Networked Cooperative Distributed Model Predictive Control Based on State Observer

Transcription

1 Apped Mathematcs, 6, 7, ubshed Onne June 6 n ScRes. Networed Cooperatve Dstrbuted Mode redctve Contro Based on State Observer Ba Su, Yanan Zhao, Jnmng Huang Coege of Engneerng, Qufu Norma Unversty, Rzhao, Chna Receved March 6; accepted 6 June 6; pubshed 9 June 6 Copyrght 6 by authors and Scentfc Research ubshng Inc. hs wor s censed under the Creatve Commons Attrbuton Internatona Lcense (CC BY. Abstract Athough dstrbuted mode predctve contro has caused sgnfcant attenton and receved many good resuts, the resuts are mosty under the assumpton that the system states can be observed. However, the states are dffcut to be observed n practce. In ths paper, a nove dstrbuted mode predctve contro s proposed based on state observer for a nd of near dscrete-tme systems where states are not measured. Frsty, an output feedbac contro aw s desgned based on Lyapunov functon and state observer. And the stabty doman s descrbed. Furthermore, the stabty doman as a termna constrant s added nto the constrant condtons of the agorthm to mae systems stabe outsde the stabty doman. he smuaton resuts show the effectveness of the proposed method. Keywords Dstrbuted System, Mode redctve Contro, Lyapunov Functon, State Observer, Stabe Doman, Cooperatve Contro. Introducton In ndustra processes, there exsts a cass of hybrd systems whch are comprsed of some subsystems whch coupe each other through energy, quaty, etc. For exampe, urban dranage networ system, transportaton system, energy power, Net system and rrgaton system. hese systems have many components, wde space dstrbuton, many constrants and many targets. We can obtan good contro performance f the centrazed contro s used to contro ths nd of systems. But ts fexbty and faut toerance are reatvey wea. If the dstrbuted contro s adopted, ts fexbty and faut toerance are better [] []. So, the probem of dstrbuted contro for these hybrd systems has become an mportant research project [3] [4]. Mode predctve contro (MC s recedng horzon contro whch can dea wth the constrants of systems How to cte ths paper: Su, B.L., Zhao, Y.N. and Huang, J.M. (6 Networed Cooperatve Dstrbuted Mode redctve Contro Based on State Observer. Apped Mathematcs, 7,

2 B. L. Su et a. states and nputs durng the desgn of optmzaton contro [5]. It adopts the strateges such as feedbac correcton, rong optmzaton [6] and has strong abty to dea wth constrants and good dynamc performance [7]. herefore, t can be more effectve to sove the optma contro probem for dstrbuted systems. hat s dstrbuted mode predctve contro [8]. In recent years, the research on the dstrbuted predctve contro method has deveoped greaty. here have been many benefca resuts about t. In Lterature [9], based on the research about the Nash optma dstrbuted predctve contro, a networed predctve contro strategy s proposed for seres connecton structure systems wth networ nformaton mode whose subsystems are couped each other. In Lterature [], for a cass of near systems wth nput-output constrants, a desgn method of stabzaton dstrbuted predctve contro s gven. But every subsystem s controer can ony optmze ths subsystem s performance ndex. he optmzaton method for the whoe system s performance ndex s not gven. In Lterature [], an teratve agorthm of stabzaton controer wth constraned nput s desgned. hs method can optmze the overa performance of the system. However, t s necessary to obtan goba nformaton, whch greaty reduces the fexbty and faut toerance of the system. Lterature [] proposes a new dstrbuted predctve coordnated contro strategy to mprove the performance of the whoe system wthout ncreasng the connectvty degree. hese references are obtaned on the assumpton that the system states can be measured. However, n the actua appcaton, the mtaton of measurng equpment n economy maes the state feedbac hard to reaze. In reference [3], a dstrbuted predctve contro agorthm s desgned n the case of the states not beng measured. But ths method can ony optmze the performance of each subsystem, not the overa performance of the system. In ths paper, a dstrbuted predctve contro method based on Lyapunov functon and state observer s desgned to optmze the overa system s performance. hs agorthm adds the quadratc functon of the decoy system's nput varabes to the performance ndex of the subsystem, expands the coordnaton degree, and optmzes the performance of the system. hs paper s arranged as foows. In the second secton, the contro probem for dstrbuted system under networ mode s descrbed n deta. he output feedbac controer based on Lyapunov functons and state observers s desgned n the thrd secton, and the stabty doman s gven. he fourth secton desgns dstrbuted predctve controer. In the ffth secton, the dstrbuted predcton controers performance s anayzed, and the steps of the agorthm desgn are gven. he smuaton resuts verfy the effectveness of the method proposed n ths paper n the sxth secton. Concuson s gven n Secton 7.. robem Formuaton Consder the dstrbuted system S whch s comprsed of m reated subsystems of S can be expressed as S. he state space descrpton n j j j j j= ( j j= ( j m m ( x + = A x + B u + A x + B u j j j= ( j x where ny satsfes u( umax, and m y = Cx + Cx =,, m ( u x R denotes the state varabe of subsystem S, u R denotes the nput varabe and y R denotes the measurabe output varabe. A, B, A, B are con- n j j stant matrces wth correspondng dmenson, respectvey. he dstrbuted structure of the system under the networ mode s shown n Fgure. Syntheszng a subsystems, we can get the system mode as: where, x + = Ax + Bu (3 Cx ( y = (4 49

