Unscented Kalman filter for learning of a solar dryer and a greenhouse

Abstract

An unscented Kalman filter can be applied for the experimental learning of the solar dryer for oranges drying and the greenhouse for crop growth to know better the processes and to improve their performances. The contributions of this document are: a) an unscented Kalman filter is designed for the learning of nonlinear functions, b) the unscented Kalman filter is applied for the experimental learning of the two mentioned processes.

Keywords

Unscented Kalman filter greenhouse solar dryer experimental learning

1 Introduction

Due to the population growth and the decrease of available spaces to treat food, it is raising the necessity to carry out research to improve some processes related to food: a solar dryer for oranges drying and a greenhouse for crop growth. Many agricultural products require a post-harvest drying process to properly preserve the products until they reach the consumer centers [1]. The use of greenhouses has become a measure taken for the controlled harvest of various vegetables [2]. Agricultural products offer the alternative to farmers when the price grows caused by overpopulation, and the learning is an important method in the current research of this issue.

The use of solar dryers, where products are directly exposed to the sun on the ground is one of oldest uses of solar energy and is still one of the agricultural processes most commonly used in many countries worldwide. Product quality is affected by the dust and insect contamination. In industrialized regions, the low cost of fuel allowed for several decades, the development of artificial drying processes on a large scale based on the use of fuels. In recent years, the shortage and price increase of fuel has sparked a new interest in the use of solar dryers, which tries to develop various techniques to solve the mentioned problems. Knowledge of solar radiation is essential to calculate various performance levels associated with solar power processes like solar dryers, solar water heaters and photovoltaic processes [3].

The employment of solar dryers through which dry air flows as an effect of natural convection can be an option for solving the problems of agricultural drying in small rural locations. These dryers can be constructed with simple materials and low skilled work, where the air flows by the pressure variations without an electric fan, and pressure variations in the air are caused by differences in temperature and humidity of the drying air and ambient air.

Several solar devices for drying food have been built and tested. In [4], a solar dryer is developed for vegetables by using natural convection to study the performance of drying in tomato and carrot, where a 50% saving in drying time is reported using a solar dryer at an average temperature of 60^oC. In [5], the behavior of dried slices of lemon by using a solar dryer combining a photovoltaic module and a dehumidifier process is studied to remove moisture in the fresh lemon slices; nevertheless, the photovoltaic module increases the total cost of drying by increasing the drying speed, results are compared to a hot air drying at 60°C.

For the control implementation of the main variables related to a greenhouse, it is necessary to have a model to represent the dynamics of the process. The model of [2, 6] is employed because it considers the most significant environmental aspects of a greenhouse.

There is some research about algorithms for the learning of nonlinear processes. In [7, 8], the deep learning method is studied. Machine learning approaches are discussed in [9 –12]. In [13, 14], unsupervised learning techniques are addressed. Even than there are several algorithms for the learning of nonlinear processes, it should be interesting to study filters as one alternative for the learning in nonlinear processes because the filters could learn faster than other algorithms.

There is some research about filters for the learning of nonlinear processes. The extended Kalman filter is considered in [15 –17]. In [18 –20] the unscented Kalman filter is focused. It shows that the filters also have been highly studied for the learning of nonlinear processes. Consequently, a filter could be a good option for the learning of nonlinear processes.

The main difference between the extended Kalman filter and unscented Kalman filter is that the unscented Kalman filter can be designed by a deterministic procedure while the extended Kalman filter can be designed by a stochastic procedure. Hence, the unscented Kalman filter could be well applied to deterministic nonlinear processes while the extended Kalman filter could be well applied to stochastic nonlinear processes. Since this study is focused in deterministic nonlinear processes, the unscented Kalman filter is the best option.

In this document, the unscented Kalman filter is employed for the experimental learning of two processes by the next steps:

The unscented Kalman filter is applied for the experimental learning of nonlinear processes. The unscented Kalman filter as the same as the extended Kalman filter represent alternatives for the minimization of a cost function which could learn faster than other algorithms [15, 16].

The unscented Kalman filter is applied for the experimental learning of the next two processes: a solar dryer for oranges drying and a greenhouse for crop growth. a) A prototype of a solar dryer with a tracking process has been built and tested. Twenty-five drying tests are conducted from July to October 2015, where those tests are performed from 10:00 hours to 16:00 hours. b) A model of a greenhouse is presented and an approximation method is applied with real data from tests to validate the model and learning.

2 Approximation model

We consider the nonlinear model: $y_{i} = f (x_{i})$ (1) $x_{i} = {[\begin{matrix} x_{1 i} & x_{2 i} & \dots & x_{di} \end{matrix}]}^{T}$ represents a vector of examples with size d, x_i also represents the inputs, d represents the number of inputs, y_i represents the output, f (·) represents a nonlinear function. i represents the examples which vary from i = 1 to i = m - 1, and m - 1 represents the number of examples.

The regression model proposed to approximate the nonlinear model (1) is ${\hat{y}}_{i} = {wx}_{i} + b$ with examples from i = 1 to i = m - 1, $w = [\begin{matrix} w_{0} & w_{1} & w_{2} & \dots & w_{d} \end{matrix}]$ represents the weights and $x_{i} = {[\begin{matrix} x_{0} & x_{1 i} & x_{2 i} & \dots & x_{di} \end{matrix}]}^{T}$ represents the examples. If w₀ = b and x₀ = 1, then the regression model becomes to: ${\hat{y}}_{i} = {wx}_{i}$ (2) $w = [\begin{matrix} w_{0} & w_{1} & w_{2} & \dots & w_{d} \end{matrix}]$ represents the weights and $x_{i} = {[\begin{matrix} x_{0} & x_{1 i} & x_{2 i} & \dots & x_{di} \end{matrix}]}^{T}$ represents the examples.

