Bernd Tibken

An iterative method for the finite-time bilinear-quadratic control problem

For bilinear control systems with quadratic cost, the so-called bilinear-quadratic problems, a feedback controller for the finite-time case is designed. An iteration procedure in close proximity to the Riccati approach is presented, and the proof of convergence is outlined. The potential of the new method is discussed, and the design procedure is illustrated for two examples.

A methodology for modeling inter-company supply chains and for evaluating a method of integrated product and process documentation

Due to the increasing complexity in products and in the resulting enterprise processes, new concepts for process optimization are necessary. One innovative concept, resulting from the requirements of the automotive industry, is the method of an integrated product and process documentation. In order to evaluate the benefits of an inter-company use of this documentation approach a simulation-based decision support system using a modular modeling concept for intra- and inter-company process chains has been developed. This concept is based on the Supply Chain Operations Reference-model (SCOR-model). The developed concept allows the evaluation of different configurations of process chains with different sets of parameters describing realistic production and inventory processes. Focussed on supply chains with high variant and complex products within the automotive industry first scenarios have been built up and the benefits of this integrated documentation approach can be visualized. This approach is the future basis for optimization methods coming from control theory.

Computing the domain of attraction for polynomial systems via BMI optimization method

Concerning a time-invariant, autonomous and polynomial system, we propose a new approach to estimate the domain of attraction (DOA) around an asymptotically stable equilibrium. Special emphasis is laid on elaborating the connections between modern results of real algebraic geometry and Lyapunovs stability theory, namely between the positive definite polynomials and the direct method of Lyapunov. The estimation problem can thereby be reduced to solving a sequence of low non-convexity-rank bilinear matrix inequalities (BMI) optimization problems. The BMI problem enables the calculation of the inner approximation to the relevant region of the DOA. We illustrate the presented approach with an example

Analysis of the dynamics in microactuators using high-speed cine photomicrography

This paper discusses the visualization of microdevice dynamics using high-speed cine photomicrograph. The visualization results in image sequences, which contain the kinematic position data of the moving parts in a microsystem. Utilizing a mathematical model, it is shown in this paper how to extract information about the forces and torques acting on the microdevice. Furthermore, the values of system parameters can be calculated if a dynamic model of the process is available and the identification of the parameters is unambiguous. In this paper, the model-based evaluation of high-speed-cinematographic image sequences is demonstrated for a microturbine and a microrelay. The measurements on the microturbine have been performed with a setup, which allows an image frequency up to 100 million frames per second for nonreproducible processes. The measurements of the microrelay have been taken with a second experimental setup using the stroboscopic principle. In the case of the microrelay, the complete identification of the dynamic model used by us is carried out. A first application of our identified model is the design of an optimal current pulse for the microrelay to damp oscillations.

Computation of subsets of the domain of attraction for polynomial systems

In this paper the asymptotic stability of polynomial nonlinear systems is investigated. Our aim is to determine a region in the state space, which is a subset of the domain of attraction. We use the Lyapunov stability theory and the theorem of Ehlich and Zeller to achieve this aim. The inequality conditions given by the theorem of Ehlich and Zeller enable us to calculate inner and outer approximations to the relevant region of attraction. Two nontrivial examples conclude the paper and show the effectiveness of the presented method.

Observability of nonlinear systems - an algebraic approach

In this contribution global observability of nonlinear systems is investigated. The main idea is to derive a criterion which allows to decide if two initial states of a system can be distinguished by the output of the system. This well known criterion is equivalent to an infinite set of nonlinear equations. If the system is globally observable this set of equations has a very special solution set which can be characterized algebraically. Based on results from commutative algebra some new algorithmic methods to describe the solution set are given. This allows a global observability analysis for polynomial systems and for some specific classes of nonlinear systems. The new method is applied to an example from the literature.

An interval arithmetic approach for the estimation of the domain of attraction

Since the analysis of asymptotic stability is not sufficient for safety-critical nonlinear systems, the analysis of the domain of attraction is the focus of current research. While several approaches for polynomial systems were presented in the last years, based on sums of squares/linear matrix inequalities (SOS/LMIs) techniques or using a sampling method, only a few approaches are available for non-polynomial systems. In this paper we present a new branch-and-bound-method using the Lyapunov stability theory. It is based on interval arithmetic and delivers lower and upper bounds for the maximum contour line of a given Lyapunov function, which bounds a subset of the domain of attraction. Our approach can applied for polynomial systems as well as for non-polynomial systems.

Simulation of controlled uncertain nonlinear systems

In this paper a new method for the simulation of uncertain nonlinear discrete time systems is given. The problem of simulation is reformulated as an optimization problem. Using methods of global optimization, namely, interval analysis, the global solution to the optimization problem stated is computed. Derivatives of the objective function of the optimization problem are computed using methods of automatic differentiation. The presented approach is implemented on a SPARC Station using the language PASCAL-XSC. For a typical example simulation results are presented and discussed.

Object Shape Recognition Using Mexican Hat Wavelet Descriptors

In this paper we present the results of object recognition using Mexican Hat wavelet descriptors. These descriptors are derived from the continuous wavelet transformation using the Mexican Hat function as mother wavelet. To describe an object shape we use an angle function derived from the extracted contour polygon. The angle function is periodical and independent from the size of the object, its position or orientation. It depends only on the starting point of the contour. The contour extraction is based here on the object oriented contour extraction method (OCE). The polygon representation uses the curvature dependent contour approximation method (CDCA). The continuous wavelet transform (CWT) is used in order to derive a suitable number of wavelet descriptors (WD).

The Ellipsoid Method for Systematic Bilinear Observer Design

Abstract Observer design for bilinear systems using Lyapunovs method leads to a system of linear matrix inequalities to be satisfied. In this paper the ellipsoid algorithm is applied to solve these inequalities and to design an observer. Emphasis lies on the case of bounded inputs. The efficiency of the new observer design is demonstrated for a continuous furnace process.