Virtual sensors for designing irrigation controllers in greenhouses

Sensors 2012,12, 15244-15266; doi:10.3390/s121115244 OPEN ACCESS sensors ISSN 1424-8220 www.mdpi.com/journal/sensors Article Virtual Sensors for Designing Irrigation Controllers in Greenhouses Jorge Antonio S´ anchez 1,⋆, Francisco Rodr´ ıguez 1, Jos´ e Luis Guzm´ an 1and Manuel R. Arahal 2 1Ingenier´ıa de Sistemas y Autom´atica, Departamento de Lenguajes y Computaci´on, The Agrifood Campus of International Excellence (ceiA3), Universidad de Almer´ıa, Ctra de la Playa s/n., Almer´ıa 04120, Spain; E-Mails: [email protected] (F.R.); [email protected] (J.L.G.) 2Departamento de Ingenier´ıa de Sistemas y Autom´atica, Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, Sevilla 41092, Spain; E-Mail: [email protected] *Author to whom correspondence should be addressed; E-Mail: jorg[email protected]; Tel./Fax: +34-950-014-536. Received: 3 August 2012; in revised form: 19 October 2012 / Accepted: 22 October 2012 / Published: 8 November 2012 Abstract: Monitoring the greenhouse transpiration for control purposes is currently a difficult task. The absence of affordable sensors that provide continuous transpiration measurements motivates the use of estimators. In the case of tomato crops, the availability of estimators allows the design of automatic fertirrigation (irrigation + fertilization) schemes in greenhouses, minimizing the dispensed water while fulfilling crop needs. This paper shows how system identification techniques can be applied to obtain nonlinear virtual sensors for estimating transpiration. The greenhouse used for this study is equipped with a microlysimeter, which allows one to continuously sample the transpiration values. While the microlysimeter is an advantageous piece of equipment for research, it is also expensive and requires maintenance. This paper presents the design and developmentof a virtual sensor to model the crop transpiration, hence avoiding the use of this kind of expensive sensor. The resulting virtual sensor is obtained by dynamical system identification techniques based on regressors taken from variables typically found in a greenhouse, such as global radiation and vapor pressure deficit. The virtual sensor is thus based on empirical data. In this paper, some effort has been made to eliminate some problems associated with grey-box models: advance phenomenon and overestimation. The results are tested with real data and compared with other approaches. Better results are obtained with the use of nonlinear Black-box virtual sensors. This sensor is based on global radiation and vapor pressure deficit Sensors 2012,12 15245 (VPD) measurements. Predictive results for the three models are developed for comparative purposes. Keywords: virtual sensor; transpiration; nonlinear model; micro-lysimeter 1. Introduction Crop growth is primarily determined by climatic variables of the environment and the amount of water and fertilizers applied through irrigation. Therefore, controlling these variables allows for control of the growth. The greenhouse environment is ideal for farming because these variables can be manipulated to achieve optimal growth and plant development. All crops need solar radiation, CO2, water, and nutrients to produce biomass (roots, stems, leaves, and fruits) through the process of photosynthesis. During this process, and when the leaves stomata are opened to capture the CO2, the plant emits water vapor through the transpiration process. This becomes a cost that the crop must make to produce dry matter. Moreover, water is lost through evaporation from the soil. The sum of these water losses is known as evapotranspiration. The losses must be compensated through irrigation. Besides, it has been demonstrated in padded greenhouses with soil covered with plastic blankets that the amount of evaporation is negligible. This happens when dealing with hydroponic cultivations [1]. According to this, water should be applied in precise amounts to cover only water losses due to crop transpiration. Excess water would mean an excessive washing out of fertilizers. In turn, it could lead to contamination of the subterranean water, or the flooding of the substratum or radicular asphyxiation. Otherwise, a hydric deficit may be provoked if irrigation does not provide enough water. This can lead to a decrease in production and can even be dangerous for the crop growth. Hence, automatic irrigation control systems are fundamental tools to supply water to the culture in the required amount and frequency. Moreover, as water is a limited resource in many agricultural areas, optimizing productivity through efficient and adequate irrigation is a basic objective. In order to design a good automatic irrigation system, the following questions must be answered: what should the frequency of the irrigations be, and how much water should be applied in the irrigation? To answer these questions, it is necessary to know how much water should be applied to replenish the losses due to the transpiration during the plant’s respiration. Measuring the water lost by transpiration is a way of obtaining the plant’s water demand. This estimation of transpiration in different