3 B. L. Su et a. Fgure. Dstrbuted schematc dagram of the system under networ pattern. A A m B B m C C m A= B C = = A A B B C C m mm m mm m mm x x x x = m u u u u = m y y y y = m he contro objectve s to desgn an output feedbac contro aw for the near dscrete-tme dstrbuted system (3 (4 based on Lyapunov functon and state observer under the premse that networ connectvty and faut toerance of the system are not added. And then the stabty doman s descrbed. Furthermore, tang the stabty doman as a termna constrant to desgn output feedbac mode predctve controer n order to mae the system stabe outsde the stabty doman. Mae sure that under the premse of nta feasbty, the system s successve feasbe. 3. Output Feedbac Contro Based on Lypunov Functon and State Observer hs secton shows the controer desgn based on Lypunov functon under the states are avaabe at frst to get the stabty doman descrpton. hen t shows the output feedbac controer desgn under the states are not avaabe. 3.. he State-Feedbac Controer Desgn Based on Lypunov Functon Consder the subsystem ( (, and structure the state feedbac controer as foows: = u Kx where K s the state feedbac gan. Gve the foowng assumpton: Assumpton (. For the subsystem, there exsts feedbac aw u( = Kx ( so that the egenvaues of Ad = A + BK are aways n the unt crce, and the system x( + = Ac x( s asymptotcay stabe, where Ac = A + BK, K = dag { K, K,, Km}. Defne the foowng matrces: { } A = dag A, A,, A d d d dm 5

4 B. L. Su et a. satsfy: where and Ao = Ac Ad ˆ Q Ao Ao + Ao Ad + AdA o < ˆ Q = Q + K RK > A A = Qˆ d d where,, Q dag Q, Q,, Qm = dag,,, m, R = dag { R, R,, Rm} Lemma. If the Assumpton ( s satsfed, there exsts a non-empty set Ω ( c { = x X : x ( x ( c, c > } as a nvarant set of the system x( + = Ac x(, and the system s stabe under the state feedbac contro aw u ( = Kx (, where c s the bggest to mae sure c K K umax. proof. Seect a Lyapunov functon canddate V = x ( x (. he dfference of V ( aong the trajectores of the cosed-oop system x( + = Ac x( s gven by V = V + V QR are postve dagona matrces, and = { }, { } Snce ˆ Q >, so V ( <. = x + x + x x = x A A x < x ˆ c c = x A A + A A + A A + A A x d d o o o d d o Qx( Snce the nput constrant u( umax, we have u( u max And snce u ( = Kx (, so c K K u. max. he proof s competed, and the set Ω s the nvarant set of the system. herefore, a states from Ω ( c can aways eep n Ω ( c and asymptotcay stabe at the orgn. hat s x Ω c, we have x( Ω ( c, and to say for the gven postve rea number d, f m sup x( hus, the stabty doman of the subsystem d s defned as foows: { :, } Ω c = x X x x c c > Suppose that at x Ω c, and the subsystems use contro aw Kx, so the system s asymptotcay stabe based on Lemma., a states of the subsystem satsfy 3.. Output Feedbac Controer Desgn Based on the State Estmaton Desgn the state observer as foows [4]: ( + = A + Bu + F y C (5 K ( uˆ = (6 where ˆx s the observer state of the system, F s the state observer gan to be dentfed. We can get the error dynamc equaton based on Equaton (3 and (5 as 5

