Passivity-based Control for Multi-Vehicle Systems Subject to StringConstraints

Steffi Knorn a , Alejandro Donaire b , Juan C. Aguero b , Richard H. Middleton b ,aUppsala University, Sweden

bThe University of Newcastle, Australia

Abstract

In this paper, we show how heterogeneous bidirectional vehicle strings can be modelled as port-Hamiltonian systems. Analysis of stabilityand string stability within this framework is straightforward and leads to a better understanding of the underlying problem. Nonlinearlocal control and additional integral action is introduced to design a suitable control law guaranteeing l2 string stability of the system withrespect to bounded disturbances.

Key words: Hamiltonian Systems, String Stability, Multi-Vehicle Systems

1 Introduction

In the field of coordinated systems, formation control is oneof many control objectives. A group of N vehicles (e. g.platoon or string) is required to follow a given referencetrajectory while the vehicles keep a prescribed distance toneighbouring vehicles. In its simplest form the vehicles inthe platoon are considered to move in a single straight line.While many different solutions to this problem have beenproposed, most researchers consider a decentralised con-trol structure where each vehicle in the string uses a localcontroller with locally available measurements instead of aglobal, centralised controller.

In most cases it is straightforward to design a local controllerto achieve stability of a string in the Lyapunov sense. How-ever, it is well known that error signals can amplify whentravelling through the string resulting in growth of the localerror norm with the position in the string. This effect is re-ferred to as string instability, e.g. in [10, 14, 17], or slinky

This paper was not presented at any IFAC meetings. Corre-sponding author: Steffi Knorn. Tel. +46-18-4717389. Fax +46-18-4717244.

Email addresses: steffi.knorn@signal.uu.se (SteffiKnorn), alejandro.donaire@newcastle.edu.au (AlejandroDonaire), juan.aguero@newcastle.edu.au (Juan C. Aguero),richard.middleton@newcastle.edu.au (Richard H.Middleton).

effect, e.g. in [3, 6, 18, 21]. A well known definition of lpstring stability has been proposed in [19].

Different approaches have been proposed to guarantee stringstability of unidirectional vehicle strings, where each localcontroller only considers information relating to a group ofvehicles in front. These approaches include: (i) introducinga time headway, [3]; (ii) local controllers that depend onthe position within the string, [8]; (iii) using the velocityor the acceleration of the lead vehicle; or, (iv) propagatingthe reference velocity to each vehicle within the platoon,e. g. [10, 20] or [2], respectively.

This paper studies heterogenous, bidirectional, nonlinearstrings of vehicles. (Heterogeneous refers to the fact thatthe dynamics of each vehicle and local controller might dif-fer between the vehicles; and bidirectional means that in-formation propagates both upstream and downstream in thestring.) It was shown in [1,17] that similar to unidirectionalstrings, linear, symmetric bidirectional strings with two in-tegrators in the open loop and constant spacing are alwaysstring unstable. The definition used in [1,17] even thoughoften not explicitly mentioned requires that the L2 normof the local error vector, i. e. e()2 =

0

Nk=0 |ek(t)|2dt

is bounded for all disturbances d in L2.

Different approaches to overcome this limitation have beenproposed. It is shown in [10] that this type of string sta-bility can be guaranteed given a sufficiently large coupling

Preprint submitted to Automatica July 18, 2014

with the leader position. In [2] a linear, bidirectional stringof N vehicles is approximated as a PDE. It is shown thatthe least stable eigenvalue of the PDE approaches the ori-gin with O(1/N2) if the string is symmetric and O(1/N) ifthe string is asymmetric. However, knowledge of the steadystate / reference velocity is needed. Both proposed solu-tions depend on either perfect communication between theleading vehicle and the rest of the string or perfect knowl-edge of the reference signal. Further, any reference signalchanges have to be communicated to every vehicle in thestring. In case sufficiently fast and lossless communicationof the necessary information cannot be guaranteed, it mightnot be possible to guarantee string stability. Thus, both con-trol structures suffer from a serious risk if the number ofvehicles increases. Therefore, it is desirable to find alterna-tives that do not depend on global communication or globalreference knowledge but local measurements and local com-munication between a small number of direct neighbours.

