Integração numérica de leis de velocidade diferenciais com o uso do SCILAB

In this work, we applied the free open source SCILAB software for the numerical integration of differential rate law equations to obtain the concentration profiles of chemical species involved in the kinetics of some complex reactions. An automated method was applied to construct the system of ordinary differential equations (ODE) from the postulated chemical models. The solutions of the ODEs were obtained numerically by standard SCILAB functions. We successfully simulated even complex chemical systems such as pH oscillators. This communication opens up the possibility of using SCILAB in simulations and modeling by our chemistry undergraduate students.

Dinâmica não-linear e reagente limitante: um exemplo de bifurcação transcrítica

The nonlinear analysis of a general mixed second order reaction was performed, aiming to explore some basic tools concerning the mathematics of nonlinear differential equations. Concepts of stability around fixed points based on linear stability analysis are introduced, together with phase plane and integral curves. The main focus is the chemical relationship between changes of limiting reagent and transcritical bifurcation, and the investigation underlying the conclusion.

Regime Switching Models for Electricity Time Series that Capture Fat Tailed Distributions

In the power market, electricity prices play an important role at the economic level. The behavior of a price trend usually known as a structural break may change over time in terms of its mean value, its volatility, or it may change for a period of time before reverting back to its original behavior or switching to another style of behavior, and the latter is typically termed a regime shift or regime switch. Our task in this thesis is to develop an electricity price time series model that captures fat tailed distributions which can explain this behavior and analyze it for better understanding. For NordPool data used, the obtained Markov Regime-Switching model operates on two regimes: regular and non-regular. Three criteria have been considered price difference criterion, capacity/flow difference criterion and spikes in Finland criterion. The suitability of GARCH modeling to simulate multi-regime modeling is also studied.

Learning Robot Motions from Visual Sensing

Learning from demonstration becomes increasingly popular as an efficient way of robot programming. Not only a scientific interest acts as an inspiration in this case but also the possibility of producing the machines that would find application in different areas of life: robots helping with daily routine at home, high performance automata in industries or friendly toys for children. One way to teach a robot to fulfill complex tasks is to start with simple training exercises, combining them to form more difficult behavior. The objective of the Master’s thesis work was to study robot programming with visual input. Dynamic movement primitives (DMPs) were chosen as a tool for motion learning and generation. Assuming a movement to be a spring system influenced by an external force, making this system move, DMPs represent the motion as a set of non-linear differential equations. During the experiments the properties of DMP, such as temporal and spacial invariance, were examined. The effect of the DMP parameters, including spring coefficient, damping factor, temporal scaling, on the trajectory generated were studied.

Methods and Models for Accelerating Dynamic Simulation of Fluid Power Circuits

The objective of this dissertation is to improve the dynamic simulation of fluid power circuits. A fluid power circuit is a typical way to implement power transmission in mobile working machines, e.g. cranes, excavators etc. Dynamic simulation is an essential tool in developing controllability and energy-efficient solutions for mobile machines. Efficient dynamic simulation is the basic requirement for the real-time simulation. In the real-time simulation of fluid power circuits there exist numerical problems due to the software and methods used for modelling and integration. A simulation model of a fluid power circuit is typically created using differential and algebraic equations. Efficient numerical methods are required since differential equations must be solved in real time. Unfortunately, simulation software packages offer only a limited selection of numerical solvers. Numerical problems cause noise to the results, which in many cases leads the simulation run to fail. Mathematically the fluid power circuit models are stiff systems of ordinary differential equations. Numerical solution of the stiff systems can be improved by two alternative approaches. The first is to develop numerical solvers suitable for solving stiff systems. The second is to decrease the model stiffness itself by introducing models and algorithms that either decrease the highest eigenvalues or neglect them by introducing steady-state solutions of the stiff parts of the models. The thesis proposes novel methods using the latter approach. The study aims to develop practical methods usable in dynamic simulation of fluid power circuits using explicit fixed-step integration algorithms. In this thesis, twomechanisms whichmake the systemstiff are studied. These are the pressure drop approaching zero in the turbulent orifice model and the volume approaching zero in the equation of pressure build-up. These are the critical areas to which alternative methods for modelling and numerical simulation are proposed. Generally, in hydraulic power transmission systems the orifice flow is clearly in the turbulent area. The flow becomes laminar as the pressure drop over the orifice approaches zero only in rare situations. These are e.g. when a valve is closed, or an actuator is driven against an end stopper, or external force makes actuator to switch its direction during operation. This means that in terms of accuracy, the description of laminar flow is not necessary. But, unfortunately, when a purely turbulent description of the orifice is used, numerical problems occur when the pressure drop comes close to zero since the first derivative of flow with respect to the pressure drop approaches infinity when the pressure drop approaches zero. Furthermore, the second derivative becomes discontinuous, which causes numerical noise and an infinitely small integration step when a variable step integrator is used. A numerically efficient model for the orifice flow is proposed using a cubic spline function to describe the flow in the laminar and transition areas. Parameters for the cubic spline function are selected such that its first derivative is equal to the first derivative of the pure turbulent orifice flow model in the boundary condition. In the dynamic simulation of fluid power circuits, a tradeoff exists between accuracy and calculation speed. This investigation is made for the two-regime flow orifice model. Especially inside of many types of valves, as well as between them, there exist very small volumes. The integration of pressures in small fluid volumes causes numerical problems in fluid power circuit simulation. Particularly in realtime simulation, these numerical problems are a great weakness. The system stiffness approaches infinity as the fluid volume approaches zero. If fixed step explicit algorithms for solving ordinary differential equations (ODE) are used, the system stability would easily be lost when integrating pressures in small volumes. To solve the problem caused by small fluid volumes, a pseudo-dynamic solver is proposed. Instead of integration of the pressure in a small volume, the pressure is solved as a steady-state pressure created in a separate cascade loop by numerical integration. The hydraulic capacitance V/Be of the parts of the circuit whose pressures are solved by the pseudo-dynamic method should be orders of magnitude smaller than that of those partswhose pressures are integrated. The key advantage of this novel method is that the numerical problems caused by the small volumes are completely avoided. Also, the method is freely applicable regardless of the integration routine applied. The superiority of both above-mentioned methods is that they are suited for use together with the semi-empirical modelling method which necessarily does not require any geometrical data of the valves and actuators to be modelled. In this modelling method, most of the needed component information can be taken from the manufacturer’s nominal graphs. This thesis introduces the methods and shows several numerical examples to demonstrate how the proposed methods improve the dynamic simulation of various hydraulic circuits.

