Optimal Control of Vehicular Formations with Nearest Neighbor Interactions

1 Optimal Contr ol of V ehicular F ormations with Near est Neighbor Interactions Fu Lin, Student Member , IEEE , Makan Fardad, and Mihailo R. Jo vano vi ´ c, Member , IEEE Abstract —W e consider the design of optimal localized feedback gains for one-dimensional f ormations in which vehicles only use information fr om their immediate neighbors. The control objective is to enhance coherence of the formation by making it behave like a rigid lattice. For the single-integrator model with symmetric gains, we establish convexity , implying that the globally optimal controller can be computed efficiently . W e also identify a class of conv ex problems f or double-integrators by restricting the controller to symmetric position and unif orm diagonal velocity gains. T o obtain the optimal non-symmetric gains for both the single- and the double-integrator models, we solve a parameterized family of optimal contr ol problems ranging from an easily solvable problem to the problem of interest as the underlying parameter increases. When this parameter is kept small, we employ perturbation analysis to decouple the matrix equations that r esult from the optimality conditions, thereby rendering the unique optimal feedback gain. This solution is used to initialize a homotopy-based Newton’s method to find the optimal localized gain. T o in vestigate the performance of localized controllers, we examine how the coherence of large- scale stochastically for ced formations scales with the number of vehicles. W e establ ish several explicit scaling relationships and show that the best performance is achieved by a localized controller that is both non-symmetric and spatially-varying. Index T erms —Conv ex optimization, formation coherence, ho- motopy , Newton’ s method, optimal localized control, pertur- bation analysis, structured sparse feedback gains, vehicular formations. I . I N T RO D U C T I O N A. Backgr ound The control of vehicular platoons has attracted considerable attention since the mid sixties [1]–[3]. Recent technological advances in developing vehicles with communication and computation capabilities have spurred rene wed interest in this area [4]–[12]. The simplest control objecti ve for the one- dimensional (1D) formation sho wn in Fig. 1 is to maintain a desired cruising velocity and to keep a pre-specified con- stant distance between neighboring v ehicles. This problem is emblematic of a wide range of technologically rele vant applications including the control of automated highways, unmanned aerial vehicles, swarms of robotic agents, and satellite constellations. Recent work in this area has focused on fundamental performance limitations of both centralized and decentralized Financial support from the National Science Foundation under CAREER A ward CMMI-06-44793 and under A wards CMMI-09-27720 and CMMI-09- 27509 is gratefully acknowledged. F . Lin and M. R. Jov anovi ´ c are with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455. M. Fardad is with the Department of Electrical Engineering and Computer Sci- ence, Syracuse Uni versity , NY 13244. E-mails: fu@umn.edu, makan@syr .edu, mihailo@umn.edu. Fig. 1: One-dimensional formation of vehicles. controllers for large-scale formations [5], [7], [9]–[12]. For centralized linear quadratic optimal control formulations based on penalizing relative position errors it was shown in [7] that stabilizability and detectability deteriorate as formation size increases. In [9], it was shown that merge and split maneuvers can exhibit poor con ver gence rates even upon inclusion of absolute position errors in cost functionals. In [5], it was shown that sensiti vity of spacing errors to disturbances increases with the number of vehicles for formations with localized symmetric controllers that utilize relative position errors between neighboring vehicles. In [11], the analysis of [5] w as expanded to include heterogeneous v ehicles, non- zero time headway , and limited communication range within the formation. The motiv ation for the current study comes from two recent papers, [12] and [10]. In [12], fundamental performance limi- tations of localized symmetric feedback for spatially in variant consensus and formation problems were examined. It was shown that, in 1D, it is impossible to hav e coherent large formations that behave like rigid lattice. This was done by exhibiting linear scaling, with the number of v ehicles, of the formation-size-normalized H 2 norm from disturbances to an appropriately defined macroscopic performance measure. In 2D this measure increases logarithmically , and in 3D it remains bounded irrespecti ve of the system size. These scalings were deriv ed by imposing uniform bounds on control energy at each vehicle. For formations on a one-dimensional lattice, it was sho wn in [10] that the decay rate (with the number of vehicles) of the least damped mode of the closed-loop system can be improv ed by introducing a small amount of ‘mistuning’ to the spatially uniform symmetric feedback gains. A large formation was modeled as a diffusi ve PDE, and an optimal small-in- norm perturbation profile that destroys the spatial symmetry and renders the system more stable was designed. Numerical computations were also used to demonstrate that the spatially- varying feedback gains hav e beneficial influence on the closed- loop H ∞ norm. The PDE approaches hav e also been found useful in the deployment of multi-agents [13], [14] and in coordination algorithms [15]. Even though traditional optimal control does not facilitate 2 T ABLE I: Summary of asymptotic scalings with the number of vehicles N for the optimal symmetric and non-symmetric position gains. The N -independent control penalty , R = r 0 I , in the quadratic performance objecti ve leads to similar gro wth with N of formation coherence and control energy (per vehicle). On the other hand, the N -dependent control penalty that provides bounded control energy yields less fa vorable coherence. Optimal position gains Control penalty R = r I Control energy (per vehicle) Formation coherence symmetric r ( N ) = r 0 = const. O ( √ N ) O ( √ N ) non-symmetric O ( 4 √ N ) O ( 4 √ N ) symmetric r ( N ) ∼ N O (1) O ( N ) non-symmetric r ( N ) ∼ √ N O (1) O ( √ N ) incorporation of structural constraints and leads to central- ized architectures, the optimal feedback gain matrix for both spatially in v ariant systems [16] and systems on graphs [17] hav e off-dia gonal decay . Sev eral recent efforts hav e focused on identification of classes of con vex distributed control prob- lems. F or spatially inv ariant controllers in which information propagates at least as fast as in the plant, conv exity was established in [18], [19]. Similar algebraic characterization for a broader class of systems was introduced in [20], and con ve xity was shown for problems with quadratically in- variant constraint sets. Since these problems are con vex in the impulse response parameters they are in general infinite dimensional. In [21], a state-space description of systems in which information propagates at most one unit in space for ev ery unit in time was provided and relaxations were used to obtain suboptimal controllers. In [22], the optimal control problem for switched autonomous systems was studied and optimality conditions for decentralization of multi-agent motions were deri ved. In [23], conv exity of the symmetric edge weight design for minimization of the mean-square de viation in distributed av erage consensus w as shown. While references [18]–[21] focus on the design of optimal dynamic distributed controllers, we dev elop tools for the design of optimal static feedback gains with pre-specified structure. Even though the framew ork of [18]–[21] does not apply to our setup, we identify a class of con vex problems which can be cast as a semi-definite program (SDP). Further- more, we show that the necessary conditions for optimality are giv en by coupled matrix equations, which can be solved by a combination of perturbation analysis and homotopy-based Newton’ s method. W e consider the design of both symmetric and non-symmetric feedback gains and show that departure from optimal symmetric design can significantly impro ve the coherence of large-scale formations. B. Pre view of key r esults W e consider the design of optimal localized feedback gains for one-dimensional formations in which each vehicle only uses relativ e distances from its immediate neighbors and its own velocity . This nearest neighbor interaction imposes structural constraints on the feedback gains. W e formulate the structured optimal control problem for both the single- and the double-integrator models. For single-inte grators, we show that the structured optimal