Fixed-Time Bearing-Based Network Localization

Nội dung text: Fixed-Time Bearing-Based Network Localization

1. Tuyển tập Hội nghị khoa học toàn quốc lần thứ nhất về Động lực học và Điều khiển Đà Nẵng, ngày 19-20/7/2019, tr. 127-131, DOI 10.15625/vap.2019000268 Fixed-Time Bearing-Based Network Localization Minh Hoang Trinh Department of Automatic Control, School of Electrical Engineering, Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung District, Hanoi, 11615, Vietnam E-mails: minh.trinhhoang@hust.edu.vn Abstract convergence time depends on the initial condition of the We consider a network localization problem in which a network system. As a result, although the target formation in [7, 8] of multiple nodes needs to estimate their positions based on can be achieved in a finite time, it is no guarantee that at measured bearing vectors and exchanging several variables. We a specified time T , the agents are in the desired propose a fixed-time bearing-based estimation law, which formation shape or not. This drawback of finite-time guarantees convergence of position estimates in a finite-time controllers can be remedied by the fixed-time control independently on the initial estimations. Simulation results are design method in [9]. In simple words, a fixed-time also provided to support the theoretical results. controller guarantees convergence of the system after a finite time T , for all initial conditions. Thus, the main Keywords: Fixed-time stability, Network localization, objective of this paper is designing a bearing-only control Bearing-only measurements, Multi-agent systems. law that solves the bearing-based network localization problem in a fixed time. The fixed-time analysis is given, and simulation results are also provided. 1. Introduction The remainder of this paper is organized as follows. In recent years, formation control and network Section 2 reviews the background on bearing rigidity, localization problems have attracted a lot of research bearing-only based network localization, and fixed-time interests [1, 2]. In a formation control problem, a group stability condition. In Section 3, we propose the control of moving autonomous agents needs to achieve a desired law and show that it guarantees a fixed-time formation shape via controlling some geometric variables convergence. Section 4 contains simulation results and regarding other agents. As a dual problem to formation Section 5 concludes the paper. control, in a network localization problem, there is a set 2. Preliminaries of stationary sensor nodes, and each node would like to estimate its position based on sensing and exchanging 2.1. Fixed-time Stability some variables with a few neighbor nodes. Consider the following system The existing works in the literature mainly focus on  designing control/estimation laws which require less xftxx==(, ), (0) x0 (1) sensing and communication resources between the where x ẻ n is the vector of system states, agents/nodes. To this end, the distance-based and the and f : ´ nn is a nonlinear function. If bearing-based approaches are advantageous in + comparison with the position-based and displacement- f (,tx ) is discontinuous, solutions of (1) are understood based approaches [3, 4]. While the distance-based in Filippov sense [10]. Assume that the system (1) has approach has been studied extensively for more than a zero equilibrium point. decade [3], the bearing-based approach has just got a Definition 2.1 ([11]) The equilibrium point x = 0 of considerable attention in recent years since bearing-only the system (1) is globally finite-time stable if it is control laws can be implemented using only a camera globally asymptotically stable and any solution xtx(, ) mounted on each agent. 