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In this work we present a second s-z mapping (ST2) as an extension of a first one, the s-z mapping (ST1), presented in a previous work done by the authors. In that work, the ST1 mapping used only one tuning parameter (csi) ξ to: 1) map the (asymptoticall stable) left half s plane in the interior of the unitary circle in z plane; and 2) attain the asymptotical stabilization of a benchmark harmonic oscillator driven by a PD discrete controller with various sampling periods. In both tests, the ST1 mapping behaved better than other mappings listed in the literature (Tustin, Backward, Shneider-Kaneshige-Groutage, etc.). In this work we use two tuning parameters csil and csi2 in the ST2 mapping to see how both tests behave, including numerical simulations comparing the ST2 mapping with those other mappings ((ST1 and Tustin).

Many interesting phenomena occur when the discrete control is applied over a flexible structure like the aliasing and hidden oscillations. In this work a particular phenomenon called our attention because was a bit strange: increasing the sampling period of a discrete proportional plus derivative closed-loop control (by Tustin or Bilinear method) over a harmonic oscillator (our flexible benchmark plant) we noted that happened regions of stability and instability. Initially we imagined that from some high value of sampling period this control system would instable and it stay instable. This work study analytically this phenomenon and show graphically this bifurcation.