and spectral analysis, march 5, 1998 - facultad - fceia › isis › inpcrasp.pdf · time [min]...

ChE 494/598, EEE 598 Intro to System Identification: Input Signals, Correlation and Spectral Analysis, March 5, 1998

© Copyright 1997, 1998 by D.E. Rivera, All Rights Reserved

Daniel E. Rivera, Associate Professor

Department of Chemical, Bio and Materials Engineeringand

Control Systems Engineering LaboratoryArizona State UniversityTempe AZ 85287-6006

[email protected], (602) 965-9476© Copyright, 1997,1998

System Identification for Process Control:A Short Course: Module Two

Principal Sources of Error in System Identification

• BIAS. Systematic errors caused by

- input signal characteristics (i.e., excitation) - choice of model structure

- mode of operation (i.e., closed-loop instead of open-loop)

• VARIANCE. Random errors introduced by the presence of noise in the data, which do not allow the model to exactly reproduce the plant output. It is affected by the following factors:

- number of model parameters

- duration of the identification test

- signal-to-noise ratio

ERROR = BIAS + VARIANCE

System Identification Structure, Revisited

white noise signal

Input Signal (Random orDeterministic)

Output Signal (random, autocorrelated)

u

yP(z)

a

++

H(z)

υDisturbance Signal (random, autocorrelated)

• persistently exciting with as much power as possible;• uncorrelated with the disturbance (desirable from a practical

standpoint)• as long as possible

How to Lower Variance

Var ∝ n

N

Φν(ω)Φu(ω)

Φu ≡ input power spectrumΦν ≡ disturbance power spectrumn ≡ number of model parametersN ≡ number of data points

Reducing the number of estimated model parameters,increasing the length of the data set , and increasing the power of the input signal all contribute to variance reduction in system identification

1



"Plant Friendly" Input Signal Design

• be as short as possible

• not take actuators to limits, or exceed move size restrictions

• cause minimum disruption to the controlled variables (i.e., low variance, small deviations from setpoint)

A plant friendly input signal should:

Note that theoretical requirements may strongly conflict with "plant-friendly" operation!

Inputs to Consider• Single Step

• Single Pulse ("Nonzero Mean" Pulse)

• Double Pulse ("Zero Mean" Pulse)

• Gaussian White Noise

• Random Binary Signal (RBS)

• Pseudo-Random Binary Signal (PRBS)

• Schroeder-phased Multisinusoidal Inputs

Single Stepu(s) =

a

su(z) =

a

1− z−1

The step input emphasizes low frequencies (steady-state/final time behavior)

0 10 20 30 40 50 60 70 80 90 100

0

0.2

0.4

0.6

0.8

1

Time [Min]

Step Input - Time Domain

10-2

10-1

100

101

10-1

100

101

102

103

Radians/Min

AR

Step Input - Power Spectral Density

Single Pulse Inputu(s) =

a(1− e−∆ts)s

u(z) =a(1− z−N)

1− z−1 N = ∆t/T

Narrowing the pulse makes the signal resemble an impulse input; widening the pulse makes the input resemble a step.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

Time [Min]

Nonzero Mean Pulse Input - Time Domain

10-1

100

101

102

10-2

10-1

100

101

102

Radians/Min

AR

Power Spectral Density for the PULSE input

We will also call this "the nonzero mean pulse"

2



Double Pulse Inputu(s) =

a(1− 2e−∆ts + e−2∆ts)s

u(z) =a(1− 2z−N + z−2N)

1− z−1 N = ∆t/T

The double pulse attenuates low frequencies, which may lead to problems if the pulse width is not made long enough.

We will also call this "the zero mean pulse"

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.5

0

0.5

1

Time [Min]

Zero Mean Pulse Input - Time Domain

10-1

100

101

102

10-4

10-2

100

102

Radians/Min

AR

Power Spectral Density for the PULSE input

White Noise, Revisited

tx = ta ta = N(0, a2σ )

1txta

0

0.5

1

1.5

0 1 2 3

Frequency [Radians]

phi

POWER SPECTRUM OF WHITE NOISE

0

0.5

1

1.5

0 2 4

Lag

rhok

AUTOCORRELATION OF WHITE NOISE

kρ = kγa2σ a

2σ = 1Φ(ω )

