A novel adaptive actuator failure compensation scheme based on multi-design integration for half-car active suspension system

Abstract

A new actuator failure compensation scheme in the presence of actuator failures with multi-uncertainties is proposed for half-car active suspension systems based on multi-design integration. A parameterized function has been introduced to denote the multiple failures occurred in the front and rear actuators. An effective feedback controller is designed to obtain asymptotic tracking and closed-loop system stability by employing backstepping technique. Based on the desired controller, multiple adaptive control signals are designed corresponding to each possible failure case. By introducing failure indicator functions and integrating these control signals, a composite controller is constructed to handle all failure patterns, which ensures both pitch and vertical motions of car body to stabilize and track the desired signals. Finally, simulation is conducted to verify the effectiveness of the controller proposed in this study.

Introduction

Recently, more and more researchers attach great importance to the study of improving the dynamic characteristics of vehicle ride comfort and handling performance.^1–3 Achieving acceptable performance in these measures is critical to enable automotive manufacturers to meet the requirements for ensuring passengers health and safety and so on. Vehicle suspension plays a fundamental role in improving passenger comfort and handling characteristics, which is generally divided into three types: active, semi-active, and passive suspensions. In active suspension systems, actuators are used to supply additional forces, so that energy can be added and dissipated from the system. The active suspension systems have advantages on good performance under different riding conditions, compared with passive and semi-active systems. Consequently, such systems have attracted extensive attention in recent years, and many diverse actuator control strategies were reported during the past few decades, such as adaptive control,^4,5 fuzzy control,^6,7 robust control,^8,9 optimal control,^10–12 and preview method. ¹³ Of all the strategies, the adaptive control method has certain advantages of guaranteeing steady-state and transient performance. Especially, with this algorithm, the unknown uncertainty can be handled with an improved capability.

However, previous experience showed that, in the event of components (such as sensors and actuators) faults, conventional feedback control designs for complex systems cannot offer satisfactory performance, and even cause instability, ¹⁴ which is totally unacceptable in complex systems. In practical applications, compared with other components faults in active suspensions, the impact of actuator failure is more severe and might lead to instability or even unexpected catastrophic accidents. For this reason, new approaches have emerged to tolerate component malfunctions, while achieving desirable performance and stability whenever the system is under faulty or fault-free conditions. Zheng et al. ¹⁵ studied infinite frequency H-infinity control for active vehicle suspension systems subjected to loss of actuator effectiveness, and further proposed controllers which can guarantee relation dynamic tire load and vehicle suspension deflections in their allowed scopes. Motivated by the fact that loss of actuator effectiveness is a common failure case in vehicle suspension systems, Moradi and Fekih^16,17 proposed two novel approaches for a full-scale vehicle dynamic model with an active suspension system to guarantee closed-loop system stability for both normal and faulted cases. Yang et al. ¹⁸ proposed a robust fault-tolerant control scheme based on $H_2/H_{\infty }$ , to solve the problem of loss of actuator effectiveness. Stabilization and tracking are two well-known control tasks. For nonlinear control systems, tracking is generally more complicated and important than stabilization, especially when the system operates under uncertain failure conditions. Many effective methods were proposed for nonlinear systems, including $H_{\infty }$ control, ¹⁹ backstepping,^20,21 adaptive control, ²² regulation control, ²³ sliding mode control, ¹⁷ and feedback linearization. Pan et al. ²⁴ applied an adaptive tracking control strategy to address parameter uncertainties for vehicle suspension systems, and improved suspension performance. In Sun et al. ²⁵ an adaptive robust controlling method was proposed, to solve the problem of actuator fault accommodation.

One limitation of these actuator failure compensation methods is the assumption that the failure conditions of all the system components are known (such as loss of actuator effectiveness, lock-in-place or Markovian jumping actuator failure). However, with the growing complexity of passenger vehicles, a variety of fault conditions are likely to be encountered, and the failure types are essentially random and unpredictable. Moreover, most of the aforementioned studies assume that the faulty state will remain during the rest of operation even if an actuator fails. However, this is unpractical, as failures usually develop in a gradual manner as time passes. In addition, the actuators in active suspension systems are likely to suffer an intermittent failure resulted from electromagnetic interference. For this reason, the failed actuators may recover from the failure case as the interference disappear and may fail again later. In a word, the actuator failure is uncertain in essence.

Besides, in practice, it is impossible to anticipate the times, patterns, types, and values of actuator failures in advance, as they are essentially uncertain in nature. That is, an actuator may switch its status among various failure patterns (e.g. failure-free case, partial or total loss of effectiveness case, lock-in-place case, time-varying failure case) randomly. In this condition, actuator failure compensation controllers which consider failure type, failure value, failure time, and failure pattern are required for active suspension systems. In addition to stability, direct adaptive control approach provides theoretically provable asymptotic tracking for systems with parameter variation and uncertainties, which is a key advantage compared with other approaches. ²⁶ Backstepping control method can link the choice of Lyapunov functions with its controls, which is usually used to design a feedback control law, which can ensure the stability of the related system. ²⁷ Therefore, adaptive control and backstepping are preferred in this study.

This article presents a novel direct adaptive controlling method for addressing the tracking control problem for active suspension systems having uncertain actuator failures. In this proposed algorithm, multiple controllers designed for individual failure cases are incorporated, as well as an integration mechanism. This article has the following contributions:

1. A parameterized mathematical function is introduced to denote multiple types of actuator failure occured in active suspension control system, especially time-varying failure case which is seldom considered in other studies of half-car active suspension control system.

2. This article developed a novel fault-tolerant controlling approach for active suspension systems to handle actuator failures with multi-uncertainties, which has a composite structure, it means that a single adaptive controller is utilized to accommodate multiple uncertainties of actuator failures.

In this approach, explicit fault detection is not needed, as multiple types of failures are estimated by using a complete controller parametrization, compared with fault diagnosis designs proposed in Yetendje et al. ²⁸ and Rizvi et al. ²⁹ Compared with the fault-tolerant control design presented in Moradi and Fekih ¹⁷ and Sun et al. ²⁵ to handle two specific failure scenarios, respectively, our adaptive control scheme does not rely on the knowledge of failure type and therefore can be used to handle multiple types of actuator failures, in addition to lock-in-place and loss of effectiveness.

The rest of this article has the following sections: The dynamic models of a half-car active suspension system and the control problems are described in the “Problem statement” section. The design procedures for the adaptive fault-tolerant controller are expended in the “Controller design” section. Simulation study of the adaptive actuator failure compensation scheme is performed in the “Illustrative example” section to demonstrate its effectiveness. Finally, the “Conclusion” section presents the conclusion.

