Assignment:

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Assignment http//www.personal.reading.ac.uk/s
is01xh/teaching/CS3assignment.htm The assignment
has three parts, all relating to least squares
system identification. Part A develops software
to do recursive least squares identification.
Part B provides some real data and allows you to
identify a mechanical "master" robot of a
master/slave tele-manipulator. Part C (worth 20
of the marks) extends the RLS algorithm to either
include instrument variables or a PLS noise model
based on the Moving Average Model, or both. You
can treat part C as optional. It is recommended
that you use Matlab for the assignment, although
you will not be penalised for using other
programming languages.
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Files WWW/Central university computer Matlab and
data files are available at WWW address
http//www.personal.rdg.ac.uk/sis01xh/teachin
g/Sysid/datafiles/Datasets.htm Here you will
also find the files loadass.m (will load the data
into a suitable vector), crib.m (available as a
skeleton for your program) and dord2.m (a way of
generating a test system) Copy this program into
an appropriate directory and run it under Matlab.
Enter a file name and you will then have a
variable y in the Matlab workspace containing the
corresponding response data.
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Skeleton code for recursive least squares
estimate (see crib.m) A skeleton for a
recursive identification algorithm. Since the
assignment assumes an ARX model this code does
likewise. You need to set Na, Nb (number of a
and b coefficients) and LN (large number). Nc
is not needed if this is an ARX model. You also
need to supply the algorithm! The code
assumes that the data is in a vector y and the
input is in a vector u theta_nminus1zeros(NaN
b,1) Initialise the estimate of theta to
zero P_nminus1LN.eye(NaNb) Initialise P
where LN is a large number Theta history
of theta starts here Step through data and for
each new point generate a new estimate for n110
Change 10 to length(y) once you have
the code working set py to the previous Na y
values pyzeros(1,Na) for in-1-1n-Na
if igt0 py(n-i)y(i) end end
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set pu to the previous Nb u values
puzeros(1,Nb) for in-1-1n-Nb if igt0
pu(n-i)u(i) end end Construct varphi from
py' and pu' varphi ... Use varphi(n), y(n)
theta(n-1) and P(n-1) to iterate the next
estimate epsilon ... P P_nminus1 - ... K
... thetatheta_nminus1 ... get ready for
the new iteration theta_nminus1theta
P_nminus1P To get a history of theta set
ThetaTheta theta' end and so it ends If
you have recorded parameter evolution you can
plot with plot(Theta)
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General form of Recursive algorithms where
is a vector of model parameters
is the difference between the measured
output and the estimated output at time n
is the scaling - sometimes known as the
Kalman Gain Supose we have all the data
collected upto time n. Then consider the
formation of the matrix at time n as
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Derivation of Recursive Least Squares Given that
is the collection Thus the
least squares solution is Now what happens
when we increase n by 1, when a new data point
comes in, we need to re-estimate this
requires repetitions calculations and
recalculating the inverse (expensive in computer
time and storage)
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Lets look at the expression
and and define
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(1)
(2)
The least squares estimate at data n
(3)
(4)
( Substitute (4) into (3) )
( Applying (1) )
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RLS Equations are
But we still require a matrix inverse to be
calculated in (8)