A theoretical examination of closed-loop properties and tuning methods of single-loop PI controllers

The dependence of the dominant closed-loop poles on the controller parameters is quantitatively elucidated by a Taylor expansion about the critical (ultimate) gain. The leading expansion coefficients are estimated from the critical (ultimate) gain and frequency and one or two closed-loop measurements of the decay ratio and frequency of system response to set-point/load changes or natural disturbances. An explicit model for the process transfer function is not required. By relating controller performance criteria to the leading poles, optimal gain settings to achieve these criteria can then be determined. In the present work, three tuning methods of increasing accuracy (the modified Ziegler-Nichols rule and Methods A and B) are constructed to satisfy a performance criterion (D.R. = 0.25) and a stability consideration. Method A is presented in a convenient chart and is especially easy to use on line. Stability robustness as measured by the Doyle-Stein index and the estimated closed-loop frequency at the proposed setting are also presented in the same chart.