Effect of network structure on the stability margin of large vehicle formation with distributed control

We study the problem of distributed control of a large network of double-integrator agents to maintain a rigid formation. A few lead vehicles are given information on the desired trajectory of the formation; while every other vehicle uses linear controller which only depends on relative position and velocity from a few other vehicles, which are called its neighbors. A predetermined information graph defines the neighbor relationships. We limit our attention to information graphs that are D-dimensional lattices, and examine the stability margin of the closed loop, which is measured by the real part of the least stable eigenvalue of the state matrix. The stability margin is shown to decay to 0 as O(1/N2/D) when the graph is “square“, where N is the number of agents. Therefore, increasing the dimension of the information graph can improve the stability margin by a significant amount. For a non-square information graph, the stability margin can be made independent of N by choosing the “aspect ratio“ appropriately. An information graph with large D may require nodes that are physically apart to exchange information. Similarly, choosing an aspect ratio to improve stability margin may entail an increase in the number of lead vehicles. These results are useful to the designer in making trade-offs between performance and cost in designing information exchange architectures for decentralized control.


I. INTRODUCTION
We consider the problem of formation control of vehicles so that neighboring vehicles maintain a constant prespecified spacing while in motion. This problem is relevant to a number of applications such as formation flying of aerial, ground, and autonomous vehicles for surveillance, reconnaissance, mine-sweeping, etc. [1], [2], [3]. A few lead vehicles are provided information on their desired trajectories that they use in computing their control actions; while the rest of the vehicles are allowed to use only locally available information. In a distributed linear control architecture, each vehicle can measure only the relative position and velocity with respect to a number of neighbors. The neighbor relationship is predefined in terms of a graph, which we call the information graph.
The one-dimensional version of this problem, in which a string of vehicles moving in a straight line have to be controlled to maintain a constant inter-vehicle separation, has been extensively studied [4], [5], [6]. The general trend of H. Hao and P. Barooah  veerman@pdx.edu the results is that the problem scales poorly with the number of vehicles: as the number of vehicles increase the sensitivity to disturbances increases [7], [8], [9] and the stability margin decays [10], [6]. The information graphs considered in the literature are usually limited to at most two neighbors, with notable exceptions such as [8], [11], [12] that consider more general information exchange architectures.
Our goal is to examine how the stability margin scales with the size of the formation and the structure of the information graph that specifies allowable information exchange between pairs of vehicles. The real part of the least stable eigenvalue is used as a measure of the stability margin. The stability margin determines the decay rate of initial formation keeping errors. Such errors arise from poor initial arrangement of the vehicles. In this paper we limit our attention to a specific class of information graphs, namely, D-dimensional (finite) lattices. These are natural choices for information graphs in 2D or 3D formation problems in which vehicles are arranged in regular pattern and relative measurements are possible among physically closest vehicles.
Each vehicle is modeled as a double integrator, and a distributed control algorithm is studied in which every vehicle (except for a few lead vehicles) use only relative position and relative velocity with respect to its neighbors in the information graph. We show that when the network is homogeneous and symmetric (all vehicles use the same control gains and information from each neighbor is given equal weight), the stability margin decays to 0 as O(1/N 2/D ) when the graph is "square". Therefore, increasing the dimension (which may need nodes physically apart to exchange information) of the information graph can improve the stability margin by a considerable amount. For non-square information graph, the stability margin can be made independent of the number of agents by choosing the "aspect ratio" appropriately. That may entail an increase in the number of lead vehicles that have access to the formation's desired trajectory.
The results in this paper are a generalization of the results in [13], which showed that the stability margin when the information graph is a 2-D lattice decays to 0 as O(1/N ). The results in [13] were obtained by using the PDE approximation by taking the continuum limit when the number of vehicles is large. In this paper we avoid such approximation, and establish the scaling laws of the stability margin for general D-dimensional lattices. In addition, [13] considered the scenario in which the desired trajectory of the formation was one with a constant velocity, and moreover, every vehicle knew this velocity. In contrast, the control law we consider requires agents to know only the desired inter-agent separation; the overall trajectory information is made available only to the lead vehicles. This makes the model more applicable to practical formation control applications in which the formation may be required to accelerate or decelerate occasionally, and the decision to do so is made available only to the lead vehicles. Our results have some interesting connections with those in [11], which are discussed at the end.
