TCP congestion control

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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)
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Modelling of dynamic systems Part 3System
Identification
Robert N. Shorten Douglas Leith The Hamilton
Institute NUI Maynooth
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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
• Validate the parameters (later)
• Solve the equations to use the model for
  prediction and analysis (now)

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
• Experiment design
• Parameter estimation
• Validate the parameters (later)
• Solve the equations to use the model for
  prediction and analysis (now)

Introduction
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What is parameter estimation?

• Parameter identification is the identification of
  the unknown parameters of a given model.
• Usually this involves two steps. The first step
  is concerned with obtaining data to allow us to
  identify the model parameters.
• The second step usually involved using some
  mathematical technique to infer the parameters
  from the observed data.

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Linear in parameter model structures

• The parameter estimation task is simple when the
  model is a linear in parameters model form.
• For example, in the equation
• the unknown parameters appear as coefficients
  of the
• variables (and offset).
• The parameters of such equations are estimated
  using the principle of least squares.

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The principle of least squares

• Karl Friedrick Gauss (the greatest mathematician
  after Hamilton) invented the principle of least
  squares to determine the orbits of planets and
  asteroids.
• Gauss stated that the parameters of the models
  should be chosen such that the sum of the
  squares of the differences between the actually
  computed values is a minimum.
• For linear in parameter models this principle can
  be applied easily.

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The principle of least squares

• Karl Friedrick Gauss (the greatest mathematician
  after Hamilton) invented the principle of least
  squares to determine the orbits of planets and
  asteroids.
• Gauss stated that the parameters of the models
  should be chosen such that the sum of the
  squares of the differences between the actually
  computed values is a minimum.
• For linear in parameter models this principle can
  be applied easily.

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The principle of least squares
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The principle of least squares The algebra

• For our example we want to minimize
• Hence, we need to solve

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The principle of least squares The algebra

• For our example we want to minimize
• Hence, we need to solve the following equations
  for the parameters a,b.

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A linear model

• Example Find the least squares line that fits
  the following data points.

X Y -1 10 0 9 1 7 2 5 3
4 4 3 5 0 6 -1
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A linear model

• Example Find the least squares line that fits
  the following data points.

X Y -1 10 0 9 1 7 2 5 3
4 4 3 5 0 6 -1
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A linear model

• Example Find the least squares line that fits
  the following data points.

X Y -1 10 0 9 1 7 2 5 3
4 4 3 5 0 6 -1
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A linear model

• Example Find the least squares line that fits
  the following data points.

X Y -1 10 0 9 1 7 2 5 3
4 4 3 5 0 6 -1
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A polynomial model

• Least squares can be used whenever we suspect a
  linear in parameters model?Find the least squares
  polynomial fit to the following data points.

X Y 1.0000 2.9218 2.0000
5.9218 3.0000 10.9218 4.0000 17.9218
5.0000 26.9218 6.0000 37.9218
7.0000 50.9218 8.0000 65.9218 9.0000
82.9218 10.0000 101.9218
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A polynomial model

• By proceeding exactly as before

X Y 1.0000 2.9218 2.0000
5.9218 3.0000 10.9218 4.0000 17.9218
5.0000 26.9218 6.0000 37.9218
7.0000 50.9218 8.0000 65.9218 9.0000
82.9218 10.0000 101.9218
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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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An exponential model (the first lecture)

• The solution to the differential equation is not
  linear in parameters.
• However, there is a change of variables to make
  it linear in parameters.

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Matrix formulation of least squares

• The least squares parameters can be derived by
  solving a set of simultaneous linear equations.
  This technique is effective but tedious for
  complicated linear in parameter models. A much
  more effective solution to the least squares
  problem can be found using matrices.
• Suppose that we wish to find the parameters of
  the following linear in parameters model and that
  we have m measurements.

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Matrix formulation of least squares

• All m-measurements can be written in matrix form
  as follows
• or more compactly as

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Matrix formulation of least squares

• The matrix is known as the matrix of
  regressors.This matrix (here a mx3 matrix) is
  usually not invertible. To find the least squares
  solution we multiply both sizes of the equation
  by the transpose of the regressor matrix.
• It can be shown that the least squares solution
  is given by the above equation.

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A linear model

• Example Find the least squares line that fits
  the following data points.

