## Continuous System Simulation and System Dynamics:

### Continuous System Simulation:

Continuous System Simulation describes systematically and
methodically how mathematical models of dynamic systems, usually described by
sets of either ordinary or partial differential equations possibly coupled
with algebraic equations, can be simulated on a digital computer.

Modern modelling and simulation environments relieve the
occasional user from having to understand how simulation really works. Once a
mathematical model of a process has been formulated, the modelling and
simulation environment compiles and simulates the model and curves of result
trajectories appear magically on the user’s screen. Yet, magic has a tendency
to fail, and it is then that the user must understand what went wrong, and why
the model could not be simulated as expected.

### System Dynamics:

System Dynamics is a computer-aided approach to policy
analysis and design. It applies to dynamic problems arising in complex
social, managerial, economic, or ecological systems, literally any dynamic
systems characterized by interdependence, mutual interaction, information
feedback, and circular causality.

**The System Dynamics Approach Involves:**

a. Defining
problems dynamically, in terms of graphs over time.

b. Striving
for an endogenous, behavioural view of the significant dynamics of a system, a
focus inward on the characteristics of a system that themselves generate or
exacerbate the perceived problem.

c. Thinking
of all concepts in the real system as continuous quantities interconnected in
loops of information feedback and circular causality.

d. Identifying
independent stocks or accumulations (levels) in the system and their inflows
and outflows (rates).

e. Formulating
a behavioural model capable of reproducing, by itself, the dynamic problem of
concern. The model is usually a computer simulation model expressed in
nonlinear equations, but is occasionally left unquantified as a diagram
capturing the stock-and-flow/causal feedback structure of the system.

f. Deriving
understandings and applicable policy insights from the resulting model.

g. Implementing
changes resulting from model-based understandings and insights.

## Continuous System Models:

Continuous system simulation is one, in which predominant
activities of the system cause smooth changes in the attributes of the system
entities. When such a system is modelled mathematically, the variable of the model
representing the attributes are controlled by continuous functions. In general,
in continuous, the relationship describes the rate at which attributes changes,
so that the model consists of differential equations.

If a system can be represented using simple differential
equation model, then it is often possible to solve the model without the use of
simulation, otherwise we use simulation to solve those models which are complex
to solve analytically.

## Differential Equations:

A differential equation is a mathematical equation that
relates some function with its derivatives where the functions usually
represent physical quantities, the derivatives represent their rates of change,
and the equation defines a relationship between the two. Because of such relations
are extremely common, differential equations play a prominent role in many
disciplines including engineering, physics, economics, and biology.

We can use the differential equation to represent the behaviour
of a continuous system. An example of a linear differential equation with constant
coefficient is one that describes the wheel suspension of an automobile.

The equation is:

The equation is:

**Mx ̈+Dx ̇+Kx=KF(t)**

Where,

*= acceleration*

**x ̈***= velocity*

**x ̇**
x = displacement

K = stiffness of spring

D = measure of
viscosity (thickness) of shock absorber

F(t) = input of system
depends on independent variable t

When more than one independent variable occurs in a differential
equation, the equation is said to be a partial differential equation. It can
involve the derivatives of the same dependent variable with respect to each of the
independent variables. An example is an equation describing the flow of heat in
a three-dimensional body. There are four independent variables, representing the
three dimensions and time, and one dependent variable, representing
temperature.

Differential equation occurs repeatedly in scientific and
engineering studies. The reason for this prominence is that most physical and
chemical process involves rates of change, which require differential equations
for their mathematical description. Since a differential coefficient can also represent
a growth rate, continuous models can also be applied to a problem of a social
or economic nature where there is a need to understand the general effect of
growth trends.

### Ordinary Differential Equations:

An ordinary differential equation (

*ODE*) is an equation containing an unknown function of one real or complex variable*x*, its derivatives, and some given functions of*x*. The unknown function is generally represented by a variable (often denoted*y*), which, therefore,*depends*on*x*. Thus*x*is often called the independent variable of the equation. The term "*ordinary*" is used in contrast with the term partial differential equation which may be with respect to*more than*one independent variable.
Linear differential equations are the differential
equations that are linear in the unknown function and its derivatives. Their
theory is well developed, and, in many cases, one may express their solutions
in terms of integrals.

Most ODE that is encountered in physics are linear, and,
therefore, most special functions may be defined as solutions of linear
differential equations.

### Partial Differential Equation (PDE):

Partial Differential Equation is a differential equation
that contains unknown multivariable functions and their partial derivatives.
(This is in contrast to ordinary differential equations, which deal with
functions of a single variable and their derivatives.) PDEs are used to formulate
problems involving functions of several variables, and are either solved in
closed form or used to create a relevant computer model.