ω

the white noise input , by definition, meets persistent excitation requirements

Gaussian White Noise vs. Random Binary Sequences

A Gaussian white noise signal fails to meet the requirements of "plant-friendliness"; the random binary

sequence represents a reasonable alternative

-3

-2

-1

0

1

2

3

0 0.05 0.1 0.15 0.2

NORMAL DISTRIBUTION [mean=0;Var=1]

x

pdf p(x)

-3

-2

-1

0

1

2

3

20 40 60 80 100

Samples

x

SAMPLES OF RANDOM NORMAL DATA +3σ

+σ

−3σ

+2σ

−σ−2σ

} }}Probability

%

68.3

95.5

99.9

Pseudo-Random Binary Sequence

1 0 1 1 0 1 1 0 1

1 nr

Exclusive OR(Modulo 2 Adder)

Test Signal

Shift Registers

The PRBS is a periodic, deterministic input which can be generated using shift registers and Boolean algebra

The main design variables are switching time (Tsw), number of shift registers (nr), and signal amplitude

3



PRBS, continued

PRBS design for Tsampl = 1, Tsw = 3, n (registers) = 4, and signal magnitude = +/- 1.0. One cycle duration is 45 minutes long.

0 5 10 15 20 25 30 35 40 45-1

-0.5

0

0.5

1

One cycle of the PRBS time input signal

Time[Min]

100

10-3

10-2

10-1

100

Radians/Min

AR

Power Spectrum of the PRBS input

PRBS Generation Info

0 1 = 10 0 = 01 1 = 01 0 = 1

XOR

Stages Feedback Registers

2 1,2 3 2,3 4 3,4 5 3,5 6 5,6 7 4,7 8 4,5,6,8 9 5,9

Note: using feeback registers other than those indicated may not result in a sequence of "maximum length"

PRBS Autocovariance Function

-Tsw +Tsw −a2 / N

+a2 +a2

0 NTsw

Ruu (kTsw)

Delay

The autocovariance function for the PRBS is periodic and "resembles" that of white noise

100

101

102

103

104

10-3 10-2 10-1 100 101

Frequency [Radians/time]

|u|

ω∗ ω∗

PRBS Design Variables

The power spectrum of a PRBS signal is given by

Φu(ω) =a2(N + 1)Tsw

N

sin(ωTsw2 )ωTsw

2

2

where a is the amplitude of the PRBS signal, Tsw is the switching time, andN = 2nr − 1, nr is the number of shift registers. The frequency range of a PRBSsignal “useful” for system excitation is

ω∗ =2πNTsw

≤ ω ≤ 2.8Tsw

= ω∗

4



100

101

102

103

104

10-3 10-2 10-1 100 101


|u|

ω∗ ω∗

PRBS Design GuidelinesPRBS Design Guidelines

ω∗ ≤1

βsτHdom≤ ω ≤ αs

τLdom≤ ω∗

τLdom ≡ low estimate of dominant time constantτHdom ≡ high estimate of dominant time constantβs ≡ factor representing settling time of processαs ≡ factor representing closed-loop speed

as a multiple of the open-loop response time

Tsw ≤2.8τLdom

αsNs = 2nr − 1 ≥ 2πβsτHdom

Tsw

• nr and Ns must be integer values;

• Tsw must be an integer multiple of the sampling time T .

Random Binary Signals/SequencesThe discrete-interval random binary sequence (RBS) is a two levelsignal which switches between the levels −a and a at regularly spacedintervals of time with probability p.function [signal] = randbin(len,p)%% RANDBIN: binary (-1 1) random signal% of length ’len’ and transition probability ’p’%state = -1; % initial statesignal = zeros(len,1);rand(’uniform’);

for i = 1:lenx = rand(1,1);if x<p

state = -state;end;signal(i,1) = state;

end;

When the transition probability is p = 0.5, the asymptotic expressionof the theoretical PSD is:

Φu(ω) = a2Tswsin2(ωTsw/2)

(ωTsw/2)2

RBS Power Spectrum - Asymptotic

100

101

102

103

104

10-3

10-2

10-1 10

0 10

1


ω∗ ω∗

RBS Example

0 50 100 150 200 250 300-1

-0.5

0

0.5

1

Realization of the Random-Binary time input signal

Time[Min]

10-3

10-2

10-1

100

10-3

10-2

10-1

100

101

Power Spectrum of the Random-Binary Input

Frequency [Radians/Min]

sampling time = 1, switching time = 3 min, 300 samples, probability of switching = 0.5, signal magnitude = +/- 1.0, window size for smoothing PSD = 32

