L§AMPLED-DATA FREQUENCY RESPONSE SYSTEM ...

L§AMPLED-DATA FREQUENCY RESPONSE

SYSTEM IDENTIFICATION FOR

LARGE SPACE STRUCTURESI

A Thesis Presented to

the Faculty of the College of

Engineering and Technology

Ohio University

In Partial Fulfillment

of the Requirements of the Degree

Master of Science

by

Thomas T. Hammon~

June, 1988

ACKNOWLEDGMENTS

First, I would like to extend my gratitude to the

Faculty and Staff of the Department of Electrical and

Computer Engineering. Their tireless efforts have aided me

immensely in my graduate studies at Ohio University.

Special thanks is due to Dr. Jerrel Mitchell, my

advisor, for suggesting the thesis topic, guiding my

research, and editing the thesis drafts. I appreciate the

input of my thesis committee, Dr. G.V.S. Raju, Dr. O.E.

Adams, Jr., and Dr. R. Dennis Irwin, to the final draft of

this thesis. Also, I would like to thank Bill Maggard for

his cooperation in the experimental stage of this work.

Finally, I would like to express my appreciation to my

family whose love and encouragement made this endeavor

possible. My mother, Harriet, and late father, Claude,

taught me the value of education and demonstrated the

perseverance required to obtain it; and my fiancee, Judy,

steadfastly supported my efforts with her love and patience.

i

TABLE OF CONTENTS

ACKNOWLEDGMENTS

LIST OF FIGURES

LIST OF TABLES

Chapter 1. INTRODUCTION

1.1 Control of Large Space structures

1.2 Design-To-Performance

1.3 One-Controller-At-a-Time

1.4 Summary

Page

i

iv

viii

1

1

3

4

8

2.1 Overview

2.5 Stochastic Analysis

2.6 Summary

2.2 Sampled-data Systems

2.3 Fourier Transforms

2.4 Deterministic Analysis

3.6 Summary

3.1 Signal Properties

3.2 unit Pulse

3.3 Increasing Magnitude

3.4 Increasing Duration

3.5 Shaping Pulses

Chapter 2.

Chapter 3.

SYSTEM IDENTIFICATION

EXCITATION ENHANCEMENT

9

9

11

14

18

22

23

24

24

26

30

32

37

44

ii

Chapter 4. EXPERIMENT DESIGN ...••...........•..•...... 46

4 • 1 Obj ective 46

4 • 2 Physical Model 4 7

4.3 Mathematical Models 49

4.4 Digital Simulation 52

4 • 5 Computations 60

4 • 6 Digital Results 62

Chapter 5. EXPERIMENT IMPLEMENTATION 71

5.1 Hybrid Experiment 71

5 • 2 Analog Computer 73

5.3 Data Acquisition/Control Unit 75

5 • 4 Digital Computer 76

5. 5 Hybrid Results 77

Chapter 6. CONCLUSION 107

6.1 Observations

6.2 Applications

107

108

RE~ERENCES 110

Appendix A. DIGITAL SIMULATION SOURCE CODE 113

Appendix B. HYBRID SIMULATION SOURCE CODE 155

iii

LIST OF FIGURES

Page

1.1 Multiple-input, mUltiple-output system 5

2.1 Typical sampled-data control system 12

2.2 Sampled-data control system modeled with ideal

sampler and zero-order-hold 12

2.3 Sampled-data system transfer function 15

2.4 Sampled-data system frequency response function 15

2.5 System with noise modeled at output 20

2.6 Unity negative feedback configuration 20

3.1 z-plane mapping of frequencies up to the half-

sampl ing frequency 27

3.2 unit digital pulse's characteristics 28

3.3 Increased magnitude pUlse's characteristics 31

3.4 Increased duration pulse's characteristics 36

3.5 Binomial triangle shaped pulse's characteristics 40

3.6 Bipolar triangle shaped signal's characteristics 43

3.7 Complementary binomial pair's spectral

characteristic 45

4 • 1 Three disk physical model 48

4.2 Three disk system sampled-data block diagram 50

4.3 Three disk system sampled-data frequency response

function 55

4.4 Three disk system continuous root locus 56

iv

4.5 Three disk system continuous root locus of 0.1 Hz.

mode 57

4.6 Three disk system sampled-data root locus for

unity gain 59

4.7 Three disk system sampled-data frequency response

block diagram 61

4.8 Frequency response function obtained using unit

digital pulse 64

4.9 Frequency response function obtained using

increased magnitude pulse ......................... 65


increased duration pulse .......................... 66


binomial triangle shaped pulse 68

4.12 Frequency response function obtained using bipolar

triangle shaped pulse ........•.................... 69

4.13 Frequency response function obtained using unity

white noise 70

5.1 Hybrid experiment hardware configuration 72

5.2 Analog computer simulation functional block

diagram 7 4

5.3 Spectral characteristics of unit digital pUlse 79

5.4 Frequency response function using unit digital

pulse ....•........................................ 80

5.5 Spectral characteristics of increased magnitude

pulse 81

v

5.6 Frequency response function using increased

magnitude pulse 82

5.7 Spectral characteristics of increased duration

pulse 83

5.8 Frequency response function using increased

duration pulse 84

5.9 Spectral characteristics of binomial triangle

shaped pulse 85

5.10 Frequency response function using binomial

triangle shaped pUlse 86

5.11 Spectral characteristics of bipolar triangle

shaped pulse 88

5.12 Frequency response function using bipolar triangle

shaped pulse 89

5.13 Frequency response function using unity white

noi se •••••••••••••••••••••••••••••••••••••••••••.• 9 a

5.14 Frequency response function using single increa.sed

magnitude record .................................. 92

5.15 Frequency response function using four increased

magnitUde records ................................. 93

5.16 Frequency response function using sixteen

increased magnitude records ....................... 94

5.17 Frequency response function using single binomial

triangle record 95

5.18 Frequency response function using four binomial

triang1e records 96

5.19 Frequency response function using sixteen binomial

vi

triangle records •.........•.•..••......••......... 97

5.20 Frequency response function using single bipolar

triangle record 99

5.21 Frequency response function using four bipolar

triangle records 100

5.22 Frequency response function using sixteen bipolar

triangle records .............•...........•........ 101

5.23 Frequency response function using sixteen spliced

triangle records 102

5.24 Frequency response function using single unity

white noise record 103

5.25 Frequency response function using four unity white

noise records ......•.............................. 104

5.26 Frequency response function using sixteen unity

white noise records 105

vii

LIST OF TABLES

Page

5.1 Analog computer potentiometer settings 75

viii

1

CHAPTER 1.

