ON QUANTIZATION AND COMMUNICATION TOPOLOGIES IN MULTI-VEHICLE RENDEZVOUS 1. Karl Henrik Johansson Alberto Speranzon,2 Sandro Zampieri

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1 ON QUANTIZATION AND COMMUNICATION TOPOLOGIES IN MULTI-VEHICLE RENDEZVOUS 1 Karl Henrik Johansson Albero Speranzon, Sandro Zampieri Deparmen of Signals, Sensors and Sysems Royal Insiue of Technology Osquldas väg 1, 1-44 Sockholm, Sweden {kallej,albspe}@s3.kh.se Diparimeno di Eleronica e Informaica Universiá di Padova via Gradenigo, 6/A, I Padova, Ialy zampi@dei.unipd.i Absrac: A rendezvous problem for a eam of auonomous vehicles ha is communicaing hrough quanized channels is considered. Communicaion opologies and feedback conrol law are presened ha solves he rendezvous problem in he sense ha a meeing poin for he vehicles is pracically sabilized. In paricular, i is shown ha uniform quanizers can someimes be replaced by logarihmic quanizers and hus reduce he need for communicaion bandwidh. Copyrigh c 5 IFAC Keywords: Conrol under limied communicaion; Quanized conrol; Muli-vehicle coordinaion; Disribued conrol; Hybrid sysems. 1. INTRODUCTION Inerplay beween coordinaion and communicaion is imporan in many muli-vehicle sysems, e.g., car plaoons on auomaed highways (Varaiya, 1993), formaions of auonomous underwaer vehicles (de Sousa and Pereira, ), and muli-robo search-and-rescue missions (Speranzon and Johansson, 3). Consrained communicaion beween vehicles sugges he deploymen of disribued (local) conrol sraegies (Lin e al., 3; Saber and Murray, 3). In many cases no only he communicaion opology is imporan, however, bu also he amoun of 1 This research is suppored by he European projec RECSYS, IST Corresponding auhor. daa being ransmied. Therefore, in his paper we sudy muli-vehicle conrol under quanized communicaion. The problem is relaed o he sabilizaion of linear plans wih quanized conrol, which has recenly been exensively sudied, see (Fagnani and Zampieri, 3) and references herein. The main conribuion of his paper is o illusraes how communicaion opologies based on uniformly and logarihmically quanized communicaion influence he soluion o a muli-vehicle rendezvous problem. A eam of auonomous vehicles wih only local posiion informaion is o mee under minimum communicaion capabiliies. We prove he exisence of several classes of soluions o his rendezvous problem. In paricular, we emphasize ha uniform quanizers can someimes

2 be replaced by logarihmic quanizers and hus reduce he need for communicaion bandwidh. The ouline of he paper is as follows. The rendezvous problem is defined in Secion. Illusraive wo-vehicle cases are sudied in deail in Secion 3. Teams of hree and more vehicles are hen considered in Secion 4. Convergence properies are invesigaed hrough simulaions in Secion 5. Some conclusions are given in Secion 6.. PROBLEM FORMULATION Consider n vehicles moving in a plane, wih dynamics described by he discree-ime sysem x + x + u (1) y + y + v () where x (x 1,..., x n ) T X R n and y (y 1,..., y n ) T Y R n, so ha (x i, y i ) denoes he posiion of vehicle i wih respec o a fixed coordinae sysem. Le U R n and V R n denoe he se of conrol values. The conrols u (u 1,..., u n ) and v (v 1,..., v n ) we are considering are feedback maps from he corresponding sae-space X and Y, respecively. Since he conrol of he x- and y-coordinaes are independen, we only consider x in he sequel..1 Quanized communicaion We furher resric he communicaion