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.

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.

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 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%.

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 c{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.

This thesis studies the Canonical Polyadic Decomposition (CPD) constrained kernel machine for large scale learning, i.e. learning with a large number of samples. The kernel machine optimization problem is solved in the primal space, such that the complexity of the problem scales linearly in the number of samples as opposed to scaling cubically in the dual space. Product feature maps are applied to transform the input data. The weights are constrained to be a CPD, so the number of weights scales linearly in the number of features. The CPD introduces a nonlinearity, so nonlinear optimization must be applied. It is studied in which situation it is more advantageous to apply iterative all-at-once opti- mization compared to Alternating Least Squares (ALS) to solve the CPD constrained kernel machine problem. Specifically, all-at-once gradient descent methods are studied. An efficient analytical algorithm for the all-at-once gradient is derived. Furthermore, it is shown that automatic differentiation (AD) can also be applied, but it is found to be slower than the analytical method. The selection of a step size is found to be challenging. It is shown that the magnitude of the gradient of the mean squared error (MSE) term decreases for an increasing number of features. As a result, selecting the step size becomes more difficult for more features. To overcome this, the Line search and the Adam method are studied. A general expression for the exact line search solution is derived. It can be applied to compute the optimal step size for any step direction and any number of features. However, the Adam method performs better in terms of loss after training, convergence and the training run time. The mini-batch Adam method is used to evaluate the performance of all-at-once optimization for large scale learning. It is found that the Adam method no longer performs well for data sets with around 16 features or more, likely due to the decrease in the magnitude of the gradient of the MSE term. On large scale data sets with fewer features, the Adam method outperforms ALS in terms of run time until convergence while achieving similar training and validation losses. The Adam method reached convergence on a data set with 11 million samples within ten minutes. Furthermore, it is shown that the scaling of the run time of the Adam method in terms of the feature map order and the CP-rank is more than an order lower than the scaling of ALS when the methods are run on a GPU. This makes to Adam method more suitable for more complex models.

This research focuses on proving the presence of coloured noise on the longitudinal, lateral and rotational velocity in steady-state cornering of a skid steer mobile robot. Furthermore, it also focuses on the creation of a Gaussian filter which is able to recreate the characteristics of the measured coloured noise. This is done to determine the characteristics of the Gaussian filter based on real-life experimental values, which defines the Gaussian filter that creates coloured process noise in the Active Inference algorithm. To accomplish the main research goal, the following is done: - Create a linear dynamical model of the jackal robot and optimize the model in such a way that its velocity states resemble the experimental values of the velocities for the given experimental inputs. - Show the presence of coloured noise in the dynamical behaviour of the jackal robot. - Create a Gaussian filter and determine its characteristics, which can recreate the coloured noise found in the velocity states of the jackal robot in steady state turning. To accomplish this, experiments are done capturing the linear acceleration, rotational ve- locities, position and heading of the skid steer mobile robot using both the internal sensors of the robot as well as an external motion capturing system. A linear model of the jackal robot is constructed and discretised as the noise is a function of the difference between the experimental value for the next step state [k + 1]exp and the prediction of the next step state [k + 1]est. The prediction is made by inputting the experimental values for step [k] into the discretised linear model together with the model input. The difference between [k + 1]exp and [k+1]est is the noise on the velocities during the steady-state cornering experiments. Optimal values for Gaussian filter are found by fitting the autocorrelations of coloured noise which is made by filtering white noise with a range of Gaussian filters on the autocorrelation of the measured coloured noise. The result of this research is proof of the presence of coloured noise in skid steer mobile robots which in turn indicates that using Active Inference on these types of robots is a worthwhile approach to control and state estimation.