0 of (1) reaches 0 at some finite time moment, i.e. In this paper, we confine our attention on a n bearing-based network localization problem. It is worth xtx(,00 )="³ 0, t Tx ( ), where T :{0} ẩ+ is noting that asymptotic convergence network localization the so-called settling-time function. control laws have been proposed in [5]. Furthermore, to Definition 2.2 ([9]) The equilibrium point x = 0 of the enhance the convergence rate, finite-time bearing-only system (1) is said to be globally fixed-time stable if it is formation control laws have been proposed in [6-8] so globally finite-time stable and the settling-time function that the agents can achieve a target formation after a Tx()Ê T , "ẻx n . finite time. Since formation control and network 0max 0 Definition 2.3 ([12]) If there exists a continuous radially localization are dual problems, it is quite straightforward n to apply the finite-time formation control laws in [7] to unbounded function V :{0} ẩ+ such that the network localization problem. However, a 1. Vx()= = 0 x 0; disadvantage of finite-time control laws is that the (finite)
2. M. H. Trinh 2. Any solution xt() of (1) satisfies the inequality as the DVxt* ( ( ))Ê-ab Vpq ( xt ( )) - V ( xt ( )) for some 1 1 ab,0> , p =-1 , q =+1 , m > 1 , 2m 2m then the origin is globally fixed-time stable for system (1) and the following estimate holds: pm n Tmax := , "ẻx 0 . ab Fig. 1: Nodes i and j measure the bearing vectors gij , g ji 2.2. Bearing-Based Network Localization and communicate their estimated positions pˆi , pˆj . Consider a sensor network of n nodes located at d pdi ẻ= (2, 3), in= 1, , , in the d-dimensional end vertex and the edge is directed from i to j . The global reference frame. Assume that each node does not incidence matrix Hh= []ij m´ n characterizes the relation know its global position, and thus it has an estimate between vertices and edges in G corresponding to this pˆ ẻ d . To localize the position, we further assume that i orientation is defined as follows: each node has its own sensing and communication ùỡ ù-=1, i f eijk ( , ) , capabilities. The sensing and communication topologies ù heji==ớ 1, if ( , ), between the nodes are characterized by a connected, kiù k ù 0, otherwise. undirected graph GVE= (, ), where Vn= {1, , } is ợù If the graph G is connected, we always have the node set, Eee=è´{1 , ,m } VV is the edge set Null() H= span (1). Let zz= [TTT , , z ] , then there [13]. An edge (,ij )ẻ E implies that two nodes i and n 1 m j can exchange their position estimates with each other, holds zHIpHp=Ä()d = , where ‘ Ä ’ denotes the and they can also sense the directional information (or Kronecker product. the bearing vector) with regard to each other. A network, denoted by (,)Gp , is described by a TTT Specifically, if eijEk =ẻ(, ) , node i (1 ÊÊin) can graph G and a configuration pp= [11 , , p ] of G sense the bearing vector in the space. The rigidity matrix of (,)Gp is defined as ppji- z ij z gg===()k ộP ự ij k ờỳg z 1 ppji- z ij k ờỳ RBdg() p=Ä=ờỳ ( H I ) diag ( P ) H . (2) to nodes j , and node j can sense the bearing vector ờỳ k ờỳPg ở m ỷ ppij- ggji==-() ij where diag() P is the block diagonal matrix of m gk ppij- projection matrices PP, , . For any bearing rigidity gg1 m to node i . Here zzkij= is the displacement vector between two nodes i and j . The projection matrix matrix, we have Null( RBnd ( p ))ấÄ Range ([1 I , p]) . associated with the bearing vector g is defined as This paper aims to design a position estimation law ij for each node based on only local information so that T TTT * PIgggdijij=- . Matrix Pg projects a vector into the ij ij they can determine a configuration ppˆˆ= [1 , , pˆn ] satisfying all measured bearing vectors between the orthogonal complement space of span{} gij . It is not * 2 T agents in a fixed time ( pˆ is different from p by only a hard to verify that PPPggg==, Pg is positive ij ij ij ij translation and a scaling). In other words, we would like semidefinite, and Null() P= span () g . to design update laws for ptˆ () so that gijij i ptˆˆ()- pt () Let NjVijEi =ẻ{|(,)} ẻ be the neighbor set ji gtˆij()== g ij , ij,,ẻ"ạ V i j, (3) of node i , then, the locally available information of a ptˆˆji()- pt () node i includes its estimated position pˆi , the set of "³tTmax >0 , and Tmax is independent on the initial measured bearing vectors {}g , and the estimated ij jẻ Ni estimations pˆ(0) . From now on, we will refer to this problem as the fixed- time bearing- based network positions {}pˆjjNẻ received from its neighbor nodes via i localization problem. wireless communication (see Fig. 1). Consider an In order to solve the problem, the following arbitrarily orientation of edges in G , that is, for each assumption on the network (,)Gp will be adopted: edge eij= (, ), we assign i as the starting vertex, j k Assumption 2.4 The network (,)Gp is infinitesimally
3. Fixed-Time Bearing-Only Network Localization bearing rigid in d , or i.e., we have pˆˆ TTT p =+ pˆ H diag()(( P sig diag ()) P gab sig ( diag ())) P g ggˆˆkk gˆ k m Null() R=Ä Range ([1, I p]). *T ab Bnd =+zPsigPgˆ ( ( ) sigPg ( ) ) ồ k gˆˆ gk gkˆ kk k In this paper, we would like to keep the background k=1 m on bearing rigidity theory as minimal as possible. For =+=zgPsigPgˆ* T ( ( )ab sigPg ( ) ) 0, ồ kkgˆˆ gk gkˆ kk k further results on bearing rigidity theory, the readers are k=1 referred to [14]. which implies that ggkmˆkk= , " = 1, , . The 3. Fixed-Time Bearing-Based Network remaining proof follows from Assumption 2.4 and a Localization similar reasoning as in [Thm. 10, 14]. ■ TTT * Let dd==-[1 , , dn ] ppˆˆ, and rpˆˆ=-Ä1 pˆ, The proposed control law since the centroid is invariant, it follows that Td d =-rrˆˆ* . For a vector vv=ẻ[1 , , vd ] , we will denote Thus, we can rewrite (5) as sig( v )aa= [sgn( v ) | v | , , sgn( v ) | v | a ]T 11 dd  T ab, (6) d =+H diag()(( Pggˆˆ sig diag ()) P g sig ( diag ())) P gˆ g and |vv |aa= [| | , ,| v | a ]T . kk k 1 d Moreover, as the scale is invariant, it follows that d The following position estimation law is proposed for evolves on the sphere d +==rrrˆˆˆ and (6) has each node in= 1, , : * pPsigPgsigPgˆ =+()ab (), (4) two equilibria in this sphere: d = 0 and d =-2rˆ . iggijgijồ jNẻ ˆˆ( ) i ij ij ij Next, we examine the stability of these equilibria. In the where a ẻ (0,1) and ba=-21 > are two control analysis, we will always assume that parameters. Note that in Eqn. (4), P can be calculated ptˆˆ()ạ"³ pt (), t 0 so that the vectors gˆ are always gˆij ij ij from pˆ and the communicated variables pˆ , and g defined. i j ij Theorem 3.3 The equilibrium d = 0 of (6) is are measured by agent i . We can rewrite the estimation asymptotically stable. law (4) in matrix form as follows: 1 2  T ab, (5) Proof. Consider the Lyapunov function V = d , pˆ =+ H diag()(( Pggˆˆ sig diag ()) P g sig ( diag ())) P gˆ g kk k 2 TTT TTT where gg= [1 , , gm ] , and ggˆˆ= [1 , , gˆm ] . Let which is positive definite, radially unbounded, and n continuously differentiable. We have 11T ppˆˆ==Ä(1 Ip ) ˆ, rpˆˆ=-Ä1 pˆ, and V= ddT nnồ ind i=1 * TT ab =-(pˆˆ p ) H diagP ()((ggˆˆ sigdiagP ()) g + sigdiagP ( ())) gˆ g srˆˆ= be the estimated centroid and estimated scale, kk k =-pˆ*TT H diag()(( P sig diag ()) P gab + sig ( diag ())) P g ggˆˆkk gˆ k respectively, we have the following lemma: m =-zgPsigPgˆ* T (( )ab + sigPg ( )) Lemma 3.1 The estimated centroid is invariant while the ồ kkgˆˆ gk gkˆ kk k k =1 scale under the estimation law (5). md =-zPgPgˆ*1(| [ ] |ab+ + | [ ] | +1 )0,Ê Proof. The result follows from ồồkgkigkiˆˆ kk ki==11  11TTT pˆˆ() t=Ä (1 I ) p =Ä (1 I ) H diag ( P )() and the inequality holds if and only if nd nd gˆk nn * ggkmˆ = , " = 1, , , or i.e., d = 0 or d =-2rˆ . 