On the Optimal Multirate Control of Networked Control Systems

Transcription

1 WSEAS RANSACIONS on SYSES On the Optimal ultiate Contol of Netwoked Contol Systems ZHANG XIANG, XIAO JIAN Key Laboatoy of agneti Suspension ehnology and aglev Vehile inisty of Eduation Shool of Eletial Engineeing Southwest Jiaotong Univesity Post Box 144#, Southwest Jiaotong Univesity, Chengdu CHINA Abstat: - Optimal multiate ontol of the netwoked ontol systems is investigated in this pape. Unde p to w σ ommuniation sequene that sensos aess ommuniation medium and m to w ρ ommuniation sequene that atuatos aess ommuniation medium, the plant of netwoked ontol system ats as a linea peiodi time-vaying system. Using lifting tehnology, a linea peiodi time-vaying system is tansfomed into a linea peiodi time-invaiant system and the linea quadati pefomane index of a linea peiodi time-vaying system is tansfomed into that of a linea peiodi time-invaiant system. Contollability and obsevability of tansfomed system ae analyzed. An optimal state feedbak ontol and an optimal output feedbak ontol ae given. he simulation esults show that the optimal multiate ontol stategies poposed in this pape ae feasible. Key-Wods: - optimal multiate ontol; netwoked ontol systems; lifting tehnology; linea quadati pefomane index; optimal state feedbak ontol; optimal output feedbak ontol 1 Intodution he netwoked ontol system (NCS) is a speial kind of ontol system distibuted ontol system, in whih multiple sensos and atuatos ae onneted to a entalized ontolle via a shaed ommuniation medium. he main diffeene to a onventional ontol system is that ontolle, sensos and atuatos tansmit infomation and ontol signal though a shaed ommuniation medium [1-4]. Samples of the ontolled vaiable, aptued by the senso devie, ae enapsulated into a fame, whih is sent to the ontolle afte the link is ganted. In the same way, ontol ations alulated by the ontolle have to be sent aoss Input hannels ρ(k) Atuato 1 Atuato m u(k) u 1 u m Shaed ommuniation medium Plant Contolle Fig. 1 A NCS with m inputs and p outputs the shaed medium, to be applied to the plant. A NCS with m sensos, p atuatos and a ontolle is shown in Fig.1. In a NCS, ommuniation paametes must be onsideed. An impotant poblem that has eeived signifiant attention in the NCS liteatue must deal with the joint seletion of a poliy fo managing ontolle-plant ommuniation and a feedbak ontolle that satisfies pefomane o stability equiements fo the losed loop system. In eent yeas, ommuniation and ontol o-design fo NCS is studied widely. he ommuniation polies of NCS ae lassified in two ategoies: stati sheduling and dynami sheduling. Unde stati sheduling, the medium aess ode of diffeent y 1 y p Senso 1 Senso p y(k) Output hannels σ(k) ISSN: Issue 7, Volume 7, July 28

2 WSEAS RANSACIONS on SYSES sensos and atuatos is detemined off-line by a peiodi ommuniation sequene. Unde dynami sheduling, the medium aess ode of sensos and atuatos is detemined on-line, based on state feedbak. If a peiodi ommuniation sequene has been hosen, the poblem that thee exists a stabilizing onstant feedbak ontolle is NP-had; If the ontolle is given in advane, only unoupled plants exist a stabilizing ommuniation sequene [5-11]. Communiation and ontol o-design ould solve the pevious poblem. Adopting o-design method, the poblem of pole-plaement output feedbak stabilization had been solved in [12]. In this pape, ommuniation and ontol o-design and multiate method ae employed to study optimal ontol of NCS. he emainde of this pape is oganized as s: In setion 2, we disuss poblem fomulation that a linea time-invaiant (LI) plant subjet to medium aess onstaints is modeled to a linea time-vaying (LV) system. Fo disussing in the ange of LI system, we use lifting tehnology to tansfom a linea peiod timevaying system into a linea time invaiant system, in setion 3. Contollability and obsevability of the linea time invaiant system given in setion 3 ae analyzed in setion 4. In setion 5, the quadati pefomane index tansfomation is given. With the quadati pefomane index given in setion 5, an optimal state feedbak ontol law and an optimal output feedbak ontol law ae given in setion 6 and setion 7, espetively. Lastly, a simulation example is given. 