The control objective is for each vehicle to maintain a constant velocity and ... straight line such that individual vehicles move at a constant desired velocity Vd ...
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D ECENTRALIZED C ONTROL OF V EHICULAR P LATOONS : I MPROVING C LOSED L OOP S TABILITY BY M ISTUNING Prabir Barooah1, Prashant G. Mehta2, and Jo˜ao P. Hespanha3 1,3
Center for Control, Dynamics and Computation, University of California, Santa Barbara, CA 93106. 2
Dept. of Mechanical Science and Engineering, University of Illinois, Urbana-Champaign, IL 61801.
Abstract We consider decentralized control of a platoon of N identical vehicles moving in a straight line. The control objective is for each vehicle to maintain a constant velocity and inter-vehicular separation using only the local information from itself and its two nearest neighbors. Each vehicle is modeled as a double integrator. To aid the analysis, we use the conservation principles to derive a continuous partial differential equation (PDE) approximation of the discrete platoon dynamics. The PDE model is used to explain the progressive loss of closed-loop stability with increasing number of vehicles, and to devise ways to combat this loss of stability. If every vehicle uses the same controller, we show that the least stable closed-loop eigenvalue approaches zero as O( N12 ) in the limit of a large number (N ) of vehicles. We then show how to ameliorate this loss of stability by small amounts of “mistuning”, i.e., changing the controller gains from their nominal values. We prove that with arbitrary small amounts of mistuning, the asymptotic behavior of the least stable closed loop eigenvalue can be improved to O( N1 ). These
conclusions are validated for the discrete platoon via numerical calculations.
I. I NTRODUCTION We consider the problem of controlling a platoon of N identical vehicles moving in a straight line such that individual vehicles move at a constant desired velocity V d with an inter-vehicular spacing of ∆; see Figure 1(a) for a schematic. Due to its relevance to an automated highway system, this problem has been extensively studied [1–6]. A controlled vehicular platoon with a constant but small inter-vehicular distance can help improve the capacity of a highway.
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Our focus in this paper is on decentralized bidirectional control architecture: the control action at a vehicle depends upon its own velocity and the relative position errors between itself and its nearest neighbors (the vehicles immediately ahead and behind it). The “decentralized” refers to the constraint that a vehicle is allowed to use only its local information, and “bidirectional” refers to the feature whereby information from the vehicles both ahead (predecessor) and behind (follower) is used. In contrast, a centralized control architecture will require information from all the vehicles to be transmitted to a central controller, making the communication overhead impractical for large platoons. A decentralized bi-directional control is also intuitively appealing when the inter-vehicular gap is small, since most human drivers use information from the preceding as well as following vehicles in such a scenario. Since the vehicles are assumed to be identical, one particularly important special case is the so-called symmetric bidirectional control, where all vehicles use identical controllers that are furthermore symmetric with respect to the predecessor and the follower position errors. Control of large vehicular platoons with a desired constant inter-vehicular spacing is challenging due to a number of reasons. Jovanovi´c et. al. have examined LQR control of a platoon of vehicles and showed that the eigenvalues of the closed-loop system approach the imaginary axis as the number of vehicles in the platoon increases without bound [3]. This progressive loss of closed-loop damping causes the closed loop performance of the platoon to become arbitrarily sluggish as the number of vehicles increases. Another challenge in the control of platoon is due to the “string instability” or “slinky-type effects.” This refers to the amplification of disturbances in a platoon (see [2, 4] and references therein). This paper provides an analysis of the progressive loss of closed-loop platoon stability with increasing N . The instability mechanism is shown to be independent of controller gains in the symmetric bi-directional architecture. More importantly, the paper also provides methods for the partial amelioration of this loss of stability while using a decentralized bi-directional architecture. Since inter-connected platoon dynamics are of primary interest, each vehicle is simply modeled as a double integrator. A double integrator vehicle model is common in the platoon control literature, as the velocity dependent drag and other non-linear terms can usually be eliminated by feedback linearization[2, 5]. There are three contributions of this work that are summarized below. In order to facilitate the analysis, we derive a linear partial differential equation (PDE) based continuous analogue of the (spatially) discrete platoon dynamics. The PDE model is inspired by the extensive
