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Active steering wheel shimmy control for electric vehicle by sampled-data output feedback
Accepted Manuscript Active steering wheel shimmy control for electric vehicle by sampled-data output feedback Qinghua Meng, Chunjiang Qian, Yong Shu
PII: DOI: Reference:
S0019-0578(18)30320-3 https://doi.org/10.1016/j.isatra.2018.08.024 ISATRA 2896
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ISA Transactions
Received date : 15 March 2018 Revised date : 28 July 2018 Accepted date : 24 August 2018 Please cite this article as: Meng Q., et al. Active steering wheel shimmy control for electric vehicle by sampled-data output feedback. ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.08.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Title page showing Author Details
Active Steering Wheel Shimmy Control for Electric Vehicle by Sampled-data Output Feedback Qinghua Menga,∗, Chunjiang Qianb , Yong Shua a School b College
of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou, 310018, China of Engineering, University of Texas at San Antonio, San Antonio, TX 78249, USA
Abstract In this paper, a new two Degree of Freedom (DOF) shimmy dynamic model of an Electric Vehicle (EV) with independent suspension subject to uncertain disturbances is built via Lagrange’s theorem firstly. Secondly, Based on the built model, an active control method is proposed to solve the shimmy problem of the steering system via a sampled-data output feedback controller. The output feedback domination approach is also used to dominate uncertain disturbances by using a scaling gain. With the help of these tools, the sampling period and scaling gain are calculated to guarantee global stability and disturbance attenuation for the closed-loop control system. Finally, simulations and tests are conducted to verify the effectiveness of the sampled-data output feedback controller for the EV’s shimmy problem under different conditions. Keywords: Shimmy Phenomenon, Electric Vehicle, Sampled-data Output Feedback, Almost Disturbance Decoupling The Electric Vehicles (EVs) driven by in-wheel motors have the potential for energy efficiency and environmental protection. In recent decades, vehicle motion control, energy optimization and performance benefits have gained the attention of more and more researchers [1, 2, 3, 4, 5]. Because of the structure 5
of the steering system, the front wheels will vibrate around the king-pins, which ∗ Corresponding
author Email addresses: [email protected] (Qinghua Meng), [email protected] (Chunjiang Qian), [email protected] (Yong Shu)
Preprint submitted to Journal of LATEX Templates
July 28, 2018
*Highlights (for review)
Highlights of this paper: (1) A shimmy dynamic model of an EV with independent suspension is presented which considers external disturbances. (2) A linear state estimate discrete-time observer is designed to construct the sampled-data control law for the shimmy control system. (3) A shimmy active ADD sampled-data output feedback controller achieving globally asymptotically stable is designed to dominate the disturbances' effect on the shimmy control system output using a scaling gain.
*Blinded Manuscript - without Author Details Click here to view linked References
Active Steering Wheel Shimmy Control for Electric Vehicle by Sampled-data Output Feedback
Abstract In this paper, a new two Degree of Freedom (DOF) shimmy dynamic model of an Electric Vehicle (EV) with independent suspension subject to uncertain disturbances is built via Lagrange’s theorem firstly. Secondly, Based on the built model, an active control method is proposed to solve the shimmy problem of the steering system via a sampled-data output feedback controller. The output feedback domination approach is also used to dominate uncertain disturbances by using a scaling gain. With the help of these tools, the sampling period and scaling gain are calculated to guarantee global stability and disturbance attenuation for the closed-loop control system. Finally, simulations and tests are conducted to verify the effectiveness of the sampled-data output feedback controller for the EV’s shimmy problem under different conditions. Keywords: Shimmy Phenomenon, Electric Vehicle, Sampled-data Output Feedback, Almost Disturbance Decoupling The Electric Vehicles (EVs) driven by in-wheel motors have the potential for energy efficiency and environmental protection. In recent decades, vehicle motion control, energy optimization and performance benefits have gained the attention of more and more researchers [1, 2, 3, 4, 5]. Because of the structure 5
of the steering system, the front wheels will vibrate around the king-pins, which is called front wheel shimmy. Shimmy occurs regardless of braking or steering, and has a significant influence on riding comfort and maneuvering stability due to its resonant characteristics. The in-wheel motors increase the unsprung mass, affecting the riding comfort and handling of the EV[6, 7, 8], which may
10
more easily cause the steering system shimmy problem than the internal comPreprint submitted to Journal of LATEX Templates
July 28, 2018
bustion engine vehicles. Since it is virtually impossible to eliminate the source of shimmy, it is necessary to study a new automatic control method to reduce shimmy phenomenon when an EV runs. In order to solve the shimmy problem, researchers did many works in theory 15
and engineering, and obtained many achievements. In [9], a simple three DOF shimmy model of a truck based on bifurcation theory was constructed, and then numerical analysis of bifurcation property, road test and numerical simulation were conducted based on the model. J.Yu et al. developed a simulation model for vehicle chassis transmissibility of steering shimmy. And brake judder was
20
used to analyze the transmissibility mechanism and quantify the relative importance of system factors through modal analysis around equilibrium point and a virtual design of experiment [10]. D. Tak´ acs and G. St´ep´ an presented a full report on the experimental investigation of the rig from the identification of system parameters to the validation of stability boundaries and vibration fre-
25
quencies of shimmy motion [11]. JW.Lu et al. established a six DOF dynamic model of a vehicle shimmy system with consideration of the clearance based on nonlinear dynamics [12]. In [13], the lateral instability of towed structures was investigated with special attention to the small amplitude lateral vibration that leads to a higher energy consumption in certain parameter domains. A
30
low DOF mechanical model of a shimmying towed tyre was proposed which describes the dynamics of the tyre-ground contact patch by the time delayed differential equation. However, the literature mentioned above focus on solid axle suspension. Now independent suspensions have been used widely in passenger cars. Only few
35
literature are about independent suspensions, and most of the achievements are the mechanism of the shimmy [14, 15]. The way to reducing the shimmy problem is to optimize the vehicle parameters in vehicle design. Once the vehicle is manufactured, it will be very difficult to deal with the shimmy problem if the shimmy phenomenon appears again.
40
With the development of control theory, more and more control methods are applied into EV [16]. Therefore, reducing the shimmy problem by active 2
vehicle control is a novel way. As aforementioned, when an EV runs under high speed, many factors will generate uncertain disturbances to the body and front steering wheels. Therefore, how to deal with uncertain disturbances is very dif45
ficult. Researchers tried some methods to deal with this problem. In [17], a continuous domination and delicate adaptive technique were proposed to cope with serious coexistence among uncertainties, including time-varying control coefficients which have unknown upper and lower bounds, nonlinear parameters and external disturbances. The authors proposed a method by combining im-
50
proved adding a power integrator method with the recursive construction to deal with serious uncertainties in [18]. In [19], the authors designed an enhanced Extended State Observer-based control strategy to deal with the disturbance attenuation problem for a class of nonintegral-chain systems subject to nonlinear mismatched uncertainties and external disturbances. An iterative learning
55
contouring controller consisting of a classical Proportional-Derivative controller and disturbance observer was proposed to deal with uncertain disturbances in [20]. In most conditions, some state parameters of an EV are not available, that means it is more difficult to control. Some controllers are designed using the state parameters acquired from equipped sensors which are expensive and
60
sometime may fail to function well. In order to solve these problems, this paper studies the problem of disturbance attenuation with internal stability by designing a new sampled-data output feedback controller to reduce the steering wheels shimmy of the EV under high speeds despite some unknown system state parameters. The control objective of the paper is to use only output feedback to
65
design a sampled-data controller which ensures almost disturbance decoupling with global asymptotic stability. The Almost Disturbance Decoupling (ADD) problem have been studied for both linear and nonlinear systems in [21, 22]. However, the aforementioned achievements are about continuous-time controllers, despite the fact that more and more systems are implemented digitally
70
in practice. Sampled-data technologies were applied into control systems in the recent years. It is necessary to solve the ADD problems by sampled-data controllers. A sampled-data observer-based output feedback fuzzy controller for 3
nonlinear systems was designed using exact discrete-time design method in [23]. Based on the input delay approach, output feedback sampled-data polynomial 75
controller for nonlinear systems was studied in [24]. A kind of sampled-data output feedback controller were designed by using the domination approach with a tunable scaling gain and a tunable sampling period, which only based on the nominal linear system in [25, 26]. Therefore, this paper presents an active control method to reduce the shimmy
80
phenomenon of an EV with independent suspension. In this method, an ADD controller based on sampled-data output feedback is constructed. The highlights of this paper lie in the following aspects. • A new two DOF steering wheel shimmy dynamic model of an EV with independent suspension is presented.
85
• A linear state estimate discrete-time observer for unknown EV parameters is designed to construct the shimmy sampled-data control law.
• A shimmy ADD sampled-data output feedback controller achieving globally asymptotically stable is designed to dominate the disturbances’ effect on the output of the shimmy control system using a scaling gain. 90
The rest of the paper is organized as follows. The two DOF steering wheels shimmy dynamic model of an EV with independent suspension subject to uncertain disturbances is presented in Section 2. Section 3 presents the state space equations of the model. The shimmy sampled-data output feedback controller is designed in Section 4. In section 5, simulations and tests are carried out using
95
the designed sampled-data output feedback method, PID method and Sliding Model Control (SMC) method for the shimmy control of the EV under different conditions. Section 6 gives the conclusion of the paper.
