Analysis and numerical solution of structured and switched differential-algebraic systems.

*(English)*Zbl 1213.65116
Berlin: TU Berlin, Fakultät II, Mathematik und Naturwissenschaften (Diss.). 281 p. (2008).

Summary: The numerical simulation of complex dynamical systems nowadays plays an important role in technical applications. Typically, the dynamical systems arising from automatic model generating tools are described by differential-algebraic equations (DAEs), i.e., by differential equations describing the dynamical behavior of the system coupled with algebraic constraints forcing these dynamics onto a specific manifold. Besides the already known difficulties in solving DAEs numerically, as e.g. order reduction of numerical methods, instabilities, or the drift-off from the solution manifold, complex systems additionally can contain higher order differential-algebraic equations, or the system can switch between different system configurations or operation modes based on certain transition conditions. Further, the coefficient matrices of the DAEs can exhibit certain structures.
In this thesis we discuss the analysis as well as the numerical solution of structured and switched differential-algebraic equations. Basically, the thesis focuses on three topics.

First, second order differential-algebraic equations are considered. It is known that the classical order reduction that is used to transform higher order ordinary differential equations into first order systems leads to a number of difficulties when applied to DAEs, as e.g. an increase in the index of the system or even the loss of solvability. In this thesis, an index reduction method for linear as well as nonlinear second order DAEs based on differentiation of the second order system is derived that allows to construct an equivalent second order system of low index in a numerical feasible way. This approach also enables the transformation into so-called trimmed first order form of low index and an explicit representation of solutions in the case of linear time-invariant second order systems.

The second topic involves structured differential-algebraic systems. As the structure of the coefficient matrices represent certain physical properties of the system the symmetry structure should be preserved during the numerical solution. In particular, linear differential-algebraic systems with symmetric and self-adjoint coefficient matrices are considered and structure preserving condensed forms for symmetric and self-adjoint linear DAEs are derived. It turns out that a structure preserving strangeness-free formulation for symmetric and self-adjoint systems only exists if the strangeness index of the system is lower or equal one. For symmetric systems we need in addition strong assumptions on the coefficient matrices in order to preserve the symmetry. Further, a structure preserving index reduction method based on so-called minimal extension is investigated that allows a structure preserving numerical treatment.

The third topic involves switched or so-called hybrid differential-algebraic systems that switch between different modes of operation based on certain transition conditions. First, we examine the formulation of hybrid systems and the existence and uniqueness of solutions after switching. Afterwards, the numerical solution of hybrid systems is considered. In particular, a consistent reinitialization after mode switching is considered that allows a continuation of the previous solution in a physical reasonable way by fixing certain components of an initial value vector, and the treatment of chattering behavior during the numerical simulation using so-called sliding mode simulation is studied. A hybrid mode controller is implemented for the numerical solution of hybrid differential-algebraic systems that organizes mode switching and allows sliding mode simulation. The functionality of the mode controller is illustrated by several examples, in particular, considering electrical circuits with switching elements and mechanical systems with dry friction phenomena. Further, the basic concepts for the control of linear hybrid descriptor systems are considered.

First, second order differential-algebraic equations are considered. It is known that the classical order reduction that is used to transform higher order ordinary differential equations into first order systems leads to a number of difficulties when applied to DAEs, as e.g. an increase in the index of the system or even the loss of solvability. In this thesis, an index reduction method for linear as well as nonlinear second order DAEs based on differentiation of the second order system is derived that allows to construct an equivalent second order system of low index in a numerical feasible way. This approach also enables the transformation into so-called trimmed first order form of low index and an explicit representation of solutions in the case of linear time-invariant second order systems.

The second topic involves structured differential-algebraic systems. As the structure of the coefficient matrices represent certain physical properties of the system the symmetry structure should be preserved during the numerical solution. In particular, linear differential-algebraic systems with symmetric and self-adjoint coefficient matrices are considered and structure preserving condensed forms for symmetric and self-adjoint linear DAEs are derived. It turns out that a structure preserving strangeness-free formulation for symmetric and self-adjoint systems only exists if the strangeness index of the system is lower or equal one. For symmetric systems we need in addition strong assumptions on the coefficient matrices in order to preserve the symmetry. Further, a structure preserving index reduction method based on so-called minimal extension is investigated that allows a structure preserving numerical treatment.

The third topic involves switched or so-called hybrid differential-algebraic systems that switch between different modes of operation based on certain transition conditions. First, we examine the formulation of hybrid systems and the existence and uniqueness of solutions after switching. Afterwards, the numerical solution of hybrid systems is considered. In particular, a consistent reinitialization after mode switching is considered that allows a continuation of the previous solution in a physical reasonable way by fixing certain components of an initial value vector, and the treatment of chattering behavior during the numerical simulation using so-called sliding mode simulation is studied. A hybrid mode controller is implemented for the numerical solution of hybrid differential-algebraic systems that organizes mode switching and allows sliding mode simulation. The functionality of the mode controller is illustrated by several examples, in particular, considering electrical circuits with switching elements and mechanical systems with dry friction phenomena. Further, the basic concepts for the control of linear hybrid descriptor systems are considered.

##### MSC:

65L80 | Numerical methods for differential-algebraic equations |

34A09 | Implicit ordinary differential equations, differential-algebraic equations |