Actuator fault-based integrated control for vehicle chassis system

Abstract

An integrated fault tolerant controller is proposed for vehicle chassis system. Based on the coupled characteristics of vertical and lateral system, the fault tolerant controller mainly concentrates on the cooperative control of controllable suspension and lateral system with external disturbances and actuator faults. A nine-DOF coupled model is developed for fault reconstruction and accurate control. Firstly, a fault reconstruction mechanism based on sliding mode is introduced; when the sliding mode achieves, actuator fault signals can be observed exactly through selecting appropriate gain matrix and equivalent output injection term. Secondly, an active suspension controller, a roll moment controller and a stability controller is developed respectively; the integrated control strategy is applied to the system under different driving conditions: when the car is traveling straightly, the main purpose of the integrated strategy is to improve the vertical performance; the lateral controller including roll moment control and stability control will be triggered when there is a steering angle input. Simulations experiments verify the performance enhancement and stability of the proposed controller under three different driving conditions.

Introduction

The performances of vehicles are influenced by the chassis system. Active control technology can effectively improve the riding comfort, maneuvering stability and safety characteristic of vehicles by using a variety of advanced control algorithms, such as controllable suspension system,^1,2 steering control systems,^3,4 electronic stability program, ⁵ etc. However, most of the individual control systems were designed under special operating conditions, simple superposition of these single control systems may result in degraded or even unstable performance. To avoid mutual interference between sub-control systems, the integrated chassis control technology has been widely used.

The key issues of the vehicle chassis system are the coupling characteristics of subsystems, the interaction between tires and the road surface, and establishment of the nonlinearly coupled dynamic models. ⁶ Based on the coupled mechanisms, some control systems containing different subsystems have been proposed. Poussot-Vassal et al. ⁷ designed a gain scheduled vehicle stability controller including steering and braking actuators, and the effectiveness of the controller was verified under critical driving situations. Yang et al. ⁸ presented an optimal guaranteed cost controller to improve the stability and tracking performance of the chassis system. Mousavinejad et al. ⁹ designed an integrated AFS and DYC control algorithm, so as to improve transient response of controllers. In view of the interaction between vertical and lateral dynamic of vehicles, Fergani et al. ¹⁰ investigated a linear parameter varying and flatness based global chassis control of vehicles to achieve collaborative control of suspension and steering/braking systems. Rodrigue Tchamna et al. ¹¹ also investigated a global control method, which combines the electronic stability control and active suspension to get superior performance of vehicles during cornering. Although the integrated control mechanism combined with multiple subsystems can achieve better performance, the number of sensors and actuators has also increased, and the structure has become increasingly complex, which may increase the probability of sensor and actuator faults. ¹² To avoid these shortcomings, fault tolerant mechanism is needed to handle system failures and maintain expected properties.

Passive fault tolerant method has been extensively used for automobile systems. Moradi and Fekih^13,14 developed a passive fault tolerant mechanism to suppress the influence of actuator faults. A robust fault tolerant method was presented by Ali et al. ¹⁵ to process the engine air path actuator faults. An adaptive fault tolerant compensation control and output feedback finite-time control of active suspension systems were designed by Pan and Sun^16,17 for stabilizing the perturbed vehicle active suspension system to improve the suspension performance. However, the robustness based passive fault tolerant control will reduce the effect of failures at the cost of system performance and cannot detect the location of faults. Based on the fault diagnosis, active fault tolerant method can reconfigure the control system through estimating the current fault. ¹⁸ Many active fault tolerant controllers have been applied to automobile systems. An Unknown Input Observer approach was used by Alain Yetendje et al. ¹⁹ to detect and identify actuator fault of active suspension systems. On the basis of fault detection and diagnosis, the robust linear parameter variable control method was investigated by Peter Gáspár et al. ²⁰ to control the active suspension of heavy vehicles containing actuator faults. A fault-based fault tolerant feedback control was utilized by Oudghiri et al. ²¹ to compensate the faults and reduce the impact of faults on the lateral stability of vehicles. The previous works on integrated chassis control such as the centralized integrated controller,^7,8,10 the characteristics of each subsystem and the coupling relationship between the subsystems were not fully considered. In this paper, a hierarchical integrated controller is developed, among which suitable controllers for each subsystem are designed, and an integrated control strategy is utilized to coordinate the vertical and lateral dynamics. In the previous works on fault tolerant control for vehicle chassis systems, passive fault tolerant control or active fault tolerant control mainly concentrate on individual chassis system such as active suspension system. However, in this study, we consider the fault tolerant control for integrated vehicle model. Besides, the sliding mode-based fault observer can also observe the unmeasured states simultaneously. The necessity of providing an integrated fault tolerant control, in the presence of external road inputs and actuator faults while ensuring stability and performance of the control system, forms the focus of the work proposed in this paper.

In this study, a nine-DOF integrated model is established, and actuator faults are taken into account. A sliding model observer is employed to estimate fault signals and system states, and a hierarchical integrated mechanism is designed to improve vehicle chassis properties. The main innovations are:

• (1) A sliding mode fault observer is employed for actuator fault signal and state estimation of the chassis model.

• (2) A hierarchical mechanism is designed for the chassis model to achieve performance improvement of the integrated vertical and lateral model.

Vehicle dynamic model

For controller design purpose, a suspension model and a four wheels lateral model are utilized to describe the coupled dynamics of the vehicle.

Vertical model

A 7-DOF suspension model with heave, pitch and roll motions is used in this paper, ²² as shown in Figure 1.

Figure 1.

Suspension model: (a) vertical model and (b) roll motion.

The terms $\theta$ and $\phi$ represent the pitch and roll angle. The damping and spring coefficients are indicated as $c_i$ and $k_i$ , and $k_{\mathrm{ti}}$ represents tire stiffness. The external disturbance is expressed as $z_{\mathrm{ri}}$ . $F_{\mathrm{ai}}$ is active forces produced by the actuator.

The suspension deflections are expressed as

$$z_{\mathrm{sui}}=z_{\mathrm{si}}-z_{\mathrm{ui}}(i=1,2,3,4)$$

(1)

The spring and damping forces are

$$\begin{array}{c}{F}_{\mathrm{si}}=k_i{z}_{\mathrm{sui}}+k_{\mathrm{ni}}{z}_{\mathrm{sui}}^3(i=1,2,3,4)\hfill \\ F_{\mathrm{di}}=c_i{\stackrel{·}{z}}_{\mathrm{sui}}(i=1,2,3,4)\hfill \end{array}$$

(2)

According to small angle assumption, the following approximate equation can be got

$$\begin{array}{c}{z}_{s1}=z_s+a\sin \left(\theta \right)+c\sin \phi ,\hfill \\ z_{s2}=z_s+a\sin \left(\theta \right)-d\sin \phi ,\hfill \\ z_{s3}=z_s-b\sin \left(\theta \right)+c\sin \phi ,\hfill \\ z_{s4}=z_s-b\sin \left(\theta \right)-d\sin \phi ,\hfill \end{array}$$

(3)

The vehicle vertical model with sprung and unsprung mass dynamics can be given as ²³

$$\begin{array}{c}{m}_s{\stackrel{··}{z}}_s+\sum (F_{\mathrm{si}}+F_{\mathrm{di}}-F_{\mathrm{ai}}+F_{\mathrm{ati}1}+F_{\mathrm{ati}2}+F_{\mathrm{ati}3}+F_{\mathrm{ati}4})=0,\hfill \\ I_y\stackrel{··}{\theta }+a(F_{s1}+F_{d1}+F_{s2}+F_{d2}+F_{\mathrm{ati}1}+F_{\mathrm{ati}2})\hfill \\ \multicolumn{1}{c}{-b(F_{s3}+F_{d3}+F_{s4}+F_{d4}+F_{\mathrm{ati}3}+F_{\mathrm{ati}4})}\\ \multicolumn{1}{c}{-F_{\theta }=0,}\\ I_x\stackrel{··}{\phi }-d(F_{s2}+F_{d2}+F_{\mathrm{ati}2}+F_{s4}+F_{d4}+F_{\mathrm{ati}4})\hfill \\ \multicolumn{1}{c}{+c(F_{s1}+F_{d1}+F_{\mathrm{ati}1}+F_{s3}+F_{d3}+F_{\mathrm{ati}3})}\\ \multicolumn{1}{c}{-F_{\phi }+m_{s}v(\stackrel{·}{\beta }+\gamma )h+m_s\mathrm{gh}\phi =0,}\\ m_{\mathrm{ui}}{\stackrel{··}{z}}_{\mathrm{ui}}-F_{\mathrm{si}}-F_{\mathrm{di}}+F_{\mathrm{ti}}-F_{\mathrm{ai}}=0(i=1,2,3,4)\hfill \end{array}$$

