Static Output Feedback Variable Structure Control for a Class of Time-delay Systems

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1 AMSE JOURNALS 04-Series: Avances C; Vol. 69; N ; pp Submie Sep. 03; Revise June 30, 04; Accepe July 5, 04 Saic Oupu Feeback Variable Srucure Conrol for a Class of ime-elay Sysems Y. ian, H. Yao, F. Yuan School of Mahemaics an Saisics, Anyang Normal Universiy 436 Xian ge Roa, , Anyang, China (yaohejun@6.com) Absrac his paper consiers he problem of saic oupu feeback variable srucure conrol for a class of ime-elay sysems. Aenion is focuse on he esign of a saic oupu feeback sliing moe surface an he variable srucure conroller. Base on Lyapunov sabiliy heorem, in erms of linear marix inequaliy (LMI), a sufficien coniion for he solvabiliy of his problem is presene which is epenen on he size of he ime elay an can be solve by LMI oolbox in MALAB. When he LMI is feasible, he explici expression of he esire saic oupu feeback sliing moe surface is also given. hen, a variable srucure conroller is obaine which make he sysems saes reach he sliing moe surface in finie ime. Finally, a numerical example is provie o emonsrae he effeciveness of he propose approach. Key wors: Delay sysems, saic oupu feeback, variable srucure conrol, linear marix inequaliy (LMI). 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]. he variable srucure conrol approach, base on using of isconinuous conrol laws, is known o be an efficien alernaive way o ackle many challenging problems of robus sabilizaion. For insance, an appropriae sliing moe sraegy can achieve sabilizaion by ominaing nonlinear erms an aiive isurbances, provie some appropriae maching coniions hol. However, he combinaion of elay phenomenon wih relay acuaors makes he 58

2 siuaion much more complex: esigning a sliing conroller wihou aking elays ino accoun may lea o unsable or chaoic behaviors or, a leas, resuls in highly chaering behaviors. In [7], Chen consiers he problem of sliing moe conrol for iscree-ime muli-inpu mulioupu sysems. Fuzzy sliing moe conroller esign for uncerain ime-elaye sysem has been suie in [8, 9,0]. Recenly, he problems of variable srucure conrol for uncerain iscreeime sysems wih elays have been suie [ 6].Sliing moe conrol for ime-elay iscree sysems wih unmache uncerainy an muli-inpu iscree sysems has been presen in [4, 5]. In [6], Zhang consiers he quasi-sliing moe variable srucure conrol for iscree linear consan sysem wih ime elay. 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 variable srucure conrol for a class of nonlinear elay sysems wih norm-boune uncerainies. Base on LMI approach, a new approach is given o esign he saic oupu feeback sliing moe surface. hen, a variable srucure 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 &() = Ax () + Ax ( ) + Bu () y () = Cx () x () = ψ () 0 () n m p where x () R is sysems sae, u () R is sysems conrol inpu, y () R is sysems oupu. n n is a sysems sae elay. ψ () is he given iniial sae on [,0]. A R n n, A R, B R m n an p n C R are known consan marices. B has full column rank. Wih Singular Value Decomposiion of B, a nonsingular ransformaion Ω B= [ U U] V 0 U = U 0 is consruce for sysems () o make B = B., () 59

3 Wih he ransformaion z () = x (), he sysems()can be rewrien as z& () z& Ax A x Bu () = = () + ( ) + () z& () = A z( ) + A z( ) + Bu( ) Insering ()ino he above formulaion, we obain z& () = U AU z () + U AU z () + U AU z ( ) + U AU z ( ) z& () = U AU z () + U AU z () + U AU z ( ) + U AU z ( ) + B u() (3) For he sysems(3), selecing he saic oupu feeback sliing moe surface as following σ () = Sy() (4) Wihσ ( ) = Sy( ) = SC z( ) = SC[ U U ] z( ) = SCU z ( ) + SCU z ( ) = 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 z& () = U AU ( U F) z () + U A ( U U F) z ( ) (5) 3. Main Resuls Y( x) W( x) Lemma [5] he LMI > 0 * Rx ( ) is equivalen o Rx ( ) > 0, Yx ( ) WxR ( ) ( xw ) ( x) > 0 where ( ) Yx= Y( x), Rx ( ) = R( x) epen on x. heorem he sliing moe equaion (5) is sable, if here exis posiive-efinie marices PQ, % R % ( n m) ( n m), marices % % % ( n m) ( n m) 3, consans, X, N, N, N R ρ ρ an marix Z R m ( n m) such ha he following LMI hols where Σ Σ Σ3 N% Σ Σ N% Σ33 N% 3 Q% 3 < 0 (6) 60