Problem statement

System description

A nonlinear half-car active suspension system shown in Figure 1 is considered in this article, which has four-degree-of-freedom (DoF).

$$\begin{array}{c}M{\stackrel{··}{z}}_c={\mathrm{\Psi }}_1(t)+u_1+u_2\hfill \\ I\stackrel{··}{\phi }={\mathrm{\Psi }}_2(t)+a{u}_1-b{u}_2\hfill \\ m_f{\stackrel{··}{z}}_1=F_{\mathrm{sf}}+F_{\mathrm{df}}-F_{\mathrm{tf}}-F_{\mathrm{bf}}-u_1\hfill \\ m_r{\stackrel{··}{z}}_2=F_{\mathrm{sr}}+F_{\mathrm{dr}}-F_{\mathrm{tr}}-F_{\mathrm{br}}-u_2\hfill \end{array}$$

(1)

where ${\mathrm{\Psi }}_1(t)=-F_{\mathrm{df}}-F_{\mathrm{dr}}-F_{\mathrm{sf}}-F_{\mathrm{sr}}$ and ${\mathrm{\Psi }}_2(t)=-a(F_{\mathrm{df}}+F_{\mathrm{sf}})+b(F_{\mathrm{dr}}+F_{\mathrm{sr}})$ . The forces produced by the nonlinear stiffening spring, the piecewise linear damper, and the tire obey $F_{\mathrm{sf}}=k_{f1}\mathrm{\Delta }{y}_f$ , $F_{\mathrm{sr}}=k_{r1}\mathrm{\Delta }{y}_r$ , $F_{\mathrm{df}}=b_{f1}\mathrm{\Delta }{\stackrel{·}{y}}_f$ , $F_{\mathrm{dr}}=b_{r1}\mathrm{\Delta }{\stackrel{·}{y}}_r$ , $F_{\mathrm{tf}}=k_{f2}(z_1-z_{o1})$ , $F_{\mathrm{tr}}=k_{r2}(z_2-z_{o2})$ , $F_{\mathrm{bf}}=b_{f2}({\stackrel{·}{z}}_1-{\stackrel{·}{z}}_{o1})$ , $F_{\mathrm{br}}=b_{r2}({\stackrel{·}{z}}_2-{\stackrel{·}{z}}_{o2})$ , $\mathrm{\Delta }{y}_f=z_c+a\sin \phi -z_1$ , and $\mathrm{\Delta }{y}_r=z_c+b\sin \phi -z_2$ . The variables shown in system (1) are given in Table 1.

Figure 1.

Schematic diagram for a half-car active suspension system.

Table 1.

Nomenclature.

Symbol	Meaning
M	Mass of the vehicle body
I	Pitch moment of inertia about the center of mass
$m_f$ , $m_r$	The front and the rear unsprung masses
$F_{\mathrm{df}}$ , $F_{\mathrm{dr}}$ , $F_{\mathrm{sf}}$ , $F_{\mathrm{sr}}$	Forces produced by the springs and dampers
$F_{\mathrm{tf}}$ , $F_{\mathrm{bf}}$ , $F_{\mathrm{tr}}$ , $F_{\mathrm{br}}$	Elasticity force and damping force of the tires
$z_c$	Vertical displacement
$\phi$	Pitch angle
$z_r$ , $z_f$	The front and the rear body displacements
$z_1$ , $z_2$	The front and the rear unsprung mass displacements
$z_{o1}$ , $z_{o2}$	The front and the rear terrain height displacements
$a$ , $b$	Distances of the front and the rear axles to the center of mass
$u_1$ , $u_2$	Control inputs
$k_{f1}$ , $k_{r1}$	The front and the rear spring coefficients
$b_{f1}$ , $b_{r1}$	The front and the rear tire spring coefficients
$b_{f2}$ , $b_{r2}$	The front and the rear tire damping coefficients
$\mathrm{\Delta }{y}_f$ , $\mathrm{\Delta }{y}_r$	The front and rear suspension spaces

Define $x_1=z_c$ , $x_2={\stackrel{·}{z}}_c$ , $x_3=\phi$ , $x_4=\stackrel{·}{\phi }$ , $x_5=z_1$ , $x_6={\stackrel{·}{z}}_1$ , $x_7=z_2$ , and $x_8={\stackrel{·}{z}}_2$ , and then system (1) becomes

$$\begin{array}{c}{\stackrel{·}{x}}_1=x_2\hfill \\ {\stackrel{·}{x}}_2=\frac{{\mathrm{\Psi }}_1}{M}+\frac{1}{M}{u}_1+\frac{1}{M}{u}_2\hfill \\ {\stackrel{·}{x}}_3=x_4\hfill \\ {\stackrel{·}{x}}_4=\frac{{\mathrm{\Psi }}_2}{I}+\frac{a}{I}{u}_1-\frac{b}{I}{u}_2\hfill \\ {\stackrel{·}{x}}_5=x_6\hfill \\ {\stackrel{·}{x}}_6=\frac{1}{m_f}(F_{\mathrm{sf}}+F_{\mathrm{df}}-F_{\mathrm{tf}}-F_{\mathrm{bf}})-\frac{1}{m_f}{u}_1\hfill \\ {\stackrel{·}{x}}_7=x_8\hfill \\ {\stackrel{·}{x}}_8=\frac{1}{m_r}(F_{\mathrm{sr}}+F_{\mathrm{dr}}-F_{\mathrm{tr}}-F_{\mathrm{br}})-\frac{1}{m_r}{u}_2\hfill \end{array}$$

(2)

Actuator failure model

The parameterized function introduced to denote the actuator failures is expressed as

$${\overline{u}}_j\left(t\right)={\overline{u}}_{j0}+\sum _{i=1}^{q_j}{\overline{u}}_{\mathrm{ji}}{f}_{\mathrm{ji}}\left(t\right),t\ge t_j$$

(3)

where failure index $j\in \{1,2\}$ , failure values ${\overline{u}}_{\mathrm{ji}}$ , $i=0,1,\dots ,q_j$ , and failure time instant $t_j$ are unknown, and signals $f_{\mathrm{ji}}(t),i=1,2,\dots ,q_j$ , $q_j\ge 1$ , are known.

Remark 1

The failure model (equation (3)) can be available to describe the following failures appeared in the active suspension system as follows:

• Stuck fault: ${\overline{u}}_j(t)={\overline{u}}_{j0}$ , $t\ge t_j$ , $j\in \{1,2\}$ ;

• Complete loss of effectiveness failure: ${\overline{u}}_j(t)=0$ , $t\ge t_j$ , $j\in \{1,2\}$ ;

• sinusoidal oscillatory failure: ${\overline{u}}_j(t)={\overline{u}}_{j1}\sin {\omega }_{j1}t$ , $t\ge t_j$ , $j\in \{1,2\}$ , in which failure time instant $t_j$ , failure value, and failure index j are all unknown.

