Ingo Hoffmann

A Case Study in Tool-Aided Analysis of Discretely Controlled Continuous Systems: The Two Tanks Problem

This case study compares the usefulness and applicability of eight computer tools with respect to the validation of logic control programs for continuous processes. Six simulation packages (Taylors Matlab-based simulator, Simulink/StateFlow, gPROMS, Shift, Dymola, and BaSiP) and two verification tools (SMV and HyTech) were applied to a single process control example with non-trivial continuous dynamics. The paper presents a detailed description of this benchmark example. Short introductions to the tools are given and the application results are decribed and discussed with emphasis on the suitability to the problem and the numerical performance.

Comparing Timed and Hybrid Automata as Approximations of Continuous Systems

We describe two approaches to derive Timed and Linear Hybrid Automata from continuous models given as systems of ordinary differential equations. A semiquantitative modeling method is applied which yields a qualitative description of the system dynamics and quantitative bounds for the residence times in the discrete states or the state variable derivatives. We discuss the problem of spurious trajectories and illustrate the two aproaches by means of a simple process engineering example. Results of a reachability analysis obtained with the tool HyTech are presented.

The entropy of 'strange' billiards inside n-simplexes

In the present work we investigate a new type of billiards defined inside n-simplex regions. We determine an invariant ergodic (SRB) measure of the dynamics for any dimension. In using symbolic dynamics, the (KS or metric) entropy is computed and we find that the system is chaotic for all cases n>2.

A Closer Look at the Uncertainty Relation of Position and Momentum

We consider particles prepared by a single slit diffraction experiment. For those particles the standard deviation σp of the momentum is discussed. We find out that σp=∞ is not an exception but a rather typical case. A necessary and sufficient condition for σp<∞ is given. Finally, the inequality σpΔx≥πℏ is derived and it is shown that this bound cannot be improved.

Identification of hybrid systems

Hybrid or continuous/discrete systems have attracted increased scientific attention in recent years. In fact, most technical systems consist of physical (sub)systems which are governed by continuous dynamics, continuous or discrete-time controllers, and logical systems which react to and trigger events in the continuous parts. Hybrid systems also arise from modeling of physical phenomena alone, e.g. in evaporation and other processes with phase transitions or in mechanical systems. Beginning with Taverninis work (1987), several rather theoretical modeling paradigms for such systems have been introduced. The dynamics of a hybrid system can simply be described as follows: Starting from an initial point the systems trajectory evolves in state space. Due to the discrete actions (phase transitions, control inputs, etc.) there exist switching manifolds in the state space. If the trajectory reaches one of these switching manifolds, the continuous dynamics of the system changes. After such a mode-change, a reinitialization of the continuous variables may be necessary, e.g. when conservation laws have to be considered. Thus the trajectory of a general hybrid dynamical system shows blips and/or jumps. In this contribution, we propose an approach to identify a model of a hybrid dynamical system from measured data.

Modular hierarchical models of hybrid systems

Hybrid or continuous/discrete systems have attracted increased scientific attention. In fact, most technical systems consist of physical (sub)systems which are governed by continuous dynamics, continuous or discrete-time controllers, and logical systems which react to and trigger events in the continuous parts. Hybrid systems also arise from modeling of physical phenomena alone, e.g. in evaporation and other processes with phase transitions or in mechanical systems with stiction/friction, hysteresis and backlash. We present a unified and general modeling paradigm which allows modularity and hierarchies of subsystems. These two features are essential for effective modeling of real systems because they allow the modeller to compose a large systems of basic blocks which represent physical subsystems or modules of the control software and can be reused in the same model as well as in other models.

Modeling hybrid dynamical systems

We present a modeling scheme for multivariate hybrid dynamical systems. From given time series embedded in appropriate state spaces we predict future outputs by making local linear fits in the neighbourhood of the actual state vectors. In particular, the proposed algorithm can be used online. Thus the quality of the forecast is improved by enclosing new measured data.