Proceedigs Agricotrol 27. 2 d IFAC Iteratioal Coferece o Modelig ad Desig of Cotrol Systems i Agriculture, 3-5 September 27, Osijek Croatia, pp. 29-34. ON-LINE ESTIMATION OF THE TRANSPIRATION IN GREENHOUSE HORTICULTURE J. Botsema *, J. Hemmig*, C. Staghellii*, P. de Visser*, E.J. va Hete*,**, J. Buddig***,T. Rieswijk*** ad S. Nieboer**** * Wageige UR Greehouse Horticulture, Wageige, The Netherlads, e-mail: ja.botsema@wur.l ** Wageige Uiversity, Farm Techology Group, Wageige, The Netherlads *** Priva B.V., De Lier, TheNetherlads **** Improvemet Cetre, Bleiswijk, The Netherlads Abstract: Usig a so-called ukow iput observer (UIO) the traspiratio of a crop i a greehouse is o-lie estimated. I this way a useful tool for the horticultural practice is developed, givig the grower istataeous isight i the status of his crop. The desig, implemetatio ad performace i practice will be show. Copyright 27 IFAC Keywords: greehouse climate, observers, traspiratio, ukow iput observer, UIO. 1. INTRODUCTION I moder greehouse horticulture the climate is cotrolled by a sophisticated greehouse climate computer. The role of the grower is to defie amog others temperature trajectories, carbo dioxide set poits ad relative humidity boudaries, i such a way that durig the growig seaso the crop i the greehouse is maitaied i a optimal coditio ad crop productio is maximised. The climate computer will the realise the climate desired by the grower. The grower observes his crop ad decides if the realised climate is good for the crop. A major parameter for the crop status is the traspiratio. A good traspiratio of the crop meas that the plats are ot stressed ad are developig well. Although traspiratio gauges (gullies) are commercially available for high-wire crops, i most cases the grower has to do a visual ispectio, which will ot always give a direct isight ito the actual traspiratio. I this paper a ew method is proposed which estimates o-lie the traspiratio of a crop i a greehouse, by usig oly climate variables, already measured by the grower s climate computer. The method is based o a observer techique for estimatig ukow iput of a system, a so-called ukow iput observer (UIO). 2. OBSERVER DESIGN It is assumed that the origial process has the followig form: x& = Ax + Bu + Ed, y = Cx, x() = x, x R, u R, d R, y R m q p (1) Where x is the state of the system, u is the cotrol iput ad d is disturbace iput actig o the system ad y is the output of the system. Here it assumed that d(t) is a measurable disturbace. I greehouse productio, most of the disturbaces actig o the process are related to the outdoor weather, which is measured. A observer, the so-called Lueberger observer, amed after his ivetor has the the followig form (Lueberger, 1966; Lueberger, 1971): x& ˆ = Axˆ + Bu + Ed + L( y yˆ ), yˆ = Cxˆ, xˆ () = xˆ (2) Where x ) ad y ) are the estimated state ad output ad L is called the observer gai. The observer is actually a copy of the model of the origial system, but sice the iitial coditios of the system ad observer are i geeral differet, the outputs will be differet. The observer is therefore drive by the differece of the outputs of the system ad the observer. If fially yˆ( t) y( t) the, uder the assumptio that (C,A) is observable, xˆ( t) x( t).
