Stabilization of NCSs: Asynchronous Partial Transfer Approach

Transcription

1 25 American Conrol Conference June 8-, 25 Porland, OR, USA WeC3 Sabilizaion of NCSs: Asynchronous Parial Transfer Approach Guangming Xie, Long Wang Absrac In his paper, a framework for sabilizaion of neworked conrol sysems wih limied daa raes is presened The key echnology is ha insead of sending he complee sae vecor(for an n-dimensional sysem, which means a daa array wih n iems) o he nework, we only send one iem of a nonsingular ransform of he sae vecor a each sep The main advanage of our mechanism is dramaically reducing he amoun of he daa sen o he nework In order o implemen such a echnology, in our framework, wo auxiliary insrumens are included ino he overall closed-loop sae feedback sysem over he nework One insrumen is used as an encoder, which runs a ransform of he sysem sae and sends he iems of he ransformed sae vecor in urn The oher akes he role as a decoder, which receives iems of he ransformed sae vecor from he nework and reconsrucs he sae vecor for he conroller I is shown ha if he plan is sabilizable via direc sae feedback conrol, hen i can be sabilized wih he same conroller under our framework over he nework I INTRODUCTION In disribued conrol sysems, a feedback conrol loop is closed hrough a nework Disribued conrol sysems wih neworks are called neworked conrol sysems(ncss) The use of a daa nework in a conrol loop has gained increasing aenion in recen years due o is cos effecive and flexible applicaions The advanages of NCSs include drasic decreasing he number and he oal lengh of cables, easiness of sysem diagnosis and mainenance, and increasing sysem agiliy Thus, many indusrial applicaions, such as roboic sysems, jacking sysems for rain cars, neworks on auomobiles, and process conrol sysems can be reconsruced and modelled as NCSs The analysis and design of NCSs are much more difficul due o he inserion of he communicaion nework in he feedback conrol loop where he convenional conrol heories wih ideal assumpions can no be applied direcly One main drawback of NCSs is he nework-induced delay Various mehodologies have been formulaed o rea he nework-induced delays A survey of recen conrol mehodologies for a closed-loop conrol sysem over a daa nework has been presened in [] Tha paper provided he overview of NCSs including sysem configuraion, nework delay characerisics, he effecs of neworked delays, and corresponding conrol mehodologies The oher main drawback of NCSs is he limiaion on bandwidh A communicaion nework has limied bandwidh which is shared by differen devices conneced o G Xie and L Wang are wih he Cener for Sysems and Conrol, he LTCS and Deparmen of Mechanics and Engineering Science, Peking Universiy, Beijing 87, China xiegming@mechpkueducn This work was suppored by Naional Naural Science Foundaion of China (No 3722, No 644 and No 6274) and Naional Key Basic Research and Developmen Program (22CB322) /5/$25 25 AACC 226 he nework To overcome he bandwidh consrain quie a lo of approaches have been proposed in recen years [2] inroduces he noion of minimum aenion conrol The basic idea is a radeoff beween open loop and closed loop conrol Open loop conrol means no uilizaion of nework bu wih poor performance, while closed loop conrol means uilizaion of nework and wih good performance The former requires no bandwidh, bu he laer requires sufficien bandwidh Then he problem is posed as an opimizaion problem [3] inroduces he noion of maximal allowable ransfer inerval, MATI In his framework, he conroller is designed wihou aking he nework ino accoun And if he ime beween ransfers of informaion from he sensor o he conroller is smaller han MATI, hen he neworked conrol sysem sill keeps sable In [4], his framework is exended o deal wih nonlinear plans In [5], based on he resuls in [3], a sae predicor for he plan o esimae he sae beween updaes is inroduced Two kinds of predicors are presened Some sufficien condiions are given for sabiliy of he NCS seup Similar o he framework in [5], [6] inroduces a compensaed model of he plan, he sae of which is updaed wih he plan s sae The compensaed model is asked o be sable Necessary and sufficien condiions for sabiliy via sae and oupu feedback conrol are derived Quanizaion heory is anoher primary approach used o achieve lower bi-raes The leading work is referred o [7], hough i was no moivaed from an NCS poin of view There are abundan resuls from sabiliy analysis o sabilizaion synhesis via differen quanizaion algorihms(see [8][9] [][][2][3] and he references herein) The common feaure of he above approaches is being minimizing he ransfer of informaion beween he sensor and he conroller/acuaor Differen from he above mechanisms, in his paper, a new framework is presened for sabilizaion of NCSs which aims o reduce he size of he daa ransferred a each ransacion The key idea of his approach is ha insead of sending he enire sae vecor(for an n-dimensional sysem, his means a daa array wih n iems) o he nework, only one iem of a nonsingular ransform of he sae vecor is sen a each ime In order o realize his approach, wo auxiliary insrumens are insered ino he overall closed-loop sae feedback sysem over he nework One insrumen is used as an encoder, which runs a ransform of he sae vecor and sends he iems of he ransformed sae vecor in urn The oher akes he role as a decoder, which receives he iems of he ransformed sae vecor from he nework and reconsrucs he sae vecor which is being used by he

