In this paper we establish the equivalence between least costly and traditional experiment design for control. We consider experiment design problems for both open and closed loop systems. In open loop, equivalence is established for three specific cases, relating to different parametrisations of the covariance expression (i.e. finite and high order approximations) and model structure (i.e. dependent and independently parameterised plant and noise models). In the closed loop setting, we consider only finite order covariance expressions. performance specifications for control are used to determine the bounds on the covariance expression for both the open and closed loop cases.