For the oscillatory failure, one case is that the failure frequency is known; in this case, the oscillatory failure can be modeled by equation (3) with ${\overline{u}}_{\mathrm{ji}}{f}_{\mathrm{ji}}(t)={\overline{u}}_{\mathrm{ji}}\sin {\omega }_{\mathrm{ji}}t$ with known ${\omega }_{\mathrm{ji}}$ . Another case is that, the accurate information about the failure frequency may be unavailable; in such case, approximations $f_{\mathrm{ji}}(t)$ of ${\omega }_{\mathrm{ji}}$ can be employed to approximate compensation of actuator failures. A set of basis functions such as Taylor series and neural networks can be employed to approximate the considered unknown nonlinear functions. The approximation error will appear in the closed-loop control system and can be handled using robust adaptive theory. Moreover, some approximate knowledge about the failure frequency can be available from some previously available data; that is, the frequency range can be known in advance; we can chose a number of sinusoidal components with the frequencies in the available frequency range to make the approximation error smaller.

The failure signal is then expressed with a compact form

$${\overline{u}}_j\left(t\right)={\theta }_j^{*T}{\varpi }_j\left(t\right)$$

(4)

where ${\theta }_j^{*}=[{\overline{u}}_{j0},{\overline{u}}_{j1},\dots ,{\overline{u}}_{j{q}_j}{]}^T\in R^{q_j+1}$ and ${\varpi }_j(t)=[1,f_{j1}(t),\dots ,f_{j{q}_j}(t){]}^T\in R^{q_j+1}$ .

With equation (4), the input signal $u(t)$ can be rewritten as

$$u\left(t\right)=(I-\sigma \left(t\right))v\left(t\right)+\sigma \left(t\right)\overline{u}\left(t\right)$$

(5)

in which $v(t)$ means applied controller and $\sigma =\mathrm{diag}\{{\sigma }_1,{\sigma }_2\}$ means uncertain actuator failure pattern matrix; if the jth actuator suffers from failure ${\sigma }_j=1$ , otherwise, ${\sigma }_j=0$ .

To facilitate the design procedure in this article, the following realistic assumption should be satisfied.

Assumption 1

System (1) is so designed that the normal actuator can still achieve the control objective by using the known failure information for any up to one actuator failure.

Assumption 2

The reference trajectories $r_z$ and $r_{\phi }$ of the system outputs $z_c$ and $\phi$ are bounded smooth functions.

Control objective. An adaptive compensation strategy to guarantee half-car suspension system (1) closed-loop stability, and the vertical and pitch motions $z_c$ and $\phi$ track the given reference trajectories $r_z$ and $r_{\phi }$ asymptotically with up to one uncertain actuator failure.

Failure patterns. To satisfy the assumption that at most one actuator failure completely failed within the system, that is, both one actuator failure and no failure case ( $u_1$ failure or $u_2$ failure) can be compensated by our control scheme. The corresponding failure patterns are described as the following: ${\sigma }_{(1)}=\mathrm{diag}\{0,0\}$ , ${\sigma }_{(2)}=\mathrm{diag}\{1,0\}$ , and ${\sigma }_{(3)}=\mathrm{diag}\{0,1\}$ .

Controller design

In this section, the adaptive failure compensation scheme is designed for half-car suspension system (1) combining with backstepping technique, to achieve the control objective even with uncertain actuator failure.

Multiple nominal controllers design

To proceed the design procedure, we rewrite system (1) as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{x}}_{13}=\left[\begin{array}{c}{x}_2\hfill \\ \multicolumn{1}{c}{x_4}\end{array}\right]}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_{24}=\left[\begin{array}{c}\frac{{\mathrm{\Psi }}_1}{M}\hfill \\ \multicolumn{1}{c}{\frac{{\mathrm{\Psi }}_2}{I}}\end{array}\right]+\mathrm{Gu}}\end{array}$$

(6)

$$\begin{array}{c}{\stackrel{·}{x}}_5=x_6\hfill \\ {\stackrel{·}{x}}_6=\frac{1}{m_f}(F_{\mathrm{sf}}+F_{\mathrm{df}}-F_{\mathrm{tf}}-F_{\mathrm{bf}})-\frac{1}{m_f}{u}_1\hfill \\ {\stackrel{·}{x}}_7=x_8\hfill \\ {\stackrel{·}{x}}_8=\frac{1}{m_r}(F_{\mathrm{sr}}+F_{\mathrm{dr}}-F_{\mathrm{tr}}-F_{\mathrm{br}})-\frac{1}{m_r}{u}_2\hfill \end{array}$$

(7)

where $x_{13}=[x_1,x_3{]}^T$ , $x_{24}=[x_2,x_4{]}^T$ , $G=\left[\begin{array}{cc}\frac{1}{M}\hfill & \frac{1}{M}\hfill \\ \multicolumn{1}{c}{\frac{a}{I}}& -\frac{b}{I}\hfill \end{array}\right]$

To develop the basic actuator failure compensation techniques for the suspension system, we need to design a desired signal $u_d\in R^2$ to meet the control equation

$$u_d=\mathrm{Gu}$$

(8)

for control action implementation.

Desired feedback control signal

Define error variables $z_{13}=x_{13}-y_m$ and $z_{24}=x_{24}-{\alpha }_1$ , where ${\alpha }_1$ is a virtual control law to be determined. Differentiating $z_{13}$ with respect to time, we have

$${\stackrel{·}{z}}_{13}={\stackrel{·}{x}}_{13}-{\stackrel{·}{y}}_m=-c_1{z}_{13}+z_{24}$$

(9)

Choosing the stabilizing function as

$${\alpha }_1=-c_1{z}_{13}+{\stackrel{·}{y}}_m$$

(10)

for a chosen constant $c_1>0$ ; and substituting equation (10) into the error system equation (9), we obtain

$${\stackrel{·}{z}}_{13}=-c_1{z}_{13}+z_{24}$$

(11)

Considering the Lyapunov function $V_1=(1/2)z_{13}^T{z}_{13}$ . The time derivative of $V_1$ is

$${\stackrel{·}{V}}_1=-c_1{z}_{13}^T{z}_{13}+z_{13}^T{z}_{24}$$

(12)

It can be found from equation (12) that the second term $z_{13}^T{z}_{24}$ can be stabilized using ${\alpha }_1$ in the next step.

• Using the second equation of equation (6)

$${\stackrel{·}{x}}_{24}=f\left(x_{24}\right)+\mathrm{Gu}$$

(13)

with $f(x_{24})=\left[\begin{array}{c}\frac{{\mathrm{\Psi }}_1}{M}\hfill \\ \multicolumn{1}{c}{\frac{{\mathrm{\Psi }}_2}{I}}\end{array}\right]$ , and $z_{24}=x_{24}-{\alpha }_1$ , we express

$${\stackrel{·}{z}}_{24}={\stackrel{·}{x}}_{24}-{\stackrel{·}{\alpha }}_1=f\left(x_{24}\right)+\mathrm{Gu}-{\stackrel{·}{\alpha }}_1$$

(14)

Considering the error measure $V_2=V_1+(1/2)z_{24}^T{z}_{24}$ . The time derivative of $V_2$ is

$${\stackrel{·}{V}}_2=-c_1{z}_{13}^T{z}_{13}+z_{24}^T(z_{13}+f\left(x_{24}\right)+\mathrm{Gu}-{\stackrel{·}{\alpha }}_1)$$

(15)

We set the control signal

$$\mathrm{Gu}=u_d=-f\left(x_{24}\right)-c_2{z}_{24}-z_{13}+{\stackrel{·}{\alpha }}_1$$

(16)

would lead to

$${\stackrel{·}{V}}_2=-c_1{z}_{13}^T{z}_{13}-c_2{z}_{24}^T{z}_{24}\le 0$$

(17)

It can be concluded that the control signal equation (16) can realize the desired control objective.

