The dynamic behavior of a Duffing-Holmes oscillator subjected to a Hybrid Position Feedback (HPF) controller is investigated. The so-called hybrid controller is a combination of two controllers, namely, the Negative Position Feedback (NPF), and Positive Position Feedback (PPF) controllers. The controller uses the inertial properties of the structure around its stable positions to achieve large displacements by effectively destabilizing the system using an NPF controller. Once the unstable equilibrium is reached, the system is stabilized to the target stable equilibrium using the PPF controller. This dynamic switch of controllers creates the HPF control concept, which specifically enables the monotonic and controlled transition between the states of bistable systems such as the Duffing-Holmes oscillator. This concept can be implemented to morphing structures such as bistable wings, wind turbine blades, and deployable structures. In this paper, a detailed response type and stability analyses of a Duffing-Holmes oscillator controlled by the HPF controller are presented. First, the response types for the components of the HPF, NPF and PPF controllers are analyzed individually. For the NPF controller, three response types are defined. These are intra-well, single cross-well, and multiple cross-well response types describing the possible responses. For the PPF controller, only two response types are defined. These are stabilized and not-stabilized, since the role of the PPF controller is to generate an attractor to the desired stable equilibrium. Finally, the complete HPF controller is analyzed in terms of response type. In this case, three response types are defined: intra-well, single cross-well and multiple cross-well. The paper summarizes all the response types with detailed analyses, and recommends controller parameters for best control performance.