Module 9
Lecture 2
System Identification
Arun K. Tangirala
Department of Chemical Engineering
IIT Madras
July 26, 2013
Module 9
Lecture 2
Arun K. Tangirala
System Identification
July 26, 2013
16
Module 9
Lecture 2
Contents of Lecture 2
In this lecture, we shall learn:
Types of inputs for identification
Pseudo-Random Binary Signal (PRBS)
Procedure for input design
Arun K. Tangirala
System Identification
July 26, 2013
17
Module 9
Lecture 2
Input design: Primary considerations
Inputs for identification
Input should be persistently exciting, i.e., should contain sufficiently many
frequencies.
A signal is persistently exciting of order n if its spectrum is non-zero at n
distinct frequencies (or its covariance matrix consisting of ACFs up to lag n
should be non-singular)
The asymptotic properties of the estimate (bias and variance) depend only on
the input spectrum - not on the actual waveform.
The input must have limited amplitude: umin ≤ u(t) ≤ umax .
Periodic inputs may have certain advantages.
Remember: Covariance matrix is typically inversely proportional to the input
power!
Arun K. Tangirala
System Identification
July 26, 2013
18
Module 9
Lecture 2
Binary symmetric signals
Crest Factor
The desired property of the waveform is defined in terms of
Cr2 =
maxt u2 (t)
N
1 X 2
lim
u (t)
N →∞ N
t=1
A good signal waveform is one that has a small crest factor.
The theoretic lower bound of Cr is 1, which is achieved for binary symmetric
signals
Arun K. Tangirala
System Identification
July 26, 2013
19
Module 9
Lecture 2
Types of inputs
There are different kinds of inputs available for identification. To each its merits
and demerits.
1
White-noise:
I Contains all frequencies uniformly.
I Theoretically a preferable input signal. Decouples the IR parameter estimation
problem. Provides uniform fit at all frequencies
I However, possesses a high crest factor.
2
Random binary:
I Generated by starting with a Gaussian sequence and then passing it through a
filter depending on the input spectrum requirements.
I The sign of the filtered signal is the RBS.
I No proper control over the spectrum. The ’sign’ operation distorts the
spectrum of the input sequence.
I Has the lowest crest factor.
Arun K. Tangirala
System Identification
July 26, 2013
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Module 9
Lecture 2
Types of inputs
3
. . . contd.
Pseudo-RBS:
I Generated using a Linear Feedback Shift Register (LFSR) of n bits. Maximum
length PRBS are 2n − 1 sequences long.
I They possess white-noise like properties.
I Frequency content can be changed by altering the clock sampling rate.
I Has the lowest crest factor.
I Disadvantage: Only maximum length PRBS possess the desired properties.
4
Multisine:
I Multisines are a combination of sinusoids of different frequencies.
I Provides very good estimates of the t.f. at those frequencies.
I However, the spectrum is not continuous. Therefore, the estimates at other
frequencies are not available.
Arun K. Tangirala
System Identification
July 26, 2013
21
Module 9
Lecture 2
PRBS
Binary signals have the lowest crest factor for a given variance.
Remember: The input signal should also satisfy the condition of persistent excitation.
Binary signals with a desired spectral shape can be generated in two ways
1
Random Binary signal: Generated by passing a random Gaussian signal
through a sign function. Disadvantage: There is little control over the
spectrum
2
Pseudo-Random Binary signal: These are deterministic binary signals
that have white-noise like properties
PRBS: u[k] = rem(a1 u[k − 1] + · · · + an u[k − n], 2)
(modulo 2)
I With n-coefficients, one can generate a 2n − 1 full length sequence (zero is
excluded)
I The choice of coefficients (which are zero / non-zero) determines if a full
length or partial length sequence is generated
Arun K. Tangirala
System Identification
July 26, 2013
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Module 9
Lecture 2
Full-length PRBS
For a n-coefficient PRBS, the maximum length sequence that can be generated without
repetition is M = 2n − 1. The table lists the {an }s that have to be non-zero.