by imposing ha communicaed daa are quanized. In paricular, uniform and logarihmic quanizaions are considered. Recall he following definiions of scalar uniform and logarihmic quanizers. Definiion 1. Le δ u be a posiive parameer. A uniform quanizer is a map q u : R R such ha x q u (x) δ u. δ u We are ineresed in how communicaion opology and quanizaion of ransmied daa influence he performance of he muli-vehicle sysem. Therefore, we consider classes of maps Ψ δ composed of various configuraions of uniform and logarihmic quanizers. For simpliciy, we suppose ha all quanizers are parameerized in a single quanizaion parameer δ.. Rendezvous We are ineresed in he convergence o a mulivehicle formaion under quanized communicaion opology. Especially, we pose he following rendezvous problem. Definiion 3. A feedback law (u) g(x) solves he rendezvous problem if for all iniial saes x X, and for all ɛ >, here exiss δ δ(ɛ) such ha x i x j B ɛ wih i < j and i, j 1,..., n. A soluion o he rendezvous problem is able o make he vehicles converge arbirarily close o a meeing poin. Noe ha he meeing poin is no pre-specified, bu is only resriced o be close o he hyperplane x 1 x n. If he communicaion is no quanized, hen he rendezvous problem is readily solved also in he case wih no communicaion. In his case, a decenralized deadbea conroller provides a soluion. We consider such a soluion rivial, since i enforces he vehicles o mee in he origin. I is insead desirable o have he meeing poin close o he iniial posiion of he vehicles. In nex secion, we presen a linear quadraic conrol problem for he wo-vehicle sysem ha address his problem and ha also sugges a naural exension o he quanized case. Noice ha he error due o he quanizaion of a variable x is bounded by δ u i.e., q u (x) x δ u. (3) Definiion. Le δ l be a posiive parameer. A logarihmic quanizer is a map q l : R R such ha q l (x) exp(q u (ln x)). The quanizaion error for a logarihmic quanizer is bounded as q l (x) x δ l x. (4) Noice ha he parameer δ l depends on δ u hrough he following expression δ l 1 e δu. 3. TWO-VEHICLES RENDEZVOUS Le z x 1 x be he oupu of he sysem (1) when n. In order o avoid aggressive soluion such as he dead-bea conrol we need o penalize he conrol inpu. Thus we consider an LQ problem wih cos J(u 1, u ) z + u 1 + u (x 1 x ) + u 1 + u. The resuling sae feedback u Kx gives he closed-loop sysem x + 1 x 1 k(x 1 x ) x + x k(x x 1 ), (5)

3 wih k 1/(1 + 3). I corresponds o z + (1 k)z, which is asympoically sable (in general for k 1), for he opimal choice of k. This means he difference x 1 x ends o zero asympoically. Noe ha (5) i is only sable, hus he wo vehicles, in general, will rendezvous o a poin differen from he origin. The linear feedback K compued above is used o design conrollers when he communicaion opology is composed of various configuraions of uniform and logarihmic quanizers. 3.1 Uniform-uniform quanizaion Proposiion 4. The feedback u 1 k ( x 1 q u (x ) ) u k ( x q u (x 1 ) ) (6) solve he rendezvous problem Proof. Le us consider he difference z x 1 x. We inroduce he following Lyapunov funcion V(z) z. Thus he incremen V(z) V(z + ) V(z) z + z is such ha V(z) z kz q u (x ) + q u (x 1 ) z k z + kδ where we have used (3). Hence V(z) < if z > δ. If we choose δ ɛ, hen z will asympoically converge o B ɛ. Thus he feedback (6) solves he rendezvous problem. Noice ha insead of posing a deerminisic LQ problem, we can consider a sochasic LQ problem where he quanizaion error of he uniform quanizer is approximaed by addiive whie noise, namely q u (x) x + e and where e is whie noise uniformly disribued in [ δ, δ] and Ee δ /1. This approach yields he same feedback conrol as above. 