(4)

The anti-sway bar forces are as

$$\begin{array}{c}{F}_{\mathrm{ati}1}=\frac{k_{\mathrm{af}}}{c+d}(\phi -\frac{z_{u2}-z_{u1}}{c+d})\hfill \\ F_{\mathrm{ati}2}=-\frac{k_{\mathrm{af}}}{c+d}(\phi -\frac{z_{u2}-z_{u1}}{c+d})\hfill \\ F_{\mathrm{ati}3}=-\frac{k_{\mathrm{r}f}}{c+d}(\phi -\frac{z_{u3}-z_{u4}}{c+d})\hfill \\ F_{\mathrm{ati}4}=\frac{k_{\mathrm{r}f}}{c+d}(\phi -\frac{z_{u3}-z_{u4}}{c+d})\hfill \end{array}$$

(5)

The meanings of specific symbols are shown in Table 1.

Table 1.

Parameters of the integrated model.

Parameter	Value	Parameter	Value
Total mass of the vehicle	2132 kg	Roll moment of inertia Ix	614 kg m2
Sprung mass ms	1592 kg	Yaw moment of inertia Iz	2488 kg m2
Unsprung mass mui	120/150 kg	Spring stiffness ki	46000 N/m
Gravity to front axle distancea	1.18 m	Damping coefficient bi	3000 Nm/s
Gravity to rear axle distanceb	1.77 m	Tire stiffness kti	230000 N/m
Centre of gravity to the left sidec	0.7875 m	Anti-roll bar stiffness kaf / krf	6695 Nm/rad
Centre of gravity to the right sided	0.7875 m	Steering coefficient Ef / Er	−0.144/0
Pitch moment of inertia Iy	2488 kg m2	Cornering stiffness kf / kr	35000 N/rad
The vertical distance fromvehicle C.G. to suspension C.G.h	0.501 m	Empirical Tire parameters C1/C2	13.098 rad−1/−0.001045 N−1rad−1

Lateral model

Vehicle lateral system shown in Figure 2 is utilized to develop the roll moment distribution controller and the stability controller. The lateral motion can be expressed as ²⁴

$$\mathrm{mu}(\stackrel{·}{\beta }+\gamma )=F_{\mathrm{yf}}+F_{\mathrm{yr}}+m_{s}h\stackrel{··}{\phi }$$

(6)

$$I_z\stackrel{·}{\gamma }=a{F}_{\mathrm{yf}}-b{F}_{\mathrm{yr}}+\mathrm{\Delta }M$$

(7)

where $\mathrm{\Delta }M$ is the direct yaw moment control (DYC).

The expected yaw rate is as ²⁵

$${\gamma }_{\mathrm{des}}=\frac{u\delta }{a+b+K_{\mathrm{us}}{u}^2}$$

(8)

where K_us is the understeer coefficient and in the neutral steering, and

$$K_{\mathrm{us}}=m\frac{b{C}_r-a{C}_f}{2(a+b)C_f{C}_r}$$

(9)

among which $C_f$ and $C_r$ are linear stiffness.

Figure 2.

Vehicle lateral model.

Tire model

Although the tire lateral force is well defined as a first order approximation, the resulting change in lateral force does not consider the change of normal force. So it is necessary to employ side slip angle and normal force to generate tire lateral force. Besides, to obtain the impact of roll moment distribution, the higher order term with normal force is included. Then the lateral force developed by each tire can be given by ²⁶ :

$$F_{\mathrm{yi}}=C_1{\alpha }_i{N}_i+C_2{\alpha }_i{N}_i^2,i=1,2,3,4$$

(10)

where ${\alpha }_i$ and $N_i$ are the side slip angle and the normal force of each tire, respectively. $C_1$ and $C_2$ are the empirical values. The front and rear tire lateral force can be written as

$$F_{\mathrm{yf}}=F_{\mathrm{y}1}+F_{y2}={\alpha }_f[C_1(N_1+N_2)+C_2(N_1^2+N_2^2)]$$

(11)

$$F_{\mathrm{yr}}=F_{\mathrm{y}3}+F_{y4}={\alpha }_r[C_1(N_3+N_4)+C_2(N_3^2+N_4^2)]$$

(12)

among which

$${\alpha }_f=\delta -\beta -\frac{a\gamma }{u}$$

(13)

$${\alpha }_r=-\beta +\frac{b\gamma }{u}$$

(14)

where $\delta$ is the front wheel angle.

When the vehicle turns, the weight of the sprung mass shifts from the inner wheel to the outer wheel, and the transferred weight is related to sprung mass, yaw rate, speed and the sprung mass center position. The transferred weight should be reacted with the suspension roll motion. Differences of the front and rear suspension characteristics would produce the roll moment distribution. By using the active control techniques, a distribution coefficient can be assigned $-1\le \epsilon \le 1$ as Williams and Haddad. ²⁷ Then the normal forces related with tires can be given as

$$N_1=m\frac{b}{2(a+b)}+m_{s}u\gamma \frac{h}{c+d}(1+\epsilon )$$

(15)

$$N_2=m\frac{b}{2(a+b)}-m_{s}u\gamma \frac{h}{c+d}(1+\epsilon )$$

(16)

$$N_3=m\frac{b}{2(a+b)}+m_{s}u\gamma \frac{h}{c+d}(1-\epsilon )$$

(17)

$$N_4=m\frac{b}{2(a+b)}-m_{s}u\gamma \frac{h}{c+d}(1-\epsilon )$$

(18)

where $\epsilon$ is the roll moment distribution coefficient. By inserting equations (15)–(18) into the empirical tire model, the lateral force of front and rear axles are as

$$\begin{array}{c}\multicolumn{1}{c}{F_{\mathrm{yf}}={\alpha }_f[C_{1}m\frac{b}{(a+b)}+\frac{1}{2}{C}_2{m}^2\frac{b^2}{{(a+b)}^2}}\\ \multicolumn{1}{c}{+2C_2{\left(m_{s}u\frac{h}{c+d}\right)}^2{(1+\epsilon )}^2{\gamma }^2]}\end{array}$$

(19)

$$\begin{array}{c}\multicolumn{1}{c}{F_{\mathrm{yr}}={\alpha }_r[C_{1}m\frac{b}{(a+b)}+\frac{1}{2}{C}_2{m}^2\frac{b^2}{{(a+b)}^2}}\\ \multicolumn{1}{c}{+2C_2{\left(m_{s}u\frac{h}{c+d}\right)}^2{(1-\epsilon )}^2{\gamma }^2]}\end{array}$$

(20)

Fault observer design

Sliding mode observer

The state variable of the integrated model is given as

$$\mathit{X}={\left(z_{u1}{z}_{u2}{z}_{u3}{z}_{u4}{\stackrel{·}{z}}_{u1}{\stackrel{·}{z}}_{u2}{\stackrel{·}{z}}_{u3}{\stackrel{·}{z}}_{u4}z\stackrel{·}{z}\theta \stackrel{·}{\theta }\phi \stackrel{·}{\phi }\beta \gamma \right)}^{\mathrm{T}}$$

(21)

Define the road input and front wheel angle as the external interferences input, the interference variables are represented as

$$\mathit{w}={\left(z_{r1}{z}_{r2}{z}_{r3}{z}_{r4}\delta \right)}^{\mathrm{T}}$$

(22)

