Output feedback H∞ control for active suspension of in-wheel motor driven electric vehicle with control faults and input delay

ISA Transactions 92 (2019) 94–108

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research article

Output feedback H∞ control for active suspension of in-wheel motor driven electric vehicle with control faults and input delay ∗

Xinxin Shao a , , Fazel Naghdy a , Haiping Du a , Hongyi Li b a b

School of Electrical, Computer & Telecommunication Engineering, University of Wollongong, Northfields Avenue, Wollongong, NSW 2522, Australia College of Information Science and Technology, Bohai University, Liaoning, China

highlights • • • •

A dynamic damping in-wheel motor driven system is employed. The motor and vehicle suspension parameters are optimized based on PSO method. A robust output feedback H∞ controller is designed for active suspension system. Actuator faults and time delay are dealt with via the output feedback controller.

article

info

Article history: Received 9 March 2018 Received in revised form 23 January 2019 Accepted 15 February 2019 Available online 25 February 2019 Keywords: Output feedback H∞ control Active suspension Electric vehicle with in-wheel motor Control faults Time delay

a b s t r a c t In this paper, an output feedback H∞ controller is proposed for active suspension of an electric vehicle driven by in-wheel motors with actuator faults and time delay. The dynamic damping in-wheel motor driven system, in which the in-wheel motor is designed as a dynamic vibration absorber (DVA), is developed to improve ride quality and isolate the force transmitted to motor bearings. Furthermore, parameters of vehicle suspension and DVA are optimized based on the particle swarm optimization (PSO) to achieve better suspension performance. As some of the states such as the DVA velocity and unsprung mass velocity are difficult to measure, a robust H∞ output feedback controller is developed to deal with the problem of active suspension control with actuator faults and time delay. The proposed controller could guarantee the system’s asymptotic stability and H∞ performance, simultaneously satisfying the performance constraints such as road holding, suspension stroke, and actuator limitation. Finally, the effectiveness of the proposed output feedback controllers is demonstrated based on the quarter vehicle suspension model under bump and random road excitations. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction Governments around the world are now convinced of the requirements to reduce air pollution and CO2 emission. Electric vehicles have been attracting a great deal of attention in recent years as they require no petrol and produce no emission. Propulsion configuration of electric vehicles can be classified as centralized motor driven layout or in-wheel motor driven layout depending on the vehicle’s architecture. The configuration, in which the motors are installed in the wheels (referred to as InWheel Motors (IWM)), has proved to be a popular research area lately. The in-wheel motor driven layout has many benefits such as high motor response, precise torque generation, simplicity and efficiency [1,2]. Moreover, in a four-IWM drive electric vehicle, the torque of each IWM can be independently and precisely controlled to achieve an improved performance of the vehicle ∗ Corresponding author. E-mail address: [email protected] (X. Shao). https://doi.org/10.1016/j.isatra.2019.02.016 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.

handling and stability [3]. However, installing the motors in the wheels can result in an increase in the unsprung mass that leads to an increase in the response of the frequency ranges around 10 Hz, which greatly deteriorates the suspension ride comfort performance and road holding ability. Furthermore, motor bearing wear is another problem that should be addressed [4]. The heavy load applied on the motor could easily result in loud noise and bearing wear and reduce the life of the motor bearing. The dynamic force acted on motor bearing can be utilized to enforce the weight constraint, which should also be reduced to decrease the IWM bearing wear [5]. The Bridgestone Company developed the so-called dynamic damping in-wheel motor driven system, in which the in-wheel motor is served as a DVA. The developed system demonstrated to have the potential to improve ride quality and road-holding performance [6]. Moreover, this kind of in-wheel motor configuration decreases the sprung mass acceleration in the range of the resonance of unsprung mass. Since the ride comfort and passenger safety are increasingly becoming critical criteria in vehicle suspension design, an active suspension

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

system will be necessary for the successful implementation of IWM driven electric vehicle. There are more studies reported in the literatures on active control of electrohydraulic suspension system and electromagnetic suspension system for improving the ride quality and vehicle handling [7–10]. Numerous control algorithms such as fuzzy control [9,11], robust H∞ control and sliding control method [10, 12] have been developed for active suspensions to improve vehicle ride comfort and handling stability. Among these control methods, the robust H∞ control strategy can deal with system complexities such as model uncertainties, external disturbance, actuator time delay and faults [13–15], which are widely utilized in vehicle suspension system. For example, unknown faults such as sensor and actuator failures may occur in the system, which could deteriorate the dynamic behaviour of the suspension. Fault detection and fault tolerant control methods were utilized for active suspension systems to detect the fault and ensure a better performance of the controlled suspension system [16,17]. A reliable fuzzy H∞ controller was designed for active suspension systems with actuator delay and faults based on Takagi–Sugeno (T–S) fuzzy model [18]. An active fuzzy fault tolerant tracking control method was proposed for a class of multi-input multi-output (MIMO) unknown nonlinear systems in the presence of unknown actuator faults, sensor failures and external disturbance [19]. An adaptive reliable H∞ optimization control method was proposed for linear time-delayed systems with time-varying actuator faults in [20]. A circuit implementation based distribute adaptive control strategy was proposed in [21] for a class of nonlinear second-order multi-agent systems against actuator faults and time-varying state/input-dependent system uncertainties and high-order nonlinear dynamics. In [22], an active fault-tolerant control method including error detection, fault diagnosis and system recovery was presented for an octorotor Unmanned Aerial Vehicle (UAV), and a pseudo-inverse control allocation approach was applied for system recovery when motor failure or rotor losses occurred. In [23], a fault tolerant compensation controller based on auxiliary system designs was proposed to solve the problem of fault tolerant tracking control for a linear time-invariant system subject to actuator faults and amplitude saturation. Moreover, a T–S fuzzy approach was applied to effectively handle the uncertainties and representing complex nonlinear systems [24]. Linear-parameter-varying technique [25–27] attracted an increasing research interest because of its effectiveness in describing nonlinearities and uncertainties. Actuator delay which occur in many real control system applications because of pneumatic and hydraulic characteristics of the actuators, may degrade the control performance and cause instability in the resulting control systems if it is not taken into account in the controller design. Various controller design schemes were proposed for linear systems with different types of delay [28–30]. A new bilinear matrix inequality (BMI) in the presence of a delay-dependent robust H∞ /L2 − L∞ output feedback controller for uncertain linear system with input time delay was proposed in [29]. External disturbance such as system friction, system uncertainties and external interference have negative effect on control system performance and stability. Control methods based on the disturbance observer could be utilized to estimate and compensate the influence of external disturbance [31–33]. However, when some of the required state variables of the suspension systems are not measurable, the aforementioned methods are not feasible, and the output-feedback based control schemes are developed to address this gap. For example, a dynamic output-feedback delay-dependent multi-object H∞ controller was proposed for active suspension system with control delay [34]. An output feedback active suspension control scheme using a recursive derivative non-singular higher order

95

terminal sliding mode approach was proposed for nonlinear suspension systems in [35]. A robust dynamic output-feedback H∞ controller was designed for a full-car active suspension with finite-frequency constraints and actuator faults in [36]. A (Q, S, R)-α -dissipative output-feedback fuzzy controller was proposed for T–S fuzzy systems with time-varying input delay and output constraints in [37]. However, the active suspension control studies reviewed so far were primarily focused on conventional vehicles and were not applied to active control of in-wheel motor electric vehicle suspension systems. Existing research investigations mainly focused on the handling stability and safety study of electric vehicle driven by in-wheel motors [38–40]. Active suspension control of IWM-EV should be deployed to improve vehicle ride quality. In-wheel vibration absorber was used in in-wheel-motor electric vehicles, and various control strategies were proposed to improve vehicle ride comfort in [4,41]. A filtered-X least mean square controller was proposed for active suspension to suppress the vibration caused by in-wheel SRM vertical force in [42]. Wang et al. [4] proposed a finite-frequency state feedback H∞ controller for active suspension of IWM-EVs. A Fault-tolerant fuzzy H∞ control design approach was proposed for active suspension of in-wheel drive electric vehicles in the presence of actuator faults in [5,43]. Among [5,43], dynamic force transmitted to the inwheel motor and motor acceleration were taken as additional controlled outputs besides the traditional optimization objectives such as sprung mass acceleration, suspension deflection and tire deflection. In order to deal with negative interaction effects between the vehicle suspension and in-wheel absorber, linear quadratic regulator (LQR) algorithm and fuzzy proportional– integral–derivative (PID) methods were developed to control the suspension damper and the in-wheel damper, respectively [41]. The parameters of in-wheel motor might not be optimal when designed by experience. Optimization these parameters could also improve ride performance of an electric vehicle driven by in-wheel motors. Sequential Quadratic Programming (SQP) optimization technique was used in [4] to optimize the parameters of dynamic vibration absorber. In general, there were several conflicting control objectives that should be optimized in active suspension control. Pareto optimal solutions could be obtained in solving multi-objective optimization problems. A multi-objective parameters optimization based on GA was proposed for active suspension of an electric vehicle driven by IWM in [44]. To the best knowledge of authors, there were no studies that analysed the dynamic output feedback H∞ control methodologies for active suspension control of in-wheel motor driven electric vehicle subject to actuator faults and delay. In this paper, the output feedback control problem of active suspension system of in-wheel motor driven electric vehicles with actuator faults and time delay is investigated. The contributions of this study can be outlined as follows: (1) In order to achieve a better vibration isolation performance, a dynamic damping in-wheel motor driven system in which the in-wheel motor serves as a DVA is employed in this paper. (2) Unlike in Ref. [4,44], the suspended motor parameters, vehicle suspension parameters are optimized based on the PSO. (3) Unlike in Ref. [5, 43], where the authors presented fault-tolerant state feedback H∞ control for active suspension system taking into account only actuator faults. In this paper, a robust output feedback H∞ controller is designed to guarantee the system’s asymptotic stability and H∞ performance in the presence of actuator faults and time delays; simultaneously satisfying the performance constraints. (4) Unlike in Ref. [18,34,45], where the authors proposed active suspension controller for conventional vehicle. In this work, a multi-objective control for an electric vehicle active suspension is developed with performance requirements such as road holding,

