**8. Results and Discussion**

The following simulations are made to observe the advantages and performance of implementing the proposed LQR-MPC-LPV algorithm described in Section 7. The algorithm was tested in the Active Suspension system described in Section 2. Table 1 shows the specifications of the Active Suspension system obtained from [7].



A discretization is made to comply with the MPC paradigm using a sampling time of *Ts* = 10 ms. A prediction horizon of *Np* = 3 was defined after several tests using different prediction horizons were made. Using a larger prediction horizon resulted in longer optimization time and more inexact variable scheduling predictions while it does not exhibit a significant improvement in control performance. The control objective is to steer all the states to the origin while complying with the following constraints.

$$-12\text{ V} \le \mu(k) \le 12\text{ V}$$

$$-1\text{ cm} \le z\_v \le 1\text{ cm}$$

The results will be divided into frequency-domain results and time-domain results.

#### *8.1. Frequency-Domain Results*

To obtain a frequency analysis in the nonlinear Active Suspension system, an algorithm similar to the process of defining the system response as a describing function is used. To produce these results, Algorithm 1 presented in [35] is issued. Figure 5 presents the frequency response of the Active Suspension deflection gain using the proposed LPV-MPC-LQR algorithm. Figure 6 presents the frequency response of the acceleration of the chassis mass using the proposed LPV-MPC-LQR algorithm; also, the frequency response plots are compared with the ones presented in [36] which use an LPV gain scheduling approach.

The results show how the suspension deflection is attenuated at every frequency, which results in better road holding and driving conditions. Additionally, the chassis acceleration stays in values which guarantee passenger comfort. Compared to the frequency responses of the work of Fialho et al. [36] the MPC-LQR-LPV approach presents an improvement especially in terms of road holding, shown in Figure 5, without affecting the passenger comfort.
