## 1 Introduction
In another work [13], PI, LQR and MPC controllers were applied to a laboratory-scale inverted pendulum. The results for the three controllers also demonstrated comparable performance. Because of the fast dynamics of the considered plant, the explicit form of MPC was used, and input constraints were accounted for in the formulation. The obtained results are expected since unconstrained MPC could be considered a form of LQR or state feedback control [14]. Similarly, the trio was compared in the control of a UAV in [15]. Considering system stability and robustness in this application, MPC outperformed LQR, which outperformed PID in simulation and real-time implementations on a Parrot Mambo Mini drone. Moreover, the tuning of LQR, in which stability is guaranteed by solving the algebraic Riccati equation, is less strenuous than PID. Additionally, LQR outperformed MPC in terms of setpoint tracking. Additionally, MPC tuning was also very strenuous for the UAV. Also, the computational requirements of MPC limited the choice of prediction horizon and sample interval. However, MPC proved very useful in accommodating control input and control rate constraints. Most of the existing studies either compared the controllers of interest on a single process or a subset of processes that do not allow for a generalization of the obtained solutions. However, in [16], five different PID controllers were compared while varying the dead-time-to-time constant ratio and the damping ratio of second-order system models. This variation of parameters allowed for the comparison of controllers for a broad class that could generalize the obtained results. Three controllers were identified to perform better for different values of damping ratio and dead-time to-time-constant ratio. These are the internal model control (IMC) based PID proposed by [PERSON] (IMC--Chien) [17], IMC with Maclaurin approximation of the dead-time (IMC--Mac) [18; 19] and PID with specified closed-loop performance (CS--PID) [20]. Despite the widespread application of PID and thousands of publications on its tuning, there is still the argument that PID tuning is poorly understood [21]. This is evident by the continuous research in the field, the inaccessibility of tuning rules (often hidden in proprietary material), non-uniform notation and no uniformly agreed controller structure. However, the three controllers adopted in [17] are applied widely across the industry, making them suitable for this work. They have also been shown to give good performances. This work adopts the approach of [16]. It compares the performance of the best of the three PID controllers highlighted earlier (IMC--Chien, IMC--Mac and CS--PID) with systematically tuned MPC controllers. Therefore, the approach will give insights into the best choice of controller for regulating the process. Hence, this work aims to determine when using either a PID or MPC on a typical second-order system, represented by (4), based on dead-time and damping ratio is more appropriate. In view of the above, the contributions of the paper could be highlighted as follows:  
1. The paper gives a systematic comparative study of industrially relevant PID controllers with an MPC formulated to simplify its tuning on SOPDT systems while considering the effects of dead time and damping. To the author's knowledge, such a study has not been undertaken before. 2. The development of guidance on the more appropriate controller to use (either PID or MPC) on second-order systems based on the combined effects of dead time and damping. 3. A departure from most works on such comparisons, this work focuses on using more realistic PID formulations routinely applied to real industrial processes. Moreover, the comparison is carried out with the best of the three PID controllers for any of the considered system models. These results are important because second-order systems represent a good percentage of controlled systems. This is true because SOPDT transfer functions can adequately represent most systems. The paper outline is as follows.