Application Predictive Control Strategies Based on Models for Optimal Irrigation of Andean Crops ^†

Abstract

Irrigation for high Andean agriculture is traditionally performed with rainwater and without the use of technology, where the influence of changes in water volumes and/or water losses is not considered. Likewise, the limited information on high Andean crops generates a lag in the use of irrigation technology. Improving the efficiency of irrigation in crops contributes substantially to the sustainable use of water. One way to perform this is by applying control strategies to irrigation processes that consider implementing a feedback logic of the water necessary for irrigation, thus satisfying the water demand of plants and minimizing waste. The article proposes a control strategy applying a model predictive control (MPC) that calculates the optimal amount of water for daily irrigation. The most important attraction of the model is the prediction and future behavior of the controlled variables as a function of the changes in the manipulated variables. The objective is to improve the productivity of the crop at minimum water consumption. For this, it will be necessary to use models that link with the Aquacrop software and which are allowed to be a source of data, as well as being used for the prediction of future values. The predictive model is evaluated in the Quinoa crop (Chenopodium Quinoa Willdenow), and the information is validated against the traditional irrigation data existing in the literature. Preliminary results indicate that the predictive model can achieve greater crop efficiency and reduce significant irrigation water supplies.

1. Introduction

Agriculture is the sector that consumes the most water on the planet, representing approximately 70% of total use in 2020 [1]. An alternative to achieve the best use of water resources in agriculture is to apply control strategies with proven performance in the industry. Classic control methodologies, such as on/off control and proportional integral derivative (PID) control, are easy to implement and have been proven to be efficient. However, they are limited and do not consider the specific conditions of the crop, given the complexity of agricultural systems (non-linearity, multivariate). MPC strategies have shown superior performance compared to classical control strategies [2,3]. This controller is based on three ideas: the use of a prediction model, the optimization on a sliding horizon, and feedback adjustment [4]. It also allows the introduction of restrictions. The literature mentions some applications of MPC in irrigation systems [5,6,7]. However, there is no report of application to high Andean crops, such as Quinoa (Chenopodium Quinoa Wildenow), among others.

In this article, a control application of MPC was implemented in a Quinoa crop to different irrigation management strategies [8,9], as well as to simulate the yield of the crop to water and evaluate the conditions in which water is a limiting factor in production [8,10]. In this investigation, the performance of the MPC controller applied to the irrigation of the Quinoa crop was evaluated. This involves taking into account AquaCrop-OpenSource (AquaCrop-OS) as a plant model and an autoregressive structure with exogenous input (ARX) as a linear prediction model. The results obtained were compared with the irrigation methods available in the Aqua-Crop-OS gallery. All simulations were performed in MATLAB 2020a.

2. Methodology

2.1. Quinoa Crop

Quinoa is a whole grain, native to the Andes of Bolivia, Chile, and Peru. This crop is tolerant to abiotic stress, hydric stress, and requires a less amount of water for its vegetative growth [11]. It has extraordinary adaptability in agro-ecological conditions from sea level to 4000 masl., being able to withstand temperatures from –4°C to 38°C and grow with relative humidity between 40% and 70% [12]. The Quinoa in traditional farming presents critical phenological stages of susceptibility and tolerance to the need for irrigation. This is reported in accordance with the Instituto Nacional de Innovación Agraria (INIA-Puno), a public institution of the Peruvian government.

2.2. Aquacrop

It was developed by the FAO in order to improve water productivity in rainfed and irrigated conditions. This simulates the yield response of arable crops to water and is particularly suitable for conditions where water is a limiting factor in crop production [8]. It has been validated for various crops, such as corn [10] and Quinoa [9]. It was developed in 2009 and its open access version AquaCrop-OS is presented in [13]. The program introduces crop information according to various characteristics: climate, type of crop, irrigation, soil, and others. The results obtained are crop growth, water balance, water content in the crop, and others. The methods provided by AquaCrop-OS are rainfed, soil moisture based, fixed interval, specified time series, and net calculation.

2.3. Model Predictive Control

The usual MPC approach is described by Equations (1)–(3).

$$\underset{u\left(k\right), \dots , u\left(k+N-1\right)}{\min }{\displaystyle \sum }_{i=0}^{N-1}\Vert y\left(k+i-1\right)-r{\left(k+i-1\right)\Vert }_Q^2+\Vert u{\left(k+i\right)\Vert }_R^2$$

(1)

Subject to:

$$y\left(k+1\right)=f\left(y\left(k\right), u\left(k\right), v\left(k\right)\right)$$

(2)

$$u_{min}\le u\left(k\right)\le u_{max},k=0,\dots ,N-1$$

(3)

where, $y$ is the control variable, $u$ is the manipulable variable, $v$ is the measurable disturbs, $r$ is the reference, and $Q$ and $R$ correspond to the weight of each term of Equation (1). The function $f$ defines the prediction model of the controller. This problem is solved at each sampling instant.