The rest of this paper is organized as follows. Section II presents the distributed formation control problem. Section III describes the technical results, including one on eigenvalues of a grounded Laplacian matrix that plays a pivotal role on establishing the main result. The main result and its implications are presented in Section IV.
II. PROBLEM STATEMENT We consider the formation control of N identical vehicles, where the position of each vehicle is a D s -dimensional vector (with D s = 1, 2 or 3); D s is referred to as the spatial dimension of the formation. Let p (d) i ∈ R be the d-th coordinate of the i-th vehicle's position, whose dynamics are modeled by a double integrator: where u ∈ R is the control input (acceleration or deceleration command). The underlying assumption is that each of the D s coordinates of a vehicle's position can be independently actuated. We say that the vehicles are fully actuated. The spatial dimension D s is 1 for a platoon of vehicles moving in a straight line, D s = 2 for a formation of ground vehicles and D s = 3 for a formation of spatial vehicles (e.g. aircrafts).
The control objective is to make the group of vehicles track a pre-specified reference trajectory while maintaining a desired formation geometry. Reference trajectory information is available only to a set of lead vehicles. This information is represented by introducing fictitious reference vehicles, one for each lead vehicle. Each reference vehicle perfectly tracks its own desired trajectory. Each lead vehicle can measure its relative position and velocity with respect to its corresponding reference vehicle, which is equivalent to lead vehicles having knowledge of the desired trajectory of the formation. Denoting the number of reference vehicles by N r , the set V := {1, . . . , N, N + 1, . . . , N + N r } is the set of all nodes in the formation, including N real vehicles and N r fictitious reference vehicles. The desired formation geometry is specified by a desired relative position vector ∆ i,j for every pair of vehicles The desired inter-vehicular spacings have to be specified in a mutually consistent fashion, i.e., we must have ∆ i,j = ∆ i,k + ∆ k,j for every triple i, j, k ∈ V. Since we are interested in rigid formations that do not change shape over time, ∆ i,j 's are constants. To maintain a rigid formation, the control must make every vehicle track its desired trajectory. The desired trajectory of a real vehicle i, denoted by p * i (t) can be uniquely determined from the trajectories of the reference vehicles and the desired formation geometry. In particular, p * i (t) = p * j (t) + ∆ i,j where j is any reference vehicle, and p * j (t) is its trajectory. Next we define an information graph that makes it convenient to describe distributed control architectures.
Definition 1: An information graph is an undirected graph G = (V, E). The set of edges E ⊂ V × V specify which pairs of nodes (vehicles) are allowed to exchange information to compute their local control actions. Two nodes i and j are called neighbors if (i, j) ∈ E, and the set of neighbors of i are denoted by N i .
In this paper we consider the following distributed control law, whereby the control action at a vehicle depends on the relative position and velocity measurements with its neighbors in the information graph: where i ∈ {1, . . . , N } on the left hand side and j ∈ V on the right hand side. The positive constants k, b are the position and velocity feedback gains, respectively. It is assumed that vehicle i knows its own neighbors (the set N i ), and the desired spacing ∆ In this paper we restrict ourselves to a specific class of information graphs, namely a finite rectangular lattice: Definition 2 (D-dimensional lattice): A D-dimensional lattice, specifically a n 1 × n 2 × · · · × n D lattice, is a graph with n 1 n 2 . . . n D nodes, denoted by Z n1×n2···×nD .
A D-dimensional lattice is drawn in R D with a Cartesian reference frame whose axes are denoted by x 1 , x 2 , . . . , x D . Note that these coordinate axes may not be related to the coordinate axes in the physical space R Ds . We also define N d (d = 1, . . . , D) as the number of real vehicles in the x d direction. Then we have the relation N 1 N 2 . . . N D = N and We assume that there is at least one boundary every node of which is a reference vehicle. Reference vehicles are only placed on the boundaries; this typically corresponds to lead vehicles being the outermost vehicles in a formation. We call such a boundary a Dirichlet boundary. A boundary of the information graph is either a Dirichlet boundary, in which case all nodes on it are reference vehicles, or none of the nodes on it are reference vehicles.
For different configuration of Dirichlet boundaries, N d and n d has a slightly different but straightforward relation. For example, in Figure 1 (c), N 1 + 1 = n 1 since the boundary perpendicular to the positive x 1 axis is a Dirichlet boundary, while N 2 = n 2 since both boundaries perpendicular to the (a) The desired formation geometry of a 1D spatial platoon with 6 vehicles and 3 reference vehicles.   The dimension D of the information graph is distinct from the spatial dimension D s . Figure 1 shows an example of two formations in space, one with D s = 1 and the other with D s = 2. The information graph for both the formations is the same 3 × 3 two-dimensional lattice, i.e., D = 2. On account of the fully actuated dynamics and independence of control gains on d, the spatial dimension D s plays no role in the results of this paper. The dimension of the information graph D, on the other hand, will be shown to play a crucial role.