2 Poblem fomulation Conside the NCS shown in Fig.1, and the dynamis of the plant is given by the ontinuous time LI system x& (t) = A x (t) + B u (t) y (t) = C x (t) (1) x (t) R n, u (t) R m and y (t) R p ae the plant s ontinuous states, inputs and outputs, espetively. Suppose that is the sample peiod, then the disete time desiption of (1) is epesented by x [(k + 1)] = A d x (k) + B d u (k) y (k) = C d x (k) (2) A A d = e, B d = e A t Bdt, C d = C. x (k) R n, u (k) R m and y (k) R p ae the plant s disete states, inputs and outputs, es- petively. he ommuniation medium onneting the sensos and the ontolle has w σ output hannels (1 w σ < p). At any one time, w σ of the p sensos an aess these hannels to ommuniate with the ontolle. At the input side, the m atuatos shae w p input hannels (1 w p < m), i.e., the ontolle an only ommuniate with w p of the m atuatos at any one time. Fo i = 1,, p, use the binay-valued funtion σ i (k) denote the medium aess status of senso i at disete time k, i.e., σ i (k): Z # {, 1}, 1 means aessing and means not aessing. he medium aess status of the p sensos ove time an be epesented by the p to w σ ommuniation sequene vaied with peiod N, σ(k) = [σ 1 (k),, σ p (k)]. We have y i (k) = σ i (k)y i (k), "i. Fo i = 1,, m, use the binay-valued funtion ρ i (k) denote the medium aess status of senso i at disete time k, i.e., ρ i (k): Z # {,1}, 1 means aessing and means not aessing. he medium aess status of the m sensos ove time an be epesented by the m to w ρ ommuniation sequene vaied with peiod, ρ(k) = [ρ 1 (k),, ρ m (k)]. We have u i (k) = ρ i (k)u i (k), "i. Given a ommuniation sequene η(k), η (k) is defined as η (k) diag(η(k)) (3) At the sensos side, if y(k) denote the input signal used by the ontolle and y (k) denote the output signal geneated by the plant, at time k, we have y(k) = σ (k)y (k) (4) At the atuatos side, if u(k) denote the output signal geneated by the ontolle and u (k) denote the input signal used by the plant, at time k, we have u (k) = ρ (k)u(k) (5) Fom the pevious disussion, beause of intoduing netwok ommuniation, we an see that the plant has been hanged and has tuned into the oiginal plant plus ommuniation netwok. he new plant has inputs u(k) and outputs y(k). σ (k) in (4) and ρ (k) in (5) ae time-vaying. We an desibe the new plant as extended plant ombining the dynamis of (1) with the medium aess status of all sensos and atuatos. Fom (2) (5), we an desibe the extended plant as the LV system x[(k + 1)] = Ax(k) + B(k)u(k) y(k) = C(k)x(k) (6) ISSN: Issue 7, Volume 7, July 28

3 WSEAS RANSACIONS on SYSES A = A d, B(k) = B d ρ (k), C(k) = σ (k)c d. 3 System tansfomation he input matix B(k) in (6) is vaied with peiod and the output matix C(k) in (6) is vaied with peiod N. Beause the input and the output of the plant given in (6) possess multiate haateisti and we employ as system sample peiod, the plant given in (6) will be a LV system. In othe wods, If we employ as basi sample peiod to ontol system (6), the digital ontolle will be a peiodially time-vaying system. Fo disussing the pevious system in the ange of LI system, we use lifting tehnology and a linea peiod time-vaying system is tansfomed into a linea time invaiant system. Supposed that the peiod of B(k) is u and the peiod of C(k) is y, then we have the s u =, y = N Let q be the least ommon multiple of and N, i.e. q = LC(, N) and let = q, then is the system yle peiod, i.e., fame peiod. We define extended input veto u e (k ) and anothe fom u * e (k ), extended output veto y e (k ), extended state veto x e (k ) as s, espetively. u( ) u e (k ) = u( + ), u[ + ( q -1) ] u1( ) u 1( + ) u u * 1[ + ( q -1) ] e (k ) =, um ( ) um ( + ) u [ k + ( 1) ] m q - y( ) y e (k ) = y( + ), y[ + ( q -1) ] x[( k 1) + ] x[( k 1) + 2 ] x e (k ) =. x[( k 1) + ( q 1) ] x( ) By