Accuracy of Fridman s estimates for sampling interval: nonlinear system case study

Transcription

1 Proceedings of h9h World Congress he Inernaional Federaion of Auomaic Conrol Cape own, Souh Africa. Augus 4-9, 14 Accuracy of Fridman s esimaes for sampling inerval: nonlinear sysem case sudy Egor V. Usik, Ruslan E. Seifullaev, Alexander L. Fradkov, aiana A. Brynseva Deparmen of heoreical Cyberneics, Sain-Peersburg Sae Universiy, Sain-Peersburg, Russia ( egor.usik@gmail.com) Deparmen of heoreical Cyberneics, Sain-Peersburg Sae Universiy, Sain-Peersburg, Russia ( ruslanspb-zeni@yandex.ru) Deparmen of heoreical Cyberneics, Sain-Peersburg Sae Universiy, Sain-Peersburg, Russia and Insiue for Problems in Mechanical Engineering, S.Peersburg, Russia ( fradkov@mail.ru) Deparmen of heoreical Cyberneics, Sain-Peersburg Sae Universiy, Sain-Peersburg, Russia ( brynseva@bk.ru) Absrac: An aemp o evaluae accuracy of Fridman s sampling inerval esimaes for nonlinear discree-coninuous sysems where he conrolled plan belongs o a class of cascade passifiable Lurie sysems. Numerical resuls obained for maser-slave configuraion of wo mobile robos demonsrae good accuracy of Fridman s esimaes: error of he sampling inerval esimae is less han 5% of he value obained from exensive simulaion. In conras, he error obained by convenional mehod from quadraic Lyapunov funcion is more han 75% of he value obained from simulaion. Keywords: Sabiliy of nonlinear sysems, Passiviy-based conrol, Neworked sysems, LMI 1. INRODUCION Recenly here was observed a srong ineres in an approach o he sampling ime evaluaion based on ransformaion of discree-coninuous sysem models o coninuous delayed sysem wih ime-varying (seesaw) delay. Efficiency of an approach has increased since i was combined wih he descripor mehod of delayed sysems analysis proposed by Emilia Fridman in 1 (Fridman (1)). he idea has become equipped wih a powerful calculaion ools based on LMI and has become a powerful design mehod allowing one o esimae maximum sampling inerval providing sabiliy of he closed loop sysem. I allows designer o seriously reduce conservaiveness of he sampling inerval esimaes (Fridman e al. (4); Fridman (1)). However he Fridman s mehod was previously developed for only o linear sysems. I was exended o a class of nonlinear Lurie sysem jus recenly Seifullaev and Fradkov (13). However conservaiveness of he esimaes has no been evaluaed. In his paper an aemp o evaluae accuracy of Fridman s esimaes for a class of nonlinear sysems is made. he problem of measuring conservaiveness of Fridman s mehod for nonlinear sysem is in ha here are no igh bounds for sampling inerval for general nonlinear sysems ha could be used o compare wih Fridman s esimaes in order o evaluae heir accuracy. herefore in his paper a class of cascade nonlinear sysems is chosen (namely, passifiable sysems Andrievsky and Fradkov (6); Fradkov e al. (1999); Polushin e al. (6)) for which convenional ype bounds for sampling inerval can be evaluaed efficienly by means of quadraic Lyapunov funcions. he conrol is chosen in such a way ha he derivaive of he Lyapunov funcion for he sysem wih he inegraor should be sricly negaive for nonzero values of he sysem sae vecor, and so by he Lyapunov heorem he asympoic sabiliy of he enire model follows. For he maserslave sysem where he param- eers