Arash Farnam

Alain Sarlette
email: alain.sarlette@inria.fr

Towards a comprehensive impossibility result for string stability

We provide a comprehensive impossibility result towards achieving string stability, i.e. keeping local relative errors in check with local controllers independently of the size of a chain of subsystems. We significantly extend existing results, from the LTI setting to any homogeneous controllers that can be nonlinear, time-varying, and locally communicating. We prove this impossibility for a set of definitions with various norm choices, including the L2-type standard and a BIBO type criterion. All results hold for a general discrete-time controller which should cover most applications.

I. INTRODUCTION

String stability roughly requires that a chain of subsystems (e.g. vehicles), controlled by local feedback from relative position measurements and subject to bounded local perturbations, keeps local deformations bounded independently of the size of the chain. In absence of further variables intervening in the system, local deformations are defined as deviations of the distances between consecutive vehicles from a constant target value. The topic has gathered attention from the observation that, for second-order integrator subsystems, using any LTI controller reacting to their predecessor's distance, the amplification of some disturbance components along the chain is unavoidable [START_REF] Swaroop | String stability of interconnected systems[END_REF], [START_REF] Swaroop | String stability of interconnected systems: An application to platooning in automated highway systems[END_REF]. The historical application is a chain of acceleration-controlled vehicles, but other distributed systems like mechanical structures [START_REF] Yamamoto | Bounded disturbance amplification for mass chains with passive interconnection[END_REF], [START_REF] Yamamoto | Design of passive interconnections in tall buildings subject to earthquake disturbances to suppress inter-storey drifts[END_REF], or more fundamental models, should also benefit from string stability insight. This has initiated a rich line of results, among others:

• Establishing various impossibility results and scalings for LTI controllers [START_REF] Swaroop | String stability of interconnected systems[END_REF], [START_REF] Sheikholeslam | Longitudinal control of a platoon of vehicles[END_REF], [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF], mostly with subsystems reacting to their immediate predecessor and immediate follower. Often PD controllers are used as a proxy for bandwidth limitation, but problems associated to integral control have also been characterized [START_REF] Barooah | Error amplification and disturbance propagation in vehicle strings with decentralized linear control[END_REF]. • Designing a passive LTI controller, with symmetric coupling to predecessor and follower, to keep in check at least in some sense the effect of disturbances acting on the leading subsystem [START_REF] Yamamoto | Bounded disturbance amplification for mass chains with passive interconnection[END_REF]. • Adding a dependence on absolute velocity into the pure double integrator, to overcome the string instability issue [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF]- [START_REF] Knorn | Passivitybased control for multi-vehicle systems subject to string constraints[END_REF]. This can take the form of strong enough drag, which e.g. for terrestrial vehicle applications are easy to ensure but would be in tradeoff with fuel efficiency; in other applications, ensuring strong drag may not always be trivial. Alternatively, the "time-headway" spacing policy makes the target inter-vehicle distance dependent on absolute velocity. For terrestrial vehicles again this appears easy to achieve, but questionable in terms of chain performance (possibly big spacing at high velocities, instead of moving efficiently like a big rigid body at all speeds); for other applications, the necessity to measure absolute velocity of each subsystem with respect to a common reference may even pose sensing questions. • Adding local communication between subsystems, as in adaptive cruise control [START_REF] Ploeg | Controller synthesis for string stability of vehicle platoons[END_REF], [START_REF] Ploeg | Lp String Stability of Cascaded Systems: Application to Vehicle Platooning[END_REF]. Remarkably, existing studies with this approach all incorporate as well a dependence on absolute velocity, as mentioned in the previous item. In the recent paper [START_REF] Besselink | String Stability and a Delay-Based Spacing Policy for Vehicle Platoons Subject to Disturbances[END_REF], a variation on time headway is proposed for a generic controlled system and in the nominal case the controller uses both absolute state information and communication with the leader.

We can further refer the reader to the recent review [START_REF] Stüdli | From vehicular platoons to general networked systems: String stability and related concepts[END_REF].

Research has also been carried out on properties related to, yet different from string stability. Without being comprehensive, we can mention the scaling of linear network eigenvalues with network size [START_REF] Barooah | Mistuning-based control design to improve closed-loop stability of vehicular platoons[END_REF], the poor robustness of large linear networks under distributed sensing [START_REF] Sarlette | Control limitations from distributed sensing: theory and Extremely Large Telescope application[END_REF], and the coherence of linear networks in presence of stochastic noise [START_REF] Bamieh | Coherence in Large-Scale Networks: Dimension Dependent Limitations of Local Feedback[END_REF]. This literature leaves several questions open. How much does each added element (dependence on absolute velocity, local communication capabilities, coupling symmetrically or asymmetrically to one or several vehicles ahead and/or behind) actually contribute towards solving the string stability issue? How much can be gained with more involved, e.g. nonlinear controller designs? What is possible with digital, quantized controllers and realistic modern communication? How much is the LTI setting actually limiting?

The objective of the present note is to answer these questions towards a more precise understanding of string stability and related issues. As a small variation, we consider a discrete-time controller, which is closer to digital controller implementations and incorporates related "natural" constraints in a direct way; although probably most current controllers are digital and discrete-time anyways, the conclusions should carry over in practice to the continuous-time setting as we discuss below. We work only with relative state measurements between neighboring subsystems, e.g. we do not allow controller dependence on absolute velocity. In accordance with the existing literature, we consider an idealized model where input disturbances must be countered while assuming perfect measurements and communication. Our main result is:

We establish that enabling nonlinear controllers, any couplings to a few vehicles in front and behind, any (e.g. nonlinear, quantized, event-based) local communication, and controller dependence on the chain length, all together do not allow to design a controller which would achieve string stability with respect to bounded disturbances acting on the subsystems of the chain. We prove this for several variants of the string stability measure, and with as only main constraints: (i) the controller uses relative state measurements between neighboring vehicles; (ii) the controller is homogeneous, i.e. each vehicle (except boundaries) reacts in the same way to its neighbors; (iii) controller discretization step ∆t remains bounded away from zero despite increasing chain length. The proof comes down to identifying a general counterexample, then working out rather basic computations; we hence believe that the contribution mainly rests on the unprecedented generality of the conclusions. Essentially: string instability in a chain of second-order integrators is an unavoidable property of distributed sensing, for a (much) larger class of controllers than LTI. The note is organized as follows. Section II describes the setting and several precise definitions of string stability. Section III gives our main result, about a (very) general impossibility of string stability; it ends with a short discussion of discrete-time vs. continuous-time controllers, and an illustrative simulation of the system behavior under a "bad" disturbance. Section IV concludes the note with an outlook on remaining open points and possible further implications.

II. PROBLEM DEFINITION

A. Model setting

We consider a chain of undamped second-order integrators, called "vehicles" for concreteness but which may represent any other (typically mechanical) subsystems coupled with local interactions. In the literature, this continuous-time system

ẍk (t) = u k (t)
with x k the position and u k the input signal of subsystem k at time t, is the standard starting point for string (in)stability [START_REF] Swaroop | String stability of interconnected systems[END_REF]- [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF], [START_REF] Stüdli | From vehicular platoons to general networked systems: String stability and related concepts[END_REF], with continuous-time e.g. LTI controllers defining the u k . We here consider its (exact) equivalent under discrete-time control. We assume that each subsystem is controlled with a discrete-time control logic, computing at each time t = n ∆t with n ∈ Z, a control signal that will be applied during the whole interval (t, t + ∆t] as input to each double integrator; thus ∆t is the time increment of the discrete-time controller. We can then integrate exactly the second-order dynamics over the time interval (t, t + ∆t] to obtain: A key constraint is that the feedback signals u k,1 , u k,2 applied between t and t + ∆t must be based on only relative displacement measurements e k = x k-1 -x k between consecutive vehicles in the chain, including possibly relative speeds ėk = v k-1 -v k , taken at time t. We then consider the following general controller, where e k: denotes for > k the set of values e k , e k+1 , ..., e :

v k (t + ∆t) = v k (t) + u k,1 (t) + d k,1 (t) (1) x k (t + ∆t) = x k (t) + v k (t) ∆t + u k,2 (t) + d k,2 (t) , where x k , v k ∈ R for k = 0,
u k,1 = f 1 (e (k-m1):(k+m2) , ė(k-m1):(k+m2) , (2)
c k,+ , c k,-, ξ k , N, t) , u k,2 = f 2 (e (k-m1):(k+m2) , ė(k-m1):(k+m2) , c k,+ , c k,-, ξ k , N, t) , c k+1,+ = g 1 (e (k-m1):(k+m2) , ė(k-m1):(k+m2) , c k,+ , c k,-, ξ k , N, t) , c k-1,-= g 2 (e (k-m1):(k+m2) , ė(k-m1):(k+m2) , c k,+ , c k,-, ξ k , N, t) , ξ k (t + ∆t) = h(ξ k (t), e (k-m1):(k+m2) , ė(k-m1):(k+m2) , c k,+ , c k,-, N, t) .
Here m 1 (resp. m 2 ) is a finite number of agents ahead (resp. behind); c k,+ , c k,-∈ R nc with n c some bounded integer are communication signals from {k -m 1 , ..., k -1} to k and from {k + 1, ..., k + m 2 } to k respectively; the ξ k ∈ R n ξ for some finite integer n ξ allow for dynamical controllers with finite memory; and f 1 , f 2 , g 1 , g 2 , h are arbitrary functions, with minimal regularity just to ensure that the solution to the dynamical system is well-defined at all times. In particular, by specifying a particular profile of u(τ ) over the discretization time span τ ∈ [t, t + ∆t) between two measurement updates, one can command f 1 = t+∆t t u(τ )dτ and f 2 = t+∆t t u(τ )dτ 2 independently. The controllers [START_REF] Swaroop | String stability of interconnected systems: An application to platooning in automated highway systems[END_REF] are applied by all vehicles k ∈ (m 1 , N -m 2 ), whereas adapted versions are applied by the m 1 leading and m 2 last vehicles. The adapted versions will play no role in the proof.

Note that m 1 and m 2 are just opportunities for a vehicle to take into account what happens several vehicles ahead and/or behind. The setting includes unidirectional chains m 2 = 0, where vehicles only get inputs from predecessors. It also includes the bidirectional symmetric chain, like encountered in homogeneous mechanical coupling [START_REF] Yamamoto | Bounded disturbance amplification for mass chains with passive interconnection[END_REF], [START_REF] Yamamoto | Design of passive interconnections in tall buildings subject to earthquake disturbances to suppress inter-storey drifts[END_REF], by imposing front-to-back symmetry on [START_REF] Swaroop | String stability of interconnected systems: An application to platooning in automated highway systems[END_REF].

