# Actuator Fault Estimation for Vehicle Active Suspensions Based on Adaptive Observer and Genetic Algorithm

Authors: Pan Jin; Wenping Xue; Kangji Li
Journal: Shock and Vibration
 (2019)

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## Abstract

This paper is concerned with the problem of actuator fault estimation (FE) for vehicle active suspension systems. First, the fast FE approach, which combines the output error term with its derivative term in the FE algorithm, is extended to the active suspension system with actuator fault and external disturbance input. Then, considering three typical kinds of actuator faults, i.e., constant gain change fault, drift fault, and stuck fault, genetic algorithm (GA) is employed to optimize the adjustable parameters in the FE algorithm, which are usually determined by trials. Finally, simulation results of FE and fault-tolerant control (FTC) are presented to illustrate the effectiveness and applicability of the proposed FE method.


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## Body

## 1. Introduction

Vehicle suspensions play a significant role in ride comfort, safety, and the overall performance of modern vehicles. According to different control forms, vehicle suspensions are commonly classified into three types: passive, semiactive, and active suspensions. The passive suspension composed of a parallel spring and damper which has been acknowledged for its simple structure and low cost [1]. The semiactive suspension [2, 3], which consists of a spring and tunable damper, can make desirable improvements over the passive one by effectively varying the rate of the energy dissipation. Compared with passive and semiactive suspensions, active suspensions are not widely used due to large energy demand and complex structure. However, they can provide high control performance in a wide frequency range and have the best potential to overcome conflicts between ride comfort and vehicle safety. As the key part for active suspensions, advanced control strategies have attracted considerable interest, such as H∞ control [4], nonlinear control [5, 6], predictive control [7], adaptive control [8, 9], fuzzy control [10, 11], and finite-time control [12]. It should be pointed out that all the abovementioned active suspension control results are under a full-functional and fault-free assumption of all system components. However, various faults, especially actuator and sensor faults, are likely to be encountered in active suspension systems. Ignorance of these faults may result in poor system performance or even instability. In this case, it is reasonable to resolve fault diagnosis and fault-tolerant control (FTC) problems, which could increase system security and reliability.Over the past few decades, fault diagnosis and FTC have been extensively investigated and many results have been obtained, see [13–18] and the references therein. Yet, for vehicle suspension systems, it seems that the corresponding results are relatively few. In [19], the fault diagnosis problem was investigated for a semiactive vehicle suspension system by using the particle filter-based approach. In [20–22], passive fault-tolerant control (PFTC) strategies were studied for active suspensions. Some active fault-tolerant control (AFTC) methods were proposed in [23–25] for full-car, quarter-car, and half-car active suspensions. Different from PFTC, AFTC guarantees the acceptable system performance through fault accommodation (FA) or system reconfiguration. As an important part in both fault diagnosis and FA, fault estimation (FE) has a strong ability to obtain the fault information, such as the size and shape [26–28]. Therefore, FE has received considerable attention, and a great number of theoretic results have been reported, e.g., based on adaptive observer [26, 29, 30], disturbance observer [27], extended state observer (ESO) [31], unknown input observer (UIO) [28, 32], and sliding mode observer (SMO) [33]. In [34], the FE-based AFTC problem was discussed for a full-car semiactive suspension system. Note that only the measurement output error was used in the adaptive observer design. In [24], the AFTC method based on FE was developed for active suspension systems in finite-frequency domain. It could be easily seen that the output error was utilized in the FE signal, which implied that only the derivative term of the output error was included in the FE algorithm. In [26, 35], it was concluded that if both the output error and the derivative of it were used in the observer design, better estimation performance can be achieved compared with the traditional proportional-integral (PI) observer. However, some parameters or matrices in the FE algorithm are often chosen by trials, which may limit the improvement of FE performance.In this paper, motivated by the above analysis, the FE problem is studied for an active suspension system based on the parameter optimization technique. The main contributions of this paper can be summarized as follows: (1) the fast FE approach, which combines the output error term with its derivative term in the FE algorithm [26], is extended to the active suspension system with the actuator faults and external disturbance input; (2) in order to improve the FE performance, genetic algorithm (GA) is used to optimize the adjustable parameters in the FE algorithm, which are usually determined by trials.The remainder of this paper is organized as follows: a quarter-vehicle active suspension model and three kinds of actuator faults are described in Section2. The actuator FE method based on an adaptive observer and GA is developed in Section 3. Simulation results and analysis are given in Section 4 to show the validity of the proposed method. Finally, the paper is concluded in Section 5.Notation.
⋅ refers to the Euclidean norm of a vector or the induced norm of a matrix. The notation X>Y (respectively, X<Y), where X and Y are real symmetric matrices, means that the matrix X−Y is positive definite (respectively, negative definite). I and 0 denote the identity and zero matrices with appropriate dimensions, respectively. λminP denotes the minimum of eigenvalues of a real symmetric matrix P. The superscript T denotes the transpose for vectors or matrices. The symbol ∗ denotes the term that is induced by symmetry. L20,∞ denotes the space of vector functions that are square integrable over 0,∞. For ωt∈L20,∞, ωt2≜∫0∞ωTtωtdt. Matrices, if not explicitly stated, are assumed to have compatible dimensions for algebraic operations.


