Invariant Sets in Saturated and Robust Vehicle Suspension Control

Abstract

This manuscript introduces a new robust control design of car active suspension systems using ellipsoidal techniques. The impact of road irregularities is regarded as an external disturbance. The suggested controller meets the following objectives: optimal passenger comfort, actuator control force limit satisfaction, effective disturbance rejection, and robustness against changes in passengers’ load. The passenger load variations cause system uncertainty that is modeled as norm-bounded. A new sufficient condition is established based on the invariant ellipsoid method and the linear matrix inequalities optimization to guarantee robust stability and performance for the system. Time-domain model of road roughness based on trigonometric functions is studied. The performance of the proposed controller is tested using a quarter-car model with an active suspension system. Comparative simulation with other techniques, e.g., H and regional pole placement, is given. Actuator dynamics and system nonlinearities are modeled and included in the design at the end.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

References

  1. 1.

    Cao, D.; Song, X.; Ahmadian, M.: Editors’ perspectives: road vehicle suspension design, dynamics, and control. Veh. Syst. Dyn. 49, 3–28 (2011)

    Article  Google Scholar 

  2. 2.

    Tseng, H.E.; Hrovat, D.: State of the art survey: active and semi-active suspension control. Veh. Syst. Dyn. 53, 1034–1062 (2015)

    Article  Google Scholar 

  3. 3.

    Fallah, M.; Bhat, R.B.; Xie, W.F.: Optimized control of semi-active suspension systems using H∞ robust control theory and current signal estimation. IEEE/ASME Trans. Mechatron. 17, 767–778 (2012)

    Article  Google Scholar 

  4. 4.

    Verros, G.; Natsiavas, S.; Papadimitriou, C.: Design optimization of quarter-car models with passive and semi-active suspensions under random road excitation. J. Vib. Control 11, 581–606 (2005)

    Article  Google Scholar 

  5. 5.

    Soliman, H.M.; Awadallah, M.A.; NadimEmira, M.: Robust controller design for active suspensions using particle swarm optimization. Int. J. Model. Identif. Control 5, 66–76 (2008)

    Article  Google Scholar 

  6. 6.

    Yagiz, N.; Hacioglu, Y.: Backstepping control of a vehicle with active suspensions. Control Eng. Pract. 16, 1457–1467 (2008)

    Article  Google Scholar 

  7. 7.

    Zuo, L.; Slotine, J.J.; Nayfeh, S.A.: Model reaching adaptive control for vibration isolation. IEEE Trans. Control Syst. 13, 611–617 (2005)

    Article  Google Scholar 

  8. 8.

    Huang, S.-J.; Chao, H.-C.: Fuzzy logic controller for a vehicle active suspension system. Proc. Inst. Mech. Eng. Part D J. Automob. Eng. 214, 1–12 (2000)

    Article  Google Scholar 

  9. 9.

    Soliman, H.M.; Bajabaa, N.: Robust guaranteed-cost control with regional pole placement of active suspensions. J. Vib. Control 19, 1170–1186 (2013)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Soliman, H.M.; Benzaouia, A.; Yousef, H.: Saturated robust control with regional pole placement and application to car active suspension. J. Vib. Control 22, 258–269 (2016)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Soliman, H.M., Al-Abri, R., Albadi, M.: Design of robust digital pole placer for car active suspension with input constraint, chapter 2. In: Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems (2018). https://doi.org/10.5772/intechopen.70587

  12. 12.

    Onat, C.; Kucukdemiral, I.; Sivrioglu, S.; Cansever, Y.G.: LPV gain scheduling controller design for a non-linear quarter-vehicle active suspension system. Trans. Inst. Meas. Control 31, 71–95 (2009)

    Article  Google Scholar 

  13. 13.

    Chen, H.; Guo, K.: Constrained H∞ control of active suspensions: an LMI approach. IEEE Trans. Control Syst. Technol. 13, 412–421 (2005)

    Article  Google Scholar 

  14. 14.

