SELF-TUNING WEIGHTING FACTOR

Transcription

1 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July SELF-UNING WEIGHING FACOR O DECOUPLING CONROL FOR INCUBAOR SYSEM M. A. Zerman, E. Fek and A. Mam 3 Department of Electrc Engneerng, Natonal School of Engneerng of uns Aymen_net@vola.fr,3 Department of Physcs, Faculty of Scence, Mathematcs, Physcs and Natural of uns, Campus Unverstes 9 El Manar,unsa ABSRAC hs paper s focused on a desgn of an algorthm for decouplng multvarable systems based on generaled predctve control (DGPC) and ts applcaton to a control of an ncubator system. Proposed control algorthm uses adaptve error weghtng factor n cost functon n order to reduce couplng between control loops. wo technques are presented and compared for predctve control of IO (wo -Input, wo-output) processes. Accordng to the frst technque the weghtng factor s tuned n synchronaton to a reference sgnal change. Accordng to the second method, the weghtng factor s set dependent on the actual control error. he algorthm s appled to ncubator clmate control; obtaned smulaton results show ts effcency. KEYWORDS Incubator Process, Decouplng Predctve Control, Adaptve Error Weghtng Factor. INRODUCION Many ndustral processes have strong cross couplng between control loops. Mostly Proportonal Integral Dfferental (PID) s the smplest soluton to control these systems. But these controllers do not offer satsfactory control because of nteractons. Also the tunng of PID s complcated when they are used n MIMO systems. Generaled predctve control appears to be attractve soluton n multvarable process control [5,]. However, these controllers cannot solve problems wth strong nteracton. In that case, decouplng can perfectly acheved by addng nteracton compensators to controllers []. But n some systems because of mproper delay structure or model msmatch, nteracton compensators cannot applcable [3]. herefore, a practcal method for decouplng s requred and compared. he frst technque, the weghtng factor s tuned n synchronaton to a reference sgnal change. Accordng to the second method, the weghtng factor s set dependent on the actual control error. Most of the ndustral systems are nfected wth dsturbance; the presence of ths term can complcate the tunng of weghtng factor. o solve ths problem we used a frst-order flter. he am of ths contrbuton s an mplementaton of the GPC control wth self-tunng of the weghtng factor ncludng constrants of the manpulated varable for control of the neonatal ncubator. DOI:.5/jtca

2 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July he ncubator process s used to produce healthful mcro-envronment n order to reduce new born heat loss by controllng temperature nsde ncubator. emperature s one of the most mportant factors that need to be mantaned wth a mnmum varaton. But only temperature control s not suffcent to provde comfortable envronment. Also, the relatve humdty control s very mportant to reduce the new born heat loss [8, 9]. For these reasons, we have developed an actve humdfcaton system based on a nebuler. herefore, ths process becomes a IO system (wo Input -wo-output). In fact, the process has tme-varyng parameters model, dependng on the external temperature and on the load under treat. Also, couplngs between temperature and humdty are very mportant that should be reduced. he paper s structured as follows. he secton s devoted to present decentraled generaled predctve control wth constrant. In secton 3, we present the frst technque of decouplng based on synchronaton of the weghtng factor wth a reference sgnal change. In Secton 4, we propose the second method of decouplng whch the weghtng factor s set dependent on the actual control error. In Secton 5 expermental setup and computer smulatons are conducted to revew the feasblty and effectveness of the proposed method, to decrease nteracton cross couplng between controlled varables (temperature and humdty) of ncubator system.. HEORY.. Decentraled Control Decentraled control (Benhammou 988, Lune 99, Gagnon 999) s based on several local controllers, nstead of a sngle controller to control a multvarable process (MIMO). Each of these controllers s responsble for controllng a sngle loop. he decentraled control scheme for a rnultvarable process s llustrated n Fg. Fgure. Structure of decentraled control (or dstrbuted) hs decentraled approach provdes some advantages, both theoretcal and practcal very nterestng to quote: Structure s easer to acheve (very smple and fast to mplement t n ndustry).hs approach s both flexble (the falure of a party system can be treated wthout affectng the rest, mantenance s also more easly). It s feasble to specfy dfferent performance and use of dfferent samplng perods for each partcular loop. Removal or addton of a control loop or manual mode settng does not affect the overall stablty of system (other loops contnue to be compensated). For these reasons, we take nto count ths structure Fg., for the concepton of the decouplng predctve control. 68