96

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

suspension stroke, the dynamic load applied on the bearings and actuator limitation; all are considered in the controller design. The remainder of this paper is organized as follows. In Section 2, the electric vehicle suspension model with a DVA is developed. In Section 3, particle swarm optimization is used to optimize the vehicle suspension and DVA parameters. In Section 4, dynamic output feedback controllers considering actuator faults and time delay are designed. The developed algorithms are validated in Section 5 and simulation results are provided. Finally, conclusions are drawn in Section 6 and future work is discussed. Notation. A−1 and AT refer to the inverse and transposition of matrix A, respectively. A > 0(< 0) means A is positive (negative) matrix. The symbol ∗ is used to stand for the symmetric part. θmax (·) represents maximal eigenvalue. RMS(·) represents root mean square function. diag {·} denotes diagonal matrix. ∥·∥2 denotes the L2 norm of vector. ∥·∥∞ denotes the H∞ norm of transfer function. Rn×n represents n × n dimensional matrix space. 2. System modelling Increased unsprung mass and motor bearing wear are two critical problems in the electric vehicle driven by in-wheel motors. In order to reduce the motor vibration, the force transmitted to the motor bearing should be isolated. In order to model the system, a quarter car active suspension with a DVA attached to an unsprung mass through a spring and a damper is developed with the in-wheel motor serving as the DVA (shown in Fig. 1). Based on Newton’s Second Law, the motion equations of this active suspension can be written as

Fig. 1. Quarter car suspension model with in-wheel DVA.

(3) Dynamic force. The maximum dynamic force applied to the in-wheel motor should not exceed the allowable maximum force. (4) Actuator saturation. To reduce the power consumption of the actuator and maintain the stability of the controlled system, the active control force provided by the active suspension system should be constrained by a threshold. Based on the performance requirements, the control outputs can be defined by the below equation and Eq. (5), given in Box I. z1 (t) = x¨ s

(4)

By defining the vehicle state vector as given in Box II, the active suspension system can be described by the following state-space equations x˙ (t) = Ax(t) + B1 w (t) + B2 u(t − d(t))

(6)

ms x¨ s (t) = ks (xu (t) − xs (t)) + cs (x˙ u (t) − x˙ s (t)) + fa (t − d(t))

(1)

z1 (t) = C1 x(t) + D1 u(t − d(t))

(7)

md x¨ d (t) = kd (xu (t) − xd (t)) + cd (x˙ u (t) − x˙ d (t))

(2)

z2 (t) = C2 x(t) + D2 u(t − d(t))

(8)

where

mu x¨ u (t) = kt (xg (t) − xu (t)) − ks (xu (t) − xs (t)) − cs (x˙ u (t) − x˙ s (t)) −kd (xu (t) − xd (t)) −cd (x˙ u (t) − x˙ d (t)) − fa (t − d(t))

⎡

(3) where xs , xu and xd denote the vertical displacements of the sprung mass, unsprung mass, and motor mass. Road disturbance is denoted by xg . The sprung mass, unsprung mass and motor mass are denoted by ms , mu and md , respectively. The suspension stiffness and damping coefficients are denoted by ks and cs , respectively. The motor stiffness and damping coefficients are denoted by kd and cd , respectively, and tyre stiffness is kt . fa (t − d(t)) denotes the actuator force considering the time-varying delay. d(t) is a time-varying continuous function and satisfies ˙ ≤ µ. The definition and value of each 0 < d(t) ≤ d and d(t) parameter is presented in Table 1. Generally speaking, suspension performance requirements include ride comfort, suspension defection, and road holding stability. The sprung mass acceleration in the vertical direction is minimized to obtain better ride performance. The following conditions are hard constraints that should be strictly satisfied: (1) Suspension deflection. The suspension deflection should not exceed its travel limit determined by the mechanical structure. (2) Road-holding stability. To ensure a firm uninterrupted contact of wheels with the road, the dynamic tyre load should not exceed the static one.

[

z2 (t) = xs − xu

kt (xu − xg )

0 ⎢ −ks ⎢

⎢ ⎢ ⎢ ⎢ A=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1 −cs

ms

ms

0

0

0

0

0

0

0 0

0

1

−kd

−cd

md

md

0

0

0 ks

0 cs

0 kd

0 cd

−kt

mu

mu

mu

mu

mu

⎤

−1 cs

0 0

−

⎥ ⎥ ⎥ ms ⎥ ⎥ −1 ⎥ ⎥, cd ⎥ ⎥ md ⎥ ⎥ ⎥ 1 cs + cd ⎦ mu

0 0 ⎢0⎥ ⎢ m1s ⎥ ⎢ ⎥ ⎢ ⎥ ⎢0⎥ ⎢ 0 ⎥ B1 = ⎢ ⎥ , B2 = ⎢ ⎥, ⎢0⎥ ⎢ 0 ⎥ ⎣−1⎦ ⎣− 1 ⎦ mu 0 0

⎡

[ C1 =

⎤

−ks

−cs

ms

ms

1 0 ks

[ C2 =

D2 = [0

0 0 cs 0

⎤

⎡

0 0 kd

0

0

0 0 cd

0 kt 0

0

cs ms 0 0

]

, D1 =

[

1 ms

]

,

] ,

−(cs + cd )

1]T , w (t) = xg , u(t − d(t)) = fa (t − d(t)).

ks (xs − xu ) + cs (x˙ s − x˙ u ) + kd (xd − xu ) + cd (x˙ d − x˙ u ) + fa

Box I.

0

]

(5)

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

[

x(t) = xs (t) − xu (t)

x˙ s (t)

x˙ d (t)

xd (t) − xu (t)

xu (t) − xg (t)

x˙ u (t)

97

]

Box II.

In practice, the state vector of a vehicle suspension system cannot be measured directly, so it is difficult to control the system by applying the state feedback H∞ controller. The output feedback H∞ controller can be defined by the following state space equations: x˙ˆ (t) = Ak xˆ (t) + Akd xˆ (t − d(t)) + Bk y(t)

(9)

ud (t) = Ck xˆ (t)

(10) n

where xˆ ∈ R is the state vector of the controller, and Ak , Akd , Bk , Ck are controller parameter matrices to be determined. The suspension deflection, motor deflection and tyre deflection can be measured by using the laser displacement sensors. However, the motor velocity, sprung mass velocity and unsprung mass velocity are difficult to measure. Therefore, y(t) is the measured output and can be written as following equation: y(t) = Cy x(t)

(11)

[

]

where Cy = diag 0 1 1 0 1 0 . Unknown faults in actuator failures can deteriorate the dynamic behaviour of the suspension. Considering the actuator faults, the real control force can be modelled as u(t) = λud (t) = (λm + N0 λ)ud (t)

λm =

λmin + λmax

λmax − λmin

N0 N0T

˙ = Acl x(t) + Bcl x(t − d(t)) + Bcl1 w(t) x(t)

(13)

z 2 (t) = Ccl2 x(t) + Dcl2 x(t − d(t))

(14)

z 1 (t) = Ccl1 x(t) + Dcl1 x(t − d(t))

(15)

u(t) = Cu x(t) where x = x Ccl2 = C 2 ,

(16)

]T

xˆ , Acl = A, Bcl1 = B1 , Bcl = B + HN0 E, Ccl1 = C 1 ,

Dcl1 = D1 + D1 N0 E , Dcl2 = D2 + D2 N0 E , Cu = 0

[

with

[ A=

A Bk Cy

[ ]

]

[ ]

[

0 B 0 ,B = 1 ,B = Ak 0 0

[ B2 H= ,E = 0 0

λCk ,

]

0 , D1 = 0

[

D2 = 0

]

[

D1 λm Ck , C 2 = C2

]

[

0 ,

]

D 2 λ m Ck .

