Reliable fuzzy H∞ control for active suspension of in-wheel motor driven electric vehicles with dynamic damping

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Reliable fuzzy H∞ control for active suspension of in-wheel motor driven electric vehicles with dynamic damping ⁎

Xinxin Shao , Fazel Naghdy, Haiping Du School of Electrical, Computer & Telecommunication Engineering, University of Wollongong, Northfields Avenue, Wollongong, NSW 2522 Australia

A R T I C L E I N F O

ABSTRACT

Keywords: Reliable fuzzy H∞ control Active suspension In-wheel motor driven electric vehicle Actuator faults

A fault-tolerant fuzzy H∞ control design approach for active suspension of in-wheel motor driven electric vehicles in the presence of sprung mass variation, actuator faults and control input constraints is proposed. The controller is designed based on the quarter-car active suspension model with a dynamic-damping-in-wheel-motor-driven-system, in which the suspended motor is operated as a dynamic absorber. The Takagi-Sugeno (T-S) fuzzy model is used to model this suspension with possible sprung mass variation. The parallel-distributed compensation (PDC) scheme is deployed to derive a fault-tolerant fuzzy controller for the T-S fuzzy suspension model. In order to reduce the motor wear caused by the dynamic force transmitted to the in-wheel motor, the dynamic force is taken as an additional controlled output besides the traditional optimization objectives such as sprung mass acceleration, suspension deflection and actuator saturation. The H∞ performance of the proposed controller is derived as linear matrix inequalities (LMIs) comprising three equality constraints which are solved efficiently by means of MATLAB LMI Toolbox. The proposed controller is applied to an electric vehicle suspension and its effectiveness is demonstrated through computer simulation.

1. Introduction Electric vehicles (EVs) provide advantages over the Internal Combustion Engine (ICE) vehicles in terms of energy efficiency and environmental friendliness and are regarded as one of the solutions to decreasing the global CO2 emission. Propulsion configurations 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 In-Wheel Motors (IWM)), has attracted an increasing research interest in recent years because of the benefits of IWMs [1]. In addition to the simplicity and efficiency, in-wheel motors could generate fast and precise torque which has no adverse effect on the driveshaft stiffness. In-wheel motors also have the capability of enhancing the performance of traction control systems (TCS), anti-lock brake systems (ABS), and electronic stability control systems (ESC) [2]. However, the development of IWMs has introduced new technological challenges. Installing the motors in the wheel can result in an increase in the unsprung mass, which greatly deteriorates the suspension ride comfort performance and road holding ability. Furthermore, the wear of the motor bearing is a problem that should be addressed in the active control of IWM EVs [3]. The motor bearing in vehicles with unsprung mass can easily wear because of heavy loads and the small gap between the motor rotor and the stator. Since the ride comfort and passenger safety are increasingly becoming critical criteria in vehicle suspension design, the influence of IWM on these criteria should be examined and addressed. In order to decrease the tyre contact force fluctuation of an EV, Bridgestone developed the so-called dynamic-damping-in-wheel-

⁎

Corresponding author.

http://dx.doi.org/10.1016/j.ymssp.2016.10.032 Received 28 July 2016; Received in revised form 14 October 2016; Accepted 28 October 2016 Available online xxxx 0888-3270/ © 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: Shao, X., Mechanical Systems and Signal Processing (2016), http://dx.doi.org/10.1016/j.ymssp.2016.10.032

