Modelling%20of%20dynamic%20systems:%20Part%202%20Solving%20differential%20equations%20using%20computers

1
Modelling of dynamic systems Part 2Solving
differential equations using computers
Robert N. Shorten Douglas Leith The Hamilton
Institute NUI Maynooth
2
Contacting me?

• How can you contact me?
• email robert.shorten_at_may.ie
• My office is in the Hamilton Institute
• When can you ask me questions?
• During lectures
• Contact hours (any time)
• Books
• Booklist on the web (www.hamilton.may.ie/course/)
  

3
Overview

• Lecture 1 Introduction
• What are ordinary differential equations?
• Why are they important?
• Why are they difficult to solve?
• Lecture 2 Solving differential equations
• Eulers method
• Lecture 3 Solving differential equations
• Runge Kutta method
• Lecture 4 Systems of differential equations,
  stiff systems, and some examples

Introduction
4
Mathematics!

• Show no fear - Maths is just another language!!
• 
  Relax, dont panic, and think about
  the equation.
• Pay attention to detail and pay particular
  attention to Theorems
• and to the proofs

5
Building our first model

• Example Malthuss law of population growth
• Government agencies use population models to
  plan.
• What do you think be a good simple model for
  population growth?
• Malthuss law states that rate of an unperturbed
  population (Y) growth is proportional to the
  population present.

Introduction
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Modelling

• Modelling is usually necessary for two reasons
  to predict and to control. However to build
  models we need to do a lot of work.
• Postulate the model structure (most physical
  systems can be classified as belonging to the
  system classes that you have already seen)
• Identify the model parameters (later)
• Validate the parameters (later)
• Solve the equations to use the model for
  prediction and analysis (now)

Introduction
10
Introduction How do we predict?

• To predict we need to solve our equations
• continuous time systems discrete time systems
  and hybrid systems.
• Sometimes, when we are very lucky, we can find an
  analytic solution to these equations.
• More often, an analytic solution cannot be found,
  and we must use a computer to approximate the
  solution to the equation.
• This is usually not a problem for discrete time
  systems. However, for continuous systems,
  modelled using differential equations, life is
  not so simple. Solving differential equations is
  the subject of the next four lectures.

Introduction
11
What are differential equations?

• Calculus, the science of change, was invented by
  Newton to help him in his investigations of
  natural phenomena.
• He discovered that lots of systems evolve
  according to rate equations.
• Example Application of Newtons law of cooling
• Experiments show that the time rate of change of
  temperature of an object and its surrounding is
  proportional to the difference between the
  temperature of the object and the surrounding
  medium

Introduction
12
What are differential equations?

• Calculus, the science of change, was invented by
  Newton to help him in his investigations of
  natural phenomena.
• Many other types of systems can be modelled by
  writing down an equation for the rate of change
  of some variable
• bandwidth utilisation in TCP networks
• acceleration of car
• mortgage repayments
• population increases
• chemical change of some kind
• flight of a football
• All of the above behaviour can be captured by
  very simple differential equations

Introduction
13
What are ODEs?

• The previous examples are examples of ordinary
  differential equations.
• Mathematically speaking, an ODE is an equation
  relating the derivatives of an unknown function
  of an dependent variable with known functions of
  an independent variable.
• ODEs are used to model many physical systems and
  are therefore capable of predicting the behaviour
  of a system for all time into the future, only
  requiring an input and some activation.
• Consider a net constant longtitudinal force F
  acting on a body of mass M. From Newtons laws F
  Ma

Introduction
14
What is a differential equations telling us to do?

• Notice that the velocity at time t depends on the
  initial velocity.
• Sometimes we are asked to find the solution of an
  ODE that satisfies some initial value this is an
  initial value problem.
• More often we are interested in the qualitative
  behaviour of the system for all initial
  conditions.

Introduction
15
Cant we solve them analytically?

• Many differential equations can be solved
  analytically. In fact, the theory underlying
  linear time-invariant systems is well-understood.
  Most linear systems can be modelled using linear
  differential equations of the following form
• Linearity means that the solution is specified by
  an equation involving only linear combinations
  derivatives of the dependent variable with
  respect to the independent variable. (the order
  of the system is given by the power of the
  highest derivative).

Introduction
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Linear time-invariant ODEs

• Consider the RL circuit depicted below. The
  source is a
• sinusoid of frequency w and of unit amplitude.

