Echelon Form Criterion for the Existence of a Regular Relative Degree of Linear Systems

For the linear time-invariant systems without a regular relative degree, this note presents a novel criterion based on the matrix echelon form for the existence of transformations making the transformed system have a regular relative degree. Moreover, a method for figuring out all of these transformations is established. Both the row and column relative degree cases are discussed. The proposed criterion is only concerned with the output controllability matrix in the row relative degree case and the input observability matrix in the column relative degree case. From a practical application perspective, the proposed criterion and computation method are easy to implement and incur low computational complexity.