The global internet protocol (IP) traffic is rising exponentially due to the increased number of bandwidth-intensive services like video-on-demand and cloud computing. To meet the demands of growing traffic, the data rates of fiber optic communication systems (FOCSs) need to be increased. In this regard, digital signal processing (DSP), which already plays a powerful role in the modern FOCSs, is being explored. Increasing the net data rate of FOCSs requires compensation for nonlinear impairments that can arise from the Kerr effect during the propagation of signal through fiber as well as from the non-ideal responses of transceiver hardware components. Furthermore, cutbacks in net data rates due to training overheads, i.e., the non-information carrying part of transmitted data consumed by DSP algorithms; need to be reduced. In this dissertation, we propose novel DSP approaches to address these problems of great practical interest.

The Kerr nonlinear effects add phase shifts to the signal, which are dependent on its instantaneous power. These phase distortions occur simultaneously with the dispersion effect of the fiber, which spreads signal pulses in time. The interplay is complicated and makes compensation of distortions challenging. The nonlinear Fourier transform(NFT), which offers immunity from the distortions of the Kerr effect, received great interest in recent years. The lossless nonlinear Schrödinger equation (NLSE), which models signal propagation in an ideal lossless optical fiber, belongs to a class of nonlinear partial differential equations known as integrable equations. These integrable equations can be solved exactly by NFT. Similar to the Fourier transform that translates a linear dispersive propagation in the time domain into phase delays in the signal spectrum, the NFT translates the nonlinear evolution of the signal governed by the lossless NLSE into trivial multiplications in the nonlinear Fourier spectrum of the signal. The NFT is exact for lossless fiber channels. In the presence of loss, the integrability property is violated. In lossy propagation, signal power reduces as it propagates. This in turn reduces the strength of the nonlinear effects along the length of the fiber. As practical fibers are lossy, the path-average approximation is often used to apply NFT on lossy fiber channels. In this approximation, the variation in the Kerr nonlinear effects due to the reduction in signal power is accounted as the variations in the Kerr-nonlinearity parameter of the fiber. Then, by approximating the varying Kerr-nonlinearity parameter with its average value over a span, a lossless fiber model is obtained. This approximation has errors associated with it which sacrifices the performance. We developed a NFT-based transmission system that is exact even in the presence of fiber loss. The proposed design eliminates errors due to loss, thus improving performance over the design that uses path-average approximation…@en

Nonlinear partial differential equations (PDEs) are generally hard to solve, but over the past 65 years the notions of Lax integrability and \textit{nonlinear Fourier transforms} (NFTs) have been developed to solve a large class of so called \textit{Lax-integrable} partial differential equations.
If we are able to find a suitable pair of linear operators, consisting of a spectral operator and a propagation operator, a so-called \textit{Lax pair}, we are able to analytically solve the nonlinear PDE.
If this pair of linear operators is a correct Lax pair for the PDE, then the spectral operator can be used to define an NFT, which transforms the signal from the physical domain to a PDE-specific spectral domain, in which all spectral components are independent, similar to the regular Fourier transform for linear problems.
In the spectral domain, the transformed signal can be propagated using the linear propagation operator, after which the propagated signal can be transformed back into the physical domain, hence solving the nonlinear PDE.

The presence of a Lax pair can thus be used to solve a nonlinear PDE, but it also yields insight into the underlying dynamics of the system. Therefore, the identification of a Lax pair for a system is of great value. However, there is no general method to find a Lax pair given an evolution equation. Systems may also be non-integrable, and therefore a corresponding Lax pair may not even exist. To the best of our knowledge, no practical general methods are known in the literature to determine whether or not a system is Lax-integrable.
Since the general problem is very complex, one may focus on more specific cases for practical purposes, in which a Lax-integrable PDE is sought using measurement data.

