Neural-Based Decentralized Robust Control of Large-Scale Uncertain Nonlinear Systems with Guaranteed H Performance

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1 Proceedngs of the 45th IEEE Conference on Decson & Control Manchester Grand Hyatt Hotel an Dego CA UA December Neural-Based Decentralzed Robust Control of Large-cale Uncertan Nonlnear ystems wth Guaranteed H Performance Hroak Mukadan eshro akaguch akayosh Umeda Yoshyuk anaka Hua Xu and osho suj Abstract hs paper nvestgates an applcaton of Neural Networks (NNs) to the decentralzed guaranteed H performance for a class of large-scale uncertan nonlnear systems In order to guarantee the adequate H performance level for the nonlnear systems nonlnear lnear matrx nequalty (NLMI) condton s derved he lnear matrx nequalty (LMI) approach nstead of the NLMI s used to construct the decentralzed local state feedback controllers wth addtve gan perturbaton he novel contrbuton s that n order to avod H performance degradaton caused by the uncertanty NNs are substtuted nto the addtve gan perturbatons Although the NNs are ncluded n the large-scale uncertan nonlnear systems t s newly shown that the closed-loop system s nternally stable and the adequate H performance bound s attaned Fnally a numercal example s gven to verfy the effcency I INRODUCION In recent years the problem of the decentralzed robust control of large-scale uncertan systems has been wdely studed (see for example and the references theren) When controllng such plant t s desrable that the control systems guarantee not only a robust stablty but also an adequate level of performance One approach to ths problem s the so-called quadratc guaranteed cost control 2 hs approach has the advantage of provdng an upper bound on a gven performance ndex Recent advance n theory of lnear matrx nequalty (LMI) has allowed a revstng of the guaranteed cost control approach 3 4 for the large-scale uncertan systems Partcularly the robust nonfragle decentralzed controller desgn for uncertan largescale nterconnected systems wth tme-delay has been establshed 3 4 However the dsturbance nputs have not been consdered n these researches In fact n order to attan the desred performance aganst the external dsturbance the consderaton of the dsturbance nputs s mportant for mplementng the actual control systems he nonlnear H control problem has been consdered extensvely 5 6 Partcularly L 2 -gan analyss of nonlnear system has been tackled 5 On the other hand the solutons to the nonlnear H control problems were characterzed n terms of the nonlnear matrx nequaltes (NLMIs) H Mukadan s wth Graduate chool of Educaton Hroshma Unversty -- Kagamyama Hgash-Hroshma Japan mukada@hroshma-uacjp akaguch Umeda Y anaka and suj are wth Graduate chool of Engneerng Hroshma Unversty -4- Kagamyama Hgash- Hroshma Japan { sakaguch umeda ytanaka tsuj }@bsyshroshma-uacjp H Xu s wth Graduate chool of Busness cences he Unversty of sukuba 39- Otsuka Bunkyo-ku okyo 2 Japan xuhua@gssmotsukatsukubaacjp 6 However n general t s very hard to solve Hamlton- Jacob nequalty or NLMIs for obtanng the controller Moreover although these approaches have the advantage of provdng an controller the H performance degradaton ncurred by the uncertanty has not been consdered Neural networks (NNs) have been utlzed for an ntellgent control system because NNs have nonlnear mappng approxmaton property he numerous control methodologes utlzng NNs have been proposed by combnng the modern control theory For example the lnear quadratc regulator (LQR) problem usng multple NNs has been nvestgated 7 However there s a possblty that NNs may result n the unstable system because the stablty of the closedloop system whch ncludes the neurocontroller has not been consdered It should be noted that the system stablty may destroy when the degree of system nonlnearty s strong 7 In order to avod ths problem the stablty of the closedloop