Adaptive Control and On-line Controller Design for Multivariable Systems in State-Space

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1 Adaptve Control and On-lne Controller Desgn for Multvarable Systems n State-Space István Harmat Department of Control Engneerng and Informaton Technology Techncal nversty of Budapest H- Budapest, Pázmány Péter sétány /D. Hungary E-mal: harmat@seeger.t.bme.hu Abstract - In many practcal cases, the system has unnown or partally unnown parameters mang the control more dffcult. The use of adaptve control strateges s one of the most promsng approaches to overcome ths problem. The paper deals wth the ndrect adaptve control of slowly varyng multvarable (MIMO) lnear systems n the statespace. Our results are ntended to exhbt the effectveness of the on-lne dentfcaton and to compare the dfferent control algorthms. In connecton wth ths subject, the possbltes of real-tme realzaton are also nvestgated. I. INTRODCTION In the more general applcatons, there are not gven enough nformaton of the process. However, some popular technques suggest successful treatments for ths problem ncludng also the dfferent adaptve methods. Indrect adaptve control descrbes a body of theoretcal and practcal engneerng prncples n whch one brngs nto unon the model-based development of feedbac controllers wth the on-lne dentfcaton based on the recursve system parameter estmaton and the state estmaton. The dentfcaton s accomplshed n a common loop n every samplng nstant. The dentfed model may be gven n the Luenberger s controllablty normal form where the structure ndces can be determned n separate off-lne experments. The estmated parameters are connected to the correspondng system parameters and the parameters of the nnovaton nose model. The parameter and the state estmaton are carred out parallel, whch rely on the state-space descrpton of the lnear MIMO system. Optmal estmaton needs the computaton of the gradent of the estmaton error, whch s related to a smple matrx recurson and the computaton of the dervatves of system matrces. After the system dentfcaton, varous type of control algorthms can be chosen. In our nvestgaton on-lne controller desgn methods are developed and mplemented ncludng pole placement, LQ optmal control, predctve control and decouplng. In order to avod the often occurred numercal and stablty problems n state-space predctve control, t s advantageous to apply numercally robust and stable predctve methods based on the results of Rosster, Kouvartas and Rce []. The mproved numercal condtons allow us to get a powerful tool for many multvarable lnear systems. Each subsystem of MIMO system nfluences on the other one. Of course, ths phenomenon needs a more sophstcated treatment n the control system. The Béla Lantos Department of Control Engneerng and Informaton Technology Techncal nversty of Budapest H- Budapest, Pázmány Péter sétány /D. Hungary E-mal: lantos@t.bme.hu decouplng of MIMO systems s a standard problem n the control engneerng. Our goal s to fnd a state feedbac n order to reduce the system to ndependent SISO channels. The algorthm conssts of dfferent steps. Frst tme, one must chec the Glbert s condton for decouplng and fnd the unstable decouplng zeros based on the canoncal form of the system decoupled n ntegrator sense. Sometmes t may be necessary to desgn a parallel compensaton for removng the unstable decouplng zeros n the model and/or to perform a precompensaton, f the Glbert s condton s not satsfed. Our method s the combnaton of some early results of Glbert [3], the controllablty condton of Haustus and some new results of Lohmann [4], extended wth modfcatons needed for numercal stablty of the desgn steps. The paper s arranged as follows. Secton II gves the survey of the concept and theory on the area of the appled adaptve control algorthms. A smlar approach can be found n []. The reader may fnd some deeper nterpretatons and a more unfed approach n [5]. The results are presented n the frame of an llustratve example n Secton III together wth the demonstraton of dfferent control strateges. One of our man ntentons s to compare ther behavors. The dentfcaton has a dstngushed role n the whole dscusson because the control performance depends above all on the precson of the model. It s evdent n the realzaton that the on-lne concept requres sutable devces and quc, smple numercal algorthms to fulfll the real tme specfcatons. On-lne controller desgn methods for slowly varyng MIMO systems are mplemented n MATLAB. Secton IV ntroduces the ACE Kt of dspace as a possble alternatve for the real-tme mplementaton. The ACE KIT s based on the board DS and has a Real-Tme Interface (RTI) to assure the connecton wth the MATLAB envronment. More nformaton about the total development envronment s descrbed n [6]. II. A SRVEY OF THE APPLIED THEORY In ths secton we gve the theoretcal bacground of the mplemented methods. Subsecton II-A setches the control system structure. The followng subsectons summarze brefly the theory of dentfcaton, pole assgnment, LQ optmal control, predctve control and the decouplng. The fnal subsecton formulates the steps of the controller desgn.

