Andreas Rauh

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. n nIn 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 ).

Verification Techniques for Sensitivity Analysis and Design of Controllers for Nonlinear Dynamic Systems with Uncertainties

Verification Techniques for Sensitivity Analysis and Design of Controllers for Nonlinear Dynamic Systems with Uncertainties Control strategies for nonlinear dynamical systems often make use of special system properties, which are, for example, differential flatness or exact input-output as well as input-to-state linearizability. However, approaches using these properties are unavoidably limited to specific classes of mathematical models. To generalize design procedures and to account for parameter uncertainties as well as modeling errors, an interval arithmetic approach for verified simulation of continuoustime dynamical system models is extended. These extensions are the synthesis, sensitivity analysis, and optimization of open-loop and closed-loop controllers. In addition to the calculation of guaranteed enclosures of the sets of all reachable states, interval arithmetic routines have been developed which verify the controllability and observability of the states of uncertain dynamic systems. Furthermore, they assure asymptotic stability of controlled systems for all possible operating conditions. Based on these results, techniques for trajectory planning can be developed which determine reference signals for linear and nonlinear controllers. For that purpose, limitations of the control variables are taken into account as further constraints. Due to the use of interval techniques, issues of the functionality, robustness, and safety of dynamic systems can be treated in a unified design approach. The presented algorithms are demonstrated for a nonlinear uncertain model of biological wastewater treatment plants.

Interval methods for simulation of dynamical systems with state-dependent switching characteristics

In this paper, an interval arithmetic simulation algorithm is introduced for simulation of continuous-time systems with state-dependent switching between different dynamical models. For that purpose, the conditions for all possible transitions between these models have to be evaluated during simulation to determine the switching times and hence to obtain guaranteed enclosures for all state variables. In contrast to other simulation techniques, all system parameters are defined as interval variables to analyze the effect of uncertainties on the switching times and the dynamical behavior of the complete system

Calculating moments of exponential densities using differential algebraic equations

This article introduces an efficient approach for calculating the moments of exponential densities. Usually, the desired moments are obtained by means of numerical integration, which is impractical due to its computational complexity and the underlying infinite integration intervals. The new approach relies on an exact conversion of these integrals into a system of ordinary differential equations with algebraic constraints. The desired moments are then obtained by solving this system of differential algebraic equations over a finite time interval. The resulting procedure is simple to implement and typically reduces the computational burden by one order of magnitude.

Interval Techniques for Design of Optimal and Robust Control Strategies

In this paper, an interval arithmetic optimization procedure for both discrete-time and continuous-time systems is presented. Besides computation of control strategies for systems with nominal parameters, robustness requirements for systems with interval bounded uncertainties are considered. Considering these uncertainties, control laws are obtained which directly take into account the influence of disturbances and deviations of system parameters from their nominal values. Compared to Bellmans discrete dynamic programming, errors resulting from gridding of state and control variable intervals as well as errors due to rounding to nearest grid points are avoided. Furthermore, the influence of time discretization errors is taken into account by validated integration of continuous-time state equations. Optimization results for a simplified model of a mechanical positioning system with switchings between models for both static and sliding friction demonstrate the efficiency of the suggested approach and its applicability to processes with state-dependent switching characteristics.

Interval Observer Design Based on Taylor Models for Nonlinear Uncertain Continuous-Time Systems

In most applications in control engineering not all state variables can be measured. Consequently, state estimation is performed to reconstruct the non-measurable states taking into account both system dynamics and the measurement model. If the system is subject to interval bounded uncertainties, an interval observer provides a guaranteed estimation of all states. The estimation consists of a recursive application of prediction and correction steps. The prediction step corresponds to a verified integration of the system model describing the system dynamics between two points of time at which measured data is available. In this paper, a Taylor model based integrator is used. Considering the state enclosures obtained in the prediction step, the correction step reconstructs states and parameters from the uncertain measurements with the help of a measurement model. The enclosures of states and parameters determined by the interval observer are consistent with both system and measurement models as well as all uncertainties.

CONSISTENCY TECHNIQUES FOR SIMULATION OF WASTEWATER TREATMENT PROCESSES WITH UNCERTAINTIES

Abstract In this paper, interval simulation methods are presented to determine guaranteed enclosures of state variables of an activated sludge process in biological wastewater treatment. This process is characterized by nonlinearities and uncertain but bounded parameters. In uncertain systems an axis-parallel interval box is mapped to a complexly shaped region in the state space that represents sets of possible combinations of state variables. The approximation of this complex region by a single interval box causes accumulation of overestimation over simulation time. The algorithm presented in this paper minimizes this so called wrapping effect by applying consistency techniques, to avoid safety-critical states.

Towards the Development of an Interval Arithmetic Environment for Validated Computer-Aided Design and Verification of Systems in Control Engineering

In this paper, an overview of the potential use of validated techniques for the analysis and design of controllers for linear and nonlinear dynamical systems with uncertainties is given. In addition to robust pole assignment for linear dynamical systems with parameter uncertainties, mathematical system models and computational techniques are considered in which constraints for both state and control variables are taken into account. For that purpose, the use of interval arithmetic routines for calculation of guaranteed enclosures of the solutions of sets of ordinary differential equations and for the calculation of validated sensitivity measures of state variables with respect to parameter variations are discussed. Simulation results as well as further steps towards the development of a general-purpose interval arithmetic framework for the design and verification of systems in control engineering are summarized.

Robust controller design for bounded state and control variables and uncertain parameters using interval methods

In this paper, a new interval arithmetic approach for analysis and design of robust controllers of systems with bounded state and control variables is presented. Based on guaranteed simulations of nonlinear dynamical systems with uncertain parameters, bounds of the admissible range of the control variables are calculated. During the computation of these bounds, time-dependant restrictions of the state variables arc considered as well as limitations of the control variables. To point out the advantages of this approach, it is applied to a subsystem model of the activated sludge process in biological wastewater treatment describing the reduction of biodegradable organic matter.