species grown in greenhouses has been developed, e.g., by Baille et al. [2] for ornamentals, Stanghellini [1], Jemaa [3], Boulard [4] and Baille [2] for tomatoes, Montero et al. [5] for geranium crops, Medrano et al. [6] for cucumbers, Suay et al. [7] for rose cultivation, Voogt et al. [8] for chrysanthemums, and Schmidt and Exarchou [9] for gerbera pots, among others. In most of these works, the microlysimeter became the basic measurement device to record the water losses in crops, subtracting the water content in an instant (t)by the water content in another instant (t−1). However, on manyoccasions, themeasurements were not continuousdue to the irrigationprocess or during the water drainage. Furthermore, it is seldom used by farmers since this device is expensive to acquire and to maintain. From an operational point of view, it is important to find alternatives to Sensors 2012,12 15246 this irrigation system gadget. Thus, virtual sensors based on transpiration become a good option to reduce total system cost, especially in the agriculture sector where profit margins are so narrow. Such virtual sensors must be based on sensors that are typically installed in greenhouses for climate control (temperature, humidity, and solar radiation), thereby reducing the installations costs. Virtual sensors become a very efficient and powerful tool that has been successfully used in other fields [10–12]. These sensors utilize models in order to estimate features from low-cost measurements. Ideally, the virtual sensors should be simple and obtainable from the collected data. It should not require extensive training. Virtual sensors are useful in replacing physical sensors, thus reducing hardware redundancy and acquisition cost, or as part of the fault detection methodologies by having their output compared with that of a corresponding actual sensor. Virtual sensors may be developed based on mathematical models obtained directly from the Physics of the system and first principles. In many cases, such mathematical models are unavailable, or their exact parameter values are unknown, or they are too complicated to be used. For this reason, the development of virtual sensors often has to be based on system identification [13]. The purpose of the current study is to develop a virtual sensor to infer transpiration from other easily measured variables. For this purpose, the use of the microlysimeter as the sensor to calculate the transpiration must be substituted, due to its high cost in acquisition and maintenance. In this paper, the development of the virtual sensor for transpiration makes use of different techniques for data preprocessing, including the selection of variables, the construction of appropriate training, the selection of test sets, the final validation and the performance assessment. The resulting virtual sensor has been validated and compared with real data and with other virtual sensors in the literature, providing promising results. The paper is organized as follows: Section 2shows a background of different sensors used to take transpiration measurements. The different virtual sensors are shown in the Section 3. Section 4gives an overview of the greenhouse where the experiments were performed, and its main characteristics, as well as the collected experimental data. The main results and discussions are summarized in Section 5. Finally, the major conclusions are drawn. 2. Crop Transpiration The irrigation control systems are essential tools to provide water to the crop in the required amount and frequency. Moreover, water is a limiting resource in many agricultural areas. In such places, it should be a basic objectiveto optimize their management and productivity through adequate and efficient irrigation. The proposed control algorithm design (Figure 1) is a hierarchical control system, consisting of two levels: •The control level uses an event-based PI controller [14], to control when a certain event occurs, either by time, by variation of a particular climatic variable such as radiation VSR), or by a particular state crop. A PI controller is used to achieve the setpoint of water supply (XQWr) as it considers the top layer of the architecture. •The setpoint generation level is based on the greenhouse climate conditions, including: (1) vapor pressure deficit (VPD, VV P D), which is a function of the temperature and the relative humidity, Sensors 2012,12 15247 (2) global radiation, and (3) the state of the crop, measured through the Leaf Area Index (LAI, XLAI ). The setpoint is fixed by the user and is defined as the crop transpiration accumulated (XET ) until an irrigation event occurs (XQW). This event could be a determined transpiration accumulated setpoint or other predetermined conditions. Figure 1. Irrigation control algorithm. The virtual sensor accumulates the amount of water lost by transpiration from the plant’s last irrigation. This measurement is compared with the fixed amount that activates irrigation (setpoint). The irrigation starts when this fixed value is exceeded, and finishes as soon as the amount of water lost has been replenished. A virtual sensor requires an accurate calibration and validation of the instantaneous