Observer-based control of vehicle platoons with random network access

Robotics and Autonomous Systems 115 (2019) 28–39

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Robotics and Autonomous Systems journal homepage: www.elsevier.com/locate/robot

Observer-based control of vehicle platoons with random network access✩ ∗

Shixi Wen a , , Ge Guo b,c a

School of Information and Engineering, Dalian University, Dalian, 116622, China State Key Laboratory of Synthetical Automation for Industrial Process, Northeastern University, Shenyang, 110004, China c School of Control Engineering, Northeastern University at Qinhuangdao, Qinhuangdao, 066004, China b

highlights • A novel observer-based platoon control model is established. • Sufficient conditions are obtained for the existence of observer-based controller. • A algorithm is proposed to determine gains of controller and observer.

article

info

Article history: Available online 16 February 2019 Keywords: Vehicle platoons control Vehicular ad hoc networks Markov MAC protocol Observer-based control String stability Zero steady-state velocity errors

a b s t r a c t This paper investigates cooperative control of vehicle platoons, focusing on the effect of medium access control (MAC) protocol and unreliable measurement on acceleration information. A Markov chain is used to describe the randomness in vehicular network access under a MAC protocol; a reduced-order observer is proposed to estimate the relative acceleration of neighboring vehicles. Based on stochastic system techniques, a series of sufficient conditions for the existence of the observer-based controller are given in the form of backward recursive Riccati Difference Equations (RDE). A controller–estimator design algorithm is derived to determine the controller and observer gains with which the disturbance of the preceding vehicle acceleration can be eliminated and string stability and zero steady-state velocity error performance can be guaranteed. Both numerical simulations and experiments with laboratory-scale Arduino cars are given to verify the effectiveness and practicability of the proposed algorithm. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The platoon of autonomous vehicles has been attracting extensive research interests from government, academia and industry since the serious traffic issues (e.g., congestion, accident, air pollution) on the road traffic [1–3]. As the one of the most important technology of the intelligent transportation systems (ITS), vehicle platoons can result in greatly increased traffic capacity, largely reduced adverse environmental effect, and a safer and more comfortable driving experience [4–6]. As a result, it has been regard as one of the most efficient way to utilize the road infrastructure of many highways and urban areas [7]. The main objective of the platoon control is to automatically organize the vehicles in a group at the same speed while maintaining a prespecified distance between any consecutive followers. Letting ✩ This work was supported by the National Natural Science Foundation of China under Grants 61273107, 61573077, 61803062, U1808205 and the Natural Science Foundation of Liaoning province of China under Grant 20180550794. ∗ Corresponding author. E-mail addresses: [email protected] (S. Wen), [email protected] (G. Guo). https://doi.org/10.1016/j.robot.2019.02.006 0921-8890/© 2019 Elsevier B.V. All rights reserved.

vehicles to run close to one another can not only improve road capacity but also save fuel consumption due to reduction in the aerodynamic drag [8,9]. The heavy-duty vehicle platoons experiments shown that the improved aerodynamics of the group leads to a reduction of fuel consumption of up to 10%. [10,11]. In general, platoon control requires support from wireless communications provided by VANETs and sensing devices installed in vehicles. By vehicular ad hoc network (VANET), vehicles in a platoon control system are dynamically coupled, hence, the spacing and velocity errors of a vehicle may affect those behind or even amplify as they propagate along the string of vehicles. This phenomenon is known as the slinky effect, which may lead to instable vehicular string [12,13], and, in turn, an uncomfortable ride experience and even rear-ending accidents [13]. Therefore string stability is the most important aspect of a platoon control system, which has been studied by many researchers, see [14] for an overview of platoon control and string stability. Generally, the existing approaches to platoon control can be divided into three types: (1) Lyapunov stability approach [15,16]; (2) Spatially invariant linear systems approach [17,18]; (3) Frequency-domain approach [19,20].

S. Wen and G. Guo / Robotics and Autonomous Systems 115 (2019) 28–39

Here, we are interested in how the MAC protocol would affect vehicle platooning. The MAC protocol in an IEEE 802.11pbased VANET usually uses a carrier-sense multiple access with collision avoidance (CSMA/CA) protocol to resolve communications conflicts among vehicles. A CSMA/CA operates on a ‘‘sense before transmitted’’ or ‘‘listen before talk’’ basis. Each vehicle can know the channel status (idle or busy) and it is permitted to transmit/receive data only when the wireless channel is idle. If the wireless network is busy, then the vehicle will wait for a random interval (random back off time) and then check again to see if it could send/receive data. Clearly, the vehicles transmit and receive information to and from neighboring vehicles in a certain random manner. According to [21–23], the medium access process of the communication nodes with CSMA/CA can be described by a Markov possess. This aspect has not been mentioned in existing results of platoon control. What is also interested in is the effect of unreliable measurement on platoon control. Most of the existing results generally assume that the spacing, velocity and acceleration can be precisely obtained. For example, the designed feedback controller in [5,6, 19,20,24] directly use the state information (i.e., position/spacing, velocity and acceleration) to compute the control input. However this assumption is not reasonable in practical platoon control systems, since some sensors might produce unreliable measurements. This is true at least for accelerometers currently available in the market. As reported in [25,26], accelerometers cannot maintain the orientation in long term deployment and calibrating an accelerometer is usually quite difficult. In addition, accelerometers inevitably have biased or drifted outputs due to variations in temperature, mechanical stresses/pressure and humidity changes [27]. One effective way to cope with the problem of unreliable or lacking measurements is to estimate them based on reliably available information [28,29]. In [28], the relative velocity and relative acceleration is derived via numerical differentiation of spacing measurements. However, the spacing measured by the distance sensor is often contaminated by measurement noise, which would degrade the control performance. In [29], an observer is introduced to estimate the acceleration of the preceding vehicle. The analysis and design of the observer-based platoon controller is performed in the frequency domain. This paper investigates vehicle platoons control focusing on the effect of random MAC protocol and unreliable measurement on acceleration information. It is assumed that the acceleration measurement of the vehicles have uncertainties, therefore a reducedorder observer is proposed to estimate the relative acceleration based on the position and velocity information of itself and the preceding vehicle. The effect of the stochastic MAC protocol is modeled as a Markov chain for each vehicle. The contributions of this paper are given as follows, (1) A novel observer-based platoon control framework is proposed, where a reduced-order observer is proposed to estimate the relative acceleration between neighboring vehicles. (2) The sufficient conditions for the existence of the observer-based controller are obtained for the platoon by the stability analysis, which are represented by a set of backward recursive RDEs. (3) An unified observed-based platoon control framework is established to determine the controller and observer gain simultaneously. The obtained observer-based controller is shown to be able to guarantee string stability and steady-state velocity errors and reject the disturbance yielded by the acceleration of the preceding vehicle. The remainder of this paper is organized as follows. Section 2 gives the observer-based control model for the vehicle platoons. It also contains the detailed analysis on the effect of Markov MAC protocol, the design of the reduced-order acceleration observer. Section 3 is the stability analysis and observer-based controller design synthesis for the platoon. Section 4 gives the useful algorithm