We define the error e_i as: $e_{i} = {\hat{y}}_{i} - y_{i} = {wx}_{i} - y_{i}$ (3)y_i represents the output (1), ${\hat{y}}_{i}$ represents the approximation of the regression model (2).

We define the cost J as: $J = \sum_{i = 0}^{m - 1} e_{i}^{2}$ (4)e_i represents the error (3).

To reach the best approximation ${\hat{y}}_{i}$ for the output y_i, we require the minimum of error $e_{i}^{2}$ , we will find the weights w to reach the minimum of the cost J, we represent it as: $\frac{\partial J}{\partial w} = 0$ (5) We make operations to reach (5) as: $\begin{matrix} \frac{\partial J}{\partial w} = \overset{m - 1}{\sum_{i = 0}} \frac{\partial J}{\partial e_{i}} \frac{\partial e_{i}}{\partial w} = 0 \\ \Rightarrow \overset{m - 1}{\sum_{i = 0}} 2 e_{i} (x_{i}) = 0 \\ \Rightarrow \overset{m - 1}{\sum_{i = 0}} ({\hat{y}}_{i} - y_{i}) (x_{i}) = 0 \\ \Rightarrow \overset{m - 1}{\sum_{i = 0}} ({wx}_{i} - y_{i}) (x_{i}) = 0 \\ \Rightarrow w \overset{m - 1}{\sum_{i = 0}} x_{i}^{T} x_{i} = \overset{m - 1}{\sum_{i = 0}} y_{i} x_{i} \\ \Rightarrow w = {[\overset{m - 1}{\sum_{i = 0}} x_{i}^{T} x_{i}]}^{- 1} \overset{m - 1}{\sum_{i = 0}} y_{i} x_{i} \end{matrix}$ (6) We find the weights w as: $w = {[\overset{m - 1}{\sum_{i = 0}} x_{i}^{T} x_{i}]}^{- 1} \overset{m - 1}{\sum_{i = 0}} y_{i} x_{i}$ (7)

2.1 Unscented Kalman filter

We use the unscented Kalman filter to minimize the cost. We use small steps in the direction of the minimum of the cost.

To reach the unscented Kalman filter, we use the weights w of (7). We define the term G_m-1 as: $G_{m - 1} = \sum_{i = 0}^{m - 1} x_{i}^{T} x_{i}$ (8) We substitute G_m-1 of (8) in (7) to reach the weights w in the iterations m and m + 1: $\begin{matrix} w_{m} = G_{m - 1}^{- 1} \overset{m - 1}{\sum_{i = 0}} y_{i} x_{i} \\ w_{m + 1} = G_{m}^{- 1} \overset{m}{\sum_{i = 0}} y_{i} x_{i} \end{matrix}$ (9) We reach the next summation: $\overset{m}{\sum_{i = 0}} y_{i} x_{i} = \overset{m - 1}{\sum_{i = 0}} y_{i} x_{i} + y_{m} x_{m}$ (10) We make the mathematical operations, we get: $\begin{matrix} \overset{m}{\sum_{i = 0}} y_{i} x_{i} = \overset{m - 1}{\sum_{i = 0}} y_{i} x_{i} + y_{m} x_{m} \\ = G_{m - 1} w_{m} + y_{m} x_{m} \\ = [G_{m - 1} w_{m} + y_{m} x_{m} \\ + x_{m}^{T} x_{m} w_{m} - x_{m}^{T} x_{m} w_{m}] \\ = [(G_{m - 1} + x_{m}^{T} x_{m}) w_{m} \\ + x_{m} (y_{m} - w_{m} x_{m})] \\ = G_{m} w_{m} - x_{m} e_{m} \\ \Rightarrow G_{m}^{- 1} \overset{m}{\sum_{i = 0}} y_{i} x_{i} = w_{m} - G_{m}^{- 1} x_{m} e_{m} \\ \Rightarrow w_{m + 1} = w_{m} - G_{m}^{- 1} x_{m} e_{m} \end{matrix}$ (11)

Then, the unscented Kalman filter is the adapting of the weights w_m+1 represented as: $w_{m + 1} = w_{m} - P_{m} x_{m} e_{m}$ (12) $P_{m} = G_{m}^{- 1}$ represents the speed which is not part of the model, but it is employed to find a solution, if P_m is big we do not reach a convergence, and if P_m is small we will reach a slow convergence. w_m+1 represents the weights (2), (7) for iteration m + 1, w_m represents the weights (2), (7) for iteration m, x_m represents the examples (2) for iteration m, and e_m represents the error (3) for iteration m.

From the definition (8), we make operations to reach: $\begin{matrix} G_{m} = \overset{m - 1}{\sum_{i = 0}} x_{i}^{T} x_{i} + x_{m}^{T} x_{m} \\ = G_{m - 1} + x_{m}^{T} x_{m} \end{matrix}$ (13) From (13), $P_{m} = G_{m}^{- 1}$ , and we use the inverse matrix lemma to reach: $\begin{matrix} G_{m}^{- 1} = {(G_{m - 1} + x_{m}^{T} x_{m})}^{- 1} \\ = G_{m - 1}^{- 1} - \frac{1}{1 + x_{m}^{T} G_{m - 1}^{- 1} x_{m}} G_{m - 1}^{- 1} x_{m} x_{m}^{T} G_{m - 1}^{- 1} \\ \Rightarrow P_{m} = P_{m - 1} - \frac{1}{1 + x_{m}^{T} P_{m - 1} x_{m}} P_{m - 1} x_{m} x_{m}^{T} P_{m - 1} \end{matrix}$ (14) Then, we find the equation to update the speed: $P_{m} = P_{m - 1} - \frac{1}{1 + x_{m}^{T} P_{m - 1} x_{m}} P_{m - 1} x_{m} x_{m}^{T} P_{m - 1}$ (15) $P_{m} = G_{m}^{- 1}$ .