4
Controller
u Steering Gear
Ksl Cs
Ksc
Figure 1: The steering wheels shimmy dynamic model of an EV driven by in-wheel motors
1. The Two DOF Shimmy Dynamic Model of an EV with Independent Suspension 100
We simplify the steering system to a two Degree of Freedom system, as shown in Figure 1. According to Lagrange’s theorem, there exists
∂T ∂U ∂D d ∂T )− + + = Qi ( dt ∂ q˙i ∂qi ∂qi ∂ q˙i
(i = 1, 2),
(1)
where 1 1 T = Il ψ˙ l2 + Ir ψ˙ r2 , 2 2 1 2 1 U = d Ksl (ψl − ψr )2 + Kz d2 sinαψl2 (sinα + f cosα) 2 2 1 2 2 + Kz d sinαψr (sinα + f cosα), 2 1 ˙2 1 1 D = Cl ψl + Cr ψ˙ r2 + d2 Cs (ψ˙ l2 − ψ˙ r2 )2 , 2 2 2 where Il is the moment of inertia of the front-left wheel around the king-pin, Ir is the moment of inertia of the front-right wheel around the king-pin, d is the length of the steering arm, ψl is the shimmy angle of the front-left wheel around 105
the king-pin, Cl is the damping coefficient of the front-left wheel around the king-pin, ψr is the shimmy angle of the front-right wheel around the king-pin, 5
Distribution of lateral force
Figure 2: The schematic drawing of the tire lateral force distribution
Cr is the damping coefficient of the front-right wheel around the king-pin, Ksl is the stiffness coefficient of the steering system, Cs is the damping coefficient of the steering system, Kz is the tire vertical stiffness, f is the tire rolling resistance 110
coefficient, α is the kingpin castor angle. The generalized forces of the front wheel shimmy system are Q1 = −Fyl (tm + rd sinα) + w
(2)
Q2 = −Fyr (tm + rd sinα) + w − u, where Fyl is the lateral force of the front-left wheel, Fyr is the lateral force of the front-right wheel, rd is the tire’s rolling radius, tm is the pneumatic trail, u is the controller. Because the tire cornering characteristics has important influence on the 6
front wheel shimmy system, the tire model is an important part of the front wheel shimmy dynamic model. In this paper, a classic tire lateral force concentration model is used to calculate the tire lateral force, as shown in Figure 2, 0
where φt is the angle between the tire motion direction and the x axle, y0 is the initial y axle value when the king-pin has no shimmy, y is the y axle value of the king-pin, y0 is the y axle value of the pneumatic trail. Therefore, Z 0 y0 = y0 − tm ψw + ydt, ˙
(3)
where ψw is the tire sideslip angle. y1 is the y axle value of the point of the application of the lateral force Fy , Z 0 y1 = vφt dt + y0 . (4) Therefore, the tire lateral force is Fy = Cα (ψw − φt ), 115
(5)
where Cα is the tire cornering stiffness. From Eq.(5), we obtain vφt = vψw − Fy v/Cα . Substituting Eq.(6) into (4) results in Z 0 y1 = (vψw − Fy v/Cα )dt + y0 .
(6)
(7)
On the other hand, the tire lateral force can be expressed by the tire lateral stiffness and lateral deformation, i.e., Fy = ρy (y1 − y0 ),
(8)
where ρy is the tire lateral stiffness. Substituting Eq.(3) and (7) into (8), we obtain Z Z (vψw − Fy v/Cα )dt + tm ψw − ydt Fy = ρy ˙ . 7
(9)
The derivative of Eq.(9) is F˙y = ρy [(vψw − Fy v/Cα ) + tm ψ˙ w − y]. ˙
(10)
Because the y˙ is very small, Eq.(10) can be rewritten as F˙y = ρy [(vψw − Fy v/Cα ) + tm ψ˙ w ].
(11)
Because the shimmy angle ψw increases or decreases with the tire sideslip angle φt proportionally, and φt is smaller than ψw , so we define k = ψw /φt .
(12)
Substituting Eq.(12) into (5) results in a new tire lateral force equation, Fy = Cα (1 − k)ψw .
(13)
Then we get the differential equation of the tire lateral force as F˙ y = Cα (1 − k)ψ˙ w . According to Eq.(11) and (14), we have C α tm Cα2 (1 − k) ˙ Fy = ψ + Cα ψ. − vρy v
(14)
(15)
Based on the aforementioned equations, we propose the steering wheel shimmy dynamics equations of an EV as following, Il ψ¨l + (Cl + d2 Cs )ψ˙ l + (d2 Ksl + Kz d2 sinα2 + Kz d2 f sinα2 )ψl − d2 Cs ψ˙ r − d2 Ksl ψr + Fyl (tm + rd sinα) − w = 0
Ir ψ¨r + (Cr + d2 Cs )ψ˙ r + (d2 Ksl + Kz d2 sinα2 + Kz d2 f sinα2 )ψr − d2 Cs ψ˙ l − d2 Ks ψl + Fyr (tm + rd sinα) − w + u = 0 K α tm K 2 (1 − k) ˙ ψl + Kα ψl − α Fyl = v vKy K α tm K 2 (1 − k) ˙ Fyr = ψr + Kα ψ r , − α vKy v 120
where w is the disturbance. 8
(16)
2. State Space Equations of System (16) According to Eq.(16), we obtain w ψ¨l + a1 ψ˙ l + a2 ψl − b1 ψ˙ r − b2 ψr = Il u w ψ¨r + a3 ψ˙ r + a4 ψr − b3 ψ˙ l − b4 ψl = − + , Ir Ir where
(17)
Kα t2m K 2 tm (1 − k) Kα tm rd sinα 1 Cl + d2 Cs + + − α v Il vKy v 2 K rd sinα(1 − k) , − α vKy 1 a2 = (d2 Ks + Kz d2 sinα2 + Kz d2 f sinα2 + Kα tm + Kα rd sinα), Il Kα t2m K 2 tm (1 − k) Kα tm rd sinα 1 Cr + d2 Cs + + − α a3 = v Ir vKy v Kα2 rd sinα(1 − k) , − vKy 1 a4 = (d2 Ks + Kz d2 sinα2 + Kz d2 f sinα2 + Kα tm + Kα rd sinα), Ir d2 Cs d2 Ks d2 Cs d 2 Ks , b2 = , b3 = , b4 = . b1 = Il Il Ir Ir a1 =
Defining Ir ψl (t) b2 Ir x2 (t) = − ψ˙ l (t) b2
x1 (t) = −
(18)
x3 (t) = −Ir ψr (t)
x4 (t) = −Ir ψ˙ r (t),
the state equations of system (17) can be written as x˙1 (t) = x2 (t)
(19)
x˙2 (t) = x3 (t) + φ1 (t, x(t)) + g1 w(t) x˙3 (t) = x4 (t) x˙4 (t) = u(t) + φ2 (t, x(t)) + g2 w(t) y = x1 (t), 9
where g1 = −
Ir , b2 Il
(20)
g2 = −1, φ1 (t, x(t)) = −a2 x1 (t) − a1 x2 (t) +
b3 x4 (t), b4
φ2 (t, x(t)) = b2 b4 x1 (t) + b2 b3 x2 (t) − a4 x3 (t) − a3 x4 (t). 3. Sampled-data Output Feedback Controller Design The objective of this paper is to design a sampled-data output feedback controller for the EV shimmy stability under different conditions such that the resulting closed-loop system is globally asymptotically stable at the origin when w(t) = 0, and also satisfies almost disturbance attenuation for every disturbance w(t) ∈ L2 to suppress the EV shimmy. Then we can design an active shimmy
ADD controller via sampled-data output feedback. Given a real number γ > 0,
a linear sampled-data feedback controller u(tk ) is designed as ζ(tk+1 ) = M ζ(tk ) + N y(tk ) u(tk ) = −Kζ(tk ), ∀t ∈ [tk , tk+1 ), tk = kT, k = 0, 1, 2, . . . ,
(21)
where the time instant tk and tk+1 are the sampling points, and T is the sampling period, M , N and K are coefficients, ζ ∈ R2 is the controller state, such that 125
the following hold:
(1) When disturbance w(t) = 0, the Eq.(19) and (21) are globally asymptotically stable at the equilibrium (x, ζ) = (0, 0). (2) For every disturbance w(t) ∈ L2 , the response of the closed-loop system (19) starting from the origin satisfies Z ∞ Z 2 |y(t)| dt 6 γ 2 0
130
∞ 0
2
kw(t)k dt.
(22)
Generally speaking, the energy of the disturbances generated to an EV is limited, R∞ 2 that means 0 kd(t)k dt is bounded. 10
From Eq.(20), it is obvious that a constant c > 0 always exists such that |φi (t, x(t))| ≤ c(|x1 (t)| + ... + |xi (t)|), i = 1, 2, ∀t ≥ 0.
(23)
On the other hand, it is also obvious that there always exists a known constant G0 > 0 such that |gi (t, x(t))| ≤ G0 , i = 1, 2, ∀t ≥ 0.
