2 Ameian ontol onfeene Maiott Watefont, Baltimoe, MD, USA June 3-July 2, 2 F4.5 An almot Anti-Winup heme fo plant with magnitue, ate an uvatue atuation Fulvio Foni, Segio Galeani, Lua Zaaian Abtat We ae the anti-winup augmentation poblem fo plant with atuation on the magnitue, ate an uvatue of the ontol input. To thi aim, given an unontaine loe-loop, we geneate a lightly moifie titly pope ontolle fo whih the eivative of the ontol ignal ae available an we olve the anti-winup poblem fo thi moifie ontol heme (namely, an almot anti-winup fo the oiginal loe-loop). Bae on thi almot appoah, we eviit an exiting Moel Reovey anti-winup olution fo ate an magnitue atuate plant an then we exten the eult to the ae of ate, magnitue an uvatue atuation, by poviing a Moel Reovey olution to thi aitional poblem. An example illutate the peuliaitie an the effetivene of the popoe olution. I. INTRODUTION Plant input atuation in othewie linea ontol ytem ha been long tuie ine the 95. In patiula the oalle anti-winup appoah aoe in thoe ealy yea a a poible epone to the nee of not aifiing the mall ignal behavio to obtain a atifatoy lage ignal behavio. While the anti-winup appoah ha been hitoially aoiate with magnitue only atuation (ee fo a uvey of ome ealy anti-winup metho an 6, 7 fo a eent uvey an a eent tutoial on thi topi), ine the late 99 quite a bit of attention ha been evote to extening the available eult to the ae whee the plant input i not only ubjet to magnitue but alo to ate atuation. Thi ontext i epeially elevant in a numbe of appliation, inluing flight ontol, ontol of Tokamak plama with upeonuting oil an many othe whee the equet to the atuato i not allowe to hange too fat (ee, e.g., 3, 9, 3, 4). Anti-winup appoahe fo magnitueate atuate plant have been given in 2 an 8 whee a plant-oe an a tati ompenation heme, epetively, ae popoe an in 6 (futhe extene in 5), whee a tati olution i given. A non-ontutive plantoe olution to the poblem wa alo given in, but key tabilizing feebak nee to be eigne fo the peial plant une onieation in that heme. Thee feebak ae not alway eay to etemine. Finally, the o-alle efeene goveno (o omman-goveno) appoahe whih ely on eeing hoizon optimal ontol iea (ee, e.g., 2, 8, 5) an be fomulate by inopoating ate atuation in the ontol eign poblem. Reently, in 4, we have haateize two Moel Reovey Anti-winup olution fo plant with ate an magnitue atuation, builing upon the peliminay eult of 7 (fo the fit appoah) an 2 (fo the eon one). In thi pape we takle a genealize poblem a ompae to the magniue an ate ae by not only impoing that the Wok uppote in pat by ENEA-Euatom an MIUR une PRIN pojet. All the autho ae with the Dipatimento i Infomatia, Sitemi e Pouzione, Univeity of Rome, To Vegata, 33 Rome, Italy foni galeani zak@ip.unioma2.it zeo (magnitue) an fit (ate) oe time eivative of the plant input nee to be boune, but alo impoing that it eon oe eivative (that we all uvatue) i boune by uitable ontant. Thi equiement genealize the magnitueate atuation equiement in impoing a plant ontol input that i vey egula beaue it oen t allow fo pike of any kin on the plant input afte ompenation. The anti-winup ahitetue that we aopt to takle the magnitueateuvatue atuaiton poblem equie the availability of the fit an eon time eivative of the ontolle output. Thi paallel the olution given in 4 whee the fit time eivative wa equie. While in 4 tit popene of the ontolle allowe fo that equiement, hee it i uneaonable that the elative egee of the ontolle i two an we iu a poible appoah to moify a linea ontol ytem to inue an abitaily mall hange in it tanfe funtion an make thoe eivative available in the moifie ontolle. A imila appoah an be aopte in the ae of a nonlinea ontolle, wheea lineaity of the plant i a key popety fo ou ontution to apply. Then, the heme popoe to geneate the moifie ontolle an alo be ue to ae anti-winup fo magnitueate only atuation with nontitly pope ontolle applying the appoah of 4 to the moifie loe-loop ytem. In all ae, the anti-winup poblem that we ae an be een a an almot anti-winup olution fo the oiginal ontol ytem, inee the moifie loe-loop that we intoue will be almot the ame a the oiginal one up to a etain fequeny of opeation. The pape i oganize a follow. In Setion II we iu how to moify an unontaine loe-loop to obtain in expliit fom N eivative of the plant input. In Setion III we illutate the almot anti-winup olution on plant with ate an magnitue atuation an in Setion IV we apply it to plant with uvatue, ate an magnitue atuation. An example i given to illutate the popoe appoahe. All poof ae omitte ue to lak of pae. II. MODIFIED LOSED-LOOPS FOR AHIEVING STRITLY PROPER ONTROLLERS The pupoe of thi etion i to how how, given a plantontolle pai (poibly, both non titly pope) an a poitive intege N, it i poible to eplae the ontolle by a titly pope one uh that the oiginal an the moifie loe loop ae abitaily loe to eah othe (in a ene to be peifie late), an moeove the output of the moifie ontolle i N time iffeentiable an it N eivative an be mae available a aitional output of the moifie ontolle. A. The oiginal loe loop W onie the following linea plant P ẋ = Ax B u u B y = y x u D y z = z x D zu u D z, () 978--4244-7425-7//$26. 2 AA 6769
whee x R n i the plant tate, u R m i the plant ontol input, y R q i the meauement output, z i the pefomane output an i a itubane input. Following the tana anti-winup appoah, we aume that a ontolle ha been aleay eigne fo plant (). We make vey few aumption on the tutue of the ontolle,, that an be eibe by the following equation: ẋ = A x B u u B y = x D u u D whee x i the ontolle tate, u i it meauement input an i an extenal efeene ignal. To guaantee exitene an uniquene of olution, we aume that loe-loop between plant () an ontolle (2) i well behave in the abene of atuation, namely with the following unontaine inteonnetion: (2) u = y, u = y. (3) Aumption : The loe-loop betwen plant () an ontolle (2) via the inteonnetion (3) i well poe an aymptotially table. Note that Aumption implie that plant () i tabilizable fom u. In the following, W will enote the oiginal loe loop ytem of (), (2) inteonnete by (3) having tanfe matix W. B. The moifie loe loop Ŵ. In oe to ahieve the availability of the fit N eivative of the ontolle output, we want to eplae the oiginal ontolle by a new ontolle whih i eentially the aae of an a filte F (ue to ompute the appoximate eivative of the ontol ignal y ) with tanfe matix F() = F () F () F N () T I I N I T = ( τ) N, (4) whee τ > i uffiiently mall an the oeffiient α,..., α N ae obtaine by witing p() = N α N N α α with p() = (τ ) N ; it i eay to ee that a τ, the output of F get loe an loe to the input of F an to it fit N eivative. While intuition ugget that, povie that τ > i uffiiently mall, the moifie loe loop will emain table an the loe loop epone with eplae by will be abitaily loe to the oiginal one, it will be hown that fo ou plan to wok it i neeay to exeie ome aitional ae. u Fig.. Let filte F be eibe by y F y, The tutue of the moifie ontolle. ẋ f = A f x f B f u f y f = f x f y (N )f = (N )f x f y Nf = Nf x D Nf u f y (5) whee A f = m(n ) m I m(n ) m(n ) α I α I α N 2 I α N I (6a) B f = I, (6b) f = τ N I, (6) (N )f = τ N I, (6) Nf = τ N α I α I α N I, (6e) D Nf = τ N I an efine by impoing the inteonnetion (ee Fig. ) (6f) u f = y, u = u y y f, y = y f. (7) Fom (2), (5), (6) an (7) the ontolle i eibe by ẋ = A x B u u B y = x y (i) = i x, i =,...,N, = N x D Nu u D N, y (N) whee y (i), i =,...,N i the i th eivative of the main ontolle output y. We will enote by y, the veto T. By efining M = (I D u ) an M = (I D u ), A,B u,b,,d u,d an be haateize a follow. A B u M B u M f A = B f (ID u M ) A f B f D u M f f B B u B = u M B M B f D u M B f D M y ()...y (n) an (note that = ) i = if, i =,...,N, N = D Nf M Nf D Nf MD u f (9) D Nu D N = D Nf MD u D Nf MD () The moifie loe loop ytem Ŵ, with tanfe matix Ŵ, i given by (), (2) an (5) inteonnete by (7) an (8) u = y, u = y ; () equivalently, Ŵ i given by (), (8) inteonnete by (). Looking at the efinition of u in (7) (ee alo Fig. ), it i lea that i not jut the aae of an F, but ontain two iet feethough tem ating with oppoing ign. The aim of uh tem (whih ae abent if =, i.e. if the plant ha no iet feethough fom u to y) an be bette unetoo looking at Fig. 2: eentially, the at the output of F (the ubytem of F having tanfe matix F ()) ha the ole of emoving the iet feethough tem fom P, o that F peeive that it i onnete to a titly pope ytem. Then, the at the output of ha the ole of guaanteeing that the oiginal loe loop (befoe the inetion of F ) i not moifie. The motivation fo thi ouble tanfomation i given in the following emak. Remak : ontol folkloe ay that intouing a uffiiently mall time ontant in a table loe loop oe not impai tability. Howeve, it may be ueful to eall that uh a tatement i tue une the aumption that the loe loop whee the time ontant i inete oe not ontain an 677
u Fig. 2. y y, F y = ũ u = ỹ P y = u Loop moifiation to enue that F ee a titly pope ytem. algebai loop 9, Se. 4.7; hene, ou inteet in enuing the onition =. The following example an be ueful in oe to laify thi point. onie a tati plant P with = 2 an a tati ontolle with D u =, whih ontitute a wellpoe, table loe loop. Intouing the filte F () = (τ) the loe loop pole ae the oot of the polynomial 2 (τ) 2 2 = τ 2 2 2τ, whih ha one poitive oot fo any hoie of τ >. Though in the above example both P an ae tati, it i eay to poue imila example whee eithe P o (o both) have a nontivial tate. Notie alo that, if, the folkloe tatement an till hol, but the aitional aumption that i uffiiently mall i neee (thi fat an be poven, even fo nonlinea ytem, by a taightfowa moifiation of the poof of 9, Popoition 4.7.2). Remak 2: The tability pat of the following popoition an be genealize to nonlinea ontolle. In patiula, 9, Popoition 4.7.2, Se. 4.7 an be ue to how that une the aumption that = an fo uffiiently mall τ >, the (loal) aymptoti tability of the loe loop ytem i peeve (an, in the linea ae, thi woul be enough to eue alo global exponential tability). The ame poof an be lightly moifie to how that the ame eult till hol povie that the feethough tem (poibly epening on x) i unifomly boune by a uffiiently mall ontant. On the othe han, genealizing the pefomane pat of the popoition i muh le taightfowa. The following Popoition ompae the loe loop epone of W an Ŵ, by howing that it i alway poible to hooe the filte F uh that the iffeene W := W Ŵ between W an Ŵ i abitaily mall up to an abitay lage fequeny. Popoition : Let Aumption hol. Fo all ε >, ω (, ) thee exit τ > uh that if τ (,τ ) then ) Ŵ i well poe an aymptotially table; 2) σ( W (jω)) < ε, fo all ω, ω): 3) moeove, σ( W (jω)) < ǫ fo all ω, ) if η Ŵ Q u D zu D =, D zu D u D y =. (2) y F µ u P u P y Fig. 3. Relation among Ŵ, Q, F an ; Ŵ eue to W when =. z y z 677 Remak 3: The alulation in the poof of the theoem how that, in geneal, a mimath will alway be intoue by filte F ; howeve, they alo laify that it i alway poible to guaantee that uh egaation will be malle than an abitay mall amount up to an abitay high fequeny, povie that τ > i hoen uffiiently mall. Howeve, the ame alulation how that, unle D zu D D u D y =, the pefomane output will be eteioate by the peene of the filte at uffiiently high fequenie, whee W (jω) D zu D D u D y inepenently fom the filte paamete τ. Example : onie the example in 2, oniting in an exponentially table plant ontolle by an integal ation plu tabilize. Fo thi example, the moifie loe loop ytem (), (2), (5), (7) oepon to A y Bu = A B u = D u.2.2.4.9.5 2 2 (3) an the filte F ha tanfe funtion F () = τ 2 2 (4) 2τ Figue 4 ummaize the iffeene among the moifie loe loop ytem, fo eveal value of τ, an the oiginal loe loop ytem. Although the tep epone of the moifie loe loop ytem ae quite imila to the tep epone of the oiginal loe loop ytem, Boe iagam of figue 4 haateize the iffeene between them. Amplitue.4.2.8.6 tep epone.4 Oiginal l Moifie l, τ=.4.2 Moifie l, τ=.2 Moifie l, τ=. Moifie l, τ=.2 2 3 Time (e) Magnitue (B) Phae (eg) 2 2 4 6 225 8 35 9 boe iagam Fequeny (a/e) Fig. 4. Example : tep epone an boe iagam of the oiginal loe loop an of the moifie loe loop, fo eveal τ. III. ALMOST ANTI-WINDUP WITH MAGNITUDE AND RATE SATURATION We ae in thi etion the poblem aiing when wanting to enue that the plant input u neve exee ome peibe magnitue boun M = (M,...,M m ) an ate boun R = (R,...,R m ). In othe wo the ontol peifiation i that the plant input u i iffeentiable almot eveywhee an that it value i boune between ±M (omponentwie) while it eivative (whih i efine almot eveywhee) i limite between ±R (omponentwie). To implify the expoition, efine at M ( ) a the eentalize ymmeti atuation funtion with boun
±M an at R ( ) a the eentalize ymmeti atuation funtion with boun ±R. In 4, two moel eovey anti-winup olution have been popoe fo thi poblem. We eviit hee the eon olution peente thee, whih aume the availability of the ontolle output eivative an onite in the inetion of a filte oniting in a opy of the plant plu n u exta tate. In 4 that olution wa given fo titly pope ontolle o that the eivative of the ontolle output wa available in expliit fom. With the moifie loe-loop of Setion II, thi i alway poible. In patiula, to takle the atemagnitue atuation ae we ue a eon oe filte in (5), o that it two output y (t) an y () (t) oepon to the moifie ontolle output an it eivative, epetively, at time t. Then the anti-winup olution onit in augmenting the moifie plant-ontolle pai (), (8) with the following filte: ẋ aw = Ax aw B u (u y ) δ = at R (y () v ) y aw = y x aw (u y ) z aw = z x aw D zu (u y ), (5) whee v i a tabilizing ignal futhe iue below an whee the following inteonnetion i ue: u = y y aw, u = at M (δ ). (6) A blok iagam epeentation of the oeponing antiwinup olution (), (8), (5), (6) i epeente in Figue 5 an will be alle anti-winup loe loop though thi etion. Fig. 5. y aw- y () y AW v Rate Satuation δ aw Magnitue Satuation δ u P y Moel eovey anti-winup with ate an magnitue atuation. When inteonneting the anti-winup ompenato (5), (6) to the moifie plant-ontolle pai (), (8), the loeloop appea into a ueful aae fom whih an be appeiate in the ooinate (x l,x,x aw,δ aw ) := (x x aw,x,x aw,δ y ). In patiula, if one make the following linea eletion of v : x v = K aw aw δ y, (7) afte ome eivation, the following tutue i obtaine: ẋ l = Ax l B u y B y l = y x l y D y z l = z x l D zu y D z (8a) ẋ = A x B u u B y = x D u u D The eletion (7) i linea fo impliity of expoition but in geneal nonlinea eletion oul lea to impove tability egion an/o pefomane. - ẋ aw = Ax aw B u (at M (δ aw y ) y ) xaw ) δ aw = at R (K aw δ aw y () y () (8b) z aw = z x aw D zu (at M (δ aw y ) y ), whee y l = y y aw an whee z aw = z z l quantifie the mimath between the atual pefomane output z of the anti-winup loe-loop ytem (), (8), (5), (6) an the eiable pefomane output of the moifie loe-loop ytem (), (8), (), whih ha been hown in the peviou etion to be loe (in a uitable ene) to the pefomane output of the oiginal loe-loop (), (2), (3). Fo the anti-winup loe-loop (), (8), (5), (6), bae on the eult in 4 an on the hange of ooinate in (8), it poible to pove the following tatement, whih illutate the eiable popetie inue by the anti-winup olution. Theoem : Given the anti-winup loe-loop (), (8), (5), (6), if x aw () = an δ () = y (), then the plant input u neve exee the ate an magnitue atuation boun. Moeove, if the eletion (7) of the ignal v guaantee loal (epetively, global) aymptoti tability of the ubytem (8b), then the following hol: ) Given any epone of the moifie loe-loop (), (8), () uh that y (t) = at M (y (t)) an y () (t) = at R (y () (t)) fo all t, then z(t) z l (t) =, fo all t, namely the epone of the anti-winup loe-loop oinie with the epone of the moifie loeloop; 2) The oigin of the anti-winup loe-loop i loally (epetively, globally) aymptotially table. Note that not muh i onveye by Theoem about the omain of attation of the ytem in the ae when v only loally tabilize the ynami (8b). A qualitative tatement i that the lage the tability egion of (8b), the lage efeene an itubane will till enue onvegene of the anti-winup loe-loop. In 4 ome eipe fo the eign of K aw in (7) wee given, in aition to aitional L 2 popetie of thi heme. In thi pape, in light of the genealization aie out in the next etion, we fou on the iffeent apet lite in Theoem an we ely on the fat that any K aw tabilizing the ynami ( ) ẋaw A Bu xaw = δ aw I K aw δ, (9) aw inue loal aymptoti tability of (8b). Inee (9) oepon to (8b) when y, y () an v ae uffiiently mall not to ativate the atuation nonlineatitie. Bae on the above eult, in ou example we will ue LQR gain fo v eigne bae on the linea ynami (9). Nevethele iffeent eletion fo K aw in (7), an of v in geneal, aime at inuing lage tability egion an exteme pefomane fom a nonlinea viewpoint in (8b) ontitute a vey inteeting poblem to takle an we ega it a futue wok. In 4, Remak 5, a fix with no guaantee of effetivene wa given to ae the ae whee the ontolle wa not titly pope. In thi fix, fo whih no tability guaantee wee given, the ontol output wa appoximately iffeentiate by a filte of the type τ, with a uffiiently mall τ. In light of the iuion of the peviou etion, thi fix an be inappliable if thee i an algebai loop between plant an ontolle, while an effetive olution an be alway obtaine by ontuting the moifie loe-loop of Setion II-B. In the following example, the anti-winup olution iue hee i illutate on the ame ae tuy ue in 4. 6772
IV. ALMOST ANTI-WINDUP WITH MAGNITUDE, RATE AND URVATURE SATURATION In thi etion, the olution given in Setion III fo the ae with magnitue an ate i extene to the moe geneal poblem aiing when in aition to equiing plant input that ae boune in magnitue an ate by, epetively, ±M an ±R, bounene of thei uvatue by anothe et of boun = (,..., m ) i alo equie. Moe peifially, the equiement on the plant input i extene hee to the fat that the plant input i twie iffeentiable almot eveywhee an that it value i between ±M, it fit eivative i between ±R an it eon eivative i between ± at all time. Simila to the peviou etion efine at ( ) a the eentalize ymmeti atuation with boun ±. Genealizing the appoah of Setion III fo atemagnitue atuation, we tat fom an oiginal loe-loop ytem (), (2), (3) an ontut the moifie loe-loop ytem (), (8), (), whee the filte F i elete with two intenal tate o that, in aition to the ontolle output y, it fit eivative y () an it eon eivative y (2) ae alo available at the output of the ontolle (8). Then the following anti-winup ompenato i eigne to augment the moifie plant-ontolle pai (), (8) ẋ aw = Ax aw B u (u y ) (2a) δ = at (y (2) v ) (2b) δ = at R(δ) R(δ (δ ) ) (2) y aw = y x aw (u y ) (2) z aw = z x aw D zu (u y ), (2e) whee, given α,α R m, the funtion at α α( ) in (2) enote the non-ymmeti eentalize atuation funtion with uppe omponentwie boun α, an lowe omponentwie boun α, an, given =... m T, the ith element of the boun R( ) an R( ) ae efine a ( R i () = min R i, ) 2 i (M i at Mi ( i )) ( R i () = max R i, ) (2) 2 i (M i at Mi ( i )) The antiwinup ompenato (2) i inteonnete to the moifie loe loop a follow: u = y y aw, u = δ. (22) One again, the ignal v in (2) i a tabilizing ignal to be efine late. A blok iagam epeentation of thi anti-winup olution i epeente in Figue 6. y (2) y y () v yaw - uvatue Satuation AW - δ δaw, δaw, Rate Satuation R(δ) - δ = u Fig. 6. Moel eovey anti-winup with uvatue, ate an magnitue atuation. P y When inteonneting the anti-winup ompenato (2), (22) to the moifie plant-ontolle pai (), (8), the loeloop appea again into a ueful aae fom, paallel to (8), whih an be appeiate in the ooinate (x l,x,x aw,δ aw,,δ aw, ) := (x x aw,x,x aw,δ y,δ y () ). In patiula, if one make the following linea eletion 2 of v : v = K aw x T aw (δ y ) T (δ y () ) T T, (23) afte ome eivation, the following tutue i obtaine: ẋ l = Ax l B u y B y l = y x l y D y z l = z x l D zu y D z (24a) ẋ = A x B u u B y = x D u u D ẋ aw = Ax aw B u δ aw, ( ) δ aw, = at R(δaw,y) R(δ aw,y ) δ aw, y () y () ( xaw ) (24b) δ aw, = at K δ aw aw, y (2) δ y (2) aw, z aw = z x aw D zu δ aw, whee y l = y y aw an whee z aw = z z l quantifie the mimath between the atual pefomane output z of the anti-winup loe-loop ytem (), (8), (2), (22) an the eiable pefomane output of the moifie loe-loop ytem (), (8), (), whih ha been hown in the peviou etion to be loe (in a uitable ene) to the pefomane output of the oiginal loe-loop (), (2), (3). Fo the anti-winup loe-loop (), (8), (2), (22), bae on the hange of ooinate in (24), it poible to pove the following tatement, whih illutate the eiable popetie inue by the anti-winup olution, an genealize the eult in Theoem. Theoem 2: Given the anti-winup loe-loop (), (8), (2), (22), if δ () M,M then the plant input u neve exee the uvatue, ate an magnitue atuation boun. Moeove, if δ () = at M (y ()), δ () = at R (y () ()), x aw () = an if the eletion (7) of the ignal v guaantee loal (epetively, global) aymptoti tability of the ubytem (8b), then the following hol: ) Given any epone of the moifie loe-loop (), (8), () uh that y (t) = at M (y (t)) an y () (t) = at R (y () (t)) an y (2) (t) = at (y (2) (t)) fo all t, then z(t) z l (t) =, fo all t, namely the epone of the anti-winup loe-loop oinie with the epone of the moifie loe-loop; 2) The oigin of the anti-winup loe-loop i loally (epetively, globally) aymptotially table. Remak 4: A key tep in the poof of Theoem 2 onit in howing that a ignal y epeting all the magnitue, ate an uvatue boun at all time mut atually epet tite boun on the ate, whih ae magnitue epenent an oepon to (2). Hene, in oe to olve the poblem with uvatue boun, ometime it i neeay to pefom an antiipatoy ation an to moify the ontolle output when it i till titly inie all the thee limit (on magnitue, ate an uvatue), beaue othewie a violation woul inevitably ou at futue time. Thi featue 2 The eletion (23) i linea fo impliity of expoition but in geneal nonlinea eletion oul lea to impove tability egion an/o pefomane. 6773
i aially iffeent fom what i foun in the ae with jut magnitue an/o ate ontaint. Note alo that if R i > 2 M i i fo ome i, then the min an max funtion in (2) will alway etun the eon agument, namely the ate ontaint will neve be ative. Paalleling the iuion afte Theoem, note that any K aw tabilizing the ynami ( A Bu = I ẋ aw δ aw, δ aw, I ) xaw K aw δ aw,, (25) δ aw, inue loal aymptoti tability of (24b). Inee (25) oepon to (24b) when y, y (), y (2) an v ae uffiiently mall not to ativate the atuation nonlineaitie. Bae on the above eult, in ou example we will ue LQR gain fo v eigne bae on the linea ynami (25). Nevethele iffeent eletion fo K aw in (23), an of v in geneal, aime at inuing lage tability egion an exteme pefomane fom a nonlinea viewpoint in (24b) ontitute an open eeah poblem. V. SIMULATION EXAMPLE The hot-peio longituinal ynami of the VISTA/MATV F-6 at Mah.2 an altitue feet (oeponing to a ynami peue value of 4.8pf) at a tim angle of attak of 28 egee i eibe loally by a eon oe plant a in () with two tate oeponing to the angle of attak an the pith ate, epetively, an two input oeponing to the eviation of the elevato efletion an of the pith thut vetoing fom the tim onition (ee 7 fo etail). A in 7, the ontolle i nonlinea an oepon to a aiy haine alloation of the input, iven by a efeene ignal fo the angle of attak. We eign the anti-winup ompenato by following the appoah of Setion III, with v a in (7) an whee K aw i an LQR gain fo (9) etemine uing weight Q = I, R = I an tate matix A B u I I intea of A B u, o that the eal pat of the eigenvalue of the loe loop ytem i foe to be le than : K aw = 5.285 4.5779 5.92 2.2466 7.367 6.445 2.2466 7.4624 Then, we eign the anti-winup ompenato by following the appoah of Setion IV, with v a in (23) an whee K aw i an LQR gain fo (25) etemine uing weight Q = I, R = I an tate matix A Bu I I I intea of I A Bu I o that the eal pat of the eigenvalue of the loe loop ytem i foe to be le than : 7.69 9.775 9.696 3.333 7.872.78 K aw = 9.859 3.673 3.32 28.852.78 9.95 Figue 7 how a ompaion among the epone of the (i) unontaine (ii) magnitue an ate anti-winup an (iii) magnitue, ate an uvatue anti-winup loe loop ytem. VI. ONLUSIONS In thi pape we popoe a olution to the moel eovey anti-winup augmentation poblem fo plant with magnitue, ate an uvatue atuation boun. The popoe ontution ae illutate on eveal imulation example. 6774 4 3 2.5.5.5 Angle of attak Refeene Unontaine Boun on magnitue an ate Boun on magnitue, ate an uvatue 5 5 Fit input 5 5.3.2...2 Seon input 5 5 Fig. 7. Repone of (i) unontaine (ii) magnitue an ate anti-winup an (iii) magnitue, ate an uvatue anti-winup loe loop ytem. M =.3665,.8727, R =.2967,.7854, =.5236,.8727. REFERENES. Babu, R. Reginatto, A.R. Teel, an L. Zaaian. Anti-winup fo exponentially untable linea ytem with input limite in magnitue an ate. In Ameian ontol onfeene, page 23 234, hiago (IL), USA, June 2. 2 A. Bempoa, A. aavola, an E. Moa. Nonlinea ontol of ontaine linea ytem via peitive efeene management. IEEE Tan. Aut. ont., 42(3):34 349, Mah 997. 3 Shifin A. Sween eek aue of Gipen ah. Aviation Week an Spae Tehnology, 39:78 79, 993. 4 F. Foni, S. Galeani, an L. Zaaian. Moel eovey anti-winup fo plant with ate an magnitue atuation. In Euopean ontol onfeene, page 324 329, Buapet (Hungay), Augut 29. 5 S. Galeani, S. 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