faculty of science and engineering
mathematics and applied mathematics
Truck platooning for a
convoy of heterogeneous trucks
Bachelor’s Project Mathematics
July 2017
Student: M. A. Huijzer
First supervisor: Dr. ir. B. Besselink Second assessor: Dr. D. Rodrigues Valesin
Abstract
The inter-vehicle time gap between trucks on a highway can be decreased using vehicle-following control systems. This results in lower fuel consumption. This thesis focuses on the development of such systems for a convoy of heterogeneous trucks, where stability is considered an essential requirement. First, the idea of a vehicle- following control system will be explained at hand of a convoy of homogeneous trucks, thereafter the differences between homogeneous and heterogeneous truck platoons will be discussed. For both the homogeneous as well as the heterogeneous case stability conditions will be derived and simulation results will be shown. For a heterogeneous convoy of trucks, the stability requirements are satisfied if the trucks are placed from small towards high values of the driveline dynamics time constant τ . Here τ can be regarded as a measure of the maximum rate-of-change of the vehicle’s acceleration. If the trucks are placed in reversed order, the control input u(t) grows and needs to be bounded due to limited engine power or braking capabilities .
Contents
1 Introduction 4
2 Homogeneous truck platoon 5
2.1 Model description . . . 5 2.2 Routh-Hurwitz stability . . . 10 2.3 Simulations . . . 13
3 Heterogeneous truck platoon 15
3.1 Model description . . . 15 3.2 Lyapunov stability . . . 19 3.3 Simulations . . . 24
4 Conclusion 36
References 38
A Matlab codes 39
A.1 Homogeneous truck platoon . . . 39 A.2 Heterogeneous truck platoon . . . 43
1 Introduction
Nowadays, the transportation of goods is an important part of our society. Most of it is done by trucks, where the truck companies want to move their goods as efficiently and cheaply as possible. Platooning is a technology that can be used in order to do so, that is, we use a vehicle-following control system to couple trucks that drive in a convoy. Such a system can decrease the inter-vehicle time gap between trucks, because it is more reliable than human drivers. Through this decreased spacing the trucks will suffer less air drag, this results in lower fuel consumption and decreased CO2 emissions [1].
Most of the research on vehicle-following control systems is done for homogeneous truck platoons, i.e., the problem is considered for a convoy consisting of exactly the same trucks, see [2], [3], and [4] for examples. This makes the platoon problem easier to analyze. The broader perspective of how platooning can be used for a convoy of heterogeneous trucks has received less attention, see [5] as example; here trucks with different manufacturers, lengths, or loads are considered. Therefore, the heterogeneous truck platoon problem will be considered in this research. To be more precise, the thesis focuses on the control design and stability analysis for a platoon consisting of trucks with varying values for the time constant τ of the driveline dynamics and a limited input function. This thesis focuses on these aspects by addressing the following two objectives.
First, in Section 2.1 the idea of a vehicle-following control system will be explained at hand of a convoy of homogeneous trucks as done as in [2] and [3]. This vehicle-following control system will be designed in such a way that vehicle i in the platoon follows its preceding vehicle i − 1 at a desired distance. Stability conditions will be derived in Section 2.2 and the research on the homogeneous vehicle platoon problem will be finished in Section 2.3, where simulation results will be shown to acquire more visual insight into its behavior.
The second objective of this thesis is the development of a vehicle-following control system for a convoy of heterogeneous trucks, see Section 3. Hereafter, stability conditions will be derived in Section 3.2 and simulation results will be shown in Section 3.3. Finally, conclusions are stated in Section 4.
2 Homogeneous truck platoon
Before describing the vehicle-following control system for a convoy of heterogeneous trucks, the system for a convoy of homogeneous trucks has to be considered. A vehicle-following control system for a homogeneous truck platoon is discussed in [2] and [3]. This system will be adopted in this section.