2 A. Control system structure The control system archtecture s shown n Fg.. It can be seen that an explct separaton between dentfcaton and control s assumed. A nown model for the controller desgn s needed n the technque of adaptve control. Snce the plant parameters and the states are n fact unnown, they are obtaned usng a recursve parameter and state estmaton algorthm. \ W + = & [W + () The obtaned results allow us to establsh a model n state space. Further on, ths model can be regarded as a sutable substtuton of the real system. In our mplementaton, the dentfcaton obtans the model n Luenberger s controllablty form, where nown controllablty ndces are supposed. C. Pole assgnment Several algorthms are avalable for multvarable feedbac desgn based on pole assgnment. One of the methods uses the Luenberger s controllablty form. The advantage of the approach s that the dentfcaton supples mmedately ths normal form. If the characterstc polynomal of the specfed closed loop s σ σ = V + D V + + D F σ ϕ () Fg.. The scheme of ndrect adaptve control then we can ntroduce the notatons The estmated parameters and states enable us to mae a model, whch s the bass of the controller desgn. For ths reason, the ey ssue of the whole concept s the performance of the dentfcaton. The dfferent controller desgn methods are elaborated n state-space. Addtonally, one also has to tae care of the statc behavors. $ ϕ F = D D O L D σ (3) B. Identfcaton The am of dentfcaton s to obtan the estmated model parameters and states from the nput and output of the system. The algorthm s descrbed n several steps. The algorthm of recursve parameter estmaton has to be executed n each samplng nstant ε W = \ W ϕ W ϑ W = \ W \ W () 3W W [, + ϕ W3W ϕ ] [,. W W] 3W. W = ϕ W () 3W = ϕ (3) ϑ W = ϑ W +. Wε W (4) then the algorthm can be contnued wth the steps of state estmaton as ϑ W $%&/ G$ G% G/. = [ W + XW + εw Gϑ Gϑ Gϑ (5) G& 9. = [W G ϑ (6) : = $ / & : + / () [ W + = $ [W + % XW + / W (8) G& 9. = [W + (9) G ϑ ϕ W + = 9 + & :.. + () $ s the matrx consstng of general rows n $ /8( % s the matrx consstng of general rows n % /8( a s the matrx belongng to general rows of $ $ ϕ F where $, % are the matrces of the dentfed model. In ths case, the state feedbac s a a = % $ (4). ϕ $ /8( /8( /8( F D. LQ optmal control The quadratc regulator problem s to fnd the control nput to mnmze the cost functonal ) [ = (5) { 4 [ [ + 5 X X } N N N N N N N = where 4 5 >. The soluton has rch and vared N N lterature. Detaled dscusson can be found for example n []. LQ optmal control s the one of most popular methods n lnear control systems. E. Predctve control n state-space Stable generalsed predctve control (SGPC) s a class of the most powerful methods n control engneerng. Our methods are based on results of Kouvartas et al.[] where some numercally robust predctve methods

3 mprovng the earler algorthms are developed. Let the model be gven by xˆ + yˆ = A xˆ = C xˆ + B uˆ (6) d : = max d D : = C A B f C B { l n : C A B =, =,,..., l } () () n r m where xˆ R, uˆ R, yˆ R. It s assumed that an ntegrator s ncluded n the model so that û s a vector of ncremental nput. Furthermore, defne the termnal constrant (whch s one of many possble cases) yˆ,, uˆ = K xˆ, N () q where uˆ, yˆ denote predcted values of nputs and outputs at the current samplng nstant and. T s an LQ optmal controller. The ey dea s to attan a control law, whch satsfy the termnal constrant and at the same tme mnmze the cost functon = { yˆ + uˆ } J = λ (8) wth respect to û (for all =,, K). The soluton can be easly obtaned f the Lagrange multpler s ntroduced nto the performance ndex (8). The man weaness of ths smple approach n tself s the poor numercal condtons for unstable plant, whch can cause naccuraces n computatons. The reason s that some powers of the matrx $ are present n the computatons. For large horzon, these powers can be small and large at the same tme. Consequently, the results may lead to the nverson of ll-condtoned matrx. An ngenous proposal brngs nto use the new control law K uˆ = q xˆ + t, =, K, N K xˆ, N q (9) where the above mentoned. T gves a stablzng state feedbac,.e. Φ = $ %. T s stable. The new form s powerful because t mples the stablty (because of the stable Φ ) and the remanng free parameters W L are able to assure the fulfllment of termnal constrant. F. Decouplng In order to elmnate the undesrable effects between channels, t s necessary to apply more ntrcate control laws. We refer to [3], as the foundaton of the algorthm used n our mplementaton. For the sae of more unfed approach, ths method s extended wth some new results based on [4] and nserted n the ndrect adaptve control envronment. The frst step for the controller desgn s to chec the Glbert s condton for decouplng. Let C be the th row of C and defne If the Glbert matrx ' = [' ' L ' ] P has maxmal ran then the Glbert s condton s satsfed and the system can be decoupled. In ths case, the followng step provdes a decoupled system n ntegrator sense by means of statc state feedbac. At ths pont, t s possble to fnd a transformaton, whch gves canoncal form of the system. The canoncal form manages to dscover the unstable decouplng zeros. If they exst then a parallel compensaton s desred. Let us now consder agan the Glbert s condton. If t s not satsfed then the plannng of an approprate precompensaton s unavodable at the current samplng nstant. The addtonal dynamc system (as a dynamc feedbac) may cause dffcultes and statenconsstency n the on-lne algorthm, snce the system order wth precompensaton dffers from that wthout precompensaton. G. Desgn algorthms In spte of the use of dfferent control methods, the steps are smlar n the general concept. Ths subsecton s ntended to gve an overvew of the sequence of desgn steps n the case of predctve control: ) Choosng the specfed closed-loop system ) Recursve parameter and state estmaton 3) Realzaton of termnal constrant 4) Prestablzator desgn 4) Calculaton of the state feedbac 5) Introducng the reference sgnal (satsfyng the statc requrements) 6) Obtan the controller output ) Lmt the controller output (f necessary). 8) Lmt the output sgnal of the dentfed model. 9) Refne the dentfcaton (f necessary). Except of step 3) and 4), the methodology s the same for every method. Step 3) and 4) depend on the current type of controller. III. ILLSTRATIVE EXAMPLE Ths Secton descrbes the appled -nput -output plant model and compares the results of the smulatons of dfferent control strateges. Identfcaton plays a dstngushed role n the whole dscusson. A. Plant model The plant model s defned n [] as a turbo generator consstng of two lnear MIMO model canddates, Model (M) and Model (M). The models are descrbed n statespace form wth nnovaton nose model. M (model ) has a zero outsde the unt crcle. The system assumed to be one of the two model canddates at a gven samplng nstant. However, the sample nstant where the change s between the model canddates s unnown for the control

4 strateges. Two model changes are mplemented n the smulaton. The swtchng from M to M s at the samplng nstant 3 and the second, from M to M s at the samplng nstant. The whole smulaton contans smulaton steps. B. Identfcaton results durng LQ optmal control The results of dentfcaton durng LQ optmal control (.e. n closed loop) are llustrated n Fgs. -6. Denote D (z) the transfer functon havng D, K, D elements. For the sae of more unfed approach, Fgs. and Fg.. The step response of the M (sold) and the dentfed (dashdot) (bottom) for nput u compare the step responses of the dentfed model wth the current model M n samplng nstant 999, whle Fgs relate to dentfed and current model M n samplng nstant 699. It s clear that the dentfcaton error s greater n the case of model M. Obvously, the man reason s that the gan of M s greater, as well. The dentfcaton of the model M shows more precse results as shown n Fg It s beyond argument that the natures of transents n the dentfed and real case are very smlar provdng an models can also cause large transent errors n the dentfcaton. The phenomenon s llustrated n Fg Fg. 4. The step response of the M (sold) and