III. STABILITY MARGIN AND GROUNDED LAPLACIAN
The dynamics of the i-th vehicle are obtained by combining the open loop dynamics (1) with the control law (2), which yields (suppressing the superscript d) To facilitate analysis, we define the following tracking error: where p * i (t) is the i-th agent's desired trajectory. Note that for a rigid formation to be possible, the desired trajectories must satisfyṗ * i −ṗ * j = 0 for every i, j, which meansṗ i −ṗ j = p i −ṗ j . Therefore, substituting (4) into (3), we havë Since the trajectory of a reference vehicle is assumed to be equal to its desired trajectory,p i = 0 if i is a reference vehicle. To express the closed-loop dynamics of the formation compactly, we define the following state: Using (5), the state-space model of the vehicle formation can now be written compactly as: where A is the closed-loop state matrix.

Definition 3:
The stability margin is the absolute value of the real part of the least stable eigenvalue of the state matrix A in (6).
To facilitate analysis, we define the matrices A 1 , A 2 and L g , where and L g is the grounded (or Dirichlet) Laplacian matrix of the information graph with reference nodes defining the grounded nodes. To precisely define this matrix recall that the Laplacian matrix of a graph G = (V, E) with n nodes is defined as where deg(i) is the number of neighbors of node i in the graph. The grounded Laplacian L g matrix of G with respect to a set of grounded nodes V g ⊂ V is the submatrix of L obtained by removing from L those rows and columns corresponding to the grounded nodes in V g . This matrix occurs in the numerical solution of PDEs with Dirichlet boundary conditions and analysis of electrical networks [14]. For example, the grounded graph Laplacian of the information graph shown in Figure 1 (c), with nodes 7, 8, 9 as the grounded nodes, is:

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It is straightforward to show that where I N is the N × N identity matrix and ⊗ is the Kronecker product. Theorem 1: The spectrum of A is where σ(·) denotes the spectrum of a matrix. The proof follows the analysis in [15], please refer to [16] for the details. The next theorem, whose proof is also provided in [16], gives an explicit formula for the eigenvalues of the grounded Laplacian for the graphs considered in this paper.
Theorem 2: The eigenvalues of the grounded graph Laplacian L g of a D-dimensional information graph Z n1×...nD are positive and are given by the following formula where ℓ d = 1, . . . , N d (d = 1, . . . , D) and the indicator function I j (x d ) (j = 0, 1, 2) is defined as: 1, if there are j Dirichlet boundaries perpendicular to x d axis, 0, otherwise. (14) It follows from Theorem 2 that the minimum eigenvalue of the grounded Laplacian is given in the following corollary.
Corollary 1: Consider the D-dimensional information graph Z n1×···×nD where D 0 is the number of axes in the information graph that have Dirichlet boundaries (either one or two) perpendicular to them. Without loss of generality, let these coordinates be x 1 , . . . , x D0 . If The minimum among them is obtained by setting ℓ d = 1 for d = 1, . . . , D, which gives Since N d ≫ 1 for each d in the summation, we use cos It is straightforward (though tedious) to repeating these calculations for the other cases (when the number of Dirichlet boundaries is not exactly one). We see from these calculations that the asymptotic dependence on N p does not change from that in (15), only the coefficients differ among the different cases. This proves the result.
The next result combines the ones establishes so far to give an explicit formula for the stability margin of the formation.
Theorem 3: Let λ min be the minimum eigenvalue of the grounded Laplacian L g . The stability margin of the closed loop with N vehicles is when N p ≫ 1, where N p is defined in Corollary 1.
Proof. From Theorem 1, it follows that the eigenvalues of state matrix A, denoted by s, satisfy: where λ ℓ ∈ σ(L g ). From Theorem 2, we see that λ ℓ is positive. Since k > 0 and b > 0, it follows that A is Hurwitz. Moreover, it follows from (17) that the least stable eigenvalue of A, denoted by s + 1 , is given by: It follows from Corollary 1 that λ min can be arbitrarily small for sufficiently large N p . For a large formation, more specifally, N p is large enough so that λ min < 4k b 2 , it makes the term inside the square root in (18) negative. Following the definition of stability margin, we obtain

IV. SCALING LAWS FOR STABILITY MARGIN
The main result of the paper is the following. Theorem 4: Consider an N -vehicle formation with a Ddimensional information graph Z n1×···×nD , with vehicle dynamics (1) and control law (2), where D 0 is the number of axes in the information graph that have Dirichlet boundaries (either one or two) perpendicular to them. The closed-loop stability margin is given by when N p ≫ 1, where N p is defined in Corollary 1.
Proof. Follows from Theorem 3 and Corollary 1.
The implication of the theorem is discussed next.