3. Uniform-logarihmic quanizaion Proposiion 5. The feedback u 1 k q l ( qu (x 1 ) x ) u k ( x q u (x 1 ) ) (7) solve he rendezvous problem Proof. Le us consider he difference z x 1 x. As in he proof of Proposiion 4, V(z) z k(q l (q u (x 1 ) x ) x + q u (x 1 )) z k z + k z δ + kδ + kδ k (δ) k 1 (δ) Fig. 1. For large quanizaion sep δ he value of k 1 (δ) ends o zero while he value of k (δ) ends o he value k (1), obained solving an opimal conrol problem wih n 1. where we used (4). V(z) < if z > δ(1 + δ). δ If we choose δ 1/ ɛ/4 4 + ɛ + ɛ /4, z converges asympoically o B ɛ. Thus he feedback (7) solves he rendezvous problem. The effec of he logarihmic quanizaion can be approximae as muliplicaive noise acing on he sysem. If, for simpliciy, we neglec he presence of uniform quanizaion, hen we have x + 1 x 1 + k 1 (x 1 x )(1 + e) x + x + k (x x 1 ) where e is whie noise uniformly disribued in [ δ, δ] and variance Ee () δ /1. Le z x 1 x, we consider an opimal conrol problem wih cos J(u 1, u ) E z + u 1(1 + e) + u E z + (u 1, u ) 1 + δ 1 ( ) u1 1 u }{{} R and dynamics ( ) z + u1 z + (11) + u }{{} u 1 e. B The feedback law is linear and given by K (R + B T P B + Ω(P )) 1 B T P where P is he soluion of a generalized Riccai difference equaion and Ω(P ) is he following marix Ω(P ) 1 + δ 1, see (Beghi and D Alessandro, 1998). The opimal feedback is k 1 (δ) k + aδ + O(δ 4 ) k (δ) k + bδ + O(δ 4 ) δ

4 where k is he same value compued solving he deerminisic opimal conrol problem and a, b are wo real values an order of magniude less han k. If we plo k 1 (δ) and k (δ) we observe ha for large value of δ he gain k 1 (δ) ends o zero, see Figure 1, meaning ha he firs vehicle, whose inpu depends on he logarihmically quanized value of he relaive disance x 1 x, does no move since x 1 x is known wih very large error. On he oher hand he second vehicle, which rely on a perfec knowledge of he posiion of he firs, applies a large inpu in order o move owards i (see Figure 1). The value of he gain for δ 1 is k (1) which he opimal gain saic gain when we have a single vehicle. 3.3 Logarihmic-logarihmic quanizaion An ineresing communicaion configuraion is he one where he daa is exchanged over logarihmic quanized channel. We assume also here ha he uniform quanizaion can be negleced. For his communicaion configuraion we can show ha for a paricular choice of k in he inerval k 1 he wo vehicles do no rendezvous in average. Also in his case we model he logarihmic quanizaion as a muliplicaive noise. Proposiion 6. Consider feedback laws u 1 kq l (q l (x 1 ) x ) u k(x q l (x 1 )) wih k 1/, hen he wo vehicle do no rendezvous in average. Proof. Modeling he quanizaion as muliplicaive noise we have ( ) ( ) x + 1/ 1/ 1/ x + e 1/ 1/ 1 x 1/ ( ) ( ) 1/ 1/ 1/ + e x + e 1 e x where we assume ha e 1 and e are random variables uniformly disribued and independen. Le Q() E ( x()x T () ) ( ) Q1 () Q () Q () Q 3 () hen we can sudy he sabiliy properies of P ( + 1) A(λ)P () (8) ( ) E(z 1 ) (1 1)P () (1 1) P () 1 }{{} C (9) where P () (Q 1 (), Q (), Q 3 ()) and where we assume E(e 1) E(e ) λ. If we consider q(z) de ( zi A(λ) ) and we sudy he polynomial p(s) q ( (1 + s)/(1 s) ) we find ou wih Rouh-Hurwiz ha he polynomial p(s), and hus q(z), is no sable. Since he sysem (9) is observable, hen we can conclude ha sysem (9) is an unsable sysem, meaning ha he variance increases, and hus he wo vehicles are likely o no rendezvous. 