Optimal response of half car vehicle model with sky-hook damper based on LQR control

  • L. V. V. Gopala RaoEmail author
  • S. Narayanan


This paper addresses the problem of determining the optimal parameters of a sky-hook damper type suspension in the control of the stationary response of half car vehicle models traversing a rough road. The optimal values of the sky-hook damper suspension parameters are obtained by equating the active suspension control force using linear quadratic regulator (LQR) with that of the sky-hook damper suspension force. Results show that the performance of half car model with optimal sky-hook damper suspension is almost close to the performance of half car model with LQR control.


Half car model LQR Sky-hook damper Optimal parameters Optimization method Random road profile 

List of symbols


Distance of front axle from the centre of gravity (CG)


Distance of rear axle from the centre of gravity (CG)


Damping coefficient

\( d_{1} \)

Font wheel excitation distribution vector

\( d_{2} \)

Rear wheel excitation distribution vector


Front random road input


Rear random road input

\( J_{1} \)

Mean square value of sprung mass acceleration,

\( J_{2} \)

Mean square value of pitch acceleration

\( J_{3} \)

Mean square value of front suspension stroke

\( J_{4} \)

Mean square value of rear suspension stroke

\( J_{5} \)

Mean square value of front suspension road holding

\( J_{6} \)

Mean square value of rear suspension road holding

\( J_{7} \)

Mean square value of front suspension control effort

\( J_{8} \)

Mean square value of rear suspension control effort

\( k_{1} \)

Front suspension spring stiffness

\( k_{2} \)

Rear suspension spring stiffness

\( k_{t1} \)

Front suspension tyre stiffness

\( k_{t2} \)

Rear suspension tyre stiffness

\( c_{s} \)

Sky-hook damper damping coefficient

\( m_{1 } \)

Mass of front wheel

\( m_{2} \)

Mass of rear wheel

\( t_{w} = L/V \)

Time lag between the front wheel and rear wheel


White noise process


State vector

\( y_{1} \)

Absolute displacement of \( m_{1 } \)

\( y_{2} \)

Absolute displacement of \( m_{2} \)

\( y_{a} \)

Front wheel absolute displacement

\( y_{b} \)

Rear wheel absolute displacement


Angle of rotation of the sprung mass about a centroidal axis

\( \alpha_{r} \)

Cut-off wave number of the road profile spectrum

\( \alpha_{s} \)

Sky-hook damper parameter

\( \rho_{i} ,\;{\text{i}} = 1, \ldots 8 \)

Weighting constants


Symmetric positive semi-definite matrix


Symmetric positive definite matrix


Excitation distribution matrix


Expectation operator


System matrix,


Control distribution matrix


Mass moment of inertia


Wheel base


Sprung mass


Zero-lag covariance matrix


Positive definite matrix


Control force vector

\( U_{1} \)

Front suspension control force

\( U_{2} \)

Rear suspension control force


Vehicle velocity


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringVignan’s Institute of Information TechnologyDuvvada, VisakhapatnamIndia
  2. 2.Professor Emeritus, Department of Mechanical EngineeringIndian Institute of Information Technology, Design and ManufacturingKancheepuram, ChennaiIndia

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