Reduced Balancing Transformations for Large Nonnormal State-Space Systems

Abstract

Model reduction of nonnormal state-space input/output systems requires a dual-state (state and adjoint state) approach. With this approach the method of balanced truncation provides the baseline of reduced balanced transformation, allowing evaluation of other more numerically efficient forms of reduced balanced transformation that are obtained by direct methods. The shortcoming of balanced truncation is the numerical intractability to generate the full-order gramians and balanced transformation for large systems. In practice, reduced balanced transformation for large systems can be obtained from empirical-based methods using empirical-based gramians. Empiricalmodelreductionsmethodsincludethewell-knownproperorthogonaldecomposition,thebalancedproper orthogonal decomposition, modified proper orthogonal decomposition, and pseudobalanced proper orthogonal decomposition. All are discussed in this paper. In our error analyses, the Karhunen–Loeve expansion based on proper orthogonal decomposition is extended to the generalized Karhunen–Loeve expansion based on balanced proper orthogonal decomposition, which is applicable for nonnormal systems. The error analyses highlight the importance of the dual-state-reduction approach. For model-reduction performance evaluation, the complex Orr–Sommerfeld input/output system is used.