Decomposition for augmented forms of large-scale systems

Abstract

Closed-loop large-scale systems in augmented format, i.e. systems of large dimension with dynamic or static state and output feedback and state and input measurements, are considered. These systems are represented by partitioned matrices and graphs on the X, U and Y vertices. Then the concept of partial subsystems of the diagonal blocks R_xx, R_UU and R_YY ∥submatrices of the reachability matrix) and the concept of the accordingly condensed reachability matrix, are introduced to achieve a strongly connected component decomposition with reduced dimensions of the matrices and graphs used in the computation. The condensed reachability matrix is compared to the reachability matrix of the condensed initial matrix, and a two-stage procedure is provided which avoids calculation of the paths in the original large-scale system. The permutation matrix to block-triangularize the system is expressed, in terms of the respective permutations PM_XX, PM_UU and PM_YY used with the R_XX, R_UU and R_YY matrices and the PM with the condensed reachability matrix. Some illustrative examples are also presented