Characterizing PID Controllers for Linear Time-Delay Systems: A Parameter-Space Approach

We focus on the proportional-integral-derivative (PID) controller design for linear time-delay systems. All the controller gains ($k_P$, $k_I$, and $k_D$) and the delay ($\tau$) are treated as free parameters and no particular constraints are imposed on the controlled plants. Such a problem (involving totally four free parameters) is of theoretical as well as practical importance, but, to the best of the authors’ knowledge, it has not been fully explored. First, we will develop an algebraic algorithm to solve the complete stability problem w.r.t. $\tau$. Consequently, for any given PID controller vector $(k_P, k_I, k_D)$, the distribution of ${NU}(\tau)$ (${NU}(\tau)$ denotes the number of characteristic roots in the right-half plane, as a function of $\tau$) can be accurately obtained and the exhaustive stability range of $\tau$ may be automatically calculated. Next, a global understanding of the distribution of ${NU}(\tau)$ over the whole $(k_P, k_I, k_D)$-space may be achieved and all structural changes regarding the ${NU}(\tau)$ distribution can be analytically determined. To achieve such a goal, a complete positive real root classification (for some appropriate auxiliary characteristic equation) will be explicitly proposed. Finally, we will give a new methodology, a new parameter-space approach, for determining the stability set in the $(k_P, k_I, k_D, \tau)$-space.