]

The output feedback H∞ control problem is to propose a controller such that: (1) the close-loop system is asymptotically stable; (2) the H∞ performance [ ∥z 1 (t) ) ∥2 ≤ γ ∥w(t)∥2 are guaranteed for all nonzero w ∈ L2 0 ∞ in the presence of actuator faults and delay; (3) the active control force should be constrained by the maximum actuator force, and the actuator saturation nonlinearity is described by umax u(t) −umax

u(t) ≥ umax umax ≥ u(t) ≥ −umax u(t) ≤ −umax

{ u(t) =

(17)

where umax is the maximum actuator control force. (4) the following control output constraints are guaranteed:

⏐ ⏐ ⏐{z 2 (t)}q ⏐ < {z 2 max }q q = 1, 2, 3. [ where z 2 max = zmax (ms + mu + md )g

(18) Fmax

]T

.

3. Parameter optimization of suspension and DVA

(12)

,λ = , ≤ I. 2 2 where ud (t) is the desired actuator force. λ is an unknown parameter due to the actuator faults. Assuming that λ is bounded by its minimum value λmin and its maximum value λmax . N0 is an unknown parameter. The fault model proposed in this paper can effectively describe different types of actuator faults. However, the sensor faults and other components faults cannot be represented by the fault model. The fault model covers three different actuator conditions such as out of order, loss of effectiveness and fault-free cases, in which are considered as follows. (1) λmax = λmin = 0, then λ = 0, which implies that the corresponding actuator has completely failed. (2) 0 < λmin < λmax < 1, which means that there exists partial fault in the corresponding actuator. (3) λmax = λmin = 1, then λ = 1, which represents the case of no fault in the actuator. Substituting (9)–(12) into (6)–(8), the following closed-loop system can be obtained

[

[

C 1 = C1

B2 λm Ck , Akd

]

] λ Ck .

To achieve better vibration isolation performance, the suspended motor parameters, vehicle suspension parameters are optimized based on the PSO method. PSO is a population-based search algorithm based on the simulation of the social behaviour of birds within a flock. The earliest attempt to use the concept for social behaviour simulation carried out by Kennedy and Eberhart [46]. The PSO technique can generate a high-quality solution with shorter calculation time and more stable convergence characteristic than other stochastic methods [47]. PSO is initialized with a group of random particles and then searches for optima by updating iterations [48]. In every iteration, each particle is updated by the best value so far (Xpbest ) and best value in the group (Xgbest ). The particle tries to modify its position using the current velocity and the distance from Xpbest and Xgbest . Let Xdk denote the position of the particle. The position of Xdk is changed by adding a velocity vdk to it Xdk+1 = Xdk + vdk+1

(19)

The velocity of the particle is defined as follows

vdk+1 = w1 vdk+1 + c1 r1 (Xpbest − Xdk ) + c2 r2 (Xpbest − Xdk )

(20)

where and v are the dth particle’s position and velocity vector, respectively. c1 and c2 are two parameters representing the particle’s confidence in itself (cognition) and in the swarm (social behaviour), respectively. A relatively high value of c1 will encourage the particles to move towards their local best experiences, while higher values of c2 will result in faster convergence to the global best position. Xpbest is the personal best position of one particle, while Xgbest is the position of the best particle of the entire swarm. r1 , r2 are random numbers uniformly distributed in the range (0,1). The inertia weight w1 is given by the following equation Xdk

w1 = win −

k d

win − wfn

k (21) iter where win , wfn are initial and final weights, iter denotes the total number of iterations. k is the current iteration number.

98

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

In this paper, PSO technique is used to search optimal suspension parameters such as motor mass md , motor stiffness kd , motor damping cd , suspension stiffness ks and suspension damping cs . The optimization objective is to suppress the sprung mass vibration, motor vibration as well as improving ride comfort, suspension deflection and road holding stability. The fitness function designed to satisfy the optimization objective can be written as follows fobj (x) = p1 RMS(

x¨ s x¨ s max

+ p3 RMS(

) + p2 RMS(

xs − xu zmax

kt (xu − xg ) (ms + mu + md )g

)

) + p4 RMS(

Fd Fd max

)

(22)

The procedure of particle swarm optimization for vehicle suspension and DVA is listed as follows [48]: (1) Initialization of algorithmic parameters. (2) Evaluate the desired fitness function of each particle. (3) Update Xpbest and Xgbest . Determine the current best value Xpbest and best value of all the particle Xgbest . (4) Update the velocity and position of each particle. (5) Terminate: The algorithm checks for stopping criteria, if the stopping criteria is satisfied, then the algorithm terminates and return the optimal [solution. ] Find vector Xgbest = md kd cd ks c s To minimize fobj (x) = p1 RMS(

x¨ s x¨ s max

+ p3 RMS(

) + p2 RMS(

xs − xu zmax

kt (xu − xg ) (ms + mu + md )g

⎧ [ md ∈ 20 ⎪ ⎪ [ ⎪ ⎪ ⎪ ⎨kd ∈ 30000 [ Subject to cd ∈ 1000 [ ⎪ ⎪ ⎪ ks ∈ 30000 ⎪ ⎪ [ ⎩ cs ∈ 1000

)

) + p4 RMS(

Fd Fd max

)

(23)

]

50

]

50000

]

where p1 , p2 , p3 , p4 are

3000

]

50000

Theorem 1. Consider the active suspension system in (13)–(16) with proposed dynamic output feedback H∞ controller in (9)–(10). For given positive [ ] scalar d,[γ , θR , and ] ρ , if there [ exist matrices ] Q11 Q12 S11 S12 R R12 Q = > 0, S = > 0, R = 11 > 0, ∗ Q22 ∗ S22 ∗ R22

[ Mi1 ˆ ˆ ˆ ˆ X > 0, Y > 0, A, Ad , B, C , M i =

]

Mi2 , Ni = Mi4

Mi3

[

Ni1 Ni3

4. Output feedback controller design

⎡ ⎤ √ Ψ 11 dM Ψ 13 Ψ 14 ε1 Ψ 15 Ψ 16 ⎢ ⎥ −R 0 0 0 0 ⎥ ⎢ ∗ ⎢ ∗ ⎥ ∗ −I 0 ε1 D1 0 ⎥<0 ⎢ ⎢ ∗ ⎥ ∗ ∗ −Ψ 44 ε1 Ψ 45 0 ⎥ ⎢ ⎣ ∗ ∗ ∗ ∗ −ε1 I 0 ⎦ ∗ ∗ ∗ ∗ ∗ −ε1 I ⎤ ⎡ √ Ψ 11 dN Ψ 13 Ψ 14 ε2 Ψ 15 Ψ 16 ⎥ ⎢ −R 0 0 0 0 ⎥ ⎢ ∗ ⎥ ⎢ ∗ ∗ −I 0 ε2 D1 0 ⎥<0 ⎢ ⎥ ⎢ ∗ ∗ ∗ −Ψ 44 ε2 Ψ 45 0 ⎥ ⎢ ⎣ ∗ ∗ ∗ ∗ −ε2 I 0 ⎦ ∗ ∗ ∗ ∗ ∗ −ε2 I ⎡ √ ˆT T ˆT T⎤ 2 2 ρ C λm C λ −µ X −µmax I ⎥ ⎢ ∗max −µ2max Y 0 0 ⎥<0 ⎢ ⎣ ∗ ∗ −I + ε2 I 0 ⎦ ∗ ∗ ∗ −ε2 I } ⎡ { 2 − z2 max X q ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ⎣ ∗ ∗

{ } − z22 max I { }q − z22 max Y

) √ ( T T ρ X C2 + Cˆ T λTm DT2 √ T ρ C2

∗ ∗ ∗

−I ∗ ∗

q

0 0

ε3 D2 −ε3 I ∗

(25)

(26)

(27)

Cˆ T λ

T⎤

⎥ ⎥ ⎥ ⎥<0 0 ⎥ ⎥ 0 ⎦

0 ⎥

−ε3 I

(28)

Lemma diagonal matrix Φ (t) = { 1 ([49]). For a time-varying } diag σ1 (t), σ2 (t), . . . , σp (t) and two matrices R and S with appropriate dimensions, if |Φ (t)| ≤ V , where V < 0 is a known diagonal matrix, then for any scalar ε > 0, we have RΦ S + S T Φ RT ≤ ε RVRT + ε −1 S T VS

[

X I

I Y

]

>0

(24)

(29)

⎡ θ 11 ⎢∗ =⎢ ⎣∗ ∗

where Ψ 11

Consider some of the state variables of the vehicle suspension system are not measurable, an output feedback controller for the active suspension of an electric vehicle with DVA structure in the presence of actuator faults and delay is developed. The proposed output feedback H∞ controller could guarantee the system’s asymptotic stability and H∞ performance in the presence of actuator faults and time delays; simultaneously satisfying the performance constraints. Furthermore, the controller formulates multi-objective control constraints such as road holding, suspension stroke, the dynamic load applied on the bearings and actuator limitation for active suspension system based on LMI technique. However, the proposed controller just focused on actuator faults, while sensor faults and other components faults are not considered. Moreover, the problem of fault estimation and external disturbance are not considered, which will be our considerations in the future work. The following lemma is needed to derive the main results.

