Mechatronic systems such as those arising in automotive applications are characterized by significant non-linearities, tight performance specifications as well as by state and input constraints which need to be enforced during system operation. This paper takes a view that model predictive control (MPC) and hybrid models can be an attractive and systematic methodology to handle these challenging control problems, even when the underlying process is not hybrid. In addition, the piecewise a fine (PWA) explicit form of MPC solutions avoids on-line optimization and can make this approach computationally viable even in situations with rather constrained computational resources. To illustrate the MPC design procedure and the underlying issues, we focus on a specific non-linear process example of a mass spring damper system actuated by an electromagnet. Such a system is one of the most common elements of mechatronic systems in automotive systems, with fuel injectors representing a concrete example. We first consider a linear MPC design for the mechanical part of the system. The approach accounts for all the constraints in the system but one, which is subsequently enforced via a state-dependent saturation element. Second, a hybrid MPC approach for the mechanical subsystem is analysed that can handle all the constraints by design and achieves better performance, at the price of a higher complexity of the controller. Finally, a hybrid MPC design that also takes into account the electrical dynamics of the system is considered.