3 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July.. Process Model he synthess of the generaled predctve controller (GPC) suggested by (Clarke et al, 987; Clarke, 988) [5,6]. hs method was used successfully n ndustral applcatons of varous forms (Rchalet et al,987; Rchalet et al,9) [,]. he approach of generaled predctve control s based on a dynamc model of type ARIMAX (Auto Regressve Integrated Movng Average wth external nputs), gven by: d ()()()() =. + A y k B u k C e () k () () Where s the number system, y () k s the system output, u () k s the system nput, e () k s the uncorrelated random sequence, () = corresponds to an ntegral acton. Its presence n the drect channel allows a ero error n steady state value. A (), B () and C () are polynomals. A() = + a + a + + a, na na nb nb B() = b + b + b + + b, nc = nc C () c c c. () Wth na, nb et nc ndcate the respectve order of these polynomals. he generaled predctve control based on the mnmaton of a quadratc crteron on a sldng horon, whch nvolves a term related to the dfference between the predcted output sequence and the sequence of future control [4]. he crteron s gven by: (3) HP Nc [()()]( ˆ) y C t= N u t= J = y k + t y k + t + u k + t wth yˆ () k s the output value predcted at tme k, yc () k s the set ponts values at tme k, u () k s the ncrement of control at tme k, N s the mnmum predcton horon, HP s the maxmum predcton horon, N s the control horon, u s the control weghtng factor and y s the error weghtng factor. c.3. Predcton of the System Output Consder the output expressed by () the output at tme nstant (k+t) wll be: B ()() C ()() A ()()() A y k+ t = u k+ t d + e k+ t (4) By applyng the Eucldean algorthm on the second term of (4) we get ()() t t = L () t + A ()()()() A C G (5) After usng (4) and (5), we assume that the term related to the dsturbance s ero, the optmal predctor of the output s: Lt ()()()() B Gt C C yˆ()( k+ t = )() u k+ t d + y k (6) ()() 69

4 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July A second Dophantne equaton decompose the predctor n two terms: a term based on the current output, old orders, the system output and a second term dependent on future orders. t + d () t H t ()() R = + (7) C ()() C Wth: ()()() = Lt B (8) he optmal predctor of the output s: t Gt ()() Rt ()() C ()() C yˆ ()()()( k + t = H ) u k + t d + y k + u k (9) Where H t () G () t R () t and Lt () are polynomal solutons to the Dophantne equatons [5]. he matrx formulaton s represented by: Yˆ = Hˆ U + Gˆ Y + Rˆ U () * * he vector of the predcted outputs Y ˆ s the sum of the predcted forced Hˆ ˆ * ˆ * GY + R U Wth: U and free responses ˆ Y = [( yˆ k/),( + k y/)( ˆ k+ k /)] yˆ, k+ HP k () * * * * = [(),()], Y y k y k y k na () U = [()( u k u)] k +, N (3) C * * * U = [( u )( k u k )] nb, d + (4) ˆ G = [()()] G, G (5) + d HP + d ˆ R = [()()] R, R (6) + d HP + d Hˆ h h h = hhp d hhp d h HP d NC (7) Where: y ()( k ) y k u k u k * * () =,( ), = C ()() C 7