When the active suspension system operates under three failure cases ${\sigma }_{(1)}=\mathrm{diag}\{0,0\}$ , ${\sigma }_{(2)}=\mathrm{diag}\{1,0\}$ , and ${\sigma }_{(3)}=\mathrm{diag}\{0,1\}$ , the input signal $u(t)=(I-\sigma (t))v(t)+\sigma (t)\overline{u}(t)$ should be designed to satisfy equation (8). By solving equation (8), we obtain three individual control laws $v_{(i)}^{*}(t)$ corresponding to each failure case with the known parameters of actuator failures.

Design for $\sigma ={\sigma }_{(1)}$

$u(t)=v(t)=v_{(1)}^{*}(t)$ for all $t\ge 0$ , while equation (8) is expressed as ${\mathrm{Gv}}_{(1)}^{*}(t)=u_d(t)$ . By solving this control equation, we have

$$v_{(1)}^{*}\left(t\right)=K_{21}{u}_d\left(t\right)$$

(18)

for $K_{21}=G^{-1}\in R^{2\times 2}$ .

Design for $\sigma ={\sigma }_{(2)}$

In such case, the signal $v_2(t)$ has no effect on the system. Hence, the signal $v_1(t)$ is set as $v_1(t)=0$ and $v_2(t)=u_2(t)$ when $u_1(t)={\overline{u}}_1(t)$ . With $G=[G_1,G_2]$ for $G_2\in R^{2\times 1}$ , $v(t)=[v_1(t),v_{0(2)}(t){]}^T$ , the control signal equation (8) becomes

$$G_1{\overline{u}}_1\left(t\right)+G_2{v}_{0(2)}\left(t\right)=u_d\left(t\right)$$

(19)

By solving this control equation, we have

$$v_{0(2)}^{*}\left(t\right)=K_{22}{u}_d\left(t\right)+K_{221}{\overline{u}}_1\left(t\right)$$

(20)

for $K_{22}=G_2^T/G_2^T{G}_2$ and $K_{221}=-(G_2^T{G}_1/G_2^T{G}_2)$ . The control signal $v(t)$ for this specific failure case is

$$v\left(t\right)=v_{(2)}^{*}\left(t\right)={[0,v_{0(2)}^{*}\left(t\right)]}^T$$

(21)

Design for $\sigma ={\sigma }_{(3)}$

In this case, $v_2(t)$ is chosen to be $0$ as $u_2(t)={\overline{u}}_2(t)$ . With $G=[G_1,G_2]$ , $v(t)=[v_{0(3)}(t),0{]}^T$ , equation (8) becomes

$$G_1{v}_{0(3)}\left(t\right)+G_2{\overline{u}}_2\left(t\right)=u_d\left(t\right)$$

(22)

Similarly, we obtain the solution $v_{0(3)}^{*}(t)$ through the solution of equation (8) as

$$v_{0(3)}^{*}\left(t\right)=K_{23}{u}_d\left(t\right)+K_{232}{\overline{u}}_2\left(t\right)$$

(23)

for $K_{23}=G_1^T/G_1^T{G}_1$ and $K_{232}=-(G_1^T{G}_2/G_1^T{G}_1)$ . And we obtain

$$v\left(t\right)=v_{(3)}^{*}\left(t\right)={[v_{0(3)}^{*}\left(t\right),0]}^T$$

(24)

Integrated nominal controller design

We first introduce the failure indicator functions

$$\begin{array}{c}{\chi }_1^{*}=\{\begin{array}{cc}1\hfill & \mathrm{if}\sigma ={\sigma }_{(1)}\hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise},\hfill \end{array}\hfill \\ {\chi }_2^{*}=\{\begin{array}{cc}1\hfill & \mathrm{if}\sigma ={\sigma }_{(2)}\hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise},\hfill \end{array}{\chi }_3^{*}=\{\begin{array}{cc}1\hfill & \mathrm{if}\sigma ={\sigma }_{(3)}\hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise}.\hfill \end{array}\hfill \end{array}$$

In addition, the above individual designs are combined here for the individual situations to construct the composite controller structure, so as to accommodate all the three failure cases

$$v^{\ast }\left(t\right)={\chi }_1^{\ast }\left(t\right)v_{(1)}^{\ast }\left(t\right)+{\chi }_2^{\ast }\left(t\right)v_{(2)}^{\ast }\left(t\right)+{\chi }_3^{\ast }\left(t\right)v_{(3)}^{\ast }\left(t\right)$$

(25)

$$\begin{array}{c}\mathrm{diag}\{{\chi }_{11}^{*}(t),{\chi }_{12}^{*}(t)\}{K}_{21}{u}_d(t)\hfill \\ \multicolumn{1}{c}{+{[0,{\chi }_2^{*}(t)K_{22}{u}_d(t)+K_{221}{{\theta }_1^{*}}_{\chi }^T(t){\varpi }_1(t)]}^T}\\ \multicolumn{1}{c}{+{[{\chi }_3^{*}(t)K_{23}{u}_d(t)+K_{232}{\theta }_{2\chi }^{*T}(t){\varpi }_2(t),0]}^T}\end{array}$$

(26)

where ${\chi }_{1i}^{*}(t)={\chi }_1^{*}(t)$ , $i=1,2$ ; ${\theta }_{1\chi }^{*}(t)={\chi }_2^{*}(t){\theta }_1^{*}$ ; ${\theta }_{2\chi }^{*}(t)={\chi }_3^{*}(t){\theta }_2^{*}$ ; ${\theta }_i^{*}=[{\overline{u}}_{i0},{\overline{u}}_{i1},\dots ,{\overline{u}}_{i{q}_i}{]}^T\in R^{q_i+1}$ ; and ${\varpi }_i(t)=[1,f_{i1}(t),\dots ,f_{i{q}_i}(t){]}^T\in R^{q_i+1}$ , $i=1,2$ .

