Robust state-feedback control design for active suspension system with time-varying input delay and wheelbase preview information

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Journal of the Franklin Institute 356 (2019) 1899–1923 www.elsevier.com/locate/jfranklin

Robust state-feedback control design for active suspension system with time-varying input delay and wheelbase preview information Hui Pang a,∗, Yan Wang a, Xu Zhang a, Zeren Xu b a School

of Mechanical and Precision Instrument Engineering, Xi’an University of Technology, Xi’an 710048, China Center for Automotive Research (ICAR), Clemson University, Greenville, SC 29607, USA

b International

Received 13 May 2018; received in revised form 27 December 2018; accepted 8 January 2019 Available online 19 January 2019

Abstract This paper develops a robust state-feedback controller for active suspension system with time-varying input delay and wheelbase preview information in the presence of the parameter uncertainties. By employing system augmentation technique, a multi-objective control optimization model is first established and then this controller design is converted to a static full-state feedback controller design with robust H∞ and generalized H2 performance, wherein the model-dependent control gain is evaluated by transforming the related nonlinear matrix inequalities into their corresponding linear matrix inequality forms based on Lyapunov theory, and then LMI (Linear-Matrix-Inequality) technique is applied to solve and obtain the desired controller. A numerical simulation case is finally provided to reveal the effectiveness and advantages of the proposed controller. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Active suspension systems are significantly effective in isolating, absorbing or dissipating vibrations transferred from various road disturbances to vehicle body, and in achieving a great improvement of ride quality and road handling [1–3]. Over the past decades, a large number ∗

Corresponding author. E-mail addresses: [email protected], [email protected] (H. Pang).

https://doi.org/10.1016/j.jfranklin.2019.01.011 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

Nomenclature ms Iy φ φ¨ muf mur msmin msmax cf cr kf kr kt f ktr a b τ min τ max Fti i Fratio v zc z¨c zmax zsf zsr zuf zur zrf zrr zi uf (t-τ (t)) ur (t-τ (t)) τ (t) ui ( t) g 02,2 I2 CG i=f

mass of vehicle body; rotary inertia of vehicle body; pitch angular displacement; pitch angular acceleration; sprung mass of front suspension; unsprung mass of rear suspension; the lower of vehicle body mass; the upper of vehicle body mass; damping coefficient of front suspension; damping coefficient of rear suspension; stiffness coefficient of front suspension; stiffness coefficient of front suspension; stiffness coefficient of front tire wheel; stiffness coefficient of rear tire wheel; distance from CG to the front suspension; distance from CG to the rear suspension; the lower of bounded input delays; the upper of bounded input delays; tire static load; ratio between the tire dynamic and static load; vehicle forward speed; vertical displacement of vehicle body; vertical velocity of vehicle body; maximum value of suspension travel limit; sprung-mass displacement of the front wheel; sprung-mass displacement of the rear wheel; unsprung-mass displacement of the front wheel; unsprung-mass displacement of the rear wheel; road disturbance of the front wheel; road disturbance of the rear wheel; suspension dynamic displacement; control force of the front actuator with input delay; control force of the rear actuator with input delay; time-varying input delay of actuator; general control force generated by actuator; gravitational acceleration; zero matrix with 2 × 2 order; identify matrix with 2 × 2 order; center of gravity; r for front and rear wheel

of control methods such as H∞ control [4–6], adaptive control [7,8], slide-mode control [9–13] and neural network control [14] have been proposed and most of the studies have been

H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

1901

applied into vehicle suspension control. Especially, the robust H∞ control has been extensively reported in [4–6,15–18], because this control approach can improve suspension’s ride comfort and restrain its safety performance constraints within a certain range. However, it is not easy to make a better trade-off between the ride comfort and safety constraints on suspension system since these performance requirements are usually conflicting, improving ride quality will lead to a larger suspension working space, which will impose some negative effects on the chassis layout and then deteriorates vehicle handling stability and maneuverability [19]. Additionally, due to the electrical and electromagnetic characteristics of various controllers and sensors, time delay is unavoidably encountered in the controlled channel of active suspension systems, which can degrade the control performance or even lead to an unstable state of the closed-loop systems. Consequently, there is a growing interest in utilizing the robust H∞ control strategy to overcome the problem, and a few research have been reported in [15–18,20,21] and the references therein. Specifically, the authors in [15,16] addressed the H∞ control problem for the active suspension system with input time delay, and the input delay was assumed to be uncertain time invariant within a known constant upper bound, while the parameter uncertainties caused by vehicle load were not taken into consideration. In [17,18], the control problem of active suspension system with the frequency band constraints and the actuator input delay has been investigated and the controller design was cast into a convex multi-objective optimization problem with LMI technique, which inspired our study. Summarizing the aforementioned studies, on the one hand, the previous work focused on H∞ controllers design with constant input delays, while the time-varying delays are usually ignored. In practice, the input delays are always time-varying and are constrained in a fixed range, and the lower of bounded input delays may not be zero. Therefore, it is still a challenging and interesting issue in developing the control strategies for active suspension systems with time delays. Moreover, for a specific active suspension system with time-varying input delays, some research achievements [22–25] have been obtained as a result of the exploration in this field, which provides some motivation for our study. On the other hand, as a popular and new control method, preview control was first proposed in wheeled vehicles [26], and this control method is generally divided into two types: the first is to install a sensor in front of the front wheel, named as look-ahead preview and the second is to estimate the road profile of the rear wheel from the response of the front wheel with assumption that the road inputs at the rear wheel are delayed after the inputs of the front wheel, which is referred to wheelbase preview. Many scholars focused on the look-ahead preview [27–31], nevertheless, the increasing cost of laser sensors and measurement complexities have been a handicap for look-ahead preview in vehicle applications [3,31–36]. It should be pointed out that the road input profile at the front wheel can be employed as the future road input excitation of the rear wheel, which will lead to a better performance than look-ahead preview control [34]. For vehicle suspension control, earlier preview control methods are first used on linear-quadratic-Gaussian (LQG) and linear quadratic regulators (LQR) control theory [37–39]. However, the possible unknown disturbances imposed on the controlled plant are not considered [27,28,32]. According to the control effect, the vehicle suspension performance with preview control was shown to be better than the suspension without preview control [31]. Besides, the multi-objective control problems have been addressed in [40–44] to achieve a better trade-off among ride quality, handling stability, as well as road holding and actuator saturation. However, the timevarying input delay, wheelbase preview information and the system parameter uncertainties are basically not taken into consideration together when designing a control scheme.

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Fig 1. Half-vehicle active suspension model.

To that end, this paper develops a robust state-feedback controller for active suspension system with time-varying input delay and wheelbase preview information in the presence of the parameter uncertainties. The main contributions of this study are described as follows: By employing the system-augmentation technique, a multi-objective control optimization model is first established with taking the time-varying input delay, the wheelbase preview information, the uncertainties of vehicle body mass, as well as the safety constraints of active suspension into consideration, simultaneously. Based on this model, a new control design procedure of the proposed state-output feedback controller with robust H∞ and generalized H2 performance was derived to ensure that the closed-loop system is asymptotically stable. Specifically, the model-dependent control gain for the designed controller is evaluated by transforming the related nonlinear matrix inequalities into their corresponding linear matrix inequality forms, and then LMI technique is used to solve and obtain the desired controller. The rest of this paper is organized as follows. In Section 2, system model and problem formulation are introduced. Section 3 presents a robust state-feedback controller design using H∞ and generalized H2 norm approach. In Section 4, simulation results are provided to illustrate the effectiveness and advantages of the proposed controller. Concluding remarks are summarized in Section 5. Notations. In this article, P is a positive definite matrix (P > 0) or a positive semi-definite matrix (P ≥ 0), sym {P } stands for P +P T , asterisk (∗) is used to denote a symmetric block matrix, and diag{…} stands for a block-diagonal matrix. 2. System model and problem formulation 2.1. Half-vehicle active suspension model To capture the essential characteristics of suspension system and simplify the controller design, a half-vehicle model with four degree-of-freedom (DOF) that is widely used in [24,31,35,36] is shown in Fig. 1. Based on Newton’s second law, the dynamic equations for this model are as follows:   Ms q¨ (t ) = GKs (zu (t ) − zs (t ) ) + GCs z˙u (t ) − z˙s (t ) + Gu (1)   Mu z¨u (t ) = Ks (zs (t ) − zu (t ) ) + Cs z˙s (t ) − z˙u (t ) + Ku (zr (t ) − zu (t ) ) − u (2)

