Model reduction and control of complex systems

Dit proefschrift behandelt een aantal vraagstukken gerelateerd aan modelreductie en regeling van twee soorten complexe systemen: `switched linear systems’ en netwerken van dynamische agenten. Eerst geven we een uitgebreide, gebalanceerde afkappingsmethode voor modelreductie van `switched linear systems’ (SLS). Vervolgens introduceren we een methode om de dynamische orde van agenten in een netwerk te reduceren, waarbij stabiliteit of synchronisatie behouden blijft in het gereduceerde orde model. Als vierde probleem analyseren we de stabiliteit en synchronisatie van netwerken waarbij de agenten een algemene, maar identieke, lineaire dynamica hebben en de onderliggende communicatietopologie willekeurig mag schakelen tussen een eindig aantal toegestane topologieën. De sterk structurele regelbaarheid van systemen die op een graaf gedefinieerd zijn, is het vijfde onderwerp van dit proefschrift. In het bijzonder tonen we aan dat er een een-op-eenrelatie is tussen de verzameling van leiders die ervoor zorgen dat het netwerk regelbaar is en de zogenaamde `zero-forcing’ verzamelingen. Tenslotte bestuderen we het probleem van storingsontkoppeling voor netwerken van dynamische agenten vanuit een graaftopologisch perspectief. We geven in termen van graphpartities voorwaarden voor de oplosbaarheid van het storingsontkoppelingsprobleem

Application of multilevel control techniques to classes of distributed parameter plants

This study concerns the application of a combination of multilevel hierarchical systems analysis techniques and Pontryagin\u27s minimum principle (multilevel control) to the problem of controlling optimally two classes of dynamic distributed parameter plants representing concentrations balances in streams, rivers and estuaries. The concentrations treated in this study are those deemed the most effective indicators of water quality, dissolved oxygen (DO) and biochemical oxygen demand (BOD). One class of plants treated in this study consists of linear continuous distributed parameter plants represented mathematically by sets of simultaneous partial differential equations. Optimal control of a plant of this class is initiated by applying spatial discretization followed by a combination of multilevel techniques and Pontryagin\u27s minimum principle for lumped parameter systems. This approach reduces the original problem of optimally controlling a distributed parameter plant to a hierarchy of subproblems comprised of ordinary differential and algebraic equations that can be solved iteratively. A general two-dimensional plant representative of a class of two-step discrete dynamic distributed parameter plants is derived from mass balances at the faces of a model of a volume element of a waterway. The resulting set of simultaneous finite-difference equations represents dynamic balances of concentrations at a finite number of spatial points in a reach of a waterway at selected time instants. Application of Pontryagin\u27s minimum principle for discrete systems in conjunction with multilevel hierarchical systems analysis techniques reduces the problem of controlling such a plant optimally to a hierarchy of subproblems to be solved iteratively. Implicit in the application of optimal control to a plant is the selection of a suitable performance index functional with which to measure the relative optimality of each solution iteration. A variety of performance indices based upon physical considerations is utilized in conjunction with several different control modes for a number of plants representative of the two classes treated in this study. Subproblem hierarchies corresponding to both continuous and discrete distributed parameter plants representing concentrations balances in waterway reaches subject to multilevel optimal control are aggregated into super hierarchies. These super hierarchies possess at least one more level than those corresponding to the single reaches and represent, in this context, the concentrations balances in multireach or regional portions of waterways. Sufficient boundary, initial and final conditions are presented for numerical solution of the subproblem hierarchies developed in this study. Flow charts for the corresponding digital computer programs also are depicted. A proof of consistency between the ordinary differential equations of the spatially discretized plant and the partial differential equations of the continuous distributed parameter plant that it approximates is developed for a representative plant. A proof of convergence of the solutions of the equations of the same spatially discretized plant also is developed. Stability analyses are conducted for representative continuous and discrete distributed parameter plants. The optimal control of the spatially discretized continuous distributed parameter plant is formulated as a linear regulator problem and the associated performance index is utilized as a Liapunov function. The optimal control of the discrete distributed parameter plant with time-varying mean volume flow rate is formulated as the problem of optimal control of a nonstationary system which is treated by transforming the nonstationary system to an equivalent stationary system. The z-transform is applied to the finite-difference equations of the plant to facilitate evaluation of the effect of the presence of transport lags. The relationship between structural characteristics and computational efficiency of subproblem hierarchies is analyzed. Multilevel hierarchical systems analysis techniques are applied to the sensitivity analysis of a spatially discretized distributed parameter plant subject to multilevel optimal control. The combination of discretization and multilevel techniques is shown to reduce the generation of trajectory sensitivity coefficients for an optimally controlled distributed parameter plant to generation of trajectory sensitivity coefficients for a series of lumped parameter plants under optimal control. A normalized performance index sensitivity function also is developed for the same plant. Numerical results of multilevel optimization are presented for various control modes and configurations applied to plants representing: single reaches of a tidal river, four contiguous reaches of a tidal river, six contiguous reaches of a tidal river with taper and waste dischargers, and single reaches of an estuary. The study culminates with the application of one of the single reach subproblem hierarchies for a discrete distributed parameter plant under multilevel optimal control and multilevel hierarchical systems analysis techniques to the problem of minimizing total treatment cost for a multireach portion of a tidal river. This demonstrates the feasibility and efficiency of the multilevel approach to the solution of dynamic systems optimization problems of regional scope

