It Joual of Math Aalyss, Vol 8, 04, o 8, 865-879 HIKARI Ltd, wwwm-hacom http://ddoog/0988/jma044368 A Epaded Method to Robustly Pactcally Output Tacg Cotol fo Uceta Nolea Systems Keyla Almha, Naohsa Otsua, School of Scece ad Egeeg,Toyo De Uvesty, Hatoyama-Mach, H-Gu, Satama 350-0394, Japa Masat N Kalmoldayev 3, Oe J Mamybayev 4 3,4 KI Satpayev Kazah Natoal Techcal Uvesty,, Satpayev steet, Almaty, Kazahsta Copyght 04 K Almha, N Otsua, MN Kalmoldaev ad OJ Mamybayev Ths s a ope access atcle dstbuted ude the Ceatve Commos Attbuto Lcese, whch pemts uestcted use, dstbuto, ad epoducto ay medum, povded the ogal wo s popely cted Abstact Ths pape studes the global pactcal output tacg poblem va state feedbac fo a class of uceta olea systems, whose chaed tegato pat has the powe of postve odd atoal umbes Sce the powe s ot estcted to be lage tha o equal to oe, the leazato of the system at the og may fal Nevetheless, t s poved ths pape that, ude mld assumptos, globally pactcal output-tacg s achevable by state feedbac cotolle Mathematcs Subject Classfcato: 93C0 Keywods: uceta olea systems, pactcal output tacg, state feedbac Itoducto I ths pape we wll cosde the pactcal output tacg poblem of olea powe tegato systems the fom
866 K Almha, N Otsua, MN Kalmoldaev ad OJ Mamybayev z& = z + φ (, t z, u, p z& = z + φ (, t z, u, p 3 z& = z + φ (, t z, u, p z& = u+ φ (, t z, u, M y = z, ( T whee z = ( z, K, z R ad u Rae the system state ad the cotol put, 0 espectvely Fo =,,, φ ( tzu,, s a uow C olea fucto of the states ad the cotol put ad p R + odd : = { q R: q> 0ad q s a ato of odd teges}, wth p obvously equal to oe (whch s ot a lmtato sce we p ca easly set v: = u the case of o-uty p The poblem of global output tacg cotol of olea systems s oe of the most mpotat ad challegg poblems the feld of olea cotol ad lots of effots have bee made dug the last decades, see [-8], [0-], as well as the efeeces thee Wth the help of the olea output egulato theoy [],[] ad the method of addg a powe tegato [3-5], sees of eseach esults have bee obtaed [6-8] Fo detals, [6], pactcal output tacg va state feedbac fo hgh-ode olea systems was cosdeed Futhe, [] ad [], the pactcal output feedbac tacg poblem was also vestgated fo a class of olea systems wth hghe-ode gowg umeasuable states, etedg the esults o stablzato [3]-[6] I these wos, t s assumed that gve olea systems have the fst appomato at the og, e, leazable at the og The poblems of global pactcal output tacg fo system ( wee vestgated [6], [8] ad [0-] oly fo p, =, K, Howeve, f 0< p <, the the system s ot leazable at each pot z R wth z + = as well as at the og, hece the desg methods [6], [8] ad [0-] 0 does ot wo The global stablzato poblem of system ( fo p > 0 (ot estcted to be lage tha o equal to oe has bee studed fo olea systems [9], [7] The techque fom [9] was ecetly eteded [8] to the pactcal output tacg poblem fo a specal case of olea systems ( (e, a lowe tagula fom ad allow fo factoal odd powes less tha oe Howeve, to ou best owledge, ths poblem of pactcal output tacg stll emas uclea ad lagely ope fo a wde class of olea systems ( cludg the tagula systems