the dentfed (dashdot) (bottom) for nput u In ths case, the plant changes from model M to model M n the samplng nstant 3 and the dentfed model gves ncorrect parameters forcng large errors wthn a small nterval. After ths, the dentfcaton yelds agan a relable model provdng acceptable control performance Fg. 5. The step response of the M (sold) and the dentfed (dashdot) (bottom) for nput u Fg. 3. The step response of the M (sold) and the dentfed (dashdot) (bottom) for nput u acceptable model for control strateges. Of course, the dentfcaton error shows great values at the begnnng of the smulaton. The extremely great values are vanshed after the transents. However, the change between the two Fg. 6. The dentfcaton error durng LQ control n the neghborhood of the model swtchng. An mportant and nterestng consequence s that the fast varyng control sgnal may be helpful n the dentfcaton. Consequently, the transents of the dentfcaton are faster

5 but ther values are greater, too. The phenomenon may easly lead to dsadvantageous output sgnals. nfortunately, ths case s typcal n the controlled loop. There s not any unfed approach to handle the mentoned trade-off. An dea s to try detectng the model changes. Snce, the behavors of the dentfcaton consderably depend on the values of the parameter vector ϑ and the covarance matrx P, t s a useful treatment to change these parameters, f necessary. The detecton of model swtchng means such a hghlghted case. An mportant concluson s that dentfcaton s a very senstve algorthm. The successfulness depends strongly on the true plant parameters and on the start values n the dentfcaton algorthm. C. Control results Ths subsecton examnes the pole assgnment, the LQ optmal, the stable generalsed predctve control and the decouplng n the above-mentoned llustratve example. To compare the results of dfferent type of control strateges, consder the Fg. 8-. A sngle pea at 3 mars the place of model swtchng from M to M values compared to other control methods then the control may lead to weaer performance. The pole assgnment (not documented n the fgures) falls often n ths crtcal class Fg. 9. The system output (sold) and the reference sgnal (dashdot) for the two subsystems by predctve control Fg.. The system output (sold) and the reference sgnal (dashdot) for the two subsystems by LQ optmal control Fg. 8. The control sgnals for the two subsystems by LQ optmal control Every result s llustrated n the same tme nterval wth the output and the control sgnals. One of the man conclusons s that the dentfcaton s the ey pont of the whole control performance that has been already dscussed above. If the control sgnal s very nhomogeneous and has great Fg.. The control sgnals for the two subsystems by predctve control To reach a hopeful compromse, many useful proposals can brng nto sght. For example, f the dentfcaton error changes n great extent that means possbly a detected model swtchng then one of the most effectve solutons n our mplementaton s to decrease the trace of the covarance matrx. The man dsadvantages of the pole assgnment are the strong couplng (greater than %), the large transents, nhomogeneous control sgnal and nose senstvty. On the other sde, ths method needs the smallest tme for the realzaton. The LQ optmum control provdes a relable, numercally robust approach on the area of ndrect adaptve control. The transents are more acceptable and the control sgnals are relatve homogeneous. The LQ control shows consderably smaller couple effects and nose senstvty than the pole assgnment. At the same tme, the real-tme realzaton requres a lttle more computatonal tme. It s evdent to see that the robustness of control strateges plays (because of the dentfcaton error) a crucal role. To mprove the mentoned property, the stable generalzed predctve control approaches suggest a good alternatve. Ther robustness promses powerful results n pont of vew of the transents. Addtonally, the most propertes are smlar to LQ optmum method, snce the ntroduced SGPC nvolves an LQ optmal controller. However, tme for realzaton ncreases forcefully wth respect to horzon,

