(363ac) Data-Driven ARX Models with Measurable Disturbances for Model Predictive Control (MPC) of Crop Irrigation

The effects of global warming and climate change have threatened the water supply in many world areas worldwide. According to a U.S. Geological Survey report, agriculture is a marked user of ground and surface water in the U.S., and irrigation accounted for 42 percent of the Nation’s total freshwater withdraws in 2015 (e.g., UN-Water, 2021). The water shortage in agriculture has promoted research on a decision support system for crop irrigation management. The conventional irrigation management approaches have been “rule-based” strategies where practitioners determine irrigation intervals and amount empirically. Recently, model-based irrigation management has drawn increased attention.

The Decision Support System for Agrotechnology Transfer (DSSAT), a software application program that comprises crop simulation models for over 42 crops, has gained wide acceptance and has been used to simulate various applications at different spatial and temporal scales (Jones et al. l, 2003). DSSAT does not have the capability of optimal control of crop irrigation in real-time as it is a simulator that requires not only soil and crop genetic information, but also daily weather data and detailed crop management over the entire crop growth period. On the other hand, irrigation can be deemed as maintaining the soil moisture above a prespecified level by replenishing water in a timely and efficient manner (Shang et al., 2019). From this perspective, simple models that describe the soil water balance in the root zone have been utilized for model-based irrigation control (Delgoda et al., 2016; Guo & You, 2018).

SML(t+1) = (1 - c) · SML(t) + IRR(t) – ET(t) + PREC(t) (1)

where SML(t) denotes the soil moisture level (SML, i.e., the amount of water in soil), IRR(t), ET(t) and PREC(t) are the irrigation, evapotranspiration, and precipitation, respectively. One drawback of this approach is that the model parameters are determined based on simplified theory. Specifically, in the model above, runoff and water percolation are assumed to be proportional to soil moisture level, modeled by a first-order autoregressive process, while the coefficients of irrigation and precipitation are assumed to be 1 and -1 respectively, indicating 100% efficiency in both irrigation and precipitation.

To address the limitations mentioned above, we propose to model soil water balance using more realistic data-driven models as the following.

A(q)SML(t)=B(q)IRR(t)+C(q)ET(t)+D(q)PREC(t)+e(t) (2)

where A(q), B(q), C(q) and D(q) are polynomials expressed in the time-shift operator q^-1. Specifically, utilizing the DSSAT model that simulates crop growth, development, and yield as a function of the soil-plant-atmosphere dynamics, we estimate a more realistic autoregressive with extra input (ARX) model where the dynamics of soil water and effects of irrigation, precipitation, and evapotranspiration are estimated using DSSAT simulations with historical data.

To formulate the model suitable for model predictive control, by taking the advantage that the evapotranspiration and precipitation are measured/estimated in real-time, the effects of precipitation and crop evapotranspiration on soil moisture can be subtracted from the soil moisture.

A(q)SML(t)- C(q)ET(t)-D(q)PREC(t)=B(q)IRR(t)+ e(t) (3)

where A(q)SML(t)- C(q)ET(t)-D(q)PREC(t) is termed “adjusted SML”. Compared to model (2), which has the effects of irrigation, precipitation, and crop evapotranspiration, the adjusted SML only has the effect of irrigation and is used to construct a model predictive control (MPC). Specifically, Eqn. (3) is further transformed into a state-space model using the canonical state-space realization. The discrepancy between the rule-based DSSAT and ARX model is estimated using the Kalman filter (Huusom et al. l, 2012). Then, the augmented state-space model is built for MPC (Wang, 2009). Various objective functions and constraints are tested and the performances in terms of crop yield and water use efficiency (WUE) are compared.

• UN-Water, 2021: Summary Progress Update 2021 – SDG 6 – water and sanitation for all, Geneva, Switzerland. pp. 21-31, 2021. (n.d.).
• Jones, J. W., Hoogenboom, G., Porter, C. H., Boote, K. J., Batchelor, W. D., Hunt, L. A., Wilkens, P. W., Singh, U., Gijsman, A. J., & Ritchie, J. T. (2003). The DSSAT cropping system model. European Journal of Agronomy, 18(3–4), 235–265.
• Shang, C., Chen, W.-H., & You, F. (2019). Robust constrained model predictive control of irrigation systems based on data-driven uncertainty set constructions. 2019 American Control Conference (ACC), 1–6.
• Delgoda, D., Malano, H., Saleem, S. K., & Halgamuge, M. N. (2016). Irrigation control based on model predictive control (MPC): Formulation of theory and validation using weather forecast data and AQUACROP model. Environmental Modelling & Software, 78, 40–53.
• Guo, C., & You, F. (2018). A Data-Driven Real-Time Irrigation Control Method Based on Model Predictive Control. 2018 IEEE Conference on Decision and Control (CDC), 2599–2604.
• Huusom, J. K., Poulsen, N. K., Jorgensen, S. B., & Jorgensen, J. B. (2012). Tuning SISO offset-free Model Predictive Control based on ARX models. Journal of Process Control, 22(10), 1997–2007.
• Wang, L. (2009). Model predictive control system design and implementation using MATLAB^®.