INTRODUCTION

1.1 Control of Large Space Structures (LSSs)

As scientists and engineers attempt to utilize the

unique properties of the Earth-orbit environment, it is

clear that larger structures must be built in space to house

these imaginative projects. Currently, there are several

projects being planned that will require a physical

structure many times larger than any previously placed in

orbit. Two notable examples are the National Aeronautics

and Space Administration's Space station and the Department

of Defense's strategic Defense Initiative's Space-Based

Lasers. It is the challenge of control systems engineers to

develop effective methods for controlling the mechanical

dynamics of these huge structures.

The construction of LSSs in orbit dictates the use of a

minimal amount of light-weight materials. Typically,

implies the interconnection of long, thin, flexible beams to

form the framework. This construction technique produces

structures that are very flexible and exhibit very little

mechanical damping. Hence, the unconstrained LSS will

vibrate for an extremely long time when disturbed.

Additionally, due to the complexity of the LSS, these

vibrations, or flexible modes, will occur at several

2

frequencies with significant magnitudes. Effective suppres

sion of these undesirable modes is essential for any LSS

control scheme.

The process of assembling LSSs will span several months

or years. During this time, the testing and use of modules

already completed requires the implementation of structural

control systems while the structure is evolving. The

implication is that the dynamics of the LSS will change

slowly over time. Similar effects will be produced by

reorienting subsystems such as solar panels, docking with

transportation vehicles, expanding and contracting due to

thermal effects, etc. Therefore, the control scheme must be

adaptable to these slowly changing dynamics.

Finally, much of the information on which the control

system is based must be extracted from the structure itself.

Ground-testing of the true structure is not possible, and

even the investigation of scaled models is of questionable

value due to gravitational effects. The method of

finite-element analysis may yield some useful information,

but models obtained in this fashion generally lack

sufficient fidelity when applied to extremely high order

systems. So the successful control scheme must be capable

of incorporating empirical information gleaned from the true

dynamics of the actual LSS.

3

1.2 Design-To-Performance (DTP)

A control system design strategy that has been tailored

to meet the stringent requirements of LSS control is the

Design-To-Performance procedure [1]. The key elements of

DTP are the following:

* Determining LSS characteristics on-orbit.

* Designing an appropriate digital controller.

The design process entails gathering structural response

data for the LSS; extracting a high-fidelity frequency

domain system model from the data; specifying a high-perfor

mance control law based on the model; and implementing the

control law with the digital controller onboard the LSS.

This process may be repeated as necessary to fine-tune the

design or to cope with changes in the dynamics of the LSS.

The DTP concept offers some important advantages in the

LSS application. other control methods often require

sensors and actuators dedicated t~ gathering data to

identify the dynamics of the LSS. The DTP method uses only

those actuators and sensors necessary to control the LSS.

Another attractive feature is the use of frequency domain

models. Methods that require state variable models are

faced with the formidable task of developing these models

for a system that may be of order 100 or more. Furthermore,

4

design procedures for frequency domain models can be used

which produce much simpler control laws than state-space

methods. A design procedure that is particularly well-suit

ed for the DTP philosophy is considered next.

1.3 One-Controller-At-a-Time (l-CAT)

One-Controller-At-a-Time is a design procedure for

mUltiple-input, mUltiple-output (MIMO) feedback control

systems [2]. A linear MIMO system may be viewed as an array

of single-input, single-output (5150) systems. For a MIMO

system with n inputs and m outputs, there are mxn 8150

systems in the model. Each MIMO output is formed by summing

the outputs of n 5I50 systems, each with a different MIMO

system input. The input-output relationships of this model

are conveniently represented with a vector equation using an

mxn coefficient matrix of 8150 system transfer functions.

This representation of a MIMO system is illustrated in

Figure 1.1, where

U1

H11

HMi

+U1

5

Figure 1.1.

U Hn in

~il\

I

I +

H UMn M

Multiple-input, multiple-output system.

VI

V 2

V m

H i l H I 2

H 21 H 22

H m I · H m 2

H In

H 2n

H mn Un

6

(1. 1)

The frequency domain description of this system is formed by

replacing the transfer functions with the corresponding

frequency response functions. In practice, these may be

obtained by exciting the system inputs one at a time,

measuring each output due to the current input, and

computing the frequency response function from the current

input to each output.

Now, the addition of control loops to this system will

couple the loops: the control loops cannot be designed

independently. However, they can be designed sequentially,

or one-at-a-time. This is the underlying theme of I-CAT.

The procedure is to close one loop, to evaluate the effects

of this closure on the next loop to be closed, and to close

the next loop based on the effects of closing the previous

loop. This process is repeated until all required loops are

closed. with prudent selection of the sequence of loop

closures, high-performance control may be achieved with this

7

step-by-step method. Used with frequency domain design

techniques, I-CAT produces low-order compensators for

high-order systems.

Another design enhancement that is compatible with the

methods outlined above is that of Modal Suppression by Phase

Stabilization (MSPS) [1]. This technique is especially

attractive for suppressing low-frequency flexible modes

encountered in LSS control. "A mode is said to be phase

stabilized if its gain, in the compensated open loop

frequency response, is greater than 1 (0 dB) and no

encirclement of the -1+jO point on the polar frequency

response occurs due to this mode. A mode is perfectly phase

stabilized if the phase of the loop frequency response is

o degrees at its peak frequency" [1J. Modes which are phase

stabilized are stabilized throughout the structure: MSPS is

a global dynamic effect, not merely a local loop phenomenon.

Phase stabilization occurs inherently in certain

hardware configurations. In particUlar, the collocation of

a force or torque actuator with the corresponding rate

sensor forms a "complementary pair" that produces modal

phase stabilization [1]. Thus, neglecting actuator and

sensor dynamics, modes excited and measured in this pair's

loop will be phase stabilized by closure of the loop; and,

as noted above, this stabilization is effective throughout

the structure. Obviously, this is an extremely nice

property to exploit whenever it is possible.

1.4 Summary

As detailed in this introductory chapter, the DTP

strategy encompasses all the salient features desirable in

LSS control. However, its effectiveness is rooted in the

fidelity of the empirical frequency domain model elicited

from the LSS. It is the purpose of this thesis to examine

some techniques of enhancing the sampled-data frequency

response functions that are characteristic of LSS dynamics.

