Stability of Linear Flocks on a Ring Road

A flock consists of a large number of moving physical objects, called agents, with their positions being controlled in such a way that they move along a prescribed path, in a prescribed and fixed configuration (or in formation). Each agent knows the position and velocity of only a few other agents, and this flux of information defines a communication graph.

In [5] graph theory and linear systems techniques are combined to provide a framework for studying the control of formations. The main tools of graph theory that are related to the problem are the directed graph Laplacians and the connectedness of the graph. Linear feedback is then used to stabilize the patterns.

More recently [6] studied a system of coupled linear differential equations describing the movement of cars in $\mathbb{R}$, where each car reacts only to its immediate neighbors, and only the movement of the first agent (the leader) is independent from the rest of the group. In the paper it was proved that When equal attention is paid to both neighbors perturbations in the orbit of the leader grow as they propagate through the flock. In fact, perturbations grow proportional to size of the flock: when the leader’s perturbation has amplitude 1, then the perturbation in the orbit of the agent furthest away from the leader will be proportional to $N$ (the size of the flock).

The aim of this paper is first of all to study the (asymptotic) stability of a family of such systems. This is answered in detail in Theorems 3.2 and 3.3. Next we assume the flock has a leader that chooses its orbit independently of the other members of the flock. We then analyze how exactly the system converges to a stable flight pattern if the leader changes its orbit. This question only makes sense when the system is already asymptotically stable, which is therefore assumed henceforth. The latter question is important in applications as too great fluctuations in the course of convergence to a coherent flight pattern will make that flock unviable.

In all of these arguments we closely follow the reasoning set forth in previous works [7, 8, 9]. However there are two important differences. The first is that the farthest member of the flock (in this work) is coupled to the leader. In the language of partial differential equations, this is akin to changing a boundary condition. The reason is twofold. Changing the boundary condition can greatly aid the mathematics, and therefore help to gain insight. The second reason is a deeper one: we do not know how these boundary condition influence the stability of these systems, and thus this note can be viewed a test case (when compared with the papers just cited). The other difference with the previous papers is that we here allow the weight of the coupling with neighbors to be negative. While at first glance this seems a little odd, there is a good reason to do so, if one hopes to study systems with more than nearest neighbor coupling. Suppose for example that one models local interaction as a discretization of a fourth derivative, a very natural idea. However the couplings to the first and second nearest neighbors will now have different signs. This goes against the grain of what one knows about Laplacian systems in general, where in general all couplings must have the same sign (see [1]). In this case we managed to overcome that problem and analyze stability also when the signs of the (nearest neighbor) interactions are different. (By necessity they must add up to 1.)

The outline of this paper is as follows. In the next section we start by specifying the model. Next we discuss the asymptotic stability of the model. Following [8] we introduce two other types of stability for flocks. These describe the effect of perturbations in the leaders motion on the outlying members of the flocks. A flock with $N$ agents is harmonically stable if the effect of a harmonic motion of the leader on the outlying members grows less than exponentially fast in $N$ (everything else held fixed). A flock is said to be impulse stable if the effect of the leader being kicked is less than exponential on the outlying members. Thus is section 4, we discuss harmonic stability of the model. The problem of impulse stability is still unsolved. We present a few comments on that problem in section 5. (The appendix contains technical results and is included for completeness.)

We begin this section establishing the model of this work. The $N+1$ agents move in $\mathbb{R}$ along orbits $x_{i}(t)$, $i\in\{0,\cdots n\}$, with velocities $x_{i}^{\prime}(t)$. When they are moving in the desired formation their velocities are equal and their relative positions are determined by $N+1$ a priori given constants $h_{i}$:

We write the equation of motion for this model in terms of

These then have the following form:

for all $i=1,\ldots,N$, and

a priori given. We will assume the feedback parameters $f$, $g$ are negative reals and the weight $\rho$ is a arbitrary real number.

It is intuitively convenient, though not necessary, to keep a particular realization of the above system in mind. Identify $x=N+1$ with $x=0$, so that the agents move on a (topological) circle. Suppose further that the offsets $h_{i}$ are given by $h_{i}=-i\mod N$. Now the desired configuration is that of $N+1$ agents moving at constant speed and uniformly distributed along a circle. (This explains our title.)

Our strategy here is primarily studying qualitative aspects of the solution of (2.3)-(2.4) as we let $N$ tend to infinity while keeping all other parameters ($\rho$, $f$, and $g$) fixed. In particular we wish to understand (1) when the system is asymptotically stable and (2) how does it converge to its equilibrium when it is asymptotically stable. This stable equilibrium is given by the two parameter family of orbits:

and

These orbits are called in formation orbits (for a more detailed discussion, cf. [6, 7, 8, 9]).

It is advantageous to write (2.3)-(2.4) in a more compact form:

The system can now be recast as a first order ordinary differential equation:

The matrix $M$ and the vector $\Gamma_{0}$ are defined below.

Setting

the matrix $P$ defined by

where $I$ is the $N$-dimensional identity matrix, is called the reduced graph Laplacian. It describes the flow of information among the agents, with the exception of the leader (hence the word ‘reduced’).