are nonidenical, he robus synchronizaion problems have also been considered Ji e al. (1); Balasubramaniam and heesar (14); Ji e al. (14). Such analyic bounds were evaluaed in Usik (1). In his paper he bounds of Usik (1) are improved and used o calculae he sampling inerval bounds numerically for an example sysem (neworks of hree mobile robos). An alernaive esimae of sampling inerval for mobile robos is made by a nonlinear version of Fridman s mehod developed in Seifullaev and Fradkov (13). he obained numerical resuls are compared. In Secion II and III he resuls of Usik (1) and Seifullaev and Fradkov (13), correspondingly are briefly exposed for compleeness. In Secion IV convenional and Friedman s esimaes for example nonlinear sysem are evaluaed numerically and compared /14 IFAC 11165

2 19h IFAC World Congress Cape own, Souh Africa. Augus 4-9, 14. CONVENIONAL ESIMAES OF SAMPLING INERVAL FOR CASCADE PASSIFIABLE SYSEMS Consider n cascade dynamical sysems of he Lurie ype wih nonlinear inpu cascades z i = Az i + Bϕ(y i ) + Bu i + (1) + α ij ϕ ij (z i z j ), () u i = ψ(u i, ) + w i, y i = Cz i, (3) where ϕ ij (x), i = 1,..., n, j = 1,..., d funcions ha describe he relaionship beween he sysems, α ij R 1. Also consider he maser sysem: z = Az + Bϕ(y ), y = Cz. (4) where z i, z are he n-dimensional vecors of he objec sae. y i, y are scalar oupus. A is he n n marix, B is he n 1 marix, C is h n marix, ϕ(y), ψ(u, ) are he coninuous nonlineariies lying in he secor. Our goal is o achieve zero asympoic sae error: z i z where, i = 1,..., n..1 Evaluaion of conrol Le us inroduce sae synchronizaion error e = z i z, and oupu synchronizaion error σ i = y i y = Ce i. And he error sysem e i = Ae i + Bξ(σ i, ) Bu i + (5) + α ij ϕ ij (e i e j ), (6) σ i = Ce i, (7) u i = ψ(u i, ) + KCAe i + KCBξ(σ i, ) + v i. (8) where ξ(σ i, ) = ϕ(σ i + y ) ϕ(y ) is a new nonlineariy and v i = ( γ KCB)u i + γkσ i. he conrol goal will ake he form lim e i =, For synhesis of he conrol we will used he mehod of backsepping Fradkov e al. (1999).. Condiions of Passificaion and Asympoic Sabilizaion o obain he condiions for achieving he goal, he following assumpions are made: (1) linear sysem ė = Ae Bu, σ = Ce is he hyperminimum-phase, i.e., he marix funcion Γ(λ) = lim λ λw (λ) is nondegenerae and posiive definie Fradkov e al. (1999), where W (λ) = C(λI A) 1 B = β(λ)/α(λ) is he ransfer funcion of he sysem. For he case wih he scalar oupu, his means: he degree of he denominaor α(λ) is equal o n. he numeraor β(λ) is of he Hurwiz degree n 1 wih posiive coefficiens; () ξ(σ, ) lies in he secor, i.e., aσ ξ(σ, )σ bσ, where a, b are he secor parameers; (3) ψ(u, ) also lies in he secor, i.e., cu ψ(u, )u du, where c, d are he secor parameers; (4) funcions ϕ ij (x), i = 1,..., n, j = 1,..., d are Lipschiz: ϕ ij (x) : ϕ ij (x) ϕ ij (x ) L ij x x, L ij >. From he hyperminimum-phase propery and he passificaion heorem Andrievsky and Fradkov (6) i follows ha he minimum disance η beween he roos of he numeraor of a ransfer funcion and he imaginary axis will be posiive. We will selec he parameers η and K in such a way ha < η < η, D P C ( max( a, ) b ) + P max( c, d ) < ηλ min, where D B =, P is KCB he posiive definie marix in he Lyapunov quadraic funcion V (x) = x P x, λ min is he leas eigenvalue of he symmeric marix P. he following resul holds Usik (1): heorem 1. Le he assumpions (1)-(4) be fulfilled and he inequaliy η i λ min (P i ) + D λ max (P i ) max( a, b ) C + + λ max (P i ) max( c, d )+ + λ max (P i ) ( L ij α ij + L ji α ji ) <, i = 1,..., n, (9) ( ) holds where D B =, P is he posiive definie KCB marix in he Lyapunov quadraic funcion V (x) = x P x, λ min, λ max are he leas and he larges eigenvalues of he given marix. hen here exis numbers K, γ, such ha he sysem (), (3) will be passive wih he quadraic sorage funcion, while he closed sysem wih he conrol v i = ( γ KCB)u i + γkσ i will be asympoically sable..3 he Discree Conroller and Condiions of Exponenial Synchronizaion Consider he discree conroller v i = ( γ KCB)u i ( k )+ γkσ i ( k ), k k+1, where k = kh are he insans of ime wih he discreizaion sep h. heorem. Consider he sysem (7) - (8) wih he discree-ime conroller. And he inequaliy η i λ min (P i ) + D λ max (P i ) max( a, b ) C + + λ max (P i ) max( c, d )+ + λ max (P i ) ( L ij α ij + L ji α ji ) <, i = 1,..., n, (1) ( ) holds where D B =, P is he posiive definie KCB marix in he Lyapunov quadraic funcion V (x) = x P x, λ min, λ max are he leas and he larges eigenvalues of he given marix. Selec he discreizaion sep saisfying he inequaliies: C κ i K e L Gh + L G e ηh C κ i K + L G, (11) for i = 1..n, where κ i is he coefficien of he esimae of he sysem oupu in erms of he Lyapunov funcion: 11166

3 19h IFAC World Congress Cape own, Souh Africa. Augus 4-9, 14 σ i κ i V, LG is he Lipschiz consan of he righ side of he sysem (7) - (8). hen he sysem under consideraion is exponenially sable, i.e. he synchronizaion error exponenially ends o zero. Proof. he iniial sysem (7) - (8) can be represened in he form ẋ i = Ãx i + Bv i + Bψ(u i, ) + Dξ(σ i, ), (1) σ i = Cx i, v i = K σ i ( k ). (13) Le s define G i (x, ) = Ãx+ Bv+ Bψ(u, )+ Dξ(σ i, ). his is a Lipschiz funcion wih consan L Gi : G i (x 1, ) G i (x, ) ( Ã + B K C + D max( a, b ) C + B max( c, d ) + n ( α ijl ij + α ji L ji )) x 1 x = L Gi x 1 x. he conroller v i represened in he following form v i = K σ i Kδ i, where δ i = σ i σ i ( k ) is he discreizaion error. he δ i saisfies he inequaliy δ i = C k G i (x, ) d ( k ) CG i (x k, k ) + k L Gi δ i d. We obain he esimaion on δ i by applying Gronwall s inequaliy: δ i ( k+1 ) CG i (x k, k ) el G i h 1 L Gi. (14) Denoe C h = el G h 1 L G. hen δ i ( k+1 ) V ik C h. For he sysem (7) - (8) we choose a Lyapunov funcion V = n x i P ix i. Calculae he derivaive of V (x). V < ηv e i P B Kδ i ηv + σ i K δ i. (15) Denoe γ i = σ i K δ i. hus inequaliy (15) can be rewrien as V η 1 V + n γ i. Fixed i: V i ηv i +γ i. Inegraing i on he inerval ( k, k+1 ) and considering σ i ( k ) κ i Vik, we obain he following inequaliy: V k+1 e η1h V k + γ k+1 η 1 γ k η 1 e η1h e η1h V k + C h K 1 Vk+1 V k. (16) Le s find he condiions when he inequaliy holds V k+1 V k. Denoe x = Vk+1 V k, α = exp ηh, β = C h K 1. In his noaion, he inequaliy (15) can be rewrien as x α 1 x + β. If β <, α < 1 β hen inequaliies x 1 and x α 1 x + β will be me. I is clear ha ha his is a condiion on he sampling sep h (11) formulaed in he heorem. herefore, x( k ) exponenially. We wrie esimaion of he error rae δ: x x( k ) C h x( k ), which is equivalen he following inequaliy x (C h + 1) x( k ). his implies he exponenial sabiliy of soluions of x, ( k, k+1 ). 