Non-linear model predictive energy management strategies for stand-alone DC microgrids

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

3D printed components for quantum devices

Since the first demonstrations of laser cooling of atomic vapors in the late 1970s, the field of ultracold atoms has seen rapid advancements in the preparation, control and measurement of atomic gases. Ultra-cold atomic systems, either as individual atoms or in larger ensembles, provide a powerful tool for experimenters. Their large deBroglie wavelengths make them particularly useful for interferometric applications, and their isotropic properties when unperturbed make them ideal candidates for frequency standards. A recent drive has seen experimenters looking to develop scalable, portable and robust atomic systems as a metrological tool outside of the typical laboratory environment. This could see unprecedented sensitivities made available for areas as diverse as GPS-free navigation, biomedical imaging and non-invasive underground mapping. Over the course of this thesis we explore additive manufacturing (3D printing) as a production technique for quantum technology. 3D-printing offers unrivalled design freedom and rapid prototyping, allowing us to develop a number of printed structures to test the viability of selective laser melting as a technique to produce metallic components that survive within, and also hold, the ultra-high vacuum environment necessary for ultracold physics. The technique has the potential to improve the efficiency and compactness of devices. We begin first by printing an Al-Si-Mg vacuum flange, which is then solution heat treated in post processing and milled to have a standard vacuum-sealing knife edge on its surface. By installing the flange on a test vacuum set-up, baking out over a week at 200­­­°C and pumping down, a pressure of 10⁻¹¹ mbar is achieved. In the same material, a conductive structure called the cylinder trap is printed as a proof of concept ultracold atom source producing the fields necessary for a magneto-optical trap. A complete cold atom experiment is constructed to test the device, including a simple microcontroller-based control system. Dissipating as little as 20mW electrical power, the atom trap generates 10⁸ atoms with an average temperature on the order of (20.1 ± 0.2)μK, whilst having no measurable effect on the vacuum pressure, measured as < 10⁻¹⁰ mbar. A next-generation device is then investigated, building on the work of the cylinder trap and consideration of contemporary work on cold-atom sources. This device would output an even colder source of atoms, with a tapered design to both act as a differential pump and for atom compression for transport to a secondary trap. With calculations on optimal trapping regimes, an Ioffe-Pritchard style magnetic trap layout is created to efficiently capture atoms from the magnetooptical trap. Atoms would then be transported through a three-dimensional funnel structure into a secondary magnetic trap where fast, evaporative cooling could occur. Simultaneously the next thermal cloud can be captured to improve the average cycle time. Encouraged by collaborative work on a additively manufactured chamber, called the coral trap, a prototype design is developed and presented consisting of the funnel structure split across a multi-chamber printed architecture

