MATLAB TOOLBOX FOR SELF-TUNING PREDICTIVE CONTROL OF TIME-DELAYED SYSTEMS

Transcription

1 MALAB OOLBOX FOR SELF-UNING PREDICIVE CONROL OF IME-DELAYED SYSEMS Radek Holiš Vladimír Bobál Department of Process Control Faclty of Applied Informatics omas Bata University in Zlin Nad Stráněmi 45 Zlin 765 Cec Repblic KEYWORDS Self-tning Control Model Predictive Control MPC ime-delay MALAB oolbox ABSRAC e designed MALAB/SIMULINK oolbox is dedicated to develop and design predictive Self-ning Control (SC) algoritm for te time-delayed systems. In practice many processes can exibit time-delay in teir dynamic beavior wic is mainly cased by a time needed for transport of te energy information or mass. In ligts of tese facts it is necessary to develop sitable algoritm and verify its correct dynamic beavior sing simlation first so tis oolbox can be sed in advantage. is paper deals wit te basic principles of Model Predictive Control (MPC) calclation of control law design process of te predictive controller and recrsive identification of control process sing Recrsive Least Sqares Metod (RLSM). ere are also many cases wen compensation of measrable distrbance is reqired so tis oolbox allows compensating of tis distrbance. INRODUCION ime-delayed systems appear in many processes in indstry and oter fields inclding economical as well as biological systems (Camaco and Normey-Rico 7). ese processes are difficlt to control sing standard feedback controllers. Wen te relative timedelay is very large or a ig performance of te control process is desired we can coose MPC as te sitable algoritm for tese types of processes. e predictive control strategy contains a model of te process in te strctre of te controller. e first time-delay compensation algoritm was sown by (Smit 957). is algoritm is known as te Smit Predictor (SP) and it contains a dynamic model of te time-delay process and it can be called as te first MPC algoritm. For more complex processes containing time-delay and affected by a measrable distrbance MPC strategy can be sed (Maciejowski ). e MPC is an attractive set of te control strategies widely sed in te indstry. e poplarity of te MPC is mostly de to its leading to a safety operation of processes nder all circmstances and ability to se constraints. e MPC is known as a control strategy were based on te measrements of plant s states at time a matematical model of te plant (often referred to as te prediction model) is being sed for prediction of te evoltion of te plant in te ftre. e MPC wit allowance to control of te time-delayed processes ability of self-tning and possibility of te measrable distrbance compensation can be powerfl and versatile algoritm for control of varios processes. is paper deals wit te se of MPC for processes wit time-delay wit possibility of measring distrbance compensation. Strategy of MPC presents a series of advantages over oter metods. e MPC can be sed to control a great variety of processes ranging from tose wit relatively simple dynamics to oter more complex ones inclding systems wit long time-delay nstable ones or nonminimm pase. e mltivariable case can easily be dealt wit. e additional advantage is tat extension to te treatment of constraints is conceptally simple and tese can be systematically inclded dring te design process. is approac of control is a totally open metodology based on certain basic principles tat is allowed for ftre extensions (Camaco and Normey- Rico 7; Rossiter 3; Haber et al. ). e MPC as been deployed on slower processes in its early days (Kvasnica 9). It was cased by te large comptational complexity of control algoritms and large time demands. rends ave expanded towards modifications of predictive control over te years. Nowadays te MPC strategy can be sed for controlling of very fast processes. ese processes can ave reqirement for comptation of control action in microseconds (e.g. explicit approac of MPC can be sed). In practice an excellent indstrial srvey reports many sccessfl applications of te MPC in varios indstry areas (Qin and Badgewell 997; Rawlings and Mayne 9). An extended version of te Generalied Predictive Control (GPC) algoritm is dedicated for design of te adaptive