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Dead Sea Hydro-power station. It is proposed to build a 400MW-hydropower station on the shore of the Dead Sea.

See attached matlab file. It might help you.
Dead Sea Hydro-power station.
It is proposed to build a 400MW-hydropower station on the shore of the Dead Sea. The plant will drain seawater from the Mediterranean through turbines and into the Dead Sea. The vertical distance between the inlet and outlet depends on...
See attached file for full problem description.

University of KwaZulu-Natal
School of Electrical, Electronic and Computer Engineering
SYSTEMS AND SIMULATION - ENEL3SS
0) INTRODUCTION
This is a first contact course in control systems. The broad aim of the course is to develop skills in
understanding the behaviour of physical dynamic systems and to understand the engineering tools
required for design of control systems. Specifically, we will cover the following main sections:
1) Modelling: We will develop the model of a physical system by writing down sets of first
order differential equations. The time varying variables of such a model are called the state
variables. The study of systems that are represented as state space models allows a very
fundamental understanding of system behaviour to be developed. State space
representations are used for system modelling and understanding, for computer simulation
and for controller design. We will start of with general (non-linear, time varying) models
but will focus on linear or linearised, time invariant systems.
2) Introduction to simulation and the Matlab/Simulink environment: The development of
affordable advanced computing hardware and software allows fast development of
simulation models of complex systems. We will use a Windows based program called
Matlab and a package called Simulink that runs under Matlab. This software is available
on the LAN and there is on-line help. You are expected to use Matlab for tutorials and the
development of these skills will be examined. We will illustrate the principle of simulation
by investigating first order explicit and implicit Euler methods. Packaged software uses
more advanced algorithms and is fairly reliable although simulation must be tested for
reasonableness as the algorithms used can and do fail on particular problems.
3) Linearisation and linear systems: Many problems either are linear or can be linearised. Our
interest in obtaining linear problems is that there are very powerful methods available for
analysing and designing linear systems. We will investigate the time response of linear
systems and the application of the Laplace transform to solving the differential equations.
4) Input-output descriptions of linear systems: In many cases we are more interested in the
input and output variables of a system than in the internal (or state) variables. As a result
we will investigate the input-output properties of systems and the output response of
systems to sinusoidal input signals. We will develop some understanding of the
relationship between the time domain behaviour and the frequency domain behaviour of
systems.
5) Discrete time and sampled data systems: The very widespread use of computer hardware
means that it is important to understand the behaviour of systems when part of the system is
digital hardware interfaced to the "real world" by A/D (analogue-digital) and D/A
converters.
Systems & Simulation - ENEL3SS 1
SS1_1.DOC
1. SYSTEM MODELLING
1.1. General non-linear system models
A fundamental principle in nature is that of conservation. Certain quantities in nature such as mass
and energy are conserved in the sense that they do not vanish without explanation. If we define a
physical system that is of interest to us by means of its boundary, the law of conservation is that the
rate of increase of a conserved quantity within the system is the rate at which that quantity crosses
the system boundary plus the rate of manufacture/transformation of the conserved quantity within
the system.
System
Crossing boundary
boundary
manufacture/
transformation
Conservation of a quantity within a system
This can be written as a general law:
dQ
= boundary + manufacture
dt
Q is the conserved quantity of interest. boundary is the rate at which the conserved quantity crosses
the boundary into the system (boundary
the rate at which the conserved quantity is manufactured/transformed from other quantities within
the system. In many processes there is no manufacture but we should look out for both
mechanisms.
Examples of conserved quantities: mass (of components or total), energy (kinetic, potential, heat,
chemical or total), sum of total mass and energy (generalised energy) in nuclear reactions, number
(of people, bank balances etc.).
Examples of unconserved quantities: temperature, pressure, value of money, social worth of people.
Simple example: Water level in a tank
The conserved quantity is the mass of water in the tank. (What is the
system boundary?) The water is incompressible and therefore the uin
volume and mass are directly related. The inflow/outflow is in
volume/time and the level is volume/area. For constant area,
dV V
= Net inflow rate = (uin-uout), h=V/A h uout
dt
The process is linear with respect to volume and hence with respect to
level if the area is constant.