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motor-driven-system [4], which suspends shaftless direct-drive motors and isolates them from the unsprung mass. The motor was designed as a vibration absorber that could offset the road vibration input. The system is shown to have the potential to improve ride quality and road-holding performance. Tyre contact force fluctuations in conventional EVs and IMW EVs with dynamic damping are compared in [5]. Although it is shown that such structures can efficiently increase the road holding performance, active suspension control should be deployed to improve vehicle ride quality. Sun et al. [1,6] investigated the dynamic effect of an in-wheel Switched Reluctance Motor (SRM) on a vehicle in terms of vibration and noise issues, and proposed corresponding control methodology to improve the EV performance. A filtered-X least mean square controller was proposed for active suspension to suppress the vibration caused by SRM vertical force in [7]. Wang and Jing [3] proposed a finite-frequency state feedback H∞ controller for active suspension of IWM EVs, and demonstrated that the deployment of a dynamic vibration absorber would significantly reduce the force applied to the in-wheel motor bearing. Jing and Wang [8] proposed a robust H∞ fault-tolerant controller for active suspension of IWM EVs to decrease the motor vibration and to reduce the dynamic load applied to the in-wheel motor bearing. In an active suspension design, a fully controlled suspension system improves the vehicle ride comfort, handling stability and safety. However, performance requirements, such as ride comfort, road-holding stability, and suspension deflection, are often conflicting design requirements. Different control methods such as fuzzy control [9,10], Linear Quadratic Regulator (LQR)/Linear Quadratic Gaussian (LQG) [11], neural network method [12], linear optimal control [13–15], robust H∞ control [16–18] and adaptive control [19–21] are proposed to deal with the trade-off between these conflicting expectations. Brezas and Smith [13] proposed a clipped-optimal control algorithm which could optimize the vehicle ride and handling behaviour for semi-active vehicle suspension. In [14], optimal control for active suspension with sinusoidal disturbance based on a modified quadratic performance index was investigated. In [15], an optimal fuzzy-PID control strategy based on an improved cultural algorithm was proposed for active suspension to suppress the vertical vibration acceleration of the quarter vehicle. In [22], a skyhook adaptive neuro active force controller consisting of four feedback control loops for active suspension was proposed and validated theoretically and experimentally. A strategy utilizing neural network and backstepping techniques for semi-active suspension systems was investigated in [23], and the neural network was used to estimate the control voltage input to the magnetorheological damper. In [24], Variational Feedback Controllers (VFC) was proposed for feedback control of a nonlinear suspension system. In [25], a fixed order non-fragile dynamic output feedback controller for a suspension system with model uncertainty and nonlinear dynamics was proposed based on the convex optimization and LMI approach. In [20], an adaptive robust control based H∞ control strategy was designed for full vehicle active suspension systems with electrohydraulic actuators in which the proposed controller could deal with actuator parametric uncertainties and uncertain actuator nonlinearities. A saturated adaptive robust control (ARC) strategy [21] and adaptive backstepping control strategy [19] were proposed for active suspension with parameter uncertainties and nonlinearities. According to the literature, the robust H∞ control strategy can deal with complexities such as sprung mass uncertainty, damper time delay and time constant uncertainty [20,24–26]. In vehicle suspension control methods used to improve vehicle performance, it is assumed that all the control components of the systems are in an ideal working condition. In practice, unknown faults in components such as sensor and actuator failures can deteriorate the dynamic behaviour of the suspension. Fault-tolerant control (FTC) method which deals with possible actuator failure has attracted attentions in recent years. Generally speaking, the FTC method can be divided into two types: passive FTC and active FTC. The controller of passive FTC is fixed while active FTC could detect the faults and compensate the effect of faults in real time. A robust finite frequency passive fault-tolerant static-output-feedback H∞ controller was designed for structure systems in [17]. In this work, the actuator faults were described by a polytopic model. A passive fault-tolerant robust LQR-based H∞ controller was proposed for four-wheel independently actuated electric vehicle using Linear Parameter-Varying (LPV) control method in [27]. FTC based on virtual sensor and virtual actuator for nonlinear system were studied in [28,29] considering both actuator and sensor faults. Adaptive sliding mode control of Markov jump nonlinear systems in the presence of actuator faults was studied in [30]. The problem of fault detection in active suspension has been researched intensively in recent years. In [31], a robust optimal sliding mode controller was proposed to deal with actuator faults and to ensure the overall stability of the full vehicle suspension system. In [32], a fault tolerant control method was proposed for the electromagnetic suspension system subject to failure of the sensors and actuators. The problem of fault detection filters for an active suspension in a finite-frequency domain was dealt with based on the generalized Kalman-Yakubovich-Popov (KYP) lemma in [33]. Robust fault-tolerant H∞ control for full-car active suspension with finite-frequency constraint was investigated in [3], where the controller was designed to reduce the heave, pitch and roll motions. Kong et al. [34] proposed a robust non-fragile H∞/L2−L∞ static output feedback controller for vehicle active suspension considering actuator time-delays and the controller gain variations. In [35,36], the non-fragile H∞ controller was proposed for the half-vehicle suspension systems in the presence of actuator uncertainty and failure. The vehicle sprung mass varies with respect to the loading conditions such as the payload and the number of vehicle occupants. Model uncertainty, such as suspension sprung and unsprung mass variations should be taken into consideration in the design of active suspensions. The active control performance of a vehicle suspension is affected if the sprung mass variation is not considered. A Takagi-Sugeno (T-S) fuzzy approach can be applied to handle the uncertainties as the T–S fuzzy model is very effective in representing complex nonlinear systems. Du and Zhang [9] presented a fuzzy H∞ static output feedback controller design approach for vehicle electrohydraulic active suspensions based on T–S fuzzy modelling techniques. A mode-dependent and fuzzy-basisdependent T-S fuzzy filter was designed for discrete-time T-S fuzzy systems using multiple packet dropouts in a network environment [37]. The fuzzy tracking control problem for uncertain nonlinear networked system based on type-2 T-S fuzzy mode was studied in [38]. The studies addressing model uncertainty, unknown actuator dead zone and control constraints, using some adaptive controllers such as adaptive fuzzy controller and adaptive neural controller for uncertain non-strict-feedback stochastic nonlinear systems were presented in [39,40]. In [41], a sampled-data H∞ control for an active suspension system with model 2

