DFG-Forschungszentrum Berlin

Simulation and control of switched systems of differential-algebraic equations*
DFG-Forschungszentrum Technische Universität Berlin


Duration: June 2006 - May 2010
Project director: Prof. Dr. V. Mehrmann

Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany

Tel: +49 (0)30 - / 314 25736

email: mehrmann@math.tu-berlin.de
Researcher: L. Scholz

Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany

Tel: +49 (0)30 - / 314 23439  

email: lscholz@math.tu-berlin.de

Cooperation: P. Kunkel (Universität Leipzig, Germany)

T. Stykel (Technische Universität Berlin, Germany)

P. Hamann (Daimler-Chrysler AG)

Vu Hoang Linh (Hanoi National University)

S. Campbell (North Carolina State University)

L. Reichel (Kent State University)
Support: DFG Research Center "Mathematics for Key Technologies"

* Continuation of the project "Model reduction for large-scale systems in control and circuit simulation" with different research focus.


Project description Publications Guests Talks

Project description.

Background.

The modeling of the dynamics of complex technical systems in almost all areas of electrical, mechanical, chemical, or traffic engineering is today highly modularized, thus allowing the easy and efficient automatic generation of mathematical models. This has many advantages, but essentially all of the potential difficulties in the simulation, control and optimization of such dynamical systems are left for the numerical methods to take care of. Typically, the dynamics of the systems is described by differential-algebraic systems (DAEs) that have been widely studied, see e.g. [2, 4, 5, 18], but many challenges remain. A particular feature of many complex dynamical systems is that they are switched systems or hybrid systems, i.e. the mathematical model itself may change with time, depending on certain indicators. Typical examples are electronic circuits, where different device models are used for different frequencies, mechanical systems such as robot manipulators or automatic gear-boxes, biological systems which act different in different day cycles or depending on the nutrition, or traffic systems, which operate different depending on delays, see e.g. [3]. The mathematical theory of switched systems of DAEs, the control theory for such systems as well as the development of efficient and accurate numerical methods is still in its infancy [3, 16, 17]. Particular challenges arise from over- or underdeterminedness of the system, whenever a switch happens and the correct initialization of the model after a switch has taken place, see e.g. [6]. Further, many DAEs arising in technical application exploit a certain symmetry structure and often are second or higher order systems. For an efficient numerical solution it is necessary to reflect these structures in the solution methods. In addition, nonlinear systems of DAEs arising e.g., from circuit simulation are often large and badly scaled and of higher index, which causes stability problems in the numerical solution of the discretized system [2,5].

Expertise.

In different cooperations in the last 10 years a new algebraic theory for linear and nonlinear differential-algebraic control systems has been developed [8,9,10,12], new numerical methods for the simulation and control of over- and underdetermined DAE systems have been derived and have been implemented in production software that is now used in industry, [1,13,14,15]. A monograph that covers the theory, control and numerical solution of general DAEs has been published [11].

Research program.

We study the analysis and numerical solution of switched control systems. Main objectives are the existence and uniqueness of solutions at switching positions, the development of methods for correct initialization at switch points, mathematical methods to avoid (numerical) chattering (with and without hystereses), and to develop efficient and accurate numerical methods for model reduction, simulation and control of large scale switched systems. A basic difficulty in switched systems is that after a switch takes place, the model dimension and structure, its properties such as the index, and the number of algebraic or differential equations or redundancies may change. In order to guarantee existence and uniqueness of solutions, the transfer of the current state to the new model has to be made in a consistent way, so that after the switch existence and hopefully also uniqueness of solutions is guaranteed. Here, the general theory of over- and underdetermined systems of differential-algebraic equations provides the right framework and is extended to switched systems of DAEs. This means, in particular, that we must find ways to compute consistent initial conditions and to deal with non-uniqueness of solutions after a switch. The theory and numerical solution techniques for DAEs have been extended for the numerical simulation of switched systems of differential-algebraic equations and numerical integration methods, consistent initialization routines and sliding mode regularization techniques have been developed. Further, index reduction methods for general linear and nonlinear differential-algebraic systems of higher order have been developed based on derivative arrays, as well as trimmed first order formulations for higher order differential-algebraic systems. Furthermore, new stabilization techniques for numerical integrators have been developed and the Lyapunov, Bohl and Sacker-Sell spectral theory has been extended to nonautonomous DAEs.

Cooperation.