Introduction
17
Cant we solve them analytically?

• It is convenient to express the output of a
  system, modelled by a differential equation, as
  the sum of two components one associated with no
  input the other associated with the input to the
  system.
• The first component is called the natural
  response of the system and is the system output
  when the input is zero.
• The second component is the component of the
  output due to the input only and is known as the
  forced response of the system.

Introduction
18
The natural response

• Consider the RL circuit depicted below. The
  source is a
• sinusoid of frequency w and of unit amplitude.
• The natural response is obtained by setting the
  input to zero.

Introduction
19
The forced response

• The forced response is obtained by setting the
  input to zero.

Introduction
20
The combined response

• What is the combined response if the current is 2
  amps when t 0,
• and if w RL1?

Introduction
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The combined response
Introduction
22
What do we do when analytical solutions cannot be
found?

• Mathematical analysis is not the grand thing it
  is made out to be!
• Maths solves only the simplest equations. As soon
  as the equations become a little more
  complicated, we are in trouble.
• Analytical solutions to differential equations
  arising in the real world can only rarely be
  found. Linear time-invariant systems are well
  understood, but most real systems are
  time-varying, non-linear and hybrid. In such
  cases we must use numerical techniques and a
  computer to investigate the system under study.

Introduction
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What do we do when we cannot find a soln?

• We begin our study of numerically approximating
  differential equations with
• What is this equation telling us to do?
• The equation is saying the following. Take any
  point in the x-t plane. If a solution to the
  above equation passes through this point then the
  tangent to the solution is given by f(t,y).

Introduction
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Numerical analysis A direction field
Introduction
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Introduction
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Introduction
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Introduction
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Introduction
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The direction field

• Another equation
• What is this equation telling us to do?
• The equation is saying the following. Take any
  point in the x-y plane. If a solution to the
  above equation passes through this point then the
  tangent to the solution is given by f(x,y).

Introduction
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Numerical analysis A direction field
Introduction
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Introduction
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Introduction
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Introduction
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Introduction
35
Numerical analysis Direction fields

• We have depicted graphically the directions in
  which solutions should proceed.
• Direction fields generate the flow associated
  with differential equations.
• For such a plot to make sense, the function
  f(x,y) should satisfy certain properties it
  should be defined in some region of the plane it
  should be single valued in this region and
  solutions to the equation in this region should
  be differentiable.

Introduction
36
Numerical analysis Word of warning

• We have been a little bit hard on mathematics.
  However, we know from maths that a solution to a
  differential equation need not always exist. It
  need not even be unique.

• Example Does a solution to the following
  equation exist?
• Example Is the solution to the following
  equation unique?

Introduction
37
Numerical analysis Word of warning

• We have been a little bit hard on mathematics.
  However, we know from maths that a solution to a
  differential equation need not always exist. It
  need not even be unique.
• Computers dont care. Always be careful about the
  output of a program. Generally, the computer will
  give you an answer not necessarily the one that
  you want.
• Use mathematical theory to verify simulation. In
  particular, verify uniqueness and existence
  conditions are satisfied.

Introduction
38
Existence and uniqueness

• Existence and uniqueness theorems for ordinary
  differential equations. You should have done
  them .What are they?
• Roughly speaking if f(t,y) is continuous in some
  region of the t-y plane then at least one
  solution to the differential equation exists if
  the partial derivative of f w.r.t y exists and is
  continuous then the solution is unique.
• Revise the concept of Lipschitzness and the
  existence and uniqueness theorems for ODEs from
  part one of the course (if it has been covered).

Introduction
39
Modelling of dynamic systems Part 2Eulers
method
Robert N. Shorten Douglas Leith The Hamilton
Institute NUI Maynooth
40
Overview

• We have already spoken about the poor job that
  mathematics has done in studying differential
  equations.
• Most engineering and scientific problems involve
  differential equations that cannot be solved
  analytically.
• We can use the direction field to get a solution
  if the order of the problem is not too high, or
  if there are not too many independent variables.
• What do we do in other cases?

Lecture 2
41
Eulers method

• There are many methods for solving ODEs
  numerically. We begin with the simplest and most
  intuitive method Eulers method.
• Eulers method is the easiest algorithm for
  numerically solving a differential equation.
• Unfortunately, it is useless. Nevertheless, it is
  useful to illustrate some pertinent points and
  introduce numerical methods. Let us assume that
  we are dealing with a differential equation that
  satisfies our existence and uniqueness
  conditions.