## 2. Quarter-Car Active Suspension with Actuator Fault

### 2.1. Quarter-Car Active Suspension Model

A simple quarter-car active suspension model is considered in this paper, as shown in Figure1. In this model, ms is the sprung mass, representing the chassis; mu is the unsprung mass, representing the mass of the wheel assembly; ut represents the active control force; zst and zut denote the displacement of the sprung mass and the unsprung mass, respectively; zrt denotes the road displacement; cs and ks denote damping and stiffness of the uncontrolled suspension, respectively; the tire is simplified as a spring and kt denotes its stiffness.Figure 1
Quarter-car active suspension model.The dynamical equations of the suspension system in Figure1 can be written as(1)msz¨st+csz˙st−z˙ut+kszst−zut=ut,muz¨ut+csz˙ut−z˙st+kszut−zzt+ktzut−zrt=−ut.Define the following state variables:x1t=zst−zut which is the suspension stroke, x2t=z˙st which is the sprung mass speed, x3t=zut−zrt which is the tire deflection, and x4t=z˙ut which is the unsprung mass speed. Choose ωt=z˙rt as the disturbance input caused by road roughness. Choose the sprung mass speed and suspension stroke as measurement outputs. Then, the dynamical equation (1) can be rewritten as the following state-space form:(2)x˙t=Axt+But+Dωt,yt=Cxt,where xt=x1tx2tx3tx4tT is the state vector. yt=z˙stzst−zutT is the output vector.(3)A=010−1−ksms−csms0csms0001ksmtcsmt−ktmt−csmt,B=01ms0−1mt,C=01001000,D=00−10.


### 2.2. Actuator Fault Description

In this paper, three typical kinds of actuator faults are considered, i.e., constant gain change, drift, and stuck [36, 37]. Then, the lumped form of the actuator output can be given as(4)uft=δut+u0,where δ represents the actuator loss in efficiency, u0 represents the value of drift or stuck failure, and ut represents the control command to the actuator. It is easily known that δ=1 and u0=0 correspond to the healthy actuator, 0<δ<1 and u0=0 correspond to the actuator with constant gain change fault, δ=1 and u0≠0 correspond to the actuator with drift fault, and δ=0 and u0≠0 correspond to the actuator with stuck fault.In order to separate the fault, the actuator output is rewritten by(5)uft=ut+δ−1ut+u0=ut+ft.Then, the actuator fault is obtained as(6)ft=δ−1ut+u0.It is assumed thatf˙t∈L20,∞. f˙t≤f1, where 0≤f1<∞. Considering the actuator fault (6), the active suspension system (2) becomes(7)x˙t=Axt+But+Bft+Dωt,yt=Cxt.


## 2.1. Quarter-Car Active Suspension Model

A simple quarter-car active suspension model is considered in this paper, as shown in Figure1. In this model, ms is the sprung mass, representing the chassis; mu is the unsprung mass, representing the mass of the wheel assembly; ut represents the active control force; zst and zut denote the displacement of the sprung mass and the unsprung mass, respectively; zrt denotes the road displacement; cs and ks denote damping and stiffness of the uncontrolled suspension, respectively; the tire is simplified as a spring and kt denotes its stiffness.Figure 1
Quarter-car active suspension model.The dynamical equations of the suspension system in Figure1 can be written as(1)msz¨st+csz˙st−z˙ut+kszst−zut=ut,muz¨ut+csz˙ut−z˙st+kszut−zzt+ktzut−zrt=−ut.Define the following state variables:x1t=zst−zut which is the suspension stroke, x2t=z˙st which is the sprung mass speed, x3t=zut−zrt which is the tire deflection, and x4t=z˙ut which is the unsprung mass speed. Choose ωt=z˙rt as the disturbance input caused by road roughness. Choose the sprung mass speed and suspension stroke as measurement outputs. Then, the dynamical equation (1) can be rewritten as the following state-space form:(2)x˙t=Axt+But+Dωt,yt=Cxt,where xt=x1tx2tx3tx4tT is the state vector. yt=z˙stzst−zutT is the output vector.(3)A=010−1−ksms−csms0csms0001ksmtcsmt−ktmt−csmt,B=01ms0−1mt,C=01001000,D=00−10.


## 2.2. Actuator Fault Description

In this paper, three typical kinds of actuator faults are considered, i.e., constant gain change, drift, and stuck [36, 37]. Then, the lumped form of the actuator output can be given as(4)uft=δut+u0,where δ represents the actuator loss in efficiency, u0 represents the value of drift or stuck failure, and ut represents the control command to the actuator. It is easily known that δ=1 and u0=0 correspond to the healthy actuator, 0<δ<1 and u0=0 correspond to the actuator with constant gain change fault, δ=1 and u0≠0 correspond to the actuator with drift fault, and δ=0 and u0≠0 correspond to the actuator with stuck fault.In order to separate the fault, the actuator output is rewritten by(5)uft=ut+δ−1ut+u0=ut+ft.Then, the actuator fault is obtained as(6)ft=δ−1ut+u0.It is assumed thatf˙t∈L20,∞. f˙t≤f1, where 0≤f1<∞. Considering the actuator fault (6), the active suspension system (2) becomes(7)x˙t=Axt+But+Bft+Dωt,yt=Cxt.


## 3. Actuator Fault Estimation

In this section, the fast FE idea [26] is introduced to the active suspension system with actuator faults and external disturbance input. Then, in order to improve the FE accuracy, GA is employed to optimize the parameters in the FE algorithm.


