Robust output-feedback based vehicle lateral motion control considering network-induced delay and tire force saturation

Author’s Accepted Manuscript Robust Output-feedback Based Vehicle Lateral Motion Control Considering Network-induced Delay and Tire Force Saturation Rongrong Wang, Hui Jing, Jinxiang Wang, Mohammed Chadli, Nan Chen www.elsevier.com/locate/neucom

PII: DOI: Reference:

S0925-2312(16)30679-8 http://dx.doi.org/10.1016/j.neucom.2016.06.041 NEUCOM17291

To appear in: Neurocomputing Received date: 18 December 2015 Revised date: 28 May 2016 Accepted date: 16 June 2016 Cite this article as: Rongrong Wang, Hui Jing, Jinxiang Wang, Mohammed Chadli and Nan Chen, Robust Output-feedback Based Vehicle Lateral Motion Control Considering Network-induced Delay and Tire Force Saturation, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.06.041 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 galley proof before it is published in its final citable 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.

Robust Output-feedback Based Vehicle Lateral Motion Control Considering Network-induced Delay and Tire Force Saturation Rongrong Wanga,∗, Hui Jinga,b , Jinxiang Wanga , Mohammed Chadlic , Nan Chena a School

of Mechanical Engineering, Southeast University, Nanjing 211189, P.R. China (e-mail:[email protected],[email protected], [email protected]) b School of Mechanical and Electrical Engineering, Guilin University of Electronic Technology, Guilin 541004, P.R. China(e-mail:[email protected]) c University of Picardie Jules Verne, MIS (E.A. 4290). 80039, Amiens, France(e-mail:[email protected])

Abstract This paper presents a robust H∞ output-feedback vehicle lateral motion control strategy considering networkinduced delay and tire force saturation. The unavoidable time delay in the in-vehicle networks degrades the control performance, and even deteriorates the system stability. In addition, the tire lateral force suffers saturation phenomenon, which also deteriorates the control effect in extreme driving conditions. To handle the network-induced control delay and tire force saturation, a robust H∞ controller is presented to regulate the vehicle lateral motion. An output-feedback control schema, which does not need the vehicle lateral velocity, is designed to achieve the desired control performance and reduce the cost of control system. The tire cornering stiffness uncertainty and external disturbances are also considered in the controller design to improve the robustness of the proposed controller. The comparative simulation results based on Carsim-Simulink joint simulation verify the effectiveness and robustness of the proposed control strategy. Keywords: Control delay, vehicle dynamics, saturation, output-feedback, robust control.

1. Introduction Advanced technologies such as active front steering (AFS), antilock brake system (ABS), and direct yawmoment control (DYC) have been utilized to improve the vehicle safety and handling increasingly[1][2][3]. To better improve the performance of the aforementioned technologies, numerous control strategies have also been presented in engineering field, such as fuzzy control [4], sliding mode control (SMC) [5], datadriven control [6], learning control [7], adaptive robust control [8], and convex programming [9][10]. However, while enjoying the convenience of these advanced technologies, the vehicle system is becoming much more complicated. In addition, to ensure the performance of each subsystems, the information among all the subsystems, controllers, sensors, and actuators should be transmitted timely. Due to the heavy tasks on information exchange and signal transmission among all the vehicle components, the in-vehicle networks such as controller area network (CAN) have attached more and more attentions in modern vehicles[11][12]. It is noteworthy that the in-vehicle networks are always working under severe environment, i.e. the varying temperature, the unreliable performance of the hardware, the limited bandwidth of the networks and the ∗ Corresponding

author Email address: [email protected] (Rongrong Wang)

Preprint submitted to Neurocomputing

July 20, 2016

noises caused by electro-magnetic effect, making the states and control signals transmitted on the network might be delayed or dropped. Therefore, the vehicle dynamics controlled through in-vehicle network can be treated as a networked control system (NCS), where the network-induced delay or dropout will probably degrade the control effect, even deteriorate the system stability [11][13]. To improve the vehicle handling and stability with delay, a robust lateral motion control of electric ground vehicles with random network-induced delays was proposed in [11]. In the paper, the random network-induced delays in both channels were modeled as two homogenous Markov chains, and statistic information of these delays were incorporated in the model-dependent tracking controller design. A controller combined AFS and DYC for four-wheel-independent-drive electric vehicles was presented in [14], where the time-varying delays were handled by expressing delay uncertainties in the form of a polytope using Taylor series expansion. In [15], the multiple successive delay components were formulated as a NCS problem, and solved by network-based H∞ control schema. In [16], an observer-based tracking controller design for networked predictive control systems was discussed, and the uncertain Markov delays were considered in the control system. In [17], the authors proposed an deterministic approach to handle the delays and packet dropout in NCS, and the maximum delays were considered in the controller design. [18] discussed the networked feedback control for systems with quantization and non-stationary random delays, and sufficient conditions for the existence of the admissible controller were established to ensure the exponential stability of the closed-loop system. In [19], a cooperative relaying schema was designed to enable train-train communications and to enhance the train control performance of communication-based train control (CBTC) systems for rail vehicles. Note that the aforementioned studies mainly focused on the NCS problem with the network-induced time delay, while no attention was paid on the actuator saturation characteristics. Actually, there definitely exists the physical limit of the actuator in real application, for example, the tire forces can be saturated in extreme driving conditions. When the actuator is saturated, the desired control effects will be decreased and thus degrades the performance of the closed-loop system. To handle the actuator saturation caused by the physical limit, [20] presented a method for estimating the domain of attraction of the origin, and a simplified condition was derived in terms of an auxiliary feedback matrix. In [21], the Takagi-Sugeno (T-S) fuzzy representation of a nonlinear system subjected to actuator saturation was presented, the problem of estimating the domain of attraction of a T-S fuzzy system were formulated and solved by optimization method. In [22], a class of singular systems with multiple time delays subjected to input saturation was discussed, and the delay-dependent sufficient condition and the robust guaranteed cost H∞ controller were proposed via Lyapunov theory, singular value theory and linear matrix inequality (LMI) approaches. In [23], the authors studied the control under quantization, saturation and delay. In the paper, the authors investigated the linear systems with given constant bounds on the quantization error and on the time-varying delay, and the decomposition of the quantization was proposed. In [24] control method for seat suspension systems with actuator saturation and time-varying input delay was discussed by employing a delay-range-dependent Lyapunov function. The property of the saturation nonlinearity was explored and