A different approach to ensure string stability was consid-ered in [5]. Modelling a symmetric bidirectional string asa mass-spring-damper system, it is shown that string stabil-ity with constant spacing can be guaranteed if the dampingcoefficients or the inverse compliances of the springs be-tween the vehicles, respectively, grow with the string lengthN. This also seems undesirable in practise. Since controllerparameters cannot be chosen infinitely high this implies thatthe string cannot be extended without bound.

It is the aim of this paper to offer an alternative approachto this problem. First, the definition of lp string stabilityproposed in [19], although very useful for unidirectionalstrings, seems uninformative and too restrictive for bidirec-tional strings. Recall that it has been shown that string sta-bility cannot be achieved for linear, symmetric, bidirectionalstrings with two integrators in the open loop, constant spac-ing and without global communication or reference signalknowledge, [1,17], and the alternatives suffer from undesir-bale risks and limitations above. Thus, we first propose adefinition of lp string stability that is less restrictive than thedefinition given in [19]. This definition proves to be usefulin guaranteeing an upper bound for the distances betweenthe vehicles at all times. Details can be found in the fol-lowing section. Second, a control algorithm is proposed thatdoes not suffer from any of the above mentioned disadvan-tages. That is, the design does not require any global com-munication or global knowledge of the reference informa-tion. Further, it is possible to choose all control parametersin a defined bound without the need to limit the number ofvehicles. We also propose a different method to model suchvehicle strings, that is to use port-Hamiltonian system the-ory, [11]. This approach offers significant advantages overmethods known in the string stability literature. The port-Hamiltonian systems description allows direct and easy tofollow stability and string stability proofs and thus offer abetter insight into the underlying problem. The method alsoallows an insight on limitations and advantages of similarsystem designs presented in the literature such as in [2, 5].Further, both linear and nonlinear systems can be described

as port-Hamiltonian systems.

The paper is organised as follows: The problem of interestin this paper (i. e. string stability of port-Hamiltonian sys-tems describing vehicle platoons) is presented in Section 2.After introducing local control to a string of vehicles in Sec-tion 3, integral action will be added in Section 4. The paperends with a numerical example in Section 5 and concludingremarks in Section 6. Some preliminary results have beenreported in [9].

2 Notation and Problem Formulation

2.1 Notation

The L2 vector norm is given by |x|2 = |x| =

xTx and theL2 and L vector function norms by x()2 =

0 |x(t)|2dt

and x() = supt0 |x(t)|, respectively. For a scalar func-tion H(x) of a vector x = [x1, x2, . . . , xn]T its gradient isdefined as H(x) =

[H(x)x1

,H(x)x2

, . . . ,H(x)xn

]T. We denote the

state, steady state and the disturbance vector by the columnvectors (generally denoted col) x(t) = col(x1(t), . . . ,xN(t)),x0 = col(x10 , . . . ,xN0 ) and d(t) = col(d1(t), . . . ,dN(t)). Thecolumn vector of ones is denoted by 1 and ~ei is the ithcanonical vector of length N. Similarly we denote the di-agonal matrix A RNN with diagonal entries a1, . . .aN asA = diag(a1, . . .aN).