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where q(t ) = [zc (t ), φ(t )]T , zs (t ) = [zs f (t ), zsr (t )]T and zu (t ) = [zu f (t ), zur (t )]T ; zr (t ) = [zr f (t ), zrr (t )]T is the road displacement input vector, u = [u f (t − τ (t )), ur (t − τ (t ))]T is the control force vector for active suspension system. In Eqs. (1) and (2), the corresponding coefficient matrices are given as follows:         0 0 0 ms 0 mu f c k , Mu = , Cs = f , Ks = f , Ms = 0 Iy 0 mur 0 cr 0 kr     0 1 k 1 ,G= . Ku = t f 0 ktr −a b Define x(t ) = [(zs (t ) − zu (t ))T , z˙s (t ), (zu (t ) − zr (t ))T ,z˙u (t )]T as the state variable vector of the closed-loop system, then the system state-space equation is given by x˙ (t ) = Ax(t ) + B1W (t ) + B2 u(t − τ (t ))

(3)

where W (t ) = [z˙r f (t ), z˙rr (t )] , and the corresponding coefficient matrices in Eq.(3) are respectively as follows: ⎡ ⎤ 02,2 I2 02,2 −I2 ⎢−GT M −1 GKs −GT M −1 GCs 02,2 GT Ms−1 GCs ⎥ s s ⎥, A=⎢ ⎣ ⎦ 02,2 02,2 02,2 I2 Mu−1 Ks Mu−1Cs −Mu−1 Ku −Mu−1Cs ⎡ ⎤ ⎡ ⎤ 02,2 02,2 ⎢02,2 ⎥ ⎢ T −1 ⎥ ⎥, B 2 = ⎢G M s G ⎥. B1 = ⎢ ⎣−I2 ⎦ ⎣ 02,2 ⎦ 02,2 Mu−1 T

Since the payloads and/or passenger numbers are often changeable, ms will accordingly change in a certain range, thus ms is selected as the uncertain parameters of active suspension system with satisfying ms min ≤ms ≤ms max . To guarantee a better dynamic performance and meet the requirements of suspension safety performance, the following aspects should be considered [30]: (1) Ride comfort:The designed controller can achieve the minimization of the vertical ac¨ thus to isolate vibrations caused by celeration (z¨c ) and pitch angular acceleration (φ) the road disturbances in the presence of the uncertain parameters ms in system (15). (2) Safety performance requirements: By employing the proposed controller, the safety performance requirements such as road holding ability, suspension dynamic displacement limit and actuator input saturation can be ensured, which are described as follows: (a) Road holding ability. The tire remains in the uninterrupted contact of wheels to rough road, in other words, the dynamic loads of the front and rear tire should not i exceed their corresponding static loads, i.e. Fratio = kti (zui − zri )/Fti < 1, wherein kti (zui − zri ) is the tire dynamic loads, Fti are expressed by Ft f = (bms g + (a + b)mu f g)(a + b)−1 (4) Ftr = (ams g + (a + b)mur g)(a + b)−1 (b) Suspension displacement limit. Due to the limitation of the suspension mechanical structure, it should be restrained within its allowable maximum value. This constraint condition is described by zi = |zsi (t ) − zui (t )| ≤ zmax (i = f , r)

(5)

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(c) Actuator input saturation. Due to the limitation of the actuator power, the actuator input saturation should be considered, that is |ui (t )| ≤ umax (i = f , r)

(6)

2.2. Comprehensive suspension model based on wheelbase preview control Theoretically, the primary factor of restraining the improvement of active suspension’s performance is a lack of sufficient road information. The basic principle of preview control for active suspension system is that the road input profile at the rear wheels can be estimated by the road input information at the front wheels. Moreover, the road input of the front suspension can be incorporated as the preview information concerning the road profile of the rear suspension into the controller design with the purpose of improving the control effect. It is obvious that there exists a time lag d = (a + b)/v for the road velocity input at the rear suspension compared to the front suspension, that is z˙rr (t ) = z˙r f (t − d ). The relationship between the front and rear road velocity input signal can be described by Laplace transfer function as

 z˙rr (t ) = e−ds L (7) z˙r f (t ) To approximate e−ds as a finite-order transfer function by using the Padé approach [3], expressed by e−ds =

Pmn (−ds) Qmn (−ds)

(m+n− j )! m! m j where Pmn (−ds ) = (m+ j=0 j!(m− j )! (−ds ) , and Qmn (−ds ) = n )! Without loss of generality, we choose n = m, such that

(8) n! (m+n )!

n

(m+n− j )! j j=0 j!(n− j )! (ds ) .

z˙rr (t ) (−1 )m sm + (−1 )m−1 bm−1 S m−1 + · · · + (−1 )b1 s + b0 = z˙r f (t ) sm + bm−1 sm−1 + · · · + b1 s + b0

(9)

2m−k )! where bk = k!((m−k (k = 0, 1, . . . , m − 1). )!d m−k For further simplicity, one can set m = 2, thus according to Eq. (9), we have

z˙rr (t ) b2 s 2 − b1 s + b0 ≈ z˙r f (t ) b2 s 2 + b1 s + b0

(10)

where b2 = 1, b1 = 6/d , b0 = 12/d 2 . Define an additional state vector ηd (t ) = [η1 (t ), η2 (t )]T , Eq. (10) can be rewritten as a state-space equation form given by η(t ˙ ) = Aη ηd (t )+Bη z˙r f (t ) (11) z˙rr (t ) = z˙r f (t − d ) = Cη ηd (t ) + z˙r f (t ) In Eq. (11), the coefficient matrices are as follows:    T 1 0 , Bη = −2b1 , 6b0 , Cη = [1, 0]. Aη = −b0 −b1 To obtain the state-space equation for active suspension system with wheelbase preview information, the weighed dynamics performances vector z1 (t ) and the normalized constraint

H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

output vector z2 (t ) for the closed-loop system (3) are respectively defined as ⎡ ⎤   zs (t ) − zu (t ) q1 z¨c (t ) ⎣ ⎦ z1 (t ) = ¨ ) , z2 (t ) = Fk (zu − zr ) . q2 φ(t u(t )

1905

(12)

where q1 and q2 are the weighted coefficients to be determined, respectively. Normally, ac√ k cording to literature [45], q1 is chosen as 1 and q2 = q1 ab, and Fk = diag( Ftt ff , Fktrtr ). Choosing xg (t ) = [x(t ), ηd (t )]T as the augmented state vector, and synthesizing Eqs. (3) and (11), the augmented control model of active suspension system with time-varying input delay and wheelbase preview information can be obtained as ⎧ ⎨x˙g (t ) = Ag (λ)xg (t ) + Bg1 w(t ) + Bg2 (λ)u(t − τ (t )) z1 (t ) = C1 (λ)xg (t ) + D1 (λ)u(t − τ (t )) (13) ⎩ z2 (t ) = C2 xg (t ) + D2 u(t − τ (t )) where the corresponding augmented coefficient matrices are given as follows:   B1 Dη A , w(t ) = z˙r f , Ag (λ) = 02,8 Aη     BE B2 , Bg1 = 1 η , Bg2 (λ) = Bη 02,2  C1 (λ) = −Ms−1 GKs ⎡ I2 02,2 C2 = ⎣02,2 02,2 02,2 02,2

−Ms−1 GCs

02,2

⎤ 02,2 02,2 02,2 Fk 02,2 02,2 ⎦, 02,2 02,2 02,2 ⎡ ⎤ 02   D1 (λ) = Ms−1 , D2 = ⎣02 ⎦. I2   0 0 , Eη = [1, 1]T . where Dη = 1 0

Ms−1 GCs

 02,2 ,

It should be pointed out that the above-mentioned matrices Ag (λ), Bg2 (λ), C1 (λ) and D1 (λ) are all matrix functions with respect to the uncertain parameter vector λ = [λ1 λ2 . . . λr ], which will be presented in Section 4. Moreover, these seven coefficient matrices are included in a convex polytope , that is ⎧ ⎫   Ag, Bg2 , C1 , D1 (λ) : ⎪ ⎪ ⎪ ⎪ ⎪ r   ⎪ ⎪ ⎪  ⎨ ⎬ Ag, Bg2 , C1 , D1 (λ) = λi Agi , Bg2i , C1i , D1i , = . i=1 ⎪ ⎪ r ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ λi = 1, λi ≥ 0, i = 1, . . . , r. ⎩ ⎭ i=1

