Coherence in Large-Scale Networks: Dimension-Dependent Limitations of Local Feedback

1 Coherence in Lar ge-Scale Networks: Dimension-Dependent Limitations of Local Feedback Bassam Bamieh, F ellow , IEEE , Mihailo R. Jov anovi ´ c, Member , IEEE , Partha Mitra, and Stacy P atterson Abstract —W e consider distributed consensus and vehicular formation control problems. Specifically we address the question of whether local feedback is sufficient to maintain coherence in large-scale networks subject to stochastic disturbances. W e define macroscopic performance measures which are global quantities that capture the notion of coherence; a notion of global order that quantifies how closely the f ormation resembles a solid object. W e consider how these measur es scale asymptotically with network size in the topologies of regular lattices in 1, 2 and higher dimensions, with vehicular platoons corresponding to the 1 dimensional case. A common phenomenon appears where a higher spatial dimension implies a more favorable scaling of coherence measures, with a dimensions of 3 being necessary to achieve coher ence in consensus and vehicular formations under certain conditions. In particular , we show that it is impossible to hav e large coherent one dimensional vehicular platoons with only local feedback. W e analyze these effects in terms of the underlying energetic modes of motion, showing that they take the form of large temporal and spatial scales resulting in an accordion-like motion of formations. A conclusion can be drawn that in low spatial dimensions, local feedback is unable to regulate large- scale disturbances, but it can in higher spatial dimensions. This phenomenon is distinct from, and unrelated to string instability issues which ar e commonly encounter ed in contr ol pr oblems for automated highways. I . I N T R O D U C T I O N The control problem for strings of vehicles (the so-called platooning problem) has been extensi vely studied in the last two decades, with original problem formulations and studies dating back to the 60’ s [1]–[5]. These problems are also intimately related to more recent formation flying and for- mation control problems [6]. It has long been observed in platooning problems that to achiev e reasonable performance, certain global information such as leader’ s position or state need to be broadcast to the entire formation. A precise analysis of the limits of performance associated with localized versus global control strategies does not appear to exist in B. Bamieh is with the Department of Mechanical Engineering, Univ ersity of California, Santa Barbara, CA 93106, USA ( bamieh@engr.ucsb.edu ). M. R. Jovanovi ´ c is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA ( mihailo@umn.edu ). Partha Mitra is with Cold Spring Harbor Laboratory , Cold Spring Harbor , NY , USA ( mitra@cshl.edu ). S. Patterson is with the Department of Electrical Engineering, T echnion - Israel Institute of T echnology , Haifa, 32000, Israel ( stacyp@ee.technion.ac.il ). This research is partially supported by NSF grants ECCS-0802008 and CMMI-0626170, and AFOSR F A9550-10-1-0143. the formation control literature. In this paper we study the platooning problem as the 1 dimensional version of a more general vehicular formations control problem on regular lat- tices in arbitrary spatial dimensions. For such problems, we in vestigate the limits of performance of any local feedback law that is globally stabilizing. In particular , we propose and study measures of the coherence of the formation. These are measures that capture the notion of how well the formation resembles a rigid lattice or a solid object. The coherence of a formation is a different concept from, and often unrelated to, string instability . In the platooning case (i.e. 1 dimensional formations), which turns out to be most problematic, a localized feedback control la w may posses string stability in the sense that the effects of any injected disturbance do not gro w with spatial location. Howe ver , as we show in this paper , it is impossible to achie ve a large coherent formation with only localized feedback if all v ehicles are subject to any amount of distributed stochastic disturbances. The net effect is that with the best localized feedback, a 1 dimensional formation will appear to behave well on a “micro- scopic” scale in the sense that distances between neighboring vehicles will be well regulated. Howe ver , if a large formation is observed in its entirety , it will appear to ha ve temporally slow , long spatial wa velength modes that are unregulated, resembling an “accordion” type of motion. This is not a safety issue, since the formation is microscopically well regulated, but it might effect throughput performance in a platooning arrangement since throughput does depend on the coherence or rigidity of the formation. The phenomenon that we discuss occurs in both consensus algorithms and v ehicular formation problems. W e therefore treat both as instances of networked dynamical systems with first order and second order local dynamics respectiv ely . Both problems are set up in the d -dimensional torus Z d N . W e be gin in section II with problem formulations of the consensus type and vehicular formations, where we view the former as a first order dynamics version of the latter . In section III, we define macroscopic and microscopic measures of performance in terms of variances of various quantities across the net- work. W e argue that the macroscopic measures capture the notion of coherence. W e also present compact formulae for calculating those measures as H 2 norms of systems with suitably defined output signals. These norms are calculated using traces of system Grammians, which in turn are related to sums inv olving eigen v alues of the underlying system and feedback gains matrices. Since the network topologies we con- 2 sider are built over T ori networks, these system matrices are multi-dimensional circulant operators, and their eigen values are calculated as the values of the Fourier symbols of the underlying feedback operators, thus allo wing for a rather direct relation between the structure of the feedback gains and the system’ s norms. Much of the remainder of the paper is de voted to establishing asymptotic (in network size) bounds for these performance measures for each underlying spatial dimension. Section IV establishes upper bounds of standard algorithms, while section V is dev oted to establishing lower bounds for any algorithm that satisfies a certain number of structural assumptions including the locality of feedback and bounded- ness of control effort. This shows that asymptotic limits of performance are determined by the network structure rather than the selection of parameters of the feedback algorithm. W e pay particular attention to the role of control effort as our lo wer bounds are established for control la ws that hav e bounded control effort in a stochastic sense. Some numerical examples illustrating the lack-of-coherence phenomenon are presented in section VI, as well as an illustration of ho w it is distinct from string instability . The interested reader may initially skim this section which numerically illustrates the basic phenomenon we study analytically in the remainder of the paper . W e end in section VII with a discussion of related work in which various versions of this phenomenon were observ ed, as well as a discussion of some open questions. Notation and Pr eliminaries The networks we consider are built o ver the d-dimensional T orus Z d N . The one-dimensional T orus Z N is simply the set of integers { 0 , 1 , . . . , N − 1 } with addition modulo N ( mo d N ), and Z d N is the direct product of d copies of Z N . Functions defined on Z d N are called arrays , and we use multi-index notation for them, as in a k = a ( k 1 ,...,k d ) to denote indi vidual entries of an array . Indices are added in the Z d N arithmetic as follows ( r 1 , . . . , r d ) = ( k 1 , . . . , k d ) + ( l 1 , . . . , l d ) m r i = ( k i + l i ) N , i = 1 , . . . , d, where () N is the operation mod N . The set Z d N and the corresponding addition operation can be visualized as a “circu- lant” graph in d-dimensional space with edge nodes connected to nodes on corresponding opposite edge of the graph. The multi-dimensional Discrete F ourier Transform is used throughout. All states are multi-dimensional arrays which we define as real or complex vector-v alued functions on the T orus Z d N . The Fourier transform (Discrete Fourier Transform) of an array a is denoted with ˆ a . W e refer to indices of spatial Fourier transforms as wavenumbers . Generally , we use k and l for spatial indices and n and m for wavenumbers. F or example, an array a ( k 1 ,...,k d ) has ( k 1 , . . . , k