Epaded method to obustly pactcally output tacg cotol 867 Poblem statemet ad Pelmaes The pupose of the pape s that let y ( t be a tme-vayg C -bouded o [0, efeece sgal ad fo ay gve postve eal umbe ε desg a state cotolle fo the system ( u = u( z, y ( t, ( such that ( all the states of the closed-loop system ( ad ( ae well-defed o [0, + ad globally bouded; ( the global pactcal output tacg s acheved, that s, fo evey z(0 R thee s a fte tme T : = T( ε, z(0 > 0, such that the output yt ( of the closed-loop system ( wth ( satsfes y( t y( t = z( t y( t < ε, t T > 0 I ode to solve the global pactcal output tacg poblem, we made the followg assumpto: Assumpto: Fo =, K,, thee est costats τ τ L τ such that + τ + τ φ (,, (,, tzu γ zk z z + L + z (4 fo a smooth fucto γ ( z, K, z wth defed as + τ =, + = > 0 (5 p Net, we wll peset seveal useful Lemmas boowed fom [4], [9] ad [0], whch wll play a mpotat ole ou late cotolle desg Lemma Fo ay postve eal umbes, y ad m, the followg equaltes holds: m ( m y+ m m y Lemma Fo all, y R ad a costat p the followg equaltes holds: (3 ( p p p p + y + y p p p ( ( + + y y
868 K Almha, N Otsua, MN Kalmoldaev ad OJ Mamybayev ( p p + y p If p ad p R + odd, the ( p p p p y y ad ( p p p p y y p Lemma3 Let cd, be postve costats The, fo ay eal-valued fucto γ ( y, > 0, the followg equalty holds: c d cd y γ(, y + γ (, y y c+ d c+ d c d c+ d c+ d Lemma4 Let,,, 0 L p> be eal umbes The, the followg equalty holds: Lemma5 If [ ] p ma( p + +, ( p p + + L L f : ab, R( a b s mootoe cotuous ad satsfes f( a = 0, the b f ( d f ( b b a a Lemma6 Let φ : R R be a C fucto wth φ (0 = 0 The, thee ests a smooth oegatve fucto γ (, K, such that φ (, K, + L+ γ (, K, 3 Global pactcal output tacg cotol by state feedbac I ths secto, we wll peset a ecusve desg appoach to costuct the tacg cotol fo system ( Theoem Let y ( t be a efeece sgal whose devatve y& ( t s also bouded The, ude Assumpto, the global pactcal output tacg poblem of the system ( s solvable by a state-feedbac cotolle of the fom ( Poof: The ductve poof eles o the smultaeous costucto of a C Lyapuov fucto whch s postve defe ad pope, as well as a homogeeous-le cotolle at each teato
Epaded method to obustly pactcally output tacg cotol 869 + Let, ρ Rodd ma ρ ma + τ,, whee τ ad ae defed as Assumpto The, the poof wll be doe by ducto Let = z y ad gve = z, =, K, The, we have p & = + φ(, t + y,, K,, u y& ( t p & = 3 + φ(, t + y,, K,, u M (6 p & = + φ (, t + y,, K,, u & = u+ φ(, t + y,, K,, u y = + y satsfy { } ad { } Ital Step We costuct the Lyapuov fucto as ( = ρ τ V ( s ds, whee 0 fo coveece Note that V s C, A dect calculato gves postve defte ad pope ( p V& ρ τ ( = + φ( t, + y,, K,, u y& ( t (7 Sce y ( t ad y& ( t ae bouded ad by Assumpto ad Lemmas-6, t ca be show that thee s a smooth fuctos γ% ( such that satsfyg ( + τ (,, K,, & γ( φ t+ y u y + y + y + M ( + τ ( + τ γ % γ + % M + M V& ( + + % γ ( + α ( + δ, whee ad ( p ( ( p p ρ τ ρ τ ρ ρ ρ ( ρ τ + τ ρ τ ( ( κ( + τ α( = ρ ρ ( ρ τ ( ρ τ δ κ % γ ( + τ = M + M Defe a smooth postve fucto such that % κ( α( + κ( The V& ρ τ ρ τ ρ ρ ( + + % κ ( + δ ( ( p p p
