Performances and Stability Analysis of Networked Control Systems

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1 Performaces ad Sabiliy Aalysis of Neworked Corol Sysems Yuaqig Xia, Li Zhou, Jie Che, Guopig Liu 2. Beijig Isiue of echology, Beijig 8,Chia Deparme of Egieerig, Uiversiy of Glamorga, Poypridd,CF37 DL,UK Absrac: he problem of performace ad sabiliy aalysis of eworked corol sysems wih radom ework delay ad daa dropou is cosidered i his paper. A ew corol scheme is proposed o overcome he effecs of ework rasmissio delay ad daa dropou, which is ermed eworked predicive corol. wo differe ways o choose corol ipu u are discussed i he paper ad he resuls of performaces ad sabiliy are preseed. Boh real-ime simulaios ad pracical experimes show he effeciveess of he corol scheme. Key Words: Neworked corol sysem, predicive corol, radom ework delay, sabiliy. INRODUCION I rece years, more ad more ework echology has bee applied o corol sysems [],much aeio has bee paid o he sudy of corol desig ad sabiliy aalysis of eworked corol sysem (NCS) [2], [3], [4], he NCS is defied as a feedback corol sysem where corol loops are closed hrough a real-ime ework [5],[6], [7], [8], [9], []. As he srucure of eworked corol sysems is differe from ha of radiio corol sysems, here exis various specific problems i eworked corol sysems, for example, ework delay, loss of daa packes, ework securiy ad safey []. Alhough much research work has bee doe i eworked corol sysems, mos work has igored a very impora feaure of eworked corol sysems. his feaure is ha he commuicaio eworks ca rasmi a packe of daa a he same ime, which is o doe i radiioal corol sysems [2], [3]. his paper makes full use of his ework feaure ad proposes a ew eworked corol scheme - eworked predicive corol, which ca overcome he effecs caused by ework delay. Furhermore, wo ways o choose corol ipu u are discussed i he paper ad he performaces have bee compared ad aalyzed. 2 NEWORKED PREDICIVE CONROL FOR SYSEMS WIH NEWORK DELAY his paper cosiders he case where he sysem coroller his work is suppored by Naioal Naure Sciece Foudaio uder Gra 6542 ad Excelle youg scholars Research Fud of Beijig Isiue of echology 26y3. is far away from he pla bu he sesor is ear o he pla. So, he ework delay i he feedback chael is o cosidered. A eworked predicive corol scheme for NCS wih radom ework delay i he forward chael is proposed. he mai par of he scheme is he eworked predicive coroller, which compesaes for he ework delay i he feedback chael ad achieves he desired corol performace. Cosider a MIMO discree sysem described i he followig sae space form x = Ax + + Bu () y = Cx m l where x R, u R, ad y R are he sae, ipu, ad oupu vecors of he sysem,respecively, * * m l* A R, B R,ad C R he sysem marices. For he simpliciy of sabiliy aalysis, i is assumed ha he referece ipu of he sysem is zero. Also, he followig assumpios are made. Assumpio : he pair, (A,B), is compleely corollable, ad he pair, (A,C) is compleely observable. Assumpio 2: he umber of cosecuive daa dropous is less ha N, where N is a posiive ieger. Assumpio 3: he upper boud of he ework delay is o greaer ha N, where N is a posiive ieger. he sae observer is desiged as xˆ $ $ + = Ax + Bu + L( y Cx ) (2) m where xˆ + R ad u R are he oe-sep ahead sae predicio ad he ipu of he observer a ime, l * respecively. he marix L R ca be desiged usig