5 B. L. Su et a. ( + = ( + ˆ ( + = Ax ( + Bu ( A ( Bu ( F ( Cx ( C ( = ( A FC e( e x x herefore, the error dynamc equaton of the state observer s regarded as a new autonomous system. hat s to say f the new system (7 s stabe, the estmaton states can trac the rea states we. Defne a quadratc functon on the observe error as foows: ( = E e e e where, = >. heorem. Consder the error dynamc equaton of the state observer (7, f there exst matrces and Y = F, and the nequaty s satsfed, then the nequaty ( ξ L A YC A YC > ( ξ ( = > E e + + E e + < e + Le + (9 s satsfed, where,, 3, ξ, are decay factors, L s postve defnte symmetrc matrx, and satsfes ξ L >. So there exsts >, such that f the nequaty above s satsfed, then e(. In other words, the observer state ˆx( converges to the rea state x(. roof. By the Schur compement emma, the nequaty (8 s equvaent to Substtutng Y =, ( ξ L A YC ( A YC > = F nto the above formua, we derve Mutpy e( + and e( have By (8, we have L A FC A FC ξ > +, =,, n the both sde of the above formua at the same tme, we e e e Le e A FC A FC e ξ > ξ e + e + e + Le + e + + e + + > the nequaty s satsfed. herefore, E( e( s regarded as the Lyapunov functon of zero nput dynamc error system, and satsfes the stabty constrants (9, then the autonomous system (7 s asymptotcay stabe. In other words, there aways exsts >, when >, e(, and observer state ˆx( utmatey converge to the rea state x(. Remar. By heorem, the state observer gan F can be computed off ne through the feasbty of the near matrx nequaty (8. hus, for the gven Ω ( c, f x Ω ( c, and x c, then the cosed oop system s asymptotcay stabe at the orgn. here exsts d >, such that mt sup x( t d. Aso, for the gven postve rea number ê, there exsts >, such that for any, we have x( ( eˆ. * Lemma [5]. For the gven any rea number c, there exst postve rea number e and set n ( ˆ { x * Ω c = x R : x( x( cˆ }, where ĉ< c, such that f x( ( eˆ, wth eˆ (, e, then Ω cˆ x Ω c. (7 (8 5