2.2 System Description

We consider a system of N vehicles with mass mi wherei = 1,2, ,N denotes the position within the string. Themotion equations of the system can be described using themomentum and position of each vehicle, i.e. pi and qi, asfollows

pi =Fi + di (1)qi =m1i pi (2)

where Fi is the control force on the vehicle, di is the dis-turbance, and the momentum satisfies pi = mivi, where vi isthe velocity. The local position error between the ith vehicleand its direct predecessor is denoted

i = qi1 + li qi. (3)

Note that li is the desired safety distance between the vehi-cles plus the length of vehicle i1 or i depending on whetherthe position of the front or the rear of each vehicle i is usedas position qi. As a minimal distance is usually required be-tween vehicles for obvious safety reasons, we assume li > 0for all i. The position q0 is the reference position availableto the first vehicle in the string. The dynamics of the stringsystem described in momenta of the vehicles and separation

2

distance between the vehicles can be described by

p

=

0 S T

S 0

H(p,) +F

0

+

d~e1v0

, (4)

where ,p RN are the displacement and momentum vec-tors, i. e. = col(1, . . . ,N), p = col(p1, . . . ,pN), and thecontrol force vector is F = col(F1, . . . ,FN). The functionH(p,) is given by

H(p,) = 12

pTM1 p. (5)

The matrix M RNN is the constant and positive definiteinertia matrix M = diag(m1, . . . ,mN). The matrix S has thebidiagonal form

S =

1 0 01 1 . . . ...0 1 . . . . . . ......

. . .. . . 1 0

0 0 1 1

. (6)

2.3 Control Objectives

The local control objective for each vehicle is to bring itslocal error to zero using local (distributed) control and onlylocally available data. The control force Fi will be chosensuch that only data from a group of neighbours of the ithvehicle (both preceding and following vehicles) are needed.The controller for the first vehicle in the string aims to followa given trajectory q0 and also minimise the local positionerror towards a group of following vehicles. In the simplestsetting the reference signal is considered to be a ramp withconstant velocity v0, i.e. q0 = v0t. Note that the vehicleswithin the string (apart from a limited group at the beginningof the string) do not have access to the reference signaland therefore have to adjust their position and momentumindirectly by forcing their local position error to zero.

The overall control objective is to achieve string stabilityor scalability, that is, the norm of the local states of thecomplete string do not grow without bound as N increasesfor nonzero disturbances or initial conditions. Consideringthe general definition of lp string stability in [19] and [15],we use the following definition of l2 string stability withrespect to disturbances:

Definition 1 Consider a system described by x = g(x,d)with states x RN , g RN satisfying g(x,0) = 0 anddisturbances d. The equilibrium x is l2 string stable withrespect to disturbances d(t), if given any > 0, there exists

1() > 0 and 2() > 0 (independent of N) such that

|x(0) x| < 1() and d()2 < 2() (7)

implies

x(t) x = supt0

|x(t) x| < N 1. (8)

3 Local Control

In this section, a local distributed controller for a bi-directional vehicle string system is designed. The localcontrol structure is motivated by previous results in the fieldof mechanical engineering. When choosing control actionsbetween the vehicles that can be modelled as virtual springsand dampers between the vehicles, the overall system caneasily be written as a port-Hamiltonian system.

The control forces consist of the spring forces Fsi , that de-pend on the position errors i, the damper forces Fri , thatdepend on the velocity differences between two neighbour-ing vehicles, and the drag forces Fdi describing the fric-tion of the vehicles towards the ground. Assume the springforce between vehicle i 1 and i is given by the functionf si (i), which is locally Lipschitz for alli within the domainof definition, strictly monotonic and satisfies f si (i)i 0and f si (i) = 0 only for i = 0 for all i = 1, . . . ,N. Thus,Fs = S T f s() where f s() is the column vector with entriesf si (i). Also assume that the condomian or target set of f sicovers the complete real axis between and . Thus, thefunction f si is invertible and f s

1i is its inverse. The vector of

inverses is denoted by f s1 .

