Time Percent Value Method Determine the approximation of the model in the last example if after examining the t t table the model order is chosen to be 4 instead of 5 t τ Table 5 values of ID: 804195
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Slide1
Slide2Homework 9
Chapter 6
Identification from Step Response
Time Percent Value Method
Determine the approximation of the model in the last example, if after examining the
t
/
t
table, the model order is chosen to be 4 instead of 5.
Slide3t
/
τ
Table
5 values of
t
i
/τ are to be located for n = 4
Result:
Solution to Homework 9
Chapter 6
Identification from Step Response
Slide4Solution
to
Homework 9
Chapter 6
Identification from Step Response
:
5
th
order approximation
:
4th order approximation
Slide5Presentation Assignment
Chapter 6
Least Squares Methods
Remaining SMI Classes:
Friday, 5 April : Lecture and Quiz 3
Wednesday, 10 April : Group Presentations
Friday, 12 April : Discussion and Review
Presentation Assignment:
Prepare a short presentation (max 6 slides). Use the SMI lecture slide as template.
The topic is “
Identification of Second Order System Using Step Response”.
The presentation must consist of theoretical explanation, formulas used, and at least one calculation example.
Each group must present different method.The weight
of the presentation will be equals to 2 homeworks.
The groups are:
Andre and DeoBurawi and JagatIndah and Arwin
Arief and Yeza
Wilbert and Keanu
Slide6Least Squares Methods
Chapter 6
Least Squares Methods
The Least Squares Methods are based on the minimization of squares of errors.
The errors are defined as the difference between the measured value and the estimated value of the process output, or between
y
(
k) and y(k
). There are two version of the methods: batch version and recursive version.
^
Slide7Least Squares Methods
Chapter 6
Least Squares Methods
Consider the discrete-time transfer function in the form of:
The aim of Least Squares (LS) Methods is to identify the parameters
a
1
, ...,
a
n
,
b1, ..., b
m from the knowledge of process inputs u(
k) and process output y(
k).As described by the transfer function above, the relation of process inputs and process outputs is:
Slide8Least Squares Methods
Chapter 6
Least Squares Methods
This relation can be written in matrix notation as:
where:
Vector of Parameters
Vector of Measured Data
Hence, the identification problem in this case is how to find
θ
based on the actual process output
y
(
k
) and the measured data from the past
m
(
k
).
Slide9Least Squares Methods
Chapter 6
Least Squares Methods
Assuming that the measurement was done for
k
times,
with the condition k ≥ n +
m, then k equations can be constructed as:
or:
Slide10If
M
is nonsingular, then the direct solution can be calculated as:
In this method, error is minimized as a linear function of the parameter vector.
The disadvantage of this solution is, that error can be abruptly larger for
t
>
k
.
Least Squares Methods
Chapter 6
Least Squares Methods
Least Error (LE) Method, Batch Version
Slide11Least Squares Methods
Chapter 6
Least Squares Methods
A better way to calculate the parameter estimate
θ
is to find the parameter set that will minimize the sum of squares of errors between the measured outputs
y
(k) and the model outputs y(k) =
mT(k)θ
^
The extreme of
J
with respect to
θ
is found when:
Slide12The derivation of
J
(θ) with respect to θ
can be calculated as:
Least Squares Methods
Chapter 6
Least Squares Methods
if
A
symmetric
Least Squares (LS) Method, Batch Version
Slide13Performing the “Second Derivative Test”,
Least Squares Methods
Chapter 6
Least Squares Methods
Second Derivative Test
If
f
’(x) = 0 and f ”(x) > 0 then
f has a local minimum at x
If f
’(x) = 0 and
f ”(
x) < 0 then f has a local maximum at x
If f
’(x) = 0 and f ”(x) = 0 then no conclusion can be drawn
Always positive definite
is a solution that will minimize the squares of errors
Slide14In order to guarantee that
M
TM is invertible, the number of row of
M
must be at least equal to the number of its column, which is again the number of parameters to be identified.
More row of
M
increase the accuracy of the calculation. In other words, the number of data row does not have to be the same as the sum of the order of numerator and denominator of the model to be identified.If possible, rows with any value assumed to be zero (because no measurement data exist) should not be used.Least Squares Methods
Chapter 6Least Squares Methods
Slide15The parameters of a model with the structure of:
Example: Least Squares Methods
Chapter 6
Least Squares Methods
are to be identified out of the following measurement data:
Perform the batch version of the Least Squares Methods to find out
a
1
,
a
2
, and
b
2
.
Hint
: n + m = 2 + 1
At least 3 measurements must be available/ utilized. Hint: If possible, avoid to many zeros due to unavailable data for u(
k) = 0 and y(k) = 0,
k < 0.
Slide16Using the least allowable data, from
k
= 2 to
k
= 4, the matrices
Y
and
M can be constructed as:
Example: Least Squares Methods
Chapter 6
Least Squares Methods
Slide17Example: Least Squares Methods
Chapter 6
Least Squares Methods
Slide18Homework 10
Chapter 6
Least Squares Methods
Redo the example, utilizing as many data as possible.
Does your result differ from the result given in the slide?
What could be the reason for that? Which result is more accurate?
Slide19Homework
10A
Chapter 6
Least Squares Methods
Redo the example,
utilizing least allowable data
, if the structure of the model is chosen to be
After you found the three parameters
a
1
,
a2, and b
1, for G2
(z), use Matlab/
Simulink to calculate the response of both G1(
z) and G2(z) if they are given the sequence of input as given before.
Compare
y
(k) from Slide 10/15
with y1(k) and y2(k) from the outputs of the transfer functions G
1(z) and G
2(z). Give analysis and conclusions.
(
Andre,
Burawi, Arwin,
Indah, Wilbert)
(
Arief,
Deo,
Keanu, Yeza
, Jagat)
Deadline: Thursday, 4 April 2019.