4. N-VEHICLES RENDEZVOUS For any pair of vehicles (i, j), i < j we define an oupu variable w i,j x i x j. Le w be a vecor collecing all subse of such oupu variables. Similar o he wo vehicles case we consider an opimal conrol problem wih cos funcion J(u) w T w + u T u 1 x T W T W x + u T u. (1) 1 Noe he marix W T W is singular, i is anyway possible o regularize he problem. Consider any subse z {z i,j } of w. Le Z R (n 1) n such ha z Zx. Then here exiss a marix L such ha ( ) I w z. L Minimizing he cos funcion (1) subjec o he dynamics (1) is equivalen o minimizing he cos J(u) z T (I + L T L)z + u T u (11) 1 subjec o dynamics z + z + Zu. The opimal conrol law for he sysem (1) is such ha u KZx wih 3 KZ k(ni 1), k n + n + 4n n( + n + n + 4n). (1) The marix W can be inerpreed as he inciden marix of a complee digraph G (V, E), wih card(v ) n, where edges beween verices represen communicaion beween vehicles. The marix Z is he inciden marix of a direced ree in he graph G. Each pair of edges (i, j) and (j, i), i < j represens a quanized communicaion channel beween vehicle i and vehicle j. 3 The marix 1 represen he n n uni marix.

5 PSfrag replacemens c 1 c c Topology 1 Topology Topology 3 Fig.. Three differen communicaion opologies for n 3. Solid lines denoe uniformly quanized communicaion channels, and dashed lines logarihmically quanized communicaion channels. 4.1 Uniform quanizaion Proposiion 7. The feedback u i k(n 1)x i + k n j1 j i q u (x j ), wih k as in (1), solves he rendezvous problem. Proof. Similar o he proof of Proposiion Uniform-logarihm quanizaion Since he logarihmic quanized channels are more efficien, because less bis need o be ransmied compared wih uniform quanized channels, we would like o have a communicaion opology wih as many link as possible. Le n 3, we consider opologies where he digraph G has a ree represening uniformly quanized channels and he remaining edges represening logarihmically quanized channels. Since we have ree vehicles he number of possible direced rees, up o a re-labeling of he verices are hree as shown in Figure. Proposiion 8. Le us define he following vecors corresponding o he hree differen opologies shown in Figure, c 1 1 ( q l (q u (x 1 ) x 3 ), q l (q u (x ) x 3 ) ) c 1 ( q l (q u (x 1 ) x 3 ), q l (q u (x ) x 3 ) ) c 1 3 ( q u (x 1 ), q u (x ) ) c 1 ( q l (q u (x 1 ) x ), q l (q u (x 1 ) x 3 ) ) c ( q u (x 1 ) x ), q l (q u (x 1 ) x 3 ) ) c 3 ( q u (x 1 ), q l (q u (x 1 ) x ) ) c 3 1 ( q l (q u (x 1 ) x ), q l (q u (x ) x 3 ) ) c 3 ( q u (x 1 ), q l (q u (x ) x 3 ) ) c 3 3 ( q l (q u (x 1 ) x ), q u (x ) ). The feedback conrol laws u j i () kj i,1 cj i,1 kj i, cj i, i 1,..., 3 j 1,..., 3 (13) wih [k j i,1, kj i,1 ], for i 1,, 3 and j 1,, 3, ih row of he marix K defined in (1) wih n 3, solve he rendezvous problem. Proof. Assume he communicaion opology is he firs one (see Figure ). We prove he saemen for such case, he oher can be proved in a similar way. In his case K k We inroduce he Lyapunov funcion V(z i,j ) z i,j, wih z i,j {z 1,3, z,3 } Le V(z i,j ) V(z + i,j ) V(z i,j), hen we have V(z 1,3 ) 3k z 1,3 + kδ + kδ z 1,3 + kδ + kδ z,3 + 3kδ V(z,3 ) 3k z,3 + kδ + kδ z,3 + kδ + kδ z 1,3 + 3kδ. Thus we have V(z 1,3 ) < and V(z,3 ) < if z 1,3 > 3δ(1 + δ) + δ z,3 3 δ z,3 > 3δ(1 + δ) + δ z 1,3. 