Cosesus Cotrol for a Class of Hgh Order System va Sldg Mode Cotrol Chagb L, Y He, ad Aguo Wu School of Electrcal ad Automato Egeerg, Taj Uversty, Taj, Cha, 300072 Abstract. I ths paper, cosesus problem for a class of hgh order mult-aget systems s studed by usg the so called sldg mode cotrol techque. For drected coected mult-aget systems, both the fxed topologes ad swtchg topologes are cosdered ths paper. By usg the sldg mode cotrol techque, a sldg mode surface s desged, each aget are coupled accordg to the prescrbed sldg surface so that they ca reach the same sldg surface fte tme ad therefore coverge to ther cosesus value. Ths cosesus cotrol techque ca be very easly used the cosesus protocol desg. Keywords: cosesus cotrol, hgh order, mult-aget systems, tme-varyg topologes. Itroducto I recet years, may researchers have draw great atteto to the cosesus cotrol of mult-aget systems. The cosesus problem has may applcatos our socety such as team formato, flockg, redezvous ad so o[-9]. There are plety results of the cosesus problems especally for the frst-order ad secod order mult-aget systems. Ufortuately, due to de dffcult exst the ukow coecto topologes ad hgh order eve olear dyamcs brg by the mult-aget systems, there are stll may challege works to do for the hgh order mult-aget systems. I [5 6], cosesus problems are cosdered for hgh order mult-aget systems wth drected ad fxed topologes. If the coecto betwee mult-aget systems s udrected or drected but balaced, [7] gave a cosesus cotrol approach for passve systems. Ths approach s also effectve f the balaced topologes are tme varyg. Recetly, by usg the compare theorem, Zhhua qu[9] gave cooperatve cotrol strategy for hgh order mult-aget systems. Cooperatvely stablty ad Lyapuov stablty ca be guarateed f the commucato topology s uformly sequetally complete over tme. O the other had, sldg mode cotrol s a smple ad effectve techque for the cotrol of olear systems. Our am of ths paper s to desg a cosesus protocol for mult-aget systems. By usg the sldg mode cotrol techque, we studed cosesus problem for a class of hgh order mult-aget systems. The commucato topology cosdered ths paper s drected, ad t may ot be balaced. Both the G. Lee (Ed.): Advaces Automato ad Robotcs, Vol., LEE 22, pp. 207 23. sprgerlk.com Sprger-Verlag Berl Hedelberg 20
208 C. L, Y. He, ad A. Wu fxed topologes ad swtchg topologes are cosdered. A sldg mode surface s desged, each aget are coupled accordg to the prescrbed sldg surface. We proved that f the mult-aget s coected, the all the agets wll reach to the same sldg surface ad therefore coverge to ther cosesus value. 2 Prelmary I ths secto, we preset some otatos ad some prelmary results of algebrac graph theory ad mult-aget system. A. Graph Theory Let () t = (, (), t ()) t be a weghted drected graph of order, where = { v,, v } s the sets of odes, () t { ( v, vj) v, vj } s the sets of edges, ad () t = aj () t s a weghted adjacecy matrx, defed as a () t = 0 ad aj () t 0, where aj () t > 0 f ad oly f ( v, vj) ( t). We assume that () t s pecewse cotuous ad bouded. The weghted drected graph () t s coected at tme t f there exsts a ode v such that all the other odes of the graph are coected to v va a path that follows the drecto of the edges of the drected graph. A ode v s sad to be coected to ode [, ] v j the teral I = ta tb f there s a path from v to v j whch respects the oretato of the edges for the drected graph ( v, t I( t), ( τ) dτ ). A weghted drected graph I () t s sad to be uformly coected f there exsts a tme horzo T > 0 ad a dex such that for all the odes vj ( j ) are coected to the ode v across tt, + T. If a drected graph () t s topology s tme varat, the t s sad to be [ ] coected f ad oly f t has a spag tree. A spag tree of a drected graph s a coected drected tree wth the same vertex set but wth a edge set whch a subset of the edge set of the drected graph. B. Mult-aget Systems Cosder agets, teractg over a etwork whose topology s gve by a graph () t = (, (), t ()) t. The dyamcs of agets are gve by () j y = A () t F( y y ) j