Systems & Simulation - ENEL3SS 2
SS1_1.DOC
[time,Uin]
+
From 1/s 1 height
Workspace +
Sum flow-volume To Workspace
1/area
[time,Uout]
From
Workspace1
time_out
Clock To Workspace1
Simulink simulation of level in a tank
Modelling is an important interest of this course and like most things, it is best learned by practice,
the main learning of system modelling will be by tutorial work. We will apply knowledge from
many branches of science and engineering and the final model will be based on exact conservation
laws and empirical/phenomenological observations. The model can usually be refined if it is found
that the reality and the model do not have satisfactory agreement and can simplified by omitting
secondary effects if it is found too cumbersome for the application and if the application can
tolerate lower fidelity. We will use the modern simulation tools that are available to validate and
visualise our model.
We will generally obtain for lumped parameter systems models in the form,
x = f ( x , u, t ) the state differential equation
& (1a)
y = g ( x , u, t ) the algebraic output equation (1b)
t is the time
x is the [n×1] state vector (of state variables)
u is the [m×1] input vector
y is the [p×1] output vector
f is an [n×1] non-linear, time varying vector equation with elements fi
g is a [p×1] non-linear, time varying equation with elements, gi
u - inputs system y - outputs
x - state variables
When we develop models of dynamical systems in this course, the model equations should always
be given in the form of eq(1) with the state, input and output variables clearly identified.
uin
Example 2 - Level in a cylindrical drum
An example would be a boiler drum.
R h uout
L
Systems & Simulation - ENEL3SS 3
SS1_1.DOC
The volume is, (show this!)
V = L R 2 arcsin(1 - h / R ) - ( R - h) 2 Rh - h2
also,
dV = 2 L 2 Rh - h2 dh
We can write the model using either the height or the volume as the state variable. (Remember
dh dV dV
that the mass and therefore the volume, is the conserved variable.) Recall that = and
dt dt dh
because solving for h as a function of V may be messy, we would prefer to use h as the state
variable. We get,
dh
= ( uin - uout ) 2 L 2 Rh - h2 state differential equation
dt
h=h algebraic output equation
Tank height simulation
[time,Uin]
+
From
Workspace + * 1/s height
Sum To Workspace
Product height rate -
[time,Uout] height
1/sqrt(2*1*u-u^2)
From
Workspace1 1/(dV/dh)
time_out
Clock To Workspace1
Simulink model for example 2 (R=1, L=1)
Example 3 - Mechanical system - Ping pong ball levitator
Assume that a nozzle makes a cone of air with velocity,
u( t )
v air ( h) = . u(t) is the position of a control valve, h is the ball vair
( h + )2
h
height above the nozzle and is a constant. This is an empirical law
based on heuristic understanding and laboratory measurement. nozzle
(v air - v ball )2
The ball experiences upward force as a result of drag, f up = c d A . A is the ball
2
cross sectional area, cd is the drag coefficient, is the density of the air. This is also an empirical
law, supported by theoretical investigations. We do not necessarily have to understand the cross-
disciplinary engineering behind empirical laws but they must be plausible i.e. they must make
sense!
Systems & Simulation - ENEL3SS 4
SS1_1.DOC
d ( mv)
The conserved variables are (i) momentum (recall correct version of Newton II - Fext = )
dt
and (ii) space.
u(t)
2
- v ball
dv ball 1 (h + )2
= c d A - mg
dt m 2
state differential equation
dh
= v ball
dt
(m is the mass and g the acceleration due to gravity.) Let us say that we are interested in the kinetic
energy of the ball and its height as outputs:
mv2ball
E= algebraic output equation
2
h=h
f(u) k_energy
Fcn2 To Workspace4
velocity
To Workspace3
Mux f(u) f(u) 1/s 1/s
Repeating XY Graph
Fcn Fcn1 velocity height
Sequence Mux Graph
height
To Workspace1
time
Clock To Workspace
Fcn = u(1)/(u(2)+0.02)^2-u(3)
Fcn1 = 0.44*1.205*0.01875^2*pi/2.5e-3*u^2*sgn(u)/2-9.8
( = 0.02, m=2.5g, r=18.75mm, cd = 0.44, =1.205)
Systems & Simulation - ENEL3SS 5
SS1_1.DOC
Example 4 - Manufacture within the system boundary - Cell growth in a biosystem.