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uncertainty was proposed. In [42], an adaptive control method was proposed for vehicle active suspension with unknown nonlinear spring and piece-wise damper dynamics. The unknown nonlinearities can be compensated by an augmented neural network online. Moreover, there are two methods, polytope with finite vertices and norm-bounded uncertainty or variation that can be used to describe uncertainties and variations in model dynamics. Multi-objective energy-to-peak controller was proposed in [18] for vehicle lateral dynamic stabilisation problem, in which the longitudinal velocity was described by a polytope with finite vertices and the nonlinear tyre model was modelled by a linear model with norm-bounded uncertainties. Robust H∞ sliding mode control combined with pole displacement technique was proposed for electrohydraulic actuator with nonlinear friction, load disturbance and external noise [16]. Furthermore, linear-parameter-varying technique [27,43,44] has attracted an increasing research interest because of its effectiveness in describing nonlinearities and uncertainties. In [44], a combined active front-wheel steering/direct yaw-moment control was proposed for uncertain vehicle model, which was described by a linear parameter-varying model with norm-bounded uncertainties. A gain-scheduling sliding mode H∞ observer was proposed for the polytopic LPV system with uncertain measurement in [43]. However, majorities of the existing control strategies are designed for conventional vehicles and rarely address active control of IWM EV suspension systems. Furthermore, the constraints on actuator control force should be considered in controller design. In spite of their importance, issues associated with the actuator nonlinear dynamics, sprung mass uncertainty, and actuator saturation have not been explicitly dealt with in any previous studies on IWM EVs suspensions. In this present work, a reliable fuzzy H∞ controller is developed for dynamic active suspension control of IWM EVs based on a T– S fuzzy model. The primary contributions of the paper are: 1) an “advanced-dynamic-damper-motor” model in which the motor itself is operated as a dynamic damper in EVs is proposed and the influence of IWMs on the ride comfort and road holding capability of EVs is investigated. 2) The T–S fuzzy model is deployed to handle the sprung mass variation, and a reliable fuzzy H∞ controller is designed to improve suspension performance in the presence of sprung mass variations and actuator faults. 3) A multi-objective control for an EV active suspension is developed to achieve vehicle ride comfort with the following control constraints such as suspension deflection, actuator saturation and dynamic load applied on the bearings are guaranteed. The remainder of the paper is structured as follows. In Section 2, the EV suspension model with an “advanced-dynamic-dampermotor” is developed. In Section 3, the fuzzy H∞ controller is designed to improve vehicle performance, ride comfort and suspension deflection. The approach is validated in Section 4 and simulation results are provided. Finally, conclusions are drawn in Section 5 and future work is discussed. 2. System modelling and problem formulation 2.1. Effects of increased unsprung mass on ride performance IWM configuration in which the motors are installed in the wheels results in an increase in the unsprung mass. The general effect of increased unsprung mass on the sprung mass acceleration, suspension deflection, and tyre deflection based on a quarter-car model (as shown in Fig. 1(a)) is investigated. The vehicle parameters are shown in Table 1. The Bode diagram of sprung mass acceleration, suspension deflection, and tyre dynamic force subjected to road disturbance are displayed in Fig. 3. As shown in Fig. 3, increasing unsprung mass has almost no effect around the natural frequency of sprung mass (around 1–2 Hz). However, an increase in the unsprung mass leads to an increase in the response of the frequency ranges around 10 Hz, which shows negative effect on suspension ride comfort performance and road holding ability. 2.2. Ride performance analysis of different IWM configurations The quarter-car suspension model with a dynamic absorber attached to the unsprung mass through a spring and a damper is shown in Fig. 1(c), in which the motor itself operates as the dynamic absorber. This system is called “advanced-dynamic-dampermotor” (ADM) [4]. Fig. 2 shows the structure of a dynamic-damping-in-wheel-motor-driven-system which is composed of tyre, flexible coupling, shaftless direct-drive motor and motor suspension [4]. The frequency responses from road disturbance to sprung mass acceleration, suspension deflection, and tyre deflection of a conventional EV (Conv-EV), In Wheel Motor driven EV (IWD-EV) and an EV with an Advanced-Dynamic-Damper-Motor system (ADM-EV) are depicted in Fig. 4. From the figure we can observe that the ADM-EV decreases the sprung mass acceleration in the range of resonance of unsprung mass. This is demonstrated that this kind of IWM configuration has the ability of improving vehicle ride comfort performance. Furthermore, the suspension deflection is

Fig. 1. Suspension model of electric vehicle with dynamic-damper-motor.

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Table 1 Suspension parameter. ms

mu

md

kt

ks

cs

kd

cd

340 kg

40 kg

30 kg

360,000 N/m

32,000 N/m

1496 N/(m/s)

41,000 N/m

1000 N/(m/s)

slightly reduced compared to Conv-EV and IWD-EV, while the tyre deflection of ADM-EV is greatly reduced compare to the IWDEV. The results show that the ADM-EV performs better than the Conv-EV and IWD-EV, especially in the range of unsprung mass resonance. 2.3. In-wheel motor electric vehicle suspension system modelling Based on Newton's Second Law, the motion equations of this active suspension can be written as

msxs̈ (t ) = − ks(xs(t ) − xu(t )) − cs(xṡ (t ) − xu̇ (t )) − fa (t )

(1)

md xd̈ (t ) = − kd (xd (t ) − xu(t )) − cd (xḋ (t ) − xu̇ (t ))

(2)

muxü (t ) = ks(xs(t ) − xu(t )) + cs(xṡ (t ) − xu̇ (t )) + kd (xd (t ) − xu(t )) + cd (xḋ (t ) − xu̇ (t )) − kt (xu(t ) − xg(t )) + fa (t )

(3)

where xs, xu and xd denote the vertical displacements of the sprung mass, unsprung mass and motor mass. xg and fa denote road disturbances and actuator force, respectively. 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. The parameter values used for this study are presented in Table 1. By defining the vehicle state vector as

x (t ) = [ xs(t ) − xu(t ) xṡ (t ) xd (t ) − xu(t ) xḋ (t ) xu(t ) − xg(t ) xu̇ (t )] the dynamic model (1) is expressed by a state space equation as

x (̇ t ) = Ax (t ) + B1w(t ) + B2u(t )

(4)

⎡ 0 1 0 0 0 −1 ⎤ ⎡ 0 ⎤ ⎢ ks cs cs ⎥ ⎡0⎤ ⎢ 1⎥ 0 0 ⎢− ms − ms 0 ms ⎥ ⎢0⎥ ⎢− ms ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 1 0 −1 ⎥ ⎢ 0 0 ⎥, B = ⎢ 0 ⎥, u(t ) = f (t ),w(t ) = x (t ). kd cd where A = ⎢ cd ⎥, B1 = ⎢ g 2 a 0 0 −m −m 0 ⎢0⎥ ⎢ 0 ⎥ md ⎥ ⎢ d d ⎢ ⎥ ⎢ ⎥ −1 0 ⎢ 0 0 0 0 0 1 ⎥ ⎢⎣ 0 ⎥⎦ ⎢ 1 ⎥ ⎢ k ⎥ k cs kd cd ⎢⎣ mu ⎥⎦ − mt cs + cd ⎥ ⎢ ms m m m m ⎣ u u u ⎦ u u u In order to reduce the power consumption of the actuator, the active control force provided by the active suspension system should be constrained by a threshold, and the actuator saturation nonlinearity is described by

⎧ umax u(t ) ≥ umax Δ ⎪ u(t ) = ⎨ u(t ) umax ≥ u(t ) ≥ − umax ⎪− u u(t ) ≤ − umax ⎩ max

(5)

where umax is the maximum actuator control force. In order to design a robust H∞ reliable state feedback load-dependent control law for the active suspension system presented, four key suspension performances, ride comfort, dynamic force applied on the wheel motor, suspension deflection and road holding (as defined below), are considered. Tire Flexible coupling Shaftless direct-drive motor Motor suspension

Fig. 2. Structure of dynamic-damping-in-wheel-motor-driven-system [4].