Our aim here is to leave as much freedom as possible on controller design. In particular, the functions f 1 , f 2 , g 1 , g 2 , h can be nonlinear and time-dependent (e.g. modulated at specific frequencies), thereby vastly extending the traditional LTI setting. Our only true restrictions on control design are:

Assumptions implied by the model:

• The controllers must be homogeneous along the chain, i.e. the functions f 1 , f 2 , g 1 , g 2 , h do not depend on vehicle index k and the internal variables are initialized with the same default values for each k. • The digital controller has a finite update time ∆t, fixed independently of chain length N .

• The control commands are based on relative state information only, i.e. there is no dependence on absolute states that would be obtained with respect to some common reference, like absolute velocity. The first point can be relaxed to approximately homogeneous in our proof; future work, if deemed relevant, could address the fully heterogeneous setting e.g. with adversarial methods. The second point is discussed after the main result. Regarding the third point, we recall that the case with control using absolute velocity has been solved [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF], [START_REF] Rogge | Vehicle Platoons Through Ring Coupling[END_REF]. It is not clear however whether relying solely on absolute velocity for achieving string stability is acceptable in all practical contexts. Without any absolute information is also the original academic question.

B. Control objective

The goal is to achieve string stability. Roughly said, this means avoiding that the performance gets unboundedly worse when the chain length N grows. Formally, several definitions have been proposed [START_REF] Swaroop | String stability of interconnected systems[END_REF]- [START_REF] Yamamoto | Bounded disturbance amplification for mass chains with passive interconnection[END_REF], [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF], [START_REF] Barooah | Error amplification and disturbance propagation in vehicle strings with decentralized linear control[END_REF], [START_REF] Ploeg | Lp String Stability of Cascaded Systems: Application to Vehicle Platooning[END_REF], [START_REF] Stüdli | From vehicular platoons to general networked systems: String stability and related concepts[END_REF]. Their common point is to focus on stabilizing the e k , i.e. the distances between consecutive vehicles. This makes sense e.g. for collision-avoidance, for maintaining a tight platoon, or to avoid too extreme accelerations on the last vehicles of the chain, and is a weaker requirement than controlling the position of every vehicle with respect to the leader. Definitions vary in the norms used to consider the distribution over vehicles, the norm over time of the signals, and the type of disturbance namely as signals or as initial conditions [START_REF] Ploeg | Lp String Stability of Cascaded Systems: Application to Vehicle Platooning[END_REF], [START_REF] Stüdli | From vehicular platoons to general networked systems: String stability and related concepts[END_REF]. We here translate the definitions in presence of input disturbances to discrete-time control.

Definition 1: For positive integers p, q, the p,q string stability requires that there exist C 1 , C 2 > 0 independent of N such that: for any disturbances satisfying

s=1,2 N k=0 n∈Z |d k,s (n ∆t) 1 ∆t s | p ∆t q/p < C 1 ,
it is ensured that n∈Z |e j (n ∆t)| p ∆t < C 2 for each j. Definition 2: For positive integers p, q, the ( p , q ) string stability requires that there exist C 1 , C 2 > 0 independent of N such that: for any disturbances satisfying

s=1,2 N k=0 n∈Z |d k,s (n ∆t) 1 ∆t s | p ∆t q/p < C 1 , it is ensured that N k=0 n∈Z |e k (n ∆t)| p ∆t q/p < C 2 .
Definition 3: For positive integer p, the ( p , ∞ ) string stability requires that there exist C 1 , C 2 > 0 independent of N such that: for any disturbances satisfying

n∈Z |d k,s (n ∆t) 1 ∆t s | p ∆t < C 1 , s = 1, 2 for all k , it is ensured that n∈Z |e j (n ∆t)| p ∆t < C 2 for all j.
Definition 4: The ( ∞ , ∞ ) string stability requires that there exist C 1 , C 2 > 0 independent of N such that: for any disturbances satisfying |d k,s (t)|/∆t s < C 1 for s = 1, 2 and all k, t, it is ensured that |e j (t)| < C 2 for all j, t.

The ∆t factors are introduced because d k,1 and d k,2 are supposed to result from integrating (respectively once and twice) a continuous-time signal d k (t) over the time interval (t, t + ∆t]. In principle we could just choose units such that ∆t = 1, but we keep ∆t for later discussion. The main point about string stability is to satisfy the constraints with constants independent of the number of vehicles N .

Definition 1 is the weakest since it imposes a bound on the vector norm of disturbance inputs, but in return it only requires each individual e j to have a bounded signal norm. It is a necessary condition for achieving stronger versions of string stability, and it has been considered a lot in the literature; e.g. until recently this was the only proven working definition for a controller depending on absolute velocity, as we exclude here [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF]- [START_REF] Knorn | Passivitybased control for multi-vehicle systems subject to string constraints[END_REF]. The other definitions each consider the same norm on input disturbances and on output errors, either summing them over vehicles (Def.2) or not (Def.3, Def.4). The most popular norm has been p = q = 2, especially the time-integration with a 2-norm has attracted a lot of attention thanks to its equivalent formulation in frequency domain. Definition 4 changes the treatment of time to formulate a BIBO type version of string stability. It seems to have attracted little attention in the literature, probably because most of the literature has focused on LTI frequency domain approaches. However, we would argue that this version is closest to practical concerns; and, as we will show that the string instability issue is not limited to the LTI setting, we will use tools that can treat this definition explicitly too. In the particular context of ( 1),( 2), or when restricting the controllers, it might well be that some of the above definitions hold strictly together; in absence of further evidence, we will treat them all.

Remark 1: The definitions were initially stated in the linear context, where C 1 and C 2 can be rescaled such that it makes no difference in which order they are chosen (e.g. variants like "for each C 1 , there exists a C 2 " become equivalent to our statement). In the nonlinear context this might differ, and we have chosen the weaker constraint: thus our impossibility results will also hold for stronger variants.

Remark 2:

The disturbance model appears quite standard: each subsystem is subject to some disturbance input, bounded in some sense, and the consequences on the system must be bounded. A possible point of discussion though is that this standard definition of string stability checks system behavior under the worst disturbance satisfying the constraints. If the worst disturbance can get rejected, then we have a strong guarantee, and this is the typical approach for studying system stability. However, if the behavior under the worst disturbance does not behave well as N gets large, it might also be that this worst disturbance becomes increasingly unlikely among all possible combinations of disturbance signals. In other words, in a probabilistic model of all possible input disturbance signals, it may be that the probability to have a problematic input signal converges to 0 as N increases to infinity. More positive results have indeed been obtained with such a probabilistic model, although not for a chain of subsytems, see e.g. [START_REF] Bamieh | Coherence in Large-Scale Networks: Dimension Dependent Limitations of Local Feedback[END_REF]. We here stick to the standard definition, thus considering the worst bounded input signal.

III. MAIN IMPOSSIBILITY RESULT

We now prove that Definitions 1-4 are all impossible to satisfy even with a general controller as allowed by [START_REF] Swaroop | String stability of interconnected systems: An application to platooning in automated highway systems[END_REF]. The main idea of the proof is to construct a disturbance input which is badly countered by any distributed controller. We take advantage of a simple construction that focuses on exactly solving the central part of the chain only, in order to give a lower bound on the induced error.

A simulation illustrating the behavior of the system under this construction can be found in Section III-D.

A. A badly countered disturbance situation

Consider disturbances of the following form:

d k,1 (t) = d k,2 (t) = 0 for all t < 0, k = 0, 1, ..., N ; d k,1 (t) = αk∆t N , d k,2 (t) = αk∆t 2 N
for all t = 0, ∆t, ..., T , k = 0, 1, ..., N ;

(3)

d k,1 (t) = d k,2 (t) = 0 for all t > T, k = 0, 1, ..., N ,
with constants α > 0 and T > 0 to be specified later.

To compute the evolution of the system under these disturbances, we exploit the finite propagation speed of signals along the chain -namely at most (m 1 , m 2 ) vehicles per time step -to restrict our attention to a central subset of vehicles.

• Consider the evolution of e k and ėk over one time step, when the N + 1 vehicles all start with the same state x k (0) = v k (0) = 0 for all k, and with controllers initialized at ξ k = c k,+ = c k,-= 0 for all k. We get

e k (∆t) = e k (0) + ∆t ėk (0) + u k-1,2 (0) -u k,2 (0) +d k-1,2 (0) -d k,2 (0) = u k-1,2 (0) -u k,2 (0) + α∆t 2 /N ; ėk (∆t) = ėk (0) + u k-1,1 (0) -u k,1 (0) +d k-1,1 (0) -d k,1 (0) = u k-1,1 (0) -u k,1 (0) + α∆t/N .
Since the e k (0) and ėk (0) are all equal, it is clear that the control inputs are all equal too, i.e. u k-1,1 = u k,1 and u k-1,2 = u k,2 , at least for all vehicles with m 1 < k < N -m 2 .

Completely irrespectively of the controller chosen, we thus have for all m 1 < k < N -m 2 : e k (∆t) = α∆t 2 /N and ėk (∆t) = α∆t/N . Also the c k,+ (∆t), c k,-(∆t) and ξ k (∆t) will be equal.

• Now consider a time t = n∆t for some integer n > 0 and assume that all the state variables satisfy equalities e k (t) = e j (t), ėk (t) = ėj (t), c k,+ (t) = c j,+ (t), c k,-(t) = c j,-(t) and ξ k (t) = ξ j (t) for all j, k ∈ [N lead , N -N tail ], for some integers N lead , N tail > 0. Slightly extending the above example, we get:

e k (t + ∆t) = e k (t) + ∆t ėk (t) + u k-1,2 (t) -u k,2 (t) (4) + d k-1,2 (t) -d k,2 (t) = e k (t) + ∆t ėk (t) + α∆t 2 /N = e j (t + ∆t) for all j, k ∈ [N lead + m 1 , N -(N tail + m 2 )] ; ėk (t + ∆t) = ėk (t) + α∆t/N = ėj (t + ∆t) for all j, k ∈ [N lead + m 1 , N -(N tail + m 2 )] ,
and similarly we maintain c k,+ (t) = c j,+ (t), c k,-(t) = c j,-(t) and ξ k (t) = ξ j (t) for that subset of vehicles.