Order
2
3
4
5
6
7
8
9
10
11
M = 2n − 1
3
7
15
31
63
127
255
511
1023
2047
Non-zero indices of {an }
1,2
2, 3
1, 4
2, 5
1, 6
3, 7
1, 2, 7, 8
4, 9
7, 10
9, 11
I Observe that the last coefficient has to be non-zero. Other choices of non-zero coefficients
also exist.
I Only full-length PRBS have white-noise like properties!
I MATLAB: uk = idinput(2047,’prbs’,[0 1],[-1 1]); % full-length PRBS
Arun K. Tangirala
System Identification
July 26, 2013
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Module 9
Lecture 2
Band-limited PRBS
To generate band-limited, for example, low-frequency content PRBS, the
full-length sequence is subjected to a simple operation
Re-sample P times faster than the frequency at which the PRBS is generated
Idea is to elongate or stretch the constant portions of PRBS
The resulting signal has the same properties as passing the PRBS through a
simple moving average filter of order P
ũ[k] =
1
(u[k] + u[k − 1] + · · · u[k − P ])
P
MATLAB: uk = idinput(1533,’prbs’,[0 0.3],[-1 1]); % full-length PRBS
Q: Why not pass the full-length PRBS through a simple low-pass filter?
Arun K. Tangirala
System Identification
July 26, 2013
24
Module 9
Lecture 2
Remarks on PRBS
For a given amplitude range, PRBS packs the maximum variance or energy
It is ideally suited only for linear systems
I Since it switches between two states, it cannot detect non-linearities
Change in the initialization only produces a shift in PRBS
I This is due to the periodicity property of PRBS
I Therefore, PRBS is not directly suited for design of uncorrelated inputs
for multivariable systems
For time-varying and non-linear systems, some modifications exist such as
Multi-valued PRBS and Amplitude-Modulated PRBS
Arun K. Tangirala
System Identification
July 26, 2013
25
Module 9
Lecture 2
Preliminary tests
Before exciting the process with the design input sequence, it is useful to perform
preliminary tests:
1. Perform a step test (3% - 10% magnitude) on the system. The step response throws
light on the gain, time constant, delay, inverse response, etc.
2. A step in one direction is insufficient. Perform a step at least test in two directions or
of different magnitudes so as to check for the effects of non-linearities and the range
of linearization.
3. From the step response, identify the effective time-constant τ of the c.t. process
4. Compute the effective bandwidth ΩBW = 1/τ .
5. Use sampling frequency Fs = 1/Ts anywhere between 10 − 20 times ΩBW and the
discrete-time input frequency range as [0, 3.5ΩBW /Fs ].
6. Design an input sequence of the appropriate type (white, rbs, multisine, prbs)
accordingly.
For systems with special frequency response characteristics, the frequency content of the
inputs have to be determined carefully.
Arun K. Tangirala
System Identification
July 26, 2013
26
Module 9
Lecture 2
Experimental design
General guidelines
Choose experimental conditions and inputs such that the predictor becomes
sensitive to parameters of interest and importance.
Choose excitation frequencies and use the input energy in those bands where a
good model is intended and/or where the disturbance activity is significant.
Open loop inputs: Binary, periodic signals with full control over the excitation
energies.
Remember Cov ĜN (eiω ) ≈
n Φvv (ω)
N Φuu (ω)
Sample 10-20 times the bandwidth frequency
Arun K. Tangirala
System Identification
July 26, 2013
27
Module 9
Lecture 2
Summary
Different types of inputs can be used in identification. While the actual
choice depends on the application, some general guidelines are available:
i.
ii.
iii.
iv.
Input should have maximum power relative to noise (high SNR)
Low amplitude to prevent the process from getting into non-linear regimes
Maximum crest factor
Periodic inputs may be advantageous in certain applications
In designing an input, the bandwidth of the system and ability to shape the
spectral content are important
Binary signals have maximum crest factor (of unity) for a given amplitude.
PRBS is widely used in linear identification.
I For non-linear and time-varying systems, variants of PRBS are used.
Some preliminary tests (unless prior process knowledge is available) are
inevitable before designing an appropriate input
Arun K. Tangirala
System Identification
July 26, 2013
28