5



Schroeder-Phased Input Signals

us(k) = λns∑i=1

√2αi cos(ωikT + φi)

T ≡ sampling timeNs ≡ sequence lengthns ≡ no. of sinusoids, ns ≤ Ns/2

αi ≡ relative power ns∑i=1αi = 1

ωi = 2πi/NsT

φi = 2πi∑

j=1jαj

λ ≡ scaling factor to insure signal meets ± usat

The Schroeder input is a periodic, deterministic signal composed of a sum of sinusoids according to:

"Low-Pass" Schroeder-Phased Input Design

u 2

2πNsT

πT

α i nonzero between ω* and ω*

α i = 0 from ω* to πT

ω*

ω*

ω∗ ≤1

βsτdom≤ ω ≤ αs

τdom≤ ω∗

τdom ≡ dominant plant time constantβs ≡ factor representing settling time of processαs ≡ factor representing closed-loop speed

as a multiple of the open-loop response time

Ns ≥2πβsτdom

T

ns ≥NsTαs2πτdom

PowerSpectrum

Input Comparison Example

Example. Consider the plant with delay

p(s) =e−s

s + 1

T = 0.3 min, τdom=1.5 min αs = 2 (i.e., 2 times the open-loop bandwidth)and βs = 3 (the 95% settling time)

Schroeder-phased Input

Ns = 96 ns = 7 One Cycle Period: 28.8 min

PRBS Input

Tsw = 2.1 Ns = 15 nr = 4 shift registers

One Cycle Period: 31.5 min

0 5 10 15 20 25 30-1

-0.5

0

0.5

1

One cycle of the PRBS time input signal

Time[Min]

100 101

10-2

100

Radians/Min

AR

Power Spectrum of the PRBS input

PRBS Design Guideline Results

6



Power Spectrum ComparisonSchroeder-phased signal with usat = 1.75 ; PRBS input

with usat = 1.0

10-2

10-1

100

101

10-1 100 101

Power spectra of Schroeder-phased and PRBS Signals

Frequency (rad/min)

Time-Domain Input Realization(1 Period)

-2

-1

0

1

2

0 5 10 15 20 25 30 35

Sample Time (min)

PRBS input signal

-2

-1

0

1

2

0 5 10 15 20 25 30 35

Schroeder-phased input signal

Sample Time (min)

PRBS has lower overall span but requires larger move sizes, in comparison to the Schroeder-phased signal

Time-Domain Output Response

-1

0

1

0 5 10 15 20 25 30 35

PRBS output signal

Sample time (min)

-1

0

1

0 5 10 15 20 25 30 35

Schroeder-phased output signal

Sample time (min)

Output variable deviations are comparable for both signals

Pulse Test ComparisonWe select the width of the "nonzero mean" pulse such that its spectrum resembles that of the Schroeder and PRBS inputs

10-2

10-1

100

101

10-1 100 101

Power Spectra of Schroeder-phased and PRBS and PULSE signals

Frequency (rad/min)

Pulse width is 2.1 minutes, height is 1.5 units

7



Nonzero-Mean Pulse: Input Comparison

-1

0

1

0 5 10 15 20 25 30 35

Pulse Input, width = dT

Time[min]

-1

0

1

0 5 10 15 20 25 30 35

PRBS input signal

Time[min]

Pulse width is 2.1 minutes, height is 1.5 units

Nonzero Mean Pulse:Output Signal Comparison

-1

0

1

0 5 10 15 20 25 30 35

PRBS output time signal

Time [min]

-1

0

1

0 5 10 15 20 25 30 35

PULSE output time signal

Time[min]

Nonzero-Mean Pulse, Revisited

-1

0

1

0 5 10 15 20 25 30 35

Pulse Input, width = dT

Time[min]

-1

0

1

0 5 10 15 20 25 30 35

PRBS input signal

Time[min]

-1

0

1

0 5 10 15 20 25 30 35

PRBS output time signal

Time [min]

-1

0

1

0 5 10 15 20 25 30 35


Time[min]

Increase nonzero mean pulse duration to 6 min, decrease pulse height to 1.0 units.