control problem is con ve x when we restrict the feedback gain to be a symmetric positi ve definite matrix. In this case, the global minimizer can be computed efficiently , and e ven analytical expressions can be derived. For double-integrators, we also identify a class of conv ex problems by restricting the controller to symmetric position and uniform diagonal velocity gains. W e then remov e this symmetric restriction for both the single- and the double-integrator models and begin the design process with a spatially uniform controller . W e dev elop a homotopy-based Ne wton’ s method that traces a continuous solution path from this controller to the optimal localized gain. Along this homotopy path, we solve a parameterized family of the structured optimal control problems and obtain ana- lytical solutions when the homotopy parameter is small. W e employ perturbation analysis to decouple the matrix equations that result from optimality conditions, thereby rendering the unique optimal structured gain. This solution is used to warm- start Newton’ s method in order to efficiently compute the desired optimal gains as the homotopy parameter is gradually increased. In the second part of the paper, we examine how the performance of the optimally-controlled formation scales with the number of vehicles. W e consider both macroscopic and microscopic performance measures based on whether attention is paid to the absolute position error of each vehicle or the relative position error between neighboring vehicles. W e note that the macroscopic performance measure quantifies the resemblance of the formation to a rigid lattice, i.e., it determines the coher ence of the formation. As sho wn in [12], ev en when local positions are well-regulated, an ‘accordion- like motion’ of the formation can arise from poor scaling of the macroscopic performance measure (formation coherence) with the number of vehicles N . Our objective is thus to enhance formation coherence by means of optimal localized feedback design. In situations for which the control penalty in the quadratic performance objecti ve is formation-size-independent we show that the optimal symmetric and non-symmetric controllers asymptotically provide O ( √ N ) and O ( 4 √ N ) scal- ings of formation coherence. Howe ver , this introduces similar growth of the control ener gy (per vehicle) with N . W e show that bounded control energy can be obtained by judicious selection of an N -dependent control penalty , leading to O ( N ) 3 and O ( √ N ) scalings of formation coherence for the optimal symmetric and non-symmetric controllers, respectiv ely . These results are summarized in T able I and the y hold for both single- and double-integrators for formations in which each vehicle has access to its own velocity ; see Sections V and VI for additional details. In addition to designing optimal localized controllers, we also provide an example of a spatially uniform non-symmetric controller that yields better scaling trends than the optimal spatially varying controller obtained by restricting design to symmetric gains. This indicates that departure from symmetry can improve coherence of large-scale formations and that the controller structure may play a more important role than the optimal selection of the feedback gains. On the other hand, our results also show that the optimal localized controller that achieves the best performance is both non-symmetric and spatially-varying . If each vehicle has access to its own velocity and to relativ e distances from its nearest neighbors, we show sim- ilarity between the optimal position gains and performance scaling trends for single- and double-integrators. The latter observation is in agreement with analytical results obtained for spatially inv ariant formations [12]. W e note that performance of controllers that rely on relativ e measurements or unidi- rectional position exchange can differ significantly for these two models. For spatially-in variant formations with r elative position and velocity measurements, it was shown in [12] that the global performance scales as O ( N 3 ) for double-inte grators and as O ( N ) for single-integrators. In Section V -B, we sho w that spatially uniform look-ahead strategy provides O ( √ N ) scaling of the global performance for the single-integrator model. On the other hand, a look-ahead strategy that is not carefully designed can introduce unfav orable propagation of disturbances through formation of double-integrators [3], [5]. The paper is organized as follo ws. W e formulate the struc- tured optimal control problem in Section II, and show con- ve xity of the symmetric gain design for the single-integrator model in Section III. For non-symmetric gains, we develop the homotopy-based Newton’ s method in Section IV. W e examine performance of localized controllers for the single- and the double-integrator models in Sections V and VI, respectiv ely , where we provide several e xplicit scaling relations. W e con- clude the paper in Section VII with a brief summary of our contributions. I I . P R O B L E M F O R M U L A T I O N A system of N identical vehicles moving along a straight line is shown in Fig. 1. All vehicles are equipped with ranging devices that allow them to measure relative distances with respect to their immediate neighbors. The objectiv e is to design an optimal controller that uses only local information (i.e., relativ e distances between the neighboring vehicles) to keep each vehicle at its global position on a grid of regularly spaced points moving with a constant velocity . W e consider both the single- and the double-integrator models of the v ehicles. The double-inte grators are employed in many studies of vehicular formations; for example, see [1]–[3], [5], [7], [9], [10], [12], [24]. On the other hand, the single- integrator (i.e., kinematic) model is simpler and perhaps more rev ealing in understanding the role of network topologies [4], [23], [25]–[28]. As we show in Section VI, the single- and the double-integrator models exhibit similar performance for formations in which each vehicle – in addition to relative positions with respect to its immediate neighbors – has an access to its own velocity . In the remainder of this section, we formulate the localized optimal control problem for both single- and double-integrators. A. Single- and double-inte grator models W e first consider the kinematic model in which control input ¯ u n directly affects the velocity , ˙ ¯ p n = ¯ d n + ¯ u n , n ∈ { 1 , . . . , N } , where ¯ p n is the position of the n th vehicle and ¯ d n is the disturbance. The desired position of the n th vehicle is giv en by p d,n = v d t + nδ, where v d is the desired cruising velocity and δ is the desired distance between the neighboring vehicles. Every vehicle is assumed to hav e access to both v d and δ . In addition, we confine our attention to formations with a known number of vehicles and leav e issue of adaptation, merging, and splitting for future study . The localized controller utilizes relative position err ors between nearest neighbors, ¯ u n = − f n ( ¯ p n − ¯ p n − 1 − δ ) − b n ( ¯ p n − ¯ p n +1 + δ ) + v d , where the design parameters f n and b n denote the forward and backward feedback gains of the n th vehicle. In deviation variables, { p n := ¯ p n − p d,n , u n := ¯ u n − v d , d n := ¯ d n } , the single-integrator model with nearest neighbor interactions is giv en by ˙ p n = d n + u n , (1a) u n = − f n ( p n − p n − 1 ) − b n ( p n − p n +1 ) , (1b) where the relative position errors p n − p n − 1 and p n − p n +1 can be obtained by ranging devices. As illustrated in Fig. 2a, fictitious lead and follow vehicles, respectiv ely indexed by 0 and N + 1 , are added to the formation. These two vehicles are assumed to move along their desired trajectories, implying that p 0 = p N +1 = 0 , and they are not considered to belong to the formation. Hence, the controls for the 1 st and the N th vehicles are gi ven by u 1 = − f 1 p 1 − b 1 ( p 1 − p 2 ) , u N = − f N ( p N − p N − 1 ) − b N p N . In other words, the first and the last vehicles have access to their own global position errors p 1 and p N , which can be obtained by equipping them with GPS devices. For the double-integrator model, ¨ ¯ p n = ¯ d n + ¯ u n , n ∈ { 1 , . . . , N } , we consider the controller that has an access to the relative position errors between the neighboring vehicles and the 4 (a) (b) Fig. 2: Formation of vehicles with localized (a) non- symmetric; and (b) symmetric gains. absolute velocity errors, ¯ u n = − f n ( ¯ p n − ¯ p n − 1 − δ ) − b n ( ¯ p n − ¯ p n +1 + δ ) − g n ( ˙ ¯ p n − v d ) , where g n denotes the velocity feedback gain. In deviation variables, { p n := ¯ p n − p d,n , v n := ˙ ¯ p n − v d , u n := ¯ u n , d n := ¯ d n } , the double-integrator model is gi ven by ¨ p n = d n + u n , (2a) u n = − f n ( p n − p n − 1 ) − b n ( p n − p n +1 ) − g n v n . (2b) In matrix form, control laws (1b) and (2b) can be written as, u = − F C p = −  F f F b   C f C T f  p, u = − F C  p v  = −  F f F b F v    C f O C T f O O I    p v  , where p , v , and u denote the position error , the v elocity error, and the control input vectors, e.g., p = [ p 1 · · · p N ] T . Fur- thermore, the N × N diagonal feedback gains are determined by F f := diag { f n } , F b := diag { b n } , F v := diag { g n } , and C f is a sparse T oeplitz matrix with 1 on the main diagonal and − 1 on the first lower sub-diagonal. For example, for N = 4 , F f =     f 1 0 0 0 0 f 2 0 0 0 0 f 3 0 0 0 0 f 4     , C f =     1 0 0 0 − 1 1 0 0 0 − 1 1 0 0 0 − 1 1     . (3) Thus, C f p determines the vector of the relative position errors p n − p n − 1 between each vehicle and the one in front of it; similarly , C T f p determines the v ector of the relati ve position errors p n − p n +1 between each vehicle and the one behind it. W e will also consider formations with no fictitious follo w- ers. In this case, the N th vehicle only uses relativ e position error with respect to the ( N − 1) th vehicle, i.e., b N = 0 implying that u N = − f N ( p N − p N − 1 ) for the single- integrator model and u N = − f N ( p N − p N − 1 ) − g N v N for the double-integrator model. B. Structured H 2 pr oblem The state-space representation of the vehicular formation is giv en by ˙ x = A x + B 1 d + B 2 u, y = C x, u = − F y . (SS) For the single-integrator model (1), the state vector is x = p , the measured output y is gi ven by the relativ e position errors between the neighboring v ehicles, and A = O , B 1 = B 2 = I , C =  C f C T f  , F =  F f F b  . (VP1) For the double-integrator model (2), the state vector is x = [ p T v T ] T , the measured output y is giv en by the relativ e position errors between the neighboring vehicles and the absolute velocity errors, and A =  O I O O  , B 1 = B 2 =  O I  , C =   C f O C T f O O I   , F =  F f F b F v  . (VP2) Here, O and I denote the zero and identity matrices, and { F f , F b , F v , C f } are defined in (3). Upon closing the loop, we have ˙ x = ( A − B 2 F C ) x + B 1 d, z =  Q 1 / 2 x r 1 / 2 u  =  Q 1 / 2 − r 1 / 2 F C  x, where z encompasses the penalty on both the state and the control. Here, Q is a symmetric positiv e semi-definite matrix and r is a positive scalar . The objective is to design the structur ed feedback gain F such that the influence of the white stochastic disturbance d , with zero mean and unit variance, on the performance output z is minimized (in the H 2 sense). This control problem can be formulated as [29], [30] minimize J = trace  P B 1 B T 1  subject to ( A − B 2 F C ) T P + P ( A − B 2 F C ) = − ( Q + r C T F T F C ) , F ∈ S (SH2) where S denotes the structural subspace that F belongs to. As shown in [29], the necessary conditions for optimality of (SH2) are given by the set of coupled matrix equations in F , P , and L ( A − B 2 F C ) T P + P ( A − B 2 F C ) = −  Q + r C T F T F C  , (NC1) ( A − B 2 F C ) L + L ( A − B 2 F C ) T = − B 1 B T 1 , (NC2) ( r F C LC T ) ◦ I S = ( B T 2 P LC T ) ◦ I S . (NC3) 5 Here, P and L are the closed-loop observ ability and control- lability Gramians, ◦ denotes the entry-wise multiplication of two matrices, and the matrix I S in (NC3) denotes the structural identity of the subspace S under the entry-wise multiplication, i.e., F ◦ I S = F, with I S = [ I I ] for the single-integrator model and I S = [ I I I ] for the double-integrator model. (For example, [ F f F b ] ◦ [ I I ] = [ F f F b ] .) In the absence of the fictitious follower , an additional constraint b N = 0 is imposed in (SH2) and thus, the structural identity for the single- and the double-integrator models are giv en by [ I I z ] and [ I I z I ] , respectiv ely . Here, I z is a diagonal matrix with its main diagonal gi ven by [ 1 · · · 1 0 ] . Remark 1: Throughout the paper, the structured optimal feedback gain F is obtained by solving (SH2) with Q = I . This choice of Q is motiv ated by our desire to enhance formation coherence, i.e., to keep the global position and velocity errors p n and v n small using localized feedbac k . Since the methods dev eloped in the paper can be applied to other choices of Q , we will describe them for general Q and set Q = I when presenting computational results. C. P erformance of optimal localized contr oller T o e valuate the performance of the optimal localized con- troller F , obtained by solving (SH2) with Q = I , we consider the closed-loop system ˙ x = ( A − B 2 F C ) x + B 1 d, ζ =  ζ 1 ζ 2  =  Q 1 / 2 s − F C  x, s = g or s = l , (4) where ζ 1 is the global or local performance output and ζ 2 is the control input. Motiv ated by [12], we examine two state performance weights for the single-integrator model • Macroscopic (global): Q g = I ; • Microscopic (local): Q l = T , where T is an N × N symmetric T oeplitz matrix with its first ro w given by [ 2 − 1 0 · · · 0 ] ∈ R N . F or e xample, for N = 4 , T =     2 − 1 0 0 − 1 2 − 1 0 0 − 1 2 − 1 0 0 − 1 2     . (5) The macroscopic performance weight Q g = I penalizes the global (absolute) position errors, ζ T 1 ζ 1 = p T Q g p = N X n = 1 p 2 n , and the microscopic performance weight Q l = T penalizes the local (relative) position errors, ζ T 1 ζ 1 = p T Q l p = N X n = 0 ( p n − p n +1 ) 2 , with p 0 = p N +1 = 0 . These state weights induce the macroscopic and microscopic performance measures [12] de- termined by the formation-size-normalized H 2 norm Π s ( N ) = (1 / N ) k G 1 k 2 2 , s = g or s = l , ( Π ) where G 1 is the transfer function of (4) from d to ζ 1 . The macroscopic performance measure Π g quantifies the resem- blance of the formation to a rigid lattice, i.e., it determines the coher ence of the formation [12]. On the other hand, the microscopic performance measure Π l quantifies ho w well regulated the distances between the neighboring vehicles are. W e will also examine the formation-size-normalized control energy (v ariance) of the closed-loop system (4), Π ctr ( N ) = (1 / N ) k G 2 k 2 2 , which is determined by the H 2 norm of the transfer function G 2 from d to ζ 2 = u . Similarly , for the double-integrator model, we use the fol- lowing performance weights • Macroscopic (global), Q g =  I O O I  ; • Microscopic (local), Q l =  T O O I  . D. Closed-loop stability: the r ole of fictitious vehicles W e next sho w that at least one fictitious v ehicle is needed in order to achieve closed-loop stability . This is because the absence of GPS devices in the formation prev ents vehicles from tracking their absolute desired trajectories. For the single-integrator model, the state-feedback gain K p = F f C f + F b C T f is a structured tridiagonal matrix satisfying K p 1 =  f 1 0 · · · 0 b N  T where 1 is the vector of all 1 ’ s. If neither the 1 st nor the N th vehicle has access to its o wn global position, i.e., f 1 = b N = 0 , then K p has a zero eigen value with corresponding eigen vector 1 . Hence, the closed-loop system is not asymptotically stable regardless of the choice of the feedback gains { f n } N n = 2 and { b n } N − 1 n = 1 . In the presence of stochastic disturbances, the average-mode (associated with the eigen vector 1 ) under goes a random walk and the steady-state v ariance of the de viation from the absolute desired trajectory becomes unbounded [12], [23], [28]. In this case, other performance measures that render this av erage- mode unobservable can be considered [12]. For the double-integrator model, the action of A cl = A − B 2 F C on [ 1 T 0 T ] T is giv en by  O I − K p − F v   1 0  =  0 − K p 1  , where 0 is the N -vector of all 0 ’ s. Thus, if f 1 = b N = 0 then A cl has a zero eigenv alue with corresponding eigenv ector [ 1 T 0 T ] T . Therefore, for both the single- and the double- integrator models, we need at least one vehicle with access to its global position in order to achieve closed-loop stability . I I I . D E S I G N O F S Y M M E T R I C G A I N S F O R T H E S I N G L E - I N T E G R ATO R M O D E L : A C O N V E X P R O B L E M In this section, we design the optimal symmetric feedback gains for the single-integrator model; see Fig. 2b. This is a special case of the localized design, obtained by restricting the forward and the backw ard gains between the neighboring vehicles to be equal to each other , i.e., f n = b n − 1 for 6 n ∈ { 2 , . . . , N } . Under this assumption, we sho w that (SH2) is a con vex optimization problem for the single-integrator model. This implies that the global minimum can be computed effi- ciently . Furthermore, in the absence of the fictitious follower , we provide analytical expr essions for the optimal feedback