(24)
Then a sampled-data output feedback controller for the system (19) can be designed. The main result is described as the following theorem. Theorem 1. The shimmy system (19) in the form of x˙ i (t) = xi+1 (t) + φi (t, x(t)) + gi (t, x(t))w(t)
(25)
x˙ n = u + φn (t, x(t)) + gn (t, x(t))w(t) y = x1 (t) can be globally stabilized by a sampled-data output feedback controller via ADD 135
in the form of u(tk ) = −ki xi (tk ) under |φi (t, x(t))| ≤ c(|x1 (t)| + ... + |xi (t)|) and |gi (t, x(t))| ≤ G0 where i = 1, 2, ∙ ∙ ∙ , n, ∀t ≥ 0, xi (t) is a state of the system,
y is the output of the system, φi (t, x(t)) and gi (t, x(t)) are linear or nonlinear items, w(t) is uncertain disturbance, G0 is a constant. Proof 140
3.1. Coordinate Transformation First of all, a coordinates transformation for the system (19) is introduced. For a constant L ≥ 1 to be determined later, we define the following coordinate transformation
zi (t) =
xi (t) u(t) , v(t) = 4 , i = 1, 2, 3, 4. i−1 L L
11
(26)
Under this coordinate transformation, the system (19) becomes z˙1 (t) = Lz2 (t) z˙2 (t) = Lz3 (t) + φˉ1 (t, z(t)) + gˉ1 w(t) (27)
z˙3 (t) = Lz4 (t) z˙4 (t) = Lv(t) + φˉ2 (t, z(t)) + gˉ2 w(t) y(t) = z1 (t),
where φˉ1 (t, z(t)) = φ1 (t, z(t))/L, φˉ2 (t, z(t)) = φ2 (t, z(t))/L3 , gˉ1 (t) = g1 (t)/L, gˉ2 (t) = g2 (t)/L3 . From Eq.(23) and (24), it can be verified that |φˉ1 (t, z(t))| ≤ c(|z1 (t)| + |z2 (t)| + |z4 (t)|)
|φˉ2 (t, z(t))| ≤ c(|z1 (t)| + |z2 (t)| + |z3 (t)| + |z4 (t)|)
G0 ≤ G0 L G0 |ˉ g 2 | ≤ 3 ≤ G0 . L |ˉ g1 | ≤
We define
z1 (t)
(28)
0
1
0
z2 (t) 0 0 1 , A = z(t) = z3 (t) 0 0 0 z4 (t) 0 0 0 0 0 ˉ gˉ φ1 (∙) , G(∙) = 1 , Φ(∙) = 0 0 φˉ2 (∙) gˉ2
0 1 0 0 0 , B = , C = , 0 0 1 0 1 0 0
(29)
under which system (27) can be rewritten as z(t) ˙ = LAz(t) + LBv(t) + Φ(t, z(t)) + Gw(t) y(t) = Cz(t).
(30)
Since only the output y(t) = z1 (t) is measurable at sampling points, a linear observer should be designed to estimate the unmeasurable state of system (30). 12
145
3.2. Linear Observer Design for System (30) An observer is designed with continuous-time states over [tk , tk+1 ), discretetime output z1 (tk ) and input v(tk ) using the same method as in [27], without involving the φˉi (t, z(t)) and gˉi (t), i = 1, 2. Therefore, the observer is zˆ˙1 (t) = Lˆ z2 (t) + La1 (z1 (tk ) − zˆ1 (t))
zˆ˙2 (t) = Lˆ z3 (t) + La2 (z1 (tk ) − zˆ1 (t)) zˆ˙3 (t) = Lˆ z4 (t) + La3 (z1 (tk ) − zˆ1 (t))
(31)
zˆ˙4 (t) = Lv(tk ) + La4 (z1 (tk ) − zˆ1 (t)), where a1 , a2 , a3 and a4 are the coefficients of the Hurwitz polynomial p1 (s) = s 4 + a 4 s 3 + a 3 s 2 + a2 s + a 1 . Defining T T zˆ(t) = [ˆ z1 (t) zˆ2 (t) zˆ3 (t) zˆ4 (t)] , H = [a1 a2 a3 a4 ] , Aˆ = A − HC,
observer (31) can be rewritten as ˆz (t) + LBv(tk ) + LHy(tk ), ∀t ∈ [tk , tk+1 ) . zˆ˙ (t) = LAˆ
(32)
3.3. Construction of the Sampled-data Output Feedback Control Law Because the state zi , i = 2, 3, 4 are not measurable, the sampled-data control law using the estimated zˆ(tk ) generated from the observer (32) is constructed as v(t) = v(tk ) = −K zˆ(tk ) = −k1 zˆ1 (tk ) − k2 zˆ2 (tk ) − k3 zˆ3 (tk ) − k4 zˆ4 (tk ), ∀t ∈ [tk , tk+1 ), k = 0, 1, 2, ...,
(33)
where k1 , k2 , k3 and k4 are the coefficients of Hurwitz polynomial p1 (s) = s4 + 150
k4 s3 + k3 s2 + k2 s + k1 and positive.
13
Substituting Eq.(33) into (30) and (32) results in B z(t) ˙ − L K(ˆ z (tk ) − z(tk )) Z(t) =LA B zˆ(t) 0 Φ(∙) G (z(tk ) − z(t)) + + w(t), + L HC c 0
(34)
where
A=
−BK
A HC
Aˆ − BK
=
A
−BK
HC
A − HC − BK
, t ∈ [tk , tk+1 ) .
As aforementioned, Aˆ = A − HC and A − BK are Hurwitz matrices. Thus
A is a Hurwitz matrix as well from the following form
A=
I
0
−I
I
−1
A − BK
−BK Aˆ
0
I
0
−I
I
.
Therefore, there is a positive definite matrix P = P T ∈ R8×8 > 0 such that
AT P + P A = −I8×8 . A Lyapunov function V (Z) = Z T P Z is constructed with h iT Z = z(t) zˆ(t) , then the derivative of V (Z) is
V˙ (Z(t)) = − LkZ(t)k2 + 2LZ T (t)P
0
BK
(z(t) − z(tk )) −HC BK Φ(∙) G + 2Z T (t)P w(t). + 2Z T (t)P 0 0
(35)
3.4. Selection of Scaling Gain and Sampling Period According to Eq.(23), the following holds kΦ(∙ )k =
q
φˉ21 + φˉ22 6 c1 kZ(t)k,
√ where c1 = c 2 + 4 ≥ 0. With the help of Eq.(36), we have T Φ(∙ ) 2Z P ≤ 2c1 kP kkZ(t)k2 . 0 14
(36)
(37)
According to Eq.(24), the following holds T G(∙ ) 2Z P w(t) ≤ 8kP k2 G20 kZ(t)k2 + 1 kw(t)k2 . 2 0
(38)
˙ kZ(t)k ≤ c2 kZ(t)k + c3 kZ(tk )k + c4 kd(t)k, ∀t ∈ [tk , tk+1 )
(39)
By Eq.(34) and (36), we can verify that
for three constants c2 , c3 and c4 , where √ c2 = L(kAk + kˆ(A)k) + c1 , c3 = L( 2kKk + kHk), c4 = 2G0 . With the help of Eq.(39), integrating Eq.(34) from tk to t yields Z t kZ(t) − Z(tk )k ≤ (c2 kZ(τ )k + c3 kZ(tk )k + c4 kw(τ )k) dτ, t ∈ [tk , tk+1 ) . tk
(40) Defining M(t) =
Z
t tk
(c2 kZ(τ )k + c3 kZ(tk )k)dτ,
(41)
taking derivative of M(t) yields ˙ M(t) =c2 kZ(τ )k + c3 kZ(tk )k ≤ c2 kZ(τ ) − Z(tk )k + (c2 + c3 )kZ(tk )k Z t kw(τ )kdτ + (c2 + c3 )kZ(tk )k. (42) ≤c2 M(t) + c2 c4 tk
Let M(tk ) = 0, Eq.(42) is solved for t ∈ [tk , tk+1 ), and there exists Z t Z τ c2 (t−tk ) c2 (τ −tk ) M(t) ≤e e c2 c4 kd(s)kdsdτ tk
+ ec2 (t−tk ) =
Z
tk
t
ec2 (τ −tk ) ec2 (t−tk ) (c2 + c3 )kZ(tk )kdτ
tk
Z t h i c 2 + c3 kZ(tk )k ec2 (t−tk ) − 1 − c4 kw(s)kds c2 tk Z t c2 (t−tk ) + c4 e e−c2 (τ −tk ) kw(τ )kdτ. tk
15
(43)
Substituting Eq.(43) into (40), the following holds kZ(t) − Z(tk )k ≤
i c2 + c3 h c2 (t−tk ) − 1 kZ(t) − Z(tk )k e c2 Z t i c2 + c3 h c2 (t−tk ) c2 (t−tk ) + e − 1 kZ(t)k) + c4 e kw(τ )kdτ. c2 tk
(44)
If the sampling period T is chosen small enough, and Δ(t) = (c2 +c3 )/c2 ec2 (t−tk ) − 1
is defined, there exists kZ(t) − Z(tk )k ≤
Δ(t − tk ) c4 ec2 (t−tk ) kZ(t)k + 1 − Δ(t − tk ) 1 − Δ(t − tk )
Z
t tk
kw(τ )kdτ.
(45)
Substituting Eq.(37), (38) and (45) into (35) results in Δ(t − tk ) kZ(t)k2 1 − Δ(t − tk ) Z t c4 ec2 (t−tk ) + 2LΓ kZ(t)k kw(τ )kdτ 1 − Δ(t − tk ) tk 1 + 2c1 kP kkZ(t)k2 + 8kP k2 G20 kZ(t)k2 + kw(t)k2 , 2
V˙ (Z(t)) ≤ − LkZ(t)k2 + 2LΓ
where
0 Γ = kP k
−HC
According to Eq.(27), we get
1 1 h 2 |y(t)| = C γ2 γ2
0
Combining Eq.(46) and (48) results in
i
BK
.