1 T kk =Ä (1Hnd I ) diag ( P gˆ )()0, = * n k By LaSalle’s invariance principle, d -{0, 2rˆ } . Since TT two equilibria are isolated, consider a neighborhood of (1)ppHdiagPˆˆ-Ä ()()gˆ   k stˆ()= d = 0 which does not contains d =-2rˆ* , then V < 0 , ppˆˆ-Ä1 for all d ạ 0 in that neighborhood. Thus, d = 0 is TT (1HpdiagPÄ ˆ )T ( )() pRˆˆ()() p  gˆ locally asymptotically stable. ■ =-B k =0, ppˆˆ-Ä11 ppˆˆ -Ä For the equilibrium d =-2rˆ* , we consider the 1 2 abfunction Vr=+d 2ˆ* and follow a similar proof as where ‘  ’ is (sig ( diag ( Pggˆˆ )) g+ sig ( diag ( P ))) g . ■ kk 2 From the invariance of the estimated centroid and scale, in Theorem 3.3, it can be proved that V ³ 0 , and thus we can then have the following lemma regarding the * equilibrium points of (5). d =-2rˆ is unstable. We thus conclude that d 0 Lemma 3.2 The system (5) has two isolated equilibria: asymptotically if d(0)ạ- 2rˆ* . * Before showing fixed-time stability of the equilibrium ppˆˆ= corresponding to ggkˆkk="=, 1, , m, and d = 0 , we state the following lemma, whose proof is ppˆˆ=  corresponding to ggkmˆ =-, " = 1, , . kk similar to [14] and will be omitted. Proof. From equation ptˆ()= 0, it follows that Lemma 3.4 Under the estimation law (6), the following inequality holds
4. M. H. Trinh zsnkˆˆk Ê-=2 1, 1, , m . (7) We can now prove the main theorem of this paper. 4. Simulation Results Theorem 3.5 Suppose that d(0)ạ- 2rˆ* , the y equilibrium d = 0 of (6) is locally fixed- time stable. Proof. Consider an arbitrarily closed neighborhood D of 1347 1 d = 0 which does not contain -2rˆ* . Let e = minztˆ* ( ) km=1, , , we can write 2 468 0 123x md VPgPg Ê-e (|[]||[]|)ab++11 + . G p ồồ gkiˆˆ gki kk ki==11 dm Fig. 2: The graph G and the true configuration p of For 0 0 , ldB+2()R is the smallest (2snˆ - 1)2 Mathematical analysis and simulations are provided to show the fixed-time convergence of the position positive eigenvalue of Rp(), and f > 0 is a constant B 0 estimates under the proposed estimation law. Fixed-time that only depends on D. The last derivation step is similar convergence is a powerful property that has not been to [Thm. 5, 8] and has been omitted for brevity. Thus, applied much in multi-agent systems. In future work, we combining this with (8) and (9), it follows that would like to design fixed-time controllers in other aa++11 bb ++ 11 problems in multi-agent systems. VVdmV Ê-ec(2 )22 - e ( )1-b (2 c ) 22. (10) References ab++11 a+1 Let pq==, , k = ec(2 ) 2 , and 221 1. Aspnes, J., et al., A theory of network localization. IEEE b+1 Transactions on Mobile Computing, 2006. 5(12): p. kdm= ec()(2)1-b 2 , then based on Lemma 2.1, 2 1663-1678. d = 0 is a fixed-time stable equilibrium of (6), i.e., we 2. Anderson, B.D.O., et al., Rigid graph control have d()ttT="³ 0, max , where architectures for autonomous formations. Control Systems pb() a pb () a Magazine, 2008. 28(6): p. 48-63. T == . ■ max ab++21 - b 3. Oh, K.K., M.C. Park, and H.S. Ahn, A survey of 2 kk 42 12 2(2)ec (dm ) multi-agent formation control. Automatica, 2015. 53: p. 424-440. 4. Zhao, S. and D. Zelazo, Bearing Rigidity Theory and its
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