d ) as the spatial index, while its Fourier transform ˆ a ( n 1 ,...,n d ) has the index ( n 1 , . . . , n d ) as the wa venumber . The wav enumber is simply a spatial frequency variable. Some elementary properties of this Fourier transform are summarized in Appendix A, Con volution operators arise naturally over Z d N . Let a be any array of numbers (or matrices) over Z d N , that is a : Z d N → C (or C n × n ). Then the operator T a of multi-dimensional circular con volution with the array a is defined as follows g = T a f = a ? f m g ( k 1 ,...,k d ) = X ( l 1 ,...,l d ) ∈ Z d N a ( k 1 ,...,k d ) − ( l 1 ,...,l d ) f ( l 1 ,...,l d ) . Note that f and g may be scalar or vector -valued (depending on whether a is scalar or matrix-valued respectively), and that the arithmetic for ( k 1 , . . . , k d ) − ( l 1 , . . . , l d ) is done in Z d N , i.e. arithmetic mo d N in each index as described abo ve. It is important to distinguish between an array a and the corresponding linear operator T a . The Fourier transform ˆ a of the array a is called the F ourier symbol of the operator T a . It is a standard fact that the eigen values of the operator T a are exactly the values of the Fourier transform ˆ a , i.e. the values of its Fourier symbol. When a is matrix valued, then the eign values of T a are the union of all eigen values of ˆ a ( n 1 ,...,n d ) as the wa venumber ( n 1 , . . . , n d ) runs through Z d N , i.e. σ ( T a ) = [ ( n 1 ,...,n d ) ∈ Z d N σ  ˆ a ( n 1 ,...,n d )  , where σ ( . ) refers to the spectrum of a matrix or operator (all finite-dimensional in our case). In this paper , we use the term dimension to refer exclusiv ely to the spatial dimension of underlying networks. T o avoid confusion with the notion of state dimension, we refer to the dimension of the state space of any dynamical system as the or der of that dynamical system. The vector dimension of signals is mostly suppressed to keep the notation from being cumbersome. For example, the state of node ( k 1 , . . . , k d ) in the d-dimensional T orus is written as x ( k 1 ,...,k d ) ( t ) . It is a scalar-valued signal for consensus problems, and vector- valued (in R d ) signal for vehicular formation problems. W e use M T to denote the transpose of a matrix M , and M ∗ to denote the complex-conjugate transpose of a matrix M or the adjoint of an operator M . Although all operators in this paper are finite dimensional, we sometimes refer to them as operators rather than matrices since we often av oid writing the cumbersome explicit matrix representations (such as in the case of multi-dimensional con volution operators). I I . P R O B L E M F O R M U L A T I O N W e formulate two types of problems, consensus and ve- hicular formations. The mathematical setting is analogous in both problems, with the main difference being that vehicular models have two states (position and v elocity) locally at each site in contrast to a scalar local state in consensus problem. This difference leads to more sev ere asymptotic scalings in vehicular formations as will be shown in the sequel. 3 A. Consensus with random insertions/deletions W e begin by formulating a continuous-time version of the consensus algorithm with additive stochastic disturbances in the dynamics [7], [8]. As opposed to standard consensus algorithms without additive disturbances, nodes do not achieve equilibrium asymptotically but fluctuate around the equilib- rium, and the v ariance of this fluctuation is a measure of ho w well approximate consensus is achieved. This formulation can be used to model scenarios such as load balancing over a distributed file system, where the additiv e noise represents file insertion and deletion, parallel processing systems where the noise processes model job creation and completion, or flocking problems in the presence of random forcing disturbances. W e consider a consensus algorithm over undirected tori, Z d N , where the deriv ativ e of the scalar state at each node is determined as a weighted av erage of the dif ferences between that node and all its 2 d neighbors. One possible such algorithm is gi ven by ˙ x k = β  x ( k 1 − 1 ,...,k d ) − x k  + · · · +  x ( k 1 ,...,k d +1) − x k  + w k , = ( − 2 dβ ) x k + β  x ( k 1 − 1 ,...,k d ) + x ( k 1 +1 ,...,k d ) + · · · + x ( k 1 ,...,k d − 1) + x ( k 1 ,...,k d +1)  + w k , (1) where we hav e used equal weights β > 0 for all the differ - ences. The process disturbance w is a mutually uncorrelated white stochastic process. W e call this the standar d consensus algorithm in this paper since it is essentially the same as other well-studied consensus algorithms [9]–[13]. The sum in the equation abov e can be written as a multimulti-dimensional con volution by defining the array O ( k 1 ,...,k d ) =    − 2 dβ k 1 = · · · = k d = 0 , β k i = ± 1 , and k j = 0 for i 6 = j, 0 otherwise . (2) The system (1) can then be written as ˙ x = O ? x + w , (3) where ? is circular con volution in Z d N . W e recall that we use the operator notation T a x := a ? x to indicate the circulant operator of con volution with any array a . W ith this notation, a general spatially in variant consensus algorithm can be written abstractly as ˙ x = T a x + w , (4) for any array a defined over Z d N . Such algorithms can be regarded as a combination of open loop dynamics ˙ x k = u k + w k , k ∈ Z d N , with the feedback “control” u = T a x , where the feedback operator array is to be suitably designed. With this point of view , consensus algorithms can be thought of as first order dynamics versions of vehicular formation problems that we introduce next. B. V ehicular F ormations Consider N d identical vehicles arranged in a d -dimensional torus, Z d N , with the double integrator dynamics ¨ x ( k 1 ,...,k d ) = u ( k 1 ,...,k d ) + w ( k 1 ,...,k d ) , (5) where ( k 1 , . . . , k d ) is a multi-index with each k i ∈ Z N , u is the control input and w is a mutually uncorrelated white stochastic process which can be considered to model random forcing. In the sequel, we will also consider the consequences of the presence of viscous friction terms in models of the form ¨ x ( k 1 ,...,k d ) = − µ ˙ x ( k 1 ,...,k d ) + u ( k 1 ,...,k d ) + w ( k 1 ,...,k d ) , (6) where µ > 0 is the linearized drag coefficient per unit mass. Each position vector x k is a d -dimensional vector with components x k =  x 1 k · · · x d k  T . The objective is to hav e the k th vehicle in the formation follow the desired trajectory ¯ x k ¯ x k := ¯ v t + k ∆ ⇔    ¯ x 1 k . . . ¯ x d k    :=    ¯ v 1 . . . ¯ v d    t +    k 1 . . . k d    ∆ , which means that all vehicles are to mov e with constant heading velocity ¯ v while maintaining their respecti ve position in a Z d N grid with spacing of ∆ in each dimension. The situation of different spacings in different directions can be similarly represented, b ut is not considered for notational simplicity . The de viations from desired trajectory are defined as ˜ x k := x k − ¯ x k , ˜ v k := ˙ x k − ¯ v . W e assume the control input to be full state feedback and linear in the variables ˜ x and ˜ v (therefore affine linear in x and v ), i.e. u = G ˜ x + F ˜ v , where G and F are the linear feedback operators. The equations of motion for the controlled system are thus d dt  ˜ x ˜ v  =  0 I G F   ˜ x ˜ v  +  0 I  w . (7) W e note that the above equations are written in operator form, i.e. by suppressing the spatial index of all the variables. Example: The operators G and F will have some very special structure depending on assumptions of the type of feedback and measurements av ailable. Consider for example a feedback control of the k th vehicle (in a one dimensional formation) of the follo wing form u k = g + ( x k +1 − x k − ∆) + g − ( x k − 1 − x k − ∆) + f + ( v k +1 − v k ) + f − ( v k − 1 − v k ) + g o ( x k − ¯ x k ) + f o ( v k − ¯ v ) , where the g ’ s and f ’ s are design constants. The first two lines represent look-ahead and look-behind position and velocity error feedbacks respectiv ely . W e refer to such terms as relative feedback since they only in volve measurements of dif ferences. On the other hand, terms in the last line require kno wledge of positions and velocities in an absolute coordinate system (a grid moving at constant velocity), and we thus refer to such 4 terms as absolute feedbac k . For later reference, it is instructive to write the feedback in the abo ve example in terms of the state variables ˜ x and ˜ v as u k = g + ( ˜ x k +1 − ˜ x k ) + g − ( ˜ x k − 1 − ˜ x k ) + f + ( ˜ v k +1 − ˜ v k ) + f − ( ˜ v k − 1 − ˜ v k ) + g o ˜ x k + f o ˜ v k . (8) C. Structural assumptions W e now list the various assumptions that can be imposed on system operators and on the control feedbacks G and F . These are structural restrictions representing