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literature on traffic dynamics; see the review [7] and references therein. In the limit of large N , the vehicles moving in a straight line are idealized as particles in a one-dimensional (traffic) flow. The spatial aspects of the dynamics are then described by the continuity and momentum equations for the flow. The continuity equation relates the flow density to the particle (or vehicle) velocity, and the momentum equation relates the acceleration to the external (control) force. In traffic modeling, to determine an appropriate form of momentum equation requires a model of the human behavior – a difficult task. With a controlled platoon, this becomes easier because the external force is simply the control action. The second contribution of this paper is to use the PDE model to derive a controller independent conclusion on stability with symmetric bi-directional architecture. In particular, the behavior of the least stable eigenvalue of the discrete platoon dynamics is predicted by analyzing the PDE spectra. We show that the least stable closed-loop eigenvalue approaches zero as O( N12 ). This prediction is confirmed by numerical computation. The third contribution and the biggest advantage of using a PDE based analysis is that the PDE reveals, better than the discrete equations do, the control structure with a decentralized architecture and suggests a mistuning-based approach to improve the stability margin. In particular, forward-backward asymmetry in the control is seen to be beneficial. The asymmetry refers to the assignment of controller gains such that a vehicle utilizes information from the preceding and following vehicles differently. Our main results, Corollary 1 and Corollary 2 shows how to achieve the best improvement in closed-loop stability by exploiting this asymmetry. In particular, we show that an arbitrary small perturbation (asymmetry) in the controller gains from their nominal (symmetric) value can improve the closed-loop damping such that the least stable eigenvalue now approaches 0 only as O( N1 ).
Numerical computations of eigenvalues in discrete platoons is used to validate these results.
The most beneficial mistuning profile for small amounts of mistuning is determined by using a perturbation based method of [8]. We note that the mistuning based approaches have been used for stability augmentation in many structural applications; see [9–11] for some recent references. The idea of improving platoon performance by using non-identical controllers has previously been considered in [12]. However, the design proposed in [12] has unbounded controller gains as N increases. In contrast, the mistuning based design proposed in this paper keeps controller gains uniformly bounded with N .
N 0 2π yi
4
yi−1 yi+1 ∆ (f )
ZN +1 (t)
ei
N
(b) ei
Z0 (t)
Zi (t) .. .
yi+1 .. .
.. .
i
yi
1
(a) A platoon with fictitious lead and follow vehicles.
0 (b) ei
yi−1 .. .
∆ (f∆) ei
2π
(b) Same platoon in y coordinates.
Fig. 1.
A platoon with N vehicles moving in one dimension.
The rest of the paper is organized as follows: section I-A states the platoon problem in formal terms; section II then describes the derivation of the PDE model. In section III the PDE is analyzed to explain the loss of stability with N , and section IV describes how to ameliorate such loss of stability by “mistuning”. Section IV-C reports simulation results that show the benefit of mistuing in time-domain. A. Problem Statement Consider a platoon of N identical vehicles moving in a straight line as shown schematically in Figure 1(a). Let Zi (t) and Vi (t) denote the position and the velocity, respectively, of the ith vehicle for i = 1, 2, . . . , N . Since the inter-connected platoon dynamics are of primary interest, a simple double integrator is used to model the essential dynamics of an individual vehicle: Z¨i = Ui , where Ui is the control (engine torque) applied on the ith vehicle. Formally, such a model arises after the velocity dependent drag and other non-linear terms have been eliminated by using feedback linearization [2, 5]. The control objective is to maintain a constant intervehicular distance ∆ and a constant velocity Vd for every vehicle. The control architecture is required to be decentralized, so that every vehicle can use locally available measurements only. In this paper, we assume a bi-directional control architecture for individual vehicles in the platoon except the first and the last vehicles. For these vehicles, we consider the following two scenarios as tabulated in Table I. In scenario I, we introduce (after [1, 3]) a fictitious
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Scenario
Length L
Leader
Follower
I
(N + 1)∆
v˜0 = 0
v˜N +1 = 0
II
N∆
v˜0 = 0
–
TABLE I T HE TWO SCENARIOS .
