DESIGN OF CONTROLLERS FOR STABLE AND UNSTABLE SYSTEMS WITH TIME DELAY

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1 DESIGN OF CONTROLLERS FOR STABLE AND UNSTABLE SYSTEMS WITH TIME DELAY P. Dotál, V. Bobál Department of Proce Control, Facult of Technolog, Toma Bata Univerit in Zlín Nám. T. G. Maarka 75, 76 7 Zlín, Czech Republic Phone: , otalp@ft.utb.cz Abtract: The paper preent one approach to eign of controller for time ela tem. The propoe metho i bae on the time ela Paé approximation. The controller are erive uing the polnomial approach an LQ control technique. Reulting proper an table controller obtaine via polnomial Diophantine equation an pectral factorization technique enure etpoint tracking a ell a loa iturbance attenuation. The proceure i evelope for table an untable firt orer time ela tem an the reult are verife b imulation in MATLAB Simulink. eor: ela tem, time ela approximation, polnomial metho, LQ control. INTRODUCTION The exitence of a time ela in input-output relation i a common propert of man technological procee. Plant ith the time ela can often not be controlle uing uual controller eigne ithout a conieration for a preence of the ea-time. The control repone uing uch controller then ten to etabilize the cloe-loop tem. It i ell knon that a an effective time-ela compenator epeciall for table tem ith long time ela, the Smith preictor can be ue. Hoever, a part of technological procee containing a time ela can be untable, uch a chemical reactor or bioreactor. Man ifferent approache have been evelope to control uch procee. While ome metho iue from everal moification of the Smith preictor (De Paor an Egan, 989), (Majhi an Atherton, 999), (Liu et al., 5), other metho emplo PI, PD an PID control trategie (Rottein an Lein, 99), (Park et al., 998) or IMC-bae metho (Huang an Chen, 997), (Tan et al., ). The paper preent one approach to control both table an untable time ela tem bae on the time ela approximation an the polnomial approach, ee, eg. (učera, 99), (Grimble, 994). The principle of the metho have been evelope for a table firt orer time ela tem (FOTDS) an publihe in (Dotál et al., a). For an untable FOTDS, the reult obtaine uing firt orer numerator an Paé time ela approximation in the DOF an DOF control tem configuration ere analze an compare in (Dotál et al., b). The reult emontrate the priorit of the Paé approximation to the numerator approximation together ith a utilization of the DOF configuration. The metho preente here i bae on the above combination. For tuning of the controller parameter, the LQ control technique i emploe, ee, e. g. (Dotál an Bobál, 999). Reulting table an proper controller obtaine via polnomial Diophantine equation an pectral factorization technique enure amptotic tracking of tep reference a ell a tep loa iturbance attenuation. Even though an metho bae on a time ela approximation cannot guarantee the control tem tabilit in general, the imulation reult ocument a uabilit of the propoe metho proviing table control repone of a goo qualit alo for higher value of the time ela. APPROXIMATE TRANSFER FUNCTIONS Conier the tranfer function of table an untable firt orer time ela tem having the form

2 G( e () + G( e () here > i the gain, > i the time contant an > i the time ela. The time ela tranform i approximate b the Paé approximation e () + Uing approximation (), approximate tranfer function take form ( b b GAS ( (4) ( + ) ( + + a + a G here AU ( b b ( (5) ( ) ( + + a a b, b, a, a + in (4), a in (5) an. Both approximate tranfer function (4) an (5) are trictl proper tranfer function b( G A ( (6) a( here b an a are coprime polnomial that fulfill the inequalit eg b < eg a. CONTROL SYSTEM DESCRIPTION The DOF control tem configuration i epicte in Fig.. Here, a controller contain next to the feeback part Q alo the feeforar part R. In the cheme, i the reference ignal, v i the loa iturbance, i the controlle output an u i the control input. Both an v are coniere to be tep function. The tranfer function of the controlle tem G repreent one from approximate tranfer function G A. For the tepie reference an the tep loa iturbance, both controller part contain an integrator an their tranfer function can be foun in the form of polnomial fraction R v q( Q (, p( here q, r, p are polnomial in. r( R ( (7) p( 4 APPLICATION OF POLYNOMIAL METHOD The controller eign ecribe in thi ection follo from the polnomial approach. The general conition require to govern the control tem propertie are formulate a follo: Strong tabilit of the control tem (in aition to the control tem tabilit, alo the controller tabilit i require). Internal properne of the control tem. Amptotic tracking of the reference. Loa iturbance attenuation. The proceure to erive amiible controller can be carrie out a follo: A feeback controller given b a olution of the polnomial Diophantine equation a ( p( + b( q( ( (8) ith a table polnomial on the right ie enure the control tem tabilit an the loa iturbance attenuation. A table polnomial p( in enominator of (7) enure the controller tabilit. Amptotic tracking of the tep reference i provie b the controller feeforar part given b a olution of the polnomial Diophantine equation t ( + b( r( ( (9) here t( i an auxiliar polnomial hich oe not enter into controller eign but hich i necear for calculation of equation (9). The control tem atifie the conition of internal properne hen the tranfer function of all component are proper. The egree of the controller polnomial then mut fulfill inequalitie eg q eg p +, eg r eg p +. () Taking into account (), the conition eg b eg a an a olvabilit of (8) an (9), the egree of polnomial q, p an r can be erive a eg q eg a, eg p eg a, eg r. () The controller parameter then follo from olution of polnomial equation (8) an (9) an epen upon coefficient of the polnomial. No, the next problem mean to fin a table polnomial that enable to obtain the acceptable tabilizing an table controller. Q - u G 5 LQ CONTROL TECHNIQUE Fig.. DOF control tem configuration. In thi ection, the polnomial i coniere a a prouct of table polnomial g an n in the form