gains. Let us denote k 1 = f 1 and k N +1 = b N and let k n = f n = b n − 1 , n ∈ { 2 , . . . , N } . (6) For the single-integrator model, the structured gain becomes a symmetric tridiagonal matrix K = F f C f + F b C T f =       k 1 + k 2 − k 2 − k 2 k 2 + k 3 . . . . . . . . . − k N − k N k N + k N +1       . (7) Consequently , A cl = − K is Hurwitz if and only if K is positiv e definite, in which case the L yapunov equation in (SH2) simplifies to K P + P K = Q + r K K . The application of [31, Lemma 1] transforms the prob- lem (SH2) of optimal symmetric design for the single- integrator model to minimize K J ( K ) = (1 / 2) trace  QK − 1 + r K  subject to K > 0 and K ∈ S K (SG) where K ∈ S K is a linear structural constraint given by (7). (Specifically , K = F f C f + F b C T f is a symmetric tridiag- onal matrix with the linear constraint (6).) By introducing an auxiliary variable X = X T ≥ Q 1 / 2 K − 1 Q 1 / 2 , we can formulate (SG) as an SDP in X and K minimize X, K (1 / 2) trace ( X + r K ) subject to K > 0 , K ∈ S K ,  K Q 1 / 2 Q 1 / 2 X  ≥ 0 , which can be solved using av ailable SDP solvers. Here, we hav e used the Schur complement [32, Appendix A.5.5] in conjunction with K > 0 to express X ≥ Q 1 / 2 K − 1 Q 1 / 2 as an LMI. Next, we exploit the structure of K to express J in (SG) with Q = I in terms of the feedback gains { k n } N +1 n = 1 between the neighboring vehicles. Since the in verse of the symmetric tridiagonal matrix K can be determined analytically [33, Theorem 2.3], the ij th entry of K − 1 is giv en by ( K − 1 ) ij = γ i ( γ N +1 − γ j ) γ N +1 , j ≥ i, γ i = i X n = 1 1 k n , (8) yielding the following expression for J J = (1 / 2) trace  K − 1 + r K  = 1 2 N X n = 1 γ n ( γ N +1 − γ n ) γ N +1 + r k 1 + k N +1 2 + N X n = 2 k n ! . The abov e expression for J is well-defined for { k n } N +1 n = 1 that guarantee positiv e definiteness of K in (7); this is because the closed-loop A -matrix is determined by A cl = − K . The global minimizer of J can be computed using the gradient method; see Appendix A. For the formations without the fictitious follower , we next deriv e explicit analytical e xpr ession for the global symmetric minimizer K = K T > 0 of (SG) with Q = I . In this case k N +1 = 0 and the ij th entry of K − 1 in (8) simplifies to ( K − 1 ) ij = γ i for j ≥ i . Consequently , the unique minimum of J = 1 2 N X n = 1 γ n + r k 1 2 + N X n = 2 k n ! = 1 2 N X n = 1 N + 1 − n k n + r k 1 2 + N X n = 2 k n ! , is attained for k 1 = p N /r , k n = p ( N + 1 − n ) / (2 r ) , n ∈ { 2 , . . . , N } . (9) W e also note that trace  K − 1  = N X n = 1 γ n = N X n = 1 N + 1 − n k n = r k 1 + 2 N X n = 2 k n ! = r trace ( K ) , (10) where the third equality follows from (9). This result is used to examine the performance of large-scale formations in Section V -C. Figure 3 sho ws the optimal symmetric gains for a formation with N = 50 v ehicles, Q = I , and r = 1 . Since the fictitious leader and the follower always mov e along their desired trajectories, the vehicles that are close to them hav e larger gains than the other vehicles. When no fictitious follower is present, the gains decrease monotonically from the first to the last vehicle; see ( × ) in Fig. 3. In other words, the farther away the vehicle is from the fictitious leader the less weight it places on the information coming from its neighbors. This is because uncorrelated disturbances that act on the vehicles corrupt the information about the absolute desired trajectory as it propagates from the fictitious leader down the formation (via relativ e information exchange between the vehicles). When both the fictitious leader and the follower are present, the gains decrease as one moves from the boundary to the center of the formation; see ( ◦ ) in Fig. 3. This can be attrib uted to the fact that the information about the absolute desired trajectories becomes noisier as it propagates from the fictitious vehicles to the center of the formation. I V . H O M OT O P Y - BA S E D N E W T O N ’ S M E T H O D In this section, we remove the symmetric feedback gain restriction and utilize a homotopy-based Newton’ s method to solve (SH2). In [29], Newton’ s method for general struc- tured H 2 problems is dev eloped. For (SH2) with the specific problem data (VP1) and (VP2), it is possible to employ a homotopy-based approach to solv e a parameterized family of 7 Fig. 3: Optimal symmetric gains for formations with follower ( ◦ ) and without follower ( × ) for N = 50 , Q = I , and r = 1 . ( × ) are obtained by e v aluating formula (9) and ( ◦ ) are computed using the gradient method described in Appendix A. problems, which ranges between an easily solvable problem and the problem of interest. In particular , we consider Q ( ε ) = Q 0 + ε ( Q d − Q 0 ) , (11) where Q 0 is the initial weight to be selected, Q d is the desir ed weight, and ε ∈ [0 , 1] is the homotopy parameter . Note that Q = Q 0 for ε = 0 , and Q = Q d for ε = 1 . The homotopy-based Newton’ s method consists of three steps: (i) For ε = 0 , we find the initial weight Q 0 with respect to which a spatially uniform gain F 0 is in versely optimal . This is equiv alent to solving problem (SH2) analytically with the performance weight Q 0 . (ii) For 0 < ε  1 , we employ perturbation analysis to determine the first few terms in the expansion F ( ε ) = P ∞ n = 0 ε n F n . (iii) For larger values of ε , we use Newton’ s method for structured H 2 design [29] to solve (SH2). W e gradually increase ε and use the structured optimal gain obtained for the pre vious value of ε to initialize the ne xt round of iterations. This process is repeated until the desired value ε = 1 is reached. In the remainder of this section, we focus on the single- integrator model. In Section VI, we solve problem (SH2) for the double-integrator model. A. Spatially uniform symmetric gain: in verse optimality for ε = 0 One of the simplest localized strategies is to use spatially uniform gain , where F f and F b are diagonal matrices with f n = f and b n = b for all n and some positive f and b . In particular , for F f = F b = I it is easy to show closed-loop stability and to find the performance weight Q 0 with respect to which the spatially uniform symmetric gain K 0 = F 0 C =  I I   C f C T f  = T is inv ersely optimal. The problem of in verse optimality amounts to finding the performance weight Q 0 for which an a priori specified K 0 is the corresponding optimal state- feedback gain [34], [35]. From linear quadratic regulator theory , the optimal state-feedback gain is given by K 0 = R − 1 B T 2 P 0 where P 0 is the positive definite solution of A T P 0 + P 0 A + Q 0 − P 0 B 2 R − 1 B T 2 P 0 = 0 . For the kinematic model (VP1), A = O and B 2 = I , with R = r I , we have K 0 = r − 1 P 0 and Q 0 − r − 1 P 0 P 0 = 0 . Therefore, the state penalty Q 0 = r K 2 0 = r T 2 guarantees in verse optimality of the spatially uniform symmetric gain K 0 . The above procedure of finding Q 0 can be applied to any structured gain F 0 that yields a symmetric positive definite K 0 , e.g., the optimal symmetric gain of Section III. B. P erturbation analysis for ε  1 W e next utilize perturbation analysis to solve (SH2) with Q ( ε ) given by (11) for ε  1 . For small ε , by representing P , L , and F as P = ∞ X n = 0 ε n P n , L = ∞ X n = 0 ε n L n , F = ∞ X n = 0 ε n F n , substituting in (NC1)-(NC3), and collecting same-order terms in ε , we obtain the set of equations (P A) with A 0 := A − B 2 F 0 C. Note that these equations are con veniently coupled in one direction, in the sense that for any n ≥ 1 , O ( ε n ) equations depend only on the solutions of O ( ε m ) equations for m ≤ n . In particular , it is easy to verify that the first and the third equations of O (1) are satisfied with K 0 = F 0 C = r − 1 B T 2 P 0 and with Q 0 = r K 2 0 identified in Section IV -A. Thus, the matrix L 0 can be obtained by solving the second equation of O (1) , and the matrices P 1 , F 1 , and L 1 can be obtained by solving the first, the third, and the second equations of O ( ε ) , respecti vely . The higher order terms F n , P n , and L n can be determined in a similar fashion. The matrix F found by this procedure is the unique optimal solution of the control problem (SH2) for ε  1 . This is because the equations (P A), under the assumption of conv ergence for small ε , giv e a unique matrix F ( ε ) = P ∞ n = 0 ε n F n . W e next provide analytical expressions for F 1 = [ F (1) f F (1) b ] obtained by solving the O ( ε ) equations in (P A) with r = 1 , Q 0 = T 2 , and Q d = I . When a fictitious follower is present, we ha ve (deriv ations are omitted for brevity) f (1) n = n ( n − N − 1)(4 n ( N + 1) − N (2 N + 7) + 1) 12 ( N 2 − 1) − 1 2 , b (1) n = n ( N + 1 − n )(4 n ( N + 1) − N (2 N + 1) − 5) 12 ( N 2 − 1) − 