A case study for the open question: disturbance decoupling problem for singular systems by output feedback

We study the disturbance decoupling problem for singular systems Ex˙=Ax+Bu+Gd, y=Cx, z=Hx with left invertibility. We give necessary and sufficient conditions for the existence of a solution to the disturbance decoupling problem with or without stability by output feedback that also makes the resulting closed-loop system regular and/or of index at most one. All results are proved based on a condensed form that can be computed using orthogonal transformations, i.e., can be computed in a numerically stable way