29

to implement the observer-based controller under the Markov chain driven MAC protocol, which is derived by the analysis of string stability and zero steady-state velocity error. Numerical simulations and experiments with laboratory-scale Arduino cars are given in Section 5 to verify the effectiveness of the proposed algorithm. The concluding remarks and future research topics are given in Section 6. Notation: Throughout the paper, Rn denotes the n-dimensional Euclidean space. For symmetric matrices P and Q, P
⎧ z˙ (t) = vr (t) ⎪ ⎨r Tr (t) 1 (ϖT ,r − CA,r vr2 (t) − mr g f ) v˙ r (t) = mr Rr ⎪ ⎩ ˙ ςr Tr (t) + Tr (t) = Tr , des (t)

(1)

where mr is the mass of the vehicle, CA,r is the lumped aerodynamic drag coefficient, g is the acceleration due to gravity, f is the coefficient of rolling resistance, Tr (t) denotes the actual driving/breaking torque, Tr , des (t) is the desired driving/breaking torque, ςr is the inertial delay of vehicle dynamics, Rr denotes the tire radius, and ϖT ,r is the mechanical efficiency of driveline. By the widely used exact feedback linearization technique [3, 4,28], the nonlinear vehicle dynamic (1) can be converted into a linear one for controller design. The output of position with relative

30

S. Wen and G. Guo / Robotics and Autonomous Systems 115 (2019) 28–39

Fig. 1. Vehicle platoons control system.

degree tree is used to construct the feedback linearization law, which is given as following: Tr , des (t) =

1

ϖT ,r

(CA,r vr (t)(2ςr v˙ r + vr ) + mr g f + mr ur )Rr

(2)

where ur is the new input signal after linearization. Then, a linear model for vehicle dynamics is obtained as z˙r (t) = vr (t) v˙ r (t) = ar (t) a˙ r (t) = −ar (t)/ςr + ur (t)/ςr

{

(3) Fig. 2. Observer-based control block diagram for vehicle platoons.

Sampling system (3) with period h > 0 and a zero-order hold, gives the following discrete-time representation of vehicle r, zr (k + 1) = zr (k) + hvr (k) vr (k + 1) = vr (k) + har (k) ar (k + 1) = b2 ar (k) + b1 ur (k)

{

which is written in the compact form as sr (k + 1) = A1 sr (k) + B1 ur (k) where b1 = 1 − e zr (k) vr (k) , ar (k)

[ sr (k) =

−h/ς

, b2 = e

]

1 0 0

[ A1 =

(4) −h/ς

h 1 0

, h is the sampling period and 0 h , b2

]

0 0 . b1

[ ] B1 =

Define the spacing error, velocity error and acceleration error, respectively, as

δr (k) = zr −1 (k) − zr (k) − L − hv vr (k) − d0 ∆vr (k) = vr −1 (k) − vr (k) ∆ar (k) = ar −1 (k) − ar (k)

(5)

where hv is the time gap, d0 is a given minimum distance, and L is the length of vehicle. Note that here, as in [30], a hybrid spacing policy is involved. Remark 1. In general, there are two types of spacing policies, constant spacing policy and variable spacing policy. The constant spacing policy means that the desired inter-vehicle distance is constant, while the desired inter-vehicle distance varies with vehicle velocity in the variable spacing policy. For safety consideration, variable spacing policy is more rational than constant spacing policy [31]. 2.2. Modeling back-off effect in vehicular MAC protocol In the IEEE 802.11p-based VANET, a CSMA/CA protocol is used to resolve the communication conflicts. When two or more vehicles request to have access to the channel, each vehicle will wait for a random back-off time in the range (0, w−1), where w is called the contention window. It is well known that the network access status of a vehicle features a Markov chain with conditional collision probability [21–23]. A binary function ρr (k) ∈ {0, 1} is used to denote the channel access status of vehicle r at time k. When ρr (k) = 1, vehicle r is allowed to use the wireless communication channel, then the position and velocity of vehicle r − 1 is transmitted to vehicle r for computing the spacing error and velocity error. Otherwise, ρr (k) = 0, vehicle r is not allowed to use the wireless communication channel, by the zero-strategy, the spacing error

and velocity error is set to be zero. For convenience, matrix Hρ (k) = diag {ρ1 (k), . . . , ρN (k)} is used to describe the channel access status of the N following vehicles. It is clear that if the ith following vehicle is allowed to access the wireless network (ρi (k) = 1), then communication matrix Hρ (k) = Hi , where matrix Hi is a diagonal matrix with the ith diagonal element being ‘1’ and the other elements ‘0’. For example, Hρ (k) = H1 = diag {1, 0, . . . , 0} if ρ1 (k) = 1; Hρ (k) = H2 = diag {0, 1, . . . , 0} if ρ2 (k) = 1. The communication matrix Hρ (k) is a Markov chain taking matrix values in a finite set H = {H1 , . . . , HN } with transmission probability matrix II = [πij ], where πij = prob{Hρ (k + 1) = Hj |Hρ (k) = Hi }, i, j = 1, . . . , N. For simplifying the forthcoming discussion, τ (k) is defined as the indicator of the Markov process, e.g., τ (k) = i if Hρ (k) = Hi , i = 1, . . . , N. 2.3. Relative acceleration estimation The observer-based control block diagram is shown in Fig. 2, in which the controller is based on the spacing error, velocity error and estimated relative acceleration. Unlike the easily available position and velocity information, the acceleration is not precisely and reliably measureable. Here, a reduced-order observer is proposed to estimate the relative acceleration on the basis of the available information of the spacing and velocity. To this end, the dynamics of the spacing error, velocity error and acceleration error in (4) are represented as below

⎧ ⎨∆ [ ar (k + 1) =] b2 ∆ar (k) [ − b1 u]r (k) + b2 ϑr (k) δr (k + 1) δr (k) (6) ⎩ ∆v (k + 1) = A11 ∆v (k) + A12 ∆ar (k) + D′ ϖr (k) r r [ ] [ ] [ ] 1 h hv −hhv ′ where A11 = , A12 = and D = , ϖr (k) = 0

1

h

0

ar −1 (k) and ϑr (k) = ur −1 (k) are the acceleration and control input of the preceding vehicle, respectively, which are treated as disturbances here. Then, the following reduced-order observer is introduced to estimate the relative acceleration

⎧ ˆ ∆ar (k + 1) = b2 ∆aˆ r (k) − b1 ur (k) ([ ] [ ]) ⎪ ⎪ ⎪ δr (k) ⎨ δˆr (k) ′ ′ + ρr (k)C Fτ (k) (k) − ∆vˆ r (k) [ ] [ ] ∆vr (k) ⎪ ⎪ ˆr (k + 1) ˆr (k) ⎪ δ δ ⎩ = A11 + A12 ∆aˆ r (k) ∆vˆ r (k + 1) ∆vˆ r (k)

(7)

S. Wen and G. Guo / Robotics and Autonomous Systems 115 (2019) 28–39

where ∆aˆ r (k) is the estimated relative acceleration, δˆ r (k + 1) and ∆vˆ r (k + 1) are the estimated spacing error and velocity error computed using the estimated acceleration error ∆aˆ r (k), C ′ = [0 0 1], Fτ′ (k) (k) ∈ R3×2 is the observer gain to be designed. The observer-based platoon controller is then given in the following form, whose design will be discussed later ur (k) = kpi (k)δr (k) + kvi (k)∆vr (k) + kvi (k)∆aˆ r (k)

(8)

Before proceeding to the main results, the platoon control system model will be reformulated in the next subsection, which is useful for simplifying the forthcoming discussions.

Fτ′ (k) (k)

31

⎡

⎤ ..

Fτ (k) (k) = ⎣

⎢

⎥ ⎦.

.

Fτ′ (k) (k)

Define e(k) = x(k) − xˆ (k) as the estimation error. Then, the dynamics of the estimation error can be obtained from (9) and (10) as follows e(k + 1) = (A − Fτ (k) (k)Hρ (k)C )e(k) + Dϖ (k)

(11)

Introducing an augmented vector ξ (k) = [x (k), e (k)] , the following augmented platoon control system is obtained T

T

ξ (k + 1) = A τ (k) (k)ξ (k) + D(k)ϖ (k) 2.4. Model transformation and the objective

[

T

(12)

here, x(k) = x(k) − C e(k) and

]

[

[

]

yˆ r (k) , yr (k) = ∆aˆ r (k)

yr (k) , xˆ r (k) = ∆ar (k)

Denote by xr (k) = [ ] [ ] [ ] y (k) δr (k) δˆ (k) , respectively as , yˆ r (k) = r and xr (k) = r ∆aˆ r (k) ∆vr (k) ∆vˆ r (k)

the state, the estimated state, measurement output, the estimated output and feedback state. Then the dynamics of platoon control system in terms of tracking errors defined in (3) can be rewritten as x(k + 1) = Ax(k) + Bu(k) + Dϖ (k) y(k) = Cx(k) + Dϖ (k)

{

(9)

with the following observer-based controller xˆ (k + 1) = Axˆ (k) + Bu(k) + Fτ (k) (k)Hρ (k)(y(k) − yˆ (k)) yˆ (k) = C xˆ (k) u(k) = Kτ (k) (k)x(k)

A τ (k) (k) =

A + BKτ (k) (k) 0

⎡

] −BKτ (k) (k)C , A − Fτ (k) (k)Hρ (k)C

⎤

C′

..

⎢ C =⎣

. C′

⎥ ⎦,

[

]

D D(k) = . Fτ (k) (k)Hρ (k)

The objective of this paper is to design the observer-based platoon controller such that the following criteria are met: (1) Individual vehicle stability: the entire platoon of vehicles (i.e., system (10)) is asymptotically stable. (2) Disturbance rejection performance: under zero initial condition, the tracking error x(k) of the platoon control system satisfies the following performance requirement

{

(10)

where ϖ (k) = [ϖ1 (k) · · · ϖN (k)]T is the preceding vehicle acceleration disturbance, u(k) = [uT1 (k) · · · uTN (k)]T is the control input, and

⎡

x1 (k)

⎤

⎡

⎤

x1 (k)

⎡

y1 (k)

⎤

⎢ .. ⎥ ⎢ . ⎥ ⎢ . ⎥ . ⎦ , x(k) = ⎣ .. ⎦ , y(k) = ⎣ .. ⎦ ,

x(k) = ⎣

xN (k)

⎡

A′

..

⎢ A=⎣

xN (k)

yN (k)

[ A11 ⎥ ′ , A = ⎦ A21

[ A12 , A21 = 0 A22

⎤ . ′

]

A

⎡ −B1 ⎢B1 B=⎢ ⎣

A22 = b2 ,

⎤ −B1 .. .

..

⎥ ⎥, ⎦

.

B1

⎡

C1

⎢ C =⎣

−B1

⎥ ⎦,

.

]

∥x(k)∥2 − γ 2 ∥ϖ (k)∥2 } − γ 2 ξ T (0)W ξ (0) ≤ 0,

(13)

k=0

for ∀(ξ (0), ϖ (0)) ̸ = 0, where γ > 0 is a given scalar, and W is a given positive definite matrix. (3) Steady-state performance: the spacing error δr (k) approaches to zero for all vehicles. (4) String stability: the transient errors are not amplifying with vehicle index under any maneuver of the leading vehicle, namely, ∥G(z)∥ ≤ 1, where G(z) = vr (z)/vr −1 (z), vr (z) is the Z-transform of the velocity vr (k). Remark 2. The advantages of using performance index (13) is that: (1) it describes the weighted average performance of the closedloop platoon control system over the possible modes when the overall control performance becomes a concern; (2) it facilitates the subsequent mathematical analysis and enhances the feasibility of the controller design problem; (3) it is useful for considering the worst-case performance for the platoon. 3. Observer-based controller design for vehicle platoons

⎤ ..

0 ,

N −1 ∑

J = E{

[

1 C1 = 0

]

0 1

0 , 0

3.1. Performance requirement guaranteed

C1

⎡

⎤

D1

..

D=⎣

⎢

⎥ ⎦,

. D1

Kτ′ (k) (k)

⎡ [ ′] D1 =

D 0

,

Kτ (k) (k) = ⎣

Kτ′ (k) (k) = [kpi (k)

⎢

kvi (k)

kai (k)],

⎤ ..

.