1 2 , (12) where f (1) n and b (1) n denote the n th diagonal entries of F (1) f and F (1) b . From (12) it follows that f (1) n = b (1) N +1 − n for n ∈ { 1 , . . . , N } . When a fictitious follower is not present, we ha ve f (1) n = ( − n 2 + ( N + 1) n − 1) / 2 , n ∈ { 1 , . . . , N − 1 } , f (1) N = ( N − 1) / 2 , b (1) n = ( n 2 − N n − 1) / 2 , n ∈ { 1 , . . . , N − 1 } , b (1) N = 0 . T o compute the optimal structured feedback gain for larger values of ε , we use F ( ε ) obtained from perturbation analysis to initialize Newton’ s method, as described in Section IV -C. 8 O (1) :      A T 0 P 0 + P 0 A 0 = − ( Q 0 + r C T F T 0 F 0 C ) A 0 L 0 + L 0 A T 0 = − B 1 B T 1 ( r F 0 C L 0 C T ) ◦ I S = ( B T 2 P 0 L 0 C T ) ◦ I S O ( ε ) :      A T 0 P 1 + P 1 A 0 = − ( Q d − Q 0 ) A 0 L 1 + L 1 A T 0 = ( B 2 F 1 C ) L 0 + L 0 ( B 2 F 1 C ) T ( r F 1 C L 0 C T ) ◦ I S = ( B T 2 P 1 L 0 C T ) ◦ I S O ( ε 2 ) :      A T 0 P 2 + P 2 A 0 = ( B 2 F 1 C ) T P 1 + P 1 ( B 2 F 1 C ) − r C T F T 1 F 1 C A 0 L 2 + L 2 A T 0 = ( B 2 F 1 C ) L 1 + L 1 ( B 2 F 1 C ) T + ( B 2 F 2 C ) L 0 + L 0 ( B 2 F 2 C ) T ( r F 2 C L 0 C T ) ◦ I S = ( B T 2 P 1 L 1 C T + B T 2 P 2 L 0 C T − r F 1 C L 1 C T ) ◦ I S . . . . . . (P A) C. Newton’ s method for lar ger values of ε In this section, we employ Newton’ s method dev eloped in [29] to solve the necessary conditions for optimality (NC1)- (NC3) as ε is gradually increased to 1 . Newton’ s method is an iterativ e descent algorithm for finding local minima in optimization problems [32]. Specifically , gi ven an initial stabilizing structured gain F 0 , a decreasing sequence of the objectiv e function { J ( F i ) } is generated by updating F accord- ing to F i +1 = F i + s i ˜ F i . Here, ˜ F i is the Newton direction that satisfies the structural constraint and s i is the step-size. The details of computing ˜ F i and choosing the step-size s i can be found in [29]. For small ε , we initialize Newton’ s method using F ( ε ) obtained from the perturbation expansion up to the first order in ε , F ( ε ) = F 0 + εF 1 . W e then increase ε slightly and use the optimal structured gain resulting from Newton’ s method at the pre vious ε to initialize the next round of iterations. W e continue increasing ε gradually until desired value ε = 1 is reached, that is, until the optimal structured gain F for the desired Q d is obtained. Since the homotopy-based Ne wton’ s method solves a family of optimization problems parameterized by ε , the optimal feedback gain is a function of ε ∈ [0 , 1] . T o see the incremental change relative to the spatially uniform gain F 0 , we consider the difference between the optimal forward gain f n ( ε ) and the uniform gain f n (0) = 1 , ˜ f n ( ε ) := f n ( ε ) − f n (0) = f n ( ε ) − 1 . Figure 4a shows the normalized profile ˜ f ( ε ) / k ˜ f ( ε ) k for a formation with fictitious follower , N = 50 , r = 1 , Q 0 = T 2 , and Q d = I . The values of ε are determined by 20 logarithmically spaced points between 10 − 4 and 1 . As ε increases, the normalized profile changes from an almost sinusoidal shape (cf. analytical expression in (12)) at ε = 10 − 4 to an almost piecewise linear shape at ε = 1 . Note that the homotopy-based Newton’ s method con ver ges to the same feedback gains at ε = 1 when it is initialized by the optimal symmetric controller obtained in Section III. Since the underlying path-graph exhibits symmetry between the edge pairs associated with f n and b N +1 − n , the optimal for - ward and backw ard gains satisfy a centr al symmetry property , f n = b N +1 − n , n ∈ { 1 , . . . , N } , for all ε ∈ [0 , 1] ; see Fig. 4b for ε = 1 . W e note that the first vehicle has a larger forward gain than other vehicles; this is because it neighbors the fictitious leader . The forward gains decrease as one mov es away from the fictitious leader; this is because information about the absolute desired trajectory of the fictitious leader becomes less accurate as it propagates down the formation. Similar interpretation can be given to the optimal backward gains, which monotonically increase as one mov es towards the fictitious follower . Since the 1 st vehicle has a ne gative backward gain (see Fig. 4b), if the distance between the 1 st and the 2 nd vehicles is greater than the desired value δ , then the 1 st vehicle distances itself ev en further from the 2 nd vehicle. On the other hand, if the distance is less than δ , then the 1 st vehicle pulls itself ev en closer to the 2 nd vehicle. This negati ve backward gain of the 1 st vehicle can be interpreted as follo ws: Since the 1 st vehicle has access to its global position, it aims to correct the absolute positions of other vehicles in order to enhance formation coherence. If the 2 nd vehicle is too close to the 1 st vehicle, then the 1 st v ehicle moves towards the 2 nd v ehicle to push it back; this in turn pushes other vehicles back. If the 2 nd vehicles is too far from the 1 st vehicle, then the 1 st vehicle mov es away from the 2 nd vehicle to pull it forward; this in turn pulls other vehicles forward. Similar interpretation can be giv en to the negati ve forward gain of the N th vehicle that neighbors the fictitious follower . Also note that the forward gain of the N th vehicle becomes positive when the fictitious follower is r emoved from the formation; see Fig. 5c. This perhaps suggests that negati ve feedback gains of the 1 st and the N th vehicles are a consequence of the fact that both of them have access to their own global positions. As sho wn in Figs. 5a and 5b, the normalized optimal gains for the formation without the fictitious follower also change continuously as ε increases to 1 . In this case, howe ver , the optimal forward and backward gains do not satisfy the central symmetry; see Fig. 5c. Since the optimal controller puts more emphasis on the vehicles ahead when the fictitious follower is not present, the forward gains hav e larger magnitudes than the backward gains. As in the formations with the fictitious 9 (a) (b) Fig. 4: Formation with fictitious follower , N = 50 , r = 1 , Q 0 = T 2 , and Q d = I . (a) Normalized optimal forward gain ˜ f ( ε ) / k ˜ f ( ε ) k changes from an almost sinusoidal shape (cf. analytical expression in (12)) at ε = 10 − 4 to an almost piecewise linear shape at ε = 1 . (b) Optimal forw ard ( ◦ ) and backward (+) gains at ε = 1 . (a) (b) (c) Fig. 5: F ormation without fictitious follower , N = 50 , r = 1 , Q 0 = T 2 , and Q d = I . Normalized optimal (a) forward and (b) backward gains. (c) Optimal forward ( ◦ ) and backward (+) gains at ε = 1 . follower , the optimal forward gains decrease monotonically as one moves aw ay from the fictitious leader . On the other hand, the optimal backward gains at first increase as one mo ves a way from the 1 st vehicle and then decrease as one approaches the N th vehicle in order to satisfy the constraint b N = 0 . V . P E R F O R M A N C E V S . S I Z E F O R T H E S I N G L E - I N T E G R A T O R M O D E L In this section, we study the performance of the optimal symmetric and non-symmetric gains obtained in Sections III and IV -C. This is done by e xamining the dependence on the formation size of performance measures Π g , Π l , and Π ctr introduced in Section II-C. Our results highlight the role of non-symmetry and spatial variations on the scaling trends in large-scale formations. They also illustrate performance improv ement achiev ed by the optimal controllers relative to spatially uniform symmetric and non-symmetric feedback gains. For the spatially uniform symmetric gain with f n = b n = α > 0 , we sho w analytically that Π g is an affine function of N . This implies that the formation coherence scales linearly with N irrespectiv e of the value of α . W e also analytically establish that the spatially uniform non-symmetric gain with { f n = α > 0 , b n = 0 } (look-ahead strategy) provides a squar e-r oot asymptotic dependence of Π g on N . Thus, sym- metry breaking between the forward and backward gains may improv e coherence of large-scale formations. Note that the forward-backward asymmetry also provides more fav orable scaling trends of the least damped mode of the closed-loop system [10]. W e then inv estigate ho w spatially varying optimal feedback gains, introduced in Sections III and IV -C, influence coherence of the formation. W e show that the optimal sym- metric gain provides a squar e-r oot dependence of Π g on N and that the optimal non-symmetric gain provides a fourth-r oot dependence of Π g on N . Even though we are primarily interested in asymptotic scal- ing of the global performance measure Π g , we also examine the local performance measure Π l and the control energy Π ctr . From Section II-C we recall that the global and local performance measures quantify the