# A Hybrid Fault-Tolerant Control for Nonlinear Active Suspension Systems Subjected to Actuator Faults and Road Disturbances

Authors: Hui Pang; Xue Liu; Yuting Shang; Rui Yao
Journal: Complexity
 (2020)

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## Abstract

This paper proposes a hybrid fault-tolerant control strategy for nonlinear active suspension subjected to actuator faults and road disturbances. First, an augmented closed-loop system model is established for the nonlinear active suspension system with the actuator faults and road disturbances. Then, based on this model, a hybrid fault-tolerant controller that consists of a nominal state-feedback controller and a robustH∞ observer is proposed to stabilize the control plant under fault-free condition and further compensate for the suspension performance loss under the actuator fault condition. Finally, a half-vehicle active suspension example is exploited to demonstrate the effectiveness of the proposed hybrid fault-tolerant controller under various running conditions.


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## Body

## 1. Introduction

As one of key components in chassis system, vehicle suspension systems connect the vehicle body and wheel to provide a support force and further to improve vehicle dynamics performance [1], and they are usually categorized into three types as passive suspension, semiactive suspension, and active suspension systems [2]. For the passive and semiactive suspension systems, both of them are limited in their ability to possess sufficient ride quality and handling stability [3–5]. While active suspension system (ASS) has the best potentials to make a well trade-off between the conflicting performance requirements [6] and to provide much better ride quality and handling capability [7–9], this is because the actuator assembled in the ASS can generate an extra control force with dissipating the kinetic energy.Up to now, a number of control algorithms such asH∞ control [10, 11], sliding-mode control [12], adaptive backstepping control [13, 14], and T-S fuzzy control [15, 16] have been proposed and employed to study vehicle ASSs. However, the existing control schemes in this research field are almost based on such an assumption that all components of vehicle ASSs are under fault-free condition. In fact, it is fairly common to encounter different faults in the real-world suspension system, especially the actuator faults. It should be noticed that the actuator faults would usually result in performance degradation, instability, or catastrophic events for the vehicle suspension system. Consequently, many scholars have devoted their efforts in developing a class of fault-tolerant controller that could deal with the performance loss caused by the actuator faults and then maintain a desirable system performance for the controlled suspension system (see [17–25] and the references therein). Among which, the design of robust H∞ fault-tolerant control for the ASSs with actuator faults has attracted great attention.For instance, anH∞ robust controller was proposed in [26] to guarantee asymptotic stability of the ASSs with three different types of actuator failures. A reliable fuzzy H∞ robust fault-tolerant controller in [27] was developed for vehicle ASSs with the actuator delay and fault via Takagi-Sugeno (T-S) fuzzy approach. In [28], an adaptive robust fault-tolerant controller was proposed to deal with the problem of fault accommodation for the unknown actuator failures in the ASSs. However, the aforementioned literatures have almost focused on developing a practically passive fault-tolerant controller with preknown fault modes. In other words, those designed fault-tolerant controllers are not very sensitive to deal with the application scenarios for the ASSs with the single or multimode actuator faults.To overcome this problem, a fault-tolerant control algorithm was developed in [29] for vehicle ASSs in the finite-frequency domain under the sinusoidal wave fault. In addition, an adaptive fault-tolerant compensation controller was proposed in [30] to enhance the output performance for a kind of vehicle suspension system, wherein the failure model was described by a scalar Markovian type function. A robust H∞ proportional-integral observer-based fault diagnosis method for the vehicle suspension system was addressed in [31], wherein the constant gain fault was diagnosed by an observer. However, these fault-tolerant control designs did not take the suspension’s nonlinearities as well as the actuator faults into account when designing the corresponding controllers; moreover, the fault estimation accuracy has to be considered and enhanced if one wants to develop an effective fault-tolerant controller. Therefore, it is still an interesting and challenging issue to design an appropriate fault-tolerant controller with higher accuracy and better performance for vehicle ASSs.In the context of the above discussions, this paper proposes a hybrid fault-tolerant control design for nonlinear active suspension subjected to the actuator faults and road disturbances. Herein, compared with the most related fault-tolerant control methods in [16, 19, 21, 31, 32], we make several steps forward. First, an augmented closed-loop system model is established for the nonlinear active suspension system with the actuator faults and road disturbances, which were rarely present in the previous papers. Second, a hybrid fault-tolerant controller (HFTC) that consists of a nominal state-feedback controller and a robust H∞ observer is proposed to stabilize the ASS under fault-free condition and compensate for the suspension performance loss under the actuator fault condition. Third, a numerical example under various running conditions is presented to reveal the advantages arising from our proposed controller, and some comparative investigations are also provided to validate the control effects of the designed HFTC.The remainder of this paper is structured as follows. In Section2, problem formulation is presented. Section 3 provides the robust H∞ observer design, and Section 4 provides the proposed HFTC synthesis. Section 5 gives the simulation investigation and discussion, and the conclusions are outlined in Section 6.Notations.
R
n and Rn×m denote the n-dimensional Euclidean space and the set of n×m real matrices, respectively. The superscript T is used to represent the transposition of the matrix. And for the symmetric matrices X and Y, X − Y is positive semidefinite for X ≥ Y and positive definite for X > Y, respectively. L2 stands for the space of square-integral vector functions. · stands for the Euclidean vector norm, and the asterisk (∗) denotes the symmetric form of a matrix. The space of square-integral vector functions over [0, +∞) is denoted by L2 [0, +∞), and for w = ∈L2 [0, +∞), its norm is given by w=∫t=0∞w2dt. Tzw denotes the transfer function of the closed-loop system from the road disturbance w to the control output z.