On non-ideal and non-linear portal frame dynamics analysis using bogoliubov averaging method

We apply the Bogoliubov Averaging Method to the study of the vibrations of an elastic foundation, forced by a Non-ideal energy source. The considered model consists of a portal plane frame with quadratic nonlinearities, with internal resonance 1:2, supporting a direct current motor with limited power. The non-ideal excitation is in primary resonance in the order of one-half with the second mode frequency. The results of the averaging method, plotted in time evolution curve and phase diagrams are compared to those obtained by numerically integrating of the original differential equations. The presence of the saturation phenomenon is verified by analytical procedures.

On state and parameter estimation in chaotic systems

State-of-the-art predictions of atmospheric states rely on large-scale numerical models of chaotic systems. This dissertation studies numerical methods for state and parameter estimation in such systems. The motivation comes from weather and climate models and a methodological perspective is adopted. The dissertation comprises three sections: state estimation, parameter estimation and chemical data assimilation with real atmospheric satellite data. In the state estimation part of this dissertation, a new filtering technique based on a combination of ensemble and variational Kalman filtering approaches, is presented, experimented and discussed. This new filter is developed for large-scale Kalman filtering applications. In the parameter estimation part, three different techniques for parameter estimation in chaotic systems are considered. The methods are studied using the parameterized Lorenz 95 system, which is a benchmark model for data assimilation. In addition, a dilemma related to the uniqueness of weather and climate model closure parameters is discussed. In the data-oriented part of this dissertation, data from the Global Ozone Monitoring by Occultation of Stars (GOMOS) satellite instrument are considered and an alternative algorithm to retrieve atmospheric parameters from the measurements is presented. The validation study presents first global comparisons between two unique satellite-borne datasets of vertical profiles of nitrogen trioxide (NO3), retrieved using GOMOS and Stratospheric Aerosol and Gas Experiment III (SAGE III) satellite instruments. The GOMOS NO3 observations are also considered in a chemical state estimation study in order to retrieve stratospheric temperature profiles. The main result of this dissertation is the consideration of likelihood calculations via Kalman filtering outputs. The concept has previously been used together with stochastic differential equations and in time series analysis. In this work, the concept is applied to chaotic dynamical systems and used together with Markov chain Monte Carlo (MCMC) methods for statistical analysis. In particular, this methodology is advocated for use in numerical weather prediction (NWP) and climate model applications. In addition, the concept is shown to be useful in estimating the filter-specific parameters related, e.g., to model error covariance matrix parameters.