formation-size-normalized H 2 norm of the transfer function from d to ζ 1 of the closed- loop system, ˙ x = − F C x + d ζ =  ζ 1 ζ 2  =  Q 1 / 2 s − F C  x, Q s =  I , s = g , T , s = l, and that Π ctr is the formation-size-normalized H 2 norm of the transfer function from d to ζ 2 . These can be determined from Π s = (1 / N ) trace ( L Q s ) , Π ctr = (1 / N ) trace  L C T F T F C  , (13) where L denotes the closed-loop controllability Gramian, ( − F C ) L + L ( − F C ) T = − I . (14) The asymptotic scaling properties of Π g , Π l , and Π ctr , for the above mentioned spatially uniform controllers and the spa- tially varying optimal controllers, obtained by solving (SH2) with Q = I and r = 1 , are summarized in T able II. For both spatially uniform symmetric and look-ahead strategies, we analytically determine the dependence of these performance measures on the formation size in Sections V -A and V -B. Furthermore, for the formation without the fictitious follower subject to the optimal symmetric gains, we provide analytical results in Section V -C. For the optimal symmetric and non- symmetric gains in the presence of fictitious followers, the scaling trends are obtained with the aid of numerical compu- tations in Section V -C. Sev eral comments about the results in T able II are given next. First, in contrast to the spatially uniform controllers, the optimal symmetric and non-symmetric gains, resulting from an N -independent control penalty r in (SH2), do not provide uniform bounds on the control energy per vehicle, Π ctr . This 10 implies the trade-of f between the formation coherence Π g and control energy Π ctr in the design of the optimal controllers. It is thus of interest to examine formation coherence for optimal controllers with bounded control energy per vehicle (see Remark 2). Second, the controller structure (e.g., symmetric or non-symmetric gains) plays an important role in the formation coherence. In particular , departure from symmetry in localized feedback gains can significantly improve coherence of large- scale formations (see Remark 3). A. Spatially uniform symmetric gain For the spatially uniform symmetric controller with f n = b n = α > 0 , we next show that Π g is an affine function of N and that, in the limit of an infinite number of vehicles, both Π l and Π ctr become formation-size-independent. These results hold irrespectiv e of the presence of the fictitious follo wer . For the single-integrator model with the fictitious follower we have K = F C = αT (see (5) for the definition of T ), and L = T − 1 / (2 α ) solves the L yapunov equation (14) [31, Lemma 1]. Since the n th diagonal entry of T − 1 is determined by (cf. (8)) ( T − 1 ) nn = n ( N + 1 − n ) / ( N + 1) , from (13) we conclude that the global performance measure Π g is an af fine function of N , and that both Π l and Π ctr are formation-size-independent, Π g = trace  T − 1  / (2 αN ) = 1 2 αN N X n = 1 n − 1 2 αN ( N + 1) N X n = 1 n 2 = N + 2 12 α , Π l = trace  T T − 1  / (2 αN ) = 1 / (2 α ) , Π ctr = trace  α 2 T T T − 1  / (2 αN ) = α. For the formation without the fictitious follower , the following expressions Π g = ( N + 1) / (4 α ) , Π l = 1 /α, Π ctr = α (3 N + 1) / (2 N ) , imply that, for the spatially uniform symmetric controller , the asymptotic scaling trends do not depend on the presence of the fictitious follower (deriv ations omitted for bre vity). B. Spatially uniform non-symmetric gain (look-ahead strat- e gy) W e next examine the asymptotic scaling of the performance measures for the spatially uniform non-symmetric gain with { f n = α > 0 , b n = 0 } . W e establish the squar e-r oot scaling of Π g with N and the formation-size-independent scaling of Π l . Furthermore, in the limit of an infinite number of vehicles, we show that Π ctr becomes N -independent. For the single-integrator model with K = F C = αC f (see (3) for the definition of C f ), the solution of the L yapunov equation (14) is gi ven by L = Z ∞ 0 e − α C f t e − α C T f t d t. (15) As shown in Appendix B, the in verse Laplace transform of ( sI + αC f ) − 1 can be used to determine the analytical expression for e − α C f t , yielding the follo wing formulae, Π g ( N ) = 1 N N X n = 1 L nn = 1 N N X n = 1 α Γ( n + 1 / 2) √ π Γ( n ) = 2 α Γ( N + 3 / 2) 3 √ π Γ( N + 1) , Π l = α, Π ctr = α − (1 / N ) L N N , with Γ( · ) denoting the Gamma function. These are used in Appendix B to show that, in the limit of an infinite number of vehicles, a look-ahead strate gy for the single-integrator model provides the square-root dependence of Π g on N and the formation-size-independent Π l and Π ctr . C. Optimal symmetric and non-symmetric controller s W e next examine the asymptotic scaling of the performance measures for the optimal symmetric and non-symmetric gains of Sections III and IV -C. For the formation without the fictitious follower , we analytically establish that the optimal symmetric gains asymptotically provide O ( √ N ) , O (1 / √ N ) , and O ( √ N ) scalings of Π g , Π l , and Π ctr , respectiv ely . W e then use numerical computations to (i) confirm these scaling trends for the optimal symmetric gains in the presence of the fictitious follo wer; and to (ii) show a fourth-root dependence of Π g and Π ctr on N and an O (1 / 4 √ N ) dependence of Π l for the optimal non-symmetric gains. All these scalings are obtained by solving (SH2) with the formation-size-independent control penalty r and Q = I . W e also demonstrate that uniform control variance (per vehicle) can be obtained by judicious selection of an N -dependent r . For the optimal symmetric and non- symmetric gains, this constraint on control energy (v ariance) increases the asymptotic dependence of Π g on N to linear and square-root, respectiv ely . For the formation without the fictitious follower , the op- timal symmetric gains are gi ven by (9). As sho wn in (10), trace ( K − 1 ) = trace ( rK ) , thereby yielding Π g = r Π ctr = 1 2 N trace  K − 1  = √ r 2 N √ N + N − 1 X n = 1 √ 2 n ! . (16) In the limit of an infinite number of vehicles, lim N → ∞ Π g ( N ) √ N = lim N → ∞ N − 1 X n = 1 r r n 2 N 1 N = Z 1 0 r r x 2 d x = r 2 r 9 , which, for an N -independent r , leads to an asymptotic square- root dependence of Π g and Π ctr on N , Π g ( N ) = r 2 rN 9 + r r 4 N , N  1 Π ctr ( N ) = r 2 N 9 r + 1 √ 4 rN , N  1 . (17) Similar calculation can be used to obtain O (1 / √ N ) asymp- 11 T ABLE II: Asymptotic dependence of Π g , Π l , and Π ctr on the formation size N for uniform symmetric, uniform non-symmetric (look-ahead strategy), and optimal symmetric and non-symmetric gains of Sections III and IV -C with Q = I and r = 1 . The scalings displayed in red are determined analytically ; other scalings are estimated based on numerical computations. Controller Π g Π l Π ctr uniform symmetric with/without follo wer O ( N ) O (1) O (1) uniform non-symmetric O ( √ N ) O (1) O (1) optimal symmetric without follo wer O ( √ N ) O (1 / √ N ) O ( √ N ) optimal symmetric with follo wer O ( √ N ) O (1 / √ N ) O ( √ N ) optimal non-symmetric with/without follo wer O ( 4 √ N ) O (1 / 4 √ N ) O ( 4 √ N ) (a) (b) Fig. 6: (a) Square-root scaling of Π g ( ∗ ) using optimal sym- metric gain of Section III, 0 . 2784 √ N + 0 . 0375 (curv e); and (b) Fourth-root scaling of Π g ( ◦ ) using optimal non-symmetric gain of Section IV -C, 0 . 4459 4 √ N − 0 . 0866 (curve). The optimal controllers are obtained by solving (SH2) with Q = I and r = 1 for the formation with the fictitious follower . totic scaling of Π l . W e next use numerical computations to study the scaling trends for the optimal symmetric and non-symmetric gains in the presence of fictitious followers. The optimal symmetric gain (cf. ( ◦ ) in Fig. 3) provides a squar e-r oot scaling of Π g with N ; see Fig. 6a. On the other hand, the optimal non- symmetric gain (cf. Fig. 4b) leads to a fourth-r oot scaling of Π g with N ; see Fig. 6b. The local performance measure Π l decreases monotonically with N for both controllers, with Π l scaling as O (1 / √ N ) for the optimal symmetric gain and as O (1 / 4 √ N ) for the optimal non-symmetric gain; see Fig. 7. For both the optimal symmetric and non-symmetric controllers, our computations indicate equiv alence between the control energy and the global performance measure when r = 1 . (For the optimal symmetric gain without the fictitious follo wer and r = 1 , we hav e analytically shown that Π ctr = Π g ; see for- mula (16).) Therefore, the asymptotic scaling of the formation- size-normalized control energy is O ( √ N ) for the optimal symmetric gain and O ( 4 √ N ) for the optimal non-symmetric gain. Finally , for the formations without the fictitious follo wer , our computations indicate that the optimal non-symmetric gains also asymptotically provide O ( 4 √ N ) , O (1 / 4 √ N ) , and O ( 4 √ N ) scalings of Π g , Π l , and Π ctr , respectiv ely . Remark 2: In contrast to the spatially uniform controllers, the optimal structured controllers of Sections III and IV -C, re- sulting from an N -independent control penalty r in (SH2), do not provide uniform bounds on the formation-size-normalized (a) (b) Fig. 7: (a) Π l ( ∗ ) using the optimal symmetric gain of Sec- tion III, 1 . 8570 / √ N + 0 . 0042 (curve); and (b) Π l ( ◦ ) using the optimal non-symmetric gain of Section IV -C, 1 . 4738 / 4 √ N + 0 . 0191 (curve). The optimal controllers are obtained by solv- ing (SH2) with Q = I and r = 1 for the formation with the fictitious follower . control energy . These controllers are obtained using H 2 frame- work in which control ef fort represents a ‘soft constraint’. It is thus of interest to examine formation coherence for optimal controllers with bounded control energy per vehicle. For formations without the fictitious follower , from (17) we see that the optimal symmetric controller with r ( N ) = 2 N / 9 asymptotically yields Π ctr ≈ 1 and Π g ≈ 2 N / 9 + 1 / (3 √ 2) . Similarly , for formations with followers, the optimal gains that result in Π ctr ≈ 1 for large N can be obtained by changing control penalty from r = 1 to r ( N ) = 0 . 08 N for the optimal symmetric gain and to r ( N ) = 0 . 175 √ N for the optimal non- symmetric gain 1 . These N -dependent control penalties provide an affine scaling of Π g with N for the optimal symmetric gain and a squar e-r oot scaling of Π g with N for the optimal non-symmetric gain; see Fig. 8. The asymptotic scalings for formations without follo wers subject to the optimal symmetric gains are obtained analytically (cf. (17)); all other scalings are obtained with the aid of computations. Remark 3: Figure 8 illustrates the global performance mea- sure Π g obtained with four aforementioned structured con- trollers that asymptotically yield Π ctr ≈ 1 for formations with fictitious follower . Note that the simple look-ahead strategy outperforms the optimal symmetric gain; O ( √ N ) vs. O ( N ) scaling. Thus, departure from symmetry in localized feedback gains can significantly improv e coherence of large-scale for- 1 Both spatially uniform symmetric and look-ahead strategies with α = 1 yield Π ctr = 1 in the limit of an infinite number of vehicles. 12 Fig. 8: Π g using four structured gains with Π ctr ≈ 1 for formations with fictitious follower: spatially uniform sym- metric (  ), N / 12 + 1 / 6 (blue curve), spatially uniform non- symmetric (  ), 2 √ N / (3 √ π ) (green curve), optimal symmet- ric ( ∗ ), 0 . 0793 N + 0 . 0493 (black curve), and optimal non- symmetric ( ◦ ), 0 . 1807 √ N − 0 . 0556 (red curve). mations. In particular, we hav e provided an example of a spatially uniform non-symmetric controller that yields better scaling trends than the optimal spatially varying controller obtained by restricting design to symmetric gains. Given the extra degrees of freedom in the optimal symmetric gain this is perhaps a surprising observation, indicating that the network topology may play a more important role than the optimal selection of the feedback gains in performance of large-scale interconnected systems. On the other hand, our results show that the optimal localized controller that achiev es the best performance is both non-symmetric and spatially-varying . V I . D O U B L E - I N T E G R ATO R M O D E L In this section, we solve (SH2) for the double-integrator model using the homotopy-based Newton’ s method. W e then discuss the influence of the optimal structured gain on the asymptotic scaling of the performance measures introduced in Section II-C. For a formation in which each vehicle – in addition to relati ve positions with respect to its immediate neighbors – has access to its own velocity , our results highlight similarity between optimal forward and backward position gains for the single- and the double-integrator models. W e further show that the performance measures exhibit similar scaling properties to those found in single-integrators. W e also establish con ve xity of (SH2) for the double-integrator model by restricting the controller to symmetric position and uniform diagonal velocity gains. The perturbation analysis and the homotopy-based New- ton’ s method closely follo w the procedure described in Sections IV -B and IV -C, respectiv ely . In particular, F 0 = [ αI αI β I ] yields K 0 = F 0 C = [ αT β I ] . As shown in [35], for positiv e α and β with β 2 > 8 α , this spatially uniform structured feedback gain is stabilizing and in versely (a) (b) Fig. 9: Double-integrator model with fictitious follo wer , N = 50 , Q = I and r = 1 . (a) The optimal forward ( ◦ ) and backward gains ( + ); (b) the optimal velocity gains (  ) . optimal with respect to Q 0 =  Q p O O Q v  , Q p = r α 2 T 2 , Q v = r ( β 2 I − 2 α T ) , r > 0 . In what follows, we choose α = 1 and β = 3 and employ the homotopy-based Ne wton’ s method to solve (SH2) for the double-integrator model. For a formation with fictitious follower , N = 50 , Q = I , and r = 1 the optimal forward and backward position gains are shown in Fig. 9a and the optimal velocity gains are shown in Fig. 9b. W e note remarkable similarity between the optimal position gains for the single- and the double-integrator models; cf. Fig. 9a and Fig. 4b. For a formation without fictitious follower , the close resemblance between the optimal position gains for both models is also observed. As in the single-integrator model, our computations indicate that the optimal localized controller , obtained by solving (SH2) with Q = I and r = 1 , provides a fourth-root dependence of the macroscopic performance measure Π g on N ; see Fig. 10a. Furthermore, the microscopic performance measure and control energy asymptotically scale as O (1 / 4 √ N ) and O ( 4 √ N ) , respectiv ely; see Fig. 10b and Fig. 10c. For comparison, we next provide the scaling trends of the performance measures for both the spatially uniform symmetric and look-ahead controllers. As in the single- integrator model, the spatially uniform symmetric gain F 0 = [ αI αI β I ] provides linear scaling of Π g with N and the formation-size-independent Π l and Π ctr , Π g ( N ) = ( N + 2) / (12 α β ) + 1 / (2 β ) , Π l = 1 / (2 αβ ) + 1 / (2 β ) , Π ctr = α/β + β / 2 . On the other hand, for the double-inte grator model the perfor- mance of the look-ahead strategy K = F C = [ αC f β I ] heavily depends on the choices of α and β . In particular, for α = 1 / 4 and β = 1 , using similar techniques as in Section V -B, we obtain Π g ( N ) = 1 √ π N X n = 1 ( N − n + 1) 2 N Γ(2 n )  8 Γ(2 n − 1 2 ) + Γ(2 n − 3 2 )  , which asymptotically leads to the formation-size-independent 13 (a) Π g (b) Π l (c) Π ctr Fig. 10: Double-integrator model with the optimal non- symmetric gain obtained by solving (SH2) with Q = I and r = 1 for formations with the fictitious follower: (a) Π g ( ◦ ) , 0 . 0736 4 √ N + 0 . 4900 (curve) (b) Π l ( ◦ ) , 1 . 1793 / 4 √ N + 0 . 0408 (curve); (c) Π ctr ( ◦ ) , 0 . 2742 4 √ N + 0 . 8830 (curv e). scaling of Π ctr and the square-root scaling of Π g with N , i.e., lim N →∞ Π g ( N ) / √ N = 16 / (3 √ 2 π ) . This is in sharp contrast to α = β = 1 which leads to an exponential dependence of Π g on N . Therefore, the design of the look-ahead strate gy is much more subtle for double-inte grators than for single-integrators. Remark 4: For the double-integrator model with K = F C = [ K p β I ] and fixed β > 0 we next show con vexity of (SH2) with respect to K p = K T p > 0 . The L yapunov equation in (SH2), for the block diagonal state weight Q with components Q 1 and Q 2 , can be rewritten in terms of the components of P =  P 1 P 0 P T 0 P 2  , K p P T 0 + P 0 K p = Q 1 + K p K p , (18a) K p P 2 − P 1 + β P 0 = β K p , (18b) 2 β P 2 = P 0 + P T 0 + Q 2 + β 2 I . (18c) Linearity of the trace operator in conjunction with B 1 = [ O I ] T and (18c) yields J = trace ( P 2 ) = trace  2 P 0 + Q 2 + β 2 I  / (2 β ) = trace  K − 1 p Q 1 + K p + Q 2 + β 2 I  / (2 β ) , where the last equation is obtained by multiplying (18a) from the left with K − 1 p and using trace ( K − 1 p P 0 K p ) = trace ( P 0 ) . For Q 1 ≥ 0 , similar argument as in Section III can be used to conclude con vexity of J with respect to K p = K T p > 0 . V I I . C O N C L U D I N G R E M A R K S W e consider the optimal control of one-dimensional for- mations with nearest neighbor interactions between the ve- hicles. W e formulate a structured optimal control problem in which local information exchange of relativ e positions between immediate neighbors imposes structural constraints on the feedback gains. W e study the design problem for both the single- and the double-integrator models and employ a homotopy-based Ne wton’ s method to compute the optimal structured gains. W e also sho w that design of symmetric gains for the single-integrator model is a con vex optimization problem, which we solve analytically for formations with no fictitious followers. For double-integrators, we identify a class of con vex problems by restricting the controller to symmetric position and uniform diagonal velocity gains. Furthermore, we in vestigate the performance of the optimal controllers by examining the asymptotic scalings of formation coherence and control energy with the number of vehicles. For formations in which all vehicles hav e access to their own velocities, the optimal structured position gains for single- and double-integrators are similar to each other . Since these two models exhibit the same asymptotic scalings of global, local, and control performance measures, we conclude that the single-integrator model, which lends itself more easily to analysis and design, captures the essential features of the optimal localized design. W e note that the tools dev eloped in this paper can also be used to design optimal structured controllers for double-integrators with relative position and velocity measurements; this is a topic of our ongoing research. As in [10], we employ perturbation analysis to determine the departure from a stabilizing spatially uniform profile that yields nominal diffusion dynamics on a one-dimensional lattice; in contrast to [10], we find the ‘mistuning’ profile by optimizing a performance index rather than by perform- ing spectral analysis. W e also show how a homotopy-based Newton’ s method can be employed to obtain non-infinitesimal variation in feedback gains that minimizes the desired objec- tiv e function. Furthermore, we establish sev eral explicit scaling relationships and identify a spatially uniform non-symmetric controller that performs better than the optimal symmetric spatially varying controller ( O ( √ N ) vs. O ( N ) scaling of coherence with O (1) control energy per vehicle). This suggests that departure from symmetry can improve coherence of large- scale formations and that the controller structure may play a more important role than the optimal feedback gain design. On the other hand, our results demonstrate that the best performance is achiev ed with the optimal localized controller that is both non-symmetric and spatially-varying. Currently , we are considering the structured feedback design for formations on general graphs [6], [8], [23], [36], [37] with the objecti ve of identifying topologies that lead to fav orable system-theoretic properties [28], [38], [39]. Even though this paper focuses on the optimal local feedback design for one- dimensional formations with path-graph topology , the de vel- oped methods can be applied to multi-agent problems with more general network topologies. A C K N OW L E D G E M E N T S The authors would like to thank anon ymous re vie wers and the associate editor for their valuable comments. 14 A P P E N D I X A. Gradient method for (SG) W e next describe the gradient method for solving (SG). Let us denote k = [ k 1 · · · k N +1 ] T . Starting with an initial guess k 0 that guarantees positiv e definiteness of K 0 , vector k is updated k i +1 = k i − s i ∇ J ( k i ) , until the norm of gradient is small enough, k∇ J ( k i ) k <  . Here, s i is the step-size determined by the backtracking line search [32, Section 9.2]: let s i = 1 and repeat s i := β s i with β ∈ (0 , 1) until a sufficient decrease in the objective function is achiev ed, J ( k i − s i ∇ J ( k i )) < J ( k i ) − α s i k∇ J ( k i ) k 2 , where α ∈ (0 , 0 . 5) . Note that J ( k i ) is defined as infinity if K in (7) determined by k i is not positive definite. For Q = I , J = 1 2 trace  K − 1 + r K  = 1 2 N X n = 1 γ n ( γ N +1 − γ n ) γ N +1 + r k 1 + k N +1 2 + N X n = 2 k n ! , the entries of the gradient ∇ J are given by ∂ J ∂ k n = r − 1 2 n − 1 X i = 1  γ i k n γ N +1  2 − 1 2 N X i = n  γ N +1 − γ i k n γ N +1  2 , n ∈ { 2 , . . . , N } , ∂ J ∂ k 1 = r 2 − 1 2 N X n = 1  γ N +1 − γ n k 1 γ N +1  2 , ∂ J ∂ k N +1 = r 2 − 1 2 N X n = 1  γ n k N +1 γ N +1  2 . B. P erformance of look-ahead strate gy W e next deriv e the analytical expressions for the perfor- mance measures Π g , Π l , and Π ctr obtained with the look- ahead strategy for the single-integrator model. The solution of the L yapunov equation (14) with F C = αC f is determined by (15). Since the i th entry of the first column of the lower triangular T oeplitz matrix ( sI + α C f ) − 1 is α i / ( s + α ) i , the corresponding entry of the matrix exponential in (15) is determined by the in verse Laplace transform of α i / ( s + α ) i , α ( αt ) i − 1 e − αt / ( i − 1)! . Thus, the n th element on the main diagonal of the matrix L in (15) is gi ven by L nn = Z ∞ 0 n X i = 1  α e − αt ( αt ) i − 1 ( i − 1)!  2 d t = α Γ( n + 1 / 2) √ π Γ( n ) = α (2 n )! 2 2 n ( n − 1)! n ! , (19) thereby yielding Π g = N X n = 1 L nn N = 2 α Γ( N + 3 / 2) 3 √ π Γ( N + 1) = 2 3 α (2 N + 2)! 2 2 N +2 N !( N + 1)! . (20) A similar procedure can be used to show that the n ( n + 1) th entry of L is determined L n ( n +1) = L ( n +1)( n +1) − α/ 2 , n = 1 , . . . , N − 1 . (21) Now , from (21) and the fact that L 11 = α/ 2 we obtain Π l = 1 N trace ( T L ) = 2 N N X n = 1 L nn − N − 1 X n = 1 L n ( n +1) ! = α, Similarly , Π ctr = (1 / N ) trace  LC T f C f  = 2 N N X n = 1 L nn − N − 1 X n = 1 L n ( n +1) ! − 1 N L N N = α − (1 / N ) L N N . Using Stirling’ s approximation n ! ≈ √ 2 π n ( n/ e) n for large n , we have lim n → ∞ L nn √ n = lim n → ∞ α √ π r n n − 1  n n − 1  n − 1 1 e = α √ π , where we used the fact that lim n → ∞ ( n/ ( n − 1)) n − 1 = e . Con- sequently , lim N → ∞ Π ctr ( N ) = α. From (19) and (20), it follo ws that Π g = (2 / 3) L ( N +1)( N +1) and thus, lim N → ∞ Π g ( N ) / √ N = (2 α ) / (3 √ π ) . W e conclude that Π g asymptotically scales as a square-root function of N and that Π ctr is formation-size- independent as N increases to infinity . R E F E R E N C E S [1] W . S. Levine and M. Athans, “On the optimal error regulation of a string of moving vehicles, ” IEEE T rans. Automat. 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Jov anovi ´ c, “On the optimal localized feedback design for multi-vehicle systems, ” in Pr oceedings of the 49th IEEE Conference on Decision and Control , 2010, pp. 5744–5749. Fu Lin (S’06) receiv ed his Bachelor of Science degree in Instrument Science and Engineering from Shanghai Jiaotong University in 2005. Currently , he is a Ph.D. candidate in the Department of Electrical and Computer Engineering at the Univ ersity of Min- nesota, Minneapolis. His primary research interests are in the analysis and design of optimal distributed controllers using tools from conv ex optimization, compressiv e sensing, and graph theory . Makan Fardad receiv ed the B.S. and M.S. degrees in electrical engineering from Sharif University of T echnology and Iran Univ ersity of Science and T echnology , respectiv ely . He received the Ph.D. de- gree in mechanical engineering from the University of California, Santa Barbara, in 2006. He was a Postdoctoral Associate at the University of Min- nesota, Minneapolis, before joining the Department of Electrical Engineering and Computer Science at Syracuse University as an Assistant Professor in August 2008. His research interests are in modeling, analysis, and optimal control of distributed and large-scale interconnected systems. Mihailo R. Jovanovi ´ c (S’00–M’05) receiv ed the Dipl. Ing. and M.S. degrees from the University of Belgrade, Serbia, in 1995 and 1998, respecti vely , and the Ph.D. degree from the University of California, Santa Barbara, in 2004. Before joining the Univer - sity of Minnesota, Minneapolis, he was a V isiting Researcher with the Department of Mechanics, the Royal Institute of T echnology , Stockholm, Sweden, from September to December 2004. Currently , he is an Associate Professor of Electrical and Computer Engineering at the Univ ersity of Minnesota, where he serv es as the Director of Graduate Studies in the interdisciplinary Ph.D. program in Control Science and Dynamical Systems. Dr . Jovano vi ´ c’ s expertise is in modeling, dynamics, and control of large- scale and distrib uted systems and his current research focuses on sparsity- promoting optimal control, dynamics and control of fluid flows, and funda- mental limitations in the control of vehicular formations. He is a member of APS and SIAM and has served as an Associate Editor of the IEEE Control Systems Society Conference Editorial Board from July 2006 until December 2010. He recei ved a CAREER A w ard from the National Science F oundation in 2007, and an Early Career A ward from the University of Minnesota Initiativ e for Renewable Energy and the Environment in 2010.