On the $\mathcal{T}$ Decomposition Method for PD Controllers of the Low Order Time Delay Process

The stabilization of Second Order Time Delay Process (SOTDP) with proportional derivative (PD) controllers is considered. A novel procedure following the line of the τ decomposition method is proposed to characterize the space of controller parameters. The division of the space of controller parameters are related to the determination of the purely imaginary roots (PIRs), and the calculation of stable intervals refers to the order of the PIRs. According to our result, analytical formulas on both topics are obtained and the stability switch (under some certain PD controller) versus delay is revealed. Finally, numerical simulations show the effectiveness and correctness.