D-STABLE ROBUST RELIABLE CONTROL FOR UNCERTAIN DELTA OPERATOR SYSTEMS

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1 Jounl of Theoeticl nd Applied nfomtion Technology 8 th Febuy 3. Vol. 48 No JATT & LLS. All ights eseved. SSN: E-SSN: D-STABLE ROBUST RELABLE CONTROL FOR UNCERTAN DELTA OPERATOR SYSTEMS BO YAO, DAN L, HAO HU College of Mthemtics nd System Science, Shenyng Noml Univesity, Shenyng, P.R. 34 Chin College of Automtion Engineeing, Nnjing Univesity of Aeonutics nd Astonutics, Nnjing, P.R. Chin, 6 ABSTRACT The D-stble obust elible contol fo uncetin delt opeto systems is minly studied by this ticle. t poposes sufficient condition of plcing poles of the closed-loop systems in specified cicul disc, in tems of line mtix inequlities by using stte feedbck. t lso gives design pocedue of such contolles. The poposed esults cn lso unify elted esults of continuous nd discete systems. Futhemoe, numeicl exmple is povided to demonstte the fesibility nd the effectiveness of the design method. Keywods: Delt Opeto, Relible Contol, Line Mtix nequlity. NTRODUCTON The stbility of line systems hs diect connection with the loction of its poles. Theefoe, pole ssignment is one of the most significnt esech subjects on line systems nlysis nd synthesis. By using the delt opeto to descibe discete system cn void the touble when we hndle poblem of bnoml condition, nd ovecome the instbility of smpling with high speed t the sme time. The model of discetiztion of delt opeto tends to continuous system model, nd this would enble the contolled studies of continuous system nd the discete systems on the issue come down to study fo delt opeto systems. Studying on delt opeto in Chin begn in the 99 of the th centuy. Zhng nd Yng [] published the fist summy ppe of delt opeto method. n ecent yes, the study of delt opeto systems contol hs got lge numbe of esults []~[5]. n ddition, lot of chievements hve been mde ieul netwok, pmete estimtion, nd signl pocessing, nd so on. Relible contol mens fo ny fult of system components (ctuto nd senso) tht my ppe, designs the ppopite contolles, ensues tht the system cn still opete coectly in the event of fult. D-stbility contol mens plcing the poles of the closed-loop system in the specified disk in the complex plne. Both eseches possess pcticl mening. Since Siljk mde elible contol fo the fist time in the 97 of the th centuy, the design method of obust contolle hs moved [6]~[8] n fct, s long s we ssign the poles of the closed-loop systems in the complex plne with n ppopite e, the system will hs some dynmic nd stbiliztion chcteistics. Fo instnce, step esponse of second ode systems with the poles: λ ζωn ± jω cn be completely specified by ntul d fequency ω λ, dmping tio ζ nd dmped n ntul fequency ω d. The ζ, n ω nd ω d cn stisfy some given bounds by limiting λ in the complex plne with n ppopite e. Consequently, we cn mke the systems hve pescibed chcteistics. This ticle minly studies the delt opeto systems with continuous fult models, nd gives the existence conditiod design method of the D- stble obust elible contolle of the line system with ctuto continuous fult.. PROBLEM FORMULATON Conside the following uncetin Delt opeto system: δ xt () ( A+ Axt ) () + ( B+ But ) () () whee δ is symbol of Delt, nd d x, T dt δ xt ( ) xt ( + T) xt ( ), T T 77

