230 research outputs found

    Mixed-Mode Oscillations in a Stochastic, Piecewise-Linear System

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    We analyze a piecewise-linear FitzHugh-Nagumo model. The system exhibits a canard near which both small amplitude and large amplitude periodic orbits exist. The addition of small noise induces mixed-mode oscillations (MMOs) in the vicinity of the canard point. We determine the effect of each model parameter on the stochastically driven MMOs. In particular we show that any parameter variation (such as a modification of the piecewise-linear function in the model) that leaves the ratio of noise amplitude to time-scale separation unchanged typically has little effect on the width of the interval of the primary bifurcation parameter over which MMOs occur. In that sense, the MMOs are robust. Furthermore we show that the piecewise-linear model exhibits MMOs more readily than the classical FitzHugh-Nagumo model for which a cubic polynomial is the only nonlinearity. By studying a piecewise-linear model we are able to explain results using analytical expressions and compare these with numerical investigations.Comment: 25 pages, 10 figure

    Screening in orbital-density-dependent functionals

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    Electronic-structure functionals that include screening effects, such as Hubbard or Koopmans' functionals, require to describe the response of a system to the fractional addition or removal of an electron from an orbital or a manifold. Here, we present a general method to incorporate screening based on linear-response theory, and we apply it to the case of the orbital-by-orbital screening of Koopmans' functionals. We illustrate the importance of such generalization when dealing with challenging systems containing orbitals with very different chemical character, also highlighting the simple dependence of the screening on the localization of the orbitals. We choose a set of 46 transition-metal complexes for which experimental data and accurate many-body perturbation theory calculations are available. When compared to experiment, results for ionization potentials show a very good performance with a mean absolute error of  0.2~0.2 eV, comparable to the most accurate many-body perturbation theory approaches. These results reiterate the role of Koopmans' compliant functionals as simple and accurate quasiparticle approximations to the exact spectral functional, bypassing diagrammatic expansions and relying only on the physics of the local density or generalized-gradient approximation

    Canards, Folded Nodes, and Mixed-Mode Oscillations in Piecewise-Linear Slow-Fast Systems

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    Canard-induced phenomena have been extensively studied in the last three decades, from both the mathematical and the application viewpoints. Canards in slow-fast systems with (at least) two slow variables, especially near folded-node singularities, give an essential generating mechanism for mixed-mode oscillations (MMOs) in the framework of smooth multiple timescale systems. There is a wealth of literature on such slow-fast dynamical systems and many models displaying canard-induced MMOs, particularly in neuroscience. In parallel, since the late 1990s several papers have shown that the canard phenomenon can be faithfully reproduced with piecewise-linear (PWL) systems in two dimensions, although very few results are available in the three-dimensional case. The present paper aims to bridge this gap by analyzing canonical PWL systems that display folded singularities, primary and secondary canards, with a similar control of the maximal winding number as in the smooth case. We also show that the singular phase portraits are compatible in both frameworks. Finally, we show using an example how to construct a (linear) global return and obtain robust PWL MMOs

    Identification of piecewise-linear mechanical oscillators via Bayesian model selection and parameter estimation

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    The problem of identifying single degree-of-freedom (SDOF) nonlinear mechanical oscillators with piecewise-linear (PWL) restoring forces is considered. PWL nonlinear systems are a class of models that specify or approximate nonlinear systems via a set of locally-linear maps, each defined over different operating regions. They are useful in modelling hybrid phenomena common in practical situations, such as, systems with different modes of operation, or systems whose dynamics change because of physical limits or thresholds. However, identifying PWL models can be a challenging task when the number of operating regions and their partitions are unknown. This paper formulates the identification of oscillators with PWL restoring forces as a task of concurrent model selection and parameter estimation, where the selection of the number of linear regions is treated as a model selection task and identifying the associated system parameters as a task of parameter estimation. In this study, PWL maps in restoring forces with up to four regions are considered, and the task of model selection and parameter estimation task is addressed in a Bayesian framework. A likelihood-free Approximate Bayesian Computation (ABC) scheme is followed, which is easy to implement and provides a simplified way of doing model selection. The proposed approach has been demonstrated using two numerical examples and an experimental study, where ABC has been used to select models and identify parameters from among four SDOF PWL systems with different number of PWL regions. The results demonstrate the flexibility of using the proposed Bayesian approach for identifying the correct model and parameters of PWL systems, in addition to furnishing uncertainty estimates of the identified parameters

    Cost-effectiveness of all-oral regimens for the treatment of multidrug-resistant tuberculosis in Korea: comparison with conventional injectable-containing regimens