6 B. L. Su et a. 4. Dstrbuted Output Feedbac Mode redctve Contro hs secton studes the desgn of mode predctve controer when states are not measured. Snce the nput constrant s reated to the observer states, states constrant s constrants of the rea states, and there are some errors between rea states and observer states, the observer errors have nfuence on the future nput and states. So the observer states are used n the performance ndex drecty to desgn the controer. In order to eep the system stabe, we adopt nfnte horzon mode predctve contro strategy. herefore, the optmzaton probem at tme s as foows: ( Q R = ˆ ( + + ˆ ( + mn J mn x u ( = ( st = A + + Bu + + F y + C + ( ( ˆ ( u + = Kx + U ( x( + X (3 where ( + s state predctve vaue. uˆ ( + s nput predctve vaue,. R = R are weght coeffcent matrces. he optmzaton probem decomposes nto two parts as Suppose the Lyapunov functon satsfes the stabty constrant N (.. ( ~ ( 3 Q R = ˆ ( + + ˆ ( + = Q = Q and J x u st (4 (.. ( ~ ( 3 Q R = ( + + ( + J x ˆ u ˆ st (5 = N ( ( + = ( + ( + = ( + Vˆ ( Q R ( ˆ ( ˆ ( ˆ ( ˆ ( ˆ ( Vˆ x + + V x + x + + u + (6 When the cosed oop system s stabe, Superpose (6 from hat s to say = N to =, we get ˆ ( ˆ =, V x = ( Q R ( ˆ ( ˆ ( ˆ ( ˆ ( ˆ ( V ˆ x + N + V x x + + u + J ( Vˆ ( ( + N herefore, the optmzaton probem (5 transforms nto mnmzng V ˆ ( x ˆ ( N = N performance ( transforms nto the foowng performance N ( ˆ ˆ ˆ ( = Q R J = x + + u + + x + N We have the foowng predctve mode based on state observer of the subsystem +, and then the 53

7 B. L. Su et a. Its parta dervatve s ( + = ( + = A ˆ ( ˆ ( ˆ x + + Bu + + Aj xj ( + f + B ˆ juj + + F Cx + + Cj xj + j j C ˆ ( ˆ x + Cj xj ( + j h h ˆ ˆ j A j j ( + j = Ax + A Bu + h + A h j j j j h + A B uˆ + + A FC e + h h j j ( + A FC e + j j j j j h h A AB j jj A Bj f ˆ + f + x + h f + = + uˆ + h + h uˆ + h uˆ + h = + Because the contro aw of the subsystem affects not ony the performance of ts own subsystem, but aso that of ts downstream subsystem, controer optmzes the performance of ts own subsystem and downstream subsystem j ( j. Here, nput and state sequences got at are made as state sequences estmatons j ( + of upstream subsystem. herefore, defne the performance of subsystem as where J N x u x W = ˆ ( + + ˆ ( + + ˆ ( + + ω ( + Q R j j Q j = j ( ˆ ( ω ( + + N + x + + W + j j j h ( + = ( ˆ ( + ˆ ( + W A A B u u j jj j j And n order to mprove the convergence of optmzaton probem, the weghtng coeffcents are added. Next, the mode predctve contro optmzaton probem of a subsystems n the dstrbuted mode predctve contro agorthm s shown as: robem. For subsystem, ĉ satsfes Lemma and. ˆ ˆ ˆ, x, u +, x+ ( + and ( +, =,,, N are nown. See contro sequence uˆ ( : + to mnmze performance: s.t. mn u( : + J ( ξ x + s x + s =,,, N (8 α ˆ ˆ s s= m ( ˆ + N x( + N (9 ˆ cˆ + x ( + =,,, N ( µn j (7 54

8 B. L. Su et a. where, uˆ + U ( cˆ ( + N Ω { } m = max number of eements n α max max λmax ( AA j AA = j j =,,, N j s the neghbourng subsystem of the subsystem, and < κ <, < ξ, µ > are the desgn parameters. x ˆ ( + s the state trac under the acton of uˆ ( +, uˆ N K N + : + N, j, =,,, N. Let ( + = ( + and j ( ˆ ( A A + N = Adx + N, =. In order to guarantee the feasbty, we defne the termna c constrant set as ˆ Ω, not Ω ( ˆ c. Gve the foowng assumpton: Assumpton (. At nta moment, there aways exsts a feasbe contro aw uˆ ( + U, =,,, N of a subsyetems bounded. 5. erformance Anayss to mae observer states ( N cˆ + Ω (, and J he dstrbuted mode predctve controer based on Lyapunov functon and state observer s desgned on the condton of nta feasbty, so the man content n ths secton s to ensure successve feasbty and stabty. 5.. Successve Feasbty hs part many studes: f the system s feasbe at tme ( >, then uˆ ˆ : + N = u( : + N s the feasbe souton of the optma probem (7-( at tme. uˆ ( : + N and uˆ ( + : + N satsfy the constrant condtons of the probem. By uˆ ( : + N = uˆ ( : + N, we get ˆ + x + = whch satsfes the stabty condton (. Lemma 3. If the Assumpton ( and ( are satsfed, and the probem (7-( have feasbe souton at any tme and satsfy ˆ ˆ c x( + N Ω ρ = λ A A, then when max d d we have ( ρ ( κ max ( κ cˆ ( + N Ω roof. Snce the probem (7-( have feasbe souton at any tme, so 55

