Discrete optimal actuator-fault-tolerant control for vehicle active suspension

1. Introduction

As an effective component for vehicle active safety technology, vehicle active suspension systems can meet the various requirements of performance criterions, such as road holding ability, ride comfort, and suspension deflection. Compared with the traditional passive and semi-passive suspensions, vehicle active suspension systems are able to store, dissipate and generate energy [1, 2]. Then, the stability can be guaranteed and the better control performance can be achieved by using the novel control strategies and advanced actuators. Many advanced control schemes have been proposed for vehicle active suspension in recent decades. For instance, a control scheme was designed to trade off among multi-objective requirements for uncertain nonlinear active suspension in [3]; an enhance adaptive self-organizing fuzzy sliding-mode controller was proposed for active suspension systems in [4]; [5] designed a discrete feedback and feedforward optimal vibration controller for vehicle active suspension with actuator time delay. Also, the art survey about active and semi-active suspension control was given in [6].

The development of actuator-fault-tolerant controller is beneficial to improve the reliability of the suspension systems. Due to the actuator faults, the generated actuator force maybe not researched the expected value. Then, the serious degradation of control performance will generate, even the suspension system becomes instability. To eliminate the influence from actuator faults and ensure the safety of the suspension systems, many researchers have paid more attention to fault-tolerant control problem. For example, an adaptive sliding fault tolerant controller was proposed for nonlinear uncertain active suspension system in [7]; considering the actuator faults, an adaptive PID-Sliding-Mode fault tolerant control scheme was developed in [8]; a reliable fuzzy controller was proposed for active suspension systems with actuator delay and fault in [9]. The above active control schemes are capable of reducing the vibration of vehicle active suspension subject to road disturbances and actuator faults. However, the rarely digital fault-tolerant control problem is considered for vehicle active suspension under computer control system. Actually, with advantages of low-cost components and conveniences of digital systems, the digital control becomes the tendency of the development of the vehicle active suspension [10]. Recently, the various methods have been applied to the actuator-fault-tolerant control problem for computer control systems, such as $H_{\mathrm{\infty }}$ control [11], stochastic control [12], and observer-based control [13]. However, there are few results about the discrete actuator-fault-tolerant controller design for vehicle active suspension in the sense of the optimal perspective.

The above considerations motivate our research in the present work. Taken the actuator faults into consideration, the discrete model of the vehicle active suspension is developed, in which the persistent road disturbances and actuator faults are described as the outputs of the designed exogenous systems with unknown initial values. Then, an optimal discrete quadratic optimal actuator-fault-tolerant controller is proposed to offset the vibration of vehicle body caused by the road disturbances and overcome the unsafely caused by the actuator faults. Meanwhile, a fault observer is designed to diagnose the actuator faults so that the physically unrealizable problem of the proposed controller is solved. Simulation results are shown that the control performance and reliability of the vehicle active suspension can be improved by using the proposed controller.

The paper is organized as follows. The discrete optimal fault-tolerant control problem of the vehicle active suspension is formulated in Section 2. Section 3 is devoted to present the main results, where an actuator-fault-observer-based discrete optimal fault-tolerant controller is proposed. Section 4 provides some simulation results to illustrate the effectiveness of the developed approaches. Finally, Section 5 gives the conclusion of this paper.

2. Problem formulation

2.1. Discrete suspension modeling with actuator faults

As shown in Fig. 1, a simple vehicle active suspension with actuator faults is considered in this paper. The global dynamic equations of the vehicle active suspension can be described as:

1

${\ddot{z}}_s\left(t\right)=\frac{c_s}{m_s}\left({\dot{z}}_u\left(t\right)-{\dot{z}}_s\left(t\right)\right)+\frac{k_s}{m_s}\left(z_u\left(t\right)-z_s\left(t\right)\right)+\frac{1}{m_s}u\left(t\right)-\stackrel{-}{\vartheta }\left(t\right),$
${\ddot{z}}_u\left(t\right)=\frac{c_s}{m_u}\left({\dot{z}}_s\left(t\right)-{\dot{z}}_u\left(t\right)\right)+\frac{k_s}{m_u}\left(z_s\left(t\right)-z_u\left(t\right)\right)+\frac{k_t}{m_u}\left(z_r\left(t\right)-z_u\left(t\right)\right)$
$+\frac{c_t}{m_u}\left({\dot{z}}_r\left(t\right)-{\dot{z}}_u\left(t\right)\right)-\frac{1}{m_u}u\left(t\right)+\stackrel{-}{\vartheta }\left(t\right),$