appopiate elementay ow tansfomation, we an tansfom u e (k ) into u * e (k ). aking use of x e (k ), u * e (k ), y e (k ) and supposed that is the sample peiod, the disete time model of the extended plant is epesented by x e [(k + 1) ] = A e x e (k ) + B * e u * e (k ) y e (k ) = C e1 x e (k ) + C e2 x e [(k + 1) ] (7) L A A A e = L 2, q L A B * e = [B * 1 B * 2 B * m ], * B j1 L B * = * * B j2 B j1 L j, O * * * B jq / B j( q / 1) L B j1 b j L B * b j + Ab j L j 1=, ξ j L ( k 2) + 1 A ξ j L B * ( k 2) + 2 jk = A ξ j L. ( k 1) A ξ j L with k = 2,, q/, ξ j = b j + Ab j + + A -1 b j, b j is the j-th element of B, L C C e1 = L, L C L L L L L L C C e2 = C. L C By the elementay ow tansfomation tansfoming u e (k ) into u * e (k ), we eaange B * e s olumns ode and tansfom B * e into B e. Based on the pevious tansfomation, the ing model is deived x e [(k + 1) ] = A e x e (k ) + B e u e (k ) y e (k ) = C e1 x e (k ) + C e2 x e [(k + 1) ] (8) Obviously, beause x(k + j) is only elated to the input u(k + i) (i j), B e in (8) is a blok lowe tiangle matix. hee exists the ing fom ISSN: Issue 7, Volume 7, July 28

4 WSEAS RANSACIONS on SYSES B11 L B 21 B22 L B e =. Bq1 Bq2 Bq3 L Bqq Although (8) using fame peiod gives a LI spae state model of the plant given by (6), it is not onvenient in use. Beause the dimension of the state x e (k ) in (8) is high, it is vey diffiult to analyze system and design ontolle. We disuss how to edue the dimension of the state x e (k ) in (8). We substitute the state equations of (8) with espet to x e [(k + 1) ] in the output equations of (8) and deive y e (k ) = C e1 x e (k ) + C e2 x e (k ) + C e2 B e u e (k ) = (C e1 + C e2 )x e (k ) + C e2 B e u e (k ) = C m x e (k ) + D m u e (k ) We define C m and D m as s C m = C e1 + C e2 A e, D m = C e2 B e. C m and D m have the ing fom L C L CA C m =, q 1 L CA L CB11 L D m = CB 21 CB22 L. CB( q 1)1 CB( q 1)2 CB( q 1)3 L hus, the state spae model given by (8) an be ewitten as s x e [(k + 1) ] = A e x e (k ) + B e u e (k ) y e (k ) = C m x e (k ) + D m u e (k ) (9) he matix A e in (8) has n(q 1) zeo eigenvalues and these eigenvalues oespond to the no obsevable modes. hese modes oespond to the pevious n(q 1) elements of the extended state veto x e (k ). he extended output veto y e (k ) is not elated to the pevious n(q 1) elements of the extended state veto x e (k ), too. Fo eduing system dimension, we delete these elements fom x e (k ). he state spae model given by (9) an be simplified as s x[(k + 1) ] = A x(k ) + B u e (k ) y e (k ) = C x(k ) + D u e (k ) (1) A = A q, B = [B q1 B q2 B qq ], C CA C =, D = D m. q 1 CA B is the last n ows of B e and C is the last n olumns of C m. Obviously, the dimension of the state spae model desibed by (1) is n, and it is same as the dimension of the oiginal plant desibed by (2) using as the sample peiod. In the state spae model desibed by (1), it is not all elements of u e (k ) that oespond to the input veto u at the sample time, and it is not all elements of y e (k ) that oespond to the input veto y at the sample time, too. Fo simplifying the state spae model given by (1), We delete all elements not oesponding to at the sample time fom u e (k ) and obtain u * (k ), i.e. u1( ) u 1( + ) u * (k u1[ + ( μ1-1) ] ) =. um ( ) um ( + ) u [ k + ( 1) ] m μm - Aoding to the ode of sample instant, we eaange u * (k ) as u (k ). If u1 u2 um 2 u1, u (k ) is as u1( ) u m ( ) u1( + u1) u (k ) =. u1( + um ) u [ + ( 1) ] m μ1 - um[ + ( μm -1) ] In matix B and D, we delete the olumns oesponding to the elements deleted fom u e (k ) and obtain matix B and D R, espetively. We ewite (1) as the ing fom x[(k + 1) ] = A x(k ) + B u (k ) y e (k ) = C x(k ) + D u (k ) We delete all elements not oesponding to at the sample time fom y e (k ) and obtain y (k ), too. In matix C and D R, we delete the ows oesponding to the elements deleted fom y e (k ) and obtain matix C and D, espetively. We an simplify the state spae model given by (1) as x[(k + 1) ] = A x(k ) + B u (k ) y (k ) = C x(k ) + D u (k ) (11) So, the extended plant, desibed by (6), whose inputs vaied with peiod and outputs vaied ISSN: Issue 7, Volume 7, July 28