2 Jounl of Theoeticl nd Applied nfomtion Technology 8 th Febuy 3. Vol. 48 No JATT & LLS. All ights eseved. SSN: E-SSN: n p ( ) R is the stte, u( t) R xt n p is the contol input; A R, B R e known constnt mtices; A nd B e uncetinty of nom bounded type witten s: [ A B] EF[ G G ] () whee E, G, G e known constnt mtices with ppopite dimension, nd define the stuctue of the uncetinty. And the pmete uncetinty F stisfy F F, whee is identity mtix with ppopite dimension. Using the stte feedbck, the fom of the contolle nd the continuous fult model espectively e: u() t Kx() t (3) f u () t MKx() t (4) Whee M dig( m, m,, m n ) is continuous fult mtix, while m i, mens outge of the i th ctuto contol signl. m i mens the noml opetion of ith ctuto contol signl. When mdi mi mui, mi, ptil fult of the i th ctuto contol signl occus, whee m m, i,, n. Let di ui M dig( m, m,, m ) d d, d, dn, Mu dig( mu,, mu,,, mun, ) M M + M, M M M then ( ) ( ) u d u d M M + M Σ M whee Σ dig ( σ, σ,, σn), σi, i,,, n The system which is composed of () nd (3) cn be descibed s: δ x() t ( A + A + BK + BK) x() t Ax() t (5) whee A A + A + BK + BK. The system which is composed of (), (3) nd (4) cn be descibed s: δ x() t ( A + A + BMK + BMK) x() t Ax () t (6) whee A A + A + BMK + BMK. The pupose of this ppe is to detemine the stte feedbck u() t Kx() t such tht the poles of the closed loop system (5) nd (6) lie in the specified disc. 3. MAN RESULTS Lemm [9] Let D nd E be given el mtices with ppopite dimension. Y is negtive definite symmetic mtix; F is time vying digonl mtix with m dimension stisfying F F, then Y + DFE + E F D < fo ll F, if nd only if thee exists positive m m definite symmetic mtix U R, such tht Y + DUD + E U E < Lemm [] Let X nd Y be mtices with ppopite dimension, F is time vying mtix with ppopite dimension stisfying F F, then, fo ny scl quntity ε >, we hve XFY + Y F X εxx + ε Y Y Lemm 3 [] λ( A) D (, ) if nd only if thee exists positive definite mtix X stisfying X + AX < X + XA Theoem Let A R, then λ( A) D (, ) if nd only if thee exists positive definite symmetic mtix P R such tht P P ( + TA) ( + TA) < T T A A ( + ) whee A T T T Poof λ ( A) D(, ) λ( A ) D(, ) A λ D(,) λ( + TA ) D(,) lemm 3. X ( + TA ) X < X ( + TA ) By Schu complements lemm, the bove inequlity is equivlent to: ( ) ( ) X + X + TA X + TA X < 77

3 Jounl of Theoeticl nd Applied nfomtion Technology 8 th Febuy 3. Vol. 48 No JATT & LLS. All ights eseved. SSN: E-SSN: X Multiplying on the both sides of the inequlity bove, nd let P X, then we get which implies ( + TA ) P( + TA ) P < P P ( + TA) ( + TA) < T T Theoem The delt opeto system (5) is sid to hve D-stble obust contolle, if thee exists positive definite symmetic mtix X >, mtix Y nd scl quntity ε > stisfying the following inequlity Π Π ε Π X + EE < Π ε (7) BY whee Π X + TA X +, Π GX+ GY Poof Accoding to theoem, we know tht the ( A) D(, ) λ if nd only if thee exists positive definite symmetic mtix tht whee A P R P P ( + TA) ( + TA) < T A ( + ) T T such Using Schu complements lemm; the bove inequlity is equivlent to: Multiplying P ( + TA ) ( + TA ) P < dig( P, ) on both sides of the inequlity bove, let X P, Y KX, it follows fom the mtix inequlity bove tht ( ) < EFGX + BY + EFGY whee X + TA + which implies Π Π + F[ ] [ ] F Π + Π < E E (8) By lemm, fo ny positive scl quntity ε >, the inequlity (8) is equivlent to: Π Π + ε ε [ ] [ ] E + Π Π < E (9) Then by using Schu complements lemm, the inequlity (9) is equivlent to the inequlity (7). Theefoe, the contolle is obtined, which cn ensue λ( A) D (, ) nd K YX. Theoem 3 The delt opeto system (6) hs D-stble obust elible contolle, if thee exists positive definite symmetic mtix X, digonl mtix U >, mtix Y nd scl quntity ε > stisfying the inequlity () whee ε 3 43 < 4 43 U 5 U BM Y X + TA X + GX+ GMY 3 ε X + EE () 4 UM B 43 UM G 5 M Y Then the contol gin mtix is given by K YX Poof Accoding to theoem, we know tht the λ( A ) D (, ) if nd only if thee exists positive definite symmetic mtix whee A P R such tht P P ( + TA ) ( + TA ) < T T A ( + ) T Using Schu complements lemm; the bove inequlity is equivlent to: 77