5 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July And D IM() Gˆ H= P d, na d +, D IM() Rˆ H= P d, nb d +, D IM() Hˆ H= P d, N, C Denote the dmenson of G ˆ, R ˆ and H ˆ, respectvely..4. Law Order We wrte the crteron J n matrx form ˆ [()()] [()()]()() ˆ J = Yk Y k Yk Y k + Uk Uk (8) Wth: C C u Y = [()()] y k + d y k + HP + d (9) C C C he optmal vector U s: ()ˆ ˆ ˆ ˆ u NC U = H H + I H Y () he optmal control law s derved from analytcal mnmaton of the prevous cost functon. Only the frst control value s fnally appled to the system. u ()()( k = u). k + u k () Whch u () k s the frst element of the vector U and IN s dagonal matrx of se N * C C NC and s dagonal matrx of se HP d * HP d I N C y =, = y ().5. Constrants Formulaton Generally, the constrants mposed on the control sgnal and ts ncrement are descrbed by nequaltes forms m n u u k u m ax su u k su (3) u mn, max Where u and su are respectvely lower threshold, the upper threshold and dervatve threshold of the control nputs. On the horon controller HC can be wrtten: 7

6 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July s u u k s u s u u k + s u, s u u k + H C s u,, (4) Or n the condensed form: I U, I Wth I s dentty matrx of dmenson (HC, HC), dmensons HC : = [ su su ],,, U = u k u k + HC (5) U and are two vectors wth the same (6) On the same horon control we obtaned: mn mn mn max (, ) u u k u k u u k u u k u k + u k+ u u k, max u u k u k + u k+ HC u u k, max Or n the condensed form: ( k ) ( k ) W U, W 3 (7) (8) Wth et 3 are two vectors wth the same dmensons HC: ( k ) = u u ( k ) u u ( k ), mn mn ( k ) = u u ( k ) u u ( k ),,. 3 max max And W s a matrx of dmenson (HC, HC). (9) W = We can rewrte the two nequaltes (3.5) and (3.8) (3) I I U, W ( k ) W ( k ) 3 (3) 7

7 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July he problem of mnmaton of the crteron J wth constrants s wrtng: U ^ ^ C C u mnj = [Y(k) Y(k)] χ[y(k) Y(k)] +λu(k) U(k) (3) U () k, (33) Wth: = I, I, W, W () k =,, k, k 3 he mnmaton of ths crteron cannot be performed analytcally. he synthess of the control law amounts to solvng an optmaton problem of a quadratc crteron under constrants lke nequaltes. Durng the smulatons proposed subsequently, the determnaton of the optmal control sequence s done usng the functon "fmncon" of Matlab. 3. DECOUPLING DESIGN OF GENERALIZED PREDICIVE CONROL 3.. Adjustng the weghtng factor synchronaton to the reference sgnal change In order to present the decouplng approach proposed, the IO system presented n Fg s used: (34) Fgure. IO process model. Assume the process to be controlled s a double-nput/double-outputs system.he transfer functons of the varous subsystems are descrbed by: dj j () j B yj ()( k = ) u k (35) A () he two output varables ( y, y ) of the process become the controlled varables, the two varable nputs ( u, u ) are the manpulated varables and, are non-measurable dsturbances. 73

8 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July A()()()( Y k = )(), B U k + k = (), A = (), B (36) u y =, U, Y. = = u y We wrte the crteron J of each controller GPC: HP ^ N [()()], C y C j = N u j = (37) J = y k + t y k + t + u k + t GPC: HP ^ N [()()] C y C j = N u j = (38) J = y k + t y k + t + u k + t he man dea of decouplng controller s the followng [3]: when some reference changes ts value (for example: set-pont of y) controller frstly ncreases only y n the second loop, and then calculate optmal control vector (3) and (33). hs control vector would mnme output devaton caused by reference changes. Values are evaluated from equaton: λ y K = + M q m r y(k) j = C j K : maxmum error weght n j loop, r y C j : reference change on j nput, M q : Polynomal of q defned by desgner. Results of smulatons n whch we take nto count the change of the set pont and the effect of dsturbances of showed the lmt of ths method. y 5 Output Y 5 5 consgne y avec découplage ysans découplage Fgure 3. Evoluton of y wth tunng weghtng factor by reference change 74

