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ORDER-REDUCED CONSTRAINED GPC ALGORITHM WITHWAVELET-BASED FEATURE EXTRACTION*Yu Shiming 1*, Wang Haiqing2and Zhang Baoyuan31. College of Information Engineering , Zhejiang University of Technology,Hangzhou, 310014, China2. Institute of Industrial Process Control, Zhejiang University, Hangzhou,310027, China3.Technical Center, Gansu Television Station, Lanzhou, 730000, ChinaAbstract: Model predictive control(MPC) has a promising prospect in metallurgical process control.In this work, the main features of a high-order process are represented by a low frequency componentof a response curve of the process based on wavelet decomposition, and then the model order of theprocess is reduced by approaching the low frequency component with a low-order model.Subsequently, the high-order process is controlled by employing the order-reduced model and aconstrained generalized predictive control(GPC) algorithm with simple feedback correction. Lastly, thefeasibility and effectiveness are confirmed by a simulation example. Copyright 2003 IFACKey words: high-order process; wavelet; feature extraction; order reduction; constrained generalizedpredictive control1. INTRODUCTIONFor a complex metallurgical process, better controlperformance can be gained by employing an MPCalgorithm than a PID method. Consequently, a greatdeal of effort has been devoted to the applications ofMPC in this area (De Keyser, 1997; Kim and Kwon,1998; Grimble and Ordys, 2001). Recently, wavelet-based modeling and identifying techniques haveattracted increasing attention. They arise in linearsystem identification (Zheng, et al., 2001; Ghanemand Romeo, 2000) as well as in nonlinear systemidentification(Ghanem and Romeo, 2001). Thewavelet transform is also being applied increasinglyin model-based predictive control (MPC). The local*Supported by the National Natural Science Foundation ofChina (20206028)*Corresponding Author: Yu Shiming (ysm).functions in both time domain and frequency domainare used to describe unstable characteristics of atime-varying system(Song, et al., 1996). Huang andJin(1997), Gu and Hu(1997) proposed a class ofpredictive control approaches based on wavelettransform and manual neural networks. Thecomputational burden of a constrained GPC may begreatly lightened by a wavelet based method(Li andXi, 2000). Binder et al. (2000) explored an adaptivediscretization scheme for dynamic optimizationproblems formulated on moving horizons. In thescheme, the finite dimensional spaces are spanned bybiorthogonal wavelets arising from B-splines. Feng etal. (1996) proposed a constrained MPC algorithmwith simultaneous identification using waveletprinciples and the on-line optimization problem issolved by solving a series of semi-definiteprogramming (SP) problems. On-line computationalrequirements for multi-time-scale systems arereduced substantially by expressing the MPCobjective and constraint equations in terms ofwavelet bases (Lee, et al.1995).It is unnecessary to circumscribe the model form ofMPC, and what is most important is the function of amodel, not its form(Xi, 1994). A process modeldirectly affects the computational burden as well asthe control performance of MPC. The low frequencyfeature component of a process can be extracted at anarbitrary scale in terms of the multi-resolutionfunction of wavelet transform. The order of a high-order process can be reduced by approaching thislow frequency feature component with a low ordermodel. The computational burden of GPC may beefficiently lightened after the order reduction. Infact, a very accurate model can hardly be obtaineddue to noises and disturbances. Therefore, a feature-extraction-based modeling method has gainedwidespread acceptance as one of promising ways.The local functions of wavelet in both time andfrequency domains are useful to separate valuablesignal information from the effects of nonstationarydisturbances and noises(Carrier and Stephanopoulos,1998). Based on wavelet decomposition, a modelorder can not only be reduced, but additive noisesand disturbances can also be removed from processresponse data by representing main features of aprocess with a low frequency component of theprocess response curve. The main purpose of this jobfocuses on the control of a high order processemploying an order-reduced model based on waveletfeature extraction and a constrained GPC algorithmin which the current error between a process outputvalue and a model output one is used to correctmodel-process mismatch. A simulation demonstratesthat the method proposed