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Fig. 3. Bode diagrams of vehicle dynamic responses for increasing unsprung mass. (a) Sprung mass acceleration. (b) Suspension deflection. (c) Tyre dynamic force.

(1) Ride comfort. The sprung mass acceleration in the vertical direction is used to quantify vehicle ride comfort performance. Therefore, the first control objective that should be minimised is the vertical accelerationxs̈ (t ). (2) Motor dynamic force. The maximum dynamic force applied to the in-wheel motor can be defined as

Fdynamic = ks(xs − xu ) + cs(xṡ − xu̇ ) + kd (xd − xu ) + cd (xḋ − xu̇ ) + fa ≤ Fmax

(6)

(3) Suspension deflection. The suspension deflection should not exceed its travel limit due to the mechanical structure. That is,

xs(t ) − xu(t ) ≤ z max

(7)

where z max is the maximum suspension deflection. (4) Road-holding stability. To ensure a firm uninterrupted contact of wheels with the road, the dynamic tyre load should not exceed the static one, that is,

kt (xu(t ) − xg(t )) < (ms + mu + md )g

(8)

To satisfy the above conditions, the vehicle body vertical acceleration is minimised to obtain a better ride performance and the latter three conditions are hard constraints that should be strictly satisfied. Thus the following output variables are defined:

z1 = xs̈ (t )

(9) T

z2 = [ xs(t ) − xu(t ) Fdynamic kt (xu(t ) − xg(t ))]

(10)

Therefore, the active suspension system can be described by the following state-space equations. 5

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Fig. 4. Comparison of frequency responses between different IWM configurations. (a) Sprung mass acceleration. (b) Suspension deflection. (c) Tyre dynamic force.

x (̇ t ) = Ax (t ) + B1w(t ) + B2u(t )

(11)

z1 = C1x (t ) + D1u

(12)

z2 = C2x (t ) + D2u

(13)

where

⎡ k c C1 = ⎢− ms − ms 0 0 0 s ⎣ s

cs ⎤ ms ⎥ ⎦

and D1 = −

1 . ms

⎡1 0 0 0 0 ⎤ 0 ⎢ ⎥ C2 = ⎢ ks cs kd cd 0 −(cs + cd )⎥ and D2 = [ 0 1 0 ]T ⎢⎣ 0 0 0 0 kt ⎥⎦ 0 The uncertainties associated with the load condition in the sprung mass should be taken into account during controller design. The sprung mass ms(t ) varies in a given range of ms ∈ [ ms min ms max ]. The uncertain sprung mass ms(t) is bounded by its minimum value msmin and its maximum value msmax, and can thus be represented by

max

1 1 1 1 = =⌢ ms, min = = ms ms(t ) ms min ms(t ) ms max

˘

1 = M1(ξ1(t ))⌢ ms + M2(ξ1(t )) ms ms(t )

˘

(14) 6

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Fig. 5. Frequency responses of the passive suspension and active suspensions.

where ξ(t ) =

1 is ms(t )

the premise variable.

The membership functions M1(ξ(t )) and M2(ξ(t ))can be calculated by 1

˘ ˘

− ms ms(t ) , M1(ξ(t )) = ⌢ ms − ms

(15)

1 ⌢ ms − m (t ) s M2(ξ(t )) = ⌢ ms − ms

(16)

˘

2

∑ Mi(ξ(t )) = 1

(17)

i =1

The membership functions are labelled as “Heavy” or “Light”, so the system with an uncertain sprung mass is represented by the following fuzzy models. Model Rule 1: If ξ(t ) is heavy. Then

x (̇ t ) = A1x (t ) + B1w(t ) + B21u(t )

(18)

z1 = C11x (t ) + D11u

(19)

z2 = C2x (t ) + D2u

(20)

Model Rule 2: If ξ(t ) is light.

x (̇ t ) = A2 x (t ) + B1w(t ) + B22u(t )

(21)

z1 = C12x (t ) + D12u

(22)

z2 = C2x (t ) + D2u

(23)

Fuzzy blending allows us to infer the overall fuzzy model as follows: 2

x (̇ t ) =

∑ Mi(ξ(t ))[Ai x(t ) + B1w(t ) + B2iu(t )]

(24)

i =1

2

z1 =

∑ Mi(ξ(t ))[C1ix(t ) + D1iu(t )]

(25)

i =1

z2 = C2x (t ) + D2u

(26) 7

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3. Fuzzy reliable H∞ controller design A robust H∞ reliable state-feedback controller for active suspension of ADM-EVs with actuator faults and parameter uncertainties is developed. The closed-loop system is asymptotically stable, and it can also ensure a prescribed gain from disturbance to performance output while the suspension stroke, maximum dynamic force applied to the suspended motor and maximum control force constraints are satisfied. The following remark and lemma are needed to derive the main results. Remark 1. [45]: For the active suspension system above, an ideal control input u obtained by the state-feedback controller is applied to the suspension system through the actuator which generates the actual input uf, which is equal to the ideal control input in the case of a faultless actuator. When actuator faults occur, u f ≠ u . Therefore, uf can be modelled by considering the actuator fault:

u f (t ) = Mau(t )

(27)

⌢ ⌢ where Ma represents the possible fault of the actuator. Ma ≤ Ma ≤ Ma , and Ma and Ma are prescribed lower and upper bounds of the actuator. Three cases are considered to describe three different actuator conditions.