⎥ ⎦,

Kτ′ (k) (k)

In order to reject the disturbance caused by the preceding vehicle’s acceleration, the performance requirement defined in (13) is analyzed for platoon control system in this subsection. It is assumed that the controller {Ki (k)}0≤k 0 and the positive definite matrix W are to be given. Define the following forward difference equation

∆Vτ (k) (k) = E{ξ T (k + 1)Pτ (k+1) (k + 1)ξ (k + 1)

− ξ T (k)Pi (k)ξ (k)|i = τ (k)},

(14)

32

S. Wen and G. Guo / Robotics and Autonomous Systems 115 (2019) 28–39

where Pi (k)(i = 1, . . . , n, 0 ≤ k < N) is a family of non-negative definite matrices. Along the state trajectory (10), ∆Vτ (k) (k) can be rewritten as

then it is clear that J = E{−

∆Vτ (k) (k) = E{[Ai (k)ξ (k) + D(k)ϖ (k)]T Pτ (k+1) (k + 1)[Ai (k)ξ (k)

(ϖ (k) − ϖ ∗ (k))T ∆τ (k) (k)(ϖ (k) − ϖ ∗ (k))

k=0

− ξ (0)(Pi (0) − γ 2 W )ξ (0)} ≤ 0 T

+D(k)ϖ (k)] − ξ T (k)Pi (k)ξ (k)|i = τ (k)}

(23)

which means that the performance requirement is guaranteed for platoon control system.

T

= E{ξ (k)T [A i (k)Pˆ i (k + 1)Ai (k) − Pi (k)]ξ (k) T +2ξ (k)T A i (k)Pˆ i (k + 1)D(k)ϖ (k)

3.2. Observer and controller gain design

T

+ϖ (k)D (k)Pˆ i (k + 1)D(k)ϖ (k)|i = τ (k)} ∑n where Pˆ i (k + 1) = j=1 πij Pj (k + 1). T

N −1 ∑

(15)

Adding the following zero term to both sides of (15)

∥x(k)∥2 − γ 2 ∥ϖ (k)∥2 − (∥x(k)∥2 − γ 2 ∥ϖ (k)∥2 )

(16)

and taking the mathematical expectation, it is clear that

∆Vτ (k) (k) = E{γ 2 ∥ϖ (k)∥2 + ξ (k)T [RT (k)R(k)

From the above discussion, if there exist solutions Pi (k) to (19) such that ∆i (k) and Pi (0) satisfy (22), then the platoon control system (12) can achieve the H∞ performance requirement. In this case, the worst-case disturbance is ϖ ∗ (k) that given in (18). In what follows, by employing the worst-case disturbance ϖ ∗ (k), a design scheme for the controller gain Ki (k) and observer gain Fi (k) in each iteration backward will be derived. For this purpose, the augmented system (10) is rewritten as follows: B(k)u(k) +˜ ℓF τ (k) (k)ζτ (k) (k) ξ (k + 1) = (˜ A τ (k) (k) + D(k)Γ τ (k) (k))ξ (k) +˜

T +A i (k)Pˆ i (k + 1)Ai (k) − Pi (k)]ξ (k) − ∥x(k)∥2 T

(24)

+2ξ (k)T A i (k)Pˆ i (k + 1)D(k)ϖ (k) − ϖ T (k)(γ 2 I T

−D (k)Pˆ i (k + 1)D(k))ϖ (k)|i = τ (k)}

(17)

T

where R(k) = [I −C ] . Applying the completing squares method results in T ∆Vτ (k) (k) = E{ξ (k)T [RT (k)R(k) + A i (k)Pˆ i (k + 1)Ai (k) − Pi (k)]ξ (k)

∗

(18)

T

T

1 where ∆i (k) = γ 2 I − D (k)Pˆ i (k + 1)D(k) and ϖ ∗ (k) = ∆− τ (k) (k)D (k)

Pˆ i (k + 1)A τ (k) (k)ξ (k). Set the following backward recursive RDE for Pi (k) as T

T

Jϖ∗ =

−E{∥x(k)∥ − γ ∥ϖ (k)∥ } 2

(20)

Taking the sum on both sides of (14) and (20) from k = 0 to k = N − 1 leads to

E{ξ T (N)Pτ (N) (N)ξ (N) − ξ T (0)P τ (0) (0)ξ (0)} N −1 ∑

T

−

(∥x(k)∥ − γ ∥ϖ (k)∥ )} 2

N −1 ∑

( ∆˜ Vτ (k) (k) + E

k=0

 +ε2 F

{N −1 ∑

∥x(k)∥2 + ε1 ∥u(k)∥2

k=0

}) 2 T  + ξ (0)Qτ (0) (0)ξ (0) τ (k) (k)ζτ (k) (k)

(27)

here, it is clear that Qτ (N) (N) = 0. Along the state trajectory (24) of the platoon control system, ∆˜ Vτ (k) (k) can be further expressed by

∗

× (˜ A i (k) + D(k)Γi (k))

k=0

2

(26)

∆˜ Vτ (k) (k) = E{ξ T (k)(˜ A i (k) + D(k)Γi (k))T Qˆ i (k + 1)

(ϖ (k) − ϖ (k)) ∆τ (k) (k)(ϖ (k) − ϖ (k)) ∗

(25)

then, the cost function can be represented as (19)

∆Vτ (k) (k) = E{(ϖ (k) − ϖ ∗ (k))T ∆i (k)(ϖ (k) − ϖ ∗ (k))|i = τ (k)} 2

2

− ξ T (k)Qi (k)ξ (k)|i = τ (k)}

which leads ∆Vτ (k) (k) equivalent to

2

}  2 ∥x(k)∥ + ε1 ∥u(k)∥ + ε2 F τ (k) (k)ζτ (k) (k) 2

where ε1 and ε2 are known constants introduced for more flexibility in the controller parameter design. In order to solve the controller and observer gain in (22), define the following forward difference equation

T

1 ˆ × D(k)∆− i (k)D (k)Pi (k + 1)Ai (k)

N −1 ∑

{N −1 ∑

∆˜ Vτ (k) (k) = E{ξ T (k + 1)Qτ (k+1) (k + 1)ξ (k + 1)

Pi (k) = Ai (k)Pˆ i (k + 1)Ai (k) + RT (k)R(k) + Ai (k)Pˆ i (k + 1)

= E{−

1 ˆ 0]T Γ τ (k) (k) = ∆− τ (k) (k)D (k)Pi (k + 1)A τ (k) (k), ˜ ℓ = [I − I ]T , ˜ Hτ (k) (k) = [0 Hρ (k)]. For platoon control system (24), the following cost function is defined

k=0

−(ϖ (k) − ϖ ∗ (k))T ∆i (k)(ϖ (k) − ϖ ∗ (k))|i = τ (k)} − E{∥x(k)∥2 − γ 2 ∥ϖ (k)∥2 }