2

the existence conditions of the desired state-feedback controller were derived. However, the aforementioned literature mainly adopted the state-feedback control, where all the system states should be obtained in the control-loop. Note that very expensive sensors such as the differential GPS, are required to measure the vehicle lateral speed and sideslip angle. Consequently, the output-feedback control is a better choice to reduce the cost for commercial cars. Numerous output-feedback control approaches for vehicle motion control have been proposed in previous research works [25]. In [26], a class of continuous-time T-S fuzzy affine dynamic systems with parametric uncertainties and input constraints were investigated. Based on a smooth piecewise quadratic Lyapunov function combined with S-procedure, some new results were developed for the continuous-time T-S fuzzy affine systems. In [27], the authors studied the linear discrete and continuous-time systems with time-invariant polytopic uncertainties, and gave sufficient conditions for static output-feedback (SOF) controller in terms of LMIs. [28] designed a robust SOF controller for vehicle dynamics subjected to external disturbance and unknown sensor faults, and a descriptor state and fault observer were designed to achieve the system state and sensor fault estimation simultaneously. In [29] the distributed robust state and output-feedback controller were designed for rendezvous of networked autonomous surface vehicles using neural networks. In [30], non-fragile multi-objective SOF control of vehicle active suspension was discussed. With a delay-dependent Lyapunov function, new existence conditions of delay-dependent nonfragile SOF H∞ controller and L2 − L∞ controller were derived in the study. [31] presented a SOF design procedure for robust emergency lateral control of a highway vehicle. A static output-feedback controller was designed for vehicle suspensions in [32], and an effective two-step computational approach was proposed in the paper. An robust H∞ output-feedback controller for vehicle active suspension systems with control delay is designed in [33], the existence of admissible controllers was formulated with Lyapunov theory and linear matrix inequality approaches. Nevertheless, as far as we know, few published literature has investigated the vehicle lateral motion control with output-feedback control schema in the presence of control delay and actuator saturation. To handle these problems, this paper proposes a robust H∞ static output-feedback controller considering vehicle parameter uncertainties and external disturbances. The main contributions and highlights of this paper are listed as follows: 1) The unavoidable control delay and actuator saturation are considered in the controller design, and both the yaw rate and sideslip angle are controlled to simultaneously improve the vehicle handling and stability; 2) A robust H∞ static output-feedback control schema is presented, thus the relative lower cost sensors can be utilized; 3) Both the tire cornering stiffness uncertainty and the external disturbances are considered in the controller design to improve the robustness of the proposed controller. The remaining of the paper is organized as follows. The vehicle lateral dynamics with parameter uncertainties in the presence of control delay and actuator saturation are discussed in Section 2. The robust H∞ output-feedback control schema is presented in Section 3. Simulation based on a high-fidelity vehicle model via CarSim-Simulink is given in Section 4, followed by the conclusion in Section 5.

3

OI

Y[

I I

OU

[

U

U

Y\

)\I

\

)\U Figure 1: Vehicle dynamics in yaw plane.

2. System Modelling and Problem Formulation 2.1. Vehicle Modelling In this section, the vehicle lateral dynamics is formulated with a bicycle model shown in Figure 1. Assuming the front-wheel steering angle is small, the vehicle’s handling dynamics in the yaw plane can be modelled as ⎧ ⎪ ⎪ ⎨v˙ y = ⎪ ⎪ ⎩r˙ =

Fyf +Fyr m

− rvx

lf Fyf −lr Fyr Iz

,

(1)

where Fyf and Fyr are the front and rear tire lateral forces, respectively; lf and lr denote the distances between the centre of gravity (CG) of vehicle to the front and rear wheel axles, respectively; m and Iz represent the mass and yaw inertia of the vehicle, respectively; vx and vy are the longitudinal and lateral velocities, respectively; r is the vehicle yaw rate. To handle the tire force saturation, the Affne Force-Input (AFI) model is adopted in the paper. The AFI model is based on the same small steering angle and constant speed assumptions. In contrast to the conventional bicycle model, the front lateral force is the input rather than substituting the front slip angle into the model and using the steering angle. This choice preserves the linear relationship between the input and the model states [34]. The rear-wheel lateral force can be formulated by the AFI model as ¯ r ), Fyr = F¯yr − C˜αr (αr − α

(2)

where F¯yr is the tire force at the operating point α ¯ r , C˜αr denotes the local cornering stiffness, and αr represents the rear-tire slip angle which can be modelled by αr =

vy lr − r. vx vx

4

(3)

Based on (2) and (3), the vehicle model (1) can be written as ⎧ ⎪ ⎪ ⎨v˙ y = ⎪ ⎪ ⎩r˙ =

˜a ( y − lr r−α Fyf +F¯yr −C ¯r ) r vx vx m v

˜a ( vy lf Fyf −lr [F¯yr −C r vx Iz

− vlrx

− rvx

r−α ¯ r )]

.

(4)

By defining the state vector of the system x(t) = [vy , r]T , the virtual control input u(t) = Fyf , and the disturbance w(t) = [d1 , d2 ]T , the state-space form of the vehicle model can be given as x(t) ˙ = Ax(t) + Bu(t) + w(t), where

⎡ ⎢a11 A=⎣ a21

with

˜ C

a11 = − mvarx , a21 = b1 = d1 =

˜a lr C r Iz vx

,

⎤

⎡

(5)

⎤

a12 ⎥ ⎢ b1 ⎥ ⎦,B = ⎣ ⎦, a22 b2

a12 = a22 =

1 b2 m, ¯ ˜ Fyr +Car α ¯r , d2 m

=

˜a lr C r mvx − vx , ˜ l2 C − Irz vaxr ,

(6)

lf Iz ,

=−

˜a α lr (F¯yr +C r ¯r ) . Iz

Due to the varying road conditions and vehicle states, the local cornering stiffness of tire is always varying and unknown. Since the local cornering stiffness is actually bounded, it can be written in the form of C˜ar = C˜a0r + λr C˜ard ,

(7)

where λr is a time-varying parameter satisfing |λr | ≤ 1, C˜a0r and C˜ard are the nominal value and variation of C˜ar , respectively. So the system matrix A can be rewritten as A = A0 + ΔA,

(8)

where A0 and ΔA are the nominal value and variation of matrix A, respectively, and can be written as ⎡ A0 =

˜a C 0r ⎢− mvx ⎣ l C˜ r a 0r

Iz vx

˜ l C −vx + rmvax0r ˜a l2 C

−

r

0r

⎤

⎡

⎥ ⎦ , ΔA =

Iz vx

˜a λr C rd ⎢− mvx ⎣ l λ C˜ r r a Iz vx

rd

˜a lr λr C rd mvx ˜a l2 λr C

−

r

Iz vx

rd

⎤ ⎥ ⎦.

Based on the definition of ΔA, we can define ΔA = M F E,

5

(9)

with

⎡ ⎢ 1 M =⎣ 0

⎡ ⎤ ˜a C rd 0 ⎥ ⎢− mvx ⎦ , F = λr , E = ⎣ l C˜ r ard 1 Iz vx

˜a lr C rd mvx 2 ˜ l Ca

−

r

rd

⎤ ⎥ ⎦.

