The mammalian cochlear amplifier displays a unique nonlinear characteristic of amplifying or compressing the acoustic stimuli based on their level. Designing control algorithms for artificial hair cells (AHCs) with active sensing capabilities could mimic such dynamics. The present work focuses on developing a novel self-sensing active AHC scheme that implements the amplification/compression function of the AHCs and extends the authors previous designs for future cochlear implants or sensor design applications. The AHC functions near a Hopf bifurcation and varies the output piezoelectric voltage by a power-law relationship with the input. It is the first time that a self-sensing AHC is modeled as a quadmorph cantilever controlled by a phenomenological cubic damping controller. The AHCs control signal is supplied to a pair of its piezoelectric layers and the output of the AHC is the sensed voltage of the second piezoelectric pair. This is in contrast to the previous studies where the AHCs tip-velocity was measured with external sensors. The requirement of permanent external sensors in the system was a strict limitation that is eliminated in this work. An in-depth study of this novel AHC is conducted in this paper and AHCs implementation is examined experimentally in Part II of the paper.