The questio is how to determie L. For this we cosider the error betwee the state of the system ad the state of the observer, e( t) = x( t) xˆ ( t), usig (1) ad (2) it follows that: e& ( t) = ( A LC) e( t) e( t ) = x xˆ (3) If L is chose i such a way that the matrix A-LC has all its eigevalues i the left half complex plae, the idepedet from x xˆ, e( t). The eigevalues of A-LC ca be chose arbitrarily, provided that (C,A) is observable. Observers ca also be defied for o-liear systems, the observer gai will the i geeral deped o the state (Dochai, 23). 3. UNKNOWN INPUT OBSERVER Observers origially were desiged to estimate the o measured states. I recet years observer desig is also used to estimate ukow iputs of a system. For simplicity we cosider a scalar system: x& ( t) = ax( t) + bu( t) + ed( t) y( t) = x( t) (4) Where u( t ) is a ukow iput ad d(t) is a measured disturbace. Furthermore it is assumed that u( t) is a slowly varyig sigal, so u& ( t). Defiig z = x ad z = u, the system ca be writte as: 1 2 obtai such a result by choosig ξ =.77 ad ω 5 to 1 times the domiat frequecy i the sigal to be estimated. 4. HUMIDITY IN A GREENHOUSE The humidity balace, expressed per m 2 soil, for a greehouse, is give by (Hete, 1994, Staghellii 1987, Staghellii ad de Jog,1995): d χ = (6) a h E V C Where h is the average height of the greehouse (m), χa is the vapour cocetratio of the greehouse air (gm -3 ), E is the crop traspiratio, C is the codesatio o the greehouse cover ad V is the moisture loss through the vetilatio widows, all i gm -2 s -1. The loss caused by vetilatio is give by out V = g ( χ χ ) (7) V a out Where χ is the vapour cocetratio i the outdoor air ad gv is the vetilatio flux (m 3 m -2 s -1 ms -1 ). The loss through codesatio is calculated by C = g e T T χ χ (8).485 T * [.2522 a c ( a out ) ( a a )] * Where χ a is the saturated vapour cocetratio that ca be approximated (1-4 C) by: d z1 a b z1 e d z = + 2 z 2 (5) Ad g C is give by: *.572Ta χ a = 5.5638e (9) This system is similar to the oe described by eq. l1 (1). Usig a observer, with observer gai L = l it 2 is easy to calculate that the trasfer fuctio from the ukow u ( z 1 ) to the estimated û ( ẑ 1 ) is give by: Defiig: bl2 G( s) = 2 s + ( a + l ) s + bl 1 2 (6) a + l1 ω = bl2, ξ = (7) 2ω the trasfer fuctio is i stadard secod order form. We ca therefore give a good recipe for tuig the observer. For a good estimate of the ukow iput sigal, which we assumed to be slowly varyig, the trasfer fuctio, for low frequecies should have a gai close to 1 ad a phase lag close to zero. Oe ca g = T T (1) 3 1/ 3 C max[,1.8 1 ( a cover ) ] which is derived from the mass trasfer theory o horizotal plates, sice (Papadakis et al., 1992) showed that the small slope (26 ) of a Velo type greehouse did ot play a role. The term betwee square brackets i eq. 8 is derived by applyig the rule of thumb for Dutch coditios that T cover, the temperature of the cover of the greehouse, ca be calculated as follows: T cover 2Tout + Ta = (11) 3 By liearizatio of eq. 9 i the iterval T a -T out the result follows. The ukow term i the humidity balace is the traspiratio E. I the ext sectio we first discuss a model for the traspiratio, which will be used for compariso with the ew approach.
5. TRANSPIRATION MODEL The traspiratio of a tomato crop i a greehouse ca be modelled by (Staghellii, 1987, Staghelllii ad De Jog, 1995): 2LAI * εrb R E = χa χa + (1 + ε ) r + r 2LAI L b s (12) Where LAI is the leaf area idex of the crop, ε is the ratio of the latet to sesible heat cotet of saturated air for a chage of 1 C i temperature. I the rage of greehouse air temperatures ε ca be approximated by:.518 a ε =.7584e T (13) R (Wm -2 ) is the et radiatio of the crop, that is the balace of itercepted ad reflected su radiatio plus the balace of icomig ad outgoig log-wave radiatio. The relatio betwee R ad the global radiatio I (Wm -2 ) is give by:.7 R.86(1 LAI e ) I = (14) Which is a simplificatio of the formula give by (Staghellii, 1987). r b (sm -1 ) is the resistace to heat trasfer of the leaf boudary layer ad (Staghellii, 1987) calculated that r b = 2 for a greehouse tomato crop. r s (sm -1 ) is the stomatal resistace ad is accordig to (Staghellii, 1987, Staghellii ad De Jog., 1995), give by: R + 4.3 r 82 2 s = LAI 1+.23( 24.5) R +.54 2LAI 2 ( Ta ) (15) This model is cosidered as oe of the best to describe traspiratio i a greehouse (Jolliet ad Bailey, 1992). 