## 2. Problem Formulation

In this section, a control-oriented half-vehicle active suspension model with four degrees of freedom (4-DOFs) shown in Figure1 is used to develop our fault-tolerant controller.Figure 1
Half-vehicle active suspension model.Based on Newton’s second law, we can easily construct the dynamics equations of this half-vehicle ASS as(1)Msq¨=GKszu−zs+GCsz˙u−z˙s−Gv1zu−zs−Gv2z˙u−z˙s+Gu,Muz¨u=Kszs−zu+Csz˙s−z˙u+Kuzr−zu+v1zu−zs+v2z˙u−z˙s−u,where q = [zc, ϕ]T is the output performance vector, zs = [zsf, zsr]T is the sprung-mass displacement vector, zu = [zuf, zur]T is the suspension displacement vector, zr = [zrf, zrr]T is the input vector of the road disturbances at the front and rear wheel, u = [utf, utr]T is the desirable control forces, and v1 (zs − zu) and v2 (z˙s − z˙u) are the nonlinear term originated from the nonlinear characteristics of vehicle suspension damper and spring, respectively.In equation (1), the corresponding coefficient matrices are given by(2)Ms=ms00Iy,Mu=muf00mur,Cs=cf00cr,Ks=kf00kr,Ku=ktf00ktr,G=11−ab.Definex = [zs − zu, z˙s, zu − zr, z˙u]T as the system state vector and let w = zr, y = [zu − zr, z˙s]T as the measured output vector and z = [q¨, zs − zu, Ku (zu − zr), u]T as the system output vector. Then, the state-space form of the ASS without the actuator faults is written as(3)x˙=Ax+B1w+B2u+D1v,y=C1x,z=C2x+D2u,wherein the corresponding coefficients matrices are described as follows:(4)A=0I0−I−GTMs−1GKs−GTMs−1GCs02×2GTMs−1GCs000IMu−1KsMu−1Cs−Mu−1Ku−Mu−1Cs,B1=00−I0,B2=0GTMs−1G0Mu−1,C1=I0,C2=−Ms−1GKs−Ms−1GCs0Ms−1GCsI00000Ku00000,D1=GTMs−1G0000Mu−1000000000Mu−1,D2=−Ms−1G000.It should be noticed thatv (x, t) = [v1 (zs − zu), v2(z˙s − z˙u)]T is the nonlinear term vector, I is an identity matrix with appropriate dimension, and the dependent variables of x and t in v (x,t) are usually omitted for brevity. Besides, to facilitate the description of the suspension system, the following two assumptions are given.Assumption 1.
In this work, it is assumed that the nonlinear termsv1 and v2 satisfy the Lipschitz condition as follows [33]:(5)v1zs−zu≤VkKszs−zu,v2z˙s−z˙u≤VcCsz˙s−z˙u,where Vk and Vc are the quantized factor that represent for the nonlinear perturbations of the spring and damper coefficients, respectively, and the values of Vc and Vk are selected as Vc = Vk = 0.1.Assumption 2.
It is assumed that the actuator faultf along with its corresponding derivatives f˙ satisfies f∈L20∞, and f˙∈L20∞.Remark 1.
It is assumed that the energy of actuator fault is bounded with satisfyingf∈L20∞, and it is accordingly inferred that the derivatives of actuator fault are satisfied with f˙∈L20∞.
With the above two assumptions, the state-space equations of this faulty ASS can be derived as(6)x˙f=Axf+B1w+Bff+B2uf+D1v,yf=C1xf+D1ff,zf=C2xf+D2uf+D2ff,where f ∈ Rn is the actuator fault vector and Bf, D1f, and D2f are the coefficient matrices with appropriate dimension, satisfying BfTD2fTT=B2TD2TT⋅M.Remark 2.
B
f
T
D
2
f
T
T
=
B
2
T
D
2
T
T
⋅
M means that the actuator faults usually occur in the same channel of a fault-tolerant controller, implying the match of the actuator faults. If the actuator faults are matched, then the negative effects caused by the faults can be compensated by the designed fault-tolerant controller. This condition may be a constraint on the hybrid fault-tolerant control method, but it is very practical and common in the fault-tolerant control community.
Furthermore, the actuator fault vectorf considered herein is subjected to an exogenous system [34, 35]. Without loss of generality, the generalized fault signal model can be described as(7)x˙ef=Awfxef+Bwfδ,f=Cwfxef,where xef∈Rnf denotes the state vector of the exogenous system, δ denotes a bounded virtual input signal, and Awf, Bwf, and Cwf denote the coefficient matrices with appropriate dimension.Remark 3.
Failures can be manifested in the breakdown of a piece of equipment (hard failure, e.g., partial blockage or stuck) or gradual degradation (incipient or soft failures, e.g., natural wear and tear and system degradation) in the active suspension system. There may be various kinds of actuator faults, but they can be theoretically modeled in a finite parameter family named as slow drift fault, constant gain fault or sinusoidal fault, and so on for the control strategy design process.
Besides, in this work, the main reasons of employing this fault expression as shown in (7) lie in the following two aspects. (1) An explicit fault diagnosis module is not needed in the proposed fault-tolerant controller synthesis. (2) This actuator fault signal model is helpful for the subsequent system augmentation since it would be intrinsically embedded in the controller design.
By combing (7) with (6) with the aforementioned assumptions and system augmentation technique, we can obtain(8)x¯˙=Aefx¯+Befu+Few˜+F,y=Cex¯,f=Cefx¯,where(9)x¯=xfxef,w˜=wδ,Aef=ABfCwf0Awf,Bef=B20,Fe=F1eF2e,F1e=B10,F2e=0Bwf,F=F1v0,Cef=0Cwf,Ce=C1D1fCwf.