lead vehicle and a fictitious follow vehicle, indexed as 0 and N + 1 respectively. Their behavior is specified by imposing a constant velocity trajectories as Z 0 (t) = Vd t and ZN +1 = Vd t−(N +1)∆. In scenario II, only a fictitious lead vehicle with index i = 0 with Z 0 (t) = Vd t is introduced. For the last vehicle in the platoon in scenario II, there is no follower vehicle and it uses information only from its predecessor to maintain a constant gap. To facilitate the analysis, consider a co-ordinate change yi = 2π(
Zi (t) − Vd t + L ), L
vi = 2π
Vi − V d , L
(1)
where L denotes the platoon length, which equals (N + 1)∆ in scenario I and N ∆ in scenario II. Figure 1(b) depicts the schematic of the platoon in the new co-ordinates. The normalization ensures that y0 (t) ≡ 2π, yi (t) ∈ [0, 2π], and yN +1 (t) ≡ 0 (yN (t) = 0) in scenario I (II). Here, we have implicitly assumed that deviations of the vehicle positions and velocities from their desired values are small. In the normalized co-ordinate, the dynamics of the ith vehicle is described by y¨i = ui , where ui := 2πUi /L. The desired spacing and velocities are δ :=
∆ , L/2π
vd :=
Vd − V d = 0, L/2π
and the desired position of the ith vehicle is (2)
yid (t) ≡ 2π − iδ. The position and velocity errors for the ith vehicle in the y co-ordinate are: y˜i (t) = yi (t) − yid (t),
v˜i = vi − vd = vi .
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We note that v˜0 = v˜N +1 = 0 for the fictitious lead and follow vehicles. For the purposes of control, it is useful to introduce the front and back relative (position) errors for the ith vehicle: (f )
ei
(b)
ei The quantity ei
(f )
Zi−1 − Zi − ∆ = yi−1 − yi − δ, L/2π Zi − Zi+1 − ∆ = yi − yi+1 − δ, = L/2π
=
for i = {1, . . . , N }.
denotes the front relative position error between the ith and its predecessor
(i − 1) vehicle, and ei
(b)
denotes the back relative position error between the ith and its
follower (i + 1) vehicle. The relative errors, including the velocity error, can be obtained in
practice by on-board devices such as radars, GPS and speed sensors. Consistent with the decentralized bidirectional linear control architecture, the control u i for the ith vehicle is assumed to depend only on 1) its velocity v˜i , and 2) the relative position errors between itself and its immediate neighbors. That is, (f )
(f )
(b)
(b)
ui = ki ei − ki ei − bi v˜i ,
(3)
where ki , bi are positive constants. The first two terms are used to compensate for any (·)
deviation away from nominal with the predecessor and the follower vehicles respectively. The third term is used to obtain a zero steady-state error in velocity. In principle, relative velocity errors between neighboring vehicles can also be incorporated into the control, but we do not examine this situation here. Equation (3) represents control using state feedback, albeit only with local (nearest neighbor) information. Analysis of this controller structure is relevant even if there are additional dynamic elements in the controller. First, a dynamic controller cannot be allowed to have a zero at the origin. The reason is that for a constant velocity reference, such a pole-zero cancellation will lead to steady-state errors that grow without bound as N increases [6]. Second, a dynamic controller cannot have an integrator either. For if it does, the closed-loop platoon dynamics become unstable for a sufficiently large value of N [6]. As a result, any allowable dynamic compensator must essentially act as a static gain at low frequencies. Furthermore, the results of [6] indicate that the principal challenge in controlling large platoons arises from the double integrator with its unbounded gain at low frequencies. Hence, the limitation and its amelioration discussed here only with the local state feedback of (3) is also relevant to the case where additional dynamic elements appear in the control.