Power Spectra Comparison

10-2

10-1

100

101

10-1 100 101


Frequency (rad/min)

Nonzero Mean Pulse Spectrum

8



Repeated "Zero-Mean" Pulse

-1

-0.5

0

0.5

1

0 5 10 15 20 25 30 35

Pulse Input, width = 2.1 min

Time[min]

-1

-0.5

0

0.5

1

0 5 10 15 20 25 30 35


Time[min]

Power Spectra Comparison

10-2

10-1

100

101

10-1 100 101


Frequency (rad/min)

Zero mean Pulse Spectrum

Zero-Mean Pulse, Longer Width

-1

-0.5

0

0.5

1

0 5 10 15 20 25 30 35

Pulse Input, width = 6 min

Time[min]

-1

-0.5

0

0.5

1

0 5 10 15 20 25 30 35


Time[min]

Increase zero mean pulse duration to 6 min, decrease overall pulse span to 1.0 units.

Power Spectra Comparison, Zero-Mean Pulse

10-2

10-1

100

101

10-1 100 101


Frequency (rad/min)

Zero mean Pulse Spectrum

9



Nonparametric Estimation Via Correlation Analysis

Benefits of Correlation Analysis:

1. "Structure-free" estimation is a useful precursor to parametric identification (plots of the impulse and step responses obtained from correlation analysis can be used to set delays and model orders in parametric estimation techniques).

2. Can be used to confirm or deny the presence of a transfer function relationship between variables.

3. Feedback relationships in the data can be predicted as well.

Correlation AnalysisCorrelation analysis can be used to estimate the FiniteImpulse Response (FIR) coefficients between u, a candi-date input, and y, a desired output

y(t) = (b0 + b1z−1 + b2z

−2 + . . .+ bnz−n)u(t) + ν(t)

ν is an unmeasured noise term. The main assumptionsbehind correlation analysis are

• stationarity of the time series ut and yt (if nonsta-tionary, one solution is to use differenced data ∇xt =xt − xt−1; differencing can be applied consecutivelyto both ut and yt until a stationary time series is ob-tained).

• independence between ut and the noise νt.

• impulse response coefficients past n are zero (i.e.,horizon for correlation analysis sufficiently long.

Correlation Analysisγuy = Γub

b = [ b0 b1 b2 . . . bn ]T

Γu =

γu(0) γu(1) . . . γu(n)

γu(1) γu(0) . . . γu(n− 1)... ... ...

γu(n) γu(n− 1) . . . γu(0)

γuy =

γuy(0)

γuy(1)...

γuy(n)

b = Γ−1u γuy

The numerical solution to correlation analysis consists of a square (n x n) linear system of equations

Persistence of Excitation Requirements for Correlation Analysis Estimates

γuy = Γub

b = [ b0 b1 b2 . . . bn ]T

Γu =

γu(0) γu(1) . . . γu(n)γu(1) γu(0) . . . γu(n− 1)

... ... ...γu(n) γu(n− 1) . . . γu(0)

γuy =

γuy(0)γuy(1)

...γuy(n)

b = Γ−1u γuy

An input signal with persistent excitation of order n is requiredsuch that Γu is nonsingular / invertible (i.e., Γ−1

u exists).

10



Input PrewhiteningThe previous result can be considerably simplified if theinput time series ut is white noise. An autocorrelated in-put signal can be “transformed” into a white noise inputthrough the process of prewhitening,

αt = φ(z−1)θ−1(z−1)utβt = φ(z−1)θ−1(z−1)yt

leading to the result

bk =γαβ(k)σ2α

or in terms of crosscorrelations

bk =ραβ(k)σβ

σαk = 0, 1, 2, . . .

Following prewhitening, the crosscorrelation function be-tween the prewhitened input and the transformed outputis directly proportional to the impulse response function.

ARMA modeling of u provides the prewhitening filter

Feedback's Process Trainer PT326

• "Hairdryer" process; input (MV) is the voltage over the heating device; output (CV) is outlet temperature

3

4

5

6

7

16 17 18 19 20 21 22 23 24

OUTPUT #1

4

6

16 17 18 19 20 21 22 23 24

INPUT #1

Described in Ljung's system identification book, pg. 440; data records included in MATLAB's System Identification toolbox demonstration M-file.