## 3. RobustH∞ Observer Design

This section contributes to establishing a robustH∞ observer to estimate the unmeasurable actuator faults and the related system states. Before proceeding, it is assumed that the pair (Aef, Ce) in (8) is observable.Accordingly, an augmented observer is constructed to accurately estimate both of the system states and the fault signals for the ASS, and the full-order observer of system (8) is expressed by(10)x¯^˙=Aefx¯^+Befuf+Ly−Cex¯^+F,f^=Cefx¯^,where x¯^∈Rn+nf is estimated state vector and L is the undetermined gain matrix for the augmented observer, respectively.Let(11)e=x¯−x¯^,(12)f˜=f−f^.Differentiating (11) yields(13)e˙=x¯˙−x¯^˙.From (12) and (13), we can derive the error system as(14)e˙=Aef−LCee+Few˜+ΔF,f˜=Cefe,where(15)ΔF=F1vx,t−F1vx^,t0.Then, with the following Definition1, Theorem 1, and Theorem 2, the existing conditions of the designed robust H∞ observer can be obtained so as to ensure the H∞ performances of the closed-loop system under zero initial condition.Definition 1.
For a givenβ > 0, the system is said to be of H∞ performance β if the system is internally asymptotically stable and the inequality f˜<βw˜ holds for ∀w˜∈L20,+∞ under zero initial conditions.Theorem 1.
For given positive scalarsα and β, the closed-loop system (13) is asymptotically stable and has a prescribed H∞ disturbance attenuation level β>0, if there exist symmetric positive definite matrix P1 with appropriate dimension and a gain matrix L such that the following inequality holds:(16)Aef−LCeTP1+P1Aef−LCe+CefTCefP1P1Fe∗−I0∗∗−βI<0.Proof.
Define the Lyapunov function as(17)V1=eTP1e.
The time derivative of (17) becomes(18)V˙1=e˙TP1e+eTP1e˙=eTAef−LCeTP1+P1Aef−LCee+2eTP1Few˜+2eTP1ΔF.
It can be derived from Lipschitz condition [33] that(19)ΔF2=Fx,t−Fx^,t2≤α2e2.
That is,(20)ΔFTΔF−α2eTe≤0.
From (18) and (20), we have(21)V˙1≤eTAef−LCeTP1+P1Aef−LCee+2eTP1Few˜+2eTP1ΔF+α2eTe−ΔFTΔF.
Let the external road disturbancew˜=0, and in terms of (21), we obtain(22)V˙1≤g1Tψ1g1,where(23)g1=eΔF,ψ1=Aef−LCeTP1+P1Aef−LCe+α2IP1∗−I.
It can be inferred from (22) that if ψ1<0, one gets V˙<0, and system (13) is asymptotically stable with w˜=0.
Next, consider theH∞ performance index β with ∀w˜∈L20,+∞; if we assume V1t=0=0, then the performance index Jw can be defined as(24)Jw=∫0∞f˜Tf˜−βw˜Tw˜dt.
BecauseVt=0=0 and V1∞≥0, in terms of (21) and (24), we can obtain(25)Jw=∫0∞f˜Tf˜−βw˜Tw˜+V˙dt−V1∞+V10≤∫0∞f˜Tf˜−βw˜Tw˜+V˙1dt≤∫0∞eTAef−LCeTP+P1Aef−LCee+2eTP1Few˜+2eTP1ΔFx,t+α2eTe−ΔFx,tTΔFx,t+eTCefTCefe−βw˜Tw˜dt.
Let(26)ζ1=eΔFx,tw˜,ψ2=Aef−LCeTP1+P1Aef−LCe+CefTCef+α2IP1P1Fe∗−I0∗∗−βI.
Ifψ2<0, then we can get(27)Jw=∫0∞f˜Tf˜−βw˜Tw˜dt≤∫0∞ζ1Tψ2ζ1dt<0.
It is equivalent to(28)f˜<βw˜.
By using Schur complement [36], one can get ψ1<0 with ψ2<0, and the condition of ψ2<0 guarantees V˙1<0; thus, system (13) is asymptotically stable. For ∀w∈L20,+∞, we obtain Jw<0 in terms of V˙1<0, i.e., f˜<βw˜. Furthermore, system (13) has H∞ performance β when it is asymptotically stable and the inequality f˜<βw˜ holds for ∀w∈L20,+∞ under ψ2<0 conditions. The proof is completed.
SinceTheorem1 involves the expression like P1LCe, it cannot be solved directly via the LMI technique, and then Theorem 2 is introduced in the following form to cope with this problem.Theorem 2.
For given positive scalarsα and β, the closed-loop system (13) is asymptotically stable and has a prescribed H∞ disturbance attenuation level β>0, if there exist symmetric positive matrices P1, Y1 with appropriate dimension and the gain matrix L such that the following inequality holds:(29)AefTP1+P1Aef−CeTY1T−Y1Ce+CefTCef+α2IP1P1Fe∗−I0∗∗−βI<0,where L=P1−1Y.Proof.
SubstitutingY1=P1L into (16), one can obtain (29).
The proof is completed.
Obviously, withTheorem2, system (13) has H∞ performance β when the system is asymptotically stable and the inequality f˜<βw˜; thus, we get f^≅f from the full-order observer in (10). Next, the proposed fault-tolerant controller synthesis will be discussed in Section 4.