The control variables of the integrated system include the additional yaw moment generated by the electronic stability control system and the active suspension force, which can be written as

$$\mathit{U}={\left(\mathrm{\Delta }M{F}_{a1}{F}_{a2}{F}_{a3}{F}_{a4}\right)}^{\mathrm{T}}$$

(23)

The system measurement outputs are selected as

$$\mathit{Y}=\left({\stackrel{·}{z}}_s\stackrel{·}{\theta }\stackrel{·}{\phi }{z}_{\mathrm{su}1}{z}_{\mathrm{su}2}{z}_{\mathrm{su}3}{z}_{\mathrm{su}4}\gamma \right)$$

(24)

Then, the system model can be written as

$$\begin{array}{c}\stackrel{·}{\mathit{X}}=\mathit{AX}+\mathit{BU}+\mathit{Gw}\hfill \\ \mathit{Y}=\mathit{CX}\hfill \end{array}$$

(25)

w is external disturbance, and $\Vert \mathit{w}\Vert \le {\beta }_0$ . The definition of matrices can be found in the Appendix . Partial actuator failure is considered here. By letting the actuator fault be a multiplication coefficient, the integrated model is as

$$\begin{array}{c}\stackrel{·}{\mathit{X}}=\mathit{AX}+\mathit{BU}\mathbf{+}\mathit{M}{\mathit{f}}_i+\mathit{Gw}\hfill \\ {\mathit{Y}}_{\mathbf{f}}=\mathit{CX}\hfill \end{array}$$

(26)

Among which $\mathit{M}=\mathit{B}$ , ${\mathbf{f}}_i\le \alpha (t,\mathit{U})$ represent the actuator faults, and ${\mathit{f}}_i=(\mathit{I}-{\mathit{f}}_i)\mathit{U}$ . $\mathit{A}$ is 16×16 state matrix, C is 8×16 output matrix. $\mathit{B}$ and G are 16×5 input matrix, respectively.

Suppose that rank $\left(\mathit{CM}\right)$ = rank $\left(\mathit{M}\right)$ , and invariant zeros of $(\mathit{A},\mathit{M},\mathit{C})$ is Hurwitz matrix. ²⁸ The coordinate change $\mathit{X}\to {\mathit{T}}_0\mathit{X}$ so that $(\mathit{A},M,C)$ can be converted to

$$\mathit{A}=\left[\begin{array}{cc}{\mathit{A}}_{11}\hfill & {\mathit{A}}_{12}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}{\mathit{A}}_{211}\hfill \\ \multicolumn{1}{c}{{\mathit{A}}_{212}}\end{array}}& {\mathit{A}}_{22}\hfill \end{array}\right],\mathit{M}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathit{M}}_2}\end{array}\right],\mathit{C}=\left[\begin{array}{cc}0\hfill & \mathit{T}\hfill \end{array}\right]$$

(27)

$\mathit{T}$ is orthogonal and ${\mathit{M}}_2={\left[\begin{array}{cc}0\hfill & {\mathit{M}}_0\hfill \end{array}\right]}^{\mathrm{T}}$ , ${\mathit{M}}_0\in {\mathit{R}}^{5\times 5}$ . ${\mathit{A}}_{11}\in {\mathit{R}}^{8\times 8}$ , ${\mathit{A}}_{211}\in {\mathit{R}}^{3\times 8}$ .

Considering equation (26), the observer that will be considered in the paper can be written as ²⁹

$$\begin{array}{c}\widehat{\stackrel{·}{\mathbf{X}}}=\mathbf{A}\hat{\mathbf{X}}+\mathbf{BU}-{\mathbf{G}}_l{\mathit{e}}_y\left(t\right)+{\mathbf{G}}_n\mathit{v}\hfill \\ {\hat{\mathbf{Y}}}_{\mathbf{f}}=\mathbf{C}\hat{\mathbf{X}}\hfill \end{array}$$

(28)

${\mathbf{G}}_l$ is linear injection item, ${\mathbf{G}}_n$ are is observer gain. v represents a discontinuous switch component which is used to induce a sliding motion, and

$$\mathit{v}=-\rho \frac{{\mathit{e}}_y\left(t\right)}{\Vert {\mathit{e}}_y\left(t\right)\Vert }({\mathit{e}}_y\left(t\right)\ne 0)$$

(29)

${\mathit{e}}_y={\mathbf{Y}}_{\mathbf{f}}-{\hat{\mathbf{Y}}}_{\mathbf{f}}$ represents the estimation error. With equations (26) and (28), the system state error $\mathit{e}\left(t\right)=\hat{\mathit{X}}-\mathit{X}$ is

$$\mathit{e}\left(t\right)={\mathit{A}}_0\mathit{e}\left(t\right)+{\mathbf{G}}_n\mathit{v}-\mathit{M}{\mathit{f}}_i-\mathit{GW}$$

(30)

among which ${\mathit{A}}_0=\mathit{A}-{\mathit{G}}_l\mathit{C}$ . The observer gain matrix ${\mathbf{G}}_n$ is as

$${\mathbf{G}}_n=\left[\begin{array}{c}-\mathit{L}\hfill \\ \multicolumn{1}{c}{{\mathit{I}}_p}\end{array}\right]{\mathit{T}}^{\mathrm{T}}{\mathit{P}}_0^{-1}$$

(31)

where $\mathit{L}=\left[\begin{array}{cc}{\mathit{L}}^0\hfill & 0\hfill \end{array}\right]$ , ${\mathit{P}}_0={\mathit{P}}_0^{\mathrm{T}}$ is design matrix. The disturbance matrix can be written as

$$\mathit{G}=\left[\begin{array}{c}{\mathit{G}}_1\hfill \\ \multicolumn{1}{c}{{\mathit{G}}_2}\end{array}\right]$$

(32)

${\mathit{G}}_1\in {\mathit{R}}^{8\times 16}$ . Suppose there is a symmetric matrix $\mathit{P}\in {\mathit{R}}^{16\times 16}$ satisfies

$$\mathit{P}=\left[\begin{array}{cc}{\mathit{P}}_1\hfill & {\mathit{P}}_1\mathit{L}\hfill \\ \multicolumn{1}{c}{{\mathit{L}}^{\mathrm{T}}{\mathit{P}}_1}& {\mathit{T}}^{\mathrm{T}}{\mathit{P}}_0\mathit{T}+{\mathit{L}}^{\mathrm{T}}{\mathit{P}}_1\mathit{L}\hfill \end{array}\right]>0$$

(33)

where ${\mathit{P}}_1\in {\mathit{R}}^{8\times 8}$ , $\mathit{P}{\mathit{A}}_0+{\mathit{A}}_0^{\mathrm{T}}\mathit{P}<0$ . The following two positive scalars are defined

$${\mu }_0=-{\lambda }_{max}(\mathit{P}{\mathit{A}}_0+{\mathit{A}}_0^{\mathrm{T}}\mathit{P}),{\mu }_1=\Vert \mathit{PQ}\Vert$$

(34)

Proposition 1 If the scalar $\rho$ in the discontinuous switch function (29) satisfies ³⁰

$$\rho \ge \Vert {\mathit{P}}_0\mathit{CM}\Vert \alpha (t,\mathit{U})+{\eta }_0$$

(35)

${\eta }_0$ denotes a positive scalar. System error $\mathit{e}\left(t\right)$ in (30) is asymptotically stable in regard to the set

$${\mathbf{O}}_{\in }=\{\mathit{e}:\Vert \mathit{e}\Vert <2{\mu }_1{\beta }_0/{\mu }_0+\in \}$$

(36)

$\in >0$ is any small scalar.