Remark 2 Note that nonlinear springs yield some signifi-cant advantages over linear spring models: First, one coulduse barrier functions on the potential energy to prevent col-lisions between the vehicles in the platoon. To do this, thestiffness of the springs have to increase when the distancesbetween vehicles decrease. For instance a good choice isto design the springs such that f si (i) as i li.Thus, in case the spatial error between the desired distanceand the actual distance between the vehicles approaches liwhich corresponds to the distance between the vehicles ap-proaching 0, the spring gets infinitely stiff to prevent a crash.Second, contrary to the linear controllers, where the controlaction is proportional to the error, nonlinear controller al-lows a large variety of possibilities to achieve the specifiedbehaviour. For example, nonlinear springs allow to boundthe control input when the position error is large, which isimpossible using linear controllers. Note that in case f si isnot defined for all i, for instance for barrier functions, thestability analysis should ensure that the initial conditionsare chosen such that f si (i(0)) exists and that the trajecto-ries of the system remain within the set of interest.

Assuming linear damping forces of the form Fri =

3

Ri(m1i1 pi1 m1i pi) and linear drag forces of the formFdi = bim1i pi, the overall control force can be described as

F = (B + R)M1 p + ~e1R1v0 + S T f s() (9)

with the constant matrices

R =

R1 + R2 R2 0 0R2 R2 + R3 R3

. . ....

0. . .

. . .. . . 0

.... . . RN1 + RN RN

0 0 RN RN

(10)

and B = diag(b1, . . . ,bN) where the entries of matrices B andR are design parameters of the controller and 0 < Ri,bi < for all i.

We will show that the system is asymptotically stable withrespect to the equilibrium (p,). However, the values ofthe displacements in steady state are undesirable and growwith the string length N in presence of a nonzero referencevelocity v0.

Lemma 3 Consider the string system (4) in closed loopwith the control law (9) and where the initial spatial devi-ation i(0), the equilibrium state i and all values of i inbetween are in the domain of definition of f si (i) for all i.Consider further that f si (i) is a strictly monotonically in-creasing function satisfying f si (0) = 0. Then, the equilibrium(p,) =

(M1v0, f s1

(S TB1v0

))is globally asymptotically

stable in the absence of disturbances, i.e. d = 0.

PROOF. Given (9) and (4) the closed loop has the port-Hamiltonian form

p

=(B + R) S

T

S 0

Hcl(p,) (11)

with the closed-loop Hamiltonian function

Hcl(p,) =12(p M1v0)TM1(p M1v0)

+

Ni=1

ii

f si (w) N

k=ibkv0

dw (12)

whereN

k=i bkv0 is the ith entry of the vector S TB1v0. Notethat the second right hand term in (12) is non-negative.The nonlinear spring functions f si (i) are strictly mono-tonically increasing and satisfy f si (0) = 0. Also note thatf s() = S T B1v0. Adding the term Nk=i bkv0 ensures that

the integrand is positive for i > i which leads to the in-tegral being positive. In case i < i the integrand is neg-ative, which leads to the integral from to being alsopositive. Hence, the integral is non-negative and only equalto zero for i = i .

Using Hcl as a Lyapunov function candidate and computingits time derivatives yields

Hcl(p,) = THcl(p,)(B + R) S TS 0

Hcl(p,)= Tp Hcl(p,)(B + R)pHcl(p,) 0 (13)

This implies Lyapunov stability. Note also that this impliesthat the spatial deviation i remains in the definition set off si (i) if the initial spatial deviation i(0), the equilibriumstate i and all values of i in between are in the domainof definition of f si (i) for all i. Asymptotic stability followsby applying the Invariance principle, which ensures that thetrajectories of the states converge to the largest invariant setS, i.e. limt p(t) = p and limt (t) = such thatHcl(p,) = 0, (see e.g. [7]).

Note that 1 = f s1

1

(Nk=1 bkv0

). Thus, for positive parame-

ters bi the steady state values of the spatial separation be-tween the vehicles is non-zero. Moreover, if a positive lowerbound on the drag and compliance coefficients exists, i. e.mini bi > b > 0 and mini ci > c > 0, the argument of f s11grows with N. Depending on the form of f s1 , might growwithout bound at the beginning of the string. In any case, is not a desired equilibrium point. This effect could beavoided by choosing parameters bi that decrease sufficientlyfast with i. Note that there are examples in the literature,where string stability can be guaranteed if control param-eters grow with the position. (Such as growing Ri in [5].)However, this implies that such a vehicle string might notbe scalable in practice and is therefore undesirable. Anotheroption is to assume that each agent has perfect knowledgeof the reference signal (in its simplest case the referencevelocity as discussed in [2]) and to use this knowledge tocompensate the influence of the drag onto the steady state.However, this requires the communication of any changesin the reference signal to all agents.