5 WSEAS RANSACIONS on SYSES with peiod N is tansfomed into the disete time model desibed by (11), sampling with fame peiod. 4 Contollability and obsevability Befoe intoduing optimal ontol of the disete time model given by (11), it is neessay to analyze ontollability and obsevability of model (11), in this setion. Fistly, we disuss ontollability and eahability of the state spae model given by (9). Beause the matix A e in (9) is singula, if eahability of system (9) holds, ontollability of system (9) holds, but, if its ontollability holds, its eahability do not ompletely holds. he dimension of state spae model desibed by (9) is n e, whih is equal to qn. Aoding to the elated onlusions in linea system theoy, we have the ing onlusion. Conlusion 1 he suffiieny and neessity ondition that system (9) is not ompletely eahable is that thee exists a non-zeo ow veto θ whih satisfies θ A e = λθ, θ B e = (12) Poof: Suffiieny If θ satisfies (12), then θ A e B e = λθ B e = Å θ n A 1 e e B e = θ n A 2 e n 1 e B e = = λ e θ B e = i.e., θ n [B e, A e B e,, A 1 e e B e ] =, thus, system (9) is not ompletely eahable. Neessity If system (9) is not ompletely eahable, it an be deomposed by ontollability as s A  e = A A B Bˆ =, e  e = PA e P -1, Bˆ e = PB e, A is the eahable pat of A e and A is the uneahable pat of A e. Let θˆ = [ z ], and z satisfies z A = λz i.e., z is the left eigenveto oesponding to the eigenvalue of A, then θˆ Â e θˆ Bˆ e =. = [ λz ] = λ θˆ, Let θ = θˆ P, then θ satisfies (12). Aoding to the above disussion, it is easy to deive the ing iteion elating to eahability of system (9). Citeion 1 he suffiieny and neessity on- dition that system (9) is ompletely eahable is ank[zi A e, B e ] = n e, z C (13) Poof: If (13) holds, then thee exist not a non-zeo ow veto θ and a sala λ that (12) holds. Oppositely, if thee exists not a non-zeo ow veto θ that (12) holds, then (13) must hold. Obviously, (12) holds, then, λ oesponding to θ is a no ontollable mode of A e. hen, we disuss ontollability of (11) whih system (9) is simplified to. Aoding to the above disussion, we have the ing theoem. heoem 1 Conside the disete time plants given by (2) and ontollability of the disete time model given by (11). If λ is a no ontollable mode of the disete time plants given by (2), λ q must be a no ontollable mode of (11). If system (2) is ompletely ontollable and at least thee exists a subsipt j that 1 + λ i + 2 i i θ b j and q uj i 1 λ + + λ (14) hold, with espet to the eigenvalues λ i (i = 1, 2,, n) of the system matix A of (2) and the left eigenvetos λ i oesponding to, then the disete time model desibed by (11) is ompletely ontollable. Poof: If λ is a no ontollable mode of (2), thee exists a non-zeo ow veto θ that θ A = λθ, θ b j =, j = 1, 2,, m (15) hold. Obviously, fom (1), θ A = θ A q = λ q θ (16) holds. On the othe hand, aoding to the above disussion in setion 3, eah non-zeo olumn in B e has the ing foms A q q uj ξ q 2quj j, A ξ j,, ζ j, q ζ j = b j + Ab j + + A uj 1 b, j = 1, 2,, m. hus, eah non-zeo element in the ow veto θ B e has the ing foms q q A uj q 2quj θ ξ j, θ A ξ j,, θ ζ j, j = 1, 2,, m. Fom (15), we have θ ζ j = θ q (b j + Ab j + + A uj 1 b ) j 1 q uj = (1 + λ + λ λ )θ b j, Å q q A uj q q θ ξ = λ uj (1 + λ + λ j j 1 q uj λ ) θ b j, j = 1, 2,, m (17) Aoding to (17), if (15) holds, then θ B e = holds. ISSN: Issue 7, Volume 7, July 28