2.1 Model description
Consider a vehicle platoon of m vehicles of which a part is shown in Figure 1. Here, di represents the distance between vehicle i and its preceding vehicle i − 1. Our aim is to construct a vehicle-following control system such that vehicle i follows vehicle i − 1 at the desired distance dr,i. This desired distance is formulated by a constant time-headway spacing policy
dr,i(t) = ri+ hvi(t) (1)
where ri is the standstill distance, h the time headway, and vi represents the deviation of the reference velocity of vehicle i. The term time headway refers to the geometric distance between vehicle i − 1 and vehicle i divided by the vehicle velocity of vehicle i.
In this section the time headway is taken independently of i, since we are considering a homogeneous vehicle platoon.
Figure 1: String of controlled vehicles.[2]
The error of the vehicle-following system is called the spacing error and is defined as ei(t) = di(t) − dr,i(t) = (si−1(t) − si(t) − Li) − (ri+ hvi(t)), (2)
where (1) is used, si(t) denotes the position of vehicle i, and Li its length. The control problem now encompasses the vehicle-following objective limt→∞ei(t) = 0 for the vehicles 2 ≤ i ≤ m in the platoon. As a basis for the control design, the following basic vehicle model is adopted:
s˙i
˙ vi
˙ ai
=
vi
ai
−1τai+ 1τui
, (3)
for all vehicles i in the platoon, where ai represents the acceleration of vehicle i. The function ui represents the external input, in other words, the desired acceleration. The time constant τ represents the driveline dynamics and can be regarded as a measure of the maximum rate-of-change of the vehicle’s acceleration. Its value is vehicle-independent for the homogeneous vehicle platoon. Figure 2 shows that bigger values of τ lead to slower convergence of the acceleration ai(t) towards the desired acceleration ui(t). In practice, this can be interpreted as higher values of τ correspond to heavier vehicles. In Section 3 a heterogeneous truck platoon will be considered, by varying the value of τ . The time argument t is omitted to improve the readability of the expression.
Figure 2: Simulation of the convergence of the acceleration ai(t) towards the desired accel- eration ui(t) for different values of the driveline dynamics time constant τ . The blue line represents the desired acceleration ui(t), the green and red line represent the accelaration ai(t) of a vehicle with τ = 1 and τ = 5, respectively.
Now, control systems will be constructed such that the vehicle-following objective limt→∞ei(t) = 0 is met for the vehicles 2 ≤ i ≤ m in the platoon [2]. Here, distributed vehicle-following control systems will be considered, since vehicles only have access to lo- cal information. The total vehicle-following control system is distributed over all vehicles.
Using (2) and (3), the error dynamics of vehicle i are given by
e1,i e2,i e3,i
:=
ei
˙ei
¨ ei
=
(si−1− si− Li) − (ri+ hvi) vi−1− vi− hai
ai−1− ai+hτai− hτui
, (4)
for 2 ≤ i ≤ m.
The third error state equation is obtained by differentiating e3,i = ¨ei, while using (3) and (4), eventually resulting in:
˙e3,i = ˙ai−1+ h τ − 1
!
˙ai− h τ ˙ui,
= −1
τai−1+1
τui−1+ h τ − 1
!
−1 τai+ 1
τui
−h τ ˙ui,
= −1
τe3,i+ 1
τui−1− h τ ˙ui− 1
τui,
= −1
τe3,i− 1 τqi+ 1
τui−1, (5)
where qi is defined as
qi = h ˙ui+ ui. (6)
The function ui can be used as the state of a dynamic controller, where qi is a new input function that requires specification. We choose qi such that it stabilizes the error dynamics and it compensates for ui−1. The last condition follows as ˙e3,i also depends on ui−1, see (5). Therefore, the control law for qi is designed as
qi = K
e1,i e2,i e3,i
+ ui−1, (7)
with K =
kp kd kdd
.