Iz vx

Remark 1: Note that the aforementioned vehicle model adopts Fyf as the system input rather than substituting the front-wheel steering angle and slip angle into the model. The advantage of the new model is that it preserves the linear relationship between the input and the system states, and thus facilitates the controller design. Remark 2: Currently there are seldom good ways to handle the saturation issue of the steering angle. When dealing with the actuation saturation issues in the motion control of ground vehicles, most of the previous works only considered the longitudinal tire forces saturation problem [36][37][38][39], and currently there is seldom good way to handle the saturation issue of the steering angle. The challenge is that the saturation limit of the steering angle is hard to obtain. That is because in different longitudinal speeds and tire-road adhesion conditions, the saturation limits of the steering angle are different, and complex to be explicitly expressed. The longitudinal tire forces also affect the saturation limit of the steering angle, according to the tire-road friction limit circle. All these problems make handling the saturation problem of steering a relatively difficult task. Since the physical limit of the front tire lateral forces Fyf can be estimated or calculated [40], we constraint the front-wheel steering angle by limiting Fyf . ¯ r at the operation point, the Remark 3: Since the absolute value of F¯yr equals approximately to C˜ar α disturbance terms d1 and d2 in (6) are small. Furthermore, together with the parameter uncertainties, these disturbance terms will be handled by the proposed robust H∞ control schema. 2.2. Problem Formulation Generally, the state-feedback control schema requires all the elements in the state vector can be measured. However, the vehicle lateral velocity vy is usually hard to be measured with low-cost sensors. In this paper, only the vehicle yaw rate r is taken as the measured output, thus we have y(t) = C1 x(t),

(10)

where C1 = [0 1]. In the vehicle lateral motion regulation, not only the handling but also the vehicle stability should be concerned. Therefore, we define the controlled output z(t) as z(t) = C2 x(t),

6

(11)

with C2 = diag{1, 1}. Then the vehicle lateral control model can be written as ⎧ ⎪ ⎪ x(t) ˙ = (A0 + ΔA)x(t) + Bu(t) + w(t) ⎪ ⎪ ⎪ ⎨ . y(t) = C1 x(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z(t) = C2 x(t)

(12)

To improve the control performance, the Proportional-Integral (PI) control schema is employed [35]. Denoting t the reference of the system states as xr , also defining ξ = [ξ1T , ξ2T ]T with ξ1 = x − xr and ξ2 = 0 (x − xr )dt, the system (12) can be rewritten as ⎧ ⎪ ˙ = Aξ(t) ˜ ˜ ⎪ ξ(t) + Bu(t) + w(t) ˜ ⎪ ⎪ ⎪ ⎨ , y(t) = C˜1 ξ(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z(t) = C˜2 ξ(t)

(13)

˜ F E, ˜ A = A0 + M F E, and with A˜ = A˜0 + M ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ ⎢A0 0⎥ ˜ ⎢B ⎥ ⎢Axr − x˙ r + w(t)⎥ A˜0 = ⎣ ˜ =⎣ ⎦ , B = ⎣ ⎦ , w(t) ⎦, I 0 0 0 ⎡ ⎡ ⎤ ⎤ ⎡ ⎤ 0 1 0 0 M 0 E 0 ⎢ ⎥ ⎥ ˜ ⎢ ⎥ ˜ ˜ =⎢ M ⎣ ⎦, ⎦,E = ⎣ ⎦ , C1 = ⎣ 0 0 0 1 0 0 0 0 C˜2 = diag{1, 1, 1, 1}. Considering the delay and actuator saturation issues, the system model (13) can be expressed as ⎧ ⎪ ˙ = Aξ(t) ˜ ˜ ⎪ ξ(t) + Bσ(u(t)) + w(t) ˜ ⎪ ⎪ ⎪ ⎨ , y(t) = C˜1 ξ(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z(t) = C˜2 ξ(t)

(14)

where the static output-feedback controller u(t) is presented as ˜ ˜ C˜1 ξ(t − τ (t)), u(t) = Ky(t − τ (t)) = K

(15)

˜ is the output-feedback control gain to be designed, τ (t) is the control delay in the system satisfying where K 0 ≤ τ (t) ≤ τ¯, with τ¯ being the maximum time delay. Generally, we assume u ∈ Rm . The actuator saturation

7

function σ(u(t)) can be described as σ(u(t)) = [σ(u1 (t)) ... σ(um (t))]T and ⎧ ⎪ ⎪ uimax , ⎪ ⎪ ⎪ ⎨ σ(ui (t)) = ui (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −uimax ,

if ui (t ) > uimax if − uimax ≤ ui (t ) ≤ uimax ,

(16)

if ui (t ) < −uimax

where i = 1, ..., m and uimax is the maximum value of ui . The control objective of vehicle lateral motion regulation is to design a robust H∞ controller, such that the following requirements can be satisfied: (1) the closed-loop system in (14) is asymptotically stable with w(t) ˜ = 0; (2) the closed-loop system in (14) has the following H∞ disturbance attenuation performance, i.e.

t 0

z T (t)z(t)dt ≤ γ 2

t

0

w ˜T (t)w(t)dt, ˜

(17)

where γ is the prescribed attenuation level. Remark 4: u ∈ Rm is assumed in the aforementioned saturation model combined with control delay. Actually, there is only one control input Fyf in this study, but this will not affect the controller design since u ∈ Rm is a general representation for the theoretical analysis in the following sections. By assuming u ∈ Rm in the controller design, the proposed controller has the potential to solve vehicle control problems with multiinputs, for example, the vehicle motion control problem taking both front wheel steering angle and direct yaw moment as the control inputs.

3. Robust Static Output-feedback Controller Design In this section, we will give the sufficient stability conditions for the closed-loop system and the design of the robust H∞ output-feedback controller. For brevity, we first investigate the state-feedback case and latter for the output-feedback case. Suppose the state signals are transmitted from sensor to controller (SC) with time delay τ1 (t), and the control signals are transmitted from controller to actuator (CA) with time delay τ2 (t). At the time instant t, we have the state-feedback controller u(t) = Kξ(t − τ1 (t)), where K is the state-feedback control gain. Then considering the time delay τ2 (t) of CA, we have u(t−τ2 (t)) = Kξ(t − τ1 (t) − τ2 (t)). Define τ (t) = τ1 (t) + τ2 (t) and suppose the actual time delay τ (t) satisfies 0 ≤ τ (t) ≤ τ¯, we have

⎧ ⎪ ˙ = Aξ(t) ˜ ˜ ⎨ ξ(t) + Bσ(u(t)) + w(t) ˜ ⎪ ⎩ u(t) = Kξ(t − τ (t))

.

(18)

˜ as bi and the ith row of K as ki . Then BK ˜ = b1 k1 + b2 k2 + · · Suppose ξ ∈ Rn . Denote the ith column of B

8

· + bm km . As for the control gain matrix K ∈ Rm×n , we define L (K) := {ξ ∈ Rn : |ki ξ| ≤ uimax , i ∈ [1, m]},

(19)

with L (K) being the region in the state space where the control input is linear in ξ. Then we will utilize the technique of auxiliary feedback matrices to reduce the conservatism of handling the actuator saturation [20][24]. For two matrices K, H ∈ Rm×n , and a vector v ∈ Rm , a matrix set is introduced as ⎤

