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 sigma-stabilising a general class of second-order linear time invariant systems with dead-time using a proportional-integral-retarded (PIR) controller is considered. The sigma-stability of a system determines the exponential decay in its response. Here the sigma-stability of the closed-loop system is ensured by assigning up to four dominant real roots at $-\sigma $-sigma. 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.