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The Control community relies heavily on good System Identification (SysID) for finding the plant models needed to develop a good controller. However over time the SysID process and controller development process have remained generally separate activities. One reason for this is that SysID and Control are disparate in their fundamental nature. For good SysID, one is faced with the challenge of persistently exciting plant dynamics; while a good control system attempts to constrain or suppress much of a plant's natural dynamics with desired dynamics. It is this inherent conflict that separates the two practices. But for many plants, their inherent instabilities makes trajectory collection difficult, thus there is a desire to perform data collection while under some simple form of control. Nevertheless, in order to perform solid SysID one must sample the very dynamics one might need to suppress; how then can this be achieved? This paper will explore the notation of persistent excitation, its relationship to phase space trajectories, and how one might recover the most nonlinear dynamics information for SysID while remaining under the linearizing based control region - the very place that those dynamics are most suppressed.