Proportional-Integral Control of First-Order Time-Delay Systems via Eigenvalue Assignment

A new design method is presented for proportional-integral (PI) controllers of first-order plants in the presence of time delays. In general, time delays can limit and degrade the achievable performance of the controlled system, and even induce instability. Thus, the PI gains should be selected carefully considering such effects of time delays. Unlike existing methods, the design method presented in this paper is based on solutions to delay differential equations, which are derived in terms of the Lambert W function. PI controllers for first-order plants with time delays are designed by obtaining the rightmost (i.e., dominant) eigenvalues in the infinite eigenspectrum of time-delay systems and assigning them to desired positions in the complex plane. The process is possible due to a novel property of the Lambert W function. The controllers designed using the presented method can improve the system performance and successfully stabilize an unstable plant. Also, sensitivity analysis of the rightmost eigenvalues is conducted, and the results compare favorably with those of a prediction-based method for eigenvalue assignment. Extension to design of PI-differential controllers is discussed with examples.