First, in Chapters Two and Three, the theoretical basis

of the experimental methods used later is examined. Next,

an experiment is designed and implemented to explore the

practical application of these methods. Finally, the

experimental findings are analyzed, and suggestions are

offered for the application and further investigation of

these methods.

8

9

CHAPTER 2.

SYSTEM IDENTIFICATION

2.1 Overview

The design of control systems requires sufficient

knowledge of the mathematical relationships that describe

the physical system that is to be controlled. Before a

system is constructed, attempts may be made at predicting or

"modeling" the system's behavior. For simple systems, the

mathematical model may be constructed by applying physical

principles known to govern the system's behavior. More

complex systems may be viewed as a combination of simpler

subsystems, and the same techniques applied. However, as

the order of the system grows, its behavior becomes more

difficult to satisfactorily predict in this manner.

After a system has been constructed, the analysis of its

input and output signals may reveal a wealth of information

concerning the dynamics of the system. "System identifica

tion" is the process of deriving a mathematical model to

describe the system's dynamic behavior from observations of

its input and output signals. As one would expect,

incorporating the system modeling information into the

system identification process often yields a more suitable

system model.

10

There are many questions to be answered before a

suitable system identification method can be prescribed.

What is to be the mathematical structure of the model (e.g.

differential or difference equations, transfer functions,

frequency response functions, etc.)? Is the system linear

or nonlinear; stationary or nonstationary; discrete,

sampled-data, or continuous; single-input or multiple-input;

etc.? What input signals are desirable and permissible to

be employed to excite the system? Proper consideration must

be given to all these questions and more, in order to

successfully identify a high-fidelity system model.

For the LSS control design procedures outlined in the

introduction, sampled-data frequency response functions are

required. It is assumed that the LSS has linear

characteristics, and excitation signals will be limited to

levels that will not evoke nonlinear behavior of the

actuators, sensors, or structure. Also, the system is

assumed to be stationary, at least during the identification

process. What input signals are desirable? This is the

question to be addressed in the sequel. First though, a

closer examination of sampled-data frequency response system

identification is necessary.

11

2.2 Sampled-data Systems

Historically, frequency response methods of system

identification have enjoyed much success. For continuous

systems, the general approach is to excite the system with

an input signal whose spectral characteristics are known.

The system's output signal due to this excitation is

observed, and its spectral characteristics are determined.

Now, the system's frequency response function may be

estimated from the input and output spectra. The same

general procedure may be extended to sampled-data systems.

A typical SISO sampled-data control system is shown in

Figure 2.1. The control loop is composed of the continuous

system that is to be controlled, an analog-to-digital (A/D)

converter, the digital computer/controller, and a digi

tal-to-analog (D/A) converter. The A/D converter samples

the system's continuous output at regular intervals of T

seconds and provides the digital computer with a number

representative of the system's output at the last sampling

time. This sampling function may be represented diagrammat

ically by an ideal sampler whose input is the system's

output, and whose output is equal to the system's output,

but only exists at the sampling instants. The D/A converter

accepts a number from the computer that represents the input

signal to be applied to the system'and applies the specified

12

Reference SaMpledInput u(t) v(t) Output

DigitalContinuous

--+ COMPuter/ H D/A H -+ AID -r-+-Controller

SysteM

~

Figure~ Typical sampled-data control system.

ReferenceInput

Digital

COMPuter/Controller

u(kT)

ZOH

u(t)

Conti nuousSysteM

SiMpl edOutput

v(kT)

Figure~ Sampled-data control system modeled with idealsampler and zero-order-hold.

13

input signal to the system until the next number in the

sequence arrives. Usually, D/A converters are used that

apply a constant input signal between numbers received from

the computer. This manner of constructing a continuous

signal from a digital sequence is described by the

zero-order-hold (ZOH) function. An equivalent representa-

tion of the sampled-data system is shown in Figure 2.2.

Now, the system to be identified is the sampled-data

system from the input of the ZOH to the output of the

sampler. So the input signal is u(kT), and the output signal

is v(kT), where k is the sample index. The relationship

between the input and output signals of this model may be

described by the discrete-time system transfer function. A

sampled-data signal is transformed to the discrete domain by

using the z-transform. The z-transform is defined as

OQ

F(z) = I f(kT)z-k ,k-O

where the relationship between the z-transform complex

(2 . 1)

variable z and the Laplace transform complex variable s is

z = eST. (2.2)

So the transfer function of the system may be expressed in

terms of the ratio of the input and output signals in the

z-domain as

H(z) = V(z).U(z)

Thus, the sampled-data system to be identified may be

represented as shown in Figure 2.3.

14

(2.3)

Finally, the relationship between the complex variable z

and the real frequency w is

(2.4)

Then, the sampled-data frequency response function for the

system is found to be

This relationship is depicted in Figure 2.4.

(2.5)

The direct computation of the input and output spectra

from empirical data is commonly accomplished using the

Discrete Fourier Transform (DFT) and its efficient

implementation with the Fast Fourier Transform (FFT). These

topics are reviewed in the next section.

2.3 Fourier Transforms

The spectra for signals in continuous control systems

are determined by applying the Fourier Transform. Similar-

ly, the spectra of sampled signals may be obtained by

U(z) ,.. H(z) U(z)

15

Figure~ Sampled-data system transfer function.

J~TH(e ) ".

Figure~ Sampled-data system frequency response function.

16

applying the Discrete Fourier Transform; and since signals

are sampled for a finite time period, or for a finite number

of samples, the finite DFT is used. The finite DFT is

defined as

N-l

F(e i GO T )=I: /CkT)e- i w k T•

k-O

(2 • 6)

Note that the spectra of sampled signals are periodic with a

period equal to the sampling frequency of

2rrw s = - ·

T

In order to implement DFTs efficiently, computer

algorithms termed Fast Fourier Transforms have been

(2.7)

developed. The effectiveness of FFTs has been a real boon

to digital spectral analysis. However, there are two

problems which arise in deriving the spectra of sampled-data

signals with FFTs. These are 1) aliasing and 2) leakage.

Both phenomena must be considered when computing sampled-da-

ta frequency spectra and are explained next.

Aliasing occurs when the sampled signal contains

frequency components that are higher than one-half the

sampling frequency. Instead of appearing in the spectrum at

the proper frequencies, these components are "folded back"

about the one-half sample frequency and added to the

17

spectral components of these lower frequencies. The obvious

solution is to sample the signal at a frequency at least

twice that of the highest component frequency. However, the

spectral content of the signal is not always well-known, and

the noise present in the signal is likely to have frequency

content that can cause aliasing. Thus, choosing a

conservatively high sampling frequency is advisable, if one

has this option.