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To describe the closed-loop dynamics of the platoon, define y ˜ := [˜ y1 , y˜2 , . . . , y˜N ]T , v ˜ := [˜ v1 , . . . , v˜N ]T . For scenario I with fictitious lead and follow vehicles, the control law (3) yields the following closed loop dynamics. " # " #" # y ˜˙ y ˜ 0 I = (f ) (b) T v ˜˙ ˜ −KI M − KI M −B v {z } |
(4)
AL−F
where KI
(f )
(f )
(f )
(f )
(b)
(b)
(b)
(b)
= diag(k1 , k2 , . . . , kN ), KI = diag(k1 , k2 , . . . , kN ), B = diag(b1 , b2 , . . . , bN )
and 1 −1
0 ... 0 1 −1
M = ...
..
.
0 1 −1 ... 0 1
.
For scenario II with a fictitious lead vehicle and no follow vehicle, the closed loop dynamics are #" # " # " y ˜ y ˜˙ 0 I = (f ) (b) T ˜ v ˜˙ −KII M − KII Mo −B v {z } |
(5)
AL
where KII = KI , KII = diag(k1 , k2 , . . . , kN −1 , 0), and 1 −1 0 ... (f )
(f )
(b)
(b)
(b)
0 1 −1
Mo = ...
(b)
..
.
0 1 −1 ... 0 0
.
Our goal is to understand the progressive loss of closed loop stability with increasing N and to devise ways to ameliorate such a loss by appropriately choosing the controller gains. While in principle this can be done by analyzing the eigenvalues of the matrix A L−F (scenario I) and of AL (scenario II), we take an alternate route. When the number of vehicles N is large, we approximate the dynamics of the discrete platoon by a partial differential equation (PDE) which is used for analysis and control design.
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II. C ONTINUOUS M ODEL
OF
V EHICLE P LATOON
In this section, we develop a continuous PDE approximation of the (spatially) discrete platoon dynamics. The PDE is derived with respect to a normalized spatial co-ordinate x ∈ [0, 2π]. We recall that the normalized location of the ith vehicle (denoted as yi ) too was defined with respect to this co-ordinate system. In effect, the two symbols x and y correspond
to the same co-ordinate representation but are used here to distinguish the continuous and discrete formulations. With respect to the normalized co-ordinate, every car is nominally assumed to lie within an interval of length δ (see Fig. 1(b)). For the purposes of continuous approximation, we smear each vehicle over its interval to get a constant mean density ρ0 ≈
1 N = 2π δ
(6)
for N vehicles in the platoon. Dynamics of the individual vehicles in the platoon create perturbations in the density, with the local density ρ(x, t) increasing (decreasing) as the cars move closer (apart). The starting point of macroscopic continuous models of traffic flow thus is the continuity equation, which relates the density ρ(x, t) (vehicles per unit characteristic length) at spatial co-ordinate x ∈ [0, 2π] and time t ∈ [0, ∞) with the velocity v(x, t): ∂ρ ∂(ρv) + = 0. ∂t ∂x
In order to analyze small perturbations about the mean, we define the perturbed quantities ρ˜, v˜ by the relations ρ(x, t) = ρ0 + ρ˜(x, t),