9 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July Output Y consgne y avec découplage y sans découplage Fgure 4. Evoluton of y wth tunng weghtng factor by reference change 5 λ y λ y λ y and λ y Fgure 5. Evoluton of the weghtng factor and y. y 5 4 u u Control output Fgure 6. Evoluton of the control output u and u he method of decouplng cannot reduce the couplng effect caused by external dsturbance. o resolve ths problem, we propose an adaptve weghtng factor dependent on the actual control error. 75

10 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July 3.. unng weghtng factor by the error observaton o overcome the drawback of the prevous method we present a new approach to decouplng the structure of the decouplng control s also descrbed n the fgure 7. Fgure 7. Control structure of generaled predctve control wth weghtng factor adjustng. he man dea of decouplng controller s the followng: he error weght factor of the output varable whose reference s kept constant should be ncreased and synchroned wth the varable whose reference s modfed. We note that a hgh value of the weghtng factor max on output error can generate a very drastc change n controlled varables and degrades the control performance, what we are tryng to avod n most cases. On the other hand, a low value of weghtng factor mn does not allow reducng the effect of the couplng caused by the other varable. o satsfy some performance (smooth control movements and mnmum couplng effect), we establsh the weghtng factors expressons max y max y =, mn + exp(()) k y max max y max y =. mn + exp(()) k y max (39) Wth: ()()(),()()(). k = y k y k k = y k y k C C A good decouplng wth better control performance seemed to be optmal when an exponental change of the weghtng factors s used and the lmt value s respected. unng of y depends on process parameters [7] and can vary n broad range [, ]. Hgher max leads to better decouplng. o check the valdaton of decouplng desgned a smulaton results are presented n Fgs 8 and 9. It should be mentoned that control horon s N = N = and the control predcton C C s H P = H P = 5. Weghtng of the control error for the frst and second manpulated varables are: 76

11 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July y y 5 y 5 =, + e x p(()) k 5 y 5 y 5 =. + e x p(()) k 5 y 5 5 Output Y 5 consgne y sans découplage y avec découplage Fgure 8. Evoluton of y wth tunng weghtng factor by error observaton and wthout decouplng 8 Output Y 6 4 consgne y sans découplage y avec découplage Fgure 9. Evoluton of y wth tunng weghtng factor by error observaton and wthout decouplng 77

12 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July 8 λ y λ y λ y and λ y Fgure. Evoluton of. the tunng weghtng factor and y by error observaton y 8 U U Control output Fgure. Evoluton of the control output u and u Fg shows that the weghtng factor dependent on the actual output error whch allow at the same tme to be powerful n dynamc and statc decouplng. Fg shows the evoluton of the control sgnal of y and y. he dsadvantage of ths method s when the controlled varable s subjected to a dsturbance. hs dsturbance affects the second varable wthout any adaptve weghts factor error. o resolve ths problem, the weghtng factor should always be dependent on the actual control error. 4. APPLICAION O AN INCUBAOR SYSEM hs secton emphases on examnng the performance of the proposed Decouplng Generaled Predctve Controller (DGPC) for temperature and humdty control of a new born ncubator system. 78

