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Model reduction with balanced realizations: An error bound and a frequency weighted generalization

An error bound for reduced order models obtained from internally balanced realizations is derived. The bound is that the infinity norm of the frequency response of the difference between the full and reduced order models is bounded by twice the sum of the balanced grammian's singular values that correspond to the truncated states of the balanced realization. The importance of a frequency weighted model reduction and the infinity norm for control system applications is discussed. A frequency weighted balanced realization which depends on specified input and output model reduction weightings is defined. Results for an example are compared using the weighted and unweighted balancing model reduction techniques.