Lecture 2
42
Eulers method

• Then, using Taylors theorem, one can write any
  function about t0 a point as
• Terms left out of the expansion are of the form
• where n is greater than 2. Hence, for h t-t0
  small enough the function can be approximated

Lecture 2
43
Eulers method

• In other words, for small h, function y(t) can be
  approximated from knowledge of the function and
  derivative at t0.
• This observation suggests the following
  approximation to the solution to a differential
  equation. Suppose that we want to approximate the
  solution to a well posed initial value problem
  over the interval a,b. To construction this
  solution we divide the interval into a number of
  mesh points

Lecture 2
44
Eulers method

• At the first mesh point we can approximate the
  solution as
• The beauty of Eulers method is that this
  procedure can be repeated at intermediate mesh
  points.

Lecture 2
45
Example

• Use Eulers method to solve approximately the
  initial value problem
• over the interval t in the interval 0,2 with
  y(t0) 1. Use a step size
• of 0.2.

k t y_eul
y_act 0 0 1.0000
1.0000 1.0000 0.2000 0.8000 0.8375
2.0000 0.4000 0.6800 0.7406 3.0000
0.6000 0.6240 0.6976 4.0000
0.8000 0.6192 0.6987 5.0000 1.0000
0.6554 0.7358 6.0000 1.2000 0.7243
0.8024 7.0000 1.4000 0.8194
0.8932 8.0000 1.6000 0.9355 1.0038
9.0000 1.8000 1.0684 1.1306 10.0000
2.0000 1.2147 1.2707
Lecture 2
46
k t y_eul y_act
0.0000 1.0000 2.0000 3.0000 4.0000
5.0000 6.0000 7.0000 8.0000 9.0000
10.0000 0.0000 0.1000 0.2000 0.3000
0.4000 0.5000 0.6000 0.7000 0.8000
0.9000 1.0000 1.0000 1.1000 1.2100
1.3310 1.4641 1.6105 1.7716 1.9487
2.1436 2.3579 2.5937 1.0000 1.1002
1.2013 1.3042 1.4097 1.5185 1.6312
1.7484 1.8707 1.9984 2.1321
Lecture 2
47
k t y_eul y_act
0.0000 1.0000 2.0000 3.0000 4.0000
5.0000 6.0000 7.0000 8.0000 9.0000
10.0000 0.0000 0.1000 0.2000 0.3000
0.4000 0.5000 0.6000 0.7000 0.8000
0.9000 1.0000 1.0000 1.1000 1.2100
1.3310 1.4641 1.6105 1.7716 1.9487
2.1436 2.3579 2.5937 1.0000 1.1002
1.2013 1.3042 1.4097 1.5185 1.6312
1.7484 1.8707 1.9984 2.1321
Lecture 2
48
k t y_eul y_act
0.0000 1.0000 2.0000 3.0000 4.0000
5.0000 6.0000 7.0000 8.0000 9.0000
10.0000 0.0000 0.1000 0.2000 0.3000
0.4000 0.5000 0.6000 0.7000 0.8000
0.9000 1.0000 1.0000 1.1000 1.2100
1.3310 1.4641 1.6105 1.7716 1.9487
2.1436 2.3579 2.5937 1.0000 1.1002
1.2013 1.3042 1.4097 1.5185 1.6312
1.7484 1.8707 1.9984 2.1321
Lecture 2
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Eulers method Geometric interpretation

• Errors accumulate over time!!!!!

50
Eulers method

• What happens if the step size is decreased? What
  happens if the step size is increased?

51
Eulers method

• What happens if the step size is decreased? What
  happens if the step size is increased?

52
Eulers method Errors

• We have seen that our approximation to the
  solution to the differential equation is not
  exact.
• Errors are introduced into the procedure as a
  result of two mechanisms
• Round-off
• Discretisation errors.
• Round-off errors arises from numbers being stored
  in computers using finite precision. This is
  usually not a concern as far more error is
  introduced as a result of truncation.

53
Eulers method Local truncation error

• There are two types of truncation error local
  truncation errors and global truncation errors.
  The local truncation error is the error after
  just one step (show)
• Hence

54
Global discretisation error and convergence

• The global error at a certain value just what we
  would ordinarily call the error the difference
  between the true value and the approximation.
  Roughly speaking, the global error can be
  written
• So halving the step size should, roughly
  speaking, halve the global discretisation error.