13 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July 4. Experment setup A neonatal ncubator s, usually, a small (approxmately:.5 x.5 x m³) cabnet wth transparent walls so that the nfant can be easly observed. Fgure. Schematc of the ncubator process wth expermental arrangement for actve humdfcaton to control temperature and Humdty. he devce ncludes an AC-powered heater, a fan to crculate the warmed ar, a contaner for water to add humdty and access ports for nursng care. he passve humdfer cannot provde a hgh humdty level at low temperature such as n the range of [ C to 38 C]. So, we replaced the passve humdfer by an external block based on ultrasonc nebuler system whch s an nstrument for convertng a lqud nto a fne spray. he actve humdfcaton system s able to ncrease the humdty to 8%. Also we developed a mcrocontroller-based system devoted to control the humdty and the heatng of the new born ncubator. he system developed measures the ar temperature and the humdty by two sensors LM35 and SY-3. hen, these data are exported to a mcrocomputer to be analyed and to develop a model for the ncubator [4],[3]. he am of ths study s to desgn and mplement a closed loop control system to regulate the temperature and humdty nsde a neonatal ncubator whch presented n Fg. he presence of couplng makes t mpossble to set one wthout affectng the other. o elmnate the couplngs between temperature and humdty, we propose a multvarable decoupled control by tunng the weghtng factor. 4. Model of the ncubator process Before proceedng wth the real-tme control, t s necessary to pass through the modelng system. For ths reason, an expermental method s proposed for modelng of the IO system. he ncubator system has wo-inputs and wo-outputs. he nputs to the system are u : control sgnal appled to the heater, u : control sgnal appled to the nebuler. he outputs are: y : temperature value output sgnal, y : humdty level output sgnal. 79

14 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July We defne each sub system by Auto Regressve Integrated Movng Average wth external nputs, (ARIMAX) model []. B j() dj C () yj ()( k = )(). u k + e k (4) A ()()() A Where d j s tme delay, j j e s whte nose and A j, B j, C are: na = na A () a a a, j nb = nb B () b b b b, j nc = nc C () c c c. We present the multvarable system as matrx form: y ()()()() k G G u k. y ()()()() k = + G G u k d B ()() d B G G A ()() A G() = G G = d B ()() d B A ()() A (5.9746e e-4) = 3 (.3698e e-4) (.3+.88) (4) (4) (43) 4.3 Computer smulatons and dscussons Before proceedng wth the real tme control, t s mportant to conduct computer smulaton wth Matlab software to check the feasblty of the proposed approach control. he prncpal parameters are set as follows: he predcton horons are HP = 4, HP =, the control horons are Nc Nc =.4, =.47. = = and the weghtng factors of the control ncrements are u u he couplng effect was suppressed by the method of changng the weghtng factor as a functon of the control error. A good decouplng could be acheved by the same dependence of the weghtng factors as n equaton (39) wth 5 y 5 y = andy =. ymn + exp(()) k y5 In Fg 7, we obtaned the ranges of the control sgnal appled to the heater and to the nebuler that scaled between % and %. he multvarable control s smulated wth constrants and at samplng tmes = seconds. Chosen the nputs constrants: % u %, % u %. he descrbed model (43) has strong nteracton between nput u and output y. For ths reason, the analye has been done by changng temperature s reference and observng output sgnal of the humdty. ow method are used and compared: tunng the weghtng factor wth error y 8

15 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July observaton and wth synchronaton to set pont change. For, t remans constant and justfed y by the low nteracton between u and G. In Fg 3, we present the smulaton result of set-pont trackng temperature whch ncreases from ntal temperature to 5 C then to 8 C and decreases to 6 C. In Fg 4, t can be notced that, there are great dfferences n the R.H. (Relatve Humdty) response due to the choce of the error weghtng factor. It s clear that adaptve weghtng factor wth reference change does not reduce the effect of couplng due to external perturbaton. But decouplng wth adaptve weght factor that dependent on the actual control error seems a good method to reduce the couplng effect due to the references change or external perturbaton. In Fg 5, we present the evoluton of the weghtng factor wth error output. We observe that the weght s very senstve to the output error. o reduce ths senstvty, we appled a frst order flter. In Fg 6, we present the evoluton of the weghtng factor whch s synchroned wth reference change. he adaptve error weghtng factor s synchroned to the reference sgnal change. In Fg. 7, we observed that an ncrease of generates a very drastc change n the manpulated varable (humdty control) y whch s an ndcator of the decouplng effects. he advantage of usng an adaptve error weghtng factor s reducng the couplng effect cause by set pont change and external perturbaton. 3 Set-pont Output-temperature 8 emperature C Fgure 3. Smulaton result of set-pont trackng temperature for predctve decouplng control Humdty% Set-pont Humdty wth tunng weght factor(error output) Humdty wth tunng weght factor(reference change) Fgure 4. Compared result of set-pont trackng humdty wth tunng weght factor by reference change and tunng weght factor by error output observaton. 8