Due to the additional controller dynamics (6), the error dynamcis must be extended with an additional equation, which can be obtained using the input definition (6) while substituting the control law (7):
˙ui = −1
hui+ 1 h
kp kd kdd
e1,i e2,i e3,i
+ 1 hui−1,
= −1
hui+ 1
h(kpe1,i+ kde2,i+ kdde3,i) + 1
hui−1. (8)
The vehicle-following control system can now be expressed in error coordinates. This will be done by combining (8) with the error states (4). As a result, the 4th order closed-loop model reads
˙e1,i
˙e2,i
˙e3,i
˙ui
=
0 1 0 0
0 0 1 0
−kτp −kτd −1+kτdd 0
kp
h kd
h
kdd
h −h1
e1,i e2,i e3,i ui
+
0 0 0
1 h
ui−1. (9)
In order to analyze the behavior of the convoy we formulate the platoon system in vehicle states instead of error states. Using (3), (4), and (8), the following homogeneous platoon model is obtained:
˙ei
˙vi
˙ai
˙ui
=
0 −1 −h 0
0 0 1 0
0 0 −1τ 1τ
kp
h −khd −kd− kdd(τ −h)hτ −kddhτh+τ
ei vi ai ui
+
0 1 0 0
0 0 0 0
0 0 0 0
0 khd khdd 1h
ei−1 vi−1 ai−1 ui−1
.
or, in short
˙xi = A0xi+ A1xi−1, (10)
for 2 ≤ i ≤ m, with the state vector xi =
ei vi ai ui
T
.
The first vehicle will regulate the speed of the convoy and therefore has a different control objective. This vehicle drives at a constant velocity, but it has to respond to velocity fluctuations of traffic driving in front of the platoon. These fluctuations will be called disturbance, denoted as w. The control objective of the first vehicle is limt→∞v1(t) = 0, where we regard v1 as the deviation of the reference velocity. The vehicle controller system will now be designed using the basic vehicle model (3) and the static feedback (6) with i = 1. As the first vehicle has no predecessor, it does not have a spacing error. The following system is obtained:
˙e1
˙v1
˙a1
˙u1
=
0 0 0 0
0 0 1 0
0 0 −1τ 1τ 0 0 0 −h1
e1 v1 a1 u1
+
0 0 0
1 h
q1+
0 0
1 τ
0
w. (11)
The aim is now to find a controller q1, such that disturbance will not lead to large variations in the velocity of the first vehicle, and it should return to its reference velocity. The control law for q1 is designed as follows:
q1 = k1v1+ k2a1 + k3u1. (12) In (12), note that there is no constant depending on e1, since its value is equal to zero by definition. Substitution of the control law (12) into the system (11) gives the closed-loop model
˙e1
˙v1
˙a1
˙u1
=
0 0 0 0
0 0 1 0
0 0 −1τ 1τ 0 kh1 kh2 k3h−1
e1 v1 a1 u1
+
0 0
1 τ
0
w,
or, in short
˙x1 = Arx1+ Brw, (13)
with the state vector x1 =
e1 v1 a1 u1
T
.
Combining the distributed vehicle-following control systems (10) and (13) we obtain the total vehicle-following control system for a platoon of m vehicles
˙x1
˙x2 ...
˙xm
=
Ar 0
A1 A0 . .. ...
0 A1 A0
x1 x2 ... xm
+
Br 0
... 0
w,
or, in short
˙x = Ax + Bw, (14)
with x =
xT1 xT2 · · · xTm
T
.
Now that we have developed the vehicle-following control system for a convoy of homo- geneous trucks, in the next section, conditions on the controller parameters will be imposed such that the system (14) has a stable equilibrium point.