]

(i = 1, 2, 3, 4) with appropriate dimensions and any positive scalars ε1 , ε2 , ε3 such that the following LMIs hold:

]

3000 weighting factors for the four performance indexes. The optimal value of the motor parameters and suspension parameters can be obtained.

Ni2 Ni4

θ 12 θ 22 ∗ ∗

θ 13 θ 23 θ 33 ∗

⎤ θ 14 θ 24 ⎥ ⎥, (See the remaining equaθ 34 ⎦ −γ 2 I

tions in Box III). Then, (1) the closed-loop   system is asymptotically stable; (2) the H∞ performance Tz 1 w ∞ < γ is minimized subject to the output constraints in (18) and maximum actuator force constraint in (17) with the disturbance energy under the bound wmax = ρ − V (0)/γ 2 . Proof. Considering the Lyapunov–Krasovskii functional as follows:

∫

T

t

∫

T

V (t) = x (t)Px(t) + t −d 0

∫

∫

t

T

x (s)Sx(s)ds t −d(t)

T

˙ x˙ (s)Rx(s)dsd α

+ −d

t

x (s)Q x(s)ds +

(30)

t +α

The derivative of V (t) along the solution of system (13)–(16) is expressed as

˙ + x (t)(Q + S)x(t) − x (t − d)Q x(t − d) V˙ (t) ≤ 2x (t)P x(t) −(1 − µ)xT (t − d(t))S T

T

T

˙ − x(t − d(t)) + dx˙ (t)Rx(t)

T

∫

t

∫

t −d(t)

T

˙ x˙ (s)Rx(s)ds

− t −d

T

˙ x˙ (s)Rx(s)ds t −d(t)

(31)

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

θ 11 = θ 12 = θ 22 = θ 13 = θ 23 = θ 33 = θ 14 = θ 24 = Mi =

[

T + Q11 + S11 AX + XAT + M11 + M11

[

B2 λm Cˆ + − M11 + N11 ˆ y + Aˆ d + M T − M13 + N13 Y T B2 λm DC 22

∗

Mi3 T

Mi4

Ni3

T

]

M3

C1

T √dXA

θ 17 =

dAT

[√

Ni4

[ ]T T T T T , N = N1 N2 N3 N4 , ]T [ ]T T T T 0 0 , Ψ 14 = θ 17 θ 27 0 θ 47 , [ ] ˆT T T = C λm D1 , T

M2

T

M4

]T

0

[√

Ψ 44

ˆ y + M T − M12 + N12 B2 λm DC 23 T ˆ y + M T − M14 + N14 , Y B2 λm DC 24

] T T T T −(1 − µ)S11 + N21 + N11 − M21 − M21 −(1 − µ)S12 + N23 + N12 − M23 − M22 , T T ∗ −(1 − µ)S22 + N24 + N14 − M24 − M24 [ T ] T M31 − N11 M33 − N12 , T T M32 − N13 M34 − N14 [ T ] T T T N31 − M31 − N21 N33 − M33 − N22 , T T T T N32 − M32 − N23 N34 − M34 − N24 [ ] T T −Q11 − N31 − N31 −Q12 − N33 − N32 , T ∗ −Q22 − N34 − N34 ] [ T B1 + M41 , T Y T B1 + M42 ] [ [ T] [ ] T T −M41 + N41 −N41 R11 R12 , θ = , R = , 34 T T T ∗ R22 −N42 + N42 −M42 ] ] [ [ Mi1 Mi2 N Ni2 , N i = i1 ,

[ T T Ψ 13 = θ 16 θ 26 [ T] XC1 , θ 26 θ 16 = T

Ψ 15

]

[

M = M1

Ψ 44

T A + Aˆ T + M12 + M13 + Q12 + S12 T T T T ˆ y + Cy Bˆ + M14 + M T + Q22 + S22 , YA + A Y + BC 14

T M21

[

θ 27 =

99

]

√

T dAˆ√ 1

√

dCˆ T λTm BT2 0

dAT Y +

√

,

dCyT Bˆ T

]

dAˆ d , θ = 47 0

[√

dBT1

√

]

dBT1 Y ,

] θR2 R11 − 2θR X θR2 R12 − 2θR I = , ∗ θR2 R22 − 2θR Y ] [ [ ]T T B = θ T18 0 0 0 , θ 18 = T 2 , Y B2 [ √ ] [ ] [ ]T T ˆT T dB2 = √ , Ψ 16 = 0 θ T29 0 0 , θ 29 = C λ . T 0 [

dY B2

Box III.

ˆ and N, ˆ the folFor any appropriately dimensioned matrices M lowing equalities hold directly according to the Newton–Leibniz formula

( ∫ ˆ × x(t) − x(t − d(t)) − 2ξ (t)M

t

) ˙ x(s)ds =0

T

(32)

t −d(t)

2ξ T (t)Nˆ ×

(

∫

t −d(t)

x(t − d(t)) − x(t − d) −

)

˙ x(s)ds

=0

(33)

t −d

where,

ξ T = [xT (t) xT (t − d(t)) xT (t − d)T w T (t)], ˆ = [M1T M2T M3T M4T ]T , Nˆ = [N1T N2T N3T M

N4T ]T

Adding the two equations in (32)–(33) into the right-hand side of (31) and after some simple calculations, the following

inequality is satisfied T 2 T w (t) + V˙ (t) [ z 1 (t ) z 1 (t) − γ w (t) ( ) −1 T ] T −1 ˆ T ˆ ˆ ≤ ξ (t) Ψ + d(t)MR M + d − d(t) NR Nˆ ξ (t) ∫ t ( ) )T ( ˆ + x˙ T (s)R R−1 ξ T (t)M ˆ + x˙ T (s)R ds ξ T (t)M − ∫ t −t −d(t) ) ( )T d(t) ( T ˆ + x˙ T (s)R ds − ξ T (t)Nˆ + x˙ (s)R R−1 ξ T (t)M t −d [ ( ) −1 T ] ˆ −1 M ˆ T + d − d(t) NR ˆ ≤ ξ T (t) Ψ + d(t)MR Nˆ ξ (t) [ ( ) ( )] d(t) d − d(t) ˆ −1 M ˆT + ˆ −1 Nˆ T ξ (t) = ξ T (t) d Ψ + dMR Ψ + dNR d

(34)

100

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

T T , + Φ14 R−1 Φ14 Ψ = Φ11 + Φ13 Φ13 ⎡ T T PAcl + PAcl + M1 + M1 + Q + S PBcl + M2T − M1 + N1 ⎢ ∗ −(1 − µ)S + N2T + N1 − M2T − M2 Φ11 = ⎢ ⎣ ∗ ∗ ∗ ∗ [√ ]T √ √ [ ]T Φ13 = Ccl1 Dcl1 0 0 , Φ14 = dRAcl dRBcl 0 dRBcl1 .

M3T − N1 M3T N3T

N3T

− −Q −

PBcl1 + M4T

− N2 − N3

−

M4T

+ −N4T −γ 2 I

∗

N4T

⎤ ⎥ ⎥, ⎦

Box IV.

where Ψ , Φ11 and Φ13 are as given in Box IV. According to inequalities (25)–(26) and Schur complement, the following inequalities can be obtained

θ13 = M3T − N1 , θ14 = PB1 + M4T ,

ˆ −1 M ˆT <0 Ψ + dMR ˆ −1 Nˆ T < 0 Ψ + dNR

(35)

θ23 = N3T − M3T − N2 , θ24 = −M4T + N4T , θ33 = −Q − N3T − N3 ,

(36)

θ34 = −N4T ,

According to inequalities (34)–(36), z 1 (t ) z 1 (t) −[ γ 2 w T (t) ) w(t) + V˙ (t) < 0 can be obtained for all nonzero w ∈ L2 0 ∞ . There∥z 1)(t)∥2 ≤ γ ∥w(t)∥2 are guaranteed fore, the H∞ performance [ for all nonzero w ∈ L2 0 ∞ in the presence of actuator faults and delay. In addition, when w (t) = 0, V˙ (t) < 0 is obtained, which means the system (13)–(16) is asymptotically stable for the actuator delay and faults. By evaluating

θ22 = −(1 − µ)S + N2T + N1 − M2T − M2 ,

T

Acl = A, Bcl1 = B1 , Bcl = B + HN0 E , Ccl1 = C 1 , Ccl2 = C 2 , Dcl1 = D1 + D1 N0 E , Cu = 0

[

λCk .

0

0

√

ˆ dM −R

∗ ∗ ∗ ∗ √

dNˆ −R ∗ ∗ ∗ ∗

Ψ14

Ψ15

Ψ16

0

0 0

0

0 0 0 0

ε√1 D1

−I ∗ ∗ ∗

−R ∗ ∗

ε1 dRH −ε1 I ∗

Ψ13

Ψ14

Ψ25

0 −I

0 0

0 ε√2 D1

∗ ∗ ∗

−R ∗ ∗

ε2 dRH −ε2 I ∗

⎤T ⎥ ⎥ ⎦

(37)

Ψ11

θ12 θ22 ∗ ∗

θ13 θ23 θ33 ∗

]

X P MT

θ11 = PA + PA + M1 +

0

[ , Ψ26 = 0

E

0

]T

0

0

]T

0

0

.