Adaptive controller design

In case of unknown actuator failures, the nominal controller $v^{*}(t)$ in equation (26) cannot be obtained to achieve the desired control objective. Therefore, an effective adaptive controller $v(t)$ should be developed for system (6) with unknown failure parameters. Replacing ${\chi }_{1i}^{*}$ , ${\chi }_2^{*}$ , ${\chi }_3^{*}$ , ${\theta }_{1\chi }^{*}$ , and ${\theta }_{2\chi }^{*}$ in $v^{*}(t)$ with their adaptive estimations, we can get the structure of adaptive controller $v(t)$ as

$$\begin{array}{cc}\multicolumn{1}{c}{v\left(t\right)}& =v_{{\chi }_1(1)}\left(t\right)+{[0,v_{{\chi }_{2}a(2)}\left(t\right)]}^T+{[v_{{\chi }_{3}a(3)}\left(t\right),0]}^T\hfill \\ \multicolumn{1}{c}{}& =\mathrm{diag}\{{\chi }_{11}\left(t\right),{\chi }_{12}\left(t\right)\}{K}_{21}{u}_d\left(t\right)\hfill \\ \multicolumn{1}{c}{}& +{[0,{\chi }_2\left(t\right)K_{22}{u}_d\left(t\right)+K_{221}{\theta }_{1\chi }^T\left(t\right){\varpi }_1\left(t\right)]}^T\hfill \\ \multicolumn{1}{c}{}& +{[{\chi }_3\left(t\right)K_{23}{u}_d\left(t\right)+K_{232}{\theta }_{2\chi }^T\left(t\right){\varpi }_2(t),0]}^T\hfill \end{array}$$

(27)

where ${\chi }_j(t)$ , ${\theta }_{1\chi }(t)$ , and ${\theta }_{2\chi }(t)$ are the estimates of ${\chi }_j^{*}(t)$ , ${\theta }_{1\chi }^{*}(t)$ , and ${\theta }_{2\chi }^{*}(t)$ , respectively.

Based on $v^{*}(t)$ in equation (26) and adaptive controller $v(t)$ in equation (27), we can derive the error equation relating $z_{24}$ to the actuator failure parameter estimation errors ${\tilde{\chi }}_j(t)$ , ${\tilde{\theta }}_{1\chi }(t)$ , and ${\tilde{\theta }}_{2\chi }(t)$ under different failure pattern conditions.

Error equations

Now, we rewrite equation (14) as

$${\stackrel{·}{z}}_{24}={\stackrel{·}{x}}_{24}-{\stackrel{·}{\alpha }}_1=-z_{13}-c_2{z}_{24}+G(I-\sigma )(v-v^{*})$$

(28)

When $\sigma ={\sigma }_{(1)}=\mathrm{diag}\{0,0\}$ , equation (28) can be expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{z}}_{24}=}& -z_{13}-c_2{z}_{24}+{\tilde{\chi }}_{11}{K}_{21}{u}_{d1}{G}_1\hfill \\ \multicolumn{1}{c}{}& +{\tilde{\chi }}_{12}{K}_{21}{u}_{d2}{G}_2+{\tilde{\chi }}_2{K}_{22}{u}_d{G}_2+{\tilde{\chi }}_3{K}_{23}{u}_d{G}_1\hfill \\ \multicolumn{1}{c}{}& +K_{221}{\tilde{\theta }}_{1\chi }^T\left(t\right){\varpi }_1\left(t\right)G_1+K_{232}{\tilde{\theta }}_{2\chi }^T\left(t\right){\varpi }_2\left(t\right)G_2\hfill \end{array}$$

(29)

When $\sigma ={\sigma }_{(2)}=\mathrm{diag}\{1,0\}$ , equation (28) can be expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{z}}_{24}=}& -z_{13}-c_2{z}_{24}+{\tilde{\chi }}_{12}{K}_{21}{u}_{d2}{G}_2\hfill \\ \multicolumn{1}{c}{}& +{\tilde{\chi }}_2{K}_{22}{u}_d{G}_2+K_{221}{\tilde{\theta }}_{1\chi }^T\left(t\right){\varpi }_1\left(t\right)G_1\hfill \end{array}$$

(30)

When $\sigma ={\sigma }_{(3)}=\mathrm{diag}\{0,1\}$ , equation (28) can be expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{z}}_{24}=}& -z_{13}-c_2{z}_{24}+{\tilde{\chi }}_{11}{K}_{21}{u}_{d1}{G}_1+{\tilde{\chi }}_3{K}_{23}{u}_d{G}_1\hfill \\ \multicolumn{1}{c}{}& +K_{232}{\tilde{\theta }}_{2\chi }^T\left(t\right){\varpi }_2\left(t\right)G_2\hfill \end{array}$$

(31)

where ${\tilde{\chi }}_{11}$ , ${\tilde{\chi }}_{12}$ , ${\tilde{\chi }}_2$ , ${\tilde{\chi }}_3$ , ${\tilde{\theta }}_{1\chi }$ , and ${\tilde{\theta }}_{2\chi }$ are the corresponding parameter errors.

Adaptive laws

The adaptive laws for ${\chi }_{\mathrm{ji}}(t)$ , ${\theta }_{1\chi }(t)$ , and ${\theta }_{2\chi }(t)$ are chosen as

$${\stackrel{·}{\chi }}_{11}\left(t\right)=-{\mathrm{\gamma }}_{11}{z}_{24}^T{K}_{21}{u}_{d1}{G}_1+f_{{\chi }_{11}}$$

(32)

$${\stackrel{·}{\chi }}_{12}\left(t\right)=-{\gamma }_{12}{z}_{24}^T{K}_{21}{u}_{d1}{G}_2+f_{{\chi }_{12}}$$

(33)

$${\stackrel{·}{\chi }}_2\left(t\right)=-{\gamma }_2{z}_{24}^T{K}_{22}{u}_d{G}_2+f_{{\chi }_2}$$

(34)

$${\stackrel{·}{\chi }}_3\left(t\right)=-{\gamma }_3{z}_{24}^T{K}_{23}{u}_d{G}_2+f_{{\chi }_3}$$

(35)

$${\stackrel{·}{\theta }}_{1\chi }\left(t\right)=-{\mathrm{\Gamma }}_{1\chi }{\varpi }_1\left(t\right)K_{221}{z}_{24}^T{G}_1+f_{{\theta }_{1\chi }}$$

(36)

$${\stackrel{·}{\theta }}_{2\chi }\left(t\right)=-{\mathrm{\Gamma }}_{2\chi }{K}_{232}{\varpi }_2\left(t\right)z_{24}^T{G}_2+f_{{\theta }_{2\chi }}$$

(37)

where adaptation gains ${\mathrm{\Gamma }}_{1\chi }={\mathrm{\Gamma }}_{1\chi }^T>0$ , ${\mathrm{\Gamma }}_{2\chi }={\mathrm{\Gamma }}_{2\chi }^T>0$ , ${\gamma }_{11}>0$ , ${\gamma }_{12}>0$ , ${\gamma }_2>0$ , and ${\gamma }_3>0$ . The $f_{{\chi }_{1i}}$ , $i=1,2$ , are standard parameter projection functions that have the desired property: $({\chi }_{1i}-{\chi }_{1i}^{*})f_{{\chi }_{1i}}\le 0$ and ${\chi }_{1i}$ are projected in the interval $0\le {\chi }_{1i}\le 1$ . Such property is also hold for ${\chi }_2$ , ${\chi }_3$ , ${\theta }_{1\chi }$ , and ${\theta }_{2\chi }$ .