## 4. Hybrid Fault-Tolerant Controller Synthesis

### 4.1. Design ofH∞ Observer-Based Fault-Tolerant Controller

To begin with, the fault-tolerant controller involved withxf and f^ is designed as(30)uf=un+uc,where uf is the desirable control force of the proposed fault-tolerant controller, un=Kxf is nominal state-feedback controller, uc=−Mf^ is H∞ observer-based compensation controller, K is the gain matrix of nominal state-feedback controller, f^ is the actuator fault estimation achieved by the designed robust H∞ observer in (10), and M is a known matrix with appropriate dimension.Substituting (30) into (6) results in (31) as(31)x˙f=A¯xf+B1w+F1v,zf=C¯2xf,where A¯=A+B2K and C¯2=C2+D2K.Additionally, according to (5), we can obtain(32)v≤VkKs0000VcCs000000000−VcCszs−zuz˙szu−zrz˙u=W1x,where(33)W1=VkKs0000VcCs000000000−VcCs.Thus far, the proposedH∞ state-feedback controller can be summarized as Theorem3 and Theorem 4.Theorem 3.
Given positive scalarsγ and λ2, the closed-loop system (31) is asymptotically stable and has a prescribed H∞ disturbance attenuation level γ>0, if there exists symmetric positive matrix P2 with appropriate dimension such that the following inequality holds:(34)P2A¯+A¯TP2+λ2W1TW1+C¯2TC¯2P2F1P2B1∗−λ2I0∗∗−γI<0.Proof.
Define the Lyapunov function as(35)V2=xP2x+∫0tλ2W1x2−λ2vx,t2dt.
Letw = 0, and taking the derivative of V gives(36)V˙2=xTP2A¯x+F1v+xTA¯T+vTF1TP2x+λ2W1x2−λ2v2,=xTP2A¯+A¯TP2+λ2W1TW1x+xTP2F1v+vTF1TP2x−λ2vTv,=g2Tχg2,where(37)g2=xv,χ1=P2A¯+A¯TP2+λ2W1TW1P2F1∗λ2I.
Thus, ifχ1<0, then V˙<0, and the closed-loop system (31) is definitely asymptotically stable. Moreover, consider the H∞ performance index γ with ∀w∈L20,+∞ and suppose V2t=0=0, if we define the performance index Jz as(38)Jz=∫0∞zTz−γwTwdt.
BecauseV2t=0=0 and V2∞≥0, from (36) and (38), we can further get(39)Jz=∫0∞zTz−γwTw+V˙dt−V2∞+V20≤∫0∞zTz−γwTw+V˙2dt≤∫0∞−γwTw+xTP2A¯+A¯TP2+λ2W1TW1+C¯2TC¯2x+xTP2F1vx,t+vTF1TP2x+wTP2B1x+xTB1TP2w−λ2vTv.
Let(40)ζ2=xvw,χ2=P2A¯+A¯TP2+λ2W1TW1+C¯2TC¯2P2F1P2B1∗−λ2I0∗∗−γI,
and ifχ2<0, then we have(41)Jz=∫0∞zTz−γwTwdt≤∫0∞ζ2Tχ2ζ2dt<0.
It is equivalent to(42)z<γw.
Similarly, it is equivalent to obtainχ1<0 through χ2<0 by using Schur complement, and χ2<0 can ensure being V˙2<0, which implies that system (31) is asymptotically stable. For ∀w∈L20,+∞, one can get Jz<0 from V˙2<0, that is, z<γw (see (41) and (42)). Furthermore, system (31) has H∞ performance β if and only if the controlled system is asymptotically stable and the inequality z<γw holds for ∀w˜∈L20,+∞ under χ2<0 conditions.
The proof is completed.
To guarantee the solvability of (34) in Theorem 3, Theorem 4 is presented herein.Theorem 4.
For given positive scalarsγ and λ2, the closed-loop system (31) is asymptotically stable and has a prescribed H∞ disturbance attenuation level γ>0, if there exist symmetric positive definite matrix Q, R with appropriate dimension and the gain matrix K such that the following inequality holds:(43)AQ+B2+QAT+RTB2TF1B1λ2QW1TQC2T+RTD2T∗−λ2I000∗∗−γI00∗∗∗−I0∗∗∗∗−I<0,where K=RQ−1.Proof.
The termχ2<0 is left multiplied by diag {P2−1, I, I, I, I} and right multiplied by diag {P2−1, I, I, I, I}; apply Schur complement and let Q = P2−1 and R = KQ; by further derivations, we can obtain (43). The proof is completed.


### 4.2. Design Procedure of the Proposed HFTC Scheme

With the aforementioned Theorem1 to Theorem 4, the actuator fault occurring at the ASS can be estimated with the designed robust H∞ observer, and then a hybrid fault-tolerant controller is further developed; the detailed design procedure is formulated as follows:Step 1. Solve Theorem2 so as to get L from (29) and estimate the actuator faults by using f^=Cefx¯^ in (10).Step 2. Construct compensation controller as uc = −MCefx¯^ = −Mf^.Step 3. Solve Theorem 4 to obtain K=RQ−1, that is, un=Kxf, and then the proposed HFTC in (30) is achieved and the corresponding control block is shown in Figure 2.Figure 2
Control block of the designed hybrid fault-tolerant controller.From Figure2, it is obvious that the proposed hybrid fault-tolerant controller is composed of a nominal state-feedback controller and a robust H∞ observer. The former one only works under the fault-free condition, and the latter one can accurately estimate the actuator faults, and the designed fault-tolerant controller can be realized by combing the nominal state-feedback controller and the robust H∞ output-feedback observer under the actuator fault condition.