2 observer desig approaches. he esimaor of he sae xˆ = x$ $ + M( y Cx ) (3) where $ x R is he sae observer. Followig he sae observer described by (2), based o he oupu daa up o + N, he sae predicios from ime + N o are cosruced as xˆ $ $ + = Ax + Bu + L( y Cx ) xˆ = Ax$ + + Bu M xˆ $ + N = Ax+ N + Bu+ N (4) I paricular, he augmeed sysem wihou ime-delay, i.e.,u=u, ca be described as follows: x+ = ( A+ BKMC) x + ( BK BKMC)ˆ x xˆ ˆ + = ( A+ BK LC BKMC) x + ( BKMC+ LC) x (5) While desigig of he Neworked Predicive coroller, corol ipu u is absoluely ecessarily. here are hree differe ways o ge follows: u,idiographic cases are aalyzed as While desigig of he Neworked Predicive coroller, corol ipu u is absoluely ecessarily. here are wo differe ways o ge u,idiographic cases are aalyzed as follows: Case : Use esimaed value u as he corol ipu, which is show i Figure. u ˆ = u = Kx (6) m * where K R is he sae feedback corol marix o be deermied usig moder corol heory. he, he corol predicios are geeraed by u = Kxˆ, for k =,,2,,N (7) + k + k Figure: he diagram of case hus, i follows from equaio (4) ha ˆ ( ) k x ˆ + k= A+ BK x ˆ (8) + Based o equaios (2), (3) ad (6), i ca be show ha k xˆ + k = ( A+ BK) ( BKMC+ LC) x+ (9) k ( A + BK) ( A + BK LC BKMC)ˆ x ad u = Kxˆ = K A BK A BK LC BKMC x + k K( A + k k BK) ( BKMC LC) x ( + k ) ( + )ˆ () Case 2: he coroller is desiged usig u k as u which is se o he pla direcly, as show i Figure 2: u = u = Kxˆ () k k Figure2: he diagram of case 2 m * where he sae feedback marix K R, K is he ework delay i forward chael. Usig prediced mehod, he oupu of he eworked predicive corol a ime is deermied by k ( ) u = u = K A+ BK ( A LC)ˆ x (2) k k k+ + K( A+ BK) LCx + K( A+ BK) Bu k k k k 3 SABILIY ANALYSIS OF CLOSED NEWORKED PREDICIVE CONROL SYSEMS Case: heorem : For he eworked predicive corol sysems wih radom ework delay i he forward chael, he closed-loop sysem () is sable if here exiss a posiive ( 2 2) ( 2 2) defiie marix P R N+ N+ such ha A ( k) PA( k) P< (3) for k =,, 2, L, N, where A L M( k) L L M2( k) L I L L L L M M L M M K M M M L M M L M M L I L L L L I L L L M M L M M L M M M L M M L M M L L I L L Ak ( ) = M3( k) L L M4( k) L L L L I L L M M L M M L M M M K M M L M M L L L I L L L L I L M M L M M L M M M L M M L M M L L L L I 2( N ) 2( N ) k, 2, L, N, Ak ( ) R + + ( ) k M k = BK ( A + BK ) ( BKMC + LC ), wih { }

3 ( ) k M 2 k = BK( A+ BK) ( A+ BK LC BKMC), M ( ) 3 k = BKMC + LC, M ( ) 4 k = A+ BK LC BKMC. (4) I he case of he closed-loop sysem wihou ime-delay, we ca ge A() as follows: M() L L M2() L L I L L L L M M L M M K M M M L M M L M M L I L L L L I L L L M M L M M L M M M L M M L M M L L I L L A() = M3() L L M4() L L L L I L L M M L M M L M M M K M M L M M L L L I L L L L I L M M L M M L M M M L M M L M M L L L L I Where A() R + + 2( N ) 2( N ), () M = A+ BKMC, M ( ) 2 k = BK BKMC, M ( ) 3 k = BKMC+ LC, M ( ) 4 k = A + BK LC BKMC heorem is he same as heorem 5 i [2]. Case2: he, he mai heorem 2 is similar o [3] which ca be saed as follows: heorem 2: For he eworked predicive corol sysems () wih radom ework delay i he forward chael, he closed-loop sysem is sable if here exiss a posiive ( 2N + mn + 2) ( 2N + mn + 2) defiie marix P R such ha Λ ( k) PΛ( k) P< (5) for k,, 2,, N = L ad Λ ( k) =Λ ( k,) which come from Λ ( k, f ) i (2) wih f =.he proof is similar o ha i [3]. 