Currently, according to the control requirements of active suspension system in (Eq. 13), the designed robust state-feedback controller is proposed as u(t ) = K xg (t )

(14)

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where K is the designed controller’s gain matrix to be calculated. Substituting Eq. (14) into Eq. (13) yields to ⎧ ⎨x˙g (t ) = Ag (λ)xg (t ) + Bg1 w(t ) + Bg2 (λ)K xg (t − τ (t )) z1 (t ) = C1 (λ)xg (t ) + D1 (λ)K xg (t − τ (t )) (15) ⎩ z2 (t ) = C2 xg (t ) + D2 K xg (t − τ (t )) Here, the robust controller design in Eq. (15) can be summarized as follows [46]: (1) The closed-loop control system (15) is asymptotically stable; (2) Given ∀w(t ) ∈ L2 [0, +∞ ), the H∞ norm of the transfer function Tz1 w from w(t ) to z1 (t ) should satisfy with Eq. (16) under zero initial conditions;   Tz w  = sup z1 (t ) 2 < γ∞ 1 ∞ w∈L 2 w(t ) 2 where γ∞ is a minimized positive constant and z1 (t ) 2 =

(16) 

∞ T 0 z1 (t )z1 (t )dt .

(3) Given ∀w(t ) ∈ L2 [0, +∞ ) and the positive constant γ2 , the generalized H2 norm of the transfer function Tz2 w from w(t ) to z2 (t ) should be satisfied under zero initial conditions, which is expressed by   Tz w  2

G H2

= sup w∈L 2

z2 (t ) ∞ < γ2

w(t ) 2

(17)

 ∞ 2 where w(t ) 2 = 0 w (t )dt and z2 (t ) ∞ = max1≤ j≤6 |z2 j (t )|, and z2 j (t ) represent for the definite constraint index in vector z2 (t ). Remark 1. For the closed-loop system (15), the robust H∞ and generalized H∞ norm can be utilized to describe the system performance indicators in the presence of the parameters uncertainties. Moreover, the H∞ and the generalized H2 norm should be restrained within a certain range under energy-bounded perturbations. 3. Robust H∞ /generalized H2 controller design To solve the robust control design problem described in the above section, the following theorem is developed. Theorem 1. The closed-loop system (15) is asymptotically stable with zero initial condition and the given performance indexes γ∞ > 0 and γ2 > 0 is satisfied with the H∞ and the generalized H2 performance requirements defined in Eqs. (16) and (17), respectively, if there exist the positive definite matrices as P¯ > 0, Q¯ 1 > 0, Q¯ 2 > 0, S¯1 > 0, S¯2 > 0, K¯ , X¯ j , Y¯ j and M¯ j (j = 1, 2,..., r) such that the following matrix inequalities hold:  ¯ <0 (18) ζ ii

 ¯ ζij

<0

(19)

H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923



 T P¯C 2 >0 γ22 I γ2 2

P¯ ∗

∞

 where ¯ ζ ii , ⎡¯ ii ⎢  ⎢∗ ¯ =⎢ ⎢∗ ζ ii ⎣∗ ∗ ⎡  ¯ ζij

⎢ ⎢ ⎢ =⎢ ⎢ ⎣

1907

(20)

 ¯

ςi j (ζ

= 1, 2; i < j, i = 1, . . . , r − 1, √ ¯ AT AI τmin X¯ i τmax − τmin ζ W gi S¯1 − 2P¯ 0n,n 0n,2n ∗ S¯2 − 2P¯ 0n,2n ¯1 ∗ ∗ Y ∗ ∗ ∗ √

¯ ii +  ¯ jj  ∗ ∗ ∗ ∗

  √ τmin X¯ i + X¯ j   2 S¯1 − 2P¯ ∗ ∗ ∗

j = i + 1, . . . , r) is expressed as ¯ zT ⎤ W 1i 0n,2 ⎥ ⎥ (21) 0n,2 ⎥ ⎥ ⎦ 0 2n,2

−I2

√ τmax − τmin ¯ ζ

 ¯ AT +W ¯ TA AI W g g

¯ zT + W ¯ zT W 1i 1j

 0n,n  2 S¯2 − 2P¯ ∗ ∗

0n,2n 0n,2n ¯1 2Y ∗

0n,2 0n,2 02n,2 −2I2

j

i

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(22) where

! "   2 ¯ ii = sym W ¯ AT WB + Z¯ iWZ + WQT QWQ1 + WQT QWQ2 − γ∞  WwT Ww , gi 1 2 1

2

    √ ¯ Agi = Agi P¯ 0n,2n Bg2i K¯ Bg1 , AI = √τmin I10 τmax − τmin I10 , W ¯ 1 = diag{−S1 − S2 }, K¯ = K P¯ , Y     ¯ ¯ ¯ z1i = C1i P¯ 02,2n D1i K¯ 0 , Z j = X j Y¯ j M¯ j , W 1 = Y¯i , 2 = M¯ i , ¯ 1 = Y¯i + Y¯ j + ¯ 2 = M¯ i + M¯ j , 

Q= 1

 Q¯ 1 ∗

   0n,n Q¯ 2 Q , = ¯ −Q1 2 ∗

0n,n −Q¯ 2



Proof. Let τ (t ) = τmin + η(t ), wherein 0 ≤ η(t ) ≤ τmax − τmin , thus the system (15) is equivalent to ⎧ ⎨x˙g (t ) = Ag (λ)xg (t ) + Bg1 w(t ) + Bg2 (λ)K xg (t − τmin − η(τ )) z1 (t ) = C1 (λ)xg (t ) + D1 (λ)K xg (t − τmin − η(τ )) (23) ⎩ z2 (t ) = C2 xg (t ) + D2 K xg (t − τmin − η(τ )) Consider the following Lyapunov–Krasovskii functional in Eq. (24) V (t ) = V1 (t ) + V2 (t ) + V3 (t ) where V1 (t ) = xTg (t )P xg (t ),

(24)

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# V2 (t ) =

#

t

t−τmax

# V3 (t ) =

xTg (s)Q1 xg (s)ds +

#

0

−τmin

t

t+θ

t−τmin

t−τmax

# x˙Tg S1 x˙g (s)dsdθ +

xTg (s)Q2 xg (s)ds, #

−τmin

−τmax

t

t+θ

x˙Tg (s)S2 x˙g (s)dsdθ .

The time derivative of Eq. (24) becomes V˙ (t ) = V˙1 (t ) + V˙2 (t ) + V˙3 (t )

(25)

where V˙1 (t ) = 2xTg (t )P x˙g (t )

V˙2 (t ) = xTg (t )Q1 xg (t ) − xTg (t − τmin )Q1 xg (t − τmin ) + xTg (t − τmin )Q2 xg (t − τmin ) − xTg (t − τmax )Q2 xg (t − τmax )

V˙3 (t ) = x˙Tg (t )[τmin S1 + (τmax − τmin )S2 ]x˙g (t ) − # −

t−τmin

t−τmax

#

t−τmin

#

x˙Tg (s )S1 x˙g (s )ds

x˙Tg (s )S2 x˙g (s )ds #

= x˙Tg (t )[τmin S1 + (τmax − τmin )S2 ]x˙g (t ) − −

t

t−τmin

t −τmax −η(t )

x˙Tg (s )S2 x˙g (s )d s −

t

x˙Tg (s )S1 x˙g (s )ds t−τmin # t −τmin −η(t ) x˙Tg (s )S2 x˙g (s )d s t−τmax

To obtain the designed controller, we need three steps to fulfill the proof of Theorem 1. Step 1: To verify the asymptotical stability of system (15), assuming w(t ) = 0, and



  choosing the matrices with appropriate dimensions as X (λ)= ri=1 λi X i , Y = ri=1 λiY i ,

 M = ri=1 λi M i , one can get the following expressions based on Newton–Leibniz formula.

