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

Article

## Abstract

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.

## Keywords

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

## List of symbols

a

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

b

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

c

Damping coefficient

$$d_{1}$$

Font wheel excitation distribution vector

$$d_{2}$$

Rear wheel excitation distribution vector

h1

h2

$$J_{1}$$

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

w(t)

White noise process

x(t)

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

θ

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

A

Symmetric positive semi-definite matrix

B

Symmetric positive definite matrix

D(t)

Excitation distribution matrix

E[.]

Expectation operator

F(t)

System matrix,

G(t)

Control distribution matrix

I

Mass moment of inertia

L

Wheel base

M

Sprung mass

P(t)

Zero-lag covariance matrix

S

Positive definite matrix

U(t)

Control force vector

$$U_{1}$$

Front suspension control force

$$U_{2}$$

Rear suspension control force

V

Vehicle velocity

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