BK
2 1 Z(t) ≤ 2 kZ(t)k2 . γ
(46)
(47)
(48)
Δ(t − tk ) 1 2 V˙ (Z(t))+ 2 |y(t)| − kw(t)k2 ≤ −LkZ(t)k2 + 2LΓ kZ(t)k2 γ 1 − Δ(t − tk ) Z t c4 ec2 (t−tk ) + 2LΓ kw(τ )kdτ kZ(t)k 1 − Δ(t − tk ) tk 1 1 + 2c1 kP kkZ(t)k2 + 8kP k2 G20 kZ(t)k2 + 2 kZ(t)k2 − kw(t)k2 , γ 2 (49) where t ∈ [tk , tk+1 ). 16
In order to keep the stability of system (35),the scaling gain L is fixed appropriately as
1 . (50) γ2 Because t ∈ [tk , tk+1 ), there exists Δ(t − tk ) ≤ Δ(T ). Substituting Eq.(50) L = 2 + 2c1 kP k + 8kP k2 G20 +
into (49) and integrating both sides of Eq.(49) from tk to tk+1 , we can obtain Z tk+1 1 2 2 ˙ V (Z(t)) + 2 |y(t)| − kw(t)k dt γ tk Z tk+1 Z tk+1 Δ(T ) kZ(t)k2 dt + 2LΓ kZ(t)k2 dt ≤−2 1 − Δ(T ) tk tk Z t Z tk+1 c 4 e c2 T + 2LΓ kZ(t)k kw(τ )kdτ dt 1 − Δ(T ) tk tk Z Z tk+1 1 tk+1 2 + 2LkP k kZ(t)k dt − kw(t)k2 dt. (51) 2 tk tk According to Cauchy-Schwarz Inequality, there exists Z t Z tk+1 kZ(t)k kw(τ )kdτ dt tk
1 ≤ 2
T = 2
Z
tk
tk+1
tk Z tk+1 tk
kZ(t)k2 dt
Z
tk+1
tk
T kZ(t)k dt + 2 2
1dt +
Z
t tk
1 2
Z
t
tk
kw(t)k2 dt
Z
tk+1
1dt
tk
kw(t)k2 dt.
Substituting Eq.(52) into (51) results in Z tk+1 1 2 2 ˙ V (Z(t)) + 2 |y(t)| − kw(t)k dt γ tk Z tk+1 2LΔ(T ) + T c4 ec2 T ≤ −2 + kZ(t)k2 dt Γ 1 − Δ(T ) tk Z tk+1 1 c 4 e c2 T kw(t)k2 dt. − + T LΓ 1 − Δ(T ) 2 tk
If T is small enough, such that 2LΔ(T ) T c 4 e c2 T −1+ + Γ < 0, 1 − Δ(T ) 1 − Δ(T ) T LΓ
1 c 4 e c2 T − < 0, 1 − Δ(T ) 2
then Eq.(53) is negative definite. Further more, we have Z tk+1 Z tk+1 1 2 2 ˙ V (Z(t)) + 2 |y(t)| − kw(t)k dt ≤ − kZ(t)k2 dt, γ tk tk 17
(52)
(53)
(54) (55)
(56)
from which it can be concluded that the closed-loop system (25) is uniformly globally asymptotically stable with the designed controller when w(t) = 0. Moreover, the following can be obtained based on Eq.(56), n−1 n−1 X Z tk+1 1 X Z tk+1 2 V (Z(tn )) − V (Z(t0 ))+ |y(t)| dt − kw(t)k2 dt ≤ 0. γ2 tk tk k=0
k=0
(57)
When Z(t0 ) = 0, V (Z(tn )) ≥ 0 and V (0) = 0, it can be concluded from
Eq.(57) that
n−1 X Z tk+1 k=0
tk
n−1 X Z tk+1 1 2 |y(t)| dt ≤ kw(t)k2 dt. γ2 tk
Because t0 = 0 , the following inequality holds when n −→ ∞, Z ∞ Z ∞ 2 |y(τ )| dτ ≤ γ 2 kw(τ )k2 dτ. 0
155
(58)
k=0
(59)
0
Then the proof ends.
According to Eq.(54) and (55), there exists 1 − T c 4 e c2 T Γ c2 T Δ(T ) ≤ min , 1 − 2T Lc4 e Γ . 1 + 2LΓ
Defining
(60)
1 − T c 4 e c2 T Γ , 1 − 2T Lc4 ec2 T Γ , 1 + 2LΓ based on Eq.(60), the sampling period T can be selected according to P = min
T ≤
1 (P + 1)c2 + c3 ln . c2 c2 + c3
(61)
4. Simulation and Test Study In this section, the simulations and tests of the shimmy control system (27) under 60km/h and 100km/h are conducted. The parameters of an EV are listed in Table 1. The original values of the system are set as z(0) = [0, 0, 0, 0]T , k1 = 44, k2 = 10, k3 = 24, k4 = 10 and L = 3. w(t) = 500 sin(20t)/(1 + t2 ) is used as the external disturbance moment. Then the controller is u(t) = u(tk ) = −44L4 zˆ1 (tk ) − 10L4 zˆ2 (tk ) − 24L4 zˆ3 (tk ) − 10L4 zˆ4 (tk ). 18
(62)
Table 1: The parameters of a steering system of an EV with independent suspension
160
Parameter
Value
Parameter
Value
Cl
14.23 Nms/rad
Il
8.48 kgm2
Cr
14.23 Nms/rad
Ir
8.48 kgm2
d2a Ksl
16 300 Nms/rad
tm
0.07m
Kα
19 200 Nms/rad
d
0.15m
d2a Cs
18.5 Nms/rad
rd
0.25m
Kz
12 000 Nms/rad
f
0.15
Ky
24 700 Nms/rad
α
4◦
In the simulations, the disturbance is not applied to the EV from time 0 to 10th second when the EV runs, and the controller is not active. The disturbance is applied to the EV from 10th second to 30th second, and the controller reacts to control the EV’s shimmy. According to Eq.(18), we map the states x1 and x3 of the controller into the EV’s parameters ψl and ψr respectively. The output of
165
the controller under different conditions are shown in Figure 3 and 4. The two shimmy angles ψl , ψr and their observations ψˆl , ψˆr are shown in Figure 5 and 6, from which we can see that the observation errors are very small, that verifies the designed observer being effective. The simulation results show the effectiveness of the sampled-data output feedback controller. According to the built two DOF
170
shimmy model, the shimmy phenomenon under different conditions are as Figure 7 and 8 which are anastomotic with other researchers’ results in [28] and [29]. We also obtain that the designed sampled-data control method can stabilize the EV’s shimmy better than PID control method under different speeds as shown in Figure 7 and 8, and the shimmy angles ψl and ψr are stabilized to
175
zero within 5 seconds. The overshoots of the shimmy angles attenuate rapidly. On the other hand, from the Figure 7 and 8, we can see that although the SMC method can stabilize the shimmy phenomenon as good as the sampled19
8 6 4 2 0 -2 -4 -6 -8 0
5
10
15 Time (s)
20
25
30
Figure 3: The output of the controller under 60km/h
data output feedback controller from time 10th second to 15th second. But the SMC method generates bigger chattering, especially after 15th second, which 180
also means that the designed sampled-data output feedback control method is aslo better than the SMC method in whole. In order to verify the proposed shimmy controller in practice, we design a hardware-in-the-loop simulation system as shown in Figure 9 which are composed of Matlab software, dSpace and EV test bed. The control algorithms
185
are programmed by MATLAB. The signals measured by angular displacement sensors are input the dSPACE through the D/A model (DS2001). Then the dSPACE calculates and outputs the control signal to the DSP controller installed on the EV to control the shimmy phenomenon. The EV test bed can simulate different roads for the EV. According to the simulation results, the
190
control results of the front-left wheel and front-right wheel are similar, so we just give the test results of the front-left wheel. The test results are shown in Figure 10 and 11. From these figures, we can obtain that the designed controller can stabilize the shimmy phenomenon within 5seconds in the tests, that is better than the PID control method. The designed controller is also better than
195
the SMC method after 13th second. The shimmy angles ψl is not stabilized to zero because the test results are influenced by many unknown factors. But the
20
10
5
0
-5
-10
-15 0
5
10
15 Time (s)
20
25
30
Figure 4: The output of the controller under 100km/h
test results can meet the practical requirement. 5. Conclusions In this paper, for an EV driven by in-wheels motors with independent suspen200
sion subject to uncertain disturbances, the authors present an automatic control method to solve the shimmy problem of the steering system via a sampled-data output feedback controller. The controller is not only more practical in practice but also more easy to design. The simulations and tests are conducted to verify the effectiveness of the sampled-data output feedback controller for the
205
EV’s shimmy problem under 60km/h and 100km/h compaired with PID and SMC methods. The results show that the designed controller can eliminate the shimmy phenomenon of the EV within 5 seconds not only in the simulations but also in the tests, and is better than the PID and SMC methods. acknowledgment
210
This research were partly supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LY16E050003 and Natural Science Foundation of China under Grant No. 61628302.
