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Characteristics of A Agricultural Engineering System And Simple

Problems

CLASSIFICATION OF AGRICULTURAL SYSTEM

For practical purposes, the following factors were used for classification:

The time scale of the system

The uncertainties of events in the system

Structure of the system

Within these classes, systems are named and classified according to the type of differential or

difference equation of the mathematical model representing system.

THE TIME SCALE

There are two classes of systems as they relate to the time scale chosen for the mathematical

model:

Continuous system

Discrete system

Continuous system:

The time scale of continuous systems is the set of non negative real numbers. Continuous

systems are called differentiable systems because they are represented by differential

equations and their solutions.

Problem 1: The following equation was fitted to the energy content of milk of a group of

cows:

Y = 2.821 + 0.965 e -0.0423t

Where y is the energy content of milk in MJ/kg and t in days after calving. Determine the

corresponding differential equation.

Solution: The following is the first derivative of the state equation:

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)0423.0(*965.0 0423.0 = tedx

dy

Where 0.965 e -0.0423t = y 2.821. After replacing values, the following is the differential

equation representing this system:

ydx

dy0423.01193.0 ==

Where 0.1193 is the energy input and 0.0423y is the energy output in MJ/kg/day.

Discrete system:

The time scale of discrete system is the set of non negative integers. Discrete state variables

cannot be fractionalized, meaning that the system cannot be represented by differential

equations. This is the case of state of variables defined as number of individuals or as

qualitative traits. Thus, the state changes are represented by difference equations.

Problem 2: A rancher sells each month 3.6 % of his feedlot steers and buys 90 new animals.

The initial number if steers is 460. Define the mathematical model of the system.

Solution: This system is discrete because the state variable, number of steers, is discrete. The

following equation represents the system:

nnn yyy 036.0901 =+

Where n is months, yn is the number of steers corresponding to the present state of the

system, yn+1 is the number of steers of the next state, x = 90 is the input as number of steers

purchased and Zn = 0.036yn is the output as number of steers sold per month. The following is

the corresponding state trajectory:

n

ny )964.0(20402500=

UNCERTAINTIES OF EVENTS

Most of the inputs reaching agricultural systems cannot be controlled and occur in a random

pattern. Therefore, the operation of all agricultural systems is subject to some kind of

uncertainties. Then, depending on whether these uncertainties are considered in the

mathematical model or are ignored, two types of systems evolve:

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Stochastic system

Deterministic system

Stochastic system:

The basic feature of a stochastic model is that state variables are defined as probability

distributions.

Problem 3: Define the state probability distributions for the citrus trees (it was found that

when the trees in a citrus farm are healthy, 20% get a disease within a year and when the trees

are diseased, 30% of the trees recover), assuming binomial distribution of events.

Solution: The following was the state joint distribution expression defined for the system, for

the initial state P0 = (1,0):

Where m is the total number of trees. This expression corresponds to a deterministic model of

the system. It is assumed that the variables have the binomial distribution, as shown below:

Where x1 and x2 are the number of healthy and diseased trees and p1 and p2 are the

corresponding probabilities. Then, by replacing the Pn values in the binomial expression, it is

possible to define the following state probability model of the system:

The probability distribution curves of diseased trees are shown below. The total number of

trees was assumed to be 10.

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Deterministic systems:

In deterministic models, the states of the system are the expected values of the outcomes.

Thus, deterministic models represent the expected or average behaviour of the system. The

first four examples in this section were all deterministic models.

A real system may be defined by a deterministic model or a stochastic model. Deterministic

model are simpler and more widely used than stochastic models.

Problem 4: The following is the deterministic model for the state equation of the previous

example, expressed as expected values:

Where x1 is the expected number of healthy trees and x2 is the expected number of diseased

trees at time n. The graphic representation of expected values is shown below. The total

number of trees is assumed to be 10.

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STRUCTURE OF SYSTEM:

The notion of structure of a system is related to how component systems are coupled to form

a more complicated system. The following classification was adopted here:

Interactive coupled systems

Conjunctive coupled systems

Interactive coupled systems:

Interacting systems may be coupled by means of interconnected differential or difference

equations, determining the interactive coupling. Interactive agricultural systems may be

arranged in two groups:

Compartmental systems

Non compartmental systems

Compartmental systems:

Components of compartmental systems are called compartments. Such compartments work

as chambers among which some material is considered to move.

Problem 5: The movement of DDT from plant to soil is 25% per month, from soil to plant

2% and carried out with ground water 5%. Define the mathematical model of the system.

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Solution: This system us represented by following figure. The following set of differential

equations defines the flow of DDT in the system.

Fot Y = (yp, ys), where yp is a state of the plant compartment, ys is a state of the soil

compartmental and t in months. Coefficients with positive signs are the compartment inputs

and coefficients with positive signs are the compartment inputs and coefficients with negative

signs are the compartmental outputs.

Non compartmental systems:

Components of non compartmental systems sometimes are called back boxes, among which

some information is considered to move.

Problem 6: The following matrix equation defines the relationship between pasture yield and

carrying capacity of Kikuyu pasture field, as affected by rainfall:

For Y = (y1,y2) where y1 is leaf growth in kg of dry matter per ha/day, y2 is the number of

cows/ha and x is the rainfall in mm/month. Determine the input and output of the system.

Solution: The mathematical model of the system has the form

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The system input is rainfall, defined by the expression Bdx/dt +Cx and the system output is

AY. The following figure shows the system black boxes exchanging information by input

output relationships.

Conjunctive coupled systems:

The idea of conjunctive coupling is that of a complex system where each of its components

operated independently. This is the case, for example, of different plots or different

experimental material, such that each plot is a component system and operates as a

replication of experiment. Grouping of experimental material determines the sources of

variation in a typical analysis of variance.

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