46-57 388 4 3 3 sariai@ee.knu. ac.ir agira@knu. ac.ir 3 labibi@knu. ac.ir (388// 388/9/8 :. -..... :. A New Approac o Boune Real Lemma Represenaion for Linear Neural Sysems Ala Sariai, Hami Reza agira an Baool Labibi Absrac: is paper is concerne wi boune real crierion for linear neural elay sysems. wo new elay-epenen boune real lemmas (BRLs are obaine in is paper, in wic, Lyapunov eory is use o erive e firs elay-epenen represenaion for BRL. Using a escripor moel ransformaion of e sysem an a new Lyapunov-Krasovskii funcional, a less conservaive boune real lemma is obaine compare o a of e firs BRL. en sufficien coniions for e sysem o possess an -norm less an a prescribe level, is given in erms of a linear marix inequaliy (LMI. e significan avanage of e erive boune real lemmas is eir efficiency in esigning conroller for e close-loop neural sysems wen elaye erm coefficiens epen on e conroller parameers. Numerical examples are given wic illusrae e effeciveness of our propose BRLs. Keywors: Boune Real Lemma (BRL, Neural sysem, escripor moel, Linear Marix Inequaliy (LMI..[4].[8]-[] -.[]-[3] Journal of Conrol, Vol. 3, No 4, Winer 388 4 3
47. K K. A j A j.... 3. 5. 4.. 6 - : x( = Ax ( + Ax ( τ + Ax( τ + Ew ( z ( = Cx ( + D ( ( ( (3 x τ + Dx τ + Dw x ( + θ = φ( θ θ [ τ,] w R p x z R q τ L [, E R np A A A R nn R nn R nn. D R q p D D C R q n R q n R q n.. N R nanb b(. R nb a(. R na :[7] X R nanb Ω a ( α Nb( α a( α X Y Na( α, Ω b( α Y N b( α X Y Ω Y >. R nanb Y R nanb - ([5]-[7]... -..[8].[8]-[] [] Xu Wang..[] Sake Friman.[3]. [8] Xu.[5] [4]. []. [4] Cen Guo : ( = ( + ( i i x Ax A x k j = j n i = ( j ( + A x + Ew A j (. A j.. [6] : ( = ( + ( τ + ( τ + ( x Ax BK x BK x Ew ( Journal of Conrol, Vol. 3, No 4, Winer 388 4 3
48 Ω PA Y Y A τy PE + C D τa A R τa A A A R C * Q D D τa A R τa A ( AA R D * * R D D τa A R τ( A + AA ( A + AA R D * * * R τ ( A ( A R * * * * D D γ I τe E R τe A E A R < (5 * * * * * γ I τe E R * * * * * * τ * * * * * * * R * * * * * * * * τ * * * * * * * * * R * * * * * * * * * * I J < zw < γ zw. (=[ w ( w ( < γ ] J < J <. J < (3 (LMI.. (3 : (4 J(w < γ> τ > P Q R R w L [, R nn X Y X Y Y > X τy > τy R nn. (5-(7 ( Ω= A P+ PA+ Y + Y + τ X + X + Y + Y + Q (3 -. V = V+ V + V3 + V4 V = x( Px( = τ+ β V x ( α x( α β (6 (7. (8 (9 ( :. ( :[8] R y Q y S y R y S y Q( y S( y < S ( y R( y ( <, ( ( ( ( <, A M = B :[9] 3 : M = [ A B] { σ( A σ( B } σ( M { σ( A σ( B } max, max, I -3 γ >. ( = ( γ J w z z w w τ (4. (3 :[] 4 z < γ (3 ( = ( γ ( = ( γ J w z z w w τ J w z z τ. (=[ w ( w ( ] γ. < zw :[] : Journal of Conrol, Vol. 3, No 4, Winer 388 4 3
49 X Y > (5 Y φ(=, [-τ,] γ >.V(q( = = z ( = ( γ J w z z τ (6 w ( (6 ( = w ( z ( = ( J w z z γ w w γ w w τ : : V( V( = = ( J z ( w = z z γ w w γ w w V + ( τ + V( V( = ( z zγ w w γ w w + V ( τ { : Jz ( w x C Cx+ x C Dx( τ + xcdx ( τ + xcdw + x ( τ DDx( τ + x ( τ DDx ( τ + x ( τ DDw + x ( τ DDx( τ + x ( τ DDw + wddwγ ww γ ww + V (} τ ζ Πζ τ (7 : J z (8 ζ = x( x( τ x( τ x( τ w( w( Σ Σ Σ3 Σ4 Σ5 Σ 6 * Σ Σ3 Σ4 Σ5 Σ 6 * * Σ33 Σ34 Σ35 Σ 36 Π = * * * Σ44 Σ45 Σ 46 * * * * Σ55 Σ56 * * * * * Σ66 (9 V = x ( α Qx( α + x ( α R x( α 3 τ τ β 4 = ( ( τ τ + η + ( / x ( α R x( α, τ V τ x α x α ηβ ( ( R R Q P. V( = X Y 4 V i i = : V x { A P + PA + τ ( X + X / + Y+ Y + τ ( Y + Y } x + x (( PA Y x ( τ + x ( τ ( PA Y x ( + x (( PA Y x ( τ + x ( τ( PA Y x ( τ x ( Y x ( τ τ x ( τ Y x ( + x A ϒAx + x A A ϒAAx + x A ϒA x ( τ + x A A ϒAAx ( τ + x A A ϒ ( A + AA x ( τ + x A ϒAx ( τ + x A A ϒAx ( τ + x ( τ( A ϒAx ( τ + x ( τ( AA ϒAAx ( τ + x ( τ( A ϒAx ( τ + x ( τ( A A ϒ ( A + AA x ( τ + x ( τ( A A ϒAx( τ + x ( τ( A ϒAx( τ + x ( τ( A + AA ϒ ( A + AA x( τ + x ( τ ( A + AA ϒAx( τ + x ( τ( A ϒAx( τ + x ( Qx ( x ( τ Qx ( τ x ( τ Rx( x ( τ( R / x ( τ + x ( PEw ( + ( Ax ( + Ax ( τ + Ax ( τ + Ew (. ϒEw ( + x ( A A ϒEw ( + x ( τ ( A A ϒEw ( + x ( τ ( A + AA ϒEw( + x ( τ ( A ϒEw ( + w ( E A ϒEw ( + w ( E ϒ Ew( (3 ϒ = R/ + ( τ / ϒ = R+τ Y > (4 Journal of Conrol, Vol. 3, No 4, Winer 388 4 3
5 Ω PA Y Y A τy PE + C D τa A R τa A A A R C * Q D D τa A R τa A ( AA R D * * R τy D D τa A R τ ( A + AA ( A + AA R D * * * R τ ( A ( A R * * * * D D γ I τe E R τe A E A R < ( * * * * * γ I τ E E R * * * * * * τ * * * * * * * R * * * * * * * * τ * * * * * * * * * R * * * * * * * * * * I (5 (4 LMI Π< (5-(7. :. A A Y. (5 LMI Y. (3 (.A ( Y = [8]-[] A (5 LMI..... (3 : (4 J(w < γ> < τ τ w L [, P Q R R nn R X R nn Y R nn. (-( τ ( / Σ = A P + PA + Y + Y + X + X + Y + Y + Aϒ A + A A ϒ AA + Q + C C Σ ϒ ϒ = PA Y Y + A A + A A AA + C D Σ3 = PA τy + A ϒA + A A ϒ ( A + AA + C D Σ4 = A A ϒA Σ5 = PE + A ϒE + A A ϒAE + C D Σ6 = A A ϒE Σ = Q + A ϒA + ( AA ϒAA + D D Σ3 = A ϒA + D D + A A ϒ ( A + AA Σ = A A ϒ A 4 Σ5 = A ϒE + ( AA ϒAE + DD Σ6 = ( AA ϒE Σ33 = R + A ϒA + D D + ( A + AA ϒ( A + AA Σ34 = ( A + AA ϒA Σ35 = A ϒE + ( AA + A ϒAE + D D Σ36 = ( A + AA ϒE Σ44 = R /+ A ϒA Σ45 = A ϒAE Σ46 = A ϒE Σ55 = E ϒE + E A ϒAE + D D γ I Σ56 = E A ϒE Σ = E ϒ Eγ I 66 J z < Π< w (, w ( L [, 4. zw < γ. z < γ < z γ X Y Y > ( X = X /, = /, R = R /, Y = τ Y Journal of Conrol, Vol. 3, No 4, Winer 388 4 3