predictive controller in tis paper. is paper is arranged as follows. e extended GPC algoritm is described in te first section. e next section sows comptation of te cost fnction for GPC and comptation of control law. Brief description of te recrsive identification procedre is introdced in te following section. e designed oolbox is briefly described afterwards. e next section contains examples of te simlation control sing designed oolbox and te last section concldes tis paper. Proceedings 3nd Eropean Conference on Modelling and Simlation ECMS Lars Nolle Alexandra Brger Cristop olen Jens Werner Jens Wellasen (Editors) ISBN: / ISBN: (CD)

2 EXENDED VERSION OF HE GENERALIZED PREDICIVE CONROL ALGORIHM e basic MPC strctre wit te extended GPC algoritm is scematically displayed in te Figre. Figre : Extended Strctre of te MPC e basic GPC algoritm minimies a cost fnction tat may be written as J ( N ) x N N ( i) [ yˆ( k i) w( k i) ] λ( i) [ Δ( k i ) ] () N were J is te fnction of te N x wic represents N N and N. e N and N are te minimm and maximm orions of cost fnction and N is te control orion of te cost fnction. is orion sold be cosen wit regard to dynamics of controlled process to andle step response (Rossiter 3; Modgalya 7). e y ˆ( k i) is an optimm prediction of te system otpt. Coefficients (i) and λ (i) are weigting coefficients and w(k i) is a vector of ftre reference seqence. e goal of te predictive control is to calclate te ftre incremental control action of te Δ ( Δ ( k )... e cost fnction J () is minimied to obtain te final control law. is is realied by minimiing wit respect to Δ were te predictions y ˆ( k i) are first expressed as a fnction of te past data and te ftre control actions Δ ( k i ). s J can be considered as a fnction of te ftre control seqence. e control orions as well as weigting factors are te tning parameters wic can be canged to modify steepness and rapidity of te control corse as reqired (Camaco and Normey-Rico 7). e orions N and N are compted as N d and N N d becase of te time-delay caracteristics of te process. In practice N and N are ardcoded in te algoritm and N is te only cangeable parameter. e modified matematical model of te Controlled Ato-Regressive Integrated Moving Average () (CARIMA) is sed by te GPC to compte te predictions. It is te typical CARIMA model extended by te vector v ( wic represents measrable distrbance sample A( ) y( ) es ( Δ d B( ) ( k ) dv D( ) v( () were d corresponds to nmber of steps of te timedelay for process dv represents nmber of steps of te time-delay for distrbance e S ( is te wite noise and Δ. e polynomial D ( ) represents caracter of distrbance and te polynomial ) describes caracter of te noise. is caracter is difficlt to determine; terefore polynomial C ( ) is cosen to be eqal to one (Camaco and Bordons 4; Fikar and Mikleš 8; Clarke et al. 987a). Consider eqation () mltiplied by Δ. en predication model can be represented as follows were otpt can be predicted as yˆ( k ) nd na a ~ y( k i) d Δv( k dv i) i i nb biδ( k d i) (3) were na nb and nd are degrees of polynomials A ( ) B ( ) and D ( ). e wite noise e S ( and its ftre vales are considered to be eqal to ero for te prediction of te ftre otpt vales. In case were time-delay is present following eqation sold be sed wen eqation (3) is applied recrsively for i... N (Clarke et al. 987b). y ˆ G H (4) Sy H V v H V v Matrices G H and S are constant matrices of dimensions N N N nb and N ( na ) respectively. Matrices H V and H V are of dimensions N ( nd ) and N N. Matrix H V can be sed only in case wen degree of polynomial D ( ) is eqal to or iger. Following eqation corresponds to te free response of te system tat is te otpt tat wold be obtained if te control signal was kept constant. f H (5) S y H Vv Forced response of te system can be written as te next eqation f G H v (6) r Vectors y v and v are defined in te following eqations. Sm of te free response (5) and te forced response (6) leads to overall response of te system defined by te following eqation f r V y ˆ f (7)