Leakage occurs if the sampled signal is aperiodic, or if

the sampling window is not precisely an integral mUltiple of

the signal's period. The finite nature of the sampled

signal may be viewed as mUltiplying an infinitely long

signal with a unity-magnitude rectangular window of the same

length as the sampled signal. The frequency domain

representation of this rectangular window contains signifi

cant "side-lobes" over frequencies adjacent to the desired

frequency. Therefore, the resultant spectrum obtained by

combining these signals in the frequency domain has coarser

resolution than would be expected, due to the spreading of

spectral information over adjacent frequencies. The usual

precaution for leakage is to use a window that is shaped to

make the time signal appear to be periodic in the window:

this suppresses the side-lobes in the frequency domain.

18

Typical time windows taper the signal to zero at each end of

the time window. One window particularly popular for system

identification is the Hanning window, which is

2nth(t)= O.5-0.5cos-, for O~t~Tr' and

T r

h(t) = 0, elsewhere,

where T r is the sampled record duration.

(2.8)

NOTE: For the remainder of this thesis in the interest

of notational simplification, (!) will be used to represent

the dependence of a signal's representation or of a function

on the complex variable exp(jwT) with the radian frequency w

replaced by the equivalent expression 2nf. This simplifica-

tion is summarized as

(2.9)

2.4 Deterministic Analysis

Input signals used for exciting systems during the

identification process may be divided into the classes of

deterministic signals and stochastic, or random, signals.

The analysis of systems differs for these two classes of

signals, and each is considered separately in this and the

succeeding section.

19

For the ideal sampled-data system outlined previously,

the frequency response function is simply

H(/) = V(/).U(!)

(2 • 10)

However, real systems must be modeled by adding a noise

signal to the measured output signal. This is necessary

because real systems and measurements differ somewhat from

the assumed ideal systems. The real system may have slight

nonlinearities, a small degree of nonstationarity, minor

disturbances, inherent sensor noise, etc. All variations

due to these nonideal characteristics are modeled as a noise

signal spectrum Nef) added to the ideal output signal

spectrum Vef) to form the measured output signal spectrum

yef). The system block diagram now is illustrated by Figure

2.5, where

Y(f}=V(f)+N(f)· ( 2 • 11)

Now, for a deterministic input signal, an estimated

frequency response function, Ref), may be calculated from

the input signal and the measured output signal by noting

that

f! _Y(/)(I) - U(!)'

(2 . 12 )

U(f)

H(f)U<f)

H(f)

20

Figure~ System with noise modeled at output.

N<f)

X(f) U(f)

H(f)

U<f)+

~< f)

Figure~ Unity negative feedback configuration.

R(f)=V(f)+N(f) andVe!) ,

fI(f)=H(f)+N(f),Ve!)

21

(2.13)

(2 • 14 )

Note the important fact that the fidelity of the estimate is

improved by increasing the relative amount of the ideal

output signal in the measured output signal, or in other

words, by increasing the signal-to-noise ratio at the

system's output. This observation is the basis of enhancing

frequency response functions by prudent selection of

deterministic input signals as detailed in Chapter Three.

Additionally, if the noise may be characterized as zero-mean

and ergodic, then the random error of an averaged magnitude

estimate, obtained by averaging nd estimates, is reduced by

a factor of l/~ [3]. This fact offers another tool for

empirical frequency response function enhancement, and its

practical effectiveness is demonstrated in the experimental

section of this thesis.

A final system configuration that needs to be examined

is that of identifying a frequency response function that is

embedded in a feedback loop. For simplicity, unity negative

feedback is assumed as depicted in Figure 2.6.

22

with knowledge of the input spectrum X(f) and the measured

output spectrum Y(f), the control error signal spectrum U(/)

is

U(·f) = X(f) - Y(f)·

In view of this relationship, the frequency response

(2 • 15)

function may be estimated using the formula developed for

the open-loop configuration, i.e. Equation 2.12.

2.5 Stochastic Analysis

The empirical analysis of systems excited by stochastic

input signals is considered in this section. Often, the

input signal is actually pseudo-random, but its statistics

are exemplary of a stochastic signal. The spectral

input-output relationships of this analysis are in terms of

one-sided spectral density functions. In particular, for

the open-loop system with output noise which is uncorrelated

with the input signal, the optimal frequency response

function estimate, in the least squares sense, is found to

be

F! ( /) = ~ u.y ( /) ,

G uuCf)

where Guy(f) is the input-output cross-spectral density

function estimate, and GuuC!) is the input auto-spectral

(2.16)

23

density function estimate [4J. In this case, the random

error of the magnitude estimate is reduced by a factor of

1/~2nd through averaging nd independent data records [3J.

For the unity negative feedback configuration, the

following relationship yields the frequency response

function estimate of the open-loop transfer function [3J

2.6 Summary

(2.17)

In this chapter, the theoretical foundations for the

methods of analysis used in the remainder of this thesis

have been reviewed. The process of system identification by

sampled-data frequency response function determination was

outlined, and the differences encountered in deterministic

and stochastic excitation methods were considered. In the

ensuing chapter, the spectral properties of several

deterministic signals are investigated to develop a

rationale for improving frequency response function

estimates using these signals.

24

CHAPTER 3.

EXCITATION ENHANCEMENT

3.1 Signal Properties

In this chapter some properties which enhance a

deterministic signal's efficacy as the excitation signal for

frequency response system identification are considered.

Then some signals are examined that offer the opportunity of

tailoring the choice of input excitation to meet constraints

of the specific system being identified. Also, comments are

included concerning the applicability of each signal.

Recall from the development of deterministic analysis in

section 2.4 that

H(!) = Y(!) andU(!) ,

Y(f)=V(f)+N(f)·

This leads to the important results that

H(!)=V(!)+N(!) andV(/) ,

H(!)=H(!)+N(!).U(f)

(3 • 1)

(3.2)

(3.3)

(3.4)

25

Equation 3.4 clearly characterizes the effect of the input

signal spectrum on the estimated frequency response function

magnitude: increasing the magnitude of the input spectrum

over frequencies of interest improves the estimate HCf).