System performance analysis

In practice, we cannot know which of the three cases the system is in, but the stability of the system to be used in the three possible failure patterns can be guaranteed by the designed control law $v(t)$ . Let ( $T_i,T_{i+1}$ ), $i=0,1$ , with $T_0=0$ , be the time intervals that $u_1$ or $u_2$ fails.

• 1. When $\sigma ={\sigma }_{(1)}=\mathrm{diag}\{0,0\}$ , we consider the positive definite function

$$\begin{array}{cc}\multicolumn{1}{c}{V_0}& =\frac{1}{2}{z}_{13}^T{z}_{13}+\frac{1}{2}{z}_{24}^T{z}_{24}\hfill \\ \multicolumn{1}{c}{}& +\frac{1}{2}[{\tilde{\chi }}_{11}^2{\gamma }_{11}^{-1}+{\tilde{\chi }}_{12}^2{\gamma }_{12}^{-1}+{\tilde{\chi }}_2^2{\gamma }_2^{-1}+{\tilde{\chi }}_3^2{\gamma }_3^{-1}+{\tilde{\theta }}_{1\chi }^T{\mathrm{\Gamma }}_{1\chi }^{-1}{\tilde{\theta }}_{1\chi }+{\tilde{\theta }}_{2\chi }^T{\mathrm{\Gamma }}_{2\chi }^{-1}{\tilde{\theta }}_{2\chi }]\hfill \end{array}$$

(38)

• 2. When $\sigma ={\sigma }_{(2)}=\mathrm{diag}\{1,0\}$ , that is, only $u_1$ fails at a finite time instant $T_1$ and is still failed during the interval $(T_1,T_2)$ , we consider the positive definite function

$$\begin{array}{c}{V}_1=\frac{1}{2}{z}_{13}^T{z}_{13}+\frac{1}{2}{z}_{24}^T{z}_{24}\hfill \\ +\frac{1}{2}[{\tilde{\chi }}_{12}^2{\gamma }_{12}^{-1}+{\tilde{\chi }}_2^2{\gamma }_2^{-1}+{\tilde{\theta }}_{1\chi }^T{\mathrm{\Gamma }}_{1\chi }^{-1}{\tilde{\theta }}_{1\chi }]\hfill \end{array}$$

(39)

• 3. When $\sigma ={\sigma }_{(3)}=\mathrm{diag}\{0,1\}$ , in this case, $u_1$ is failure-free, only $u_2$ fails at time $T_1$ and is still failed during the interval $(T_1,T_2)$ , we consider the positive definite function

$$\begin{array}{cc}\multicolumn{1}{c}{V_2=}& \frac{1}{2}{z}_{13}^T{z}_{13}+\frac{1}{2}{z}_{24}^T{z}_{24}\hfill \\ \multicolumn{1}{c}{}& +\frac{1}{2}[{\tilde{\chi }}_{11}^2{\gamma }_{11}^{-1}+{\tilde{\chi }}_3^2{\gamma }_3^{-1}+{\tilde{\theta }}_{2\chi }^T{\mathrm{\Gamma }}_{2\chi }^{-1}{\tilde{\theta }}_{2\chi }]\hfill \end{array}$$

(40)

Then, with equations (11) and (32)–(37), the time derivative of $V_0$ is ${\stackrel{·}{V}}_0=-c_1{z}_{13}^T{z}_{13}-c_2{z}_{24}^T{z}_{24}\le 0$ , $t\in (T_0,T_1)$ ; it follows that $V_0\in L^{\infty }$ , $x_i(t)\in L^{\infty }$ , $i=1,2,3,4$ , $u_d(t)\in L^{\infty }$ , ${\tilde{\chi }}_{11}(t)\in L^{\infty }$ , ${\tilde{\chi }}_{12}(t)\in L^{\infty }$ , ${\tilde{\chi }}_2(t)\in L^{\infty }$ , ${\tilde{\chi }}_3(t)\in L^{\infty }$ , ${\tilde{\theta }}_{1\chi }(t)\in L^{\infty }$ , ${\tilde{\theta }}_{2\chi }(t)\in L^{\infty }$ , and $z_{13}(t)\in L^{\infty }$ . The function $V_0$ has a finite value, and its derivative is ${\stackrel{·}{V}}_0=-c_1{z}_{13}^T{z}_{13}-c_2{z}_{24}^T{z}_{24}\le 0$ , which means $z_{13}(t)\in L^2$ . Equation (27) indicates that $v(t)$ is bounded. For this reason, the closed-loop system stability is guaranteed. Furthermore, equation (11) implies ${\stackrel{·}{z}}_{13}(t)\in L^{\infty }$ . Applying the Barbâlat lemma, we have $\underset{t\to \infty }{lim}{z}_{13}(t)=0$ . Therefore, the closed-loop stability and asymptotic tracking properties can be achieved for $\sigma ={\sigma }_{(1)}$ . Similar properties can also be obtained for failure pattern $\sigma ={\sigma }_{(2)}=\mathrm{diag}\{1,0\}$ and $\sigma ={\sigma }_{(3)}=\mathrm{diag}\{0,1\}$ .

The original system is an eighth-order system; the above adaptive design focus on the first four states; next, we analyze the stability of zero dynamics consisting of the last four states. We set $z_{13}=z_{24}=0$ and obtain

$$\begin{array}{c}{u}_1=\frac{1}{a+b}(-b{\mathrm{\Psi }}_1-{\mathrm{\Psi }}_2+\mathrm{bM}{\stackrel{··}{y}}_{m1}+I{\stackrel{··}{y}}_{m2})\hfill \\ u_2=\frac{1}{a+b}(-a{\mathrm{\Psi }}_1+{\mathrm{\Psi }}_2+\mathrm{aM}{\stackrel{··}{y}}_{m1}-I{\stackrel{··}{y}}_{m2})\hfill \end{array}$$

(41)

Substituting equation (41) into dynamic equations of ${\stackrel{·}{x}}_6$ and ${\stackrel{·}{x}}_8$ , we have

$$\stackrel{·}{x}=\mathrm{Ax}+B{z}_o+B_r{y}_m$$

(42)

where

$$\begin{array}{c}x=\left[\begin{array}{c}{x}_5\hfill \\ \multicolumn{1}{c}{x_6}\\ \multicolumn{1}{c}{x_7}\\ \multicolumn{1}{c}{x_8}\end{array}\right]A=\left[\begin{array}{cccc}0\hfill & 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\frac{k_{f2}}{m_f}}& -\frac{b_{f2}}{m_f}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -\frac{k_{r2}}{m_r}\hfill & -\frac{b_{r2}}{m_r}\hfill \end{array}\right]\hfill \end{array}$$

$$\begin{array}{c}B=\left[\begin{array}{cccc}0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\frac{k_{f2}}{m_f}}& \frac{b_{f2}}{m_f}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \frac{k_{r2}}{m_r}\hfill & \frac{b_{r2}}{m_r}\hfill \end{array}\right]\hfill \\ z_o=\left[\begin{array}{c}{z}_{o1}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{z}}_{o1}}\\ \multicolumn{1}{c}{z_{o2}}\\ \multicolumn{1}{c}{{\stackrel{·}{z}}_{o2}}\end{array}\right]B_r=\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\frac{\mathrm{bM}}{m_f(a+b)}}& -\frac{I}{m_f(a+b)}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{-\frac{\mathrm{bM}}{m_r(a+b)}}& \frac{I}{m_r(a+b)}\hfill \end{array}\right]\hfill \\ y_m=\left[\begin{array}{c}{\stackrel{··}{y}}_{m1}\hfill \\ \multicolumn{1}{c}{{\stackrel{··}{y}}_{m2}}\end{array}\right]\hfill \end{array}$$