6 WSEAS RANSACIONS on SYSES Fom(16), λ q is a no ontollable mode of system (1). As the no ontollable modes of system (11) ae same to those of system (1), λ q is a no ontollable mode of system (11), too. On the othe hand, If system (2) is ompletely ontollable and supposed that λ i is a abitay eigenvalue of A and θ is the left eigenveto λ i oesponding to, then thee exist seveal subsipts j that θi b j holds. Futhemoe, if (14) holds, then θ B e holds. Beause λ i is abitay, system (1) is ompletely ontollable, i.e., system (11) is ompletely ontollable. Lastly, we disuss obsevability and onstutibility of the state spae model given by (9). Simila to ontollability and eahability, beause the matix A e in (9) is singula, obsevability and onstutibility of system (11) ought to be investigated, espetively. Simila to eahability, we have the ing onlusion elating to onstutibility of (9). Conlusion 2 he suffiieny and neessity ondition that system (9) is not ompletely onstutible is that thee exists a non-zeo ow veto θ whih satisfies A e θ = λθ, C e θ = (18) Poof: he poof of onlusion 2 is ompletely simila to the poof of onlusion 1. Using onlusion 2, it is easy to deive the ing iteion elating to obsevability of (9). Citeion 2 he suffiieny and neessity ondition that system (9) is ompletely obsevable is zi Ae ank C = n e, z C (19) m Poof: he poof of iteion 2 is ompletely simila to the poof of iteion 1. Using iteion 2, it is easy to deive the ing theoem. heoem 2 System (9) isn t ompletely obsevable, and must be a no obsevable mode of (9). Poof: Aoding to iteion 2, we an onstut the ing matix. zi L A 2 zi L A zi Ae q C = L zi A m CA C (2) L L q 1 L CA Obviously, if z =, the ank of the matix shown in (2) is less than n e. Besides eigenvalue A q, matix A e in (9) inludes n e n = (q 1)n zeo eigenvalues, whih ae all the no obsevable modes of system (9). hen, we analyze obsevability of simplified system given by (11). Using iteion 2, it is easy to deive the ing theoem. heoem 3 Conside the disete time plants given by (2) and obsevability of the disete time model given by (11). he ing onlusions hold. If system (2) has a no obsevable mode λ, then λ q must be a no obsevable mode of system (11). If system (2) is ompletely obsevable, then system (11) is ompletely obsevable. Poof: Aoding to the above disussion in setion 3 and the definition of C in (1), C in (11) has the ing fom. C = C H the submatix C is ompose of the ows in C oesponding to the elements of the output sample instants of y e (k ) and the submatix H is ompose of CA, CA 2,, CA q 1 being appopiately deleted some ows. Suppose that λ is a no obsevable mode of system (2), thee must exists a non-zeo veto θ whih satisfies Aθ = λθ, Cθ = (21) So, we have A θ = A q θ = λ q θ (22) On the othe hand, if (21) holds, then CAθ = λcθ =, CA 2 θ = λ 2 Cθ =, Å CA q 1 θ = λ q 1 Cθ = hold. Consequently, Hθ = hold. hus, we have C θ = θ H C = (23) θ i.e., λ q is a no obsevable mode of system (11). On the othe hand, if thee exists not a veto θ that (23) holds, thee exists not a veto θ that (22) and (23) hold simultaneously. hus, if system (2) is ompletely obsevable, system (11) is ompletely obsevable. 5 Quadati pefomane index tansfomation Optimal ontol is a main kind of ontol method. he ente of optimal ontol is: with given onditions and fo etain plants, a ontol law is ISSN: Issue 7, Volume 7, July 28

7 WSEAS RANSACIONS on SYSES deided by some pedefined pefomane indexes. Unde this ontol law, the losed-loop system has optimal value. Appopiately seleting eah weight matix in the pefomane indexes given by system, seveal equiements on system, fo example, apidity, peision, stability, sensitivity, and so on, ae satisfied. In this setion, aoding to the multiate haateisti of the extended plan desibed by (6) and based on the state spae model given by (11), we give a disete fom of integato pefomane index. Conside linea quadati optimal ontol poblem of the ontinuous time plant given by (1). Supposed that the quadati integato index of the system epesented by (1) is given as t f J = [ x ( t) Q x ( t) + u ( t) R u ( t)] dt (24) Q is a semi-definite onstant matix and R is a definite onstant matix. If a zeo-ode-holde is employed and the sample peiod is the basi sample peiod, the disete fom fo (24) is given as N J = [ x ( k) QJ x( k) + 2 x ( k ) J k = u ( k) + u ( k ) RJ u ( k)] (25), weight matix Q J, J and R J ae given as s, espetively. Q J = α ( t) Q α( t) dt, J = R J = α ( t) Q β( t) dt, β ( t) Q β( t) dt + R, α(t) = e At, β(t) = t α ( τ) Bdτ. Using two extended veto