3 ( g(. () The firt polnomial g i obtaine b pectral factorization ( a( ) ( a( ) + b ( b( g ( g( ϕ () here the aterik enote a conjugate polnomial. Remark: It i ell knon from the LQ control theor that the polnomial g i ue to minimization of the quaratic cot function { e t) + ϕu ( t) } J ( & t (4) here e (t) i the tracking error, u& (t) i the control input erivative an ϕ > i the eighting coefficient. The econ polnomial n, enuring properne of the controller, i given for a table FOTDS a n ( a(. (5) For an untable FOTDS, thi polnomial i etermine in to ifferent a. In the firt cae, thi polnomial i given a a table part of pectral factorization n ( a ( a(. (6) Then, the proceure lea to a trictl proper controller. The egree of on the right ie of (8) an (9) i given a eg ( eg[ g( ] ega( +. Taking into account () an the relation eg [ p( ] eg ( eg a( eg a( +, trict properne of (7) i evient. In the econ cae, the polnomial n i choen a a table part of the polnomial a n a + ( ) ( +. (7) No, the proceure lea to a nontrictl proper feeback part of the controller. Since equalitie eg ( eg [ g( ] eg a( an eg[ p( ] eg ( eg a( eg a( hol, Q( in (7) i nontrictl proper. A preference of uch etermination of polnomial lie in the fact that all controller parameter can be tune b onl ingle electable parameter. Taking into account that all parameter except ϕ in () are given b propertie of the controlle tem, the coefficient of g epen upon ingle electable parameter ϕ. Since polnomial n oe not contain an electable parameter, alo the coefficient of polnomial, an, the controller parameter given b olution of (8) an (9) epen next to fat given parameter, an onl upon ϕ. 6 CONTROLLER DESIGN polnomial g ha the form g ( + g + g + g (8) ith coefficient g ϕ g g (9) ϕ ϕ g +. g + 4 ϕ In behalf of hortne of the riting, other important equation an erive formula for coniere tem are introuce in the form of table in the folloing orer: Form of ue polnomial n. Formula for computation of coefficient n. Tranfer function of the reulting controller. Formula for computation of controller parameter. Conition of the reulting controller tabilit. Tab.. Stable FOTDS n ( + n + n n, n + q + q + q Q(, + p + p ) r R( + p + p) p g + g, p g q g, + q g, q g p >, p > for all Tab.. Untable FOTDS Strictl proper controller n ( + n + n n, n + p q + q + q Q(, + p + p ) r R( + p + p ) 4g + ( + ) g + g p g +, q g 4 + For both table an untable FOTDS, norme

4 q [ p + g + ( + ) g ] q ( p g) g + p > for all, p > for < Tab.. Untable FOTDS Nontrictl proper feeback part of the controller n ( + n n q + q + q Q(, + p ) q p g r R( + p g + g + g g p > for < +, q [ p + ( g + )] q g [ ( p ) ] 7 EXAMPLES All imulation experiment in thi ection ere performe b MATLAB-Simulink tool. In all cae, the reference ignal (t) (t) a ue. 7. Stable FOTDS Conier a table firt orer time ela tem ith tranfer function () here an 4. In realize imulation, the tep iturbance v(t). a incorporate into the controlle tem at time t v. The value of t v ma be een belo each figure. The tep reference an loa iturbance repone for 4 ( ), an ( ) an 4 ( 6) are hon in Fig., an 4. The repone in all figure clearl ocument an effect of the parameter ϕ upon the control repone. An increaing ϕ improve the control tabilit an b chooing of it higher value aperioic repone can be obtaine. The repone in Fig. 5 emontrate the robutne of the propoe metho againt change of. The controller parameter ere compute for a nominal moel ith 4 an ubequentl ue for perturbe moel ith the ± % etimation error in value ( 8.8 an 9.). 7. Untable FOTDS Let in tranfer function () an 4. The repone for ( /.5) are hon in Fig. 6. Thee ocument a control of goo qualit b both trictl an nontrictl controller for a mall value ). The repone to the tep loa iturbance uing both controller are compare in Fig. 7. Thee ho that a nontrictl controller give approximatel halflength overhoot. Such a in the above cae, an effect of parameter ϕ upon the control repone i hon in Fig. 8 an 9. Again, it can be een that an increaing value of ϕ reult in aperioic repone ithout overhoot uring tracking a ell a uring iturbance attenuation. Repone for 5 ( /.5) are hon in Fig.. Epeciall, the etpoint ϕ ϕ Fig.. Step reference an loa iturbance repone ( 4, t v 5) ϕ ϕ 5 ϕ 5 5 Fig.. Step reference an loa iturbance repone (, t v ) ϕ ϕ Fig. 4. Step reference an loa iturbance repone. ( 4, t v ).