Adaptive Markov chain Monte Carlo and Bayesian filtering for state space models

This thesis is concerned with the state and parameter estimation in state space models. The estimation of states and parameters is an important task when mathematical modeling is applied to many different application areas such as the global positioning systems, target tracking, navigation, brain imaging, spread of infectious diseases, biological processes, telecommunications, audio signal processing, stochastic optimal control, machine learning, and physical systems. In Bayesian settings, the estimation of states or parameters amounts to computation of the posterior probability density function. Except for a very restricted number of models, it is impossible to compute this density function in a closed form. Hence, we need approximation methods. A state estimation problem involves estimating the states (latent variables) that are not directly observed in the output of the system. In this thesis, we use the Kalman filter, extended Kalman filter, Gauss–Hermite filters, and particle filters to estimate the states based on available measurements. Among these filters, particle filters are numerical methods for approximating the filtering distributions of non-linear non-Gaussian state space models via Monte Carlo. The performance of a particle filter heavily depends on the chosen importance distribution. For instance, inappropriate choice of the importance distribution can lead to the failure of convergence of the particle filter algorithm. In this thesis, we analyze the theoretical Lᵖ particle filter convergence with general importance distributions, where p ≥2 is an integer. A parameter estimation problem is considered with inferring the model parameters from measurements. For high-dimensional complex models, estimation of parameters can be done by Markov chain Monte Carlo (MCMC) methods. In its operation, the MCMC method requires the unnormalized posterior distribution of the parameters and a proposal distribution. In this thesis, we show how the posterior density function of the parameters of a state space model can be computed by filtering based methods, where the states are integrated out. This type of computation is then applied to estimate parameters of stochastic differential equations. Furthermore, we compute the partial derivatives of the log-posterior density function and use the hybrid Monte Carlo and scaled conjugate gradient methods to infer the parameters of stochastic differential equations. The computational efficiency of MCMC methods is highly depend on the chosen proposal distribution. A commonly used proposal distribution is Gaussian. In this kind of proposal, the covariance matrix must be well tuned. To tune it, adaptive MCMC methods can be used. In this thesis, we propose a new way of updating the covariance matrix using the variational Bayesian adaptive Kalman filter algorithm.

Determining vehicle drivetrain dynamics using system identification

Target of this book is to propose an approach for modelling drivetrain dynamics in order to design further a vibration control system of a hybrid bus. In this thesis two approaches are examined and compared. First model is obtained by theoretical means: drivetrain is represented as a system of rotating masses, which motion is described with differential equations. Second model is obtained using system identification method: mathematical description of the dynamic behavior of a system is formed based on measured input (torque) and output (speed) data. Then two models are compared and an optimal approach is suggested.

Multicomponent diffusion during Prato cheese ripening: mathematical modeling using the finite element method

The partial replacement of NaCl by KCl is a promising alternative to produce a cheese with lower sodium content since KCl does not change the final quality of the cheese product. In order to assure proper salt proportions, mathematical models are employed to control the product process and simulate the multicomponent diffusion during the reduced salt cheese ripening period. The generalized Fick's Second Law is widely accepted as the primary mass transfer model within solid foods. The Finite Element Method (FEM) was used to solve the system of differential equations formed. Therefore, a NaCl and KCl multicomponent diffusion was simulated using a 20% (w/w) static brine with 70% NaCl and 30% KCl during Prato cheese (a Brazilian semi-hard cheese) salting and ripening. The theoretical results were compared with experimental data, and indicated that the deviation was 4.43% for NaCl and 4.72% for KCl validating the proposed model for the production of good quality, reduced-sodium cheeses.

Mathematical modeling of microbial growth in milk

A mathematical model to predict microbial growth in milk was developed and analyzed. The model consists of a system of two differential equations of first order. The equations are based on physical hypotheses of population growth. The model was applied to five different sets of data of microbial growth in dairy products selected from Combase, which is the most important database in the area with thousands of datasets from around the world, and the results showed a good fit. In addition, the model provides equations for the evaluation of the maximum specific growth rate and the duration of the lag phase which may provide useful information about microbial growth.

Bayesian analysis of SEIR epidemic models

This thesis concerns the analysis of epidemic models. We adopt the Bayesian paradigm and develop suitable Markov Chain Monte Carlo (MCMC) algorithms. This is done by considering an Ebola outbreak in the Democratic Republic of Congo, former Zaïre, 1995 as a case of SEIR epidemic models. We model the Ebola epidemic deterministically using ODEs and stochastically through SDEs to take into account a possible bias in each compartment. Since the model has unknown parameters, we use different methods to estimate them such as least squares, maximum likelihood and MCMC. The motivation behind choosing MCMC over other existing methods in this thesis is that it has the ability to tackle complicated nonlinear problems with large number of parameters. First, in a deterministic Ebola model, we compute the likelihood function by sum of square of residuals method and estimate parameters using the LSQ and MCMC methods. We sample parameters and then use them to calculate the basic reproduction number and to study the disease-free equilibrium. From the sampled chain from the posterior, we test the convergence diagnostic and confirm the viability of the model. The results show that the Ebola model fits the observed onset data with high precision, and all the unknown model parameters are well identified. Second, we convert the ODE model into a SDE Ebola model. We compute the likelihood function using extended Kalman filter (EKF) and estimate parameters again. The motivation of using the SDE formulation here is to consider the impact of modelling errors. Moreover, the EKF approach allows us to formulate a filtered likelihood for the parameters of such a stochastic model. We use the MCMC procedure to attain the posterior distributions of the parameters of the SDE Ebola model drift and diffusion parts. In this thesis, we analyse two cases: (1) the model error covariance matrix of the dynamic noise is close to zero , i.e. only small stochasticity added into the model. The results are then similar to the ones got from deterministic Ebola model, even if methods of computing the likelihood function are different (2) the model error covariance matrix is different from zero, i.e. a considerable stochasticity is introduced into the Ebola model. This accounts for the situation where we would know that the model is not exact. As a results, we obtain parameter posteriors with larger variances. Consequently, the model predictions then show larger uncertainties, in accordance with the assumption of an incomplete model.