5 x ( = Ax ( + Ax ( τ + Ax ( τ + Ew ( : Λ J z ( ζ Πζ + x (( τy x( + x ( τ( τy x( τ τ J z < Y.Π< (4 LMI = Y < (7 ζ Π ζ <.Π<. ( ( (5 LMI τ Y = Y ( τ ( τ X Y Y > : : X = X / ( τ ( ( τ X Y / Y > ( τ ( ( τ ( τ ( ( τ (8 X Y / Y (9 X Y / Y > ( τ ( ( τ X Y / Y >. (8 (3 ( :3 V τ -.. :II - 4 (3 (3. : ( = y( = y( + Ax( + A x( τ + A y( τ + Ew( x X τ Y > τ Y ( ( τ Ω= + + + + + + + A P PA Y Y τ X X Y Q ( - :. (8 -(. V( x (( τy x( τ Y = Y. Y= Y < V (3 x ( τ( τy x( τ : 4 i i= V( = V x { A P+ PA+ τ ( X + ( / X X Y ( } (( ( τ + Y+ Y + Y+ Y x+ x PA Y x + x ( PA x ( τ + x ( PEw( + ( x ( + x( τ ( τy ( x ( + x( τ x ( Yx( τ x ( τ Yx( + ( τ / x ( x( + ( / x ( Rx ( ( / x ( τ Rx ( τ + ( Ax + Ax ( τ + Ax ( τ + Ew( τ. ( Ax + Ax ( τ + Ax ( τ + Ew( + x ( Qx( x ( τ Qx( τ + x ( Rx ( x ( τ Rx ( τ + x ( ( τy x( + x ( τ( τy x( τ = Λ Y > X Y > Y Λ = Λ + x ( ( τy x( + x ( τ( τy x( τ (3 (4 (5. (3 Λ (6 Λ x( = ( / ( Ax( + Ax( τ + Ax ( τ + Ew( Journal of Conrol, Vol. 3, No 4, Winer 388 4 3
5 Q C Φ Y + P P A A E R τ A R τa Q D * R A R τ A D * * R A R τ A * * * D D γ I * * * * γ I E R τe < Q * * * * * R * * * * * * τ * * * * * * * R * * * * * * * * τ * * * * * * * * * I x( V = x ( y ( EnP y ( τ + β + τ+ β V = y ( α y( α β y ( α y( α β (36 (37 V3= x ( α Qx( α + y ( α Ry( α τ τ (38 + y ( α R y( α τ R R Q I n P,, En = P P P = = > P P 3 V V = x ( Px ( x ( = x ( y ( P y ( = x ( y ( P (39 Ax ( A x ( τ ( + + x ( y ( P y ( Ay ( τ ( + = x ( y ( P Ew ( τ τ x( τ = x( x ( α y ( τ = y ( y ( α - (4 (4 ( ( ( ( ( τ ( τ x I x En = y A I y x + w A A + y E I n En = ( (33 (3 (3.. (3.3 (4 J(w < γ> τ > w L [, P Q R R nn R X Y P P 3 R nn X Y Y. (33 (34 I A Φ = P + P+ τ X + Y + Y A I I I (34 P. P = P P3. (3 (3. (3 - V = V + V + V3 (35 Journal of Conrol, Vol. 3, No 4, Winer 388 4 3
53 Q x( ( ( R y( Q x( τ ( τ ( τ R y( τ ( ( ( τ ( τ (44 ( τ ( τ ( ( τ ( ( τ ( τ ( τ ( ( ( τ Ew( + y ( τ A REw( ( ( ( τ ( τ V 3 = x y x y y A R Ay y A R A y + + + y A R A y + y A R A y + y A R A y + y A R A y + y A R Ew + y A R + w E R Ew y R y (43 (4 V = V + V + V : V 3, (44 I x ( V x ( y ( P A Iy ( x( x( + τ x ( y ( X x ( y ( Y y ( + y ( x ( τ + x ( y (( Y N + y ( τ + x ( y ( P w( E Q x x ( y ( ( + R+ τ+ A ( R+ τ A y ( Q x ( τ x ( τ y ( τ R y ( τ + y ( τ A ( R+ τ Ay( τ + y ( τ A ( R+ τ Ay( τ + y ( A ( R+ τ A y( τ + y ( A ( R+ τ A y ( τ + y ( τ A ( R+ τ Ay ( τ + y ( A ( R+ τ Ew ( + y ( τ A ( R+ τ Ew ( + y ( τ A ( R+ τ Ew( + w ( E ( R + τ Ew( y ( τ R y( τ (45 J ( w z zγ w w γ w w + V ( τ z = ( ξ Πξτ (46 ξ = x ( y ( x ( τ y ( τ y( τ w ( w( x ( V = x ( Px ( = x ( y ( P I x ( = x ( y ( P A A I A y ( + + x ( α x ( y ( P A A τ y ( α + x ( y ( P w ( E a( α = x ( b( α = cl xcl ( = x ( y ( τ x ( α N = P y ( α A A I V xcl P x cl A A I A + + X Y N xcl ( + x cl ( x cl ( α τ Y N x cl ( α + xcl ( P w ( E I = xcl P xcl + τ xcl ( Xxcl ( (4 A I + x ( Yx ( + x (( Y + N x ( τ cl cl cl cl y ( α + y ( α y ( α τ y ( α + xcl ( P w ( E X Y V 3 Y (4 V = = x ( ( ( τ τa A y ( + + τy ( τ A Ay( τ + τy ( τ A Ay( τ + τy ( A A y( τ + τy ( A A y ( τ (43 + τy ( τ A Ay ( τ + τy ( A Ew( + τy ( τ A Ew( + τy ( τ A Ew( + τw ( E Ew ( V x y y( α ( α ( α α y ( α y y τ Journal of Conrol, Vol. 3, No 4, Winer 388 4 3
54 Q C Φ Y + P P A A E R τ A R τ A Q D * R A R τ A D * * R A R τ A * * * D D γ I * * * * γ I E R τe < (47 Q * * * * * R * * * * * * τ * * * * * * * R * * * * * * * * τ * * * * * * * * * I 3 :4. LMI. (3. (45 τ.. (3 : w L [, J(w τ < τ( < τ X Y R nn P Q R R R nn (47 P P 3 R nn X Y Y I A Φ = P + P+ τ X + Y + Y A I I I. (48 (48 P. P = P P3 :5. Φ Y P P + A A E Q * Π = R * * R * * * DD γ I * * * * γ I + Λ Λ< I A Φ = P + P+ τ X + Y + Y A I I I Q C R τ A R τa D Λ = AR τ A D AR τ A ER τ E Q R τ = R τ I (33 LMI..J< Π< =. (34 (4. Journal of Conrol, Vol. 3, No 4, Winer 388 4 3
55.4. A = A = A = 5.5. E= [.5 ] C = [ ] D=.4 D = [. ] D = [.3.7] A D D 3 γ. τ γ (eorem γ (eorem 3. τ γ / /45 /64 / 3/ /4 /3 3/97 /55 /4 5/5 3/9 /5 6/47 3/97 /6 8/37 4/94 (3 H 3 3.. γ γ τ (eorem τ (Corollary. 3 τ τ /5 / /9 3 /8 /37 3/5 /5 /44 4 /3 /5 4/5 /35 /56 γ 3 5 /39 /6.. -5 (3 :. A = A = E = [.5 ].9 C = [ ] [8] [].D = D = A =. [] γ τ=/846.. Sake, Yaes an e Souza [] Du an ang [] Friman an Sake [] eorem eorem 3 γ /. / / / 3 γ [] /364. [] 3 γ. [8] [] [].. [8] γ 3. (3 3 : Journal of Conrol, Vol. 3, No 4, Winer 388 4 3
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57 [] Du, H., ang, N., 7, "H conrol of acive veicle suspensions wi acuaor ime elay", Journal of Soun an Vibraion, 3, -, 36-5. Submie o Journal of Opimizaion eory an Applicaions. [] Sake, U., Yaes, I. an e Souza, C., 998, "Boune Real Crieria for Linear ime-delay Sysems", IEEE ransacion on Auomaic Conrol, 43, 7, 6-. Journal of Conrol, Vol. 3, No 4, Winer 388 4 3