Electro-optical modulators are effectively used for ultrafast pulse lasers operation control. The scheme of picosecond pulse-periodic high-peak-power Nd:YAG lasers is composed of an active-passive mode-locked and negative feedback-controlled master oscillator and regenerative amplifier based on common end-diode-pumped Nd:YAG crystal. A double-crystal thermally compensated Pockels cell based on KTP crystals of the Y-cut direction is employed as a key control element. The cell was assembled using a pair of equal-length crystals grown according to high-resistivity technology. The scheme provides output pulses with energy up to 1.6 mJ, a duration of 25 ps at repetition rates tunable from 0 to 200 Hz. The laser operation stages are analyzed in detail. The scheme looks attractive and promising for developing advanced ultrafast laser systems with higher repetition rates, peak and, accordingly, average power levels. The Pockels cell based on KTP crystals expands the line of available fast electro-optical control elements, along with the previously used RTP ones. The factors limiting laser pulse energy and repetition rate are discussed. Parasitic nonlinear conversion in the crystals of the Pockels cell along the axis may play an essential role. The results of comparative measurements of the second and third harmonics made with the Pockels cells based on KTP and RTP crystals of both X-cut and Y-cut directions are presented. The minimum second and third harmonics efficiency levels observed in the Y-cut Pockels cells of the KTP crystal seem to be their important advantage.