Tuning of a time‐delayed controller for a general class of second‐order linear time invariant systems with dead‐time

In this study, the problem of ‐stabilising a general class of second‐order linear time invariant systems with dead‐time using a proportional–integral–retarded (PIR) controller is considered. The ‐stability of a system determines the exponential decay in its response. Here the ‐stability of the closed‐loop system is ensured by assigning up to four dominant real roots at . For the tuning of the controller gains an extensive analysis of all the possible allocations of the gains according to the response of the closed‐loop system is presented. The D‐partition method is used to provide important insight into the problem. As a consequence of this analysis, to achieve the desired decay rate, exact analytic expressions for tuning the PIR controller parameters are given. To illustrate the theoretical results obtained, the under‐actuated mechanical system called inverted pendulum is used.