system wth the neurocontroller has been consdered 8 9 However snce much effort has been made towards fndng the guaranteed cost controller wth addtve gan perturbatons the guaranteed H performance bound has not been dscussed In ths paper the guaranteed H performance analyss and control synthess of the decentralzed robust control for the large-scale uncertan nonlnear systems wth the neurocontroller s dscussed he crucal dfference between the exstng results 8 9 and the proposed method s that the dsturbance nput s newly consdered As a result the L 2 -gan condton of the large-scale uncertan nonlnear systems s satsfed he novel contrbutons are as follows Frst n order to guarantee the adequate H performance level NLMI condton s establshed econd a class of the fxed state feedback controller of the large-scale uncertan nonlnear systems wth the gan perturbatons s derved by means of the LMI As a result snce the LMI s used nstead of the NLMI t s easy to obtan the fxed gans Fnally n order to compensate for the degradaton of the gven dsturbance attenuaton level caused by the parameter uncertantes NNs are used Although the neurocontrollers are ncluded n the large-scale uncertan nonlnear systems t s newly shown that the robust nternal stablty of the closed-loop system and the adequate H performance bound are both attaned In order to verfy the effectveness of our desgn approach the numercal example s gven he notatons used n ths paper are farly standard he superscrpt denotes the matrx transpose I n R n n denotes the dentty matrces z 2 denotes the square norm of a vector z R k he notaton L 2 ( ) wll be also used /6/$2 26 IEEE 6348

2 for vector-valued functon 5 Let us ntroduce the followng parttoned matrces II PRELIMINARY Consder the contnuous-tme large-scale uncertan nonlnear systems that s a specal knd of the multmachne power systems whch consst of N subsystems of the followng form ẋ (t) =A + A (t) x (t)+b + B (t) u (t) +G + G (t) g (x(t)) + H d (t) (a) u (t) =K + K (t) x (t) (b) z (t) =C x (t)+d u (t) = N (c) for the th subsystem x R n x(t) := x (t) x N (t) R n s the state u R m s the nput z R p s the controlled output d R q s the external dsturbance respectvely he matrces A B G H C and D are known real constant matrces of approprate dmensons that descrbe the nomnal model K s the fxed gan matrx A (t) B (t) and G (t) are real tme varyng parameter uncertantes K (t) s the addtve gan such as neural nputs g (x) R l s unknown nonlnear vector functons that represent nonlnearty between the th subsystem and the nteractons of other subsystems he uncertan matrces A (t) B (t) and G (t) and the addtve gan K (t) are assumed to be of the followng structure: A (t) B (t) = L F (t) E A E B (2a) G (t) =L G F G (t)e G (2b) K (t) =L K N (t)e K (2c) wth F (t) R r s F G (t) R t l and N (t) R v w are unknown matrx functons wth Lebesgue measurable elements and satsfyng F (t)f (t) I s F G (t)f G (t) I l N (t)n (t) I w L E A E B L G E G L K and E K are known real constant matrces wth approprate dmensons It may be noted that the systems () nclude the dsturbance nput compared wth the exstng results hat s the consdered systems () are an extenson of Wthout loss of generalty the followng assumptons concernng the unknown nonlnear vector functons are made Assumpton : here exst known constant matrces W j such that for all x j R nj g (x) W j x j (3) j= for all j and for all t u (t) d (t) u(t):= R m d(t) := R p u N (t) d N (t) z (t) g (x) z(t):= z N (t) R q g(x(t)) := g N (x) R l A := A + LF (t)e A B := B + LF (t)e B G := G + L G F G (t)e G K := K + L K N(t)E K A O L O A := L := O A N O L N E A O B O E A := B := O E AN O B N E B O G O E B := G := O E BN O G N L G O E G O L G := E G := O L GN O E GN H O C O H := C := O H N O C N D O K O D := K := O D N O K N L K O E K O L K := E K := O L KN O E KN F (t) O F (t):= O F N (t) F G (t) O F G (t):= O F GN (t) N (t) O N(t):= n := n m := m O N N (t) p:= p q := q l := l j l := l j= j Usng the above notatons the large-scale uncertan nonlnear 6349

