Collisionless and Decentralized Formation Control for Strings
CCollisionless and Decentralized Formation Control for Strings (cid:63)
Young-Pil Choi a , Dante Kalise b , Andr´es A. Peters c a Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 03722, Republic of Korea b School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2QL, United Kingdom c Faculty of Engineering and Sciences, Universidad Adolfo Ib´a˜nez, Santiago 7941169, Chile
Abstract
A decentralized feedback controller for multi-agent systems, inspired by vehicle platooning, is proposed. The closed-loop resulting from the decentralized control action has three distinctive features: the generation of collision-freetrajectories, flocking of the system towards a consensus state in velocity, and asymptotic convergence to a prescribedpattern of distances between agents. For each feature, a rigorous dynamical analysis is provided, yielding a character-ization of the set of parameters and initial configurations where collision avoidance, flocking, and pattern formationis guaranteed. Numerical tests assess the theoretical results presented.
Keywords:
Multi-agent systems, decentralized control, nonlinear control, consensus and pattern formation control,autonomous vehicles.
1. Introduction
Multi-agent systems (MAS) have proven to be a versatile framework for studying diverse scalability problems inScience and Engineering, such as dynamic networks [23], autonomous vehicles [4], collective behaviour of humansor animals [29, 30], and many others. Mathematically, MAS are often modelled as large-scale dynamical systemswhere each agent can be considered as a subset of states, updated via interaction forces such as attraction, repulsion,alignment, etc., [18, 14] or through the optimization of a pay-o ff function in a control / game framework [21, 20].In this work, we approach the study of MAS from a control viewpoint. We study a class of sparsely inter-connected agents in one dimension, interacting through nonlinear couplings and a decentralized control law. Theelementary building block of our approach is the celebrated Cucker-Smale model for consensus dynamics [14], whichcorresponds to a MAS where each agent is endowed with second-order nonlinear dynamics for velocity alignment,and where the influence of neighbouring agents decays with distance. The Cucker-Smale model and variants canrepresent the physical motion of agents on the real line, inspired by autonomous vehicle formations in platooningwith a nearest-neighbour interaction scheme [28, 31]. The couplings to be studied are motivated by the more generalsetting of the Cucker-Smale dynamics in arbitrary dimension. The original Cucker-Smale dynamics considers fullnetwork connectivity in the agent interactions, generating flocking dynamics capable of exhibiting emergent consen-sus behavior, that is, agents that may reach a common velocity in steady state, without the action of external forces.This framework has been extended in several directions, being most notable the inclusion of forcing terms and con-trol [13, 17, 7, 19], optimal control [3, 6, 2] formation control [24, 25, 11], leadership [15] and collision-avoidancecapabilities [12, 1, 5, 10], the latter being an increasingly sought after property of formation control schemes forautonomous fleets of vehicles.The main contribution of this paper is to propose a nearest neighbour interaction of agents on the real line whichexhibits emergent consensus and collision avoidance under the action of a simple decentralised control law. Theproposed feedback enforces a desired inter-agent distance in steady state and is inspired by formation control in (cid:63) This paper was not presented at any IFAC meeting. Corresponding author D. Kalise Tel. +
44 738 039 1139.
Email addresses: [email protected] (Young-Pil Choi), [email protected] (Dante Kalise), [email protected] (Andr´es A. Peters)
Preprint submitted to Elsevier March 1, 2021 a r X i v : . [ m a t h . O C ] F e b ehicular platoons [28, 26]. Such a model has the potential to achieve the goals of platooning applications, that is, thecoordinated, scalable, secure and e ffi cient travel of automated vehicles [16], while at the same time o ff ering flocking,pattern formation, and collision avoidance features from the non-linear dynamics. Moreover, we provide collision-avoidance guarantees that, from a safety viewpoint, do not rely on traditional concepts in platooning such as stringstability. For the derivation of collision-avoidance results we consider the framework developed in [8, 9, 22], whichuses singular interaction kernels that blow-up whenever two agents are located at the same position. We modify thissetup to also consider agents with a volume by including a threshold inter-agent distance where the kernel becomessingular.Our main result is a rigorous characterization of flocking, collision-avoidance, and platooning behaviour for theproposed nonlinear model, in terms of the initial configuration of the system, interaction and control law parameters.Cucker-Smale models usually consider full state measurements of every agent, available to every agent, which reflectsa free interaction between them giving rise to self-organization properties (see for example the recent work [32] wherethe full interaction Cucker-Smale model is characterised on the real line). We show that these properties are stillpresent when a highly sparse nearest neighbour interaction is considered in the 1D case.The remainder of the paper is structured as follows. In Section 2 we present the proposed model to be studied, aCucker-Smale model with nearest neighbor singular interactions and a decentralized feedback control. We also definehere a total energy functional E ( x , v ) for the model and show that it is not increasing in time. In Section 3 we give theresults ensuring the collision-avoidance behaviour of the controlled system, and Section 4 includes a flocking estimateshowing that the velocity alignment between individuals and the inter-agent distances are uniformly bounded in time.In Section 5 we present the main formation control result. We provide di ff erent numerical experiments that illustrateour theoretical results in Section 6 along with some concluding remarks.