3 Control Law Comptation and Cost Fnction e predicated otpt ŷ expressed in te previos paragrap is part of te eqation (). It is evident tat J is te cost fnction of y and. e individal elements of te smmation of te cost fnction in te eqation () can be written in a matrix form. is cost fnction can be defined as J (G H Q λ Sy w) Q (G H Sy w) (8) were Q and Q λ are te diagonal weigting matrices of sie N N wit elements ( j) and λ ( j) respectively. Altog in practice te most common coice is to set ( j) and λ ( j) constants on te orion. In a fact te vales of tese weigting factors mst be normalied in order to obtain a correct weigting of te different errors and controller otpts. After some maniplations J can be written as J ( H ( Q λ Sy G Q G) ( H w) Q ( H Sy Sy w) Q w) G J Minimiing J wit respect to it means leads to M P y P Pw () (9) were M G G Q λ is of dimension N N P G S of dimension N ( na ) P G H of dimension N nb and P G of dimension N N. In a receding orion algoritm only te crrent vale of te Δ ( is compted so if m is te first row of matrix M ten Δ ( is given by Δ ( mp y mp mpw () Eqations (8) () deal wit te case of no distrbance rejection. For te expression of te final control law containing compensation of te measrable distrbance Δ ( is compted as te following control law form Δ ( mp () y mp mpw mpvv mpv v were PV G H V is of dimension N ( nd ) and PV G H V of dimension N N. H V and H V are matrices inclding te coefficients of te system step response to te distrbance. Ftre vales of te distrbance can be determined only in certain cases e.g. be measrement or generally in case wen it is related to te process load. In oter cases it can be predicted sing means trends past data oter information or by combination of specified items. If tis is te case te term corresponding to ftre deterministic distrbance can be compted (Scwar et al. ). After introdcing vectors y w v and v final control law is defined as yˆ( k d) Δ( k ) yˆ( k d ) Δ( k ) Δ( mp mp yˆ( k d na) Δ( k nb) w( k d ) Δv( k ) w( k d ) Δv( k ) mp mp (3) V w( k d N ) Δv( k nd ) Δv( Δv( k ) mp V Δv( k N ) If te ftre load distrbance is constant and eqal to te last measred vale (i.e. Δv ( ) te last term of te eqation (3) vanises (Pawlowskaa et al. ). It is evident tat matrices H V and H V are dependent on te relative difference between nmber of steps of timedelay of inpt-otpt and distrbance-otpt wic is defined as ρ d dv (4) is leads to tree types of strctres for matrices H V and H V based on te vale of ρ : ρ < and ρ H VX ρ > H VX N ρ ρ N ρ ρ ρ ρ N ρ ρ ρ ρ (5) (6) ρ were i are te coefficients of H V and H V matrices obtained from te delay free distrbance response moved p/down according to vale of ρ (Pawlowskaa et al. ). RECURSIVE IDENIFICAION e identification of systems deals wit te problem of creating matematical models of dynamical systems based on data observed from te system. It is an

4 alternative procedre for obtaining a model in case wen it is not possible to determine a set of differential eqations tat describes te dynamic beavior of te system. e MPC reqires an internal model of te system; terefore really precise model of process is necessary for correct beavior of predictive algoritm. Identification of control processes can be divided into two grops wic are sed most often. e first grop is Offline (one-time) Identification Metods (OfIM) and te second is Online (ongoing) Identification Metods (OnIM). e OfIM type as well as te OnIM type can be sed dring te real-time control of processes. e estimated parameters obtained from te OfIM are sally selected as a starting point for te SC. ey can be also cosen as te internal model trogot te control procedre wen te process does not cange its dynamic beavior mc and adaptive control is not reqired. ese SCs can tilie an ato-tning or adaptive approac in many practical applications (Bitmead et al. 99). e most known adaptive approac is to se OfIM recrsively. Offline Identification Metods e well known OfIM is te MALAB fnction fminsearc. It finds te minimm of te entered fnction witot restricting conditions. e entered fnction can be single variable or mltivariable type. is fnction ses te simplex searc metod for finding te minimm of a fnction. is is a direct searc metod tat does not se nmerical or analytic gradients. However te most known metod for te identification