PDEs can be used to describe a wide variety of phenomena in
nature such as sound, heat, electrostatics, electrodynamics, fluid flow,
elasticity, or quantum mechanics. These seemingly distinct physical phenomena
can be formalized similarly in terms of PDEs. Just as ordinary differential
equations often model one-dimensional dynamical systems, partial differential
equations often model multidimensional systems. PDEs find their generalization
in stochastic partial differential equations.

### Non-Linear Differential Equations:

Non-linear differential equations are formed by the

*products of the unknown function and its derivatives*are allowed and its degree is > 1. Nonlinear differential equations can exhibit very complicated*behaviour over extended time intervals.*

### Linear Differential Equations:

A linear differential equation with constant coefficients is
always of this form, although derivatives of any order may other forms, such as
being raised to a power, or are combined in any way- for example, by being
multiplied together, the differential equation is said to be non-linear.

## Analog Computers:

Analog computers are generally used to solve continuous
model but sometimes are also used to solve static models. Some device whose
behaviour is equivalent to a mathematical operation such as addition or
integration is combined together in a manner specified by a mathematical model
of a system to allow the system to be simulated. That combination is used
in the simulation of a continuous system is referred to as an analogue computer or when
they are used to solve differential equation they are referred to as
differential analyzer.

Simulation with an analog computer is more properly
described as being based on a mathematical model than as being a physical
model. The most widely used form of analog computers is the electronics analog computers based on the operational amplifiers. Voltages in the computers are
equated to mathematical variables and the operational amplifiers can add and
integrate the voltage.

With appropriate circuits, an amplifier can be made to add
several input voltages, each representing a variable of model, to produce a
voltage representing the sum of the input variables. Different scale factors
can be used on the input to represent the coefficient of the model equations. Such
amplifiers are called summer. Another circuit arrangement produces an integrator
for which the output is the integral with respect to time of single input
voltage or the sum of several input voltages. All voltages can be positive or
negative to correspond to the sign of the variable represented. To satisfy the
equation of the model, it is sometimes necessary to use a sing inverter.

#### Advantages:

a. Parallel operation: many signal values can
be computed simultaneously.

b. Computation can be done for some applications without the requirement for transducers to convert the inputs/outputs to/from digital electronic form.

c. Setup requires the programmer to scale the problem for the dynamic range of the computer. This can give insight into the problem and the effects of various errors.

b. Computation can be done for some applications without the requirement for transducers to convert the inputs/outputs to/from digital electronic form.

c. Setup requires the programmer to scale the problem for the dynamic range of the computer. This can give insight into the problem and the effects of various errors.

#### Disadvantages:

a. Computation elements have a limited useful
dynamic range, usually not much more than 120 dB, about 6 significant digits of
accuracy.

b. Useful solution to problems of any size can take an inordinate amount of setup time (though modern analog computers have interfaces that make setup substantially easier than it used to be).

c. For a given size (mass) and power consumption, digital computers can solve larger problems.

d. Solutions appear in real (or scaled) time, and maybe difficult to record for later use or analysis.

e. The range of useful time constants is limited. Problems that have components operating on vastly different time scales are difficult to deal with accurately.

b. Useful solution to problems of any size can take an inordinate amount of setup time (though modern analog computers have interfaces that make setup substantially easier than it used to be).

c. For a given size (mass) and power consumption, digital computers can solve larger problems.

d. Solutions appear in real (or scaled) time, and maybe difficult to record for later use or analysis.

e. The range of useful time constants is limited. Problems that have components operating on vastly different time scales are difficult to deal with accurately.

### Components of Analog Computer:

#### 1. Adder:

With an appropriate circuit, an amplifier made to
add several input voltage each representing the variable of the model to
produce a voltage each representing a sum of the input voltage.

#### 2. Subtractor:

With an appropriate circuit, an amplifier made to subtract
several input voltage each representing the variable of the model to produce the voltage each representing a difference of input voltage.

#### 3. Differentiator:

An op-amp differentiator or a differentiating the amplifier is a circuit configuration which produces output voltage amplitude
that is proportional to the rate of change of the applied input voltage.

#### 4. Integrator:

The circuit arrangement for which the output is integral
with respect to time of single input voltage or the sum of several input
voltage.

#### 5. Invertor:

It is an amplifier designed to cause the output to reverse the
sign of the input.

#### 6. Scale Factor:

This circuit multiplies each input by a factor
(the factor is determined by circuit design) and then adds these values
together. The factor that is used to multiply each input is determined by the
ratio of the feedback resistor to the input resistor.