2. Problem description and preliminary results
We consider a string of N agents each characterized by a pair ( x i ( t ) , v i ( t )) in R evolving in time t through second-order dynamics of the form dx i ( t ) dt = v i ( t ) , i = , . . . , N , t > , dv i ( t ) dt = I i ( t ) + u i ( t ) , (1)subject to initial data ( x i (0) , v i (0)) = : ( x i , v i ) for i = , . . . , N . (2)Here, the term I i describes nonlocal velocity interactions between individuals which are weighted by a singular Figure 1: Diagram at a particular instant of three consecutive agents for the considered MAS. The singular interactions, providing barriers to theagents, are indicated with the radii δ i of the semicircles. ψ ( r ) : R + −→ R , I = ψ ( | x − x | − δ )( v − v ) , I k = ψ ( | x k − x k − | − δ k − )( v k − − v k ) + ψ ( | x k + − x k | − δ k )( v k + − v k ) , I N = ψ ( | x N − x N − | − δ N − )( v N − − v N ) , k = , . . . , N − , where the parameters δ i > u i serves as a decentralized feedback control depending on weight function φ ( r ) : R + −→ R and is given by u = − φ ( | x − x − z | )( x − x − z ) , u k = φ ( | x k − − x k − z k − | )( x k − − x k − z k − ) − φ ( | x k − x k + − z k | )( x k − x k + − z k ) , u N = φ ( | x N − − x N − z N − | )( x N − − x N − z N − ) , k = , . . . , N − . This feedback also depends on a vector of relative distances z : = ( z , . . . , z N − ) ∈ R N − . The objective of this controllaw is to induce the formation of a string pattern characterized by z . The complete setting is depicted in Figure 1. Forthe sake of clarity, the weight functions ψ and φ are chosen as ψ ( r ) = r α , and φ ( r ) = + r ) β , α, β > . (3)However, the results we will state in the forthcoming sections can extended to the case in which φ is bounded andLipschitz continuous. We will establish conditions under which the string (1) converge to a consensus state with a pre-scribed formation while avoiding collisions between agents. For this, we begin by stating an a priori energy estimatewhich will be significantly used for estimating consensus emergence. We first define the total energy functional E ( x , v ) : = E ( v ) + E ( x ) = N N (cid:88) i , j = | v i − v j | + N (cid:88) i = (cid:90) | x i − − x i − z i − | φ ( r ) dr , and its dissipation rate D ( x , v ) : = N (cid:88) i = ψ ( | x i − x i − | − δ i − )( v i − v i − ) Lemma 1.
Let { ( x i , v i ) } Ni = be a smooth solution to the system (1) on the time interval [0 , T ] . Then:(i) the mean velocity is conserved in time: v c ( t ) : = N N (cid:88) i = v i ( t ) = v c (0) . (ii) the total energy is not increasing in time:ddt E ( x ( t ) , v ( t )) + D ( x ( t ) , v ( t )) = . P roof . (i) A straightforward computation yields N (cid:88) i = I i = N (cid:88) i = u i = . ddt v c ( t ) = , i.e. v c ( t ) = v c (0) (4)for t ∈ [0 , T ].(ii) Let us first begin with the estimate for the kinetic energy:14 N ddt N (cid:88) i , j = ( v i − v j ) = N N (cid:88) i , j = ( v i − v j ) (cid:32) dv i dt − dv j dt (cid:33) = N N (cid:88) i , j = (cid:32) v i dv i dt + v j dv j dt (cid:33) = N (cid:88) i = v i dv i dt = N (cid:88) i = v i ( I i + u i ) , (5)where we used (4). Here, we use the same idea of [11, Lemma 3.1] to obtain N (cid:88) i = v i u i = − ddt N (cid:88) i = (cid:90) | x i − − x i − z i − | φ ( r ) dr . (6)On the other hand, we estimate the term with I i as N − (cid:88) i = v i I i = N − (cid:88) i = v i ( ψ ( | x i − x i − | − δ i − )( v i − − v i ) + ψ ( | x i − x i + | − δ i )( v i + − v i )) = N − (cid:88) i = v i ( ψ ( | x i − x i − | − δ i − )( v i − − v i )) + N (cid:88) i = v i − ( ψ ( | x i − x i − | − δ i − )( v i − v i − )) = v ψ ( | x − x | − δ )( v − v ) − N − (cid:88) i = ψ ( | x i − x i − | − δ i − )( v i − − v i ) + v N − ψ ( | x N − x N − | − δ N − )( v N − v N − ) = − v I − v N − I N − N − (cid:88) i = ψ ( | x i − x i − | − δ i − )( v i − − v i ) . This asserts N (cid:88) i = v i I i = v I + N − (cid:88) i = v i I i + v N I N = − ( v − v ) I − N − (cid:88) i = ψ ( | x i − x i − | − δ i − )( v i − − v i ) − ( v N − − v N ) I N = − N (cid:88) i = ψ ( | x i − x i − | − δ i − )( v i − − v i ) . (7)Combining the estimates (5), (7), and (6), we conclude the desired result. (cid:3)
3. Global and local existence of solutions
In this section, we show that system (1), under certain parametric and initial conditions, exhibits a non-collisionalbehaviour, which together with Cauchy-Lipschitz theory, subsequently provides global-in-time existence and unique-ness of smooth solutions to the system (1)-(2). We present two results regarding existence of non-collisional trajec-tories. The first theorem requires a prescribed ordering for the initial datum x i and the power α in (3) to be α ≥ α ≥
2, but the initial ordering assumption is removed from the string. A third resultcharacterizes a pathological case, where a 2-agent string blows up in finite time.
Theorem 1.
Suppose that α ≥ and the initial configuration x satisfies x i + > x i + δ i for all i = , . . . , N − . Then,there exists the global unique smooth solution to the system (1) - (2) satisfying x i + ( t ) > x i ( t ) + δ i for all i = , . . . , N − and all t > . roof . We first notice that ψ ( x i + − x i − δ i ) is regular as long as x i + > x i + δ i , and thus there exists a unique smoothsolution to the system (1). For a fixed T ∈ (0 , ∞ ), let us assume that there is t ∗ ∈ (0 , T ] where the smoothness ofsolutions breaks down for the first time, i.e. there is an index (cid:96) such that x (cid:96) + ( t ) − x (cid:96) ( t ) > δ (cid:96) for t ∈ (0 , t ∗ ) and lim t → t ∗ − x (cid:96) + ( t ) − x (cid:96) ( t ) = δ (cid:96) . (8)We denote by [ (cid:96) ] the set of such indices and set i ∗ = min[ (cid:96) ]. We first claim i ∗ ≥
2. If i ∗ =
1, then for t ∈ (0 , t ∗ ) weestimate ddt Ψ ( x − x − δ ) = ψ ( x − x − δ )( v − v ) = I = ddt ( v − v c ) − u , where we used (4) and Ψ is the primitive of ψ , i.e. Ψ ( r ) = ln( r ) for α = , − α r − α for α > Ψ ( x ( t ) − x ( t ) − δ ) = Ψ ( x − x − δ ) = ( v ( t ) − v c ( t )) − ( v − v c (0)) − (cid:90) t u ( s ) ds (9)for t ∈ [0 , t ∗ ). On the other hand, by H¨older’s inequality we find | v ( t ) − v c ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N (cid:88) k = ( v ( t ) − v k ( t )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:118)(cid:117)(cid:116) N N (cid:88) k = ( v ( t ) − v k ( t )) and | u ( s ) | ≤ (cid:107) φ (cid:107) L ∞ | x − x − z | ≤ (cid:107) φ (cid:107) L ∞ (cid:32) | z | + | x − x | + (cid:90) s | v ( τ ) − v ( τ ) | d τ (cid:33) . These observations together with the energy estimate in Lemma 1 imply that the right hand side of (9) is bounded onthe time interval (0 , t ∗ ), and subsequently t (cid:55)→ Ψ ( x ( t ) − x ( t ) − δ ) is bounded on the time interval [0 , t ∗ ). This is acontradiction to (8) and thus the claim follows. By the definition of i ∗ , there exists a constant c i ∗ > x i ∗ ( t ) − x i ∗ − ( t ) − δ i ∗ − > c i ∗ (10)for all t ∈ (0 , t ∗ ). Similarly as above, we now estimate ddt Ψ ( x i ∗ + − x i ∗ − δ i ∗ ) = ψ ( x i ∗ + − x i ∗ − δ i ∗ )( v i ∗ + − v i ∗ ) = I i ∗ + ψ ( x i ∗ − x i ∗ − − δ i ∗ − )( v i ∗ − v i ∗ − ) = ddt ( v i ∗ − v c ) + ψ ( x i ∗ − x i ∗ − − δ i ∗ − )( v i ∗ − v i ∗ − ) − u i ∗ , and thus Ψ ( x i ∗ + ( t ) − x i ∗ ( t ) − δ i ∗ ) = Ψ ( x i ∗ + − x i ∗ − δ i ∗ ) + ( v i ∗ ( t ) − v c ( t )) − ( v i ∗ − v c (0)) + (cid:90) t ψ ( x i ∗ ( s ) − x i ∗ − ( s ) − δ i ∗ − )( v i ∗ ( s ) − v i ∗ − ( s )) ds − (cid:90) t u i ∗ ( s ) ds (11)for t ∈ (0 , t ∗ ). Here the boundedness of the second and fourth terms can be obtained by using almost the sameargument as above. We also use (10) to obtain | ψ ( x i ∗ ( s ) − x i ∗ − ( s ) − δ i ∗ − )( v i ∗ ( s ) − v i ∗ − ( s ) | ≤ c − α i ∗ NE ( v ( t )) ≤ c − α i ∗ NE ( x , v ) . , t ∗ ), so is the left hand side. This leads to acontradiction and thus the unique smooth solution can be actually exists up to an arbitrary finite time T >
0. Thiscompletes the proof. (cid:3)
We next present the second existence theorem whose proof is based on the energy estimate. For this, we firstintroduce a function L α − δ with α ≥ L α − δ ( t ) = N − (cid:88) i = ( | x i ( t ) − x i + ( t ) | − δ i ) − ( α − for α > , N − (cid:88) i = log( | x i ( t ) − x i + ( t ) | − δ i ) for α = | L α − ( t ) | < ∞ for t ∈ [0 , T ] for some α ≥ x i ( t ) and x i + ( t ) arestrictly greater than δ i for all i = , . . . , N − t ∈ [0 , T ]. Theorem 2.
Suppose that α ≥ and that the initial configuration x satisfies | x i − x i + | > δ i for all i = , . . . , N − . Then, there exists the global unique smooth solution to the system (1) - (2) where the distancesbetween agents satisfy | x i ( t ) − x i + ( t ) | > δ i for all i = , . . . , N − and all t > . P roof . We first introduce the maximal life-span T = T ( x ) of the initial datum x : T : = sup (cid:8) s ∈ R + : ∃ solution ( x ( t ) , v ( t )) for the system (1)-(2) in a time-interval [0 , s ) (cid:9) By the assumption, T >
0. We then claim T = ∞ . First, note that it follows from Lemma 1 that N − (cid:88) i = (cid:90) t ( v i + ( s ) − v i ( s )) ( | x i ( s ) − x i + ( s ) | − δ i ) α ds ≤ E ( x , v ) . (12)Let us prove the claim above by dealing with two cases separately: α = α > α =
2: A straightforward computation gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddt N − (cid:88) i = log( | x i ( t ) − x i + ( t ) | − δ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − (cid:88) i = ( x i ( t ) − x j ( t )) · ( v i ( t ) − v j ( t )) | x i ( t ) − x j ( t ) | ( | x i ( t ) − x i + ( t ) | − δ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N − (cid:88) i = | v i ( t ) − v j ( t ) || x i ( t ) − x j ( t ) | − δ i for t ∈ [0 , T ). This yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − (cid:88) i = log( | x i ( t ) − x i + ( t ) | − δ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − (cid:88) i = log( | x i − x i + | − δ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + N − (cid:88) i = (cid:90) t | v i ( s ) − v i + ( s ) || x i ( s ) − x i + ( s ) | − δ i ds . On the other hand, by using the H¨older inequality and (12), we estimate N − (cid:88) i = (cid:90) t | v i ( s ) − v i + ( s ) || x i ( s ) − x i + ( s ) | − δ i ds ≤ √ t N − (cid:88) i = (cid:32)(cid:90) t | v i ( s ) − v i + ( s ) | ( | x i ( s ) − x i + ( s ) | − δ i ) ds (cid:33) / ≤ (cid:112) t ( N − E ( x , v ) . Thus, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − (cid:88) i = log( | x i ( t ) − x i + ( t ) | − δ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − (cid:88) i = log( | x i − x i + | − δ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:112) t ( N − E ( x , v ) , (13)6or t ∈ [0 , T ).(ii) α >
2: Taking the time derivative to L α − δ , we get (omitting the time arguments) dL α − δ dt = − ( α − N − (cid:88) i = ( | x i − x i + | − δ i ) − α + (cid:104) x i − x i + , v i − v i + (cid:105)| x i − x i + |≤ C N − (cid:88) i = ( | x i − x i + | − δ i ) − α + | v i − v i + |≤ C N − (cid:88) i = | x i − x i + | − δ i ) α − + C N − (cid:88) i = | v i − v i + | ( | x i − x i + | − δ i ) α = CL α − ( t ) + C N − (cid:88) i = | v i − v i + | ( | x i − x i + | − δ i ) α , for t ∈ [0 , T ), where we used Young’s inequality. Applying Gronwall’s inequality to the above, we have L α − δ ( t ) ≤ L α − δ (0) e Ct + Ce Ct N − (cid:88) i = (cid:90) t | v i ( s ) − v i + ( s ) | ( | x i ( s ) − x i + ( s ) | − δ i ) α ds ≤ e Ct (cid:16) L α − δ (0) + CE ( x , v ) (cid:17) , (14)for t ∈ [0 , T ), due to (12). Since the right hand sides of (13) and (14) are uniformly bounded in the time interval[0 , T ), the life-span T should be infinity, i.e., T = ∞ . This completes the proof. (cid:3) Remark 1.
In Theorem 1, it is crucially used the fact that the system is posed in one dimension. However, Theorem2 can also deal with higher dimensional problems, see [9].
We conclude this section with a negative result characterizing a pathological configuration with 2 agents wherethe system blows up in finite time.
Theorem 3.
Let α ∈ (0 , and N = . Furthermore, we assume that δ , z , and the initial data { ( x i , v i ) } i = satisfy δ + z ≥ , x > x + δ , and v − v = − α ( x − x − δ ) − α . (15) Then, the smoothness of solutions to the system (1) - (2) breaks down in finite time. P roof . For the proof, it su ffi ces to show that there exists a finite time t ∗ < ∞ such that x ( t ∗ ) + δ = x ( t ∗ ). Fornotational simplicity, we set x : = x − x and v : = v − v . Then we easily find that x and v satisfy dx ( t ) dt = v ( t ) , dv ( t ) dt = − I ( t ) + u ( t )) = − ψ ( x ( t ) − δ ) v − φ ( | x ( t ) + z | )( x ( t ) + z ) . Note that the smooth solutions exist as long as x ( t ) > δ , and this and the assumption δ + z ≥ x ( t ) + z ≥ φ ≥
0, this implies that dv ( t ) dt ≤ − ψ ( x ( t ) − δ ) v = − ddt Ψ ( x ( t ) − δ ) . Here Ψ is the primitive of ψ , i.e. Ψ ( r ) = − α r − α . We then solve the above di ff erential inequality to get d ( x ( t ) − δ ) dt = v ( t ) ≤ − Ψ ( x ( t ) − δ ) = − − α ( x ( t ) − δ ) − α (16)7ue to (15). We notice that the above di ff erential inequality is sub-linear, and thus there exists t ∗ < ∞ such that x ( t ∗ ) − δ =
0. Indeed, we obtain from (16) that( x ( t ) − δ ) α ≤ ( x − δ ) α − α − α t . Hence we have t ∗ ≤ ( x − δ ) α α (1 − α ) , thus completing the proof. (cid:3)
4. Time-asymptotic behavior
Having characterized the well-posedness of the trajectories of the system (1), we now turn our attention to thestudy of flocking emergence within the controlled string. In a flocking configuration, all agents travel with the sameconstant velocity, and as a direct consequence the distance between agents remain constant. We provide a rigorousasymptotic flocking estimate for the system (1).
Theorem 4.
Suppose that either assumptions of Theorems 1 or 2 hold. Furthermore, we assume (cid:90) ∞ φ ( r ) dr > N N (cid:88) i , j = | v i − v j | + N (cid:88) i = (cid:90) | x i − − x i − z i − | φ ( r ) dr . (17) Then, the string converges asymptotically towards a flocking state, that is sup ≤ t ≤∞ max i , j = ,..., N | x i ( t ) − x j ( t ) | < ∞ and max i , j = ,..., N | v i ( t ) − v j ( t ) | → as t → ∞ . P roof . From Theorems 1 or 2, it follows existence and uniqueness of a smooth solution globally in time. (Uniform-in-time boundedness): It follows from the energy estimate in Lemma 1 that E ( x ( t )) ≤ E ( x , v ) , i.e. N (cid:88) i = (cid:90) | x i − ( t ) − x i ( t ) − z i − | | x i − − x i − z i − | φ ( r ) dr ≤ N N (cid:88) i , j = | v i − v j | for t ≥ . (18)On the other hand, under our main assumptions, we can find some constant ρ > N N (cid:88) i , j = | v i − v j | + N (cid:88) i = (cid:90) | x i − − x i − z i − | φ ( r ) dr ≤ (cid:90) ρ φ ( r ) dr . This together with (18) yields 0 ≤ N N (cid:88) i , j = | v i − v j | ≤ (cid:90) ρ | x i − − x i − z i − | φ ( r ) dr for all i = , . . . , N . This implies that | x i − ( t ) − x i ( t ) − z i − | ≤ ρ for i = , . . . , N . (19)For any i < j , by telescoping and the triangle inequality we estimate | x i − x j | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j − (cid:88) (cid:96) = i ( x (cid:96) − x (cid:96) + ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ j − (cid:88) (cid:96) = i | x (cid:96) − x (cid:96) + | ≤ j − (cid:88) (cid:96) = i | x (cid:96) − x (cid:96) + − z (cid:96) | + j − (cid:88) (cid:96) = i | z (cid:96) | , | x i − x j | ≤ | j − i | ρ + j − (cid:88) (cid:96) = i | z (cid:96) | ≤ ( N − ρ + N − (cid:88) i = | z i | < ∞ , given the boundedness of distances between agents at all times. (Velocity alignment behavior): From the bound above, we find | x i − x i − | − δ i − ≤ | x i − x i − − z i − | + | z i − | − δ i − ≤ ρ + | z i − | + δ i − ≤ ρ + max i = ,..., N − ( | z i | + δ i ) . Since ψ is monotonically decreasing, we obtain ψ m : = min ≤ i ≤ N ψ ( | x i − x i − | − δ i − ) ≥ ψ (cid:32) ρ + max i = ,..., N − ( | z i | + δ i ) (cid:33) > . This implies that the dissipation rate D is bounded from below by D ( x ( t ) , v ( t )) = N (cid:88) i = ψ ( | x i − x i − | − δ i − )( v i − − v i ) ≥ ψ m N (cid:88) i = ( v i − − v i ) . Then, by Lemma 1, we get N (cid:88) i = (cid:90) ∞ ( v i − ( t ) − v i ( t )) dt < ∞ , and subsequently, this leads to (cid:90) ∞ E ( v ( t )) dt = N N (cid:88) i , j = (cid:90) ∞ | v i ( t ) − v j ( t ) | dt < ∞ . Indeed, by telescoping, for any i < j | v i − v j | ≤ j − (cid:88) (cid:96) = i | v (cid:96) − v (cid:96) + | ≤ (cid:112) | i − j | (cid:118)(cid:117)(cid:116) j − (cid:88) (cid:96) = i | v (cid:96) − v (cid:96) + | , and thus N (cid:88) i , j = | v i − v j | ≤ c N N (cid:88) i = | v i − − v i | , where c N : = N (cid:88) i , j = | i − j | . (20)Moreover, we also find (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) i = v i u i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) i = φ ( | x i − − x i − z i − | ) (cid:104) x i − − x i − z i − , v i − − v i (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ N (cid:88) i = | v i − − v i | ≤ C (cid:112) E ( x , v ) , (21)where C is independent of t and we usedmax ≤ i , j ≤ N | v i ( t ) − v j ( t ) | ≤ (cid:118)(cid:117)(cid:116) N (cid:88) i , j = | v i ( t ) − v j ( t ) | ≤ (cid:112) NE ( x , v ) . Furthermore, note that14 N N (cid:88) i , j = | v i − v j | = (cid:90) t ( − D ( x ( s ) , v ( s ))) ds + N (cid:88) i = (cid:90) t v i ( s ) u i ( s ) ds + N N (cid:88) i , j = | v i − v j | . D is integrable and thus the first term on the right side of the above equality is absolutely con-tinuous. Regarding the second term, its time-derivative is uniformly bounded in time, see (21), from where it followsthat it is Lipschitz continuous. This implies that the E ( v ( t )) is the sum of an absolutely continuous function and aLipschitz continuous function. Thus, we obtain that E ( v ( t )) is uniformly continuous. Since E ( v ( t )) is also integrable, E ( v ( t )) → t → ∞ . This completes the proof. (cid:3) Remark 2. If β ≤ , then φ is not integrable, thus the left hand side of (17) becomes infinity. This implies that theassumption (17) automatically holds. On the other hand, if β > , we obtain (cid:90) ∞ φ ( r ) dr = (cid:90) ∞ + r ) β dr = β − , and thus (17) can be rewritten as E ( x , v ) < β − . (22) From these equivalences, it becomes evident that the fulfilment of the flocking condition depends only on twoparameters of the model, namely, the number of agents N in the string and the control interaction constant β > ,which regulates the strength of the control action. The constant α does not play a role on the condition. Having fixeda number of agents and β , flocking solely depends on the cohesiveness of the initial configuration. Remark 3.
If we define Φ by the primitive of φ , then it is clear that r (cid:55)→ Φ ( r ) is strictly increasing, and thus theconstant ρ appeared in (19) can be expressed by ρ = (cid:118)(cid:117)(cid:116) Φ − N N (cid:88) i , j = | v i − v j | + Φ ( | x i − − x i − z i − | ) . (23)
5. Exponential emergence of pattern formation and velocity alignment
In this section, we conclude our characterization of the string trajectories by studying the exponential emergenceof pattern formation and velocity alignment behavior under additional assumption on the solutions. We first providean auxiliary result, a modification of Young’s inequality, which can be proved by a similar argument as in [11, Lemma6.1]. We thus omit its proof here.
Lemma 2.
Let a , . . . , a N − be a set of vectors in R d and b , . . . , b N − be a set of positive scalars. Then − N − (cid:88) i = b i | a i | + N − (cid:88) i = b i (cid:104) a i , a i + (cid:105) ≤ − (cid:15) N − (cid:88) i = b i | a i | , where (cid:15) ∈ (0 , is a su ffi ciently small number. We now state our main result, which provides non-collisional behavior, flocking, and an exponential decay esti-mate towards the string configuration encoded in the relative distance vector z . Proposition 1.
Suppose that the assumptions of Theorem 4 are satisfied. Furthermore, we assume that inf t ≥ min ≤ i ≤ N − ( | x i ( t ) − x i + ( t ) | − δ i ) > . (24) Then, we have max i = ,..., N | x i − ( t ) − x i ( t ) − z i − | + max i , j = ,..., N | v i ( t ) − v j ( t ) | → exponentially fast as t → ∞ . roof . We first notice that the energy estimate in Lemma 1 only provides a dissipation rate for the velocity. In orderto have a complete exponential decay estimate, it is required to obtain the dissipation rate associated to the positions.For this, we consider the following quantity: N − (cid:88) i = ( x i − x i + − z i )( v i − v i + ) . Note that the total energy is bounded from below and above by14 N N (cid:88) i , j = | v i − v j | + φ m N − (cid:88) i = | x i − x i + − z i | ≤ E ( x , v ) ≤ N N (cid:88) i , j = | v i − v j | + N − (cid:88) i = | x i − x i + − z i | , where we used φ m : = min s ∈ [0 , ρ ] φ ( s ) ≤ φ ( r ) ≤ . This shows that a modified total energy E γ , defined as E γ ( x , v ) : = γ E ( x , v ) + N − (cid:88) i = ( x i − x i + − z i )( v i − v i + ) , has similar upper and lower bound estimates as the one for E ( x , v ) when γ > γ > (cid:112) N /φ m ,we readily find E γ ( x , v ) ≥ γφ m N − (cid:88) i = | x i − x i + − z i | + (cid:32) γ N − γφ m (cid:33) N (cid:88) i , j = | v i − v j | ≥ c γ N − (cid:88) i = | x i − x i + − z i | + N (cid:88) i , j = | v i − v j | , (25)where c γ > c γ : = min (cid:40) γφ m , γ N − γφ m (cid:41) . The upper bound on E γ can be easily obtained. On the other hand, it follows from (1) that ddt N − (cid:88) i = ( x i − x i + − z i )( v i − v i + ) = N − (cid:88) i = ( v i − v i + ) + N − (cid:88) i = ( x i − x i + − z i )( I i − I i + ) + N − (cid:88) i = ( x i − x i + − z i )( u i − u i + ) . (26)Since | I | ≤ ψ M | v − v | , | I N | ≤ ψ M | v N − − v N | , and | I i | ≤ ψ M ( | v i − − v i | + | v i − v i + | )for i = , . . . , N −
1, we easily find N − (cid:88) i = | I i − I i + | ≤ | I | + N − (cid:88) i = | I i | + | I N | ≤ ψ M N − (cid:88) i = | v i − v i + | , where ψ M : = sup r ∈ [0 , ∞ ) ψ ( r ) < ∞ , which can be defined by the assumption (24). Thus, by using Young’s inequalitywe have N − (cid:88) i = ( x i − x i + − z i )( I i − I i + ) ≤ (cid:15) N − (cid:88) i = | x i − x i + − z i | + ψ M (cid:15) N − (cid:88) i = | v i − v i + | , (27)11here (cid:15) will be determined later. Regarding the term with u i , we estimate N − (cid:88) i = ( x i − x i + − z i )( u i − u i + ) = − N − (cid:88) i = φ ( | x i − x i + − z i | ) | x i − x i + − z i | + N − (cid:88) i = φ ( | x i − x i + − z i | )( x i − x i + − z i )( x i + − x i + − z i + ) . We then use Lemma 2 with b i = φ ( | x i − x i + − z i | ) and a i = x i − x i + − z i to get N − (cid:88) i = ( x i − x i + − z i )( u i − u i + ) ≤ − (cid:15) N − (cid:88) i = φ ( | x i − x i + − z i | ) | x i − x i + − z i | ≤ − (cid:15) φ m N − (cid:88) i = | x i − x i + − z i | , where (cid:15) is given in Lemma 2. This together with (26), (27), and choosing (cid:15) = (cid:15) φ m implies ddt N − (cid:88) i = ( x i − x i + − z i )( v i − v i + ) ≤ − (cid:15) φ m N − (cid:88) i = | x i − x i + − z i | + ψ M (cid:15) φ m N − (cid:88) i = | v i − v i + | . Thus the modified total energy E γ satisfies ddt E γ ( x , v ) ≤ − γψ m − ψ M (cid:15) φ m N − (cid:88) i = | v i − v i + | − (cid:15) φ m N − (cid:88) i = | x i − x i + − z i | . By taking γ > max { (4 ψ M ) / ( (cid:15) φ m ψ m ) , (cid:112) N /φ m } and using (20) and (25), we further estimate ddt E γ ( x , v ) ≤ − c N γψ m − ψ M (cid:15) φ m N (cid:88) i , j = | v i − v j | − (cid:15) φ m N − (cid:88) i = | x i − x i + − z i | ≤ − c γ min c N γψ m − ψ M (cid:15) φ m , (cid:15) φ m E γ ( x , v ) . Applying the Gr¨onwall’s lemma to the above gives the exponential decay of the modified total energy E γ . Moreover,the relation (25) concludes the desired result. (cid:3) Remark 4.
The a priori assumption (24) imposes some constraints on z i . For instance, if we fix the order for the initialpositions as x i + δ i < x i + for i = , . . . , N − , then by Theorem 2, we have x i ( t ) + δ i < x i + ( t ) for all t ≥ . This impliesthat in order to have the time-asymptotic pattern formation z i and δ i should satisfy z i > δ i for all i = , . . . , N − .On the other hand, if we assume z i − > ρ + δ i − , i = , . . . , N , where ρ is given as in (23) , then | x i ( t ) − x i − ( t ) | − δ i − ≥ z i − − ρ − δ i − > for all t ≥ and all i = , . . . , N. Indeed, it follows from (19) that | x i ( t ) − x i − ( t ) | = | x i ( t ) − x i − ( t ) − z i − + z i | ≥ z i − − ρ, thus subtracting δ i − from the both sides gives the desired result. igure 2: Positions over time of 5 particles on the line. Left: case 1) agents initially at rest and located at non-collided positions. Right: case 2)agents with non-zero initial velocities and located at close proximity but not collided. Thin lines represent the volumes of the agents. No collisionsoccur due to the singular interactions and the desired formation is acquired in steady state.
6. Numerical examples
In the following, we further elucidate the applicability of our results through two numerical experiments illustrat-ing string flocking to a pattern formation, as well as energy evolution.
For simplicity in the visualization, we will first consider a collection of N = α = . β = . δ =
2. We will consider two cases for the agents initialconditions: 1) the agents are initially at rest and located at non-collided positions, 2) the agents have di ff erent initialvelocities with v c (0) = z i = δ + δ is added to avoid configurations that are collided in steady state). In such a scenario,the agents should reposition themselves reaching the consensus velocity of zero and the desired pattern. Figure 2illustrates both cases. As predicted by Theorems 2 and 4, and given that α > β < v c (0) = − .
2. We can observe that the statement of 1 is satisfied over the time evolution of the dynamics, that is, the totalenergy of the system is non-increasing and the dissipation rate is entirely determined by the interaction term (seeFigure 3 bottom-left. Some agents are very close to the interaction limit determined by δ =
2, at t = sec ], which ishighlighted by Figure 3 top-right. The agents 2 and 3 then approach each other for a short period of time and around t = sec ] all the agents begin to spread out to reach the desired formation, with an average velocity of − . β > N =
10 agents with α = . δ =
0. For the same initial conditions with v c (0) =
0, weconsider two cases: 1) β = . β = . β influences the value of the initial energies, as noted earlier. However, the kinetic energy is positive and thesame in both cases, that is, the agents are not initially moving with the same velocity. For β = .
1, condition (17) ofTheorem 4 is not satisfied, as the value 0 . / (4 . − ≈ . x i ( t ) − x i + ( t ) − z i igure 3: 5 particles on the line with non-zero average velocity. Left-Top: Energy decomposition; Left-Bottom: Dissipation; Right-Top: Minimaldistance between agents; Right-Bottom: Positions over time of the particles achieving a consensus speed and the desired spatial formation as thesystem evolves. Note that the energies satisfy statement (ii) in Lemma 1. diverge and we can appreciate clustering. On the other hand, for β = . | x i ( t ) − x i + ( t ) − z i | → u i ( t ) = t , i . Conclusions
We have presented a control system for platooning composed by a string of agents interacting under nonlinear sin-gular dynamics and a decentralized feedback law. The resulting closed-loop exhibits important features for platooningcontrol, namely, collision-avoidance, velocity flocking, and asymptotic pattern formation. The derivation of rigorousenergy estimates allow the characterization of conditions under which the aforementioned features are guaranteed.Energy estimates are governed by: the number of agents in the string, the strength of the control interaction termexpressed through the parameter β in (3), and the cohesiveness of the initial configuration. In particular, the depen-dence with respect to the number of agents is a relevant topic of interest for future research. Although our results areasymptotic, we have observed the transient behaviour of the control system and it exhibits similarities to linear-timeinvariant platooning, namely, slow transients as the number of agents increases. The energy analysis we presented canbe extended to study mean field dynamics arising when N → ∞ and the system is characterized by an agent densityfunction [27]. Although the applicability of the mean field framework seems inadequate from a safety viewpoint ascollision-avoidance is an eminently microscopic phenomenon, it can be a powerful mathematical method to furtherunderstand the large-scale structure of the control system. Acknowledgements
YPC has been supported by NRF grant (No. 2017R1C1B2012918) and Yonsei University Research Fund of2020-22-0505. DK was supported by a public grant as part of the Investissement d’avenir project,reference ANR-11-LABX-0056-LMH, LabEx LMH, and by the UK Engineering and Physical Sciences Research Council (EPSRC)grants EP / V04771X /
1, EP / T024429 /
1, and EP / V025899 / igure 4: 10 particles on the line with zero average velocity when β = .
1. Left-Top: Positions over time of the particles when the control is used;Left-Bottom: Energy decomposition and flocking condition for β >
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