−ε1 I ⎤ Ψ26 0 ⎥ ⎥ 0 ⎥ ⎥<0 ⎥ 0 ⎥ 0 ⎦

F1 = (38)

[

]

M . Z

]

I I = 0 0

[

⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎦

[

Y . NT

By defining

⎤

]

X MT

[

]

I I , F2 = 0 0

Y . NT

then PF1 = F2 , and the following equations can be obtained:

[

F1T PAF1 = F2T AF1 =

F1T QF1

[

F1T RF1 =

AX Aˆ

]

A

ˆ y Y T A + BC

]

Q11

=

(39)

[

Q12 S , F1T SF1 = 11 Q22 ∗

∗

[

R11

]

[

]

[

, F1T PBF1 =

]

S12 , S22

]

R12 Mi1 , F1T Mi F1 = R22 Mi3

∗

B2 λm Cˆ Aˆ d

[

Mi2 , Mi4

−ε2 I F1T Ni F1

⎤ θ14 θ24 ⎥ , θ34 ⎦ 2 −γ I M1T

]T

0

[ , Ψ25 = ε2 H T P

N X , P −1 = W MT

[

F1T PF1 = F2T F1 =

+ Q + S , θ12 = PB +

M2T

− M1 + N1 ,

ˆT F1T E T = C λ 0

T

]

[

]

B B Ni2 , F1T PB1 = T 1 , F1T PH = T 2 , Ni4 Y B1 Y B2

N = i1 Ni3

[

T

0

where X , Y ∈ Rn×n are symmetric matrices. The following equation is achieved

[

where

⎡ θ11 ⎢∗ =⎣ ∗ ∗

]T

0

]

Y P = NT

dRH

Ψ13

E

[

According to inequalities (35)–(36), then the following LMIs hold:

⎡ Ψ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎣ ∗ ∗ ⎡ Ψ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎣ ∗ ∗

0

There are several nonlinear variables presented in the above equations that cannot be eliminated by the change in the variables. By using the variable substitution method proposed in [34]. Since the matrix P is nonsingular, we partition matrix P and its inverse as follows

The equation in (35) can be written as

dRH

[ Ψ15 = ε1 H T P [ Ψ16 = 0

]

⎡ ⎤ √ ˆ Ψ13 Ψ14 Ψ11 dM ⎢ ⎥ −R 0 0 ⎥ ˆ −1 M ˆT =⎢ ∗ Ψ + dMR ⎣ ∗ ∗ −I 0 ⎦ ∗ ∗ ∗ −R ⎡ ⎤ ⎡ ⎤T ⎡ ⎤ ⎡ Ψ15 Ψ15 Ψ16 Ψ16 ⎢ ⎢ 0 ⎥ ⎢ 0 ⎥ 0 0 ⎥ ⎢ T ⎢ ⎥ +⎢ ⎣√D1 ⎦ N0 ⎣ 0 ⎦ + ⎣ 0 ⎦ N0 ⎣√D1

[ ]T Ψ13 = C 1 D1 0 0 , ]T [√ √ √ Ψ14 = dRA dRB 0 dRB1 ,

]

[

X I

]

I , Y

[ , C 1 F1 = C1 X

C1 , D1 F1 = D1 λm Cˆ

]

[

0 .

]

]

0 , 0

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

We define the variables Aˆ = Y T AX + NBκ Cy X + NAκ M T

(40)

101

where Ak , Bk , Ck , λ, λm , and λ are defined in (9)–(10) and (12). Substituting (54)–(56) into (51)–(53), the closed-loop system is obtained

˙ = Acl x(t) + Bcl1 w(t) x(t) Aˆ d = Y B2 λm Cκ M + NAκ d M T

T

T

(41)

Bˆ = NBκ

(42)

Cˆ = Cκ M T

(43)

{

)T ( Γ1 = diag F1 , F1 , F1 , I , F1 , I , PR−1 F1 , I , I

}

Given matrices X , Y and M , N, we can determine Aκ , Aκ d , Bκ , Cκ from Aˆ , Aˆ d , Bˆ , Cˆ . Performing congruence transformation by multiplying full rank matrices Γ1T on the left and Γ1 on the right, respectively. The inequalities (38)–(39) are equivalent to the inequalities (25)–(26) in Theorem 1. In what follows, we will show that hard constraints in (27)–(28) are guaranteed. From the definition of the Lyapunov function in (30), we know that T x (t)Px(t) < ρ with V = γ 2 wmax + V (0). Similarly, the following inequalities hold:

⏐2

⏐

max ⏐z 2 (t)q ⏐ t >0

  ≤ max xT (t) {Ccl2 + Dcl2 }Tq {Ccl2 + Dcl2 }q x(t)2 q = 1, 2, 3. t >0 ( 1 ) −2 T − 12 < ρ · θmax P {Ccl2 + Dcl2 }q {Ccl2 + Dcl2 }q P (45)

  max |u(t)| ≤ max xT (t) {Cu }T {Cu } x(t)2 t >0 t >0 ( 1 ) 1 < ρ · θmax P − 2 {Cu }T {Cu } P − 2 2

(46)

where θmax (·) represents maximal eigenvalue. Consequently, the output feedback controller parameters are obtained

)−1

Cκ = Cˆ M T

(

Bκ = N Aκ d = N

(47)

−1 ˆ

B

−1

(48)

(

Aˆ d − Y T B2 λm Cκ M T

(

)(

Aκ = N −1 Aˆ − Y T AX − NBκ Cy X

M

)(

T −1

)

MT

)−1

(49)

(50)

Remark 1. Theorem 1 presents an output feedback controller for an active suspension system with control delay and faults. When there are only control faults in the quarter-car model, the fault-tolerant controller is presented for active suspension system of electric vehicle with DVA based on the output feedback H∞ control method. The active suspension system can be described by the following state-space equations: x˙ (t) = Ax(t) + B1 w (t) + B2 u(t)

(51)

z2 (t) = C2 x(t) + D2 u(t)

(52)

z1 (t) = C1 x(t) + D1 u(t)

(58)

z 1 (t) = Ccl1 x(t)

(59)

u(t) = Cu x(t)

where x = x xˆ , Acl = A + HN0 E, Bcl1 = B, Ccl1 = C 1 + D1 N0 E, Ccl2 = C 2 + D2 N0 E with

x˙ˆ (t) = Ak xˆ (t) + Bk y(t)

(55)

u(t) = λud (t) = λCk xˆ (t) = (λm + N0 λ)Ck xˆ (t)

(56)

[ ]

] [ λ Cκ , C 1 = C1

[

E= 0

]

[ ]

D 1 λ m Cκ ,

]

D 2 λ m Cκ , Cu = 0

[

C 2 = C2

] λCκ .

[

Employing a similar method to which is proposed in Theorem 1, the following Corollary 1 is obtained for the active suspension system with only actuator faults. Corollary 1. Given positive scalars γ and ρ , a dynamic output feedback controller in the form of (55)–(56) exists, such that the closed-loop system in (57)–(60) is asymptotically stable with w (t) = 0, and H∞ performance [ ∥z 1)(t)∥2 ≤ γ ∥w(t)∥2 are guaranteed for all nonzero w ∈ L2 0 ∞ , while the control output constraints in (17)–(18) are guaranteed with the disturbance energy under the bound wmax = ρ− V (0)/γ 2 , if there exist symmetric matrices X > 0, ˆ B, ˆ Cˆ with appropriate Y > 0, positive scalars ε1 , ε2 , ε3 , and A, dimensions satisfying the following LMIs: ⎤ ⎡ Λ11 Λ12 Λ13 Λ14 Λ15 2I ⎥ ⎢ ∗ −γ 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ −I ε1 D1 0 ⎥<0 ⎢ ∗ ⎥ ⎢ ⎣ ∗ ∗ ∗ −ε1 I 0 ⎦ ∗ ∗ ∗ ∗ −ε1 I ⎡ 2 ⎤ √ ˆT T 2 −µmax X −µmax I ρ C λm Cˆ T λT ⎢ ⎥ ∗ −µ2max Y 0 0 ⎥ ⎢ ⎢ ⎥<0 ⎣ ∗ ∗ − I + ε2 I 0 ⎦ ∗ ∗ ∗ −ε2 I } { } ) ⎡ { 2 √ ( T T − z2 max X − z22 max I ρ X C2 + Cˆ T λTm DT2 q q { } ⎢ √ T ⎢ ∗ − z22 max Y ρ C2 ⎢ q ⎢ ⎢ ∗ ∗ −I ⎢ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗

(61)

(62)

0 0

ε3 D2 −ε3 I ∗

Cˆ T λ

T⎤

⎥ ⎥ ⎥ ⎥<0 0 ⎥ ⎥ 0 ⎦

0 ⎥

−ε3 I

(63) [

X

I

I

Y

] >0

(64)

⎡

(

T ˆ ˆ ⎢AX + XA + B2 λm C + B2 λm C

Λ11 = ⎣

[

(54)

Considering the actuator faults, the output feedback H∞ controller can be defined by the following state space equation:

]

A A= Bκ Cy

(53)

where A, B1 , B2 , C1 , C2 , Cy , D1 , and D2 are defined in (6)–(8) and (11).