870 K Almha, N Otsua, MN Kalmoldaev ad OJ Mamybayev p If we tae the vtual cotolle as ( % κ (, = + p ( + τ = β ( + τ (8 the t follows that V& ρ ( + + δ (9 ( ρ τ ρ p p Iductve Step Suppose at the - th step, thee ae a C, postve defte ad V, K,, whch s postve defte ad pope, pope Lyapuov fucto ad a set of L 0 p p C vtual cotolles,, defed by = 0, ξ = = ξ β (, ξ = p p M M = ξ β (, K,, ξ =, p p whee β,, K,,, ae smooth postve fuctos such that ρ ρ (,, ( + ( ξ + + ξ p p ξ ( V& K L ( ρ τ ρ + + ( δ (0 We clam that (0 also holds at Step To pove ths clam, cosde the Lyapuov fucto (, K, = ( K, + (, K, V V U ( ρ τ = K + The fucto V (,,, V,, s ds K ca be show to be C, pope ad postve defte wth the followg popety: fo =, K,, ( ( ρ τ U ρ τ = ( s ds, ( U = ξ ( = ( ρ τ ρ τ (3
Epaded method to obustly pactcally output tacg cotol 87 ad thee s a ow costat N > 0, such that ( U N (4 ρ τ Poofs of these popetes poceed just the same way as the poofs fo [4, popostos ad ] ad [5] whee the set of postve odd teges s cosdeed stead of ou R odd Wth these popetes, we obta ρ ρ ( ρ τ p ( p,, ( + ξ + + ξ + ξ ρ ( ρ τ p + ( δ + ξ ( + + φ( t, + y,, K,, u U ( ρ τ p p + & + ξ ( + + V& K L = fo a vtual cotolle p + to be detemed late I ode to poceed futhe, a boudg estmate fo each tem the ght had sde of (5 s eeded The tems (5 ca be estmated usg the followg popostos whose poofs ae smla to that of [7] ad theefoe s omtted hee Poposto: Thee ests a postve smooth fuctos a(, K, such that ( ρ τ p p ρ ρ ξ ( ξ + a(, K, ξ 3 Poposto: Thee ests a postve smooth fuctos b(, K, such that ( ρ τ ρ ρ ρ ξ φ(, t+ y,, K,, u ( ξ + ξ + L+ ξ ρ ρ ρ + ξ + b(, K, ξ + δ 3 Poposto3: Thee ests a postve smooth fuctos c(, K, such that U c ( ρ ρ ρ ρ ρ ρ ξ + ξ + + ξ + ξ + (,, ξ + δ = & L 3 K Substtutg the esults of the pevous to (5, we ave at (5
87 K Almha, N Otsua, MN Kalmoldaev ad OJ Mamybayev ρ ρ (,, V& K + ξ + L+ ξ + ξ ( ρ τ p + ρ τ ρ ρ p p ( + + + % κ (, K, ξ + ξ + δ, (6 whee % κ (, K, a (, K, + b (, K, + c (, K, s a smooth postve fucto Theefoe, f we tae the vtual cotol + as {( % κ (, K, } + = ξ + + p + p = K + p : ξ β (,,, (7 the, we obta ρ ρ ρ (,,, ( + ( ξ + + ξ + ξ ( ρ τ p p ρ + ξ ( + + + δ V& K L whch poves the ductve agumet At the th step, by applyg the feedbac cotol u = = ξ + = ξ + p ( + τ β (, K, β (, K, wth the costucted va the ductve pocedue, we ave at V& K L (9 (8 C, pope ad postve defte Lyapuov fucto V (, K, ( ( ρ ρ ρ ρ,,, ξ + + ξ + ξ + δ Recall that V(,,, K, = U(,,, K,, whee U s ae defed = ( The, t follows fom Lemma 4 that fo ay τ > 0, whee τ = τ τ V (,,, K, c U (,,, K,,,,, K, R, (0 c: = ma(, Moeove, we have
Epaded method to obustly pactcally output tacg cotol 873 ( ρ τ U(,,, K, = s ds ρ ( ξ ( ρ τ ( ρ τ ξ = ξ ( ρ τ ξ ( ρ τ λ =, ( whee λ = ( ρ τ ρ Theefoe, V& λ (,,, K, V (,,, K, + δ ρ λ ( ( K λ ρ,,,, = V + δ ( It wll be show that the state ( t of closed-loop system (6 s well-defed o [ 0, + ad globally bouded Fst, toduce the followg set { } ρ λ t V δ Ω= : R 4, (3 ad let ( t be the tajectoy of (6 wth a tal state (0 If t Ω, the t follows fom (3 that V& ( ( t V ( ( t + δ ρ δ < 0 λ ρ (4 Ths mples that, as log as t ( Ω, V ( ( t s stctly deceasg wth tme t, ad hece ( t must ete the complemet set stay thee foeve Theefoe, (4 leads to R Ω a fte tme T 0 ad t V ( ( t V ( (0 = V& ( ( t dt 0 ρ ( δ [ T < 0, t 0, V ( ( t < 4, t [ T, λ (5 whch shows V L, ad so do ad U