9 B. L. Su et a. ( + ˆ ( + N x N ( ˆ + N x( + N herefore, ( ˆ + N x( + N + + N = A + N, we derve From d ( + N = A ( + N d = A A + N d d λmax ( A ˆ d Ad x ( + N + cˆ ( κ cˆ ( κ + thus ( κ cˆ ( + N Ω Lemma 4. If the Assumpton ( and ( are satsfed, and the probem (7-( have feasbe souton at any tme and satsfy m α h λ mn then for a =,,, N, we have ˆ x ( ˆ + x( + (3 and ( : + N satsfes (8 (9. roof. When =,,, N, snce uˆ ˆ : + N = u( : + N, so from the predctve mode, we derve ˆ x ˆ ˆ ˆ ˆ A Ax Bu Aj xj Bjuj j j ( + = ( + ( + ( + ( + F Ce + Ce j j + A Bu + h j h ˆ ( h ˆ j j j j j j h h ( ˆ ( + A A x + h + A B u + h + A h FC e + h + A FC e + h j j j ˆ ˆ ˆ ˆ A Ax Bu Aj xj Bjuj j j ( + = ( + ( + ( + ( + F Ce + Ce j j + A Bu + h j h ( ˆ ( h ˆ j j j j j j h ( ˆ ( + A A x + h + A B u + h + h j j j h ( + + ( + A FC e h A FC e h 56

10 B. L. Su et a. he above two formuas subtract, we have ( + ˆ ( + x j j ( ˆ ( ˆ ( = A A x + h x + h h j j j ( ( ( + A FC e + h e + h h j j j By heorem, there aways exsts a >, when x. herefore, + + eventuay converges to j j λmax ( ˆ ( ˆ ( A A x + h x + h Obvousy, (3 s satsfed. Furthermore, where, ξ m λmn ( herefore, ( When = N, h j j j >, we have e(, and the observer state ˆx( ( ( ( + A FC e + h e + h h j j j h h ˆ ( ˆ ( j j j j m A A A A x + h x + h κc m α ˆ h xj( + h j + h α mn ˆ ( + ˆ ( + h x h h α + h x + h λ λ = α h, ξ α h mn m <. ( ˆ ( h ξ + satsfes the constrant (8. ( + = ( + + ( + + ( + x ˆ N A N A N B uˆ N d j j j j j j + F Ce + N + Cjej + N j ( ( ( + = ( + + ( + + ( + N A N A N B uˆ N d j j j j j j he above two formuas subtract, we have herefore, (3 s satsfed. + F Ce + N + Cjej + N j ( ( ( + ˆ ( + = ˆ ( + ˆ ( + N x N A x N x N d 57