˘

˘

⌢ 1. Ma = Ma = 0 , so that Ma = 0 , which means that the corresponding actuator u f (t ) has completely failed. ⌢ 2. Ma = Ma = 1, so that we have Ma = 1, representing the case of no fault in the actuator. ⌢ 3. 0 < Ma = Ma < 1, which indicates that there is a partial fault in the corresponding actuator.

˘ ˘

˘

Then we introduce the following scalars which will be used in the design of the controller.

⌢ Ma0 = (Ma + Ma )/2.

(28)

La = (Ma − Ma0 )/ Ma0 .

(29)

˘

⌢ ⌢ Ja = (Ma − Ma )/(Ma + Ma ).

˘˘

(30) T

JaT Ja

≤ I. Then we haveMa = Ma0(I + La ), andLa La ≤ Hence, the overall fuzzy reliable control law considering the parameter variations and actuator faults can be represented by 2

u f (t ) = Mau(t ) =

∑ MaMj(ξ1(t ))Kajx(t ).

(31)

j =1

where Kaj is the corresponding actuator fault-tolerant feedback control gain matrix to be determined. Lemma 1. [17]. For a time-varying diagonal matrix Φ(t ) = 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.

RΦS + S T ΦT RT ≤ εRVRT + ε−1S T VS

(32)

Theorem 1. Consider the suspension system (24–26) with reliable fuzzy H∞ state feedback controller (31), the closed-loop system is asymptotically stable and satisfies z1(t ) 2 < γ w(t ) 2 for all w if there exist matrices Qj > 0 ,Kaj and any scalar εaij > 0 , εbj > 0 such that matrix inequalities (33), (34) and (35) are satisfied.

⎡ sym(A Q i j ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎡− I ⎢ ⎢⎣ *

Bwi

QjC1Ti + K ajTMaT0D1Ti

εaijB2i

* *

−γ 2I *

0 −I

0 εaijD1i

*

*

*

−εaijJa−1

*

*

*

*

+ B2iMa0Kaj )

K ajTMaT0 ⎤ ⎥ 0 ⎥ ⎥ 0 ⎥ < 0 i = 1, 2, j = 1, 2, ⎥ 0 ⎥ −1⎥ −εaijJa ⎦

ρ QjC2i ⎤ ⎥ < 0 i = 1, 2, j = 1, 2, 2 Q ⎥⎦ −z max

⎡− I + ε J bj a ⎢ ⎢ * ⎢ ⎢⎣ *

ρ Ma0Kaj 2 Q −umax

*

0

(34)

⎤ ⎥

ρ K ajTMaT0 ⎥ −εbjJa−1

(33)

⎥ ⎥⎦

< 0 j = 1, 2, (35)

Proof. Consider the Lyapunov–Krasovskii function as given by: 8

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V (t ) =

∑ xT (t )Pxj (t ) (36)

j =1

We take the derivative of V (t ) along the solution of the system: 2

V ̇ (t ) =

2

∑ ∑ Mi(ξ1(t ))xT (t ){[Ai

T T + B2iMa0(I + La )Kaj ]T Pj + Pj[Ai + B2iMa0(I + La )Kaj ]}x (t ) + x T (t )PB j wiω(t ) + ω (t )BwiPx j (t )

(37)

i =1 j =1

In order to establish that system (24–26) is asymptotically stable with a disturbance attenuation γ > 0 , it is required that the associated Hamiltonian

H (x, w, t ) = z1T (t )z1(t ) − γ 2ωT (t )ω(t ) + V ̇ (t ) < 0

(38)

H (x , w , 2

t) =

2 2 T T T T T + B2iMa0(I + La )Kaj ]T Pj + Pj[Ai + B2iMa0(I + La )Kaj ]}x (t ) + x T (t )PB j wiω(t ) + ω (t )BwiPx j (t ) − γ ω (t )ω (t ) + x (t )

∑ ∑ {[Ai i =1 j =1

[C1i + D1iMa0(I + La )Kaj ]T [C1i + D1iMa0(I + La )Kaj ]x (t ) ⎡ [A + B M (I + L )K ]T P + P [A + B M (I + L )K ] ⎤ 2i a 0 2i a 0 a aj j j i a aj ⎢ i ⎥ PB j wi ⎥ T ⎢ ξ (t ) = ∑ ∑ ξ (t ) +[C1i + D1iMa0(I + La )Kaj ] [C1i + D1iMa0(I + La )Kaj ] ⎢ ⎥ i =1 j =1 2 ⎥ T ⎢ BwiPj −γ I ⎦ ⎣ 2

2

T

T

T

(39)

T

where ξ(t ) = [ x (t ) ω (t )] , and the requirement that H (x, w, t ) < 0 for all ξ(t ) ≠ 0 is obtained

⎡ [A + B M (I + L )K ]T P + P [A + B M (I + L )K ] ⎤ 2i a 0 a aj j j i 2i a 0 a aj ⎥ ⎢ i PB j wi ⎢ +[C + D M (I + L )K ]T [C + D M (I + L )K ] ⎥<0 1 i 1 i a 0 a aj 1 i 1 i a 0 a aj ⎢ ⎥ T ⎢ Bwi Pj −γ 2I ⎥⎦ ⎣