T

diag {A, A}, ˜ B(k) = [BT

Jϖ∗ = E

+(ϖ (k)) ∆i (k)ϖ (k) T

∗

where u(k) = K τ (k) (k)˜ ℓT ξ (k), ζτ (k) (k) = ˜ Hτ (k) (k)ξ (k), ˜ A τ (k) (k) =

2

−ξ T (k)Qi (k)ξ (k) + 2ξ T (k)(˜ Ai (k) + D(k)Γi (k))T Qˆ i (k + 1)u(k) (21)

k=0

Since Pi (N) = 0, if ∆i (k) and Pi (0)in (19) satisfy the following conditions

{ T ∆i (k) = γ 2 I − D (k)Pˆ i (k + 1)D(k) > 0 Pi (0) ≤ γ 2 W

(22)

+uT (k)Qˆ i (k + 1)u(k) +2ξ T (k)(˜ A i (k) + D(k)Γi (k))T Qˆ i (k + 1)˜ ℓFi (k)ζτ (k) (k) +ζ Tτ (k) (k)FiT (k)˜ ℓT Qˆ i (k + 1)˜ ℓF i (k)ζτ (k) (k) +2ξ T (k)uT (k)Qi (k + 1)˜ ℓF i (k)ζτ (k) (k)|i = τ (k)}

(28)

S. Wen and G. Guo / Robotics and Autonomous Systems 115 (2019) 28–39

∑n

where Qˆ i (k + 1) = j=1 πij Qj (k + 1). With the aim to complete the square with respect to u(k) and ζi (k) in the cost function J ϖ ∗ , here, let Qi (k) satisfy the following backward recursive RDE Qi (k) = (˜ A i (k) + D(k)Γ i (k))T Qˆ i (k + 1)(˜ A i (k) + D(k)Γ i (k))

+RT (k)R(k) + ˜ ℓK T (k)˜ BT (k)Qˆ i (k + 1)˜ ℓFi (k)˜ H(k) i

T +Θ (k) + ˜ B(k)Ki (k)˜ ℓT Qˆ i (k + 1)˜ H T (k)Fi (k)˜ ℓT

(29)

where

Θ (k) = Θ1 (k) + Θ2 (k) + Θ3 (k) + Θ4 (k) i

Θ2 (k) = ˜ A (k)Qˆ i (k + 1)˜ ℓ∆−i21 (k)˜ ℓT Qˆ i (k + 1)˜ A i (k) T

i

T

B(k)Ki (k)˜ ℓT Θ3 (k) = Γ (k)D (k)Qˆ i (k + 1)˜ i

+˜ ℓK T (k)˜ BT (k)Qˆ i (k + 1)D(k)Γi (k) i

Lemma 1 ([32]). Let U , V and W be known nonzero matrices with appropriate dimensions. The solution X to arg minX ∥U X W − V ∥F is ◦ ◦ U VW . By Lemma 1, it is clear that the controller and observer gains can respectively be chosen as Ki (k) = −Pi (k)R◦ (k) Fi (k) = −Qi (k)˜ H ◦ (k)

(34)

T

ℓFi (k)˜ H(k) Θ4 (k) = Γ (k)D (k)Qˆ i (k + 1)˜ T

4. String stability and observer-based controller design algorithm

i

+˜ H T (k)FiT (k)˜ ℓT Qˆ i (k + 1)D(k)Γi (k), ∆i1 (k) = ˜ BT (k)Qˆ i (k + 1)˜ B(k) + ε1 I ,

4.1. String stability analysis

∆i2 (k) = ˜ ℓQˆ i (k + 1)˜ ℓ T + ε2 I

In this section, the controller obtained by backward recursive RDE will be constrained to satisfy the string stability and zero steady-state error requirements. Firstly, the result that ensures string stability is given. Then, the issue of zero steady-state velocity error is analyzed by using the obtained results. Denote the velocity error ∆vr (k) = [zr −1 (k) − zr (k) − L − d0 ]/h, then the spacing error can be represented as

then, the following expression for J ϖ ∗ is derived J ϖ ∗ = E ξ T (0)Qτ (0) (0)ξ (0)

{

+E

Then the solution for the optimization problem (33) can be solved according to the following Lemma.

Remark 3. A method of relative acceleration estimation has been proposed in [33] by using Laplace transform method. However, the observer gain needs to be given a priori. Otherwise, the proposed observer may be no longer applicable. Different from [33], a framework of observer-based control has been established in this paper, where the observer can be jointly designed with feedback controller. To the best of author’s knowledge, observer-based control strategy has not been found in the available published results for platoon control.

Θ1 (k) = ˜ AT (k)Qˆ i (k + 1)˜ B(k)∆−i11 (k)˜ BT (k)Qˆ i (k + 1)˜ A i (k).

T

33

{N −1 ∑

}

(u(k) + u∗ (k))T ∆i1 (k)(u(k) + u∗ (k))+

k=0

δr (k) = h∆vr (k) − hv vr (k)

(F τ (k) (k)ζτ (k) (k) + ζ ∗τ (k) (k))T ∆i2 (k)(F τ (k) (k)ζτ (k) (k)

For analysis the string stability, the following relationship is used

} + ζ τ (k) (k))|i = τ (k) ∗

(30)

−1

(35)

−1

BT (k)Qˆ i (k + 1)˜ A i (k)ξ (k) and ζi∗ = ∆ i2 (k) where u∗ (k) = ∆ i1 (k)˜

˜ ℓQˆ i (k + 1)˜ A i (k)ξ (k).

{

ar (k) = [vr (k) − vr (k − 1)]/h ∆ar (k) = [∆vr (k) − ∆vr (k − 1)]/h

Substituting (35) and (36) into the dynamic equation (4) of the platoon, it follows that

[vr (k + 1) − vr (k)]/h =

Consider the following inequality on J ϖ ∗ J ϖ ∗ ≤ E ξ T (0)Qτ (0) (0)ξ (0)

b2 [vr (k) − vr (k − 1)]/h + b1 {ki (k)[h∆vr (k) − hv vr (k)]

{ N −1 ∑

+ kvi (k)∆vr (k) + kai (k)[∆vr (k) − ∆vr (k − 1)]/h}

{

+E

}

p

  ∥Ki (k)R(k) + Pi (k)∥2F ∆i1 (k)F ∥ξ (k)∥2 + } (31)

G(z) =

k=0

−1

−1

B(k)Qˆ i (k + 1)˜ A i (k) and Qi (k) = ∆ 21 (k) where Pi (k) = ∆ i1 (k)˜

˜ ℓT Qˆ i (k + 1)˜ A i (k) with { ∆i1 (k) > 0 ∆i2 (k) > 0

i

=

vr (z) vr −1 (z) [b1 kpi (z)h2 + b1 kvi (z)h + b1 kai (z)] − z −1 b1 kai (z) p z − [1 + b2 − b1 ki (z)h(hv + h) − b1 kvi (z)h − b1 kai (z)] + z −1 [b2 − b1 kai (z)]