### 3.1. Adaptive Fault Estimation Design

For system (7), consider the following adaptive observer:(8)x^˙t=Ax^t+But+Bf^t−Ly^t−yt,y^t=Cx^t,where x^t and y^t are the state vector and the output vector of the observer, respectively, f^t is the estimation of the fault, and L is the gain matrix to be designed. Denote(9)ext=x^t−xt,eft=f^t−ft,eyt=y^t−yt.We can obtain the following error dynamics:(10)e˙xt=A−LCext+Beft−Dωt,eyt=Cext.The adaptive FE algorithm is presented as(11)f^˙t=−ΛNe˙yt+σeyt,where Λ=ΛT>0 is the given learning rate, N is a suitable dimensional matrix, and σ>0 is the weight scalar of the integral item.Theorem 1.
For system (10), given matrix Λ>0, scalars σ>0 and γ>0, if there exist matrices P>0, M, and N, such that the following conditions hold:(12)Φ=ATP+PA−MC−CTMT+CTC∗∗∗−1σBTPA+1σBTMC−2σBTPB∗∗0−1σΛ−1−γ2I∗−DTP1σDTPB0−γ2I<0,(13)BTP=NC,where M=PL. Then, ext and eft are uniformly ultimately bounded with ωt=0. Moreover, under the zero initial condition,(14)eyt2<γf˙tωt2,holds for all nonzero ωt∈L20,∞.Proof.
Choose the following Lyapunov function:(15)Vt=exTtPext+1σefTtΛ−1eft.
From (10), (11), and (13), we can get(16)V˙t=e˙xTtPext+exTtPe˙xt+2σefTtΛ−1e˙ft=A−LCext+Beft−DωtTPext+exTtPA−LCext+Beft−Dωt+2σefTtΛ−1−ΛNe˙yt+σeyt−f˙t=exTtPA−LC+A−LCTPext−2exTtPDωt−2σefTtNCA−LCext−2σefTtNCBeft+2σefTtNCDωt−2σefTtΛ−1f˙t.
Whenωt=0, we have(17)V˙t=ξ′TtΦ′ξ′t+γ2f˙Ttf˙t≤ξ′TtΦ′ξ′t+γ2f12,where ξ′t=exTtefTtf˙TtT and(18)Φ′=PA−LC+A−LCTP∗∗−1σNCA−LC−2σNCB∗0−1σΛ−1−γ2I.
It follows from (12) and (13) that Φ′<0 holds. Letting κ=λmin−Φ′, we obtain(19)V˙t≤−κξ′t2+γ2f12.
From (19), it is seen that V˙t<0 when ξ′t2>γ2f12/κ. Therefore, with ωt=0, ext and eft are uniformly ultimately bounded.
Moreover, define(20)J≜∫0∞eyTteyt−γ2f˙tωtTf˙tωtdt.
According to (13) and M=PL, under the zero initial condition, we obtain(21)J=∫0∞eyTteyt−γ2f˙tωtTf˙tωt+V˙tdt−V∞=∫0∞exTtPA−LC+A−LCTP+CTCext−2exTtPDωt−2σefTtNCA−LCext−2σefTtNCBeft+2σefTtNCDωt−2σefTtΛ−1f˙t−γ2f˙Ttf˙t−γ2ωTtωtdt−V∞=∫0∞exTtATP+PA−MC−CTMT+CTCext−2exTtPDωt−2σefTtBTPA−BTMCext−2σefTtBTPBeft+2σefTtBTPDωt−2σefTtΛ−1f˙t−γ2f˙Ttf˙t−γ2ωTtωtdt−V∞=∫0∞ξTtΦξtdt−V∞,where ξt=exTtefTtf˙TtωTtT.
From (12), it is known that Φ<0. Then, for nonzero ωt∈L20,∞, we have(22)J<−VT≤0.
Hence, we obtain(23)∫0∞eyTteytdt≤γ2∫0∞f˙tωtTf˙tωtdt,which is equivalent to (14). The proof is completed.Remark 1.
Note that the conditions in Theorem1 are not LMIs (linear matrix inequalities) due to the existence of (13). Fortunately, (13) can be transformed into the following approximate condition [26]:(24)ηIrBTP−NCBTP−NCTηIn≥0,where η is a sufficiently small positive constant. By solving LMIs (12) and (24), we can easily obtain the gain matrix L in the adaptive observer (8): L=P−1M.


### 3.2. Parameter optimization

For the FE algorithm (11), the adaptive learning rate Λ and the integral weight σ are undetermined parameters which are usually selected by trials. It is known that these two parameters may influence the accuracy and rapidity of the FE. For example, if a small Λ is selected, the tracking speed of the observer will be slow. Otherwise, if a large Λ is selected, a large overshoot on the FE may be generated. Therefore, in order to improve FE performance, Λ and σ can be viewed as parameters that need to be optimized.GA [38] is a random, global searching, and population-based optimization method. After decades of research and development, GA is known as a kind of effective and robust optimization method which can be employed to solve a variety of optimization problems [39–41], including problems in which the objective function is discontinuous, nondifferentiable, stochastic, or highly nonlinear. In this study, GA is used to optimize Λ and σ in the FE algorithm. For the quarter-car suspension system (7), it is easily seen that the learning rate Λ>0 is a scalar. Hence, the optimization vector is defined as α=Λ,σ. Three common kinds of actuator faults, which are, respectively, constant gain change, drift, and stuck, are all considered in the construction of the objective function for the sake of comprehensiveness. The initial expression of the objective function is described as(25)Qmapeα=13∑j=13∑k=1njf^k−fkfk×100%nj,where f^k and fk, respectively, represent the estimated fault and actual fault at the kth sampling point. j=1, j=2, and j=3, respectively, represent the constant gain change, drift, and stuck faults. nj represents the number of corresponding data points. In (25), the mean absolute percentage error (MAPE) is chosen as the error indicator. However, when the actual fault value is equal or very close to zero, the MAPE value will be too large, which is not suitable for the parameter optimization. In order to avoid this situation, the objective function is modified as(26)Qα=13∑j=13∑k=1njf^k−fkfk+1×100%nj.From (26), it is clearly seen that the relative error is employed to construct the objective function if the absolute value of the actual fault is far greater than 1. In contrast, the absolute error is employed if that of the actual fault is far less than 1.Based on GA, the parameter optimization process is described as follows.Step 1.
Set main parameters for GA, such as population size, maximum generation, crossover fraction, and so on.Step 2.
Randomly generate the initial population:α10,α20,…,αn0.Step 3.
Obtain observer (8) by solving the following convex optimization problem:(27)minγs.t.12,24.Step 4.
Calculate the objective function (26) with the obtained observer.Step 5.
If the stopping criteria are satisfied (maximum number of iterations is reached or objective function is achieved with certain accuracy), exit with the optimized vectorα∗. Otherwise, go to Step 6.Step 6.
Form a new generation ofα by “selection,” “crossover,” and “mutation” operations, and go to Step 3.