S/V
Bugs grow at a rate, r = µ max cells/cell/unit time (This is known as Monod law). µmax
S / V + Ks
is the maximum growth rate, S/V is the concentration of substrate (i.e. bug food!) which changes
with time as the bugs consume it. KS is a constant - the so-called limiting substrate concentration.
If S/V=KS the growth rate is half the maximum. If S/V>>KS the growth rate is equal to µmax. Cells
consume the substrate by ×r (kg/cell/unit time) to grow.
The conserved variables of interest are the number of cells, and the amount of substrate. Assuming
that the reaction is in a fixed volume, conservation of quantities and their concentration is
equivalent. With C as the number of cells and uin the substrate feed rate, we get a model with state
differential equation,
dC S/V
= µ max C cell growth
dt S / V + Ks
dS S/V
= - µ max C + uin substrate conservation
dt S / V + Ks
Mux f(u) 1/s
Mux Fcn cells
Graph
-0.1*u(1) 1/s
Fcn1 substraste
Graph1
Systems & Simulation - ENEL3SS 6
SS1_1.DOC
Example 5 ­ Aids
(See: Jeffrey AM, Xia X and Craig I, "On attaining maximum and durable suppression of the viral
load", African Control Conference, Cape Town, December 2003.)
Simple model is a predator-prey model (human CD4+ cells are the prey) and anti-retroviral drugs
are a control. The drugs are never 100% effective and the real system is very complicated.
Model:
dT
= s + pT (1 - T / Tm ) - T T - VT
dt
dT1
= q1 VT - 1T1 - kT1
dt
dT2
= q 2 VT - 2 T2 + kT1
dt
dV
= N 2 T2 - cV1
dt
State variables, x = [T, T1, T2, V]T are concentrations of uninfected CD4+ cells, latently infected
CD4+ cells, actively infected CD4+ cells, and free virus particles respectively.
s 10/mm3/day source production rate of uninfected cells from the body
p 0.03/day proliferation rate
Tm 1500/mm3 steady state cell count
T 0.02//day death rate constant of healthy cells
7.5×10-6/mm3/day infection effectiveness
q1 0.05 production rate constant of latently infected CD4+ cells
q2 0.55 production rate constant of actively infected CD4+ cells
1 0.02/day death rate constant of latently infected cells
2 0.5/day death rate constant of actively infected cells
k 0.075 activation rate of latently infected cells
N 2000 virons/cell production of free virus particles per actively infected cell
c 5/day death rate constant of free virus particles
The two classes of commonly used anti-retroviral agents are Reverse Transcriptase (RT) Inhibitors
and Protease Inhibitors (PI). Both agents work within the CD4+ T cell because they cannot prevent
the virus from entering the cell. Reverse Transcriptase Inhibitors reduce successful infection of the
CD4+ T cell by the virus by reducing the values of q1 and q2. Perfect inhibition would occur if the
rates were zero but in practice, perfect inhibition is not attainable. Protease Inhibitors block the
protease enzyme so that the virus particles that are produced are mostly non-infectious. There are
therefore two types of virus particles when protease inhibitors are used. The first type is the
infectious virus particles that still continue to infect CD4+ T cells and the other is the non-
infectious type. Similarly, perfect inhibition occurs when all virus particles that are produced are
noninfectious. Current therapies use a combination of Reverse Transcriptase and Protease
Inhibitors and the combined therapy model can be presented as,
Systems & Simulation - ENEL3SS 7
SS1_1.DOC
dT
= s + pT (1 - T / Tm ) - T T - VT
dt
dT1
= µ RT q1 VT - 1T1 - kT1
dt
dT2
= µ RT q 2 VT - 2 T2 + kT1
dt
dV I
= µ PI N 2 T2 - cV I
dt
dV N
= (1 - µ PI ) N 2 T2 - cV N
dt
µRT is the control input for reverse transcriptase and µPI is the control input for protease inhibitors
and the free virus particles have been split into infectious and non-infectious.
Comments
1) Build the model in Simulink and test its performance at various values of 0
2) No drug has µ· = 0 and µ· depends on patient, drug and virus.
3) CD4+ cell counts of around 200/mm3 are regarded as critically low levels.
Systems & Simulation - ENEL3SS 8
SS1_1.DOC

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