A semi-active Tri-D algorithm combining Coulomb, Reid and viscous damping mechanisms has recently been developed by the authors to drive piezoelectric friction dampers. The objective of this study is to analytically compare its performance with those of bang-bang control, clipped-optimal control, modulated homogeneous control, and a modified clipped-optimal control. Two far-field and two near-field historical earthquake records with various intensities and dominant frequencies were used in this study. All algorithms were evaluated with a ¼ scale 3-story frame structure in terms of reductions in peak inter-story drift ratio and peak floor acceleration. A piezoelectric friction damper was considered to be installed between a bracing support and the first floor of the frame structure. Both advantages and disadvantages of each control algorithm were discussed with numerical simulations. At near resonance, both bang-bang and clipped-optimal algorithms are more effective in drift reduction, and the modified clipped-optimal algorithm is more effective in acceleration reduction than both Tri-D and modulated homogeneous algorithms. But the latter requires less control force on the average. For a non-resonant case, the Tri-D and modulated homogeneous algorithms are more effective in acceleration reduction than others even with less control force required. Overall, the Tri-D and modulated homogeneous controls are effective in response reduction, adaptive, and robust to earthquake characteristics.