5 repone ithout an overhoot ocument the uabilit of the propoe metho. Note that for intance De Paor an Egan metho (989) provie table etpoint repone for / <.56 an Majhi an Atherton (999) require thi ratio a / <. Simulation reult in Fig. ho a robutne of the metho. Here, a nominal moel ith the exact value ( /.75) ha been ue for a ± % etimation error in (. an.7). The repone emontrate a ufficient robutne of the controller alo for a relativel higher ratio beteen an, though the enitivit to an etimation error in gro up accoring to an increaing ratio / Fig. 5. Nominal an perturbe tem repone ( 4 (),8.8 (), 9. (), ϕ 4, t v ) Fig. 8. Nontrictl proper controller: Step reference an loa iturbance repone (, ϕ 5 (), (), 4 (), v -., t v 5) Fig. 6. Step reference an loa iturbance repone for nontrictl () an trictl () proper controller (, ϕ 5, v -., t v 5) Fig. 9. Nontrictl proper controller: Step etpoint an loa iturbancerepone ( 4, ϕ 5 (), (), 4 (), v -., t v 5) Fig. 7. Step loa iturbance repone for nontrictl () an trictl () proper controller ( 4, ϕ, v., t v ). 4 5 Fig.. Nontrictl proper controller: Step reference an loa iturbance repone ( 5, ϕ 6, v -., t v 5).

6 Fig.. Nominal an perturbe tem repone ( (),. (),.7 (), ϕ 4, v -., t v 5). 8 CONSLUSION The problem of control eign for table an untable time ela tem ha been olve an analze. The propoe metho i bae on the firt orer Paé time ela approximation. The controller eign ue the polnomial metho an a controller etting emplo reult of the LQ control theor. The preente proceure provie atifactor control repone in the tracking of the tep reference a ell a in tep loa iturbance attenuation. The preente reult have emontrate the uabilit of the metho an the control of a goo qualit for relativel high ratio beteen the time ela an the time contant ( / 6 for a table an /.5 for an untable FOTDS). The proceure make poible a tuning of the controller parameter b a ingle electable parameter. Uing erive formula, the controller parameter can be automaticall compute. From thi reaon, the metho coul alo be ue for an aaptive control. ACNOWLEDGMENTS Thi ork a upporte in part b the Minitr of Eucation of the Czech Republic uner grant MSM 7885 an b the Grant Agenc of the Czech Republic uner grant No. //7 an No. /5/7. REFERENCES De Paor, M. an R.P.. Egan (989). Extention an partial optimiation of a moifie Smith preictor an controller for untable procee ith time ela. Int. J. Control, 5, 5-6. Majhi, S. an D.P. Atherton (999). Moifie Smith preictor an controller for procee ith time ela. IEE Proc. Control Theor Appl., 46, Rottein, G.E. an D.R. Lein (99). Simple PI an PID tuning for open-loop untable tem. In. Eng. Chem. Re.,, Park, J.H., S.W. Sung an I. Lee (998). An enhance PID control trateg for untable procee. Automatica, 4, Liu, T., Y.Z. Cai, D.Y. Gu an W.D. Zhang (5). Ne moifie Smith preictor cheme for integrating an untable procee ith time ela. IEE Proc. Control Theor Appl., 5, Huang, P. an C.C. Chen (997). Control tem nthei for open-loop untable proce ith time ela. IEE Proc. Control Theor Appl., 44, pp. 4-46, (997). Tan, W., H.J. Marquez an T.W. Chen (). IMC eign for untable procee ith time ela. J. Proce Control,, -. učera, V. (99). Diophantine equation in control A urve. Automatica, 9, Grimble, M.J. (994). Robut inutrial control. Optimal eign approach for polnomial tem. Prentice Hall, Lonon. Dotál, P., V. Bobál an R. Prokop (a). A ne methoolog of imple controller eign for time ela tem. Part one table tem. Journal of Electrical Engineering, 5, 4-8. Dotál, P., V. Bobál an R. Prokop (b). The eign of imple controller for untable time ela tem uing LQ control theor. In: Proc. European Control Conference ECC, Porto, Portugal, 6-. Dotál, P. an V. Bobál (999). The uboptimal tracking problem in linear tem. In: Proc. 7 th IEEE Meiterannean Conference on Control an Automation, Haifa, Irael,