where $m_s$ and $m_u$ represent the sprung mass and unsprung mass, respectively; $k_t$ and $c_t$ are the compressibility and damping of the pneumatic tire; $k_s$ and $c_s$ stand for the stiffness and damping of the passive suspension system, respectively. $z_s\left(t\right)$ and $z_u\left(t\right)$ denote the displacements of the sprung and unsprung masses. $u\left(t\right)$ is the control force of the vehicle active suspension, ${\dot{z}}_r\left(t\right)$ and $\stackrel{-}{\vartheta }\left(t\right)$ represent the road disturbances and the actuator fault acting on the vehicle active suspension, respectively.

Define the following state variables:

2

${\stackrel{-}{x}}_1\left(t\right)=z_s\left(t\right)-z_u\left(t\right),{\stackrel{-}{x}}_2\left(t\right)=z_u\left(t\right)-z_r\left(t\right),{\stackrel{-}{x}}_3\left(t\right)={\dot{z}}_s\left(t\right),{\stackrel{-}{x}}_4\left(t\right)={\dot{z}}_u\left(t\right),$

where ${\stackrel{-}{x}}_1\left(t\right)$ is the suspension deflection, ${\stackrel{-}{x}}_2\left(t\right)$ denotes the tire deflection, ${\stackrel{-}{x}}_3\left(t\right)$ is the sprung mass velocity, and ${\stackrel{-}{x}}_4\left(t\right)$ denotes the unsprung mass velocity.

Let $\stackrel{-}{x}\left(t\right)={\left[\begin{array}{cc}{\stackrel{-}{x}}_1^T\left(t\right)& {\stackrel{-}{x}}_2^T\left(t\right)\end{array}\begin{array}{cc}{\stackrel{-}{x}}_3^T\left(t\right)& {\stackrel{-}{x}}_4^T\left(t\right)\end{array}\right]}^T$, the state-space model of the vehicle active suspension (1) subject to the actuator faults can be formulated as:

3

$\dot{\stackrel{-}{x}}\left(t\right)=\stackrel{-}{A}\stackrel{-}{x}\left(t\right)+\stackrel{-}{B}u\left(t\right)+\stackrel{-}{D}\stackrel{-}{v}\left(t\right)+{\stackrel{-}{D}}_1\stackrel{-}{\vartheta }\left(t\right),$
${\stackrel{-}{y}}_c\left(t\right)=C\stackrel{-}{x}\left(t\right)+Eu\left(t\right)+E_1\stackrel{-}{\vartheta }\left(t\right),$
${\stackrel{-}{y}}_m\left(t\right)=\stackrel{-}{C}\stackrel{-}{x}\left(t\right),$

where:

4

$\stackrel{-}{A}=\left[\begin{array}{cccc}0& 0& 1& -1\\ 0& 0& 0& 1\\ -\frac{k_s}{m_s}& 0& -\frac{c_s}{m_s}& \frac{c_s}{m_s}\\ \frac{k_s}{m_u}& -\frac{k_t}{m_u}& \frac{c_s}{m_u}& -\frac{c_s+c_t}{m_u}\end{array}\right],\stackrel{-}{B}=\left[\begin{array}{c}\begin{array}{c}0\\ 0\end{array}\\ \frac{1}{m_s}\\ -\frac{1}{m_u}\end{array}\right],\stackrel{-}{D}=\left[\begin{array}{c}\begin{array}{c}0\\ -1\end{array}\\ 0\\ \frac{c_t}{m_u}\end{array}\right],{\stackrel{-}{D}}_1=\left[\begin{array}{c}\begin{array}{c}0\\ 0\end{array}\\ -1\\ 1\end{array}\right],\stackrel{-}{C}=I,$
$C=\left[\begin{array}{cccc}-\frac{k_s}{m_s}& 0& -\frac{c_s}{m_s}& \frac{c_s}{m_s}\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right],E=\left[\begin{array}{c}\frac{1}{m_s}\\ 0\\ 0\end{array}\right],E_1=\left[\begin{array}{c}-1\\ 0\\ 0\end{array}\right],\stackrel{-}{v}\left(t\right)={\dot{z}}_r\left(t\right).$

Fig. 1Simple vehicle active suspension

Remark 1: The sprung mass acceleration ${\ddot{z}}_s\left(t\right)$, the suspension deflection $z_s\left(t\right)-z_u\left(t\right)$, and the tire deflection $z_u\left(t\right)-z_r\left(t\right)$ are viewed as the main performance criteria, which directly relate to the ride comfort, the road holding ability, and driving safety ability directly. Hence, we design the controlled output ${\stackrel{-}{y}}_c\left(t\right)$ as the second formula of (3).