transpiration measurements at each sampling instant. The measurement of transpiration includes direct (observation, porometer, lysimeter, etc.), and indirect methods (water budget, energy balance, etc.). The indirect methods are more complicated for prediction and verification of the transpiration values because of the meteorological factors involved. The transpiration data can be collected through direct methods such as: •The porometer [15] allows the determination of the leaf conductivity as an index of the stomatal opening and the closing of stomata. It measures the flow of gases or diffusion that takes place through the stomata. The latest porometers allow computerized records. •The bag method [16] collects the water transpired by introducing a branch in a clear plastic bag. The transpired water condenses inside the bag. The total water lost by transpiration corresponds with the weight of the placed water. The time between measurements is undefined, as it depends on the water collected. •In the cobalt chloride method [17], the transpiration is indicated by a color change of a piece of filter paper impregnated with a 3% solution of cobalt chloride. It is applied on a leaf and held in place with a clip. It is blue when dry and pink when wet. The speed at which the paper changes color is an indication of the rate of transpiration. This method can be used to measure the relative rates of transpiration of different species. Sensors 2012,12 15248 •The microlysimeter [1,4] is used in plants growing in pots completely closed. The plants are first weighed before measuring, and then they are weighed again at convenient time intervals. Soil evaporation is avoided by covering it with waterproof material. This method can be used with small plants and crops in soilless culture. The results are expressed in grams or milliliters of water transpired per leaf area per unit time. The microlysimeter is the basic measurement device used to continuously record the water losses in crops. Contrarily, the rest of the described sensors take measurements at intervals as they modify the working conditions of the plants. The porometer and microlysimeter are expensive to acquire and maintain, and are difficult for the farmers to manage. 3. Virtual Sensors This section is devoted to describing the main features of three different types of virtual sensors for transpiration designed to replace the microlysimeter in the automatic irrigation system. The aim of such virtual sensors is to substitute the expensive microlysimeter in measuring the transpiration and controlling the irrigation. Figure 2shows the input variables of a virtual sensor for crop transpiration: (1) global (solar) radiation, (2) Vapor Pressure Deficit (VPD), which is a function of the temperature (T) and the relative humidity (RH), and (3) Leaf Area Index (LAI). These variables are measured by two different sensors typically installed in commercial greenhouses: a psychrometer for the temperature and the relative humidity (VPD), and a pyranometer for the solar radiation. Figure 2. Virtual sensor scheme. The transpiration model needs to calculate the LAI. The LAI measurements are taken in a noncontinuous way. In this paper, a simplified TOMGRO model [18] adapted to the Mediterranean conditions is utilized to estimate the LAI. The simplified TOMGRO model needs the temperature as the input. Sensors 2012,12 15249 Based on the architecture of Figure 2, three different virtual sensors are considered in this paper. The first one is based on a pseudo-physical structure. The others have a more empirical structure with linear and nonlinear behaviors, respectively. All three kinds are described in the following. 3.1. Pseudo-Physical (Grey-Box) Virtual Sensor Most transpiration estimators are based on the Penman–Monteith equation. In 1948, Penman derived an equation that combined the energy balance and the convective transport of vapor. Afterwards, this model was adapted by Monteith to estimate actual evapotranspiration from plants [19]. This equation essentially combines the equation for heat transfer between the crop and the mass of the surrounding air. In this way, various authors have obtained new formulations without satisfactory results for various crops. The tomato evapotranspiration studies of [20] have as a main drawback the estimation of the leaf stomatal and aerodynamical resistances. Jolliet and Bailey [15] concluded that a layer model proposed in [1] predicts accurately the transpiration in tomatoes. These authors observedthe tomato crop transpiration (XET ) increases linearly with the radiation (VSR), the vapor pressure deficit (VV P D), and the wind speed inside the greenhouse. They also pointed out that transpiration can be regulated by not only humidity but also radiation and wind speed. These same authors simplified the Penman–Monteith equation [21]. to describe the transpiration (XET ) as a process based on two main variables: the solar radiation (VSR) arriving at a particular depth in the canopy plant, and the vapor pressure deficit VV P D Equation (1). The reduced virtual sensor is shown in the following equation: CλXET =VSRCA+VV P DCB(1) where the coefficients CAand CBare the