The signals considered for use in the sequel are finite

duration sequences that are non-zero for only a portion of

the identification time window. The start of the

identification window is coincident with the first non-zero

element of the sequence, and the input sequence will be

permitted to assume non-zero values for M elements, or

sample periods. Thus, the signals of interest may be

expressed by their z-transforms as

M-I

U(z) = I u(kT)z-k.k-O

Furthermore, by invoking the relationship z=exp(jwT) the

spectrum of the signal is shown to be

M-l

U(e i GlJ T) = I u(kT)e- i GlJ kT

•k-O

(3 .5)

(3 .6)

Note that Equation 3.6 is the finite DFT of the signal as

introduced in section 2.3. The foregoing relationships will

be used to investigate the signal sequences proposed in this

chapter.

26

A closer look at the relationship z=exp(jwT) will provide

some needed insight into the signals presented. In the

z-plane this relationship describes a unit circle centered

at z=o. Real frequencies of interest are restricted to the

range of zero to the half-sampling rate, or O~W ~ws/2. This

describes the half of the unit circle that begins at z=l, or

w=o, and proceeds in the counter-clockwise direction to

z=-l, or w=w s/2. This relationship helps to reconcile the

two signal representations above, and it is graphically

depicted in Figure 3.1.

3.2 unit Pulse

An obvious choice for launching this signal investiga

tion is the unit digital pulse. The unit digital pulse is

defined by

u(kT)= 1, for k=O, and

u( kT) = 0, otherwise.

(3.7)

From Equation 3.6, the spectrum of the unit digital pulse is

found to be

U( e}WT) = u(O) = 1 . (3.8)

The unit digital pulse's characteristics are graphically

displayed in Figure 3.2. A shorthand notation will

occasionally be used in the sequel to denote the peak

1M

27

Re

Figure 3. 1. z-plane mapping of 0 ~ w s w .rz •

Uni t Digi tal Pulse (1 xl)

Sequence

28

10

8

6

4OJ'0 2::::l+J

t: 0t::n~ -2:2

-4

-6

-8

• - = - - - - =- - - - - - -

•Input

-10o 2 4 6 8 10 12 14

40

20

o

-20

-40

-60

-80

-100

Sample Index k

Uni t Digi tal Pulse (1 xl)

Spectrum

111111

Input

1/16 1/8 1/4 1/2

Figure .l..:..L..

Fraction of Sampling Frequency

Unit digital pulse's characteristics.

29

magnitude and duration in sampling periods of an input

signal. The form of this notation is (peak magnitude x

duration), which for the unit pulse is (lx1) as shown in

Figure 3.2.

The most convenient result of Equation 3.8 coupled with

Equation 3.1 shows that the frequency response function

estimate is simply the measured output spectrum

Hef) = yet)· (3.9)

Two advantages of unit digital pulse excitation are the

simplicity of computation and uniformity of magnitude across

the spectrum of interest. Additionally, this signal and

estimated frequency response function provide convenient

references for comparison with other excitation methods.

The unity amplitude of this pulse is assumed to be

sufficiently small so that no structural response

constraints are violated, e.g. actuator saturation,

non-linear structural behavior, etc. However, the unity

spectral magnitude does not enhance the estimated frequency

response function by suppressing the noise component as

suggested by Equation 3.4.

30

3.3 Increasing Magnitude

Suppose the magnitude of an input signal is increased by

the factor a. That is

ua(kT) = au(kT). (3 . 10)

since the DFT is a linear transformation, the effect of this

factor on the signal's spectrum is

V a(/) = aV(!).

So application of Equation 3.4 yields

H(!)=H(!)+ N(!).aUe/)

(3 • 11)

(3 • 12)

In the particular case of u(kT) being the unit digital pulse

H(!) = H(!)+ N(!).a

(3 • 13 )

This equation shows the effect of increasing the amplitude

of the unit pulse excitation by a factor of a. The

resulting input signal is noted in the shorthand introduced

as (aXl) , and its characteristics are shown in Figure 3.3

for the signal (lOxl). Clearly, the estimated frequency

response function is improved by increasing the magnitude of

the input.

Increased Magnitude Pulse (lOx 1)

Sequence

31

10

8

6

4

2

o

-2

-4

-6

-8

-

- - - - - - - - -- - - - - - - -

•Input

-10o 2 4 6 8 10 12 14

Sample Index k

Increased Magni tude Pulse (lOx 1)

Spectrum

40

20

co 0'0

ca,) -20"0:;:j

1-1

c: -400)

<0

2 -60

-80

-100

1/16 1/8 1/4 1/2

Input

Figure~


Increased magnitude pulse's characteristics.

32

This section demonstrates the possibility of enhancing

the system identification results by increasing the

excitation magnitude. However, as alluded to in the

previous section, there are limits on the signal's

magnitude; and hence, there are limits on the improvement

possible in this manner. On the other hand, the improvement

is still uniform across the spectrum, which is quite

desirable.

3.4 Increasing Duration

In this section the effects of increasing the duration

of a constant amplitude pulse are considered. A constant

amplitude pulse that extends over M sample periods has the

z-transform

U-l'\ -k

Uu(z)= L az ·k-O

The expansion of the summation yields

U u(z)= a[l +Z-l + .•• +Z-CU-2)+Z-CU-I)].

(3 • 14 )

(3 . 15 )

Now multiplying the summation by the unity factor of ZU-l/ZM-l

gives

(3.16)

33

The effect of extending the duration of this pulse at

low frequencies may be shown by evaluating Equation 3.16 for

z=l, which corresponds to w=o. This indicates that UMCl)=aM

, so the low-frequency magnitude spectrum is increased by a

factor equal to the number of sampling periods over which

the pulse extends. Hence, increasing the pulse duration

increases the input signal's magnitude spectrum in this

region and enhances the estimated frequency response

function in the manner explained previously.

Further consideration of Equation 3.16 is necessary to

determine the effects at higher frequencies. This represen

tation of the signal points out that nulls may occur in the

signal's spectrum if roots of the numerator lie on the unit

semi-circle of interest. Note that the roots of the

denominator are all at z=o, and these do not affect the real

frequency spectrum. The roots of the numerator may be

investigated by simplifying the expression by mUltiplying it

by (z-l) as suggested in [5], which adds a root at z=l that

may be discarded later. This analysis proceeds as follows:

zA/-l =0, and

(3.17)

(3 • 18)

(3.19)

34

Now, the frequencies of the spectral nulls may be determined

by sUbstituting the expression z=exp(jwT) and solving for

frequencies that are in the range 0 s w ~ w .r: as

(. T)Me 1 W = 1, or

e}wMT = 1 .