Neopentyyliglykolin ja propionihapon välisen esteröintireaktion kinetiikka

Glykolien esterit ovat haluttuja pintareaktiivisia aineita. Niitä voidaan valmistaa esteröintireaktiolla karboksyylihappojen kanssa katalyytin läsnä ollessa, jolloin toivottu reaktiotuote on yleensä muodostuva monoesteri. Monoesterin saannon lisäämiseksi reaktiossa muodostuvaa vettä voidaan poistaa jatkuvasti reaktiosta. Reaktion tasapainotilan tutkiminen on kuitenkin tärkeää, jotta reaktion kinetiikka tunnettaisiin mahdollisimman hyvin. Tällöin reaktiotuotteita ei poisteta reaktioseoksesta reaktion aikana. Glykolit esteröityvät happojen kanssa kahdessa vaiheessa. Ensimmäisessä vaiheessa muodostuu monoesteriä ja vettä ja toisessa vaiheessa diesteriä ja vettä. Kokeiden perusteella ensimmäinen vaihe on selvästi toista vaihetta nopeampi reaktio. Kirjallisuudessa on esitetty myös kaksi sivureaktiota, transesteröityminen ja disproportionaatio. Reaktion kinetiikka voidaan kuvata ilman näitä pieniä sivureaktiota, mutta täydellisen kuvaamisen vuoksi on ne myös otettava huomioon. Reaktion kinetiikan tutkimiseksi suoritettiin viisi laboratoriokoetta eri lämpötiloissa neopentyyliglykolilla ja propionihapolla homogeenisen para-tolueenisulfonihapon toimiessa katalyyttina. Lähtöaineiden ja tuotteiden konsentraatioita seurattiin ajan funktiona ja saatujen tulosten perusteella sovitettiin reaktiomekanismin differentiaaliyhtälöiden reaktionopeusvakiot. Nopeusvakioiden lämpötilariippuvuutta tutkittiin Arrheniuksen yhtälön avulla. Lisäksi määritettiin tasapainovakiot kullekin osareaktiolle.

Stochastic particle models: mean reversion and burgers dynamics. An application to commodity markets

The aim of this study is to propose a stochastic model for commodity markets linked with the Burgers equation from fluid dynamics. We construct a stochastic particles method for commodity markets, in which particles represent market participants. A discontinuity in the model is included through an interacting kernel equal to the Heaviside function and its link with the Burgers equation is given. The Burgers equation and the connection of this model with stochastic differential equations are also studied. Further, based on the law of large numbers, we prove the convergence, for large N, of a system of stochastic differential equations describing the evolution of the prices of N traders to a deterministic partial differential equation of Burgers type. Numerical experiments highlight the success of the new proposal in modeling some commodity markets, and this is confirmed by the ability of the model to reproduce price spikes when their effects occur in a sufficiently long period of time.

Sampling Interval and estimated Betas : Implications for the Presence of Transitory Components in Stock Prices

We provide a theoretical framework to explain the empirical finding that the estimated betas are sensitive to the sampling interval even when using continuously compounded returns. We suppose that stock prices have both permanent and transitory components. The permanent component is a standard geometric Brownian motion while the transitory component is a stationary Ornstein-Uhlenbeck process. The discrete time representation of the beta depends on the sampling interval and two components labelled \"permanent and transitory betas\". We show that if no transitory component is present in stock prices, then no sampling interval effect occurs. However, the presence of a transitory component implies that the beta is an increasing (decreasing) function of the sampling interval for more (less) risky assets. In our framework, assets are labelled risky if their \"permanent beta\" is greater than their \"transitory beta\" and vice versa for less risky assets. Simulations show that our theoretical results provide good approximations for the means and standard deviations of estimated betas in small samples. Our results can be perceived as indirect evidence for the presence of a transitory component in stock prices, as proposed by Fama and French (1988) and Poterba and Summers (1988).