The orbit of the leader is assumed to be beforehand given and therefore only appears in the forcing term $\Gamma_{0}(t)$. We will refer to this agent as an independent leader. Analyzing (2.3)-(2.4) and assuming without loss of generality that $h_{0}=0$, one gathers that:

In order to define $M$ matrix of (2.5) in terms of these quantities, we use the Kronecker product, $\otimes$,

where $A$ and $K$ the $2\times 2$ matrices:

The advantage of this somewhat roundabout way of defining the matrix $M$ is that in the eigenvalues of the reduced Laplacian $P$ can be given explicitly. From that the eigenvalues of $M$ can then be derived.

The system defined in (2.3)-(2.4) is called asymptotically stable if all eigenvalues of $M$ have negative real part. Assuming the $\Gamma_{0}(t)=0$, for $t>t_{0}$, the solution of the system tends to $0$ exponentially fast (in $t$) if and only if the system is asymptotically stable. This corresponds to the classical notion of asymptotic stability.

The study of the eigenvalues of the $N\times N$ matrix $Q_{\rho}$ defined in (2.6) constitutes a special case of results given in [2, 3]. They are given by:

for all real $\rho$. These eigenvalues are all real if and only if $\rho\in[0,1]$ and imaginary otherwise, and the locus of the set of eigenvalues is invariant under multiplication by $-1$. We have:

One can show that the eigenvalues of $M=I\otimes A+P\otimes K$ are the solutions $\nu_{\ell\pm}$ of the equation

where $\lambda_{\ell}$ runs through the spectrum of $P$ (cf. [4, 5, 6, 7]). So we have:

These expressions are the same as the corresponding ones for a slghtly differnt one dimensional flock given in [8]. There it was assumed that $\rho\in[0,1)$. We extend that research by looking at real values of $\rho$ outside the interval $[0,1]$. This may at first seem obscure. Here however is the motivation. Suppose for a moment that one allows each agent to interact with two neighbors on either side, then one could be tempted to model this interaction as a discretization of a fourth derivative in the spatial variable. In that case some of the weights of the interaction would have negative values.

The system is harmonically unstable roughly if oscillatory or harmonic perturbations in the orbit of the leader (that is: of the form $e^{i\omega t}$) have their amplitude magnified by a factor that is exponentially large in $N$ (cf. [8]).

We first need some notation. It will often be convenient to replace $\rho$ by a different constant:

or, equivalently,

We also define (for $\rho\neq 0$):

where

We will assume here without further proof that fluctuations of the leader that are propagated through the system are largest for the agents furthest away from the leader, i.e. halfway in the flock. For simplicity we only consider in this section the response of the agent $k=\frac{N+1}{2}$ where $N$ is odd (and large).

Recall that by Theorems 3.2 and 3.3 the system is asymptotically stable if and only if $\rho\in[0,1]$ and $f$ and $g$ negative or else $\rho\in\mathbb{R}\backslash[0,1]$ and both of the following hold:

• •

  $f$ and $g$ are negative, and

• •

  $f+g^{2}\geq 0$ or else $4|\rho(1-\rho)|\leq\frac{-g^{2}}{f+g^{2}}$.

Recall from the proof of Theorem 4.1 that if $z_{0}(t)$ equals $e^{i\omega t}$ then $z_{k}$ is asymptotic to $a_{k}\,e^{i\omega t}$. Thus the amplification at the $k$-th agent of the leader’s signal is given $a_{k}(\omega)$. We need to determine whether $\max_{k}\sup_{\omega}|a_{k}(\omega))|$ is exponential in $N$ (instability) or less than exponential (stability). We may assume $k=M$. So let

Following [8], we call a system harmonically stable if it is asymptotically stable and if

Suppose now that the flock is in a stable equilibrium (i.e. moves stably in formation) when the leader suddenly and quickly changes its velocity. Roughly speaking we call the system impulse stable when the physical response of the other agents (i.e., the acceleration, or the velocity, or the position) is less than exponential in $M$. As observed in the proof of Theorem 4.3, the case $\rho=\frac{1}{2}$ is extremely similar to the problem studied in [7], and the solution is in fact similar to the one given in that case. The calculations there indicate responses that are ‘proportional’ to $M$. Thus may we conclude that here also:

The cases $\rho\neq 1/2$ lead to problems similar to the one that remained unsolved in [9]. It seems very likely at this point that most of these cases are impulse unstable, though this is by no means obvious or known. More specifically, as we vary $f$, $g$ and $\rho\neq 1/2$, we do not even qualitatively understand the large $N$ behavior of the motion of these flocks. This question is relevant because velocity changes of the leader are a natural context in which stability plays an important role for the cohesion of the flock. We might think for example of a lead car accelerating when a traffic light turns green or a large flock of animals changing course because outlying members spotted and try evade a predator.

The mathematical problem boils down to an inverse Fourier transform of $a_{M}(\omega)$ where $M$ is large. Current standard integration techniques do not readily give asymptotic (in $M$) expressions for such integrals. We do not address this challenging question any further here, leaving it as an open problem for future research.

In this section we gather some technical results which we exhibited partially before in [8]. The results there were proved only for $\rho\in[0,1]$. Some of the calculations extend verbatim (or almost) to all $\rho\in\mathbb{R}$. The proof of the main result, Proposition 6.3, had to be modified substantially however.

The complicated looking conditions in the following proposition are nothing but the conditions that insure asymptotic stability (see Theorems 3.2 and 3.3).