of te discrete transfer fnction model parameters is te Least Sqares Metod (LSM) based on te idea of linear regression. is identification algoritm can be carried ot in a recrsive manner as well in an order to se it for SC. e LSM is based on minimiing te sm of sqared sbtraction of measred and model otpt vale. e LSM is defined as te vector Θˆ tat minimies te qadratic error Θˆ ( F F) F y (7) Note tat Θˆ is a vector of estimated model parameters wic as dimension n F is a matrix of dimension N n d n y is a data vector of dimension N n d were N is a nmber of measred data n is an order of system and d is a nmber of steps of timedelay. e F depends on past inpts and otpts and tat tis condition can be flfilled if te inpt signal seqence is adeqately cosen in sc a way tat te obtained vectors are linearly independent. (Camaco and Normey-Rico 7). Online Identification Metods e OnIM are mainly sed to adjst te estimates of te process parameters from initial estimates in te eac sampling period. Since te approac wen calclation of estimated parameters is performed eac sampling period tese metods are capable to react on canges in a dynamic beavior of system as well as tey are able to compensate sligtly non-liner beavior of te system. One of te advantages of te process parameter estimation sing te LSM is fact tat tis algoritm can be sed recrsively. e parameter vector compted at step k can be compted as a fnction of te parameter vector estimated at step k. e recrsive least sqares metod (RLSM) is te most known recrsive metod and it ses te AtoRegressive exogenos (ARX) model (Bobál et al. 5). y( Θ ( e ( were Θ is a vector of model parameters Θ ( k ) and Φ is a regression vector [ [ a a a b b b ] n s n (8) (9) Φ ( y( k ) y( k ) y( k n) ( k d ) ( k d ) ( k d n)] () Final RLSM algoritm can be defined as ˆ ˆ k ) Θ ( Θ( k ) eˆ( () ξ ( were C is covariance matrix and ξ ( Φ ( k ) () e RLSM can be modified by weigting of te past data and forgetting of tem to always work wit te most actal and relevant data. Application of te RLSM wit exponential forgetting reslts in a more realistic sitations. Parameters of te control law are being continosly adjsted in order to track time-varying properties of te controlled plant (Bobal et al. 5; Skormin 6). Final algoritm is defined as ˆ ˆ k ) Θ ( Θ( k ) eˆ( (3) ϕ ξ( were covariance matrix is defined as follows k ) Φ ( k ) C ( k ) (4) ϕ ϕ ξ ( e RLSM wit adaptive directional forgetting eliminates disadvantages of te RLSM wit exponential forgetting. It forgets old information only in te direction in wic new data bring new information wic also elps to avoid te estimator windp effect. e RLSM wit exponential forgetting as well as wit adaptive directional forgetting as been cosen as an algoritm for te SC algoritm sed in te introdced oolbox (Bobal et al. 5; Skormin 6). OOLBOX DESCRIPION e oolbox for te SC GPC of time-delayed processes wit measrable distrbance compensation is depicted in te Figre. is oolbox was developed sing te MALAB R4b. Basic setting of oolbox

5 is possible in te init.m file wic is an initialiation rotine. is rotine is exected atomatically once te simlation is started sing te MALAB/SIMULINK. te Normal Measrement Distrbance (NMD) means tat distrbance is measred every samplee interval and otpt is compensated basedd on te present and past data of distrbance. Second option is to se te Predicted Vector of Distrbance (PVD) were corse of te distrbance over time is known dring te wole simlated control process. Moreover distrbance is measred for overcompensation of invalid data as well. For example distrbance d can be first measred and sed for te control c or distrbance vector can be statistically compted based on te past data etc. Usage of te PVD significantly s improves wole control process in terms of control qality. Figre : SC GPC oolbox MALAB Sceme e oolbox consists of te following parts: GPC controller block RLSM identification block (sed forr SC) 3 Controlled process model 4 Distrbance model 5 Reslting carts of control corses and a predicatess 6 Estimated parameters of process from RLSM 7 8 Noise signal and distrbance signal 9 System parameters setting Ambient condition setting signal