## 4.1. Design ofH∞ Observer-Based Fault-Tolerant Controller

To begin with, the fault-tolerant controller involved withxf and f^ is designed as(30)uf=un+uc,where uf is the desirable control force of the proposed fault-tolerant controller, un=Kxf is nominal state-feedback controller, uc=−Mf^ is H∞ observer-based compensation controller, K is the gain matrix of nominal state-feedback controller, f^ is the actuator fault estimation achieved by the designed robust H∞ observer in (10), and M is a known matrix with appropriate dimension.Substituting (30) into (6) results in (31) as(31)x˙f=A¯xf+B1w+F1v,zf=C¯2xf,where A¯=A+B2K and C¯2=C2+D2K.Additionally, according to (5), we can obtain(32)v≤VkKs0000VcCs000000000−VcCszs−zuz˙szu−zrz˙u=W1x,where(33)W1=VkKs0000VcCs000000000−VcCs.Thus far, the proposedH∞ state-feedback controller can be summarized as Theorem3 and Theorem 4.Theorem 3.
Given positive scalarsγ and λ2, the closed-loop system (31) is asymptotically stable and has a prescribed H∞ disturbance attenuation level γ>0, if there exists symmetric positive matrix P2 with appropriate dimension such that the following inequality holds:(34)P2A¯+A¯TP2+λ2W1TW1+C¯2TC¯2P2F1P2B1∗−λ2I0∗∗−γI<0.Proof.
Define the Lyapunov function as(35)V2=xP2x+∫0tλ2W1x2−λ2vx,t2dt.
Letw = 0, and taking the derivative of V gives(36)V˙2=xTP2A¯x+F1v+xTA¯T+vTF1TP2x+λ2W1x2−λ2v2,=xTP2A¯+A¯TP2+λ2W1TW1x+xTP2F1v+vTF1TP2x−λ2vTv,=g2Tχg2,where(37)g2=xv,χ1=P2A¯+A¯TP2+λ2W1TW1P2F1∗λ2I.
Thus, ifχ1<0, then V˙<0, and the closed-loop system (31) is definitely asymptotically stable. Moreover, consider the H∞ performance index γ with ∀w∈L20,+∞ and suppose V2t=0=0, if we define the performance index Jz as(38)Jz=∫0∞zTz−γwTwdt.
BecauseV2t=0=0 and V2∞≥0, from (36) and (38), we can further get(39)Jz=∫0∞zTz−γwTw+V˙dt−V2∞+V20≤∫0∞zTz−γwTw+V˙2dt≤∫0∞−γwTw+xTP2A¯+A¯TP2+λ2W1TW1+C¯2TC¯2x+xTP2F1vx,t+vTF1TP2x+wTP2B1x+xTB1TP2w−λ2vTv.
Let(40)ζ2=xvw,χ2=P2A¯+A¯TP2+λ2W1TW1+C¯2TC¯2P2F1P2B1∗−λ2I0∗∗−γI,
and ifχ2<0, then we have(41)Jz=∫0∞zTz−γwTwdt≤∫0∞ζ2Tχ2ζ2dt<0.
It is equivalent to(42)z<γw.
Similarly, it is equivalent to obtainχ1<0 through χ2<0 by using Schur complement, and χ2<0 can ensure being V˙2<0, which implies that system (31) is asymptotically stable. For ∀w∈L20,+∞, one can get Jz<0 from V˙2<0, that is, z<γw (see (41) and (42)). Furthermore, system (31) has H∞ performance β if and only if the controlled system is asymptotically stable and the inequality z<γw holds for ∀w˜∈L20,+∞ under χ2<0 conditions.
The proof is completed.
To guarantee the solvability of (34) in Theorem 3, Theorem 4 is presented herein.Theorem 4.
For given positive scalarsγ and λ2, the closed-loop system (31) is asymptotically stable and has a prescribed H∞ disturbance attenuation level γ>0, if there exist symmetric positive definite matrix Q, R with appropriate dimension and the gain matrix K such that the following inequality holds:(43)AQ+B2+QAT+RTB2TF1B1λ2QW1TQC2T+RTD2T∗−λ2I000∗∗−γI00∗∗∗−I0∗∗∗∗−I<0,where K=RQ−1.Proof.
The termχ2<0 is left multiplied by diag {P2−1, I, I, I, I} and right multiplied by diag {P2−1, I, I, I, I}; apply Schur complement and let Q = P2−1 and R = KQ; by further derivations, we can obtain (43). The proof is completed.


## 4.2. Design Procedure of the Proposed HFTC Scheme

With the aforementioned Theorem1 to Theorem 4, the actuator fault occurring at the ASS can be estimated with the designed robust H∞ observer, and then a hybrid fault-tolerant controller is further developed; the detailed design procedure is formulated as follows:Step 1. Solve Theorem2 so as to get L from (29) and estimate the actuator faults by using f^=Cefx¯^ in (10).Step 2. Construct compensation controller as uc = −MCefx¯^ = −Mf^.Step 3. Solve Theorem 4 to obtain K=RQ−1, that is, un=Kxf, and then the proposed HFTC in (30) is achieved and the corresponding control block is shown in Figure 2.Figure 2
Control block of the designed hybrid fault-tolerant controller.From Figure2, it is obvious that the proposed hybrid fault-tolerant controller is composed of a nominal state-feedback controller and a robust H∞ observer. The former one only works under the fault-free condition, and the latter one can accurately estimate the actuator faults, and the designed fault-tolerant controller can be realized by combing the nominal state-feedback controller and the robust H∞ output-feedback observer under the actuator fault condition.