 # t

T

1 = 2ξ (t )X (λ) xg (t ) − xg (t − τmin ) − x˙g (s )ds = 0 (26) t−τmin

#

T

2 = 2ξ (t )Y (λ) xg (t − τmin ) − xg (t − τmin − η(t ) ) −

t−τmin

t −τmin −η(t )

#

T

3 = 2ξ (t )M (λ) xg (t − τmin − η(t ) ) − xg (t − τmax ) −

 x˙g (s )ds = 0

t −τmin −η(t )

 x˙g (s )ds = 0

t−τmax

T

where ξ (t ) = [xTg (t ) xTg (t − τmin ) xTg (t − τmax ) xTg (t − τmin − η(t ) )]. Synthesizing Eqs. (25)–(28) and scaling these formulas properly, we have

(27)

(28)

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1909



T

T

T

−1 V˙ (t ) ≤ ξ (t ) (λ) + τmin X (λ)S−1 1 X (λ) + η(t )Y (λ)S2 Y (λ) 

T +(τmax − τmin − η(t ) )M (λ)S−1 M λ) ξ(t ) ( 2  T   T  # t

ds − ξ (t )X (λ) + x˙Tg (s )S1 S−1 X λ) ξ( t + S x ˙ s ( ) ( ) 1 g 1 t−τmin   T  # t−τmin  T

ds − ξ (t )Y (λ) + x˙Tg (s )S2 S−1 Y λ) ξ( t + S x ˙ s ( ) ( ) 2 g 2 t−τmin t −τmin −η(t )

# −

t−τmax

 T   T 

T −1 ξ (t )M (λ) + x˙g (s )S2 S2 M (λ)ξ(t ) + S2 x˙g (s ) ds

(29)

By further transformation, we obtain 

T

T

T

−1 ˙ V (t ) ≤ ξ (t ) (λ) + τmin X (λ)S−1 1 X (λ) + η(t )Y (λ)S2 Y (λ) 

T −1 + (τmax − τmin − η(t ) )M (λ)S2 M (λ) ξ(t ) 



T

T

T

η(t ) −1 −1 = ξ (t ) (λ) + τmin X (λ)S1 X (λ) + (τmax − τmin )Y (λ)S2 Y (λ) ξ(t ) τmax − τmin 

T

T τmax − τmin − η(t ) + ξ (t ) (λ) + τmin X (λ)S−1 1 X (λ) τmax − τmin 

T + (τmax − τmin )M (λ)S−1 M λ) ξ(t ) (30) ( 2 where

T $



T ˆ QT Qˆ 2W ˆ Q2 (λ) = sym W Ag PW B + Z (λ)W Z + W Q1 Q1W Q1 + W 2



+ Wˆ ATg [τmin S1 + (τmax − τmin )S2 ]W Ag ,  Q1 Q1 = ∗

 

0n,n Q2 , Q2 = −Q1 ∗

 W Ag = Ag (λ)

0n,2n

%

Z (λ) = X (λ)

Y (λ)

 0n,n , −Q2

  Bg2 (λ)K , W B = In

 0n,3n ,

&

M (λ) ,

⎡

⎤    In − In 0n,2n

In 0n,3n 0n,n ⎣ ⎦ , W Q2 = W Z = 0n,n In 0n − In , W Q1 = 0n,n In 0n,2n 0n,2n 0n,2n − In In

Since

η(t ) τmax −τmin

> 0,

τmax −τmin −η(t ) τmax −τmin

T

In In

 0n,2n . 0n,n

> 0, if the following inequalities hold:

T

−1 (λ) + τmin X (λ)S−1 1 X (λ) + (τmax − τmin )Y (λ)S2 Y (λ) < 0

(31)

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H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923



T



T

−1 (λ) + τmin X (λ)S−1 1 X (λ) + (τmax − τmin )M (λ)S2 M (λ) < 0

(32)

We have V˙ (t ) < 0 with w(t)=0, and according to Lyapunov theory, system (15) can reach its asymptotical stability state. Step 2: To ensure that H∞ performance index of the control system (15) satisfies with

Tz1 w ∞ < γ∞ under zero initial condition, given ∀w(t ) ∈ L2 [0, +∞ ), it is assumed that V (t )|t=0 = 0 and consider the following index Jz1 w as # ∞  T  2 T z1 (t )z1 (t ) − γ∞ Jz1 w = w (t )w (t ) dt (33) 0

We need to prove that # ∞  T  2 T z1 (t )z1 (t ) − γ∞ Jz1 w ≤ w (t )w (t ) dt + V (t )|t=∞ − V (t )|t=0 #0 ∞  T  2 T z1 (t )z1 (t ) − γ∞ = w (t )w (t ) + V˙ (t ) dt < 0

(34)

0

This is equivalent to validate 2 T V¯ (t )= zT1 (t )z1 (t ) − γ∞ w (t )w (t ) + V˙ (t ) < 0

(35)

From Eqs. (29) and (30), we have  2 T V¯ < ξ T (t ) (λ) + WzT1 Wz1 − γ∞ WwT Ww + τmin X (λ)S−1 1 X (λ)

 T −1 T + η(t )Y (λ)S−1 2 Y (λ) + (τmax − τmin − η(t ) )M (λ)S2 M (λ) ξ(t ) # t  T   T  T ξ (t )X (λ) + x˙Tg (s )S1 S−1 − 1 x (λ)ξ(t ) + S1 x˙g (s ) ds t−τmin t−τmin

# −

t −τmin −η(t ) t−τmin

# −

t −τmin −η(t )

 T   T  T ξ (t )Y (λ) + x˙Tg (s )S2 S−1 2 Y (λ)ξ(t ) + S2 x˙g (s ) ds  T   T  T ξ (t )M (λ) + x˙Tg (s )S2 S−1 2 M (λ)ξ(t ) + S2 x˙g (s ) ds

(36)

By further deriving Eq. (36), we can get  2 T V¯ ≤ ξ T (t ) (λ) + WzT1 Wz1 − γ∞ WwT Ww + τmin X (λ)S−1 1 X (λ)

 T −1 T + η(t )Y (λ)S−1 2 Y (λ) + (τmax − τmin − η(t ) )M (λ)S2 M (λ) ξ(t )   η(t ) 2 (λ) + WzT1 Wz1 − γ∞ = ξ T (t ) WwT Ww τmax − τmin  T −1 T +τmin X (λ)S−1 1 X (λ) + (τmax − τmin )Y (λ)S2 Y (λ) ξ(t )  τmax − τmin − η(t )  2 ξ T (t ) WwT Ww (λ) + WzT1 Wz1 − γ∞ τmax − τmin  T −1 T +τmin X (λ)S−1 1 X (λ) + (τmax − τmin )M (λ)S2 M (λ) ξ(t ) < 0

where

 ξ T (t ) = xTg (t )

xTg (t − τmin )

xTg (t − τmax )

xTg (t − τmax − η(t ) )

 w (t ) ,

(37)

H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

1911

! "

¯ Q1 + WQT Q2WQ2 (λ) = sym WATg PWB + Z (λ)WZ + WQT1 Q1W 2 + WATg [τmin S1 + (τmax − τmin )S2 ]WAg ,  Z (λ) = X (λ)

  M (λ) , WAg = Ag (λ)

Y (λ)

0n,2n

Bg2 (λ)K

  Bg1 , WB = In

 0n,3n+1 ,

⎡

⎤     In − In 0n,2n+1 I 0 I 0 0 , WQ2 = n,n n n,2n+1 , Wz = ⎣0n,n In 0n − In 0n,1 ⎦, WQ1 = n n,3n+1 0n,n In 0n,2n+1 0n,2n In 0n,n+1 0n,2n − In In 0n,1     Wz1 = C1 (λ) 02,2n D1 (λ)K 0 , Ww = 01,4n 1 If the following inequalities of Eqs. (38) and (39) hold 2 T −1 T (λ) + WzT1 Wz1 − γ∞ WwT Ww + τmin X (λ)S−1 1 X (λ) + (τmax − τmin )Y (λ)S2 Y (λ) < 0

(38)