Equation 3.21 is satisfied for all w that satisfy

wi'vlT=2nrr , n an integer, or

2nllW=--.

u r

(3.20)

(3 • 21)

(3.22)

(3.23)

By sUbstitution of the relationship T=2nlw s , the equation

becomes

nw sW=--.

M

(3.24)

Application of the constraints on w leads to constraints on

n of

w ~ 0 =? n ~ 0, and

ui , Mw~-~n~-.

2 2

(3.25)

(3.26)

35

Note that n = 0 =+ w = 0 =+ z = 1 and corresponds to the root added

earlier,and discarded now. Therefore, the admissible

numerator roots correspond to the values of n

Mn= 1,2, ... ,-, for M even, and

2

M-1n= 1,2, ... ,--, for M odd.

2

(3.27)

The values of n listed in Equation 3.27 in conjunction

with Equation 3.24 identify frequencies at which magnitude

nulls appear in the signal's spectrum. These nulls prohibit

identification of the frequency response function at these

frequencies. However, with proper precautions, signals of

constant amplitude and extending over more than one sampling

period may be used in certain circumstances. An example of

this type of signal is illustrated in Figure 3.4 for a pulse

of (lxlO).

First, observe that the lowest frequency at which a null

occurs is for ri = 1, or w = w sl M • So for frequencies

sufficiently below this null, the increased duration pulse

may be used successfully. Also, note that for signals of

length M and M+l, the nulls are distinct from each other.

This suggests that using this pair of signals and averaging

the results may successfully identify modes that would

otherwise be masked by the nulls of one signal or the other

[5]. However, for pulses extending over several sampling

Increased Duration Pulse (1 x 10)

Sequence

36

10

8

6

4

2

o

-2

-4

-6

-8

• • .1 • .1 • .1 • .1 • - - -. - - -

•Input

-10o 2 4 6 8 10 12 14

Sample Index k

Increased Duration Pulse (1 x 10)

Spectrum

40

20

co 0"C

C

Q) -20"C::l

~ -40t::0><'0

:2 -60

-80

-100

Input

1/16 1/8 1/4 1/2

Figure~


Increased duration pUlse's characteristics.

37

periods, the nulls cause much difficulty as the frequency

approaches the half-sampling rate. Of course, the same

considerations of structural limitations must be considered

as were in the increased magnitude case of section 3.3.

Finally, note that a combination of increasing magnitude and

duration may better fit specific system constraints.

3.5 Shaping Pulses

The drawbacks associated with the nulls encountered in

the previous section may be overcome to a large extent by

shaping the input signal. The strategy is to avoid nulls

below the half-sampling rate. Consider a signal with the

z-transform representation [5]

(Z+l)M-lUu(z)= M-l •

Z

Examination of Equation 3.28 shows that nulls on the

(3.28)

semi-circle of interest occur only at z = - 1 , or w = w 5/2,

which is the half-sampling rate. This will avoid much of

the problem associated with using a protracted constant

amplitude pulse. Further analysis of Equation 3.28 will

reveal the input sequence that has this z-transform.

First, expansion of the numerator and expression of the

fraction as a series yield the following:

M- 1 b M-2 b 1Z + M-2Z + ... + lZ+UM(z)= M-l ' and

Z

- 1Uu(Z)= 1+b M - 2 z +

b -(U-2) -(M- 1)+ lZ +Z ·

38

(3.29)

(3.30)

So the sequence desired is shown in the expression of the

z-transform

(3 • 31)

Note that this sequence is the ordered coefficients of the

binomial expansion of (z+ 1) using (M - 1) factors in the

expansion. Therefore, this signal will be referred to as a

binomial weighted triangular pulse, or simply a binomial

triangle pulse.

The increased spectral magnitude at low frequencies

offered by this signal may be ascertained from Equation 3.28

evaluated at z=l, or w=o, which gives

(3.32)

This result indicates that each sampling period extension of

this set of signals yields a doubling of the low-frequency

magnitude. Thus the magnitude increases exponentially with

increments in M, in contrast to linear increments in the

case of extending a constant amplitude pulse. However, due

to the increasing peak magnitude of the binomial triangle,

39

these effects may be restricted to a few sampling periods.

If it is desirable to use the binomial excitation over a

longer time, the magnitude of the signal may be scaled so

that system constraints are observed. Of course, this will

reduce the spectral magnitude as well, but the nulls are

still located at the half-sampling rate. This method

provides the designer with some freedom in tailoring the

excitation to the system at hand. A binomial triangle pulse

with peak magnitude normalized to unity and spanning ten

sampling periods (lxlO) is shown in Figure 3.5.

The problem with spectral nulls was diminished

considerably by the introduction of the binomial triangle

signal. However, the null at the half-sampling rate may

still mask system modes at frequencies in this region. This

leads to the consideration of a signal which is

complementary to the binomial triangle signal [6]. Note

that the nulls of the following expression all occur at z=l,

or w = 0/

(z-l)M-lUM(z)= M-l •

Z

(3.33)

The sequence which has this z-transform may be identified in

a like manner to the binomial triangle analysis as

Binomial Triangle Shaped Pulse (1 x 10)

Sequence

40

10

8

6

4

2

o

-2

-4

-6

-8

•• • II • .1- - • ;;;- - - ~-~

•Input

-10o 2 4 6 8 10 12 14

Sample Index k

Binomial Triangle Shaped Pulse (1 x 10)

Spectrum

40

20

o

-20

-40

-60

-80

-100

~-.\\\\

1IIITllfTII

Input

Figure~

1/16 1/8 1/4 1/2


Binomial triangle shaped pulse's characteristics.

U ( M-l ( M-2M Z)= Z + -1)b M _ 2 z + ...

41

(3.34)

(3.35)

(3.36)

So in this case, the sequence is seen to be the ordered

coefficients of the binomial expansion of (z- 1) using (1\-1- 1)

factors in the expansion. Note that the elements of this

sequence alternate signs. Therefore, this signal will be

referred to as a bipolar binomial weighted triangular

signal, or simply a bipolar triangle. Additionally, the

cryptic notation used will include a minus sign with the

peak magnitude to distinguish it from the binomial triangle,

i.e. (-lx10) indicates a bipolar triangle with peak

magnitude normalized to unity and spanning ten sample

periods.