## 3.1. Adaptive Fault Estimation Design

For system (7), consider the following adaptive observer:(8)x^˙t=Ax^t+But+Bf^t−Ly^t−yt,y^t=Cx^t,where x^t and y^t are the state vector and the output vector of the observer, respectively, f^t is the estimation of the fault, and L is the gain matrix to be designed. Denote(9)ext=x^t−xt,eft=f^t−ft,eyt=y^t−yt.We can obtain the following error dynamics:(10)e˙xt=A−LCext+Beft−Dωt,eyt=Cext.The adaptive FE algorithm is presented as(11)f^˙t=−ΛNe˙yt+σeyt,where Λ=ΛT>0 is the given learning rate, N is a suitable dimensional matrix, and σ>0 is the weight scalar of the integral item.Theorem 1.
For system (10), given matrix Λ>0, scalars σ>0 and γ>0, if there exist matrices P>0, M, and N, such that the following conditions hold:(12)Φ=ATP+PA−MC−CTMT+CTC∗∗∗−1σBTPA+1σBTMC−2σBTPB∗∗0−1σΛ−1−γ2I∗−DTP1σDTPB0−γ2I<0,(13)BTP=NC,where M=PL. Then, ext and eft are uniformly ultimately bounded with ωt=0. Moreover, under the zero initial condition,(14)eyt2<γf˙tωt2,holds for all nonzero ωt∈L20,∞.Proof.
Choose the following Lyapunov function:(15)Vt=exTtPext+1σefTtΛ−1eft.
From (10), (11), and (13), we can get(16)V˙t=e˙xTtPext+exTtPe˙xt+2σefTtΛ−1e˙ft=A−LCext+Beft−DωtTPext+exTtPA−LCext+Beft−Dωt+2σefTtΛ−1−ΛNe˙yt+σeyt−f˙t=exTtPA−LC+A−LCTPext−2exTtPDωt−2σefTtNCA−LCext−2σefTtNCBeft+2σefTtNCDωt−2σefTtΛ−1f˙t.
Whenωt=0, we have(17)V˙t=ξ′TtΦ′ξ′t+γ2f˙Ttf˙t≤ξ′TtΦ′ξ′t+γ2f12,where ξ′t=exTtefTtf˙TtT and(18)Φ′=PA−LC+A−LCTP∗∗−1σNCA−LC−2σNCB∗0−1σΛ−1−γ2I.
It follows from (12) and (13) that Φ′<0 holds. Letting κ=λmin−Φ′, we obtain(19)V˙t≤−κξ′t2+γ2f12.
From (19), it is seen that V˙t<0 when ξ′t2>γ2f12/κ. Therefore, with ωt=0, ext and eft are uniformly ultimately bounded.
Moreover, define(20)J≜∫0∞eyTteyt−γ2f˙tωtTf˙tωtdt.
According to (13) and M=PL, under the zero initial condition, we obtain(21)J=∫0∞eyTteyt−γ2f˙tωtTf˙tωt+V˙tdt−V∞=∫0∞exTtPA−LC+A−LCTP+CTCext−2exTtPDωt−2σefTtNCA−LCext−2σefTtNCBeft+2σefTtNCDωt−2σefTtΛ−1f˙t−γ2f˙Ttf˙t−γ2ωTtωtdt−V∞=∫0∞exTtATP+PA−MC−CTMT+CTCext−2exTtPDωt−2σefTtBTPA−BTMCext−2σefTtBTPBeft+2σefTtBTPDωt−2σefTtΛ−1f˙t−γ2f˙Ttf˙t−γ2ωTtωtdt−V∞=∫0∞ξTtΦξtdt−V∞,where ξt=exTtefTtf˙TtωTtT.
From (12), it is known that Φ<0. Then, for nonzero ωt∈L20,∞, we have(22)J<−VT≤0.
Hence, we obtain(23)∫0∞eyTteytdt≤γ2∫0∞f˙tωtTf˙tωtdt,which is equivalent to (14). The proof is completed.Remark 1.
Note that the conditions in Theorem1 are not LMIs (linear matrix inequalities) due to the existence of (13). Fortunately, (13) can be transformed into the following approximate condition [26]:(24)ηIrBTP−NCBTP−NCTηIn≥0,where η is a sufficiently small positive constant. By solving LMIs (12) and (24), we can easily obtain the gain matrix L in the adaptive observer (8): L=P−1M.