X Y -1 10 0 9 1 7 2 5 3
4 4 3 5 0 6 -1
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A linear model

• The regressor is given by
• Hence

reg -1 1 0 1 1 1 2
1 3 1 4 1 5 1 6
1
reg'reg 92 20 20 8
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Summary Linear least squares

• To do a least squares fit we start by expanding
  the unknown function as a linear sum of basis
  functions
• We have seen that the basis functions can be
  linear or non-linear. The linear parameters can
  be found using

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Discrete time dynamic systems

• Our examples work beautifully for static systems.
  What about identifying the parameters of dynamic
  systems. Dynamic systems are in principle not any
  different to static systems. We define our
  regressors and solve the regression problem.
• Consider the following problem. We wish to build
  a model of the relationship between the throttle
  and the speed of an automobile. We begin by
  collecting data from an experiment.

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Discrete time dynamic systems
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Discrete time dynamic systems

• A good choice for the model structure is first
  order
• We can solve for the parameters by solving
• yielding

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Recursive identification

• The algorithms that we have looked at so far are
  called batch algorithms.
• Sometime we want to estimate model parameters
  recursively so that the parameters can be
  estimated on-line.
• Also, if system parameters change over time, then
  we need to continually estimate and verify the
  model parameters.

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Recursive least squares

• The least squares algorithm invented by Gauss can
  be arranged in such a way such that the results
  obtained at time index k-1 can be used to obtain
  the parameter estimates at time index k. To see
  this we use
• and note that

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Recursive least squares

• With a little manipulation (show) we get
• where
• More complicated versions of the algorithm are
  available that avoid matrix inversion.

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Recursive least squares (car example)
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Recursive least squares (car example)
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The matrix inversion lemma

• One not-so-nice feature of the RLS formula is the
  presence of a matrix inversion at each step. This
  can be removed using the matrix inversion lemma
  (the Sherman-Morrisson formula).

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The RLS algorithm

• Application of the lemma results in the standard
  RLS algorithm.

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Time-varying systems

• Much of the appeal of the RLS algorithm is that
  we can potentially deal with time-varying
  systems.
• Example Suppose that a rocket ascends from the
  surface of the earth propelled a thrust force
  generated through the ejection of mass. If we
  assume that the rate of change of mass of the
  fuel is um and the exhaust velocity is ve, then
  the physical equations governing the rocket is

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Forgetting factors

• For time varying systems we must estimate the
  parameters recursively. How can we modify the
  basic RLS algorithm?
• To estimate time-varying parameters we would like
  to forget past data points. The only place in the
  above formula that depends on past data points is
  the covariance matrix.

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Forgetting factors

• For time varying systems we must estimate the
  parameters recursively. How can we modify the
  basic RLS algorithm?
• This corresponds to minimising the time-varying
  cost function

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The RLS algorithm

• Application of the matrix inversion lemma results
  in the standard RLS algorithm with a forgetting
  factor.

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Example

• Consider the dynamic system
• where the parameters ak, bk vary as shown.

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Example
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Numerical issues

• RLS algorithm is of great theoretical importance.
  However, it suffers from one very big
  disadvantage. It is numerically unstable.
• The numerical instability stems from the
  equation
• If no information enters the systems, P becomes
  singular and the estimator returns garbage.

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Numerical issues
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Persistence of excitation

• One final thought Persistence of excitation
• Persistence of excitation has a strict
  mathematical definition.
• Roughly speaking, PE means that the input signal
  has been chosen such that the least squares
  estimate is unique.
• The really interested student should consult
  Astrom for more on this topic.

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Error surfaces and gradient methods

• All the examples that we have looked at so far
  involved linear in parameter models. In this case
  finding the least squares solution was easy
  because the error surface is quadratic.
• Huh! What is meant by a quadratic cost function.
• Consider the examples of the line fitting. We
  were trying to minimize

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Least mean squares and gradient methods

• To make life simple, lets assume that we have
  two observations (m2) and that we assume that b
  0. Then.
• Remember we are trying to find the parameter a
  that minimises this function. But the function is
  quadratic in a.

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Least mean squares and gradient methods

• The quadratic surface looks like the following
  for a single parameter.

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Least mean squares and gradient methods

• With two parameters we get some thing like

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A word on gradient methods

• Another way of estimating the best parameters is
  to estimate the parameters in an interative
  manner in the direction of the gradient.
• For linear in parameter structures the batch
  version of least squares is better. However, the
  above idea can be extended to deal with model
  structures that are not linear in parameters
  (Doug will tell you all about this).