Cosesus Cotrol for a Class of Hgh Order System va Sldg Mode Cotrol 209 C fucto k : σ > y σ σ apart from Defto : A s locally passve, f there exst 0 ad σ + + > 0 such that yf( y ) > 0 for all, y = 0, where F (0) = 0. Defto 2: Cosder the mult-aget system (). If for ay tal state y 0, the states of agets satsfy lm j y ( t) y ( t) = 0 t for all, j, the we say the mult-aget system solves a cosesus problem asymptotcally. Furthermore, f there exsts y such that lm y ( t) y = 0 t for all, the we call y as the cosesus state of the mult-aget system. ow, we study the cosesus problem of mult-aget system (). Frst, we cosder a tme varat topology case,.e. the mult-aget system () has a fxed topology. The, we have the followg theorem. Proposto (Fxed Topologes): Cosder a drected etwork of agets uder dyamcs equato () wth fxed topologes ad F() s local passve. The, the system globally coverges to the equlbrum pot (ts cosesus value) f, ad oly f the weghted drected graph s coected (has a spag tree). Moreover, covergece occurs fte-tme f there exsts a costat ε > 0 such that ( ) ε, 0. yf y y Proof: Defe a Lyapuov fucto caddate as ( ) max( k V y = y ) m( y ) = y y (2) k ote that the Lyapuov fucto s ot smooth ad dfferetable, but t stll ca be used to coclude the attractvty propertes of equlbra. The rght had sde D dervatve of V( y ) of the system ca be defed as ( ) = lmsup ( ( + )) ( ( )) max m [ ] V y V y t r V y t + r 0 r v s oe of the aget wth the maxmum value at tme t, the the D Suppose dervatve of the caddate Lyapuov fucto alog () s V ( y) = y y max m j j max j = A ( t) F( y y ) A ( t) F( y y ) jk m
20 C. L, Y. He, ad A. Wu Sce Aj () t > 0 ad Ajk () t > 0, we have V ( y) = ymax y m 0 mmedately. ote that V ( y) = ymax y m = 0 holds f ad oly f all the paret odes of v ad v k hold both the maxmum value ad the mmum value, also so do ther respectve parets. ow we prove the fte tme stablty of system (). Let m{ A } deote the mmum postve elemet of A j, the we have V ( y) = ymax y m < m{ A} ε holds all the tme except all paret odes ad ther respectve parets hold both the maxmum value ad the mmum value, form the fte-tme stablty preseted [2], we ca kow the cosesus reached fte tme. Remark : The fte tme cosesus protocol preseted Proposto s a dscotuous cosesus strategy. Actually, cotuous but o-smooth cosesus strategy such as F() = sg( y j y ) y j y α, 0< α < s also vald to guaratee fte tme cosesus. ext, we cosder a swtchg topologes case for mult-aget system (). Proposto 2 (Swtchg Topologes): Cosder a etwork wth a tme varyg topology () t = (, (), t ()) t. If the swtchg topology s uformly coected ad F() s local passve, the the system globally coverges to the equlbrum pot (ts cosesus value) f, ad oly f the weghted drected graph has a spag tree. Moreover, covergece occurs fte-tme f there exsts a ε > 0 such that yf( y) ε, y 0. Proof: Cosder Lyapuov fucto Eq. (2) for the swtchg topology case. Although the set of eghbors ad the drected weght matrx chage wth tme, accordg to chage the topology, every tme terval [ tt, + T] or ts subdoma, the etwork s coected. Therefore, there s at least oe y max or y m chaged accordg to ts eghbors status a sub-doma of [ tt, + T] (ot ecessarly the whole tme terval). As a result, V ( y) = ymax y m s decreasg alog the sub-doma (or the whole tme terval[ tt, + T] ) ad cosequetly V ( y) = ymax y m 0, whch meas the Lyapuov fucto Eq. (2) vashes ad the system acheves covergece. V ( y) = y y < m A ε every tme terval Smlarly, f we have max m { } [ tt, T] +, the cosesus reached fte tme. Ths completes the proof.
Cosesus Cotrol for a Class of Hgh Order System va Sldg Mode Cotrol 2 3 Ma Theoretcal Results I ths secto, we gve the cosesus cotrol strategy of a class of hgh order multaget systems. Cosder a class of detcal lear dyamc systems, whch ca be of ay order, are gve as where 2 u x = x2 x 2 = x3 x = f( x, t) + u + d ( x, t) x = ( x, x,, x ) R represets the state vector of the -th ode; R s the cotrol put. We assume the cosesus state vector to be x. Defe the cosesus error state of aget v as e = x x. The cosesus cotrol objectve s to acheve e = 0, =,,. The frst step s to desg a sldg surface fucto. Let the sldg- surface fucto be selected as T S = C e (4) C = [ c, c,, c ] T. If a approprate cosesus cotrol law s avalable, where 2 the the sldg mode wll be obtaed fte tme. I the sldg mode, the error dyamcs of ode v wll be ce c e ce + + + = 0. Those c, c2,, c are postve costats such that the polyomal φλ ( ) = c λ + c λ + + c s Hurwtz. The choce of C decdes the 2 decayg rate of the cosesus error. The expected cosesus dyamc should be a stable dyamc. The secod step s to desg a cosesus cotrol law such that all agets ca reach ad sustaed to the same sldg mode a fte tme. From (3) ad (4), we have S = C e = ce + c e + + ce = c f x t + u + d x t + c x + + c x T ( (, ) (, )) 2 (3)