2 T −1 T (λ) + WzT1 Wz1 − γ∞ WwT Ww + τmin X (λ)S−1 1 X (λ) + (τmax − τmin )M (λ)S2 M (λ) < 0 (39)

It is definitely ensured that V¯ (t ) < 0 holds with w(t) = 0, and we subsequently have V˙ (t ) < 0. Based on Lyapunov theory, the closed-loop system (15) is asymptotically stable. Moreover, for the non-zero external disturbance w(t)∈ L2 [0, ∞ ), we have Jz1 w < 0 according to V¯ (t ) < 0, 2 i.e. z1 (t ) 22 < γ∞

w(t ) 22 , which guarantees that the system (15) has a given attenuation level γ∞ under zero initial conditions. To obtain Eqs. (21) and (22), using Schur complement yields to the following inequalities: ⎡ ⎤ √ √ (λ) τmin X (λ) τmax − τminY (λ)  ⎦<0 −S1 0n,n (40) (λ) = ⎣ ∗ 1 ∗ ∗ −S2 ⎡ ⎤ √ √ (λ) τmin X (λ) τmax − τmin M (λ)  ⎦<0 −S1 0n,n (41) (λ) = ⎣ ∗ 2 ∗ ∗ −S2 2 wherein (λ) = (λ) + WzT1 Wz1 − γ∞ WwT Ww . By further deriving, we obtain

 1

 2

(λ) =

r 

λ2i

i=1

(λ) =

r  i=1



+

1ii

λ2i

 2ii

r−1  r 

λi λ j

i=1 j=i+1

+

r−1  r  i=1 j=i+1



(42)

1i j

λi λ j



(43)

2i j

Note that the inequalities of Eqs. (42) and (43) are equivalent to  < 0, ζ = 1, 2, ζ ii ζ i j < 0, i < j, i, j = 1, 2, . . . , r.

(44)

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H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

where

⎡

ii ζ ii = ⎣ ∗ ∗  ζij

 ⎤ √ τmax − τmin  ζ ⎦ 0n,n −S2

√

τmin Xi −S1 ∗

⎡

  √ τmin Xi + X j −2S1 ∗

ii +  j j =⎣ ∗ ∗

(45)

⎤ √ τmax − τmin ζ ⎦ 0n,n −2S2

(46)

In (45) and (46),

' ( ii = sym WATi PWB + ZiWZ + WQT1 Q1WQ1 + WQT2 Q2WQ2 + WATgi [τmin S1 + (τmax − τmin )S2 ]WAgi

2 + WATg j [τmin S1 + (τmax − τmin )S2 ]WAg j + WzT1 Wz1 − γ∞ WwT Ww ,

    Zi = [Xi Yi Mi ], WAgi = Agi 0n,2n Bg2i K Bg1 , Wz1i = C1i 02,2n D1i K 0 





 1 = Yi ,  2 = Mi ,  1 = Yi + Y j ,  2 = Mi + M j . By applying Schur complement into Eqs. (45) and (46), we have ⎡ ⎤ √ √ ii τmin Xi τmax − τmin ζ WATgi AI WzT1i ⎢∗ −S1 0n,n 0n,2n 0n,2 ⎥  ⎢ ⎥ ⎢ =⎢∗ ∗ −S2 0n,2n 0n,2 ⎥ ⎥<0 ζ ii ⎣∗ ∗ ∗ Y1 02n,2 ⎦ ∗ ∗ ∗ ∗ −I2 ⎡     √ √ ii +  j j τmin Xi + X j τmax − τmin ¯ ζ WATgi ⎢ ⎢  ∗ −2S1 0n,n ⎢ =⎢ ∗ ∗ −2S2 ζij ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗

(47)

 WzT1i + WzT1 j 0n,2 0n,2 02n,2 −2I2

+ WATg j AI 0n,2n 0n,2n 2 Y1 ∗

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (48)





2 In (47) and (48), ii = sym{WATi PWB + ZiWZ } + WQT1 Q1WQ1 + WQT2 Q2WQ2 − γ∞ WwT Ww , √ √ −1 −1 τmax − τmin I10 ]. Y1 = diag{−S1 − S2 }, AI = [ τmin I10 Define J1 = diag{P −1 P −1 ... P −1 1}, J2 = diag{P −1 P −1 }, J3 = d iag{S1 S2 }, J4 = d iag{J1 , J2 J3 1}, J5 = diag{P −1 P −1 . . . P −1 }, pre- and postmultiplying the inequality (47) and (48) by J4T and J4 , we further obtain √ ⎡¯ √ ¯ AT AI W ¯ zT ⎤ ii τmin X¯ i τmax − τmin ζ W 1i gi ⎢∗ −P¯ S1 P¯ 0n,n 0n,2n 0n,2 ⎥  ⎢ ⎥ ˜ ⎥<0 ¯ S2 P¯ =⎢ (49) ∗ ∗ − P 0 0 n, 2n n, 2 ⎢ ⎥ ζ ii ⎣∗ ⎦ ¯ ∗ ∗ Y1 02n,2 ∗ ∗ ∗ ∗ −I2

H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

    ¯ ii +  ¯ j j √τmin X¯ i +X¯ j √τmax − τmin ¯ ζ W ¯ AT +W ¯ zT + W ¯ TA AI W ¯ zT  gj 1i 1j gi ⎢ ¯ S1 P¯ ⎢  ∗ −2 0 0 0 P n,n n, 2n 2, 2 ˜ ⎢ =⎢ ∗ ∗ −2P¯ S2 P¯ 0n,2n 0n,2 ζij ⎢ ⎣ ¯1 ∗ ∗ ∗ 2Y 02n,2 ∗ ∗ ∗ ∗ −2I2 ⎡

1913

⎤ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎦ (50)

where P¯ = P −1 , Q¯ 1 = P −1 Q1 P −1 , J1 [X (λ) Y (λ) M(λ)]J5 . Since S1 > 0 and S2 > 0, we have )   −1  ¯ ¯ S−1 1 − P S1 S1 − P ≥ 0  −1    S − P¯ S2 S−1 − P¯ ≥ 0 2



Q¯ 2 = P −1 Q2 P −1 ,

[X¯ (λ)

Y¯ (λ)

M¯ (λ)] =

(51)

2

By further transformation, we get

¯ −P¯ S1 P¯ ≤ S−1 1 − 2P −1 ¯ ¯ −P S2 P ≤ S2 − 2P¯

(52)

Thus, one can obtain Eqs. (21) and (22) by performing transformation of Eqs. (49) and (50). Step 3: In order to ensure that generalized H2 performance index of system (15), H2 (GH2 ) satisfies with z2 (t ) ∞ < γ2 w(t ) 2 , it is observed from Eq. (23) that z1T (t )z1 (t ) ≥ 0 guarantees that 2 T V˙ (t ) < γ∞ w (t )w (t )

(53)

Integrating the left and right part of Eq. (53) from 0 to arbitrary t>0, we have # t 2 V (t ) < γ∞ wT (s )w (s )ds

(54)

0

According to the previous discussion, the last two terms of Eq. (24) are positive definite, thus we have # t 2 xTg (t )P xg (t ) < γ∞ wT (s )w (s )ds (55) 0

Multiplying Eq. (55) by γ22 , we have # t γ22 T 2 x t P x t < γ wT (s )w (s )ds ( ) ( ) g 2 2 g γ∞ 0

(56)

Only if the following inequality of Eq. (57) holds with t ∈ [0, ∞ ) and z2 (t ) 2∞ ≥

z2T (t )z2 (t )

z2 (t ) 2∞ <

γ22 T x (t )P xg (t ) 2 g γ∞

(57)

We get z2T (t )z2 (t )

=

xTg (t )C2T C2 xg (t )

<

xTg (t )

γ22 P xg (t ) < γ22 2 γ∞

#

t

wT (s )w (s )ds 0

(58)

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H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

If Eq. (58) holds, it is just needed to be satisfied with C2T C2 <

γ22 P 2 γ∞

By Schur complement, we obtain   P C2T >0 γ2 ∗ γ 22 I2

(59)

(60)

∞

Multiplying Eq. (60) by diag{P −1 , I2 } and by means of congruent transformation in matrix, we have   T P¯ P¯C 2 >0 (61) γ2 ∗ γ 22 I2 ∞