By iterating this argument we get the following property.

Lemma 1: Consider the system (1),(2) subject to the particular disturbance (3) and zero initial conditions. Then for any (well-defined) controller, the solution satisfies:

e k (t) = t(t + ∆t) α / (2N )
(5) ėk (t) = t α/N , for all t ∈ [0, T ] and all k ∈ ( t ∆t m 1 , N -t ∆t m 2 ). Proof: The main argument is provided by the explanations preceding the statement. From (4), the ėk is obtained as a sum of t/∆t times the bias α∆t/N . Then replacing this into the expression of e k in (4), one observes that the increment of e k at time n = t/∆t is linear in n, so the standard formula for a linearly progressing series gives the result.

To be useful at time t, the solution (5) of Lemma 1 should cover at least 1 vehicle, i.e. N -t ∆t (m 1 + m 2 ) ≥ 1. For fixed m 1 , m 2 and ∆t, we can ensure to have a valid solution for at least N/2 vehicles over the interval [0, T ], by taking T = N ∆t 2(m1+m2) . As m 1 and m 2 are constants independent of N , we essentially suggest to select the duration T of the "bad" input disturbance to be of order N ∆t.

B. Consequences for string stability

We now investigate what the above construction implies for string stability. First take Definition 1. For the proposed disturbance, the condition on d k,1 , d k,2 becomes: 2T q/p ( α N ) q N k=0 k q < C 1 . For large N , the dominating term in the sum is N q+1 such that we need in fact 2T q/p α q N < C 1 or in other words, α of order 1 / (N 1/q T 1/p ). This fixes the allowed disturbance amplitude as a function of its duration and of N .

For the vehicles covered by Lemma 1, we then have

n∈Z |e j (n ∆t)| p ∆t ≥ T /∆t n=0 |e j (n ∆t)| p ∆t ( α 2N ) p T /∆t n=0 (n∆t) 2p ∆t ( α 2N ) p ∆t 2p+1 ( T ∆t ) 2p+1
. Towards the last line we have again used the dominating term in the sum. Combining this with the just obtained bound on α and with taking T of order N ∆t as suggested at the end of Section III-A, we obtain that n∈Z |e j (n ∆t)| p is at least of order N p-p/q ∆t 2p .

A similar argument can be repeated for the other definitions of Section II, yielding the following results.

Theorem 2: For the system (1),(2), there exist disturbances d k,1 and d k,2 satisfying the required respective bounds according to the definitions of Section II and such that, irrespectively of any (well-defined) controller choice, for large N :

[Definition 1]: n |e j (n ∆t)| p grows as N p-p/q ∆t 2p ;

[Definition 2]: N k=0 ( n |e k (n ∆t)| p )
q/p grows as N q ∆t 2q ;

[Definition 3]: n |e j (n dt)| p grows as N p ∆t 2p ;

[Definition 4]: |e j (t)| grows as N ∆t 2 .

Proof: We will always assume T of order N ∆t and consider the output errors for t ∈ [0, T ]. The computation for Def.1 is given above. For Def.2 it is the same, but taking the power q/p and summing the disturbance over the number of vehicles for which Lemma 1 is valid -this can be of order N as mentioned in the last paragraph of Section III-A. For Def.3, the disturbance can be larger i.e. α of order 1/T 1/p , and with respect to the Def.1 computation this adds a factor N p/q to the output error. For Def.4, we can have α of order 1, and since the result of Lemma 1 is valid for T of order N ∆t we can have e k (t) of order T (T + 1) α/N ∼ N 2 ∆t 2 /N .

For ∆t fixed and N growing to infinity, this result establishes impossibility to satisfy any of the definitions of string stability given in Section II, except possibly Def.1 with q = 1 (which does not appear to have any practical significance, see comments about the Definitions in Section II). This impossibility is established in a very general setting, allowing unidirectional or bidirectional symmetric or asymmetric coupling, looking a number of vehicles ahead and behind (as long as that number is independent of N ), communicating with neighbors with any encoding/decoding schemes with possibly packets and event-based logic, and processing all this in an arbitrary nonlinear control system with memory. In particular, even perfect local communication among the vehicles is not sufficient, on its own, to ensure string stability. It is thus no wonder that vehicle chain controllers with realistic communication channels have so far required an additional feedback from absolute velocity to achieve string stability [START_REF] Ploeg | Controller synthesis for string stability of vehicle platoons[END_REF]- [START_REF] Besselink | String Stability and a Delay-Based Spacing Policy for Vehicle Platoons Subject to Disturbances[END_REF].

C. How telling is the discrete-time controller setting?

The reader will have noticed that the above impossibility breaks down if we let ∆t converge to zero fast enough as N grows to infinity. While this does not look like a practical solution, it may express a relevant tradeoff; and, it does create a gap with the pure continuous-time literature. We will thus briefly comment on the comparison of this result with the literature on continuous-time, typically LTI systems.

First, note that since our result only follows a sufficient construction, Theorem 2 proves that it is necessary -yet possibly not even sufficient -to let ∆t go to zero with increasing N in order to satisfy string stability. And indeed, for e.g. PD coupling with nearest neighbors, we know from LTI results that our disturbance would yield string instability in continuous-time too. This may not be too surprising as a lowfrequency disturbance appears to cause most of the problem. Our argument leading to Thm.2 thus indeed appears to be quite too optimistic still. Now let us try to discuss which controller features would typically go with a very small ∆t:

• One obvious effect of smaller ∆t is faster communication across the vehicle chain. If one could communicate arbitrarily fast, perfectly and without measurement errors, then each vehicle k could get very fast knowledge of e 1 + e 2 + ... + e k = x k -x 0 . One can then obviously achieve string stability: just control each x k -x 0 independently to stabilize each vehicle with respect to the leader. The "distributed system" setting and chain size N play no role anymore. Of course this idealized situation is unrealistic. In reality, precision of a message (and in fact of a measurement) is in a clear tradeoff to update speed. As long as the communication bandwidth per signal remains bounded when N increases, the imperfections resulting from smaller ∆t are likely to counterbalance the apparent benefits of smaller ∆t from our perfectcommunication model. • Setting sensing and communication aside, in practice, the controller's discretization step ∆t is chosen as the desired dwell-time before vehicle k reacts to a measurement; thus in practice ∆t converging to zero would mean, controller bandwidth tending to infinity, pointing towards controllers with gain increasing as a function of N . It is known indeed that academically speaking, this can provide string stability: in continuous-time, without communication, a LTI controller whose gain increases fast enough with N , can ensure string stability. However, as the control gain keeps increasing towards infinity, effects of unmodeled system limitations and imperfections cannot be neglected forever and practical problems are likely to appear. • Theorem 2 thus shows anyways that string stability is, at best, not robust to time-discretization. This is important to know towards system simulations, where situations that work only for infinitesimal ∆t are quickly considered non-robust for all practical purposes. In a sense, testing robustness to finite ∆t can even be mathematically compared to the traditional requirement of "no poles cancellation" in the continuous-time setting. Indeed, allowing a decreasingly small ∆t without any measurement noises can be compared to allowing precise computation of lim ∆t→0 s(t+∆t)-s(t) ∆t for a signal s, i.e. evaluating pure derivatives. For a double-integrator, this implies the possibility of pole cancellation at zero frequency, which is almost always excluded.

• The dependence on ∆t is rooted in the fact that we analyze the system before the signals from the edges of the chain reach all the vehicles and make a detailed analysis harder. This does not mean of course that the vehicle chain would automatically be stabilized as soon as the signals from the edges have crossed the chain, see e.g. the simulations in Section III-D. In this sense, it appears that the boundary controllers would play a key role towards string stabilizing the system with infinitesimal ∆t, similarly to PDE control.

These arguments give strong indications to conjecture that string stability would be impossible with any "reasonable" homogeneous, possibly nonlinear, and communicating controllers, in continuous-time too. At this point of detail, we might argue as well that the digital-controller model is in fact closer to applications, than the traditional continuous-time one.

Remark 3: To further connect this result to existing work, we can look at how the chain reacts to disturbances acting on the first subsystem only. This has indeed been considered in several continuous-time LTI studies, which we first review now. For unidirectional coupling, reaction to leader-disturbances is sometimes viewed as a major indicator of general behavior [START_REF] Swaroop | String stability of interconnected systems[END_REF], [START_REF] Swaroop | String stability of interconnected systems: An application to platooning in automated highway systems[END_REF]. For bidirectional chains under symmetric coupling, although impossibility results are known from e.g. [START_REF] Barooah | Error amplification and disturbance propagation in vehicle strings with decentralized linear control[END_REF], under the condition of disturbance restricted to the leading vehicle, 2,2 string stability has been established in [START_REF] Yamamoto | Bounded disturbance amplification for mass chains with passive interconnection[END_REF]. Pushing further the idea of [START_REF] Barooah | Mistuning-based control design to improve closed-loop stability of vehicular platoons[END_REF] and of [START_REF] Herman | Disturbance scaling in bidirectional vehicle platoons with different asymmetry in position and velocity coupling[END_REF], [START_REF] Herman | Nonzero bound on Fiedler eigenvalue causes exponential growth of H-infinity norm of vehicular platoon[END_REF] about possible advantages of slight mistuning in the controller symmetry, we have proved in [START_REF] Farnam | String Stability towards Leader thanks to Asymmetric Bidirectional Controller[END_REF] that a sufficiently asymmetric bidirectional PD controller is sufficient to ensure also the stronger versions of string stability, with respect to disturbances restricted to a fixed number of leading vehicles. The proof uses an analytic almost-inversion of the system equations based on forward and backward flows, loosely inspired from [START_REF] O'connor | Wave-based analysis and control of lump-modeled flexible robots[END_REF].

This line of work can be related in two ways to the result of the present paper. First, we have checked that a discrete-time version of our string stability result in [START_REF] O'connor | Wave-based analysis and control of lump-modeled flexible robots[END_REF] can be worked out perfectly well for the model ( 1), [START_REF] Swaroop | String stability of interconnected systems: An application to platooning in automated highway systems[END_REF], see [START_REF] Farnam | Towards impossibility and possibility results for string stability of platoon of vehicles[END_REF]. This suggests that our discrete-time model does enable positive results when the continuous-time model does, i.e. it adds evidence in favor of (1),( 2) not being essentially more constraining than the more standard continuous-time approach. Second, the disturbance proposed in Section III-A is extensively distributed along the vehicle chain, as a function of N . This is consistent with a picture of two regimes: when disturbances act on a few vehicles (at known places!), it may be possible to reject them in a string stable way; however when they are distributed along the whole chain, there is no way to achieve string stability on the basis of relative measurements only.