Activation/deactivation switces of 9 and e GPC controller block contains tree tabs see Figre 3. First tab is sed for te setting of te Sample time Dead times (time delays) Control orion and Weigting parameters. Second tab is intended to design desired trajectory and last tab can set properties of distrbance compensation. Figre 4 : GPC G Distrbance Compensation Window e RLSM identification block is allowed for te self- tning prposes by te Identification Switc. Oterwise identified parameters of process are constants dringg te control process. e RLSM block allows to set Sample/dead time in te first tab. Second tab is designed for setting of f te ype of identificationn (RLSM RLSM wit exponential forgetting or RLSM wit adaptive directional d forgetting). Oter boxes can modify te RLSM parameters. Figre 3 : Main Settings of GPC Window e measrable distrbance compensation can be enabled/disabled based on te ceckbox as it is visible on te Figre 4. If disabled No Distrbance Compensation (NDC) approac is sed. In case wen distrbance compensationn is enabled ser can coose one of te two possible ways of its compensation. First Figre 5 : RLSM Settings for te SC Window o execte te simlation set parameters of te controlled system in te init.m file first. en enable/disable Noise N and external Distrbance sing Switces. Coose C SC wit RLSM or non SC algoritm sing te Identification Switc. Set Ambient temperatre if reqired. r Set GPC controller block and RLSM identification block according to description above. Rn te simlation s andd display carts sing 5. SIMULAIONN VERIFICAION OF OOLBOX e designed SC GPC oolbox was verified by simlation on tree exampless GPC algoritm witott SC SC GPCC algoritm and GPC algoritm wit distrbance compensation.

6 e Controlled process model (item 3 of te Figre ) is represented as te following continos transfer fnction wic was sed for all simlations. G S e following second order linear discrete transfer fnction was sed for te simlation prposes as a model for estimated parameters of controlled process for GPC and RLSM. G ( ) S B( s) ( s) e A( s) B( A( s d ) ) s b d a 9s b a e s d ( (5) (6) e Figre 7 sows tat wen identification process is nderestimated and parameters are not accrate wole control process becomes nstable and it cannot be effectively controlled sing GPC. SC GPC Algoritm Disadvantages of o inaccrate parameters estimation can be eliminated byy SC GPC algoritm. ee RLSM wit adaptive directional forgettingg and te controlled system model (8) was sed for te simlation prposes. GPC Algoritm witot SC First te GPC algoritm witot SC was verified. Very precisee estimated model (7) of te controlled system was sed. G S ( ) Following parameters weree sed for all simlations: s d 5 λ N 6 N 5 S s (7) and N 55 Figre 8 : SC GPC Control Corses Figre 6 : GPC witot SC Exact Identificationn From te Figre 6 it is obvios tat te control qality is very good after start-p pase and oversoots are not significant. Next te exact estimated parameters were canged sbseqently G S ( ) were inaccrate identification is performed. 5 (8) Figre 9 : SC S GPC Parametrs Evoltion e Figre 8 depicts tat wen GPC SC is sed even initial inaccratee system parameter estimation does not prevent ig control qality. System parameters evoltion is captred on te Figre 9. GPC Algoritmm Distrbance Compensation e controlled system model (7) was sed to verify fnctionality of te GPC wit NDC NMD and PVD. Control corses are depicted on te Figre. Figre 7 : GPC witot SC Inaccrate Identification Figre : GPC Distrbance Compensation

7 CONCLUSION is paper as presented an extended SC GPC oolbox wit possibility to compensatee te measrable distrbance. e oolbox as been created in te MALAB/SIMULINK environment wit a prposee to create a simlation sitable for te t design and verification of adaptive control of time-delay systems wit sage of te MPC strategy. e GPC algoritm as itself witot sing te SCC is able to control processes in a really good qality after start-p pase and oversoots are not significant. However tis is possible only in case wen process parameters are estimated very precisely. Oterwise in case wen tese parameters are nott estimated wit sfficient precision control