Within the DFG Research Center we cooperate with the projects "Passivation of linear time invariant systems arising in circuit simulation and electric field computation" (C. Mehl, V. Mehrmann) and we employ methods from the previous project "Numerical solution of large unstructured linear systems in circuit simulation" (V. Mehrmann). Further cooperation exits with the projects "Control and numerical methods for coupled systems" (T. Stykel), "Numerical solution of large nonlinear eigenvalue problems" (C. Mehl, V.Mehrmann) concerning structured eigenvalue problems, "Reduced-Order Modelling and Optimal Control of Robot Guided Laser Material Treatments" (D. Hömberg, T. Stykel) concerning the simulation of mechanical systems, and "Adaptive solution of parametric eigenvalue problems for partial differential equations" (C. Carstensen, C. Mehl and V. Mehrmann).

References.

[1] M. Arnold, V. Mehrmann and A. Steinbrecher: Index Reduction in Industrial Multibody System Simulation. MATHEON, DFG Research Center "Mathematics for Key Technologies", Berlin, Germany, No. 146, 2004.
[2] K. E. Brenan, S. L. Campbell and L. R. Petzold: Numerical Solution of Initial-Value Problems in Differential Algebraic Equations. Volume 14 of Classics in Applied Mathematics. SIAM, Philadelphia, PA, second edition, 1996.
[3] D. Chen: Stabilization of planar switched systems. Systems Control Lett., Vol. 51, pp. 79-88, 2004. 
[4] E. Eich-Soellner and C. Führer: Numerical Methods in Multibody Systems. B. G. Teubner Stuttgart, 1998.
[5] E. Hairer and G. Wanner: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, second edition, 1996.
[6] P. Hamann: Modellierung und Simulation von realen Planetengetrieben. Diplomarbeit, TU Berlin, in cooperation with DaimlerChrysler AG, 2003.
[7] P. Hamann and V. Mehrmann: Numerical solution of hybrid systems of differential-algebraic equations. Comput. Methods Appl. Mech. Engrg., Vol. 197 (6-8), pp. 694-705, 2008.
[8] A. Ilchmann and V. Mehrmann: A behavioural approach to linear time-varying descriptor system. Part 1. General Theory. SIAM J. Cont., 2005.
[9] A. Ilchmann and V. Mehrmann: A behavioural approach to linear time-varying descriptor system. Part 2. Descriptor Systems. SIAM J. Cont., 2005.
[10] P. Kunkel and V. Mehrmann: Analysis of over- and underdetermined nonlinear differential-algebraic systems with application to nonlinear control problems. Math. Control, Signals, Sys., Vol. 14, pp. 233-256, 2001.
[11] P. Kunkel and V. Mehrmann: Differential-Algebraic Equations - Analysis and Numerical Solution. EMS Publishing House, Zürich, Switzerland, 2006.
[12] P. Kunkel, V. Mehrmann and W. Rath: Analysis and Numerical solution of control problems in descriptor form. Math. Control Signal, Sys., Vol. 14, pp. 29-61, 2001.
[13] P. Kunkel, V. Mehrmann, W. Rath and J. Weickert: GELDA: A Software Package for the Solution of General Linear Differential Algebraic equations. SIAM J. Sci. Comput., Vol. 18, pp. 115-138, 1997.
[14] P. Kunkel, V. Mehrmann and S. Seidel: A MATLAB package for the numerical solution of General Nonlinear Differential-Algebraic equations. Preprint, Institut für Mathematik, TU Berlin, 2005.
[15] P. Kunkel, V. Mehrmann and I. Seufer: GENDA: A software package for the numerical solution of General Nonlinear Differential-Algebraic equations. Preprint 730, Institut für Mathematik, TU Berlin, 2002.
[16] J. Lygeros, G.J. Pappas and S. Sastry: An Introduction to Hybrid System Modeling, Analysis, and Control. Preprints of the First Nonlinear Control Network Pedagogical School, Athens, Greece, pp. 307-329, 1999.
[17] S. McIlraith, G. Biswas, D. Clancy and V. Gupta: Hybrid System Diagnosis. Third Intl. Workshop on hybrid systems, pp. 282-295, 2000.
[18] P. J. Rabier and W. C. Rheinboldt: Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint. SIAM, Philadelphia, PA 19104-2688, USA, 2000.

Publications.

Publications of previous project phase.

Guests.


Talks.