A. Stability Margin with Square Information Graphs
In interpreting Theorem 4, it is useful to start with the special case of a square information graph, which has equal number of real vehicles/nodes along each coordinate axis in the drawing of the information graph.

B. Stability Margin with Non-square Information Graphs
For ease of description, we describe the idea for nonsquare information graph with only one Dirichlet boundary. The information graph with other boundary configurations can be interpreted in a similar manner. The following corollary is immediate from Theorem 4.
Corollary 3: Suppose only one of the boundaries of the information graph has lead vehicles, and let this boundary be perpendicular to x 1 axis, without loss of generality. Then, the stability margin is given by S = π 2 b/(8N 2 1 ). It follows from this result that by choosing the structure of the information graph in such a way that N 1 increases slowly in relation to N , the loss of the stability margin as a function of N can be slowed down. In fact, when N 1 is held at a constant value independent of N , the stability margin is a constant independent of the total number of vehicles! More generally, if N 1 = O(N c ), where c ∈ [0, 1] is a fixed constant, it follows from Corollary 3 that S = O(1/N 2c ) as N → ∞. If c < 1 D , the resulting reduction of S with N is slower than that obtained for a square lattice; cf. Corollary 2. This shows that within the class of D dimensional lattices (for a fixed D), certain information graphs provide better scaling of the stability margin than others. The price one pays for improving stability margin by reducing N 1 is an increase in the number of lead vehicles. This is because the number of lead vehicles, N r , is related to N 1 (under the assumptions in Corollary 3) by N r = N/N 1 . There is thus a trade-off between improved stability margin and cost of having a large number of lead vehicles.
It is important to stress that not all non-square graphs are advantageous. For example, if N 1 = O(N ), which means N 2 through N D are O(1), it follows from Corollary 3 that the stability margin is S = O(1/N 2 ). This is the same trend as in a 1-D information graph. In this case, we can say that the D dimensional information graph effectively behaves as a one dimensional graph. Figure 2 shows a few examples of information graph that are relevant to the discussion above. Figure 3 provides numerical corroboration of the discussion above. It is clear from the figure that the prediction from Corollary 3 and Theorem 4 match very well with numerical computed eigenvalues of the state matrix A.

V. CONCLUSION AND DISCUSSION
We study the problem of distributed control of a large network of double-integrator agents with D-dimensional information graph. The controller used is a linear PD controller which depends on information on relative position and velocity from its neighbors to compute its own control.
x 1 We showed that the stability margin scales as O(1/N 2/D ) for a D-dimensional square information graph. Therefore, increasing the dimension of the information graph can improve the stability margin by a considerable amount. For non-square information graph, the stability margin can be made independent of the number of agents by choosing the "aspect ratio" appropriately. However, it should be taken into account that increasing the dimension of the information graph or choosing a beneficial aspect ratio may require long range communication or entail an increase in the number of lead vehicles. Thus, a larger stability margin can be achieved by designing the graph (and its boundary conditions) appropriately, but that may be accompanied by the increased cost of long-range communication or large number of lead vehicles. These results are therefore useful to the designer in making trade-offs between performance and cost in designing information exchange architectures for decentralized control.
Our results for square D-lattices are complementary to those of [11], in which the effect of graph dimension on the response of the closed loop to stochastic disturbances is quantified in terms of "microscopic" and "macroscopic" measures. It was shown in [11] that for D > 5, these performance measures become independent of N , while for smaller D, the performance becomes worse without bound as the number of vehicles increase. In contrast, we showed that the stability margin decays to 0 as N increases in every D. Though the decay is slower for larger D, it is never independent of N . To achieve a size-independent stability margin, the graph needs to be non-square. Since the analysis of [11] is done in the spatial Fourier domain, it is not clear if non-square lattices with boundaries can be handled in that framework.