6 whch can njure the real-tme specfcatons. Although, the predctve algorthms can easly lead to numercal problems, the numercally mproved SGPC algorthm s able to avod the troubles. In order to elmnate the couple effects, one has to apply decouplng algorthms. In our mplementaton, the nomnal multvarable MIMO system has been reduced nto ndependent SISO systems. nfortunately, the plant s unnown from the vew of pont of the algorthm. The effectveness of the ndrect adaptve control based on decouplng algorthm depends strongly on the accuracy of the dentfed model. An accurate model guarantees good performance n the transents, n tracng the reference sgnals and n the nose senstvty. However, the method s very senstve to the model swtchng. The results are demonsrateded n the Fg. and. Because of the complexty, the decouplng algorthm can be regarded as the computatonally most complcated method n the real tme specfcaton Fg.. The system output (sold) and the reference sgnal (dashdot) for the two subsystems by decouplng Fg.. The control sgnals for the two subsystems by decouplng In subsecton II-F was mentoned the possblty of statenconsstency whle swtchng n/out the precompensator. Theoretcally, t s not guaranteed to avod the stuaton. Correspondng wth our experences, f the Glbert s condton s satsfed for the nomnal plant then t s also vald for the dentfed model. A further nterestng experence that the dentfed model often satsfes the Glbert s condton, even f the nomnal plant does not. IV. IMPLEMENTATION CONCEPT The dspace DS [6] s a sngle board system, whch s specfcally desgned for development of hgh-speed multvarable dgtal controllers and real-tme smulatons n varous felds of control engneerng. The board s based on the Texas Instruments TMS3C3 floatng pont Dgtal Sgnal Processor (DSP). All of the mplemented algorthms hold the real-tme specfcatons. The development envronment allows us to record the tme hstory of the varables and tune the real-tme parameter. It s especally advantageous n the ndrect adaptve control, where the theory for choosng of parameters has not been elaborated yet. A typcal development and test sesson for real-tme applcaton consst of some mportant steps whch can be realzed by usng MATLAB, Smuln, dspace I/O board lbrary, and specal tools of dspace TRACE and COCKPIT. VI. CONCLSION Indrect adaptve control strateges provde a powerful tool for the multvarable lnear system. The ey pont of the whole concept s the performance of the dentfcaton. Dfferent type of control methods has been examned. The pole assgnment s a quc and smple algorthm wth wea performance. The LQ controller s an acceptable compromse between the error transents and the real tme specfcatons. In order to ncrease the robustness, the numercally robust predctve controllers suggest an adequate alternatve. The decouplng of multvarable systems damps consderably the effects between the subsystems. All of the algorthms satsfy the real tme requrement n the dspace development envronment. VI. ACKNOWLEDGEMENT Support for the research of ntellgent control systems s provded by the Hungaran Natonal Research Programs under grant No. OTKA T 9 and FKFP 4/99 VII. REFERENCES [] Nazaruddn, Y.Y.: Adaptve Regelung von En- und Mehrgröβensystemen auf der Bass der Zustandsraumdarstellung. Dssertaton, Ruhr nverstät, Bochum, 994. [] Rosstter, J.A., Kouvartas, B., Rce, M. J.: A numercally robust state-space approach to stablepredctve control strateges. Automatca, Vol 34, No, pp.65-3, 998. [3] Glbert, E.G.: The decouplng of multvarable systems by state feedbac. SIAM Journal of Control, Vol, No, 969, pp.5-63 [4]Lohmann, B.: Vollständge und telwese Führungsentopplung m Zustandsraum. Fortschrtt-Berchte der VDI-Zetschrften, Rehe 8, Nr.44, VDI-Verlag, Düsseldorf. Dssertaton, nverstät Karlsruhe, 99. [5] Sastry, S, Bodson M.: Adaptve Control. Prentce-Hall, Inc. Englewood Clffs, New Jersey,989 [6]dSPACE DS Software Envronment (Reference Gude) [] Kalath, T., Lnear Systems, Prentce-Hall, Englewood Clffs, New Jersey, 98