parameters dependent on the crop. In [2], it was observed that the coefficient CBincreases with the XLAI , and, furthermore, the coefficient adopted different values during the day due to oscillations in stomatal resistance Equation (2). These daily oscillations was corrected with two different parameters for diurnal (CBD) and nocturnal (CBN) periods. CλXET =e(−CkXLAI )VGRCA+VDP V XLAI CB(2) where XET is the crop evapotranspiration (gm−2min−1), Cλis the latent heat of evaporation (kg/oC), Ckis the light extinction coefficient for crops (it is related to the leaf inclination angle and the leaf arrangement with regard to the Leaf Area Index, and provides an indication of the plant’s efficiency on intercepting the solar radiation), XLAI is the leaf area index (m2m−2),VV P D is the vapor pressure deficit (KP a), and VGRis the global radiation reaching the crop (Wm−2). The coefficients CA(unitless) and CB(kgm−2h−1kP a−1) are constants dependent on the crop. To obtain more reliable results of the virtual sensor, the parameter CBis obtained for diurnal (CBD) and nocturnal (CBN) periods through calibration. 3.2. Black-Box (Empirical) Virtual Sensors The identification system tries to solve the problem of constructing mathematical models of dynamic systems based on the data they observe [22]: inputs u(t)and outputs y(t). The goal is to infer the Sensors 2012,12 15250 relationship between sampled outputs and inputs. The identification process was carried out based on prior knowledge of dynamic behavior of gases [14] and the behavior of transpiration ([1,2,6]). VPD and global radiation was selected as climatic inputs and the LAI as crop growth input. This method is used with the objective of knowing how the dynamics of the systems work and responding to some of the questions raised while drawing up this document. 3.2.1. Linear Black-Box Virtual Sensors The parametric virtual sensors (or black box) are diagrams capable of representing any system without having any knowledge about the physical process dynamics. Parametric virtual sensors are not obtained (at least not completely) from the application of physical laws. These virtual sensors are constructed using observations carried out on the system so as to select a concrete value of the parameters. This value is chosen in such a way that the virtual sensor can accommodate the results to the acquired data. This process is called identification. The adjustment of the parameters is the simplest part of the problem of identification [23]. Online identification concerns an algorithm that efficiently uses the measured information when it is obtained from the plant in real time. In this way, it is possible to detect the changes in the dynamics of the system and adjust the virtual sensor conveniently. Under some circumstances, these methods can be rather simple (e.g., the method of minimal recurrent squares developed in this section). Figure 3. Graphical user interface of “ident”, a part of the System identification Matlab-Toolboxr[24]. The black-box virtual sensors are developed using a system identification technique shown in [23]. The virtual sensors family contains 32 possible formulations based on Equation (3). To obtain each Sensors 2012,12 15251 one of the structures, it is necessary to determine the polynomial order as well as the coefficient of the numerator and the denominator for each transfer function [23]. The effects to the inputs, output, and the disturbances are defined as: A(z)y(t) = B(z) F(z)u(t−nk) + C(z) D(z)e(t)(3) where, y(t)is the transpiration, u(t)is the different input variables, and e(t)is the estimation error. A(z), B(z), C(z), D(z), E(z), and F(z)represent the polynomials that define the output (transpiration), inputs, and the estimation error. The order of the polynomial equations is defined by regressors, where na outlines the outputs, nb is the order of the input, and nk is the delay of the radiation solar (nkVGR) and the vapor pressure deficit (nkVV P D ). System Identification Matlab’s Toolbox [24] was used for the identification process. The ARX, ARMAX, OUTPUT ERROR, BOX JENKINS, and FIR formulations were tried out. The differences among these virtual sensors are the way in which the inputs, outputs and disturbances are defined with parametric equations. The System Identification Toolbox of Matlab(R) software was used to obtain the virtual sensor. Figure 3shows the main interface of the Toolbox and the whole process to obtain a virtual sensor. The top of the figure displays the calibration and validation process interface, and the bottom shows the output analysis process. The toolbox allows to process data, estimate the parameters of different types of structures, and validate virtual sensors using different strategies. For the identification process, only the data from the experiments are required. The data can be handled in the time domain or frequency domain, and the experiments can have one or multiple inputs and/or outputs [25]. 