Conditions on Decomposing Linear Systems With More Than One Matrix to Block Triangular or Diagonal Form

This technical note provides necessary and sufficient conditions to determine that a linear system with more than one matrix in its state-space representation can be decomposed into cascade or separate sub-systems. In order to perform such decomposition, one needs to determine a linear transformation matrix. Furthermore, the given conditions are adapted to a simple but effective condition to derive all possible scalar sub-systems for a given linear system. Numerical examples are provided to demonstrate the applicability of the presented results.