2 Acuaor Plan x( + ) = Ax( ) + Bu( ) Sensor x() Asynchronous Parial Transfer T y () = [ y y] = Qx () n Acuaor Plan x( + ) = Ax( ) + Bu( ) Sensor x() y () y () n Swiching Nework Nework Nework Conroller u () = Kx () Fig Sandard NCS model Nework Conroller u () = Kx ˆ() Reconsrucor z( + ) = Aˆ z( ) + Bˆ y ( ) r ( ) r ( ) r ( ) xˆ( ) = Cˆ z( ) + Dˆ y ( ) r () r () r () y () r() Fig2 NCS model wih asynchronous parial ransfer conroller I is shown ha if he plan is sabilizable via direc sae feedback conrol, hen i can be sabilized wih he same conroller under our framework over he nework The main advanage of our mechanism is dramaically reducing he amoun of he daa sen o he nework and decreasing he load of he nework This paper is organized as follows Secion II formulaes he problem and presens he preliminary resuls Secion III is he main resuls Secion IV conains a numerical example Finally, he conclusion is provided in Secion V Noaions: We use sandard noaions hroughou his noe R is he se of real number, C is he of complex number I n is he uni marix of dimension n II PROBLEM FORMULATION In his paper, we consider a sandard discree-ime linearime-invarian sysem described as follows: x( +)=Ax()+Bu(),=,, 2, () where x R n is he sae vecor, u R p is he conrol inpu A R n n, B R n p are known consan marices Differen from general framework of NCSs(shown in Fig), wo auxiliary insrumens are added ino he overall closed-loop feedback sysem over he nework(shown in Fig2): Afer he sensor, an auxiliary insrumen, named as asynchronous parial ransfer, is insered o upload signals o he communicaion nework, where a sae ransformaion is made as y() =[y (),,y n ()] T = Qx(), where Q is a consan marix o be designed A each ime insan, one and only one iem of y() is uploaded o he nework A swiching signal r() :{,, 2, } {, 2,,n} is inroduced o describe when and which iem of y() is uploaded In deail, r() =i means ha he i-h iem of 227 y(), y i (), is sen o he nework The swiching signal r() is also a conrol facor o be designed Before he conroller, anoher auxiliary insrumen, named as reconsrucor, is inerposed o received he signal from he nework In he reconsrucor, a dynamic esimaor of he sysem sae is given as follows: z( +) ˆx() =Âr()z()+ ˆB r() y r() () = Ĉr()z()+ ˆD r() y r() () where z() is he sae of he esimaor, y r() () is he inpu and ˆx() is he oupu The consan marices Âi, ˆB i, Ĉi, ˆD i are given as follows: Â i = Q(A + BK)Q F i, ˆBi = Q(A + BK)Q e i, Ĉ i = Q F i, ˆDi = Q e i, i =,,n (3) Moreover, he vecors e,,e n R n and marices F,,F n R n n are defined as follows: e,,e n, E i = e i e T i, F i = I n E i, i =,,n We ake he iniial sae of he esimaor o be zero, ie, z() = and he marix K is he feedback gain marix o be designed To avoid unnecessary complexiy, we assume ha here is no ime delay in our framework Here, we are ineresed in he synhesis problem for he NCS in Fig2 which is formulaed as follows: Problem To design a ransform marix Q, a feedback gain K and a swiching signal r() such ha he overall closed-loop sysem is asympoically sable (2) (4)

