Static Output Feedback Sliding Mode Control for Nonlinear Systems with Delay

Transcription

1 AMSE JOURNALS 04-Series: Avances C; Vol. 69; N ; pp 8-38 Submie July 03; Revise April 5, 04; Accepe May, 04 Saic Oupu Feeback Sliing Moe Conrol for Nonlinear Sysems wih Delay H. Yao, F. Yuan School of Mahemaics an Saisics, Anyang Normal Universiy, Anyang, Henan, 45500, China (yaohejun@6.com) Absrac: he problem of saic oupu feeback sliing moe conrol for a class of nonlinear elay sysems wih norm-boune uncerainies is consiere in his paper. Base on Linear marix inequaliy approach, a new approach is given o esign he saic oupu feeback sliing moe surface. hen, a sliing moe conroller is obaine which make he sysems saes reach he sliing moe surface in finie ime. All he coniions are expresse in erms of LMI. Finally, a numerical example is given o emonsrae he valiiy of he resuls. Key wors Delay sysems, saic oupu feeback, sliing moe conrol. Inroucion ime elay is frequenly encounere in various engineering, communicaion, an biological sysems. he characerisics of ynamic sysems are significanly affece by he presence of ime elays, even o he exen of insabiliy in exreme siuaions. herefore, he suy of elay sysems has receive much aenion, an various analysis an synhesis mehos have been evelope over he pas years [ 6]. As is known, base on using of isconinuous conrol laws, he sliing moe conrol approach is known o be an efficien alernaive way o ackle many challenging problems of robus sabilizaion. Li consiere he problem of aapive fuzzy sliing moe conrol for a class of nonlinear ime elay sysems [7]. Kown gave an improve elay-epenen coniion o esign robus conroller for uncerain ime-elay sysems [8]. Base on LMI approach, Chen consiere he problem of exponenial sabiliy for uncerain sochasic sysems wih muliple elays [9]. Xia an Qu esigne he robus sliing moe conroller for uncerain sysems wih elays by using 8

2 LMI approach [0,]. he problem of iscree-ime oupu feeback sliing moe conrol for imeelay sysems wih uncerainy is researche in []. Bu he resuls abou saic oupu feeback sliing moe conrol for elay sysems have never been presene. his paper presens he problem of saic oupu feeback sliing moe conrol for a class of nonlinear elay sysems wih norm-boune uncerainies. Base on Linear marix inequaliy approach, a new approach is given o esign he saic oupu feeback sliing moe surface. hen, a sliing moe conroller is obaine which make he sysems saes reach he sliing moe surface in finie ime. Noaions: hroughou he paper,. Problem Formulaion n R enoes he n imensional Euclien space. Consier he following nonlinear sysems wih elay x () = ( A+Δ A ()) x () + ( A +ΔA()) x ( ) + Bu () y () = Cx () x () = ψ () 0 () n m Where x () R is sysems sae, u () R is sysems conrol inpu, y () R p is sysems n n oupu. is a sysems sae elay. ψ () is he given iniial sae on [,0]. A R, n n m n A R, B R an p n C R are known consan marices. B has full column rank. n n ΔA () R n n an ΔA () R are unknown marices represening he uncerainies an saisfying [ ΔA ( ) Δ A( )] = GD ( )[ H H ] () where GHan, H are consan marices wih appropriae imensions, D () is unknown marix saisfying Wih Singular Value Decomposiion of B, D () D() I Ω B= [ U U] V 0 U 0 a nonsingular ransformaion = is consruce for sysems () o make B = U B. Wih he ransformaion z () = x (), he sysems()can be rewrien as z () z A A x A A x Bu () = = ( +Δ ()) () + ( +Δ ()) ( ) + () z () = + Δ + + Δ + ( A A( ) ) z( ) ( A A( ) ) z( ) Bu( ), 9

