A new procedure for transforming of time-varying linear systems to a companion canonical form via D-similarity transformations

The authors discuss the transformation of time-varying vector linear dynamical systems of the form dx/dt=A( t)x, where A(t) is an n×n `time-varying´ matrix, to a companion canonical form dz/dt=C(t)z, where C(t) is a companion canonical matrix, using a D-similarity transformation. An improved procedure for obtaining the transformation matrix is developed, which does not rely on quasicompanion forms. It is noted that the results obtained have immediate applications in the derivation of analytical solution techniques and stability criteria for general time-varying linear systems dx/dt=A(t)x