In this thesis, we therefore focus on data-driven identification of Lax-integrable partial differential equations (PDEs), specifically those of the AKNS-type.
The primary objective is to find a Lax-integrable PDE that best explains given experimental measurement data, enabling a comprehensive analysis using the nonlinear Fourier transform (NFT).
While real-world systems may not be exactly be Lax-integrable due to imperfections, many are well approximated by such equations, including scenarios such as fiber optical wave propagation, surface wave propagation in shallow water canals, and mechanical wave propagation in coupled pendulums.@en

In this work we investigate two boundary-value control algorithms for the heat equation on the finite interval. The algorithms we discuss here are based on the unified transform method (UTM), a method invented by A. S. Fokas to solve boundary value problems of partial differential equations. We first inspect the boundary-value control algorithm presented in Kalimeris et al.. This algorithm is originally constructed to find the Neumann boundary value on the right side for nullcontrol of the heat equation. In this work the algorithm has been expanded to allow arbitrary control objective, with either the Dirichlet or Neumann boundary values from both sides. Other improvements have also been made.
Furthermore, a second algorithm is constructed, which aims for a lower computational cost than the first algorithm. This second algorithm uses the same principles as the algorithm used in [10], but stems from another part of the derivation of the UTM. Both algorithms were tested with various control objectives, and show promising results.

Improving the energy efficiency of electric vehicles has various significant benefits, such as increasing the driving range. The Thermal Management System (TMS) plays a large role in optimizing the vehicle energy consumption and battery lifetime. In this thesis, a nonlinear Model Predictive control (MPC) strategy is presented to regulate the battery, motor and inverter temperatures to minimize vehicle energy consumption and maximize battery lifetime, two conflicting objectives traded off by a single tunable parameter, whilst staying
within temperature limits to ensure safety. The control dynamics are nonlinear and discontinuous, due to valves in the system that can only be fully opened or fully closed, leading to a nonlinear mixed-integer optimization problem. A nonlinear model of the TMS including actuators and electric components is formulated that is validated by using simulation results over three drive cycles for moderate and hot ambient conditions. Mean temperature deviations between -0.22 and 0.25 °C are achieved for the battery, and between 0.14 and 1.48 °C for the motors, depending on the drive cycle. Using outer convexification, the optimization problem is reformulated as a continuous problem, which can be solved efficiently. The control strategy is tested in a simulation environment for both moderate and hot ambient temperatures and three different drive cycles. The strategy is compared to a benchmark strategy that uses a finite-state machine and PID-based control loops. A decrease in power consumption between 7% and 11% is achieved for 5 out of 6 use cases whilst additionally decreasing the ageing rate by 0% to 7%. For one use case, an increase in
energy consumption is achieved by 2.5%, but the relative ageing rate is decreased by 44%. At hot temperatures, improvements are mostly achieved due to finding an energy-efficient battery cooling trajectory. At moderate temperatures, improvements are mostly achieved by increased motor cooling to take advantage of the temperature-dependent motor losses. The results obtained using a continuous solver are also compared to those obtained using a
mixed-integer solver. Minimal loss in performance is seen compared to the mixed-integer solver, whilst requiring a significantly lower computation time that is within the sample time, making the MPC strategy fast enough for real-time control in the simulation environment. Finally, the effect of using noisy forecast information is used to test the robustness of the controller. Using noisy forecast information instead of perfect forecast information, the energy consumption and ageing rate increase by 0.0% to 1.2%.

The nonlinear Fourier transform for the focusing periodic nonlinear Schrodinger equation is investigated. This paper is focused on the approximation of the spines in the nonlinear spectrum using results from Floquet theory. Algorithms for the numerical computation of the spines based on the Fourier collocation method are being examined and a new algorithm is presented. The new algorithm developed during the project computes the spines by tracking sign changes of the function ς=(Δ(.)) in the area ℜ<( Δ (.))| < 1, where delta is the Floquet discriminant. The new algorithm is successfully applied to examples where both the modified Fourier collocation method and the method implemented in the FNFT software library fail. In addition, the spine points that are numerically computed by the new algorithm are equally distributed along the curve, while using the other algorithms the computed points are clustered around the periodic eigenvalues. Finally, the algorithm provides information on which spectrum points belong to the same spine. The pseudocode and the MATLAB source code of the algorithm developed are provided.