4 Integral Action

Studying a bidirectional vehicle string in the previous sec-tion showed that introducing local control based on a mass-spring-damper system can guarantee stability of an equilib-rium point which is not the desired equilibrium. Thus, inte-gral action will be introduced in this section to ensure stringstability of the desired equilibrium. Also, it will be shownthat using suitable integral action allows to reject constant

4

unknown disturbances. The following theorem studies sucha controller with additional integral action.

Theorem 4 Assume the disturbances d include a constantcomponent dc and a dynamical component dd(t) such thatd = dc + dd(t) and there exists a constant D < satisfyingdd()2 D. Consider the string system (4) with a referencesignal with constant velocity v0, disturbance d in closed loopwith a controller obtained by adding the control in Lemma 3and the additional dynamic control force FIA, i. e.

F = (B + R)M1 p + ~e1R1v0 + S T f s() + FIA (14)FIA = ApM1 p + MKS T f s() (B + R + Ap)Kz3 (15)

z3 = S T f s(). (16)

where Ap is a constant diagonal matrix Ap = diag(ap1 , . . . ,apN )with 0 api < for all i, K RNN is a constant diagonalpositive matrix K = diag(k1, . . . ,kN) with 0 < ki < forall i. Then

(1) for dd(t) = 0 the desired equilibrium point

(p,,z3) =(M1v0,0,K1(B + R + Ap)1

(dc

(B + Ap

)1v0

))(17)

is globally asymptotically stable (despite the presenceof constant unknown disturbances dc),

(2) the system is passive with input dd, output y = z1 Hz(z)and storage function Hz(z) and constant disturbancesdc are rejected, and

(3) the system is l2 string stable with respect to the dynamicdisturbances dd(t).

PROOF. (1): Consider the following change of coordinates

z1 =p M1v0 + MK(z3 ), (18)z2 = (19)

with = K1(B + R + Ap)1(dc

(B + Ap

)1v0

). Hence,

z1 =S T f s() RM1 p + ~e1R1v0 BM1 p + d ApM1 p+MKS T f s() (B + R + Ap)Kz3 MKS T f s()

= (B + R + Ap)M1z1 + S T f s() (20)

and

z2 = S M1(p M1v0) = S M1z1 + S K(z3 ).

Thus, the closed loop dynamics have the port Hamiltonianform

z1

z2

z3

=

(B + R + Ap) S T 0

S 0 S0 S T 0

Hz(z) (21)

with the Hamiltonian function

Hz(z1,z2,z3) = 12zT1 M

1z1 +N

i=1

z2i0

f si (w)dw

+12

(z3 )TK(z3 ). (22)

Note that it can be shown in a similar way as below (12) thatHz(z) is anon-negative function. As the spring functions f siare strictly monotonically increasing and satisfy f si (0) = 0the integrand of the second right hand term in (22) is positivefor z2i > 0 and negative for z2i > 0. Hence, the integral isnon-negative and only equal to zero for z2i = 0. It can beshown in a similar way as in the proof of Lemma 3 that thesystem is asymptotically stable by using the Hamiltonianas a Lyapunov function and considering that B + R + Ap ispositive definite.

(2): Note that with constant disturbances dc and additionaldynamic disturbances dd(t) the system description changesto

z1

z2

z3

=

(B + R + Ap) S T 0

S 0 S0 S T 0

Hz(z) +

dd(t)

00

. (23)

with Hz(z) given in (22). Using Hz(z) as a Lyapunov func-tion candidate, taking its derivative and setting y = z1 Hz(z)yields

Hz(z) min(B + R + Ap)|y|2 + yTdd(t). (24)

As B+R+Ap is positive definite, the system is passive, [12].