in this paper is not onlyfeasible, but also efficient.2. WAVELET-BASED FEATUREEXTRACTION AND ORDER-REDUCED MODELSuppose that )()(2RLt is a scale function,)()(2RLt is a wavelet transform function;)2(2)(2/,kttjjkj= and)2(2)(2/,kttjjkj= , Zkj , (the set of allpositive integers); _,kjkjspanV = is a subspacelinearly spanned by ),2,1(,L=kkj . For anarbitrary jjVtf )( , the following relationship holds=kkjkjjtctf )()(, （1）where j is a scale parameter; k is a local parameter;jV expresses a scale space; kjc,is a scalingcoefficient. The jV can be decomposed as thefollowing manner11 +=jjjWVV （2）with 11 +jjWV ; 1+jV and 1+jW denote a lowfrequency subspace of jV representing coarseinformation and a high frequency subspace of jVrepresenting fine detail, respectively; denotes thedirect sum.For a high process with an input )(tu and an output)(ty , assume 0)( Vty , where 0V is a functionspace. To extract the low frequency features of )(ty ,the space 0V is decomposed in terms of the Eq.(2).When the low frequency space jV is determined, thelow frequency component )(tyjof )(ty projectedin the subspace jV can be derived by employing theformula (1). Using )(tu ,)(tyjas the input andoutput signal, respectively, the order-reduced modelof the high order process can be gained throughidentification.The least squares(LS) identification algorithmexploits the time-discrete values of )(tu and )(tyjas the identification signals expressed as)1(u , )2(u , )(mu and )1(jy ,)2(jy , )(myj,respectively. Assume that the order-reduced modelgained through identification is given by)()2()1()(21nkyakyakyakyjnjjj+= L+)()2()1(21nkubkubkubn+ L + )(kw （3）where )(kw denotes noise. Since wavelet transformcan effectively attenuate noise, )(kw may beapproximately regarded as a weak white noise withzero mean. Thus the least squares form of Eq.(3) canbe expressed as the follows)()()( kwkkyTj+= h （4）where),(,),2(),1()( nkykykykjjj= LhTnkukuku )(,),2(),1( L ,()Tnnbbbaaa ,2121LL= .Let Tjjjjmyyy )(,),2(),1( L=y，Tmwww )(,),2(),1( L=w ,=)()2()1()2()0()1()1()1()0(nmymymynyyynyyyjjjjjjjjjLMLMMLLH)()2()1()2()0()1()1()1()0(nmumumumuuumuuuLMLMMLL.Then, the estimated value of can bedetermined by employing the following weightedleast squares formulationjTTyHHH1)(= (5)where denotes a weighting diagonal matrix.3. ORDER-REDUCED GPCALGORITHMThe model of a high order process and its order-reduced model are given respectively by)()1()(1nkykykyn+= L /)()()1(1knkukun+ L （6）and)()1()(1mkzakzakzm+= L/)()()1(1kwmkubkubm+ L（7）where )(),( kzky express the outputs of the abovehigh order process and order-reduced model,respectively ; )(ku is the input; n and m are theorders of the process and the order-reduced model,respectively; )(k and )(kw express the additivenoises imposed onto the process output and the order-reduced model output, respectively; is adifferencing operator; and 0n , 0ma , mn .Due to the efficient denoising capabilities of wavelettransform, only weak noise is considered in this job,and )(kw is regarded approximately as a whitenoise. Based on the above wavelet-based featureextraction, the parameters of order-reduced model isdetermined by replacing )(),( kukyjof Eq.(5) withtheir increment forms )(),( tutyj .Let M be a control horizon and P be a predictivehorizon. By solving Diophantine equations (Wang,1998) in terms of Eq.(7), we have the following P-step predictive output equationduGz += p（8）where ( )TpPkzkzkz )(,),2(),1( += Lz denotesthe P-step predictive output vector of the model,( )TMkukuku )1(,),1(),( += Lu is acontrol increment vector to be solved；G is a M matrix which is derived by solving Diopantineequations and decided by the coefficientsmaa ,1L and mbb ,1L；d is a column vectordecided by known input and output increments.It is inevitable that the model does not match theprocess due to the reduction of the model order. Toovercome this mismatch, we introduce a feedbackcorrection action in Eq. (8), similar to the feedbackprinciple of nonparametric MPC, that ishduGz )(kep+= （9）where )()()( kzkyke = ; h is a feedbackcorrection vector.The P-step referential trajectory vector is representedas( )TrrrrPkzkzkz )(,),2(),1( += Lz (10)where spkylkzllr)1()()( +=+ ; is asmoothing factor, and sp is a set point value.The k-th moving-horizon optimal criterion is givenby( ) ( )rpTrpJ zzQzz = + uu T（11）where Q denotes a symmetrical positive weightingmatrix. Let TP)1,1,1( L= be a P-dimension vectorand TM)1,1,1( L= be an M-dimensionvector，then the output constraints are expressed ashduGhd )()(maxminkezkezPP （12）where maxmin, zz denote the lower bound and theupper bound of the process