21
0.3
0.2
0.1
0
-0.1
-0.2
-0.3 0
5
10
15 Time (s)
20
25
30
20
25
30
(a) Left wheel 0.3
0.2
0.1
0
-0.1
-0.2
-0.3 0
5
10
15 Time (s)
(b) Right wheel Figure 5: States and their observation values of the left and right wheels under 60km/h and initial excitation 10◦
22
0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0
5
10
15 Time (s)
20
25
30
20
25
30
(a) Left wheel 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0
5
10
15 Time (s)
(b) Right wheel Figure 6: States and their observation values of the left and right wheels under 100km/h and initial excitation 10◦
23
0.4 Without control PID SMC Sampled-data output feedback
0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0
5
10
15
20
25
30
Time (s)
(a) Left wheel 0.4 Without control PID SMC Sampled-data output feedback
0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0
5
10
15
20
25
30
Time (s)
(b) Right wheel Figure 7: Shimmy control of the left and right wheels under 60km/h and initial excitation 10◦
24
0.5 Without control PID SMC Sampled-data output feedback
0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0
5
10
15
20
25
30
Time (s)
(a) Left wheel 0.5 Without control PID SMC Sampled-data output feedback
0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0
5
10
15
20
25
30
Time (s)
(b) Right wheel Figure 8: Shimmy control of the left and right wheels under 100km/h and initial excitation 10◦
25
Matlab
Compiling
Driving
dSpace
Data transmit ting by sensors
Figure 9: The hardware-in-the-loop simulation system
0.4 Without control PID SMC Sampled-data output feedback
0.3
0.2
0.1
0
-0.1
-0.2 0
5
10
15 Time (s)
20
25
30
Figure 10: Shimmy control test of the left wheel under 60km/h and initial excitation 10◦
26
0.6 Without control PID SMC Sampled-data output feedback
0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0
5
10
15 Time (s)
20
25
30
Figure 11: Shimmy control test of the left wheel under 100km/h and initial excitation 10◦
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PII: DOI: Reference:
S0019-0578(18)30320-3 https://doi.org/10.1016/j.isatra.2018.08.024 ISATRA 2896
To appear in:
ISA Transactions
Received date : 15 March 2018 Revised date : 28 July 2018 Accepted date : 24 August 2018 Please cite this article as: Meng Q., et al. Active steering wheel shimmy control for electric vehicle by sampled-data output feedback. ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.08.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Title page showing Author Details
Active Steering Wheel Shimmy Control for Electric Vehicle by Sampled-data Output Feedback Qinghua Menga,∗, Chunjiang Qianb , Yong Shua a School b College
of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou, 310018, China of Engineering, University of Texas at San Antonio, San Antonio, TX 78249, USA
Abstract In this paper, a new two Degree of Freedom (DOF) shimmy dynamic model of an Electric Vehicle (EV) with independent suspension subject to uncertain disturbances is built via Lagrange’s theorem firstly. Secondly, Based on the built model, an active control method is proposed to solve the shimmy problem of the steering system via a sampled-data output feedback controller. The output feedback domination approach is also used to dominate uncertain disturbances by using a scaling gain. With the help of these tools, the sampling period and scaling gain are calculated to guarantee global stability and disturbance attenuation for the closed-loop control system. Finally, simulations and tests are conducted to verify the effectiveness of the sampled-data output feedback controller for the EV’s shimmy problem under different conditions. Keywords: Shimmy Phenomenon, Electric Vehicle, Sampled-data Output Feedback, Almost Disturbance Decoupling The Electric Vehicles (EVs) driven by in-wheel motors have the potential for energy efficiency and environmental protection. In recent decades, vehicle motion control, energy optimization and performance benefits have gained the attention of more and more researchers [1, 2, 3, 4, 5]. Because of the structure 5
of the steering system, the front wheels will vibrate around the king-pins, which ∗ Corresponding
author Email addresses: [email protected] (Qinghua Meng), [email protected] (Chunjiang Qian), [email protected] (Yong Shu)
Preprint submitted to Journal of LATEX Templates
July 28, 2018
*Highlights (for review)
Highlights of this paper: (1) A shimmy dynamic model of an EV with independent suspension is presented which considers external disturbances. (2) A linear state estimate discrete-time observer is designed to construct the sampled-data control law for the shimmy control system. (3) A shimmy active ADD sampled-data output feedback controller achieving globally asymptotically stable is designed to dominate the disturbances' effect on the shimmy control system output using a scaling gain.
*Blinded Manuscript - without Author Details Click here to view linked References
Active Steering Wheel Shimmy Control for Electric Vehicle by Sampled-data Output Feedback
Abstract In this paper, a new two Degree of Freedom (DOF) shimmy dynamic model of an Electric Vehicle (EV) with independent suspension subject to uncertain disturbances is built via Lagrange’s theorem firstly. Secondly, Based on the built model, an active control method is proposed to solve the shimmy problem of the steering system via a sampled-data output feedback controller. The output feedback domination approach is also used to dominate uncertain disturbances by using a scaling gain. With the help of these tools, the sampling period and scaling gain are calculated to guarantee global stability and disturbance attenuation for the closed-loop control system. Finally, simulations and tests are conducted to verify the effectiveness of the sampled-data output feedback controller for the EV’s shimmy problem under different conditions. Keywords: Shimmy Phenomenon, Electric Vehicle, Sampled-data Output Feedback, Almost Disturbance Decoupling The Electric Vehicles (EVs) driven by in-wheel motors have the potential for energy efficiency and environmental protection. In recent decades, vehicle motion control, energy optimization and performance benefits have gained the attention of more and more researchers [1, 2, 3, 4, 5]. Because of the structure 5
of the steering system, the front wheels will vibrate around the king-pins, which is called front wheel shimmy. Shimmy occurs regardless of braking or steering, and has a significant influence on riding comfort and maneuvering stability due to its resonant characteristics. The in-wheel motors increase the unsprung mass, affecting the riding comfort and handling of the EV[6, 7, 8], which may
10
more easily cause the steering system shimmy problem than the internal comPreprint submitted to Journal of LATEX Templates
July 28, 2018
bustion engine vehicles. Since it is virtually impossible to eliminate the source of shimmy, it is necessary to study a new automatic control method to reduce shimmy phenomenon when an EV runs. In order to solve the shimmy problem, researchers did many works in theory 15
and engineering, and obtained many achievements. In [9], a simple three DOF shimmy model of a truck based on bifurcation theory was constructed, and then numerical analysis of bifurcation property, road test and numerical simulation were conducted based on the model. J.Yu et al. developed a simulation model for vehicle chassis transmissibility of steering shimmy. And brake judder was
20
used to analyze the transmissibility mechanism and quantify the relative importance of system factors through modal analysis around equilibrium point and a virtual design of experiment [10]. D. Tak´ acs and G. St´ep´ an presented a full report on the experimental investigation of the rig from the identification of system parameters to the validation of stability boundaries and vibration fre-
25
quencies of shimmy motion [11]. JW.Lu et al. established a six DOF dynamic model of a vehicle shimmy system with consideration of the clearance based on nonlinear dynamics [12]. In [13], the lateral instability of towed structures was investigated with special attention to the small amplitude lateral vibration that leads to a higher energy consumption in certain parameter domains. A
30
low DOF mechanical model of a shimmying towed tyre was proposed which describes the dynamics of the tyre-ground contact patch by the time delayed differential equation. However, the literature mentioned above focus on solid axle suspension. Now independent suspensions have been used widely in passenger cars. Only few
35
literature are about independent suspensions, and most of the achievements are the mechanism of the shimmy [14, 15]. The way to reducing the shimmy problem is to optimize the vehicle parameters in vehicle design. Once the vehicle is manufactured, it will be very difficult to deal with the shimmy problem if the shimmy phenomenon appears again.
40
With the development of control theory, more and more control methods are applied into EV [16]. Therefore, reducing the shimmy problem by active 2
vehicle control is a novel way. As aforementioned, when an EV runs under high speed, many factors will generate uncertain disturbances to the body and front steering wheels. Therefore, how to deal with uncertain disturbances is very dif45
ficult. Researchers tried some methods to deal with this problem. In [17], a continuous domination and delicate adaptive technique were proposed to cope with serious coexistence among uncertainties, including time-varying control coefficients which have unknown upper and lower bounds, nonlinear parameters and external disturbances. The authors proposed a method by combining im-
50
proved adding a power integrator method with the recursive construction to deal with serious uncertainties in [18]. In [19], the authors designed an enhanced Extended State Observer-based control strategy to deal with the disturbance attenuation problem for a class of nonintegral-chain systems subject to nonlinear mismatched uncertainties and external disturbances. An iterative learning
55
contouring controller consisting of a classical Proportional-Derivative controller and disturbance observer was proposed to deal with uncertain disturbances in [20]. In most conditions, some state parameters of an EV are not available, that means it is more difficult to control. Some controllers are designed using the state parameters acquired from equipped sensors which are expensive and
60
sometime may fail to function well. In order to solve these problems, this paper studies the problem of disturbance attenuation with internal stability by designing a new sampled-data output feedback controller to reduce the steering wheels shimmy of the EV under high speeds despite some unknown system state parameters. The control objective of the paper is to use only output feedback to
65
design a sampled-data controller which ensures almost disturbance decoupling with global asymptotic stability. The Almost Disturbance Decoupling (ADD) problem have been studied for both linear and nonlinear systems in [21, 22]. However, the aforementioned achievements are about continuous-time controllers, despite the fact that more and more systems are implemented digitally
70
in practice. Sampled-data technologies were applied into control systems in the recent years. It is necessary to solve the ADD problems by sampled-data controllers. A sampled-data observer-based output feedback fuzzy controller for 3
nonlinear systems was designed using exact discrete-time design method in [23]. Based on the input delay approach, output feedback sampled-data polynomial 75
controller for nonlinear systems was studied in [24]. A kind of sampled-data output feedback controller were designed by using the domination approach with a tunable scaling gain and a tunable sampling period, which only based on the nominal linear system in [25, 26]. Therefore, this paper presents an active control method to reduce the shimmy
80
phenomenon of an EV with independent suspension. In this method, an ADD controller based on sampled-data output feedback is constructed. The highlights of this paper lie in the following aspects. • A new two DOF steering wheel shimmy dynamic model of an EV with independent suspension is presented.
85
• A linear state estimate discrete-time observer for unknown EV parameters is designed to construct the shimmy sampled-data control law.