The world population is growing rapidly and the demand for healthy food grows with it. Greenhouse cultivation provides an efficient way to grow crops in a protected and controlled environment. In the past, many greenhouse control algorithms have been developed. How- ever, the majority of these algorithms rely on an explicit parametric model description of the greenhouse. These models are often based on physical laws such as conservation of mass and energy and contain many parameters which should be identified. Due to the complex and highly non-linear dynamics of greenhouses, these models might not be applicable to control greenhouses other than the one for which the model has been designed and identified. Hence, in current horticultural practice these control algorithms are scarcely used. Therefore, the need rises for a control algorithm which does not rely on a parametric system representa- tion but rather on input/output data of the greenhouse system, hereby establishing a way to control the system with unknown or unmodeled dynamics. A recently proposed algo- rithm, Data-Enabled Predictive Control (DeePC), is able to replace system identification, state estimation and future trajectory prediction by one single optimization framework. The algorithm exploits a non-parametric model constructed solely from input/output data of the system. This algorithm is employed in order to control the greenhouse climate. It is shown that in numerical simulation the DeePC algorithm is able to control the greenhouse climate. A comparison is made with the conventional Nonlinear Model Predictive Control (NMPC) algorithm in order to show the differences between a predictive control algorithm that has direct access to the non-linear greenhouse simulation model and a purely data-driven predic- tive control algorithm. Both algorithms are compared based on reference tracking accuracy and computational time. Furthermore, it is shown in numerical simulation that the DeePC algorithm is able to cope with changing dynamics within the greenhouse system throughout the crop cycle.

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 ever increasing global connectivity, the demand on the fiber-optic communication infrastructure is projected to keep increasing rapidly. A major factor currently limiting transmission capacity is the fiber nonlinearity. Some researchers have suggested the application of nonlinear Fourier transforms to exploit the fiber nonlinearity rather than ignoring or mitigating it. Nonlinear Fourier transforms allow us to solve certain nonlinear partial differential equations by transforming the complex evolution of the solution in the time-domain to a simple multiplication with a nonlinear frequency response in the nonlinear Fourier domain. This method is analogous to solving linear partial differential equations using the Fourier transform. The nonlinear Schrödinger equation is a suitable model for the propagation of light through a single-mode optical fiber. Its lossless version is solvable through a nonlinear Fourier transform. In recent years, several nonlinear Fourier transform based communication systems have been proposed. Such systems require numerical algorithms to compute the nonlinear Fourier transforms as nonlinear Fourier spectra are known analytically for only a handful of signals, and linear superposition cannot be used to compute the spectrum of a more complex signal. Computationally efficient algorithms are therefore not only essential for the real-time operation of nonlinear Fourier transform based communication systems, but are also important for their simulation. One common way to improve the spectral efficiency of a communication system is to increase the signal power in order to reduce the impact of noise. Another is to increase the signal duration in order to reduce the impact of information-free guard intervals that are inserted between transmissions to deal with the channel memory. Longer signals however require more resources to process them. The numerical problem of computing nonlinear Fourier transforms furthermore gets harder for both higher power and longer durations. Hence in the literature, we observe that the inability to perform efficient communication in these regimes is typically attributed to numerical problems of existing algorithms. In this dissertation we develop new algorithms that require shorter computation times for achieving similar accuracies as existing algorithms. Furthermore, we theoretically investigate whether some of the problems that are commonly attributed to numerical difficulties could occur in the absence of numerical effects. The nonlinear Fourier transform for signals that decay sufficiently fast is currently the most commonly used transform in nonlinear Fourier transform based communication systems. We developed new algorithms for computing the continuous nonlinear Fourier spectrum which is one part of the nonlinear Fourier spectrum for decaying signals. We demonstrated significant improvements over existing algorithms in multiple numerical benchmarks, and implemented the algorithms in the open source software library FNFT. We also developed NFDMLab, which is a Python based open source simulation environment for nonlinear Fourier transform based communication systems that relies on FNFT. The developed forward nonlinear Fourier transform algorithms are fast higher-order methods with a complexity of O(D log2D) for computing the continuous nonlinear Fourier spectrum from D samples of a decaying signal. In the numerical benchmarks, we introduced the trade-off between accuracy and computation time as a new way to compare nonlinear Fourier transform algorithms and found that the newly proposed algorithms perform significantly better than prior work in this regard. We also provided the first counting analysis of a fast nonlinear Fourier transform algorithm. There is also interest in using the nonlinear Fourier transform for periodic signals, as it is closer to the method used in conventional orthogonal frequency division multiplexing communication systems. The definition of the nonlinear Fourier transform for periodic signals is different from that of decaying signals. Communication systems based on nonlinear Fourier transforms for periodic signals make use of so-called finite-genus solutions of the nonlinear Schrödinger equation. Riemann theta functions are the traditional way to realize inverse nonlinear Fourier transforms that are used to synthesize finite-genus solutions. They are multi-dimensional Fourier series and their numerical computation suffers from the curse of dimensionality. This limits the genus of the signals used in the communication systems and is seen as a major bottleneck. We derived new bounds on the series truncation error and proposed two tensor-train based and a hyperbolic cross index set based algorithms for computing high-dimensional Riemann theta functions. We compared them to existing algorithms in multiple numerical benchmarks. The bounds that we derived on the truncation error of the Riemann theta functions allowed us to rule out several of the existing approaches for the high-dimension regime. We demonstrated that the algorithm based on the hyperbolic cross can compute Riemann theta functions upto 60 dimensions with moderate accuracy which is significantly higher than what was previously feasible. We also tried to improve the performance of nonlinear Fourier transform based communication systems known as b-modulators in the highly nonlinear regime using improved numerical algorithms. When we did not see improvements, we conducted a theoretical analysis of b-modulation systems. The analysis allowed us to prove theoretically that nonlinear bandwidth, signal duration and power are coupled when singularities in the nonlinear spectrum are avoided. When the nonlinear bandwidth is fixed, the coupling results in an upper bound on the transmit power. The power bound decreases with increasing signal duration which consequently decreases the signal-to-noise ratios for long signals, which explains the observed performance degradation in this regime without resorting to numerical difficulties. This result is the first of its kind as such a behaviour is not known from conventional linear systems. We also demonstrated numerically that the transmit powers achieved by an exemplary b-modulated system are close to its theoretical limits. Fiber-optic communication systems based on nonlinear Fourier transforms have been proposed to potentially tackle fiber nonlinearity, which is a major factor currently limiting transmission capacity. Efficient numerical algorithms are essential for real-time operation as well as efficient simulations of nonlinear Fourier transform based fiber-optic communication systems. The algorithms presented in this dissertation potentially make already published nonlinear Fourier transform based communication systems more practical and also allow for development of new designs which were previously infeasible. In this dissertation furthermore a limitation on communication system design imposed by the structure of the nonlinear Fourier transform was identified. It can be used to explain the inability to perform efficient communication with long duration signals, which was previously attributed to numerical problems, and guide the design of future systems. @en