From (2), (3), (25), and (28), the unscented Kalman filter is [18 –20]: $\begin{matrix} {\hat{y}}_{m} = w_{m} x_{m} \\ e_{m} = {\hat{y}}_{m} - y_{m} \\ P_{m} = P_{m - 1} - \frac{1}{1 + x_{m}^{T} P_{m - 1} x_{m}} P_{m - 1} x_{m} x_{m}^{T} P_{m - 1} \\ w_{m + 1} = w_{m} - P_{m} x_{m} e_{m} \end{matrix}$ (16)P_m represents the speed, e_m represents the error, y_m represents the output, ${\hat{y}}_{m}$ represents the approximation output, x_m represents the examples, and w_m represents the weights.

3 Case 1: Orange solar drying

3.1 Model of the solar drying

The ability to approximate the behavior of food drying using solar energy can help producers to change parameters in their production methods. The orange, is a fruit rich in vitamin C and essential oils, its pulp is typically formed of eleven segments of juice filled with flavor goes from sweet to acid. One of the varieties of orange grown in Mexico is the valence. In 2014 citrus harvest was four million tons, which placed Mexico as the fourth largest producer in the world. A solar dryer with tracking process was built for this research. Several tests were conducted in July and October 2015. A model is proposed and compared with real data. A model is applied to approximate the behavior of the drying process.

The most abundant component in food is water. Determining the moisture content in food is one of the most important parts of the model which indicates the amount of water involved in the composition thereof. Values of moisture content in all samples became dimensionless, assuming the final moisture content is the equilibrium moisture content.

The moisture ratio M_R, as dimensionless moisture content can be calculated according to: $M_{R} = \frac{M_{t} - M_{e}}{M_{0} - M_{e}}$ (17)

M₀, M_t and M_e are the initial moisture content, the moisture content, and the equilibrium moisture content, respectively. The equilibrium moisture content is extremely small compared to the moisture content, then the equilibrium moisture content can be omitted. Different models which represent this process from experimental data are shown in Table 1.

Table 1

Models for the drying process

Model	Equation
Page [21]	MR = e^-ktⁿ
Newton [22]	MR = e^-kt
Modified Page [23]	MR = e^-(kt)ⁿ
Exponential [24]	MR = ae^-kt
Logarithmic [25]	MR = ae^-kt + c

One of the models used to determine the moisture content of agricultural products during the drying process is the exponential model, which from Newton’s Law of Cooling, we have: $\dot{Q} = α A (T - T_{a})$ (18)

α is the exchange heat coefficient, A is the product area, T is the product temperature, and T_a is the ambient temperature. Furthermore, if a small amount of heat dQ is transferred between a process with mass m and its environment, the process undergoes a small change of temperature dT, then the specific heat capacity is defined as c, the model is: $c = \frac{1}{m} \frac{dQ}{dT}$ (19)

Then, $mc \frac{dT}{dt} = α A (T - T_{a})$

Simplifying, $\dot{T} = k (T (t) - T_{a})$ (20)

In which $k = \frac{α A}{mc}$ is a proportionality constant. As it is assumed that the product is cooling, the process must satisfy T > T_a and k < 0. If T₀ = T (t = 0) ≠ T_a, then: $\dot{T} = k (T (t) - T_{a}) \neq 0$

Consequently, there are no temperature variation. If k > 0 and T₀ > T_a then $\dot{T} > 0$ , then the temperature increases and the constant k is negative, rewriting (20) for this case: $\dot{T} = - k (T (t) - T_{a})$

Thus, (20) with initial conditions is rewritten as: $\dot{T} = - k (T (t) - T_{a}), T (0) = T_{0}$ (21)

A solution of (21) is a function T (t) which lets its satisfaction. First, we must find a function T (t) as a solution of $\dot{T} = - k (T (t) - T_{a})$ . We must find functions whose derivative is the product of -k and T (t) - T_a. If: $P (t) = T (t) - T_{a}$

Then, $\dot{P} = \dot{T}$

We must be able to find a solution of (21) if we a find solution of $\dot{P} = - kP (t)$ , and vice versa. If P (t) is a solution of $\dot{P} = - kP (t)$ then $\dot{P} (t) = - kP (t)$ . But $\dot{P} (t) = \dot{T} (t)$ , $\dot{T} (t) = - kP (t) = - k (T (t) - T_{a})$ , then T (t) is a solution of (21), and vice versa, if T (t) is a solution of $\dot{P} = - kP (t)$ , then as $\dot{P} = \dot{T}$ we have $\dot{P} (t) = \dot{T} (t) = - k (T (t) - T_{a}) = - kP (t)$ .

Now, it is easier to find a solution of $\dot{P} = - kP (t)$ because we need to find a function whose derivative match. We know that a function that satisfies this property is the exponential function. Consequently, if x (t) = e^t then $\dot{x} (t) = e^{t} = x (t)$ . This means that x (t) = e^t is a solution of $\dot{x} = x (t)$ .