⎡ ⎢ ⎢ W (v, K, H) = ⎢ ⎢ ⎣

v1 f1 + (1 − v1 )h1 .. . vm fm + (1 − vm )hm

⎥ ⎥ ⎥, ⎥ ⎦

(20)

where hi is the ith row of H, and the auxiliary matrix H satisfies |hi ξ| ≤ uimax , i ∈ [1, m]. Let V := {v ∈ Rm : vi = 1 or 0}. It can be seen that there are 2m elements in the V . Then a v ∈ V will be adopted to choose from the rows of K and H to establish a new matrix W (v, K, H). That is, if vi = 1, then the ith row of W (v, K, H) is ki , and if vi = 0, then the ith row of W (v, K, H) is hi . Remark 5: Note that the delays from sensor-to-controller (SC) and controller-to-actuator (CA) might not be the same, and may be handled independently. However, since the proposed controller is static in this study, based on the parallel distributed compensation (PDC) technology [17], delays from SC and CA are lumped together into the control delay to facilitate the controller design. Based on the above analysis, we give the following theorem to present the existence conditions of the desired control gains. The following Lemma is given in advance. ˜ and E ˜ be the real matrices with proper dimensions, and F satisfies Lemma 1: [35] [41]: Let Π = ΠT , M F T F < I, then the following condition: ˜ F E˜ + (M ˜ F E) ˜ T < 0, Π+M

(21)

holds if and only if there exists a positive scalar  > 0 such that ˜ + −1 M ˜M ˜ < 0. ˜T E Π + E

(22)

Theorem 1: Given positive constants γ, τ¯, and uimax , the closed-loop system in (14) is asymptotically ˜ ∈ [0, ∞), if there exist symmetric stable with w(t) ˜ = 0, and satisfies the H∞ performance index (17) for all w(t)

9

positive definite matrices P and Q, general matrices H, S1 , S2 , S3 , Wv , and Ws such that ⎤ ⎡ ⎢Λ11 Λ12 ⎥ ⎦ < 0, ⎣ ∗ Λ22 ⎡ ⎤ P −1 2 I H( ) −u ⎢ imax ⎥ ρ ⎣ ⎦ ≤ 0, i = 1, ..., m P −1 ∗ −( ρ ) I

(23a)

(23b)

with

Λ11

⎡ ⎢Π11 ⎢ =⎢ ⎢ ∗ ⎣ ∗

Π12 Π22 ∗

⎡ ⎤ Π13 ⎥ ⎢P ⎢ ⎥ ⎢ Π23 ⎥ ⎥ , Λ12 = ⎢ 0 ⎣ ⎦ Π33 P

C˜2T 0 0

⎤ S1 ⎥ ⎥ S2 ⎥ ⎥, ⎦ S3

Λ22 = diag[−γ −2I − I − τ¯−1 Q], ˜ v − S1 + S T , Π13 = A˜T P + S T , Π22 = −S2 − S T , where Π11 = P A˜ + A˜T P + S1 + S1T , Π12 = P BW 2 3 2 ˜ s )T − S T , Π33 = τ¯Q − 2P , and [•] denotes • + •T . Π23 = (P BW 3 s Proof : Define a Lyapunov function for the system in (14) as V (t) = V1 (t) + V2 (t), where V1 (t) = ξ T (t)P ξ(t),

0 t ˙ V2 (t) = ξ˙T (α)Qξ(α)dαdβ, −¯ τ

t+β

with P and Q being symmetric positive definite matrices. Computing the time derivative of the above Lyapunov function along the trajectory of system (14), we can get V˙ (t) = V˙1 (t) + V˙2 (t)

  T T ˙ ˙ ˙ = ξ (t)P ξ(t) + τ¯ξ (t)Qξ(t) − s

t

t−¯ τ

Since

ξ(t) − ξ(t − τ (t)) −

t

t−τ (t)

˙ ξ˙T (s)Qξ(s)ds.

(24)

˙ ξ(s)ds = 0,

then for any matrices Si (i = 1, 2, 3) with proper dimensions, we get 

T

2η (t)S ξ(t) − ξ(t − τ (t)) −

t

t−τ (t)

 ˙ξ(s)ds = 0,

(25)

where η T (t) = [ξ T (t), ξ T (t − τ (t)), ξ˙T (t), w ˜ T (t)], S = [S1T , S2T , S3T , 0]T . By some algebraic manipulations,

10

(25) can be written as  T

T

η (t) [ S[1 − 1 0 0] ]s η(t) − η (t)S

t

t−τ (t)

 ˙ ξ(s)ds

= 0.

(26)

s

Since ˙ = Aξ(t) ˜ ˜ ξ(t) + Bσ(Kξ(t − τ (t))) + w(t), ˜ for any matrix T > 0 with proper dimension, we have ˙ + Aξ(t) ˜ ˜ 2Θ[−ξ(t) + Bσ(Kξ(t − τ (t))) + w(t)] ˜ = 0,

(27)

˜ ˜ where Θ = ξ T (t)T + ξ˙T (t)T . The terms of 2ξ T (t)T Bσ(Kξ(t − τ (t))) and 2ξ˙T (t)T Bσ(Kξ(t − τ (t))) in (27) can be written as ˜ 2ξ T (t)T Bσ(Kξ(t − τ (t))) = 2

m 

ξ T (t)T bi σ(ki ξ(t − τ (t))),

i=1

˜ − τ (t))) = 2 2ξ˙T (t)T Bσ(Kξ(t

m 

ξ˙T (t)T bi σ(ki ξ(t − τ (t))).

i=1

According to the saturation function (16), for each term 2ξ T (t)T bi σ(ki ξ(t − τ (t))), we get 1. If ξ T (t)T bi ≥ 0 and ki ξ(t − τ (t)) ≤ −uimax , then for −uimax ≤ hi ξ(t − τ (t)), we have 2ξ T (t )Tbi σ(ki ξ(t − τ (t ))) = − 2ξ T (t )Tbi uimax ≤2ξ T (t)T bi hi ξ(t − τ (t)), 2. If ξ T (t)T bi ≥ 0 and ki ξ(t − τ (t)) ≥ −uimax , then σ(ki ξ(t − τ (t))) ≤ ki ξ(t − τ (t)), we have 2ξ T (t )Tbi σ(u(t − τ (t ))) ≤ 2ξ T (t )Tbi ki ξ(t − τ (t )), 3. If ξ T (t)T bi ≤ 0 and ki ξ(t − τ (t)) ≥ uimax , then σ(ki ξ(t − τ (t))) = uimax , for uimax ≥ hi ξ(t − τ (t)), we have 2ξ T (t )Tbi σ(ki ξ(t − τ (t ))) = 2ξ T (t )Tbi uimax ≤ 2ξ T (t)T bi hi ξ(t − τ (t)), 4. If ξ T (t)T bi ≤ 0 and ki ξ(t − τ (t)) ≤ uimax , then σ(ki ξ(t − τ (t))) ≥ ki ξ(t − τ (t)), we have 2ξ T (t )Tbi σ(ki ξ(t − τ (t ))) ≤2ξ T (t )Tbi ki ξ(t − τ (t )). Combining the aforementioned four cases, we can conclude that   2ξ T (t )Tbi σ(ki ξ(t − τ (t ))) ≤ max 2ξ T (t )Tbi hi ξ(t − τ (t )), 2ξ T (t )Tbi ki ξ(t − τ (t )) .

11

(28)

Note that if 2ξ T (t )Tbi hi ξ(t − τ (t )) ≤ 2ξ T (t )Tbi ki ξ(t − τ (t )), then vi = 1, otherwise vi = 0. Thus, we have 2ξ T (t )Tbi σ(ki ξ(t − τ (t ))) ≤ 2(vi ξ T (t )Tbi ki ξ(t − τ (t )) + (1 − vi )ξ T (t )Tbi hi ξ(t − τ (t ))).