Defining a positive function $V_3=x^T\mathrm{Px}$ , with $P$ as a positive matrix, the time derivative of $V_3$ is ${\stackrel{·}{V}}_3={\stackrel{·}{x}}^T\mathrm{Px}+x^{T}P\stackrel{·}{x}=x^T(A^{T}P+\mathrm{AP})x+2x^T\mathrm{PB}{z}_o+2x^{T}P{B}_r{x}_r$ . We can verify that the eigenvalues of matrix A have negative real parts and $A^{T}P+\mathrm{AP}=-Q$ , with $Q>0$ . It is worth noting that

$$\begin{array}{c}2x^T\mathrm{PB}{z}_o\le \frac{1}{{\eta }_1}{x}^T\mathrm{PB}{B}^T\mathrm{Px}+{\eta }_1{z}_o^T{z}_o,\hfill \\ 2x^{T}P{B}_r{z}_r\le \frac{1}{{\eta }_2}{x}^{T}P{B}_r{B}_r^T\mathrm{Px}+{\eta }_2{z}_r^T{z}_r\hfill \end{array}$$

$$z_o^T{z}_o\le z_{\mathrm{omax}},y_m^T{y}_m\le y_{\mathrm{mmax}}$$

where ${\eta }_1>0$ and ${\eta }_2>0$ . Choosing matrices $P$ , $Q$ and tuning values ${\eta }_1$ , ${\eta }_2$ properly, then ${\stackrel{·}{V}}_3$ can be written as

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{V}}_3\le }& -x^T\mathrm{Qx}+\frac{1}{{\eta }_1}{x}^T\mathrm{PB}{B}^T\mathrm{Px}\hfill \\ \multicolumn{1}{c}{}& +{\eta }_1{z}_o^T{z}_o+\frac{1}{{\eta }_2}{x}^{T}P{B}_r{B}_r^T\mathrm{Px}+{\eta }_2{z}_r^T{z}_r\hfill \\ \multicolumn{1}{c}{}& \le [-{\lambda }_{\min }\left(P^{-\frac{1}{2}}Q{P}^{-\frac{1}{2}}\right)+\frac{1}{{\eta }_1}{\lambda }_{\max }\left(P^{-\frac{1}{2}}B{B}^T{P}^{-\frac{1}{2}}\right)\hfill \\ \multicolumn{1}{c}{}& +\frac{1}{{\eta }_2}{\lambda }_{\max }\left(P^{-\frac{1}{2}}{B}_r{B}_r^T{P}^{-\frac{1}{2}}\right)]V_3+{\eta }_1{z}_o^T{z}_o+{\eta }_2{z}_r^T{z}_r\hfill \\ \multicolumn{1}{c}{}& \le -{\epsilon }_1{V}_3+{\epsilon }_2\hfill \end{array}$$

(43)

where ${\epsilon }_2={\eta }_1{z}_{\mathrm{omax}}+{\eta }_2{y}_{\mathrm{mmax}}$ . It can be derived from equation (43) that

$$V_3\left(t\right)\le (V_3\left(0\right)-\frac{{\epsilon }_2}{{\epsilon }_1})e^{-{\epsilon }_{1}t}+\frac{{\epsilon }_2}{{\epsilon }_1}$$

(44)

which shows us that $|x_k|\le \sqrt{q/{\lambda }_{\min }(p)}$ , $k=5,6,7,8$ with

$$q=\{\begin{array}{cc}{V}_3\left(0\right)\hfill & \mathrm{if}{V}_3(0)\ge \frac{{\epsilon }_2}{{\epsilon }_1}\hfill \\ \multicolumn{1}{c}{\frac{2{\epsilon }_2}{{\epsilon }_1}-V_3\left(0\right)}& \mathrm{otherwise}\hfill \end{array}$$

From the previous analysis, we know all the signals of the zero dynamics are bounded. In short, the following theorem is obtained.

Theorem 1

Considering that active suspension system (1) with unknown actuator failure (4) satisfies Assumption 1 and Assumption 2, the proposed scheme synthesizes multiple controllers by using integration mechanism to form a composite control law (27) updated by laws (32)–(37). This scheme can effectively handle multiple uncertain failure cases, and guarantee closed-loop systems stability as well as the asymptotic output tracking performance.

Illustrative example

The effectiveness of our proposed adaptive actuator failure compensation approach is illustrated with an example in this section. In this paper, the classic bump road input is used to verify the proposed controller, whose ground displacement is expressed as

$$z_{o1}=\{\begin{array}{cc}\frac{h_b[1-\cos \left(8\pi t\right)]}{2}\hfill & 1s\le t\le 1.25s\hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise}\hfill \end{array}$$

where $h_b$ denotes the height of the bump road input. The half-car model parameters are given as follows: $M=1200\mathrm{kg}$ , $I=600\mathrm{kg}{\mathrm{m}}^2$ , $m_f=100\mathrm{kg}$ , $m_r=100\mathrm{kg}$ , $a=1.2\mathrm{m}$ , $b=1.5\mathrm{m}$ , $k_{f1}=k_{r1}=15,000\mathrm{N}/\mathrm{m}$ , $b_{f1}=b_{r1}=2500\mathrm{Ns}/\mathrm{m}$ , $b_{f2}=b_{r2}=1000\mathrm{Ns}/\mathrm{m}$ , $h_b=2\mathrm{cm}$ , $V=20\mathrm{m}/\mathrm{s}$ .

The road condition $z_{o2}$ for both the rear and front wheel are assumed the same with a time delay $(a+b)/V$ in the simulation study, where $V$ is the vehicle forward velocity.

Remark 2

The external disturbance $z_{\mathrm{oi}}(t)$ , $i=1,2$ , can come in many forms. ³⁰ For our main topic of interest, $z_{\mathrm{oi}}(t)$ is a combination of a constant and sinusoidal functions, which can be described by

$$\begin{array}{cc}\multicolumn{1}{c}{z_{\mathrm{oi}}\left(t\right)}& =c_{i0}+\sum _{j=1}^{n_i}{c}_{\mathrm{ij}}\sin ({\omega }_{\mathrm{ij}}t+{\mathrm{\Upsilon }}_{\mathrm{ij}})\hfill \\ \multicolumn{1}{c}{}& =c_i+\sum _{j=1}^{n_i}{a}_{\mathrm{ij}}\sin {\omega }_{\mathrm{ij}}t+\sum _{j=1}^{n_i}{b}_{\mathrm{ij}}\cos {\omega }_{\mathrm{ij}}t\hfill \end{array}$$

(45)

where $c_i$ , $a_{\mathrm{ij}}$ , and $b_{\mathrm{ij}}$ are arbitrarily unknown amplitudes; ${\mathrm{\Upsilon }}_{\mathrm{ij}}$ are unknown phase angles; and ${\omega }_{\mathrm{ij}}$ are known frequencies.