The increase in spectral magnitude offered at the

half"-sampling rate is determined by evaluating Equation 3.33

at z=-l, or w=w s / 2 , which yields

(3.37)

This result verifies that the bipolar triangle is

complementary to the binomial triangle. This suggests using

42

these signals as a pair to span a wider range of frequencies

if necessary. A rationale for combining the results

obtained with these two signals is developed next. A

bipolar triangle signal is presented in Figure 3.6 that

corresponds to the binomial triangle signal in Figure 3.5.

since the spectral magnitude of the binomial signal

decreases as frequency increases, and the spectral magnitude

of the bipolar signal increases as frequency increases, it

is reasonable to find the frequency at which these spectra

are equal, and to use the binomial results for lower

frequencies and the bipolar results for higher frequencies.

The boundary frequency may be found by identifying the

frequency on the semi-circle in the z-plane at which the

z-transforms of the two signals have the same magnitude. By

equating the magnitudes of Equations 3.28 and 3.33 the

expression obtained is

(3.38)

In terms of frequency, this expression becomes

(3.39)

The solution of Equation 3.39 for the range of admissible

frequencies yields the boundary frequency of

Bipolar Triangle Shaped Pulse (- 1 x 10)

Sequence

43

10

8

6

4Q)

'0 2='~

0Ct:nco -2:2

-4

-6

-8

•• .1 ••- - - - -- - • • - - - -•

•Input

-10o 2 4 6 8 10 12 14

Sample Index k

Bipolar Triangle Shaped Pulse (- 1 x 10)

Spectrum

40

20

a::l 0'0

....Q) -20"0:::3~

c: -40t:nC'O

:2 -60

-80

-100

-:/

//

//

Input

1/16 1/8 1/4 1/2

Figure bh


Bipolar triangle shaped signal's characteristics.

W sW=-.

4

44

(3.40)

Thus, for a given sampling frequency, the system may be

identified by using the binomial excitation signal results

for frequencies below the quarter-sampling frequency, and

using the bipolar excitation signal results for frequencies

between the quarter-sampling frequency and the half-sampling

frequency. Of course, this doubles the identification

effort, but it is a method for consideration in particular

systems. The effective excitation spectrum using this

strategy is obtained by splicing the appropriate spectral

sections of Figures 3.5 and 3.6, and the resulting spectrum

is shown in Figure 3.7.

3.6 Summary

In this chapter several methods of altering determinis-

tic signals to enhance their effectiveness in system

identification applications have been considered. These

methods include altering a signal's magnitude, duration,

shape, and using complementary signal pairs. The method or

combination of methods that will provide the best results

depends primarily upon the system constraints and the

frequency range of interest. The practical aspects of using

these methods are highlighted in the experiment which

follows.

45Shaped Complementary Pulses (1 x 10)

Spliced Spectra

40

20

o

-20

-40

-60

-80

-100

-:~

<;

Input

1/16 1/8 1/4 1/2


Figure~ Complementary binomial pair's spectral characteristic.

46

CHAPTER 4.

EXPERIMENT DESIGN

4.1 Objective

A prime objective of this thesis is the experimental

investigation of the effectiveness of the methods of

estimated frequency response function enhancement outlined

in the previous chapter. Since the identification of LSS

systems is the ultimate goal, the experimental model should

display LSS characteristics, i.e. lightly damped flexible

modes. In addition, some of the concepts associated with

DTP, I-CAT, and MSPS, which were introduced in Chapter One,

will be demonstrated in the hypothetical system analysis.

For example, a simple control loop will be employed to

stabilize the system to permit application of an excitation

signal, which is a fundamental step in the DTP procedure.

This control loop will incorporate the property of MSPS in

order to ensure global system stability. Furthermore, the

system selected for the experiment may be viewed as one of

the single-input, mUltiple-output subsystems in Figure 1.1.

Thus, the estimated frequency response function is exemplary

of an element in the matrix in Equation 1.1, which is the

foundation of the 1-CAT methodology.

47

In order to add credence to the simulation, the

continuous structural dynamics will be modeled on an analog

computer, and the stabilizing compensation will be provided

by a digital controller to form a hybrid simulation. The

use of the analog computer introduces limits on system

dynamics that may be viewed as structural constraints; and

these must be carefully observed, as the admonishments of

Chapter Three indicate. In particular, excitation signals

must be scaled so that the system behavior remains linear,

i.e. that the analog computer amplifiers do not saturate.

So it is clear that the ensuing experiment embodies many of

the practicalities encountered in an actual system

identification endeavor.

4.2 Physical Model

As an aid in visualizing the physical concepts

discussed, a hypothetical structure is adopted, which is the

same configuration as that used by Maggard [7J. This

structure consists of three rigid disks joined at their

centers by two massless rods as depicted in Figure 4.1. The

motion of this structure is restricted to rotation about the

axis through the centers of the disks. However, the two

rods exhibit torsional spring constants of k} and k 2 about

this axis, so that the system dynamics include a rigid-body

mode and two flexible modes. In the interest of

Disk-3

Dis](-2

Dis](-1

48

Figure 4.1. Three disk physical model.

49

computational simplification, the mass polar moments of

inertia of the disks are all deemed to be unity. Thus, the

specification of the two torsional spring constants will

determine the frequencies of the two flexible modes. Also,

since LSS modes are very lightly damped, no damping is

modeled in the structure.

A torque actuator and a rotational velocity sensor are

collocated on the first disk. This complementary pair will

be used to form the MSPS control loop to stabilize the

structure during the identification process. Additionally,

a rotational velocity sensor is placed on the third disk,

and the sampled-data frequency response function to be

identified is from the digital torque input to the Disk-3

sampled velocity. A block diagram of the sampled-data

system is presented in Figure 4.2.

4.3 Mathematical Models

The mathematical models presented in this section are in

the form of transfer functions. Since the goal is to

implement these transfer functions efficiently on an analog

computer, they are stated in forms which facilitate

programming the analog computer. The transfer functions

derived here may be transformed to differential equations

that lead to simple analog computer implementations.

... H3

( s)" T

.,.(kT)

II- ZOH Hi (s)...T

50

Figure 4.2. Three disk system sampled-dat~ block diagram.

