Optimal Vehicle Suspensions: A System-Level Study of Potential Benefits and Limitations

Abstract

Fundamental ride and handling aspects of active and semi-active suspensions are presented in a systematic way starting with simple vehicle models as basic building blocks. Optimal, mostly Linear-Quadratic (H2), principles are used to gradually reveal and explore key system characteristics where each additional model Degree-of-Freedom (DoF) brings new insight into potential system benefits and limitations. The chapter concludes with practical considerations and examples including some that go beyond the more traditional ride and handling benefits.

Appendix

In this appendix we establish the LQG-optimal trade-off line for the 1 DoF model of Fig. 7. We start with the covariance (Lyapunov) equation

$$ \left( {A - BK} \right)X + X\left( {A - BK} \right)^{T} = - G\Gamma G^{T} $$

(61)

where in our case G^T= [−1 0] and Γ = 1 since we are dealing with normalized covariance,

$$ X = \left[ {\begin{array}{*{20}c} {X_{1} } & {X_{3} } \\ {X_{3} } & {X_{2} } \\ \end{array} } \right] $$

(62)

where,

$$ X_{1} = \left( {X_{1,rms,norm} } \right)^{2} , {\text{ and }}X_{2} = \left( {X_{2,rms,norm} } \right)^{2} $$

(63)

Define

$$ A_{CL} = A - BK = \left[ {\begin{array}{*{20}c} 0 & 1 \\ { - k_{1} } & { - k_{2} } \\ \end{array} } \right] $$

(64)

with optimal control gains

$$ k_{1} = r^{ - 1/2} ,\quad \, k_{2}^{{}} = \sqrt 2 r^{ - 1/4} $$

(65)

then the covariance equation becomes

$$ A_{CL} X + XA_{CL}^{T} = -GG^{T} $$

(66)

or

$$ \left[ {\begin{array}{*{20}c} {2X_{3} } & {X_{2} - k_{1} X_{1} - k_{2} X_{3} } \\ {X_{2} - k_{1} X_{1} - k_{2} X_{3} } & { - 2k_{1} X_{3} - 2k_{2} X_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {-1} & 0 \\ 0 & 0 \\ \end{array} } \right] $$

(67)

Solving for X₁, X₂, and X₃,

$$ X_{1} = \frac{3}{2\sqrt 2 }r^{1/4} ,\quad X_{2} = \frac{1}{2\sqrt 2 }r^{-1/4} ,\quad \, X_{3} = -\frac{1}{2} $$

(68)

from where we get the normalized rms rattlespace

$$ x_{1,rms,norm} = \sqrt {X_{1} } = \frac{\sqrt 3 }{{\sqrt {2\sqrt 2 } }}r^{1/8} $$

(69)

The normalized rms sprung mass acceleration then follows from

$$ U = KXK^{T} = \frac{1}{2\sqrt 2 }r^{ - 3/4} $$

(70)

so that

$$ U = \frac{27}{64}X_{1}^{ - 3} $$

(71)

resulting in the normalized rms acceleration versus rattlespace equation

$$ u_{rms,norm} = \frac{3\sqrt 3 }{8}x_{1,rms,norm}^{ - 3} $$

(72)

which was used to plot the corresponding optimal trade-off line in Fig. 13.