Energy Efficiency of Pulsed Actuations on Linear Resonators

Abstract

The objective of this paper is to show that under some circumstances, the sign of a sampled sinusoid sequences, briefly <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> , is optimal to provide maximum energy transfer to linear resonators in the context of discrete-time pulsed actuation at periodic times with bounded sequences. It will be proven that there is an optimal <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> sequence which maximizes the resonator amplitude at any given finite time, and that under some conditions, there is a sufficiently high time above which any <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> sequence at the resonant frequency of the resonator also provides a locally unique maximum, in the case of a lossless or leaky resonator. The tool used to prove this last result is a theorem of quadratic programming. Since pulsed digital oscillators (PDOs) under certain conditions produce <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> sequences, a variation of the standard PDO topology that simplifies these conditions is also proposed. It is proved that except for a set of initial conditions of the resonator of zero Lebesgue measure, the bitstream at the output of this topology produces a locally unique maximum in the total energy transferred to the resonator.