## Analog Method:

The general methods by which analog computer are applied can be demonstrated using the second-order differential equation given as:

**Mx ̈+Dx ̇+Kx=KF(t)**

Solving the equation for the highest order derivate gives:

**Mx ̈=KF(t)-Dx ̇-Kx**

Suppose a variable representing the input F(t) is supplied, and assume for the time being that there exist variables representing

**-x**and**-x ̇**. These three variables can be scaled and added with a summer to produce a voltage representing**Mx ̈**. Integrating this variable with a scale factor of**1/M**produce**( x) ̇**. Changing the sign produce**x ̇**, which supplies one of the variables initially assumed; and further integration produces**-x**, which was other assumed variables. For convenience, a further sign inverter is included to produce**+x**as an output.
A block diagram to solve the problem in this manner is shown
below. The symbols used in the figure are standard symbols for drawing block
diagrams representing analog computer arrangements. The circle indicates scale
factors applied to the variable. The triangular symbol at the left of the
figure represents the operating of the adding variables. The triangular symbol
with a vertical bar represents integration, and the containing a minus sign is a
sign changer.

**Draw the analog model of the Liver with following set of equations:**

(dx_1)/dt=-k12x_1+k21x_2

(dx_2)/dt=k12x_1-(k21+k23) x_2

(dx_3)/dt=k23x_2

## Hybrid Computers:

The term hybrid computers have emerged to describe the
combination of traditional analog computer elements (that gives the smooth
continuous output and carry out non-linear operations) as well as the circuit
components that have the capacity of storing values, switching operations and
performing a logical operation. The scope of analog computers has been considerably
extended by developments of a solid-logic electronic device. Hybrid computer may
be used to simulate a system that is mainly continuous and also have some
digital elements. For e.g. an artificial satellite for which both the
continuous equation of motion and the digital controls signal must be
simulated. A hybrid computer is useful when a system that can be adequately
represented by an analog computer model is subject of a repetitive study.

## Digital Analog Simulators:

To avoid the disadvantage of analog computers, many digital computer programming languages have written to produce digital-analogue simulator.
These allow a continuous model to be programmed on a digital computer in the same
way as it is solved on an analog computer. These languages contain macro
instructions that carry out the action of address, integrators and sign chargers.

A program uses these macro-instructions to link them
together in essentially the same way as operational amplifiers are connected in
analog computers. Later more powerful techniques of applying digital computers
to the simulation of the continuous system have been developed. Due to these
digital-analogue simulators are not now in common uses.

## Continuous System Simulation Language:

It is a restriction to keep the digital computer within the
limit to a routine that represents as it is done with a digital-analog simulator. To
remove the restriction a number of continuous system simulation language have
been developed. They use familiar statement type of input for a digital computer,
allowing a problem to be programmed directly from the equation of mathematical the model rather than requiring the equation to be broken down in functional
elements.

A CSSL include macros or subroutines that forms the function
of specific analog elements so that it is possible to incorporate the convenience of an analog simulator. To allow the users to define special-purpose
elements that correspond to an operation that are particularly important in a specific type of application.

It includes a variety of algebraic and logical expression to
describe the relation between variable. Therefore, they remove the orientation
towards linear differential equation which characterizes analog computer. One
particular CSSL that illustrates the nature of these languages is the Continuous
System Modeling Program.

## CSMP III (Continuous System Modeling Programming III):

A CSMP III program is constructed from three general types
of statements:

### Structural Statement:

It defines the model. They consist of FORTRAN like statement
and functional block designed for an operation that frequently occurs in a model
definition. It can make use of the operation of addition, subtraction,
multiplication, division, and exponential using the same notation and rules used in
FORTRAN.

### Data Statement:

It assigns numeric values to the parameter constant and
initial condition.

### Control Statement:

It specifies the option in the assembly and execution of
program and the choice of output.

For example, the model includes the equation: X=6Y/W+(Z-2)^2

The following statement would be used: X=6.0*Y⁄W+(Z-2.0)**2.0

Note that real constants are specified in decimal notation.
Exponent notation may also be used; for example, 1.2E-4 represents 0.00012.
Fixed value constants may also be declared. Variable names may have up to six
characters.