3.2.2. Nonlinear Black-Box Virtual Sensors A nonlinear component was added to the transpiration virtual sensor to get better fitting. This component is introduced as result of the strong nonlinear behavior in the system inputs. Moreover, these nonlinearities add complexity to the virtual sensor. This increase in complexity is not always translated into higher performance. In System Identification the mathematical relationships between the system’s inputs u(t), and outputs y(t)can be computed. Such outputs, inputs, and nonlinearities are introduced in an ad hoc form, relying on a priori knowledge about the system. An important step in system identification is to choose a structure, and generally start testing the simpler structures, and lower order. The first structure tested, and in the end chosen as virtual sensors, was the nonlinear ARX (4). Also the Hammerstein–Wiener virtual sensor was tried out, which are very useful in the case of the nonlinearities affect to sensors, and actuators, such as dead zones or saturation [24]. On the other hand, Nonlinear ARX (NonARX) is more flexible [24]. The general structure for Nonlinear ARX virtual sensor is [26]: y(t) = f(y(t−1), y(t−2), y(t−3), ..., u(t), u(t−1), u(t−2), ..)(4) where y(t) is the output variable in ttime; uand yare the different input and output variables (regressors); and fis the nonlinear function. The current transpiration value is predicted as a weighted sum of past values, and current and past inputs values. With such information the equation becomes: Sensors 2012,12 15252 y(t) = [a1, a2, ..., ana, b1, b2, ..., bnb][y(t−1), y(t−2), ..., y(t−na), u(t), u(t−1), ..., u(t−nb −1)]T (5) where y(t−1), y(t−2), ..., y(t−na), u(t), u(t−1), ..., u(t−nb −1are the regressors, the so-called delayed inputs and outputs. Nonlinear ARX regressors can be both delayed input–output variables and more complex nonlinear expressions of delayed variables. The nonlinearity estimator block maps the regressors to the virtual sensor output using a combination of nonlinear and linear functions (Figure 4). Figure 4. The Nonlinear ARX computes the output y in two stages [24]. u(t) are the model inputs, and y(t) are the model outputs. Available nonlinearity estimators can be selected from a canopy of different structures such as neural networks, tree-partition networks, wavelet networks and piecewise polynomial approximation. The nonlinearity estimator block can include linear and nonlinear blocks in parallel [24]. 4. Greenhouse Environment The research data used in this work have been obtained from greenhouses located in the Experimental Station of Cajamar Foundation, in El Ejido, in the province of Almeria, Spain (2◦43’W, 36◦48’N, and 151 m elevation). The crops grows in a multispan “Parral-type” greenhouse (Figure 5); [18]. The greenhouse has a surface of 877 m2(37.8 ×23.2 m), polyethylene cover, automated ventilation [27] with lateral windows in the northern and southern walls, flap roof window in each span, mesh-protected anti-trips “bionet” of 20 ×10 thickness, and night heating applied with a 95 kW hot air heater that is programmed to maintain the minimumtemperature above14 ◦C. The greenhouse orientation is east–west with the crop rows aligned north–south. Cropping conditions and crop management are very similar to those in commercial greenhouses. Climatic parameters are continuously monitored within the greenhouse. Outside the greenhouse, a meteorological station was installed, in which air temperature, relative humidity, solar and photosynthetic active radiation (PAR), rain detector, wind direction, and velocity measurements were taken. The cover temperature sensors were located on the faces oriented to the east (two sensors), and west (two sensors). During the experiments, the inside climate variables were also taken, among which stand out: air temperature, and relative humidity with a ventilated psychrometer (model MTH-A1, ITC, Almeria, Spain), solar radiation with a pyranometer (model MRG-1P, ITC, Almeria, Spain), and Photosynthetic Sensors 2012,12 15259 radiation reaching a given depth in canopy ([2]; Equation (6)) was tried out, but with the same results, such results was the expected: VGRe(−CkXLAI )(6) where VGRis the global radiation reaching the crop (W m−2), −Ckis the extinction coefficient of radiation (unitless), and XLAI is the leaf area index (m2 cropm−2 soil). As noted above, LAI remains constant during a chosen day as the parameter (Ck), which means the regressor is constant during a day, depending exclusively on the radiation. This virtual sensor was also evaluated by using an estimated VDP as a function of the inside temperature and the relative humidity. Despite this estimation, the virtual sensors obtained with these two variables show no improvement compared with those calibrated only using VPD. In the end, the radiation and the vapor pressure deficit remain as the unique inputs. Table 3. Results of the nonlinear virtual sensor calibration: na is the parametric equation’s order of outputs; nb is the order of the input; and nk is the delay of the solar radiation (nkVGR)and of the vapor pressure deficit (nkVV P D ). Virtual sensor na nb nkVGRnkVV P D NonARX2230 2 2 4 1 NonARX4430 4 4 4 1 In order to obtain the number of terms in the regressors, it is necessary to obtain a model to try out different combinations. In addition to the number of terms obtained, the nonlinearities were estimated