B2 λm Cκ B B ,B = 1 ,H = 2 , Aκ 0 0

[

Λ12 =

y(t) = Cy x(t)

(60)

]T

[

(44)

(57)

z 2 (t) = Ccl2 x(t)

(

⎥ )T ⎦ ,

]

B1 Y T B1

Λ13 =

⎤

A + Aˆ T

ˆ y + BC ˆ y AT Y + Y T A + BC

∗

[(

)T

,

C1 X + D1 λm Cˆ C1T

)T ] , Λ12 =

[

ε1 B2

ε1 Y T B2

]

[ ] ˆT T , Λ15 = C λ . 0

The Corollary 1 can be easily proved following the proof of Theorem 1. In this case, the output feedback controller parameters are obtained Cκ = Cˆ M T

(

)−1

(65)

102

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

Table 1 Optimization results of vehicle suspension parameters.

5.2. Proposed control methods validation

Definition

Symbol

Without DVA

With DVA

Sprung mass Mass of tyre, rim of wheel Mass of motor Stiffness of suspension Damping of suspension Stiffness of tyre Suspended motor stiffness Suspended motor damping

ms mu md ks cs kt ka2 ca2

350 kg 50 kg 30 kg 42 288 N/m 1686 N s/m 250 000 N/m – –

350 kg 50 kg 30 kg 42 288 N/m 1686 N s/m 250 000 N/m 30 542 N/m 1157 N s/m

Table 2 RMS comparison of vehicle dynamic responses. Vehicle type

Body acceleration (m/s2 )

Suspension stroke (mm)

Tyre dynamic force (N)

IWM-EV DVA-EV

0.1806 0.1570

0.9382 0.8806

132.5611 98.6614

Bκ = N −1 Bˆ Aκ = N

−1

(

(66) T

Aˆ − Y AX

)(

M

T −1

)

(

− B κ Cy X M

T −1

)

−N

Y B2 λm Cκ

−1 T

(67) Substitute (65)–(67) into (55)–(56), the active suspension control force can be obtained. 5. Simulation results 5.1. Parameter optimization results Particle swarm optimization results for vehicle suspension and DVA and effectiveness of DVA are illustrated in this section. The parameters of the quarter car model with DVA and without DVA are listed in Table 1. In the PSO algorithm, the value of win , wfn , c1 , c2 and iter are defined as follows: win = 0.9, wfn = 0.1, c1 = 1.3, c2 = 1.7, iter = 20. Since the effect of sprung mass acceleration is more important for the vehicle ride performance than other three hard constraints, so we choose four performance indexes as: p1 = 0.4, p2 = 0.2, p3 = 0.2, p4 = 0.2. The optimized parameters are listed as follows: md = 30 kg, ks = 42288 N/m, cs = 1686 N s/m, ka2 = 30542 N/m, ca2 = 1157 N s/m. Random road excitation is used to demonstrate the effectiveness of DVA configuration. The class B road profile with constant vehicle speed of 40 km/h is used in this section. The ‘‘IWM-EV’’ represents the in-wheel motor driven electric vehicle without DVA, while the ‘‘DVA-EV’’ denotes the in-wheel motor driven electric vehicle with DVA. Fig. 2 shows the random responses of sprung mass acceleration and tyre dynamic force, from which it can be seen that both sprung mass acceleration and tyre dynamic force of the DVA-EV are smaller than those of IWM-EV. Frequency responses of the sprung mass acceleration and tyre deflection are shown in Fig. 3, it can be seen in the figure that the DVA-EV decreases the sprung mass acceleration around 10 Hz. Furthermore, the suspension deflection of DVA-EV is greatly reduced around 10 Hz compare to the IWM-EV, which shows that the DVA-EV performs better than the IWM-EV, especially in the range of unsprung mass resonance. As a result, this kind of DVA-EV configuration has the ability of improving vehicle ride comfort performance and road holding ability. Table 2 shows the Root Mean Square (RMS) comparison of vehicle dynamic responses in terms of body acceleration, suspension stroke and tyre dynamic load under random road excitation. It is clear that the DVA-EV has better performance than IWMEV, which indicate that the DVA-EV structure has the potential to improve ride quality and road-holding performance.

In this section, bump road excitation and random road excitation are used to illustrate the effectiveness of the proposed controller. The quarter vehicle with in-wheel DVA model is used and vehicle parameters are listed in Table 1. We assume that the maximum suspension deflection zmax = 50 mm, the maximum control force umax = 2000 N and the dynamic force applied on the bearings Fmax = 3000N. The ‘‘Passive’’ denotes quarter car passive suspension system with in-wheel DVA structure. ‘‘Output feedback controller I’’ is the output feedback controller for the active suspension with only actuator faults while the ‘‘output feedback controller II’’ is the output feedback controller for the active suspension with actuator faults and time delay. The dynamic output-feedback controller I for the active suspension systems in (57)–(60) with only control faults can be derived by using Corollary 1. In addition, we can obtain the minimum guaranteed closed-loop H∞ performance index γ is 4.52. ⎡ −0.0007 ⎢ ⎢ 0.0023 ⎢ ⎢ 0.001 Ac = 1×105 ⎢ ⎢−0.0219 ⎢ ⎢ ⎣−0.0124 −4.513

−0.0001 0.0004 −0.0001 −0.0118 −0.0265 −0.1491

⎡

0 ⎢ 0 ⎢ ⎢ 0 Bc = 1 × 106 ⎢ ⎢0.0004 ⎣0.0002 0.0497 Cc = −214.04

63.12

[

0

0

0

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

0

0

0

0

−0.0001 0.001 0.0011 −0.0134

0.0001

−0.0001 −0.001 −0.0026 0.0301

0

0.0017 0.0048

−0.0463

0 0 0.0002 −0.0018 −0.0013 −0.1267

0 0 0 0.0037 0.0003 0.7373

409.22

−144.05

0 0 0 0 0 0

0 0

−0.0087

⎤

0 0 0 0.0039 −0.0071 6.0597

36.56

0 0⎥ ⎥ 0⎥ ⎥, 0⎥ 0⎦ 0

] −5.09 .

We compare the proposed output feedback controller I with those in references [4] and [43]. In [4], a finite-frequency H∞ state-feedback controller was proposed for active suspension equipped in in- wheel motor driven electric ground vehicle. Based on the Theorem 2 in [4], the control gain matrix is obtained as follows: Kfinite = 104

[ × 3.895

1.004

−0.195

0.018

−4.352

0.073 ,

]

In [43], A fault-tolerant fuzzy H∞ control design approach was proposed for active suspension of in-wheel drive electric vehicle. Based on the Corollary 3 in [43], the control gain matrix is obtained as follows: Kc = 104

[ × −0.681

0.487

−0.012

0.025

−6.885

] −0.029 ,

Furthermore, the dynamic output-feedback controller for the active suspension systems with control delay and faults can be derived by using Theorem 1. In this paper, ρ = 1, θR = 1, upper bound d of actuator delay d(t) is d = 5 ms. The dynamic outputfeedback controller in (9)–(10) for the upper bound d = 5 ms is obtained, and it can be found that the minimum guaranteed closed-loop H∞ performance index γ is 6.36. ⎡ −0.0037 ⎢ ⎢−0.0371 ⎢ ⎢ 0.0919 Ac = 1×104 ⎢ ⎢ 1.0097 ⎢ ⎢ ⎣−2.0433 −4.0055

−0.0002 −0.0059 −0.0208 −0.0488 −0.244 −1.0079

0

0

0

0

0

0

0

0

−0.0178 0.0002 −0.0005 −0.0073

0

0

−0.0178 −0.0001 −0.0052

−0.0179 −0.0012

0

⎤

⎥ ⎥ ⎥ ⎥ 0 ⎥, ⎥ 0 ⎥ ⎥ −0.0001⎦ −0.0192

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

Fig. 2. Random responses of sprung mass acceleration and tyre dynamic force.

Fig. 3. Frequency responses of sprung mass acceleration and tyre deflection.

Fig. 4. Bump responses of passive and active suspensions with 60% actuator thrust loss.

103

104

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

Fig. 6. Frequency responses of passive and active suspensions with 60% actuator thrust loss.

Fig. 5. Active forces with 60% actuator thrust loss.

