# Half-car active suspension system

## Model description:

Consider the half-car active suspension system with disturbances shown in the attached image. The dynamic equations are given as follows:

\begin{align*} \dot{x}_1 &=x_2 \\ \dot{x}_2 &= \dfrac{1}{m_S}[-(B_f + B_r)x_2 + (aB_f - bB_r)x_4 \cos{x_3} -k_fx_5 \\ & + B_fx_6 - k_rx_7 + B_rx_8 + (f_f + f_r)] \\ \dot{x}_3 & = x_4 \\ \dot{x}_4 &= \dfrac{1}{J_y}[(aB_f - bB_r)x_2\cos{x_3} \\ & - (a^2B_f + b^2B_r)x_4\cos^2{x_3} + ak_fx_5\cos{x_3}\\ & -aB_fx_6\cos{x_3}-bk_rx_7\cos{x_3} \\ & +bB_rx_8\cos{x_3} + (-af_r+bf_r)\cos{x_3}] \\ \dot{x}_5 &= x_2 - ax_4\cos{x_3} - x_6 \\ \dot{x}_6 &= \dfrac{1}{m_{uf}}[-K_{tf}x_1 + B_fx_2 + aK_{tf}\sin{x_3} \\ & - aB_fx_4 \cos{x_3} +(k_f + K_{tf})x_5 - B_f x_6 + K_{tf}z_{rf} - f_t] \\ \dot{x}_7 &= x_2 + bx_4\cos{x_3} - x_8 \\ \dot{x}_8 &= \dfrac{1}{m_{ur}}[-K_{tf}x_1 + B_rx_2 - bK_{tf}\sin{x_3} \\ & + bB_rx_4 \cos{x_3} +(k_r + K_{tr})x_7 - B_r x_8 + K_{tr}z_{rr} - f_r] \\ y_1 &= x_1 + x_2 :=h_1 \\ y_2 &= x_3 + x_4 := h_2, \end{align*}

where $x_1 = z$ is the displacement of the center of gravity, $x_2 = \dot{z}$ is the payload velocity, $x_3 = \theta$ is the pitch angle, $x_4=\dot{\theta}$ is the pitch velocity, $x_5 = z_{sf} - z_{uf}$ the front wheel suspension travel, $x_6 = \dot{z}_{uf}$ is the front unsprung mass velocity, $x_7 = z_{sr} - z_{ur}$ is rear wheel suspension travel and $x_8=\dot{z}_{ur}$ is the rear unsprung mass velocity.

The physical parameters are defined as:

 $m_s$ Mass of the car body $575$ $kg$ $B_f$ and $B_r$ Front and rear damping coefficients $1000$ $N/m/s$ $a$ Distance between front axle and centre of gravity $1.38$ $m$ $b$ Distance between rear axle and centre of gravity $1.36$ $m$ $J_y$ Centroidal moment of inertia $769$ $kg/m^2$ $m_{uf} = m_{ur}$ Unsprung masses on the front and rear wheels $60$ $kg$ $K_{tf} = K_{tr}$ Front and rear tire spring coefficients $190 000$ $N/m$ $k_f = k_r$ Front and rear spring coefficients $16812$ $N/m$ $z_{rf} = z_{rr}$ Front and rear terrain height disturbances $\mu(1-\cos{8\pi t})$, $\mu_r = 0.05$ $m$

## Publication details:

 Title Application of feedback linearisation to the tracking and almost disturbance decoupling control of multi-input multi-output nonlinear system Publication Type Journal Article Authors Chen, C.C, and Lin Y.-F