(38) (32)

For the purpose of suppressing the cost function (25), the controller and observer gain in (31) can be selected in each iteration backward as follows:

 ∗  ⎧ K (k)R(k) + Pi (k) ⎪ i ⎨Ki (k) = argK ∗ min F (k) i  ∗  min Fi (k)˜ H(k) + Qi (k)F ⎪ ⎩Fi (k) = argF ∗ (k)

(37)

By collecting the similar terms together and applying Z-transform for (37), one can obtain that

k=0 N −1 ∑  2   Fi (k)˜ H(k) + Qi (k)F ∆i2 (k)F ∥ξ (k)∥2

(36)

(33)

jw h

Using z = e = cos(wh)+j sin(wh), then (38) can be rewritten as Eq. (39) which is given in Box I. If we impose the condition |vr (z)/vr −1 (z)| ≤ 1 for any w, then the following inequality on w can be obtained. (hw )6 [−1 + b2 − b1 kai (z)]2 /36 + w 4 {h4 [1 + b2 − b1 kai (z)]2 /4

− h2 [−1 + b2 − b1 kai (z)]2 /3} + w2 h2 {[−3 + b2 − b1 kpi (z)hv ][1 + b2 − b1 kai (z)]

34

S. Wen and G. Guo / Robotics and Autonomous Systems 115 (2019) 28–39

G(z) =

Ξi (z) − cos(w h)b1 kai (z) − j sin(w h)b1 kai (z) vr (z) = p a vr −1 (z) cos(w h)[b2 − b1 ki (z) + 1] − j sin(w h)[−1 + b2 − b1 kai (z)] − [1 + b2 − b1 ki (z)hhv − Ξi (z)]

(39)

Box I.

− (1 + b2 )Ξi (z) + 4} + b1 hkai (z)[b1 kai (z)(2 − h) − 2Ξi (z)] ≥ 0 (40) where the second-order Taylor approximations is used to denote sin(w h) and cos(w h), e.g., sin(w h) ≈ w h − (w h)3 /3!

and

cos(w h) ≈ 1 − (w h)2 /2!.

Therefore, the sufficient condition for the non-negativity of the second-order polynomial on the left-hand side of (40) is h2 [1 + b2 − b1 kai (z)]2 /4 − [−1 + b2 − b1 kai (z)]2 /3 ≥ 0,

[3 − b2 +

p b1 ki (z)hv

[kai (z)]2 +

p 2kai (z)ki (z)

where Ξi (z) =

(41b)

+ 2kai (z)kvi (z) ≤ 0,

p b1 ki (z)h2

v

+ b1 ki (z)h +

(41c)

b1 kai (z).

4.2. Zero steady-state velocity errors analysis In what follows, the zero steady-state velocity error is analyzed which can be guaranteed by the designed controller. Let velocity error ∆v1 (k) = v0 (k) − v1 (k) denote the first follower with respect to the lead vehicle’s velocity change. Applying the Z -transform of both sides of (38), the relative velocity error of the first follower with respect to the lead vehicle’s speed change is obtained as follows p

Φ (z) =

z − [1 + b2 − b1 ki (z)hhv ] + z −1 b2 ∆v1 (z) = p v1 (z) [b1 ki (z)h2 + b1 kvi (z)h + b1 kai (z)] − z −1 b1 kai (z) (42)

It is assumed that the velocity error ∆v1 (k) can reach its final steady-state within a finite time kf , ∆v1 (k) − ∆v1 (kf ) = 0 for all k ≥ kf . Thus,

∆v1 (z) =

∆v1 (kf ) z

lim ∆vr (z) = lim (z − 1) z →1

Notation

Value

Inertial lag sampling period Length of vehicle Actuator time delay Desired spacing H∞ disturbance attenuation level

ς

0.25 s 0.1 s 6m 0.1 s 5m 1.25

h L

τ δd γ

gain Ki (k) and observer gain Fi (k) can be solved by (34), and go to the next step, else jump to Step 7. Step 3. Restrict the obtained controller gain Ki (k) by the conditions given in (41a)–(41c). If this is feasible, then the resulting controller gain can be used for string stabilizing control and go to the next step, else jump to Step 7. Step 4. Compute the matrix ∆i (k)in (18). If ∆i (k) is positive define, then step to the next procedure, else jump to Step 7. Step 5. Solve the backward RDEs of (19) and (29) to get Pi (k) and Qi (k), respectively. Step 6. If k ̸ = 0, set k = k − 1, i = τ (k) and go back to Step 2, else turn to the next step. Step 7. If the condition {∆i1 (k) > 0, ∆i2 (k) > 0, ∆i (k) > 0, Pi (0) ≤ γ 2 W } is not satisfied, this algorithm is infeasible. Stop. Remark 4. It can be observed from Algorithm 1 that in the observer-based controller design procedure, all the important factors contributing to the system complexity have been reflected which include: (1) time-varying platoon control model; (2) the transition probabilities of communication matrix; (3) string stability requirement of the platoon control; (4) the prescribed disturbance attenuation level. 5. Simulations and experiments 5.1. Numerical example

.

(43)

By using the final value theorem and (37)–(39) yields

k→∞

Parameters

(41a)

][1 + b2 − b1 kai (z)]

+ (1 + b2 )Ξi (z) − 4 ≤ 0,

Table 1 Parameters in the simulation.

∆vr (z) vr (z) v2 (z) ∆v1 (z) ··· vr (z) vr −1 (z) v1 (z) Φ (z)

= lim (z − 1)Gr −1 (z)∆v1 (z) = 0, z →1

(44)

for vehicle r, r = 1, . . . , N, namely, zero steady-state errors are achieved. 4.3. Observed-based controller design algorithm According to the observer-based controller design in Section 3, the concrete implement procedure process can be listed as follows Algorithm 1. The observed-based platoon control algorithm: Step 1. Set k = N − 1, i = τ (N − 1), i = 1, . . . , n. Then Pi (N) = 0 and Qi (N) = 0 are available. Step 2. Compute the matrices ∆i1 (k) and ∆i2 (k) in (29), respectively. If ∆i1 (k) and ∆i2 (k) are all positive define, then the controller