6. THE GREENHOUSE DYNAMICS The absolute humidity defied by the humidity balace, eq. 6, together with the traspiratio model, eq. 12, has bee simulated with data from a real greehouse (Gree Q, Moster, The Netherlads) ad compared with the measured absolute humidity. The results are give i fig. 1. Vapour cocetratio (g/m 3 /m 2 ) 24 22 2 18 16 14 12 simulatio measured 1 6 8 1 12 14 16 18 Fig. 1. The simulated ad measured absolute humidity i a greehouse o 19 th September 26. Sice there is a good resemblace betwee the measured ad simulated value it ca be cocluded, that the model o which the ukow iput observer is based, is good. 7. THE TRANSPIRATION MONITOR The estimatio of the traspiratio usig a ukow iput observer, as described i sectio 3 goes as follows. Eq. 6 is i the form of the system described by eq. 4., where E is the ukow iput to be estimated. The measured disturbaces are the codesatio C ad the outside humidity. If eq. 6 is rewritte i the form of eq. 4, the parameters i eq. 4 are as follows: gv 1 gv 1 a =, b = ad ed( t) = χout C (16) h h h h The vetilatio flux g V is measured by the vetilatio moitor, described i (Botsema et al., 25) The vetilatio flux is time varyig, ad so will be a. The tuig of the observer ca be doe i a similar way as for time ivariat parameters. I the experimets with a commercial greehouse tomato crop the vetilatio rate ad the traspiratio were calculated with the vetilatio moitor ad the traspiratio moitor, by meas of the stadard data of the iside ad outside climate, as collected by the climate computer of the grower. 8. MEASUREMENT OF TRANSPIRATION The ew approach will ot oly be compared with the model from sectio 5, but also with the traspiratio determied with the aid of a traspiratio measuremet gully, a Priva Groscale. The measuremet gully is show i figure 2. The iformatio available from the climate computer, is M the mass of the measuremet gully, D the drai of the surplus water i the gully ad S the supply of water
through the drips. The mass balace for the measuremet gully is, with E the traspiratio of the crop: dm ( t ) = S ( t ) E ( t ) D ( t ) (17) From this balace of the measuremet gully the traspiratio of the crop ca be calculated as: dm E( t) = S( t) D( t) ( t) (18) Sice the ucorrected ad ufiltered sigal gives o isight ito the traspiratio the sigal is firstly corrected for outliers. The result is show i figure 4. Clearly already oe ca see what the traspiratio will be, although the sigal is still rather oisy. Therefore the sigal of figure 4 is filter usig a movig average over 1 hour. The result is show i figure 5..1.8.6.4.2 -.2 -.4 -.6 -.8 5 1 15 2 25 Fig. 4. The traspiratio o 19 th September 26, corrected for outliers. Fig. 2. The Priva Groscale. Usig equatio (18) gives the followig result for the traspiratio o 19 th September 26. 1-1 -2-3 -4.1.8.6.4.2 5 1 15 2 25 Fig. 5. The traspiratio o 19 th September 26 filtered by a movig average of 1 hour. -5 5 1 15 2 25 Fig. 3. The traspiratio calculated from the data from the measuremet gully o 19 th September 26. The calculated traspiratio is ot corrected for outliers ad is ot filtered. The large deviatio at 2 o clock i the morig is caused by a automatic reset o the sigals for the drai ad supply. The positive peaks are due to the drai ad the egative peaks are due to the supply. 9. RESULTS I fig 6. the model of sectio 5 is compared with the results of the measuremet gully of sectio 8. I the greehouse, were the measuremets were performed, some leaf area measuremets were doe, o differet days. O these differet days the average leaf area of the crop was 1.7 ad 2.2. With these two values the traspiratio was calculated with the climate data o September 19 th 26. Clearly the LAI has a cosiderable ifluece o the outcome of the model. If we compare the outcome of the model with the recostructed traspiratio from the measuremet gully, it ca be cocluded that the model ad the