process becomes nstable. First option is to se sitable identification process for parameters estimation or se te SC algoritm. e SC algoritms ave several advantages e.g. initial parameter estimation can be only raw sligtly s nonlinear process can be controlled sing te SC and inflence of an nexpected conditions dring tee control process can be eliminated. e simlation sows tat RLSM wit adaptive directional forgetting can be sitable SC algoritm for te MPC in general. Incorporationn of te distrbance compensation into te control law can ave really positive effect e on overall control processes. An oversoot cased by te distrbance can be eliminated wen it is measred and predicted. e GPC wit NMD and PVD improves te control qality and redces te oversoot in comparison wit te GPC wit NMC. oolbox is maintained by te Department of Process Control Faclty of Applied Informatics omas Bata University in Zlin and for its downloading feel freee to contact ators or mentioned department. ACKNOWLEDGEMEN is work was spported by te Cec Repblic Ministry of Edcation - grant IGA/Cebiaec/8/. REFERENCES Bitmead R.R. Gevers M. and V. Hert. 99.Adaptive optimal control. e tinking man s GPC Prentice Hall Englewood Cliffs New Jersey. Bobál V. Böm J. Fessl J. and J. Macáček. 5. Digital Self-tning Controllers: Algoritms Implementation and Applications. Springer-Verlag London. Camaco E.F. and C. Bordons. 4. Model predictive control Springer Verlag London. Camaco E.F.. and J.E. Normey-Rico. 7.Control of dead- time processes Springer-Verlag London. Clarke D.W. Motadi C. and P.S. ffs. 987a. Generalied predictivee control part I: te basic algoritm Atomatica Clarke D.W.; C. Motadi and P.S. ffs. 987b. Generalied predictive control part II: extensions and interpretations Atomatica Fikar M. and J. Mikleš. 8. Process modelling optimisation and control Springer-Verlag Berlin. Haber R. Bars R.and R U. Scmit.. Predictive control in process engineering: From te basics to te applications.. Weinaim: Willey-VCH W Verlag. Kvasnica M. 9. Real-ime Model Predictive Control via Mlti-Parametric Programming: eory and ools VDM Verlag. Maciejowski J. J M..PredictiveControl wit ConstraintsPearson Edcation. Modgalya K.M. 7.Digital control. Cicester: Jon Wiley. Pawlowskaa A. Gmána J. L. Normey-RicoJ. E. and M. Berengela.. Improving feedforward distrbance compensation capabilities in Generalied Predictivee Control. Jornal of Process Control Qin S. J. and. A. Badgewell Ann overview of indstrial model predictive control tecnology. In: Cemical Process Control V volme 93 no AICe Symposim S Series American Institte of Cemical Engineers. Rawlings J. B. and a D. Q. Mayne. 9. Model Predictivee Control: eory and Design.. Nob Hill Pb. Rossiter J.A. 3.Model based predictive control: a practical approac CRC Press. Scwar M. H. Cox C. S. andd J. Börcsök.. A Filtered ning Metod for a GPC Controller. In: University of Kassel Germany Skormin A. V. 6. Introdction to Process Control: Analysis Matematical M Modeling Control and Optimiation Springer Verlag London. Smit O. J Closed control of loops wit dead time. Cem. Eng. Progress AUHOR BIOGRAPHIES RADEK HOLIŠ stdied at te omas Bata University in Zlín Cec Repblic were e obtained is master s degree in Atomatic Control and Informatics in 4. He now attends P.D. stdy in te Departmentt of Process Control Faclty of Applied Informatics of te t omas Bata University in Zlín. His researc interestt focses on modeling and simlation of discrete tecnological processes adaptivee control and model predictive control. He crrently works at Honeywell HS-CZ Brno in Aerospacee division as Software Design Engineer.. His l address is rolis@fai.tb.c. VLADIMÍR V BOBÁL gradated in 966 from te Brno University of ecnology Cec Repblic. He received r is P.D. degree in ecnical Cybernetics C at Institte of ecnical Cybernetics C Slovak Academy of Sciences S Bratislava Slovak Repblic. He is now Professor at te Department of Processs Control Facltyy of Applied Informatics of te omas Bata University in Zlín Cec Repblic. His researc interests are adaptive and predictive control system identification and CAD for atomatic control systems. Yo can contactt im on l address bobal@fai.tb.c..