If we define x (t) = e^at we have $\dot{x} (t) = {ae}^{at} = ax (t)$ , then: $P (t) = e^{- kt}$ (22)

P (t) is a solution of $\dot{P} = - kP (t)$ . Thus $\dot{P} = - {ke}^{- kt} = - kP (t)$ . This is not a unique solution because if P (t) = ce^-kt where c is a constant, then $\dot{P} (t) = c (- {ke}^{- kt}) = - k ({ce}^{- kt}) = - kP (t)$ . So $\dot{P} (t) = - kP (t)$ . Then we conclude that: $M_{R} = P (t) = {ce}^{- kt}$ (23)

To evaluate the fit of the model with the experimental data, the next statistical coefficients is used: chi-squared distribution χ², mean squared error MSE, and root mean square error RMSE. Those are represented by: $χ^{2} = \frac{\sum_{i = 1}^{N} {(M_{R_{exp, i}} - M_{R_{pre, i}})}^{2}}{N - z}$ (24) $MSE = \frac{1}{N} \sum_{i = 1}^{N} {(M_{R_{pre, i}} - M_{R_{exp, i}})}^{2}$ (25) $RMSE = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(M_{R_{pre, i}} - M_{R_{exp, i}})}^{2}}$ (26)

M_{R
_exp,i} represents the moisture ratio which is gotten experimentally, M_{R
_pre,i} are the approximations made by models, N is the number of data, and z is the number of constants.

The solar radiation that reach the ground on a horizontal surface is formed by direct radiation and diffuse radiation which together form the global solar radiation, their units are $\frac{W}{m^{2}}$ [3]. To obtain the global solar radiation we have used a pyranometer EPPLEY 8-48, which has a constant of 9.60 × 10 $^{- 6} \frac{W}{m^{2}}$ .

3.2 Experimental prototype of solar drying

The prototype is built and presented in Fig. 1 where 4 are trays, 5 is the mechanical elevator, 6 are covers, 7 is the pivot base, and 8 are castors. The solar dryer with tracking process is a direct type dryers which operates by natural convection. The dimensions of the drying chamber are 300mm × 600mm × 1300mm, and it contains five trays constructed of 22 gauge galvanized sheet. On the upper surface it has two decks of clear glass with thick of 3mm, with a spacing between them of 50mm. The inclination of the drying chamber is 30^oC to the horizontal plane and is oriented south-north to capture maximum solar energy.

Fig. 1

Grading process.

The experimental test is: the orange used for drying tests are orange-valence type, which are being slicedwith about 5mm thickness and placed in each of the five trays on the drying chamber.

Each of the tests are conducted for a period of 12.5 hours, the hours in which tests are conducted were from 7:00 hours to 19:30 hours. In this period of time the data is gotten at intervals of 5 seconds, four temperatures are gotten at the same time: the ambient temperature, inlet temperature of the drying chamber, temperature inside of the drying chamber, and outlet temperature of the drying chamber. Also, global radiation is recorded. Four temperatures are considered in the solar dryer, T₁ is the inlet temperature of the drying chamber, T₂ is the outlet temperature of the drying chamber, T₃ is the temperature inside the drying chamber, and T_a is the ambient temperature, which are shown in Figs. 2 and 3.

Fig. 2

Temperatures in the solar dryer prototype with tracking process.

Fig. 3

Experimental test with slices of orange inside.

The experimental test is: the orange used for drying tests are orange-valence type, which are being slicedwith about 5mm thickness and placed in each of the five trays on the drying chamber.

To evaluate moisture removal, the product mass is gotten in 10 seconds intervals. The average moisture content corresponding to the tests in July is of 55.746% and tests in October is of 52.906% of moisture removal. The data and results are presented in section Results.

4 Case 2: A greenhouse

4.1 Model of the greenhouse

For the model of crop growth, we represent the process as a model with four state variables, based on the work of [6]: x_d is the crop dry weight in [kg m^-2], x_c is the carbon dioxide concentration in [kg m^-3], x_T is the air temperature in [^oC], and x_h is the absolute humidity in [kg m^-3].

The representation of each of these variables is shown in (27), (29), (31), and (34). $\dot{x_{d}} = C_{α β} φ_{phot, c} - C_{resp, d} x_{d} 2^{(0.1 x_{T} - 2.5)}$ (27)

Defining C_αβ = C_αC_β as the yield factor in which C_α corresponds to a factor that converts assimilated carbon dioxide into absorbed sugar equivalent, C_β is a performance factor that counts synthesis and respiration losses during the conversion from carbohydrates to structure and whose value is between zero and one; as well as φ_phot,c which is the photosynthesis rate in [kg m^-2 s^-1] represented in (28): $\begin{matrix} φ_{phot, c} = [(1 - e^{- C_{pl, d} x_{d}}) \\ * \frac{C_{rad, phot} ν_{rad} (- C_{{co}_{2, 1}} x_{T}^{2} + C_{{co}_{2, 2}} x_{T} - C_{{co}_{2, 3}}) (x_{c} - c_{Γ})}{C_{rad, phot} ν_{rad} + (- C_{{co}_{2, 1}} x_{T}^{2} + C_{{co}_{2, 2}} x_{T} - C_{{co}_{2, 3}}) (x_{c} - c_{Γ})}] \end{matrix}$ (28)

Also C_pl,d is the effective greenhouse surface in [m² kg^-1], C_rad,phot is the light use efficiency in [kg J^-1], ν_rad is the solar radiation outside the greenhouse in [W m^-2], C_{co
_2,1} in [m s^-1 ^oC^-2], C_{co
_2,2} in [m s^-1 ^oC^-1], and C_{co
_2,3} in [m s^-1] represents the temperature influence on photosynthesis, c_Γ is the carbon dioxide compensation point in [kg m^-3], C_resp,d is the respiration rate expressed in terms of the amount of respired dry matter in [s^-1]. $\begin{matrix} \dot{x_{c}} = [\frac{1}{C_{cap, c}} \\ * (- φ_{phot, c} + C_{resp, c} x_{d} 2^{(0.1 x_{T} - 2.5)} + u_{c} - φ_{vent, c})] \end{matrix}$ (29)

