  This paper derives new formulations for designing dominant pole placement
based proportional-integral-derivative (PID) controllers to handle second order
processes with time delays (SOPTD). Previously, similar attempts have been made
for pole placement in delay-free systems. The presence of the time delay term
manifests itself as a higher order system with variable number of interlaced
poles and zeros upon Pade approximation, which makes it difficult to achieve
precise pole placement control. We here report the analytical expressions to
constrain the closed loop dominant and non-dominant poles at the desired
locations in the complex s-plane, using a third order Pade approximation for
the delay term. However, invariance of the closed loop performance with
different time delay approximation has also been verified using increasing
order of Pade, representing a closed to reality higher order delay dynamics.
The choice of the nature of non-dominant poles e.g. all being complex, real or
a combination of them modifies the characteristic equation and influences the
achievable stability regions. The effect of different types of non-dominant
poles and the corresponding stability regions are obtained for nine test-bench
processes indicating different levels of open-loop damping and lag to delay
ratio. Next, we investigate which expression yields a wider stability region in
the design parameter space by using Monte Carlo simulations while uniformly
sampling a chosen design parameter space. Various time and frequency domain
control performance parameters are investigated next, as well as their
deviations with uncertain process parameters, using thousands of Monte Carlo
simulations, around the robust stable solution for each of the nine test-bench
processes.
