(186g) PID Design for Optimal Closed-Loop Response with Specified Gain and Phase Margins for Second Order Systems

The Proportional-Integrative-Derivative(PID) control has been widely used in industry and
large amount of research work has been done on PID design. Most modern PID tuning methods
are model-based and there is always model mismatch. Thus, robustness plays an important role
in PID design. In practice, gain and phase margins (GPM) serve as important indicators of
system robustness. Intensive research has been done on designing PID to meet gain and phase
margin specifications.

In general, the GPM based PI design process usually leads to 4 equations and 4 unknowns.
Numerical methods can be used to directly solve the problem, or approximation on arctan
function can be adopted to simplify the problem and then the analytical solution can be obtained.
For PID, there are 5 unknowns for total 4 equations, so extra approximations on derivative term
are usually made to reduce the number of unknowns. The extra degree of freedom can also be
used to achieve another objective, such as the minimization of the integral square error (ISE).
The internal model control (IMC) tuning algorithm can also be combined with GPM, which
significantly simplifies the problem since there is only one tuning parameter for IMC. The
analytical solution can be easily obtained and suitable for online application such as adaptive
control. However, the down side is that there would be only 3 unknowns for 4 equations, thus
gain margin and phase margin are not independent in such case and the robustness criterion
can not be exactly met. So far, most research is based on first-order plus time-delay (FOPTD)
model, and very few work has been done on second-order plus time-delay (SOPTD) system
due to its complexity. Ho et al. explored the PID tuning method with specified GPM for the
under-damped second-order system with approximations on natural frequency of the controller
ω_c and the arctan function.

Besides robustness, good closed-loop controller performance is also required in PID tuning,
and usually there is a trade-off between performance and robustness. The maximum value
of closed-loop amplitude ratio M_T and bandwidth ω_bare important closed-loop performance
criteria. A larger M_T leads to a faster response, but also implies a larger overshoot and a lower
robustness level, so it should be properly bounded. The bandwidth ω_b, however, should be
as large as possible, since larger bandwidth values result in faster closed-loop responses and
smaller settling time. However, there is no report yet on directly applying performance criteria
M_T and ω_bwith robustness criteria GPM in PID tuning. Thus, the motivation of this work
is to develop a novel robust PID tuning method by using M_T and ω_bas performance criteria
combined with the robustness criteria GPM for SOPTD systems. In this work, M_T and ω_b
are adopted as performance criteria combined with GPM as robustness criteria in PID tuning
for the first time. Then the PID design for SOPTD systems is formulated into a nonlinear
optimization problem to maximize the bandwidth with constraints on both GPM and M_T , so
that the settling time is minimized, and criteria on robustness and closed-loop performance are
both satisfied simultaneously.

  In the theory part of this work, the open-loop amplitude ratio and the phase equations
are explicitly given for SOPDT model and PID in parallel form without any approximation,
so that gain and phase margin can be calculated. Further more, the closed-loop amplitude
ratio equation is also derived to calculate M_T and ω_b. Then the PID tuning method based on
nonlinear optimization is proposed and some simulation results would be discussed in the end
of the paper.