3 Insering ()ino he above formulaion, we obain z () = ( U AU + U GDHU ) z () + ( U AU + U GDHU ) z () + ( U A U + U GDH U ) z ( ) + ( U AU + U GDH U ) z ( ) z () = ( U AU + U GDHU ) z () + ( U AU + U GDHU ) z () + ( U A U + U GDH U ) z ( ) + ( U AU + U GDH U ) z ( ) + B u( ) (3) For he sysems(3), selecing he saic oupu feeback sliing moe surface as following Wih σ σ () = Sy() (4) ( ) Sy( ) SC z( ) SC[ U U] z( ) SCU z( ) SCUz( ) 0 = = = = + =, by he assumpion ha SCU is nonsingular, we obain where F = ( SCU ) SCU. obaine z () = ( SCU ) SCU z () = Fz () Insering he above formulaion ino he sysems(3), he sliing moe equaion is where z () = Az () + A z ( ) (5) A= U AU ( U F) + U GDH( U U F) A = U A ( U U F) + U GDH ( U U F) 3. Main Resuls Lemma [3] For known consan ε > 0 an marices DEF,, saisfying following marix inequaliy is hol DEF + E F D εdd + ε E E Y( x) W( x) Lemma [5] he LMI > 0 * Rx ( ) is equivalen o Rx ( ) > 0, Yx ( ) WxR ( ) ( xw ) ( x) > 0 FF I, hen he where ( ) Yx= Y( x), Rx ( ) = R( x) epen on x. heorem For he given consanα > 0, he sliing moe equaion is sable, if here exis posiive-efinie marices PQ, % R % ( n m) ( n m), marices % % % ( n m) ( n m) 3, X, N, N, N R consans ρ, ρ an marix 3 Z R m ( n m) such ha he following linear marix inequaliy hols 30

4 where Σ Σ Σ3 N% Σ 5 Σ Σ N% Σ Σ Q% 0 α I N% < Σ = N% + N% U A( U X UZ) ( U X UZ ) A U + αu GG U Σ = N% N% U A ( U X U Z) ρ ( U X U Z) A U + αρ U GG U Σ = P% + N% + X ρ ( U X U Z) A U + αρu GG U Σ 5 = ( UX UZ ) H Σ = N% N% ρu A ( U X U Z) ρ ( U X U Z) A U + αρ U GG U Σ = N% + ρ X ρ ( U X U Z) A U + αρ ρu GG U Σ 5 = ( UX UZ ) H Σ = Q% + ρ X + ρ X + αρu GG U We can Design he sliing moe surface σ () = Sy() where marix S saisfying (6) SC( U F U ) = 0, F = ZX Proof: Selecing Lyapunov funcional such as where PQ, are wo posiive-efinie marices. 0 + θ V () z () Pz () z () sqz () ss θ = + hen, along he soluion of sysem (5) we have V ( ) = z ( ) Pz ( ) + z ( ) Qz ( ) z ( s) Qz ( s) s + ( z ( ) N 3 + z ( ) N + z ( ) N )( z ( ) z ( ) z ( s) s) + ( z ( ) M + z ( ) M + z ( ) M )( Az ( ) A z ( ) + z ( )) 3 z ( ) Pz ( ) + z ( ) Qz ( ) z ( s) Qz ( s) s + ( z ( ) N + z ( ) N + z ( ) N )( z ( ) z ( )) + ( z ( ) M + z ( ) M 3 + z ( ) M )( Az ( ) A z ( ) + z ( )) + ( z ( ) N 3 + z ( ) N + z ( ) N ) Q ( z ( ) N + z ( ) N + z ( ) N ) z = ξ ( ) Ξξ( ) ( s) Qz ( s) s 3

5 where N, N, N 3, M, M, M 3 are consan marices wih appropriae imensions o be confirme. he inequaliy is equivalen o where N ξ () = () ( ) () z z z Ξ Ξ Ξ3 Ξ= Ξ Ξ3 Ξ 33 Ξ = N + N M A A M + N Q N Ξ = N N A M M A + N Q N Ξ = P + N A M + M + N Q N Ξ = N N M A A M + N Q N Ξ = N A M M + N Q N 3 Ξ = Q + M + M + N Q N Ξ< 0 (7) MU G N Q N N N 3 MUG D H U UF H U UF N 3 MU 3 G [ ] Ξ=Θ+ 0 ( ) ( ) 0 0 MU G [ HU ( UF ) H ( U UF ) 0] D MU G < 0 MU 3 G Θ Θ Θ Θ= Θ Θ Θ Θ = N + N MU AU ( U F) ( U U F) A U M Θ = N N MU A ( U U F) ( U U F) A U M Θ = P+ N + M ( U U F) A U M Θ = N N M U A ( U U F) ( U U F) A U M Θ 3 = N3 + M ( U UF) A UM3 Θ = Q + M + M Wih lemma,we know ha he following inequaliy hols for given consanα > 0 3

