Variational ensemble Kalman filtering applied to data assimilation problems in computational fluid dynamics

One challenge on data assimilation (DA) methods is how the error covariance for the model state is computed. Ensemble methods have been proposed for producing error covariance estimates, as error is propagated in time using the non-linear model. Variational methods, on the other hand, use the concepts of control theory, whereby the state estimate is optimized from both the background and the measurements. Numerical optimization schemes are applied which solve the problem of memory storage and huge matrix inversion needed by classical Kalman filter methods. Variational Ensemble Kalman filter (VEnKF), as a method inspired the Variational Kalman Filter (VKF), enjoys the benefits from both ensemble methods and variational methods. It avoids filter inbreeding problems which emerge when the ensemble spread underestimates the true error covariance. In VEnKF this is tackled by resampling the ensemble every time measurements are available. One advantage of VEnKF over VKF is that it needs neither tangent linear code nor adjoint code. In this thesis, VEnKF has been applied to a two-dimensional shallow water model simulating a dam-break experiment. The model is a public code with water height measurements recorded in seven stations along the 21:2 m long 1:4 m wide flume’s mid-line. Because the data were too sparse to assimilate the 30 171 model state vector, we chose to interpolate the data both in time and in space. The results of the assimilation were compared with that of a pure simulation. We have found that the results revealed by the VEnKF were more realistic, without numerical artifacts present in the pure simulation. Creating a wrapper code for a model and DA scheme might be challenging, especially when the two were designed independently or are poorly documented. In this thesis we have presented a non-intrusive approach of coupling the model and a DA scheme. An external program is used to send and receive information between the model and DA procedure using files. The advantage of this method is that the model code changes needed are minimal, only a few lines which facilitate input and output. Apart from being simple to coupling, the approach can be employed even if the two were written in different programming languages, because the communication is not through code. The non-intrusive approach is made to accommodate parallel computing by just telling the control program to wait until all the processes have ended before the DA procedure is invoked. It is worth mentioning the overhead increase caused by the approach, as at every assimilation cycle both the model and the DA procedure have to be initialized. Nonetheless, the method can be an ideal approach for a benchmark platform in testing DA methods. The non-intrusive VEnKF has been applied to a multi-purpose hydrodynamic model COHERENS to assimilate Total Suspended Matter (TSM) in lake Säkylän Pyhäjärvi. The lake has an area of 154 km2 with an average depth of 5:4 m. Turbidity and chlorophyll-a concentrations from MERIS satellite images for 7 days between May 16 and July 6 2009 were available. The effect of the organic matter has been computationally eliminated to obtain TSM data. Because of computational demands from both COHERENS and VEnKF, we have chosen to use 1 km grid resolution. The results of the VEnKF have been compared with the measurements recorded at an automatic station located at the North-Western part of the lake. However, due to TSM data sparsity in both time and space, it could not be well matched. The use of multiple automatic stations with real time data is important to elude the time sparsity problem. With DA, this will help in better understanding the environmental hazard variables for instance. We have found that using a very high ensemble size does not necessarily improve the results, because there is a limit whereby additional ensemble members add very little to the performance. Successful implementation of the non-intrusive VEnKF and the ensemble size limit for performance leads to an emerging area of Reduced Order Modeling (ROM). To save computational resources, running full-blown model in ROM is avoided. When the ROM is applied with the non-intrusive DA approach, it might result in a cheaper algorithm that will relax computation challenges existing in the field of modelling and DA.

Quantum Gate and Quantum State Preparation through Neighboring Optimal Control

Successful implementation of fault-tolerant quantum computation on a system of qubits places severe demands on the hardware used to control the many-qubit state. It is known that an accuracy threshold Pa exists for any quantum gate that is to be used for such a computation to be able to continue for an unlimited number of steps. Specifically, the error probability Pe for such a gate must fall below the accuracy threshold: Pe < Pa. Estimates of Pa vary widely, though Pa ∼ 10−4 has emerged as a challenging target for hardware designers. I present a theoretical framework based on neighboring optimal control that takes as input a good quantum gate and returns a new gate with better performance. I illustrate this approach by applying it to a universal set of quantum gates produced using non-adiabatic rapid passage. Performance improvements are substantial comparing to the original (unimproved) gates, both for ideal and non-ideal controls. Under suitable conditions detailed below, all gate error probabilities fall by 1 to 4 orders of magnitude below the target threshold of 10−4. After applying the neighboring optimal control theory to improve the performance of quantum gates in a universal set, I further apply the general control theory in a two-step procedure for fault-tolerant logical state preparation, and I illustrate this procedure by preparing a logical Bell state fault-tolerantly. The two-step preparation procedure is as follow: Step 1 provides a one-shot procedure using neighboring optimal control theory to prepare a physical qubit state which is a high-fidelity approximation to the Bell state |β01⟩ = 1/√2(|01⟩ + |10⟩). I show that for ideal (non-ideal) control, an approximate |β01⟩ state could be prepared with error probability ϵ ∼ 10−6 (10−5) with one-shot local operations. Step 2 then takes a block of p pairs of physical qubits, each prepared in |β01⟩ state using Step 1, and fault-tolerantly prepares the logical Bell state for the C4 quantum error detection code.