By introducing a new coordinate transformation

$${\mathit{T}}_L=\left[\begin{array}{cc}{\mathit{I}}_{8\times 8}\hfill & \mathit{L}\hfill \\ \multicolumn{1}{c}{0}& \mathit{T}\hfill \end{array}\right]$$

(37)

$(\mathit{A},M,C)$ can be transformed to

$$\mathbf{A}=\left[\begin{array}{cc}{\mathbf{A}}_{11}\hfill & {\mathbf{A}}_{12}\hfill \\ \multicolumn{1}{c}{{\mathbf{A}}_{21}}& {\mathbf{A}}_{22}\hfill \end{array}\right],\mathbf{M}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathbf{M}}_2}\end{array}\right],\mathit{C}=\left[\begin{array}{cc}0\hfill & {\mathit{I}}_8\hfill \end{array}\right]$$

(38)

where ${\mathbf{A}}_{11}={\mathit{A}}_{11}+{\mathit{L}}^0{\mathit{A}}_{211}$ . The nonlinear gain matrix can be converted to

$${\mathbf{G}}_n=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathit{P}}_0^{-1}}\end{array}\right]$$

(39)

The disturbance matrix has the form

$$\mathbf{G}=\left[\begin{array}{c}{\mathbf{G}}_1\hfill \\ \multicolumn{1}{c}{{\mathbf{G}}_2}\end{array}\right]=\left[\begin{array}{c}{\mathbf{G}}_1+\mathit{L}{\mathbf{G}}_2\hfill \\ \multicolumn{1}{c}{\mathit{T}{\mathbf{G}}_2}\end{array}\right]$$

(40)

and the Lyapunov matrix can be written as

$$\mathbf{P}={\left({\mathit{T}}_L^{-1}\right)}^{-1}{\mathit{PT}}_L^{-1}=\left[\begin{array}{cc}{\mathit{P}}_1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathit{P}}_0\hfill \end{array}\right]$$

(41)

The estimation error in new coordinate can be partitioned as

$$\begin{array}{c}{\stackrel{·}{\mathit{e}}}_1\left(t\right)={\mathbf{A}}_{11}{\mathit{e}}_1\left(t\right)+({\mathbf{A}}_{12}-{\mathbf{G}}_{l,1}){\mathit{e}}_y\left(t\right)-{\mathbf{G}}_1\mathit{W}\hfill \\ {\stackrel{·}{\mathit{e}}}_y\left(t\right)={\mathbf{A}}_{21}{\mathit{e}}_1\left(t\right)+({\mathbf{A}}_{22}-{\mathbf{G}}_{l,2}){\mathit{e}}_y\left(t\right)+{\mathit{P}}_0^{-1}\mathit{v}\hfill \\ \multicolumn{1}{c}{-{\mathbf{M}}_2{\mathbf{f}}_i-{\mathbf{G}}_2\mathit{W}}\end{array}$$

(42)

Proposition 2 If the scalar $\rho$ in the discontinuous switch function (29) satisfies ³⁰

$$\begin{array}{c}\multicolumn{1}{c}{\rho \ge 2\Vert {\mathit{P}}_0{\mathbf{A}}_{21}\Vert {\mu }_1{\beta }_0/{\mu }_0+\Vert {\mathit{P}}_0{\mathbf{G}}_2\Vert {\beta }_0}\\ \multicolumn{1}{c}{+\Vert {\mathit{P}}_0{\mathbf{M}}_2\Vert \alpha (t,\mathit{U})+{\eta }_0}\end{array}$$

(43)

then the sliding mode will occur in finite time.

Robust fault observer

To estimate fault signals exactly in the presence of interferences, assume that sliding motion in 3.1 has achieved, by selecting appropriate output injection term ${\mathit{v}}_{\mathrm{eq}}$ and the gain matrix ${\mathit{L}}^0$ , the impact of disturbances on the fault reconstruction can be minimized. The reconstruction signal can be defined as ³⁰

$$\hat{\mathbf{f}}\stackrel{\mathrm{def}}{=}{\mathit{W}}_{\mathrm{sc}}{\mathit{T}}^{\mathrm{T}}{\mathit{P}}_0^{-1}{\mathit{v}}_{\mathrm{eq}}$$

(44)

${\mathit{W}}_{\mathrm{sc}}=\left[\begin{array}{cc}{\mathit{W}}_1\hfill & {\mathit{M}}_0^{-1}\hfill \end{array}\right]$ , ${\mathit{W}}_1\in {\mathit{R}}^{5\times 3}$ . Rewriting the estimation error in new coordinate and pre-multiplying W _sc T ^T implies

$$\hat{\mathbf{f}}=\mathbf{f}\mathbf{+}\hat{\mathbf{G}}\left(\mathrm{s}\right)\mathit{W}$$

(45)

where $\hat{\mathbf{G}}\left(\mathrm{s}\right)$ is given as

$$\begin{array}{c}\multicolumn{1}{c}{\hat{\mathbf{G}}\left(\mathrm{s}\right)={\mathit{W}}_{\mathrm{sc}}{\mathit{A}}_{21}{(s{\mathit{I}}_8-({\mathit{A}}_{11}+\mathit{L}{\mathit{A}}_{21}))}^{-1}({\mathit{G}}_1+{\mathit{G}}_2)}\\ \multicolumn{1}{c}{+{\mathit{W}}_{\mathrm{sc}}{\mathit{G}}_2}\end{array}$$

(46)

Then the objective is to keep ${\mathit{L}}_2$ gain of the transfer function from the effect of external disturbance to the fault reconstruction results not exceed $\gamma \in {\mathit{R}}_{+}$ . Define ${\mathit{D}}_1\in {\mathit{R}}^{8\times 8}$ , ${\gamma }_0\in {\mathbf{R}}_{+}$ , and also define the following matrices

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{B}}_d=\left[\begin{array}{cc}0\hfill & \mathit{G}\hfill \end{array}\right],{\mathit{D}}_d=\left[\begin{array}{cc}{\mathit{D}}_1\hfill & 0\hfill \end{array}\right],\mathit{H}=\left[\begin{array}{cc}0\hfill & {\mathit{W}}_{\mathrm{sc}}{\mathit{G}}_2\hfill \end{array}\right],}\\ \multicolumn{1}{c}{\mathit{E}=\left[\begin{array}{cc}-{\mathit{W}}_{\mathrm{sc}}{\mathit{A}}_{21}\hfill & {\mathit{E}}_2\hfill \end{array}\right]}\end{array}$$

(47)

The Lyapunov matrix satisfies

$$\mathit{P}=\left[\begin{array}{cc}{\mathit{P}}_{11}\hfill & {\mathit{P}}_{12}\hfill \\ \multicolumn{1}{c}{{\mathit{P}}_{12}^{\mathrm{T}}}& {\mathit{P}}_2\hfill \end{array}\right],{\mathit{P}}_{12}=\left[\begin{array}{cc}{\mathit{P}}_{121}\hfill & 0\hfill \end{array}\right],{\mathit{P}}_{121}\in {\mathit{R}}^{8\times 3}$$

(48)

and ${\mathit{W}}_1,{\mathit{E}}_2$ and $\gamma$ satisfy the Bounded Real Lemma

$$\left[\begin{array}{ccc}\mathit{PA}+{\mathit{A}}^{\mathrm{T}}\mathit{P}-{\gamma }_0{\mathit{C}}^{\mathrm{T}}{\left({\mathit{D}}_d{\mathit{D}}_d^{\mathrm{T}}\right)}^{-1}\mathit{C}\hfill & -\mathit{P}{\mathit{B}}_d\hfill & {\mathit{E}}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{-{\mathit{B}}_d^{\mathrm{T}}\mathit{P}}& -{\gamma }_0\mathit{I}\hfill & {\mathit{H}}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{\mathit{E}}& \mathit{H}\hfill & -{\gamma }_0{\mathit{I}}_5\hfill \end{array}\right]<0$$

(49)

$$\left[\begin{array}{ccc}{\mathit{P}}_{11}{\mathit{A}}_{11}+{\mathit{A}}_{11}^{\mathrm{T}}{\mathit{P}}_{11}+{\mathit{P}}_{12}{\mathit{A}}_{21}+{\mathit{A}}_{21}^{\mathrm{T}}{\mathit{P}}_{12}^{\mathrm{T}}\hfill & ★\hfill & ★\hfill \\ \multicolumn{1}{c}{-{({\mathit{P}}_{11}{\mathit{G}}_1+{\mathit{P}}_{12}{\mathit{G}}_2)}^{\mathrm{T}}}& -\gamma \mathit{I}\hfill & ★\hfill \\ \multicolumn{1}{c}{-{\mathit{W}}_{\mathrm{sc}}{\mathit{A}}_{21}}& {\mathit{W}}_{\mathrm{sc}}{\mathit{G}}_2\hfill & -\gamma {\mathit{I}}_5\hfill \end{array}\right]<0$$