D. Illustrative simulation

We can of course only illustrate the string instability for a particular choice of controller. However, trusting in the simple analysis of Section III-A, our main argument is independent of controller choice. Thus, we will just show how indeed the solution of Lemma 1 appears for a simple linear controller without communication. We choose this simplicity to avoid selecting too many elements in the controller design "arbitrarily"since according to Theorem 2, any attempt is anyways doomed to fail. Complementarily the simulation shows what happens once the "boundary effects" have propagated throughout the chain i.e. when the solution of Lemma 1 is not valid anymore; this will depend on the choice of controller, but it falls outside the scope of the present paper. We thus suggest that the reader should not draw too strong conclusions from what happens outside the scope of Lemma 1 with this particular controller.

We take a hint from [START_REF] Herman | Disturbance scaling in bidirectional vehicle platoons with different asymmetry in position and velocity coupling[END_REF], [START_REF] Herman | Nonzero bound on Fiedler eigenvalue causes exponential growth of H-infinity norm of vehicular platoon[END_REF] and select a PD controller having bidirectional coupling, with gain on position feedback symmetric towards the preceding and following vehicle, but gain on velocity asymmetric. Considering a simple sampleand-hold digital actuation, we will thus assume that u k (τ ) =

f (t, k) := b 1 (v k-1 (t) -v k (t)) + b 2 (v k+1 (t) -v k (t)) +a(x k-1 (t) -x k (t)) + a(x k+1 (t) -x k (t))
for all τ ∈ (t, t + ∆t] in continuous-time. After exact integration this yields the exact discrete-time model [START_REF] Swaroop | String stability of interconnected systems[END_REF] with

u k,1 (t) = f (t, k) ∆t , u k,2 (t) = f (t, k) ∆t 2 /2 .
The simulation takes arbitrary values a = 1, b 1 = 2, b 2 = 0.5, and ∆t = 0.1; for the first and last vehicle, we just drop from the feedback law the term associated to the missing neighbor.

The precise scaling of input disturbances to apply and of output signals to monitor, depends on the definition of string stability that one wishes to consider. We will illustrate the BIBO type scaling of Definition 4, with α = 1 independent of N . As the illustrated controller is linear, it is just a matter of scaling to translate the simulation to other definitions.

To better illustrate the effect of N only, we will make a small variation on the applied disturbance. Indeed, since our analysis in Lemma 1 only goes up to time t = T , for a causal system, it does not matter which disturbance we apply for t > T . Therefore, we take an input disturbance of the form (3), but instead of stopping it at t = T which scales with N , we apply this same nonzero input for all t > 0, i.e. we are examining a sort of step response. In this way, the input disturbance acting on vehicle say k = N/2 becomes independent of N , as illustrated on Fig. 1. The impossibility to achieve BIBO string stability via Lemma 1 will be visible as the fact that, when N grows, the maximum deviation |e N/2 (T )| at time T = N ∆t/5 grows unboundedly.

On Figure 2, we show the simulated spacing error between consecutive vehicles under this model, up to t = T and for two different chain lengths N = 10 and N = 50. The black squares are the solution given by Lemma 1. One indeed observes that many (central) vehicles follow this solution, while others are progressively affected by the boundary effects and behave differently. Note that as N increases, in accordance with Lemma 1, the error at a given time becomes lower; this is due to the consecutive vehicles' disturbance inputs becoming more similar as N increases. However, in return, the solution of Lemma 1 remains valid for a longer time T and therefore overall the error at time T , that we can easily compute, keeps increasing unboundedly with N .

We can also have a look at the behavior of the chain for t > T , see Figure 3. A striking observation is that the errors keep increasing way beyond the point covered by our analysis (black dot very close to the origin). However, as this depends on the chosen controller we must be careful about further conclusions. The main conclusion might thus just be that for a fixed N , the chosen controller indeed stabilizes the errors to bounded values, i.e. it does effectively stabilize the system. Only, it does not so uniformly in N , and this is what string instability essentially means.

IV. CONCLUDING DISCUSSION This paper significantly extends the scope of an impossibility result regarding string stability towards input disturbances For this time interval, a significant number of e k (t) is supposed to follow the solution described by Lemma 1; the latter is plotted as black dots which indeed superimpose with a number of simulated curves. (marked with a black dot), the result of Lemma 1 no longer holds and the system behavior does depend on the particular controller choice. We observe that, with the chosen controller: (i) for fixed N indeed each error eventually stabilizes to a bounded value; and (ii) the analysis of Lemma 1, i.e. with errors taken into account only up to time t = T , is very optimistic.

acting on all subsystems. Indeed, while existing results have centered on LTI systems, we here allow controllers to be nonlinear, N -dependent, time-varying, thus possibly modulated and digitally quantized -as well as using any type of perfect local communication at finite speed. The analysis involves no complicated elements once the setting and example are identified, but as the search for alternative controllers had remained open so far, it appears to give a definite answer clearly narrowing down the options towards achieving string stability. Essential features for the impossibility are:

• Second-order integrator model for individual subsystems: if the dynamics was first-order, our counterexample would not work; • Relative measurements: variations that do solve string stability by adding an absolute velocity term are known, see e.g. time-headway policies [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF]- [START_REF] Knorn | Passivitybased control for multi-vehicle systems subject to string constraints[END_REF]. With respect to this criterion, academically, string instability appears more than ever as a property of distributed sensing. In practice, using absolute velocity in the feedback controller or damping becomes a question of hardware and application tradeoff. We must mention that the string instability issue is not directly linked to the low observability for long-range modes in distributed systems with relative measurements [START_REF] Sarlette | Control limitations from distributed sensing: theory and Extremely Large Telescope application[END_REF]. Indeed, here the target variables are not the absolute displacements x k , for which indeed there would be an observability issue, but rather the relative displacements e k , which are directly measured. Also see the previous point. Also note that we have only identified one particular, badly rejected disturbance input. There may be disturbance inputs for which the system reaction is even worse.

With this we believe to have given at least a much more comprehensive picture of what can be done on the standard academic property of string stability. If this string stability property appears critical in some key applications, those results should help guide a possible search for very particular controllers to achieve it, if it is feasible at all. A point that we did not study is string stability with respect to disturbances on the initial state, instead of on input signals; a similar analysis might be possible.

A different option for the future is to acknowledge that the academic definition of string stability is too strong to be useful, thus even in an extended framework with nonlinear controllers and so on. In that sense, we can think of two reasonable variations on string stability.

• One option is to consider the tradeoff in a more integrated picture for finite N : to have a given acceptable error, what are the best possible combinations of absolute-velocitydependence h, control+communication bandwidths 1/∆t, possibly nonlinear effects, and associated gains in presence of other noises, as a function of chain length N ? This, knowing that the limit for infinite N will not work, but will also not be essential for most applications. • Another approach would be, as already mentioned in Remark 2, to acknowledge that the worst-case formulation of string stability is too strong: as the worst-case disturbance could become more and more unlikely with increasing N , it might be more telling to take the limit N → ∞ with a probability distribution over disturbances. In [START_REF] Bamieh | Coherence in Large-Scale Networks: Dimension Dependent Limitations of Local Feedback[END_REF], precisely this approach is taken for the behavior of a lattice of simple linearly coupled systems.

As a final word, we may reflect on the more profound implications of our impossibility result. The investigation of [START_REF] Bamieh | Coherence in Large-Scale Networks: Dimension Dependent Limitations of Local Feedback[END_REF] for instance is motivated by the stability of physical matter, which after all appears to be governed by forces depending on relative states. Implications are also expected for the numerical simulation of related PDEs. It may be an important theoretical aim to pin down what essential element in the system structure leads to this impossibility.

A first point in this direction is that the analysis towards our Theorem 2 can be easily extended to other spatial interconnection structures, e.g. a D-dimensional lattice of N possibly nonlinear systems:

• We can keep our counterexample with d k increasing along one dimension of the lattice from 0 to α with steps α N 1/D , and constant along the other dimensions. • Computing the acceptable T and α for each case, we get the relevant error growing like N β where β ≥ 0 depends on the lattice dimension and on the choice of definition, but it is always > 0 for Definitions 2-4. Compared to [START_REF] Bamieh | Coherence in Large-Scale Networks: Dimension Dependent Limitations of Local Feedback[END_REF], we thus generalize the setting by allowing any nonlinear, time-varying local interactions towards improving the situation, but we obtain a more negative result by considering the worst bounded disturbance, over time and over subsystem indices. If the aim is to break a system into parts, this particular disturbance may be useful insight. In contrast, to understand the stability of lattices in a natural environment, one may have to acknowledge that bad disturbances in fact become negligibly probable with increasing N . This sets a maybe unexpected link between distributed systems and error correcting codes, where scaling to larger codes must essentially rely on the increasing unlikelihood of uncorrectable errors [START_REF] Macwilliams | The theory of error-correcting codes[END_REF].

Fig. 1 .Fig. 2 .

 12 Fig. 1. Disturbance inputs applied to the vehicles, according to (3) but without limiting the time window to t < T (see main text), for N = 10 and for N = 50 vehicles respectively. The figure is showing d k,1 (t) as the corresponding d k,2 (t) are just the same multiplied by ∆t.

Fig. 3 .

 3 Fig.3. Evolution of the e k (t) for larger times t, i.e. the plots of Figure2are in fact zooms on the beginning of the present plots. For O(t) > T (marked with a black dot), the result of Lemma 1 no longer holds and the system behavior does depend on the particular controller choice. We observe that, with the chosen controller: (i) for fixed N indeed each error eventually stabilizes to a bounded value; and (ii) the analysis of Lemma 1, i.e. with errors taken into account only up to time t = T , is very optimistic.

  1, 2, ..., N denote position and velocity respectively; the control inputs u k,1 , u k,2 result from integrating the control signal applied during the interval (t, t + ∆t] respectively once and twice; and the disturbances d k,1 , d k,2 result from similarly integrating the continuoustime input disturbances. Model (1) is an exact discrete-time equivalent because u k (t) over the interval (t, t + ∆t] does not depend on the state values after time t.