16 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July 9 8 λ y not fltred λ y fltred 7 weght factor Fgure 5. Evoluton of the fltered weghtng factor. y 5 weght factor Fgure6. Evoluton of the weghtng factor synchroned wth change set pont Control% Uh wth tunng weghtng factor by error observaton Uh wth tunng weght factor by set-pont change 5 5 Fgure 7. Compared results of decouplng control humdty by set-pont and error output tunng weghtng factor. 8

17 Internatonal Journal of Informaton echnology, Control and Automaton (IJICA) Vol., No.3, July 5. CONCLUSIONS hs paper has developed a systematc desgn approach for Decouplng Generaled Predctve temperature and humdty Control (DGPC) of an ncubator process. Smulatons results have been demonstrated that decouplng by tunng weghtng factor wth synchronaton n set-pont change does not allow a good decouplng relatve to the tunng error observaton. Proposed concept of adaptve weghtng leads to smple decouplng controller and t s not applcable only to predctve control but to all optmal multvarable controllers based on some cost functon mnmaton. he smulaton and expermental results have shown that the proposed control method s capable of gvng satsfactory temperature and humdty control performance for the ncubator system. REFERENCES [] A. K Abbas and S. Leonhardt, (9) System Identfcaton of neonatal ncubator based on adaptve ARMAX technque, IFMBE Proc, vol., pp [] A. Ogunnake and W. H. Ray, (994) Process Dynamcs, Modelng, and Control, Oxford Unv. Press. [3] Bego, O. Perc, N. I. Petrovc, () Decouplng multvarable GPC wth reference observaton, th Medterranean Electrotechncal Conference, vol., pp [4] C. C. sa and C. H. Huang, Model reference adaptve pred- ctve control for a varable-frequency ol-coolng machne, IEEE rans. Ind. Electron, 5,vol., pp , 4. [5] D. W. Clarke, C. Mohtad and P. S. uffs, (987) Generaled Predctve Control - Part I and II, Automatca, 3(), pp [6] D. W. Clarke, (988) Applcaton of Generaled Predctve Control to Industral Processes, IEEE Control System Magane, vol. 8, pp [7] F. Arous, (9) Predctve Control Algorthms for Lnear and non Lnear Processes, hess,. [8] F. elle, V. Bach, S. Delanaud, H. M. Baye, A. Leke, A. Apdoh and M. Abdche, (999) Influence du nveau d humdté de l ar sur le sommel du nouveau-né en ncubateur, RBM. Revue européenne de botechnologe médcale, vol., pp [9] Gler and M. Burunkaya, () Humdty control of an ncubator usng the mcrocontroller-based actve humdfer system employng an ultrasonc nebuler, Journal of Medcal Engneerng and echnology, vol.6, pp [] G. L, D. P. Stoten and J.-Y. u, () Model pred ctve control of dynamcally substructured systems wth applcaton to a servohydraulcally actuated mechancal plant, IE Control heory Appl., 4(), pp [] J. Rchalet, D. O Donavan, (9) Elementary Predctve Functonal Control, Sprng er Verlag, Berln, -5. [] J.. J. Rchalet, A. Rault and J. Papon, (978) Model predctve heurstc control: Applcatons to ndustal processes, Automatca, vol.4, pp [3] M. A. Zerman, E. Fek and A. Mam, () Applcaton of Ge netc Algorthms n dentfcaton and control of a new system humdfcaton nsde a newborn ncubator, Internatonal Conference on Communcatons, Computng and Control Applcatons, pp. -6. [4] M. A. Zerman, E. Fek and A. Mam, () Applcaton of Adaptve Predctve Control to a Newborn Incubator, Amercan J. of Engneerng and Appled Scences, 4 (), pp