Ekaterina Auer

VALENCIA-IVP: A Comparison with Other Initial Value Problem Solvers

Validated integration of ordinary differential equations with uncertain initial conditions and uncertain parameters is important for many practical applications. If guaranteed bounds for the uncertainties are known, interval methods can be applied to obtain validated enclosures of all states. However, validated computations are often affected by overestimation, which, in naive implementations, might even lead to meaningless results. Parallelepiped and QR preconditioning of the state equations, Taylor model arithmetic, as well as simulation techniques employing splitting and merging routines are a few existing approaches for reduction of overestimation. In this paper, the recently developed validated solver ValEncIA-IVP and several methods implemented there for reduction of overestimation are described. Furthermore, a detailed comparison of this solver with COSY VI and VNODE, two of the most well- known validated ODE solvers, is presented. Simulation results for simplified system models in mechanical and bio- process engineering show specific properties, advantages, and limitations of each tool.

Validated Modeling of Mechanical Systems with SmartMOBILE: Improvement of Performance by ValEncIA-IVP

Computer simulations of real life processes can generate erroneous results, in many cases due to the use of finite precision arithmetic. To ensure correctness of the results obtained with the help of a computer, various kinds of validating arithmetic and algorithms were developed. Their purpose is to provide bounds in which the exact result is guaranteed to be contained. Verified modeling of kinematics and dynamics of multibody systems is a challenging application field for such methods, largely because of possible overestimation of the guaranteed bounds, leading to meaningless results. In this paper, we discuss approaches to validated modeling of multibody systems and present a template-based tool SmartMOBILE , which features the possibility to choose an appropriate kind of arithmetic according to the modeling task. We consider different strategies for obtaining tight state enclosures in SmartMOBILE including improvements in the underlying data types (Taylor models), modeling elements (rotation error reduction), and focus on enhancement through the choice of initial value problem solvers ( ValEncIA-IVP ).

Interval Methods for Control-Oriented Modeling of the Thermal Behavior of High-Temperature Fuel Cell Stacks

Abstract Solid oxide fuel cells (SOFCs) can be used as decentralized energy supply devices for providing electricity and heat directly by converting chemical energy. In such applications of SOFCs, the electric power demand is commonly varying over time. Therefore, all processes in fuel cell systems are typically instationary. For example, the heating and cooling phases for starting up and shutting down the fuel cell system as well as the response to varying electrical load demands characterize the instationarity of the operating conditions for the thermal subprocess. In contrast to most existing control approaches, which only cover stationary operating strategies, our work aims at controlling SOFC systems in instationary operating regions. This means that a mathematical system model for these regions is necessary. Such control-oriented models for the temperature distribution in a fuel cell stack module can be obtained by the method of finite volume discretization. On the basis of the first law of thermodynamics, local energy balances are derived for each volume element, which leads to a system of coupled nonlinear ordinary differential equations. In this paper, parameter identification routines are compared which are based both on classical floating point techniques and on verified interval arithmetic approaches. In particular, interval techniques are employed to deal with imperfect system knowledge expressed by bounded parameter uncertainties and to search for globally instead of locally optimal system parameterizations.

Thermal behavior of high-temperature fuel cells: reliable parameter identification and interval-based sliding mode control

In this contribution, we present interval methods for mathematical modeling, for parameter identification, and for control design of dynamical systems. The corresponding approaches are applied to the thermal subsystem of a high-temperature solid oxide fuel cell (SOFC) which is available as a test rig at the Chair of Mechatronics at the University of Rostock. In practice, most internal parameters of SOFC stack modules cannot be measured directly. Therefore, system characteristics such as heat capacities or internal thermal resistances cannot be identified exactly, but only bounded. For this reason, intervals represent a good first approach to dealing with parameter uncertainty. In the first part of the paper, we present interval methods for the parameter identification aiming at the computation of globally optimal parameterizations. In comparison with classical local optimization procedures, the approximation quality is improved by the presented identification approach. The corresponding bounds for admissible domains are used to design a robust sliding mode control law for arbitrary operating points compensating the impact of disturbances and parameter uncertainties in a reliable way. In the second part of the paper, we show a simple approach to handling non-smoothness appearing in SOFC models based on ordinary differential equations in a verified way. We use a generalized derivative definition for a certain type of non-smooth functions inside the algorithm of the verified solver ValEncIA-IVP to be able to compute solutions to non-smooth initial value problems. The applicability of our method is demonstrated using the designed sliding mode controller.