v(x, t) = 0 + v˜(x, t),
where the mean velocity is zero because of our choice of the co-ordinate system (see (1)). Even though v and v˜ are the same, we use v˜ to draw attention to the fact that the velocity is a small perturbation of the mean value. For such perturbations, the linearized continuity equation is given by ∂ ρ˜ ∂˜ v + ρ0 =0 ∂t ∂x
⇒
∂˜ v 1 ∂ ρ˜ =− ∂x ρ0 ∂t
(7)
This equation is consistent with the physical intuition whereby a positive gradient in velocity (due to say the predecessor speeding up or the follower slowing down) will cause the local
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density to decrease. In order to study density perturbations, one thus needs to specify the velocity which here arises due to the linearized momentum balance: ∂˜ v ∂v = F (x, t) ⇒ = u(x, t), ∂t ∂t
(8)
where F (x, t) is the acceleration due to control u(x, t) and possibly disturbance. Here, we focus only on the control. Using (3), the control for the ith vehicle in the platoon is of the form: (pf )
ui (t) = ui |
where (pf )
ui
(pb)
(t) − ui {z (p)
ui (t)
(f )
(pb)
(t) := ki (yi−1 (t) − yi (t) − δ),
ui
(v)
(t) +ui (t), } (b)
(t) := ki (yi (t) − yi+1 (t) − δ),
are the position dependent f ront and back control terms, and, (9)
(v)
ui (t) := −b˜ vi (t). Corresponding to this discrete control law, we derive a continuous approximation u(x, t) = u(pf ) (x, t) − u(pb) (x, t) +u(v) (x, t), | {z }
(10)
u(p) (x,t)
such that u(pf ) (yi , t) = ui
(pf )
(pf )
ui
(pb)
(t), u(pb) (yi , t) = ui
(t), and u(v) (yi , t) = ui (t). Now, (v)
δ (f ) (f ) = ki (yi−1 − yi − δ) = ki (1 − )(yi−1 − yi ) yi−1 − yi Z yi−1 ρ(x) ≈ kpf (x)(1 − )dx, ρ0 yi
where the approximation is obtained by smearing the control action over the interval [y i , yi−1 ] and substituting kf (x) for the discrete control gain ki . Since ρ = ρ0 + ρ˜, we have Z 1 yi−1 1 (pf ) ui (t) ≈ − kf (x)˜ ρ(x, t)dx = − [kf ρ˜](x+ , t)δ, ρ 0 yi ρ0 (f )
by the Mean Value Theorem, where x+ ∈ [yi , yi−1 ]. Since ui
(pf )
(t) = u(pf ) (yi , t) and δ = 1/ρ0
(see (6)), we take
u(pf ) (x, t) = −
1 [k ρ˜](x+ , t), 2 f ρ0
and
u(pb) (x, t) = −
1 [k ρ˜](x− , t), 2 b ρ0
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where x+ ∈ [yi , yi−1 ] and x− ∈ [yi+1 , yi ]. Using (10), we have u(p) (x, t) = u(pf ) (x, t) − u(pb) (x, t) 1 = − 2 [kf ρ˜](x+ , t) − [kb ρ˜](x− , t) . ρ0 In order to specify the control, one thus needs to approximate the terms on the right hand side as functions of (x, t). For a small perturbation about a nominally symmetric bi-directional architecture, a valid approximation is obtained by taking x+ − x− ≈ δ which yields δ ∂[k (+) ρ˜] 1 (p) (−) (x, t) u (x, t) ≈ − 2 [kp ρ˜](x, t) + ρ0 2 ∂x 1 1 ∂ = − 2 k (−) ρ˜ − 3 (k (+) ρ˜), ρ0 2ρ0 ∂x where k (+) := kf (x) + kb (x),
k (−) := kf (x) − kb (x).
(11)
(12)
The velocity feedback term in (9) has a continuous counterpart u(v) (x, t) = −b(x)˜ v (x, t).