Arash Farnam
email: arash.farnam@ugent.be

Alain Sarlette
email: alain.sarlette@inria.fr

About string stability of a vehicle chain with unidirectional controller

This paper deals with the problem of string stability in a chain of accelerationcontrolled vehicles. It is known that string stability cannot be achieved, with any linear controller, when the vehicles' control inputs are based only on relative distances to a fixed number of predecessors. We extend the set of characteristics under which string stability is impossible by including elements like dynamic sensor parts and local intervehicular communication, as in cooperative adaptive cruise control. It is also known that a weaker form of string stability is achievable by adding absolute velocity measurements (e.g. "time-headway" policy). We show that a stronger property can also be achieved, provided steady-state control gain is infinite e.g. by using integral control.

Introduction

Grouping vehicles into tight platoons is a method for increasing the capacity of roads by automated highway systems [START_REF] Chu | Decentralized control of high-speed vehicular strings[END_REF]). The distances between vehicles is decreased by ensuring safety thanks to automatic controllers, enabling many vehicles to accelerate or brake simultaneously and eliminating the distance needed for human reaction. The most fundamental platoon is the vehicle chain, where all vehicles are aligned after each other. During the recent years numerous works have considered different control strategies to stabilize each vehicle at a desired distance from its predecessor and follower [START_REF] Chu | Decentralized control of high-speed vehicular strings[END_REF]; [START_REF] Stüdli | From vehicular platoons to general networked systems: String stability and related concepts[END_REF]; [START_REF] Sheikholeslam | Longitudinal control of a platoon of vehicles[END_REF]; [START_REF] Levine | On the optimal error regulation of a string of moving vehicles[END_REF]; [START_REF] Rogge | Vehicle platoons through ring coupling[END_REF]).

When disturbance inputs affect the vehicle chain, its a priori cooperative coupling can lead to new types of instabilities. In particular, string instability is a situation where the spacing error between consecutive vehicles grows unbounded as the number of vehicles increases to infinity. Since its definition in [START_REF] Swaroop | String stability of interconnected systems[END_REF]; [START_REF] Swaroop | String stability of interconnected systems: An application to platooning in automated highway systems[END_REF], string (in)stability has spurred a lot of discussion and research. Indeed, it is well known since [START_REF] Swaroop | String stability of interconnected systems: An application to platooning in automated highway systems[END_REF] that string stability cannot be achieved in a homogeneous string of interconnected secondorder integrators (e.g. acceleration-controlled vehicles), with any controller that is linear and whose local control actions are determined from the relative distance to a few directly preceding vehicles: some perturbations will unavoidably grow unbounded along the chain. This has attracted attention as a prototypical, unavoidable shortcoming of linear systems [START_REF] Sheikholeslam | Longitudinal control of a platoon of vehicles[END_REF]; [START_REF] Levine | On the optimal error regulation of a string of moving vehicles[END_REF]). When the controller reacts to the just preceding vehicle only, the unavoidability of string instability follows essentially from the Bode integral for the transfer function from vehicle i -1 to i, which takes the form of a complementary sensitivity function. This has led to first observations about how a disturbance gets amplified from one vehicle to the next, without bound as the chain gets longer. However, it has soon been noted that the problem is more general and that also with other types of couplings where the dynamics cannot be reduced to propagation of disturbances from i-1 to i -e.g. with bidirectional coupling [START_REF] Barooah | Error amplification and disturbance propagation in vehicle strings with decentralized linear control[END_REF]; [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF]; Herman et al. ( 2017) -small disturbances on initial conditions or in the inputs can lead to big errors on inter-vehicle spacing, more precisely unbounded errors as the number of vehicles tends to infinity. This has led to the more general concept of string instability. The latter has also become a relevant concept beyond the strict application to road vehicles, for the distributed control of chains of abstract subsystems, with particular types of open-loop dynamics and control inputs depending on local relative output measurements. We will use the road vehicle chain as a guiding example, but keeping in mind that the relevance of possible settings or assumptions should not be restricted to this particular application.

To solve the string instability issue, a notable alternative setting is to allow the use of absolute velocity in the controller, see [START_REF] Rogge | Vehicle platoons through ring coupling[END_REF], [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF], [START_REF] Knorn | Passivity-based control for multivehicle systems subject to string constraints[END_REF]. It has been shown that the effect of a disturbance from the leading vehicle can then be kept in check for arbitrarily long chains. Another natural feature is to allow local communication, as in Cooperative Adaptive Cruise Control (CACC, see Ploeg et al. (2014), Öncü et al. (2014), Ploeg et al. (2014), [START_REF] Ploeg | Graceful degradation of Cooperative Adaptive Cruise Control[END_REF], [START_REF] Milanés | Modeling cooperative and autonomous adaptive cruise control dynamic responses using experimental data[END_REF]). Remarkably, the literature considering communication also always assumes the use of absolute velocity in the controller, and it thus remains unclear what can be done with communication but without using absolute velocity.

The aim of the present paper is to add essentially two types of precision to this picture. First, we consider countering the effect of disturbances acting possibly on all vehicles in the (L 2 , l 2 ) sense. This is in agreement with the more general concept of string stability, and with the practical concern of avoiding catastrophic effects in long chains due to small errors in all the vehicles' inputs. For this, we give a positive result using PID control and absolute velocity. Second, we clarify how alternative settings behave in absence of absolute velocity or integral action in the controller. For this, we provide a series of negative results; in particular the use of absolute velocity appears necessary even in presence of CACC-type communication. A more detailed statement of state of the art and of our contributions is given in Section 3, after clarifying the setting.

While the present work focuses on unidirectional vehicle chains -i.e. vehicles react to their predecessors only -a whole line of work has been developed for bidirectional chains as well, where vehicles react to predecessors and followers. Impossibilities to satisfy string stability have been obtained for both symmetric [START_REF] Barooah | Error amplification and disturbance propagation in vehicle strings with decentralized linear control[END_REF]; [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF]) and asymmetric (Herman et al. (2017)) couplings. Conversely, [START_REF] Hao | Stability and robustness of large platoons of vehicles with doubleintegrator models and nearest neighbor interaction[END_REF]; [START_REF] Yamamoto | Bounded disturbance amplification for mass chains with passive interconnection[END_REF] have identified the possibility to avoid unbounded growth of a disturbance that acts on the leader only, without resorting to absolute velocity; this result thus fundamentally differs from the impossibility implied by the Bode integral for unidirectional chains. Our viewpoint on the bidirectional setting can be found in [START_REF] Farnam | Towards impossibility and possibility results for string stability of platoon of vehicles[END_REF].

The paper is organized as follows. Section 2 describes the string stability property to be investigated and the various control settings which we analyze. Section 3 gives the summary of our results. The proofs are given in Appendix.

2 Problem Setting 2.1 Open-loop model and control objective Consider a chain of N + 1 subsystems (e.g. vehicles) with respective configuration (e.g. deviation from nominal position) x i ∈ R, for i = 0, 1, 2, ..., N . They move according to the second-order integrator dynamics, expressed in Laplace domain:

s 2 x i (s) = u i (s) + d i (s) (1)
where u i is the control input, and d i is a disturbance input signal. These disturbance signals can also be an indirect way to model nonzero initial conditions, which are the focus in other papers. The pure second-order integrator is a standard idealization, valid in good approximation for space vehicles, vacuum transit, after other dynamics has been compensated by local feedback, or when it can be included (e.g. by linearization) in the form of u i (s) that we further specify below.

The goal of stability, on a system like (1) with fixed N , is to ensure that arbitrary input signals d i are not amplified unboundedly in the state or output. String stability further checks what happens when N becomes infinite; thus if the system is stable for each N but with stability bounds depending on N in a bad way, then string stability may fail. The standard definition of string stability [START_REF] Swaroop | String stability of interconnected systems[END_REF], [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF]) considers how the relative distances between consecutive vehicles are affected. In the "timeheadway" extension [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF], the target distance between vehicles is made proportional to their absolute velocity, such that the configuration error ("output signal" of interest) is

e i (s) = x i-1 (s) -x i (s) -hsx i (s) , (2)
for some h ≥ 0. Different variants of string stability, see e.g. [START_REF] Stüdli | From vehicular platoons to general networked systems: String stability and related concepts[END_REF], consider how this error signal is affected either by initial conditions or by disturbance inputs, and measure the signals with different norms. We here focus on the effect of disturbance inputs, and consider two types of 2-norms. For a vector v of signals for any situation with bounded disturbance signal d i at some i and zero disturbances on all k = i, and for all N .

v i (t), i = 1, 2, ..., N , the L 2 norm of v i is defined as v i (.) 2 = +∞ -∞ (v i (t)) 2 dt and the (L 2 , l 2 ) norm of v is defined as v(.) 2 = N i=1 +∞ -∞ (v i (t)) 2 dt ,
The difficulty, for both definitions, is to make the bound uniform in N . Definition 2, considered in e.g. [START_REF] Barooah | Mistuning-based control design to improve closed-loop stability of vehicular platoons[END_REF]; [START_REF] Hao | Stability and robustness of large platoons of vehicles with doubleintegrator models and nearest neighbor interaction[END_REF]; [START_REF] Yamamoto | Bounded disturbance amplification for mass chains with passive interconnection[END_REF]; [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF]; [START_REF] Knorn | Passivity-based control for multivehicle systems subject to string constraints[END_REF]Herman et al. (2017), essentially requires that the transfer function from any d i to any e k is H-infinity bounded independently of N . This appears to be a desirable necessary property for practical applications, but not strictly sufficient. Indeed, L 2 string stability does not bound independently of N what happens when disturbances are present at the inputs of all the vehicles: if each individual disturbance d i can induce a nonzero bounded error on each e k , without further constraints, then when disturbances act on all the vehicles the sum of their effects could make e k grow unbounded with N . Avoiding this is the motivation behind the stronger Definition 1, which is in line with the definitions considered by [START_REF] Stüdli | From vehicular platoons to general networked systems: String stability and related concepts[END_REF]; [START_REF] Barooah | Error amplification and disturbance propagation in vehicle strings with decentralized linear control[END_REF]; [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF] among others. Indeed, Definition 1 requires that a bounded disturbance input vector (times N ) should lead to a bounded output error vector (times N ). The true practical concern should probably involve infinity norms both over time and over vehicle indices, in a BIBO sense. The corresponding 2-norms are used as a standard proxy to take advantage of their basis-independence, like in many other control-theoretic performance studies.