u e (k ) and x e (k ), the quadati integato index (25) an be equivalently tansfomed into the ing fom with sample peiod (fame peiod) N J = { x e [( k + 1) ] Qe xe[( k + 1) ] k = QJ Q e = + 2 x e [( k + 1) ] e ue ( ) + u e ( ) Re ue ( )} (26) QJ L L, L QJ J L J L e =, L J ( A 1 L ) J RJ 2P1 P2 P3 L Pq 1 Pq P 2 RJ L R e =, Pq 1 RJ Pq L RJ with P k = B qk (A -1 ) J, k = 1,, q A and B qk stated by (1). he speial stutue of the fist olumn in weight matix e and the speial stutue of the fist ow and the fist olumn in weight matix R e ae podued by a oss-tem x e (k ) J u e (k ). Employing the state spae model given by (1), we have the ing expession x(k ) = A -1 {x e[(k + 1) ] B u e (k )} hen, thee exsits the ing equation x e (k ) J u e (k ) = {x e [(k + 1) ] B u e (k )} (A -1 ) J u e (k ) = x [(k + 1) e ](A -1 ) J u e (k ) u (k e )B (A -1 ) J u e (k ) Aoding to the above expession, the speial stutue of the fist olumn in weight matix e and the speial stutue of the fist ow and the fist olumn in weight matix R e ae podued. he pefomane index J defined in (26) oesponds to the multiate model desibed by (8). Beause the dimension of state veto x e (k ) in (8) is high, omputation is high. hus, we simplify the model desibed by (8) and deive simplified model given by (1). Fo utilizing (11), u(k ) and x(k ) substitute u e (k ) and x e (k ) in (25), espetively. We disuss how to implement this tansfomation. Fom (1), we an deive as x e [(k + 1) ] = A e x e (k ) + B e u e (k ) = A e x e (k ) + B u (k ) A 2 A = x e (k ) + B u (k ) q A = A E x(k ) + B u (k ) (27) ISSN: Issue 7, Volume 7, July 28

8 WSEAS RANSACIONS on SYSES A 2 A A E =. q A aking use of (27), we an deive the ing fom x [(k + 1) e ]Q e x e [(k + 1) ] = x (k )A Q E ea E x(k ) + 2x (k )A Q E e B u (k ) + u (k )B Q eb u (k ) (28) x [(k + 1) e ] e u e (k ) = x e [(k + 1) ] u (k ) = x (k )A E u (k ) + u (k )B u (k ) (29) Using (28) and (29), the pefomane index J an be ewitten as N J = [ x ( ) Q x( ) + 2 x ( ) = u ( ) + u ( ) R u ( )] (3) Q = A Q E ea E, = A E + A Q E eb, R = B Q eb + 2B + R. Fom the quadati index given by (3), the plant desibed by (11) an be optimized. With linea quadati pefomane index given by (3), utilizing uent method, optimal ontol of the plant given by (11) an be solved, effetively. 6 Optimal state feedbak In this setion, with linea quadati pefomane index given by (3), we disuss optimal ontol poblem of the plant desibed by (11). We have the below-mentioned theoem. heoem 4 finite-time optimal ontol poblem Aoding to the quadati index given by (3), the plant desibed by (11) has optimal ontol law as u (k ) = [K (k) + R -1 ]x(k ) (31), K (k) is solved by the ing eusion K (k) = [R + B S (k + 1)B ] 1 B S (k + 1) A ~ (32) and matix S (k) is solution of disete time Riati equation as S (k) = Q ~ + A ~ S (k + 1) A ~ - A ~ S (k + 1)B [R + B S (k + 1)B ] 1 B S (k + 1) A ~ (33) with bounday ondition S (N ) = (34) he optimal pefomane index funtion is J min = x ()S ()x() (35) Q ~ = Q - R -1, A ~ = A - B R -1. Poof: Let u (k ) = u (k ) + R -1 x(k ), then, the state equation of (11) is equivalent to x[(k + 1) ] = A ~ x(k ) + B u (k ) (36) and the quadati index given by (3) is equivalent to J ~ = ~ [ x ( ) Q x( ) + u ( ) R = u (k )] (37) Beause the pefomane index J ~ has no oss-item, aoding to the onlusion with espet to finite-time linea quadati optimal ontol of disete time system, we an obtain the optimal ontol law (31). Now, with the below-mentioned quadati pefomane index J = [ x ( ) Q x( ) + 2 x ( ) = u ( ) + u ( ) R u ( )] (38) we disuss infinite-time linea quadati optimal ontol fo the plant given by (11). Utilizing the above-mentioned method and appopiately defining extended vetos and weighed maties, the ontinuous time pefomane index J = [ x ( t) Q x ( t) + u ( t) R u ( t)] dt (39) an be disetely tansfomed into the pefomane index J given by (38). he below-mentioned theoem gives solution to the infinite-time linea quadati optimal ontol (ILQOC) poblem. heoem 5 infinite-time optimal ontol poblem Aoding to the quadati index given by (3), the plant desibed by (11) has infinite-time optimal ontol law as u (k ) = [K + R -1 ]x(k ) (4) K = [R + B S B ] 1 B S A ~ (41) and matix S (k) is solution of algeba Riati equation as S = Q ~ + A ~ S A ~ - A ~ S B [R + B S B ] 1 B S A ~ (42) Poof: Let u (k ) = u (k ) + R -1 x(k ), then, the state equation of (11) is equivalent to x[(k + 1) ] = A ~ x(k ) + B u (k ) and the quadati index given by (37) is equivalent ISSN: Issue 7, Volume 7, July 28