(13)
With the feedback control u(x, t) = u(p) (x, t) + u(v) (x, t), where u(p) (x, t) and u(v) (x, t) are given by (11) and (13), the linearized momentum equation (8) becomes 1 (−) 1 ∂ ∂˜ v (+) = − 2 k ρ˜ + 3 (˜ ρk ) + b˜ v . ∂t ρ0 2ρ0 ∂x
Upon differentiating both sides with respect to t and using the continuity equation (7) we obtain the PDE that describes small velocity perturbations v˜ due to the inter-connected platoon dynamics:
∂2 ∂ +b 2 ∂t ∂t
v˜ =
v 1 ∂ ∂˜ v 1 (−) ∂˜ (kp ) + 2 (kp(+) ) ρ0 ∂x 2ρ0 ∂x ∂x
(14)
The boundary conditions for the PDE depend upon the dynamics of the first and the last vehicles in the platoon. For scenario I with a constant velocity fictitious and lead vehicles, the appropriate boundary conditions are of the Dirichlet type on both ends: v˜(0, t) = v˜(2π, t) = 0.
∀t ∈ [0, ∞)
(15)
For scenario II with the only a fictitious lead vehicle, the appropriate boundary conditions are of Neumann-Dirichlet type: ∂˜ v (0, t) = v˜(2π, t) = 0. ∂x
∀t ∈ [0, ∞)
(16)
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A. Eigenvalue comparison For preliminary validation purposes, we consider the simplest case where the position control gains are constant for every vehicle, i.e., kf (x) = kb (x) = k0 and b(x) = b0 . In such a case k (−) (x) ≡ 0, k (+) (x) ≡ 2k0 and the governing PDE (14) simplifies to 2 ∂ k0 ∂ 2 ∂ + b0 − 2 2 v˜ = 0 ∂t2 ∂t ρ0 ∂x
Note that this is a damped wave equation with a wave speed of
√ k0 . ρ0
(17)
The wave equation
is consistent with the physical intuition that a symmetric bidirectional control architecture causes a disturbance to propagated equally in both directions. Figure 2 compares the closed loop eigenvalues of a discrete platoon with N = 25 vehicles and the PDE (17). The eigenvalues of the platoon are obtained by numerically evaluating the eigenvalues of the matrices AL−F and AL (defined in (4) and (5)). The eigenvalues of the PDE are also computed numerically after using a Galerkin method [13]. The figure shows that the two sets of eigenvalues are in good match. In particular, the least stable eigenvalues are well-captured by the PDE. Additional validation appears in the following sections, where we present and compare results for analysis and control design. III. A NALYSIS
OF LOSS OF STABILITY
In this section, we analyze the stability of a discrete platoon by evaluating the eigenvalues of the PDE
where x ∈ [0, 2π], ρ0 =
N 2π
2 ∂ ∂2 2 ∂ + b − a 0 0 ∂t2 ∂t ∂x2
is the mean density and a20 :=
v˜ = 0,
k0 ρ20
(18)
(19)
is the wave speed. The PDE corresponds to the platoon with symmetric and constant control gains: kf (x) = kb (x) ≡ k0 and b(x) ≡ b0 . On taking the Laplace transform, one obtains the characteristic equation
s2 + b0 s − a20 λ = 0,
(20)
where λ is an eigenvalue of the Laplacian, i.e., d2 η = λη(x), dx2
(21)
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2
2 platoon pde
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
(a)
−0.4
−0.3
Scenario
−0.2 Real
I
(
platoon pde
1.5
Imaginary
Imaginary
1.5
−2
−0.1
Dirichlet-
−0.5
−0.4
−0.3
−0.2 Real
−0.1
0
(b) Scenario II ( Neumann-
Dirichlet )
Dirichlet )
Fig. 2. Comparison of closed loop eigenvalues of the platoon dynamics and the spectrum of the corresponding PDE (18) for the two different scenarios: (a) platoon with fictitious lead and follow vehicles, and correspondingly the PDE (18) with Dirichlet boundary conditions, (b) platoon with fictitious lead vehicle, and correspondingly the PDE (18) with NeumannDirichlet boundary conditions. For ease of comparison, only a few of the eigenvalues are shown. Both plots are for N = 25 vehicles; the controller parameters are ki
(f )
(b)
= ki
and b(x) ≡ 0.5.
