23 research outputs found

    Identification of weakly coupled multiphysics problems. Application to the inverse problem of electrocardiography

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    This work addresses the inverse problem of electrocardiography from a new perspective, by combining electrical and mechanical measurements. Our strategy relies on the defini-tion of a model of the electromechanical contraction which is registered on ECG data but also on measured mechanical displacements of the heart tissue typically extracted from medical images. In this respect, we establish in this work the convergence of a sequential estimator which combines for such coupled problems various state of the art sequential data assimilation methods in a unified consistent and efficient framework. Indeed we ag-gregate a Luenberger observer for the mechanical state and a Reduced Order Unscented Kalman Filter applied on the parameters to be identified and a POD projection of the electrical state. Then using synthetic data we show the benefits of our approach for the estimation of the electrical state of the ventricles along the heart beat compared with more classical strategies which only consider an electrophysiological model with ECG measurements. Our numerical results actually show that the mechanical measurements improve the identifiability of the electrical problem allowing to reconstruct the electrical state of the coupled system more precisely. Therefore, this work is intended to be a first proof of concept, with theoretical justifications and numerical investigations, of the ad-vantage of using available multi-modal observations for the estimation and identification of an electromechanical model of the heart

    Stability of linear and non-linear Kalman filters

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    Kalmanin suodin ja sen approksimatiiviset yleistykset epälineaarisille systeemeille ovat stokastisten dynaamisten systeemien tilaestimoinnin perustyökaluja. Oleellinen kysymys näitä suotimia käytettäessä on, ovatko ne jossakin mielessä stabiileja. Lineaarisella Kalmanin suotimella on vahvoja eksponentiaalisia stabiilisuusominaisuuksia, mutta epälineaarisille Kalmanin suotimille osoitetut stabiilisuustulokset ovat hyvin heikkoja. Näiden tulosten tarkentaminen ja vahvistaminen on avoin tutkimuskohde. Tässä tutkielmassa esitellään lineaarisen Kalmanin suotimen merkittävimmät stabiilisuustulokset ja olemassa olevat tulokset epälineaarisille Kalmanin suotimille. Lineaarisen Kalmanin suotimen stabiilisuus vaatii säätötekniset oletukset systeemin havaittavuudesta (englanniksi detectability) ja stabiloituvuudesta. Nämä oletukset mahdollistavat epäoptimaalisen lineaarisen tilaestimaattorin, johon liittyvä virhekovarianssimatriisi on rajoitettu, konstruoimisen, jolloin Kalmanin suotimen lineaarisen minimivarianssin ominaisuutta voidaan hyödyntää. Varsinaisessa stabiliisuustodistuksessa käytetään erästä Ljapunovin stabiilisuusteorian yleistystä. Epälineaarisista Kalmanin suotimista tutkielmassa käsitellään pääasiassa laajennettua Kalmanin suodinta ja hajustamatonta Kalmanin suodinta. Molemmille suotimille todistetaan stokastisia stabiilisuustuloksia erittäin tiukin oletuksin, jotka eivät mahdollista stabiilisuuden toteamista etukäteen. Tulokset saadaan erään stokastista stabiilisuutta koskevan lemman melko suoraviivaisella soveltamisella, vaikkakin hajustamattomalle Kalmanin suotimille todistettaviin tuloksiin vaaditaan myös eräiden approksimaatioiden muuntamista yhtälöiksi diagonaalisten satunnaismatriisien avulla. Hajustamatonta Kalmanin suodinta koskevat tulokset voidaan yleistää kaikilla epälineaarisille Kalmanin suotimille (tai gaussisille suotimille). Nämä tulokset ovat kuitenkin hyvin kvalitatiivisia ja niiden ainoa konkreettinen anti on kohinakovarianssimatriisien virittämisen vaikutuksen selventäminen, mistä tutkielmassa esitetään muutamia yksinkertaisia numeerisia esimerkkejä. Tutkielman lopussa kartoitetaan eräitä mahdollisesti lupaavia menetelmiä, joita ei ole tähän mennessä käytetty epälineaaristen Kalmanin suotimien stabiilisuuden tutkimiseen. Näitä menetelmiä ovat Fourier'n-Hermiten sarjakehitelmä ja teleskooppisummamenetelmä, jonka avulla on aikaisemmin tutkittu partikkelisuotimien tasaista konvergenssia

    Estimation for Linear and Semi-linear Infinite-dimensional Systems

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    Estimating the state of a system that is not fully known or that is exposed to noise has been an intensely studied problem in recent mathematical history. Such systems are often modelled by either ordinary differential equations, which evolve in finite-dimensional state-spaces, or partial differential equations, the state-space of which is infinite-dimensional. The Kalman filter is a minimal mean squared error estimator for linear finite-dimensional and linear infinite-dimensional systems disturbed by Wiener processes, which are stochastic processes representing the noise. For nonlinear finite-dimensional systems the extended Kalman filter is a widely used extension thereof which relies on linearization of the system. In all cases the Kalman filter consists of a differential or integral equation coupled with a Riccati equation, which is an equation that determines the optimal estimator gain. This thesis proposes an estimator for semi-linear infinite-dimensional systems. It is shown that under some conditions such a system can also be coupled with a Riccati equation. To motivate this result, the Kalman filter for finite-dimensional and infinite-dimensional systems is reviewed, as well as the corresponding theory for both stochastic processes and infinite-dimensional systems. Important results concerning the infinite-dimensional Riccati equation are outlined and existence of solutions for a class of semi-linear infinite-dimensional systems is established. Finally the well-posedness of the coupling between a semi-linear infinite-dimensional system with a Riccati equation is proven using a fixed point argument