## 3.2. Parameter optimization

For the FE algorithm (11), the adaptive learning rate Λ and the integral weight σ are undetermined parameters which are usually selected by trials. It is known that these two parameters may influence the accuracy and rapidity of the FE. For example, if a small Λ is selected, the tracking speed of the observer will be slow. Otherwise, if a large Λ is selected, a large overshoot on the FE may be generated. Therefore, in order to improve FE performance, Λ and σ can be viewed as parameters that need to be optimized.GA [38] is a random, global searching, and population-based optimization method. After decades of research and development, GA is known as a kind of effective and robust optimization method which can be employed to solve a variety of optimization problems [39–41], including problems in which the objective function is discontinuous, nondifferentiable, stochastic, or highly nonlinear. In this study, GA is used to optimize Λ and σ in the FE algorithm. For the quarter-car suspension system (7), it is easily seen that the learning rate Λ>0 is a scalar. Hence, the optimization vector is defined as α=Λ,σ. Three common kinds of actuator faults, which are, respectively, constant gain change, drift, and stuck, are all considered in the construction of the objective function for the sake of comprehensiveness. The initial expression of the objective function is described as(25)Qmapeα=13∑j=13∑k=1njf^k−fkfk×100%nj,where f^k and fk, respectively, represent the estimated fault and actual fault at the kth sampling point. j=1, j=2, and j=3, respectively, represent the constant gain change, drift, and stuck faults. nj represents the number of corresponding data points. In (25), the mean absolute percentage error (MAPE) is chosen as the error indicator. However, when the actual fault value is equal or very close to zero, the MAPE value will be too large, which is not suitable for the parameter optimization. In order to avoid this situation, the objective function is modified as(26)Qα=13∑j=13∑k=1njf^k−fkfk+1×100%nj.From (26), it is clearly seen that the relative error is employed to construct the objective function if the absolute value of the actual fault is far greater than 1. In contrast, the absolute error is employed if that of the actual fault is far less than 1.Based on GA, the parameter optimization process is described as follows.Step 1.
Set main parameters for GA, such as population size, maximum generation, crossover fraction, and so on.Step 2.
Randomly generate the initial population:α10,α20,…,αn0.Step 3.
Obtain observer (8) by solving the following convex optimization problem:(27)minγs.t.12,24.Step 4.
Calculate the objective function (26) with the obtained observer.Step 5.
If the stopping criteria are satisfied (maximum number of iterations is reached or objective function is achieved with certain accuracy), exit with the optimized vectorα∗. Otherwise, go to Step 6.Step 6.
Form a new generation ofα by “selection,” “crossover,” and “mutation” operations, and go to Step 3.


## 4. Results and Analysis

### 4.1. Fault Estimation Analysis

In this section, we will employ the above FE method to a quarter-car suspension model described in Section2. In addition, the disturbance input ωt is represented as a white-noise signal whose power spectral density (PSD) is given by Gzrf=4π2G0n02v, where G0 is the road roughness coefficient, n0 is the reference spatial frequency, and v is the vehicle forward velocity. Some parameters of the quarter-car suspension system are listed in Table 1.Table 1
Quarter-car suspension parameters.
Parameter
Description
Value
m
s
Sprung mass
320
kg
m
u
Unsprung mass
40
kg
k
s
Spring stiffness
18
kN
/
m
k
t
Tire stiffness
200
kN
/
m
c
s
Damping coefficient
1
kN
⋅
s
/
m
g
Gravitational acceleration
9.8
m
/
s
2
G
0
Road roughness coefficient
256
×
10
−
6
m
3
n
0
Reference spatial frequency
0.1
m
−
1
v
Vehicle forward velocity
20
m
/
sFor the fault-free situation, theH2/H∞ control method [42] is employed to improve the ride comfort as well as guarantee hard constraints. These hard constraints usually include good road holding ability, allowable suspension stroke, and maximal active force [4]. Considering the above control objective, the quarter-car active suspension system can be described as(28)x˙t=Axt+But+Dωt,z1t=C1xt+D1ut,z2t=C2xt+D2ut,where xt, ωt, A, B, and D have been defined in (2). z1t is the constrained output which reflects the hard constraints mentioned above. z2t is the control output which reflects the ride comfort.(29)C1=1zmax00000ktgms+mu00000,D1=001umax,C2=−ksms−csms0csms,D2=1ms,where zmax and umax are upper bounds for the suspension stroke and active control force, respectively.Choosingzmax=0.08m and umax=1.5kN and applying Theorem 4.3.1 in [42] with γ1=20, the following state feedback controller can be obtained:(30)ut=Kxt=103×9.1022−1.0908−3.8023−0.5576xt.For the optimization ofΛ and σ in the FE algorithm (11), three types of actuator faults are supposed to be generated to evaluate the actual fault fk in the object function (26). In this simulation, every type of fault lasts 2.5 seconds. δ and u0 in (6) are listed in Table 2. For convenience, the fault in Table 2 is called the reference actuator fault.Table 2
Reference actuator fault.
Fault values
Constant gain change
Drift
Stuck
Delta
0.6
1
0
u
0
0
−200 N
300 NThen, GA is used to optimize the vectorα=Λ,σ. Main parameters of GA are given in Table 3. Letting η=1×10−12 and utilizing the optimization procedure in Section 3.2 (the procedure is run 5 times), we can get the optimized results shown in Table 4. It is easily seen that the best Qα∗=0.03275% and the average one is 0.04817%. To verify the availability of the proposed FE method, several groups of actuator faults, which are different from the reference one, are selected and corresponding results are given in Table 5. In Table 5, α∗=28601,180 and Qjα∗ is defined as(31)Qjα∗=∑k=1njf^k−fkfk+1×100%nj,where f^k, fk, j, and nj have the same meaning as in (26). From Table 5, it can be seen that the calculated Qjα∗ are very small under different fault situation, which indicates that the estimated fault can accurately track the actual fault.Table 3
Main parameters of GA.
Parameters
Population size
Evolutionary generation
Crossover fraction
Mutation fraction
Value
20
10
0.8
0.2Table 4
GA optimization results.
Run no.
α
∗
Q
α
∗ (%)
1
[29329, 148]
0.05521
2
[22038, 244]
0.05512
3
[29150, 178]
0.04646
4
[28601, 180]
0.03275
5
[12873, 151]
0.05129Table 5
Estimation results for different faults.
Constant gain changeu0=0
Q
1
α
∗ (%)
Driftδ=1
Q
2
α
∗ (%)
Stuckδ=0
Q
3
α
∗ (%)
δ
=
0.4
0.0573
u
0
=
−
100
N
0.0015
u
0
=
200
N
0.0523
δ
=
0.6
0.0531
u
0
=
−
200
N
0.0015
u
0
=
300
N
0.0436
δ
=
0.8
0.0427
u
0
=
−
300
N
0.0016
u
0
=
400
N
0.0429Since the vehicle velocity varies frequently during practical driving, the optimized observer is applied under different vehicle velocity conditions. Estimation results are presented in Table6. These results prove that the proposed FE method has good robustness against uncertain vehicle velocity, which is applicable to practical applications.Table 6
Estimation results for different vehicle velocity.
Velocitym/s
Q
1
α
∗ (%)
Q
2
α
∗ (%)
Q
3
α
∗ (%)
10
0.0495
0.0015
0.0417
20
0.0531
0.0015
0.0436
30
0.0553
0.0016
0.0503To further demonstrate the effectiveness of the proposed FE method, fault tracking effects are compared for three types of actuator faults. Three FE methods are employed: the proposed one, the one without optimization ofΛ and σ, the one in which only the output error ey term is included in the FE algorithm. For the second method, Λ and σ are fixed as 10 and 1 [26]. For the third method, solve the LMIs (24) in [34] and obtain the corresponding observer. Simulation results obtained by using the above three methods are denoted as “optimized observer,” “nonoptimized observer,” and “derivative-free observer,” respectively. Fault tracking effects with constant gain change fault, drift fault, and stuck fault are, respectively, depicted in Figures 2∼4. According to these simulation results, we can see that the proposed FE method is well suited for different types of actuator faults. Compared with other two methods, the proposed one can effectively improve the FE accuracy and rapidity.Figure 2
Tracking effects of the constant gain change fault (random road).Figure 3
Tracking effects of the drift fault (random road).Figure 4
Tracking effects of the stuck fault (random road).Similarly, another type of disturbance input (usually called shock road disturbance) caused by an isolated bump is also considered in this paper. The ground displacement can be referred to (25) in [4], where the height and length of the bump are chosen as 0.1 m and 5 m. The vehicle forward velocity is chosen as 20 m/s. Fault tracking effects with constant gain change fault, drift fault, and stuck fault are, respectively, depicted in Figures 5∼7. These results show good robustness of the proposed FE method against different types of disturbance input.Figure 5
Tracking effects of the constant gain change fault (bump road).Figure 6
Tracking effects of the drift fault (bump road).Figure 7
Tracking effects of the stuck fault (bump road).