3 systems () can be rewrtten as ẋ(t) =A x(t)+b u(t)+g g(x(t)) + Hd(t)(4a) u(t) =K x(t) (4b) z(t) =Cx(t)+Du(t) (4c) It s assumed that the performance of large-scale uncertan nonlnear systems (4) s measured n term of L 2 -gan n ths paper It wll be gven the followng defnton of fnte L 2 -gan 5 Defnton : Let γ ystem (4) wth ntal state x() = s sad to have L 2 -gan less than or equal to γ f z(t) 2 dt γ 2 d(t) 2 dt (5) for all and d(t) L 2 ( ) he followng lemma wll play an mportant role n solvng the nonlnear state feedback H control problem 5 Lemma : Consder the class of nonlnear system gven by (6) ẋ(t) = f(x)+g(x)d(t) (6a) z(t) =h(x) f(x )= h(x )= (6b) x R n d R q z R p and g(x) R n q Let γ > If there exsts a smooth soluton V of Hamlton-Jacob nequalty V x (x)f(x)+ V 2 γ 2 x (x)g(x)g (x) V x (x) + 2 h (x)h(x) V(x )= (7) then the nonlnear system (6) has L 2 -gan less than or equal to γ he followng theorem ndcates the suffcent condton for exstence of the H control synthess for uncertan nonlnear systems heorem : For a gven contnuous matrx-valued functon P (x) suppose that the followng nonlnear matrx nequalty holds for the large-scale uncertan nonlnear systems () Φ P (x)g G P < = N (8) (x) I l Φ = P (x)(a +B K )+(A +B K ) P (x) + γ 2 P (x)h H P (x) +(C + D K ) (C + D K )+W W = N j= W j W j > If such condton s met the large-scale uncertan nonlnear systems (4) have L 2 -gan less than or equal to γ In order to prove heorem the followng nequalty s needed N x j Wj W j x j g g (9) j= It should be noted that t s easy to verfy that the above nequalty holds under Assumpton Now let us prove heorem Proof: uppose now there exsts the postve defnte contnuous matrx-valued functon P (x) such that V (x(t)) = x (t)p (x) x(t) () P (x) O P (x) = O P N (x) P (x) > = N () Comparng (4) and vector form of (6) t follows that f(x)=(a + B K ) x(t)+g g(x(t)) (2a) g(x)=h (2b) h(x)=(c + DK ) x(t) (2c) V x (x)=x (t)p (x) (2d) Applyng (2) to (7) the followng equaton holds N = 2 x (t)p (x) (A + B K ) x(t)+g g(x(t)) + (A + B K ) x(t)+g g(x(t)) P (x)x(t) γ 2 x (t)p (x)hh P (x)x(t) + 2 x (t)(c + DK ) (C + DK ) x(t) Moreover t s easy to verfy the followng equalty M = N j= ( x j W j W j x j g g ) = x (t)wx(t) g g Addng the above equalty to N and usng (9) t follows that N = N + 2 (x (t)wx(t) g g M) < x(t) Φ P (x)g x(t) 2 g(x(t)) G P (x) I l g(x(t)) (3) Φ = P (x)(a + B K )+(A + B K ) P (x) + γ 2 P (x)hh P (x) +(C + DK ) (C + DK )+W 635

4 Hence the followng nonlnear matrx nequalty (4) results n N < Φ P (x)g G < P (x) I l Next the matrx nequalty (4) s wrtten by the followng form Φ O P (x)g O O Φ N O P N (x)g N G P (x) O I l O O G NP N (x) O I ln < (4) Usng chur complement for the equaton (4) t s easy to verfy that the nonlnear matrx nequalty (4) holds ff the followng reduced-order nonlnear matrx nequalty (5) holds Φ P (x)g G P < N (5) (x) I l herefore snce the condton (7) holds when the nonlnear matrx nequaltes (8) are satsfed the consdered systems (4) have L 2 -gan less than or equal to γ he proof of heorem s completed III LMI CONDIION FOR HE EXIENCE OF HE CONROLLER In vew of the prevous secton the guaranteed H performance analyss nvolve solvng the NLMIs (8) hs property also mples that the complcated computatonal effort s needed However t s well-known that t s very hard to solve the NLMIs In ths secton the LMI condton wll be establshed for the guaranteed H performance analyss and control synthess nstead of the NLMIs heorem 2: Consder the large-scale uncertan nonlnear systems (a) and addtve gan perturbaton (b) Now suppose that there exst the matrces X R n n Y R m n and postve constants ε ε 2 and ε K satsfyng the LMI (6) for all uncertan functons F (t) and F j (t) and the arbtrary functon N (t) as the neural nput hen the fxed gan matrx K = Y X attans L 2 -gan less than or equal to γ Proof: Let us ntroduce