    Modeling and Simulation Methods of Neuronal Populations and Neuronal Networks

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    This thesis presents numerical methods and modeling related to simulating neurons. Two approaches to the simulation are taken: a population density approach and a neuronal network approach. The first two chapters present the results from the population density approach and its applications. The population density approach assumes that each neuron can be identified by its states (e.g., membrane potential, conductance of ion channels). Additionally, it assumes the population is large such that it can be approximated by a continuous population density distribution in the state space. By updating this population density, we can learn the macroscopic behavior of the population, such as the average firing rate and average membrane potential. The Population density approach avoids the need to simulate every single neuron when the population is large. While many previous population-density methods, such as the mean-field method, make further simplifications to the models, we developed the Asymmetric Particle Population Density (APPD) method to simulate the population density directly without the need to simplify the dynamics of the model. This enables us to simulate the macroscopic properties of coupled neuronal populations as accurately as a direct simulation. The APPD method tracks multiple asymmetric Gaussians as they advance in time due to a convection-diffusion equation, and our main theoretical innovation is deriving this update algorithm by tracking a level set. Tracking a single Gaussian is also applicable to the Bayesian filtering for continuous-discrete systems. By adding a measurement-update step, we reformulated our tracking method as the Level Set Kalman Filter(LSKF) method and find that it offers greater accuracy than state-of-the-art methods. Chapter IV presents the methods for direct simulation of a neuronal network. For this approach, the aim is to build a high-performance and expandable framework that can be used to simulate various neuronal networks. The implementation is done on GPUs using CUDA, and this framework enables simulation for millions of neurons on a high-performance desktop computer. Additionally, real-time visualization of neuron activities is implemented. Pairing with the simulation framework, a detailed mouse cortex model with experiment-determined morphology using the CUBIC-Atlas, and neuron connectome information from Allen's brain atlas is generated.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169840/1/nywang_1.pd

    Joint state and parameter estimation for distributed mechanical systems

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    We present a novel strategy to perform estimation for a dynamical mechanical system in standard operating conditions, namely, without ad hoc experimental testing. We adopt a sequential approach, and the joint state-parameter estimation procedure is based on a state estimator inspired from collocated feedback control. This type of state estimator is chosen due to its particular effectiveness and robustness, but the methodology proposed to adequately extend state estimation to joint state-parameter estimation is general, and - indeed -applicable with any other choice of state feedback observer. The convergence of the resulting joint estimator is mathematically established. In addition, we demonstrate its effectiveness with a biomechanical test problem defined to feature the same essential characteristics as a heart model, in which we identify localized contractility and stiffness parameters using measurements of a type that is available in medical imaging

    The constrained discrete-time state-dependent Riccati equation technique for uncertain nonlinear systems

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    The objective of the thesis is to introduce a relatively general nonlinear controller/estimator synthesis framework using a special type of the state-dependent Riccati equation technique. The continuous time state-dependent Riccati equation (SDRE) technique is extended to discrete-time under input and state constraints, yielding constrained (C) discrete-time (D) SDRE, referred to as CD-SDRE. For the latter, stability analysis and calculation of a region of attraction are carried out. The derivation of the D-SDRE under state-dependent weights is provided. Stability of the D-SDRE feedback system is established using Lyapunov stability approach. Receding horizon strategy is used to take into account the constraints on D-SDRE controller. Stability condition of the CD-SDRE controller is analyzed by using a switched system. The use of CD-SDRE scheme in the presence of constraints is then systematically demonstrated by applying this scheme to problems of spacecraft formation orbit reconfiguration under limited performance on thrusters. Simulation results demonstrate the efficacy and reliability of the proposed CD-SDRE. The CD-SDRE technique is further investigated in a case where there are uncertainties in nonlinear systems to be controlled. First, the system stability under each of the controllers in the robust CD-SDRE technique is separately established. The stability of the closed-loop system under the robust CD-SDRE controller is then proven based on the stability of each control system comprising switching configuration. A high fidelity dynamical model of spacecraft attitude motion in 3-dimensional space is derived with a partially filled fuel tank, assumed to have the first fuel slosh mode. The proposed robust CD-SDRE controller is then applied to the spacecraft attitude control system to stabilize its motion in the presence of uncertainties characterized by the first fuel slosh mode. The performance of the robust CD-SDRE technique is discussed. Subsequently, filtering techniques are investigated by using the D-SDRE technique. Detailed derivation of the D-SDRE-based filter (D-SDREF) is provided under the assumption of Gaussian noises and the stability condition of the error signal between the measured signal and the estimated signals is proven to be input-to-state stable. For the non-Gaussian distributed noises, we propose a filter by combining the D-SDREF and the particle filter (PF), named the combined D-SDRE/PF. Two algorithms for the filtering techniques are provided. Several filtering techniques are compared with challenging numerical examples to show the reliability and efficacy of the proposed D-SDREF and the combined D-SDRE/PF
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