4 REALIME SIMULAION AND PRACICAL EXPERIMEN Example : o impleme eworked corol sysems, a es rig was buil, based o a ARM9 embedded sysem. he forward chael is hrough a ework ad he commuicaio proocol bewee he coroller ad he sesor is UDP. he kerel chip of he embedded board is AMEL s A9RM92, which is a cos-effecive, high-performace 32-bi RISC microcoroller for Ehere- based embedded sysems. A M/M self-adapive ework coroller is iegraed i he chip ad he chip also has a high compuig performace ad ca work a speeds up o 8 MHz. 2-chael 6-bi high speed digial-aalog (D/A) coverers ad 8-chael 6-bi high speed aalog-digial (A/D) coverers i he coroller board provide I/O ierfaces for he corolled pla. I order o validae he proposed mehod, a servo moor corol sysem which cosiss of a DC moor, peripheral equipme, speed sesors is cosidered. he model of he moor corol pla a samplig period.4 secod was ideified o be A(z ) G(z ) = = B(z ) *z.586*z.9977*z *z.3538*z.359*z.2932*z *z he sysem ca also be wrie as he sae space form wih he followig sysem marices A =,B= C = [ ] he marices K ad L were desiged o be K = [ ], L=, M = [ ] which esure he close-loop sysem wihou ime delay is sable. o illusrae he operaio of he proposed eworked predicive corol scheme, wo cases were cosidered: a) Local corol. here is o ework i he closed-loop sysem, i.e., he oupu sigal from he coroller is direcly coeced o he pla. So, he ework delay is zero. he desig of marices K, L ad M has esured ha he closed-loop sysem is sable. b) Irae based corol. I his case, he oupu sigal was physically rasmied bewee wo Irae IP addresses ad which were boh locaed o he Chiese Academy of Scieces, Beijig. I was measured ha he maximum ework delay was.2 secod. As he samplig period is.4 secod, he upper boud N = 3. For k =,, 2, 3, all eigevalues of marices A (k) are sable ad a commo posiive defiie marix P saisfyig iequaliies (3) was foud usig he LMI oolbox. ha meas he closed-loop sysem is sable wih K,L ad M give above. c) Iere based corol. I his case, he oupu sigal was rasmied bewee he same wo Iere IP addresses ad which were boh locaed o he Chiese Academy of Scieces, Beijig. From [3], we ge ha he maximum ework delay was measured o be.32 secod.so we add.2 secod radom delay o simulae he iere delay. he samplig period was sill.4 secod. So, he upper boud N = 8. Similar o he Irae corol case, for k =,, 2,..., 8, all eigevalues of marices Λ( k ) are sable ad also a commo posiive defiie marix P saisfyig iequaliies (5) was derived usig he LMI oolbox. ha implies ha

4 he closed-loop sysem is sable for give K, L ad M above. o evaluae he performace of he eworked predicive corol scheme, oe real-ime simulaio ad oe real-ime experime were carried ou. Case: u ˆ = u = Kx ) Real-ime simulaio. I his simulaio, he servo moor pla o be corolled is represeed by is model bu he ework was a real oe. he simulaios were performed usig Malab/Simulik/Real ime Workshop. he real-ime simulaio diagram is show i Figure.he referece ipu is a square wave geeraed by he pulse geeraor block, which chages bewee o rpm wih period 5s. he coroller block Necrl is he eworked predicive coroller. Blocks