*8.2. Time-Domain Results*

To obtain time-domain results using the proposed LPV-MPC-LQR control algorithm, the system was simulated using two different disturbances. Figures 7–9 present the suspension behavior when a bump disturbance of 5 cm is introduced. Figures 10–12 show the suspension behavior when driving through a sinusoidal road. The system was simulated using Matlab®; also, the software package YALMIP [37] using QP-solver SDPT3 was used for the MPC optimization. The results presented by [7] are included to make a comparison. Additionally, the results using the MPC with a frozen scheduling parameter approach without using the RLS to show the effect of the scheduling variable prediction in control performance are included.

**Figure 5.** Frequency response of the Active Suspension deflection gain.

**Figure 6.** Frequency response of the chassis acceleration.

**Figure 7.** Chassis displacement—Bump Disturbance (Blue—disturbance, Red—MPC-LQR, Yellow— Passive, Purple—H2, Green—MPC-Frozen).

**Figure 8.** Suspension Deflection—Bump Disturbance (Blue—disturbance, Red—MPC-LQR, Yellow— Passive, Purple—H2, Green—MPC-Frozen).

The results of both displacement and deflection show a better performance, which results in better road holding while maintaining passenger comfort. Additionally, the comfort exhibits improvement in terms of chassis acceleration as shown in Figure 9. Additionally, to express the results numerically, both the RMS value and the maximum value of the displacement of the chassis, the suspension deflection and the acceleration of the chassis are presented in Tables 2 and 3 respectively.


**Table 2.** RMS Values performance.


**Figure 9.** Chassis Acceleration—Bump disturbance (Blue—MPC-LQR, Red—Passive, Yellow—H2, Purple—MPC-Frozen).

Similar to the bump disturbance case, the proposed LPV-MPC-LQR control strategy exhibits better performance in both displacement and deflection, which results in better road holding. In terms of comfort, the acceleration of the chassis presented in Figure 12 shows a major improvement. Table 4 presents the peak values for the displacement of the chassis, the suspension deflection, and the acceleration of the chassis.



As shown in the previous figures, the proposed LPV-MPC-LQR control algorithm presents a better performance when compared with the H2 control strategy in both disturbance cases (bump disturbance and sinusoidal road disturbance). The RLS prediction of the future scheduling parameters have improved the control performance as well. Additionally, the proposed algorithm shows an appropriate optimization time with a worst optimization time of 930 ms and an average optimization time of 93 ms.

**Figure 10.** Chassis displacement—Sinusoidal Disturbance (Blue—disturbance, Red—MPC-LQR, Yellow—Passive, Purple—H2, Green—MPC-Frozen).

**Figure 11.** Suspension Deflection—Sinusoidal Disturbance (Blue—disturbance, Red—MPC-LQR, Yellow—Passive, Purple—H2, Green—MPC-Frozen).

**Figure 12.** Chassis Acceleration—Sinusoidal disturbance (Blue—MPC-LQR, Red—Passive, Yellow— H2, Purple—MPC-Frozen).

#### **9. Conclusions**

In this paper, a novel LPV-MPC-LQR control algorithm ensuring Quadratic Stability and with the inclusion of attraction sets was presented. This method runs an RLS algorithm to obtain the prediction of the future scheduling parameter values, which simplifies the prediction of the future states while ensuring Quadratic Stability. This application can cope with nonlinear systems that can be embedded into LPV representation and therefore reduce the complexity of the algorithm and allow fast execution times. This control strategy was designed and tested on a nonlinear Active Suspension system. The results show improvements to the performance of the Active Suspension in terms of road holding and passenger comfort. Future research works should deal with recursive feasibility analysis based on stability conditions, and robustness analysis. Optimization of the LPV-MPC-LQR algorithm to achieve faster execution times using techniques of the embedded systems will also be considered in future works as well.

**Author Contributions:** All Authors D.R.-G., A.F.-C., F.B.-C., D.S. and C.S. have contributed as follows: Conceptualization, D.R.-G., A.F.-C., F.B.-C., D.S. and C.S.; Methodology, D.R.-G., A.F.-C., F.B.-C., D.S. and C.S.; Software, D.R.-G., D.S. and C.S.; Validation, D.R.-G., A.F.-C., F.B.-C., D.S. and C.S.; Formal analysis, D.R.-G., A.F.-C., F.B.-C., D.S. and C.S.; Investigation, D.R.-G., A.F.-C., F.B.-C., D.S. and C.S.; Writing—original draft preparation, D.R.-G. and A.F.-C.; Writing—review and editing, D.R.-G., A.F.-C., F.B.-C., D.S. and C.S.; supervision, A.F.-C. and F.B.-C.; project administration A.F.-C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not Applicable.

**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.

**Acknowledgments:** The authors would like to thank Consejo Nacional de Ciencia y Tecnología (CONACyT) and Tecnológico de Monterrey for the financial support to conduct the present research. Additionally, thanks go to the Sensors and Devices Research Group and the Robotics Research Group from the School of Engineering and Sciences of Tecnológico de Monterrey for the support given to develop this work.

**Conflicts of Interest:** The authors declare no conflict of interest.
