TIME DELAY BASEDUNKNOWN INPUT OBSERVER DESIGN FOR NETWORK CONTROL SYSTEM

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1 TIME DELAY ASEDUNKNOWN INPUT OSERVER DESIGN FOR NETWORK CONTROL SYSTEM Siddhan Chopra J.S. Laher Elecrical Engineering Deparen NIT Kurukshera (India Elecrical Engineering Deparen NIT Kurukshera (India ASTRACT Many odern conrol syses use counicaion neworks o exchange inforaion aong syse s coponens. Due o inserion of counicaion nework hese syses are ofen subjeced o disurbances perurbaion ec. This paper deals wih he observer design proble for he nework conrol syses wih unknown inpus. In his paper he exising approach wih one delay in feedback sysehas been exended by insering wo counicaion neworks wih individual respecive delays in observaion and conrol channels. Nework effec is anifesed in ers of iedelayin signals exchanged beween coponens.a nuerical exaple is also presened and siulaed using MATLA/SIMULINK o deonsrae he approach. Index Ters: Nework Conrol Syse Observer Siulink Tie DelayUnknown Inpu.. INTRODUCTION A conrol syse in which is coponens such as plan sensors acuaors are spaially disribued is called Nework Conrol Syse (NCS. In such syses counicaion akes place via a band-liied counicaion nework. One of he os proinen feaure of he NCS is ha exchange of inforaion beween syse s coponens akes place in for of inforaion packes. NCSs has been an area of exensive research over he pas few decades due o is increased flexibiliy lower coss siple insallaion and easy ainenance ec. On he oher hand here has been soe serious issues due o he use of counicaion nework such liied bandwidh nework induced delay packe dropou. These probles us be look ino as hese can have adverse effec on syse sabiliy. The ypical NCS srucure is shown in Fig... Lieraure review The roos of observer design can be raced back o lae 960s when Luenberger firs designed observer in 966 []. Since hen differen ypes of observers have been developed which are being used in various engineering applicaions. As he large scale syses are ore suscepible o high disurbance and fauls he sudy of UIOs becoe ore pivoal []. Mos of he UIOsproposed in lieraure are for non-neworked syses []. haacharya [4] provided he soluion o he observer design proble for syses subjeced o unknown inpus or 8 P a g e

2 disurbances.one siple ehod was proposed by Darouach e al. [5] for designing he full order observer he necessary and sufficien condiions for his observer were given. Kobayashi and Nakaizo [6] suggesed an approach based on Silveran s inverse ehod. On he oher hand Hui and Zak [7] proposed a projecion operaor approach. Recenly an UIO was designed by Taha e al. [] for nework conrol syse sabiliy guaranee bounds for he delay and perurbaion induced by nework were also esablished. Plan Acuaor Sensor Counicaion Nework Observer and Conroller Fig Typical Nework Conrol Syse II SYSTEM MODELLING Proposed archiecure for a neworked unknown inpu observer schee is shown in Fig. The inpu o he observer block are heknown inpu u and delayed version of he plan s oupu ha is ŷ. We have assued ha hese wo quaniies are known o he observer block. The innovaion block is differen for differen observers []. Esiae of plan sae ha is x is he oupu of he observer. Reference inpu (V ˆ ref and x acs as inpus o he conroller. The ˆ unknown inpu for he plan is aken as u ha can be any non-lineariy or any unknown plan disurbance.. Review of unknown inpu observer for non-neworked syses: Zak & Hui [] inroduced a projecion operaor approach based on which observer is designed in his paper. We have considered he Linear-Tie invarian (LTI class of syses. The dynaics of plan is given ( x A x u u as: where nn A n n x is plan sae u is known inpu u is unknown inpu y is oupu respecively. I is assued ha A are all 84 P a g e

3 known syse paraeers. For he non-neworked syse he dynaics of unknown inpu observer (UIO as presened in [] are: z I MC A z A My u L y C z C My ( x z My ( ˆ ( ( ( Plan Model û x x Counicaion (0 Nework ( x A x u u ( Sensor y Counicaion u v ref Nework ( Conroller u kx v x ˆ ˆ ref Unknown Inpu Observer [ z Y ] x I ˆ z z D z Y z A z Y z z z ŷ Y f ( yˆ Innovaion lock z (0 Fig. Neworked UIO Archiecure where M n p is chosen such ha ( ( I MC 0 (4 and L is he addiional gain o iprove he convergence rae []. The iniial condiions for he observer for he above z(0 ( I MC xˆ (0 dynaical syse are (5 The designed observer for he non-neworked syse converges esiaion error o zeroas under he assupion ha he pair (C A is deecable[]. In order o analyse he effec of counicaion nework on sae esiaion he dynaics of he observer by aking x c zcan be wrien as follows: x I MC A x A My u L y C x C My (6 ( ( ( c c c ( x A x y u (7 c c c c c where 85 P a g e