55
Eulers method Errors

• For the previous example

56
Improving Eulers method

• The modified Eulers method is based on including
  more terms in the Taylor expansion.
• We can approximate the first two terms in the
  expansion. How do we get the higher order term?
  We approximate this term as

57
Improving Eulers method

• With this approximation for the second derivative
  we obtain the following approximation
• It can be shown that for this method

58
Improving Eulers method
59
Improving Eulers method

• Investigate the effect of step-size on the
  accuracy of the solution.
• Show for the example that the global truncation
  error is approximately
• What order is the local truncation error?

60
Modelling of dynamic systems Part
2Runge-Kutta methods
Robert N. Shorten Douglas Leith The Hamilton
Institute NUI Maynooth
61
Overview

• In the last lecture we have looked at the Euler
  method. Eulers method is usually not good
  enough. The main reason is that the error is
  global O(h).
• An improvement in the Euler method was obtained
  through the modified Euler method.
• Lets take a another approach to finding the
  numerical solution of the differential equation.

Lecture 2
62
Second order RK method
Lecture 2
63
Fourth order RK method

• This results in the following algorithm

Lecture 2
64
Fourth order RK method

• This results in the following algorithm
• This is the second order Runge-Kutta method. This
  algorithm is an improvement on what we have
  looked at already. However, it is based on simple
  trapezoidal integration. There are more
  sophisticated numerical integration algorithms.
  One of these is Simpsons rule. Application of
  Simpsons rule leads to the commonly used 4th
  order Runge-Kutta method.

Lecture 2
65
Fourth order RK method

• The fourth order Runge-Kutta yields
• where

Lecture 2
66
Example

• Use the RK method to solve approximately the
  initial value problem
• over the interval t in the interval 0,1 with
  y(t0) 1. Use a step size
• of 0.1.

k t y_rk y_act
0 0 1.0000 1.0000
1.0000 0.2000 0.8375 0.8375 2.0000
0.4000 0.7406 0.7406 3.0000 0.6000
0.6976 0.6976 4.0000 0.8000 0.6987
0.6987 5.0000 1.0000 0.7358
0.7358 6.0000 1.2000 0.8024 0.8024
7.0000 1.4000 0.8932 0.8932 8.0000
1.6000 1.0038 1.0038 9.0000
1.8000 1.1306 1.1306 10.0000 2.0000
1.2707 1.2707
Lecture 2
67
k t y_eul
y_act 0 0 1.0000
1.0000 1.0000 0.2000 0.8000 0.8375
2.0000 0.4000 0.6800 0.7406 3.0000
0.6000 0.6240 0.6976 4.0000
0.8000 0.6192 0.6987 5.0000 1.0000
0.6554 0.7358 6.0000 1.2000 0.7243
0.8024 7.0000 1.4000 0.8194
0.8932 8.0000 1.6000 0.9355 1.0038
9.0000 1.8000 1.0684 1.1306 10.0000
2.0000 1.2147 1.2707
Lecture 2
68
k t y_eul
y_act 0 0 1.0000
1.0000 1.0000 0.2000 0.8000 0.8375
2.0000 0.4000 0.6800 0.7406 3.0000
0.6000 0.6240 0.6976 4.0000
0.8000 0.6192 0.6987 5.0000 1.0000
0.6554 0.7358 6.0000 1.2000 0.7243
0.8024 7.0000 1.4000 0.8194
0.8932 8.0000 1.6000 0.9355 1.0038
9.0000 1.8000 1.0684 1.1306 10.0000
2.0000 1.2147 1.2707
Lecture 2
69
Lecture 2
70
Runge-Kutta method

• What happens if the step size is decreased? What
  happens if the step size is increased?

71
Fourth order Runge-Kutta method

• The fourth order Runge-Kutta is very popular for
  a variety of reasons.
• 1) It is simple to program.
• 2) It is very accurate. The global truncation
  error is O(h4). The local truncation error
  is O(h5).
• 3) It is very efficient.

72
A few words on stability

• All of the methods that we have considered were
  of the form
• This is a first order difference equation and
  will be unstable (when)?
• If we keep the step size small enough then we can
  usually ensure stability.