This guarantees the generalized H2 performance requirement on the normalized safety constraints. The proof is completed. Remark 2. From Theorem 1 and its proof, the minimum value of γ∞ can be obtained through the optimization solution that guarantees the smaller H∞ performance indicators of closedloop system (15). Thus, the minimum solution is converted into solving a corresponding convex optimization problem, specifically described as Theorem 2. Theorem 2. For the closed-loop system (15), the given performance indexes γ2 > 0 and the upper and lower bounds of time-varying input delay satisfying τmax > τmin > 0, if there exists ∗ ¯ ∗ ¯ ∗ ¯ ∗ ¯∗ ¯∗ ¯ ∗ ¯ ∗ ¯ ∗ ¯ ∗ a set of feasible solutions denoted as (γ∞ , P , Q1 , Q2 , S1 , S2 , K , X j , Y j , M j )( j = 1, . . . , r) for the convex optimization problem expressed by min γ∞ s.t.(18 ), (19 ), (20 )

(62)

Then the control law K ∗ = K¯ ∗ P¯ ∗ will minimize the H∞ performance of system (15), which is the optimal H∞ performance. 4. Simulation study and discussion Due to the changeable number of passengers or dynamic payloads, the vehicle body mass ms will accordingly change in a certain range, so it is determined as the uncertain parameter of the active suspension system, and assume that ms has a perturbation of ±10%, then the relationship between the uncertain parameter vector λ = [ λ1 λ2 ] and ms are as follows: λ1 =

1/ms − 1/ms max 1/ms min − 1/ms , λ2 = 1/ms min − 1/ms max 1/ms min − 1/ms max

where λ1 ≥ 0, λ2 ≥ 0, λ1 + λ2 = 1,ms min , ms max are the lower and upper bound of ms , and the interval of λ is calculated as λ = [ 0.45 0.55 ]. Moreover, a pronounced bump and random road surfaces are adopted as input excitations to perform the simulation investigations for the vehicle active suspension system shown in Fig. 1. The half-vehicle model parameters are listed in Table 1, and Theorem 1 is applied to derive the designed controller. During the controller design and validation process, it should be satisfied with the following four requirements:

H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

1915

Table 1 Model Parameters of 4-DOF active suspension system. ms

Iy kg·m2

690 kg kr 22,000 N/m

1222 ktf 2000,000 N/m

muf

msf

a

b

kf

40 kg ktr 2000,000 N/m

45 kg Ftf 4014.5 N

1.3 m Ftr 3580.5 N

1.5 m cf 1000 Nsm−1

18,000 N/m cr 1000 Nsm−1

(1) The vertical and pitch angular acceleration converge to zero within a finite time, re¨ ) → 0. spectively; that is, z¨c (t ) → 0 and φ(t (2) The front and rear suspension dynamic deflections zi (i = f, s), are less than the maximal value of suspension travel zmax =0.1 m. (3) For the riding safety, the load ratios of the front and rear suspensions represented by i Fradio (i = f, s) should be less than one. (4) The actuator control force of the front and rear suspensions represented by ui (t) (i = f, s) should be restrained in the saturation limitation, i.e. umax =1500 N. It is assumed that the actuator input delay of the active suspension system is expressed as τ (t) = 7.5 + 2.5sin(t/25) ms [47], and the lower and upper bound of the input delay is τ min =5 ms and τ max =10 ms, respectively. Based on Theorem 2, the minimal value of H∞ performance index is calculated as γ min = 8.5394, thus the proposed robust controller with wheelbase preview for the active suspension with time-varying input delay can be obtained and denoted as Controller Ⅰ with the control gain matrix K1 as   0.2424 0.0268 −2.8172 −0.2496 0.0134 0.0012 −0.0131 −0.0012 −0.0006 −0.0001 K1 = 10 × . 0.0276 0.3520 −0.3079 −3.4087 0.0015 0.0159 −0.0015 −0.0155 −0.0022 −0.0002 5

Simultaneously, applying Theorem 2 to the closed-loop system (3) yields to the robust controller without wheelbase preview for the active suspension with time-varying input delay, denoted as Controller Ⅱ with the control gain matrix K2 as   0.5587 −0.4670 −6.1985 4.2726 0.0309 −0.0211 −0.0313 0.0213 4 . K2 = 10 × 0.5575 −0.4620 −6.1828 4.2300 0.0308 −0.0209 −0.0311 0.0209 According to literature [20] and generalized bounded real lemma [45], the robust controller with wheelbase preview for active suspension without time-varying input delay can be solved and denoted as Controller III with control gain matrix K3 given by 

K3 = 104 ×

 −0.1696 −0.0494 5.8767 0.1608 −0.2574 −0.0526 −0.0460 −0.0096 −0.0682 −0.0094 . −0.0514 0.5371 0.0358 4.2435 −0.0455 −0.1806 −0.0043 −0.0480 −0.5883 −0.0603

4.1. Simulation analysis in frequency domain According to ISO 2361 criteria, the human body is sensitive about 4–8 Hz in the vertical vibration and 1–2 Hz in the pitch vibration directions. Fig. 2 shows the comparisons of the maximum singular value for vehicle body acceleration in the vertical and pitch motions of suspension system under the different four control schemes. It is obvious that, compared to Uncontrolled, Controller Ⅱ and Controller Ⅲ, the proposed Controller Ⅰ can achieve a better control performance overall, especially in the frequency ranges being sensitive to human body, and the vertical and pitch acceleration responses of active suspension system with Controller

1916

H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

Fig 2. The comparisons of the maximum singular value for (a) vehicle body acceleration, (b) pitch angular acceleration with different control schemes.

Fig 3. The comparisons of the maximum singular value for (a) vehicle body acceleration, (b) pitch angular acceleration with ms perturbation in Controller I and Uncontrolled suspension.

Ⅰ have a smaller gain values of vibration energy, which implies that the proposed control scheme can well improve the riding quality in the concerned frequency range. Additionally, to verify the robustness of Controller Ⅰ, Fig. 3 shows the comparisons of the maximum singular value for the vertical and pitch angular accelerations of active suspension system with Controller Ⅰ and Uncontrolled suspension when ms has a perturbation of ±10% over its nominal value of ms , i.e. 0.9 ms and 1.1 ms . It is observed that the perturbations of vehicle body mass have little effects on the ride quality of vehicle suspension. 4.2. Simulation analysis in time domain 4.2.1. Bump road response In general case, bump road input is taken as a certain disturbance, which is an isolated disturbance on a smooth road surface. Herein a classical bump disturbance function extracted from [15] is used to mimic the actual bump road excitation as given by h b (1 − cos(5πt )), 1 ≤ t ≤ 0.4 zr f = 2 (63) 0, otherwise where hb is the height of bump road, although the front and rear wheel have the same road input profile, yet there still exists a time delay (a + b)/v. The value of hb and v are extracted form literatures [41] as hb = 0.1 m and v = 45 (km/h).

H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

1917

Fig 4. The response comparisons of (a) vertical acceleration, (b) pitch angular acceleration, (c) suspension dynamic displacement of front wheel, (d) suspension dynamic displacement of rear wheel, (e) dynamic load ratio of front wheel, (f) dynamic load ratio of rear wheel, (g) control force of front wheel and (h) control force of rear wheel under condition of τ (t) = 0 and bump road.

Fig. 4 displays the time evaluation of the vertical and pitch angular accelerations in vehicle body, suspension dynamic deflections, tire dynamic loads and actuator control forces at the front and rear wheels for active suspension system with Uncontrolled, Controller Ⅰ and Controller Ⅱ when τ (t)=0. It can be confirmed from Fig. 4(a) and (b) that, compared to both of Uncontrolled and Controller Ⅱ, the vertical and pitch angular accelerations of vehicle body under Controller Ⅰ can be significantly reduced, and reach dynamic stability within a shorter time. As shown in Fig. 4(c) and (d), the suspension dynamic deflections of active suspension system with Controller I have the relatively smaller positive peak value that is less than zmax ; analyzing Fig. 4(e) and (f) gives that the load ratios of the front and rear suspension are always less than one, illustrating that the dynamic loads are less than its static loads, which ensures the firm uninterrupted contact of wheels to road. Additionally, it can be inferred from Fig. 4(g) and (h) that the front and rear actuator control forces uf , ur are always less than the maximal control force umax , which satisfies the actuator input saturation requirement. Fig. 5 shows the simulation results of vehicle suspension performances including the vertical and pitch angular accelerations of vehicle body, suspension dynamic deflections, tire dynamic loads and actuator control forces at the front and rear wheels for active suspension system with Uncontrolled, Controller I and Controller III when there exists time-varying input