Due to the continuously increasing volume of road freight transportation, there is a need to aid or automate truck-trailer driving. Truck docking, the process of parking a truck-trailer combination at a loading dock, is one of the most difficult manoeuvres for professional drivers. In this work, we propose a Model Predictive Control (MPC)-based truck docking driver assistance system. The objective of the system is to support the truck driver while parking the vehicle by means of visual instructions. MPC is an advanced control method that can be used to control Multiple-Input and Multiple-Output (MIMO) systems based on a finite horizon optimization. The control structure allows one to formulate a multi-objective control strategy with explicit constraints. This work has been conducted within the scope of the VIsion Supported Truck docking Assistant (VISTA) project and the practical application of the proposed system is to experiment with an MPC-based approach for the VISTA system currently in development. To determine the optimal control action, the proposed MPC-based driver assistant makes use of a kinematic model describing the motion of the vehicle as well as a second-order linear time-invariant model representing the driver’s behaviour. The system is examined by testing four performance categories: path tracking error, robustness, computational speed, and driver acceptance. The performance of the system is tested with professional truck drivers and people without a truck driver’s licence in a Virtual Reality (VR) simulation. The results suggest that the proposed MPC-based driver assistant is able to perform well regarding reference path tracking and is robust against driver errors, with satisfactory computational speed. The feedback and comments from the professional drivers during testing also indicate definite possible advantages of using the system. As of now, the MPC-based solution is preferred over previous concepts of the VISTA system.

In order to improve the autonomy of construction robots, a Simultaneous Localization and Mapping (SLAM) solution is needed that localizes the robot using solely on-board sensing with sub-5 mm accuracy. As pointed out by the results of the Hilti SLAM Challenge, the state-of-the-art in SLAM currently does not offer a solution that comes close to this requirement. The winning algorithm FAST-LIO2 achieved an accuracy typically around 4-10 cm. When considering the literature, it can be noted that the state-of-the-art SLAM algorithms are designed with generality in mind, and limited attention has been paid to SLAM that focuses on increasing accuracy by targeting a specific scene structure.

In order to improve SLAM accuracy, this thesis proposes a novel 2D Lidar SLAM algorithm focused on indoor environments, hereafter called Ray-SLAM. The algorithm is centered around the assumption that walls are available, which is valid in many cases on the building construction site. Ray-SLAM introduces several novelties which are (1) a sparse map representation to facilitate a joint pose-map optimization scheme, (2) observation-to-map alignment using a non-iterative procedure (contrary to the Iterative Closest Point algorithm) and (3) a stop-and-go strategy to prevent Lidar motion distortion to corrupt the map.

The algorithm was extensively tested using six real-world indoor datasets recorded with the Ouster OS0 Lidar in rooms with various shapes and sizes and with a total trajectory length of 307 m. The motion was constrained to the horizontal plane and a stop-and-go motion pattern was applied. Ground-truth was recorded at static points to evaluate the accuracy. In the majority of the test cases, Ray-SLAM was able to estimate the trajectory successfully. Failure cases led back to either a lack of unoccluded walls in the scene or violated model assumptions about these walls (e.g. incorrect referencing to clutter close to the wall or to a door being opened).

Ray-SLAM's accuracy was compared with two 3D Lidar SLAM algorithms, LOAM and FAST-LIO2 respectively. The compared algorithms solved a harder 6DOF estimation problem, but had access to 64 Lidar scan rings instead of one and (optionally) the IMU. An overall Mean Absolute Error of 5.7 mm is reported by Ray-SLAM in the successful cases, which is 5.0x more accurate than LOAM and 2.4x more accurate than FAST-LIO2. Further research is suggested to improve the robustness of Ray-SLAM and extend it to a full 3D SLAM algorithm.

The ability to compute models that correctly predict the trajectories of a nonlinear system can become a significant challenge in systems and control. The introduction of Koopman operator theory helped to deal with this challenge. The Koopman operator is a composition operator that globally describes a nonlinear system in an infinite-dimensional linear framework. To implement this theory, the usual approach is to approximate the Koopman operator through data-driven methods. These algorithms use measurements of the nonlinear system to compute the approximated operator. Generally, noise can be present in real-world scenarios. Noisy measurements can have a considerable deteriorating effect on the data-driven approximation of Koopman operators. The approximation of this operator in presence of noisy training data is a necessary step for its implementation to a wider spectrum of real-world applications. Many robust numerical methods were designed to solve this issue. Koopman subspace identification (KSI) is a promising approach. As the name suggests, this algorithm employs subspace identification modeling to compute the matrix approximation of the Koopman operator. In this work, we test KSI against other state-of-the-art techniques. Additionally, we improve its performance in predicting the state trajectories of the nonlinear system in presence of noisy measurements. To this end, we propose a reducing-order routine that computes the most robust model against measurement noise. Furthermore, a randomized singular value decomposition is adopted to reduce computational times. The improved KSI is then compared against the other state-of-the-art algorithms in the presence of noisy data sets. We will show that the upgraded KSI outperforms most of the other techniques.

With inland transportation increasing every passing day, vehicle platooning offers a good solution towards travelling more efficiently. Along with reducing traffic congestion on roads, platooning also leads to better fuel consumption among vehicles, fewer accidents, and most importantly, vehicle platoons can be made autonomous using optimization-based control techniques, such as Model Predictive Control (MPC). Much research has been done towards optimise fuel consumption through slip-streaming and by using topological data. A separate research field also looks into performing lanes changes and avoiding obstacles. However, both these research fields are disjointed and use a different model and MPC formulations to employ control. This project aims at developing an unified system that can implement longitudinal control (fuel optimisation), formation reconfiguration (lane changes) and collision avoidance using one model and MPC formulation. The platoon switches between these behaviours depending on the environment around the platoon. This report also highlights the limitations of the controller and gives concrete recommendations on how to deal with the shortcomings.

Optical fibers form the backbone of our global data transmission infrastructure. As demands on global data transmission grow the capacity of these systems needs to be increased. The behaviour of light waves through these optical fibers is described by the Manakov Equation (ME), a system of nonlinear partial differential equations.

The ME is an integrable system, which can be solved analytically using Nonlinear Fourier Transforms. Recently, fiber-optic communication systems based on the Nonlinear Fourier Transform (NFT) of the ME have been proposed. Similar to the linear Fourier Transform, which decomposes a signal in linear frequency components, the NFT decomposes a signal in nonlinear frequency components. This nonlinear spectrum consists of a continuous and a discrete part. The continuous spectrum in general constitutes the whole real line. The discrete spectrum consists of distinct points in the complex plane which correspond to so-called solitons, which are stable wave forms. The evolution of the nonlinear spectrum along the fiber is trivial.

The nonlinear spectrum however cannot be computed analytically for most signals and therefore numerical methods are needed. The existing numerical methods have a high computational complexity of O(D^2) for computing the continuous spectrum, with D the number of time samples of the signal. For the Nonlinear Schrödinger Equation (NSE), a simplification of the ME, more efficient numerical methods exist with a computational complexity of O(D log^2(D)). In this thesis we present an extension of these so-called fast NFT methods to the ME. The resulting algorithms are second and fourth-order algorithms based on second and fourth order exponential integration methods respectively.

We developed open source software implementing the fast NFT algorithms for the ME and integrated them in the already existing Fast Nonlinear Fourier Transform (FNFT) software library. We provide detailed documentation and examples which allow other researchers to use the algorithms as tools or as a base for developing new algorithms. We furthermore test the accuracy of the developed algorithms against analytic examples. Of these examples, the rectangle signal and secant hyperbolic signal are new analytic examples for the ME to the best of our knowledge.