9 WSEAS RANSACIONS on SYSES to J ~ = [ = x ( k ~ ) Q x( k ) + u ( k ) R u (k )] (43) Beause the pefomane index J ~ has no oss-item, aoding to the onlusion with espet to infinite-time linea quadati optimal ontol of disete time system, we an obtain the optimal ontol law (4). Based on the above-mentioned theoem 4 and theoem 5, we an obtain design aithmeti of the quadati optimal state feedbak ontolle fo the plant desibed by (11). Fo the infinite-time linea quadati optimal ontol (ILQOC) poblem, although the egulato given by (4) is linea time-invaiant, the state-feedbak gain is timevaying at eah sample time in fame peiod. 7 Optimal output feedbak We disuss the optimal output feedbak ontol law whih substitutes the optimal state feedbak ontol law given by (31) and (4), in this setion. Fo sampleness, we only disuss the optimal output feedbak ontol law whih substitutes the linea time-invaiant optimal state feedbak ontol law given by (4). We intodue the extenal efeene inputs (t), (t) R v and the sample peiod of (t) ae as j = q j (j = 1, 2,, v). hus, the extended efeene inputs an be defined as u( ) e (k ) = u( + ). u[ + ( q -1) ] In e (k ), we delete the elements not oesponding to the sample time (k + i j ) of the efeene inputs and obtain veto (k ). With the efeene inputs (k ), the state feedbak ontol law given by (4) an be ewitten as u (k ) = K (k ) Kx(k ) (44) K = K + R 1. Coesponding to the state feedbak ontol law given by (44), the output feedbak ontol law is as u (k ) = L (k ) L y y (k ) (45) Beause of ausality ondition, L y must be a blok lowe tiangle matix. We substitute the output equation of (1) into (44) and an deive u (k ) = L (k ) L y [C x(k ) + D u (k )] (46) We define W = (I + L y D ) 1 (47) hen, equation (46) is equivalent to u (k ) = WL (k ) WL y C x(k ). Obviously, if L and L y satisfy WL = K (48) WL y C = K (49) espetively, the output feedbak ontol law given by (45) is equivalent to the state feedbak ontol law given by (44). Equation (48) an be ewitten as L y (C D K) = K (5) In (49), we solve L y whih satisfies ausality ondition. We substitute L y into (48) and an deive L = (I + L y D )K (51) Beause of ausality ondition, L must be a blok lowe tiangle matix, too. Beause L y and D ae blok lowe tiangle matix, L is a blok lowe tiangle matix while K is a blok lowe tiangle matix. If auate solution of L y an not be solve, linea least squaes an be utilized to solve appoximate solution of L y. 8 Simulation example Suppose that plant is as x& (t) = Ax(t) + Bu(t) y(t) = Cx(t) A =, B = 1 1, C = Design the disete time output feedbak ontol law whih demands the linea quadati pefomane index J = ( x + x +.1u +.1u dt ) to be optimal. Whee, system sample peiod is =.1s and the initial value of the states ae given as x() = [1. 1.]. System samples y 1 (t), y 2 (t) and outputs u 1 (t) at the fist sample instant. Also, samples y 2 (t) and outputs u 2 (t) at the seond sample instant. Aoding to (12), the disete time desiption of the plant is epesented as x[(k + 1)] = Fx(k) + Gu(k) y(k) = C d x(k) F = 1, G =.1, C d = ISSN: Issue 7, Volume 7, July 28