### 4.2. Fault-Tolerant Control Based on Fault Estimation

Based on the proposed FE method, a simple FTC strategy is adopted for the active suspension system (7). Fault tolerance is realized by adjusting the actuator output with the estimated fault.According to (6), let a=δ−1 and b=u0, we have(32)ft=aut+b.The following linear functions are given to describe the relationship between the estimated fault value and the normal output of the actuator:(33)f^t=a^ut+b^,where a^ and b^ are approximations of a and b, respectively.Sincef^t and ut are linearly correlated, we can obtain a and b by linearly fitting the sampling values of f^t and ut over a period of time. Based on the fitting results of a and b, we can qualitatively identify the actuator fault type. Detailed criteria are shown in Table 7, where ε and ε′ are given small positive constants (also called the thresholds of a^ and b^ in fault identifying).Table 7
Criteria for fault-type identification.
Fault type
a
,
b
a
^
,
b
^
Fault-free
a
=
0
,
b
=
0
−
ε
≤
a
^
≤
ε
,
−
ε
′
≤
b
^
≤
ε
′
Gain change
−
1
<
a
<
0
,
b
=
0
−
1
+
ε
<
a
^
<
−
ε
,
−
ε
′
≤
b
^
≤
ε
′
Drift
a
=
0
,
b
≠
0
−
ε
⩽
a
^
⩽
ε
,
b
^
<
−
ε
′
or
b
^
>
ε
′
Stuck
a
=
−
1
,
b
≠
0
−
1
−
ε
≤
a
^
≤
−
1
+
ε
,
b
^
<
−
ε
′
or
b
^
>
ε
′Based on the FE results in Figures2∼4, the fault type can be obtained with the aid of the least square fitting approach. Detailed fitting results under different types of faults are given in Table 8. Then, the FTC strategy can be adopted to compensate the corresponding fault.Table 8
Fitting results under different types of faults.
Fault type
a
,
b
a
^
,
b
^
Fault-free
0, 0
3.8832
×
10
−
11
,
−
1.5347
×
10
−
8
Gain change
−0.6, 0
−
0.6001
,
−
3.6204
×
10
−
3
Drift
0, −100
−
5.7261
×
10
−
11
,
−
100.0000
Stuck
−1, 400
−1.0002, 400.0117Taking the gain change fault as an example, according to (4), the ideal fault-tolerant controller for the gain change fault can be expressed as(34)uct=1δut=1a+1ut.Then, based on the fitting results, the fault-tolerant controller to be designed can be expressed as(35)u^ct=1a^+1ut.In simulation, the FTC is applied for the gain change fault. Simulation time interval is set from 0 to 3 s (second). The gain change fault starts from 0.5 s. The FTC is employed from 1.5 s. Response of the active suspension system is depicted in Figure8. It is easily seen that, from 0.5 s to 1.5 s, during which the fault exists while the FTC is not employed, the body acceleration attenuation effect of the suspension system with fault gets worse than that of the fault-free system. At about 1.7 s (0.2 s after the FTC is employed), the body acceleration response is similar to that of the fault-free system. Moreover, it is known that the suspension stroke and actuator output force are guaranteed to be within the required constraints when the FTC is adopted. The relative dynamic tire load (RDTL) may exceed the upper bound 1 occasionally and briefly, which is acceptable in some driving situations [4].Figure 8
Active suspension system response with the constant gain change fault (blue solid line) and fault-free (red dotted line).Comparison of root-mean-square (RMS) values of body accelerationz¨s from 2 s to 3 s is given in Table 9. For the active suspension system, RMS values of z¨s with FTC and without FTC are denoted as “Active (FTC)” and “Active (WFTC),” respectively. From Table 9, it is easily seen that, for the active suspension with the gain change fault, the RMS value of body acceleration obtained with FTC is reduced than that obtained from the passive system or without FTC. In addition, by applying the FTC strategy, the RMS value of body acceleration is almost equal to that of the fault-free system. The FTC results also demonstrate the applicability and effectiveness of the developed FE method.Table 9
Comparison of RMS values ofz¨sm/s2.
Passive
Active (WFTC)
Active (FTC)
Fault-free
Gain change
Fault-free
Gain change
1.2141
0.6490
0.9474
0.6490
0.6487