Beamforming is a signal processing technique used in highly directional antennas. An array of antenna elements transmits the same signal, but with a different time delay for each element. By providing the right time delays for each antenna element, the whole array transmits a high-powered signal in one desired direction. This technique can be used for example to provide satellite television and Internet connections on board of aircrafts. Recently, developments in the field of integrated microwave photonics have paved the way for broadband, low-loss, and low-weight beamformer systems. These photonic beamformers convert the signals to be transmitted to the optical domain, provide the correct time delays with tunable optical delay lines, and then convert the signal back to the radio frequency domain. The main challenge here lies in tuning the actuators of the tunable optical delay lines in such a way that they provide the desired time delays. Challenges like actuator crosstalk, parameter sensitivity, noise and model errors cause complications when traditional tuning algorithms are used, such as nonlinear optimization routines. All results obtained with these photonic beamformers in the literature so far have been achieved by tuning the whole system by hand, or by applying nonlinear optimization techniques to a simplified simulation of the system rather than the actual system. In order to find a practical way of tuning a photonic beamformer in real time, this thesis takes a data-driven approach. Instead of relying on perfectly accurate physical models, a surrogate function is used that approximates the relation between the system actuators and a cost function, namely the difference between the measured and desired time delay of each antenna element. By performing nonlinear optimization techniques on this surrogate cost function and by continuously updating the approximation as new measurements are obtained, the time delays of each antenna element should converge towards the desired values. The Data-based Online Nonlinear Extremum-seeker (DONE) algorithm is used to update and optimize the surrogate function in real time. This algorithm is especially designed to optimize cost functions that are costly to evaluate (for example in terms of time), that contain noise, and for which derivatives cannot be easily computed or approximated. The DONE algorithm is applied to a simulation of a photonic beamformer and to the real system, as well as to several other applications. It is shown that the algorithm outperforms comparable methods on several fronts, especially computation time. Furthermore, the theory behind the algorithm is investigated, but practical results are also given, for example rules of thumb for choosing the hyper-parameters. Finally, variations to the DONE algorithm have been developed that are easier to use, can be implemented more efficiently, and can deal with time-varying objective functions.@en