3. SAMPLING INERVAL ESIMAION BASED ON FRIDMAN S MEHOD AND LMI Le us describe an alernaive approach o esimaion of he sampling inerval based on he resuls proposed in Seifullaev and Fradkov (13). Consider a nonlinear sysem N ẋ = Ax + q i ξ i + Bu, σ i = ri x, ξ i = ϕ i (σ i, ), i = 1,..., N, (17) where x R n is he sae vecor, u R m is he conrol funcion, A R n n, B R n m are consan marices, q i R n, r i R n are consan vecors. Assume ha ξ i = ϕ i (σ i, ) are nonlinear funcions saisfying µ 1i σ i σ i ξ i µ i σ i, i = 1,..., N (18) for all, where µ 1i < µ i are real numbers. Given a sequence of sampling imes = < 1 <... < k <... and a piecewise consan conrol funcion u = u d ( k ), k < k+1, (19) where lim k =. k Assume ha h R (h > ) and k+1 k h, k () and consider a sampled-ime conrol law u = Kx( k ), k < k+1, (1) where K R m n. he law (1) can be rewrien as follows: u = Kx( τ), () where τ = k, k < k+1. I is required o analyze he influence of he upper bound h of sampling inervals on he closed-loop sysem exponenial sabiliy: N ẋ = Ax + BKx( τ) + q i ξ i, σ i = ri x, ξ i = ϕ i (σ, ), i = 1,..., N, τ = k, [ k, k+1 ). (3) hus, insead of he radiional reducion o discreeime sysem an alernaive mehod was used: he effec of sampling is considered as delay followed by he consrucion and use of Lyapunov-Krasovskii funcional Fridman (1). Wih S-procedure Yakubovich e al. (4) he esimaion of sampling sep is reduced o feasibiliy analysis of linear marix inequaliies (LMI). he following resul is obained based on he resuls obained in Seifullaev and Fradkov (13). heorem 3. Given α >, le here exis marices P R n n (P > ), Q R n n (Q > ), P R n n, P 3 R n n, X R n n, X 1 R n n, R n n, Y 1 R n n, Y R n n, Y (i) 3 R n n (i = 1,..., N), and 11167

4 19h IFAC World Congress Cape own, Souh Africa. Augus 4-9, 14 posiive numbers {κ i } N, {κ 1i} N such ha following LMIs are feasible: P + h X + X Φ + 11 Φ+ 1 Φ+ 13 hx 1 hx hx 1 hx 1 + h X + X Φ 11 Φ 1 Φ 13 (1) Φ Φ (N) 14 Φ Φ 3 Φ (1) 4... Φ (N) 4 Φ 33 Φ (1) Φ (N) 34 Φ (1) Φ (N) 44 (1) Φ Φ + (N) 14 hy1 Φ + Φ+ 3 Φ (1) 4... Φ (N) 4 hy Φ + 33 Φ (1) Φ (N) 34 h Φ + (1) hq1 Y (1) Φ + (N) 44 hqn Y (N) 3 hqe αh where Φ 11 = A P + P A + αp Y 1 Y (1 αh) X + X >, (4) <, (5) 1 N κ i µ 1i µ i r i ri, Φ + 11 = A P + P A + αp Y 1 Y X + N X κ 1i µ 1i µ i r i ri, 1 Φ 1 = P P + A P 3 Y + h X + X, Φ + 1 = P P + A P 3 Y, <, (6) Φ 13 = Y 1 + P BK + (1 αh)(x X 1 ), Φ + 13 = Y 1 + P BK + (X X 1 ), Φ (i) 14 = P + 1 κ i(µ 1i + µ i )r i, Φ + (i) 14 = P + 1 κ 1 i (µ 1i + µ i )r i, Φ = P 3 P 3 + hq, Φ + = P 3 P 3, Φ 3 = Y + P 3 BK h(x X 1 ), Φ + 3 = Y + P 3 BK, Φ (i) 4 = P 3 q i, Φ (i) 34 = Y (i) 3 q i, Φ 33 = + (1 αh) X + X X 1 X 1 Φ + 33 = + X + X X 1 X1, Φ (i) 44 = κ i, Φ + (i) 44 = κ 1i. hen sysem (3) is exponenially sable wih decay rae α., 4. EXAMPLE. HREE MOBILE ROBOS We will compare wo mehods of esimaion of sampling sep by he example of a model nonlinear sysem consising of hree mobile hree-wheeled robos in maser-slave configuraion. Considering ha he robos move a low speed, one can resric he discussion o he kinemaic model of he driving and he driven vehicles. he model can be represened in he following way Laombe (1991): ẋ 1 = v cos(ϕ 1 ), ẏ 1 = v cos(ϕ ), ẋ = v sin(ϕ 1 ), ẏ = v sin(ϕ ), ϕ 1 = ω, ϕ = u, (7) ż 1 = v cos(ϕ 3 ), ż = v sin(ϕ 3 ), ϕ 3 = r, where u, r are he conrol funcions, ω is he fixed angular velociy, v