# 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 $\mu(1-\cos{8\pi t})$, $\mu_r = 0.05$ $m$

## Attachment: halfcar.PNG

## 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 Year of Publication 2006 Authors Chen, C.C, and Lin Y.-F Journal IEE Proceedings of Control Theory and Applications Volume 153 Issue 3 Start Page 331 Pagination 331-341 Date Published 05-2006 ISSN 1350-2379 Accession Number 8827360 Keywords closed loop systems, feedback, linearisation techniques, MIMO systems, nonlinear control systems, stability, suspensions (mechanical components) Abstract The tracking and almost disturbance decoupling problem of multi-input multi-output nonlinear systems based on the feedback linearisation approach are studied. The main contribution of this study is to construct a controller, under appropriate conditions, such that the resulting closed-loop system is valid for any initial condition and bounded tracking signal with the following characteristics: input-to-state stability with respect to disturbance inputs and almost disturbance decoupling, that is, the influence of disturbances on the L2 norm of the output tracking error can be arbitrarily attenuated by changing some adjustable parameters. One example, which cannot be solved by the first paper of the almost disturbance decoupling problem on account of requiring some sufficient conditions that the nonlinearities multiplying the disturbances satisfy structural triangular conditions, is proposed to exploit the fact that the tracking and the almost disturbance decoupling performances are easily achieved by the proposed approach. To demonstrate the practical applicability, a famous half-car active suspension system has been investigated. DOI 10.1049/ip-cta:20050025