• A shimmy ADD sampled-data output feedback controller achieving globally asymptotically stable is designed to dominate the disturbances’ effect on the output of the shimmy control system using a scaling gain. 90
The rest of the paper is organized as follows. The two DOF steering wheels shimmy dynamic model of an EV with independent suspension subject to uncertain disturbances is presented in Section 2. Section 3 presents the state space equations of the model. The shimmy sampled-data output feedback controller is designed in Section 4. In section 5, simulations and tests are carried out using
95
the designed sampled-data output feedback method, PID method and Sliding Model Control (SMC) method for the shimmy control of the EV under different conditions. Section 6 gives the conclusion of the paper.
4
Controller
u Steering Gear
Ksl Cs
Ksc
Figure 1: The steering wheels shimmy dynamic model of an EV driven by in-wheel motors
1. The Two DOF Shimmy Dynamic Model of an EV with Independent Suspension 100
We simplify the steering system to a two Degree of Freedom system, as shown in Figure 1. According to Lagrange’s theorem, there exists
∂T ∂U ∂D d ∂T )− + + = Qi ( dt ∂ q˙i ∂qi ∂qi ∂ q˙i
(i = 1, 2),
(1)
where 1 1 T = Il ψ˙ l2 + Ir ψ˙ r2 , 2 2 1 2 1 U = d Ksl (ψl − ψr )2 + Kz d2 sinαψl2 (sinα + f cosα) 2 2 1 2 2 + Kz d sinαψr (sinα + f cosα), 2 1 ˙2 1 1 D = Cl ψl + Cr ψ˙ r2 + d2 Cs (ψ˙ l2 − ψ˙ r2 )2 , 2 2 2 where Il is the moment of inertia of the front-left wheel around the king-pin, Ir is the moment of inertia of the front-right wheel around the king-pin, d is the length of the steering arm, ψl is the shimmy angle of the front-left wheel around 105
the king-pin, Cl is the damping coefficient of the front-left wheel around the king-pin, ψr is the shimmy angle of the front-right wheel around the king-pin, 5
Distribution of lateral force
Figure 2: The schematic drawing of the tire lateral force distribution
Cr is the damping coefficient of the front-right wheel around the king-pin, Ksl is the stiffness coefficient of the steering system, Cs is the damping coefficient of the steering system, Kz is the tire vertical stiffness, f is the tire rolling resistance 110
coefficient, α is the kingpin castor angle. The generalized forces of the front wheel shimmy system are Q1 = −Fyl (tm + rd sinα) + w
(2)
Q2 = −Fyr (tm + rd sinα) + w − u, where Fyl is the lateral force of the front-left wheel, Fyr is the lateral force of the front-right wheel, rd is the tire’s rolling radius, tm is the pneumatic trail, u is the controller. Because the tire cornering characteristics has important influence on the 6
front wheel shimmy system, the tire model is an important part of the front wheel shimmy dynamic model. In this paper, a classic tire lateral force concentration model is used to calculate the tire lateral force, as shown in Figure 2, 0
where φt is the angle between the tire motion direction and the x axle, y0 is the initial y axle value when the king-pin has no shimmy, y is the y axle value of the king-pin, y0 is the y axle value of the pneumatic trail. Therefore, Z 0 y0 = y0 − tm ψw + ydt, ˙
(3)
where ψw is the tire sideslip angle. y1 is the y axle value of the point of the application of the lateral force Fy , Z 0 y1 = vφt dt + y0 . (4) Therefore, the tire lateral force is Fy = Cα (ψw − φt ), 115
(5)
where Cα is the tire cornering stiffness. From Eq.(5), we obtain vφt = vψw − Fy v/Cα . Substituting Eq.(6) into (4) results in Z 0 y1 = (vψw − Fy v/Cα )dt + y0 .
(6)
(7)
On the other hand, the tire lateral force can be expressed by the tire lateral stiffness and lateral deformation, i.e., Fy = ρy (y1 − y0 ),
(8)
where ρy is the tire lateral stiffness. Substituting Eq.(3) and (7) into (8), we obtain Z Z (vψw − Fy v/Cα )dt + tm ψw − ydt Fy = ρy ˙ . 7
(9)
The derivative of Eq.(9) is F˙y = ρy [(vψw − Fy v/Cα ) + tm ψ˙ w − y]. ˙
(10)
Because the y˙ is very small, Eq.(10) can be rewritten as F˙y = ρy [(vψw − Fy v/Cα ) + tm ψ˙ w ].
(11)
Because the shimmy angle ψw increases or decreases with the tire sideslip angle φt proportionally, and φt is smaller than ψw , so we define k = ψw /φt .
(12)
Substituting Eq.(12) into (5) results in a new tire lateral force equation, Fy = Cα (1 − k)ψw .
(13)
Then we get the differential equation of the tire lateral force as F˙ y = Cα (1 − k)ψ˙ w . According to Eq.(11) and (14), we have C α tm Cα2 (1 − k) ˙ Fy = ψ + Cα ψ. − vρy v
(14)
(15)
Based on the aforementioned equations, we propose the steering wheel shimmy dynamics equations of an EV as following, Il ψ¨l + (Cl + d2 Cs )ψ˙ l + (d2 Ksl + Kz d2 sinα2 + Kz d2 f sinα2 )ψl − d2 Cs ψ˙ r − d2 Ksl ψr + Fyl (tm + rd sinα) − w = 0
Ir ψ¨r + (Cr + d2 Cs )ψ˙ r + (d2 Ksl + Kz d2 sinα2 + Kz d2 f sinα2 )ψr − d2 Cs ψ˙ l − d2 Ks ψl + Fyr (tm + rd sinα) − w + u = 0 K α tm K 2 (1 − k) ˙ ψl + Kα ψl − α Fyl = v vKy K α tm K 2 (1 − k) ˙ Fyr = ψr + Kα ψ r , − α vKy v 120
where w is the disturbance. 8
(16)
2. State Space Equations of System (16) According to Eq.(16), we obtain w ψ¨l + a1 ψ˙ l + a2 ψl − b1 ψ˙ r − b2 ψr = Il u w ψ¨r + a3 ψ˙ r + a4 ψr − b3 ψ˙ l − b4 ψl = − + , Ir Ir where
(17)
Kα t2m K 2 tm (1 − k) Kα tm rd sinα 1 Cl + d2 Cs + + − α v Il vKy v 2 K rd sinα(1 − k) , − α vKy 1 a2 = (d2 Ks + Kz d2 sinα2 + Kz d2 f sinα2 + Kα tm + Kα rd sinα), Il Kα t2m K 2 tm (1 − k) Kα tm rd sinα 1 Cr + d2 Cs + + − α a3 = v Ir vKy v Kα2 rd sinα(1 − k) , − vKy 1 a4 = (d2 Ks + Kz d2 sinα2 + Kz d2 f sinα2 + Kα tm + Kα rd sinα), Ir d2 Cs d2 Ks d2 Cs d 2 Ks , b2 = , b3 = , b4 = . b1 = Il Il Ir Ir a1 =
Defining Ir ψl (t) b2 Ir x2 (t) = − ψ˙ l (t) b2
x1 (t) = −
(18)
x3 (t) = −Ir ψr (t)
x4 (t) = −Ir ψ˙ r (t),
the state equations of system (17) can be written as x˙1 (t) = x2 (t)
(19)
x˙2 (t) = x3 (t) + φ1 (t, x(t)) + g1 w(t) x˙3 (t) = x4 (t) x˙4 (t) = u(t) + φ2 (t, x(t)) + g2 w(t) y = x1 (t), 9
where g1 = −
Ir , b2 Il
(20)
g2 = −1, φ1 (t, x(t)) = −a2 x1 (t) − a1 x2 (t) +
b3 x4 (t), b4
φ2 (t, x(t)) = b2 b4 x1 (t) + b2 b3 x2 (t) − a4 x3 (t) − a3 x4 (t). 3. Sampled-data Output Feedback Controller Design The objective of this paper is to design a sampled-data output feedback controller for the EV shimmy stability under different conditions such that the resulting closed-loop system is globally asymptotically stable at the origin when w(t) = 0, and also satisfies almost disturbance attenuation for every disturbance w(t) ∈ L2 to suppress the EV shimmy. Then we can design an active shimmy
ADD controller via sampled-data output feedback. Given a real number γ > 0,
a linear sampled-data feedback controller u(tk ) is designed as ζ(tk+1 ) = M ζ(tk ) + N y(tk ) u(tk ) = −Kζ(tk ), ∀t ∈ [tk , tk+1 ), tk = kT, k = 0, 1, 2, . . . ,
(21)
where the time instant tk and tk+1 are the sampling points, and T is the sampling period, M , N and K are coefficients, ζ ∈ R2 is the controller state, such that 125
the following hold:
(1) When disturbance w(t) = 0, the Eq.(19) and (21) are globally asymptotically stable at the equilibrium (x, ζ) = (0, 0). (2) For every disturbance w(t) ∈ L2 , the response of the closed-loop system (19) starting from the origin satisfies Z ∞ Z 2 |y(t)| dt 6 γ 2 0
130
∞ 0
2
kw(t)k dt.
(22)
Generally speaking, the energy of the disturbances generated to an EV is limited, R∞ 2 that means 0 kd(t)k dt is bounded. 10
From Eq.(20), it is obvious that a constant c > 0 always exists such that |φi (t, x(t))| ≤ c(|x1 (t)| + ... + |xi (t)|), i = 1, 2, ∀t ≥ 0.
(23)
On the other hand, it is also obvious that there always exists a known constant G0 > 0 such that |gi (t, x(t))| ≤ G0 , i = 1, 2, ∀t ≥ 0.