(40)

and it is further equivalent to

⎡[A + B M K ]T P + P [A + B M K ] PB [C + D M K ]T ⎤ 2i a 0 aj j j i 2i a 0 aj j wi 1i 1i a 0 aj ⎥ ⎢ i ⎢* ⎥ −γ 2I 0 ⎢ ⎥ ⎣* ⎦ * −I ⎛⎡ PB ⎞T ⎡ PB ⎤ ⎤ j 2i ⎜ ⎢ j 2i ⎥ ⎟ ⎢ ⎥ +⎢ 0 ⎥La[ Ma0La 0 0] + ⎜⎢ 0 ⎥La[ Ma0La 0 0]⎟ < 0 ⎜ ⎟ ⎢⎣ D1i ⎥⎦ ⎝⎣⎢ D1i ⎥⎦ ⎠

(41)

By using Lemma 1, the inequality is obtained by

⎡ [A + B M K ]T P + P[A + B M K ] ⎤ 2i a 0 aj j i 2i a 0 aj T T T T ⎥ ⎢ i PB [ C + D M K ] + ε PB J D K M j wi 1i 1i a 0 aj aij j 2i a 1i aj a 0 ⎥ ⎢ +ε PB J B T P ⎢ aij j 2i a 2i j ⎥ ⎢ ⎥<0 * −γ 2I 0 0 ⎢ ⎥ T * * −I + εaijD1iJaD1i 0 ⎢ ⎥ ⎢ ⎥ * * * −εaijJ −1a ⎥⎦ ⎢⎣

(42)

⎧ ⎫ DefiningQj = Pj −1,Kaj = KajQj , and performing congruence transformation by pre- and post- multiplying diag ⎨ Pj −1 I I I ⎬, then ⎭ ⎩ the inequality (42) is equivalent to the inequality (33) in Theorem 1. Then we show that the suspension stroke, maximum dynamic force applied on the suspended motor, dynamic tyre force and maximum control force constraint are guaranteed. max z2(t ) 2 = max x T (t )(C2i + D2iKaj )T (C2i + D2iKaj )x (t ) t >0

t >0

−1

1 −1

2

−1

1

= max x T (t )Pj2 Pj 2 (C2i + D2iKaj )T (C2i + D2iKaj )Pj 2 Pj2 x (t ) t >0

<ρ 2

−1

2 ⋅θmax(Pj 2 (C2i + D2iKaj )T (C2i + D2iKaj )Pj 2 ) < z max I

max u(t ) 2 = max x T (t )[Ma0(I + La )Kaj ]T Ma0(I + La )Kajx (t ) t >0

t >0

(43) −1

2

−1

2 = ρ⋅θmax(Pj 2 [Ma0(I + La )Kaj ]T Ma0(I + La )KajPj 2 ) < umax I

(44)

where θmax(⋅) represents the maximal eigenvalue. Considering the Schur complement and Lemma 1, the inequalities (43) and (44) are 9

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equivalent to the inequalities (34) and (35) in Theorem 1. Therefore, all the conditions in Theorem 1 are satisfied. The proof is completed. Based on the above conditions, the fuzzy reliable H∞ controller can be designed with the minimal γ by solving the following convex optimization problem. min γ subject to LMIs (33), (34) and (35). The feedback gain matrix for the controller can be given by

⎞−1 ⎞⎛ 2 ⎛ 2 ⎜ ⎟ ⎜ K = Ma⎜∑ Mj (ξ1(t ))Kaj ⎟⎜∑ Mj (ξ1(t ))Qj ⎟⎟ ⎠ ⎠⎝ j =1 ⎝ j =1

(45)

If we assume that there is no actuator fault in the suspension system of the ADM-EV, the fuzzy H∞ controller is presented for the active suspension system based on the T-S fuzzy model method and then we have the following corollaries, which can be easily proved following the proof of Theorem 1. Corollary 2. Assume ρ is a prescribed positive scalar. Consider system (24–26) with the fuzzy controller, and then the closed-loop system is asymptotically stable and satisfies z1(t ) 2 < γ w(t ) 2 for all w if there are matrices Qj > 0 and Kaj satisfying. T T T⎤ ⎡ sym(A Q + B K ) B i j 2i aj wi Qj C1i + K ajD1i ⎥ ⎢ 2 ⎢ ⎥ < 0, i = 1, 2, j = 1, 2, * −γ I 0 ⎢ ⎥ ⎣ ⎦ * * −I

(46)

⎡− I ⎢ ⎢⎣ *

(47)

ρ (QjC2i + KajD2i )⎤ ⎥ < 0, i = 1, 2, j = 1, 2, 2 ⎥⎦ Q −z max

⎡− I ρ Kaj ⎤ ⎢ ⎥ < 0, j = 1, 2, 2 ⎢⎣ * −umax Q ⎥⎦

(48)

Moreover, if the fuzzy controller has a feasible solution, the feedback gain matrix for the controller can be given by

⎞−1 ⎞⎛ 2 ⎛ 2 ⎜ ⎟ ⎜ K = ⎜∑ Mj (ξ1(t ))Kaj ⎟⎜∑ Mj (ξ1(t ))Qj ⎟⎟ . ⎠ ⎠⎝ j =1 ⎝ j =1

(49)

Furthermore, if we assume that there is no variation of sprung mass and actuator faults, the conventional H∞ controller is presented by Corollary 3. Corollary 3. Assume ρ is a prescribed positive scalar. Consider the system in (11–13) with the conventional H∞ controller. The closed-loop system is asymptotically stable and satisfies z1(t ) 2 < γ w(t ) 2 for all w if there are matrices Q > 0 and Ka satisfying.