(29)

Summing up both sides of (29) from 1 to m, we have ˜ σ(K ξ(t − τ (t ))) ≤2 2ξ T (t )T B

m 

(vi ξ T (t)T bi ki ξ(t − τ (t)) + (1 − vi )ξ T (t)T bi hi ξ(t − τ (t)))

i=1

=2ξ T (t)T [

m 

(30)

bi (vi ki + (1 − vi )hi )]ξ(t − τ (t))

i=1

˜ (v, K, H)]ξ(t − τ (t)), =2ξ T (t)T [BW that is ˜ σ(K ξ(t − τ (t ))) ≤ 2 ξ T (t )T B ˜ W (v , K , H )ξ(t − τ (t )). 2ξ T (t )T B

(31)

˜ σ(K ξ(t − τ (t ))) ≤ 2 ξ˙T (t )T B ˜ W (s, K , H )ξ(t − τ (t )), 2ξ˙ T (t )T B

(32)

Similarly, we can get

where W (s, K, H) and s have similar definitions with W (v, K, H) and v, respectively. For brevity, in the following sections, we represent W (v, K, H) and W (s, K, H) as Wv and Ws , respectively. By adding (26), (27), (31) and (32) to (24), after some algebraic manipulations, we have

t   ˙ − ˙ V˙ (t) ≤ ξ˙T (t)P ξ(t) + τ¯ξ˙T (t)Qξ(t) ξ˙T (s)Qξ(s)ds s t−¯ τ  

t T ˙ − η (t)S ξ(s)ds + η T (t) [ S[1 − 1 0 0] ]s η(t) t−τ (t)

s

      ˙ + ξ T (t) T A˜ ξ(t) + ξ T (t)T BWv ξ(t − τ (t)) + ξ T (t)T w(t) ˜ − ξ T (t)T ξ(t) s s s     ˜ − ξ˙T (t)2T ξ˙T (t) + ξ˙T (t)T Aξ(t) + ξ˙T (t)T BWs ξ(t − τ (t)) + ξ˙T (t)T w(t) ˜ s s

  t ˙ + τ¯η T (t)SQ−1 ST η(t) − ≤ ξ˙T (t)P ξ(t) + τ¯ξ˙T (t)Qξ(t) [η T (t)S s

t−τ (t)

(33)

+ ξ˙T (s)Q]Q−1 [η T (t)S + ξ˙T (s)Q]T ds + η T (t) [ S[1 − 1 0 0] ]s η(t)       ˙ − ξ T (t)T ξ(t) + ξ T (t) T A˜ ξ(t) + ξ˙T (t)T BWv ξ(t − τ (t)) + ξ T (t)T w(t) ˜ s

s

s

    ˜ − ξ (t)2T ξ (t) + ξ˙T (t)T Aξ(t) + ξ˙T (t)T BWs ξ(t − τ (t)) + ξ˙T (t)T w(t) ˜ s s

t =η T (t)[Ψ + Ψs + τ¯SQ−1 ST ]η(t) − [η T (t)S + ξ˙T (s)Q]Q−1 [η T (t)S + ξ˙T (s)Q]T ds ˙T

˙T

t−τ (t)

≤ η T (t)[Ψ + Ψs + τ¯SQ−1 ST ]η(t), that is V˙ (t) ≤ η T (t)[Ψ + Ψs + τ¯SQ−1 ST ]η(t),

12

(34)

where ⎡ ⎢ ⎢ ⎢ ⎢ Ψ=⎢ ⎢ ⎢ ⎣

T A˜ + A˜T T

˜ v T BW

P − T + A˜T T

∗

0

˜ s )T (T BW

∗

∗

τ¯Q − 2T

∗

∗

∗

⎤ T

⎥ ⎥ 0⎥ ⎥ ⎥, T⎥ ⎥ ⎦ 0

Ψs = S[1 − 1 0 0] + [1 − 1 0 0]T ST . With the definition T = P , Ψ can be rewritten as ⎡ ˜ ˜T ⎢P A + A P ⎢ ⎢ ∗ ⎢ Ψ=⎢ ⎢ ∗ ⎢ ⎣ ∗

˜ v P BW

A˜T P

0

˜ s )T (P BW

∗

τ¯Q − 2P

∗

∗

⎤ P⎥ ⎥ 0⎥ ⎥ ⎥. ⎥ P⎥ ⎦ 0

(35)

Then, we will give the asymptotic stable condition for the system (14) with w(t) ˜ = 0. The time derivative of the Lyapunov function (34) can be rewritten as ˜ η (t), V˙ (t) ≤ η˜T (t)Π˜

(36)

˜ T , η˜T (t) = [ξ T (t), ξ T (t − τ (t)), ξ˙T (t)], and S ˜ −1 S ˜ = [S1T , S2T , S3T ]T with ˜ =Ψ ˜ +Ψ ˜ s + τ˜SQ where Π ⎡ ˜ ˜T ⎢P A + A P ⎢ ˜ =⎢ Ψ ∗ ⎢ ⎣ ∗

˜ v P BW 0 ∗

A˜T T

⎤

⎥ ⎥ T⎥, ˜ (P BWs ) ⎥ ⎦ τ¯Q − 2P

˜T . ˜ − 1 0] + [1 − 1 0]T S ˜ s = S[1 Ψ ˜ < 0. It can be observed that the robust asymptotical stability of the system (14) can be guaranteed if Π ˜ < 0. Thus we can get that the system (14) is robust Note that the condition (23a) in Theorem 1 ensures Π asymptotically stable with time-delay and actuator saturation. Next, we will prove that the H∞ performance of the system (14) under zero initial condition can be ˜T (t)w(t) ˜ to both sides of (34), we can get guaranteed. Adding the term z T (t)z(t) − γ 2 w V˙ (t) + z T (t)z(t) − γ 2 w ˜T (t)w(t) ˜ ≤ η T (t)Πs η(t),

13

(37)

where Πs = Ψh + Ψs + τ¯SQ−1 ST and ⎡ Φ ⎢ s ⎢ ⎢∗ ⎢ Ψh = ⎢ ⎢∗ ⎢ ⎣ ∗

⎤

˜ v P BW

A˜T P

P

0

˜ s )T (P BW

0

∗

τ¯Q − 2P

P

∗

∗

−γ −2 I

⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

(38)

with Φs = P A˜ + A˜T P + C˜2T C˜2 . Similarly, the H∞ performance of the system (14) can be guaranteed if Πs < 0. By Schur complement, Πs < 0 is equivalent to ⎡ ⎢Ψh + Ψs ⎣ ∗

⎤ S

⎥ ⎦ < 0, −¯ τ −1 Q

(39)

and the above inequality can be further rewritten as ⎡ ⎢Π11 ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎣ ∗

⎤

Π12

Π13

P

C˜2T

Π22

Π23

0

0

S2

∗

Π33

P

0

S3

∗

∗

−γ −2 I

0

0

∗

∗

∗

−I

0

∗

∗

∗

∗

−¯ τ −1 Q

S1

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(40)

which is equivalent to (23a) in Theorem 1. Therefore,the H∞ performance of the system (14) can be ensured. Finally, we will establish the LMI condition (23b). By the Theorem 1 in [20], an ellipsoid of the form Ω(P, ρ) := {ξ ∈ Rn : ξ T P ξ ≤ ρ},