Additionally, C_cap,c is the greenhouse volume in [m], C_resp,c is the respiration coefficient expressed in terms of the amount of carbon dioxide produced in [s^-1], u_c is the supply rate of carbon dioxide in [kg m^-2 s^-1], φ_vent,c is the mass exchange of carbon dioxide through the vents in [kg m^-2 s^-1], and is represented by (30). $φ_{vent, c} = (u_{v} + C_{leak}) (x_{c} - ν_{c})$ (30)

In which u_v is the ventilation rate through the vents in [m s^-1], C_leak is the leakage through the cover in [m s^-1], and ν_c is the carbon dioxide concentration outside the greenhouse in [kg m^-3]. $\dot{x_{T}} = \frac{1}{C_{cap, q}} (u_{q} - Q_{vent, q} + Q_{rad, q})$ (31)

C_cap,q is the heat capacity of the greenhouse air in [J m^-2 ^oC^-1], u_q is the energy supply by the heating process in [W m^-2], Q_vent,q is the energy exchange with the outdoor air due the ventilation and transmission through the cover in [W m^-2] governed by (32), Q_rad,q is the heat load due to the solar radiation in [W m^-2], and is expressed by (33). $Q_{vent, q} = (C_{cap, q, v} u_{v} + C_{ai, ou}) (x_{T} - ν_{T})$ (32)

In which C_cap,q,v is the heat capacity per volume unit of greenhouse air in [J m^-3 ^oC^-1], C_ai,ou represents the overall heat transfer through the cover in [W m^-2 ^oC^-1], and ν_T is the outside air temperature in [^oC]. $Q_{rad, q} = C_{rad, q} ν_{rad}$ (33)

C_rad,q is the heat load coefficient due to solar radiation. $\dot{x_{h}} = \frac{1}{C_{cap, h}} (φ_{transp, h} - φ_{vent, h})$ (34)

In which C_cap,h is the greenhouse volume in [m], φ_transp,h is the transpiration in [kg m^-2 s^-1] represented in (35), and φ_vent,h is the mass exchange of water vapor through the vents in [kg m^-2 s^-1] . $\begin{matrix} φ_{transp, h} = [(1 - e^{- C_{pl, d} x_{d}}) \\ * C_{v, pl, ai} (\frac{C_{v, 1}}{C_{R} (x_{T} + C_{T, abs})} e^{\frac{C_{v, 2} x_{T}}{x_{T} + C_{v, 3}}} - x_{h})] \end{matrix}$ (35)

The term $\frac{C_{v, 1}}{C_{R} (x_{T} + C_{T, abs})} e^{\frac{C_{v, 2} x_{T}}{x_{T} + C_{v, 3}}}$ represents the saturated water vapor content at temperature x_T in the upper region of the greenhouse in [kg m^-3], C_v,pl,ai is the mass transfer coefficient in [m s^-1], C_v,1 in [J m^-3], C_v,2 and C_v,3 in [^oC] are values that represent the saturation water vapor pressure in the greenhouse, C_R is the gas constant in [J K^-1 kmol^-1], C_T,abs represents the conversion of temperature from ^oC to ^oK, φ_vent,h is the mass exchange of water vapor through the vents in [kg m^-2 s^-1], which we represent by (36). $φ_{vent, h} = (u_{v} + C_{leak}) (x_{h} - ν_{h})$ (36)

ν_h is the humidity concentration outside the greenhouse in [kg m^-3].

If the state variables are defined as x₁ = x_c, x₂ = x_T, x₃ = x_h, x₄ = x_d, control inputs as u₁ = u_c, u₂ = u_v, u₃ = u_q, and functions as f_a = φ_phot,c, f_b = 2^(0.1x_T-2.5), f_c = φ_transp,h, then we rewrite (27), (29), (31), (34) as: $\begin{matrix} \dot{x_{1}} & = [\frac{1}{C_{cap, c}} (- f_{a} + C_{resp, c} x_{4} f_{b} - C_{leak} (x_{1} - ν_{c})) \\ + \frac{1}{C_{cap, c}} u_{1} - \frac{(x_{1} - ν_{c})}{C_{cap, c}} u_{2}] \\ \dot{x_{2}} & = [\frac{1}{C_{cap, q}} (Q_{rad, q} - C_{ai, ou} (x_{2} - ν_{T})) \\ - \frac{C_{cap, qv} (x_{2} - ν_{T})}{C_{cap, q}} u_{2} + \frac{1}{C_{cap, q}} u_{3}] \\ \dot{x_{3}} & = \frac{1}{C_{cap, h}} (f_{c} - C_{leak} (x_{3} - ν_{h})) - \frac{(x_{3} - ν_{h})}{C_{cap, h}} u_{2} \\ \dot{x_{4}} & = C_{α β} f_{a} - C_{resp, d} f_{b} x_{4} \end{matrix}$ (37)

With: $\begin{matrix} f_{a} = [(1 - e^{- C_{pl, d} x_{4}}) \\ * \frac{C_{rad, phot} ν_{rad} (- C_{{co}_{2, 1}} x_{2}^{2} + C_{{co}_{2, 2}} x_{2} - C_{{co}_{2, 3}}) (x_{1} - c_{Γ})}{C_{rad, phot} ν_{rad} + (- C_{{co}_{2, 1}} x_{2}^{2} + C_{{co}_{2, 2}} x_{2} - C_{{co}_{2, 3}}) (x_{1} - c_{Γ})}] \\ f_{b} = 2^{(0.1 x_{2} - 2.5)} \\ f_{c} = [(1 - e^{- C_{pl, d} x_{4}}) \\ * C_{v, pl, ai} (\frac{C_{v, 1}}{C_{R} (x_{2} + C_{T, abs})} e^{\frac{C_{v, 2} x_{2}}{x_{2} + C_{v, 3}}} - x_{3})] \\ Q_{rad, q} = C_{rad, q} ν_{rad} \end{matrix}$

Model parameters are listed in Table 2. Parameters of Table 2 represent the best values such that the model (37) represent in the best form a greenhouse, they were validated by experiments in [2, 6].

