System Identification
6.435
SET 4
–Input Design
–Persistence of Excitation
–Pseudo-random Sequences
Munther A. Dahleh
Lecture 4
6.435, System Identification
Prof. Munther A. Dahleh
1
Input Signals
• Commonly used signals
– Step function
– Pseudorandom binary sequence (PRBS)
– Autoregressive, moving average process
– Periodic signals: sum of sinusoids
• Notion of “sufficient excitation”. Conditions !
• Degeneracy of input design.
• Relations between PBRS & white noise.
• Frequency domain properties of such signals.
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6.435, System Identification
Prof. Munther A. Dahleh
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Examples
• Step input
• A Pseudorandom binary sequence
– periodic signal
– switches between two levels in a certain fashion
– levels = ± a period = M
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6.435, System Identification
Prof. Munther A. Dahleh
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• Autoregressive moving average
is a random sequence
{ARMA process}
•
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6.435, System Identification
Prof. Munther A. Dahleh
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Spectral Properties
• PRBS [takes on 0,1]
all calculations are mod 2.
are either 0 or 1
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6.435, System Identification
Prof. Munther A. Dahleh
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• Covariance function
and evaluating
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,
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Prof. Munther A. Dahleh
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• ARMA • Sum of sinusoids
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6.435, System Identification
Prof. Munther A. Dahleh
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PRBS vs. WN
• Given any smooth function
• Approximate the integral
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by Riemann sum.
6.435, System Identification
Prof. Munther A. Dahleh
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• The spectrum of PRBS approximate WN as distributions. • Homework: = WN
= PRBS
Compare
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6.435, System Identification
Prof. Munther A. Dahleh
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Persistent Excitation
Definition:
A quasi-stationary input, u, is persistently exciting of order n
if the matrix
is positive definite.
• Recall: The correlation method in the time-domain required
the inversion of
to estimate M-parameters of the
impulse response.
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6.435, System Identification
Prof. Munther A. Dahleh
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Relation to the Spectrum
Theorem:
Let u be a quasi-stationary input of dimension nu, with
spectrum
. Assume that
for at least n
distinct frequencies. Then u is p.e. of order n.
Proof:
Let
be a n
nu
that
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xnu row vector such
nu
. Define
6.435, System Identification
Prof. Munther A. Dahleh
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Then,
But
frequencies
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at n distinct
at these frequencies = g = 0.
6.435, System Identification
Prof. Munther A. Dahleh
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Theorem (Scalar):
If u is p.e of order n
for at least n-points.
Proof:
Suppose
for at most
points.
Let g be any vector with
Pick a vector
at the
points where
at some other frequency.
then
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6.435, System Identification
Prof. Munther A. Dahleh
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Examples
• Step input: persistently exciting of order 1
• PRBS:
(Verify!)
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6.435, System Identification
Prof. Munther A. Dahleh
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singular
1st and last row
are the same.
PRBS is p.e. of order M.
• ARMA Process is p.e. of any order.
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6.435, System Identification
Prof. Munther A. Dahleh
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• Sum of sinusoids
is non-zero at exactly n-points, where
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6.435, System Identification
Prof. Munther A. Dahleh
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Theorem:
is quasi-stationary. Define
Then
is persistently exciting of order n.
Proof:
Define
equivalence established.
Lecture 4
6.435, System Identification
Prof. Munther A. Dahleh
17
Spectrum of Filtered Signals
Generation of a process
with a given Covariance:
SISOO
given
If
signal
then x is the
output of a filter with a WN
signal as an input. The Filter is
the spectral factor of
SISO
stable minimum phase.
Lecture 4
6.435, System Identification
Prof. Munther A. Dahleh
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Important relations
A model for noisy outputs:
quasi-stationary
constant.
•
•
•
Very Important relations in system ID.
Correlation methods are central in identifying an unknown plant.
Proofs: Messy; Straight forward.
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6.435, System Identification
Prof. Munther A. Dahleh
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Ergodicity
•
is a stochastic process
Sample function
or a realization
•
Sample mean
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6.435, System Identification
Prof. Munther A. Dahleh
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•
Sample Covariance
• A process is 2nd-order ergodic if
mean
covariance
•
Sample averages
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the sample mean of any realization.
the sample covariance
of any realization.
Ensemble averages
6.435, System Identification
Prof. Munther A. Dahleh
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A general ergodic process
+
WN Signal
uniformly stable
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quasi–stationary signal
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Prof. Munther A. Dahleh
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w.p.1
w.p.1
w.p.1
Remark:
Most of our computations will depend on a given realization of
a quasi-stationary process. Ergodicity will allow us to make
statements about repeated experiments.
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6.435, System Identification
Prof. Munther A. Dahleh
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