= 1 and bi = 0.5 for i = 1, 2, . . . , N , and for the PDE kf (x) ≡ 1
and η is an eigenfunction satisfying appropriate boundary conditions – (15) for scenario I and (16) for scenario II. The eigen-solutions for the two scenarios are given by the following simple Lemma. Lemma 1: Consider the eigenvalue problem (21) for the Laplacian with boundary conditions (15) and (16) corresponding to the scenarios I and II respectively. The eigenvalues and the eigenfunctions for the two scenarios are given in the Table II. The eigenfunctions for either scenario provide a basis of L2 ([0, 2π]).
Proof. It is a simple calculation to verify that the eigenvalues and eigenfunctions given in the table satisfy the eigenvalue problem. Any eigenfunctions of the Laplacian on [0, 2π] are known to provide a basis for L2 ([0, 2π]) [14]. To see the effect of N on stability, we evaluate the eigenvalues (roots of the characteristic equation (20)) for the Dirichlet boundary conditions (scenario I). Using Table II, the l th eigenvalue is given by s± l
=
−b0 ±
p b20 − a20 l2 , 2
(22)
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eigenvalue λl
boundary condition
eigenfunction ψl (x)
l
sin( lx ) 2
l = 1, 2, . . .
) cos( (2l−1)x 4
l = 1, 2, . . .
η(0) = η(2π) = 0 2
(Dirichlet - Dirichlet)
∂η (0) ∂x
− l4
= η(2π) = 0 − (2l−1) 16
(Neumann - Dirichlet)
2
TABLE II T HE EIGEN - SOLUTIONS FOR THE L APLACIAN WITH TWO DIFFERENT BOUNDARY CONDITIONS .
Im
Re
PSfrag replacements
0
PSfrag replacements
Im
s− l
s+ l
Re
0
s+ l s− l
Fig. 3.
(a) Eigenvalues move toward zero
(b) Mistuning “exchanges” stability
with increasing N .
− between s+ l and sl .
A schematic explaining the loss of stability as N increases and how mistuning ameliorates this loss.
where l = 1, 2, . . .. The real part of the eigenvalue depends upon the discriminant D(l, N ) = (b20 −a20 l2 ), where the wave speed a0 depends both on control gain k0 and number of vehicles N (see (19)). For a fixed control gain, there are two cases to consider:
1) If D(l, N ) < 0, the roots s± l are complex with the real part given by −b0 ,
− + 2) If D(l, N ) > 0, the roots s± l are real with sl + sl = −2b0 .
∂ In the former case, the damping is determined by the velocity feedback term b 0 ∂t , while in
+ the latter case one eigenvalue (s− l ) gains damping at the expense of the other (s l ) which + − looses damping. When s± l are real, the eigenvalue sl is closer to the origin than sl ; so th we call s+ l the l less-stable eigenvalue. The following lemma gives the dependence of this
eigenvalue on the number of vehicles N . Lemma 2: Consider the eigenvalue problem for the linear PDE (18) with boundary conditions (15) and (16), corresponding to scenarios I and II respectively. The l th less-stable
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s+ l for l 16 to ensure that 1 = 0, π 2 where < ·, · > denotes the standard inner product in L2 (0, 2π). Explicitly, this leads to an equation
l (2r0 + b0 )r1 = 4πρ0
Z
2π 0
l2 km (x) sin(lx)dx + 8πρ20
Z
2π
ks (x) cos2 ( 0
lx )dx 2
(36)
± For values of r0 = s± l (0), where sl (0) is given by (22), the equation above leads to an
expression for perturbation in the two eigenvalues. We denote these perturbations as r 1± . For r0 = s + k (0), we have from from Lemma (2) that b0 >> |2r0 | when l