(41)

is chosen as a subset of the set L (H), that is Ω(P, ρ) ⊂ L (H). Then we have H(

P −1 T ) H ≤ u2imax . ρ

(42)

By Schur complement, the above inequality can be rewritten as ⎡ 2 ⎢−uimax ⎣ ∗

⎤ H( Pρ )−1 −( Pρ )−1 I

⎥ ⎦ ≤ 0,

(43)

which is equivalent to (23b) in Theorem 1. This completes the proof. Now we are in the position to give the robust H∞ output-feedback controller design approach. Theorem 2: Given positive constants γ, τ¯, and u1max , the closed-loop system in (14) is stochastically ˜ ∈ [0, ∞), if there exists a stable with w(t) ˜ = 0, and satisfies the H∞ performance index (17) for all w(t) 14

¯ general matrices S¯1 , S¯2 , S¯3 , Y¯i , i = 1, 2, and a scalar symmetric positive definite matrices XU , XR , and Q,  > 0 such that ⎤ ⎡ ˜ ˜ ⎢Λ11 Λ12 ⎥ ⎦ < 0, i = 1, 2 ⎣ ˜ 22 ∗ Λ ⎡ ⎤ 2 ˜ Y1 ⎥ ⎢−u1max I ⎣ ⎦ ≤ 0, ˜ −1 ∗ −Xρ where

˜ 11 Λ

˜ 12 Λ

(44a)

(44b)

⎡ ˜2 ˜3 ˜ Ξ Ξ ⎢Ξ1 ⎢ ⎢ ∗ −S¯2 − S¯T Y˜ T B T − S¯T 2 3 i ⎢ =⎢ ⎢ ¯ − 2X ˜ ∗ −¯ τQ ⎢∗ ⎣ ∗ ∗ ∗ ⎤ ⎡ T T T ¯ ˜ ˜ ˜ M X C2 S1 XE ⎥ ⎢ ⎥ ⎢ T ⎢ 0 ¯ 0 0 ⎥ S2 ⎥ ⎢ =⎢ ⎥, ⎢ 0 0 M ⎥ S¯3T ⎥ ⎢ ⎦ ⎣ 0 0 0 0

⎤ I ⎥ ⎥ 0 ⎥ ⎥ ⎥, ⎥ I ⎥ ⎦ 2 −γ I

¯ − I − I], ˜ 22 = diag[−I − τ¯−1 Q Λ   ˜ 1 = XA ˜ T0 + S¯1 , Ξ ˜ 2 = B Y˜i − S¯1 + S¯2T , Ξ s

˜3 = Ξ

˜ T XA 0

+

˜ S¯3T , X

= U XU U T + RXR RT ,

Y˜i = Y¯i RT , Y¯1 = YH , Y¯2 = YR . Moreover, if the above LMI has feasible solutions with XU , XR and YR , the static output-feedback control ˜ can be obtained as K ˜ = YR X −1 . gain K R ¯ v = Wv P −1 , ¯ = P −1 QP −1 , S¯1 = P −1 S1 P −1 , S¯2 = P −1 S2 P −1 , S¯3 = P −1 S3 P −1 , W Proof : Define Q ¯ s = Ws P −1 , and P¯ = P −1 , performing a congruence transformation with Γ = diag{P, ¯ P ¯, P ¯, I, I, P ¯ , } to W (23a), we have ⎡ ¯ ⎢Π11 ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎣ ∗

¯ 12 Π

¯ 13 Π

P¯ C˜2T

S¯1

¯ 22 Π

¯ 23 Π

0

0

S¯2

∗

¯ 33 Π

I

0

S¯3

∗

∗

−γ −2 I

0

0

∗

∗

∗

−I

0

∗

∗

∗

∗

¯ −¯ τ −1 Q

I

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(45)

¯ 12 = B ˜W ¯ v − S¯1 + S¯T , Π ¯ 13 = P¯ A˜T + S¯T , Π ¯ 22 = −S¯2 − S¯T , Π ¯ 23 = ¯ 11 = A˜P¯ + P¯ A˜T + S¯1 + S¯T , Π where Π 1 2 3 2 ¯ 33 = τ¯Q ¯ − 2P¯ . ˜W ¯ s )T − S¯T , Π (B 3

15

˜ F E, ˜ we have As A˜ = A˜0 + M ⎡ ˜ ⎢Π11 ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎣ ∗

¯ 12 Π

˜ 13 Π

I

P¯ C˜2T

¯ 22 Π

¯ 23 Π

0

0

∗

¯ 33 Π

I

0

∗

∗

−γ −2 I

0

∗

∗

∗

−I

∗

∗

∗

∗

⎤ ⎡⎡ ⎤ ˜ ⎥ ⎥ ⎢⎢M ⎥ ⎥ ⎥ ⎢⎢ ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ¯ S2 ⎥ ⎢⎢ 0 ⎥ ⎥ ⎥ ⎥ ⎢⎢ ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ˜⎥ ⎥ S¯3 ⎥ ⎢⎢M ˜ P¯ 0 0 0 0 0]⎥ < 0, ⎥ + ⎢⎢ ⎥ F [E ⎥ ⎥ ⎢⎢ ⎥ ⎥ ⎢⎢ ⎥ 0 ⎥ ⎥ ⎥ ⎢⎢ 0 ⎥ ⎥ ⎥ ⎢⎢ ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ 0 ⎥ ⎢⎢ 0 ⎥ ⎥ ⎦ ⎦ ⎣⎣ ⎦ ¯ 0 −¯ τ −1 Q S¯1

⎤

(46)

s

˜ 11 = A˜0 P¯ + P¯ A˜T0 + S¯1 + S¯1T , Π ˜ 13 = P¯ A˜T0 + S¯3T . It follows from Lemma 1 that (46) is equivalent to where Π ⎤ ˆ Λ12 ⎥ ⎦ < 0, ˜ 22 Λ

⎡ ˆ ⎢Λ11 ⎣ ∗

(47)

with ⎡

ˆ 12 Λ

˜ 11 Π

⎢ ⎢ ⎢ ∗ ⎢ =⎢ ⎢ ∗ ⎢ ⎣ ∗

⎤

¯ 12 Π

˜ 13 Π

I

¯ 22 Π

¯ 23 Π

0

∗

¯ 33 Π

I

∗

∗

−γ −2 I

⎥ ⎥ ⎥ ⎥ ˆ ⎥ , Λ11 ⎥ ⎥ ⎦

⎡ P¯ C˜ T ⎢ 2 ⎢ ⎢ 0 ⎢ =⎢ ⎢ 0 ⎢ ⎣ 0

S¯1

˜T P¯ E

S¯2

0

S¯3

0

0

0

˜ M

⎤

⎥ ⎥ 0 ⎥ ⎥ ⎥. ˜⎥ M ⎥ ⎦ 0

Here, we choose Ws = Wv . As only there is only one control input u = Fyf in this study, we have m = 1. Based on the definition of Wv , we have Wv = {H} when v = 0, and Wv = {K} when v = 1. Define new ¯v = H ¯ (v = 0), Y2 = W ¯v = K ¯ (v = 1), with H ¯ = H P¯ and K ¯ = K P¯ . Inspired by variables as X = P¯ , Y1 = W the two steps static output-feedback controller design schema proposed in [32], this paper adopts the similar algorithm to obtain the static output-feedback gain. The main steps of the algorithm include: (1) First, we will calculate the state-feedback control gain. Solving the LMI of (47), if there exists a feasible solution set: X and Yi , then we can proceed the following step. (2) As the C˜1 in (13) is defined as