Table 2

Greenhouse parameters

Parameter	Parameter
C_resp,d = 2.65 × 10^-7s^-1	C_αβ = 0.544
C_ai,ou = 6.1Wm^-2oC^-1	C_rad,q = 0.2
c_Γ = 5.2 × 10^-5kgm^-3	C_T,abs = 273.15K
C_resp,c = 4.87 × 10^-7s^-1	C_pl,d = 53m²kg^-1
C_cap,q = 30000Jm^-2oC^-1	C_v,1 = 9348Jm^-3
C_cap,q,v = 1290Jm^-3oC^-1	C_v,2 = 17.4
Cco_2,1 = 5.11 × 10^-6ms^-1oC^-2	C_v3 = 239^oC
Cco_2,2 = 2.30 × 10^-4ms^-1oC^-2	C_v4 = 10998Jm^-3
Cco_2,3 = 6.29 × 10^-4ms^-1	C_cap,c = 4.1m
C_leak = 0.75 × 10^-4ms^-1	C_cap,h = 4.1m
x_1max = 2.75 × 10^-3Kgm^-3	x_1min = 0Kgm^-3
C_R = 8314JK^-1kmol^-1	x_2min = 6 .5^oC
C_rad,phot = 3.55 × 10^-9kgJ^-1	x_2max = 40^oC
C_v,pl,ai = 3.6 × 10^-3ms^-1

4.2 Experimental prototype of the greenhouse

The prototype of the is shown in Fig. 4. We represent the variables of the process in the past subsection.

Fig. 4

Prototype of the greenhouse.

4.3 State feedback control for the crop growth process

The model (37) can be expressed in matrix form according to [26, 27] as: $[\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \\ {\dot{x}}_{3} \\ {\dot{x}}_{4} \end{matrix}] = [\begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \end{matrix}] + [\begin{matrix} B_{11} & B_{12} & 0 \\ 0 & B_{22} & B_{23} \\ 0 & B_{32} & 0 \\ 0 & 0 & 0 \end{matrix}] [\begin{matrix} u_{1} \\ u_{2} \\ u_{3} \end{matrix}]$ (38) With: $\begin{array}{l} A_{1} = \frac{2^{01 x_{2} - 25} C_{r e s p, c} x_{4} - C_{l e a k} (x_{1} - ν_{c})}{C_{c a p, c}} - \frac{\frac{A_{11 n}}{A_{11 d}}}{C_{c a p, c}} \\ A_{11 n} = [(1 - e^{- C_{p l, d} x_{4}}) C_{r a d, p h o t} (x_{1} - c_{Γ}) * (- C_{c o_{2, 1}} x_{2}^{2} + C_{c o_{2, 2}} x_{2} - C_{c o_{2, 3} %}) ν_{r a d}] \\ A_{11 d} = [C_{r a d, p h o t} ν_{r a d} + (x_{1} - c_{Γ}) (- C_{\begin{array}{l} c o_{2, 1} \end{array}} x_{2} %^{2} + C_{c o_{2, 2}} x_{2} - C_{c o_{2, 3}})] A_{2} = \frac{C_{r a d, q} ν_{r a d} - C_{a i, o u} (x_{2} - ν_{T})}{C_{c a p, q}} \\ A_{3} = \frac{A_{33 n}}{C_{c a p, h}} A_{33 n} = [- C_{l e a k} (x_{3} - ν_{h}) + (1 - e^{- C_{p l, d} x_{4}}) C_{v, p l, a i} (\frac{e^{\frac{C_{v, 2} x_{2}}{C_{v, 3} + x_{2}}} C_{v, 1}}{C_{R} (C_{T, a b s} + x_{2})} - x_{3})] \\ A_{4} = - 2^{01 x_{2} - 25} C_{r e s p, d} x_{4} + \frac{A_{44 n}}{A_{44 d}} \\ A_{44 n} = [(1 - e^{- C_{p l, d} x_{4}}) C_{r a d, p h o t} % C_{α β} (x_{1} - c_{Γ}) (- C_{c o_{2, 1}} x_{2}^{2} + C_{c o_{2, 2}} x_{2} - C_{c o_{2, 3}}) ν_{r a d}] \\ A_{44 d} = [C_{r a d, p h o t} ν_{r a d} + (x_{1} - c_{Γ}) (- C_{c o_{2, 1}} x_{2} %^{2} + C_{c o_{2, 2}} x_{2} - C_{c o_{2, 3}})] \\ B_{11} = \frac{1}{C_{c a p, c}}, B_{12} =- \frac{(x_{1} - ν_{c})}{C_{cap, c}}, \\ B_{22} = - \frac{C_{c a p, q v} (x_{2} - ν_{T})}{C_{c a p, q}} \\ B_{23} = \frac{1}{C_{c a p, q}}, B_{32} =- \frac{(x_{3} - ν_{h})}{C_{cap, h}} % \end{array}$ (39)

For the nonlinear model (39), we employ a control law as: $u = {[B^{T} B]}^{- 1} B^{T} [- A + \dot{x_{s}} - \frac{1}{2} k_{s} e_{s}]$ (40) e_s = x - x_s, k_s represents a positive constant, $u = {[\begin{matrix} u_{1} & u_{2} & u_{3} \end{matrix}]}^{T}$ represents the input. We employ the state feedback control to reach $\dot{e_{s}} = - \frac{1}{2} k_{s} e_{s}$ . If (40) is replaced in (38), (39), then the tracking error of closed loop process of the state feedback control applied to the nonlinear process is exponentially convergent $\dot{e_{s}} = - \frac{1}{2} k_{s} e_{s}$ [26, 28]. If we define the Lyapunov functions as $v_{s} = \frac{1}{2} e_{s}^{T}$ and solve de derivative of v_s as $\dot{v_{s}} = e_{s}^{T} \dot{e_{s}} = - \frac{1}{2} k_{s} e_{s}^{T} = - k_{s} v_{s}$ , the solution is v_s = e^-k_stv_si, in which v_si is the initial condition of v_s, which can be expressed as $v_{s} = e^{- k_{s} t} v_{si} = \frac{1}{2} e^{- k_{s} t} ∥ e_{si} ∥^{2}$ , where e_si is the initial condition of the tracking error e_s.