(3): Extending the argument in (24) leads to

Hz(z) min(B + R + Ap)

2|y|2 + 1

2min(B + R + Ap) |dd(t)|2

min(B + R + Ap)2

y 1

min(B + R + Ap)dd(t)2

12min(B + R + Ap) |dd(t)|

2. (25)

Hence,

Hz(z(t)) Hz(z(0)) + 12min(B + R + Ap) dd()2. (26)

5

Note that since R is positive semidefinite and B is positivedefinite, it can be shown using Gershgorins Theorem thatmin(B + R + Ap) mini(bi + api ). Thus,

Hz(z(t)) (2 min

i(mi)

)1|z1(0)|2 +

Ni=1

z2i (0)0

f si (w)dw

+maxi ki

2|z3(0) |2 +

(2 min

i

(bi + api

))1d()2

(2 min

i(mi)

)1|z1(0)|2 +

Ni=1

Li|z2i (0)|2

+maxi ki

2|z3(0) |2 +

(2 min

i

(bi + api

))1d()2

(27)

where Li is the Lipschitz constant for f si (w) for w [0,z2i(0)].(Since f si is monotonically increasing, it is equivalent torequire that for every w = w there exists a constant c suchthat f si (w) = cw.) Since the mass mi, the drag coefficient biand the integral action control parameters api and ki for eachvehicle are positive, Hz(z) is bounded for all N if |z(0)| andd()2 do not increase with N. Given the terms including z1and z3 in (22) are quadratic and f si is strictly monotonicallyincreasing, an upper bound on Hz(z) implies that the states zare also bounded. Further, note that = z2 and p is a linearcombination of z1 and z3 plus two constant offsets terms.Hence, p and are also bounded if an upper bound for Hz(z)exists. Therefore, the system is string stable.

Note that the integral action introduces more design param-eters in the controller by adding the matrices Ap and K inthe control law.

Remark 5 The procedure to design the integral action fol-lows ideas proposed in [4,13,16]. The change of coordinates(18)-(19) is fundamental for this design. The procedure is asfollows: First, extend the state vector by adding new statesz3, and choose their dynamics to be driven by the displace-ments . This will result in integral action on the displace-ments. Second, choose the change of coordinates (19) topreserve the variables on which the additional integral ac-tion is required. Third, compute the derivative with respectto time of (19), and replace the derivative of the states bytheir state equations from the open loop system and desiredclosed loop. Then solve this equation for z1 to obtain thechange of coordinate (18). Finally, compute the time deriva-tive of (18), replace the derivative of the state by their cor-responding state equations, and solve the resulting equationfor the control law FIA. The block-matrices and functions ofthe closed loop (21) have to be chosen to satisfy the limitedinformation available in each vehicle, to guarantee stringstability and to ensure that the control law does not dependon the unknown disturbance.

N = 10

Velo

city

v i(t)

Time t0 100 200 300 400 500

29.5

30

30.5

31

31.5

32

32.5

33

33.5

34

Figure 1. Velocities vi over time for a string of 10 vehicles fori = 1 (red) ... i = 10 (purple)

N = 10

Dist

ance

i(t

)

Time t0 100 200 300 400 500

4

2

0

2

4

6

8

10

Figure 2. Displacements i over time for a string of 10 vehiclesfor i = 1 (red) ... i = 10 (purple)

5 Example

Ten homogeneous strings of length N = 10, 20, . . . ,100 withlocal control and integral action control (mi = 1, f si (i) =i + 0.12i , bi = 0.1, ri = 20 and Ki = 100 for all i =1,2, . . . ,N) have been simulated. Nonzero initial conditionsfor p and have been used for the first vehicle in the string.Figures 1-4 show the evolution of the velocities and inter-vehicle distances for a string of 10 and 100 vehicles. Itcan be observed that the disturbance is not amplified whentraveling through the longer string. Instead, all deviationsfrom the steady state values remain bounded independentlyof the string size and the position within the string.