4 Σ = N% + N% U AU ( X UZ) ( UX UZ) A U Σ = N% N% U A ( UX UZ) ρ( UX UZ) A U Σ 3 = P% + N% 3 + X ρ( UX UZ) A U Σ = N% N% ρu A ( U X U Z) ρ ( U X U Z) A U Σ 3 = N% + ρx ρ( U X UZ) AU Σ = Q% + ρ X + ρ X 33 We can esign he sliing moe surface σ () = Sy() where marix S saisfying SC( U F U ) = 0, F = ZX he proof is given in Appenix. 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 u = SCB SCAx + SCA x + k + sign (9) () ( ) [ () ( ) σ() ε 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. Proof Along he soluion of sysem () we have σ () & σ() = σ () SCAx() + σ () SCA x( ) σ ( ) SCAx( ) σ ( ) SCA x( ) σ ( ) kσ( ) 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 &() = Ax () + 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: 6

5 U Wih he nonsingular ransformaion =, he sliing moe equaion will be wrien as U z& () = U AU ( U F) z () () Corollary For he given consanα > 0, he sliing moe equaion () is sable, if here exis n m posiive-efinie marices X R hols ( ) ( n m) We can esign he sliing moe surface where marix S saisfying he proof is omie., marix Z R m ( n m) such ha he following LMI U AU ( X U Z) + ( U X U Z) A U < 0 (3) σ () = Sy() (4) SC( U F U ) = 0, F = ZX Corollary For he sysems (), wih he conroller u () = ( SCB) [ SCAx () + kσ() + εsignσ()] (5) 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() ξ k Where a =, a =, b=, > 0 is ime elay, ξτ,,k are known consans. τ τ τ I is impossible o avoi he exernal isurbance an ime elay. We consier he following ime-elay sysems x &() = Ax () + Ax ( ) + Bu () y () = Cx () x () = ψ () 0 where 6

6 A= =, A = =, B=, C = [ 0 ], = 0. a a 0 5 Wih Singular Value Decomposiion of B, i is easy o obain he nonsingular ransformaion = By solving he linear marix inequaliy (6), we can obain he sliing moe surface gain marix S = 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 63

7 Fig.. Sae x () response of sysem In he above figures, one can see ha he sysem is well sabilize wih he saic oupu feeback variable srucure conrol. 5. Conclusion he problem of he saic oupu feeback variable srucure conrol for a class of ime-elay sysems is consiere in his paper. Compare wih raiional sliing moe surface esign approach which nees known sysems saes, by using LMI echnique, we obain a new approach o esign he sliing moe surface which only nees known sysems oupu. An hree slack marices are inrouce in orer o obain a more complicae sufficien coniion which can achieve he aim of furher reucing conservaism. hen he variable srucure conroller is esigne o make he saes reach sliing moe surface in finie ime. A numerical example is provie o emonsrae he effeciveness of he propose approach. Acknowlegmens he auhors woul like o hank he 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 ; 64

8 Naure Science Founaion of Henan Province uner Gran ; he Eucaion Deparmen of Henan Province Key Founaion uner Gran 3A003,4A Appenix Proof of heorem 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 + z ( ) N ( ) 3)( ( ) ( ) & ( ) ) ( ( ) ( ) + 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 + z& ( ) M )( Az ( ) 3 3 Az( ) + z& ( )) + ( z ( N ) + z 3 & & + z ( ) N + z ( ) N ) + z ( s) Qz ( s) s = ξ ( ) Ξξ( ) ( ) N + z ( ) N ) Q ( z ( ) N 3 where N, N, N 3, M, M, M 3 are consan marices wih appropriae imensions o be confirme. he inequaliy is equivalen o 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) 65

9 where N N Ξ=Θ+ Q N N N3 0< 0 N 3 0 Θ Θ Θ Θ= Θ Θ Θ Θ = N + N MU AU ( U F) ( U U F) Θ = N N MU A ( U U F) ( U U F) Θ = P+ N + M ( U U F) Θ = N N M U A ( U U F) ( U U F) Θ 3 = N3 + M ( U UF) A UM3 Θ = Q + M + M Wih lemma, we know ha he inequaliy(7)is equivalen o Where Δ Δ Δ Δ Δ Δ 3 3 Δ= < N N N Q Δ = N + N MU AU ( U F) ( U U F) Δ = N N MU A ( U U F) ( U U F) Δ = P+ N + M ( U U F) Δ = N N M U A ( U U F) ( U U F) Δ = N + M ( U U F) Δ = Q + M + M Pre- an Pos-muliplying he inequaliy (8) by iag{ M, M, M, M, I} an giving some ransformaions iag{ M, M, M, M, I} an M = M, M = ρm, M = ρ M, X = M, Z = FX, P% = XPX, Q% = XQX, where ρ, ρ 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)

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