4 A ( I MC ( A LC ( I MC ( A M L LC M ( I MC c c ( c dynaics of he UIO will ge changed due he presence of he counicaion nework beween plan and observer and can be wrien as follows: ( x A x yˆ u (8 xˆ x Myˆ (9 c c c c c c we assue sae feedback conrol is used: u kxˆ k( x Myˆ c (0 The III NETWORK EFFECT AS PURE TIME DELAY. Closed Loop Dynaics As can be seen in figure ( we have insered wo counicaion neworks one beween unknown inpu observer (UIO and is inpus and anoher one beween conroller and plan due o which plan ges he delayed inpu. In his yˆ y( secion we odel he counicaion nework effec as pure ie-delay. So i is assued ha uˆ u( where and are he ie-delays occurring due o nework effec. Rewriing he dynaics of he observer and conroller in he disinc for of NCS observer/conroller while assuing ha he innovaion funcion of he observer is ebedded in UIO dynaics : ( ( x ( A x ( y( u ( u C x D y c c c c c c c c ( ( ( ( where Cc k and D c km Sae dynaics of plan and conroller can be wrien as: x ( A x ( u ( u ( (4 ( A x ( C x ( D C x ( ( u ( ( c c c x ( A x ( y( u ( ( c c c c c ( ( (5 A x ( C x ( C x ( D C x ( c c c c c c c c Cobining he above wo equaions o find x ( x ( x ( x ( c (6 y siplifying (6 we ge x ( x( x( x( x( ( u ( 0 4 (7 86 P a g e

5 where A ( A C c c c 0 0 C D C 0 c c c ( 0 C 0 0 c D C c 4 (. NCS and Approxiaion of Tie-Delay Fro [] we have x ( 0 x ( 0 c 0 Taking he second derivaive of x( and subsiuing he above approxiaion we have x( x ( u ( (8 0 4 Using he following aylor series expansion for x(-a: n n a ( n x( a ( x ( (9 n! n0 Neglecing he higher order ers we ge x x x x ( ( ( ( (0 x( x( x( x( x( x( x ( x( ( where Subsiuing (0( in (7 we x ( x( [ x( x ( x( ] [ x( x ( x( ] [ x( x ( x( ] u ( ( 0 4 ge Subsiuing (8 in above equaion x x x x x u x x x u ( ( [ ( ( ( ( ( ] [ ( ( ( ( ( ] ( [ x( x ( ( x ( u ( ] u ( Rearranging he above ers we ge x( x( u ( u ( 0 where (5 (4 87 P a g e

6 ( 0 0 ( ( I ( Tie-Delay ased UIO Design for Neworked Syse In his secion conroller and observer is designed in order o iniize he effec of uliple delay ers as well as of he uknown inpu fro he dynaics of he closed loop syse. We know D C c 4 0 and 4 ( D C ( c The observer is o be designed in such a way ha effec of uliple delay ers and unknown inpu is nullified. So ( his eans should resul equal o zero i.e 0 D C 0 or D 0 (6 Subsiuing he 4 4 c c value of Dc km in he above equaion we ge km 0 and we know ha he lone condiion for he ( ( I MC 0 (7 exisence of UIO for he non-neworked syses as in []: So o design he observer ( km 0 (8 ( I MC 0 we have he following equaions o solve: variable o be solved for he design of he Neworked UIO []. (9 As we don have uch conrol over k so M is he p Now uliplying equaion (9 by a non-singular arix H and hen adding i in equaion (8 we ge ( ( km H MC H 0 n p (0 we ge he quadraic equaion in ers of M i.e Neworked UIO design variable. nn Le R k hen above equaion can be represened ( ( as RM H MC H 0 ( n p Above equaion can be wrien as follows: MA M C 0 n p ( Hence he Neworked UIO can be designed by solving equaion (. IV UNKNOWN INPUT OSERVER DESIGN EXAMPLE FOR A NETWORKED SYSTEM In his secion we deonsrae he exaple for he proposed Neworked UIO design following he algorih and algorih given in []. We assue ha he given syse has one known inpu one unknown inpu and one oupu. 88 P a g e