(50)

The stars in inequality represent the matrix is symmetric. If the gain matrices of the observer are selected as

$$\begin{array}{c}{\mathit{L}}^0={\mathit{P}}_{11}^{-1}{\mathit{P}}_{121}\hfill \\ {\mathit{G}}_l={\gamma }_0{\mathit{P}}^{-1}{\mathit{C}}^{\mathrm{T}}{\left({\mathit{D}}_d{\mathit{D}}_d^{\mathrm{T}}\right)}^{-1}\hfill \end{array}$$

(51)

then $\mathit{P}{\mathit{A}}_0+{\mathit{A}}_0^{\mathrm{T}}\mathit{P}<0$ is satisfied and $\Vert \widehat{\mathbf{G}}\left(\mathrm{s}\right)\Vert \le \gamma$ , and ${\mathit{P}}_1={\mathit{P}}_{11}$ , ${\mathit{L}}^0={\mathit{P}}_{11}^{-1}{\mathit{P}}_{121},{\mathit{P}}_0=\mathit{T}({\mathit{P}}_{22}-{\mathit{P}}_{12}^{\mathrm{T}}{\mathit{P}}_{11}^{-1}{\mathit{P}}_{12}){\mathit{T}}^{\mathrm{T}}$ .

Stand-alone controller design

Suspension controller design

To improve system stability and achieve desirable suspension performance, a quantization based sliding mode controller is utilized. In the designed control system, the controller is approximately considered as a continuous system. The control diagram is shown in Figure 3. The switch block in Figure 3 refers to the conversion relationship between the input voltage and output force of the electro-hydraulic actuator.

Figure 3.

Diagram block of the proposed controller.

Quantization can be seen as a process of encoding. When analog signal is converted to digital signal, the processor maps it to a finite precision value according to a certain coding mode. Quantization control theory includes control problems based on static quantizer and dynamic quantizer, among which the dynamic quantizer expands the quantization range by adjusting the quantization level to achieve asymptotic stability of the system. ³¹ This section mainly concentrates on the connection channel between the suspension model and the controller; both static quantizer and dynamic quantizer are used, and traditional sliding mode controller is also adopted. The quantization system is shown in Figure 4.

Figure 4.

Flow chart of the quantization system.

Assuming the suspension states and road profiles are bounded. Selecting state variables as

$$[x_1{x}_2{x}_3{x}_4{x}_5{x}_6{]}^{\mathrm{T}}=[z_s{\stackrel{·}{z}}_s\theta \stackrel{·}{\theta }\phi \stackrel{·}{\phi }{]}^{\mathrm{T}}$$

The suspension system model can be expressed as

$$\begin{array}{c}{\stackrel{·}{x}}_1=x_2,\hfill \\ {\stackrel{·}{x}}_2=f_z+G_z{F}_z,\hfill \\ f_z=-\frac{1}{m_s}\sum (F_{\mathrm{si}}+F_{\mathrm{di}}),G_z=\frac{1}{m_s},\hfill \\ {\stackrel{·}{x}}_3=x_4,\hfill \\ {\stackrel{·}{x}}_4=f_{\theta }+G_{\theta }{F}_{\theta },\hfill \\ f_{\theta }=-\frac{1}{I_y}\hfill \\ \multicolumn{1}{c}{[a(F_{s1}+F_{d1}+F_{s2}+F_{d2})-b(F_{s3}+F_{d3}+F_{s4}+F_{d4})],}\\ \multicolumn{1}{c}{G_{\theta }=\frac{1}{I_y},}\\ {\stackrel{·}{x}}_5=x_6,\hfill \\ {\stackrel{·}{x}}_6=f_{\phi }+G_{\phi }{F}_{\phi },\hfill \\ f_{\phi }=-\frac{1}{I_x}[c(F_{s1}+F_{d1}+F_{s3}+F_{d3})\hfill \\ \multicolumn{1}{c}{-d(F_{s2}+F_{d2}+F_{s4}+F_{d4})],G_{\phi }=\frac{1}{I_x}.}\end{array}$$

(52)

the above functions are rewritten as ³²

$$\mathit{x}\left(t\right)=A_0\stackrel{·}{\mathit{x}}\left(t\right)+B_j[{\mathit{U}}_{\mathrm{force}}+{\mathit{d}}_j\left(t\right)]$$

(53)

where $\mathit{x}\left(t\right)=[x_i\left(t\right);x_{i+1}\left(t\right)]$ ; $i=1,3,5$ , $j=1,2,3$ ; ${\mathit{U}}_{\mathrm{force}}=[F_z;F_{\theta };F_{\phi }]$ is the inner force generated by the active suspension. $A_0=[\begin{array}{cc}0\hfill & 1\hfill \end{array};\begin{array}{cc}0\hfill & 0\hfill \end{array}]$ ; $B_1=[0;G_z]$ , $B_2=[0;G_{\theta }]$ , $B_3=[0;G_{\phi }]$ ; $d_1\left(t\right)=[0;\frac{f_z}{G_z}]$ , $d_2\left(t\right)=[0;\frac{f_{\theta }}{G_{\theta }}]$ , $d_3\left(t\right)=[0;\frac{f_{\phi }}{G_{\phi }}]$ .

Assumption 1: ${\mathit{d}}_j\left(t\right)$ satisfy $|{\mathit{d}}_j(t)|\le d_1+d_2|\mathit{x}(t)|+d_3|{\mathit{U}}_{\mathrm{force}}|\le d_4$ , where $d_1\ge 0$ , $d_2\ge 0,d_4>0,0\le d_3\le 1$ , and $d_1,d_2,$ $d_3,d_4$ are constants.

As shown in Figure 4, a static quantizer $q_{{\mu }_s}(\mathit{v})$ is used in the uplink channel, where v is the control law designed by the quantized state, ${\mu }_s$ is a fixed number; the down channel utilized a dynamic quantizer $q_{\mu }(\mathit{x})$ to transfer system quantized states, and $\mu$ is the quantization parameter adjusted statically. Define a function that takes the nearest integer, then the uniform quantizer with the quantization level parameter $\mu$ is as

$$q_{\mu }(\mathit{x})\stackrel{\mathrm{def}}{=}{\mu }_j\mathrm{round}\left(\frac{\mathit{x}}{{\mu }_j}\right),{\mu }_j>0;j=1,2,3$$

(54)

The uniform quantization error is

$${\mathit{e}}_{\mu }\stackrel{\mathrm{def}}{=}{q}_{\mu }(\mathit{x})-\mathit{x}$$

(55)

then

$${\mathit{e}}_{\mu }\stackrel{\mathrm{def}}{=}|q_{\mu }(\mathit{x})-\mathit{x}|\le \Gamma {\mu }_j$$

(56)

$\Gamma =\sqrt{p}/2,p$ is the dimension of the vector x .

With respect to dynamic quantizer, a special quantization parameter ${\mu }_j=0$ is given when the system runs to the sliding surface

$$q_{\mu }(\mathit{x})\stackrel{\mathrm{def}}{=}0,{\mu }_j=0$$

(57)

Defined the sliding surface

$$s(\mathit{x}(t))=C\mathit{x}(t)=0$$

(58)

where C is a given vector to ensure the system has stable eigenvalues, and $C{B}_j\ne 0$ .