measuremet gully give similar results. Notice that whe the traspiratio is decreasig, it seems that the measured traspiratio from the gully is decreasig faster, tha the calculated traspiratio from the model..12.1.8.6.4.2 traspiratio model, LAI=1.7 traspiratio model, LAI=2.2 Priva Groscale 5 1 15 2 25 Fig. 6. The traspiratio o 19 th September 26, calculated from the model of sectio 5 ad measured by the Priva Groscale. I fig. 7 the results of the traspiratio moitor, based o the UIO-approach are compared with the results from the measuremet gully..1.8.6.4.2 traspiratio moitor Priva Groscale 5 1 15 2 25 Fig. 7. The traspiratio o 19 th September 26, determied with the traspiratio moitor ad measured by the Priva Groscale. moitor is ot very accurate whe the differece betwee iside ad outside temperature is small. From fig. 6 it ca be see that the model of sectio 5 is rather sesitive for the leaf area idex (LAI). This LAI is i ormal practice i greehouse ot measured. So whe usig this model, oe should take care of the fact that the LAI has to be estimated ad this will ifluece the accuracy of the calculated traspiratio. I order to discuss the differeces betwee the approaches some climate data o 19 th September are give. Radiatio (W/m 2 ) 7 6 5 4 3 2 1 5 1 15 2 25 Fig. 8. The global radiatio o 19 th September 26. The sudde chage i the radiatio aroud 16. hours, results, accordig to the model of sectio 5, also i a chage i traspiratio. However the measuremet gully is hardly detectig this chage, see fig. 6. The traspiratio moitor detects, just as the model, this chage i traspiratio. The measuremet gully detects a sudde chage i traspiratio, aroud 12. hours ad a similar chage aroud 19. hours. These chages are also detected by the traspiratio moitor, but ot by the model. The chage is probably caused by rapidly opeig ad closig of the vetilatio widows, as ca be see i figure 9. It seems that rapidly opeig of the widow causes a decrease i traspiratio, probably caused by coolig of the leaves. Agai the resemblace betwee the two sigals is good. Also here the sigal of the moitor is slower tha the sigal of the measuremet gully, whe the traspiratio is decreasig. If we compare the three methods it ca be said that all three methods (model, measuremet gully ad traspiratio moitor) give similar results. The traspiratio moitor is producig a positive traspiratio durig ight, this is due to the fact that the traspiratio moitor is based o the vetilatio moitor (Botsema et. al., 25). This vetilatio
Widow aperture (%) 1 8 6 4 2 lee side wid side 5 1 15 2 25 Fig. 9. The widow apertures 19 th September 26. 1. CONCLUSIONS A ukow iput observer is a good ad simple method for estimatio of the traspiratio of a greehouse crop, the so-called traspiratio moitor. The method uses oly measuremets which are already available i the climate computer of most growers, though ot all growers measure the outside humidity, which is required here. Such sesors are commercially available for the horticulture practice. The traspiratio moitor is a successful ad useful expasio of the vetilatio moitor, which is based o the same cocept of a UIO. Hete, E. J. v. (1994). Greehouse climate maagemet: a optimal cotrol approach, PhD Thesis Wageige Uiversity. Jolliet, O. ad B.J. Bailey (1992). The effect of climate o tomato traspiratio i greehouse: measuremets ad model compariso. Agricultural ad Forest Meteorology, 58, 43-62. Lueberger, D. G. (1966). Observer for Multivariable Systems. IEEE Trasactios o Automatic Cotrol, AC-11, 19-197. Lueberger, D. G. (1971). A Itroductio to Observers. IEEE Trasactios o Automatic Cotrol, AC-16, 596-62. Papadakis, G., A. Fragoudakis ad S. Krytsis (1992). Mixed, forced ad free covectio heat trasfer at the greehouse cover. J. Agr. Eg. Research, 51, 191-25 Staghellii, C. (1987). Traspiratio of greehouse crops, PhD Thesis Wageige Uiversity. Staghellii, C. ad T. de Jog (1995). A model of humidity ad its applicatio i a greehouse. Agricultural ad Forest Meteorology, 76, 129-148. The method is easily implemeted ad tued for a particular greehouse. Furthermore the ew method gives the grower valuable isight ito the status of his crop. The traspiratio moitor developed here is a good example of a itelliget or soft sesor. ACKNOWLEDGEMENTS This research was sposored by the Dutch Miistry of Agriculture, Nature ad Food Quality ad by the Dutch Product Board for Horticulture (Productschap Tuibouw) uder project umber, PT 1242. REFERENCES Botsema, J., E.J. va Hete, J. G. Koret, J. Buddig ad T. Rieswijk (25). O-lie estimatio of the vetilatio rate of greehouses. Proceedigs 16 th IFAC World Cogress, Prague, Czech Republic. Dochai, D. (23). State ad parameter estimatio i chemical ad biochemical processes: a tutorial. Joural of Process Cotrol, 13, 81-818.