Production and Oscillations of a Bose Einstein Condensate on an Atom Chip

This thesis describes production of and experiments with a Bose-Einstein condensate of approximately 2 × 10[superscript 4] [superscript 87]Rb atoms, trapped at the surface of an atom chip. In the first half of this thesis I describe the process of trapping and cooling the atomic vapour close to the surface of an atom chip. This process, which cools the vapour by over 9 orders of magnitude, involves a highly complex sequence of events which I implemented and optimised over the first two years of my PhD. In the early stages of this process, the atomic vapour is laser cooled and magneto-optically trapped. The vapour is then transferred to a highly elongated magnetic trap produced by high field gradients a few hundred microns from the surface of the atom chip. Here the vapour is evaporatively cooled to below the transition temperature where a Bose-Einstein condensate emerges. A simple existing analytic model of evaporative cooling is extended in this work to account for the shape of our highly elongated trap. Predictions of this model are presented here along with experimental observations with which it has good agreement. The second part of my thesis investigates some of the characteristics of the condensate, and dynamics of its low energy collective oscillations in the trap, based on experimental measurements taken in the final 18 months of my PhD. In particular, measurements taken of the centre of mass oscillations of the condensate along the long axis of the trap are presented. In the zero temperature limit the condensate is expected to behave as a perfect superfluid, and these low energy oscillations should go undamped. However, at finite temperature where not all atoms in the gas are condensed, damping is observed. In our experiment significant damping is found with an 1/e decay rate which varies between 2s[superscript -1] and 8s[superscript -1], depending on the fraction of non-condensed atoms in the gas. A finite temperature formalism is then used to describe the likely damping mechanism - Landau damping. We use a simple model of this formalism which estimates the temperature dependence of the damping rate γ(T), but find this gives a significant overestimation of the rates we measure. However, we argue that a straightforward adaptation to this model reduces the predicted damping rate significantly, and suggests a functional form of γ(T) that is in much better agreement with our experimental measurements.Open Acces

Electromagnetic fast-transients in LV networks with ubiquitous small-scale embedded generation

Small-scale embedded generation projects rated below 16A per phase are being integrated into low-voltage distribution networks in ever increasing numbers. Seen from the network operator's perspective as little more than negative load, the commissioning of such generators is subject to compliance with the Fit and Forget connection requirements of ENA Engineering Recommendation G83/1. This thesis has sought to quantify the electromagnetic switching transient implications of integrating very large volumes of embedded generation into the UK's low-voltage supply networks. Laboratory testing of a converter-interfaced PV source has been undertaken to characterise typical switching transient waveshapes, and equivalent representative source models have been constructed in EMTP-ATP. A detailed frequency-dependent travelling wave equivalent of the DNO-approved Generic UK LV Distribution network model has been developed and, by means of extensive statistical simulation studies, used to quantify the cumulative impact of geographically localised generators switching in response to common network conditions. It is found that the magnitude of generator-induced voltage and current transients is dependent on the number of concurrently switched generators, and on their relative locations within the network. A theoretical maximum overvoltage of 1.72pu is predicted at customer nodes remote from the LV transformer terminals, for a scenario in which all households have installed embedded generation. Latent diversity in switch pole closing and inrush inception times is found to reduce predicted peak transient voltages to around 25-40% of their theoretical maxima.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Stochastic Real-time Optimal Control: A Pseudospectral Approach for Bearing-Only Trajectory Optimization

A method is presented to couple and solve the optimal control and the optimal estimation problems simultaneously, allowing systems with bearing-only sensors to maneuver to obtain observability for relative navigation without unnecessarily detracting from a primary mission. A fundamentally new approach to trajectory optimization and the dual control problem is developed, constraining polynomial approximations of the Fisher Information Matrix to provide an information gradient and allow prescription of the level of future estimation certainty required for mission accomplishment. Disturbances, modeling deficiencies, and corrupted measurements are addressed with recursive updating of the target estimate with an Unscented Kalman Filter and the optimal path with Radau pseudospectral collocation methods and sequential quadratic programming. The basic real-time optimal control (RTOC) structure is investigated, specifically addressing limitations of current techniques in this area that lose error integration. The resulting guidance method can be applied to any bearing-only system, such as submarines using passive sonar, anti-radiation missiles, or small UAVs seeking to land on power lines for energy harvesting. Methods and tools required for implementation are developed, including variable calculation timing and tip-tail blending for potential discontinuities. Validation is accomplished with simulation and flight test, autonomously landing a quadrotor helicopter on a wire