A Restarted Lanczos Algorithm For Model Reduction

When we design and reseach a control system, we need to build a dynamic mathematic model for the system in order to analysis the dynamic properties. Models of dynamical systems are useful primarily for two reasons: first for simulation and second for control. As the development of science and technology, many mathematic models arising from applications have high orders, such as the microcircuit simulation whose order can reach 10~6 and the International Space Station(ISS) which is a complex structure composed of many modules, Furthermore each module is described in terms of n ≈ 10~3 state variables. Because the simulation and control of large-scale systems cost huge storage and computational speed, and these problems are usually ill-conditioned, we need to simplify the model of the large-scale system in order to do simulation and control in short time. This simplification is called model reduction.In this paper, we proposed an implicitly restarted Lanczos algorithm for linearly time invariant stable systems , the propertites of the reduced model is revealed. For such systems, it is well known that oblique projections onto a Krylov subspace may generate unstable partial realizations when using it to get a transfer function of order m, where n >> m. Another limitation of classical Krylov subspace methods is that they generally generate partial realizations that contain nonessential modes. The new method proposed in this paper can greatly remedy these difficulties.Five sections are folded in this thesis, Section 1 gives us an introduction of the basic conceptions of reduced-order modeling and looks back the former works of this area. In section 2, we briefly introduce the classical Krylov subspace method in model reducing and in section 3 we discuss the restarted Lanczos algorithm for model reducing. Section 4 we analysis and prove some of the properties of the re-duced model . Numerical results are given in the last section.