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    Background Regimens for the treatment of multidrug-resistant tuberculosis (MDR-TB) have been changed from injectable-containing regimens to all-oral regimens. The economic effectiveness of new all-oral regimens compared with conventional injectable-containing regimens was scarcely evaluated. This study was conducted to compare the cost-effectiveness between all-oral longer-course regimens (the oral regimen group) and conventional injectable-containing regimens (the control group) to treat newly diagnosed MDR-TB patients. Methods A health economic analysis over lifetime horizon (20 years) from the perspective of the healthcare system in Korea was conducted. We developed a combined simulation model of a decision tree model (initial two years) and two Markov models (remaining 18 years, six-month cycle length) to calculate the incremental cost-effectiveness ratio (ICER) between the two groups. The transition probabilities and cost in each cycle were assumed based on the published data and the analysis of health big data that combined country-level claims data and TB registry in 2013–2018. Results: The oral regimen group was assumed to spend 20,778 USD more and lived 1.093 years or 1.056 quality-adjusted life year (QALY) longer than the control group. The ICER of the base case was calculated to be 19,007 USD/life year gained and 19,674 USD/QALY. The results of sensitivity analyses showed that base case results were very robust and stable, and the oral regimen was cost-effective with a 100% probability for a willingness to pay more than 21,250 USD/QALY. Conclusion This study confirmed that the new all-oral longer regimens for the treatment of MDR-TB were cost-effective in replacing conventional injectable-containing regimens

    Computational Methods for Optimal Control of Hybrid Systems

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    This thesis aims to find algorithms for optimal control of hybrid systems and explore them in sufficient detail to be able to implement the ideas in computational tools. By hybrid systems is meant systems with interacting continuous and discrete dynamics. Code for computations has been developed in parallel to the theory. The optimal control methods studied in this thesis are global, i.e. the entire state space is considered simultaneously rather than searching for locally optimal trajectories. The optimal value function that maps each state of the state space onto the minimal cost for trajectories starting in that state is central for global methods. It is often difficult to compute the value function of an optimal control problem, even for a purely continuous system. This thesis shows that a lower bound of the value function of a hybrid optimal control problem can be found via convex optimization in a linear program. Moreover, a dual of this optimization problem, parameterized in the control law, has been formulated via general ideas from duality in transportation problems. It is shown that the lower bound of the value function is tight for continuous systems and that there is no gap between the dual optimization problems. Two computational tools are presented. One is built on theory for piecewise affine systems. Various analysis and synthesis problems for this kind of systems are via piecewise quadratic Lyapunov-like functions cast into linear matrix inequalities. The second tool can be used for value function computation, control law extraction, and simulation of hybrid systems. This tool parameterizes the value function in its values in a uniform grid of points in the state space, and the optimization problem is formulated as a linear program. The usage of this tool is illustrated in a case study

    MODEL ORDER REDUCTION OF NONLINEAR DYNAMIC SYSTEMS USING MULTIPLE PROJECTION BASES AND OPTIMIZED STATE-SPACE SAMPLING

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    Model order reduction (MOR) is a very powerful technique that is used to deal with the increasing complexity of dynamic systems. It is a mature and well understood field of study that has been applied to large linear dynamic systems with great success. However, the continued scaling of integrated micro-systems, the use of new technologies, and aggressive mixed-signal design has forced designers to consider nonlinear effects for more accurate model representations. This has created the need for a methodology to generate compact models from nonlinear systems of high dimensionality, since only such a solution will give an accurate description for current and future complex systems.The goal of this research is to develop a methodology for the model order reduction of large multidimensional nonlinear systems. To address a broad range of nonlinear systems, which makes the task of generalizing a reduction technique difficult, we use the concept of transforming the nonlinear representation into a composite structure of well defined basic functions from multiple projection bases.We build upon the concept of a training phase from the trajectory piecewise-linear (TPWL) methodology as a practical strategy to reduce the state exploration required for a large nonlinear system. We improve upon this methodology in two important ways: First, with a new strategy for the use of multiple projection bases in the reduction process and their coalescence into a unified base that better captures the behavior of the overall system; and second, with a novel strategy for the optimization of the state locations chosen during training. This optimization technique is based on using the Hessian of the system as an error bound metric.Finally, in order to treat the overall linear/nonlinear reduction task, we introduce a hierarchical approach using a block projection base. These three strategies together offer us a new perspective to the problem of model order reduction of nonlinear systems and the tracking or preservation of physical parameters in the final compact model

    A new methodology for obtaining piecewise affine models using a set of linearisation points and voronoi partitions

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    To understand complex dynamical systems, approximations are often made by a linearisation about an operating point of interest. The drawback of this linear approximation is that it only describes the system locally, around the operating point. One possible solution to overcome this drawback is to approximate the complex nonlinear dynamical system with a piecewise affine (PWA) system. Approximating nonlinear dynamical systems is very important in system theory where one is interested in simplifying its analysis and numerical simulation. PWA modelling is a very powerful tool to represent nonlinear systems as a collection of a finite number of linear systems. In the literature of PWA, a uniform grid (UG) approximation is the current method being used to approximate a nonlinear system as a PWA system. The drawback of this method is the potential large amount of regions required to obtain a desired accuracy, which is most evident for systems with more than one variable in the domain of the nonlinearity. In order to reduce the number of regions, the proposed research will develop a new methodology for obtaining PWA models using a set of linearisation points (SLP) and the Voronoi partition. First, in order to generate a partition based on a SLP, the curvature of the nonlinearity is used as a tool for selecting appropriate locations for the linearisation points. Next, an algorithm is proposed to automate the SLP approximation for both curves and surfaces. The SLP and UG approximation methods are then compared over several simple examples. Finally, the newly proposed approximation methodology is applied to three case studies: modelling of a nonlinear mechanical system and modelling and control of an unmanned aerial vehicle (UAV) and a micro air vehicle (MAV
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