Firstly, a numerical example is given to show how the observerbased controller can be used for a six-vehicle platoons control system with the Markov chain driven stochastic MAC protocol. In the simulation, the needed parameters for the vehicle control are shown in Table 1. The Markov transition probability matrix II = [πij ], i, j = 1, 2, . . . , 6 for the communication matrix Hρ (k) is given as follows 0.3 ⎢0.25 ⎢ ⎢ 0.2 II = ⎢ ⎢0.18 ⎣0.24 0.12

⎡

0.2 0.15 0.1 0.12 0.16 0.21

0.1 0 .1 0.2 0 .2 0 .1 0.37

0.1 0.2 0.1 0.15 0.25 0.1

0.2 0.15 0.3 0.05 0.15 0.1

0.1 0.15⎥ ⎥ 0.1 ⎥ ⎥. 0 .3 ⎥ 0 .1 ⎦ 0.1

⎤

According to this transition probability matrix II, the random network access status yielded by the Markov chain is shown in Fig. 3. By the proposed observer-based platoon algorithm, a series of controller and observer gains iteratively in a backward manner (which are omitted to save space) is obtained. In the simulation,

S. Wen and G. Guo / Robotics and Autonomous Systems 115 (2019) 28–39

35

Fig. 5. Estimation errors of the relative acceleration.

Fig. 3. Random network access status of the six-vehicle platoons control system stimulated by a Markov chain.

it is assumed that the acceleration profile of the leading vehicle is

⎧ 0 m/s ⎪ ⎪ ⎪ ⎨0.3 m/s a0 (k) = 0 m/s ⎪ ⎪ ⎪ ⎩−0.3 m/s 0 m/s

0 ≤ k < 15 15 ≤ k < 30 30 ≤ k < 45 . 45 ≤ k < 60 k ≥ 60

Then the velocity profile of the leading vehicle is derived by v0 (k + 1) = v0 (k) + ha0 (k). Since the measurements of the practical platoon are inevitable subject to measurement noise or measurement

error, the position information and velocity information used in the observer-based controller are added with white noise, respectively. Using the obtained observer-based controller the position, velocity, acceleration and ∥G(z)∥ = ∥vr (z)/vr −1 (z)∥ of a six-vehicle platoons under the random network access shown is shown in Fig. 4. The estimation errors of acceleration errors are given in Fig. 5. From those figures, it is clear that the proposed observerbased controller can guarantee the string stability of the platoon subject to unmeasurable acceleration and random network access in VANET. Remark 5. Since string stability is defined as the transient errors are not amplifying with vehicle index due to any maneuver of the lead vehicle, both acceleration and deceleration are added for the leading vehicle. In the simulation, it is found that the unsuitable value (or initial value) of velocity and acceleration for leading vehicle and each following vehicle may cause the negative velocity

Fig. 4. Six-vehicles platoons control with the observer-based controller and random network access. (a) Position. (b) Velocity. (c) Acceleration. (d) ∥G(z)∥ ∥vr (z)/vr −1 (z)∥ < 1.

=

36

S. Wen and G. Guo / Robotics and Autonomous Systems 115 (2019) 28–39 Table 2 Parameters of the three Arduino cars. i=0 i=1 i=2

Ai (cm2 )

dmi (N)

σ (m/s3 )

cdi

15 12 12

0.5 0.4 0.4

0.98 0.98 0.98

0.15 0.15 0.15

5.2. Experiment with Arduino cars Arduino cars experiment (see in Fig. 8(b)) is given to shown that the proposed observed-based control is effective for practical platoon control system with random network access and unavailable acceleration. The structure of each Arduino car is shown in Fig. 8(a), whose dynamics is

Fig. 6. Accelerator inaccuracy status.

z˙r (t) = vr (t) v˙ r (t) = ar (t) a˙ r (t) = −0.25ar (t) + 0.25ur (t),

{ in the transient process with the designed controller. However, negative velocity implies that vehicles reverse the movement in platoon. This should be avoided because that negative velocity may yield collision. By using the numerical method, Matlab software can be used to help finding the suitable value of velocity and acceleration for each vehicle. In the next, the comparative results will be given with the controller designing in [34]. A platoon control problem is investigated in [34] with network access constraints and fading channels in the context of networked systems. Here, the effect of the inaccurate acceleration measurement on the proposed controller in [34] will be considered. By using the designed method in [34] with the same simulation parameters used in the above numerical example, the feedback gain of the following feedback controller is obtained as kpi = 35, kvi = 41 and kai = 7.5. ur (k) = kpi δr (k) + kvi ∆vr (k) + kai ∆ar (k)

(45)

The model of the acceleration measurement inaccuracy is described by afr (k) = θr ar (k) where θr is a coefficient denoting the acceleration inaccuracy. As in [34], each following vehicle r receives the inaccurate acceleration measurement afr −1 (k) from its preceding vehicle via the communications network with fading channels. Consider the effect of acceleration measurement inaccuracy, the feedback controller in (45) has the following form ur (k) = kpi δr (k) + kvi ∆vr (k) + kai ∆afr (k) where ∆afr (k) = afr −1 (k) − afr (k). In the simulation, it is assumed that all the vehicles are subject to the same acceleration inaccuracy, namely, θr = θ . A random process is used to produce the acceleration inaccuracy coefficient over time interval [0, 200], which is shown in Fig. 6. Under the proposed feedback controller in [34], the simulation results of the position, velocity, acceleration and frequency responses of a sixvehicles platoons with inaccurate acceleration measurement are shown Fig. 7. It can be clearly seen from Fig. 7 that the vehicular platoon cannot maintain string stability under the effect of inaccurate acceleration measurement. Due to the fact that acceleration measurement is inaccurate which has not been analyzed in the controller design. The existence of fluctuations in the velocity and acceleration may lead to uncomfortable driving experience or even rear-end accidents. Compared with the obtained results in Fig. 6 with the proposed method of this paper, it is clear that the proposed method has advantages.

(46)

where the parameters of Arduino car is shown in Table 2. The Arduino car is driven and steered by two front wheels. The longitudinal speed can be detected by an incremental encoder sensor installed on the shaft of the rear wheels. An Arduino processor onboard each car is served as the real-time computing and control unit. In the experiments, Arduino cars run very close to each other, e.g., the spacing is actually as small as several centimeters. Therefore, it is impractical to measure the spacing of cars using the GPS device; instead, we use infrared sensors GP2D12 produced by Sharp Corporation for distance measurement. GP2D12 has an integrated function of signal processing and analog voltage output. Its maximum sensing range is 80 cm. Each Arduino car transmits the velocity to its successive car via wireless communication modules APC220. The maximum velocity of each Arduino car is about 40 cm/s. The half-duplex communication of the wireless communication modules APC220 leading each car do not send and receive measurements at the same time. For example, when car 1 send the measurement to car 2, it cannot simultaneously receive the measurement from the leading car. Under such situation, the communication channel between car 1 and car 2 is working while the communication channel between leading car and car 1 keeps silent. Car 2 can receive the velocity information from the car 1 but car 1 cannot receive the velocity information from the leading car. In the experiment, the random network access stimulated by Markov chain for car 2 with the same Markov transmission probability matrix II as in the numerical simulation experiment. In the experiments, the velocity of leading car is shown in Fig. 10. The desired spacing between cars is set to be 20 cm. The spacing error and velocity of each Arduino car by using the proposed observer-based controller under random network access are shown in Figs. 9 and 10, respectively. In Fig. 9, the blue solid line represents the spacing δ1 between the lead Arduino car and the first follower, the green dashed line represents the spacing δ2 − δ1 between the first two follower Arduino cars. It is clearly seen from Figs. 9–10 that spacing keeping and velocity tracking is achieved by the proposed observer-based controller, regardless of the random network access and unavailable acceleration information. More important, the proposed algorithm can guarantee the string stability. The comparison between the measured relative acceleration (by using the accelerator installed on the top of the car) and estimated relative acceleration of car 1 is shown in Fig. 11. It is clear that the relative acceleration estimated by the proposed observer is acceptable. Remark 6. It is seen from Fig. 11 that there exists peaks in the estimated value of the relative acceleration (see at time 1 s and 5 s). The reason for the peaks may be that the proposed observer uses the

S. Wen and G. Guo / Robotics and Autonomous Systems 115 (2019) 28–39

37

Fig. 7. Six-vehicles platoons with inaccurate acceleration measurement using in the feedback controller proposed in [34]. (a) Position. (b) Velocity. (c) Acceleration. (d) ∥G(z)∥ = ∥vr (z)/vr −1 (z)∥ > 1.

Fig. 8. Arduino cars experiment: (a) Arduino car; (b) A platoon of Arduino cars in the traffic control experiment platform.

inaccurate spacing measurement. In practice, the measurements of infrared sensors GP2D12 are inevitable subject to measurement noise and measurement error [35]. In the next, the Arduino cars comparative experiments are given to show the advantage of the proposed observer-based controller. For simplicity, the leading car keeps uniform motion with velocity 20 cm/s. During the time interval [0, 7 s), the relative acceleration information referred in the feedback controller is estimated by the observer designed in this paper. While in the time interval [7 s, 10 s), the relative acceleration is estimated by the numerical differentiation of spacing measurements that is used in [23]. The observer-based controller is used in the experiment under the random network access, which has the same structure and value as it used in the aforementioned experiment. The spacing of the platoon is given in Fig. 12. As shown in Fig. 12, after 7 s, the output of the infrared sensor GP2D12 is suddenly increase to the

unreasonable value. This phenomenon is causing by the rangelimited of the infrared sensor GP2D12. For example, if the distance of the vehicle is closer than 10 cm, then the measurement of the infrared sensor GP2D12 is inaccurate and it outputs a series of increased measurements. In our experiment, during the time interval [7 s, 10 s), the Arduino cars has a collision. The main reason is that the spacing measurements implemented by infrared sensor GP2D12 are often contaminated by measurement noise and disturbance. The measurement noise and disturbance leads the unstable measurement of the spacing (which can be seen from Fig. 8). When the spacing error is divided by a smaller time interval (e.g., the time interval in the experiment is set 0.01 s), it will yield a larger and varying relative acceleration. Under this case, the controller of platoon cannot guarantee the string stability of the cars. From this point of view, that the relative acceleration is estimated by the

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S. Wen and G. Guo / Robotics and Autonomous Systems 115 (2019) 28–39

Fig. 9. Spacing δ1 −δ0 and δ2 −δ1 of Arduino cars using the observer-based feedback controller.

Fig. 10. Velocity of three Arduino cars using the observer-based feedback controller.

Fig. 12. Spacing δ1 − δ0 and δ2 − δ1 of Arduino cars under different acceleration estimation methods.

can be guaranteed. According to those obtained RDE, the algorithm is proposed for the platoon to implement the observer-based controller, which can simultaneously decide the controller and observer gain, reject the disturbance generated by the preceding vehicle’s acceleration and guarantee the string stability and zero steady-state velocity errors. Both numerical simulations and experiments with laboratory-scale Arduino cars have demonstrated the effectiveness and practicability of the presented observedbased control algorithm. In order to deal with the effect of random medium access, the dynamic equation of the vehicle in this paper is simplified by using feedback linearization. In practical vehicle platoon, vehicle are inevitable suffer to uncertainty (e.g., the mass of passenger vehicles or the engine time constant) and disturbance (e.g., the lead vehicle’s acceleration and wind gust), leading the vehicle has nonlinear dynamic behavior. In the further research, the proposed framework in this paper will be extended for the vehicles platoon who has the nonlinear dynamic behavior. On the other hand, the feedback control constraint problem and velocity optimal problem will be investigated for the vehicle platoons, which aim to avoid the negative velocity during the transient process. Due to the fact that vehicle with negative velocity (e.g., reverse moment) may cause collisions in platoon. References

Fig. 11. Measured value and estimated value of the relative acceleration of car 1.

numerical differentiation of spacing measurements is not suitable directly used in the feedback controller.

6. Conclusion In this paper, an observer-based controller is proposed for the platoon control to solve random network access and unmeasurable acceleration information. By employing the stochastic analysis techniques, a series of backward recursive RDE is derived as the sufficient conditions for the existence of the time varying controller and observer such that the performance of the platoon

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Shixi Wen received the ME and Ph.D. degrees in Control Science and Control Engineering, respectively, in 2011 and 2015, from Dalian Maritime University, China. From 2015 to 2017, he was a post-doctoral research fellow in Dalian University of Technology. And now, he is a lecturer in Dalian University. His research interests include networked control systems, vehicular cooperative control in intelligent vehicle highway system

Prof. Ge Guo received the B.E. and Ph.D. degrees in 1994 and 1998, respectively, from Northeastern University, China. He joined Lanzhou University of Technology in 1999, where he was an associate professor, the director of the Institute of Intelligent Control and Robots and dean of the Department of Electric Engineering from 2000 to 2004. Since 2005, he has been a professor in Dalian Maritime University. His research interests include networked control systems, intelligent highway vehicle systems and process control. He is the Managing Editor of International Journal of Systems, Control and Communications, an Associate editor of Information Sciences and an Editorial Board Member of ACTA Automatica Sinica. He was the honoree of the New Century Excellent Talents in Universities, China, the nominee of Gansu Top Ten Excellent Youths, Gansu Province, China, and the recipient of Fok Ying Tong Education Foundation, China.