The following exogenous system is introduced to describe the dynamic characteristics of the actuator fault $\stackrel{-}{\vartheta }\left(t\right)$:

5

$\dot{\stackrel{-}{\phi }}\left(t\right)=\stackrel{-}{G_0}\stackrel{-}{\phi }\left(t\right),t\ge t_0,\stackrel{-}{\phi }\left(0\right)={\stackrel{-}{\phi }}_0,$
$\stackrel{-}{\phi }\left(t\right)=0,0\le t\le t_0,$
$\stackrel{-}{\vartheta }\left(t\right)=H\stackrel{-}{\phi }\left(t\right),$

where $\stackrel{-}{\phi }\left(t\right)$ is the state vector of the actuator faults, ${\stackrel{-}{G}}_0$ and $H$ are constant matrices with appropriate dimensions. The actuator fault $\stackrel{-}{\phi }\left(t\right)$ occurs at $t_0$, in which the initial fault time $t_0$ and the initial fault state ${\stackrel{-}{\phi }}_0$ are unknown.

Let $T$ denotes the sampling period, the discrete model of vehicle active suspension system (3) can be expressed as:

6

$x\left(k+1\right)=Ax\left(k\right)+Bu\left(k\right)+Dv\left(k\right)+D_1\vartheta \left(k\right),$
$y_c\left(k\right)=Cx\left(k\right)+Eu\left(k\right)+E_1\vartheta \left(k\right),$
$y_m\left(k\right)=\stackrel{-}{C}x\left(k\right),$

where $A=e^{\stackrel{-}{A}T}$, $B={\int }_0^T{e}^{\stackrel{-}{A}T}\stackrel{-}{B}dt$, $D={\int }_0^T{e}^{\stackrel{-}{A}T}\stackrel{-}{D}dt$, and $D_1={\int }_0^T{e}^{\stackrel{-}{A}T}{\stackrel{-}{D}}_{1}dt$.

The discrete representation of the actuator fault Eq. (5) can be obtained:

7

$\phi \left(k+1\right)=G_0\phi \left(k\right),k\ge t_0,\phi \left(0\right)={\phi }_0,$
$\phi \left(k\right)=0,0\le k\le t_0,$
$\vartheta \left(k\right)=H\phi \left(k\right),$

where $G_0=e^{\stackrel{-}{G_0}T}$.

Remark 2: Various kinds of the actuator faults can be described as Eq. (7) with known dynamic characteristics, and unknown magnitudes and phases, such as periodic faults, sinusoidal faults, and other actuator faults in the discrete form.

Assumption 1: The pair $\left(A,B\right)$ is controllable and the pair $\left(A,C\right)$ is observable.

Assumption 2: The pair $\left(H,G_0\right)$ is completely observable, and all the eigenvalues ${\lambda }_i\left(G_0\right)$ of $G_0$ satisfy $\left|{\lambda }_i\left(G_0\right)\right|\le 1$.

3. Design of discrete optimal fault-tolerant controller via an actuator fault observer

In this section, we propose a discrete optimal fault-tolerant controller first and prove the existence and uniqueness of the designed controller. Then, a discrete fault observer is designed to solve the physically unrealizable problem caused by the status vector of actuator faults.

4. Simulation results

In this section, we will apply the proposed optimal actuator-fault-tolerant control law to a vehicle active suspension. The parameters of a vehicle active suspension are shown in Table 1.

Fig. 2The system structure of vehicle active suspension

Then, defining the sampling period $T=$0.08 s, the matrices in Eq. (3) can be obtained:

The road displacement input is generated from a ground displacement PSD (8), where $S_g\left({\mathrm{\Omega }}_0\right)=$256×10^-6 m³ with $n_1=$ 2 and $n_2=$ 1.5. Under $v_0=$ 20 m/s and $l=$ 400 m, the road disturbances $v\left(t\right)$ acting on the vehicle active suspension can be computed, which is depicted in Fig. 3.

Two types of actuator faults are considered in this section, which include attenuation type and oscillation type. Assume that the actuator fault occurs at $t_0=$4 s, and the initial value of the actuator fault ${\phi }_0={\left[\begin{array}{cc}0.3& 0.3\end{array}\right]}^T$.