Sensitivity-based feedforward and feedback control for uncertain systems

In many control applications, we are interested in accurate trajectory tracking. This is especially true for cases in which exact analytic solutions are not available because initial states are not consistent with the desired state or output trajectories or because parameters are significantly uncertain. In these cases, control strategies can be derived on the basis of a verified sensitivity analysis. For that purpose, we have to define suitable performance indices which describe the deviation between the actual and desired trajectories. In this paper, an overview of different sensitivity-based control procedures is given. These procedures include tracking control for systems with bounded parameter uncertainties as well as measurement errors described by interval variables. Moreover, we present a first verified approach to automatic path following by means of an automatic modification of desired output trajectories. This procedure is necessary in cases in which exact trajectory tracking is not possible due to the violation of control constraints.

A verified realization of a Dempster–Shafer based fault tree analysis

Fault tree analysis is a method to determine the likelihood of a system attaining an undesirable state based on the information about its lower level parts. However, conventional approaches cannot process imprecise or incomplete data. There are a number of ways to solve this problem. In this paper, we will consider the one that is based on the Dempster–Shafer theory. The major advantage of the techniques proposed here is the use of verified methods (in particular, interval analysis) to handle Dempster–Shafer structures in an efficient and consistent way. First, we concentrate on DSI (Dempster–Shafer with intervals), a recently developed tool. It is written in MATLAB and serves as a basis for a new add-on for Dempster–Shafer based fault tree analysis. This new add-on will be described in detail in the second part of our paper. Here, we propagate experts’ statements with uncertainties through fault trees, using mixing based on arithmetic averaging. Furthermore, we introduce an implementation of the interval scale based algorithm for estimating system reliability, extended by new input distributions.

Modeling, Design, and Simulation of Systems with Uncertainties

To describe the true behavior of most real-world systems with sufficient accuracy, engineers have to overcome difficulties arising from their lack of knowledge about certain parts of a process or from the impossibility of characterizing it with absolute certainty. Depending on the application at hand, uncertainties in modeling and measurements can be represented in different ways. For example, bounded uncertainties can be described by intervals, affine forms or general polynomial enclosures such as Taylor models, whereas stochastic uncertainties can be characterized in the form of a distribution described, for example, by the mean value, the standard deviation and higher-order moments. The goal of this Special Volume on Modeling, Design, and Simulation of Systems with Uncertainties is to cover modern methods for dealing with the challenges presented by imprecise or unavailable information. All contributions tackle the topic from the point of view of control, state and parameter estimation, optimization and simulation. Thematically, this volume can be divided into two parts. In the first we present works highlighting the theoretic background and current research on algorithmic approaches in the field of uncertainty handling, together with their reliable software implementation. The second part is concerned with real-life application scenarios from various areas including but not limited to mechatronics, robotics, and biomedical engineering.

Interval Algorithms in Modeling of Multibody Systems

We will show how a variety of interval algorithms have found their use in the multibody modeling program MOBILE. This paper acquaints the reader with the key features of this open source software, describes how interval arithmetic help to implement new transmission elements, and reports on interval modeling of dynamics, which is an inherent part of multibody simulations. In the latter case, the interval extension of MOBILE enhanced with an interval initial value problem solver (based on VNODE) is presented. The functionality of this application is shown with some examples. We provide insights into techniques used to enhance already existing modeling software with interval arithmetic concepts.

A verified method for solving piecewise smooth initial value problems

Abstract In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview of possibilities to formulate non-smooth problems and point out connections between the traditional non-smooth theory and interval analysis. Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain practically relevant subclass of such systems. We implement the approach for the solver VALENCIA-IVP by introducing into it a specialized template for enclosing the first-order derivatives of non-smooth functions. We demonstrate the applicability of our technique using a mechanical system model with friction and hysteresis. We conclude the paper by giving a perspective on future research directions in this area.

Uses of GPU Powered Interval Optimization for Parameter Identification in the Context of SO Fuel Cells

Abstract In this paper, we discuss parameter identification for models based on ordinary differential equations in the context of solid oxide fuel cells. In this case, verified methods (e.g. interval analysis), which provide a guarantee of correctness for the computed result, can be of great help for dealing with the appearing uncertainty and for devising accurate control strategies. Moreover, interval arithmetic can be used to discard infeasible areas of parameter space in a natural way and so to improve the results of traditional numerical algorithms. We describe a simulation environment interfacing different verified and floating point based approaches and show how the interchangeability between different techniques enhances parameter identification. Additionally, we give details on a possible parallelization of our version of the global interval optimization algorithm on the CPU and the GPU. The applicability of the method and the features of the environment are demonstrated with the help of different fuel cell models.