Publication Date

Summer 2011

Degree Type


Degree Name

Master of Science (MS)


General Engineering


Peter Reischl


PID Tuning, Root Locus, Time Delay

Subject Areas

Electrical engineering; Mechanical engineering; Chemical engineering


This thesis research uses closed-loop pole analysis to study the dynamic behavior of proportional-integral-derivative (PID) controlled feedback systems with time delay. A conventional tool for drawing root loci, the MATLAB function rlocus() cannot draw root loci for systems with time delay, and so another numerical method was devised to examine the appearance and behavior of root loci in systems with time delay.

Approximating the transfer function of time delay can lead to a mismatch between a predicted and actual response. Such a mismatch is avoided with the numerical method developed here. The method looks at the angle and magnitude conditions of the closed-loop characteristic equation to identify the true positions of closed-loop poles, their associated compensation gains, and the gain that makes a time-delayed system become marginally stable. Predictions for system response made with the numerical method are verified with a mathematical analysis and cross-checked against known results.

This research generates tuning coefficients for proportional-integral (PI) control of a first-order plant with time delay and PID control of a second-order plant with time delay. The research has applications to industrial processes, such as temperature-control loops.