For attenuation type of actuator fault, the matrices $G_0$ and $F$ is given by:

Designing the poles of the actuator fault observer as ${\lambda }_{\mathrm{1,2}}=$0.48±0.5i, the gain weighting matrix $K_e$ is obtained as ${\left[\begin{array}{cc}0.95& 5.4\end{array}\right]}^T$.

For oscillation type of actuator fault, the matrices $G_0$ and $F$ are given as follows:

Fig. 3The curve of random road disturbances

Designing the poles of the actuator fault observer as ${\lambda }_{\mathrm{1,2}}=$0.3±0.19i, the gain weighting matrix $K_e$ is obtained as ${\left[\begin{array}{cc}1.4& 6.5\end{array}\right]}^T$. The comparison results between real actuator faults and the actuator fault observer are shown in Fig. 4. It can be seen clearly that the actuator fault $\vartheta \left(k\right)$, can be diagnosed in 4s exactly by using the designed actuator fault observer Eq. (30).

Setting the initial status $x_0={\left[\begin{array}{cc}\begin{array}{cc}0.1101& 0.1101\end{array}& \begin{array}{cc}1.3769& 1.3769\end{array}\end{array}\right]}^T$, the weight matrices $Q=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left\{3\times {10}^7,5\times {10}^7,{10}^7\right\}$ and$R=$ 1, by solving the discrete Riccati Eq. (20) and Stein Eqs. (21), respectively, the feedforward and feedback optimal actuator-fault-tolerant controller (FFOAFTC) under two types of actuator faults Eqs. (36) and (37) can be obtained. Meanwhile, the feedback optimal actuator-fault-tolerant controller (FOAFTC) without feedforward item of actuator faults is used to compare with the proposed FFOAFTC, which is described as:

The curves of FFOAFTC and FOAFTC under attenuation type and oscillation type of actuator fault are shown in Fig. 5.

When the optimal FFOAFTC acts on the vehicle active suspension, the response curves of the sprung mass acceleration (SMA), sprung deflection (SD), and tire deflection (TD) are shown in Figs. 6-8, respectively, in which the results are compared with ones of FOAFTC and open-loop (OL) systems. From these figures, one can be seen that during the actuator faults period, the values of SMA, SD, and TD converge to smaller values by using the FFOAFTC.

In order to show the ability of controller to reduce the vibration amplitude, the frequency spectrums of SMA, SD, and TD in presence of both actuator faults are shown in Figs. 9-11 compared with FOAFTC and uncontrolled ones.

Meanwhile, in order to show the effectiveness of the proposed FFOAFTC clearly, the peak values and root-mean square (RMS) values of SMA, SD, and TD are compared with FOAFTC and uncontrolled ones in Table 2. For example, under the attenuation type of actuator fault, the peak value of SMA is reduced about 23.3 % compared to the OL one by using the proposed FFOAFTC, the RMS value of the SMC is reduced about 70.4 %, the RMS of sprung deflection is reduced about 69.3 %, and the RMS of tire deflection is reduced about 60.4 % less than that of OL system.

Fig. 4The comparison between real actuator fault and actuator fault observer

Fig. 5The curves of control law for attenuation type and oscillation type of actuator fault

Fig. 6The curves of SMA under different actuator fault types

Fig. 7The curves of SD under different actuator fault types

Fig. 8The curves of TD under different actuator fault types

Fig. 9The frequency spectrums of SMA under different actuator fault types

Fig. 10The frequency spectrums of SD under different actuator fault types

Fig. 11The frequency spectrums of TD under different actuator fault types

From Table 2 and Figs. 4-11, it can be found that the designed observer can diagnose the actuator fault exactly, and the proposed optimal actuator-fault-tolerant controller can effectively compensate the actuator fault and offset the vehicle body’s vibration caused by road disturbances. The control performance of vehicle active suspension is improved significantly.

5. Conclusions

This paper has researched the problem of discrete actuator-fault-tolerant control problem for a class of vehicle active suspension. Different from the previous approaches, with regarded the actuator faults as an output of exosystem with unknown initial value and occurred time, the discrete model of vehicle active suspension with actuator faults was established first. Furthermore, an optimal actuator-fault-tolerant control scheme was devised from discrete Riccati equation and Stein equations. What’s more, an actuator fault observer was designed to make the control scheme physical realizability. Then, the implementability of the vehicle active suspension is guaranteed, and the control performance can be improved significantly. Our future study would focus on extending the obtained results in this paper to more complicated cases for nonlinear active suspensions with delayed measurements and inputs.