5 Results

In this section, the unscented Kalman filter (29) is applied for the experimental learning of the next two processes: a solar dryer for oranges drying and a greenhouse for crop growth. The unscented Kalman filter of (29) called Unscented Kalman Filter is compared with the extreme machine learning of [11] called Machine Learning 1 and with the extreme learning machine of [12] called Machine Learning 2. We compare the data of y_m with ${\hat{y}}_{m}$ . We use the determination coefficient R², it is a parameter which determines the quality of the experimental learning: $R^{2} = 1 - \frac{\sum_{mf} {({\hat{y}}_{m} - {\bar{y}}_{m})}^{2}}{\sum_{mf} {(y_{m} - {\bar{y}}_{m})}^{2}}$ (41) ${\hat{y}}_{m}$ represents the approximation output, y_m represents the output, ${\bar{y}}_{m}$ represents the mean of the output y_m, and mf represents the final time. R² generates values from 0 to 1. If an algorithm has good performance R² has values near to 1, and if an algorithm has bad performance R² has values near to 0.

5.1 Learning of the solar drying

Temperatures T₁, T₂, T₃, T_a are renamed as x₁, x₂, x₃, x₄. Those features affect the drying behavior M_R renamed as y. Dataset from sensors has 8000 lectures covering 12.5 hours from 7:00 to 19:30 during a sunny day. It is desired the experimental learning during the drying process $\hat{y}$ using an hypothesis to define the necessary parameters to have an acceptable drying process.

In Fig. 5 we represent the results found for the average percentage of moisture removal on drying orange slices with dimensionless units versus the time in seconds, equivalent to each one of the five tests conducted using c = 59.243, k = 0.411, χ² = 0.00067, MSE = 0.994, and RMSE = 0.010 for the model, in which c represents the higher percentage of moisture removal on all tests and k represents a constant offset for fitting. In the Unscented Kalman Filter, we employ the speed α = 0.01 and a total of 1000 iterations. We apply several near values of the initial conditions in the unscented Kalman filter as shown in Fig. 6.

Fig. 5

Comparison between the unscented Kalman filter and a model for the learning of orange drying.

Fig. 6

Unscented Kalman filter for the learning of orange drying.

The comparison of the Unscented Kalman Filter, Machine Learning 1, and Machine Learning 2 is show in the Table 3 for the R² of equation (41).

Fig. 6 and Table 3 yield that the that the Unscented Kalman Filter reaches better learning than both Machile Learning 1 and Machile Learning 2 due to the Unscented Kalman filter obtains the biggest R².

Table 3

Comparison results

	R ²
Machine Learning 1	0.9020
Machine Learning 2	0.9217
Unscented Kalman Filter	0.9314

5.2 Learning of the greenhouse

The state variables carbon dioxide concentration x₁, air temperature x₂, and absolute humidity inside the greenhouse x₃ affect the behavior of the crop dry weight y. It is desired the experimental learning during the crop growth $\hat{y}$ the using a hypothesis to define the necessary parameters to have an acceptable harvest.

Taking the data and using the Unscented Kalman Filter make possible the experimental learning of the behavior of the carbon dioxide concentration in [kg m^-3], the air temperature in [^oC], and the absolute humidity in [kg m^-3] inside the greenhouse versus the time in seconds as shown in Figs. 7 –9, respectively.

Fig. 7

Unscented Kalman filter for the learning of carbon dioxide concentration.

Fig. 8

Unscented Kalman filter for the learning of air temperature.

Fig. 9

Unscented Kalman filter for the learning of absolute humidity.

The comparison of the Unscented Kalman Filter, Machine Learning 1, and Machine Learning 2 is show in the Table 4 for the R² of equation (41).

Table 4

Comparison results

	R ²
Machine Learning 1	0.9141
Machine Learning 2	0.9314
Unscented Kalman Filter	0.9420

Figures 7 –9, and Table 4 yield that the that the Unscented Kalman Filter reaches better learning than both Machile Learning 1 and Machile Learning 2 due to the Unscented Kalman filter obtains the biggest R².

6 Conclusions

The results gotten in this document show that it is possible to dry orange slices using a direct type solar dryer through natural convection, earning up to 59.243% moisture removal of the product in a time of six hours. A model represented accurately the drying process with a success of 84%. The unscented Kalman filter was programmed for the experimental learning of the behavior during the drying process of orange slices with a success of 0.93. Furthermore, the unscented Kalman filter was programmed for the experimental learning of the behavior during the crop growth with a success of 0.94. In the forthcoming work, other alternative algorithms for the experimental learning will be designed, and other physical processes will be studied.

Footnotes

Acknowledgment

The authors are grateful to the editors and reviewers for their valuable comments and insightful suggestions, which helped to improve this research significantly. The authors thank the Instituto Politécnico Nacional, the Secretaría de Investigación y Posgrado, Comisión de Operación y Fomento de Actividades Académicas, and Consejo Nacional de Ciencia y Tecnología for their help in this study.

References

Mendoza-Medina

J.L.

, Martínez-Soto

, Alcántara-González

M.L.