the structure of open loop dynamics and measurements, and the type of feedback control av ailable respecti vely . (A1) Spatial In variance. All operators are spatially in- variant with respect to Z d N . This implies that they are con volution operators. For instance, the operation Gx can be written as the con volution (over Z d N ) of the array x with an array { G ( k 1 ,...,k d ) } ( Gx ) k = X l ∈ Z d N G k − l x l , (9) where the arithmetic for k − l is done in Z d N . For each k , the array element G k is a d × d matrix ( G is then an N d × N d operator). Note that in the absence of spatial inv ariance, each term of the sum in (9) would need to be written as G k,l x l . That is, one would require a two-indexed array of matrices G k,l rather than a single-index ed array . In the example abo ve of a one dimensional circular forma- tion, the array elements for position feedback are gi ven by  g o − g + − g −  , g + , 0 , . . . , 0 , g −  . (A2) Relative vs. Absolute Feedback. W e use the term Relativ e Feedback when given feedback in volves only dif- ferences between quantities. For example, in position feed- back, this implies that for each term of the form α x ( k 1 ,...,k d ) in the con volution, another term of the form − αx ( l 1 ,...,l d ) occurs for some other multi-index l . This implies that the array G has the property X k ∈ Z d N G k = 0 . (10) W e use the term Absolute Feedback when gi ven operator does not satisfy this assumption. Note that in the example above, relative position feedback corresponds to g o = 0 , and in this case, condition (10) is satisfied. (A3) Locality . The feedbacks use only local information from a neighborhood of width 2 q , where q is independent of N . Specifically , G ( k 1 ,...,k d ) = 0 , if k i > q , and k i < N − q for any i ∈ { 1 , . . . , d } . (11) The same condition holds for F . (A4) Reflection Symmetry . The interactions between ve- hicles have mirror symmetry . This has the consequence that the arrays representing G and F hav e even symmetry , e.g. for each nonzero term like αG ( k 1 ,...,k d ) in the array there is a corresponding term αG ( − k 1 ,..., − k d ) . This in particular implies that the Fourier symbols of G and F are real valued. In the example above, this condition giv es g + = g − and f + = f − . (A5) Coordinate Decoupling . For d ≥ 2 , feedback control of thrust in each coordinate direction depends only on mea- surements of position and v elocity error vector components in that coordinate. This is equi valent to imposing that each array element G k and F k are d × d diagonal matrices. For further simplicity we assume those diagonal elements to be equal, i.e. G k = diag { g k , . . . , g k } , F k = diag { f k , . . . , f k } . (12) This in effect renders the matrix-vector con volution in (9) into d decoupled scalar con volutions. Assumptions (A1) through (A3) appear to be important for subsequent dev elopments, while assumptions (A4) and (A5) are made to simplify calculations. I I I . P E R F O R M A N C E M E A S U R E S W e will consider how v arious performance measures scale with system size for the consensus and vehicle formations problems. Some of these measures can be quantified as steady state v ariances of outputs of linear systems dri ven by stochastic inputs, so we consider some generalities first. Consider a general linear system dri ven by zero mean white noise with unit cov ariance ˙ x = Ax + B w , y = H x. Since we are interested in cases where A is not necessarily Hurwitz (typically due to a single unstable mode at the origin representing motion of the mean), the state x may not have finite steady state variances. Howe ver , in all cases we consider here the outputs y do hav e finite variances, i.e. the unstable modes of A are not observable from y . In such cases, the output does have a finite steady state variance, which is quantified by the square of the H 2 norm of the system from w to y V := X k ∈ Z d N lim t →∞ E { y ∗ k ( t ) y k ( t ) } , (13) where the index k ranges over all “sites” in the lattice Z d N . W e are interested in spatially in variant problems over dis- crete T ori. This type of in v ariance implies that the v ariances of all outputs are equal, i.e. E { y ∗ k y k } is independent of k . Thus, if the output v ariance at a giv en site is to be computed, it is simply the total H 2 norm di vided by the system size E { y ∗ k y k } = 1 M X l ∈ Z d N E { y ∗ l y l } = V M , (14) where M is the size of the system ( M = N d for d -dimensional T ori). W e refer to quantities like (14) as individual output variances . 5 Next, we define sev eral dif ferent performance measures and giv e the corresponding output operators for each measure for both the consensus and vehicular formation problems. In the vehicular formation problem, we assume for simplicity that the output in volv es positions only , and thus the output equation has the form y =  C 0   ˜ x ˜ v  , i.e. H =  C 0  , where C is a circulant operator . A consensus problem with the same performance measure has a corresponding output equation of the form (with the same C operator) y = C x. P erformance Measur es: W e now list the three different performance measures we consider . (P1) Local error . This is a measure of the dif ference between neighboring nodes or vehicles. For the consensus problem, the k th output (in the case of one dimension) is defined by y k := x k − x k − 1 . For the case of vehicular formations, local error is the difference of neighboring vehicles positions from desired spacing, which can equi valently be written as y k := ˜ x k − ˜ x k − 1 . The output operator is then giv en by C := ( I − D ) , where D is the right shift operator , ( Dx ) k := x k − 1 . In the case of d dimensions, we define a v ector output that contains as components the local error in each respective dimension, i.e. C := 1 √ 2 d  I − D 1 · · · I − D d  T , (15) where D r is the right shift along the r th dimension, i.e. ( D r x ) ( k 1 ,...,k r ,...,k d ) := x ( k 1 ,...,k r − 1 ,...,k d ) , and 1 / √ 2 d is a con venient normalization factor . This operator is closely related to the standard consensus operator O in Eq. (2) by the follo wing easily established identity C ∗ C = − 1 2 dβ O . (16) (P2) Long range deviation (Disorder). In the consensus problem, this corresponds to measuring the disagreement between the two furthest nodes in the network graph. As- sume for simplicity that N is even and we are in dimension 1. Then, the most distant node from node k is N 2 hops away , and we define y k := x k − x k + N 2 . In the vehicular formation problem, long range deviation corresponds to measuring the de viation of the distance between the two most distant vehicles from what it should be. The most distant vehicle to the k th one is the vehicle index ed by k + N 2 . The desired distance between them is ∆ N 2 , and the de viation from this distance is y k := x k − x k + N 2 − ∆ N 2 = ˜ x k − ˜ x k + N 2 . (17) W e consider the variance of this quantity to be a measure of disorder , reflecting the lack of “end-to-end rigidity” in the vehicle formation. Generalizing this measure to d dimensions yields an output operator of the form C := T ( δ 0 − δ ( N/ 2 ,...,N/ 2) ) , (18) i.e. the operator of con volution with the array 1 δ 0 − δ ( N/ 2 ,...,N/ 2) . (P3) Deviation from a verage. F or the consensus problem, this quantity measures the deviation of each state from the av erage of all states, y k := x k − 1 M X l ∈ Z d N x l . (19) In operator form we ha ve y = ( I − T ¯ 1 ) x , where ¯ 1 is the array of all elements equal to 1 / M . In vehicular formations, this measure can be interpreted as the deviation of each vehicle’ s position error from the av erage of the ov erall position error y = ( I − T ¯ 1 ) ˜ x . W e note that performance measures (P1) through (P3) are such that C can be represented as a con volution with an array { C k } which has the property P k ∈ Z d N C k = 0 . This condition causes the mean mode at zero to be unobservable, and thus guarantees that all outputs defined abov e have finite variances. W e refer to the performance measure (P1) as a microscopic err or since it inv olves quantities local to any given site. This is in contrast to the measures (P2) and (P3) which inv olve quanti- ties that are far apart in the network, and we thus refer to these as macr oscopic err ors . W e consider the macroscopic errors as measures of disorder or equiv alently , lack of coherence. As we will show in the sequel, both macroscopic measures scale similarly asymptotically with system size, which justifies using either of them as a measure of disorder . F ormulae for variances: Since we consider spatially inv ari- ant systems and in particular systems on the discrete T ori Z d N , it is possible to deri ve formulae for the abo ve defined measures in terms of the Fourier symbols of the operators K , F and C . Recall the state space formula for the H 2 norm V defined in (13) V = tr  Z ∞ 0 B ∗ e A ∗ t H ∗ H e At B dt  . When A , B and H are circulant operators, traces can be rewritten in terms of their respective Fourier symbols (see (47)) as V = tr X n Z ∞ 0 ˆ B ∗ n e ˆ A ∗ n t ˆ H ∗ n ˆ H n e ˆ A n t ˆ B n dt ! (20) = X n tr  ˆ B ∗ n ˆ P n ˆ B n  , (21) 1 By a slight abuse of notation, we define the shifted Kronecker delta δ l k := δ k − l , where δ k = 1 for k = 0 , and zero otherwise, is the standard Kronecker delta. With this notation, δ 0 is also the standard Kronecker delta. 