2.2 Routh-Hurwitz stability
We have seen that the behavior of a convoy of homogeneous trucks is described by the system (14). In this subsection, we will consider the stability of the equilibrium points of this system. We notice that the system (14) is linear and that the state matrix A is lower-block triangular, which means that the system has asymptotically stable equilibrium points if and only if the matrices Ar (13), and A0 (10) are Hurwitz (i.e., all eigenvalues have negative real parts). The Routh-Hurwitz stability criterion will be used to find conditions on the control parameters such that the matrices Ar and A0 are Hurwitz. After finding these conditions, the theory of pole-placement will be used to assign the eigenvalues of the matrices. Here, the first vehicle and follower vehicles are considered separately.
First Vehicle
First, the stability of the first vehicle, that is, the matrix Ar, will be considered. We recall that the control objective of the first vehicle is given by limt→∞vi(t) = 0, where vi is
regarded as a deviation of the reference velocity. Then, the closed-loop system is stable if all the eigenvalues of the state matrix Ar lie in the open left-half complex plane. The matrix Ar is block diagonal, see (13); it can be split into two blocks, as
0 0 0 0
0 0 1 0
0 0 −1τ τ1 0 kh1 kh2 k3h−1
,
where the solid lines indicate the block diagonal structure of this matrix. As a result, the upper block has a zero eigenvalue, which corresponds to the spacing error e1. This eigenvalue is not of interest in the stability analysis, since the first vehicle has no predecessor and thus no spacing error. The characteristic polynomial of the lower block matrix is given by
λ3 + 1
τ + 1 − k3 h
!
λ2 + 1 − k2− k3 hτ
!
λ − k1
hτ = 0. (15)
The Routh-Hurwitz stability criterion can now be used to find conditions on the controller parameters of (12), for which the eigenvalues of the lower block matrix lie in the open-half complex plane. The Routh-Hurwitz table for the characteristic polynomial (15) is given
1 1−khτ2−k3
1
τ +1−kh3 −kh1
1−k2−k3
hτ +h+τ (1−kk1τ
3) 0
−kh1 0
. (16)
The eigenvalues of the matrix Ar lie in the open left-half complex plane if all values in the first column of (16) are positive. More information about the Routh-Hurwitz stability cri- terion can be found in [6]. Thus, the controller q1 stabilizes the velocity of the first vehicle for any h > 0, τ > 0, and with any choice for k1 < 0, k3 < h+τ , and k2 < 1−k3+h+τ (1−kk1hτ2
3). Next, pole-placement will be used to assign the eigenvalues of the lower block matrix.
These eigenvalues should lie in the open left-half of the complex plane as stated before; it
would be even better if the eigenvalues are real, the complex part of eigenvalues will cause overshoot. The closed loop poles should not be chosen far away from the open loop poles, otherwise it will demand high control effort. Therefore, the poles of the closed loop state matrix Ar will be placed at −1, that is, the characteristic polynomial (15) should equal
(λ + 1)3 = λ3+ 3λ2+ 3λ + 1 = 0, (17) which is the case for
k1 = −hτ, k2 = 3h(1 − τ ) − h
τ, k3 = 1 + h
τ − 3h. (18) Vehicle 1 < i ≤ m
For the follower vehicles, the control objective is no longer drive at a constant velocity, but to follow its predecessor. Equivalently, the controller parameters, see the matrix K in (7), should be chosen such that limt→∞ei(t) = 0, that is the eigenvalues of the state matrix (9) should lie in the open left-half complex plane. This matrix is recalled as
0 1 0 0
0 0 1 0
−kτp −kτd −1+kτdd 0
kp
h kd
h
kdd
h −h1
,
where the solid lines indicate the lower block-triangular structure of this matrix. As a result, the spectrum of this matrix is the same as the union of the spectra of the lower and upper block matrix. The eigenvalue of the lower block matrix is −h1, which is the result of the choice of the spacing policy (1). The eigenvalue of the lower block matrix lies in the open left-half complex plane, since the time-headway is chosen as h > 0 by definition (see Section 2). The eigenvalues of the upper block matrix are the roots of the characteristic polynomial
λ3+1 + kdd
τ λ2 +kd
τ λ + kp
τ = 0. (19)
The Routh-Hurwitz stability criterion can be used to find conditions on the controller parameters of (7) for which the eigenvalues of the lower block matrix lie in the open- half complex plane. This is the case if and only if all entries in the first column of the Routh-Hurwitz table
1 kτd
1+kdd
τ
kp
τ kd
τ − 1+kkp
dd 0
kp
τ 0
are positive. Therefore, the dynamic controller (6) with the feedback (7) stabilizes the error-states for any time headway h > 0, and with any choice for kp, kd > 0, kdd > −1, such that (1 + kdd)kd> kpτ .