Model predictive control applied to a supply chain management problem

Supply chains are ubiquitous in any commercial delivery systems. The exchange of goods and services, from different supply points to distinct destinations scattered along a given geographical area, requires the management of stocks and vehicles fleets in order to minimize costs while maintaining good quality services. Even if the operating conditions remain constant over a given time horizon, managing a supply chain is a very complex task. Its complexity increases exponentially with both the number of network nodes and the dynamical operational changes. Moreover, the management system must be adaptive in order to easily cope with several disturbances such as machinery and vehicles breakdowns or changes in demand. This work proposes the use of a model predictive control paradigm in order to tackle the above referred issues. The obtained simulation results suggest that this strategy promotes an easy tasks rescheduling in case of disturbances or anticipated changes in operating conditions. © Springer International Publishing Switzerland 2017

Brain emotional learning based control of a SDOF structural system with a MR damper

This paper describes the application of a Brain Emotional Learning (BEL) controller to improve the response of a SDOF structural system under an earthquake excitation using a magnetorheological (MR) damper. The main goal is to study the performance of a BEL based semi-active control system to generate the control signal for a MR damper. The proposed approach consists of a two controllers: a primary controller based on a BEL algorithm that determines the desired damping force from the system response and a secondary controller that modifies the input current to the MR damper to generate a reference damping force. A parametric model of the damper is used to predict the damping force based on the piston motion and also the current input. A Simulink model of the structural system is developed to analyze the effectiveness of the semi-active controller. Finally, the numerical results are presented and discussed.

Optimal control of non-linear systems through hybrid cell-mapping/artificial-neural-networks techniques

In this paper, a real-time optimal control technique for non-linear plants is proposed. The control system makes use of the cell-mapping (CM) techniques, widely used for the global analysis of highly non-linear systems. The CM framework is employed for designing approximate optimal controllers via a control variable discretization. Furthermore, CM-based designs can be improved by the use of supervised feedforward artificial neural networks (ANNs), which have proved to be universal and efficient tools for function approximation, providing also very fast responses. The quantitative nature of the approximate CM solutions fits very well with ANNs characteristics. Here, we propose several control architectures which combine, in a different manner, supervised neural networks and CM control algorithms. On the one hand, different CM control laws computed for various target objectives can be employed for training a neural network, explicitly including the target information in the input vectors. This way, tracking problems, in addition to regulation ones, can be addressed in a fast and unified manner, obtaining smooth, averaged and global feedback control laws. On the other hand, adjoining CM and ANNs are also combined into a hybrid architecture to address problems where accuracy and real-time response are critical. Finally, some optimal control problems are solved with the proposed CM, neural and hybrid techniques, illustrating their good performance.

Feedback control of underactuated cable-driven parallel robots

In this work an Underactuated Cable-Driven Parallel Robot (UACDPR) that operates in the three dimensional Euclidean space is considered. The End-Effector has 6 degrees of freedom and is actuated by 4 cables, therefore from a mechanical point of view the robot is defined underconstrained. However, considering only three controlled pose variables, the degree of redundancy for the control theory can be considered one. The aim of this thesis is to design a feedback controller for a point-to-point motion that satisfies the transient requirements, and is capable of reducing oscillations that derive from the reduced number of constraints. A force control is chosen for the positioning of the End-Effector, and error with respect to the reference is computed through data measure of several sensors (load cells, encoders and inclinometers) such as cable lengths, tension and orientation of the platform. In order to express the relation between pose and cable tension, the inverse model is derived from the kinematic and dynamic model of the parallel robot. The intrinsic non-linear nature of UACDPRs systems introduces an additional level of complexity in the development of the controller, as a result the control law is composed by a partial feedback linearization, and damping injection to reduce orientation instability. The fourth cable allows to satisfy a further tension distribution constraint, ensuring positive tension during all the instants of motion. Then simulations with different initial conditions are presented in order to optimize control parameters, and lastly an experimental validation of the model is carried out, the results are analysed and limits of the presented approach are defined.

Simulação da varredura da superfície por um cortador de base segmentado

Base cutting and feeding into harvesters of plants lying close to the ground surface require an efficient sweeping action of the cutting mechanism. It is not the case of conventional sugarcane harvesters which have rigid blades mounted on discs capable to contaminate the cane with dirt as well as damage the ratoons. The objective of this work was to simulate the sweeping performance of a segmented base cutter. The model was developed using the laws of dynamic. Simulation included two rotational speeds (400 and 600 rpm), two cutting heights (0.12 and 0.13 m) and two disk tilting angles (-10º and -12º). The simulated sweeping angle varied between 56º and 193º, which are very promising as a mean to cutting and feeding cane sticks lying on the ground. Cutting height was the variable that affected sweeping action the most. This behavior indicates the need to have an automatic control of the cutting disk height in order to keep good sweeping performance as the harvester moves forward.