874 K Almha, N Otsua, MN Kalmoldaev ad OJ Mamybayev By z = + y ad y L, we coclude L as well Notg ( % κ ( = + p ( + τ ( + τ = β p ad κ% ( s smooth fucto of, we have we have ( L ad L L Sce U Iductvely, we ca pove L, = 3,4, L, ad so do ( t L ad (4, Thus, the soluto ( t of the system (6 s well-defed ad globally bouded o[ 0, + Net, t wll be show that y( t y ( t = z ( t y ( t < ε, t T > 0 (6 Ths s easly show fom (4, (5 ad by tug the paamete δ as follows: yt ( y ( t = ( t V ( ( t ρ ( 4δ < ε Theefoe, fo ayε > 0, thee s globally pactcal output-tacg such that (6 holds Ths completes the poof of Theoem Rema By the defto of ( ma{ } λ we ow Hece, t ca be cocluded that the cotolle u, as defed (8, ca be guaateed to be at least C whe ( + τ Futhemoe, whe, =, K,, ae teges ad ( + τ =, a smooth cotolle s costucted by ths desg method 4 A Illustatve Eample I ths secto, we gve a smple umecal eample to llustate the coectess ad effectveess of the theoetcal esults by cosdeg the followg olea system
Epaded method to obustly pactcally output tacg cotol 875 35 z& = z, z& = u+ dtzz (, dt ( (7 z = y Clealy, the system s of the fom ( ad s p = 35 So that global pactcal poblem caot be solved as [6] ad [8] Howeve, usg ou desg method descbed the poof of Theoem, t s possble Ou objectve s to desg a pactcal output-tacg cotolle such that the output of the system (7 tacs a desed efeece sgal y = s t, ad all the states of the system (7 ae globally bouded But by Theoem ad Assumpto, the output y of the system (7 ca be globally tacs the y = s t by selectg τ =, τ = 0 ad =, = 5 wth = 5 Assumpto stll holds sce 53 53 53 53 zz ( + z + z z + z = γ ( z, z z + z fo a smooth, o-egatve fucto γ ( z, z = ( + z + z + Let = ρ = 5 Rodd ad defe the eo sgal = z y ad gve = z The, we have (8 Net, we choose whee = 0 The & = y& (, t 35 & = u+ d( t + y, (9 = y y 5 5 75 V ( = s ds, 0 V& ( = & 7 ( & = y ( t, 7 3 5 usg y ( t ad y& ( t ad Lemmas-6, we have
876 K Almha, N Otsua, MN Kalmoldaev ad OJ Mamybayev V& ( + + 7 35 7 35 35 7 3 ξ + ξ ( + δ + δ 75 35 75 35 35 0 67 whee s a vtual cotol ad the last equalty follows fom the completo of squaes, fo ay eal costat δ > 0 Obseve that the vtual cotolle 3 = + δ = ξ β, 3 5 3 67 ( 35, whee edes 5 3 ξ =, β( : = + > 0 67 δ V& ( ξ + ξ + δ 7 5 3 5 3 5 Net, defe ξ =, 5 5 ξ = ad choose V (, = V + U (, : = V + s ds A staghtfowad calculato gves 7 5 3 5 3 5 U V& (, ξ + ξ ( + δ + ξu+ ξd( t ( + y + & (30 Obseve that ad ξ ( ξ + 75ξ 3 75 35 35 ξdt (( + y ξ + b(, ξ + δ, 3 whee
Epaded method to obustly pactcally output tacg cotol 877 γ (, b (, = + γ (, ( + ξ + 9 β ( ( + γ (, + ξ δ 43 03 3 ad whee U & ξ + c ξ + δ 3 (,, 0 3 8 5 4 5 4 5 5 β ( ξ ( + ξ c(, = 5 β ( ξ + δ Substtutg the esults of the pevous to (30, we ave at V& (, ξ + ξ u + + 75 + b (, + c (, ξ + δ Hece, choosg { 75 (, (, } u = + + b + c ξ (3 yelds V& (, ξ + ξ + δ (3 The equalty (3 mples that all the solutos z ( t ad z ( t of the closed-loop system (7 ad (3 ae globally bouded ad well defed [0, + Moeove, fo ayε > 0, thee ests a costat δε = ε ad a fte tme T such that z y < ε, t T (33 Ths mples that the tacg eo ca be made abtaly small afte a fte tme by tug the paamete δ 5 Coclusos I ths pape, we have developed a systematc appoach to costuct a pactcal output tacg cotolle fo olea systems ae ot the stct feedbac fom, whose chaed tegato pat has the powe of postve odd atoal umbes Such a cotolle guaatees that the states of the closed-loop system ae globally bouded, whle the tacg eo ca be bouded by ay gve postve umbe afte a fte tme
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