are used in this work to simulate and plot the results. The simulation results are agreed with the experimental results, which show that the chaotic synchronization dynamic of the polarization modes can be achieved by VPOF.

Due to substantial generation and demand fluctuations in stand-alone green micro-grids, energy management strategies (EMSs) are becoming essential for the power sharing purpose and regulating the microgrids voltage. The classical EMSs track the maximum power points (MPPs) of wind and PV branches independently and rely on batteries, as slack terminals, to absorb any possible excess energy. However, in order to protect batteries from being overcharged by realizing the constant current-constant voltage (IU) charging regime as well as to consider the wind turbine operational constraints, more flexible multivariable and non-linear strategies, equipped with a power curtailment feature, are necessary to control microgrids. This dissertation work comprises developing an EMS that dynamically optimises the operation of stand-alone dc microgrids, consisting of wind, photovoltaic (PV), and battery branches, and coordinately manage all energy flows in order to achieve four control objectives: i) regulating dc bus voltage level of microgrids; ii) proportional power sharing between generators as a local droop control realization; iii) charging batteries as close to IU regime as possible; and iv) tracking MPPs of wind and PV branches during their normal operations. Non-linear model predictive control (NMPC) strategies are inherently multivariable and handle constraints and delays. In this thesis, the above mentioned EMS is developed as a NMPC strategy to extract the optimal control signals, which are duty cycles of three DC-DC converters and pitch angle of a wind turbine. Due to bimodal operation and discontinuous differential states of batteries, microgrids belong to the class of hybrid dynamical systems of non-Filippov type. This dissertation work involves a mathematical approximation of stand-alone dc microgrids as complementarity systems (CSs) of Filippov type. The proposed model is used to develop NMPC strategies and to simulate microgrids using Modelica. As part of the modelling efforts, this dissertation work also proposes a novel algorithm to identify an accurate equivalent electrical circuit of PV modules using both standard test condition (STC) and nominal operating cell temperature (NOCT) information provided by manufacturers. Moreover, two separate stochastic models are presented for hourly wind speed and solar irradiance levels.

Chaotic systems have properties such as ergodicity, sensitivity to initial conditions/parameter mismatches, mixing property, deterministic dynamics, structure complexity, to mention a few, that map nicely with cryptographic requirements such as confusion, diffusion, deterministic pseudorandomness, algorithm complexity. Furthermore, the possibility of chaotic synchronization, where the master system (transmitter) is driving the slave system (receiver) by its output signal, made it probable for the possible utilization of chaotic systems to implement security in the communication systems. Many methods like chaotic masking, chaotic modulation, inclusion, chaotic shift keying (CSK) had been proposed however, many attack methods later showed them to be insecure. Different modifications of these methods also exist in the literature to improve the security, but almost all suffer from the same drawback. Therefore, the implementation of chaotic systems in security still remains a challenge. In this work, different possibilities on how it might be possible to improve the security of the existing methods are explored. The main problem with the existing methods is that the message imprint could be found in the dynamics of the transmitted signal, therefore by some signal processing or pattern classification techniques, etc, allow the exposition of the hidden message. Therefore, the challenge is to remove any pattern or change in dynamics that the message might bring in the transmitted signal.

Matching networks are widely used to enhance active power transfer when radio frequency generators drive complex loads. The tuning of the network for varying loads typically involves searching for optimum matching conditions. However, improving the matching condition of the network does not necessarily indicate an increase in active power transfer. As an example, a 71 network with three adjustable elements may achieve comparable matches for a variety of elements' settings, each matching triple exhibiting a different transferred active power. Furthermore, the influence of the transmission lines used to connect the matching network to its source and load is rarely taken into account. The purpose of the work is to optimise the power gain of a narrowband matching sys-tem in the frequency range of 1.8 — 30 MHz. The system consists of a source, a match-ing network, a load and two interconnecting lines whose characteristic impedance is complex. The optimisation process involves optimum choice of the transmission lines' lengths and development of a matching strategy. Its objective is to ensure automatic and continuous adjustment of the matching network for optimum ac¬tive power transfer to its load while matching the network's input impedance to a resistive source. The network topologies employed are limited to the most common 71 and T networks consisting of two variable capacitors and one central inductor. Losses are assumed to be mainly caused by the inductor. An appropriate simple and synthetic model of the losses is proposed which is suitable for active power transfer optimisation. The model is validated against losses of inductors derived by different works. After choosing a proper network parametrisation and exact inclusion of the losses during network design, the losses of a network terminated by a resistance and de¬signed to match (exactly) a source resistance at its input are derived. Then its power gain is optimised by a proper choice of the network's parameter and the impact of changing the purely resistive termination to impedances exhibiting capacitive or in¬ductive imaginary parts is considered. An explicit solution is calculated for networks with a constant Q factor central inductor, its differences from the approximate solu¬tion (network elements designed as if the network would be lossless) are considered. Example diagrams are given illustrating those differences and power gain contour Smith charts are drawn for typical ranges of the L, iv, and T networks' elements. Combining the results of the different approaches yields an optimum matching strat¬egy. The losses of transmission lines connecting source and network of load and network are determined, where the lines' complex characteristic impedance is taken into account. Those losses are included in a power gain optimisation of the complete matching system. Finally, an experimental setup is designed under which the matching strategy of the network is tested and validated.