"Hairdryer" Process Example

load dryer2

% Select the 300 first values for building a model:zu = u2(1:300) - mean(u2(1:300));zy = y2(1:300) - mean(y2(1:300));z2 = [zy zu];

% Plot the interval from sample 200 to 300: (0.08 sec is the sampling interval)idplot(z2,200:300,0.08)

"Hairdryer" Data Correlation Analysis - MATLAB commands

% If you wish to remove nonstationarity in the data, use the following command:%% zuf = filter([1 -1],[1],zu);% zyf = filter([1 -1],[1],zy);% z2 = [zyf zuf];%% Let us now estimate the impulse response of the system% via correlation analysis (CRA) to get some idea of time constants% and the like; the dash-dot lines on the impulse response coefficient plot% indicates standard error bounds corresponding to 99% significance level%% Command format: ir = cra(z2,M,NA)

M = 20; % 20 lags (default value)NA = 10; % Order for AR prewhitening filter (default value)

ir = cra(z2,M,NA,2);

11



Correlation Analysis Results, Hairdryer Data

0

0.05

0.1

0.15

-20 -10 0 10 20

Covf for filtered y

-0.5

0

0.5

1

1.5

-20 -10 0 10 20

Covf for prewhitened u

-0.2

0

0.2

0.4

0.6

-20 -10 0 10 20

Correlation from u to y (prewh)

-0.05

0

0.05

0.1

0.15

-20 -10 0 10 20

Impulse response estimate

Step Response, Hairdryer Data

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Time (seconds)

Out

put

Step Response from Correlation Analysis

% We can form the corresponding step response by integrating% the impulse response:

stepr = cumsum(ir);slen = 0.08*[0:size(stepr)-1];plot(slen,stepr)

Spectral analysis consists of taking the Fourier trans-forms of the “smoothed” sample estimators for covari-ance and crosscovariance. The smoothing process re-duces variability at the expense of adding bias to theestimate. The resulting estimate is

p̃(eiω) =Φ̂yu(ω)Φ̂u(ω)

where Φ̂yu and Φ̂u denote the “smoothed” cross- andauto-spectra between u and y, described by the formulas

Φ̂yu(ω) =M∑

k=−Mcyu(k)w(k)e−iωk Φ̂u(ω) =

M∑k=−M

cu(k)w(k)e−iωk

w(k) is the lag “window” that accomplishes smoothing.A common window choice is the Tukey (“Hamming”) lagwindow

w(k) =

12(1 + cos(πkM )) |k| ≤M

0 |k| > M

where M is the truncation parameter for the window.

Spectral AnalysisSpectral Analysis Spectral Analysis, continued

One useful byproduct of spectral analysis is an estimateof the noise power spectrum,

Φ̂ν(ω) = Φ̂y(ω)− |Φ̂yu(ω)|2

Φ̂u(ω)

The estimated noise spectrum can in turn be relatedto uncertainty in the frequency response estimate (fromJenkins and Watts):

|p(eiω)− p̃(eiω|2 ≤ 2$ − 2

Φ̂ν(ω)Φ̂u(ω)

f2,$−2

f2,$−2 is the two-way Fisher statistic for a user-specifiedconfidence level. Other measures for obtaining uncer-tainty (based on deterministic, as opposed to statisticalarguments) are found on pages 155-156 of Ljung.

12



% We now obtain the frequency response via spectral analysis; % the arguments to SPA are% [g phiv] = spa(z2,M2,w);% The default frequency vector w is% w = [1:128]/128*pi;% M2 is the truncation parameter for the Hamming lag window with % default value M2 = min(30,length(z2)/10);

w = [1:128]/128*pi;M2 = min(30,length(z2)/10);

[gs phiv] = spa(z2,M2,w);% setting the sampling interval

gs = sett(gs,0.08); % We get a Bode Plot representation and 99.7 % confidence intervals/three standard deviations via% bodeplot([gs phiv],3)bodeplot([gs phiv],3), pause

Spectral Analysis Using MATLABSpectral Analysis Using MATLAB Spectral Analysis Results - Hairdryer Data

10-2

10-1

100

10-1 100 101 102

frequency (rad/sec)

AMPLITUDE PLOT, input # 1 output # 1

-600

-400

-200

0

10-1 100 101 102

PHASE PLOT, input # 1 output # 1

frequency (rad/sec)

phas

e

dashed-dotted lines indicate 99% confidence bounds on the estimates

M2 = 30

13

and spectral analysis, march 5, 1998 - facultad - fceia › isis › inpcrasp.pdf · time [min]...

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