• Lemma 1 ³³ : There is a fixed constant $\tau >0$ , assume $\mu >0$ which satisfies

$$\mu \le \frac{|C\mathit{x}|}{(\tau +1)|C|\Gamma }$$

(59)

then the following inequation can be obtained

$$|C{\mathit{e}}_{\mu }|\le |C|\Gamma \mu \le \frac{1}{\tau }|C{q}_{\mu }(\mathit{x})|$$

(60)

By considering the uplink and down channel quantize, the following control law can be designed ³³

$$\begin{array}{c}\mathit{v}=-\frac{\tau +1}{\tau -\chi }(C{B}_j{)}^{-1}[|\mathrm{CA}{q}_{\mu }(\mathit{x}(t))|+|\mathrm{CA}|\Gamma {\mu }_j\hfill \\ \multicolumn{1}{c}{+|C{B}_j|{\Gamma }_1{\mu }_s+|C{B}_j|(d_1+d_2|q_{\mu }(\mathit{x}(t))|}\\ +d_2\Gamma {\mu }_j+d_3{\Gamma }_1{\mu }_s)]\mathrm{sgn}[C{q}_{\mu }(\mathit{x}(t))]-\frac{\tau \rho }{\tau -\chi }(C{B}_j{)}^{-1}\hfill \\ \multicolumn{1}{c}{\mathrm{sgn}[C{q}_{\mu }(\mathit{x}(t))]}\end{array}$$

(61)

$${\mathit{U}}_{\mathrm{force}}=q_{{\mu }_s}(\mathit{v})$$

(62)

where $\chi =1+(\tau +1)d_3,\tau >\frac{1+d_3}{1-d_3},\rho >0$ . The control law can keep the system states reach and maintain on the sliding surface $s(\mathit{x}(t))=0$ .

A simple and efficient adjustment law is as follows ³⁴

$$\{\begin{array}{c}{\mu }_j=\frac{\lfloor |C\mathit{x}(t)|\rfloor }{(\tau +1)|C|\Gamma },|C\mathit{x}(t)|\ge 1\hfill \\ \multicolumn{1}{c}{{\mu }_j=\frac{{\alpha }^{\lambda }}{(\tau +1)|C|\Gamma },0<|C\mathit{x}(t)|<1}\\ \multicolumn{1}{c}{{\mu }_j=0,|C\mathit{x}(t)|=0}\end{array}$$

(63)

where $\alpha \in (0,1),{\alpha }^{\lambda }<|C\mathit{x}|<{\alpha }^{\lambda -1}$ .

Roll moment controller

In order to obtain favorable handling characteristics in the stability region, the appropriate roll moment distribution coefficient $\epsilon$ should be got. According to the input-output linearization method, ³⁵ the yaw rate tracking error $e_{\mathrm{yaw}}$ can be regarded as the output. By differentiating the tracking error, the roll moment distribution coefficient $\epsilon$ appears explicit in the equation

$${\stackrel{·}{e}}_{\mathrm{yaw}}=\frac{u}{a+b}\stackrel{·}{\delta }-\stackrel{·}{\gamma }=\frac{u}{a+b}\stackrel{·}{\delta }-\frac{a{F}_{\mathrm{yf}}-b{F}_{\mathrm{yr}}}{I_z}$$

(64)

where when in the stability region, $\mathrm{\Delta }M=0$ . To get the control input $\epsilon$ , define

$${\stackrel{·}{e}}_{\mathrm{yaw}}+K{e}_{\mathrm{yaw}}=0$$

(65)

then the tracking error of the yaw rate will converge to zero, and system is asymptotically stable. Introducing the notation $C_f=C_{1}m\frac{b}{(a+b)}+\frac{1}{2}{C}_2{m}^2\frac{b^2}{{(a+b)}^2},C_{\epsilon }=2C_2{\left(m_{s}u\frac{h}{c+d}\right)}^2$ , $C_r=C_{1}m\frac{b}{(a+b)}+\frac{1}{2}{C}_2{m}^2\frac{b^2}{{(a+b)}^2}$ , where $C_f$ and $C_r$ are the linear tire stiffness. Then equation (65) can be further written as

$$\begin{array}{c}-{\epsilon }^2\frac{C_{\epsilon }}{I_{z}u}{\gamma }^2[\delta \mathrm{au}-\gamma (a^2+b^2)+u\beta (b-a)]\hfill \\ \multicolumn{1}{c}{-\epsilon \frac{2C_{\epsilon }}{I_{z}u}{\gamma }^2[\delta \mathrm{au}-\gamma (a^2-b^2)-u\beta (b+a)]}\\ -\frac{C_{\epsilon }}{I_{z}u}{\gamma }^2[\delta \mathrm{au}-\gamma (a^2+b^2)+u\beta (b-a)]\hfill \\ \multicolumn{1}{c}{+\frac{1}{I_{z}u}[\delta \mathrm{au}{C}_f-\gamma (a^2{C}_f+b^2{C}_r)+u\beta (b{C}_r-a{C}_f)]}\\ +\frac{u}{a+b}\stackrel{·}{\delta }+K(\frac{u}{a+b}\delta -\gamma )=0\hfill \end{array}$$

(66)

By solving the above equation, the roll moment distribution coefficient $\epsilon$ can be obtained and the range of $\epsilon$ is set as [–1, 1].

As the distances between left and right suspensions to the vehicle centerline is equal, the force transformations of active suspension can be written as

$$\left[\begin{array}{c}{F}_z\hfill \\ \multicolumn{1}{c}{F_{\theta }}\\ \multicolumn{1}{c}{F_{\phi }}\end{array}\right]=\left[\begin{array}{cccc}1\hfill & 1\hfill & 1\hfill & 1\hfill \\ \multicolumn{1}{c}{a}& a\hfill & -b\hfill & -b\hfill \\ \multicolumn{1}{c}{c}& -c\hfill & c\hfill & -c\hfill \end{array}\right]\left[\begin{array}{c}{F}_{a1}\hfill \\ \multicolumn{1}{c}{F_{a2}}\\ \multicolumn{1}{c}{F_{a3}}\\ \multicolumn{1}{c}{F_{a4}}\end{array}\right]$$

(67)

where $F_{a1}$ , $F_{a2}$ , $F_{a3}$ , $F_{a4}$ are the active forces generated by the active suspensions. Through inverse matrix operation, the expressions can be obtained as

$$\left[\begin{array}{c}{F}_{a1}\hfill \\ \multicolumn{1}{c}{F_{a2}}\\ \multicolumn{1}{c}{F_{a3}}\\ \multicolumn{1}{c}{F_{a4}}\end{array}\right]=\left[\begin{array}{ccc}\frac{b}{2a+2b}\hfill & -\frac{1}{2a+2b}\hfill & \frac{1}{4c}\hfill \\ \multicolumn{1}{c}{\frac{b}{2a+2b}}& -\frac{1}{2a+2b}\hfill & -\frac{1}{4c}\hfill \\ \multicolumn{1}{c}{\frac{a}{2a+2b}}& \frac{1}{2a+2b}\hfill & \frac{1}{4c}\hfill \\ \multicolumn{1}{c}{\frac{a}{2a+2b}}& \frac{1}{2a+2b}\hfill & -\frac{1}{4c}\hfill \end{array}\right]\left[\begin{array}{c}{F}_z\hfill \\ \multicolumn{1}{c}{F_{\theta }}\\ \multicolumn{1}{c}{F_{\phi }}\end{array}\right]$$

(68)

The active forces of front and rear axles are generated by the active suspensions, and can be given as

$$F_f=\frac{1+\epsilon }{2}{F}_{\phi },F_r=\frac{1-\epsilon }{2}{F}_{\phi }$$

The equations can then be given as

$$\left[\begin{array}{c}{F}_{a1}\hfill \\ \multicolumn{1}{c}{F_{a2}}\end{array}\right]=\left[\begin{array}{ccc}\frac{b}{2a+2b}\hfill & -\frac{1}{2a+2b}\hfill & \frac{1}{4c}\hfill \\ \multicolumn{1}{c}{\frac{b}{2a+2b}}& -\frac{1}{2a+2b}\hfill & -\frac{1}{4c}\hfill \end{array}\right]\left[\begin{array}{c}{F}_z\hfill \\ \multicolumn{1}{c}{F_{\theta }}\\ \multicolumn{1}{c}{\frac{1+\epsilon }{2}{F}_{\phi }}\end{array}\right]$$

(69)

$$\left[\begin{array}{c}{F}_{a3}\hfill \\ \multicolumn{1}{c}{F_{a4}}\end{array}\right]=\left[\begin{array}{ccc}\frac{a}{2a+2b}\hfill & \frac{1}{2a+2b}\hfill & \frac{1}{4c}\hfill \\ \multicolumn{1}{c}{\frac{a}{2a+2b}}& \frac{1}{2a+2b}\hfill & -\frac{1}{4c}\hfill \end{array}\right]\left[\begin{array}{c}{F}_z\hfill \\ \multicolumn{1}{c}{F_{\theta }}\\ \multicolumn{1}{c}{\frac{1-\epsilon }{2}{F}_{\phi }}\end{array}\right]$$

(70)

Control of sprung mass motions is changed into control the four subsystems by utilizing the decoupling calculation.

