Reduction of large-scale systems via generalized Gramians

System analysis and control design of large-scale dynamic systems are important in many applications. However, due to high dimensionality of the system, practical implementation of the theoretic results in large-scale systems is a difficult and expensive task, even with the aid of modern computers. In this paper, a different approach is undertaken to overcome the computational difficulty which is often associated with the nearly singular large-scale mathematical models. Instead of using the standard dynamic equations to represent a large-scale system, mathematical models in the descriptor form (generalized dynamic equations) are used. It is shown that via numerically reliable algorithms, the generalized dynamic model can be balanced in the sense that the balanced model is equally controllable and observable. Moreover, the balancing transformation can be obtained by solving generalized Lyapunov equations for observability and controllability Gramians, Based upon the balanced model, reduced-order models which closely match the input-output behavior of the system can be derived. Computer simulations of a power machine system will be presented to illustrate the effectiveness of the model reduction algorithm.