3 δ For any ɛ > here exiss δ 1/ ɛ/ ɛ + ɛ / such ha z 1,3 and z,3 asympoically converges o he ball B ɛ. Thus he feedback (13) solve he rendezvous problem. 5. SIMULATION RESULTS A simulaion sudy have been done for he hreevehicle case. In Figure 3 wih are shown he rajecories of he hree vehicles for he hree differen communicaion opologies of Figure. In doed line is shown he rajecory of he vehicles when he communicaion is wihou quanizaion (perfec). In solid line is shown he rajecories of he vehicles communicaing using opology 1, in dashed line he rajecories when opology is used and in dashed-doed he rajecories when opology 3 is used. We can noice he hree vehicles rendezvous (a he poin marked wih a

6 Trajecories of he hree vehiles on he plane x 1 x Perfec communicaion Topology 1 Topology Topology x x 3 Perfec communicaion Topology 1 Topology Topology Fig. 3. Trajecories for hree vehicles for he hree differen opologies of figure. In he simulaions we assumed he uniform quanizaion error equal o zero. square, diamond and sar), bu rajecories are very differen depending on he opology. If we consider he ime evoluion of z 1, x 1 x and z,3 x x 3 and shown in Figure (4(a)) (in Figure (4(b)) are shown similar ime evoluions for he relaive disances in he y-coordinae), we can noice ha vehicles communicaing using he opology 1 (cf., Figure ) rendezvous slowly han when communicaing using he oher wo opologies. This behavior can be explained considering ha he hird vehicle knows wih higher accuracy he posiion of he wo vehicles V 1, V while hese wo las have a very rough informaion of heir relaive disance o he vehicle V 3, due o he logarihmically quanized communicaion. This resuls in slower performance, compare o he remaining opologies. 6. CONCLUSIONS In his paper we have considered he mulivehicle rendezvous problem under quanized communicaion opologies. In paricular resuls have been derived for wo and hree vehicles sysems for differen opologies and various configuraions of uniform and logarihmic quanizers. Some simulaion resuls showing he behavior of he differen opologies have been sudied in order o verify he resuls. The rajecories followed by he vehicles seem o depend upon he communicaion opology used. REFERENCES Beghi, A. and D. D Alessandro (1998). Discreeime opimal conrol wih conrol-dependen noise and generalized Riccai difference equaion. Auomaica 34(8), (a) Differences x 1 () x () and x () x 3 () for he hree vehicles corresponding o he rajecories of Figure 3. 5 y 1 y y y 3 Perfec communicaion Case 1 Case Case 3 Perfec communicaion Case 1 Case Case (b) Differences y 1 () y () and y () y 3 () for he hree vehicles corresponding o he rajecories of Figure 3. Fig. 4. Performance comparison of he difference communicaion opologies. de Sousa, J. B. and F. L. Pereira (). On coordinaed conrol sraegies for neworked dynamic conrol sysems: an applicaion o AUV s. In: Proc. MTNS. Fagnani, F. and S. Zampieri (3). Sabiliy analysis and synhesis for scalar linear sysems wih quanized feedback. IEEE Trans. Auoma. Conrol 48(9), Lin, J., A. S. Morse and B. D. O. Anderson (3). The muli-agen rendezvous problem. In: Proc. IEEE CDC. Saber, R. O. and R. M. Murray (3). Flocking wih obsacle avoidance: cooperaion wih limied communicaion in mobile neworks.. In: Proc. IEEE CDC. Speranzon, A. and K. H. Johansson (3). On some communicaion schemes for disribued pursui evasion games. In: Proc. IEEE CDC. Varaiya, P. (1993). Smar cars on smar roads: Problems of conrol. IEEE Trans. Auoma. Conrol 38(),