We saw that one of the eigenvalues of the state matrix (9) is −1h, the other eigenval- ues can be placed at −1 using pole-placement. This is the case, when the characteristic polynomial (19) equals (17). This results in the controller parameters
kp = τ, kd= 3τ, kdd = 3τ − 1. (20)
In this section, we found conditions on the parameters of the controllers of the first and follower vehicles, respectively, such that the closed-loop system (33) is stable and its poles are placed −1. To get more insight into the actual behavior of the system, simulation results will be shown in the next section.
2.3 Simulations
In Section 2.1, we developed a vehicle-following control system for a convoy of homogeneous trucks. This system was constructed such that every vehicle in the platoon follows its predecessor at the desired distance dr,i, defined in (1), where the first vehicle leads the convoy and drives at a constant velocity. We saw that the control objective of the first vehicle is given limt→∞v1(t) = 0. Namely, the vehicle should return to this reference velocity after small disturbances caused by the traffic driving in front of the platoon. The control objective of the follower vehicles is given limt→∞ei(t) = 0 and is satisfied for the
controller parameters (20). In this subsection, the a convoy of homogeneous trucks will be simulated.
Consider a homogeneous truck platoon consisting of four trucks. The assumption is that all the four trucks have zero initial spacing errors and accelerations, and have an initial velocity of 20 m/s. At t = 0, the first vehicle experiences a disturbance from the traffic in front of the platoon, and has to slow down with an acceleration 1τ for two seconds. Next, the first vehicle returns to its reference speed of 20 m/s.
Figure 3: Simulation results obtained for a homogeneous four-vehicle platoon with h = 0.5 and τ = 1. The first vehicle drives at a reference speed of 20 m/s, where a disturbance is added for 0 < t < 2. The results are based on (14) with the controller (12), (18) and the dynamic controller (6) with the feedback (7), (20) for the first and follower vehicles, respectively.
In Figure 3, a simulation result of (14) is given. The controller parameters (18) and (20) satisfy the Routh-Hurwitz stability criterion for the values of h = 0.5 and τ = 1, see Section 2.2, and the freely selectable eigenvalues are placed −1. In this figure, the first vehicle indeed returns to its reference speed after the disturbance in the first two seconds.
The follower vehicles encounter some spacing error, but the vehicles satisfy their control approach, that is, the spacing error converges towards zero.
3 Heterogeneous truck platoon
In the previous section, we developed a vehicle-following control system for a convoy of homogeneous trucks. In this section, the vehicle-following control system for a convoy of heterogeneous trucks will be explored. The parameters τ and h are no longer vehicle independent; we have to consider τi and hi for each vehicle i in our convoy. The spacing error ei(t) will be defined in order to develop the vehicle-following control system. Also, the control law can be designed by formulating the error dynamics. But, we cannot define the feedback as we did for the homogeneous case (7). Namely, ˙e3,iwill not lead to a similar expression as (5); ˙e3,i now depends on τi as well as τi−1, and these terms do not cancel.
Therefore, in this section we will develop a vehicle-following control system with a static feedback controller rather than dynamic feedback controller as done as in Section 2.
