# Parameterschätzung und Modellevaluation für komplexe Systeme

by Jan Schumann-Bischoff

Date of Examination:2016-04-06

Date of issue:2016-10-17

Advisor:Prof. Dr. Ulrich Parlitz

Referee:Prof. Dr. Ulrich Parlitz

Referee:Prof. Dr. Florentin Wörgötter

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## Abstract

### English

Mathematical models in form of dynamical systems play an important role in many disciplines, such as system biology, engineering, or physics. They are widely used to simulate the time evolution of the internal states of systems like cardiac muscle cells, neurons, electrical circuits, or the weather, to give just a few examples. Often the models are nonlinear. To obtain realistic results the parameterization of the model equations must be adjusted beforehand so that the model describes the real world system one aims to simulate as accurate as possible. Another task where dynamical systems play an important role is when full states of a real world system are required but only the time evolution of some state variables are actually measured in form of time series. This is a common situation in climatology. There, the temperature, barometric pressure, humidity, etc. are measured at a much smaller number of locations than it would be necessary to obtain the full state of the atmosphere. Nevertheless, the latter is required for example to initialize a weather forecast. In both situations a state and parameter estimation algorithm can be used to estimate full states of the system and model parameters given some measured data time series and a mathematical model of the system. If states and parameters are estimated from data, then one is typically also interested in the uncertainty of the estimated values. Whether states and parameters can be uniquely estimated from the data can by analyzed by investigating the observability of the mathematical model. In this thesis methods for state and parameter estimation as well as for analyzing the uncertainty and observability are presented and discussed. The first part is about the implementation of state and parameter estimation algorithms which are based on solving high dimensional optimization problems. To solve these optimization problems many optimization methods rely on accurate information about derivatives. It is discussed that a technique called automatic differentiation has the capability to provide numerically exact values for derivatives by requiring only a minimum of additional runtime of the estimation method. Using this technique an error prone implementation of derivatives in a computer program is not necessary. In the following parts a method is presented and evaluated which provides a measure for the uncertainty in state and parameter estimation. Among other things results based on this method are compared to results based on an optimization based state and parameter estimation algorithm. It is also investigated how the initialization of the estimation algorithm and the number and combination of measured model variables affect the accuracy of the estimated states and parameters. Furthermore, a method is presented which can be used to identify redundant model parameters and state variables which can not be uniquely estimated. This method also provides information about which and how many parameters may be removed from the estimation task so that all remaining state variables and parameters can be uniquely estimated. Finally, a nonlinear electronic circuit which may exhibit chaotic behavior is realized and used as an experimental system. Based on measured data from this circuit the previously presented methods are evaluated and verified.**Keywords:**State and Parameter Estimation; Observability; Uncertainty Analysis; Automatic Differentiation; Data Assimilation; Nonlinear Modelling