  This paper studies the class of logarithmically completely monotonic (LCM)
functions. These functions play an important role in characterising externally
positive linear systems which find applications in important control problems
such as non-overshooting reference tracking. Conditions are proposed to ensure
a rational function is LCM, a result that enables the known space of linear
continuous-time externally positive systems to be enlarged and an efficient and
optimal pole-placement procedure for the monotonic tracking controller
synthesis problem to be developed. The presented conditions are shown to be
less conservative than existing approaches whilst being computationally
tractable.
