## 1 Introduction
In order to adapt the behavior of semi-active or active suspensions according to the road conditions, it is often assumed that all vehicles are equipped with sufficient sensors to accurately measure their respective states, which is not a realistic assumption, as it is impractical to expect mass-produced vehicles to have such advanced features [26]. In [27], Lyapunov function-control barrier function-quadratic programming is used for active suspension control. In [28], a novel optimal control strategy for regenerative active suspension system to enhance energy harvesting is described. These works consider that some states, such as the vertical displacement and velocity of the sprung and unsprung mass, as well as tire deflection, to be measurements that can be readily obtained. In order to address the issue of incomplete state information, it is possible to incorporate an observer to estimate those vehicle states that cannot be directly measured. In [29], an active suspension system uses an unknown input observer to estimate the unmeasured vehicle states. In [30], an observer that estimates the effect of the road disturbance is presented. Nevertheless, the incorporation of supplementary observation stages leads to an increase in overall system complexity. In order to provide a more straightforward approach, it is possible to design controllers that rely solely on the data obtained from the available sensors, at the cost of a more complex design stage [31]. In this particular case, the majority of research findings are related to active suspension systems. This is due to the fact that, from a theoretical standpoint, they have fewer limitations and complexity than semi-active suspensions. In [32; 33; 34; 35], different methodologies for active suspensions are presented, where the actuator force can be generated as a function of the vertical acceleration of the sprung mass and/or suspension deflection. In general terms, there is a notable lack of research on semi-active suspensions that considers the inherent incompleteness of vehicle state information. In [36], a hybrid damping control of magnetorheological semi-active suspension is used; however, a filter is used to estimate the unmeasured vehicle states. In [37], an adaptive backstepping control design for semi-active suspension with MR dampers is presented; however, the totality of the vehicle states is assumed to be measured. In [38], a MR damper-based semi-active suspension system that assumes all vehicle states available is designed using an LQR controller incorporating NAGA-II algorithm. In consideration of the aforementioned reasons, this paper presents a methodology for the design of semi-active suspensions utilizing MR dampers, with due consideration of the limited availability of vehicle sensors. The main contributions are summarized as follows:An improved model for the MR damper behavior is presented and validated from experiments. The model accurately describes the hysteretic nature and the effect of the applied current on the MR damper, demonstrating superior performance compared to other state-of-the-art damper models, such as Bingham or bi-viscous models. * Convex conditions are obtained to design a robust static output feedback MR damper current controller, only utilizing suspension deflection information, which can be practically measured by a Linear variable differential transformer (LVDT). Stability and robustness of the proposed semi-active suspension system are guaranteed under Lyapunov and \(\mathcal{H}_{\infty}\) criteria, respectively. An evaluation of the vehicle behavior under two road scenarios demonstrates that the proposed MR damper-based semi-active suspension system exhibits enhanced performance compared to a passive suspension system. The remainder of this article is structured as follows. In Section 2, the problem to be addressed is formulated, with the proposed MR damper model presented in Section 2.1, the semi-active suspension model shown in Section 2.2, and the MR current controller design described in Section 2.3.