ESP controller design

The ESP controller improves the lateral dynamics through generating corrective yaw moment. A terminal sliding mode control ³⁶ is utilized for the ESP. According to equation (7), following equation can be got

$$\stackrel{··}{\psi }=\stackrel{·}{\gamma }=\frac{M_z}{I_z}+\frac{\mathrm{\Delta }M}{I_z}$$

(71)

Where $M_z=a{F}_{\mathrm{yf}}-b{F}_{\mathrm{yr}}$ . The tracking error of the yaw rate is

$$\tilde{\psi }=\psi -{\psi }_{\mathrm{des}},\tilde{\gamma }=\gamma -{\gamma }_{\mathrm{des}}$$

(72)

Define the sliding surface as

$$s_{\gamma }=\tilde{\psi }+{\alpha }_{\gamma 1}\mathrm{sig}{\mathrm{n}}^{{\gamma }_1}(\tilde{\psi })+{\alpha }_{\gamma 2}\mathrm{sig}{\mathrm{n}}^{{\gamma }_2}(\tilde{\gamma })$$

(73)

among which, ${\alpha }_{\gamma 1}>0$ , ${\alpha }_{\gamma 2}>0$ , $1<{\gamma }_2<2$ , ${\gamma }_1>{\gamma }_2$ . The deviation of the sliding surface is expressed as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{s}}_{\gamma }=\tilde{\gamma }+{\alpha }_{\gamma 1}{\gamma }_1{\left|\tilde{\psi }\right|}^{{\gamma }_1-1}\tilde{\gamma }+{\alpha }_{\gamma 2}{\gamma }_2{\left|\tilde{\gamma }\right|}^{{\gamma }_2-1}\stackrel{·}{\stackrel{~}{\gamma }}}\\ =(\gamma -{\gamma }_{\mathrm{des}})+{\alpha }_{\gamma 1}{\gamma }_1{|\psi -{\psi }_{\mathrm{des}}|}^{{\gamma }_1-1}(\gamma -{\gamma }_{\mathrm{des}})\hfill \\ \multicolumn{1}{c}{+{\alpha }_{\gamma 2}{\gamma }_2{|\gamma -{\gamma }_{\mathrm{des}}|}^{{\gamma }_2-1}(\stackrel{·}{\gamma }-{\stackrel{·}{\gamma }}_{\mathrm{des}})}\\ =(\gamma -{\gamma }_{\mathrm{des}})+{\alpha }_{\gamma 1}{\gamma }_1{|\psi -{\psi }_{\mathrm{des}}|}^{{\gamma }_1-1}(\gamma -{\gamma }_{\mathrm{des}})\hfill \\ \multicolumn{1}{c}{+{\alpha }_{\gamma 2}{\gamma }_2{|\gamma -{\gamma }_{\mathrm{des}}|}^{{\gamma }_2-1}(\frac{M_z}{I_z}+\frac{\mathrm{\Delta }M}{I_z}-{\stackrel{·}{\gamma }}_{\mathrm{des}})}\end{array}$$

(74)

The control law of the ESP controller is formulated as

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }M=I_z[(1/{\alpha }_{\gamma 2}{\gamma }_2){\mathrm{sign}}^{2-{\gamma }_2}\left(\tilde{\gamma }\right)(1+{\alpha }_{\gamma 1}{\gamma }_1{\left|\tilde{\psi }\right|}^{{\gamma }_1-1}}\end{array})-\frac{M_z}{I_z}+{\stackrel{·}{\gamma }}_{\mathrm{des}}+K_{\gamma 1}\mathrm{sign}(s_{\gamma })+K_{\gamma 2}{s}_{\gamma }]$$

(75)

where

$$K_{\gamma 1},K_{\gamma 2}>0.$$

Define the Lyapunov function as

$$V_{\gamma }=\frac{1}{2}{s}_{\gamma }^2$$

(76)

the time derivative of (76) is as

$${\stackrel{·}{V}}_{\gamma }=s_{\gamma }((\gamma -{\gamma }_{\mathrm{des}})+{\alpha }_{\gamma 1}{\gamma }_1{|\psi -{\psi }_{\mathrm{des}}|}^{{\gamma }_1-1}(\gamma -{\gamma }_{\mathrm{des}})+{\alpha }_{\gamma 2}{\gamma }_2{|\gamma -{\gamma }_{\mathrm{des}}|}^{{\gamma }_2-1}(\frac{M_z}{I_z}+\frac{\mathrm{\Delta }M}{I_z}-{\stackrel{·}{\gamma }}_{\mathrm{des}}))$$

(77)

By implementing the control law (75) into (77) can obtain

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{\gamma }=-K_{\gamma 1}{\alpha }_{\gamma 2}{\gamma }_2{|\gamma -{\gamma }_{\mathrm{des}}|}^{{\gamma }_2-1}\left|s_{\gamma }\right|}\\ \multicolumn{1}{c}{-K_{\gamma 2}{\alpha }_{\gamma 2}{\gamma }_2{|\gamma -{\gamma }_{\mathrm{des}}|}^{{\gamma }_2-1}{s}_{\gamma }^2}\\ \multicolumn{1}{c}{\le -K_{\gamma 2}{\alpha }_{\gamma 2}{\gamma }_2{|\gamma -{\gamma }_{\mathrm{des}}|}^{{\gamma }_2-1}{s}_{\gamma }^2}\\ \multicolumn{1}{c}{=-2K_{\gamma 2}{\alpha }_{\gamma 2}{\gamma }_2{|\gamma -{\gamma }_{\mathrm{des}}|}^{{\gamma }_2-1}{V}_{\gamma }}\end{array}$$

(78)

For $V_{\gamma }>0$ , $K_{\gamma 2},{\alpha }_{\gamma 2},{\gamma }_2>0$ , so it can be concluded that ${\stackrel{·}{V}}_{\gamma }\le 0$ . The discontinuity of the sign function can be modified as

$$\mathrm{sign}{s}_{\beta }=\frac{s_{\gamma }}{\left|s_{\gamma }\right|+{\delta }_{\gamma }}$$

(79)

where ${\delta }_{\gamma }$ is an arbitrary small positive constant.

Integrated control strategy

When the car travels straightly, the active suspension controller generates active control force to minimizing sprung mass motion; the roll moment controller and ESP controller do not work. As the brake-based ESP controller is not suitable for the vehicle in the stable area for the ESP controller directly affects the longitudinal motion of the vehicle. When there is a steering angle input, coordination of roll moment controller and ESP controller determines which controller is used. The boundary of the phase plane is defined as ³⁷

$$\mathrm{\Omega }=|2.5\stackrel{·}{\beta }+10\beta |<1$$

(80)

In the stability area, the roll moment controller forces the yaw rate to track desired yaw rate through controlling lateral tire forces. When the system excessed the stable lines, ESP controller affords additional yaw moment to keep the car running in stability region through the differential braking system. The switching between roll moment controller and ESP controller can be expressed as

$$M_{\mathrm{yaw}}={\rho }^{*}{M}_z+(1-{\rho }^{*})\mathrm{\Delta }M$$

(81)

where ${\rho }^{*}$ is the switching gain which is varying according to the stability region, and ${\rho }^{*}$ can be expressed as

$${\rho }^{*}=\{\begin{array}{c}1,\left|\mathrm{\Omega }\right|\le 0.8\hfill \\ \multicolumn{1}{c}{-5\left|\mathrm{\Omega }\right|+5,0.8\le \left|\mathrm{\Omega }\right|<1}\\ \multicolumn{1}{c}{0,\left|\mathrm{\Omega }\right|\ge 1}\end{array}$$

(82)

Simulation results

The control goal of the designed controller is to improve the chassis characteristics. In this section, the effect of the fault tolerant mechanism is validated through three different 3operating conditions. The front wheel inputs are shown in Figure 5, and the external road disturbances of four wheels are given in Figure 6. The simulation parameters are listed in Table 1.