is he fixed linear velociy. he sysem (7) can be represened in he form ẋ 1 = v + v (cos(ϕ 1 ) 1), ẋ = v ϕ 1 + v (sin(ϕ 1 ) ϕ 1 ), ẏ 1 = v + v (cos(ϕ ) 1), ẏ = v ϕ + v (sin(ϕ ) ϕ ), ϕ = u. ż 1 = v + v (cos(ϕ 3 ) 1), ż = v ϕ 3 + v (sin(ϕ 3 ) ϕ 3 ), ϕ 3 = r, (8) hus, a small values of angle ϕ i, i = 1,, 3, he moion along axes x 1, y 1, z 1 can be negleced. Inroduce he following noaion: = x y, e = x z, (9) ε 1 = ϕ 1 ϕ 3, ε = ϕ 1 ϕ, (3) ξ 1 = sin(ϕ 1 ) sin(ϕ ) + ϕ ϕ 1, (31) ξ = sin(ϕ 1 ) sin(ϕ 3 ) + ϕ 3 ϕ 1. (3) Using hese noaion, we can rewrie he sysem model as follows: ė 1 = vε 1 + vξ 1 (ε 1, ), ė = vε + vξ (ε, ) ε 1 = w 1, ε = w, (33) where ξ i (ε i, ) = cos ϕ 1 + ϕ i+1 sin ε i ε i, i = 1,. (34) Denoe α i = cos ϕ1+ϕi+1 and rewrie (34) as follows ξ i (ε, ) = α i sin ε i Nonlineariies (35) saisfy ε i, i = 1,. (35) ε i ε i ξ i, i = 1, (36) for all (see Fig.1). Verify, ha he assumpions of heorem 1 are fulfilled. ransfer funcions of each sysem are equal v/λ. he 11168

5 19h IFAC World Congress Cape own, Souh Africa. Augus 4-9, ε 1 e ε Fig. 1. Secor nonlineariies,,. degree of denominaor is 1, he degree of numeraor is, v >. Indeed, here exiss feedbacks in he form ε i = Ke i, K <, such ha he linear sysem is asympoically sable. Nonlinearies ξ i are in secor, as i noed in (38). Consider he Lyapunov quadraic funcions V i (e i ) = e i H ie i and compare he assumpion 4) wih following inequaliy: V i = e i P v(ε i + ξ i ) (e i ) HvK(1 + max( a, b )). his inequaliy hold wih K <. Using he backsepping mehod, we synhesize he discree conrollers w i = γ(ε i ( k ) Ke( k )) + Kvε i ( k ), i = 1,. (37) Represen sysems (33), (37) in he following way: X i = AX i + Du i + F ξ i (, σ), u = KX i ( k ), [ ] [ ] ei v where X i =, A =, D = ε i (38) [ [ v, F =, 1] ] K = [γk Kv γ], Consider he Lyapunov quadraic funcions V i (X i ) = Xi P ix i. For applying heorem, calculae he parameer η i < : V i = ((A + B K)X i + F ξ i (ε i, )) P X i + + X i P i ((A + B K)X i + F ξ i (ε i, )) (max Re(A + B K) + max(, )v)v i = η i V i. Define he vehicle moion velociy v =.1 m/s. hen (39) η i = (max( γ,.1k) +.) <. (4) Selec γ, K such ha (4) holds:k = 5, γ =.6, hen η i =.6. Evaluae in Malab P i, P i = 6.936, so we can esimae parameer κ i : X i κ i Vi κ P i X i. (41) Assume, ha κ i = 1/ P i =.181. Fig. 7 illusrae he resuls of heorem. For applying heorem 3, represen sysems (33), (37) wih defined parameers in he following way: Ẋ = AX + Du + F 1 ξ 1 (, σ) + F ξ (, σ), u = KX( k ),.1 (4) ε where X = 1 1 e, A =.1, D =, ε 1.1 [ ] F 1 =, F =.1, K =, Fig.. Errors e i and ε i, i = 1, in sysem (33), (37). wih sampling inerval h = Fig. 3. Errors e i and ε i, i = 1, in sysem (33), (37). wih sampling inerval h = Fig. 4. Errors e i and ε i, i = 1, in sysem (33), (37). wih sampling inerval h = 1.5 In abl here are values of maximum upper bound h when (4) is exponenially sable wih a small enough decay rae. heorem heorem 3 Simulaion h =.16 h = 1.3 h = h, 1.8 < h <1.8 abl. Maximum upper bound on he variable sampling In figures () - (6) we can see he sysem (33) for various sampling inerval. 5. CONCLUSIONS An aemp o evaluae accuracy of Fridman s sampling inerval esimaes for a class of nonlinear discree-coninuous ε 1 e ε ε 1 e ε 11169