(24)
Then a sampled-data output feedback controller for the system (19) can be designed. The main result is described as the following theorem. Theorem 1. The shimmy system (19) in the form of x˙ i (t) = xi+1 (t) + φi (t, x(t)) + gi (t, x(t))w(t)
(25)
x˙ n = u + φn (t, x(t)) + gn (t, x(t))w(t) y = x1 (t) can be globally stabilized by a sampled-data output feedback controller via ADD 135
in the form of u(tk ) = −ki xi (tk ) under |φi (t, x(t))| ≤ c(|x1 (t)| + ... + |xi (t)|) and |gi (t, x(t))| ≤ G0 where i = 1, 2, ∙ ∙ ∙ , n, ∀t ≥ 0, xi (t) is a state of the system,
y is the output of the system, φi (t, x(t)) and gi (t, x(t)) are linear or nonlinear items, w(t) is uncertain disturbance, G0 is a constant. Proof 140
3.1. Coordinate Transformation First of all, a coordinates transformation for the system (19) is introduced. For a constant L ≥ 1 to be determined later, we define the following coordinate transformation
zi (t) =
xi (t) u(t) , v(t) = 4 , i = 1, 2, 3, 4. i−1 L L
11
(26)
Under this coordinate transformation, the system (19) becomes z˙1 (t) = Lz2 (t) z˙2 (t) = Lz3 (t) + φˉ1 (t, z(t)) + gˉ1 w(t) (27)
z˙3 (t) = Lz4 (t) z˙4 (t) = Lv(t) + φˉ2 (t, z(t)) + gˉ2 w(t) y(t) = z1 (t),
where φˉ1 (t, z(t)) = φ1 (t, z(t))/L, φˉ2 (t, z(t)) = φ2 (t, z(t))/L3 , gˉ1 (t) = g1 (t)/L, gˉ2 (t) = g2 (t)/L3 . From Eq.(23) and (24), it can be verified that |φˉ1 (t, z(t))| ≤ c(|z1 (t)| + |z2 (t)| + |z4 (t)|)
|φˉ2 (t, z(t))| ≤ c(|z1 (t)| + |z2 (t)| + |z3 (t)| + |z4 (t)|)
G0 ≤ G0 L G0 |ˉ g 2 | ≤ 3 ≤ G0 . L |ˉ g1 | ≤
We define
z1 (t)
(28)
0
1
0
z2 (t) 0 0 1 , A = z(t) = z3 (t) 0 0 0 z4 (t) 0 0 0 0 0 ˉ gˉ φ1 (∙) , G(∙) = 1 , Φ(∙) = 0 0 φˉ2 (∙) gˉ2
0 1 0 0 0 , B = , C = , 0 0 1 0 1 0 0
(29)
under which system (27) can be rewritten as z(t) ˙ = LAz(t) + LBv(t) + Φ(t, z(t)) + Gw(t) y(t) = Cz(t).
(30)
Since only the output y(t) = z1 (t) is measurable at sampling points, a linear observer should be designed to estimate the unmeasurable state of system (30). 12
145
3.2. Linear Observer Design for System (30) An observer is designed with continuous-time states over [tk , tk+1 ), discretetime output z1 (tk ) and input v(tk ) using the same method as in [27], without involving the φˉi (t, z(t)) and gˉi (t), i = 1, 2. Therefore, the observer is zˆ˙1 (t) = Lˆ z2 (t) + La1 (z1 (tk ) − zˆ1 (t))
zˆ˙2 (t) = Lˆ z3 (t) + La2 (z1 (tk ) − zˆ1 (t)) zˆ˙3 (t) = Lˆ z4 (t) + La3 (z1 (tk ) − zˆ1 (t))
(31)
zˆ˙4 (t) = Lv(tk ) + La4 (z1 (tk ) − zˆ1 (t)), where a1 , a2 , a3 and a4 are the coefficients of the Hurwitz polynomial p1 (s) = s 4 + a 4 s 3 + a 3 s 2 + a2 s + a 1 . Defining T T zˆ(t) = [ˆ z1 (t) zˆ2 (t) zˆ3 (t) zˆ4 (t)] , H = [a1 a2 a3 a4 ] , Aˆ = A − HC,
observer (31) can be rewritten as ˆz (t) + LBv(tk ) + LHy(tk ), ∀t ∈ [tk , tk+1 ) . zˆ˙ (t) = LAˆ
(32)
3.3. Construction of the Sampled-data Output Feedback Control Law Because the state zi , i = 2, 3, 4 are not measurable, the sampled-data control law using the estimated zˆ(tk ) generated from the observer (32) is constructed as v(t) = v(tk ) = −K zˆ(tk ) = −k1 zˆ1 (tk ) − k2 zˆ2 (tk ) − k3 zˆ3 (tk ) − k4 zˆ4 (tk ), ∀t ∈ [tk , tk+1 ), k = 0, 1, 2, ...,
(33)
where k1 , k2 , k3 and k4 are the coefficients of Hurwitz polynomial p1 (s) = s4 + 150
k4 s3 + k3 s2 + k2 s + k1 and positive.
13
Substituting Eq.(33) into (30) and (32) results in B z(t) ˙ − L K(ˆ z (tk ) − z(tk )) Z(t) =LA B zˆ(t) 0 Φ(∙) G (z(tk ) − z(t)) + + w(t), + L HC c 0
(34)
where
A=
−BK
A HC
Aˆ − BK
=
A
−BK
HC
A − HC − BK
, t ∈ [tk , tk+1 ) .
As aforementioned, Aˆ = A − HC and A − BK are Hurwitz matrices. Thus
A is a Hurwitz matrix as well from the following form
A=
I
0
−I
I
−1
A − BK
−BK Aˆ
0
I
0
−I
I
.
Therefore, there is a positive definite matrix P = P T ∈ R8×8 > 0 such that
AT P + P A = −I8×8 . A Lyapunov function V (Z) = Z T P Z is constructed with h iT Z = z(t) zˆ(t) , then the derivative of V (Z) is
V˙ (Z(t)) = − LkZ(t)k2 + 2LZ T (t)P
0
BK
(z(t) − z(tk )) −HC BK Φ(∙) G + 2Z T (t)P w(t). + 2Z T (t)P 0 0
(35)
3.4. Selection of Scaling Gain and Sampling Period According to Eq.(23), the following holds kΦ(∙ )k =
q
φˉ21 + φˉ22 6 c1 kZ(t)k,
√ where c1 = c 2 + 4 ≥ 0. With the help of Eq.(36), we have T Φ(∙ ) 2Z P ≤ 2c1 kP kkZ(t)k2 . 0 14
(36)
(37)
According to Eq.(24), the following holds T G(∙ ) 2Z P w(t) ≤ 8kP k2 G20 kZ(t)k2 + 1 kw(t)k2 . 2 0
(38)
˙ kZ(t)k ≤ c2 kZ(t)k + c3 kZ(tk )k + c4 kd(t)k, ∀t ∈ [tk , tk+1 )
(39)
By Eq.(34) and (36), we can verify that
for three constants c2 , c3 and c4 , where √ c2 = L(kAk + kˆ(A)k) + c1 , c3 = L( 2kKk + kHk), c4 = 2G0 . With the help of Eq.(39), integrating Eq.(34) from tk to t yields Z t kZ(t) − Z(tk )k ≤ (c2 kZ(τ )k + c3 kZ(tk )k + c4 kw(τ )k) dτ, t ∈ [tk , tk+1 ) . tk
(40) Defining M(t) =
Z
t tk
(c2 kZ(τ )k + c3 kZ(tk )k)dτ,
(41)
taking derivative of M(t) yields ˙ M(t) =c2 kZ(τ )k + c3 kZ(tk )k ≤ c2 kZ(τ ) − Z(tk )k + (c2 + c3 )kZ(tk )k Z t kw(τ )kdτ + (c2 + c3 )kZ(tk )k. (42) ≤c2 M(t) + c2 c4 tk
Let M(tk ) = 0, Eq.(42) is solved for t ∈ [tk , tk+1 ), and there exists Z t Z τ c2 (t−tk ) c2 (τ −tk ) M(t) ≤e e c2 c4 kd(s)kdsdτ tk
+ ec2 (t−tk ) =
Z
tk
t
ec2 (τ −tk ) ec2 (t−tk ) (c2 + c3 )kZ(tk )kdτ
tk
Z t h i c 2 + c3 kZ(tk )k ec2 (t−tk ) − 1 − c4 kw(s)kds c2 tk Z t c2 (t−tk ) + c4 e e−c2 (τ −tk ) kw(τ )kdτ. tk
15
(43)
Substituting Eq.(43) into (40), the following holds kZ(t) − Z(tk )k ≤
i c2 + c3 h c2 (t−tk ) − 1 kZ(t) − Z(tk )k e c2 Z t i c2 + c3 h c2 (t−tk ) c2 (t−tk ) + e − 1 kZ(t)k) + c4 e kw(τ )kdτ. c2 tk
(44)
If the sampling period T is chosen small enough, and Δ(t) = (c2 +c3 )/c2 ec2 (t−tk ) − 1
is defined, there exists kZ(t) − Z(tk )k ≤
Δ(t − tk ) c4 ec2 (t−tk ) kZ(t)k + 1 − Δ(t − tk ) 1 − Δ(t − tk )
Z
t tk
kw(τ )kdτ.
(45)
Substituting Eq.(37), (38) and (45) into (35) results in Δ(t − tk ) kZ(t)k2 1 − Δ(t − tk ) Z t c4 ec2 (t−tk ) + 2LΓ kZ(t)k kw(τ )kdτ 1 − Δ(t − tk ) tk 1 + 2c1 kP kkZ(t)k2 + 8kP k2 G20 kZ(t)k2 + kw(t)k2 , 2
V˙ (Z(t)) ≤ − LkZ(t)k2 + 2LΓ
where
0 Γ = kP k
−HC
According to Eq.(27), we get
1 1 h 2 |y(t)| = C γ2 γ2
0
Combining Eq.(46) and (48) results in
i
BK
.