We write the disturbance model (equation (45)) in a compact form

$$z_{\mathrm{oi}}\left(t\right)={\theta }_{\mathrm{di}}^{*T}{\varpi }_{\mathrm{di}}\left(t\right)$$

(46)

where ${\theta }_{\mathrm{di}}^{*}=[c_{i0},a_{i1},\dots ,a_{i{n}_i},b_{i1},\dots ,b_{i{n}_i}{]}^T\in R^{2n_i+1}$ and ${\varpi }_{\mathrm{di}}\left(t\right)=[1,\sin {\omega }_{i_1}t,\sin {\omega }_{i_2}t,\dots ,\sin {\omega }_{i_{n_i}}t,\cos {\omega }_{i_1}t,\cos {\omega }_{i_2}t,\dots ,\cos {\omega }_{i_{n_i}}t{]}^T\in R^{2n_i+1}$ .

The unknown disturbances caused by poor road condition can be regarded as an unknown additional term in the system input signal; if the unknown time-varying disturbances can be parameterizable and modeled by function (46), the adaptive failure compensation scheme proposed in this article can be extended to compensate the unknown disturbances completely so as to ensure the closed-loop system stable and asymptotic output tracking. If the unknown bounded disturbances are unparameterizable, it cannot be compensated completely, which influences the precision of the adaptive parameter estimation and vehicle transient performance; however, the control scheme design for the parameterizable disturbances can also be extended to improve the vehicle transient performance to the most extent by properly adjusting the initial value and tuning parameters. For the adaptive control design, in our article, the applied parameter projection design can guarantee that the parameter estimation errors are bounded, so the closed-loop system signals are bounded and the asymptotic output tracking can be ensured.

Desired attitude command: In this article, the desired reference trajectory parameter $y_m(t)$ is set to zero for the output $y(t)$ of active suspension system.

Failure scenario 1: (1) both actuators function healthily over the time interval $0\le t<3\mathrm{s}$ ; (2) only actuator $u_1$ get stuck at a constant value $u_1(t)=10\mathrm{N}$ for $t\ge 3\mathrm{s}$ ; (3) then $u_1$ becomes failure-free, both actuators function healthily for $t\ge 3\mathrm{s}$ ; (4) only actuator $u_2$ get stuck at a constant value $u_2(t)=-10\mathrm{N}$ for $t\ge 7\mathrm{s}$ ; and (5) then $u_2$ becomes normal, both actuators function normally for $t\ge 9\mathrm{s}$ .

Failure scenario 2: (1) both actuators function healthily over the time interval $0\le t<3\mathrm{s}$ ; (2) only actuator $u_1$ is failed during the time interval $t\ge 3\mathrm{s}$ with $u_1=7.5\sin (5t)\mathrm{N}$ ; (3) then $u_1$ becomes normal, both actuators function healthily for $t\ge 3\mathrm{s}$ ; (4) only actuator $u_2$ is failed during the time interval $t\ge 7\mathrm{s}$ with $u_2=5\sin (10t)\mathrm{N}$ ; and (5) both actuators function normally for $t\ge 9\mathrm{s}$ .

Simulation conditions

The initial value are $x_1(0)=10\mathrm{cm}$ and $x_3(0)=0.2\mathrm{rad}$ , while zero is used in the rest values. For simulation, the design parameters are chosen as $c_1=12$ , $c_2=10$ ; ${\gamma }_{1i}=100,i=1,2$ ; ${\gamma }_2=100$ ; ${\gamma }_3=100$ ; ${\mathrm{\Gamma }}_{1\chi }=100I_{2\times 2}$ ; ${\mathrm{\Gamma }}_{2\chi }=100I_{2\times 2}$ . Basic functions in equation (4) are ${\varpi }_1(t)=[1,\sin (5t){]}^T\in R^2$ and ${\varpi }_2(t)=[1,\sin (10t){]}^T\in R^2$ .

Simulation results

The adaptive backstepping control method was compared with our proposed adaptive actuator failure compensation method, with the simulation results shown as follows. Figure 2(a) and (b) presents the vertical displacement and acceleration of the active suspension system, respectively; Figure 3(a) and (b) presents the pitch angle displacement and acceleration of the active suspension system, respectively. Comparing with the adaptive backstepping control method designed for solving stuck failure case, our proposed fault tolerant control (FTC) scheme significantly improves suspension performances. It is shown that the vertical and pitch angle displacements as well as their acceleration signal almost approach zero successfully after the simulation starts. A transient response occurs in the output when $u_1$ is failed at the instant $t=3\mathrm{s}$ before subsequently recovering to normal at $t=5\mathrm{s}$ , and the system output asymptotically converges to zero over time. Similar desired performance can be obtained when $u_2$ is stuck at the instant $t=7\mathrm{s}$ before functioning normally again at $t=9\mathrm{s}$ . Moreover, the corresponding control inputs have been presented in Figure 4(a) and (b), respectively. The result of the system with uncertain time-varying failure between the adaptive backstepping control method and the new FTC approach is presented in Figures 5–7, which illustrates that the proposed method can achieve satisfied performance in the active suspension system with different failure cases.

Figure 2.

(a) Vertical displacement and (b) vertical acceleration for failure scenario 1.

Figure 3.

(a) Pitch angle displacement and (b) pitch angle acceleration for failure scenario 1.

Figure 4.

Control input signals for failure scenario 1: (a) control input signals with FTC method and (b) control input signals for adaptive backstepping control method.

Figure 5.

(a) Vertical displacement and (b) vertical acceleration for failure scenario 2.

Figure 6.

(a) Pitch angle displacement and (b) pitch angle acceleration for failure scenario 2.

Figure 7.

Control input signals for failure scenario 2: (a) control input signals with FTC method and (b) control input signals for adaptive backstepping control method.

Summarizing all the cases (normal case, lock-in-place failure case, and time-varying failure case), it is seen that uncertain actuator failures can be compensated by using the adaptive control law, while maintaining the tracking capability of vehicle active suspension systems.

Conclusion

This article proposes an adaptive actuator failure compensation method which can be used for attitude tracking control of vehicle active suspension systems. The feature of actuator failure studied here is that the knowledge of failure values, failure patterns, and failure time instants is unknown in advance. After all possible failures are completely parametrized and an integration mechanism is introduced, we combine multiple controllers to construct a composite failure compensator, by which the uncertain actuator failures can be solved. The proposed scheme can ensure the desired tracking property and stability of the half-car active suspension system. The desired adaptive actuator failure compensation performance was verified by simulation.