6 MU G MU GD HU UF H U UF MU 3 G [ ( ) ( ) 0] MU G [ HU ( UF ) H ( U UF ) 0] D MU G MU 3 G [ HU ( UF) H ( U UF) 0 ] [ HU ( UF) H ( U UF) 0] α MU G MU G + α MU G MU G MU 3 G M3UG Wih lemma,we know ha he inequaliy(7)is equivalen o Where Δ Δ Δ3 N Δ5 N Δ Δ Δ Δ= Δ < Q 0 α I N 0 0 Δ = N + N M U A( U U F) ( U U F) A U M + αm U GG U M Δ = N N M U A ( U U F) ( U U F) A U M + αm U GG U M Δ = P + N + M ( U U F) A U M + αm U GG U M Δ = ( U U F) H 5 Δ = N N MU A ( U UF) ( U UF) A U M + αm U GG U M Δ = N + M ( U U F) A U M + αm U GG U M Δ = ( U U F) H 5 Δ = Q + M + M + αm U GG U M Pre- an Pos-muliplying he inequaliy (8) by iag{ M, M, M, M, I} an giving some ransformaions an iag{ M, M, M, M, I} M = M, M = ρ M, M = ρ M, X = M, Z = FX, P% = XPX, Q% = XQX, where ρ, ρ 3 are consans o be obaine, we know ha he inequaliy (8) is equivalen o (6). From he inequaliy(6),we obain he sliing moe equaion is sable. V () ξ () Ξ ξ() < 0 (8) 33

7 Remark Compare wih raiional sliing moe surface conroller esign approach, hree marices N%, N%, N% are inrouce as slack variable in orer o obain less conservaive resuls 3 an more complicae coniion. Meanwhile, we have propose a new approach obain he saic oupu feeback sliing moe surface coniion which can achieve he aim of furher reucing conservaism. heorem For he nonlinear elay sysems (), wih he conroller SC σ ( ) u( ) = ( SCB) [ SCAx( ) + SCAx( ) + ( GH + GH σ ( ) (9) + B ρ( )) + kσ( ) + εsignσ( )] ω where k, ε are consans saisfying k > 0, ε > 0, hen he sysems saes will reach he sliing moe surface(4)in finie ime. Proof: Along he soluion of sysem () we have σ () σ() = σ () SC( A+Δ A) x() + σ () SC( A +ΔA ) x( ) + σ ( ) SCB ω( ) σ ( ) SCAx( ) σ ( ) SCA x( ) ω σ ( ) SC σ( ) ( ( )) GH + GH + Bω ρ σ ( ) kσ ( ) σ ( ) σ ( ) εsignσ( ) σ ( ) kσ( ) σ ( ) εsignσ( ) < 0 (0) Wih he conroller(9)an he above equaion(0), we know ha he reaching coniion is saisfie. Remark In he following, when he ime elay = 0, he sysem () will be simplifie as x () = ( A+Δ A ()) x () + Bu () () I is obvious ha he propose marix ransformaion meho can be sill applie for he sysem (). As a resul, he saic oupu feeback conroller esign schemes will be propose as follows: U Wih he nonsingular ransformaion =, he sliing moe equaion will be wrien as U where z () = Az () () A= U AU ( U F) + U GDH( U U F) Corollary For he given consanα > 0, he sliing moe equaion () is sable, if here exis n m posiive-efinie marices X R ( ) ( n m), marix marix inequaliy hols 34 Z R m ( n m) such ha he following linear

8 U AU ( X U Z) + ( U X U Z) A U + αu GG U ( U X U Z) H α I < 0 (3) We can Design he sliing moe surface σ () = Sy() (4) where marix S saisfying SC( U F U ) = 0, F = ZX he proof is omie. Corollary For he nonlinear sysems (), wih he conroller SC GH σ ( ) u () = ( SCB) [ SCAx () + + kσ() + εsignσ()] (5) σ ( ) where k, ε are consans saisfying k > 0, ε > 0, hen he sysems saes will reach he sliing moe surface(4)in finie ime. Wih he proofs of heorem an heorem, Corollary an Corollary can be easily obaine. 4. Numerical Example he emperaure conrol sysem for polymerizaion reacor is a ineria link wih ime elay. he sae space moel of polymerizaion reacor is usually wrien as [6] x () = x () x () = a x () a x () + bu() y () = x() I is impossible o avoi he exernal isurbance an ime elay. We consier he nonlinear elay sysem wih norm-boune uncerainies as following x () = ( A+Δ A ()) x () + ( A +ΔA()) x ( ) + Bu () y () = Cx () x () = ψ () 0 where A=,,, [ 9.5 ], ( ) 0.sin,, 0 0 A = 0.5 B= C = ω = G= 0 H = [ ], H = [ ], ψ ( ) =, = 0., D( ) = cos Wih Singular Value Decomposiion of B, I is easy o obain he nonsingular ransformaion 35