Information Filtering and Array Algorithms for Discrete-Time Markovian Jump Linear Systems

This technical note develops information filter and array algorithms for a linear minimum mean square error estimator of discrete-time Markovian jump linear systems. A numerical example for a two-mode Markovian jump linear system, to show the advantage of using array algorithms to filter this class of systems, is provided.

Robust model predictive controller with output feedback and target tracking

A model predictive controller (MPC) is proposed, which is robustly stable for some classes of model uncertainty and to unknown disturbances. It is considered as the case of open-loop stable systems, where only the inputs and controlled outputs are measured. It is assumed that the controller will work in a scenario where target tracking is also required. Here, it is extended to the nominal infinite horizon MPC with output feedback. The method considers an extended cost function that can be made globally convergent for any finite input horizon considered for the uncertain system. The method is based on the explicit inclusion of cost contracting constraints in the control problem. The controller considers the output feedback case through a non-minimal state-space model that is built using past output measurements and past input increments. The application of the robust output feedback MPC is illustrated through the simulation of a low-order multivariable system.

A Separation Principle for the Continuous-Time LQ-Problem With Markovian Jump Parameters

In this paper, we devise a separation principle for the finite horizon quadratic optimal control problem of continuous-time Markovian jump linear systems driven by a Wiener process and with partial observations. We assume that the output variable and the jump parameters are available to the controller. It is desired to design a dynamic Markovian jump controller such that the closed loop system minimizes the quadratic functional cost of the system over a finite horizon period of time. As in the case with no jumps, we show that an optimal controller can be obtained from two coupled Riccati differential equations, one associated to the optimal control problem when the state variable is available, and the other one associated to the optimal filtering problem. This is a separation principle for the finite horizon quadratic optimal control problem for continuous-time Markovian jump linear systems. For the case in which the matrices are all time-invariant we analyze the asymptotic behavior of the solution of the derived interconnected Riccati differential equations to the solution of the associated set of coupled algebraic Riccati equations as well as the mean square stabilizing property of this limiting solution. When there is only one mode of operation our results coincide with the traditional ones for the LQG control of continuous-time linear systems.

Some remarks on static-feedback linearization for time-varying systems

This work summarizes some results about static state feedback linearization for time-varying systems. Three different necessary and sufficient conditions are stated in this paper. The first condition is the one by [Sluis, W. M. (1993). A necessary condition for dynamic feedback linearization. Systems & Control Letters, 21, 277-283]. The second and the third are the generalizations of known results due respectively to [Aranda-Bricaire, E., Moog, C. H., Pomet, J. B. (1995). A linear algebraic framework for dynamic feedback linearization. IEEE Transactions on Automatic Control, 40, 127-132] and to [Jakubczyk, B., Respondek, W. (1980). On linearization of control systems. Bulletin del` Academie Polonaise des Sciences. Serie des Sciences Mathematiques, 28, 517-522]. The proofs of the second and third conditions are established by showing the equivalence between these three conditions. The results are re-stated in the infinite dimensional geometric approach of [Fliess, M., Levine J., Martin, P., Rouchon, P. (1999). A Lie-Backlund approach to equivalence and flatness of nonlinear systems. IEEE Transactions on Automatic Control, 44(5), 922-937]. (C) 2008 Elsevier Ltd. All rights reserved.

A generalized multi-period mean-variance portfolio optimization with Markov switching parameters

In this paper, we deal with a generalized multi-period mean-variance portfolio selection problem with market parameters Subject to Markov random regime switchings. Problems of this kind have been recently considered in the literature for control over bankruptcy, for cases in which there are no jumps in market parameters (see [Zhu, S. S., Li, D., & Wang, S. Y. (2004). Risk control over bankruptcy in dynamic portfolio selection: A generalized mean variance formulation. IEEE Transactions on Automatic Control, 49, 447-457]). We present necessary and Sufficient conditions for obtaining an optimal control policy for this Markovian generalized multi-period meal-variance problem, based on a set of interconnected Riccati difference equations, and oil a set of other recursive equations. Some closed formulas are also derived for two special cases, extending some previous results in the literature. We apply the results to a numerical example with real data for Fisk control over bankruptcy Ill a dynamic portfolio selection problem with Markov jumps selection problem. (C) 2008 Elsevier Ltd. All rights reserved.

Asymptotic limits of a self-dual Ginzburg-Landau functional in bounded domains

In the author's joint paper [HJS] with Jest and Struwe, we discuss asymtotic limits of a self-dual Ginzburg-Landau functional involving a section of a line bundle over a closed Riemann surface and a connection on this bundle. In this paper, the author generalizes the above results [HJS] to the case of bounded domains.