Today in the scientific and technological world, physical and artificial processes are often described by mathematical models which can be used for simulation, optimization or control. As the mathematical models get more detailed and different coupling effects are required to include, usually the dimension of these models become very large. Such large-scale systems lead to large memory requirements and computational complexity. To handle these large models efficiently in simulation, control or optimization model order reduction (MOR) is essential. The fundamental idea of model order reduction is to approximate a large-scale model by a reduced model of lower state space dimension that has the same (to the largest possible extent) input-output behavior as the original system. Recently, the system-theoretic method Balanced Truncation (BT) which was believed to be applicable only to moderately sized problems, has been adapted to really large-scale problems. Moreover, it also has been extended to so-called descriptor systems, i.e., systems whose dynamics obey differential-algebraic equations. In this thesis, a BT algorithm is developed for MOR of index-1 descriptor systems based on several papers from the literature. It is then applied to the setting of a piezo-mechanical system. The algorithm is verified by real-world data describing micro-mechanical piezo-actuators. The whole algorithm works for sparse descriptor form of the system. The piezo-mechanical original system is a second order index-1 descriptor system, where mass, damping, stiffness, input and output matrices are highly sparse. Several techniques are introduced to reduce the system into a first order index-1 descriptor system by preserving the sparsity pattern of the original models. Several numerical experiments are used to illustrate the efficiency of the algorithm.

Today in the scientific and technological world, physical and artificial processes are often described by mathematical models which can be used for simulation, optimization or control. As the mathematical models get more detailed and different coupling effects are required to include, usually the dimension of these models become very large. Such large-scale systems lead to large memory requirements and computational complexity. To handle these large models efficiently in simulation, control or optimization model order reduction (MOR) is essential. The fundamental idea of model order reduction is to approximate a large-scale model by a reduced model of lower state space dimension that has the same (to the largest possible extent) input-output behavior as the original system. Recently, the system-theoretic method Balanced Truncation (BT) which was believed to be applicable only to moderately sized problems, has been adapted to really large-scale problems. Moreover, it also has been extended to so-called descriptor systems, i.e., systems whose dynamics obey differential-algebraic equations. In this thesis, a BT algorithm is developed for MOR of index-1 descriptor systems based on several papers from the literature. It is then applied to the setting of a piezo-mechanical system. The algorithm is verified by real-world data describing micro-mechanical piezo-actuators. The whole algorithm works for sparse descriptor form of the system. The piezo-mechanical original system is a second order index-1 descriptor system, where mass, damping, stiffness, input and output matrices are highly sparse. Several techniques are introduced to reduce the system into a first order index-1 descriptor system by preserving the sparsity pattern of the original models. Several numerical experiments are used to illustrate the efficiency of the algorithm.