9 = By solving he linear marix inequaliy (6), we can obain he sliing moe surface gain marix S = 0.8 an he saic oupu feeback sliing moe surface as following σ () = Sy() Wih he saic oupu feeback variable srucure conroller (9) in heorem, an choosing he iniial coniions he simulaion resuls are shown in figs. - ψ () = [ 0.5] Fig.. Sae x () response of sysem 36

10 Fig.. Sae x () response of sysem In he above figures, one can see ha he sysem is well sabilize wih he saic oupu feeback sliing moe conroller. 5. Conclusion his paper consiers he problem of he saic oupu feeback sliing moe conrol for a class of nonlinear elay sysems wih norm-boune uncerainies. A saic oupu feeback sliing moe surface is esigne by using linear marix inequaliy approach. hen he sliing moe conroller is esigne o make he saes reach sliing moe surface in finie ime. Acknowlegmens he auhors woul like o hank he associae eior an he anonymous reviewers for heir consrucive commens an suggesions o improve he qualiy an he presenaion of he paper. his work is suppore by Naional Naure Science Founaion of China uner Gran ; Naure Science Founaion of Henan Province uner Gran ; he Eucaion Deparmen of Henan Province Key Founaion uner Gran 3A003,4A

11 References. F. Gouaisbau, M. Dambrine, J. P. Richar (00) Robus conrol of elay sysems: a sliing moe conrol esign via LMI, Sysems Conrol Leers, vol.46, pp. 46: K. K. Shyu,W. J. Liu, K. C. Hsu(005) Design of large-scale ime-elaye sysems wih ea-zone inpu via variable srucure conrol, Auomaica, vol.4, pp S. W. Kau, Y. S. Liu (005) A new LMI coniion for robus sabiliy of iscree-ime uncerain sysems, Sysems Conrol Leers, vol.54, pp S.Y.Xu,.W.Chen (004) Robus H conrol for uncerain iscree-ime sysems wih imevarying elays via exponenial oupu feeback conroller, Sys. Conr. Le., vol. 5, no.4, pp H.Yao, F.Yuan(0) Variable Srucure Conrol(VSC) for a Class of Uncerain Discree- ime Large-scale Sysems wih Delays, Avances in Moeling C Auomaic Conrol, vol.66, no., pp H.P.Pang, C.J.Liu an A.Z.Liu (006) Sliing moe conrol for uncerain ime-elay sysems an is applicaion research, Journal of sysem simulaion, vol. 8, no., pp W. L. Li, H. J. Yao (006) Aapive fuzzy sliing moe conrol for a class of nonlinear ime-elay sysems, Journal of Henan Normal Universiy, vol. 34, pp O. M. Kwon, J. H. Park (004) On improve elay-epenen robus conrol for uncerain ime-elay sysems, IEEE rans. Auomaic Conrol, vol.49, pp W. H. Chen, Z. H. Guan, X. M. Lu (005) Delay-epenen exponenial sabiliy of uncerain sochasic sysems wih muliple elays: an LMI approach, Sysems an Conrol Leers, vol.54, pp Y. Q. Xia, Y. M. Jia (003) Robus sliing-moe conrol for uncerain ime-elay sysems: an LMI approach, IEEE rans. Auomaic Conrol, vol.48, pp S. C. Qu, X. Y. Wang (007) Sliing-moe variable srucure conrol for uncerain inpuelay sysems, Proceeing of he 4h inernaional conference on impulsive an hybri ynamical sysems, July 007, Nanning, China, pp S. Janarhanan, B. Banyopahgay, V. K. hakar (004) Discree-ime oupu feeback sliing moe conrol for ime-elay sysems wih uncerainy, Proceeings of he IEEE Inernaional Conference on Conrol Applicaions, April 004, Las Vegas, Nevaa, pp