22 C. L, Y. He, ad A. Wu Therefore, the cotroller ca be desged as follows: where ad u = f( x, t) c ( c x + + c x ) 0 2 s = j ( j ) u A F S S where () The, we have the followg theorem. u = u0 + u s (5) F s a local passve fucto ad ε > 0, yf( y) ε, y 0 Theorem : Cosder mult-aget system (3) wth drected ad fxed topologes. Gve the sldg fucto (4) ad cotrol law (5), the sldg mode s guarateed to be reached fte tme f ad oly f the graph s coected. Therefore, the multaget system wll reach cosesus by cosesus protocol (5). Proof: From (3), (4) ad (5), we have j S = A () t F( S S ) j = AFS ( S) j j ote that F() s local passve ad ε > 0, yf( y) ε, y 0 From proposto, t easly to obta that all S for all =,, wll coverge to ther cosesus value S fte tme. After that, all agets wll reach to the same sldg mode ad coverge to ther cosesus state. Ths completes the proof. Remark 2: It s worth otg that the passve fucto F() has plety of choces. j j F( ) = k ( S S ) + k sg( S S ) ca be chose as a caddate For example, 2 fucto. Remark 3: The fal cosesus value of system (3) over protocol (5) s ukow prevously ad ca be determed by may factors. The tal value of all agets, the sldg surface selected, the commucato topologes wll all affect the fal cosesus value. The fal cosesus value of the mult-aget systems wll be x = ( x,0,,0). I the followg, we cosder the drected topology s chagg over tme. Smlarly to Proposto 2, we have the followg theorem for drected mult-aget systems wth swtchg topologes. Theorem 2: Cosder mult-aget system (3) wth drected ad swtchg topologes. Gve the sldg fucto (4) ad cotrol law (5), the sldg mode s guarateed to
Cosesus Cotrol for a Class of Hgh Order System va Sldg Mode Cotrol 23 be reached fte tme f ad oly f the graph s coected. Therefore, the multaget system wll reach cosesus by cosesus protocol (5). Proof: The proof s smlar to Proposto 2 ad s omtted here. 4 Coclusos Ths paper s cocered wth the cosesus cotrol of a class of hgh order olear system. Both the fxed drected topologes ad swtchg drected topologes are cosdered. By usg the so called sldg mode cotrol techque, we proposed a ew cosesus cotrol protocol whch pledged all agets ca reach to the same cosesus value uder fxed ad swtchg drected topologes. The fal cosesus value s determed by both the tal state of the mult-aget system ad the pre-desged sldg surface. Our future objectve s to desg a cosesus dyamc for hgh order mult-aget systems over applcable partal commucato topologes. Refereces. Papachrstodoulou, A., Jadbabae, A., Müz, U.: Effects of delay mult-aget cosesus ad oscllator sychrozato. IEEE Tras. Autom. Cotrol 55(6), 47 477 (200) 2. Hu, Q., Haddad, W.M., Bhat, S.P.: Fte-tme semstablty, Flppov systems, ad cosesus protocols for olear dyamcal etworks wth swtchg topologes. olear Aalyss: Hybrd Systems 4(3), 557 573 (200) 3. L, P., Ja, Y.: Cosesus of a Class of Secod-Order Mult-Aget Systems Wth Tme- Delay ad Jotly-Coected Topologes. IEEE Tras. Autom. Cotrol 55(3), 778 784 (200) 4. Scardov, L., Sepulchre, R.: Sychrozato etworks of Idetcal Lear Systems. Automatc 45, 2557 2562 (2009) 5. L, Z., Dua, Z., Che, G., Huag, L.: Cosesus of Multaget Systems ad Sychrozato of Complex etworks: A Ufed Vewpot. IEEE Tras. Crc. Syst. I 57(), 23 224 (200) 6. Seo, J.H., Shm, H., Back, J.: Cosesus of hgh-order lear systems usg dyamc output feedback compesator: Low ga approach. Automatca 45, 2659 2664 (2009) 7. Chopra,., Spog, M.W., Lozao, R.: Sychrozato of blateral teleoperators wth tme delay. Automatca 44(8), 242 248 (2008) 8. Jag, F., Wag, L.: Cosesus seekg of hgh-order dyamc mult-aget systems wth fxed ad swtchg topologes. Iteratoal Joural of Cotrol - It. J. Cotr. 83(2), 404 420 (200) 9. Qu, Z.: Cooperatve Cotrol of Dyamcal Systems. Sprger (2009) 0. Utk, V.I.: Sldg modes cotrol ad optmzato. I: CCES. Sprger, ew York (992). Huag, J.Y., Gao, W., Huag, J.C.: Varable structure cotrol: A survey. IEEE Tras. Id. Elect. 40, 2 22 (993) 2. Huag, Y.-J., Kuo, T.-C., Chag, S.-H.: Adaptve Sldg-Mode Cotrol for o-lear Systems Wth Ucerta Parameters. IEEE Trasactos o Systems 38(2), 534 539 (2008)