## 5. Simulation Investigation and Discussion

In this section, to verify the effectiveness and advantages of the proposed HFTC, the performance analysis and comparative simulations are conducted for the following three types of vehicle ASSs described as follows:(i)
NSFC-without fault: only the nominal state-feedback controller works under the actuator fault-free condition.(ii)
NSFC-with fault: only the nominal state-feedback controller works under the actuator fault condition.(iii)
HFTC-with fault: the designed hybrid fault-tolerant controller works under the actuator fault condition.Besides, the controller gain is calculated through MATLAB 2016a® LMI toolbox and FEASP Solver (MathWorks, Inc., Natick, MA, USA), and the simulation is conducted under MATLAB 2016a® Simulink (MathWorks, Inc., Natick, MA, USA). The half-vehicle model parameters are listed in Table1 [37].Table 1
Parameters of half-vehicle suspension.
Parameter
Value
m
s/kg
1235
m
uf(mur)/kg
49
I
y/(kg·m2)
1731
k
f(kr)/(N·m−1)
22,741
k
tf/(N·m−1)
492,982
k
tr/(N·m−1)
302,342
c
f/(N·s·m−1)
1221
c
r/(N·s·m−1)
1228
a/m
1.2
b/m
1.5


### 5.1. Performance Analysis under the General Actuator Fault

The random road disturbance is assumed as a vibration signal that is consistent and typically specified as a white noise expressed by [38](44)z˙r=−2πf0zr+2πn0ωGqn0v,where Gq(n0) is selected as 1024 × 10−6m3, v = 20 m/s, the lower cutoff frequency of road profile f0 = 0.1, the reference spatial frequency n0 = 0.1 (1/m), and ω (t) is zero mean white Gaussian noise with identity power spectral density with the sample time of 0.01s.The fault signal and its corresponding fault estimation are revealed in Figure3(a), and the fault estimation error is displayed in Figure 3(b), respectively. Note that the estimation error of the actuator fault is calculated by Δf=f−f^.Figure 3
The variations of (a) fault estimation and (b) fault estimation error for the general fault signal.
(a)
(b)It is observed from Figure3 that the estimation error of the actuator fault has a peak value less than ±0.05, implying that the designed H∞ observer can accurately estimate the actuator fault. Besides, Figure 4 is presented to show the performance comparisons of z¨c, ϕ¨, Δyf and Δyr, and Ftf and Ftr for the three types of ASSs in time domain.Figure 4
The response comparisons of ASS with the general fault signal under random road disturbance.
(a)
(b)
(c)
(d)
(e)
(f)Additionally, it can be concluded from Figure4(a) that the response of z¨c becomes obviously deteriorated in case of the NSFC-without fault, yet z¨c of the HFTC-with fault can yield better ride comfort performance in comparisons with the NSFC-with fault, while the responses of ϕ¨ in Figure 4(b) remain basically unchanged for these three ASSs. Moreover, it can be inferred from Figures 4(c) to 4(f) that Δyf and Δyr and Ftf and Ftr are effectively reduced for the faulty ASS in case of the proposed HFTC-with fault, which can prevent suspension breakdown when suspension deflection is out of its limitation and can further improve tire life span and vehicle handling stability simultaneously.Moreover, Figure5 plots the control forces of the front and rear actuators with the designed HFTC. It is seen from Figure 5 that the control forces vary with the change of fault signals on time, which illustrates that the proposed HFTC can generate the compensative control forces to reduce the negative effects of the actuator fault on the closed-loop system.Figure 5
The control forces of (a) the front actuator and (b) rear actuator under the random road disturbance.
(a)
(b)


### 5.2. Performance Analysis under the Sinusoidal Actuator Fault

To further evaluate the proposed HFTC, the sinusoidal signal is used to mimic the actuator faults and validate the effectiveness of the proposed HFTC. Figure6 shows the variations of the front actuator fault and the corresponding estimation as well as the fault estimation error. It is easily found that the designed robust H∞ observer can accurately estimate the actuator fault with the fault estimation error of −0.02 N to 0.02 N.Figure 6
The variations of (a) fault estimation and (b) fault estimation error for the sinusoidal fault signal.
(a)
(b)Figure7 shows the response comparisons of z¨c and ϕ¨, Δyf and Δyr, and Ftf and Ftr for the three types of ASSs under the random road disturbance, respectively. It is observed that z¨c of the HFTC-with fault is significantly improved as compared with the NSFC-without fault, while ϕ¨ stays the same in case of these three ASSs. Additionally, the safety constraint performance indicators of Δyf and Δyr and Ftf and Ftr for the ASS become deteriorated in case of the NSFC-without fault. However, these four performance indicators can remain stable for the faulty ASS with the proposed HFTC.Figure 7
The response comparisons of ASS with the sinusoidal fault signal under the random road disturbance.
(a)
(b)
(c)
(d)
(e)
(f)Figure8 shows the variations of control forces at the front and rear actuator, which illustrates that our proposed HFTC can produce the desirable control forces to compensate for the performance penalties of the ASS in the presence of the sinusoidal actuator fault.Figure 8
The control forces of (a) the front actuator and (b) rear actuator under the random road disturbance.
(a)
(b)Additionally, in order to make a comprehensive performance comparisons of the faulty ASS with these three control schemes, Table2 summarizes the root-mean-square (RMS) values of z¨c, ϕ¨, Δyf, Δyr, Ftf, and Ftr under the sinusoidal fault signal. It is concluded from Table 2 that compared to the NSFC-without fault, the performance indicators of z¨c, ϕ¨, Δyf and Δyr, and Ftf and Ftr are, respectively, increased about 61.8%, 3.20%, 26.6%, 69.2%, 16.7%, and 63.3% in case of the NSFC-with fault. However, all the performance indicators for the ASS with the proposed HFTC can remain basically unchanged as compared with the NSFC-with fault.Table 2
The comparisons of RMS values under the random road disturbance.
NSFC-without fault
NSFC-with fault
HFTC-with fault
z
¨
c/(m/s2)
0.1293 (—)
0.2577 (↑61.8%)
0.1293
ϕ
¨/(rad/s2)
0.1853 (—)
0.1913 (↑3.20%)
0.1853
Δyf/m
0.0075 (—)
0.0950 (↑26.6%)
0.0075
Δyr/m
0.0052 (—)
0.0880 (↑69.2%)
0.0052
F
tf
/N
202.1314 (—)
237.7155 (↑16.7%)
203.8815
F
tr
/N
150.4045 (—)
245.5683 (↑63.3%)
150.1407


