System Identification
6.435
SET 10
– Instrumental Variable Methods
– Identification in Closed Loop
– Asymptotic Results
Munther A. Dahleh
Lecture 10
6.435, System Identification
Prof. Munther A. Dahleh
1
Instrumental Variable Method
• Connects parametric methods and correlation methods.
• Least squares revisited:
satisfies
• We can arrive to this by correlating both sides with
Lecture 10
6.435, System Identification
Prof. Munther A. Dahleh
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• If
is not white, the above method will yield a biased estimate.
• Main Idea: Correlate with a vector
from
•
which is uncorrelated
Instrument
• If experiment is open loop:
is constructed from u.
• In closed loop,
Lecture 10
is constructed from reference signals.
6.435, System Identification
Prof. Munther A. Dahleh
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• Detail:
• Define:
• Extended IV
Filtered error
Lecture 10
6.435, System Identification
Prof. Munther A. Dahleh
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some function
• In LS
• If
Lecture 10
has an inverse, then
6.435, System Identification
Prof. Munther A. Dahleh
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Choice of Instrument
System:
“v may not be white”
Open Loop Operation
u & v are
independent
Define the instrument as a function of u
Lecture 10
6.435, System Identification
Prof. Munther A. Dahleh
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To solve for
Recall:
If
order
Lecture 10
, we need the inverse of
. The inverse exists if u is p.e. of
(or order
if
)
6.435, System Identification
Prof. Munther A. Dahleh
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Convergence Results
• Assume Data generated in closed loop as before. Define
(Past Data)
• Results:
w.p.1
uniformly in
w.p.1
Lecture 10
6.435, System Identification
Prof. Munther A. Dahleh
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Consistency
Ass.
Let
Question: Is
True if
?
is non singular
Conditions under which
Lecture 10
is non singular are hard to find.
6.435, System Identification
Prof. Munther A. Dahleh
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Theorem: Suppose the true system
has degrees
Suppose the model
, with degrees
Also, suppose
the instrument is given by
with
Then:
1) If
is singular
2) If
is singular.
with
Lecture 10
6.435, System Identification
Prof. Munther A. Dahleh
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Proof: Notice that
&
where
is
(noiseless)
Since
(pole/zero cancellation at
is singular
Lecture 10
[ similarly for
6.435, System Identification
Prof. Munther A. Dahleh
)
]
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Result:
If
(or u is p.e.)
Then
is generically non singular.
Proof: The set of parameters that satisfy singularity have
measure zero.
Lecture 10
6.435, System Identification
Prof. Munther A. Dahleh
12
Asymptotic Distribution
Suppose
is non singular
Lecture 10
6.435, System Identification
Prof. Munther A. Dahleh
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Lecture 10
6.435, System Identification
Prof. Munther A. Dahleh
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Frequency Domain Characterization
⇒ a different kind of
fit!
Lecture 10
6.435, System Identification
Prof. Munther A. Dahleh
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