Recv ad Sed are he receiver ad seder of he UDP commuicaio proocol. All of hem were desiged usig Malab S-Fucios. he simulaed pla ad he coroller were execued i a ARM 9 embedded sysem. he real ework (Irae or Iere) was bewee UDP commuicaio blocks Recv ad Sed i Figure 3. Figure 3: he diagram of he real-ime simulaio Four real ime simulaios were coduced: local corol (i.e., o ework), Irae based corol wihou delay compesaio, Iere based corol wih delay compesaio ad Irae based corol wih delay compesaio ad he real-ime simulaio resuls are show i Figures 5 ad 6.he Iere based corol wihou delay compesaio was also coduced, bu i was foud ha he sysem was o loger sable due o he large ework delay, which was bewee.2-.3s. Ipu(rpm) Local corol Irae based corol wih compesaio Irae based corol wihou compesaio Iere based corol wih compesaio Figure5: Ipus of servo pla (Simulaio) 2) Real-ime experime. he differece bewee he real-ime simulaio ad real-ime experimes is ha he pla model of he servo moor i he real-ime simulaio is replaced by D/A block Dac ad A/D block Adc ad he real servo moor. he diagram of he real-ime experime is show i Figure 7. he wo blocks Dac ad Adc were he driver of he A/D ad D/A chaels i he embedded sysem ad were desiged i Malab S-Fucio. Similarly, four real-ime experimes were made: local corol (i.e., o ework), Irae based corol wihou delay compesaio, Irae based corol wih delay compesaio ad Iere based corol wih delay compesaio. he real-ime experime resuls are show i Figures 8 ad 9. Also, i was foud ha he Iere based corol wihou delay compesaio was usable. Figure6: he diagram of he real-ime experime Oupu(rpm) Local corol 2 Irae based corol wih compesaio Irae based corol wihou compesaio Iere based corol wih compesaio Figure 4: Oupus of servo pla (Simulaio) Oupu(rpm) Local corol 2 Irae based corol wih compesaio Irae based corol wihou compesaio Iere based corol wih compesaio Figure 7: Oupus of servo pla (Experime)

5 3 25 Local corol Irae based corol wih compesaio Irae based corol wihou compesaio Iere based corol wih compesaio 2) Real-ime experime. 2 Ipu(rpm) Figure 8: Ipus of servo pla (Experime) u = u = Kxˆ Case2: ) Real-ime simulaio k k Figure2: he diagram of he real-ime experime 4 2 Oupu(r) Local corol 2 Irae based corol wih compesaio Irae based corol wihou compesaio Iere based corol wih compesaio Figure 3: Oupus of servo pla (Experime) Figure 9: he diagram of he real-ime simulaio Local corol Irae based corol wih compesaio Irae based corol wihou compesaio 2 2 Oupu(rpm) Ipu(rpm) 5 5 Ipu(rpm) 2 Local corol Irae based corol wih compesaio Irae based corol wihou compesaio Figure : Oupus of servo pla (Simulaio) Local corol Irae based corol wih compesaio Irae based corol wihou compesaio Figure : Ipus of servo pla (Simulaio) Figure 4: Ipus of servo pla (Experime) From he resuls of real-ime simulaios ad real-ime experimes, i is clear ha he ework rasmissio delay degrades he performace of NCS. Bu he eworked predicive corol scheme proposed i he paper ca compesae for he ework delay acively. Is performace is very close o ha of he local corol scheme (i.e., o ework). he resuls also shows he performaces of wo differe ways o ge u.we ca see ha case 2 is ideal, i whichu is go from he pla direcly. However, i is a piy ha i is impossible i real-ime experime; bu, we fid ha he effec of case is very similar o case 2. Alhough i is hard o make he model of he servo moor pla be exacly he same as he real pracical oe, he resuls of he real-ime experimes are very similar o