### 5.3. Comparative Investigation

To demonstrate the differences between the proposed HFTC and the most related FTC approach in [32], the comparison investigation is performed. Herein, the bump road is used as the road disturbance, which is expressed by [7](45)zr=Am21−cos2πuLt,0≤t≤Lu,0,t>Lu,where Am, L, and u represent the height and length of bump road and vehicle forward speed, respectively. Note that their corresponding values are extracted from [7] as Am = 100 (mm), L = 5 m, and u = 45 (km/h).In [31], a proportional-integral observer (PIO) was used to estimate the actuator fault in the active suspension system, which is different from our proposed robust H∞ observer (RHO). To reveal the differences for the PIO and RHO, the fault estimation and the corresponding fault estimation errors are presented in Figure 9. It can be obtained from Figure 9 that our designed RHO has higher estimation accuracy as compared with the PIO in [31]. However, only the design approach of the PIO is included in [31] for the suspension system, while the fault-tolerant controller design is not included.Figure 9
The comparison of (a) fault estimation and (b) fault estimation error.
(a)
(b)Moreover, in [32], both the PIO and the corresponding fault-tolerant controller were addressed, so it is very suitable to conduct the comparative studies on the proposed HFTC and the fault-tolerant controller (FTC) in [32] for the ASS. The response comparisons of the simulation results are shown in Figure 10. It can be observed that z¨c of the HFTC-with fault is improved in comparison with the FTC in [32]. However, the pitch angular acceleration ϕ¨ stays nearly the same for the proposed HFTC and the FTC in [32].Figure 10
The response comparisons of the ASS with random fault signal under bump road disturbance.
(a)
(b)
(c)
(d)
(e)
(f)Besides, we can conclude that the tire dynamic loads ofFtf and Ftr are reduced by using the proposed HFTC approach, which can prevent suspension breakdown and simultaneously improve the tire life span and vehicle handling stability when encountering a certain actuator fault. Overall, the proposed HFTC has better control performances.


## 5.1. Performance Analysis under the General Actuator Fault

The random road disturbance is assumed as a vibration signal that is consistent and typically specified as a white noise expressed by [38](44)z˙r=−2πf0zr+2πn0ωGqn0v,where Gq(n0) is selected as 1024 × 10−6m3, v = 20 m/s, the lower cutoff frequency of road profile f0 = 0.1, the reference spatial frequency n0 = 0.1 (1/m), and ω (t) is zero mean white Gaussian noise with identity power spectral density with the sample time of 0.01s.The fault signal and its corresponding fault estimation are revealed in Figure3(a), and the fault estimation error is displayed in Figure 3(b), respectively. Note that the estimation error of the actuator fault is calculated by Δf=f−f^.Figure 3
The variations of (a) fault estimation and (b) fault estimation error for the general fault signal.
(a)
(b)It is observed from Figure3 that the estimation error of the actuator fault has a peak value less than ±0.05, implying that the designed H∞ observer can accurately estimate the actuator fault. Besides, Figure 4 is presented to show the performance comparisons of z¨c, ϕ¨, Δyf and Δyr, and Ftf and Ftr for the three types of ASSs in time domain.Figure 4
The response comparisons of ASS with the general fault signal under random road disturbance.
(a)
(b)
(c)
(d)
(e)
(f)Additionally, it can be concluded from Figure4(a) that the response of z¨c becomes obviously deteriorated in case of the NSFC-without fault, yet z¨c of the HFTC-with fault can yield better ride comfort performance in comparisons with the NSFC-with fault, while the responses of ϕ¨ in Figure 4(b) remain basically unchanged for these three ASSs. Moreover, it can be inferred from Figures 4(c) to 4(f) that Δyf and Δyr and Ftf and Ftr are effectively reduced for the faulty ASS in case of the proposed HFTC-with fault, which can prevent suspension breakdown when suspension deflection is out of its limitation and can further improve tire life span and vehicle handling stability simultaneously.Moreover, Figure5 plots the control forces of the front and rear actuators with the designed HFTC. It is seen from Figure 5 that the control forces vary with the change of fault signals on time, which illustrates that the proposed HFTC can generate the compensative control forces to reduce the negative effects of the actuator fault on the closed-loop system.Figure 5
The control forces of (a) the front actuator and (b) rear actuator under the random road disturbance.
(a)
(b)