output, respectively.Let ( )TMkukuku )1(,),1(),( += Lu，thenMku uRu )1( += （13）and the control constraints are give byMMkuukuu uR )1()1(maxmin (14）where maxmin, uu represent the lower bound andupper bound of the manipulative variable,MM=1111110LOMMR .The changing rate of the manipulative variable isconstrained bymaxmaxminminu uu （15）where maxmin, uu are the lower bound and theupper bound of the rate , respectively.Substituting (9）into(11), the criterion can be cast as aquadratic programming form. Letrke zhdg += )(， and obviously g is a constantcolumn vector, then()()uuguGQguG TTJ +=Since Q is a symmetrical matrix, we can rewrite theabove criterion asQgguQGguIQGGuTTTTJ += )(2)(Let )(2 IQGGS +=T，)(2 QGgfT= . Since thethird term at the right side of the above criterion is aconstant, the criterion at the time instant k isequivalent to the following quadratic programmingufuSu +=TJ21(16)The constraint equations given by (12） and （14）are rewritten as the standard form in therelationship “”，then the above receding horizoncriterion of GPC is cast as the following standardquadratic programming with the lower and upperboundsufuSu +=TJ21min+MMMMPxPuuukukuuzkeukezutsuuRuRhdGhdGmaxminminmaxminmax)1()1()()(.4. SIMULATION EXAMPLESuppose that the parameters of a 5-th order processare given by the vectorTp),(5151 LL=0.19245,-1.0416,5501,2.5912.7306,-3.(= ,T0.3107)2,-0.0786,6943,0.2860.8065,-0. .At the scale 2=j，we get an order-reduced modelwith the order 2 through wavelet based featureextraction and identification. For the sake ofconvenience, take15010010050500101)(=kkkkuas input signals for identification. The parameters ofthe model gained through identification is expressedas=Tmbbaa ),(2,121T4,0.9036)3534,0.2881.2081,-0.(.The step responses of the process and the order-reduced model is pictured in Fig.1. It can be observedthat the main features of the process is sufficientlydescribed by the order-reduced model based onwavelet based feature extraction.The simulation parametric values are given asfollows:=minz 0；=maxz 70%；=minu -50%；=maxu 50%；minu =-5%；=maxu 5%,sp =60%In the simulation, the satisfactory results can beobtained only by taking T)1,1,1( L=h as asimple form of h .The controlled variable is shown in Fig.2. When7061 k , a disturbance with an amplitude of 6%of the full scale is imposed to the controlled variable;when 150141 k , the amplitude of thedisturbance is 6% of the full scale. Fig.3 and Fig.4denote the manipulated variable and its changing rate,0 10 20 30 40 500246810ProcessModelFigure 1. Step response of processand modely(k),z(k)kk/TFigure 2. Controlled variabley(k)0 100 20000.81krespectively, and both of them satisfy all constraints.Therefore the high order process can be effectivelycontrolled by employing the low order model derivedby wavelet based feature extraction.5. CONCLUSIONSThe computational burden of an MPC algorithm isconsiderably heavy due to multi-step prediction ofprocess output and online receding horizonoptimization. How to lighten the computationalburden is quite a significant job. This study showsthat the order-reduced model combined with thesimple feedback correction can not only lighten thecomputational burden, but also control the high orderprocess efficiently.REFERENCESBinder, T.，L. Blank，W. Dahmen， and W.Marquardt (2000). Adaptive multiscale method forreal-time moving horizon optimization. In:Proceedings of the American Control Conference,Chicago, Illinois, USA, pp.4234-4238.Carrier, J.F. and G. Stephanopoulos(1998). Wavelet-based modulation in control-relevant processidentification. AIChE Journal, 44(2), pp.341-360.De Keyser, R. M. C. (1997). Improved mould-levelcontrol in a continuous steel casting line, ControlEngineering Practice, 5(2), pp.231-237.Feng, W., H. Genceli, and M. Nikolaou(1996).Constrained model predictive control withsimultaneous identification using wavelets.Computers & Chemical Engineering, 20( Suppl ptB), pp.S1011-S1016.Ghanem, R., and F. Romeo(2000). Wavelet-basedapproach for the identification of linear time-varying dynamical systems, Journal of Soundand Vibration, 234(4), pp.555-567.Ghanem, R., and F. Romeo(2001). Wavelet-basedapproach for model and parameter identificationof non-linear system. International Journal ofNon-Linear Mechanics, 36(5), pp.835-859.Grimble, M. and A.W. Ordys (2001). Forpredictive control industrial applications, AnnualReviews in Control, 25, pp. 13-24.Gu, D.B. and H.S. Hu (2000). Wavelet neuralnetwork based predictive control for mobile robots.In: Proceedings of the IEEE InternationalConference on Systems, Man and Cybernetics,Tennessee, USA, pp.3544-3