through different possibilities: wavelet network, tree partitions, and sigmoid network. Of all these tested ways to obtain the nonlinearities, the wavelet network gave the best fit joining a nonlinear block and a linear block. Table 3shows the parameters of the resulting virtual sensor. This virtual sensor was obtained with prediction aim. 5.3. Virtual Sensors Validation All available data (more than one million for each variable) has been used for the validation. In total, nine different spring–summer and autumn–winter seasons were used. Figures 9,10 and 11 show an example of the results obtained in the validation process of the grey-box, as well as the linear and nonlinear dynamic virtual sensors in the different cycles. The validation of the virtual sensors can be seen in Figure 9. The presence of delay is demonstrated in Figure 10. In the end, Figure 11 samples a day detailed with the three virtual sensors. In some instances, the system dynamics is not well captured by the virtual sensors, as happens in Figure 10. This is caused by the difficulty in calculating the transpiration by using the microlysimeter. Furthermore, as Figure 9shows, the transpiration behaves similarly to the sunlight: rising in the morning, lowering in the afternoon, and remaining almost constant at night. The grey-box virtual sensor shows good dynamics. A small overestimation exists in the start and the end of the crop seasons but does not reach a significant level, with which in the first analysis it could be concluded that a correlation would not be necessary, although it is recommended. Furthermore, Sensors 2012,12 15260 Figure 10 shows the delay phenomenon, which characterizes the advancement of the virtual sensor dynamics over the underlying physics, as other authors have asserted ([6,29]). Figure 9. Five days in 2005 spring. Figure 10. One full day with delay in the autumn–winter of 2007–2008. The figure starts at midnight. Sensors 2012,12 15261 Figure 11. One full day in 2008 spring. One of the characteristics of the calibration of linear and nonlinear black-box virtual sensor is that the first validation is done during the identification, since the trial and error processes must be done to choose the virtual sensor. Figures 9,10 and 11 show that both black-boxes follow the dynamics of the tomato crop transpiration, presenting some adjustment problems in nocturnal areas. The problem of night setting is not of interest because it is sought. The night transpiration estimated through the virtual sensor obtains values higher that the pseudo-physical sensor. This overestimation is remarkable, but it is not very important for the final result, due to the fact that night transpiration measurements are very low in comparison with the daylight ones. The transpiration is so low, in fact, that it is difficult to take it with the microlysimeter, due to its 1 g sensor precision. Moreover, this dynamic virtual sensor does not show the anticipation phenomenon, as is evident in the static virtual sensor. It is the resistance of the plant transpiration resulting in a delay of about the same processes that produce it, graphically shown as a delayed action on transpiration estimated. For all the seasons included in this work (from the autumn in 2004 to the autumn in 2008–2009), the virtual sensor’s goodness was obtained (the calibration seasons are marked with ∗in the tables). This goodness for a data series is calculated through the minimum mean square error (MMSE). In all cases, the dynamic virtual sensor obtained good results (MMSE<6%), as can be seen in the three validation errors (see Table 4and Figure 12). Tables 5,6and 7show a full review of the errors of each model. Sensors 2012,12 15262 Table 4. Virtual sensor error comparison, in g m−2min−1. Season Samples Grey-box Linear Black-box Nonlinear Black-box E interval A-w 2004–2005 104,831 0.395 0.316 0.305 [0,9.40] S-s 2005 ∗125,704 0.355 0.320 0.310 [0,12.87] A-w 2005–2006 164,790 0.485 0.308 0.296 [0,9.60] S-s 2006 89,000 0.416 0.332 0.358 [0,10.27] A-w 2006–2007 104,817 0.457 0.440 0.421 [010.73] S-s 2006 149,807 0.543 0.421 0.428 [0,11.33] A-w 2007–2008 123,657 0.557 0.380 0.364 [0,10.30] S-s 2008 132,654 0.493 0.418 0.396 [0,11.90] A-w 2008–2009 106,855 0.472 0.434 0.456 [0,10.45] Figure 12. Virtual sensor error comparison, minimum mean square error in g m−2min−1. Table 5. Grey-virtual sensor error calculation in g m−2min−1. Season Samples Max error Mean Error σ λE interval A-w 2004–2005 104,831 2.12 0.395 0.491 [0,9.40] S-s 2005 ∗125,704 3.65 0.355 0.500 [0,12.87] A-w 2005–2006 164,790 2.79 0.485 0.740 [0,9.60] S-s 2006 89,000 3.01 0.416 0.747 [0,10.27] A-w 2006–2007 104,817 2.86 0.457 0.598 [0,10.73] S-s 2007 149,807 3.28 0.543 0.812 [0,11.33] A-w 2007–2008 123,657 2.92 0.557 0.602 [0,10.30] S-s 2008 132,654 3.44 0.493 0.612 [0,11.90] A-w 2008–2009 106,855 3.16 0.472 0.581 [0,10.45] Sensors 2012,12 15263 Table 6. Empirical (black-box) virtual sensors error calculation in g m−2min−1. Season Samples Max error Mean Error σ λE interval A-w 2004–2005 104,831 1.42 0.305 0.398 [0,9.40] S-s 2005 ∗125,704 2.56 0.310 0.406 [0,12.87] A-w 2005–2006 164,790 2.41 0.296 0.392 [0,9.60] S-s 2006 89,000 2.21 0.358 0.607 [0,10.27] A-w 2006–2007 104,817 2.75 0.421 0.576 [010.73] S-s 2006 149,807 1.98 0.428 0.639 [0,11.33] A-w 2007–2008 123,657 2.02 0.364 0.523 [0,10.30] S-s 2008 132,654 2.21 0.396 0.498 [0,11.90] A-w 2008–2009 106,855 2.69 0.434 0.512 [0,10.45] Table 7. Nonlinear virtual sensor minimum mean square error in g m−2min−1. Season Samples Max error Mean Error σ λE interval A-w 2004–2005 104,831 1.21 0.316 0.387 [0,9.40] S-s 2005 ∗125,704 2.33 0.320 0.407 [0,12.87] A-w 2005–2006 164,790 2.19 0.308 0.410 [0,9.60] S-s 2006 89,000 2.42 0.332 0.601 [0,10.27] A-w 2006–2007 104,817 2.75 0.440 0.579 [010.73] S-s 2006 149,807 1.96 0.421 0.657 [0,11.33] A-w 2007–2008 123,657 2.02 0.380 0.539 [0,10.30] S-s 2008 132,654 2.20 0.418 0.496 [0,11.90] A-w 2008–2009 106,855 2.40 0.456 0.519 [0,10.45] 6. Conclusions Significantly, the research was conducted over three years, for a total of eight cycles of cultivation in those which took the different climatic and physiological variables of a tomato crop. All data were taken at intervals of one minute to give approximately over one million input data for each variable. The main objective, as noted at the beginning of this paper, is to implement the virtual transpiration sensor of tomato crops for the design of irrigation controllers. For this occasion, we had a measurement system based on transpiration weighing the difference that occurs on a scale from one moment to the next, and to measure water loss with a good approximation. The next task was to seek a virtual sensor that would replace the system based on the weight difference caused by the loss of water by the crop. One was based on sensors that are typically installed in greenhouses: temperature, humidity, and solar radiation. The aim is to reduce total installation costs and to avoid the constant maintenance that the scales require. After preliminary assessment of some of these virtual sensors, Nonlinear ARX had a better fit and, in the end, was the best election for the irrigation virtual sensor proposed. This virtual sensor had good Sensors 2012,12 15264 results in the calibration and validation of the virtual sensor. An average error of 5%, for all cycles taken, shows how the choice of this virtual sensor was successful. However, it also presents some problems, such as the overestimation at night which occurs. On the other hand, linear and nonlinear black-box virtual sensors have demonstrated the absence of a advance phenomenon. It was translated to an advancement of the dynamics of the estimated values for real ones. System identification techniques were chosen to obtain a dynamic virtual sensor. Matlabrsoftware package was a good option to work with the identification techniques. A large number of different nonlinear dynamic virtual sensors structures was tested. The proposed dynamic for virtual sensors require only two inputs, global radiation and vapor pressure deficit, thus eliminating the inclusion of the XLAI . This virtual sensor presents the possibility of using modern control algorithms that cannot be used with the grey-box sensor. As summary: •Grey-box virtual sensor has a good fixing as advantage, but some disadvantages such as the overestimation at different moments of the year, a higher final error result, and the advance phenomenon. •The black-box virtual sensors obtain better results and also allows the elimination of the grey-box problems: advance phenomenon and overestimation. An overestimation only appears during nocturnal periods. •Nonlinear black-box had the best results. This paper has dealt with the transpiration from an industrial point of view, as a process in which there are entries and exits. The crop itself, and some aspects of the climate inside the greenhouse, are considered disturbances affecting the dynamics. This shifts the focus away from the strictly agronomic, agronomy classic, which studies the exchanges that occur in the greenhouse by static virtual sensors based on fundamental principles, without including the system dynamic effect. Acknowledgments This work is framed within the “Controlcrop” Project, PIO-TEP-6174, supported by the Ministry of Economy, Innovation and Science Government of Andalusia (Spain), and by the National Plan Projects DPI2010-21589-C05-04, and DPI2011-27818-C02-01 of the Spanish Ministry of Science and Innovation and FEDER funds. The authors are grateful for the invaluable contributions of the University of Almer´ıa Investigation Group of Automatic, Electronic and Robotic (TEP197), the University of Seville Investigation Department of Systems Engineering and Automatic, the Cajamar Foundation Experimental Station “Las Palmerillas”, and the Agrifood Campus of International Excellence (ceiA3). References 1. Stanghellini, C. Transpiration of Greenhouse Crops. Ph.D. Thesis, Wageningen University, Wageningen, The Netherlands, 1987. Sensors 2012,12 15265 2. Baille, M.; Baille, L.; Laury, J.C. A simplified model for predicting evapotranspiration rate of nine ornamental species vs. climate factors and leaf area. Sci. Hort. Amsterdam 1994,59, 217–232. 3. Jemaa, R.; Boulard, T.; Baille, A. Some results on water and nutrient consumption of a greenhouse tomato crop grown in rockwool. Acta Horticul. 