Figure 5.

Front wheel inputs: (a) step steering input and (b) SLC steering input.

Figure 6.

Four-wheel road disturbances.

Step steering input

The step steering input plotted in Figure 5(a) is considered in this section, and the vehicle travels at a speed of 60 km/h.

The observed results under step steering are given in Figure 7(a) to (d). The figures show that the designed observer can track the system states exactly.

Figure 7.

Observations under step input: (a) sprung mass pitch velocity, (b) sprung mass roll velocity, (c) sprung mass vertical velocities, and (d) unsprung mass velocities.

The responses of the nominal model to the controller are given in Figure 8(a) to (e). Figure 8(a) indicates that the controller can suppress the sprung mass motion when there is no steering input, and the vibration performance of the sprung mass has been effectively improved; when there is a step steering input, the suspension performance is degraded. When there is steering angle input, the suspension and tire deflections are decreased simultaneously, as shown in Figure 8(b) and (c). Figure 8(d) and (e) indicate that during step steering, the yaw rate and lateral acceleration are reduced significantly with the integrated controller. The results are in line with the design goals.

Figure 8.

Control results under step input: (a) sprung mass accelerations, (b) suspension deflections, (c) tire deflections, (d) lateral acceleration, and (e) yaw rate.

SLC steering input

This section mainly discusses the control results of the fault tolerant controller under SLC steering input condition.

Figure 9(a) to (d) shows the observed results of unmeasured states under SLC steering input. The results show that the designed method can follow system states exactly.

Figure 9.

Observations under SLC input: (a) sprung mass pitch velocity, (b) sprung mass roll velocity, (c) sprung mass vertical velocities, and (d) unsprung mass velocities.

The time responses of the vertical and lateral dynamics to the integrated controller are shown in Figure 10(a) to (e). Figure 10(a) shows that the vibration performance improves significantly when there is no steering input, and the performance closes to the passive system with steering input. Obviously, the suspension and tire deflections in Figure 10(b) and (c) are almost consistent with no steering input. The lateral dynamics are given in Figure 10(d) and (e), and the lateral performance has been improved greatly under the SLC input.

Figure 10.

Control results under SLC input: (a) sprung mass accelerations, (b) suspension deflections, (c) tire deflections, (d) lateral acceleration, and (e) yaw rate.

Actuator fault

Loss of actuator effectiveness is discussed in this section. Assume that there is a virtual actuator that produces a yaw moment, so the actuator fault can be taken into account. The failure of four suspension actuators and one DYC actuator is analyzed here, and the considered faults are defined as follows:

$$\begin{array}{c}{\eta }_{\mathrm{fl}}=\{\begin{array}{c}1t<2\hfill \\ \multicolumn{1}{c}{-0.25t+1.52\le t<4}\\ \multicolumn{1}{c}{0.54\le t<6}\\ \multicolumn{1}{c}{16\le t}\end{array}{\eta }_{\mathrm{fr}}=\{\begin{array}{c}1t<4\hfill \\ \multicolumn{1}{c}{0.54\le t<6}\\ \multicolumn{1}{c}{16\le t}\end{array}\hfill \\ {\eta }_{\mathrm{rl}}=\{\begin{array}{c}1t<4\hfill \\ \multicolumn{1}{c}{0.54\le t<6}\\ \multicolumn{1}{c}{16\le t}\end{array}{\eta }_{\mathrm{rr}}=\{\begin{array}{c}1t<4\hfill \\ \multicolumn{1}{c}{0.64\le t<6}\\ \multicolumn{1}{c}{16\le t}\end{array}\hfill \\ {\eta }_v=\{\begin{array}{c}1t<3\hfill \\ \multicolumn{1}{c}{0.53\le t<5}\\ \multicolumn{1}{c}{0.25t-0.755\le t\le 7}\\ \multicolumn{1}{c}{17\le t}\end{array}\hfill \end{array}$$

(83)

The simulation is carried out under a step input. The real fault signals and estimated fault signals of active suspension and DYC actuators are given in Figure 11(a) to (e). The estimation results indicate that the designed observer can reconstruct actuator fault signals exactly when the actuator faults occur, and the estimated values of the actuators are basically consistent with the real values.

Figure 11.

Actuator faults and estimations: (a) FL ASS actuator fault, (b) FR ASS actuator fault, (c) RL ASS actuator fault, (d) RR ASS actuator fault, and (e) DYC actuator fault.

To evaluate the control performance with actuator faults, the RMS values under step steering and SLC input are presented in Tables 2 and 3, respectively. It can be deduced from tables that the proposed active fault tolerant controller provides efficient performance under fault conditions.

Table 2.

Performance evaluation under step steering input.

Case/Step	Nominalmodel	Faultmodel
Vertical acceleration	0.3820	0.3896
Pitch acceleration	0.3550	0.3614
Roll acceleration	0.5048	0.5102
Front left suspension deflection	0.0052	0.0053
Lateral acceleration	2.2647	2.3099
Yaw rate	0.1350	0.1371

Table 3.

Performance evaluation under SLC input.

Case/SLC	Nominalmodel	Faultmodel
Vertical acceleration	0.3465	0.3520
Pitch acceleration	0.2756	0.2801
Roll acceleration	0.3721	0.3795
Front left suspension deflection	0.0045	0.0046
Lateral acceleration	1.5328	1.5936
Yaw rate	0.0925	0.0994

Comparisons between $H_{\infty }$ fault tolerant control and the designed controller are presented to further illustrate the effectiveness of the proposed method, and the simulation was carried out under straight driving condition. In the presence of actuator faults, the dynamic response of the vertical system under and the designed controller is plotted in Figure 12(a) to (c).

Figure 12.

Fault tolerant performance with $H_{\infty }$ and the designed controller: (a) sprung mass accelerations, (b) suspension deflections, and (c) tire deflections.

It can be seen from Figure 12(a) that the vertical, pitch and roll accelerations with the designed controller can be well controlled in the interval of actuator faults, while the $H_{\infty }$ controller is greatly affected by the faults, as shown by the blue dotted line. Similarly, from Figure 12(b) and (c), it can be found that in the actuator fault interval, the suspension deflection and tire deflection performances with the designed controller are better than those of the $H_{\infty }$ fault tolerant controller.

Conclusions

This study presented a fault observer-based hierarchical integrated control algorithm to process the actuator faults and coupling properties of vehicle chassis model. The observer was designed for the integrated model with respect to unmeasured states and actuator faults, to provide accurate control parameters and improve the reliability of the system. Then, a hierarchical integrated control mechanism was proposed for the integrated model to improve the chassis performance with interferences and failures. Simulation results indicate that the controller can perfectly estimate actuator fault signals and effectively improve the vehicle chassis performance. Conclusions of the study are:

• (1) The proposed observer can track the system states and estimate the actuator faults accurately;

• (2) The hierarchical control strategy can promote the vehicle performance with different steering input;

• (3) The fault tolerant control mechanism can provide excellent characteristics for vehicle chassis system.

The active fault tolerant method was verified under different steering inputs and actuator faults. Future work would possibly concern the following aspects:

1. Chassis system modeling with consideration of component nonlinearities and longitude dynamic study for further potential exploration of the integrated control system.

2. Multiple sensor and actuator faults should be considered simultaneously.

3. More advanced fault tolerant control strategies that could be suitable for the complete failure condition being required for vehicle performance improvement.