**Write a CSMP program of following differential equation.**

Mx ̈+Dx ̇+Kx=KF(t)

Here,

**Structural Statement**

Mx ̈=KF(t)-Dx ̇-Kx

x ̈=(KF(t)-Dx ̇-Kx)/M

x_2 dot= 1.0/M [KF(t)-Dx ̇-Kx]

x_2 dot= 1.0/M [K*F(t)-D*x ̇-K*x]

x_2 dot=(1.0/M)*[K*F(t)-D*x ̇-K*x]

x_2 dot=INTGRL (O.O,x_2 dot)

x=INTGRL(0.0,x_1 dot)

**Data Statement**

M = 3.0

F(t) = 1.0

K = 4.0

**Control Statement**

DELT (Integral Interval) = 0.05

FINTIME (Finish Time) = 1.5

PRDEL (Integral at which to print result) = 2

### Hybrid Statement:

The system to be studied is either continuous or discrete and we have to select the analog or digital computer for the study of the system. There are many advantages and disadvantages of analog and digital computer. To achieve the advantages of both (analog and digital computer) we can combine both analog/digital computer system into a single form and simulate the system through it.

In this case, one computer is simulating the system being studied
while others providing the simulation of the environment in which the system is to
operate. The hybrid simulation requires some extra technical improvement, high-speed converters are used to convert the signal from one form to another.

## Feedback System:

Feedback systems have a closed-loop structure that brings results
from past action of the system back to control future action, so feedback
systems are influenced by their own past behaviour. Extending the blind control
example, a feedback system would be a system that not only opens the blinds
when the sun rises but also adjusts the blinds during the day to ensure the
room is not subjected to direct sunlight.

Even though the open system can consist of many parts and thus
become very complex (these systems have high detail complexity), experience
shows that the behaviour of even small feedback systems consisting of only a few
parts (and thus low detail complexity) can be very difficult to predict in
practice: despite low detail complexity, these systems have high dynamic
complexity. In business prototyping, we deal with both kinds of systems; system
dynamics is particularly good at capturing the dynamics of feedback systems.

## Interactive Systems:

Interactive systems are computer systems characterized by
significant amounts of interaction between humans and the computer. Most users
have grown up using Macintosh or Windows computer operating systems, which are
prime examples of graphical interactive systems. Editors, CAD-CAM (Computer
Aided Design-Computer Aided Manufacture) systems and data entry systems are
all computer systems involving a high degree of human-computer interaction.
Games and simulations are interactive systems. Web browsers and Integrated
Development Environments (IDEs) are also examples of very complex interactive
systems.

Some estimates suggest that as much as 90 percent of computer
technology development effort is now devoted to enhancements and innovations in
interface and interaction. To improve efficiency and effectiveness of computer
software, programmers, and designers not only need a good knowledge of
programming languages, but a better understanding of human information
processing capabilities as well. They need to know how people perceive screen
colors, why and how to construct unambiguous icons, what common patterns or
errors occur on the part of users, and how user effectiveness is related to the
various mental models of systems people possess.

## Real-Time Simulation:

Real-time simulation refers to a computer model of a physical system that can execute at the same rate as the actual "wall clock" time.
In other words, the computer model runs at the same rate as the actual physical
system. For example, if a tank takes 10 minutes to fill in the real-world, the
simulation would take 10 minutes as well.

Real-time simulation occurs commonly in computer gaming, but
also is important in the industrial market for operator training and off-line
controller tuning. Computer languages like LabVIEW, VisSim, and Simulink allow
quick creation of such real-time simulations and have connections to industrial
displays and Programmable Logic Controllers via OLE for process control or
digital and analog I/O cards. Several real-time simulators are available on the
market like xPC Target and RT-LAB for mechatronic systems and using Simulink,
eFPGAsim and eDRIVEsim for power electronic simulation and eMEGAsim, HYPERSIM and
RTDS for power grid real-time (RTS) simulation.

## Predator-Prey Model:

It is also called the parasite-host model. An environment consists of two population i.e. predator and prey. It is also a mathematical model. The prey is passive but the predator depends on the prey for their source of food.

Let,

x(t) = number of prey population at time t.

y(t) = number of predator population at time t.

r.x(t) = rate of growth of prey for some +ve ‘r’, where r = natural birth and death rate.

Because of the interaction between predator and prey, it will be reasonable to assume that the death rate of prey is proportional to the product of two population size x(t).y(t) or the death rate of prey is a.x(t).y(t). Therefore the overall rate of change of prey population, dx/dt is given by, dx/dt=r.x(t)-a.x(t).y(t).

Where a is positive constant of proportion. Also, the predator population depends on the prey for their existence, the rate of the predator in the absence of prey is –s.y(t) for some positives.

The interaction between two population cause predator population to increase at a rate of proportion x(t).y(t). Thus, overall change predator population dy/dt=-s.y(t)-b.x(t).y(t). Where, b is positive constant.

As the predator population increases the prey population decreases. This cause a decrease in the rate of an increased predator, which eventually results in a decrease in the number of predators. These in turns cause the number of prey population to increase.

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