⎡

0.0243

⎢ ⎢ −7.6245 ⎢ ⎢−17.1847 Ad = ⎢ ⎢ 39.2780 ⎢ ⎢ ⎣ 169.6451 154.0348 ⎡

0.0021

0.0091

−0.6448 −1.4625 3.3321 14.3083 13.0703

−2.8443 −6.4091 14.6446 63.2485 57.4290

0

⎢ ⎢ 0.0006 ⎢ ⎢ 0.0015 Bc = 1 × 106 ⎢ ⎢−0.0022 ⎢ ⎢ ⎣ 0.2548 −0.2908

Cc = −971.85

[

−0.0023 0.7191 1.6194 −3.6969 −15.9698 −14.4991

0.0003

−0.0818 −0.1847 0.4227 1.7480 1.6982

0

0

0

0

0.0004 −0.0003 −0.0124 0.0137 0.0117

0

−0.0035 0.0028 0.0148 −0.0472 −1.2374

−82.16

0 0 0 0

−362.48

91.64

−0.4929

⎤

0

−0.0027 0.0006 0.0166 −0.0027

⎤ −0.0001 ⎥ 0.0253 ⎥ ⎥ 0.0569 ⎥ ⎥, −0.1301⎥ ⎥ ⎥ −0.5525⎦

0

0⎥

⎥ ⎥ ⎥, 0⎥ ⎥ ⎥ 0⎦ 0⎥

0

−10.43

3.22 ,

]

The following bump road profile is used to validate effectiveness of proposed dynamic output-feedback controllers:

x(t) =

⎧ 2π v0 a ⎪ ⎨ (1 − cos( t)), 2

⎪ ⎩

l

0,

0≤t≤ t>

l

l

v0

(68)

v0

where a is the height of the bump and l is the length of the bump. Here we choose a = 0.1 m, l = 2 m and the vehicle forward velocity of v0 = 25 km/h. Fig. 4 shows the bump responses for four suspensions, i.e., passive suspension, active suspension with state feedback H∞ controller, active suspension with finite-frequency H∞ controller and active suspension with the proposed fault-tolerant output feedback controller when 60% actuator thrust loss occurs. The responses, including the body acceleration, suspension deflection, dynamic tyre load, and motor dynamic force, are plotted. It can be seen from Fig. 4 that the proposed fault-tolerant output feedback controller reduces the body acceleration significantly compare to the passive suspension when 60% actuator thrust loss occurs. The active suspension with the proposed controller also achieves better suspension performance when compared to the other two active suspensions. In addition, the suspension deflection, tyre dynamic force and maximum control force constraints are guaranteed simultaneously. Fig. 5 shows the active forces with 60% actuator thrust loss. The time-domain responses show that the proposed output feedback controller provides better vehicle performance than the passive suspension and the other two active controllers with partial faults in the actuator. Fig. 6

shows the frequency responses of passive and active suspension systems with 60% actuator thrust loss. The natural frequencies of vehicle sprung mass and unsprung mass are about 1.8 Hz and 10 Hz, respectively. In the frequency domain, it can be seen that the active suspension with the designed fault tolerant controller performs significantly better in the natural frequencies of vehicle sprung mass. This confirms that the proposed control method can realize good vehicle ride performance despite the change of the actuator thrust loss. Figs. 7 and 8 show the bump responses of passive suspension and active suspensions with two different output feedback controllers in the presence of control delay and actuator faults. The upper bound of actuator delay is 5 ms. Fig. 9 shows the active forces of two output feedback controllers under 30% and 60% actuator thrust losses. As observed in these diagrams, the proposed dynamic output feedback controller II achieves better suspension and motor performance than those of the passive suspension and the output feedback controller I when actuator faults and delay occur. This clearly demonstrates the effectiveness of the output feedback controller II in the presence of control delay. Moreover, with actuator time delay, the output feedback controller I achieves the worst performance among the three types of suspensions. This is because the output feedback controller I become instability when actuator delay occurs in the system. Vehicle performance of the output feedback controller I becomes worse when the actuator time delay occurs, which shows that the controller I is not robust to actuator delay. Furthermore, with an increase in the actuator thrust loss, the output feedback controller II reveals marginally better performance than the output feedback controller I and the passive suspension in spite of actuator faults and time delay under bump road excitation. Fig. 10 illustrates the frequency responses for passive suspension and active suspension with output feedback controller II under 60% actuator thrust loss and 5 ms actuator delay, from which we can observe that ride performance of the active suspension is much better than that of the passive one around the resonant frequency of sprung mass, demonstrating better robustness of the proposed output feedback controller II. Furthermore, the random road excitation is used to demonstrate the effectiveness of the proposed control system. The Power Spectral Density (PSD) of the random road excitation can

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

Fig. 7. Bump responses of passive and active suspensions with 30% actuator thrust loss and 5 ms time delay.

Fig. 8. Bump responses of passive and active suspensions with 60% actuator thrust loss and 5 ms time delay.

105

106

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

Fig. 9. Active forces. (a) 30% actuator thrust loss. (b) 60% actuator thrust loss.

be expressed as the following equation: n Gq (n) = Gq (n0 )( )−ω n0

(69)

The PSD of the road velocity can be expressed as the following equation: Gq˙ (n) = (2π n)2 Gq (n)

(70)

The random road excitation is determined by feeding a white noise through a linear first-order filter: x˙ g + 2π f0 xg = 2π

Gq (n0 )vw

√

(71)

where n is the spatial frequency and n0 is the reference spatial frequency. Gq (n0 ) is PSD for the reference spatial frequency. ω is frequency index, usually ω = 2. v is the vehicle speed, and w stands for the white noise disturbance of the road. f0 is cut off frequency, which could avoid overestimating the low frequency component of the road. The class B road profile with constant vehicle speed of 40 km/h is used to test the system. The RMS comparison of the vehicle dynamic responses under random road excitation is shown in Table 3. The active suspension with the fault-tolerant controller achieves marginally better performance than the passive suspension in the presence of 60% actuator thrust loss. Suspension deflection, actuator force maximum force applied to the motor bearing and tyre dynamic force are guaranteed simultaneously. The RMS comparison of the passive suspension, the active suspensions with the output feedback controller I and the output feedback controller II in the presence of control delay and different actuator faults are shown in Table 4. Suspension performance of the output feedback controller I is impaired when the actuator delay occurs. The active suspension with the output feedback controller II reveals much better performance than those with the output feedback controller I and the passive suspension; which shows that the output feedback controller II is able to guarantee better suspension and motor performance in spite of actuator faults and delays under random road excitation.

Fig. 10. Frequency responses of passive and active suspensions with 60% actuator thrust loss and time delay.

ability around 10 Hz. Parameters of vehicle suspension and DVA were optimized based on PSO method. In order to achieve better ride comfort and reduce the force applied on the in-wheel motor bearing, a robust H∞ dynamic output feedback controller was derived such that the close-loop system was asymptotic stability and simultaneously satisfied the constraint performances such as road holding, suspension stroke, dynamic load applied to the bearings and actuator limitation. Finally, the simulation results demonstrated the effectiveness of the proposed output feedback controllers in improving suspension performance in spite of actuator faults and time delay. Meanwhile, the proposed faulttolerant output feedback H∞ controller achieved a better vehicle and motor performance than those of the passive suspension for different actuator thrust losses. When different actuator thrust losses and time delay occurred, the proposed output feedback controller II revealed much better performance than the output feedback controller I and the passive suspension.

6. Conclusions In this paper, the problem of output feedback H∞ control for active suspensions deployed in in-wheel motor driven electric vehicles with actuator faults and time delay was investigated. A quarter car active suspension system with in-wheel motor served as DVA was established, and this kind of configuration was demonstrated to improve ride performance and road holding

Acknowledgements This research is supported under Australian Research Council’s Discovery Projects funding scheme (project number DP140100303) and Chinese Scholarship Council (CSC), China scholarship.

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

107

Table 3 The RMS comparison of vehicle dynamic responses with 60% actuator thrust loss. Suspension types

Body acceleration (m/s2 )

Suspension deflection (m)

Dynamic tyre load (N)

Motor dynamic force (N)

Active force (N)

Passive State-feedback controller Finite-frequency H∞ controller Output-feedback Controller I

0.162 0.131

8.996e−4 7.351e−4

110.726 103.454

75.012 66.017

– 13.455

0.129

7.024e−4

104.133

64.891

15.285

0.129

7.143e−4

111.552

69.267

16.559

Table 4 The RMS comparison of vehicle dynamic responses in the presence of control delay and different actuator faults. Suspension types

Body acceleration (m/s2 )

Suspension deflection (m)

Relative dynamic tyre load

Motor dynamic force (N)

Active force (N)

Passive Controller I with 30% actuator thrust loss Controller II with 30% actuator thrust loss Controller I with 60% actuator thrust loss Controller II with 60% actuator thrust loss