Constraints on controller design

We will consider different constraints on the feedback controller design, all revolving around the unavailability of absolute or long-range position measurements. The most general controller that we consider, and motivated in the next paragraph, writes:

u i (s) = K(s)e i (s) + H(s)r i (s) (3) r i (s) = W (s) q i-1 (s) q i (s) = F (s)e i (s) + G(s)r i (s) e i = M (r) (s)x i-1 (s) -M (f ) (s)x i (s) -hsx i (s) ,
for i = 1, 2, ..., N . Here K(s) represents the linear controller's transfer function depending on measured configuration error e i , and satisfying K(0) = 0 (impossibility to have a perfect derivative of the measurement output in order to cancel the pole at s = 0 in the open-loop system (1)). Furthermore, unlike in Li and De Schutter (2012), we do not allow K(s) to grow unboundedly with N and we assume all transfer functions independent of N . For h = 0, the e i are the measured inter-vehicle distances and they may differ from the true inter-vehicle distance x i-1 (s)-x i (s). Examining a simple physical sensor model, without a global reference, even shows that the typical situation does not necessarily correspond to e i (s) = M (s)e i (s) for h = 0, i.e. a relative distance measurement is not just proportional to the true relative distance. Indeed, consider a distance sensor that measures the time-of-flight between a sensor part mounted at the front of vehicle i and its companion part mounted on the rear of vehicle i-1. The mounts of these sensor parts can have some dynamics, that are themselves sensitive to the relative position of a sensor part p (r)

i or p (f ) i
with respect to the vehicle on which it is mounted x i . Thus, we should have s 2 p i = K (.) (s) (x i -p i ) , which yields:

M (r) = K (r) s 2 + K (r) , M (f ) = K (f ) s 2 + K (f ) , (4)
with K (r) (0) = 0 and K (f ) (0) = 0, respectively for the sensor parts mounted on the rear and front ends of the vehicles. In particular, at the limit of infinitely stiff mounts

K (r) , K (f ) → ∞, we get M (r) (s) = M (f ) (s) = 1
and the controller u i (s) just depends on the configuration error e i (s); if furthermore h = 0 it reduces to true relative position e i (s) = x i-1 (s) -x i (s). In addition, u i relies on a communication signal r i (s) received from the preceding vehicle. Imperfections in the communication channel are taken into account by a simplified linear model, with the bounded transfer function | W (jω) | not exactly known. The signal q i (s) sent into this channel by vehicle i is computed as a linear function, on the same basis as the control signal, with H(s), F (s) and G(s) controller transfer functions to be designed. We make the following assumptions on the communication channel.

Assumption 1 [communication channel]:

• The poles of F, G, H, W must all have negative real parts.

• G(jω), F (jω) and H(jω) K(jω) are bounded for all ω.

• System behavior must be robust to replacing W by any W close to W , without adapting the controller F, G, H, K.

Let us briefly motivate the various features of (3) and the associated Assumption 1. -The most basic setting, with h = 0, M (r) (s) = M (f ) (s) = 1 and W = 0 (thus no communication H = F = G = 0), was the initial focus of [START_REF] Swaroop | String stability of interconnected systems[END_REF] and is known to be string unstable with any linear controller. To check how alternative settings could help, we stay in the linear realm.

-Just introducing the possibility of h > 0, was shown to enable satisfying Definition 2 [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF]). However, the time-headway policy makes the effective inter-vehicle distance velocity-dependent, where absolute velocity of the chain would be defined via some other controller, to be carefully interfaced, and where high velocity would automatically imply large(r) inter-vehicle distance, unlike what a high-capacity vehicle "train" should do. For some vehicle types -think of a platoon of space vehicles -or for abstract subsystems, it also remains to be seen exactly how (accurately) the absolute velocity of each individual would be measured in practice towards implementing (3) with h > 0. Therefore, it is relevant to check whether other features could allow us to achieve string stability without using absolute velocity in the controller, i.e. with h = 0.

-The use of dynamic sensor mounts M (r) (s), M (f ) (s) is another way of breaking the symmetry of all input commands depending just on e i = x i-1 -x i . Tuning these dynamics in some beneficial way appears as a less invasive solution, than assuming with h > 0 that a global reference is available for absolute velocity measurement. We thus see this as another potential opportunity to break the impossibility observed in [START_REF] Swaroop | String stability of interconnected systems[END_REF]; [START_REF] Swaroop | String stability of interconnected systems: An application to platooning in automated highway systems[END_REF].

-The use of local communication is a natural feature, considered e.g. in Ploeg et al. (2014); Öncü et al. (2014); Ploeg et al. (2014[START_REF] Ploeg | Graceful degradation of Cooperative Adaptive Cruise Control[END_REF]; [START_REF] Milanés | Modeling cooperative and autonomous adaptive cruise control dynamic responses using experimental data[END_REF], with h > 0. We want to see how it fares with h = 0. To give a meaning to "local communication" in continuous-time, it is necessary to specify a communication model of finite capacity. If instead communication was assumed perfect and instantaneous, then communicating r 1 = x 0 -x 1 , r i = r i-1 + e i for i > 1, vehicle i would in fact receive r i = x 0 -x i . In other words, vehicle i has access to the perfect value of x 0 -x i , as if the coupling with the leader was global. On this basis each vehicle can just stabilize its position with respect to the leader with some stable controller u i (s) = H(s)r i (s) = H(s)(x 0 -x i ), independently of all the other vehicles in the chain. However, assuming perfect transmission of arbitrary signals throughout the whole vehicle chain is not what a realistic communication channel can do. Assumption 1 formulates a few conditions to make the behavior robust to imperfect communication: the communication system must be internally stable (first item); the encoding and decoding filters must avoid unbounded amplification of possible communication noise, both towards generating the next communicated signal and before applying the controller K (second item); and every controller should work under the premise that we know the communication channel only imperfectly (third item). Another typical limitation would be a finite bandwidth in W associated with additive communication noise, but we do not even need this assumption for our results.

Summary of results

We now summarize more accurately the known results from the literature and new results of the present paper, see Table 1. Our proofs are given in appendix.

Regarding known results, the first observations of string instability [START_REF] Swaroop | String stability of interconnected systems: An application to platooning in automated highway systems[END_REF]) were made when each control input u i is reacting just to the relative distance x i-1 -x i with the vehicle in front, thus corresponding to (1),( 2),(3) with M (r) = M (f ) = 1, h = W = 0. It was observed that L 2 string stability is impossible (Definition 2), and a fortiori (L 2 , l 2 ) string stability (Definition 1) as well, with any linear controller that avoids pole cancellation (K(0) = 0). Let us repeat the short proof of this first observation, which has guided researchers in their search for alternative control strategies. Let u i (s) = K(s)e i (s). From (1), the closed-loop equation for the e i and with a disturbance on the first subsystem only writes

e i = T (s) i-1 1 s 2 + K(s) d 0 , with T (s) = K(s) s 2 +K(s) = R(s)
1+R(s) with R(s) = K(s)/s 2 , taking the form of a complementary sensitivity function. To guarantee L 2 string stability, with N unboundedly large, it is then necessary in particular that |T (jω)| ≤ 1 at all frequencies ω. This is impossible for a stable system, from the statement of Bode's Complementary Sensitivity integral, which we recall below. Lemma 1: Assume that the loop transfer function R(s) of a system has (at least) a double pole at s = 0. If the associated feedback system is stable, then the complementary sensitivity function T (s) = R(s) 1+R(s) must satisfy:

∞ 0 ln | T (jω) | .dω/ω 2 = π k 1 q (T ) k ≥ 0 ,
where {q This impossibility can be circumvented by using a controller with h > 0 (and still W = 0 i.e. no communication, M (r) = M (f ) = 1 i.e. no special sensors), as proposed in [START_REF] Swaroop | String stability of interconnected systems: An application to platooning in automated highway systems[END_REF] and analyzed clearly in [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF]. In that case the closed-loop dynamics writes

e i = K(s) s 2 + (1 + hs)K(s) e i-1 (5)
+ 1 s 2 + (1 + hs)K(s) (d i-1 -(1 + hs)d i )
and we would have T (s) = R(s) 1+R(s) with R(s) = K(s) s 2 +hsK(s) , which does not have a double pole at s = 0 and thus circumvents Lemma 1. For completeness and later comparison, we give the following result comparable to [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF]. Proposition 1: The norm at s = jω of transfer function T (s) = K(s) s 2 +(1+hs)K(s) in (5) is < 1 at all frequencies ω = 0, and its H ∞ norm equals T (0) = 1, if and only if one of the following equivalent conditions hold: (a) If one chooses K(s) = K(s)(1 + hs) first and then derives K(s) from h, then we should ensure that h satisfies

h > max ω R(jω) 1+ R(jω) 2 -1 ω 2 (6) in which R(s) = K(s)/s 2 . (b) If one chooses K(s) first, then the criterion becomes h > max ω K R (jω) (2 -ω 2 K R (jω)) + ωK J (jω) (7)
where

K R (jω) = 1 2 ( 1 K(jω) + 1 K(jω) * ), K J (jω) = 1 2j ( 1 K(jω) -1 K(jω) * )
, and the maximization runs over all ω for which the argument of the square root is positive.

For particular controllers one can get easy criteria, e.g. for a PD controller K(s) = bs + a, it is not hard to see that if a > 2b 2 the right hand side in case (b) is decreasing with ω, and the condition becomes h > 2/a.

A direct consequence of Prop.1 is that one can avoid amplifying a disturbance d 0 (s) along the vehicle chain, and satisfy Definition 2 of string stability. However, remarkably, a corresponding result about Definition 1, i.e. the stronger version of string stability as considered in e.g. [START_REF] Stüdli | From vehicular platoons to general networked systems: String stability and related concepts[END_REF]; [START_REF] Barooah | Error amplification and disturbance propagation in vehicle strings with decentralized linear control[END_REF]; Seiler et al. ( 2004), appears to be missing in the literature. The present paper provides the following results in this direction. Theorem 1: Consider the vehicle chain system (1),( 2),(3) with M (r) = M (f ) = 1 and W = 0. There exists no pair (K(s), h) , where h ≥ 0 is any constant time-headway and K(s) any stabilizing linear controller with K(0) bounded, that would achieve (L 2 , l 2 ) norm string stability (Def.1). Theorem 2: Consider the vehicle chain system (1),( 2),(3) with M (r) = M (f ) = 1 and W = 0. A stabilizing PID controller K(s) with headway h satisfying (7), can ensure (L 2 , l 2 ) norm string stability (Def.1).