4 Jounl of Theoeticl nd Applied nfomtion Technology 8 th Febuy 3. Vol. 48 No JATT & LLS. All ights eseved. SSN: E-SSN: P ( + TA ) ( + TA ) P < Multiplying dig( P, ) to the bove inequlity fom the left nd ight espectively, let X P, Y KX, it follows fom the mtix inequlity bove tht Γ < Γ whee EFGX + BMY + EFGMY Γ X + TA + which implies tht X Σ Σ + F[ ] [ ] F E Λ + Λ < E () BMY whee Σ X + TA X +, Λ GX + GMY Accoding to lemm, fo ny positive scl quntity ε >, we hve Σ ε + + X E E Σ ε [ ] [ ] Λ Λ < () Fom Schu complements lemm, the inequlity () is equivlent to: Σ Λ Σ Λ < ε (3) Replcing M M + M Σ M in (3), then the inequlity (3) cn be descibed s: 3 + ε 3 BM Σ Σ BM < GM GM Fom lemm, thee exists positive definite m m symmetic mtixu R, such tht 3 + ε 3 BM UU U BM + GM GM 5 U 5 (4 < (5) by Schu complements lemm, the inequlity (5) is equivlent to the inequlity (), the poof is completed. Theefoe, the contolle is obtined, which cn ensue λ( A ) D (, ) nd K YX. Remk As the model of discetiztion of delt opeto tends to continuous system model when the smpling intevl T, nd then this would enble the contolled studies of continuous systems nd the discete systems on the issue boil down to study fo delt opeto systems. Theefoe, the theoems in this ppe e suitble fo both continuous system nd discete system. 4. NUMERCAL EXAMPLE Conside the delt opeto system (5) nd (6) with pmetes: A.6.5.3, B E..3, G G M dig(.65,.85,.55,.9) M dig(.45,.45,.45,.3) x [,,] ) 773

5 Jounl of Theoeticl nd Applied nfomtion Technology 8 th Febuy 3. Vol. 48 No JATT & LLS. All ights eseved. SSN: E-SSN: The cicul egion is chosen to be D (, ) D(3, ) nd the smpling intevl T.. By computing, we cn obtin X Y ε K fo system (5), nd we lso get X Y U dig(59.589,6.43, 75.79,38.39) ε K fo system (6). Fig shows the pole distibution of delt opeto system (5) with D-stble obust contolle. All of its poles e in the specified cicul disc. Fig shows the pole distibution of delt opeto system (5) with ctuto continuous fult. The D-stble obust contolle is out of wok in this condition. The poles of system (5) e ptly out of the specified cicul disc. Fig 3 shows the pole distibution of delt opeto system (6) with D-stble obust elible contolle. Fig. The Pole Distibution Of System (5) Fig. The Pole Distibution Of Delt Opeto System (5) With Actuto Continuous Fult Fig 3. The Pole Distibution Of System (6) 5. CONCLUSON Genelly, in pcticl system design, designes wnt the system s elible s possible. And this ppe gives method to design the D-stble obust elible contolle of the line system with ctuto continuous fult. As the model of discetiztion of 774

6 Jounl of Theoeticl nd Applied nfomtion Technology 8 th Febuy 3. Vol. 48 No JATT & LLS. All ights eseved. SSN: E-SSN: delt opeto tends to continuous system model, the contolle designed by this ppe is suitble fo both discete systems nd continuous systems. Finlly, n exmple shows the effectiveness nd the fesibility of this method. REFERENCES [] Zhng D J nd Yng C W,. Delt opeto theoy fo feed-bck contol systems: A suvey. Contol Theoy & Applictions, 5(): [] ZHANG Dun-jin, WANG Zhong-yong nd WU Jie, 3. Suvey on system contol nd signl pocessing using the delt opeto. Contol nd Decision, 8(4): [3] LU Mn, JNG Yun wei nd ZHANG Siying, 4. D-Stble Robust Fult-Tolent Contol fo Delt Opeto Systems. Jounl of Nothesten Univesity(Ntul Science), 5(8): [4] Yn Junong, Guo Xjid Gui Xi, 6. Robust Contol of Pole Resticted in Cicul Region fo Delt Opeto System. Jounl of Nnjing Univesity of Aeonutics & Astonutics, 38(Suppl): [5] Xio Min-qing, 9. Robust D-stbiliztion fo Delt opeto systems with senso filue. Contol Theoy & Applictions, 6(): [6] Veillette R J, Mednic J V nd Pekins W R, 99. Design of elible contol system. EEE Tns Automt Contol, 37(): [7] Yng G H, Wng J L nd Soh Y C, Relible LQG contol with senso filues EEE Poceedings-Pt D, Contol Theoy nd Applictions, 47: [8] Yng G H, Wng J L nd Soh Y C,. Relible H design fo line system. Automtic, 37: [9] Xio Minqing, 8. An Unified LM Appoch to Relible D-Stbiliztion Contolle Design fo Line Systems. Poceedings of the 7 th Chinese Contol Confeence, pp [].R. Petesen, 987. A stbiliztion lgoithm fo clss of uncetin line Systems. Syst.Cont. Let, 8: [] Chilli M nd Ghinet P, 996. H design with pole plcement constints: n LM ppoch. EEE Tns. Automt Contol, 4: