Robust Disturbance Rejection with Time Domain Specifications in Control Systems Design


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1 Robust Disturbace Rejectio with Time Domai Speciicatios i Cotrol Systems Desig Fabrizio Leoari Mauá Istitute o Techology, Electrical Egieerig Departmet Praça Mauá, o.1 CEP São Caetao o Sul SP Brazil José Jaime Da Cruz Automatio a Cotrol Lab., Telecom a Cotrol Departmet Uiversity o Sao Paulo Av. Pro. Luciao Gualberto, Travessa 03, o. 158 CEP São Paulo SP Brazil Abstract. The robust esig o compesators aimig isturbace rejectio with time omai speciicatios is iscusse i this paper rom a perspective o loop shapig. The costraits to provie robust isturbace rejectio are erive as uctios o the isturbace reerece moel, which cotais the associate time omai speciicatios. It is show that the larger is the istace betwee the omial plat moel a the reerece moel to be ollowe the more restrictive the esig costraits are. The problem ca be pose as a moel trackig compesator a the propose proceure may reuce the coservativeess ormally associate with its esig. The plat moel is assume subject to ustructure ucertaities a the esig speciicatios are writte i the usual orm o loop shape costraits. Hece techiques like H or LQG/LTR ca be applie as esig tools. I orer to illustrate the applicatio o the propose methoology we cosier a multivariable mixture tak as a example. Keywors. Disturbace Rejectio, Moel Matchig, 2D Cotrol, Robust Cotrol, Loop Shapig, Time Speciicatios 1. Itrouctio Nowaays most multivariable liear cotrol esig techiques are carrie out i the requecy omai. May textbooks cotai a exhaustive presetatio o the theme (Gree, 1995; Helto, 1998; Skogesta, 1996; Skogesta, Nevertheless i may practical situatios part o the speciicatios is give i the time omai. The use o H 2 a H theories requires that speciicatios i the time omai be expresse i the requecy omai beore they ca be applie. For servo problems, time omai costraits are ituitive. They are also quite atural i may regulatory problems. For SISO systems these speciicatios ca ote be traslate ito the requecy omai, but this is ot the case or MIMO systems i geeral Moel matchig a 2D cotrol are two approaches that ca be use to iirectly hale time omai speciicatios to solve servo problems a also may be aapte to treat the isturbace rejectio with time omai costraits. However, eve whe applie to servo problems, the loop gai is ormally mae too high i orer to make the iput/output traser matrix close eough to the ietity matrix so that the overall system behaves approximately as the reerece moel cotaiig the give time speciicatios. The objective o the esig approach presete i this paper is to aapt the Moel matchig a 2D cotrol schemes erivig coitios to guaratee robust isturbace rejectio with a prescribe time omai behavior. Attetio is restricte to the case where the isturbaces ca be measure. I particular, we propose a esig techique a argue that it may be less coservative tha usually. Coitios to guaratee setpoit trackig over give requecy rages withi prescribe accuracy are well kow a will be omitte here. Although the coitios or o measure isturbace rejectio a measuremet error rejectio are also well kow they are iclue i this paper just or the sake o completeess sice together with the measure isturbace rejectio they orm the so calle regulatory problem. The plat moel is assume subject to ustructure ucertaities a the esig speciicatios are writte i the usual orm o loop shape costraits. Hece techiques like H or LQG/LTR (Athas, 1986; Doyle, 1981 ca be applie as esig tools. I orer to illustrate the applicatio o the propose methoology we cosier a multivariable mixture tak as a example. The paper is structure as ollows. Sectio 2 cotais a prelimiary iscussio o how the isturbace rejectio problem with time omai speciicatios ca be pose as a moel trackig problem. I sectio 3 the esig coitios require or robust isturbace rejectio with time omai speciicatios are writte i both the orm o loop sesitivity costraits a i terms o costraits o the loop gai shape. A aalysis o the cotrol magitue associate to moel ollowig is perorme i sectio 4. A umerical example is presete i sectio 5 to illustrate the propose methoology. Sectio 6 cotais the coclusios o the paper.
2 2. Prelimiary Discussio Cosier the system represete i Fig. (1. P ( a K ( are respectively the plat a compesator traser matrices. The traser matrix Σ ( is a ilter a will be calle isturbace reerece moel. ( is the o measure isturbace relecte at the plat output, ( is the measure isturbace relecte at the plat iput, ( is the measuremet error, r ( is the system iput a y ( is the system output. All sigal a tras er matrices are assume to have compatible imesios. ( Σ ( ( ( r( K( u( P( y( e( ( Figure 1. Cotrol problem. Whe there is o measure isturbace ( ( = 0, the esig problem turs ito the classical oe a the coitios to guaratee setpoit trackig, o measure isturbace rejectio a measuremet error rejectio over give requecy rages withi prescribe accuracy are well kow. O the other ha, takig oly the measure isturbace, i.e., or r ( = ( = 0, the block iagram i Fig. (1 becomes as show i Fig. (2. This igure is equivalet to the oe cosiere i reerece (Jockheere, 1999 i orer to solve a aircrat propulsio cotrol problem. ( P( ( u( y( e( K( ( Figure 2. Moel trackig structure. Notice that the cotrol problem epicte i Fig. (2 is a twoegree o reeom problem. By simple block maipulatio it becomes the iagram i Fig. (3 the 2D cotrol with a preilter (Leoari, 2002a. ( K( v( e( P( ( y( K( Figure 3. The 2D cotrol structure. ( Desig aimig measure isturbace rejectio with time omai speciicatios ca thus be viewe as a moel matchig problem a i this case it ca be reuce to makig the closeloop traser matrix rom v ( to y ( close to K 1 ( s i the requecy rage where the matchig betwee Σ ( a the traser matrix rom r ( to y ( is sought (Kwakeraak, 1996.
3 I reerece (Maciejowski, 1989 (see page 14 he writes about the choice o the preilter: "we ca cosier irst the problem o esigig K ( to obtai esire S ( a T (, a subsequetly esig Σ ( to give a suitable" traser matrix rom ( to y (. A high loop gai is require i orer to get a closeloop traser matrix close to K 1 ( s a the choice o how high it is mae may give rise to a coservative esig i it is take higher tha ecessary. As propose i (Leoari, 2002b, the moel trackig problem is uerstoo i this paper i the ollowig sese. We look or a compesator K ( such that the orm o the traser matrix rom ( jω to e ( jω (see ig. 2 be below some prescribe value i a give requecy rage this requiremet will be calle moel ollowig. Aitioally we wish that the cotributios to the output y ( o both the o measure isturbace ( a the measuremet error ( be below give values i give requecy sets. I what ollows we aopt the perspective o a set membership or traser matrices o the plat moel cosierig, i particular, the multiplicative represetatio o the moelig error. We assume that a upper bou is give or the spectral orm o the multiplicative error matrix i the orm o a scalar uctio e ( ω (Doyle, Loop Shapig M I what ollows the symbol represets the Eucliea orm o complex vectors. i[, mi [ a [ ith, the miimum a the imum sigular values o [, respectively. eote the 3.1. Nomial Plat Moel Assume that the plat yamics are give by their omial moel. The, or the systems represete i Fig. (3, the ollowig set o equatios hol: ( I K( ( S( P( K ( (, y( = S( ( S( P( s (1 e ( = S ( ( ( S( (, (2 ( I K ( ( S( K ( (. u( = S( K( ( S( s (3 S ( = I P( K ( is the loop sesitivity matrix. (s P (s a K ( where ( 1 At this poit, sice we are obviously assumig that Σ is stable, it shoul be clear that system stability is etermie solely by the closeloop cotaiig. Because this is a classical situatio, stability is ot aresse i the paper Measure Disturbace Rejectio Assume that > 0 α (typically α << 1 measure isturbace i a give set o requecies { ω ω } is a give umber which expresses the esire accuracy o the rejectio o Ω i the sese that e jω ( jw α ( ω Ω Ω = R : ω or a give ω. Assumig that ( = ( = 0 ollowig coitio rom Eq. (2: (. Typically, to accomplish moel ollowig we get the α [ S( ( ω Ω (4 Hece, as it shoul be expecte, the sesitivity must ecrease as the istace 1 betwee the plat a the isturbace reerece moel icreases. The same occurs as α ecreases. Nevertheless, epeig o the speciic problem at ha, this coitio may be ot so restrictive a the sesitivity may be ot ecessarily low. 1 Distace is obviously uerstoo here as measure by the spectral orm o the ierece o the two traser matrices.
4 Whe the rightha sie o (4 is much smaller tha 1, this coitio ca be rewritte approximately as mi K ( ( ω Ω. (5 α Similarly as above, this coitio shows that the loop gai shoul icrease with both the istace betwee P a Σ a the iverse oα No Measure Disturbace Rejectio Suppose that = { ω ω } Ω R : ω is a give requecy set where the o measure isturbace ( ( = a ( = 0. For a give α > 0 (typically α << 1 has its eergy. Assume also that 0 measure isturbace rejectio coitio as preomiatly we express the o e( ( α ( ω Ω. (6 From Eq. (2 we the get the ollowig wellkow suiciet coitio: [ S( α ( ω Ω, (7 which leas to the ollowig approximate coitio (Da Cruz, 1996: mi 1 α jω K ( ( ω Ω (8 whe α << Measuremet Error Rejectio Suppose that = { ω ω } Assume also that 0 rejectio coitio as Ω R : ω is a give requecy set where the measuremet error preomiatly has its eergy. ( = a ( = 0. For a give α > 0 (typically α << 1 we express the measuremet error y( ( α ( ω Ω. (9 where From Eq. (1 we the get the coitio o measuremet error rejectio: [ T( α ( ω Ω, (10 T ( = S( P( K (. (11 Alteratively, rom Eq. (1 it ollows the ollowig approximate orm (Da Cruz, 1996: K ( α ( ω Ω. (12
5 whe α << Moel Ucertaities First o all, recall that or the multiplicative moel error aopte the stability robustess coitio is give by (Doyle, 1981 [ T ( e ( ω ( ω < M Assumig that ( ω < 1 e M or Ω Ω. (13 ω, the coitios (4 a (7 above moiy respectively to (Gree, 1995: [ 1 em ( ω N ( α [ S( ( ω Ω (14 a [ S( α [ 1 ( ω ( ω Ω. (15 e M Assumig that α << 1, coitio (10 ca be rewritte i the ollowig approximate orm (Da Cruz, 1996: α [ T ( ( ω Ω. (16 1 e ( ω M As expecte the eect o moel ucertaity is to make the costraits o S a T more restrictive. From Eq. (1 we ca see that S( P( ( I K( P (, Σ ( a K ( exhibit low gais. Hece, the overall traser matrix is approximately equal to P ( value o ω is ot arbitrary but relate to both P ( a Σ ( has o arbitrary yamics at high requecies where, i geeral,. I view o this the. This meas that the moel ollowig coitio (14 oes ot ecessarily implies moel matchig. This is the reaso why we call the proceure moel trackig. Simulatios carrie out by ow iicate it is reasoable to expect ice matchig up to oe ecae above the reerece moel bawith. I geeral this is eough to esure goo moel ollowig. It shoul be emphasize that (14(16 together with (13 are the key coitios to esig robust compesators i orer to achieve moel trackig. To close this sectio recall that the coitios or robust esig ca also be expresse i terms o the loop gai i the usual way (Doyle, Cotrol Magitue Aalysis or Moel Followig I this sectio we restrict the aalysis to the omial plat. From Eq. (3 it is straightorwar to see that u( ( = K ( 1 [ I P( K ( [ P( ( (17 Assume that P, Σ a it ollows that [ P ( K ( 1 α are such that α 1 mi >> >> or ω Ω. I this case Eq. (18 leas to. Thus i coitio (5 hols, u( ( P( 1 [ P( ( (18 i we assume that both P a K are square a From Eq. (19 it the ollows immeiately that 1 P exists.
6 sup 0 u( ( ( 1 [ P ( [ P( (19 This equatio shows that the worstcase relative icremet o the cotrol magitue is approximately the same as the relative istace betwee both the plat a the isturbace reerece moels. Hece isturbace reerece moels that are istat rom the plat moel require a large cotrol magitue to be ollowe. This is i accorace with coitio (14 which shows that the larger is the is tace betwee the plat a the isturbace reerece moels the more restrictive the coitio o moel ollowig is. 5. Numerical Example I orer to illustrate the applicatio o the propose methoology we cosier the stirre tak o reerece (Kwakeraak, Its omial liearize state moel is give by 0, x& ( t = ( 0 0,02 x t 0,25 0,01 0 y = x( t u( t 0,75 (20 Sice the mai cotributio o this work ocus o the isturbace rejectio with time omai speciicatios, other esig speciicatios are omitte i this examp le. Besies, as moel ucertaities turs the esig costraits just more restrictive, they are omitte i this istace a may be cosiere iclue i the esig speciicatios. Cosier as cotrol system speciicatios that isturbace at both cotrol chaels shall have its iluece o the output rejecte as i iltere by a seco orer system 0.25 s Σ ( = ( i = 1,2 2 s 0.7s 0.25 ii, (21 withi a tolerace o 10% (i.e., α = 0. 1 i the requecy rage that extes up to oe ecae above the reerece moel bawith. From the time omai speciicatios we thus have Σ11( = 0 0 Σ ( 22 s s, (22 The H mixesesitivity ramework will be use to obtai K ( (Gree, 1995; Helto, 1998; Skogesta, 1996; Skogesta, To simpliy we assume that moel ucertaities have alreay bee take ito accout i the eiitio o the weightig matrix W1 ( 10 s =. (23 Oce the plat P ( is i the rightha sie o (4 its iput variables may be scale i such a way as to get it close to the ietity i low requecies. I this case coitio (4 ca be rewritte as α [ S( ( ω Ω, (24 where S ( u S u is a square osigular matrix with compatible imesios 2. 2 Obviously, S u is a part o the compesator.
7 For simplicity S u is cosiere here as a costat matrix a equal to the plat iverse at low requecies, amely, S u = lim P 1 (. (24 s 0 Notice that scalig the plat iput variables oes ot chage the multiplicative moelig error. Hece both perormace a stability robustess coitios are ot aecte. Figure (4 shows the sigular value Boe plots o matrices relevat or the moel ollowig esig. W ( is the weightig matrix or the mixesesitivity proceure. 1 s 40 b 20 0 S ( i u 20 [ W ( j i 1 ω 40 [ ( i Σ ra/sec Figure 4. Boe plots or Moel Followig. Notice that i [ W1 ( is situate 20B above. With the sake o illustratio, the time respose o the closeloop system a the cotrol variables have bee evaluate or the moel ollowig esig Σ y( ( s ( sec Figure 5. Closeloop time respose.
8 Two uit steps isturbaces applie at istats 10 sec. a 30 sec. with positive a egative sigs, respectively, have bee cosiere. Simulatio results have bee plotte i Fig. (5. As it ca be see, the process outputs ollow closely the correspoig oes o the isturbace reerece moel. The cotrol time history is plotte i Fig. ( u 1 ( t 5 u 2 ( t 0 sec Figure 6. Cotrol time history. 6. Coclusios This work iscusse the robust cotrol esig or the rejectio o measure isturbace with time omai speciicatios. The moel trackig problem has bee pose as a atural way o ealig with time omai speciicatios i a requecy esig cotext. It has bee show that the moel ollowig coitio epes irectly o the istace betwee the reerece moel a the plat omial moel the larger the istace betwee both moels the more restrictive the coitio is. It has bee show that the relative icrease i the cotrol magitue to attai moel ollowig is approximately the same as the relative istace betwee the plat a the isturbace reerece moels. Hece, as expecte, the cotrol magitue icreases with the istace betwee the omial a isturbace reerece moels. Sice the isturbace reerece moel Σ ( is a explicit part o the compesator, small ajustmets may geerally be oe ater the esig has bee complete. This possibility ca be useul i practical applicatios where ie tuig is require urig cotrol systems startup. The mixe sesitivity ormulatio o the H cotrol theory has bee use i a umerical example to illustrate the applicatio o the methoology. Nevertheless, sice the esig coitios are ultimately expresse as costraits o Boe Diagrams o the system, ay loopshapig techique coul be equally use. 7. Ackowlegmets F. Leoari is grateul to Mauá Istitute o Techology or its partial iacial support. 8. Reereces Athas, M., 1986, "A Tutorial O The LQG/LTR Metho". Proceeigs o America Cotrol Coerece, Seattle, WA. Da Cruz, J. J., 1996 "Multivariable Robust Cotrol", Eusp, São Paulo, SP, (i Portuguese. Doyle, J. C.; Stei, G., 1981 "Mutivariable Feeback Desig: Cocepts or a Classical/Moer Sythesis". IEEE Trasactios o Automatic Cotrol, Vol. 26, pp Leoari, F.; Da Cruz, J. J., 2002 "Robust Moel Trackig a 2D Cotrol Desig". Proceeigs o the 10 th Meiterraea Coerece o Cotrol a Automatio, Lisbo. Leoari, F., 2002 "Robust Multivariable Cotrol System Desig with Time Domai Speciicatios". PhD. Thesis Epusp. São Paulo, SP (i Portugues e.
9 Gree, M.; Limebeer, D. J. N., 1995 "Liear Robust Cotrol". Pretice Hall, New Jersey. Helto, J.W.; Merio, O., 1998 "Classical Cotrol Usig H Methos", SIAM, Philaelphia. Jockheere, E. A.; Yu, G. R., 1999 "Propulsio Cotrol o Cripple Aircrat by H Moel Matchig". IEEE Trasactios o Cotrol Systems Techology, Vol. 7(2, pp Kwakeraak, H., 1996 "How Robust are H Optimal Cotrol Systems?", Proceeigs o the XI Brazilia Coerece o Automatics, São Paulo, pp Kwakeraak, H.; Siva, R., 1970 "Liear Optimal Cotrol Systems", Wiley, New York. Maciejowski, J. M., 1989 "Multivariable Feeback Desig". Aiso Wesley, Wokigham. Skogesta,, S.; Postlethwaite, I., 1996 Multivariable Feeback Cotrol: Aalysis a Desig, Joh Wiley & Sos. Zhou, K.; Doyle, J. C., 1998 "Essetials o Robust Cotrol". Pretice Hall, New Jersey.
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