6 where the indi vidual integrals are defined as ˆ P n := Z ∞ 0 e ˆ A ∗ n t ˆ H ∗ n ˆ H n e ˆ A n t dt. (22) If ˆ A n is Hurwitz, then ˆ P n can be obtained by solving the L yapuno v equation ˆ A ∗ n ˆ P n + ˆ P n ˆ A n = − ˆ H ∗ n ˆ H n . (23) For wa venumbers n for which ˆ A n is not Hurwitz, ˆ P n is still finite if the non-Hurwitz modes of ˆ A n are not observable from ˆ H n . In this case we can analyze the integral in (22) on a case by case basis. The L yapunov equations are easy to solve in the Fourier domain. Equation (23) is a scalar equation in the Consensus case and a 2 d × 2 d matrix equation in the V ehicular case 2 . The two respectiv e calculations are summarized in the next lemma. The proof is gi ven in the Appendix. Lemma 3.1: The output variances (13) for the consensus and vehicular problems satisfying assumptions (A1) - (A5) are giv en by V c = − 1 2 X n 6 =0 , n ∈ Z d N | ˆ c n | 2 < (ˆ a n ) , (24) V v = d 2 X n 6 =0 , n ∈ Z d N | ˆ c n | 2 ˆ g n ˆ f n , (25) where < ( ˆ a n ) is the real part of ˆ a n , ˆ c is the Fourier symbol of the output operator corresponding to the performance index under consideration, and ˆ a , ˆ g and ˆ f are the Fourier symbols of the consensus operator (4), and the position and velocity feedback operators (12) respecti vely . These e xpressions can then be worked out for the variety of output operators C representing the different performance measures defined earlier . The next result presents a summary of those calculations for the six dif ferent cases. Cor ollary 3.2: The follo wing are performance measures (P1) , (P2) and (P3) expressed in terms of the Fourier symbols ˆ g , ˆ f and ˆ a , of the operators G , F , and T a defining vehicular formations and consensus algorithms which satisfy assump- tions (A1) - (A5) . The array O is that of the standard consensus algorithm (2). 1) Consensus a) Local Error: V loc c = 1 4 d 1 β X n 6 =0 , n ∈ Z d N ˆ O n < (ˆ a n ) (26) b) Long Range Deviation: V lr d c = − 2 X n 1 + ··· + n d odd , n ∈ Z d N 1 < (ˆ a n ) (27) c) Deviation from A verage: V dav c = − 1 2 X n 6 =0 , n ∈ Z d N 1 < (ˆ a n ) . (28) 2 Note that in d dimensions, the transformed state vector is of dimension 2 d for each wa venumber n . 2) V ehicular F ormations a) Local Error: V loc v = − 1 4 1 β X n 6 =0 , n ∈ Z d N ˆ O n ˆ g n ˆ f n (29) b) Long Range Deviation: V lr d v = 2 d X n 1 + ··· + n d odd , n ∈ Z d N 1 ˆ g n ˆ f n (30) c) Deviation from A verage: V dav v = d 2 X n 6 =0 , n ∈ Z d N 1 ˆ g n ˆ f n . (31) I V . U P P E R B O U N D S U S I N G S TA N DA R D A L G O R I T H M S In this section we derive asymptotic upper bounds for all three performance measures of both the consensus and vehicular problems. These bounds are derived by exhibiting simple feedback laws similar to the one in the standard consensus algorithm (2). In the case of vehicular formations, we make a distinction between the cases of relativ e versus absolute position and velocity feedbacks, and deriv e bounds for all four possible combinations of such feedbacks. The behavior of the asymptotic bounds has an important dependence on the underlying spatial dimension d . For the purpose of cross comparison, all of the upper bounds deri ved in this section are summarized in T able I. For later reference, we note that the Fourier transform of the array O in Eq. (2) is a quantity that occurs often, and can be easily calculated as ˆ O n = − 2 dβ + d X r =1  β e − i 2 π N n r + β e i 2 π N n r  = − 2 β d X r =1  1 − cos  2 π N n r  . (32) A. Upper bounds in the consensus case W e consider the standard consensus algorithm (1). In this case the array a is exactly O , and thus e xpression (26) for the local error immediately simplifies to V loc c = 1 4 dβ X n 6 =0 , n ∈ Z d N 1 = 1 4 dβ ( M − 1) , which then implies the following upper bound for the individ- ual local error at each site V loc c M ≤ 1 4 dβ . Thus, the individual local error measure for the standard consensus algorithm is bounded from abov e for any network size in any dimension d . The deriv ation of the macroscopic error upper bounds are a little more in v olved. First we observe that V lr d c ≤ 4 V dav c . This is easily seen since first, the sums in (27) and (28) in volve terms that are all of the same sign (since ˆ a n ≤ 0 ), and second, 7 that the sum in (27) is taken over a subset of the terms in (28). It therefore suf fices to deriv e the upper bounds for V dav c . W e begin with a simplifying observ ation. Because the arrays a we consider are real, their Fourier symbols ˆ a have ev en symmetry about all the mid axes of Z d N . More precisely ˆ a ( n 1 ,...,n r ,...,n d ) = ˆ a ( n 1 ,...,N − n r ,...,n d ) , for any of the dimension indices r . Assume for simplicity that N is odd, and define ¯ N := ( N + 1) / 2 . The even symmetry property implies that the discrete hyper-cube Z d N can be divided into 2 d hyper-cubes, each of the size of Z d ¯ N , and over which the v alues of ˆ a can be generated from its values over Z d ¯ N by appropriate reflections. Consequently , a sum like (28) can be reduced to V dav c = − 1 2 X n 6 =0 , n ∈ Z d N 1 < (ˆ a n ) = − 2 d 2 X n 6 =0 , n ∈ Z d ¯ N 1 < (ˆ a n ) . W e now calculate an upper bound on the deviation from av erage measure (28) for the Fourier symbol (32) of the standard consensus algorithm V dav c = 1 4 β X n 6 =0 , n ∈ Z d N 1 P d r =1  1 − cos  2 π N n r  = 2 d 4 β X n 6 =0 , n ∈ Z d ¯ N 1 P d r =1  1 − cos  2 π N n r  ≤ 2 d 32 β N 2 X n 6 =0 , n ∈ Z d ¯ N 1 ( n 2 1 + · · · + n 2 d ) , (33) where the first equality follows from reflection symmetry , and the inequality follo ws from (49), and noting that the denominator is made up of d terms of the form 1 − cos  2 π N n r  ≥ 2 π 2  2 π N n r  2 = 8 N 2 n 2 r , where the inequality is v alid in the range n r ∈ [0 , ( ¯ N − 1)] . The asymptotics of sums in Eq. (33) are presented in Ap- pendix B. Using those expressions, we calculate the individual deviation from average measure at each site V dav c N d ≤ 2 d 32 β N 2 − d X n 6 =0 , n ∈ Z d ¯ N 1 ( n 2 1 + · · · + n 2 d ) ≈ 2 d 32 β N 2 − d    1 d − 2 ( ¯ N d − 2 − 1) d 6 = 2 log( ¯ N ) d = 2 ≤ C d 1 β    N d = 1 log( N ) d = 2 1 d ≥ 3 , (34) where we have used ¯ N ≤ N , and C d is a constant that depends on the dimension d , but is independent of N or the algorithm parameter β . W e note that the upper bounds ha ve exactly the same form when written in terms of the network size M = N d . B. Upper bounds for vehicular formations T o establish upper bounds in this case, we use a feedback control law which is similar to (8). This law can be most compactly written in operator notation as u = T O ˜ x + T O ˜ v + g o ˜ x + f o ˜ v, where T O is the operator of con volution with the array O defined in the consensus problem (2). Note that in the multi- dimensional case, all signals are d -vectors, and thus T O abov e is our notation for a diagonal operator with T O in each entry of the diagonal. The last two terms represent absolute position and velocity error feedbacks respecti vely . The first two terms represent a feedback where each vector component of u k is formed by a law like (1) from the corresponding vector components of ˜ x k and ˜ v k and all 2 d immediate neighbor sites in the lattice. W ith the abov e feedback law , the closed loop system (7) has the following e xpressions for the Fourier symbols of G and F ˆ g n = g o + ˆ O n , ˆ f n = f o + ˆ O n , (35) where ˆ O is the Fourier symbol (32). W e impose the additional conditions that g o ≤ 0 and f o ≤ 0 since otherwise the closed loop system will hav e an increasing number of strictly unstable modes as N increases. When g o 6 = 0 (resp. f o 6 = 0 ) we refer to that feedback as using absolute position (resp. velocity) feedback. There are four possible combinations of such feedback scenarios. W e now use these expressions for the symbols ˆ g and ˆ f to calculate upper bounds on performance measures (P1) , (P2) and (P3) for all four feedback scenarios. W e begin with the local error (29) which in this case is gi ven by V loc v = − 1 4 β X n 6 =0 , n ∈ Z d N ˆ O n ( g o + ˆ O n )( f o + ˆ O n ) . (36) In the case of relativ e position and velocity error feedback, which corresponds to g o = 0 and f o = 0 , the sum in Eq. (36) becomes − P 1 / ˆ O n . This has the same form as V dav c in Eq. (28) for the standard consensus problem, and thus will grow asymptotically as deriv ed in Eq. (34). For this scenario, the final answer is listed as V loc v in T able I after multiplying by the extra 1 /β factor . In the case of relative position and absolute velocity feedback, the sum in Eq. (36) becomes P − 1 / ( f o + ˆ O n ) . Each term is bounded from abov e by − 1 / ( f o + ˆ O n ) ≤ − 1 /f o since f o < 0 and ˆ O n ≤ 0 . Thus the entire sum has an upper bound that scales like M , which yields a constant bound for the individual local error once divided by the network size M . An e xactly symmetric argument applies to the case of absolute position but relativ e velocity feedback. Finally , in the case of both absolute position and v elocity feedback f o < 0 and g o < 0 implying a uniform bound on each term in the sum. Similarly the entire sum scales like M and thus is uniformly bounded upon division by the network size. All of these four cases for the local error scalings are summarized in T able I. 8 T ABLE I Summery of asymptotic scalings of upper bounds in terms of the the total network size M and the spatial dimensions d . Performance measures are classified as either microscopic (local error), or macroscopic (de viation from av erage or long range deviation). There are four possible feedback strategies in vehicular formations depending on which combination of relative or absolute position or velocity error feedback is used. Quantities listed are up to a multiplicativ e factor that is independent of M or algorithm parameter β . Microscopic Macroscopic Consensus 1 /β 1 β    M d = 1 log( M ) d = 2 1 d ≥ 3 V ehicular Formations Feedback type: abs. pos. & abs. vel. 1 /β 1 V ehicular Formations Feedback type: rel. pos. & abs. vel. or abs. pos. & rel. vel. 1 /β 1 β    M d = 1 log( M ) d = 2 1 d ≥ 3 V ehicular Formations Feedback type: rel. pos. & rel. vel. 1 β 2    M d = 1 log( M ) d = 2 1 d ≥ 3 1 β 2          M 3 d = 1 M d = 2 M 1 / 3 d = 3 log( M ) d = 4 1 d ≥ 5 W e now consider the case of the deviation from average measure (31) which for our specific algorithm is V dav v = d 2 X n 6 =0 , n ∈ Z d N 1 ( g o + ˆ O n )( f o + ˆ O n ) . When g o < 0 and f o < 0 , each term in the sum is bounded and the entire sum scales as M . Thus, the indi vidual de viation from a verage at each site is bounded in this case. When either f o = 0 or g o = 0 , then the sums scale like − P 1 / ˆ O n (since the other factor in the fraction is uniformly bounded), i.e. like the de viation from average in the consensus case (34). The only case that requires further examination is that of relativ e position and relative velocity feedback ( g o = f o = 0 ). In this case V dav v = d 2 X n 6 =0 , n ∈ Z d N 1 ˆ O 2 n ≤ d 2 d 2 8 1 β 2 N 4 X n 6 =0 , n ∈ Z d ¯ N 1 ( n 2 1 + · · · + n 2 d ) 2 , where the inequality is deri ved by the same ar gument used in deriving the inequality (33). Dividing this expression by the network size N d and using the asymptotic expressions (52) yields V dav v N d ≤ C d 1 β 2    1 d − 4 (1 − N 4 − d ) d 6 = 4 log( N ) d = 4 , (37) where C d is a constant depending on the dimension d b ut independent of N or the algorithm parameter β . Rewriting these bounds in terms of the total network size M = N d giv es the corresponding entries in T able I, where the other cases are also summarized. W e finally point out that V lr d v ≤ 4 V dav v due to an argument identical to that employed in the consensus case. W e thus conclude that the upper bounds just deri ved apply to the case of the long range de viation measure as well. The r ole of viscous friction: It is interesting to observe that in vehicular models with viscous friction (6), a certain amount of absolute velocity feedback is inherently present in the dynamics. The model (6) with a feedback control of the form (8) has the following Fourier symbol for the velocity feedback operator F ˆ f n = − µ + f o + ˆ O n . W e conclude that e ven in cases of only relati ve v elocity error feedback (i.e. when f o = 0 ), the viscous friction term µ > 0 provides some amount of absolute velocity error feedback. Thus, in an en vironment which has viscous damping, perfor- mance in vehicle formation problems scale in a similar manner to consensus problems. These comments are also applicable to the lo wer bounds dev eloped in the next section. The r ole of contr ol effort: A common feature of all the asymptotic upper bounds of the standard algorithms just presented is their dependence on the parameter β . If this parameter is fixed in advance based on design considerations, then the algorithm’ s performance will scale as shown in T able I. Howe ver , it is possible to consider the redesign of the algorithms as the network size increases. For example, it is possible to increase β proportionally to M in consensus algorithms to achieve bounded macroscopic errors ev en for one dimensional networks. As can be seen from (1), this has the effect of increasing the control feedback gains unboundedly (in M ), which would clearly be unacceptable in any realistic control problem. Thus, any consideration of the fundamental limits of performance of more general algorithms must account for some notion of control effort, and we turn to this issue in the next section. V . L OW E R B O U N D S A natural question arises as to whether one can design feedback controls with better asymptotic performance than the standard algorithms presented in the previous section. In this section we analyze the performance of any linear static state feedback control algorithm satisfying the structural assumptions (A1) - (A5) , and subject to a constraint on control effort. A standard measure of control ef fort in stochastic 9 settings is the steady state variance of the control signal at each site E { u ∗ k u k } , (38) which is independent of k due to the spatial in variance assump- tion. W e constrain this quantity and deri ve lo wer bounds on the performance of any algorithm that respects this constraint. The basic conclusion is that lower bounds on performance scale like the upper bounds listed in T able I with the control ef fort replacing the parameter β . In other words, any algorithm with contr ol ef fort constraints will not do better asymptotically than the standard algorithms of section IV -B. This is somewhat surprising giv en the extra degrees of freedom possible through feedback control design, and it perhaps implies that it is primarily the netw ork topology and the structural constraints, rather than the selection of the algorithm’ s parameters that determine these fundamental limitations. W e now turn to the calculation of lower bounds on both mi- croscopic and macroscopic performance measures. F or brevity , we include only the calculations for the de viation from av erage macroscopic measures. These calculations are a little more in volv ed than those for the upper bounds since they need to be valid for an entire class of feedback gains. Howe ver , the basic ideas of utilizing H 2 norms are similar, and this is what we do in the sequel. In addition, a new ingredient appears where the control ef fort bound, combined with the locality property , implies a uniform bound on the entries of the feedback arrays. This is stated precisely in the next lemma whose proof is found in the Appendix. These bounds then finally impose lo wer bounds on the performance of control- constrained local algorithms. Lemma 5.1: Consider general consensus (4) and vehicular formation (7) algorithms where the feedback arrays a , g and f posses the locality property (A3). The following bounds hold k a k ∞ ≤ B a E  u 2 k  k g k ∞ ≤ B g  E  u 2 k  2 (39) k f k ∞ ≤ B f E  u 2 k  , where B a , B g and B f are constants independent of the network size. A. Lower bounds for consensus algorithms W e start with the de viation from average measure for a stable consensus algorithm subject to a constraint of bounded control v ariance at each site E  u 2 k  ≤ W . (40) W e first observ e a bound on < (ˆ a n ) that can be established from the definition of the Fourier transform < (ˆ a n ) = <   X k ∈ Z d N a k e − i 2 π N ( n.k )   = X k ∈ Z d N a k cos  2 π N n · k  = X k ∈ Z d N a k  1 −  1 − cos  2 π N n · k  = X k ∈ Z d N ( − a k )  1 − cos  2 π N n · k  where the last equality is a consequence of the condition P k ∈ Z d N a k = 0 . For lower bounds on P 1 / < ( − ˆ a n ) , upper bounds on < ( − ˆ a n ) are needed. Observe that |< ( − ˆ a n ) | =       X k ∈ Z d N a k  1 − cos  2 π N n · k        ≤ X k ∈ Z d N | a k |  1 − cos  2 π N n · k  ≤ 4 π 2 N 2 X k ∈ Z d N | a k | ( n · k ) 2 , where the second inequality follo ws from (48). The last quantity can be further bounded by recalling the locality property (11), which has the consequence X k ∈ Z d N | a k | ( k 1 n 1 + · · · + k d n d ) 2 = X k ∈ Z d N , | k i |≤ q | a k | ( k 1 n 1 + · · · + k d n d ) 2 ≤ X 0 6 = k ∈ Z d N , | k i |≤ q | a k | ( q n 1 + · · · + q n d ) 2 = q 2 ( n 1 + · · · + n d ) 2 X 0 6 = k ∈ Z d N , | k i |≤ q | a k | . Now the locality property can be used again to bound the abov e sum using the the control effort bounds (39) and (40) X 0 6 = k ∈ Z d N , | k i |≤ q | a k | ≤ (2 q ) d k a k ∞ ≤ (2 q ) d B a W . (41) Putting the abov e together giv es V dav c = X n 6 =0 , n ∈ Z d N 1 −< (ˆ a n ) ≥ N 2 π 2 (2 q ) d +2 B a W X n 6 =0 , n ∈ Z d N 1 ( n 1 + · · · + n d ) 2 ≥ C d W N 2 X n 6 =0 , n ∈ Z d N 1 ( n 2 1 + · · · + n 2 d ) , where the last inequality follo ws from (50), and C d is a constant independent of N . Finally , utilizing (51) and dividing by the network size M = N d , a lower bound on the deviation from a verage is obtained V dav c N d ≥ C d W N 2 − d X n 6 =0 , n ∈ Z d N 1 ( n 2 1 + · · · + n 2 d ) ≈ C d W    1 d − 2 (1 − N 2 − d ) d 6 = 2 log( N ) d = 2 , ≥ C d 1 W    N d = 1 log( N ) d = 2 1 d ≥ 3 , (42) where by a slight ab use of notation, we use C d to denote different constants in the expressions abo ve. W e observe ho w 10 the lower bounds (42) hav e the same asymptotic form as the upper bounds for the standard consensus algorithm (34), but with the control ef fort bound W replacing the parameter β . B. Lower bounds for vehicular formations W e recall the de velopment of the upper bounds for v ehicular formations in Section IV -B. The Fourier symbols of general feedback gains G and F hav e a similar form to (35), and can be written as ˆ g n = g o + ˆ γ n , ˆ f n = f o + ˆ φ n , (43) where g o , f o and ˆ γ , ˆ φ are the absolute and relative feedback terms respectiv ely . As before, we impose the conditions that g o , f o ≤ 0 . W e assume that we have a control effort constraint of the form (40). The case of absolute position and absolute v elocity feedback has upper bounds which are finite, and the question of lower bounds is moot. For the other three cases, lower bounds on (31) are established using upper bounds on the symbols ˆ g and ˆ f which can be deriv ed as follows k ˆ f k ∞ ≤ k f k 1 ≤ (2 q + 1) k f k ∞ ≤ (2 q + 1) B f W , where the inequalities follo w from (46), the locality property , and (39) respecti vely . For g we similarly hav e k ˆ g k ∞ ≤ (2 q + 1) B g W 2 . Consider now the case of relativ e position and absolute velocity feedback. A lo wer bound is established by V dav v = d 2 X n 6 =0 , n ∈ Z d N 1 | ˆ g n || ˆ f n | ≥ d 2(2 q + 1) B f 1 W X n 6 =0 , n ∈ Z d N 1 | ˆ g n | . Now a lower bound on the sum can be established in exactly the same manner as (42) in the consensus case since ˆ g is a symbol of a local relative feedback operator . The case of relativ e velocity and absolute position feedback is similar with the exception that the factor of 1 W is replaced by 1 W 2 . The final case to consider is that of relativ e position and relativ e velocity feedback. One can repeat the same arguments made in the consensus case up to equation (41) for both ˆ g n and ˆ f n to state X 0 6 = n ∈ Z d N 1 | ˆ g n || ˆ f n | ≥ c 1 N 4 k g k ∞ k f k ∞ X 0 6 = n ∈ Z d N 1 ( n 1 + · · · + n d ) 4 ≥ c 2 N 4 W 3 X 0 6 = n ∈ Z d N 1 ( n 1 + · · · + n d ) 4 , where c 1 and c 2 are some constants independent of N and W . The asymptotic behavior of this expression (divided by the network size) was gi ven earlier in (37). W e thus conclude that the lower bounds in this case are exactly like the upper bounds shown in T able I for relativ e position and relative velocity feedback, but with the 1 β 2 term replaced by 1 W 3 . V I . E X A M P L E S A N D M U LT I S C A L E I N T E R P R E TA T I O N Numerical simulations of cases where macroscopic mea- sures grow unboundedly with network size show a particular type of motion for the entire formation. In the one dimensional case, it can be described as an accordion-like motion in which large shape features in the formation fluctuate. Figure 1 shows the results of a simulation of a 100 vehicle platoon with both relativ e position and relativ e velocity error feedbacks. This corresponds to a control strategy of the type for which upper bounds were calculated in section IV -B with g o = f o = 0 . M . J O V A N O V I ´ C , U M N 2 All vehic les subject to stoc hastic disturbances ( N = 100 ) long vs. shor t rang e de viations Figure 2: Absolute positions of a ll v ehicles . Fig. 1. V ehicle position trajectories (relativ e to vehicle number 1) of a 100 vehicle formation all of which are subjected to random disturbances. T op graph is a “zoomed out” vie w exhibiting the slow accordion-like motion of the entire formation. Bottom graph is a zoomed in view showing that v ehicle- to-vehicle distances are relatively well regulated. An interesting feature of these plots is the phenomenon of lack of formation coherence. This is only discernible when one “zooms out” to view the entire formation. The length of the formation fluctuates stochastically , but with a distinct slo w temporal and long spatial wav elength signature. In contrast, the zoomed-in view in Figure 1 sho ws a relati vely well regulated vehicle-to-v ehicle spacing. In general, it appears that small scale (both temporally and spatially) disturbances are well regulated, while large scale disturbances are not. An intuitive interpretation of this phenomenon is that local feedback strategies are unable to re gulate against large scale disturbances. In this paper we have not directly analyzed the temporal and spatial scale dependent disturbance attenuation limits of performance. Howe ver , it appears that our microscopic and macroscopic measures of performance do indeed correspond to 11 small and lar ge scale (both spatially and temporally) motions respectiv ely . W e next outline a more mathematical argument that connects these measures. Mode shapes: T o appreciate the connection between H 2 norms and mode shapes in our system, consider first a general linear system dri ven by a white random process ˙ x = Ax + w. When A is a normal matrix, it is easy to sho w (by diagonaliz- ing the system with the orthonormal state transformation made up of the eigenv ectors of A ) that the steady state variance of the state is lim t →∞ E { x ∗ ( t ) x ( t ) } = X i 1 2 <{ λ i } , where the sum is taken ov er all the eigenv alues λ i of A . Thus we can say that under white disturbance excitation, the amount of energy each mode contains is in versely proportional to its distance from the imaginary axis. In other words, slower modes are more energetic. Now , all the systems we consider in this paper are diagonalizable (or block-diagonlizable) by the spatial Discrete Fourier T ransform. In addition, for the standard algorithms, we ha ve the situation that slow temporal modes corr espond to long spatial wavelengths . This pro vides an explanation for the observ ation that the most energetic motions are those that are temporally slow and hav e long spatial wa velengths. String instability: While string instability is sometimes an issue in formation control, the phenomenon we study in this paper is distinct from string instability . The example presented in this section is that of a formation that does posses string stability . For illustration, we repeat the simulation but with disturbances acting only on the first vehicle. The resulting vehicle trajectories are sho wn in Figure 2. It is interesting to note that temporally high frequency disturbances appear to be very well regulated, and do not propagate far into the formation, while lo w temporal frequency disturbances appear to propagate deep into the formation. What is not sho wn in the figure is that low frequency disturbances are ev entually regulated for vehicles far from the first. This is consistent with the intuitive notion discussed earlier that local feedback is relati vely unable to regulate lar ge scale disturbances. 2 Onl y fir st vehic le subject to stoc hastic disturbance ( N = 100 ) x n ( t ) x n ( t ) Fig. 2. V ehicle position trajectories (relative to leader) of the first few of a 100 vehicle formation. Only lead v ehicle is subjected to random disturbances. V ehicle trajectories exhibit regulation against that disturbance, indicating the absence of string instability . Multi-scale pr operties of disturbance r ejection: An intrigu- ing explanation of the abov e example and our scaling results is as follows. The macroscopic error measures capture how well the network regulates against large-scale disturbances. In large, one dimensional networks, local feedback alone is thus unable to re gulate against these large-scale disturbances, and global feedback is required to achiev e this. This seems rather intuitiv e. Perhaps surprisingly , in large networks with higher spatial dimensionality , local feedback alone can indeed regulate against lar ge-scale disturbances. This follo ws for networks for which the macroscopic error measure is bounded irrespectiv e of network size. The “critical dimension” needed to achiev e this depends on the order of the node dynamics as well as the type of feedback strategy as sho wn in T able I (e.g. dimension 3 for relativ e position and absolute velocity feedback, and dimension 5 for relative position and velocity feedback in cases of vehicular formations). V I I . D I S C U S S I O N A. General networks The networks considered in this paper are ones which can be built on top of a T orus network. Some concepts, such as coherence and microscopic and macroscopic errors are easily generalized to arbitrary networks. The correct generalization of the concept of spatial dimension ho wev er is more subtle. For any network of dynamical systems for which a distance metric is defined between nodes (e.g. from an imbedding of the network in R n ), the notion of long range de viation can be defined as done in this paper . The calculation of that quantity in volv es system Grammians and may even be written in terms of the underlying system matrices for certain structures. Thus coherence measures can be calculated numerically for such networks. Howe ver , more explicit calculations to uncover scaling laws as network size increases will clearly require more analytical expressions for the system norms in such networks. T o generalize the present results, one would require a notion of how to grow the network size while preserving certain topological properties such as the spatial dimension. Preliminary results on self-similar and fractal networks have been obtained [14]. The proper notion of spatial dimension to capture coherence in general graphs remains a research topic at this time. B. Distributed estimation and r esistive lattices The results presented here have a strong resemblance to results on performance limitations of distributed estimation algorithms based on network topology [15], [16], where asymptotic bounds similar to (34) first appeared in the controls literature (see also [17] where a consensus problem with noisy observations is analyzed yielding performance bounds like the consensus upper bounds we have in the present paper). In that work, the arguments are based on an analogy with effecti ve resistance in resisti ve lattices and certain imbeddings of their graphs in d-dimensional space [18]. It is not clear how the resistiv e analogy can be generalized to cover the case of second order dynamics (i.e. vehicular formations), or the 12 lower bounds on more general control laws. W e hav e therefore av oided the resistiv e network analogy in this paper by directly using the multi-dimensional Fourier transform and reducing all calculations to sums of the form (51) resulting in a self- contained argument. It is interesting to note that the original arguments for the asymptotic behavior in resistive lattices [19] in the physics literature are based on approximations of the Green’ s function of the diffusion operator in d -dimensions, for which the underlying techniques are approximations like (54). C. Order of local dynamics W e hav e attempted to keep the development general enough that it is applicable to networked dynamical systems whose dynamics are not necessarily those of v ehicles in any particular physical setting. What we refer to in this paper as consensus and v ehicular formations problems respecti vely represent net- works where the local dynamics (at each site) are first and second order chains of integrators respectively . The dynamical models are such that the stochastic disturbance enters into the first integrator , and the performance objecti ves in volve variances of the outputs of the last integrators at each site. One generalization of this set up is where the local dynamics is a chain of n integrators. It is then possible to show that (by retracing the ar guments for the vehicular formations case and generalizing (52)) the cutoff dimension to ha ve bounded macroscopic measures with only local relativ e state feedback is 1 + 2 n . D. LQR designs It was observed in [20] that optimal LQR designs for vehic- ular platoons suf fer from a fundamental problem as the platoon size increases to infinity . These optimal feedback laws are almost local in a sense described by [21], where control gains decay exponentially as a function of distance. The resulting optimal feedbacks [20] suffer from the problem of having underdamped slo w modes with long spatial wa velengths. Thus, the same incoherence phenomenon occurs in these optimal LQR designs where the performance objectiv e is composed of sums of local relativ e errors (leading to feedback laws with exponentially decaying gains on relative errors). E. Measuring performance in larg e scale systems In spatial dimensions where performance scalings are bounded, the underlying system eigen v alues still limit towards zero, suggesting ultimate instability in the limit as M → ∞ . Howe ver , measures of performance remain bounded in these cases. In such cases the locations of internal eigen values are not a good indication of the system’ s performance in the limit of large netw orks. T ake the consensus problem ov er Z d N as an example. The “least damped eigenv alue” (other than zero) quantifies the con ver gence time of deviation from a verage (in the absence of stochastic disturbances), and it scales as 1 | λ 2 | = Θ  N 2 /d  , (44) as can be shown by explicit eigen v alue calculations [12], [13]. If we use this quantity as a measure of performance, it indicates that performance becomes arbitrarily bad (in the limit of large N ) in any spatial dimension d . On the other hand, consider the use of a macroscopic error measure like the variance of the deviation from average (19) in the presence of stochastic disturbances. That quantity can be expressed in terms of the system eigen v alues as V dav c = 1 2 N d X n 6 =0 1 | λ n | , (45) where the sum is taken ov er all the system’ s eigenv alues other than zero. Note that this sum is just (28) rewritten to emphasize the contrast with (44). The important observation is that (44) indicates that as network size increases, the system eigen v alues approach the stability boundary , indicating an ev entual catastrophic loss of performance in any spatial dimension d . On the other hand, (45) is uniformly bounded in dimensions d ≥ 3 (as shown in (34)), thus implying well beha ved systems as quantified by the macroscopic performance measures. A similar point to the abov e has been recently made [22]. The least damped eigen value is traditionally used as an important measure of performance. The examples in this paper demonstrate that for lar ge scale systems, it is not a very meaningful measure of performance, and that the general question of ho w to measure performance in large scale systems is a subtle one. F . Detuning/mistuning designs It is shown in [23] that spatially-inv ariant local controllers for platoons hav e closed loop eigenv alues that approach the origin at a rate of O ( 1 N 2 ) . A “mistuning” design modification is proposed [23], resulting in spatially-v arying local controllers where the closed loop eigen values approach the origin at the better rate of O ( 1 N ) . In this paper , we have not used the real part of the least damped eigen value as a measure of performance b ut rather the v ariance of certain system outputs. This amounts to using an H 2 norm as the measure of performance. It was shown in [21] that for spatially- in variant plants, one can not improve H 2 performance with spatially-varying controllers. The resulting controllers ho wev er hav e exponentially decaying gains rather than completely local gains. The problem of designing optimal H 2 controllers with a prescribed neighborhood of interaction remains an open and non-conv ex one. It is an interesting and open question as to whether mistuning designs for the H 2 measures we use in this paper can yield local controllers with better asymptotic performance than spatially-inv ariant ones. It was also shown [23] that a mistuning design can improv e H ∞ performance for platoon problems. This shows that there is perhaps an important distinction between H ∞ and H 2 measures of performance for large scale systems. A point that is worthy of further inv estigation. 