Application of $\tau$ Decomposition Method in PID Controllers of Second Order Time Delay Process

The stabilization of second order time delay process with proportional-integral-derivative (PID) controllers is investigated in the paper. The
$\tau$
decomposition method is utilized to decompose and characterize the controller parameter space. The division of controller parameter space and the decomposition of
$\tau$
axis are related to purely imaginary roots, which also affect the calculation of each stable interval. Based on the analytical results, the relevant analytical formulas are obtained, and the relationship between the controllers of different stability types and the initially stable is shown. Finally, the accuracy and validity were verified with experimental data analysis.