Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020 Network-based deployment of the second-order multi agents: a PDE approach ? Maria Terushkin ∗ Emilia Fridman ∗∗ ∗ School of Electrical Engineering, Tel Aviv University, Israel. (e-mail: [email protected]). ∗∗ School of Electrical Engineering, Tel Aviv University, Israel. (e-mail: [email protected]) Abstract: Deployment of a second-order nonlinear multi agent system over a desired open smooth curve in 2D or 3D space is considered. We assume that the agents have access to their velocities and to the local information of the desired curve and their displacements with respect to their closest neighbors, whereas in addition a leader is able to measure his absolute position. We assume that a small number of leaders transmit their measurements to other agents through a communication network. We take into account the following network imperfections: the variable sampling, transmission delay and quantization. We propose a static output-feedback controller and model the resulting closed-loop system as a disturbed (due to quantization) nonlinear damped wave equation with delayed point state measurements, where the state is the relative position of the agents with respect to the desired curve. To manage with the open curve we consider Neumann boundary conditions. We derive linear matrix inequalities (LMIs) that guarantee the input-to-state stability (ISS) of the system. The advantage of our approach is in the simplicity of the control law and the conditions. Numerical example illustrates the efficiency of the method. Keywords: Distributed parameter systems, multi-agent systems, network-based control, time-delay. 1. INTRODUCTION deployment formations along predefined spatial-{temporal paths by means of boundary control were contemplated in Deployment of large-scale multi agent system (MAS), Meurer and Krstic (2011). Pilloni et al. (2015) address where a group of agents rearrange their positions into a the problem of driving the state of a network of agents, target spatial configuration in order to achieve a common modeled by boundary controlled heat equations, toward a goal, has attracted attention of many researchers in the re- common steady state profile. Formation tracking control cent years Mesbahi and Egerstedt (2010), Oh et al. (2015). of a MAS, where the collective dynamics was modeled This is due to their vast applications, such as cooperative by a wave PDE was studied in Tang et al. (2017). Qi movement of robots or vehicles Ren et al. (2007), biochemi- et al. (2019) considered the control of collective dynamics cal reaction networks, animal flocking behavior (see Olfati- of a large scale MAS moving in a 3D space under the Saber (2006)), search-and-rescue, environmental sensing occurrence of arbitrarily large boundary input delay. and monitoring Dunbabin and Marques (2012), etc. The In the case of measurements of the leaders' absolute po- majority of the existing work in the field of MAS is con- sitions, the majority of PDE-based results employ the centrated on deploying of interconnected agents, modeled PDE observer for output-feedback control. The latter may by ordinary differential equations (ODEs) that provides be difficult for implementation. Recently a simple static efficient methods when the number of agents is low. output-feedback controller was suggested in Wei et al. When the number of agents is large, a methodology (2019), where it was proposed to transmit the leader abso- based on partial differential equations (PDEs) becomes lute displacement with respect to the desired curve to all efficient. In Frihauf and Krstic (2010); Meurer (2012), the the agents by using communication network. The network- agents were treated as a continuum, and the collective based results of Wei et al. (2019) were confined to the dynamics was modeled by a reaction-diffusion PDE, un- first-order integrators and to deployment onto the closed der the boundary control. Feedforward control combined curves. Among the network imperfection, the quantization with backstepping-based boundary controller was imple- effects were neglected. Moreover, in the case of several mented in Freudenthaler and Meurer (2016), where the leaders a common delay (i.e. synchronized transmissions collective dynamics was modeled by a modified viscous in the same time with the same network-induced delay) Burger's equation. Finite-time transitions between desired was considered that may be restrictive. ? This work was supported by Israel Science Foundation (grant no. 673/19) and by C. and H. Manderman Chair at Tel Aviv University. Copyright lies with the authors 7707 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020 2 In this paper we study deployment of the second-order 0 is given. If there exists 0 < α1 < α0 and γ such that for nonlinear agents onto the open curves. We assume that all t ≥ t0 the following holds the agents have access to their velocities and to the local V_ (t)+2α V (t)−2α sup V (t+θ)−γ2jw(t)j2 ≤ 0; t ≥ t ; information of the desired curve and their displacements 0 1 0 −τM ≤θ≤0 with respect to their closest neighbors, whereas in addition then a leader is able to measure his absolute displacement with 2 respect to the desired curve. As in Wei et al. (2019) we γ 2 V (t) ≤ exp −2α(t−t0) sup V (t0+θ)+ ∆w; t ≥ t0; propose to transmit the leaders absolute displacements to −τM ≤θ≤0 " other agents by using communication network. However, (2) these transmissions are not synchronous that leads to where " = 2(α0 − α1) > 0; and α > 0 is a unique positive multiple delays in the closed-loop system. Moreover, we solution of take into account the quantization effect (see Liberzon α = α0 − α1 exp(2ατM ): (2003)). 2. MAIN RESULTS By applying the time-delay approach to networked control systems (see Chapter 7 of Fridman (2014)), we model the resulting closed-loop system as a disturbed (due to 2.1 Problem formulation quantization) nonlinear damped wave equation with the delayed point state measurements under the Neumann Consider a group of N agents, described by the second- n boundary conditions. The state is the relative position of order dynamics, that can move in space R ; n 2 f1; 2; 3g: 2 the agents with respect to the desired curve. Note that Our aim is to deploy N agents onto a C curve Γ : [0;L] −! n the existing results under the point delayed measurements R . If Γ(0) 6= Γ(L), the curve Γ is open. For simplicity, we are confined to one delay and to unperturbed systems assume that the desired curve does not evolve over time. (see Fridman and Blighovsky (2012); Kang and Fridman We neglect collision avoidance as we assume agents of zero (2019); Terushkin and Fridman (2019)). Here, for the volume operating within a large workspace. Furthermore, first time for such systems, we analyze the ISS of the we assume that no static or moving obstacles are present closed-loop system by combining the Lyapunov-Krasovskii in the operating workspace. method with the generalized Halanay's inequality (Wen The dynamics of each agent, in each dimension n; n = et al. (2008)). Moreover, we treat the case of multiple f1; 2; 3g, is given by delays. We derive LMIs that guarantee ISS. The advantage z¨ (t) = u (t) + f(z ; t); i = 1; : : : ; N; t ≥ t : (3) of our approach is in the simplicity of the control law i i i 0 and the conditions. Numerical example of deployment onto where zi(t) 2 R; ui 2 R are components of the position smooth open curve in a 3D space illustrates the efficiency and control of agent i respectively, and the acceleration of the method. nonlinearities f are of class C1. For brevity, the notation of dimension is omitted. We assume that the derivative fz Notation Throughout the paper the notation P > 0 is uniformly bounded by a constant ρ1 > 0 : with P 2 n×n stands for a symmetric and positive R jf (z; t)j ≤ ρ ; 8(z; t) 2 × [0;L] × [t ; 1): (4) definite matrix, with the symmetric elements denoted by z 1 R 0 ∗: Functions, continuous (continuously differentiable) in N points are assigned on the desired curve with constant 1 L all arguments, are referred to as of class C (of class C ). spacing h = N ; namely Γ(h);:::; Γ(hN) which will give L2(0;L) is the Hilbert space of square integrable functions the final desired position of each agent. 2 R L 2 z(ξ); ξ 2 [0;L] with the norm kzkL2 = 0 z (ξ)dξ. The leader-enabled deployment of mobile agents is consid- H 1(0;L) is the Sobolev space of absolutely continuous ered under the following assumptions: dz 2 2 scalar functions z : [0;L] ! R with dξ 2 L (0;L): H (0;L) (1) Agents i = 2;:::;N − 1 measure their displacements is the Sobolev space of scalar functions z : [0;L] ! R with with respect to the closest neighbors i−1; i+1. They dz d2z 2 have access to Γ((i − 1)h); Γ(ih) and Γ((i + 1)h). The absolutely continuous dξ and with dξ2 2 L (0;L). boundary agents with i = 1 and i = N measure 1.1 Mathematical preliminaries the relative positions of the agents 2 and N − 1 and have access to Γ(h); Γ(2h) and Γ((N − 1)h); Γ(Nh) The following inequalities will be useful: respectively. (2) All the agents measure their own velocityz _ with Lemma 1.1. Wirtinger's inequality Hardy et al. (1988). i 1 respect to the global coordinate system. However, no Let z 2 H [a; b] be a scalar function, with the boundary agent can measure its global position z except for the values stated below.