1918

H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

Fig 5. The response comparisons of (a) vertical acceleration, (b) pitch angular acceleration, (c) suspension dynamic displacement of front wheel, (d) suspension dynamic displacement of rear wheel, (e) dynamic load ratio of front wheel, (f) dynamic load ratio of rear wheel, (g) control force of front wheel and (h) control force of rear wheel under condition of τ (t) = 7.5 + 2.5sin (t/25) and bump road.

delay as τ (t) = 7.5 + 2.5sin(t/25) ms. It is observed from Fig. 5 that the control performances of active suspension system with Controller III become deteriorated even when a smaller time delay of τ (t) is introduced. However, if the proposed Controller I is applied to active suspension system, the performance deterioration caused by actuator input delay can be compensated with a better improvement of ride quality and handling stability under bump road profile. 4.2.2. Random road response To further validate the control effect of the designed Controller I, assuming that vehicle is running across B-class random road profile described by Gaussian white noise signal, which is given by [44] √ z˙r f (t ) = −2π f0 zr f (t ) + 2π n0 G0 vω(t ) (64) where f0 is the lower cut-off frequency of road profile, n0 is the reference spatial frequency, G0 is the road roughness coefficient and ω(t) is white Gaussian noise with zero mean value. In this simulation, we choose n0 = 0.1(1/m), G0 = 64 × 10−6 m3 and v = 45 (km/h). Similarly, Figs. 6 and 7 show the time evaluation of vehicle suspension performances including the vertical and pitch angular accelerations of vehicle body, suspension dynamic deflections, tire dynamic loads and actuator control forces at the front and rear wheels for

H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

1919

Fig 6. The response comparisons of (a) vertical acceleration, (b) pitch angular acceleration, (c) suspension dynamic displacement of front wheel, (d) suspension dynamic displacement of rear wheel, (e) dynamic load ratio of front wheel, (f) dynamic load ratio of rear wheel, (g) control force of front wheel and (h) control force of rear wheel under condition of τ (t) = 0 and random road.

active suspension system with Uncontrolled, Controller Ⅰ and Controller Ⅱ without time input delay, respectively. And the time-varying input delay as τ (t)=7.5 + 2.5sin(t/25) ms. It is obviously observed that, like the bump road response, the proposed Controller I reveals the most desirable time evaluation of vehicle dynamic and safety constraint performances whether considering the actuator time input delay or not under random road profile. Specifically, it is seen from Fig. 6(a) and (b) that the vertical and pitch angular accelerations of vehicle body for active suspension system with Controller Ⅰ have obvious improvements compared to the uncontrolled suspension system; Compared with Controller Ⅱ, the Controller Ⅰyields a better performance. However, it can be confirmed from Fig. 7(a) and (b) that, due to the time-varying input delay τ (t), the performance responses of active suspension with Controller Ⅲ are deteriorated, yet active suspension system with Controller Ⅰ has a better control effect. Moreover, under random road profile, the safety performance responses of active suspension system with Controller Ⅰ have the similar variation trend as that under bump road, implying that the proposed controller can well satisfy the safety constraint requirements of active suspension system. Additionally, the root mean square (RMS) values of vehicle body acceleration are closely related to the ride comfort, which are used to validate the effectiveness of the proposed controller. To this end, according to literature [6], the RMS calculation expression of x(t) is

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Fig 7. The response comparisons of (a) vertical acceleration, (b) pitch angular acceleration, (c) suspension dynamic displacement of front wheel, (d) suspension dynamic displacement of rear wheel, (e) dynamic load ratio of front wheel, (f) dynamic load ratio of rear wheel, (g) control force of front wheel and (h) control force of rear wheel under condition of τ (t) = 7.5 + 2.5sin(t/25) and random road. Table 2 RMS comparisons of z¨c and φ¨ under random road profile without input delay. Performance indexes

Control scheme

z¨c φ¨

given by RMSx =

*

#

T

(1/T )

Uncontrolled

Controller Ⅰ

Controller Ⅱ

0.3872 0.0945

0.1368(↓64.66%) 0.0318(↓66.36%)

0.2823(↓27.09%) 0.0655(↓30.75%)

x T (t )x(t )dt

(65)

0

After some calculations based on Eq. (67), Tables 2 and 3 summarize the RMS values ¨ accelerations of vehicle body using Uncontrolled, of the vertical (z¨c ) and pitch angular (φ) Controller Ⅰ, Controller Ⅱ and Controller Ⅲ under various conditions, respectively. As shown in Table 2, compared to the uncontrolled suspension system, the RMS values of z¨c and φ¨ for active suspension system with Controller Ⅰ and Controller Ⅱ can be reduced about 64.66%, 66.36% and 27.09%, 30.75% under random road profile, respectively. In addition,

H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

1921

Table 3 RMS comparisons of z¨c and φ¨ under random road profile with time-varying input delay. Performance indexes

z¨c φ¨

Control scheme Uncontrolled

Controller Ⅰ

Controller Ⅲ

0.3872 0.0945

0.2685(↓30.66%) 0.0469(↓50.37%)

0.3521(↓9.060%) 0.0754(↓20.21%)

Fig 8. The bar charts of vehicle performance improvements for active suspension system (a) without time input delay and (b) with time-varying input delay with respect to the uncontrolled suspension system.

as shown in Table 3, the RMS values of z¨c and φ¨ for active suspension system with Controller I can be reduced about 30.66%, 50.37%, respectively, and the corresponding dynamic performances with Controller Ⅲ can be augmented about 9.06%, 20.21% under random road profile, respectively. It is evident that, in terms of control effect, vehicle active suspension with Controller Ⅰ has better performance compared to Controller Ⅲ. Additionally, based on the results in Tables 2 and 3, the corresponding bar charts of the vertical and pitch angular accelerations for active suspension system are presented in Fig. 8 to better display vehicle ¨ performance improvements, which illustrate the vertical (z¨c ) and pitch angular acceleration (φ) performance improvements with respect to the uncontrolled suspension system when there is no time-varying input delay, respectively. From the aforementioned discussions, one can get that active suspension system with Controller I can provide the desired ride comfort and handling stability, which further indicates the effectiveness of the proposed controller for active suspension system in the presences of the parameter uncertainties. 5. Conclusions This paper presents a robust state-feedback controller design for active suspension system with time-varying input delay and wheelbase preview information in the presence of the parameter uncertainties and external road disturbances. Based on the established multi-objective control problem of active suspension system using system-augmentation technology, the designed state-output feedback controller with robust H∞ and generalized H2 performance level is evaluated through transforming the nonlinear matrix inequalities into a convex optimization problem using LMI technique, which guarantees the safety performance constraints such as suspension deflection limit, tire dynamic load and actuator input saturation, simultaneously. Finally, a simulation investigation is provided to verify the effectiveness of the proposed controller design for active suspension system under bump and random road conditions. This