matrces X = P and Y = K P nce heorem 2 s proved by usng the chur Complement t s omtted (see eg 9) IV NEURAL NEWORK FOR ADDIIVE GAIN PERURBAION In general the exstence of the parameter uncertantes and the gan perturbatons yeld the performance degradaton he man purpose of ths paper s to mprove the degradaton of H performance usng NNs It should be noted that the proposed neurocontroller regulates ts outputs n real-tme under the nternal stablty guaranteed by the LMI approach Fxed gan K Neurocontroller K Decentralzed controller Fxed gan K 2 Neurocontroller K 2 Decentralzed controller Fxed gan K 3 Neurocontroller K u (t) d (t) u 2 (t) d 2 (t) u 3 (t) Uncertan plant wth nterconnecton ubsystem Uncertan plant wth nterconnecton ubsystem 2 Uncertan plant wth nterconnecton ubsystem 3 Decentralzed controller d 3 (t) Fg Block dagram of proposed system composed of three-dmensonal subsystems he decentralzed neurocontroller for the large-scale uncertan nonlnear systems s consdered As a specfc example the block dagram of the proposed control systems that have three-dmensonal subsystems s gven by Fg Fg shows that each neurocontroller uses only the state values of each subsystem as ts nput It should be noted that ths example s also used n the next secton In order to calculate the control sgnal for the largescale uncertan nonlnear systems the equaton () should be descrbed to the dscrete-tme system hus the contnuoustme dynamcs () s transformed to the followng dscretetme system x (k +)= Ā + Ā(k) x (k)+ B + B (k) u (k) + Ḡ + Ḡ (k) g (x)+ H d (k) (6a) u (k)=k + K (k)x (k) (6b) z (k)=c x (k)+d u (k) (6c) Ā := c A +I n Ā(k) := c A (k) B := c B B (k) := c B (k) Ḡ := c G Ḡ(k) := c G (k) H := c H and k s number of step c Rs a suffcent small samplng perod n dscrete-tme systems It should be noted that Euler approxmaton s used as a dscrete-tme approxmaton to smplfy the systems descrpton For each subsystem the NNs should be traned n real-tme so that the norm of the state dscrepancy that s gven by x (k +) between the operatng pont x = and the present state value becomes as small as possble at each step k z (t) z 2 (t) z 3 (t) 635

5 Ψ G H Θ Ξ X EK X G I l EG H γ 2 I q Θ I p + ε K D L K L K D ε K D L K L K EB Ξ ε K E B L K L K D ε I s + ε K E B L K L K EB E G ε 2 I s E K X ε K I v X W Ψ = A X + B Y +(A X + B Y ) + ε L L + ε 2 L G L G + ε K B L K L K B Θ =(C X + D Y )+ε K D L K L K B Ξ =(E A X + E B Y )+ε K E B L K L K B s = j=j s j < N (k) of the equaton (7) can be expressed as a nonlnear functon of the state x (k) the weght coeffcent w (k) of NN and the threshold θ (k) as follows U (k) nput layer w ( ) (k) hdden layer w 2 ( )(k) output layer O (k) N (k) =f (x (k)w (k)θ (k)) (7) For each subsystem an energy functon E (k) s defned as the square norm of the state dscrepancy At each step the weght coeffcents are modfed so as to mnmze E (k) gven by E (k) := 2 x (k +)x (k +)= x (k +) 2 (8) 2 U h (k) U H (k) h H threshold w (s h)(k) w ( H) (k) θ ()(k) s threshold w 2 (t s)(k) w 2 ( ) (k) θ 2 ()(k) t O t (k) O (k) E (k) can be calculated by usng the observed state value x (k +) herefore t s not necessary to consder the behavor of the uncertan matrces F (k) and F j (k) IfE (k) can be mnmzed as small as possble for each subsystem the dscrepancy x (k +) 2 would also be mnmzed In the learnng of NN the modfcaton of weght coeffcent w (k) s gven by E (k) w (k +)=w (k) η w (k) (9a) E (k) w (k) = E (k) N (k) N (k) w (k) (9b) η 2 3 s the learnng rato he term E (k) N (k) n the equaton (9b) can be calculated from the energy functon (8) as follows E (k) N (k) = x (k+) B + c L F (k)e B L K E K x (k) (2) It should be noted that snce F (k) s the unknown matrx functon (2) that s used to learn for the NN can not be calculated o remove ths problem suppose there exsts the parameter α(k) such that B + c L F (k)e B α (k) B α (k) s the matrx value functon and ts elements are postve scalar hen the above