11 B. L. Su et a. Next we can derve that ( N + satsfes (9. Lemma 5. If the Assumpton ( and ( are satsfed, and the probem (7-( has feasbe souton at any tme, then uˆ ( + U, =,,, N. roof. Snce the probem (7~ ( has feasbe souton at any tme, so uˆ + U =,,, N. hen we ony need to proof that when N 3 and 4, and trange nequaty, we derve ( + x ˆ N =, we have ˆ ( ( ( ( x ˆ + N + N + + N ( κ cˆ + < cˆ then, ( ( ˆ + N Ω c. herefore, ˆ ( ˆ ( u + N = K x + N U. u + U. By Lemma Lemma 6. If the Assumpton ( and ( are satsfed, and the probem (7-( has feasbe souton at any tme, then ˆ cˆ x ( + N Ω. roof. By trange nequaty, we have then, ( N ( + x ˆ N ( ( ( x ˆ + N + N + + N ( κ cˆ cˆ + cˆ + Ω. Remar. Accordng to Lemma to 6, f the assumpton ( and ( are satsfed, then uˆ ˆ +, x ( +, =,,, N are feasbe souton of (7-(. Snce ( ˆ + N Ω c, so x + N Ω c. by Lemma,we can derve that the cosed oop system states satsfy 5.. Stabty heorem. If the Assumpton ( and ( are satsfed, the contro aw satsfes the constrant condton (8- (, and desgn parameters κ, µ satsfy the foowng nequaty ( N κ + < µ then the system asymptotcay stabe at the orgn. roof. When ˆx( gets nto Ω ( ĉ, we adopt state feedbac contro to mae system asymptotcay stabe. Next, we ony need to prove that when ( X \ Ω ( cˆ, the system asymptotcay stabe to the orgn. Defne so By the constrant (, we have N = ˆ ( + V x = ˆ cˆ + x ( + µn 58

12 B. L. Su et a. For V (, we mae dfference as Snce ( X \ ( cˆ N ˆ cˆ V x ( + + µ = N N cˆ V V x x = µ = cˆ ( + + ( + N µ Ω, so By heorem, we have By Lemma 4, we have ( ˆ ( + + ˆ ( + = N ( ( ( = Substtute (5-(7 nto (4, we derve therefore, when the system s asymptotcay stabe Agorthm Steps (4 ˆ > c (5 ˆ cˆ x ( + N (6 ( ( ( ( N N + + (7 ( N ( N cˆ cˆ κ V ( V ( < cˆ = cˆ + µ µ ( N κ + < µ We gve the dstrbuted mode predctve contro agorthm based on Lyapunov functon and state observer. Agorthm Off-ne part:. Gve decay coeffcent ξ, stabe matrx L;. By heorem, we obtan observer gan F = Y. On-ne part:. Choose the approprate parameter QR,, and Lyapunov functon V = x ( x (, and obtan the stabty doman estmaton by cacuaton (Here s ony the form, snce the states are unavaabe, rea to use s { } ( cˆ x X : x ( x ( cˆ, cˆ. Intaze ˆx(, uˆ ( +, =,,, N, to satsfy Assumpton (. At, If ( Ω ( cˆ, adopt feedbac contro uˆ ( = K (, or cacuate ( Ω = > ; for any and downstream subsystems; 3. Receve (, j ( +, j ˆ S ; 4. Let +, repeat step., then + +, then send to upstream. If ( Ω ( cˆ, choose the feedbac contro aw ˆ Kx, or sove the optma probem, we get uˆ ( : + N, and then appy u 6. Numerca Exampe Consder the dstrbuted system under networed contro as foows: ˆ u = to the subsystem 59

13 B. L. Su et a. where: ( + = + x Ax Bu = Cx ( y A A B B C C A= B C A A = = B B C C that s ths system has two subsystems. Subsystem : x + = A x + B u + A x + B u subsystem : where, = + y C x C x ( + = x A x B u A x B u = + y C x C x A = A A A = = = B = B = B = B = C = C = C = C = Let the subsystem contro constrant as U = U = { u : u, =, } [ ]. We use the Matab smuatontoos to smuate the agorthm proposed n ths paper: By the agorthm above, we can obtan that the stabty doman of the subsystem and shown n Fgure, Fgure 3. Choose the nta states x = [, ], x = [, ], the states trac of the subsystem and are shown n Fgures 4-7, - and * are rea states and estmaton states, respectvey. Fgure 8, Fgure 9 show the nput trac of a the subsystems. From the smuaton resuts, we can see the agorthm can guarantee that estmaton stats trac the rea states we, and asymptotcay stabe to the orgn. We can aso see that the contro ow satsfed the constrant and stabe eventuay. 7. Concuson For a nd of the dstrbuted systems wth nput and state constrant and unavaabe states under networed con- Fgure. he stabty doman of the subsystem. 6