BK
2 1 Z(t) ≤ 2 kZ(t)k2 . γ
(46)
(47)
(48)
Δ(t − tk ) 1 2 V˙ (Z(t))+ 2 |y(t)| − kw(t)k2 ≤ −LkZ(t)k2 + 2LΓ kZ(t)k2 γ 1 − Δ(t − tk ) Z t c4 ec2 (t−tk ) + 2LΓ kw(τ )kdτ kZ(t)k 1 − Δ(t − tk ) tk 1 1 + 2c1 kP kkZ(t)k2 + 8kP k2 G20 kZ(t)k2 + 2 kZ(t)k2 − kw(t)k2 , γ 2 (49) where t ∈ [tk , tk+1 ). 16
In order to keep the stability of system (35),the scaling gain L is fixed appropriately as
1 . (50) γ2 Because t ∈ [tk , tk+1 ), there exists Δ(t − tk ) ≤ Δ(T ). Substituting Eq.(50) L = 2 + 2c1 kP k + 8kP k2 G20 +
into (49) and integrating both sides of Eq.(49) from tk to tk+1 , we can obtain Z tk+1 1 2 2 ˙ V (Z(t)) + 2 |y(t)| − kw(t)k dt γ tk Z tk+1 Z tk+1 Δ(T ) kZ(t)k2 dt + 2LΓ kZ(t)k2 dt ≤−2 1 − Δ(T ) tk tk Z t Z tk+1 c 4 e c2 T + 2LΓ kZ(t)k kw(τ )kdτ dt 1 − Δ(T ) tk tk Z Z tk+1 1 tk+1 2 + 2LkP k kZ(t)k dt − kw(t)k2 dt. (51) 2 tk tk According to Cauchy-Schwarz Inequality, there exists Z t Z tk+1 kZ(t)k kw(τ )kdτ dt tk
1 ≤ 2
T = 2
Z
tk
tk+1
tk Z tk+1 tk
kZ(t)k2 dt
Z
tk+1
tk
T kZ(t)k dt + 2 2
1dt +
Z
t tk
1 2
Z
t
tk
kw(t)k2 dt
Z
tk+1
1dt
tk
kw(t)k2 dt.
Substituting Eq.(52) into (51) results in Z tk+1 1 2 2 ˙ V (Z(t)) + 2 |y(t)| − kw(t)k dt γ tk Z tk+1 2LΔ(T ) + T c4 ec2 T ≤ −2 + kZ(t)k2 dt Γ 1 − Δ(T ) tk Z tk+1 1 c 4 e c2 T kw(t)k2 dt. − + T LΓ 1 − Δ(T ) 2 tk
If T is small enough, such that 2LΔ(T ) T c 4 e c2 T −1+ + Γ < 0, 1 − Δ(T ) 1 − Δ(T ) T LΓ
1 c 4 e c2 T − < 0, 1 − Δ(T ) 2
then Eq.(53) is negative definite. Further more, we have Z tk+1 Z tk+1 1 2 2 ˙ V (Z(t)) + 2 |y(t)| − kw(t)k dt ≤ − kZ(t)k2 dt, γ tk tk 17
(52)
(53)
(54) (55)
(56)
from which it can be concluded that the closed-loop system (25) is uniformly globally asymptotically stable with the designed controller when w(t) = 0. Moreover, the following can be obtained based on Eq.(56), n−1 n−1 X Z tk+1 1 X Z tk+1 2 V (Z(tn )) − V (Z(t0 ))+ |y(t)| dt − kw(t)k2 dt ≤ 0. γ2 tk tk k=0
k=0
(57)
When Z(t0 ) = 0, V (Z(tn )) ≥ 0 and V (0) = 0, it can be concluded from
Eq.(57) that
n−1 X Z tk+1 k=0
tk
n−1 X Z tk+1 1 2 |y(t)| dt ≤ kw(t)k2 dt. γ2 tk
Because t0 = 0 , the following inequality holds when n −→ ∞, Z ∞ Z ∞ 2 |y(τ )| dτ ≤ γ 2 kw(τ )k2 dτ. 0
155
(58)
k=0
(59)
0
Then the proof ends.
According to Eq.(54) and (55), there exists 1 − T c 4 e c2 T Γ c2 T Δ(T ) ≤ min , 1 − 2T Lc4 e Γ . 1 + 2LΓ
Defining
(60)
1 − T c 4 e c2 T Γ , 1 − 2T Lc4 ec2 T Γ , 1 + 2LΓ based on Eq.(60), the sampling period T can be selected according to P = min
T ≤
1 (P + 1)c2 + c3 ln . c2 c2 + c3
(61)
4. Simulation and Test Study In this section, the simulations and tests of the shimmy control system (27) under 60km/h and 100km/h are conducted. The parameters of an EV are listed in Table 1. The original values of the system are set as z(0) = [0, 0, 0, 0]T , k1 = 44, k2 = 10, k3 = 24, k4 = 10 and L = 3. w(t) = 500 sin(20t)/(1 + t2 ) is used as the external disturbance moment. Then the controller is u(t) = u(tk ) = −44L4 zˆ1 (tk ) − 10L4 zˆ2 (tk ) − 24L4 zˆ3 (tk ) − 10L4 zˆ4 (tk ). 18
(62)
Table 1: The parameters of a steering system of an EV with independent suspension
160
Parameter
Value
Parameter
Value
Cl
14.23 Nms/rad
Il
8.48 kgm2
Cr
14.23 Nms/rad
Ir
8.48 kgm2
d2a Ksl
16 300 Nms/rad
tm
0.07m
Kα
19 200 Nms/rad
d
0.15m
d2a Cs
18.5 Nms/rad
rd
0.25m
Kz
12 000 Nms/rad
f
0.15
Ky
24 700 Nms/rad
α
4◦
In the simulations, the disturbance is not applied to the EV from time 0 to 10th second when the EV runs, and the controller is not active. The disturbance is applied to the EV from 10th second to 30th second, and the controller reacts to control the EV’s shimmy. According to Eq.(18), we map the states x1 and x3 of the controller into the EV’s parameters ψl and ψr respectively. The output of
165
the controller under different conditions are shown in Figure 3 and 4. The two shimmy angles ψl , ψr and their observations ψˆl , ψˆr are shown in Figure 5 and 6, from which we can see that the observation errors are very small, that verifies the designed observer being effective. The simulation results show the effectiveness of the sampled-data output feedback controller. According to the built two DOF
170
shimmy model, the shimmy phenomenon under different conditions are as Figure 7 and 8 which are anastomotic with other researchers’ results in [28] and [29]. We also obtain that the designed sampled-data control method can stabilize the EV’s shimmy better than PID control method under different speeds as shown in Figure 7 and 8, and the shimmy angles ψl and ψr are stabilized to
175
zero within 5 seconds. The overshoots of the shimmy angles attenuate rapidly. On the other hand, from the Figure 7 and 8, we can see that although the SMC method can stabilize the shimmy phenomenon as good as the sampled19
8 6 4 2 0 -2 -4 -6 -8 0
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Figure 3: The output of the controller under 60km/h
data output feedback controller from time 10th second to 15th second. But the SMC method generates bigger chattering, especially after 15th second, which 180
also means that the designed sampled-data output feedback control method is aslo better than the SMC method in whole. In order to verify the proposed shimmy controller in practice, we design a hardware-in-the-loop simulation system as shown in Figure 9 which are composed of Matlab software, dSpace and EV test bed. The control algorithms
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are programmed by MATLAB. The signals measured by angular displacement sensors are input the dSPACE through the D/A model (DS2001). Then the dSPACE calculates and outputs the control signal to the DSP controller installed on the EV to control the shimmy phenomenon. The EV test bed can simulate different roads for the EV. According to the simulation results, the
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control results of the front-left wheel and front-right wheel are similar, so we just give the test results of the front-left wheel. The test results are shown in Figure 10 and 11. From these figures, we can obtain that the designed controller can stabilize the shimmy phenomenon within 5seconds in the tests, that is better than the PID control method. The designed controller is also better than
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the SMC method after 13th second. The shimmy angles ψl is not stabilized to zero because the test results are influenced by many unknown factors. But the
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Figure 4: The output of the controller under 100km/h
test results can meet the practical requirement. 5. Conclusions In this paper, for an EV driven by in-wheels motors with independent suspen200
sion subject to uncertain disturbances, the authors present an automatic control method to solve the shimmy problem of the steering system via a sampled-data output feedback controller. The controller is not only more practical in practice but also more easy to design. The simulations and tests are conducted to verify the effectiveness of the sampled-data output feedback controller for the
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EV’s shimmy problem under 60km/h and 100km/h compaired with PID and SMC methods. The results show that the designed controller can eliminate the shimmy phenomenon of the EV within 5 seconds not only in the simulations but also in the tests, and is better than the PID and SMC methods. acknowledgment
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This research were partly supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LY16E050003 and Natural Science Foundation of China under Grant No. 61628302.
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(b) Right wheel Figure 6: States and their observation values of the left and right wheels under 100km/h and initial excitation 10◦
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(b) Right wheel Figure 7: Shimmy control of the left and right wheels under 60km/h and initial excitation 10◦
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Figure 9: The hardware-in-the-loop simulation system
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Figure 10: Shimmy control test of the left wheel under 60km/h and initial excitation 10◦
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0.6 Without control PID SMC Sampled-data output feedback
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Figure 11: Shimmy control test of the left wheel under 100km/h and initial excitation 10◦
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