## 4.1. Fault Estimation Analysis

In this section, we will employ the above FE method to a quarter-car suspension model described in Section2. In addition, the disturbance input ωt is represented as a white-noise signal whose power spectral density (PSD) is given by Gzrf=4π2G0n02v, where G0 is the road roughness coefficient, n0 is the reference spatial frequency, and v is the vehicle forward velocity. Some parameters of the quarter-car suspension system are listed in Table 1.Table 1
Quarter-car suspension parameters.
Parameter
Description
Value
m
s
Sprung mass
320
kg
m
u
Unsprung mass
40
kg
k
s
Spring stiffness
18
kN
/
m
k
t
Tire stiffness
200
kN
/
m
c
s
Damping coefficient
1
kN
⋅
s
/
m
g
Gravitational acceleration
9.8
m
/
s
2
G
0
Road roughness coefficient
256
×
10
−
6
m
3
n
0
Reference spatial frequency
0.1
m
−
1
v
Vehicle forward velocity
20
m
/
sFor the fault-free situation, theH2/H∞ control method [42] is employed to improve the ride comfort as well as guarantee hard constraints. These hard constraints usually include good road holding ability, allowable suspension stroke, and maximal active force [4]. Considering the above control objective, the quarter-car active suspension system can be described as(28)x˙t=Axt+But+Dωt,z1t=C1xt+D1ut,z2t=C2xt+D2ut,where xt, ωt, A, B, and D have been defined in (2). z1t is the constrained output which reflects the hard constraints mentioned above. z2t is the control output which reflects the ride comfort.(29)C1=1zmax00000ktgms+mu00000,D1=001umax,C2=−ksms−csms0csms,D2=1ms,where zmax and umax are upper bounds for the suspension stroke and active control force, respectively.Choosingzmax=0.08m and umax=1.5kN and applying Theorem 4.3.1 in [42] with γ1=20, the following state feedback controller can be obtained:(30)ut=Kxt=103×9.1022−1.0908−3.8023−0.5576xt.For the optimization ofΛ and σ in the FE algorithm (11), three types of actuator faults are supposed to be generated to evaluate the actual fault fk in the object function (26). In this simulation, every type of fault lasts 2.5 seconds. δ and u0 in (6) are listed in Table 2. For convenience, the fault in Table 2 is called the reference actuator fault.Table 2
Reference actuator fault.
Fault values
Constant gain change
Drift
Stuck
Delta
0.6
1
0
u
0
0
−200 N
300 NThen, GA is used to optimize the vectorα=Λ,σ. Main parameters of GA are given in Table 3. Letting η=1×10−12 and utilizing the optimization procedure in Section 3.2 (the procedure is run 5 times), we can get the optimized results shown in Table 4. It is easily seen that the best Qα∗=0.03275% and the average one is 0.04817%. To verify the availability of the proposed FE method, several groups of actuator faults, which are different from the reference one, are selected and corresponding results are given in Table 5. In Table 5, α∗=28601,180 and Qjα∗ is defined as(31)Qjα∗=∑k=1njf^k−fkfk+1×100%nj,where f^k, fk, j, and nj have the same meaning as in (26). From Table 5, it can be seen that the calculated Qjα∗ are very small under different fault situation, which indicates that the estimated fault can accurately track the actual fault.Table 3
Main parameters of GA.
Parameters
Population size
Evolutionary generation
Crossover fraction
Mutation fraction
Value
20
10
0.8
0.2Table 4
GA optimization results.
Run no.
α
∗
Q
α
∗ (%)
1
[29329, 148]
0.05521
2
[22038, 244]
0.05512
3
[29150, 178]
0.04646
4
[28601, 180]
0.03275
5
[12873, 151]
0.05129Table 5
Estimation results for different faults.
Constant gain changeu0=0
Q
1
α
∗ (%)
Driftδ=1
Q
2
α
∗ (%)
Stuckδ=0
Q
3
α
∗ (%)
δ
=
0.4
0.0573
u
0
=
−
100
N
0.0015
u
0
=
200
N
0.0523
δ
=
0.6
0.0531
u
0
=
−
200
N
0.0015
u
0
=
300
N
0.0436
δ
=
0.8
0.0427
u
0
=
−
300
N
0.0016
u
0
=
400
N
0.0429Since the vehicle velocity varies frequently during practical driving, the optimized observer is applied under different vehicle velocity conditions. Estimation results are presented in Table6. These results prove that the proposed FE method has good robustness against uncertain vehicle velocity, which is applicable to practical applications.Table 6
Estimation results for different vehicle velocity.
Velocitym/s
Q
1
α
∗ (%)
Q
2
α
∗ (%)
Q
3
α
∗ (%)
10
0.0495
0.0015
0.0417
20
0.0531
0.0015
0.0436
30
0.0553
0.0016
0.0503To further demonstrate the effectiveness of the proposed FE method, fault tracking effects are compared for three types of actuator faults. Three FE methods are employed: the proposed one, the one without optimization ofΛ and σ, the one in which only the output error ey term is included in the FE algorithm. For the second method, Λ and σ are fixed as 10 and 1 [26]. For the third method, solve the LMIs (24) in [34] and obtain the corresponding observer. Simulation results obtained by using the above three methods are denoted as “optimized observer,” “nonoptimized observer,” and “derivative-free observer,” respectively. Fault tracking effects with constant gain change fault, drift fault, and stuck fault are, respectively, depicted in Figures 2∼4. According to these simulation results, we can see that the proposed FE method is well suited for different types of actuator faults. Compared with other two methods, the proposed one can effectively improve the FE accuracy and rapidity.Figure 2
Tracking effects of the constant gain change fault (random road).Figure 3
Tracking effects of the drift fault (random road).Figure 4
Tracking effects of the stuck fault (random road).Similarly, another type of disturbance input (usually called shock road disturbance) caused by an isolated bump is also considered in this paper. The ground displacement can be referred to (25) in [4], where the height and length of the bump are chosen as 0.1 m and 5 m. The vehicle forward velocity is chosen as 20 m/s. Fault tracking effects with constant gain change fault, drift fault, and stuck fault are, respectively, depicted in Figures 5∼7. These results show good robustness of the proposed FE method against different types of disturbance input.Figure 5
Tracking effects of the constant gain change fault (bump road).Figure 6
Tracking effects of the drift fault (bump road).Figure 7
Tracking effects of the stuck fault (bump road).


