WATER RESOURCES RESEARCH, VOL. 26, NO. 4, PAGES 647-669, APRIL 1990 An Optimal Control Method for Real-Time Irrigation Scheduling ANGELOS L. PROTOPAPAS 1 AND ARIS P. GEORGAKAKOS Georgia Institute of Technology, Atlanta In this paper a systematic methodology for makingreal-timeirrigation decisions is presented.A physically based representation of the dynamics of the soil-crop-atmosphere system is used.The variables characterizingthe crop and soil statusare concurrentlysimulatedwith an integrated state space model.Soil moisture and salinity conditions, whichsynergistically control the plant water uptake, areobtained by using lumped parameter mass balance models for therootzone.Cropyield is predicted by explicitly modeling the plant growth processes, such as assimilation, respiration, and transpiration, which are driven by the climatic inputs. The control model is an analytical optimization method for multistage multidimensional sequential decision-making problems. It is suitable for systems with nonlinear dynamics and objective functions. The method is based on local iterative approximations of the nonlinear problem witha linear quadratic problem. Thisapproach is evaluated in a series of casestudies,where optimalirrigation schedules are obtained on an hourly basisover the growingseason. 1. INTRODUCTION Agricultural activities areimportant for the economies of most countries. The demandfor agriculturalproductsin- creases continuously as a result of growing populations, higher incomes, andnewuses of traditional products. It is estimated that production will increase by expansion of arable landand, mainly, by intensified use and better man- agement of the production factors. There is therefore an immediate need for efficient use of resources, in particularof water, in the productionprocess. Modem agriculture is a specialized andhighly mechanized industry, in which solar energy is transformed to useful organic products. During the growing season the farmer makes important operational decisions whichaffect the final yield. Competitive and efficient agriculturerequires such decisions to be made optimally. In the short run, maximiza- tion of a performance index is sought by exhaustive alloca- tion of the limiting production factors, that is, water, land, and nutrients among different crops.The index is usually net benefits, although in some cases,when water is scarce,the objective mayshiftto maximizing produced cropweight per unit of water used.In the longrun, objectives are relatedto societal issues suchas welfare, malnutrition,deforestration and climatic change, environmental pollution, income redis- tribution and employment, soil conservation, and energy consumption. Even if optimal use of resources can lead to short-term gain in cropyield, the long-term effects canbe immeasurable. Yet, in current practice the farmers act with the short-term objective of maximizing the benefits of the production process by prudentuse of the available re- sources. This is also thephilosophy of our study, which aims inintroducing modern decision-making methodologies for water use in agriculture. •Now at Metcalf and Eddy, Incorporated, Wakefield, Massachu- setts. Copyright 1990 bythe American Geophysical Union. Paper number 89WR03250. 0043- ! 397/90/89 WR-03250505.00 2. THE IRRIGATION SCHEDULING PROBLEM: LITERATURE REVIEW Irrigation in a broad sense is the human effort to control the soil-crop-atmosphere continuum.In particular, the irri- gationscheduling problemis to determinethe optimal tim- ing, quantity, and qualityof artificially supplied water to the soil in order to control the crop yield. Given that for six major U.S. grain crops theaverage annual yieldis 20-35%of the record yield, there is much to be done to increase the solarenergyconversion efficiency, which is directly related to the water transpired by the crop. In currentirrigation practicemany farmersirrigateby the calendar,that is, in a predecided schedule based on experi- ence, while others receive water at scheduled times and use it whetherthe crop needs it or not. As Hillel [ 1987] reports, when rule of thumb methods are not in use, irrigation scheduling techniques are classified as soil,plant, or climate based. In the first case, measurements of soil moisture and salinity are taken at a site. When these parameters reach certain critical values, irrigation is applied. In the second case,variables characterizing the status of the plant, such as leaf temperature, water content, and color, are measured or visually inspected. In some cases, functions of these param- eters, called stress indices,have been usedto indicatethe necessity of irrigation. In the third case,measurements of climatic variables, such as total absorbed radiation, air temperature, relative humidity, and windspeed, areused to estimateevapotranspiration during a given time period. Irrigation is thenapplied whenever the estimated waterloss exceeds a threshold value. The central idea in our researchis to concurrently predict all variables characterizing the plant and soil status,using a model of the soil-crop-climate sys- tem, and then make more informed real-time irrigation decisions on the basis of modernoptimal control techniques. Two distinct approaches can be used to tacklethe irriga- tion scheduling problem. The physicallybased (causal or physiological) approach emphasizes the need for modeling theprocesses resulting in crop growth and theirrelation to climate and soil. The statistical (regressionor correlative) approach relies onextensive fieldexperimentation and uses statistical analysis of field measurements. Several studies in the literature mix the two approaches in different propor- 647