⎡ sym(AQ + B K ) B QC1T + K aT D1T ⎤ 2 a w ⎢ ⎥ 2 ⎢ ⎥ < 0, * −γ I 0 ⎢⎣ ⎥⎦ * * −I

(50)

⎡− I ⎢ ⎢⎣ *

(51)

ρ (QC2 + KaD2 )⎤ ⎥ < 0, 2 ⎥⎦ −z max Q

⎡− I ρ Ka ⎤ ⎢ ⎥ < 0, 2 ⎢⎣ * −umax Q ⎥⎦

(52)

The feedback gain matrix function for the controller can be given by

K = KaQ−1

(53)

4. Simulation results First, we consider a conventional H∞ controller with a guaranteed H∞ performance without taking into account the sprung mass variation and fault-tolerant control method. The minimum guaranteed closed-loop H∞ performance index γ = 8.875 and the controller gain matrix is given by

Kc = 104 × [−1.649 0.092 −2.448 −0.052 0.329 0.113]

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Next we apply T-S fuzzy H∞ controller Kf for the uncertain active suspension system defined in Corollary 2. It is shown that the minimum guaranteed closed-loop H∞ performance index isγ = 8.73, and the fuzzy H∞ controller gain matrix is

Kf = 104 × [−1.294 0.237 −2.236 −0.038 −0.230 0.111] Finally, we propose a reliable fuzzy H∞ controller Krf designed for the active suspension in ADM-EVs with actuator faults and uncertain parameters. The minimum guaranteed closed-loop H∞ performance index is obtained as γ = 29.6 and the reliable fuzzy controller gain matrix regarding 0%, 30%, 60%, and 90% actuator thrust loss are

Krf 1 = 104 × [ − 0. 386 0.994 −2.094 −0.041 −1.540 0.130 ]

Krf 2 = 104 × [ − 0. 270 0.696 −1.466 −0.029 −1.077 0.091] Krf 3 = 104 × [ − 0. 154 0.398 −0.838 −0.016 −0.616 0.052 ]

Krf 4 = 103 × [ − 0. 386 0.993 −2.095 −0.041 −1.540 0.013] The effectiveness of the proposed reliable fuzzy H∞ control method is validated through computer simulation using a bumpy road disturbance and random road excitation. The parameters of the ADM-EV are set as listed in Table 1. We assume that the maximum suspension deflectionz max = 100 mm, the maximum control force umax = 3000 N and the dynamic force applied on the bearings mai = 1. Fmax = 3000 N. Furthermore, the sprung mass is assumed to be within the range 238–442 kg andmai = 0.1, ⌢

˘

Fig. 6. Vehicle dynamic responses under bump road excitation. (a) Vehicle body acceleration response. (b) Suspension deflection. (c) Actuator force. (d) Tyre dynamic force.

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Fig. 7. In-wheel motor dynamic responses under bump road excitation. (a) Motor acceleration response. (b) Motor dynamic force.

Fig. 8. Bump responses of passive suspension and active suspension with T-S fuzzy controller for different sprung masses. (a) Frequency response. (b) Vehicle body

4.1. Bump road excitation In this work, the bumpy road excitation is introduced to analyse the dynamic response characteristics of the EV suspension, which is given by 2πV0 ⎧a l ⎪ 2 (1 − cos( l t )), 0 ≤ t ≤ V0 x 0 (t ) = ⎨ l ⎪ 0, t> V ⎩ 0

(54)

where a is the height of the bump and l is the length of the bump. Here we choose a=0.1 m, and l=2 m, and the vehicle forward 12

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Fig. 9. Tzw

∞

of active suspensions with T-S fuzzy controller and reliable fuzzy controller versus the uncertain parameter.

velocity at V0=30 km/h. Fig. 5 shows the frequency responses for a passive suspension, the active suspension with a conventional H∞ controller Kc and an active suspension with the T-S fuzzy controller Kf. It is observed from Fig. 5 that the two active suspensions perform much better than the passive suspension, and the active suspension with the T-S fuzzy controller performs the best among the three suspensions. Bump responses for the three suspensions are illustrated in Fig. 6. We can observe from Fig. 6(a) that the sprung mass acceleration of active suspensions is greatly reduced when compared with that of passive suspension. This indicates that the suspension performance is significantly improved compared to the passive suspension. The active suspension with T-S fuzzy H∞ controller achieves suspension performance that is very similar to the active suspension with conventional H∞ controller. In addition, it can be seen from Fig. 6(b–d) that the suspension deflection, tyre dynamic force and maximum control force constraints are guaranteed simultaneously thus demonstrating the advantage of the T-S fuzzy H∞ control method with the sprung mass variation. The in-wheel motor dynamic responses of the of passive suspension and active suspension systems with a conventional H∞ controller and the T-S fuzzy H∞ controller are shown in Fig. 7. It is clear that the active suspension systems controlled by the conventional controller and T-S fuzzy controller significantly reduce the dynamic force applied on the motor bearings, which suggests that motor wear could be reduced. Fig. 7(a) shows the in-wheel motor acceleration of three suspensions, i.e., passive suspension and active suspension with conventional H∞ controller, and active suspension with the T-S fuzzy H∞ controller. From the figure we can see that the in-wheel motor vibrations of the two active controllers are the same and marginally larger than that of passive suspension system. Active suspension has a small adverse influence on the in-wheel motor performance that can be ignored. In order to illustrate the effect of sprung mass variation, Fig. 8 illustrates the frequency response and bump response for active suspension with the T-S fuzzy H∞ controller and for passive suspension for two values of the sprung mass (238 kg and 442 kg). It is observed from Fig. 8(a) that maximum singular values of active suspensions with the T-S fuzzy controllers are much smaller than those of passive suspensions, demonstrating the effectiveness of T-S fuzzy controller in the presence of parameter uncertainty. Furthermore, Fig. 8(b) shows that the sprung mass accelerations of active suspensions have lower peaks and shorter settling times than those of passive suspensions. This indicates that the T-S fuzzy controller performs significantly better than the passive 13

