Invariant Sets in Saturated and Robust Vehicle Suspension Control


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.

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  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).

  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 

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Correspondence to Hisham M. Soliman.



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

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

Subject to (9).

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

$$Y = KP$$

Hence, (9) reduces to

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

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

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

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

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

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.

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Soliman, H.M., El-Metwally, K. & Soliman, M. Invariant Sets in Saturated and Robust Vehicle Suspension Control. Arab J Sci Eng (2020).

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