Model reduction for periodic descriptor systems
The first part of this project is to analyze periodic descriptor systems both in continuous and discrete-time case. Simulations and analysis of such periodic systems can be unacceptably expensive and time-consuming when the systems are very large. Hence, model reduction is an efficient tool which helps scientists and engineers to replace such a large periodic model by a smaller model which is amenable to fast and efficient simulation and still preserve the input-output behavior of the original large model as good as possible. A core part of this project work is to develop numerical algorithms for efficient model reduction for these periodic descriptor systems. Mainly the projection based model reduction approaches are considered in this project work.
We develop efficient implementations of Krylov subspace based projection methods for model order reduction of problems that result from the linearization of nonlinear circuit problems and the resulting models are linear periodic time-varying (LPTV). The algorithmic realization of the method employs recycling techniques for shifted Krylov subspaces and their invariance properties. The efficiency and accuracy of the developed algorithm is illustrated by real-life numerical examples and compared to other Krylov based projection methods used for model reduction.
Balancing based projection technique is applied for model reduction of these large sparse discrete-time LPTV descriptor systems. The projected periodic Lyapunov equations that appear in the stability analysis and in model reduction of these periodic descriptor systems are first solved analytically (using direct solvers). For large scale systems, these projected periodic Lyapunov equations are solved using iterative techniques, e.g., ADI method, Smith method. The main task of these iterative computations is to preserve the cyclic structure of system matrices in each iteration steps of the periodic solutions. Our proposed algorithms analyze the cyclic structure of the matrices arising in the iterative computations of the periodic solutions of the projected discrete-time periodic Lyapunov equations. Low-rank version of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations. A further development of these algorithms is in progress.