3.1 Model description
Consider a heterogeneous vehicle platoon of m vehicles. The objective is to construct a vehicle-following control system such that vehicle i follows vehicle i − 1 at the desired distance dr,i. The time-headway spacing policy is adopted from Section 2, but now vehicle dependent values of the time-headway h need to be considered. Therefore, the desired distance is formulated as
dr,i(t) = ri+ hivi(t),
for every vehicle i in the platoon. The spacing error is defined as before, leading to ei(t) = di(t) − dr,i(t) = (si−1(t) − si(t) − Li) − (ri+ hivi(t)). (21) The control problem now encompasses the vehicle-following objective limt→∞ei(t) = 0 for the vehicles 2 ≤ i ≤ m and limt→∞v1(t) = 0 for the first vehicle in the platoon. As a basis
for the control design, the following basic vehicle model is adopted from Section 2:
˙ si v˙i
˙ ai
=
vi ai
−τ1
iai+ τ1
iui
, (22)
where we consider vehicle dependent values of the driveline dynamics time constant τ . The vehicle-following control system can now be developed using the error dynamics.
These can be calculated by substitution of (22) into the derivatives of the spacing error (21) and is given
e1,i e2,i e3,i
=
ei
˙ei
¨ ei
=
(si−1− si− Li) − (ri+ hivi) vi−1− vi− hiai
ai−1− ai+ hτi
iai− hτi
iui
, (23)
for 2 ≤ i ≤ m.
Further derivation of the error dynamics will lead to an expression which depends on τi−1. Therefore, we cannot choose a feedback controller (7) to stabilize the error as we did in Section 2. Instead, the input function ui is chosen such that it stabilizes the error dynamics and it compensates for ai−1 and ai.
Hence, the control law for ui is designed as follows:
ui = τi hi
ai−1− ai+hi
τiai+ K
e1,i e2,i
,
= τi
hiai−1+
1 − τi hi
ai+ τi hiK
e1,i e2,i
,
= τi
hiai−1+
1 − τi
hi
ai+ τi
hik1,ie1,i+ τi
hik2,i(vi−1− vi− hiai) ,
= τi k2,i
hi vi−1+ 1
hiai−1+k1,i
hi e1,i− k2,i hi vi +
1 τi − 1
hi − k2,i
ai
!
, (24)
where K =
k1,i k2,i
is the feedback matrix that remains to be chosen.
Next, we derive the closed-loop system in terms of the original vehicle coordinates, which can be used to analyze the behavior of the convoy. As a result of (22), (23), and (24), the system reads
˙ei
˙vi
˙ai
=
0 −1 −hi
0 0 1
0 0 −τ1
i
ei vi
ai
+
0 1 0 0 0 0 0 0 0
ei−1 vi−1
ai−1
+
0 0
1 τi
ui (25)
=
0 −1 −hi
0 0 1
k1,i
hi −kh2,i
i −h1
i − k2,i
ei
vi ai
+
0 1 0
0 0 0
0 kh2,i
i
1 hi
ei−1
vi−1 ai−1
. (26)
The closed-loop system (26) does not depend on τi, resulting that for every value of τi, the feedback matrix K can be chosen such that the closed-loop system is stable. For some values of τi this will lead to high entries in the feedback matrix K (see (24)), and so high values of ui. But, due to limited engine power or braking capabilities the control input ui of a vehicle has a limit. This limit can be encountered in our vehicle-following control system by introducing a saturation function fi(ui) expressed as
fi(ui) =
−¯ui if ui < −¯ui, ui if − ¯ui ≤ ui ≤ ¯ui, u¯i if ui > ¯ui,
(27)
where the limit of the control input ui will be denoted by ¯ui. Figure 4 shows that the saturation function fi(ui) takes the value ui (24) if this is between the the limits −¯u and
¯
u, else it takes on the value of the limit.
Every instance of ui will now be replaced by fi(ui), which results in the basic vehicle model
˙ si
˙ vi
˙ ai
=
vi ai
−τ1
iai+ τ1
ifi(ui)
. (28)
