Multi-agent deployment under the leader displacement measurement: a PDE-based approach Jieqiang Wei, Emilia Fridman, Anton Selivanov, Karl H. Johansson Abstract— We study the deployment of a first-order multi- In this paper, we consider a formation control problem agent system over a desired smooth curve in 3D space. We which is referred to as deployment. This can be seen as assume that the agents have access to the local information a combination of a displacement-based and position-based of the desired curve and their displacements with respect to their closest neighbors, whereas in addition a leader is able to formation control method. Each agent measures the relative measure his absolute displacement with respect to the desired positions (displacements) of its neighboring agents with curve. In this paper we consider the case that the desired respect to a global coordinate system. The desired formation curve is a closed C2 curve and we assume that the leader is specified by the desired displacements between pair of transmit his measurement to other agents through a commu- agents. Then the agents, without knowing their absolute nication network. We start the algorithm with displacement- based formation control protocol. Connections from this ODE positions, achieve the desired formation by controlling the model to a PDE model (heat equation), which can be seen as displacements of their neighboring agents. This will lead the a reduced model, are then established. The resulting closed- agents to the desired formation up to a constant distance. loop system is modeled as a heat equation with delay (due to As pointed out in [15], in order to move the agents to the the communication). The boundary condition is periodic since prescribed absolute positions, a small number of agents able the desired curve is closed. By choosing appropriate controller gains (the diffusion coefficient and the gain multiplying the to measure their absolute positions are needed. For existing leader state), we can achieve any desired decay rate provided ODE methods we refer to [1], [20] and the references within. the delay is small enough. The advantage of our approach is in Here we review some related work on the multi-agent the simplicity of the control law and the conditions. Numerical deployment using PDE models. In [7] and [17], the agents example illustrates the efficiency of the method. dynamics are modeled by reaction-advection-diffusion PDEs. Index Terms— Distributed parameters systems; Lyapunov method; Time delays; Multi-agent systems; Deployment. By using the backstepping approach to boundary control, the agents are deployed onto families of planar curves and 2D manifolds, respectively. In [14], the authors consider I. INTRODUCTION finite-time deployment of MAS into a planar formation, via Most of the existing work on multi-agent systems (MAS) predefined spatial-temporal paths, using a leader-follower consider interconnected agents modeled using ordinary dif- architecture, i.e., boundary control. The same problem of ferential equations (ODEs) or difference equations, and de- deployment into planar curves using boundary control is sign the control for each agent depending either on global or considered in [19] and [3] by using non-analytic solutions local information. Besides these studies, there has been some and a modified viscous Burger’s equation, respectively. In work using partial differential equations (PDEs) to describe [16], the authors proposed a boundary control law for a the spatial dynamics of multi-agent systems, e.g., [3], [4], MAS, which is modeled as the heat equation, to achieve [7], [17], [19]. This approach is especially powerful when state consensus. the number of the agents is large. One of the advantages of The main contributions of the paper is that we propose a framework which connects a ODE formation control protocol arXiv:1903.09193v1 [math.OC] 21 Mar 2019 using PDE models for MAS is to reduce a high-dimensional ODE system to a single PDE. Reversely, given a desired and a PDE model for the deployment of mobile agents onto PDE model, the corresponding performance and the control arbitrary closed C2 curves. Furthermore, in this framework protocol for the individual agents (in ODE form) can be we assume only leader measures its absolute position and designed by proper discretization. In principle, this procedure use simple static output-feedback control. More precisely, the is independent with respect to the number of agents, provided leader calculates its displacement with respect to the desired this number is large enough. curve. Then the leader sends the value of its displacement to all the agents by using a communication network which *This work is supported by Knut and Alice Wallenberg Foundation, results in time-varying delay due to sampling and communi- Swedish Research Council, and Swedish Foundation for Strategic Research. cation [4]. The other agents, which are referred as followers, Jieqiang Wei, Anton Selivanov and Karl H. Johansson are with the Department of Automatic Control, School of Electrical Engineer- have access only to the local information of the desired curve ing and Computer Science. KTH Royal Institute of Technology, SE- and displacements with respect to their neighbors. Since the 100 44 Stockholm, Sweden. fjieqiang, [email protected], desired formation is a closed curve, the MAS is modeled [email protected]. Emilia Fridman is with the School of Electrical Engineering, Tel Aviv as a diffusion equation with periodic boundary condition. University, Israel. [email protected]. The method used in this paper is based on [6] and [18] which deal with Dirichlet and mixed boundary conditions. where h = 2π=N. Consider the following displacement- We derive linear matrix inequality (LMI) conditions with based formation control protocol arbitrary delay for desired convergence rate. Compared to (z (t) − z (t)) + (z (t) − z (t)) the ODE MAS with communication delay, e.g., [13], the z_ (t) =a i−1 i i+1 i i h2 LMI conditions derived in this paper are simpler with lower (γ((i − 1)h) − γ(ih)) + (γ((i + 1)h) − γ(ih)) dimension, and they are always feasible. − a h2 The paper is organized as follows. In Section II, some i = 1; : : : ; N: useful inequalities are recalled. The MAS deployment prob- (5) lem using sampled control is formulated in Section III. The 3 where z0 = zN ; zN+1 = z1, zi 2 R is the position of the main results are included in Section IV and V. In Section agent vi, and a > 0, guarantees that all agents converge to IV, we derive LMI conditions to guarantee the deployment the formation on the closed C2 curve for the desired decay rate without communication delay. In Section V, the similar type of the E := f(z1; : : : ; zN ) j zi − zj = γ(ih) − γ(jh)g; (6) result is obtained for the case with delay. Simulations are which is the desired curve up to constant translations [15]. presented in Section VI. The paper is concluded in Section VII. Remark 1. The implementation of the system (5) includes Notations. With R>0 we denote the set of non-negative firstly the agents align the local coordination system, then real numbers, respectively. L2(a; b) is the Hilbert space of the agent compare the displacement (to its neighbors) with square integrable functions φ(ξ); ξ 2 [a; b] with the corre- respect to the desired displacement continuously. It can be q R b 2 1 proved that the formation of the agents converges to the sponding norm given as kφkL2 = a z dξ. H (a; b) is the Sobolev space of absolutely continuous scalar functions formation given by desired displacements asymptotically up dφ 2 to a constant translation [15]. φ :[a; b] ! R with dξ 2 L2(a; b) . H (a; b) is the Sobolev space of scalar functions φ :[a; b] ! R with absolutely As suggested in [2], when N is large, the model (5) is an dφ d2φ continuous dξ and with dξ2 2 L2(a; b). approximation of II. PRELIMINARIES zt(x; t) = a(zxx(x; t) − γxx(x)): (7) Lemma 1 (Wirtinger’s inequality [11]). For f 2 H1(a; b), By denoting the error e(x; t) = z(x; t) − γ(x), the error 2(b − a) dynamic of (7) is given as the following heat equation kfk ≤ kf 0k if f(a) = 0 or f(b) = 0. π et(x; t) = aexx(x; t); x 2 [0; 2π]: (8) Lemma 2 (Halanay’s inequality, [10], [4]). Let 0 < δ1 < 3 2δ0 and let V :[t0 − τM ; 1) ! [0; 1) be an absolutely Notice that the components of e(x; t) 2 R are decoupled. continuous function that satisfies It can be seen that system (7) cannot drive the agents onto the desired curve γ, but up to a constant translation. In fact, _ V (t) 6 −2δ0V (t) + δ1 sup V (t + θ); t > t0: (1) z∗ = γ + c is an equilibrium of system (7) for any constant −τ θ 0 M 6 6 c. This is consistent with the displacement-based formation Then control in [15]. In order to solve this problem, we shall −2δ(t−t0) employ additional control input to guarantee the convergence V (t) 6 e sup V (t0 + θ); t > t0; (2) −τM 6θ60 to the desired curve. More precisely, we assign leader agents where δ > 0 is the unique positive solution of who can measure the absolute positions of themselves and of their targets. 2δτM 2 δ1e Since the desired curve γ is closed and is C , it is natural δ = δ − : (3) 0 2 to consider the multi-agent system with periodic boundary III. PROBLEM FORMULATION condition z(0; t) = z(2π; t) We consider N agents in R3 governed by (9) zx(0; t) = zx(2π; t): z_i = ui; i 2 f1;:::;Ng; (4) Furthermore, we assume, without loss of generality, that the 3 3 leader is located at x = π and it can measure z(π; t) − where zi 2 R are the states and ui 2 R are the control inputs.