73
Efficiency of the RK schemes

• We have said that the 4th order RK method is more
  efficient than other techniques that we have
  considered.
• How do me measure efficiency? There are two
  components to consider.
• Computational effort
• Truncation errors.
• We get a rough measure of computational effort by
  counting how many times the function f(t,y) is
  evaluated as we make a prediction to a desired
  accuracy.

74
Efficiency of the RK schemes

• The 4th order Runge Kutta method is much more
  efficient than the Euler method as we usually
  require much less computational effort to obtain
  the same accuracy.
• For our running example, the error of the RK4
  method with a step size of 0.1 is 9.3771e-006.
  There total number of computations to obtain this
  estimate is 40. To obtain a similar error with
  Euler we need a step-size of 1e-5. The total
  computational requirement is therefor 1e5.

75
Runge-Kutta-Fehlberg

• The big problem when using any method is to
  estimate the accuracy of the approximation (the
  global truncation error).
• Normally we do not have access to the solution so
  the problem is not straight forward.
• One approach to estimate the accuracy of the
  solution was proposed by Fehlberg.

76
Runge-Kutta-Fehlberg

• Fehlbergs basic idea was to run two RK methods
  of different order. For example, using a 4th
  order RK method, the error introduced by
  truncation may be estimated as
• If we use a 5th order RK method
• Hence an approximation to the truncation error
  (small h)

77
Runge-Kutta-Fehlberg

• At each time step we can estimate the truncation
  error. If this is too large, we simple decrease
  the step size. If it is too small, we can
  increase the step size.
• In fact, given a tolerance C, the new step size
  can be estimated as (show)

78
Multi-step methods A qualitative discussion

• All of the methods that we have considered have
  been of the form
• It is of course possible to develop methods where
  the prediction depends on more than one previous
  value of prediction. These are called multi-step
  methods.

79
Adams-Bashforth Moulton method

• The ABM predictor-corrector method is a
  multi-step method. It is obtained by
  approximating the integral in the formula
• by an interpolation polynomial of third
  degree. This polynomial is chosen so that is
  passes through the points
• This produces an ABM prediction

80
Adams-Bashforth Moulton method

• Having obtained the prediction, a second
  polynomial is constructed to fit the points
• and the point
• This polynomial is then integrated to obtain the
  final prediction at time index k1.

81
Adams-Bashforth Moulton method

• The ABM formulas are
• The advantage of this technique is that the
  difference between the predictor and corrector
  gives an estimate of the truncation error.

82
Adams-Bashforth-Moulton method

• Disadvantages of the technique include
• (1) The first 4 predictions must be obtained
  using some other method.
• (2)The technique results in a high order
  difference equation. Care must be taken to
  ensure that this equation remains stable
  (zeros inside the unit circle).

83
Adams-Bashforth-Moulton method
84
Adams-Bashforth-Moulton method
85
Examples

• Eulers method
• Modified Euler method
• 4th order Runge-Kutta method

86
Modelling of dynamic systems Part 2Systems of
differential equations
Robert N. Shorten Douglas Leith The Hamilton
Institute NUI Maynooth
87
Is the world first order?

• Up to now we have considered approximating the
  solution to first order ordinary differential
  equations using numerical methods. Our methods
  work for non-linear systems and for time-varying
  systems.
• What about second order systems?
• What about higher order systems in general?
• What about systems of ODEs?

88
A system of differential equations

• The predator-prey model of Volterra (Foxes and
  Rabbits)
• Let x denote the rabbit population and let y
  denote the fox population. Volterra postulated
  the following. In the absence of foxes we have
• and in the absence of rabbits we have

89
A system of differential equations

• Now let the populations of rabbits and foxes
  intermingle with one another. We assume that the
  number of contacts is proportional to the total
  population
• We now have a system of differential equations.

90
The predator-prey model of Volterra

• Lets assume that all parameters are unity.
  Assume that the initial condition is (2,4).

91
Higher order equations
92
How do we simulate complicated systems?

• Fortunately, systems of differential equations
  can be treated in the same way. To illustrate the
  concepts consider the initial value problem

93
How do we simulate complicated systems?

• Formally speaking, a solution to the system of
  differential equations is a pair of function
  f(x,y,t) and g(x,y,t) that satisfy
• How do we numerically approximate such solutions?

94
How do we simulate complicated systems?

• A numerical solution to the system of
  differential equations over some alttltb interval
  is found by considering the differentials
• Can you guess Eulers method for systems of
  differential equations?