In a second set of simulations, ten heterogeneous strings oflength N = 10, 20, . . . ,100 with local control and integralaction control have been simulated. The parameters werechosen randomly in the ranges mi [0.8,1.2], f si (i) = cii+

6

N = 100Ve

loci

tyv i

(t)

Time t0 100 200 300 400 500

29.5

30

30.5

31

31.5

32

32.5

33

33.5

34

Figure 3. Velocities vi over time for a string of 100 vehicles fori = 1 (red) ... i = 100 (purple)

N = 100

Dist

ance

i(t

)

Time t0 100 200 300 400 500

4

2

0

2

4

6

8

10

Figure 4. Displacements i over time for a string of 100 vehiclesfor i = 1 (red) ... i = 100 (purple)

0.12i with ci [0.8,1.2], bi [0.08,0.12], ri [18,22]for all i = 1,2, . . . ,N. The maximal point wise norm of thecomplete state vector z(t), i.e. maxt |z(t)|, for all string sizesis shown for the homogeneous strings in Figure 5 and forthe heterogeneous strings in Figure 6. One can observe thatthe maximal deviation from the steady state value does notgrow with string size but instead settles on a constant valuefor N 40 for the set of homogeneous strings. As thesystem parameters of the heterogeneous strings are chosenrandomly in a range around the nominal value used in thehomogeneous setting, the values of maxt |z(t)| differ and donot settle on a constant value for sufficiently long strings.However, it can be observed in Figure 6 that despite thevariation in maxt |z(t)|, the values are around the same valueas for the set of homogeneous strings and remain boundedindependently of the string size N.

max

t|z(

t)|

N10 20 30 40 50 60 70 80 90 100113.9

114

114.1

114.2

Figure 5. Velocities vi over time for a string of 100 vehicles

max

t|z(

t)|

N10 20 30 40 50 60 70 80 90 100110

112

114

116

118

Figure 6. Displacements i over time for a string of 100 vehicles

6 Conclusions

Modelling bidirectional, heterogeneous vehicle strings asport-Hamiltonian systems offers significant advantages overtraditional system descriptions involving ordinary differen-tial equations. It was shown that this system description notonly incorporates both linear and nonlinear systems, but alsoleads to an easy to follow and straightforward stability andstring stability analysis.

The local control law proposed in this paper consists of vir-tual nonlinear springs and dampers between the vehicles anddrag towards ground. Suitable integral action control wasadded to guarantee string stability. The advantage of this ap-proach is that it only relies on decentralised control and lo-cally measurable data of a set of direct neighbours. No globalcommunication with the leading vehicle or global knowl-edge of the reference signal are necessary and all controlparameters can be chosen in defined bounds independent ofthe string size.

It has been shown in [17] that the common strict form ofstring stability (requiring the l2 of all states to be boundedfor any l2 bounded disturbance) cannot be achieved for sym-metric homogeneous bidirectional strings with tight spacingand two poles in the open loop of each vehicle in the string.Thus, a different more informative definition has been used.The definition proves to be useful to guarantee point wise(in time) bounded states in bidirectional vehicle strings.

So far only nonlinear springs have been investigated in thiswork. A more detailed analysis is necessary to study nonlin-ear dampers, drag or more general nonlinear systems. At thecurrent stage the port-Hamiltonian description based stabil-ity analysis is only suitable to cover symmetric communi-cation settings. If the virtual springs and dampers betweentwo agents are modified to allow for unbalanced forces attheir ends, an extension of the existing approach is needed.

The local data such as the distances and velocity differencestowards neighbouring vehicles are assumed to be available

7

without noise, delay or other real world inaccuracies. A moredetailed analysis is needed to investigate the effects of com-munication or measurement limitations on string stability.

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