In light of Thm.1, the lack of positive result about (L 2 , l 2 ) string stability in the timeheadway literature can thus be attributed to their focus on bounded stabilizing controllers K(s), excluding e.g. controllers with integral action. Theorem 2 further clarifies that a PID controller indeed does allow to achieve this stronger version of string stability.

The result of Thm.2 obviously covers as well the case where communication is allowed on top of time headway h > 0. This is the standard setting considered in papers like Öncü et al. (2014); [START_REF] Ploeg | Graceful degradation of Cooperative Adaptive Cruise Control[END_REF], where the communication is meant as a way to improve performance rather than just achieving string stability. Those papers also stay with the weaker notion of Def.2, and they impose a more precise communication structure, called Cooperative Adaptive Cruise Control (CACC), by assuming that the message sent by vehicle i to its follower i + 1 is a filtered version of the input command u i :

q i (s) = 1 B(s) K(s)e i (s) + H(s)r i (s) , (8)
or in other words (3) with F = K/B and G = H/B. By Thm.2, we have thus established that with h > 0 and PID control, there exists a communication (namely the trivial one F = H = G = 0) which does achieve the stronger string stability of Def.1 as well. A remaining question is then, how much can communication allow us to weaken some assumptions and still achieve string stability? A negative answer is expressed by the following results. Theorem 3: Consider the vehicle chain system (1),( 2),(3) with M (r) = M (f ) = 1, with the structure (8) of CACC communication. (a) There exists no h > 0 and W, B, H, K satisfying Assumption 1 with K(0) bounded, allowing to satisfy (L 2 , l 2 ) string stability (Def.1). (b) For h = 0, there exists no W, B, H, K satisfying Assumption 1 and allowing to satisfy L 2 (Def.2) and thus a fortiori (L 2 , l 2 ) string stability (Def.1). Theorem 4: Consider the vehicle chain system (1),( 2),(3) with M (r) = M (f ) = 1 and h = 0. (a) For K(0) bounded, there exist no W, F, G, H, K satisfying Assumption 1, with each q i possibly an n-dimensional vector signal for some n ≥ 1, allowing to satisfy (L 2 , l 2 ) string stability (Def.1). (b) For q i scalar signals, there exist no W, F, G, H, K satisfying Assumption 1 and allowing to satisfy L 2 (Def.2) and thus a fortiori (L 2 , l 2 ) string stability (Def.1).

From Thm.3a, CACC-type communication together with time headway does not exempt us of requiring unbounded K(0), i.e. a result equivalent to Thm.1 still holds. Moreover, Theorems 3b and 4 indicate that communication does not allow us to avoid the necessity of time headway; most importantly, in a CACC-type setting or generalized scalar setting, even Def.2 cannot be satisfied. The assumptions a priori still leave a loophole for unbounded K(0) and communicating vector signals q i , but we believe that this is just a technical issue and in fact we conjecture that no communication model satisfying Assumption 1 would allow to achieve string stability with (1),( 2),(3) and h = 0.

As a final attempt, we have checked whether we could replace the requirement of time headway h > 0, by using a relative position measurement that is not just proportional to the true relative position thanks to the dynamics M (r) , M (f ) . Unfortunately, here too the result is an impossibility. Theorem 5: Consider the vehicle chain system (1),( 2),( 3) with h = 0 and no communication (W = G = F = H = 0). There exists no choice of stabilizing K(s) and of M (r) , M (f ) under the form (4), allowing to satisfy L 2 string stability (Def.2), and thus a fortiori (L 2 , l 2 ) string stability (Def.1).

Conclusion

We have identified impossibilities to achieve string stability with linear controllers in several extended settings -communication, sensor dynamics, time-headway with bounded DC gain -and one possible solution, namely a PID controller with sufficient time headway. While Table 1: known results and clarifications provided by the present paper ; with M (r) = M (f ) = 1 unless specified.

Standard impossibility; [START_REF] Swaroop | String stability of interconnected systems[END_REF]; [START_REF] Swaroop | String stability of interconnected systems: An application to platooning in automated highway systems[END_REF]; Lemma 1 any W = 0, scalar q i , h = 0 Def.1 and Def.2 fail Theorem 5 W = 0, h = 0, tuning M (r) , M (f ) Def.1 and Def.2 fail the proofs do not involve complicated techniques, they do complete the picture about string stability in the strong sense ((L 2 , l 2 ) bounded reaction to simultaneous perturbations on all the vehicles) and weak sense (L 2 bounded reaction of all the vehicles to a disturbance input acting on a single one). We have focused on the standard double-integrator model, i.e. Newton's equation with inputs as only forces, which is not already internally stable; other such models, including MIMO ones, may be worth considering in future work. Regarding extended settings, the paper narrows down the options towards achieving string stability without using absolute velocity. In particular:

h = W =
• About the use of inter-vehicle communication on a linear channel W (s), we have shown that while this may allow to jointly improve some other performance criteria, it is not a fix on its own for string instability: in all the settings that we have studied, whenever there exists a controller achieving L 2 or (L 2 , l 2 ) string stability with W = 0, then there also exists a controller achieving it with W = 0. (The converse of that statement is trivial: if a satisfactory controller is found without using a communication channel, then the presence of a channel W = 0 cannot hurt since the controller can just choose to ignore the communication messages.)

• Checking typical sensor details, one observes that a distance measurement does not just correspond to a transfer function applied to the actual distance. In other words, actual outputs are not the perfect relative measurements on which the standard string instability proofs rely in an essential way. One might thus wonder whether modeling some sensor details could solve string stability, similarly to and perhaps more gently than the use of absolute velocity. We show though that modeling such sensor details does not change the picture for either definition of string stability.

• Regarding bounded DC gain, the bad behavior close to zero frequency is a basic feature and a versatile focus, which should allow to similarly complete the picture for a.o. bidirectional controllers, see [START_REF] Farnam | Towards impossibility and possibility results for string stability of platoon of vehicles[END_REF].

These results also highlight the necessity to carefully check in applications whether Def.1, Def.2 or possibly some other string stability notion is the correct proxy for what applications really require. This point has indeed attracted surprisingly little discussion in the literature, although it does appear to change the possible conclusions.

We must end with a short outlook on options that remain open for the controller design itself. One main assumption in this line of literature, including our paper, is of course the use of linear models; nonlinearities and in particular quantization in digital controllers are a priori not covered by the tight impossibility results. However, it appears that this is not a key point. We indeed address this, together with the remaining loophole of vector communication, in ongoing work. Another point regarding communication, is the assumption of local message transmission. If instead all-to-all communication was allowed, e.g. via broadcasting over a communication bus, the picture could be changed. Access to sending over the bus would have to be managed with event-driven decision logic, calling for a comprehensive cyberphysical systems treatment of string stability. One would still have to investigate though, which breakthrough useful information such bus could transmit, when only local distances are measured. I. Herman, S. Knorn and A. Ahlén, "Disturbance scaling in bidirectional vehicle platoons with different asymmetry in position and velocity coupling," Automatica, vol. 82, pp. 13-20, 2017.

A Proofs

Proof of Proposition 1: For case (a), we reformulate

T (s) = 1 1+hs K(s)/s 2 1+ K(s)/s 2 . Then writing |T (jω)| 2 = 1 1 + ω 2 h 2 R(jω) 1 + R(jω) 2
< 1 for all ω = 0 directly yields the expression, where the Bode integral (Lemma 1) ensures that the max inside the square root will be non-negative. For case (b), we just write 1/|T (jω)| 2 = | -ω 2 /K(jω) + (1 + hjω)| 2 > 1 and we group real and imaginary parts to isolate h.

Proof of Theorem 1:

We take in particular a disturbance input d 0 that affects the leading vehicle only. From ( 5), such disturbance leads to

e i = T (s) i-1 1 s 2 + (1 + hs)K(s) d 0 , i = 1, 2, ..., N, with T (s) = K(s) s 2 +(1+hs)K(s) as in Proposition 1. Then N i=1 |e i (s)| 2 = N -1 i=0 | T (s) | 2i • |d 0 (s)| 2 2 |s 2 + (1 + hs)K(s)| 2 . ( 9
)
Consider a disturbance concentrated at low frequencies, such that -|d 0 (jω

)| 2 dω ≥ 1 2 +∞ -∞ |d 0 (jω)| 2 dω for some 1. Writing e(.) 2 2 ≥ - N i=1 |e i (jω)| 2 dω , (10)
we lower-bound the right hand side thanks to continuity at ω = 0. Indeed, in (9), for K(0) finite, there exist δ, α > 0 such that

1 |s 2 +(1+hs)K(s)| 2 | s=jω > α for all ω ∈ (-δ, δ).
Moreover, since T (0) = 1, we can also make min ω∈[0, ) |T (jω)| arbitrarily close to 1 by taking close enough to 0. Therefore, we can make the right hand side of (10) arbitrarily close to N α 2 d 0 (.) 2 2 by concentrating d 0 on low enough frequencies < δ. Thus the factor relating e(.) 2 2 to d 0 (.) 2 2 = d(.) 2 2 cannot be bounded independently of the disturbance signal d(.) and of N .

Proof of Theorem 2: For this positive result we must prove that we can tune the gains such that the system is stable, and simultaneously we can guarantee || e(.) || 2 < C 0 || d(.) || 2 in which the constant C 0 is bounded independently of number of vehicles N . We split the proof in two parts. First, we show that there exists a set of PID controller tuning and of values of h, for which the system is stable while satisfying Proposition 1 (i.e. the L 2 stability criterion). Second, we show that this also allows to satisfy (L 2 , l 2 ) string stability.

For stability under the conditions of Proposition 1: Considering the first criterion in Proposition 1, we would fix some tuning of the polynomial K(s) which makes the system stable (just checking always the same denominator s 3 + s(1 + h s) K(s) = s 3 + s K(s)). Once K(s) has been selected, we would then choose h according to the related criterion, while adapting the other parameters in order to maintain K(s) fixed as selected. For this to be possible, the only essential element is to prove that h in the first criterion of Proposition 1 always remains bounded for a stable PID controller.