1995,408, 137–146. 4. Boulard, T.; Wang, S. Greenhouse crop transpiration simulation from external climate conditions. Agric. For. Meteorol. 2000,100, 25–34. 5. Montero, J.; Ant´on, A.; Munoz, P.; Lorenzo, P. Transpiration from geranium grown under high temperatures and low humidities in greenhouses. Agric. For. Meteorol. 2001,107, 323–332. 6. Medrano, E.; Lorenzo, P.; S´anchez-Guerrero, M.C.; Montero, J.I. Evaluation and modelling of greenhouse cucumber-crop transpiration under high and low radiation conditions. Sci Hort. Amsterdam 2005,105, 163–175. 7. Suay, R.; Mart´ınez, P.; Roca, D.; Mart´ınez, M.; Herrero, J.; Ramos, C. Measurement and estimation of transpiration of a soilless rose crop and application to irrigation management. Acta Horticul. 2003,614, 625–630. 8. Voogt, W.; Kipp, J.; de Graaf, R.; Spaans, L. A fertigation model for glasshouse crops grown in soil. Acta Horticul. 2000,537, 495–502. 9. Schmidt, U.; Exarchou, E. Controllingof irrigation systems of greenhouse plants by usingmeasured transpiration sum. Acta Horticul. 2000,537, 487–494. 10. Li, H.; Braun, J.E. Decoupling features and virtual sensors for diagnosis of faults in vapor compression air conditioners. Int. J. Refrig. 2007,30, 546–564. 11. Ponsart, J.; Theilliol, D.; Aubrun, C. Virtual sensors design for active fault tolerant control system applied to a winding machine. Control Eng. Pract. 2010,18, 1037–1044. 12. Thies, S.; Tresadern, P.; Kenney, L.; Howard, D.; Goulermas, J.; Smith, C.; Rigby, J. A virtual sensor tool to simulate accelerometers for upper limb FES triggering. J. Biomech. 2006, 39(suppl. 1), S80. 13. Heredia, G.; Ollero, A. Virtual sensor for failure detection, identification and recovery in the transition phase of a morphing aircraft. Sensors 2010,10, 2188–2201. 14. Sanchez, J.; Visioli, A.; Dormido, S. A two-degree-of-freedom PI controller based on events. J. Process Control 2011,21, 639–651. 15. Jolliet, O.; Bailey, B. The effect of climate on tomato transpiration in greenhouses: measurements and models comparison. Agric. For. Meteorol. 1992,58, 43–62. 16. Saugier, B.; Granier, A.; Pontailler, J.Y.; Dufrˆene, E.; Baldocchi, D.D. Transpiration of a boreal pine forest measured by branch bag, sap flow and micrometeorological methods. Tree Physiol. 1997,17, 511–519. 17. Van Bavel, C.H.M.; Nakayama, F.S.; Ehrler, W.L. Measuring transpiration resistance of leaves. Plant Physiol. 1964,40, 535–540. 18. Ram´ırez, A.; Rodr´ıguez, F.; Berenguel, M.; Heuvelink, E. Calibration and validation of complex and simplified tomato growth models for control purposes in the southeast of Spain. Acta Horticul. 2004,654, 147–154. 19. Monteith, J. Evaporation and environment. In The State and Movement of Water in Living Organism; Cambridge University Press: Cambridge, UK, 1965; Volume 19, pp. 205–234. Sensors 2012,12 15266 20. Stanghellini, C. Evapotranspiration in greenhouses with special reference to mediterranean conditions. Acta Horticul. 1995,335, 295–304. 21. Yoo, S.H.; Choi, J.Y.; Jang, M.W. Estimation of design water requirement using FAO Penman-Monteith and optimal probability distribution function in South Korea. Agric. Water Manag. 2008,95, 845–853. 22. Ruiz-Arahal, M.; Berenguel, M.; Rodr´ıguez, F. T´ ecnicas de predicci´ on con aplicaciones en la ingenier´ ıa; Secretariado de publicaciones: Universidad de Sevilla, Seville, Spain, 2006; pp. 117–119. 23. Ljung, L. System Identification: Theory for the User; 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, USA, 1999. 24. Ljung, L. System Indentification Toolbox TM 7 Getting Started Guide; MathWorks: Natick, MA, USA, 2010. 25. Guzman, J.; Rivera, D.; Dormido, S.; Berenguel, M. An interactive software tool for system identification. Adv. Eng. Softw. 2012,45, 115–123. 26. Ljung, L.; Glad, T. Modeling of Dynamic Systems; Prentice Hall: Englewood Cliffs, NJ, USA, 1994. 27. Berenguel, M.; Rodr´ıguez, F.; Guzm´an, J.; Lacasa, D.; P´erez-Parra, J. Greenhouse diurnal temperature control with natural ventilation based on empirical models. Acta Horticul. 2006, 719, 57–64. 28. Rodr´ıguez, F.; Guzm´an, J.; Berenguel, M.; Arahal, M. Adaptive hierarchical control of greenhouse crop production. Int. J. Adapt. Control Signal Process. 2008,22, 180–187. 29. G´azquez, J.; L´opez, J.; Baeza, E.; P´arez-Parra, J.; Fern´andez, M.; Baille, A.; Gonz´alez-Real, M. Effects of vapour pressure deficit and radiation on the transpiration rate of a greenhouse sweet pepper crop. Acta Horticul. 2008,797, 259–265. 30. Seginer, I. The Penman–Monteith Evapotranspiration Equation as an Element in Greenhouse Ventilation Design. Biosyst. Eng. 2002,82, 423–439. 31. Jones, J.; Dayan, E.; Allen, L.V.; Challa, H. A dynamic tomato growth and yield model (Tomgro). Trans. ASAE 1991,34, 663–672. 32. Jones, J.W.; Kening, A.; Vallejos, C.E. Reduced state-variable tomato growth model. Trans. ASAE 1999,42, 255–265. 33. Gallardo, M.; Thompson, R.; Rodriguez, J.; Rodriguez, F.; Fernandez, M.; Sanchez, J.; Magan, J. Simulation of transpiration, drainage, N uptake, nitrate leaching, and N uptake concentration in tomato grown in open substrate. Agric. Water Manag. 2009,96, 1773–1784. c 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).