3 Before given he main resul, we firs give some basic lemmas Lemma : [4] Given a marix G C n n (orr n n ), here exiss a nonsingular marix H C n n (orr n n ) such ha he marix produc HGH o be a riangular marix Lemma 2: Given a riangular marix G C n n or R n n, all he eigenvalues of he marix produc GF GF n are zero, where F,,F n are defined by (4) Proof: See Appendix A III STABILIZATION DESIGN In his secion, we consider he sabilizaion design problem formulaed in he previous secion By he definiion of E,,E n, i is easy o verify ha for any swiching signal r(), e r() y r() () E r() y() I follows ha ˆB r() y r() () ˆB r() e T r() y(), ˆD r() y r() () ˆD r() e T r() y() Thus, we can rewrie (2) as follows: z( +) =Âr()z()+ B r() y() ˆx() = Ĉr()z()+ D (5) r() y() where Âi, Ĉ i is given by (3) and B i = Q(A + BK)Q E i, Di = Q E i,i=,,n (6) I is easy o see ha he esimaor (5) is a sandard swiched linear sysem(see [5] for deails on swiched sysems) I follows ha ˆx() =Q (F r() z()+e r() y()) (7) Applying he sae feedback u() =K ˆx() o he plan, we ge he closed-loop sysem as follows: x( +)=Ax()+BKQ (F r() z()+e r() y()) z( +)=Q(A + BK)Q (F r() z()+e r() y()) (8) Since y() =Qx(), we ge y( +)= QAQ y()+qbkq (F r() z()+e r() y()) z( +)= Q(A + BK)Q (F r() z()+e r() y()) (9) Denoe [ ] y() ξ() = () y() z() I is easy o prove ha augmen variable ξ saisfies he following dynamic equaion: ξ( +)=Γ r() ξ() () 228 where [ ] Q(A + BK)Q QBKQ Γ i = F i QAQ, F i (2) i =,,n The sysem () is also a sandard swiched linear sysem and i is obvious ha such a sysem is asympoically sable implies ha he closed-loop NCS in Fig2 is asympoically sable Theorem : Assume ha he sysem marix A has only real eigenvalues, if he pair (A, B) is sabilizable, hen here exis real marices Q, K and a periodically swiching signal r() such ha he NCS in Fig2 is asympoically sable Proof: Firs, by Lemma, we selec suiable nonsingular marix Q such ha QAQ is in riangular form Nex, we selec suiable marix K such ha A + BK is Schur sable Finally, we selec he periodically swiching signal r() given by r(kn + i) =n i +,i=,,n,k =,, (3) I is obvious ha he period of r() is n Thus, we ge ξ((k +)n) =Γξ(kn),k =,, (4) where [ ] Q(A + BK) Γ= n Q QAQ F QAQ F n (5) Since QAQ is in riangular form, by Lemma 2, we know ha he eigenvalues of QAQ F QAQ F n are all zero I follows ha he marix Γ is Schur sable Thus, he sysem () is asympoically sable This means ha he NCS in Fig2 is asympoically sable Corollary : Assume ha he sysem marix A has complex eigenvalue, and he pair (A, B) is sabilizable, hen here exis a complex marix Q, a real marix K and a periodically swiching signal r() such ha he NCS in Fig2 is asympoically sable Proof: The proof is similar o ha of Theorem Remark : In Theorem, since all eigenvalues of he marix QAQ F QAQ F n are zero, i follows ha (QAQ F QAQ F n ) n = n n This implies ha Qx() z() =y() z(), >n 2 I means ha he sae of he esimaor can converge o he exac value of he sae in finie seps This fac shows ha he esimaor is a deadbea observer Remark 2: In Theorem, if (A, B) is reachable, we can selec suiable K such ha all eigenvalues of A + BK are zero I follows ha all he eigenvalues of he marix Γ are zero Thus, we ge Γ 2n = 2n 2n This implies ha he plan can be sabilized in finie seps This fac shows ha he NCS in Fig2 can realize deadbea conrol when he sysem is reachable

4 IV EXAMPLE To illusrae he efficiency of he proposed approach, a numerical example is presened Example : Consider he plan model wih 2 A = 2 3 4,B = 4 (6) By simple calculaion, we know ha he eigenvalues of A are, 3 and -2, and (A, B) is reachable Thus, by Theorem, he sysem can be sabilized In fac, we ake such ha Q = QAQ = 2 3 is a riangular marix Nex, we ake K = [ ] such ha A + BK is Schur sable Finally, applying he periodic swiching signal (2) wih iniial sae x() = [,, 5] T, he simulaion resuls show ha he closedloop sysem is sable indeed(see Fig 3) The numerical simulaion is finished wih Malab 65 V CONCLUSION In his paper, a new framework for sabilizaion of neworked conrol sysems wih limied daa raes has been presened The approach is based on reducing he size of he daa ransferred a each ransacion In our framework, wo auxiliary insrumens are included ino he overall closed-loop sae feedback sysem over he nework One insrumen is used as an encoder, which sends only a scalar value each sep The oher akes he role as a decoder, which receives signals from he nework and reconsrucs he sae vecor for he conroller By adding hese wo insrumens, i has been shown ha if he plan is sabilizable via direc sae feedback conrol, hen i can be sabilized wih he same conroller under our framework over he nework The fuure work includes considering nework-induced delay ino he framework x x 2 x 3 z z 2 z 3 y z y 2 z 2 y 3 z 3 ACKNOWLEDGMENTS The auhors are graeful o he anonymous reviewers for heir helpful and valuable commens and suggesions APPENDIX A Proof: [Proof of Lemma 2] Supposing g g 2 G, g n Fig3 Simulaion resuls of Example

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