13 A P P E N D I X A. Multi-dimensional Discrete F ourier T r ansform W e define the Discrete F ourier T ransform for functions { f k } ov er Z d N by ˆ f n := X k ∈ Z d N f k e − i ( 2 π N n · k ) , where n · k := n 1 k 1 + · · · + n d k d . The in verse transform is giv en by f k := 1 M X n ∈ Z d N ˆ f k e i ( 2 π N n · k ) , where M = N d . An immediate consequence of the definitions are the follo wing bounds k ˆ f k ∞ ≤ k f k 1 , k f k ∞ ≤ 1 M k ˆ f k 1 . (46) Let δ be the Kronecker delta on Z d N . It’ s transform is the array 1 , which is the array of all elements equal to 1 . The transform of 1 is M δ . W e use the symbol ¯ 1 to denote the array of all elements equal to 1 M . If T f denotes the circulant operator of circular conv olution with f , then the eigen v alues of T f are just the numbers { ˆ f n } , and consequently the trace of T f is gi ven by the sum tr ( T f ) = X n ∈ Z d N ˆ f n . (47) B. Bounds and asymptotics of sums The following facts are useful in establishing asymptotic bounds. 1) For an y x ∈ R and any y ∈ [ − π , π ] 1 − cos( x ) ≤ x 2 , (48) 1 − cos( y ) ≥ 2 π 2 y 2 . (49) 2) Gi ven d integers n 1 , . . . , n d , ( n 1 + · · · + n d ) 2 ≤ (2 d + 1) ( n 2 1 + · · · + n 2 d ) . (50) Proof: X i n i ! 2 = X i n 2 i + X i X j 6 = i n i n j . Using n i n j ≤ (max { n i , n j } ) 2 ≤ n 2 i + n 2 j , we get the bound X i n i ! 2 ≤ X i n 2 i + d X i =1 X j 6 = i ( n 2 i + n 2 j ) ≤ X i n 2 i + 2 d X i n 2 i . 3) In the limit of lar ge N , X n 6 =0 n ∈ Z d N 1 ( n 2 1 + · · · + n 2 d ) ≈    1 d − 2 ( N d − 2 − 1) d 6 = 2 log( N ) d = 2 (51) X n 6 =0 n ∈ Z d N 1 ( n 2 1 + · · · + n 2 d ) 2 ≈    1 d − 4 ( N d − 4 − 1) d 6 = 4 log( N ) d = 4 (52) (53) where f ( N ) ≈ g ( N ) is notation for c g ( N ) ≤ f ( N ) ≤ ¯ c g ( N ) , for some constants ¯ c and c and all N ≥ ¯ N for some ¯ N . Proof: W e be gin with (51). Upper and lower bounds on this sum can be deriv ed by viewing it as upper and lo wer Rieman sums for the integral Z · · · Z 1 x 2 1 + · · · + x 2 d dx 1 · · · dx d , ov er the region ∆ ≤ r ≤ 1 for the lower bound, and ∆ ≤ r ≤ √ d for the upper bound. Here ∆ = 1 N , and the asymptotic behavior is determined by the lower limit on the integrals, so both upper and lo wer bounds behav e the same asymptotically . Using the grid points x 1 = n 1 ∆ , . . . , x d = n d ∆ , and using the volume increment ∆ d , we get Z · · · Z 1 x 2 1 + · · · + x 2 d dx 1 · · · dx d ≈ ∆ d X n 6 =0 , n ∈ Z d N 1 ((∆ n 1 ) 2 + · · · + (∆ n d ) 2 ) = ∆ d − 2 X n 6 =0 , n ∈ Z d N 1 ( n 2 1 + · · · + n 2 d ) . (54) Now the integral can be ev aluated using hyperspherical coor - dinates by Z 1 x 2 1 + · · · + x 2 d dx 1 · · · dx d = Z 1 ∆ Z π 0 · · · Z 2 π 0 1 r 2 r d − 1 sin d − 2 ( φ 1 ) . . . sin( φ d − 2 ) dr dφ 1 . . . dφ d − 1 = C d Z 1 ∆ r d − 3 dr , where C d is a constant that depends only on the dimension d (and can be expressed in terms of the v olume of the unit sphere in R d ). Ev aluating this integral, using ∆ = 1 N and equation (54) gi ves the result (51). The proof of (52) is very similar to the abov e, with the exception that one approximates the integral of 1 ( x 2 1 + ··· + x 2 d ) 2 = 1 r 4 instead. The details are omitted for bre vity . C. Proof of Lemma 3.1 For the consensus problem, the state equation is (4), and thus the L yapunov equation (23) becomes ˆ a ∗ n ˆ p n + ˆ p n ˆ a n = − ˆ C ∗ n ˆ C n , 14 where we hav e used H = C (and the choice of C depends on the particular performance measure being considered). Since all quantities are scalars, this equation is immediately solved for ˆ p n = − ˆ C ∗ n ˆ C n 2 < (ˆ a n ) for n 6 = 0 . In the case n = 0 , we look at the inte gral definition (22), conclude that ˆ C 0 = 0 implies that ˆ p 0 = 0 . Thus, the sum (21) is calculated to be (24). For the vehicular problem, the state equation is (7) with the output equation H =  C 0  . The L yapunov equation (23) becomes  0 ˆ G ∗ n I ˆ F ∗ n   ˆ X n ˆ Z n ˆ Z ∗ n ˆ Y n  +  ˆ X n ˆ Z n ˆ Z ∗ n ˆ Y n   0 I ˆ G n ˆ F n  =  − ˆ C ∗ n ˆ C n 0 0 0  , where each of the submatrices is of size d × d . From the above, we extract the follo wing matrix equations ˆ G ∗ n ˆ Z ∗ n + ˆ Z n ˆ G n = − ˆ C ∗ n ˆ C n (55) ˆ G ∗ n ˆ Y n + ˆ X n + ˆ Z n ˆ F n = 0 ˆ Z n + ˆ F ∗ n ˆ Y n + ˆ Z ∗ n + ˆ Y n ˆ F n = 0 . (56) Since we are only interested in the quantity tr  ˆ B ∗ n ˆ P n ˆ B n  =  0 I   ˆ X n ˆ Z n ˆ Z ∗ n ˆ Y n   0 I  = tr  ˆ Y n  , then only equations (55) and (56) are relev ant. The coordinate decoupling assumption (A5) on the operators G , F and C implies that the matrices ˆ G n , ˆ F n and ˆ C n are all diagonal. It follows that ˆ Z n , ˆ X n and ˆ Y n are also diagonal, and the abov e matrix equations are tri vial to solve. ˆ Z n has the solution Z = − 1 2 ˆ G − 1 n ˆ C ∗ n ˆ C n . Similarly , equation (56) is solved to yield Y = 1 2 ( ˆ G n ˆ F n ) − 1 ˆ C ∗ n ˆ C n , for n 6 = 0 . For the unstable mode at n = 0 , the integral (22) can be easily ev aluated to yield ˆ Z 0 = 0 (since ˆ C 0 = 0 for all the performance measures we consider). Adding in the assumption that all matrices are diagonal with equal elements, we obtain in summary the total H 2 norm of the vehicle formation problem (7) is gi ven by V v = d 2 X n 6 =0 n ∈ Z d N ˆ c ∗ n ˆ c n ˆ g n ˆ f n , (57) where the multiplicative factor of d comes from taking the trace of ˆ Y n . Pr oof of Cor ollary 3.2 1) Consensus: The local error measure output operator C is giv en by (15), for which C ∗ C = − 1 2 dβ O by the identity (16). Combining this with Lemma 3.1 gi ves the result for V loc c . The long range de viation measure has the output operator defined in (18), which has the F ourier symbol ˆ c n = 1 − e − iπ ( n 1 + ··· + n d ) , from which we conclude that ˆ c n =  0 ( n 1 + · · · + n d ) ev en 2 ( n 1 + · · · + n d ) odd . Combining this with Lemma 3.1 gi ves the result for V lr d c . The deviation from average output operator is C = I − T 1 , or equiv alently , the con volution operator T δ 0 − 1 . The corre- sponding Fourier symbol is the Fourier transform of the array δ 0 − 1 , which gi ves ˆ c n = 1 n − δ n =  0 n = 0 1 n 6 = 0 . Putting this in the general formula (24) yields the result for V dav c 2) V ehicular formations: The deriv ations for this case are very similar to those for the consensus problem and are therefore omitted for bre vity . Pr oof of Lemma 5.1 W e re write the dynamics of the consensus algorithm so that u is an output ˙ x = T a x + w u = T a x. The present task is then to calculate the H 2 norm from w to u . Applying formula (24) with T a as the C operator yields X k ∈ Z d N E  u 2 k  = − 1 2 X n 6 =0 , n ∈ Z d N ˆ a 2 n ˆ a n = 1 2 X n ∈ Z d N ( − ˆ a n ) , after observing that ˆ a n is real and ˆ a 0 = 0 . Furthermore, our stability condition requires that for all n , ( − ˆ a n ) ≥ 0 , implying that the sum abov e is the ` 1 -norm of { ˆ a n } . Putting this together with the bound (46) gi ves k a k ∞ ≤ 1 M k ˆ a k 1 = 2 E  u 2 k  . (58) In the vehicular formations case, the dynamics are giv en by (7) together with the output equation u =  G F   ˜ x ˜ v  . Formula (25) is not applicable here since the output depends on all states, but the H 2 norm for this system can be calculated in a manner similar to the proof of Lemma 3.1. They L yapuno v equation in this case becomes  0 ˆ G ∗ n I ˆ F ∗ n   ˆ X n ˆ Z n ˆ Z ∗ n ˆ Y n  +  ˆ X n ˆ Z n ˆ Z ∗ n ˆ Y n   0 I ˆ G n ˆ F n  = −  ˆ G ∗ n ˆ G n ˆ G ∗ n ˆ F n ˆ F ∗ n ˆ G n ˆ F ∗ n ˆ F n  , from which we extract the matrix equations ˆ G ∗ n ˆ Z ∗ n + ˆ Z n ˆ G n = − ˆ G ∗ n ˆ G n ˆ G ∗ n ˆ Y n + ˆ X n + ˆ Z n ˆ F n = − ˆ G ∗ n ˆ F n ˆ Z n + ˆ F ∗ n ˆ Y n + ˆ Z ∗ n + ˆ Y n ˆ F n = − ˆ F ∗ n ˆ F n . Only the first and last equation need be solved since we are only interested in tr ( ˆ Y n ) . All of the abov e are d × d diagonal 15 matrices with equal entries, so we solve the equations in terms of a single entry as 2 ˆ g n ˆ z n = − ˆ g 2 n ⇒ ˆ z n = − 1 2 ˆ g 2 ˆ f n ˆ y n = − ˆ f 2 n − 2 ˆ z n ⇒ ˆ y n = 1 2  − ˆ f n + 1 ˆ f n ˆ g n  . The H 2 norm of the system is then X 0 6 = n ∈ Z d N tr ( ˆ Y n ) = d 2 X 0 6 = n ∈ Z d N  − ˆ f n + 1 ˆ f n ˆ g n  = d 2  k ˆ f k 1 + k 1 ˆ f ˆ g k 1  , where the last equation follows from the stability conditions ˆ f n ≤ 0 , ˆ g n ≤ 0 . This inequality has two consequences after observing that P n ∈ Z d N tr ( ˆ Y n ) = M E  u 2 k  and using (46) k f k ∞ ≤ 2 d E  u 2 k  (59) k 1 ˆ f ˆ g k 1 ≤ 2 d M E  u 2 k  . (60) The second inequality can be used to bound k g k ∞ as follo ws. First k 1 ˆ f ˆ g k 1 ≥ k ˆ g k 1 min n | 1 ˆ f n | = k ˆ g k 1 1 k ˆ f k ∞ . An upper bound on k ˆ f k ∞ is deri ved from k ˆ f k ∞ ≤ k f k 1 ≤ (2 q ) d k f k ∞ , where the last inequality follows from the locality assumption on f . Combining these last two bounds with (60) yields 2 d M E  u 2 k  ≥ 1 (2 q ) d k f k ∞ k g k 1 ≥ 1 (2 q ) d k f k ∞ M k g k ∞ , which when combined with (59) gi ves k g k ∞ ≤ 2(2 q ) d d k f k ∞ E  u 2 k  ≤ B g  E  u 2 k  2 . R E F E R E N C E S [1] S. 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