95
Eulers method for systems of ODEs

• We approximate the differentials as
• By subdividing the time interval into M
  sub-intervals we obtain the recursive formulae

96
The predator-prey model of Volterra

• Lets assume that all parameters are unity.
  Assume that the initial condition is (2,4).

97
The predator-prey model of Volterra

• We can also plot x against y - giving us the
  phase plane (more about this later).

98
Runge-Kutta for systems of ODEs

• Virtually all of the ODE solvers that we have
  looked at can be used to solve systems of ODEs.
• The Runge-Kutta solver for a pair of ODEs is
  analagous to the solver for a first order ODE.

99
Runge-Kutta for systems of ODEs
100
Systems of differential equations

• Virtually all ODE solvers can be extended to
  solve systems of differential equations.
• We have considered (briefly) systems of two
  equations. Exactly the same methodology applies
  to systems of more than two equations.
• When comparing ODE solvers, the issues that are
  relevant for first order equations are also
  relevant for systems of equations. Can you think
  of any issues that are special to systems of
  equations?

101
Higher order equations

• We have dealt with systems of equations. How do
  we deal with higher order differential equations?
  
• The equation
• can also be written
• i.e. as a system of differential equations.

102
Some more examples

• Systems of two first order equations (second
  order systems) are very special. Consider again
• The plane with x and y as co-ordinates is known
  as the phase plane. The phase plane can be
  derived from the direction field of

103
The phase plane

• Consider the system
• This system is unstable.

104
The phase plane

• Consider the system
• This system is stable and has a fixed point at
  the origin.

105
A word of warning Stiff systems

• We have already encountered stability problems
  when looking at first order ODEs.
• Roughly speaking, stability is maintained by
  choosing a very small step size.
• However, very small step-sizes lead to large
  computational burdens.

106
A word of warning Stiff systems

• Consider the differential equation
• We can write
• This decays very fast and therefore

107
A word of warning Stiff systems

• So, given that y(t) is not that important, we
  should choose the step-size to approximate
• WRONG!!!!! The stability of the numerical
  integration is determined by the largest
  eigenvalue of the system.Always choose a
  step-size to guarantee stability.
• This is an example of a stiff systema number of
  variables evolving at different time scales.
  There are special solvers for such systems.

108
Examples

• To finish this part of the course we give two
  examples where numerical solutions to systems of
  ODEs are required.
• 1) Physics Modelling a pendulum
• A non-linear continuous time system
• 2) Computer Science Network congestion
  control
• A system with continuous/discrete state A
  hybrid systems model (Hespanha)

109
Modeling pendulums

• Pendulums appear everywhere!

110
Modeling pendulums

• This is a second order differential equation.
  Hence, it can be written in the form.

111
Modeling pendulums
112
TCP congestion control

• Mathematical modelling of the internet is a hot
  topic!! Hopefully, we can use these models to
  understand what is going on and avoid congestion.
  Typically we model and analyse simple scenarios!

113
TCP congestion control

• Roughly speaking, TCP operates as follows
• Data packets reaching a destination are
  acknowledged by sending an appropriate message
  to the sender.
• Upon receipt of the acknowledgement, data sources
  increase their send rate, thereby probing the
  network for available bandwidth, until congestion
  is encountered.
• Network congestion is deduced through the loss of
  data packets (receipt of duplicate ACKs or non
  receipt of ACKs), and results in sources
  reducing their send rate drastically (by half).

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TCP congestion control

• Congestion control is necessary for a number of
  reasons, so that
• catastrophic collapse of the network is avoided
  under heavy loads
• each data source receives a fair share of the
  available bandwidth
• the available bandwidth B is utilised in an
  optimal fashion.
• interactions of the network sources should not
  cause destabilising network side effects such as
  oscillations or instability

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TCP congestion control

• Hespanhas hybrid model of TCP traffic.
• Loss of packets caused by queues filling at the
  bottleneck link.
• TCP sources have two modes of operation
• Additive increase
• Multiplicative decrease
• Packet-loss detected at sources one RTT after
  loss of packet.

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TCP congestion control
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TCP congestion control
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Modelling the queue not full state

• The rate at which the queue grows is easy to
  determine.
• While the queue is not full

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Modelling the queue full state

• When the queue is full
• One RTT later the sources are informed of
  congestion

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TCP congestion control
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TCP congestion control Example (Hespanha)
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TCP congestion control Example (Fairness)