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study provides a unified framework and technical foundation for the implementation of active suspension control design. Acknowledgments This work is supported by National Natural Science Foundation of China under Grants 51675423 and 51305342, and Primary Research & Development Plan of Shaanxi Province under Grant 2017GY-029. References [1] H. Kim, H. Yang, Y. Park, Improving the vehicle performance with active suspension using road-sensing algorithm, Comput. Struct. 80 (18) (2002) 1569–1577. [2] N. Yagiz, Y. Hacioglu, Y. Taskin, Fuzzy sliding-mode control of active suspensions, IEEE Trans. Ind. Electron. 55 (11) (2008) 3883–3890. [3] A. Liberzon, D. Rubinstein, P.O. Gutman, Active suspension for single wheel station of off-road track vehicle, Int. J.Robust Nonlinear Control 11 (10) (2001) 977–999. [4] M. Zapateiro, F. Pozo, H.R. Karimi, N. Luo, Semi-active control methodologies for suspension control with magnetorheological dampers, IEEE/ASME Trans. Mechatron. 17 (2) (2012) 370–380. [5] T. Sande, B. Gysen, I. Besselink, J. Paulides, E. Lomonova, H. Nijmeijer, Robust control of an electromagnetic active suspension system: simulations and measurements, Mechatronics 23 (2) (2013) 204–212. [6] S. Xu, J. Lam, C. Yang, Robust H∞ control for uncertain singular systems with state delay, Int. J. Robust Nonlinear Control 13 (13) (2003) 1213–1223. [7] W. Sun, H. Gao, O. Kaynak, Adaptive backstepping control for active suspension system with hard constraints, IEEE/ASME Trans. Mechatron. 18 (3) (2013) 1072–1079. [8] H. Li, J. Yu, C. Hilton, H. Liu, Adaptive sliding-mode control for nonlinear active suspension vehicle systems using T-S fuzzy approach, IEEE Trans. Ind. Electron. 60 (8) (2013) 3328–3338. [9] V.S. Deshpande, B. Mohan, P.D. Shendge, S.B. Phadke, Disturbance observer based sliding mode control of active suspension systems, J. Sound Vib. 333 (11) (2014) 2281–2296. [10] B. Zhang, Q. Han, X. Zhang, Recent advances in vibration control of offshore platforms, Nonlinear Dyn. 89 (2) (2017) 1–17. [11] B. Zhang, Q. Han, X. Zhang, X. Yu, Sliding mode control with mixed current and delayed states for offshore steel jacket platforms, IEEE Trans. Control Syst. Technol. 22 (5) (2014) 1769–1783. [12] Y. Kao, J. Xie, C. Wang, H.R. Karimi, A sliding mode approach to H∞ math Container Loading Mathjax, non-fragile observer-based control design for uncertain Markovian neutral-type stochastic systems, Automatica 52 (2015) 218–226. [13] Y. Han, Y. Kao, C. Gao, Robust sliding mode control for uncertain discrete singular systems with time-varying delays and external disturbances, Automatica 75 (4) (2017) 210–216. [14] I. Eski, S. Yıldırım, Vibration control of vehicle active suspension system using a new robust neural network control system, Simul. Model. Pract. Theory 17 (5) (2009) 778–793. [15] H. Gao, W. Sun, P. Shi, Robust sampled-data H∞ control for vehicle active suspension systems, IEEE Trans. Control Syst. Technol. 18 (1) (2010) 238–245. [16] H. Du, N. Zhang, J. Lam, Parameter-dependent input-delayed control of uncertain vehicle suspensions, J. Sound Vib. 317 (2008) 537–556. [17] H. Chen, K. Guo, Constrained H∞ control of active suspensions: an LMI approach, IEEE Trans. Control Syst. Technol. 13 (3) (2005) 412–421. [18] W. Sun, Y. Zhao, J. Li, Active suspension control with frequency band constraints and Actuator input delay, IEEE Trans. Ind. Electron. 59 (1) (2011) 530–537. [19] L. Guo, L. Zhang, Robust H∞ control of active vehicle suspension under non-stationary running, J. Sound Vib. 331 (2012) 5824–5837. [20] Y. Kong, D. Zhao, B. Yang, Robust non-fragile H∞ /L2 −L∞ control of uncertain linear system with time-delay and application to vehicle active suspension, Int. J. Robust Nonlinear Control 25 (2015) 2122–2141. [21] M. Rehan, N. Iqbal, K.S. Hong, Delay-range-dependent control of nonlinear time-delay systems under input saturation, Int. J.Robust Nonlinear Control 26 (8) (2016) 1647–1666.

H. Pang, Y. Wang and X. Zhang et al. / Journal of the Franklin Institute 356 (2019) 1899–1923

1923

[22] H. Li, H. Liu, S. Hand, Design of robust H∞ controller for a half-vehicle active suspension system with input delay, Int. J. Syst. Sci. 44 (4) (2013) 625–640. [23] Y. Zhao, W. Sun, H. Gao, Robust control synthesis for seat suspension systems with actuator saturation and time-varying input delay, J. Sound Vib. 329 (21) (2010) 4335–4353. [24] H.D. Choi, C.K. Ahn, M.T. Lim, Dynamic output-feedback H∞ control for active half-vehicle suspension systems with time-varying input delay, Int. J. Control Autom. Syst. 14 (1) (2016) 59–68. [25] H.D. Choi, C.K. Ahn, P. Shi, Dynamic output-Feedback dissipative control for T-S fuzzy systems with time– varying input delay and output constraints, IEEE Trans. Fuzzy Syst. 25 (3) (2017) 511–526. [26] E.K. Bender, Optimum linear preview control with application to vehicle suspension, ASME J. Basic Eng. 90 (2) (1968) 213–221. [27] M. Elmadany, Z. Abduljabbar, M. Foda, Optimal preview control of active suspensions with integral constraint, J. Vib. Control 9 (12) (2003) 1377–1400. [28] A.G. Thompson, B.R. Davis, Computation of the RMS state variable and control forces in a half-car model with preview active suspension using spectral decomposition methods, J. Sound Vib. 285 (3) (2005) 571–583. [29] I. Youn, A. Hac, Preview control of active suspension with integral action, Int. J. Automot. Technol. 7 (5) (2006) 547–554. [30] L.V.V. Rao, S. Narayanan, Preview control of random response of a half car vehicle model traversing a rough road, J. Sound Vib. 310 (1) (2008) 352–365. [31] R.S. Prabakar, C. Sujatha, S. Narayanan, Optimal semi-active preview control response of a half car vehicle model with magnetorheological damper, J. Sound Vib. 326 (3) (2009) 400–420. [32] S.H. Ha, S.B. Choi, G.S. Lee, Design of a denoising hybrid fuzzy-PID controller for active suspension systems of heavy vehicles based on model adaptive wheelbase preview strategy, J. Vibroeng. 17 (2) (2015) 883–904. [33] X. Zhang, J. Zhang, Performance analysis of limited bandwidth active suspension with preview based on a discrete time model, WSEAS Trans. Syst. 9 (8) (2010) 834–843. [34] S. Ryu, Y. Park, M. Suh, Ride quality analysis of a tracked vehicle suspension with a preview control, J. Terramech. 48 (6) (2011) 409–417. [35] P. Li, J. Lam, K.C. Cheung, Multi-objective control for active vehicle suspension with wheelbase preview, J. Sound Vib. 333 (21) (2014) 5269–5282. [36] P. Li, J. Lam, K.C. Cheung, Velocity-dependent multi-objective control of vehicle suspension with preview measurements, Mechatronics 24 (5) (2014) 464–475. [37] S. Senthil, S. Narayanan, Optimal preview control of a two-DOF vehicle model using stochastic optimal control theory, Veh. Syst. Dyn. 25 (6) (1996) 413–430. [38] E.M. Elbeheiry, D.C. Karnopp, Optimal control of vehicle random vibration with constrained suspension deflection, J. Sound Vib. 189 (5) (1996) 547–564. [39] H.S. Roh, Y.J. Park, Stochastic optimal preview control of an active vehicle suspension, J. Sound Vib. 220 (2) (1999) 313–330. [40] W. Sun, H. Gao, O. Kaynak, Finite frequency H∞ control for vehicle active suspension systems, IEEE Trans. Control Syst. Technol. 19 (2) (2011) 416–422. [41] W. Sun, Z. Zhao, H. Gao, Saturated adaptive robust control for active suspension systems, IEEE Trans. Ind. Electron. 60 (9) (2013) 3889–3896. [42] W. Sun, H. Pan, Y. Zhang, Multi-objective control for uncertain nonlinear active suspension systems, Mechatronics 24 (4) (2014) 318–327. [43] H. Gao, J. Lam, C. Wang, Multi-objective control of vehicle active suspension systems via load-dependent controllers, J. Sound Vib. 290 (3) (2006) 654–675. [44] M. Gobbi, F. Levi, G. Mastinu, Multi-objective stochastic optimization of the suspension system of road vehicles, J. Sound Vib. 298 (4) (2006) 1055–1072. [45] H. Du, N. Zhang, Constrained H∞ control of active suspension for a half-car model with a time delay in control, Proc. Inst. Mech. Eng. Part D J. Automob. Eng. 222 (5) (2008) 665–684. [46] B. Zhang, Q. Han, X. Zhang, Event-triggered H∞ reliable control for offshore structures in network environments, J. Sound Vib. 368 (2016) 1–21. [47] H. Li, H. Liu, Multi-objective H∞ control for vehicle active suspension systems with random actuator delay, Int. J. Syst. Sci. 43 (12) (2012) 2214–2227.