14 B. L. Su et a. Fgure 3. he stabty doman of the subsystem. Fgure 4. he state components of the subsystem x ( - representatves the rea state, * representatves the estmaton state ˆx. Fgure 5. he state components of the subsystem x ( - representatves the rea state, * representatves the estmaton state ˆx. 6

15 B. L. Su et a. Fgure 6. he state components of the subsystem x ( - representatves the rea state, * representatves the estmaton state ˆx. Fgure 7. he state components of the subsystem x ( - representatves the rea state, * representatves the estmaton state ˆx. Fgure 8. he contro ne of the subsystem. 6

16 B. L. Su et a. Fgure 9. he contro ne of the subsystem. tro patten, we consder the desgn and stabty probem of the output feedbac predctve controer based on Lyapunov functon and state observer. he man dea s: For the consdered system, use Lyapunov functon and states reconstructon to desgn output feedbac controer n order to get the stabty doman. Furthermore, the stabty doman as a termna constrant, the dstrbuted mode predctve controer s desgned. he controer s successve feasbty under the condton of nta feasbty. he smuaton resuts verfy the effectveness of the method proposed n ths paper. References [] Scatton, R. (9 Archtectures for Dstrbuted and Herarchca Mode redctve Contro A Revew. Journa of rocess Contro, 9, [] Chrstofdes,.D, Scatton, R., Muñoz de a eña, D. and Lu, J.F. (3 Dstrbuted Mode redctve Contro: A utora Revew and Future Research Drectons. Computers & Chemca Engneerng, 5, [3] Mn, H.B., Lu, Z.G., Lu, Y., Wang, S.C. and Yang, Y.L. (3 Dstrbuted Coordnaton Contro of EuerLagrange System under Swtchng Networ opoogy. Acta Automatca Snca, 39, 3-. [4] Cha,.Y., L, S.Y. and Wang, H. (3 Modeng and Controng of Compex Industra rocesses under the Networ Informaton Mode. Acta Automatca Snca, 39, [5] Su, B.L., L, S.Y. and Zhu, Q.M. (9 redctve Contro of the Inta Stabe Regon for Constraned Swtched Nonnear Systems. Scence n Chna ress, 39, [6] Zhu, J. ( Integent redctve Contro and Its Appcaton. Zhejang Unversty ress, Hangzhou. [7] Kong, X.B. and Lu, X.J. (3 Nonear Mode redvtve Contro for DFIG-Based Wnd ower Generaton. Acta Automatca Snca, 39, [8] Camponogara, E., Ja, D., Krogh, B.H. and audar, S. ( Dstrbuted Mode redctve Contro. IEEE Contro Systems,, [9] L, S.Y. (8 Mode redctve Contro and Appcaton of Goba Operatng System. Scence ress, Bejng. [] A-Gherw, W., Budman, H. and Eame, A. ( A Robust Dstrbuted Mode redctve Contro Agorthm. Journa of rocess Contro,, [] Farna, M. and Scatton, R. ( Dstrbuted redctve Contro: A Non-Cooperatve Agorthm wth Neghbor-to- Neghbor Communcaton for Lnear Systems. Automatca, 48, [] Zheng, Y., L, S.Y. and L, N. ( Dstrbuted Mode redctve Contro over Networ Informaton Exchange for Large-Scae Systems. Contro Engneerng ractce, 9, [3] Zheng, Y. and L, S.Y. (3 Networed Cooperatve Dstrbuted Mode redctve Contro for Dynamc Coupng Systems. Acta Automatca Snca, 39, [4] Zhao, M. (3 redctve Contro Strategy for LV Systems Based on Offne State Observer. Contro Engneerng 63

17 B. L. Su et a. of Chna,, 7-3. [5] E-Farra, N.H. and Chrstofdes,.D. (3 Bounded Robust Contro of Constraned Mutvarabe Nonnear rocess. Chemca Engneerng Scence, 58, Submt your manuscrpt at: 64