equaton (2) can be rewrtten as follows E (k) N (k) α(k)x (k +) B L K E K x (k) (2) However snce there exsts the functons α (k) t s dffcult to decde the learnng rule of the NN hen the modfcaton Fg 2 tructure of the multlayered neural networks of the weght coeffcent of the equaton (9b) s newly defned as follows w (k) µ x (k +) B L K E K x (k) N (k) w (k) (22) µ := η α (k) s defned as a new learnng rato ε s used nstead of decdng η accordng to α (k) On the N (k) other hand can be calculated usng the chan rule w (k) on the NN As a result usng (2) NN can be traned so as to decrease the guaranteed H performance on-lne he utlzed NN are of a three-layer feed-forward network as shown n Fg 2 A lnear functon s utlzed n the neurons of the nput and the hdden layers and a sgmod functon n the output layer For each subsystem nputs and outputs of each layer can be descrbed as follows U y (k) s y q (k) = {q = (nput layer)} w (yz) (k)o z (k) {q = 2(hdden layer)} w (yz) 2 (k)o z 2(k) {q = 3(output layer)} s y (k) {q = (nput layer)} o y q (k) = s y 2 (k)+θy (k) {q = 2(hdden layer)} e ( sy 3 (k)+θy 2 (k)) {q = 3(output layer)} +e ( sy 3 (k)+θy 2 (k)) s y q (k) and oy q (k) are the nput and output of neuron y n the qth layer at step k w (yz) q (k) ndcates the weght 6352

6 coeffcent from neuron z n the qth layer to neuron y n the (q +)th layer U y (k) s the nput of NN θy q (k) s a postve constant for the threshold of neuron y n the (q +)th layer As the addtve gan perturbatons defned n the formula (b) the outputs of NN are chosen adaptvely n the range of V NUMERICAL EXAMPLE In order to demonstrate the effcency of the proposed algorthm a numercal example s gven Consder the nterconnected uncertan nonlnear systems () composed of three two-dmensonal subsystems he system matrces and the nonlnear functons wth the uncertantes are gven as follows 2 A = B 5 = L = E A = 5 E B =G = 9 2 L G = E G = 2 2 L K = 8 8 E K = H = C = D = 5 A 2 = B 2 2 = L 2 = E A2 = 2 E B2 =7 G 2 = 3 L G2 = E G2 = 4 5 L K2 = 7 7 E K2 = H 2 = C 2 = D 2 = 8 8 A 3 = B = L 3 = 7 E A3 = E B3 = G 3 = 2 3 L G3 = E 5 G3 = 6 L K3 = 6 6 E K3 = H 3 = C 3 = D 3 = x (t) N (t) x (t) = N (t) = N 2 (t) ( ) 2π F (t) =F G2 (t) =sn 5 t F G (t) =F 3 (t) = ( ) 2π F 2 (t) =F G3 (t) =cos 5 t W = W 22 = W 33 = W 2 = W 3 = W 2 = W 23 = W 3 = W 32 =5 W sn ( ) x g (x) = + + WN sn ( ) x N W sn ( ) x + + WN sn ( ) x N x () = x 2 () = x 3 () = he unknown functons g (x) satsfy g (x) = W x + + W N x N he three basc quanttes for the subsystem are γ = γ 2 = 733 and γ 3 = respectvely hus for every boundary value γ > γ = max { γ γ 2 γ 3 } = the reduced-order LMIs (6) have the solutons he dsturbance attenuaton level s chosen as γ = In ths case the fxed state feedback gan K that s based on the proposed LMI (6) s gven by K = K 2 = K 3 = In order to compare the obtaned result a dfferent gan s obtaned he nterconnected nonlnear systems that gnore the uncertanty are gven as follows x(t) =A x(t)+bũ(t)+gg( x(t)) + Hd(t) (23a) ũ(t) = K x(t) z(t) =C x(t)+dũ(t) (23b) (23c) ũ (t) x(t) := R n ũ(t) := ũ 2 (t) R m ũ 3 (t) z (t) K O z(t) := z 2 (t) R q K = K 2 z 3 (t) O K3 Usng the smlar technque used n heorem 2 the state feedback gan K by means of LMI that s based on (24) s gven below K = K 2 = K 3 = K := Ỹ X Φ = A X +B Ỹ +(A X +B Ỹ ) Φ G H (C X + D Ỹ ) X G I l H γ 2 I q < C X + D Ỹ I p X W (24) For each subsystem the neurocontroller s composed of 3 neurons n the hdden layer On the other hand there exst two neurons n the nput and the output layers respectvely he state varables of each subsystem are used as the NN s nputs and the learnng rato µ =8 µ 2 =6 and µ 3 = 6 he ntal weghts are randomly chosen n the range of 5 5 Moreover the external dsturbances are chosen as follows ( ) 2π d (t)=sn 3 t d 2 (t)=sn ( ) ( ) 2π 2π 6 t d 3 (t)=sn 9 t he smulaton results are shown n Fg 3-5 It s easy to verfy that the nfluence of the external dsturbance to the 6353