## 5.2. Performance Analysis under the Sinusoidal Actuator Fault

To further evaluate the proposed HFTC, the sinusoidal signal is used to mimic the actuator faults and validate the effectiveness of the proposed HFTC. Figure6 shows the variations of the front actuator fault and the corresponding estimation as well as the fault estimation error. It is easily found that the designed robust H∞ observer can accurately estimate the actuator fault with the fault estimation error of −0.02 N to 0.02 N.Figure 6
The variations of (a) fault estimation and (b) fault estimation error for the sinusoidal fault signal.
(a)
(b)Figure7 shows the response comparisons of z¨c and ϕ¨, Δyf and Δyr, and Ftf and Ftr for the three types of ASSs under the random road disturbance, respectively. It is observed that z¨c of the HFTC-with fault is significantly improved as compared with the NSFC-without fault, while ϕ¨ stays the same in case of these three ASSs. Additionally, the safety constraint performance indicators of Δyf and Δyr and Ftf and Ftr for the ASS become deteriorated in case of the NSFC-without fault. However, these four performance indicators can remain stable for the faulty ASS with the proposed HFTC.Figure 7
The response comparisons of ASS with the sinusoidal fault signal under the random road disturbance.
(a)
(b)
(c)
(d)
(e)
(f)Figure8 shows the variations of control forces at the front and rear actuator, which illustrates that our proposed HFTC can produce the desirable control forces to compensate for the performance penalties of the ASS in the presence of the sinusoidal actuator fault.Figure 8
The control forces of (a) the front actuator and (b) rear actuator under the random road disturbance.
(a)
(b)Additionally, in order to make a comprehensive performance comparisons of the faulty ASS with these three control schemes, Table2 summarizes the root-mean-square (RMS) values of z¨c, ϕ¨, Δyf, Δyr, Ftf, and Ftr under the sinusoidal fault signal. It is concluded from Table 2 that compared to the NSFC-without fault, the performance indicators of z¨c, ϕ¨, Δyf and Δyr, and Ftf and Ftr are, respectively, increased about 61.8%, 3.20%, 26.6%, 69.2%, 16.7%, and 63.3% in case of the NSFC-with fault. However, all the performance indicators for the ASS with the proposed HFTC can remain basically unchanged as compared with the NSFC-with fault.Table 2
The comparisons of RMS values under the random road disturbance.
NSFC-without fault
NSFC-with fault
HFTC-with fault
z
¨
c/(m/s2)
0.1293 (—)
0.2577 (↑61.8%)
0.1293
ϕ
¨/(rad/s2)
0.1853 (—)
0.1913 (↑3.20%)
0.1853
Δyf/m
0.0075 (—)
0.0950 (↑26.6%)
0.0075
Δyr/m
0.0052 (—)
0.0880 (↑69.2%)
0.0052
F
tf
/N
202.1314 (—)
237.7155 (↑16.7%)
203.8815
F
tr
/N
150.4045 (—)
245.5683 (↑63.3%)
150.1407


## 5.3. Comparative Investigation

To demonstrate the differences between the proposed HFTC and the most related FTC approach in [32], the comparison investigation is performed. Herein, the bump road is used as the road disturbance, which is expressed by [7](45)zr=Am21−cos2πuLt,0≤t≤Lu,0,t>Lu,where Am, L, and u represent the height and length of bump road and vehicle forward speed, respectively. Note that their corresponding values are extracted from [7] as Am = 100 (mm), L = 5 m, and u = 45 (km/h).In [31], a proportional-integral observer (PIO) was used to estimate the actuator fault in the active suspension system, which is different from our proposed robust H∞ observer (RHO). To reveal the differences for the PIO and RHO, the fault estimation and the corresponding fault estimation errors are presented in Figure 9. It can be obtained from Figure 9 that our designed RHO has higher estimation accuracy as compared with the PIO in [31]. However, only the design approach of the PIO is included in [31] for the suspension system, while the fault-tolerant controller design is not included.Figure 9
The comparison of (a) fault estimation and (b) fault estimation error.
(a)
(b)Moreover, in [32], both the PIO and the corresponding fault-tolerant controller were addressed, so it is very suitable to conduct the comparative studies on the proposed HFTC and the fault-tolerant controller (FTC) in [32] for the ASS. The response comparisons of the simulation results are shown in Figure 10. It can be observed that z¨c of the HFTC-with fault is improved in comparison with the FTC in [32]. However, the pitch angular acceleration ϕ¨ stays nearly the same for the proposed HFTC and the FTC in [32].Figure 10
The response comparisons of the ASS with random fault signal under bump road disturbance.
(a)
(b)
(c)
(d)
(e)
(f)Besides, we can conclude that the tire dynamic loads ofFtf and Ftr are reduced by using the proposed HFTC approach, which can prevent suspension breakdown and simultaneously improve the tire life span and vehicle handling stability when encountering a certain actuator fault. Overall, the proposed HFTC has better control performances.


## 6. Conclusions

This paper has presented a hybrid fault-tolerant controller design for a class of nonlinear ASSs in the presence of the actuator faults and road disturbances. With the help of system augmentation technique, we established the augmented closed-loop system model of faulty ASS. Based on this model, we further proposed the hybrid fault-tolerant controller consisting of a nominal state-feedback controller and a robustH∞ observer, which can not only achieve the asymptotic stability of this ASS under fault-free condition but also reduce the negative effects resulting from the unknown actuator faults and road disturbances under the fault condition. In addition, the designed hybrid fault-tolerant controller has been validated to be effective and feasible by using the more convincing numerical simulation results. Future work will focus on the controller design and verification for a repetitive control system [39, 40] with multimode actuator faults; meanwhile, the actuator input delay and saturation constraint will be considered in the corresponding fault-tolerant controller design.



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*Source: 1874212-2020-01-22.xml*