2.4. ARX Model

The ARX model is represented in the form of a difference equation for multiple inputs as follows:

$$A\left(z\right)y\left(k\right)=\left[\begin{array}{ccc}{B}_1(z^{-1})& B_2(z^{-1})& B_2(z^{-1})\end{array}\right]\left[\begin{array}{c}u\left(k\right)\\ v_1\left(k\right)\\ v_2\left(k\right)\end{array}\right]+e\left(k\right)$$

(4)

where:

$$A\left(z\right)=1+a_1{z}^{-1}+\dots +a_n{z}^{-n}$$

(5)

$$B_i\left(z\right)=b_{i,1}{z}^{-1}+\dots +b_{i,m}{z}^{-m}$$

(6)

where $y\left(k\right)$ is the system output, $u\left(k\right)$ is the system input, $v_i$ is the measurable disturbs, $e\left(k\right)$ is the system disturbance, $d$ is the system delay, $n$ is the degree of $A\left(z\right)$, $m$ is the degree of $B_i\left(z\right)$ and $i$ is the number of inputs. Equation (4) in regression form is:

$$\begin{array}{ll}y\left(k\right)=a_{1}y\left(k-1\right)& +\dots +a_{n}y\left(k-n\right)+b_{1,1}u\left(k-1\right)+\dots +b_{1,m}u\left(k-m\right)\\ & +b_{2,1}{v}_1\left(k-1\right)+\dots +b_{2,m}{v}_1\left(k-m\right)+b_{3,1}{v}_2\left(k-1\right)+\dots \\ & +b_{3,m}{v}_2\left(k-m\right)+e\left(k\right)\end{array}$$

(7)

For this work, it is considered that $y\left(k\right)$ is the water deficit, $u\left(k\right)$ is the irrigation, $v_1\left(k\right)$ is the evapotranspiration, and $v_2\left(k\right)$ is the precipitation.

3. Results

The research was carried out by simulating the conditions of the Quinoa crop in the high Andean phytogeographic domain in the region of Puno, Peru. It is located between 3812 and 5500 m above sea level. The model for response to water deficit was performed by simulation in Aquacrop-OS, a pseudorandom binary sequence (PRBS) input of irrigation (mm), and real meteorological data from the years 1964–2021 taken from Servicio Nacional de Hidrología y Meteorología (SENAMHI), and Quinoa data obtained from INIA-Puno were used. Normalized mean square error (NMSE) was used as evaluation criterion to estimate the parameters and validation of the model, obtaining the best performance in 2017 (NMSE = 5.7212 × 10⁻⁴) and the worst for 1984 (NMSE = 0.0164) for the identification experiment. For validation, NMSE = 5.9894 × 10⁻⁴ and NMSE = 0.0202, respectively.

Model Predictive Control in Quinoa Crop

The control variable for this work is the water deficit (mm), and the manipulable variable is irrigation (mm). The measurable disturbances are evapotranspiration (mm) and precipitation (mm). Figure 1 shows the simulation result of the MPC controller for a simulation of a growing season using an ARX structure as a prediction model.

Table 1. Field yield and total irrigation.

Method	Field Yield (Ton/He)	Total Irrigation (mm)
Rainfed	4.31	0
Soil moisture-based	4.72	164.63
Fixed interval	4.72	289.26
Specified time series	4.72	320
Net calculation	4.72	160.08
MPC	4.72	80.17

shows the crop yield per hectare (ton/h) and the total irrigation (mm) of the simulated methods.

It was observed that the method that consumes the least amount of water is the rainfed method, followed by the MPC; but, the former has the worst field yield. The MPC has similar field yield of the simulated methods with lower water consumption.

4. Conclusions

In this work, the problem of MPC control of water deficit applied in a quinoa crop model using AquaCrop-OS was presented. An ARX structure with multiple inputs and one output was proposed as a prediction model. The proposed irrigation methodology presents the best performance among the simulated methods. The future work of this research should be the implementation of the virtual model of Quinoa in a real irrigation system to determine the true efficiency and viability, according to the irrigation recommendations proposed by MPC and Aquacrop.