10 WSEAS RANSACIONS on SYSES As q = LC( u, y ) = 2, fame peiod = q =.2s. Suppose that sample peiod is, the disete time model of the plant is x[(k + 1) ] = A x (k ) + B u (k ) y(k ) = C x(k ) + D u (k ) u1( k ) u (k ) = u ( + ) 2 k A = 1, B =. 1. 1, C = 2 1, D = Fom (12), we an obtain Q J = , J = , R J = Fom (17), we an obtain Q = , = ,.913 R = Fom (26), (27) and (3), we solve the output ontol gain L y = he zeo-input state esponse of the losedloop ontol system is shown in Fig.2. As shown in Fig.2, by the ontol of the designed ontolle, the esponse of the losed-loop ontol system is asymptotially stable. We obtain a quite satisfatoy ontol esult. x(t) x 1 (t) x 2 (t) t (s) Fig. 2 Zeo-input state esponse of the losed-loop ontol system 9 Conlusion Optimal multiate ontol of the netwoked ontol system is mainly studied in this pape. Using lifting tehnology, a linea peiodi time-vaying system is tansfomed into a linea peiodi timeinvaiant system, the linea quadati pefomane index of a linea peiodi time-vaying system is tansfomed into that of a linea peiodi timeinvaiant system. An optimal state feedbak ontol and an optimal output feedbak ontol ae given. he simulation esults show that the optimal output feedbak ontol law poposed in this pape is effetive. Poposed multiate solutions in this pape have been extensively tested in simulation envionment. In the futue, a patial expeiment will be need to test poposed multiate solutions in this pape. In addition, the seletion of ontol and ommuniation poliies in obust will be investigated. Refeenes: [1] L. G. Bushnell. Netwoks and Contol[J]. IEEE Contol System agazine, Vol.21, No.1, 21, pp [2] G. C. Walsh, Y. Hong, L.G. Bushnell. Stability Analysis of Netwoked Contol Systems[J]. IEEE ansations on Contol Systems ehnology, Vol.1, No.3, 22, pp [3] Y. Halevi, A. Ray. Integated ommuniation and ontol systems: Pat I Analysis[J]. Jounal of Dynami Systems, easuement and Contol, Vol.11, No.4, 1988, pp [4] Y. Halevi, A. Ray. Pefomane analysis of integated ommuniation and ontol system netwoks[j]. Jounal of Dynami Systems, easuement and Contol, Vol.112, No.3, 199, pp [5] R. W. Bokett. Stabilization of moto netwoks[c]. In Poeedings of the 34th IEEE Confeene on Deision and Contol, 1995, pp [6] D. Histu, K. ogansen. Limited ommuniation ontol[j]. Systems and Contol Lettes. Vol.37, No.4, 1999, pp [7]. S. Baniky, S.. Phillips, W. Zhang. Sheduling and Feedbak Co-Design fo Netwoked Contol Systems[C]. In Poeedings of the 41st IEEE Confeene on Deision and Contol, 22, pp [8] D. Histu. Feedbak Contol Systems as Uses ISSN: Issue 7, Volume 7, July 28

11 WSEAS RANSACIONS on SYSES of a Shaed Netwok: Communiation Sequenes that Guaantee Stability[C]. In Poeedings of the 4th IEEE Confeene on Deision and Contol, 21, pp [9] H. Ishii, B. A. Fanis. Stabilization with Contol Netwoks[J]. Automatia, Vol.38, No.1, 22, pp [1] B. Linoln, B. Benhadsson. Effiient Puning of Seah ees in LQR Contol of Swithed Linea Systems[C]. In Poeedings of the 39th IEEE Confeene on Deision and Contol, 2, pp [11] H. Rehbinde,. Sanfidson. Sheduling of a Limited Communiation Channel fo Optimal Contol[C]. In Poeedings of the 39th IEEE Confeene on Deision and Contol, 2, pp [12] L. Zhang, D. Histu. Communiation and Contol Co-Design fo Netwoked Contol Systems[J]. Automatia, Vol.42, No.1, 26, pp [13] J. Xiao. ultiate digital ontol system[]. Siene Pess, 23. [14] Z. Q. Sun. Compute ontol theoy and appliation[]. Qinghua Univesity Pess, [15] W. Haoping,V. Chistian, C. Afzal and K. Vladan. Sampled taking fo linea delayed plants using pieewise funtioning ontolle [J]. WSEAS ansations on System and Contol, Vol.2, No.8, 27, pp [16] C. Botan, F. Ostafi. A Solution to the Continuous and Disete ime Linea Quadati[J]. WSEAS ansations on System and Contol, Vol.3, No.2, 28, pp [17] L. ihai, L. Romulus. Optimal Contol of Flying Objets ove Afte Estimated State Veto Using a Redued Ode Obseve[J]. WSEAS ansations on Ciuits and Systems, Vol.7, No.6, 28, pp [18] Y. C. Chang, S. S. Chen. Stati Output Feedbak Simultaneous Stabilization of Inteval ime-delay Systems [J]. WSEAS ansations on Systems, Vol.7, No.3, 28, pp [19] V. Casanova, J. Salt. ultiate ontol implementation fo an integated ommuniation and ontol system[j]. Contol Engineeing Patie II, Vol.11, No.11, 23, pp [2] L. L. Yang, S. H. Yang. ultiate ontol in intenet-based ontol systems[j]. IEEE ansations on Systems, an, and Cybenetis Pat C: Appliations and Reviews, Vol.37, No.2, 27, pp ISSN: Issue 7, Volume 7, July 28