With data-driven control it is possible to design a controller for systems with non-parametric models. The intermediate step of modelling or identification of the system is not necessary, because a non-parametric model of the system can be obtained by means of experimental data. In the linear time-invariant (LTI) framework, these non-parametric models are well defined in the frequency-domain, they are represented as frequency response functions (FRFs). Data-driven control in the frequency-domain has gained momentum in the past decades, but still relatively few methods have been developed for multiple-input-multiple-output (MIMO) systems. One of the objectives of this thesis is to develop a controller synthesis method for MIMO systems. While the LTI framework has many advantages, such as the vast available literature and the relatively simple theory, it has its limitations. In reality, all systems have nonlinearities and some systems, especially position dependent systems (which are common in mechatronics), are less suitable to be modelled as LTI systems. The nonlinear dynamics are most likely interpreted as an uncertainty for which a robust controller has to be designed at the cost of performance. By taking into account the scheduling variable, in this case the position dependency, it is possible to improve the performance. To do so, the system is modelled as a linear parameter-varying (LPV) system for which an LPV controller is designed that takes the scheduling variable into account. This thesis is concerned with developing a controller synthesis approach in the LPV framework using non-parametric models of MIMO systems using the local approach. This implies that an LPV controller is designed for the nonlinear system at the operating points of interest, at these points the system is assumed to exhibit LTI behaviour. What follows is an interpolation of the multiple LTI controllers to obtain a global parametrisation of the controller in the entire operating range, such that local stability and performance is guaranteed at every operating point. Two novel data-driven LPV controller synthesis methods, based on LTI methods, are presented in this thesis. These methods improve the performance of a parameter dependent system compared to an LTI approach. The improvements are shown through simulations with nonlinear systems. From these simulations it can be concluded that the LPV controllers improve the performance compared to a similar LTI controller. Furthermore, an example is shown where an LPV controller is crucial to guarantee closed-loop stability of the nonlinear system.

As a nonlinear alternative to the linear interpretation of arterial blood pressure waveform, soliton theory has been proposed to model arterial blood pressure by interpreting the pulsatile nature of pressure pulses in the viewpoint of soliton transmission. The existing solitary wave literature supports this interpretation by deriving Korteweg-de Vries (KdV) type dynamics from 1-D Navier-Stokes equations. In this paper, we explain and discuss the derivation of KdV type dynamics for arterial blood pressure from basics of fluid motion. As original work, we provide two verification tests for two of the existing KdV models in three case studies which are considered to be interconnected sections of a simplified arterial network. Finally,
using both KdV models and considering realistic inlet boundary conditions, we study arterial blood pressure waveforms using nonlinear Fourier analysis to extract physical information.

The non-linear Fourier transform may be considered an extension of Fourier analysis to non-linear problems. It has applications in many different scientific and engineering disciplines, for instance as a solution to the non-linear Schrödinger equation which describes how electro-magnetic waves travel through ideal non-linear media. Hence, the non-linear Fourier transform is fundamental for communication schemes that take optical fiber non-linearities into account.
Wahls et al. proposed an algorithm for a fast inverse non-linear Fourier transform based around a discretization of the non-linear Schrödinger equation. But one of the steps in the algorithm — known as synthesis — only had heuristic solutions. However, the problem closely resembles a Nevanlinna-Pick interpolation with degree constraint, a problem known to have a guaranteed solution. Indeed, synthesis is a limit case of Nevanlinna-Pick interpolation with degree constraint on the boundary of the region of analyticity. Although there are solvers to the boundary Nevanlinna-Pick interpolation with degree constraint available in the literature, they do not scale to the numerical difficulty or size required, and are based around simplifying assumptions that do not hold for synthesis.
In this thesis I propose two different numerical continuation solvers for synthesis. The methods have lower computational complexity than other guaranteed solvers by using fast linear Fourier transforms to perform polynomial operations. Moreover, the convergence speed of the algorithms is improved by approaching the solution in a novel trajectory. However, the solvers only offer a quicker convergence rate than the heuristic methods for very difficult problems, and may not be effective for real-time applications, as the high-demands in error precision required means a very high computational cost. This seems to be a limitation of the theory, as the high computational cost is a consequence of the large number of iterations required even when the cost of each iteration is low.