0.1617 0.2539

8.9964e−04 0.0016

0.0282 0.0312

75.0124 174.7807

– 102.1168

0.1301

7.0113e−04

0.0293

68.5019

21.1690

0.2204

0.0013

0.0299

114.5852

53.0063

0.1369

7.2105e−04

0.0285

69.3987

15.2151

Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Murata S. Innovation by in-wheel-motor drive unit. Veh Syst Dyn 2012;50:807–30. [2] Watts A, Vallance A, Whitehead A, Hilton C, Fraser A. The technology and economics of in-wheel motors. SAE Int J Passeng Cars Electron Electr Syst 2010;3:37–55. [3] Shuai Z, Zhang H, Wang J, Li J, Ouyang M. Combined AFS and DYC control of four-wheel-independent-drive electric vehicles over CAN network with time-varying delays. IEEE Trans Veh Technol 2014;63:591–602. [4] Wang R, Jing H, Yan F, Karimi HR, Chen N. Optimization and finitefrequency H∞ control of active suspensions in in-wheel motor driven electric ground vehicles. J Franklin Inst 2015;352:468–84. [5] Jing H, Wang R, Li C, Wang J, Chen N. Fault-tolerant control of active suspensions in in-wheel motor driven electric vehicles. Int J Veh Des 2015;68:22–36. [6] B. Corporation BC, Bridgestone Dynamic-Damping In-wheel Motor Drive System, http://enginuitysystems.com/files/In-Wheel_Motor/; 2016. [7] Moradi M, Fekih A. Adaptive PID-sliding-mode fault-tolerant control approach for vehicle suspension systems subject to actuator faults. IEEE Trans Veh Technol 2014;63:1041–54. [8] Sun W, Pan H, Gao H. Filter-based adaptive vibration control for active vehicle suspensions with electrohydraulic actuators. IEEE Trans Veh Technol 2016;65:4619–26. [9] Su X, Yang X, Shi P, Wu L. Fuzzy control of nonlinear electromagnetic suspension systems. Mechatronics 2014;24:328–35. [10] Lin J, Cheng KWE, Zhang Z, Cheung NC, Xue X. Adaptive sliding mode technique-based electromagnetic suspension system with linear switched reluctance actuator. IET Electr Power Appl 2014;9:50–9. [11] Wen S, Chen MZ, Zeng Z, Yu X, Huang T. Fuzzy control for uncertain vehicle active suspension systems via dynamic sliding-mode approach. IEEE Trans Syst Man Cy-S 2017;47:24–32. [12] Ozer HO, Hacioglu Y, Yagiz N. High order sliding mode control with estimation for vehicle active suspensions. Trans Inst Meas Control 2018;40:1457–70. [13] Karimi HR. Robust delay-dependent H∞ control of uncertain time-delay systems with mixed neutral, discrete, and distributed time-delays and markovian switching parameters. IEEE Trans Circuits-I 2011;58:1910–23. [14] Zhang H, Huang X, Wang J, Karimi HR. Robust energy-to-peak sideslip angle estimation with applications to ground vehicles. Mechatronics 2015;30:338–47.

[15] Chadli M, Karimi HR. Robust observer design for unknown inputs Takagi–Sugeno models. IEEE Trans Fuzzy Sys 2013;21:158–64. [16] Moradi M, Fekih A. A stability guaranteed robust fault tolerant control design for vehicle suspension systems subject to actuator faults and disturbances. IEEE Trans Control Syst Technol 2015;23:1164–71. [17] Sun W, Pan H, Yu J, Gao H. Reliability control for uncertain half-car active suspension systems with possible actuator faults. IET Control Theory A 2014;8:746–54. [18] Li H, Liu H, Gao H, Shi P. Reliable fuzzy control for active suspension systems with actuator delay and fault. IEEE Trans Fuzzy Syst 2012;20:342–57. [19] Bounemeur A, Chemachema M, Essounbouli N. Indirect adaptive fuzzy fault-tolerant tracking control for MIMO nonlinear systems with actuator and sensor failures. ISA Trans 2018. [20] Ye D, Su L, Wang J-L, Pan Y-N. Adaptive reliable H∞ optimization control for linear systems with time-varying actuator fault and delays. IEEE Trans Syst Man CY-S 2017;47:1635–43. [21] Jin X, Wang S, Qin J, Zheng WX, Kang Y. Adaptive fault-tolerant consensus for a class of uncertain nonlinear second-order multi-agent systems with circuit implementation. IEEE Trans Circuits-I 2018;65:2243–55. [22] Saied M, Lussier B, Fantoni I, Shraim H, Francis C. Fault diagnosis and FaultTolerant control of an octorotor UAV using motors speeds measurements. IFAC PapersOnline 2017;50:5263–8. [23] Jin X, Qin J, Shi Y, Zheng WX. Auxiliary fault tolerant control with actuator amplitude saturation and limited rate. IEEE Trans Syst Man CY-S 2017. [24] Du H, Zhang N. Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint. IEEE Trans Fuzzy Syst 2009;17:343–56. [25] Wang R, Zhang H, Wang J. Linear parameter-varying controller design for four-wheel independently actuated electric ground vehicles with active steering systems. IEEE Trans Control Syst Technol 2014;22:1281–96. [26] Zhang H, Zhang G, Wang J. Observer design for LPV systems with uncertain measurements on scheduling variables: Application to an electric ground vehicle. IEEE-ASME Trans Mech 2016;21:1659–70. [27] Zhang H, Wang J. Vehicle lateral dynamics control through AFS/DYC and robust gain-scheduling approach. IEEE Trans Veh Technol 2016;65:489–94. [28] Khazaee M, Markazi AH, Omidi E. Adaptive fuzzy predictive sliding control of uncertain nonlinear systems with bound-known input delay. ISA Trans 2015;59:314–24. [29] Kong Y, Zhao D, Yang B, Han C, Han K. Robust non-fragile H∞ / L2 -L∞ control of uncertain linear system with time-delay and application to vehicle active suspension. Int J Robust Nonlinear 2015;25:2122–41. [30] Li L, Liao F. Robust preview control for a class of uncertain discrete-time systems with time-varying delay. ISA Trans 2018;73:11–21. [31] Ginoya D, Shendge P, Phadke S. Sliding mode control for mismatched uncertain systems using an extended disturbance observer. IEEE Trans Ind Electron 2014;61:1983–92. [32] Ning D, Sun S, Zhang F, Du H, Li W, Zhang B. Disturbance observer based Takagi-Sugeno fuzzy control for an active seat suspension. Mech Syst Signal Process 2017;93:515–30.

108

X. Shao, F. Naghdy, H. Du et al. / ISA Transactions 92 (2019) 94–108

[33] Ning D, Sun S, Li H, Du H, Li W. Active control of an innovative seat suspension system with acceleration measurement based friction estimation. J Sound Vib 2016;384:28–44. [34] Li H, Jing X, Karimi HR. Output-feedback-based H∞ control for vehicle suspension systems with control delay. IEEE Trans Ind Electron 2014;61:436–46. [35] Rath JJ, Defoort M, Karimi HR, Veluvolu KC. Output feedback active suspension control with higher order terminal sliding mode. IEEE Trans Ind Electron 2017;64:1392–403. [36] Wang R, Jing H, Karimi HR, Chen N. Robust fault-tolerant H∞ control of active suspension systems with finite-frequency constraint. Mech Syst Signal Process 2015;62:341–55. [37] Choi HD, Ahn CK, Shi P, Wu L, Lim MT. Dynamic output-feedback dissipative control for T–S fuzzy systems with time-varying input delay and output constraints. IEEE Trans Fuzzy Syst 2017;25:511–26. [38] Shuai Z, Zhang H, Wang J, Li J, Ouyang M. Lateral motion control for four-wheel-independent-drive electric vehicles using optimal torque allocation and dynamic message priority scheduling. Control Eng Pract 2014;24:55–66. [39] Geng C, Mostefai L, Denaï M, Hori Y. Direct yaw-moment control of an inwheel-motored electric vehicle based on body slip angle fuzzy observer. IEEE Trans Ind Electron 2009;56:1411–9. [40] Ifedi CJ, Mecrow BC, Brockway ST, Boast GS, Atkinson GJ, Kostic-Perovic D. Fault-tolerant in-wheel motor topologies for high-performance electric vehicles. IEEE Trans Ind Appl 2013;49:1249–57.

[41] Liu M, Gu F, Zhang Y. Ride comfort optimization of in-wheel-motor electric vehicles with in-wheel vibration absorbers. Energies 2017;10:1647. [42] Wang Y, Li Y, Sun W, Yang C, Xu G. Fxlms method for suppressing inwheel switched reluctance motor vertical force based on vehicle active suspension system. J Control Sci Eng 2014;2014:6. [43] Shao X, Naghdy F, Du H. Reliable fuzzy H∞ control for active suspension of in-wheel motor driven electric vehicles with dynamic damping. Mech Syst Signal Process 2017;87:365–83. [44] Shao X, Naghdy F, Du H. Enhanced ride performance of electric vehicle suspension system based on genetic algorithm optimization. In: Proceedings of the IEEE 2017 20th international conference on electrical machines and systems (ICEMS). 2017, p. 1–6. [45] Li H, Jing X, Lam H-K, Shi P. Fuzzy sampled-data control for uncertain vehicle suspension systems. IEEE Trans Cybernetics 2014;44:1111–26. [46] Kaveh A. Advances in metaheuristic algorithms for optimal design of structures. SPRINGER; 2016. [47] Shi Y, Eberhart R. A modified particle swarm optimizer. In: Proceeding of the 1998 IEEE international conference on computational intelligence. 1998, p. 69–73. [48] Rajeswari K, Lakshmi P. Pso optimized fuzzy logic controller for active suspension system. In: Proceeding of 2010 international conference on the advances in recent technologies in communication and computing (ARTCom). 2010, p. 278–83. [49] Zhao Y, Zhao L, Gao H. Vibration control of seat suspension using H∞ reliable control. J Vib Control 2010;16:1859–79.