    Li, H.; Jing, X.; Karimi, H.R.: Output-feedback-based H control for vehicle suspension systems with control delay. IEEE Trans. Ind. Electron. 61, 436–446 (2014)

    Article  Google Scholar 

  15. 15.

    Akbari, A.; Lohmann, B.: Output feedback HN/GH2 preview control of active vehicle suspensions: a comparison study of LQG preview. Veh. Syst. Dyn. 48, 1475–1494 (2010)

    Article  Google Scholar 

  16. 16.

    Polyak, B.T.; Topunov, M.V.: Suppression of bounded exogenous disturbances: output feedback. Autom. Remote Control 69, 801–818 (2008)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Blanchini, F.; Miani, S.: Set-Theoretic Methods in Control. Birkhauser, Basel (2008)

    Google Scholar 

  18. 18.

    Dahleh, M.A.; Pearson, J.B.: L1-optimal feedback controllers for MIMO discrete-time systems. IEEE Trans. Autom. Control 32, 314–322 (1987)

    Article  Google Scholar 

  19. 19.

    Du, H.; Zhang, N.: Fuzzy control for nonlinear uncertain electro-hydraulic active suspensions with input constraint. IEEE Trans. Fuzzy Syst. 17, 343–356 (2009)

    Article  Google Scholar 

  20. 20.

    Vaddi, P.K.R.; Kumar, C.S.: A non-linear vehicle dynamics model for an accurate representation of suspension kinematics. Proc. IMech. E Part C J. Mech. Eng. Sci. 229(6), 1002–1014 (2015)

    Article  Google Scholar 

  21. 21.

    Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. Society for Industrial and Applied Mathematics, Philadelphia (1994)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hisham M. Soliman.

Appendix

Appendix

Proof of Theorem 2

The original minimization problem in terms of the ellipsoid \({\mathcal{E}}_{z} = z^{\prime } P^{ - 1} z, z = \left( {C + DK} \right)x\) can be written as

$$\hbox{min } tr\left[ {\left( {C + DK} \right)P\left( {C + DK} \right)} \right]$$
(A1)

Subject to (9).

The matrix Eq. (9) is nonlinear in the variables K, and P. To linearize it, the following new matrix is introduced

$$Y = KP$$
(A2)

Hence, (9) reduces to

$$\left( {PA + BY + *} \right) + \alpha P + \frac{1}{\alpha }EE' < 0$$
(A3)

The objective function (A1) to be minimized is rewritten as

$$\hbox{min } J\left( {P,Y} \right) = tr\left[ {CPC^{\prime } + \left( {CPK^{\prime } D^{\prime } + *} \right) + DYP^{ - 1} Y^{\prime } D^{\prime } } \right]$$
(A4)

Equation (A4) is nonlinear in Y. To reduce it to minimizing a linear function, another matrix is introduced as follows.

$$Z \ge YP^{ - 1} Y^{\prime } \Leftrightarrow \left[ {\begin{array}{*{20}c} Z & * \\ {Y^{\prime } } & P \\ \end{array} } \right] \ge 0$$
(A5)

Minimizing \(J \left( {P, Y} \right)\) is equivalent to minimizing (13) subject to (15).

To include system uncertainty, the matrices \(A, B\) are replaced by \(A + \Delta A\), \(B + \Delta B\), respectively, in (A3) to get (12). Applying the Schur complement \(\varPsi \Delta \left( t \right)\varPhi + *\left\langle {\varepsilon \varPsi \varPsi^{\prime } + \varepsilon^{ - 1} \varPhi^{\prime } \varPhi ,\varepsilon } \right\rangle 0\) [21] to eliminate the uncertainty matrices Δ(t), Theorem 2 is obtained.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Soliman, H.M., El-Metwally, K. & Soliman, M. Invariant Sets in Saturated and Robust Vehicle Suspension Control. Arab J Sci Eng (2020). https://doi.org/10.1007/s13369-020-04703-3

Download citation

Keywords

  • Active suspension modeling
  • Disturbance rejection control
  • Invariant ellipsoids
  • Saturated control
  • Optimal control