We thus consider s 3 + s K(s) to be any third-order polynomial with roots in the open left half plane. Then in the criterion,

R 1 + R = s K(s) s 3 + s K(s)
remains bounded for all s = jω and we must only investigate the behavior for ω close to 0. From the inverse triangle inequality

| R 1+ R | 2 -1 ≤ |( R 1+ R ) 2 -1| a sufficient criterion for Proposition 1(a) is h > (jω) 2 K2 (jω) -[(jω) 3 + (jω) K(jω)] 2 ω 2 [(jω) 3 + (jω) K(jω)] 2 , which just comes down to h > ω 4 -2ω 2 K(jω) (jω K(jω) -jω 3 ) 2 .
For ω close to 0 and K(jω) a PID controller, the dominating term is h > |2ω k I /k 2 I |, with k I the integral gain. This imposes a bounded constraint on h and it is thus possible indeed to satisfy stability and the criterion of Proposition 1 simultaneously with a PID controller.

For (L 2 , l 2 ) string stability, we write in matrix form: with the induced matrix norms D 2 = λ max (D * D) where * is the complex conjugate transpose. The proof now comes down to proving a bounded norm, independent of N and s = jω, for each of the three terms in front of d(s) 2 in (11).

For the first term, since A * A = diag(0, 1, 1, 1, ..., 1), we immediately have |L(s)| A 2 = |L(s)|, and the latter can be bounded independently of s = jω for a stable system.

For the second term, we have B * B = diag 1-|T (s)| 2N 1-|T (s) 2 | , 0, 0, ..., 0 . Under the conditions of Prop.1, the numerator is lower than 1 and |L(jω

)| B(jω) 2 ≤ |L(jω)| 2 1-|T (jω)| 2 .
The unbounded DC gain ensures that, when T (jω) -1 converges to 0 at ω = 0, so does L(jω). Analysis close to ω = 0 yields

|L(jω)| 2 1 -|T (jω)| 2 ω 2 k 2 I • 1 h 2 ω 2 = 1 k 2
I h 2 i.e. the limit for ω → 0 of |L(jω)| B(jω) 2 is bounded, independently of N . It is then easy to find a bound that is valid at all frequencies ω, independently of N .

To bound the third term, we can use the Gershgorin disk theorem on the matrix |P | 2 C * C. For |T (jω)| ≤ 1 we can bound finite sums of powers of |T (jω)| by an infinite geometric series, and trivially check that the eigenvalues are bounded independently of N , for all ω outside a neighborhood of the origin ω = 0. The latter is indeed the only place where |T (jω)| = 1, and an expansion for ω 1 shows that in fact |P (jω)| 2 C(jω) 2 2 converges to zero for ω → 0. This concludes the proof.

Proof of Theorem 3: Defining z i = [e i ; q i-1 -q i ], the closed-loop dynamics with d 0 = 0 only is described by:

z i+1 = T(s) z i , z 1 = 1 s 2 +K(1+hs) -K/B s 2 +K(1+hs) d 0 with T(s) = K s 2 +K(1+hs) HW s 2 +K(1+hs) K B • s 2 s 2 +K(1+hs) HW B • s 2 s 2 +K(1+hs)
.

The key simplification implied by CACC is that T(s) is singular for all s, since the right column equals HW/K times the left column. Thus the single nonzero eigenvalue of T (s) equals its trace, trace(T(s)) = K+ HW B s 2 K(1+hs)+s 2 . (a) For any h > 0, trace(T(s)) approaches 1 when s approaches 0. Thus like in the proof of Thm.1, a uniform bound over ω, N cannot be found for e 2 , if the corresponding mode with s = 0 has a nonzero component in z 1 . The corresponding eigenvector at s = 0 is z i ∝ [1 ; 0], while the zero eigenvector is z i ∝ [1 ; -K(0)/HW (0)]. Thus z 1 will have a nonzero component on the "bad" mode unless z 1 ∝ [1 ; -K/B(0)] ∝ [1 ; -K(0)/HW (0)], i.e. either K(0) unbounded which we exclude by assumption, or HW (0) = B(0) exactly. The latter is forbidden by the last requirement of Assumption 1.

(b) For h = 0, we can rewrite trace(T(s)) = R 1+R with R = K+ HW B s 2 s 2 (1-HW/B) . Since HW B is bounded, the denominator decays at least as s 2 for s close to 0 and we are in the conditions to apply Lemma 1 (Bode complementary sensitivity integral); this implies that there will be a range of frequencies ω where T(jω) has an eigenvalue with norm |trace(T(jω))| larger than 1. As for case (a), with the last requirement of Assumption 1 the system will unavoidably have a component of z 1 (ω) on this mode, which unavoidably makes the system string unstable in the sense of Def.2.

Proof of Theorem 4: Similarly to the case of CACC, and somewhat simplified thanks to h = 0, by defining z i = [e i ; q i-1 ] we can reformulate the dynamics as:

z i+1 = T(s)z i + 1 s 2 +K 0 (d i -d i+1 ) with T(s) = K-HW F s 2 +K HW (I-GW ) s 2 +K F GW .
Here I is the identity matrix, emphasizing that q i might be a vector and F, G, H, W appropriate matrices.

For case (a) the proof follows the same lines as Thm.3(a), after checking that T(0) has an eigenvalue 1.

For case (b) we use a Routh-Hurwitz type criterion for discrete systems, see e.g. [START_REF] Serban | Multidimensional systems: BIBO stability test based on functional Schur coefficients[END_REF]. For a two-dimensional state matrix A, it states that the eigenvalues belong to the unit circle provided Since s 2 1-|det| 2 is real negative for s = jω and |det| < 1, the above equation cannot be satisfied if (GW -1)/(s 2 + K) takes a real negative value for some s = jω. Indeed, for any c 1 , c 2 real negative and c 3 complex but of norm smaller than one, we have that 1+c 1 c 2 (1-c 3 ) lies outside the unit disk. Thus to conclude the proof, there remains to show that (GW -1)/(s 2 + K) will always take a real negative value for some s = jω.

Since s 2 + K has two more zeros than poles, and all zeros must satisfy stability, we have that the phase Bode plot of 1/(s 2 + K) goes down at least by 180 degrees, to end at -180 degrees for ω tending to infinity. In contrast, GW -1 has as many zeros as poles; all poles are stable, implying 90 degrees down in the phase Bode plot, such that overall with GW -1 we either go down or stay, and again we end at -180 degrees for ω tending to infinity. Now assume as a first possibility, that GW -1 starts at another value than -180 degrees. In this case, it must go down nontrivially, i.e. we must go sown by strictly more than 180 degrees to end up at -360 degrees: somewhere in between, there will be a 180 degree phase, proving impossibility. (Note indeed that we forbid any perfect cancellation with GW = 1 at a target value of ω.) So the only choice left is that GW starts at -180 degrees. Then for K(0) finite we would have a negative real phase at s = 0, thus impossible. There remains the case with K having a pole of order m > 0 at s = 0. In this case, 1/(s 2 + K) has m of its zeros at s = 0, and 1 s m 1 s 2 +K has a phase Bode plot going down by (180 + m90) degrees overall. This means, GW -1 s 2 +K would start with a phase of -180 + m90 degrees at s = 0, then go down by 180 + m90 degrees to end up at -360 degrees for ω tending to infinity, with m > 0. Again, this implies a phase of -180 degrees for some intermediate ω. There are no possibilities left, so the proof is concluded. Proof of Theorem 5: The error dynamics write

e i = M (r) • 1 M (f ) • T (s) e i-1 =: A(s) e i-1 (12)
where T (s) = K (s) s 2 +K (s) with K (s) = M (f ) (s) • K(s). With (4), M (r) , M (f ) and T all take the form of complementary sensitivity functions satisfying Lemma 1. We thus have

∞ 0 ln|A(jω)|.dω/ω 2 = k 1 q (M (r) ) k - 1 q (M (f ) ) k + 1 q (T ) k ,
with the q's denoting the respective zeros of the loop transfer functions in the right half plane.

Having a control effect requires |A(jω)| < 1 at some frequencies, while having |A(jω)| > 1 at any frequency would imply that the system is string unstable. Combining these two features requires that the right hand side be negative. The only way to obtain this is if K (f ) (s)/s 2 has zeros in the open right half plane, without having the same zeros in the other terms. However, the latter would mean that M (f ) (s) has zeros in the right half plane, unmatched by the other transfer functions, and by ( 12) this would imply that the vehicle chain has a pole in the right half plane i.e. it is unstable.

k

  } are the zeros of R(s) in the open right half plane. In particular, if |T (jω)| < 1 at some frequencies, then necessarily |T (jω)| > 1 at other frequencies.

  h > 0 with PID controller (a fortiori also for any W = 0) Def.1 and Def.2 hold CACC with time-headway; Öncü et al. (2014); Ploeg et al. (2015) W = 0, h > 0 Def.2 holds with possibly better scaling, lower h Theorem 3a CACC-type W = 0, h > 0, K(0) bounded Def.1 fails Theorem 3b CACC-type W = 0, h = 0 Def.1 and Def.2 fail Theorem 4a any W = 0, h = 0, K(0) bounded Def.1 fails Theorem 4b

.

  e(s) = -L(s)A + L(s)B(s) + P (s)C(s) d(s)with P (s) = s 2 +hsK(s) (s 2 +(1+hs)K(s)) 2 , L(s) =1 s 2 +(1+hs)K(s) and the N × (N + 1) matrices By the triangle inequality, e(s) 2 ≤ |L(s)| A 2 + |L(s)| B(s) 2 (11) +|P (s)| C(s) 2 d(s) 2

  • |det(A)| ≤ 1 and • det(A) * trace(A) -trace(A) * | ≤ 1 -|det(A)| 2 . The determinant of T(s) imposes |det| = GW K -HW F s 2 + K =: |A| |s 2 + K| ≤ 1 ,where we have defined A = (GK -HF )W . Next,

  which by Parseval both have equivalent expressions in frequency domain. When evaluating the vector norm at a specific value of t in time domain or of s in frequency domain we write e.g. v(s) 2 = The chain (1), controlled with feedback signals u i to be designed, is called L 2 string stable if there exists a constant value c 1 such that e k (.) 2 ≤ c d i (.) 2 for all k ,

	N i=1 |v i (s)| 2 .
	Definition 1 [(L 2 , l 2 ) string stability]: The chain (1), controlled with feedback signals u i
	to be designed, is called (L 2 , l 2 ) string stable if there exists a constant value c 1 such that
	e(.) 2 ≤ c 1 d(.) 2
	for all bounded signals d, and all chain lengths N .
	Definition 2 [L 2 string stability]: