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.
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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)−γ2|w(t)|2 ≤ 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 ε other agents by using communication network. However, (2) these transmissions are not synchronous that leads to where ε = 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 ∈ {1, 2, 3}. 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 {1, 2, 3}, 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) ∈ R, ui ∈ 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 ∈ n×n stands for a symmetric and positive R |f (z, t)| ≤ ρ , ∀(z, t) ∈ × [0,L] × [t , ∞). (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 ∈ [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 ∈ 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 ∈ 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 ∈ H [a, b] be a scalar function, with the boundary agent can measure its global position z except for the values stated below. Then i leader agents labeled zi , m ∈ {1,...,M}. Z b 2 Z b 2 m 2 (b − a) dz(ξ) (3) The agents form a chain topology, and the adjacent σ z (ξ)dξ ≤ 2 dξ (1) a π a dξ agents keep their order. where Our objective is to deploy the agents onto the desired curve 1, if z(a) = z(b) = 0; σ = Γ by exploiting M << N leaders. 1/4, if z(a) = 0 or z(b) = 0. Lemma 1.2. Generalized Halanay’s inequality Wen et al. 2.2 Controller design and PDE model + (2008). Let V :[t0 −τM , ∞) −→ R be a locally absolutely continuous function, and w :[t0, ∞) −→ C be a bounded We propose a leader-follower displacement-based control, continuous function satisfying kw(t)k ≤ ∆w, where ∆w > where the position measurements of the leader agents
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are transmitted through communication network to other The leader zim sends his absolute (relative to Γ) position agents. Define zim − Γ(imh) to all the agents zi located on [xm−1, xm) z (t) = z (t), z (t) = z (t), through communication network. The measurements are 0 2 N+1 N−1 (5) Γ(0) = Γ(2h), Γ((N + 1)h) = Γ((N − 1)h). subject to quantization effect Liberzon (2003), Fridman and Dambrine (2009). A quantizer is a piecewise constant Consider the following static output-feedback controller: function q : R → R such that υ2 u (t) = [z (t) − 2z (t) + z (t)] (6) |q(y) − y| ≤ ∆q, (10) i h2 i+1 i i−1 where ∆ is the quantization error bound. Here for sim- υ2 q plicity we assume that the quantizer has no constraints on − 2 Γ (i + 1)h − 2Γ ih + Γ (i − 1)h h the quantization range. − βz˙i(t) − f(Γ(ih), t) +u ¯i(t), i = 1,...,N. m m Let s0 < s1 < ... be the sampling times of the measure- Hereu ¯i will be found below as the product of a constant m m gain K > 0 on the corresponding leaders’ position mea- ments with limk→∞ sk = ∞ and ηk are network-induced surements. Note that the proposed controller (6) contains delays. We assume that sk + ηk < sk+1 + ηk+1 for all 3 gains: υ2 (larger υ2 allows to reduce the number of the k. The agents zi from the interval [xm−1, xm] employ the leaders Wei et al. (2019)), K (compensates the destabi- controller m lizing effect of the nonlinearity f) and β (improves the u¯i(t) = Kq(zim (sk )−Γ(imh)), t ∈ [sk +ηk, sk+1 +ηk+1), convergence). Given υ2, we aim to achieve the deployment (11) with as small as possible number M of leaders. whereu ¯i(t) = 0 for t < t0. Denote the error By using the time-delay approach to network-controlled systems (see Chapter 7 of Fridman (2014)), denote ei(t) = zi(t) − Γ(ih), u = 0,...N + 1, t ≥ t0. m m m m m m m We have τ =τ (t)=t−sk , t ∈ [sk +ηk , sk+1 +ηk+1), k = 0, 1,... m f(zi, t) − f(Γ(ih), t) = ρ(ei, t)ei(t), where τ (t) ≤ τM ∀m = 1, ..., M and τM is the sum of Z 1 maximum transmission interval and maximum allowable ρ(ei, t) = fz (θei + Γ(ih), t)) dθ, delay. Then the controller (11) can be presented as 0 m m u¯i(t) = Keim (t − τ ) + Kwm(t − τ ), t ≥ t0, where due to (4) m m m (12) wm(t − τ ) = q(ei (t − τ )) − ei (t − τ ) |ρ| ≤ ρ ∀(e , t) ∈ × [t , ∞). m m 1 i R 0 with Then the closed-loop system (3), (6) can be presented as: m |wm(t − τ )| ≤ ∆q, ∀t ≥ t0. (13) υ2 e¨ (t) = [e (t) − 2e (t) + e (t)] (7) i h2 i+1 i i−1 System (7), (12) can be considered as the discretization in the spatial variable of the damped wave equation − βz˙i(t) + ρ(ei, t)ei(t) +u ¯i(t), i = 1,...,N. 2 ett(x, t) = υ exx(x, t) − βet(x, t) + (ρ(e, t) − K)e(x, t) We further treat the large-scale MAS (3) as a continuum M with a spatial domain x ∈ [0,L]. Following Fridman and X m m + K χm [e(ˆxm, t − τ ) + wm(t − τ )] , (14) Blighovsky (2012); Wei et al. (2019), we divide x ∈ [0,L] m=1 into M sampling intervals x ∈ (0,L), t ≥ t0 0 = x0 < x1 < . . . < xM = L, (8) under the Neumann boundary conditions where the length of the interval is bounded ex(0, t) = ex(L, t) = 0, (15) x − x = ∆ < ∆, m = 1, ..., M. (9) m m−1 m and initial conditions defined by the difference between of Since we aim to minimize M, we can always assume that the initial agents positions and their target positions on Γ. L ∆ ≤ M+1 . Similar to Terushkin and Fridman (2019), we Here χm are given by place the leader agent approximately in the middle of each 1, x ∈ [xm−1, xm] interval (see Fig. 1) such that χm(x) = , m = 1,...,M. (16) 0, x 6∈ [xm−1, xm] ∆ ∆ xˆm − xm−1 ≤ , xm − xˆm ≤ . Note that (5) with e = e and e = e corresponds 2 2 0 2 N+1 N−1 to spatial discretization under the Neumann boundary Note that if in discretization, the number Nm of agents conditions. Due to (4) located on [xm−1, xm] is even, then leader may be located |ρ| ≤ ρ ∀(e, t) × [t , ∞). in such a way that he has 0.5Nm − 1 agents on [xm−1, xm] 1 R 0 from the left (or right) and 0.5Nm from the right (or left). Consider initial conditions for (14), (15) as e(x, t) ≡ e(x, t ), e (x, t) ≡ e (x, t ), t ≤ t . leader 0 t t 0 0 zi m Well-posedness of (14), (15) can be proved similar to Terushkin and Fridman (2019). The state is presented as ∆m T T 0 L ζ(t) = [ζ0(t) ζ1(t)] = [e et(t)] . Denote 0 I h 0 i x0 xm−1 xˆm xm xM 2 A = 2 ∂ ,F (ζ, t) = . υ −βI F1(ζ0, t) ∂x2 1 2 Fig. 1. MAS: leader location Here F1 : H (0,L) × [t0, ∞) → L (0,L) is given by
7709 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020
2 2 F1(ζ0, t) = f(z, t) − f(z − ζ0, t) c0(kex(·, t)k 2 + ket(·, t)k 2 ) ≤ exp − 2α(t − t0) M L L X m m γ2 − K χm(x)[ζ0(ˆxm, t − τ ) + wm(t − τ )] . 2 2 2 × kex(·, t0)kL2 + ket(·, t0)kL2 + ∆qL, (22) m=1 ε The resulting differential equation where ε = 2(α0 − α1) and α > 0 is a unique positive ˙ solution of α = α0 − α1 exp(2ατM ). ζ(t) = A ζ(t) + F (ζ(t), t), t ≥ t0 (17) is considered in the Hilbert space H = H 1(0,L) × Moreover, if the strict inequalities (18)-(21) are feasible 2 2 2 2 with α = α > 0, then the error system (14), (15) is ISS L (0,L), where and kζkH = kζ0xkL2 + kζ1kL2 . 0 1 with a small enough decay rate. The operator A with the dense domain nζ o Proof Denote ( )= 0 ∈ 2(0,L)× 1(0,L) ζ (0)=ζ (L)=0 D A ζ H H 0x 0x m m 1 ν1 = e(x, t)−e(x, t−τ ), ν2 = e(x, t−τ )−e(x, t−τM ). generates an exponentially stable semigroup (see Pazy By employing the relations m m (1983)). The times sk + ηk , k = 0, 1, ..., m = 1, ..., M are m e(x, t − τ ) = e(x, t) − ν1(x, t), (23) ordered as t0, t1, ... By employing the step method for t ∈ Z x [t0, t1], t ∈ [t1, t2] and applying Theorems 6.1.2 and 6.1.5 e(ˆxm, t) = e(x, t) − eζ (ζ, t)dζ from Pazy (1983) (see Terushkin and Fridman (2019)), we xˆm find that a unique mild solution exists in C([t0, ∞), H ) for the error dynamics can be represented as T (14), (15), initialized by [e(·, t0)et(·, t0)] ∈ H . Moreover, 2 T ett = υ exx − βet + (ρ − K)e (24) if [e(·, t0) et(·, t0)] ∈ D(A )), then there exists a unique M Z x 1 X m m classical solutionC ([t0,∞), H )with ζ(t)∈D(A ) for t>t0. + K χm ν1 + eζ (ζ, t − τ )dζ − wm(t − τ ) . m=1 xˆm 2.3 ISS analysis For deriving the ISS conditions of (7), we use a Lyapunov functional similar to Terushkin and Fridman (2019) For the choice of the controller gains β and K we follow Remark 3.1 of Terushkin and Fridman (2019), where larger V (t) = V0(t)+Vs(t)+Vr(t), t ∈ [tk, tk+1), k = 0, 1, 2,... 2 β and K = ρ + β lead to faster convergence. (25) 1 4 where V (t) is given by Theorem 2.1. Consider the damped wave equation (14) 0 Z L Z L under the Neumann boundary conditions (15) initialized 2 2 T 1 V0(t)=p3υ exdx+ [e et]P0[e et] dx, (26) for t ≤ t0 by e(·, t) ≡ e(·, t0) ∈ H (0,L), et(·, t) ≡ 0 0 2 et(·, t0) ∈ L (0,L) with the bounds τM , ρ1, ∆q, ∆ in (13) 2 with P0 given by (18), and and (9). Given α > α > 0, υ2, β > 0,K = ρ + β , let 0 1 1 4 M x t there exist γ, s > 0, r > 0, q and p , p , p that satisfy X Z m Z 12 1 2 3 V (t) = s e2α0(s−t)e2(x, s)ds dx, (27) the LMIs s m=1 xm−1 t−τM M p1 p2 Z xm Z 0 Z t P0 = > 0, (18) X 2α0(s−t) 2 ∗ p3 Vr(t) = rτM e es(x, s)ds dθ dx, xm−1 −τM t+θ α0p3 − p2 ≤ 0, (19) m=1 r q with some scalars s, r ≥ 0. Here, V and V treat time-delay R = 12 ≥ 0, (20) s r ∗ r terms. This functional is defined on the mild solutions of (14), (15), and due to (18) it is positive definite with 0 2 2 0 and V (t) ≥ c (kex(·, t)kL2 + ket(·, t)kL2 ) for some c > 0. Ψ|ρ=±ρ = 1 As in Fridman (2013), we consider first ψ11 ψ12 ψ13 ψ14 Kp2 −2αM p2 −Kp2 2 T ∗ ψ22 Kp2 + τM r 0 Kp2 0 −Kp2 [e(·, t) ≡ e(·, t0), et(·, t) ≡ et(·, t0)] ∈ D(A ). ∗ ∗ ψ33 ψ34 0 2αM p2 0 Then we can differentiate V (t) along the classical solutions ∗ ∗ ∗ ψ44 0 0 0 ≤ 0, of the wave equation. By differentiating Vs and Vr we have ∗ ∗ ∗ ∗ ψ55 0 0 M Z xm ∗ ∗ ∗ ∗ ∗ −2αM p3 0 X V˙ +2α V = s e2(x, t)−e−2α0τM e2(x, t−τ ) dx ∗ ∗ ∗ ∗ ∗ ∗ −γ2 s 0 s M m=1 xm−1 (21) (28) α1 where αM = M and and M ψ11 = 2p2(ρ − K) + 2(α0 − αM )p1 + s(1 − exp(−2α0τM )) Z xm ˙ 2 X 2 ψ12 = p1 + p2(2α0 − β) + p3(ρ − K) Vr + 2α0Vr ≤ τM r et dx (29) m=1 xm−1 ψ13 = Kp2 + s exp(−2α0τM ) + 2αM p1 M Z xm Z t ψ = s exp(−2α τ ), ψ = 2p + 2p (α − β) −2α0τM X 2 14 0 M 22 2 3 0 − τM re es(x, s)ds dx. ψ33 = −(s + r) exp(−2α0τM ) − 2αM p1 m=1 xm−1 t−τM ψ34 = −(s + q12) exp(−2α0τM ) Then, under (20) by Lemma 3.4 of Fridman (2014) we find 2 2 2 M M ψ44 = −(s + r) exp(−2α0τM ), ψ55 = −2αM p3υ (π /∆ ). Z xm Z t Z xm X 2 X T Then, (14) is ISS, namely there exists c0 > 0 such that for −τMr esds dx≤− [ν1 ν2]R[ν1 ν2] dx. x t−τ x all t ≥ t0 m=1 m−1 M m=1 m−1
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Z x We proceed with differentiating (25) along (14). Note m m m that integration by parts and substitution of boundary η = [e et ν1 ν2 eζ (ζ, t−τ )dζ et(x, t−τ ) wm(t−τ )]. xˆ conditions leads to m Z L Z L Then, by taking into account (19), 2 2 2 M 2υ (p2e + p3et)exxdx = −2υ p2ex + p3exext dx. Z L Z xm 0 0 2 2 X T W ≤2υ (α0p3 − p2) exdx + ηΨη dx ≤ 0, Then 0 m=1 xm−1 Z L Z L ˙ 2 2 T if Ψ ≤ 0, where Ψ is given by (21). Note that Ψ is affine in V0 + 2α0V0 ≤2υ (α0p3 −p2) exdx+ [e et]G[e et] dx ρ. Thus, it is sufficient to verify (21) in the vertices ±ρ1. M 0 0 X Z xm + 2 K(p2e + p3et) Thus, under (18)-(21), due to Halanay’s inequality, the xm−1 classical solutions of the wave equation satisfy (22). Since m=1 Z x m m D(A ) is dense in H , inequality (22) remains true (by × ν1 + eζ (ζ, t − τ )dζ − wm(t − τ ) dx continuous extension) for the mild solutions originated in xˆ m . where H ∆ 2p (ρ − K) + 2α p p + p (2α − β) + p (ρ − K) The feasibility of strict LMIs with α0 = α1 = 0 implies G = 2 0 1 1 2 0 3 . ∗ 2p2 + 2p3(α0 − β) their feasibility with a slightly largerα ¯0 = α0 + δ > 0, where δ > 0 is small, that completes the proof. By (13), we have Remark 2.1. Differently from the existing works on dis- M Z xm X m 2 2 tributed sampled-data control under point measurements |wm(t − τ )| dx ≤ ∆qL. (30) Fridman and Blighovsky (2012); Wei et al. (2019); m=1 xm−1 Terushkin and Fridman (2019), we consider the point mea- m We will further apply the generalized Halanay’s inequality surements (12) under the different delays τ that leads to more restrictive conditions via Halanay’s inequality with (2) for some 0 < α1 < α0. Note that α1 M αM = instead of α1 for equal delays τ1 = ... = τ . Note M M Z L Z xm that still it is easier to satisfy the resulting conditions for 2 1 X 2 sup ex(x, t+θ)dx≥ sup ex(x, t+θ)dx M >> 1 than for M = 1 since the stabilizing term with the −τ ≤θ≤0 M −τ ≤θ≤0 M 0 m=1 M xm−1 coefficient − α1 p υ2 in (32) after application of Wirtinger’s M M 3 1 X Z xm inequality in (33) leads to ≥ e2 (x, t − τ m)dx. M x 2 m=1 xm−1 α1 2 π α1(M + 1) 2 π − p3υ ≤ − p3υ =⇒ −∞. (31) M ∆2 M L2 M→∞ Then ∆ ˙ 2 2 2.4 Numerical simulations W = V (t) + 2α0V (t) − 2α1 sup V (t + θ) − γ ∆qL −τM ≤θ≤0 M Z xm In the sequel, we validate the proposed control approach in ˙ 2 X 2 m ≤ V (t)+2α0V (t)−2αM p3υ ex(x, t−τ )dx a simulation. Consider a group of N = 49 agents, governed m=1 xm−1 by (3) with a linear f = z, where ρ1 = 1 in (4). Our M x T X Z m+1 e(x, t − τ m) e(x, t − τ m) objective is deployment from initial position of Γ0 onto a − 2αM m P0 m dx smooth open curve Γ, parameterized in the interval [0, π] x et(x, t − τ ) et(x, t − τ ) m=1 m h π π i M Z xm Γ0 = sin i , cos i , 0 , i = 1,...,N (33) 2 X m 2 N N − γ |wm(t − τ )| dx. (32) h π 2π π π 2π i xm−1 Γ = sin i +2 cos i , cos i +2 sin i , 2+cos i . m=1 N N N N N By Wirtinger’s inequality (1) with (b − a) = (∆)/2 and We design a controller with the gains σ = 1/4 we have υ2 = 4.1, β = 3,K = 1 + β2/4. (34) Z xm 2 m For the linear system, LMI (21) of Theorem 2.1 is verified ex(x, t − τ )dx = only in one vertex ρ = ρ1 = 1. We find that the LMIs of xm−1 M Theorem 2.1 are feasible for M ≥ 2 leaders. By further h Z xˆm Z xm i X 2 m 2 m verifying the LMIs of Theorem 2.1 with α = α = 0.4, ex(x, t − τ )dx + ex(x, t − τ )dx 0 1 m=1 xm−1 xˆm we find that the system with two leaders (M = 2) is ISS 2 2 M xˆm provided τ ≤ 0.52. Note that increasing υ till 5.1, results π X h Z M ≥ [e(x, t − τ m) − e(ˆx , t − τ m)]2dx in 1 leader being sufficient for deployment if τ ≤ 0.29. ∆2 m M m=1 xm−1 We further show simulations of the deployment for M = 2, Z xm i + [e(x, t − τ m) − e(ˆx , t − τ m)]2dx τM = 0.52, where the network induced delays are bounded m by ηm ≤ 0.02, and quantization error is bounded by xˆm k M 2 Z xm ∆q = 0.01. The agents are divided into two groups: 1) π X m m 2 ≥ 2 [e(x, t − τ ) − e(ˆxm, t − τ )] dx agents z1, ..., z24 with the leader zi1 = z13 and 2) agents ∆ xm−1 m=1 z25, ..., z49 with the leader zi2 = z37. Fig. 2 depicts the M 2 π2 X Z xm Z x transitions of a system driven by two leaders from initial = e (ζ, t − τ m)dζ dx. ∆2 ζ (marked blue) to final (marked pink) open curves, given by m=1 xm−1 xˆm (33). Trajectories of the leaders are shown in cyan, whereas Denote the trajectories of the followers are shown in grey. Fig.
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3(a) demonstrates the convergence of the error energy for Fridman, E. (2014). Introduction to time-delay systems: each dimension. The error e for one of the dimensions is analysis and control. Birkhauser, Systems and Control: illustrated by Fig. 3(b). Foundations and Applications. Fridman, E. and Blighovsky, A. (2012). Robust sampled-
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2 Fridman, E. and Dambrine, M. (2009). Control under
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