The relation between the input of an ideal optical single-mode fiber and the corresponding fiber output constitutes a nonlinear system that can be described using the nonlinear Schrödinger equation. This nonlinear system has the interesting property that it can be solved analytically using nonlinear Fourier transforms. To utilize this property, new methods of optical communication are being developed by embedding information in the nonlinear Fourier domain and employing fast nonlinear Fourier transforms. Many of the recent works use a specialized form of inverse nonlinear Fourier transform to generate information-bearing fiber inputs in the form of so-called multi-soliton pulses. Recently, multiple fast inverse nonlinear Fourier transform algorithms that can generate multi-solitons have been proposed. The goal of this thesis is to study and improve these algorithms, in particular, with respect to their computational complexity.
Based on the literature survey, discrete Darboux transform combined with other discrete techniques is studied and a new algorithm is proposed. The algorithm employs a single-start approach in which discrete Darboux matrix is computed at only one sample point and rest of the samples are computed by evolution of the Darboux matrix. The algorithm is hence named as discrete Darboux evolution algorithm (DDE). The errors in the generated signal and run-time are studied by comparison with the classical Darboux transform (CDT). The DDE algorithm is shown to have floating point operations complexity of O(KN) for K eigenvalues and N samples. However, in a limited precision environment the number of samples that can be generated is found to be limited. To better understand the effects of machine precision, both the CDT and DDE algorithms are studied in a multi-precision environment. Certain insights from the study are used to develop two modifications to overcome the limitations. The first modification computes the signal using multiple single-start runs while the second one uses a multi-start approach. The second modification is shown to have errors comparable with other fast algorithms in literature. Additionally, in a qualitative comparison it is shown to be potentially faster than existing algorithms in a certain regime.

We study the important mathematical problem of approximating the inverse of low Kronecker-rank matrices in this same form. A traditional alternating least squares (ALS) scheme for solving such problems is presented, and we discuss two efficient solutions to the subproblems arising in the corresponding iterations. The first relies on a least-squares formulation while the second on a gradient-based solution from the literature. The former new approach is slightly less efficient but more robust as it does not involve forming the normal equations. We also advocate for employing a Nesterov-type acceleration in the higher level \ac{ALS} scheme. Usage of the resulting algorithm is evaluated in the context of approximating inverses in low Kronecker-rank form in order to preserve this structure for its continued exploitation in applications such as the matrix sign iterations and preconditioning linear systems.

Our theoretical study is motivated by two real-life practical applications in the field of adaptive optics (AO). We address each of these in terms of exploiting the Kronecker products featured within their problem formulations.

A minimum variance control scheme of wavefront control for atmospheric turbulence correction leads to a large-scale Kronecker structured constrained least-squares problem whose efficient solution is crucial for real-time implementation. Potential approaches based on the alternating direction method of multipliers (ADMM), projected alternating Barzilai-Borwein (PABB), and active sets (AS) methods are analyzed and compared in the context of exploiting the Kronecker structure of the system matrices using data from a partially realistic numerical study. The PABB approach is shown to be the most competitive, also with respective to alternative solutions exploiting only the sparsity of the system matrices instead of their Kronecker form as well.

The optimal design of a novel wavefront sensor called a sparse aperture mask (SAM) is also addressed in our work. Here Kronecker products appear when propagating wavefronts using the matrix-form of evaluating two dimensional Fourier transforms. The design problem is formulated in terms of a trade-off between the achievable light throughput of the mask and its ability to distinguish so-called Zernike modes that form a basis for the reconstructed wavefront. Two nonlinear optimization approaches for the solution are presented and discussed in terms of exploiting the Kronecker products by evaluating matrix-vector multiplications with them using a large but sparse philosophy. A final framework is proposed to combine the strengths of these in order to tackle the design problem.