⎡ ⎢ C˜1 = ⎣

⎤ 0 1 0 0

0 0 ⎥ ⎦, 0 1

by computing the nullspace of C˜1 , we obtain the matrix: ⎡ ⎢ 1 U =⎣ 0

⎤T 0 0 0 1

0 ⎥ ⎦ , 0

where U is a matrix whose columns are the basis of the nullspace of C˜1 . Let R = C˜1† + U L, 16

with L = U † X C˜1T (C˜1 X C˜1 )−1 , ˜ = U XU U + where C˜1† and U † are the Moose-Penrose pseudoinverse of C˜1 and U , respectively. Then define X RXR R and Y˜i = Y¯i RT , with XU , XR and Y¯i being matrices with proper dimensions. By replacing the X and ˜ and Y˜i , respectively, we can get (44a). Yi in (47) with X ¯ = H P¯ , inequality (23b) is equivalent to Next, with the definition X = P¯ and Y1 = H ⎤

⎡ 2 ⎢−u1max I

⎣

∗

Y1 −Xρ−1

⎥ ⎦ ≤ 0.

(48)

˜ and Y˜1 , respectively, we can get (44b). Based on (44a) and (44b), By replacing the X and Y1 in (48) with X ˜ can be calculated as the matrices XR and Y¯i can be solved, then the static output-feedback control gain K ˜ = YR X −1 . This completes the proof. K R Note that the lateral front-tire force Fyf is directly adopted as the control input in the controller design. However, the front-wheel steering angle δf is the actual control input of a vehicle. Since the real lateral front-tire force can be directly measured [42] or can be estimated [43], the front-wheel steering angle δf can be obtained with the following controller:

δf = p 1 e + p 2

0

t

e dt,

(49)

where p1 and p2 are positive scalars, e = Fyf − Fyf , with Fyf being the actual lateral force of the front-tire.

4. Simulation Results In this section, two simulation cases including the single-lane change maneuver and double-lane change maneuver are presented to verify the effectiveness of the proposed control method. The simulations are implemented based on CarSim with a high-fidelity, full-vehicle model. The parameters used in the simulations are listed in Table 1. The parameter uncertainty on the local tire cornering stiffness is assumed to be 40% of the nominal value. The desired vehicle lateral velocity is supposed to be zero for maintaining vehicle stability. The desired vehicle yaw rate rd can be generated as [41] rd =

vx δsw , l(1 + kus vx2 )

(50)

where l = lf + ls is the distance between the front and rear axles, δsw is the driver’s request of the front-wheel steering angle, and kus = 0.002 is the stability factor. In order to show the effectiveness of the proposed method, the performance of a LQR controller [40] is also given and compared with that of the proposed controller. Here, we denote the proposed controller as Controller 1 and the LQR controller as Controller 2. 17

Table 1: The vehicle parameters used in the simulation Symbol

Definition

Value

m

Vehicle mass

1500 kg

Iz ˜r C

Inertia moment of the vehicle about yaw axis

2500 kg · m2

Nominal local cornering stiffness of rear tire

45000 N/rad

ls

Half of the track width

0.8 m

lf

Distance of CG from front axle

1.3 m

lr

Distance of CG from rear axle

1.4 m

τ¯

Maximum delay time

0.02 s

μ

Road adherence factor

0.3

u1max

Maximum lateral force of the front-tire

1125 N

1.5 1

δsw (deg)

0.5 0 −0.5 −1 −1.5

0

1

2

3 Time (sec)

4

5

6

Figure 2: The driver’s front-wheel steering angle request in the single-lane change maneuver.

4.1. Single Lane-Change Maneuver In this simulation case, the vehicle is made to complete a single-lane change maneuver on the low-adherence road (μ = 0.3) at speed of 30 m/s. The driver’s steering request on the front-wheel is plotted in Figure 2. The desired front-wheel steering angle δf is shown in Figure 3. In this figure, when network condition is ideal with no time delay, there are no fluctuations for the two controllers. When the time delay occurs, i. e. τ¯ = 0.02 s, there are no significant changes for the δf generated by Controller 1. However for Controller 2, the control input begins to oscillate at 0.8 s, and δf can not converge even when δsw returns to zero. Note that the actual front-tire lateral force is strictly bounded due to the physical constraints, i.e. Fymax = μFz . Then from Figure 4 one can see that the performance of the Controller 1 is satisfied even in conditions of time delay and actuator saturation, i.e. Fyf for the vehicle with Controller 1 is reasonable and within the friction limit. In contrast, Fyf for the vehicle with Controller 2 fluctuates and has been forced to be within 15 0.65

δf (deg)

10

0.6 0.55 1.6

5

1.7

1.8

1.9

0 Controller 1(τ=0.00s) Controller 2(τ=0.00s)

−5

Controller 1(τ=0.02s) Controller 2(τ=0.02s)

−10

0

1

2

3 Time (sec)

4

5

Figure 3: The front-wheel steering angle in the single-lane change maneuver.

18

6

1500 1000

0

yf

F (N)

500

−500

−1000

Controller 1(τ=0.00s) Controller 2(τ=0.00s) Controller 1(τ=0.02s)

−1500

Controller 2(τ=0.02s)

−2000

0

1

2

3 Time (sec)

4

5

6

Figure 4: The actual front-tire lateral force in the single-lane change maneuver.

30.8 Controller 1(τ=0.00s)

30.7

Controller 2(τ=0.00s)

vx (m/s)

30.6

Controller 1(τ=0.02s) Controller 2(τ=0.02s)

30.5 30.4 30.3 30.2 30.1 30

0

1

2

3 Time (sec)

4

5

6

Figure 5: The longitudinal velocity in the single-lane change maneuver.

0.6 Controller 1(τ=0.00s) Controller 2(τ=0.00s)

0.4

Controller 1(τ=0.02s)

vy (m/s)

Controller 2(τ=0.02s)

0.2 0 −0.2 −0.4

0

1

2

3 Time (sec)

4

5

6

Figure 6: The lateral velocities in the single-lane change maneuver.

10

5.5 5 4.5

5 r (deg/s)

4 1.6

1.7

1.8

1.9

2

0 Reference Controller 1(τ=0.00s)

−5

Controller 2(τ=0.00s) Controller 1(τ=0.02s) Controller 2(τ=0.02s)

−10

0

1

2

3 Time (sec)

4

5

Figure 7: The vehicle yaw rate in the single-lane change maneuver.