6 hose of he real-ime simulaios. 5 CONCLUSIONS A ew eworked predicive corol scheme i a sae-space form has bee proposed for MIMO eworked disribued corol sysems wih radom ework delay ad he sabiliy of he closed loop eworked predicive corol sysems has also bee discussed i his paper. Based o he ework feaure of rasmiig a se of daa each ime, he proposed eworked predicive coroller cosiss of he corol predicio geeraor ad he ework delay compesaor. he former provides a se of fuure corol predicios o saisfy he sysem performace requiremes. he laer compesaes he radom ework rasmissio delay. wo impora heorems o he sabiliy of he closed-loop eworked predicive corol sysem have give he aalyical sabiliy crieria for differe u, respecively. REFERENCES [] J. W. Oversree ad A. zes, A iere-based real-ime corol egieerig laboraory, IEEE Corol Sysems Magazie,vol. 9, pp. 9 34, 999. [2] M. Y. Chow ad Y. ipsuwa, Nework-based corol sysems: A uorial, i Proc. IECON : 27h Au. Cof. IEEE Idusrial Elecroics Soc., 2, pp [3] J. Nilsso, B. Berhardsso, ad B. Wiemark, Sochasic aalysis ad corol of real-ime sysems wih radom ime delays, Auomaica,vol. 34, pp , 998. [4] W. Zhag, M. S. Braicky, ad S. M. Phillips, Sabiliy of eworked corol sysems, IEEE Corol Sys. Mag., vol. 2, pp , Feb.2. [5] W. Zhag, M. S. Braicky, ad S. M. Philips, Sabiliy of eworked corol sysems, IEEE Corol Sysems Magazie, vol. 2(), pp , 2. [6] G. C. Walsh, O. Beldima, ad L. G. Bushell, Asympoic behavior of oliear eworked corol sysems, IEEE rasacios o Auomaic Corol, vol. 46, o. 7, pp , 2. [7] J. K. Yook, D. M. ilbury, ad N. R. Soparkar, A desig mehodology for disribued corol sysems o opimize performace i he presece of ime delays, Ieraioal Joural of Corol, vol. 74, o., pp , 2. [8] H. S. Park, Y. H. Kim, D. S. Kim, ad W. H. Kwo, A schedulig mehod for ework based corol sysems, IEEE ras. o Corol Sysems echology, vol., pp , 22. [9] Y. ipsuwa ad M.-Y. Chow, gai adapaio of eworked mobile robo o compesae QoS egiaio ad deerioraio, vol. 4. Sevilla, Spai, 22. [] H. Li, G. Zhai, ad P. J. Asaklis, Robus sabiliy ad disurbace aeuaio aalysis of a class of eworked corol sysems, i Proceedigs of 42d IEEE Coferece o Corol, (Maui, Hawaii USA), pp , 23. [] S. H. Yag, X. che, D. W. Edwards, ad J. L. Aly, Desig issues ad implemeaio of iere based process corol, Corol Egieerig Pracices, vol. (6), pp , 23. [2] Yuaqi Xia, J. Che, G. P. Liu ad D. Rees, Sabiliy Aalysis of Neworked Predicive Corol Sysems wih Radom Nework Delay. IEEE Ieraioal Coferece O Neworkig, Sesig ad Corol. April, 27, Lodo, UK. [3] Guo-Pig Liu, Yuaqig Xia ad David Rees, Desig ad Sabiliy Crieria of Neworked Predicive Corol Sysems wih Radom Nework Delay i he Feedback Chael, IEEE rasacios o Sysems, ma ad Cybereics,vol.37, No. 2, 27 [4] R. A. Decarlo, M. S. Braicky, S. Peersso, ad B. Learso, Perspecives ad resuls o he sabiliy ad sabilizabiliy of hybrid sysems, Proceedigs of he IEEE, vol. 88, pp ,2. [5] S. Boyd, L. E. Ghaoui, E. Fero, ad V. Balakrisha, Liear marix iequaliies i sysems ad corol heory. Philadelphia: SIAM: Sudies i Applied Mahemaics, 994.