7 5 0 A C ( Neworked UIO exisence: rank( C ( =; rank( = ( As boh ranks are equal o so he UIO exis. Copue he design arix M : Following he algorih given in [] we ge M Copue P : P I MC n Find Q : Q Copue A and C : ( A A ( C A A C C A ( Copue eigen values of eigen( ( A : ( A = [ ] 7 Since he eigen values of deecable. ( A are negaive which eans ha he arix is sable hen ( A C ( is 8 Design gain L:.448 A C L= place ( 0 89 P a g e

8 ( 9 Copue NCS paraeers i.e A : c c c A P ( A LC P ( A M L LC M c c P ( c 0 UIO observer for he neworked syse can be wrien as follows: x A x yˆ u ( c c c c c xˆ x Myˆ c We assue he sae iniial condiion of plan sae and unknown inpu observer sae as used in [] i.e x 0.56 ( and 0.79 x c 6.66 ( and he unknown inpu is aken as u ( 0.5sin(. Fig. ( and Fig (4 shows he esiaion of plan saes. V SIMULATION RESULTS 5. For Non-Neworked Syse ( 0 Fig. shows he esiaion of plan saes for non-neworked syse wih unknown inpus. Fig (a 90 P a g e

9 Fig. (b Fig. (c Fig. Siulaion Resuls of Esiaion of Saes for Non-Neworked UIO 5. For Neworked Syse ( 0 0 Fig. 4 shows he esiaion of plan saes for neworked conrol syse. Fig. 4(a 9 P a g e

10 Fig. 4(b 4(c. Fig. 4 Siulaion Resuls of Esiaion of Saes For Neworked UIO V CONCLUSION NCS has been an acive research area over he years due o wide variey of applicaions. In his paper a ie delay based unknown inpu observer for neworked conrol syse has been proposed.the presence of counicaion neworks adds ie delay o he syse. For differen counicaion neworks individual respecive ie delays are considered. The approach of unknown inpu observer design [] wihou ie delays (non-neworked syse has been reviewed o exend he approach o neworked syse. The closed loop dynaics for nework conrol syse wih unknown inpu observer has been proposed o iniize he effec of ie delay and effec of unknown inpu on sae esiaion. The siulaed resuls of nuerical exaple shows ha he esiaed saes are alos converging o original saes using he proposed approach. For fuure work we would like o exend our resuls o larger delays plus we would like o ake ino coun he copuaion ie of conrol law. 9 P a g e

11 REFERENCES. S. Hui S.H. Zak Observer Design for Syse wih Unknown inpus Inernaional Journal of Applied Maheaics and Copuer Science Vol. 5 No.4 pp Ahad F. Taha Ahed Elahdi Jiesh H. PanchalDengfeng Sun Unknown Inpu Observer Design and Analysis for Neworked Conrol Syses Inernaional Journal of Conrol 05.. D. G. Luenberger Observers for ulivariable syses. IEEE Transacions on Auoaic Conrol vol. AC- no. pp S. P. haacharyya Observer design for linear syses wih unknown inpus IEEE Transacions on Auoaic Conrol vol. no. pp M. Darouach M. Zasadzinski S. J. Xu Full-order observers for linear syses wih unknown inpus IEEE Transacions on Auoaic Conrol vol. 9 no. pp N. Kobayashi and T. Nakaizo "An Observer design for linear syses wih unkown inpus" Inernaional Journal of Conrol vol. 5 pp S. Hui and S. H. Zak Observer design for syses wih unknown inpus Inernaional Journalof Applied Maheaics and Copuer Science vol. 5 no. 4 pp Ahad F. Taha Ahed Elahdi Jiesh H. PanchalDengfeng Sun Neworked Unknown Inpu Observer Analysis and Design for Tie Delay Syses IEEE Inernaional Conference on Syses Man and Cyberneics04. 9 P a g e

Introduction to Numerical Analysis. In this lesson you will be taken through a pair of techniques that will be used to solve the equations of.

Introduction to Numerical Analysis. In this lesson you will be taken through a pair of techniques that will be used to solve the equations of. Inroducion o Nuerical Analysis oion In his lesson you will be aen hrough a pair of echniques ha will be used o solve he equaions of and v dx d a F d for siuaions in which F is well nown, and he iniial