## 4.2. Fault-Tolerant Control Based on Fault Estimation

Based on the proposed FE method, a simple FTC strategy is adopted for the active suspension system (7). Fault tolerance is realized by adjusting the actuator output with the estimated fault.According to (6), let a=δ−1 and b=u0, we have(32)ft=aut+b.The following linear functions are given to describe the relationship between the estimated fault value and the normal output of the actuator:(33)f^t=a^ut+b^,where a^ and b^ are approximations of a and b, respectively.Sincef^t and ut are linearly correlated, we can obtain a and b by linearly fitting the sampling values of f^t and ut over a period of time. Based on the fitting results of a and b, we can qualitatively identify the actuator fault type. Detailed criteria are shown in Table 7, where ε and ε′ are given small positive constants (also called the thresholds of a^ and b^ in fault identifying).Table 7
Criteria for fault-type identification.
Fault type
a
,
b
a
^
,
b
^
Fault-free
a
=
0
,
b
=
0
−
ε
≤
a
^
≤
ε
,
−
ε
′
≤
b
^
≤
ε
′
Gain change
−
1
<
a
<
0
,
b
=
0
−
1
+
ε
<
a
^
<
−
ε
,
−
ε
′
≤
b
^
≤
ε
′
Drift
a
=
0
,
b
≠
0
−
ε
⩽
a
^
⩽
ε
,
b
^
<
−
ε
′
or
b
^
>
ε
′
Stuck
a
=
−
1
,
b
≠
0
−
1
−
ε
≤
a
^
≤
−
1
+
ε
,
b
^
<
−
ε
′
or
b
^
>
ε
′Based on the FE results in Figures2∼4, the fault type can be obtained with the aid of the least square fitting approach. Detailed fitting results under different types of faults are given in Table 8. Then, the FTC strategy can be adopted to compensate the corresponding fault.Table 8
Fitting results under different types of faults.
Fault type
a
,
b
a
^
,
b
^
Fault-free
0, 0
3.8832
×
10
−
11
,
−
1.5347
×
10
−
8
Gain change
−0.6, 0
−
0.6001
,
−
3.6204
×
10
−
3
Drift
0, −100
−
5.7261
×
10
−
11
,
−
100.0000
Stuck
−1, 400
−1.0002, 400.0117Taking the gain change fault as an example, according to (4), the ideal fault-tolerant controller for the gain change fault can be expressed as(34)uct=1δut=1a+1ut.Then, based on the fitting results, the fault-tolerant controller to be designed can be expressed as(35)u^ct=1a^+1ut.In simulation, the FTC is applied for the gain change fault. Simulation time interval is set from 0 to 3 s (second). The gain change fault starts from 0.5 s. The FTC is employed from 1.5 s. Response of the active suspension system is depicted in Figure8. It is easily seen that, from 0.5 s to 1.5 s, during which the fault exists while the FTC is not employed, the body acceleration attenuation effect of the suspension system with fault gets worse than that of the fault-free system. At about 1.7 s (0.2 s after the FTC is employed), the body acceleration response is similar to that of the fault-free system. Moreover, it is known that the suspension stroke and actuator output force are guaranteed to be within the required constraints when the FTC is adopted. The relative dynamic tire load (RDTL) may exceed the upper bound 1 occasionally and briefly, which is acceptable in some driving situations [4].Figure 8
Active suspension system response with the constant gain change fault (blue solid line) and fault-free (red dotted line).Comparison of root-mean-square (RMS) values of body accelerationz¨s from 2 s to 3 s is given in Table 9. For the active suspension system, RMS values of z¨s with FTC and without FTC are denoted as “Active (FTC)” and “Active (WFTC),” respectively. From Table 9, it is easily seen that, for the active suspension with the gain change fault, the RMS value of body acceleration obtained with FTC is reduced than that obtained from the passive system or without FTC. In addition, by applying the FTC strategy, the RMS value of body acceleration is almost equal to that of the fault-free system. The FTC results also demonstrate the applicability and effectiveness of the developed FE method.Table 9
Comparison of RMS values ofz¨sm/s2.
Passive
Active (WFTC)
Active (FTC)
Fault-free
Gain change
Fault-free
Gain change
1.2141
0.6490
0.9474
0.6490
0.6487


## 5. Conclusions

The FE problem is addressed in this paper for the quarter-car active suspension system with actuator faults. In the design process of the adaptive observer, the output error term and its derivative term are included in the FE algorithm. Then, GA is used to optimize some parameters in the FE algorithm, which are commonly selected by trials. Three typical kinds of actuator faults are taken into consideration for the GA optimization. Simulation results show that, for every kind of actuator fault, the proposed method can improve the rapidity and accuracy of FE. Furthermore, the proposed FE method is successfully utilized in the FTC of the active suspension system to improve the ride comfort and guarantee hard constraints. The experimental tests of the proposed FE method, which are important for practical applications, will be studied in the future.



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*Source: 1783850-2019-12-18.xml*