6 19h IFAC World Congress Cape own, Souh Africa. Augus 4-9, ε 1 e ε ACKNOWLEDGEMENS his work was suppored by Russian Foundaion for Basic Research (projec ), and Russian Federal Program Cadres (agreemens 8846, 8855) Fig. 5. Errors e i and ε i, i = 1, in sysem (33), (37). wih sampling inerval h = Fig. 6. Errors e i and ε i, i = 1, in sysem (33), (37). wih sampling inerval h = 1.8 L(h) h Fig. 7. Funcion L(h) = C κ i K e LGh + L G e ηh C κ i K + L G, for esimaion discreizaion sep h. sysems is made. he proposed approach is applicable o he cascade passifiable Lurie sysems. Numerical resuls obained for maser-slave configuraion of wo mobile robos demonsrae good accuracy of Fridman s esimaes: error of he sampling inerval esimae is less han 5% of he value obained from exensive simulaion. In conras, he error obained by convenional mehod from quadraic Lyapunov funcion is more han 75% of he value obained by simulaion. Fuure sudy will be devoed o evaluaion of accuracy of sampling inerval esimaes for oher classes of nonlinear sysems. Also we will apply our approach o even riggered sysems (Xie e al. (13); Yu and Ansaklis (13)), by inroducing deadband according o he values some riggered funcion ha can be considered as a goal funcion in a version of he speed-gradien mehod. ε 1 e ε REFERENCES Andrievsky, B. and Fradkov, A. (6). Mehod of passificaion in adapive conrol, esimaion, and synchronizaion. Auom. Remoe Conrol, 11, Balasubramaniam, P. and heesar, S.J.S. (14). Secure communicaion via synchronizaion of lure sysems using sampled-daa conroller. Circuis, Sysems, and Signal Processing, 33, Fradkov, A., Miroshnik, I., and Nikiforov, V. (1999). Nonlinear and Adapive Conrol of Complex Sysems. Dordrech, he Neherlands, Kluwer. Fridman, E. (1). New lyapunov-krasovskii funcionals for sabiliy of linear rearded and neural ype sysems. Sysems & Conrol Leers, 43(4), Fridman, E. (1). A refined inpu delay approach o sampled-daa conrol. Auomaica, 46(8), Fridman, E., Seure, A., and Richard, J. (4). Robus sampled-daa sabilizaion of linear sysems: an inpu delay approach. Auomaica, 4(8), Ji, D.H., Park, J.H., Lee, S.M., Koo, J.H., and Won, S.C. (1). Synchronizaion crierion for lure sysems via delayed pd conroller. Journal of Opimizaion heory and Applicaions, 147, Ji, X., Ren, M., and Su, H. (14). On designing ime-delay feedback conroller for maser-slave synchronizaion of lur e sysems. Asian Journal of Conrol, 16, Laombe, J.C. (1991). Robo Moion Planning. Kluwer Academic Publishers, Boson. Polushin, I., Fradkov, A., and Hill, D. (6). Passiviy and passificaion of nonlinear sysem (review). Auom. Remoe Conrol, 3, Seifullaev, R. and Fradkov, A. (13). Sampled-daa conrol of nonlinear oscillaions based on lmis and fridman s mehod. In 5h IFAC Inernaional Workshop on Periodic Conrol Sysems. Caen, France. Usik, E. (1). Synchronizaion of nonlinear lurie sysems on he basis of passificaion and backsepping. Auom. Remoe Conrol, 73, Xie, D., Yuan, D., Lu, J., and Zhang, Y. (13). Consensus conrol of second-order leader-follower muli-agen sysems wih even-riggered sraegy. ransacions of he Insiue of Measuremen and Conrol, 35, Yakubovich, Leonov, and Gelig (4). Sabiliy of saionary ses in conrol sysems wih disconinuous nonlineariies. World Scienific, 37(5), Yu, H. and Ansaklis, P.J. (13). Even-riggered oupu feedback conrol for neworked conrol sysems using passiviy: Achieving l sabiliy in he presence of communicaion delays and signal quanizaion. Auomaica, 49,