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Module 9 Lecture 2

System Identification

Arun K. Tangirala

Department of Chemical Engineering

IIT Madras

July 26, 2013

Module 9

Lecture 2Arun K. Tangirala System Identification July 26, 2013 16
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Module 9 Lecture 2

Contents of Lecture 2

In this lecture, we shall learn:

Types of inputs for identificationPseudo-Random Binary Signal (PRBS)

Procedure for input design

Arun K. Tangirala System Identification July 26, 2013 17
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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!

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Module 9 Lecture 2

Binary symmetric signals

Crest Factor

The desired property of the waveform is defined in terms of

C2r =maxt u

2(t)

limN

1N

N

t=1

u2(t)

A good signal waveform is one that has a small crest factor.

The theoretic lower bound ofCr is 1, which is achieved for binary symmetric

signals

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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:

Contains all frequencies uniformly.

Theoretically a preferable input signal. Decouples the IR parameter estimation

problem. Provides uniform fit at all frequencies

However, possesses a high crest factor.

2 Random binary:

Generated by starting with a Gaussian sequence and then passing it through a

filter depending on the input spectrum requirements.

The sign of the filtered signal is the RBS.

No proper control over the spectrum. The sign operation distorts the

spectrum of the input sequence.

Has the lowest crest factor.


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Module 9 Lecture 2

Types of inputs . . . contd.

3 Pseudo-RBS:

Generated using a Linear Feedback Shift Register (LFSR) ofn bits. Maximum

length PRBS are 2n 1 sequences long.

They possess white-noise like properties.

Frequency content can be changed by altering the clock sampling rate.

Has the lowest crest factor.

Disadvantage: Only maximum length PRBS possess the desired properties.

4 Multisine:

Multisines are a combination of sinusoids of different frequencies.

Provides very good estimates of the t.f. at those frequencies.

However, the spectrum is not continuous. Therefore, the estimates at other

frequencies are not available.


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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

spectrum2 Pseudo-Random Binary signal: These are deterministic binary signals

that have white-noise like properties

PRBS: u[k] = rem(a1u[k 1] + + anu[k n], 2) (modulo 2)

With n-coefficients, one can generate a 2n 1 full length sequence (zero is

excluded)

The choice of coefficients (which are zero / non-zero) determines if a full

length or partial length sequence is generated


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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 M = 2n 1 Non-zero indices of{an}

2 3 1,2

3 7 2, 3

4 15 1, 4

5 31 2, 5

6 63 1, 6

7 127 3, 7

8 255 1, 2, 7, 8

9 511 4, 9

10 1023 7, 10

11 2047 9, 11

Observe that the last coefficient has to be non-zero. Other choices of non-zero coefficients

also exist.

Only full-length PRBS have white-noise like properties!

MATLAB: uk = idinput(2047,prbs,[0 1],[-1 1]); % full-length PRBSArun K. Tangirala System Identification July 26, 2013 23

Module 9 Lecture 2
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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

u[k] =

1

P(u[k] + u[k 1] + u[k 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?


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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

Since it switches between two states, it cannot detect non-linearities

Change in the initialization only produces a shift in PRBS

This is due to the periodicity property of PRBS

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


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u 9 u

Preliminary tests

Before exciting the process with the design input sequence, it is useful to performpreliminary 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 of the c.t. process

4. Compute the effective bandwidth BW = 1/.

5. Use sampling frequency Fs = 1/Ts anywhere between 10 20 times BW and the

discrete-time input frequency range as [0, 3.5BW/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.


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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 GN(ei)

n

N

vv()

uu()

Sample 10-20 times the bandwidth frequency


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Summary

Different types of inputs can be used in identification. While the actualchoice depends on the application, some general guidelines are available:

i. Input should have maximum power relative to noise (high SNR)

ii. Low amplitude to prevent the process from getting into non-linear regimes

iii. Maximum crest factor

iv. 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.

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

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