, López-Orozco

and Mercado-Flores

, Modelos aplicados al proceso de secado del chile poblano. VII Congreso Nacional de Ciencia de los Alimentos y III Foro de Ciencia y Tecnología de Alimentos, Guanajuato, Gto, 2003, pp. 416–424.

Henten

, Greenhouse climate management, an optimal control approach, Tesis doctoral. Wageningen Agricultural University, Netherlands, 1994.

Mghouchi

, Bouardi

, Choulli

and Ajzoul

, Models for obtaining the daily direct, diffuse and global solar radiations, Renewable and Sustainable Energy Reviews 56 (2016), 87–99.

Eke

B.A.

, Development of small scale direct mode natural convection solar dryer for tomato, okra and carrot, International Journal of Engineering and Technology 3(2) (2013), 199–204.

Chen

, Hernandez

C.E.

and Huang

, A study of the drying effect on lemon slices using a closed-type solar dryer, Solar Energy 78 (2005), 97–103.

Rubio

J.J.

, Gutierrez

, Pacheco

and Perez

, Comparison of three proposed controls to accelerate the grownth of crop, International Journal of Innovative Computing, Information and Control 7 (2011), 4097–4114.

Duan

, Xie

, Wang

and Bai

, Deep learning enabled intelligent fault diagnosis: Overview and applications, Journal of Intelligent & Fuzzy Systems 35(5) (2018), 5771–5784.

Lee

S.-H.

and Kang

M.-S.

, Intelligent noise prediction scheme with pattern analysis and deep learning technique, Journal of Intelligent & Fuzzy Systems 35(6) (2018), 5867–5879.

Narayan

, Mathew

and Chatterji

, sEMG signal classification with novel feature extraction using different machine learning approaches, Journal of Intelligent & Fuzzy Systems 35(5) (2018), 5099–5109.

10.

Owolabi

T.O.

and Gondal

M.A.

, Quantitative analysis of LIBS spectra using hybrid chemometric models through fusion of extreme learning machines and support vector regression, Journal of Intelligent & Fuzzy Systems 35(6) (2018), 6277–6286.

11.

Pacheco

A.G.C.

, Krohling

R.A.

and da Silva

C.A.S.

, Restricted Boltzmann machine to determine the input weights for extreme learning machines, Expert Systems With Applications 96 (2018), 77–85.

12.

Uzair

, Shafait

, Ghanem

and Mian

, Representation learning with deep extreme learning machines for efficient image set classification, Neural Computing & Applications 30 (2018), 1211–1223.

13.

Kukker

and Sharma

, Neural reinforcement learning classifier for elbow, finger and hand movements, Journal of Intelligent & Fuzzy Systems 35(5) (2018), 5111–5121.

14.

Tiwari

, Saxena

M.K.

, Mehendiratta

, Vatsa

, Srivastava

and Gera

, Market segmentation using supervised and unsupervised learning techniques for E-commerce applications, Journal of Intelligent & Fuzzy Systems 35(5) (2018), 5353–5363.

15.

Rubio

J.J.

, Lughofer

, Meda-Campaña

J.A.

, Paramo

L.A.

, Novoa

J.F.

and Pacheco

, Neural network updating via argument Kalman filter for modeling of Takagi-Sugeno fuzzy models, Journal of Intelligent & Fuzzy Systems 35(2) (2018), 2585–2596.

16.

Rubio

J.J.

, Stable Kalman filter and neural network for the chaotic systems identification, Journal of the Franklin Institute 354(16) (2017), 7444–7462.

17.

Yousefizadeh

, Bendtsen

J.D.

, Vafamand

, Khooban

M.H.

, Dragicevic

and Blaabjerg

, EKF-based predictive stabilization of shipboard DC microgrids with uncertain time-varying load, IEEE Journal of Emerging and Selected Topics in Power Electronics 7(2) (2019), 901–909.

18.

Safarinejadian

and Vafamand

, Kalman randomized joint UKF algorithm for dual estimation of states and parameters in a nonlinear system, Journal of Electrical Engineering & Technology 10 (2015), 742–750.

19.

Vafamand

, Arefi

M.M.

and Khayatian

, Nonlinear system identification based on Takagi-Sugeno fuzzy modeling and unscented Kalman filter, ISA Transactions 74 (2018), 134–143.

20.

Kardan

M.A.

, Asemani

M.H.

, Khayatian

, Vafamand

, Khooban

M.H.

, Dragicevic

and Blaabjerg

, Improved stabilization of nonlinear DC microgrids, cubature kalman filter approach, IEEE Transactions on Industry Applications 54(5) (2018), 5104–5112.

21.

Agrawal

Y.C.

and Singh

R.D.

, Thin layer drying studies for short grain rice, Journal of Agricultural Engineering 21 (1977), 41–47.

22.

O’Callaghan

J.R.

, Menzies

D.J.

and Bailey

P.H.

, Digital simulation of agricultural drier performance, Journal of Agricultural Engineering Research 16 (1971), 223–244.

23.

Overhults

D.G.

, White

G.M.

, Hamilton

H.E.

and Ross

I.J.

, Drying soybeans with heated air, Transactions of the ASAE 16 (1973), 112–113.

24.

Westerman

P.W.

, White

G.M.

and Ross

I.J.

, Relative humidity effect on the high-temperature drying of shelled corn, Transactions of the ASAE 16 (1973), 1136–1139.

25.

Henderson

S.M.

, Progress in developing the thin layer drying equation, Transactions of the ASAE 16 (1974), 1167–1168.

26.

Khalil

H.K.

, Nonlinear systems, Prentice Hall, 3ra Edición, 2002.

27.

Slotine

J.E.

and Li

, Applied nonlinear control, MacMillan, 1991.

28.

Ioannou

P.A.

and Sun

, Robust adaptative control, Prentice Hall, 1996.