7 tate varables x Control nput u (a) Wth neurocontroller Wthout neurocontroller Nonlnear system Wth neurocontroller Wthout neurocontroller Nonlnear system 2 (c) 3 me s 3 me s tate varables x2 tate feedback gan wth addtve gan (b) - -4 Wth neurocontroller Wthout neurocotroller Nonlnear system K 2 +8N 2 (t) K +8N (t) -5 2 (d) Fg 3 mulaton results of subsystem 3 me s 3 me s Control nput u (a) tate varables x Wth neurocontroller Wthout neurocontroller Nonlnear system Wth neurocontroller Wthout neurocontroller Nonlnear system (c) 3 me s 3 me s tate varables x32 tate feedback gan wth addtve gan (b) (d) Fg 5 mulaton results of subsystem 3 - Wth neurocontroller Wthout neurocontroller Nonlnear system K 32 +6N 32 (t) K 3 +6N 3 (t) 3 me s 3 me s Control nput u2-2 (a) tate varables x Wth neurocontroller Wthout neurocontroller Nonlnear system Wth neurocontroller Wthout neurocontroller Nonlnear system -5 2 (c) 3 me s 3 me s tate varables x22 tate feedback gan wth addtve gan (b) -5 5 K 2 +7N 2 (t) K 22 +7N 22 (t) Wth neurocontroller Wthout neurocontroller Nonlnear system 2 (d) Fg 4 mulaton results of subsystem 2 3 me s 3 me s proposed system that s based on the NLMI s smaller than that of the large-scale systems usng the gan K Moreover the dsturbance attenuaton of the proposed neurocontroller s observed herefore t s shown that the proposed NLMI method wth the neurocontroller result n the effectveness on the suppresson of the dsturbance Fnally t s easy to verfy that L 2 -gan of the proposed closed-loop system s 766 that s less than γ = hus t s shown that the proposed method satsfy the boundary of L 2 -gan hat s t s emphaszed that the desred dsturbance attenuaton level can be attaned VI CONCLUION In ths paper the guaranteed H performance analyss and control synthess of the large-scale uncertan nonlnear systems wth the neurocontroller has been studed Usng the NLMIs approach the suffcent condton that s related to the exstence of the guaranteed H controller for the large-scale uncertan nonlnear systems wth the addtve gan perturbatons has been derved Moreover the class of the fxed state feedback controller has been newly establshed by means of the LMIs As another mportant feature the robust stablty of the closed-loop system s guaranteed even f the systems nclude NNs Moreover the guaranteed H control synthess wth NNs has succeeded n reducng the degradaton of the gven dsturbance attenuaton level that s caused by the parameter uncertantes Fnally the numercal example have shown the excellent result REFERENCE Y Guo DJ Hll and Y Wang Nonlnear decentralzed control of large-scale power systems Automatca vol36 no9 pp IR Petersen and DC McFarlane Optmal guaranteed cost control and flterng for uncertan lnear systems IEEE rans Automatc Control vol39 no9 pp ept JH Park Robust nonfragle decentralzed controller desgn for uncertan large-scale nterconnected systems wth tme-delays rans AME vol24 no4 pp JH Park Robust non-fragle guaranteed cost control of uncertan large-scale systems wth tme-delays n subsystem nterconnectons Int J ystems cence vol35 no4 pp AJ van der chaft L 2 -gan analyss of nonlnear systems and nonlnear state feedback H control IEEE rans Automatc Control vol37 no6 pp W-M Lu and JC Doyle H control of nonlnear systems: a convex characterzaton IEEE rans Automatc Control vol4 no9 pp ept Y Igun H aka and H okumaru A nonlnear regulator desgn n the presence of system uncertantes usng multlayered neural networks IEEE rans Neural Networks vol2 no4 pp4-47 July 99 8 H Mukadan Y Ish Y anaka N Bu and suj LMI based neurocontroller for guaranteed cost control of dscrete-tme uncertan system n Proc 43rd IEEE Conference on Decson and Control Bahamas 24 pp H Mukadan Y Ish suj Decentralzed guaranteed cost control for dscrete-tme uncertan large-scale systems usng neural networks n Proc IFAC World Congress Prague Czech Republc 25 K Zhou Essentals of robust control New Jersey: Prentce Hall