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Fig. 10. Frequency responses of active suspensions with T-S fuzzy controller and reliable fuzzy controller.

suspension despite the sprung mass variation. In order to illustrate the effectiveness of the reliable fuzzy H∞ controller, frequency analysis and bump analysis of the passive suspension, the active suspension with fuzzy controller and an active suspension with the reliable fuzzy controller are shown with 0%, 30%, 60%, and 90% actuator thrust loss. Fig. 9 presents the Tzw ω of the active suspension with the T-S fuzzy controller Kf and reliable fuzzy H∞ controller Krf versus the variation of sprung mass parameter ms with 0%, 30%, 60%, and 90% actuator thrust loss, respectively. It can be observed from Fig. 9 that the reliable fuzzy controller Krf always yields a smaller closed-loop H∞ norm than the fuzzy controller Kf despite the change of the actuator thrust loss. This shows that the designed reliable fuzzy controller Krf achieves significantly better closed-loop H∞ performance than the fuzzy H∞ controller Kf when actuator failure occurs. However, when the actuator thrust loss increases, the maximum singular value of both controllers is increased; which indicates a poorer performance of the suspension when actuator thrust loss occurs. Figs. 10 and 11 show the bump responses of the closed-loop systems with the fuzzy controller Kf and reliable fuzzy controller Krf with 0%, 30%, 60%, and 90% actuator thrust loss in frequency and time domains, respectively. The simulation results show that the designed reliable fuzzy controller is capable of providing better performance than the fuzzy controller can in both frequency and time domains, and the reliable fuzzy controller can maintain a satisfactory performance with the variational sprung mass and the partial fault in the actuator.

4.2. Random road excitation A random road excitation is used to demonstrate the effectiveness of the proposed reliable fuzzy H∞ control method applied on EV suspension. The disturbance is assumed to be zero-mean white noise with identity power spectral density. Figs. 12 and 13 show the vehicle dynamic responses and in-wheel motor responses of passive suspension, active suspension with conventional H∞ controller and T-S fuzzy H∞ controller under random road excitation. From Fig. 12 we can see that active suspensions greatly reduce the vehicle sprung mass acceleration compared to passive suspension. The active suspension performance with T-S fuzzy controller is marginally better than that of conventional H∞ controller. Suspension deflection, actuator force and tyre dynamic force are guaranteed simultaneously. The motor acceleration of active suspensions is a little bigger than the passive suspension, which has small adverse influence on the vehicle performance. Dynamic force applied to the in-wheel motor of 14

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Fig. 11. Body acceleration of active suspensions with T-S fuzzy controller and reliable fuzzy controller under bump road excitation.

active suspension is marginally reduced, demonstrating better motor performance than that of passive suspension. The Root Mean Square (RMS) comparison of vehicle dynamic response under random road excitation is shown in Table 2. It is clear that the active suspension with T-S fuzzy controller achieves marginally better suspension performance than the passive suspension, which is a little better than the active suspension conventional H∞ controller. Random responses of passive suspension, active suspension with T-S fuzzy controller and reliable fuzzy controller are shown in Fig. 14 in the presence of 0%, 30%, 60%, and 90% actuator thrust loss. The body acceleration of active suspension with reliable fuzzy controller is much the same with that of T-S fuzzy H∞ controller when there is 0% actuator thrust loss. Table 3 shows the RMS comparison of sprung mass acceleration with different actuator thrust losses. We can observe that the proposed reliable fuzzy controller is robust to parameter uncertainties and actuator faults according to Table 3. With an increase in the actuator thrust loss, reliable fuzzy controller reveals better performance than the T-S fuzzy controller, which shows that reliable fuzzy controller is able to guarantee a better performance in spite of actuator faults and parameter uncertainties under random road excitation.

5. Conclusion In this paper, in order to deal with the suspension vibration and in-wheel motor bearing wear, a multi-objective robust H∞ reliable fuzzy control for active suspension system of IWM EV with dynamic damping is proposed. First, the mathematical EV suspension system model was developed and the variation of sprung mass was represented by a T-S fuzzy model. Then a reliable fuzzy state feedback controller was designed for the T-S fuzzy model to cope with possible actuator faults, sprung mass variation and control input constraints. Finally, the effectiveness of the proposed controller was demonstrated through computer simulation. The results show that the suspension performance of EV with “advanced-dynamic-damper-motor” (ADM-EV) is better than a conventional in-wheel motor driven EV (IWD-EV). Comparison of the performance of the active suspension and the passive suspension shows that the proposed reliable fuzzy H∞ controller offers the best suspension performance. Meanwhile, when different actuator thrust losses occur, the proposed reliable fuzzy H∞ controller achieves a significantly better closed-loop H∞ performance than the fuzzy H∞ controller does. 15

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Fig. 12. Vehicle dynamic responses under random road excitation. (a) Vehicle body acceleration response. (b) Suspension deflection. (c) Actuator force. (d) Tyre dynamic force.

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Fig. 13. In-wheel motor dynamic responses under random road excitation. (a) Motor acceleration response. (b) Motor dynamic force.

Table 2 RMS comparison of vehicle response under random road excitation. Suspension type

Passive Conventional H∞ controller T-S fuzzy controller

Vehicle performance (RMS)

ẍs

ẍd

xs(t ) − xu(t ) zmax

kt (xu(t ) − xg(t )) (ms + mu + md )g

Fdynamic Fmax

u u max

2.413 0.987 0.865

17.124 23.938 23.839

0.167 0.174 0.177

0.468 0.556 0.550

0.402 0.294 0.295

0.283 0.278

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Fig. 14. Body acceleration of active suspensions with T-S fuzzy controller and reliable fuzzy controller under random road excitation.

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