19

6

3 2

y

2

a (m/s )

1 0 −1 Controller 1(τ=0.00s)

−2

Controller 2(τ=0.00s) Controller 1(τ=0.02s)

−3

Controller 2(τ=0.02s)

−4

0

1

2

3 Time (sec)

4

5

6

Figure 8: The lateral acceleration in the single-lane change maneuver.

the friction limit, resulting in undesirable consequences, i.e. large fluctuations on lateral velocity, yaw rate, and lateral acceleration, as shown in Figure 6, Figure 7, and Figure 8, respectively. The longitudinal velocity of the vehicle is given in Figure 5. It should be noted that the control objective in this paper is to enhance the stability and handling performances, such that the lateral velocity and yaw rate should be well controlled. From Figure 6, it can be identified that the lateral velocity by Controller 1 is similar to the driver’s request, no matter when the time delay occurs or not. However, Controller 2 can only work well under the ideal network condition, and the time delay and actuator saturation will seriously deteriorate the control performance. From Figure 7, it is easy to see that the yaw rate obtained by Controller 1 is close to the reference, despite of the time delay and actuator saturation. As for Controller 2, when there is no time delay, the performance of yaw rate response is satisfied. However, the yaw rate for Controller 2 unavoidably oscillates when the time delay becomes 0.02 s. In addition, the aforementioned simulations are tested under severe maneuvers. From Figure 8 it can be observed that the lateral acceleration ay = 2.5 m/s2 , being close to μg, which means that the friction limit has been reached. Under such circumstance, lateral force of the front-tire has been fully utilized. 4.2. Double-Lane Change Maneuver In this simulation case, the vehicle is supposed to complete a double-lane change maneuver with the same adherence road and the longitudinal speed. The driver’s request on the front-wheel is plotted in Figure 9. The front-wheel steering angle is shown in Figure 10. In this figure, when there is no time delay, the performances of the two controllers are satisfied. When the time delay τ¯ = 0.02 s, the performance of the Controller 1 is still good, which shows strong robustness of Controller 1. However, the front-wheel steering angle generated by Controller 2 fluctuates, which means that the performance of the Controller 2 is poor on the worsen network condition. The actual front-tire lateral force Fyf is shown in Figure 11, from which one can see when there is no time delay, Fyf obtained by Controller 1 and 2 are all smooth and maintained in reasonable regions. However, when the time delay occurs, Fyf obtained by Controller 2 fluctuates sharply. As shown in Figure 11, due to the physical limitation of the front-tire lateral force on low adherence road, Fyf is strictly bounded. For the vehicle with Controller 2, the fluctuations result in oscillations on the lateral velocity, lateral acceleration, and 20

1

δsw (deg)

0.5

0

−0.5

−1 0

1

2

3

4

5 Time (sec)

6

7

8

9

10

Figure 9: The driver’s request of front-wheel steering angle in the double-lane change maneuver.

15

0.6

10

0.55 0.5

δf (deg)

5

1.6

1.8

2

0 −5 Controller 1(τ=0.00s)

−0.55

Controller 2(τ=0.00s)

−10 −15

−0.6

Controller 1(τ=0.02s)

3.8

0

4

1

Controller 2(τ=0.02s)

4.2

2

3

4

5 Time (sec)

6

7

8

9

10

Figure 10: The front-wheel steering angle in the double-lane change maneuver.

1500 1000

0

yf

F (N)

500

−500 Controller 1(τ=0.00s)

−1000

Controller 2(τ=0.00s) Controller 1(τ=0.02s)

−1500

Controller 2(τ=0.02s)

−2000

0

1

2

3

4

5 Time (sec)

6

7

8

9

10

Figure 11: The front-tire lateral force in the double-lane change maneuver.

31.5 Controller 1(τ=0.00s) Controller 2(τ=0.00s) Controller 1(τ=0.02s)

31 vx (m/s)

Controller 2(τ=0.02s)

30.5

30

0

1

2

3

4

5 Time (sec)

6

7

8

9

Figure 12: The longitudinal velocity in the double-lane change maneuver.

21

10

0.3 0.2

0

y

v (m/s)

0.1

−0.1 −0.2 Controller 1(τ=0.00s)

−0.3

Controller 2(τ=0.00s)

−0.4

Controller 1(τ=0.02s) Controller 2(τ=0.02s)

−0.5

0

1

2

3

4

5 Time (sec)

6

7

8

9

10

9

10

Figure 13: The lateral velocity in the double-lane change maneuver.

10 5

8

4.5

6

4 1.6

r (deg/s)

4

1.8

2

2 0 −2

Reference Controller 1(τ=0.00s)

−4

Controller 2(τ=0.00s)

−6 −8

Controller 1(τ=0.02s) Controller 2(τ=0.02s)

0

1

2

3

4

5 Time (sec)

6

7

8

Figure 14: The vehicle yaw rate in the double-lane change maneuver.

3 2

ay (m/s2)

1 0 −1 −2

Controller 1(τ=0.00s) Controller 2(τ=0.00s)

−3

Controller 1(τ=0.02s) Controller 2(τ=0.02s)

−4

0

1

2

3

4

5 Time (sec)

6

7

8

9

Figure 15: The lateral acceleration in the double-lane change maneuver.

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10

yaw rate. On the other hand, as for Controller 1, the front-tire lateral forces have not exceeded the friction limit. In Figure 13, lateral velocity of the vehicle with Controller 1 is small and shows similar trend with the control input, despite of the time delay and actuator saturation. However for the vehicle with Controller 2, fluctuations of the lateral velocity inevitably appear as the time delay becomes 0.02 s. From Figure 14, one can see that when there is no time delay, the yaw rates controlled by both Controller 1 and 2 are close to their references. However, when the time delay is 0.02 s, the performance of Controller 1 is still satisfied, while that of Controller 2 is deteriorated. Note that the maximum lateral accelerations of the vehicles are close to μg, which means that both of the vehicles controlled by Controller 1 and 2 have reached the friction limit. What is more, compared with the LQR controller, the proposed controller based on the output-feedback schema does not need information of the vehicle lateral velocity, which is more practical in real applications. Consequently, it can be concluded that the comparative simulation results clearly verify the proposed controller’s good performance, and the robustness to network time-delay issues, actuator saturation, parameter uncertainties, and external disturbances.

5. Conclusion To regulate the vehicle lateral dynamics in the presence of network-induced delay and actuator saturation, a robust H∞ output-feedback control strategy is designed in this paper. Both of the vehicle handling and vehicle stability are chosen as the control objectives. Considering the costs of currently available sensors are topically high for vehicle lateral velocity measurement, a robust H∞ static output-feedback controller is designed. The robustness of the proposed controller is enhanced by considering the tire cornering stiffness uncertainty and external disturbances in the controller design. The comparative simulation results with a LQR controller have verified the effectiveness of the proposed control schema. This work assumes that the maximum lateral tire force is known and constant. However, the maximum lateral force of the front-tire is actually dependent on the road adherence, the varying maximum lateral tire force makes the controller design more complicated, which will be left for our future works.

Acknowledgement This work was partly supported by NSFC (51375086, 51505081), Jiangsu Province Science Foundation for Youths, China (Grant no. BK20140634), the Opening Project of Guangxi Colleges and Universities Key Laboratory of UAV Remote Sensing(WRJ2015KF02), Foundation of Guangxi key laboratory of Manufacturing System and Advanced Manufacturing Technology(15-040-30-004Z), and the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1542).

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