In this thesis a number of open problems in the theory of ordinary differential equations (ODEs) and dynamical systems are considered. The intention being to address current problems in the theory of systems control and synchronisation as well as enhance the understanding of the dynamics of those systems treated herein. More specifically, we address three central problems; the determination of exact analytical solutions of (non)linear (in)homogeneous ODEs of order 1 and 2, the determination of upper/lower bounds on solutions of nonlinear ODEs and finally, the synchronisation of dynamical systems for the purposes of secure communication. With regard to the first of these problems we identify a new solvable class of Riccati equations and show that the solution may be written in closed-form. Following this we show how the Riccati equation solution leads us quite naturally to the identification of a new solvable class of 2nd order linear ODEs, as well as a yet more general class of Riccati equations. In addition, we demonstrate a new alternative method to Lagrange's variation of parameters for the solution of 2nd order linear inhomogeneous ODEs. The advantage of our approach being that a choice of solution methods is offered thereby allowing the solver to pick the simplest option. Furthermore, we solve, by means of variable transforms and identification of the first integral, an example of the Duffing-van der Pol oscillator and an associated ODE that connects the equations of Lienard and Riccati. These fundamental results are subsequently applied to the problem of solving the ODE describing a lengthening pendulum and the matter of bounded controller design for linear time-varying systems. In addressing the second of the above problems we generalise an existing GrOnwall-like integral inequality to yield several new such inequalities. Using one of the new inequalities we show that a certain class of nonlinear ODEs will always have bounded solutions and subsequently demonstrate how one can numerically evaluate the upper limits on the square of the solution of any given ODE in this class. Finally, we apply our results to an academic example and verify our conclusions with numerical simulation. The third and final open problem we consider herein is concerned with the synchronisation of chaotic dynamical systems with the express intention of exploiting that synchronisation for the purposes of secure transmission of information. The particular issue that we concern ourselves with is the matter of limiting the amount of distortion present in the message arriving at the receiver. Since the distortion encountered is primarily a due to the presence of noise and the message itself we meet our ends by employing an observer-based synchronisation technique incorporating a proportional-integral observer. We show how the PI observer used gives us the freedom to reduce message distortion without compromising on synchronisation quality and rate. We verify our results by applying the method to synchronise two parameter-matched Duffing oscillators operating in a chaotic regime. Simulations clearly show the enhanced performance of the proposed method over the more traditional proportional observer-based approach under the same conditions. The structure of thesis is as follows: first of all we describe the motivation behind object of study before going on to give a general introduction to the theory of ODEs and dynamical systems. This lead-in also includes a brief history of the theory of ODEs and dynamical systems, a general overview of the subject (as wholly as is possible without getting into the mathematical detail that is left to the appendices) and concludes with a statement of the scope of the thesis as well as the contributions to knowledge contained herein. We then go on to state and prove our main results and contributions to the solution of those problems detailed above starting with the solution of ODEs.

It is well-known that there are no general approaches for observer and controller design for nonlinear systems. Instead, focus is placed upon design for classes of systems. On the other hand, a wide variety of dynamical systems belong to the class of state-affine systems. Amongst these are biological wastewater treatment processes, which are essential in order to prevent pollution in the environment and prevent disease in the consumption of recycled water. An interesting aspect found in biological wastewater treatment systems, and many typical industrial processes, are time-delays. In almost all systems there are time-delays and nonlinearities and it is not surprising that time-delay and nonlinear systems have received a great deal of attention in mathematics and control engineering. This project introduces new methodologies for the design of controllers and observers for a class of state-affine systems and a class of linear time-delay systems. Firstly, new observable and controllable canonical forms are introduced. These are then used to establish new controller and observer design methodologies for a class of state¬affine systems. In particular, an adaptive observer design is established. The methodologies are simple since they are based upon linear techniques. Secondly, a full-state controller and a separation principle are established for a class of single-input single-output linear time-delay systems. The designs are based on a new stability criterion and are derived from first principles. Finally, the new observer design methodology for the class of state-affine systems is used to produce observers for the estimation of biomass concentration in a biological wastewater treatment bioreactor. The observers are applied in theory and in simulation, where a full and a partial knowledge of the kinetic rate of reaction of biomass are considered. In addition, the performances are shown both in the absence and in the presence of measurement noise for a variety of influent flow characteristics.