51

Application of the laws governing rotational motion

yields the differential equations describing the behavior of

the structure. Manipulation of these equations and Laplace

transformation leads to the pertinent transfer functions

(4. 1)

s4+(k 1 +2k 2)S2+k 1k 2

s[s4+2(k 1 +k 2)S2+3k 1k 2 ] '

(4.2)

The determination of the flexible modes' frequencies is

now considered. An analysis of the dynamics of a version of

the proposed Space station indicates that significant

flexible modes will span the frequency range of 0.1 to 1.5

Hz. [1]. So these two extreme frequencies are chosen to be

the flexible modes of this model. These frequencies are

characteristic of the physical model adopted when k 1=O.2636

and k 2 = 44.35 N-m/rad. Finally, the expression of the transfer

functions in terms of the modal components yields

0.3333s

0.3348s 1.488x 10- 3s

-----+S2 + 0.3948 S2 + 88 .83

and(4.3)

0.3333 0.6667s---+-----

s s2+0.3948

5.194XIO- 6s

S2 + 88 .83

(4.4)

In general' terms, Equation 4.3 may be written as

52

(4.5)

where B3 R is the rigid body component of the transfer

function, and B3 L and B3 H are the low- and high-frequency

flexible modes' components, respectively.

Note that the transfer function of the first disk may be

expressed as a linear combination of the modal components of

the third disk transfer function. This permits implementa-

tion of the third disk transfer function on the analog

computer and formation of the first disk transfer function

by using the existing third disk modal components. This

relationship may be expressed as

(4 • 6)

4.4 Digital simulation

As a precursor to the hybrid simulation suggested in

this chapter, a digital simulation of the experiment was

conducted. Several benefits accrue from this tack. First,

the analog computer realization may be modeled, and the

range of values assumed by program variables under various

experimental conditions may be readily ascertained. This

information is necessary to "magnitude-scale" the analog

computer variables and also provides a basis for

53

establishing the "structural constraints." Also, the

effects of changing parameters of the identification

process, e.g. sampling period or time record window, may be

quickly evaluated. In addition,· the results of the digital

simulation identification process provide reference data to

validate the implementation of the hybrid experiment.

The digital simulation was created using the MatrixX

control system analysis and design software on an IBM PC-XT

microcomputer. The MatrixX programs written to effect the

investigation are presented in Appendix A. As a first step,

the CALCFRSP program was developed to display the continuous

and sampled-data frequency response functions corresponding

to the Disk-3 velocity transfer function. Next, the RTLOCUS

program was instituted to demonstrate the stabilizing

effects of closing the Disk-l loop in either the continuous

or sampled-data domains. Finally, several subprograms were

written and combined to provide an interactive system

identification program titled SYSTEMID.

In order to generate sampled-data results, a sampling

period must be chosen. According to the sampling theorem,

the sampling rate must be at least twice the highest

frequency of interest in the sampled signal [4J. Since

there is a mode of 1.5 Hz. in the structure, a minimum

sampling rate of 3 Hz. is required. However, to provide

54

some tolerance for modal uncertainty and to abate the

aliasing of noise, as described in Chapter Two, a somewhat

higher sampling rate is necessary. On the other hand, a

higher rate increases the number of data points collected

for a fixed time record window, and thereby complicates the

data collection and analysis. In view of the foregoing, a

sampling rate of 5 Hz. was selected for the experiment.

Using this rate, the CALCFRSP program produces the ideal

Disk-3 velocity sampled-data frequency response function

shown in Figure 4.3.

The RTLOCUS program provides some insight into the MSPS

stabilizing effects due to closure of the Disk-l loop.

First, for the continuous system, Figure 4.4 shows the root

locus. Note that the single pole at the origin will migrate

to the left along the real axis with increasing gain. The

four poles on the imaginary axis move toward the adjacent

zeros on this axis. Closer examination of a typical path

taken by these poles is provided by the magnified view of

Figure 4.5. This shows that the entire locus for the 0.1

Hz. mode is confined to the left-half plane for all

closed-loop gains. Thus, the global effect of MSPS inherent

in the complementary actuator-sensor pair configuration is

graphically confirmed.

4e

20

e

-20

-40

-60

-80

-100

55

~

...~

- ~\~

c-~.... ""~~ -

---... -.~

~

"~ ~- ~,

~ '"N }~~~~~

"-

--

I

-~~~

....

• 1

FREQUENCY IN HZ

1

Figure~ Three disk system sampled-data frequencyresponse function.

1~

12

9

6

-3

-9

-12

-1~

-15 -10 -5 e

REAL

5 10

56

15

Figure~ Three disk system continuous root locus.

57

-.2 -.1~ -.1

REAL

-.05 o

Figure~ Three disk system continuous root locus of 0.1Hz. mode.

58

Now, the feedback compensator gain used to stabilize the

modes is selected by investigation of the sampled-data root

locus. Some care must be taken in choosing this gain in the

sampled-data case. If the gain is too large, MSPS can be

lost in the sampled-data system, and the system becomes

unstable. Also, careless gain choices may unduly suppress

the closed-loop time responses. On the other hand, gains

that are too low may not provide sufficient modal damping

for limiting responses to desirable excitation signals.

Another consideration for deterministic identification is

that it is desirable to have the responses due to the pulse

inputs to settle within the time record window. This

obviates the need for using a tapering window to suppress

side-lobe leakage due to the input responses in the Fourier

analysis. In consideration of these constraints, a feedback

gain of unity was deemed to be sufficient, and the resulting

sampled-data root locus generated by the RTLOCUS program is

shown in Figure 4.6. Note that all the roots for this gain

are within the unit circle, so system stability is assured

in the unity negative feedback configuration, which was

analyzed in Chapter Two.

The system identification program SYSTEMID was designed

to permit the user to have much flexibility in the

identification process. The program provides the capability

to specify any of the deterministic signals discussed in

59

1.2

-18e o

-90

Figure~ Three disk system sampled-data root locus forunity gain.

60

Chapter Three and stochasti~ identification is possible

using pseudo-random white noise. Also, the user may alter

the sampling rate and the time record window. However, due

to memory constraints, the number of data samples are

limited to 256 for each record, and averaging of mUltiple

records is not implemented. Additionally, no noise is added

to the output "measurements", so the results are idealized.

4.5 Computations

At this point, a few comments are in order concerning

the computation of the Disk-3 velocity open-loop frequency

response function estimate. The analysis methods of Chapter

Two are sufficient to find the oisk-l response function

estimate, if this is desired. with reference to the system

block diagram shown in Figure 4.7, the oisk-3 response

function may be estimated by noting the following

relationships for deterministic excitations:

T(f)=X(f)-8 1(f), and

(4.7)

(4.8)

(4.9)

N1

(£)

1+X(£) ,.(£) 81

( £)

HI (f) t

Figure~ Three disk system sampled-data frequencyresponse block diagram.

61

62

Equation 4.9 shows the computations necessary for

determining the estimate in terms of known or measurable

signals' spectra.

A parallel analysis for stochastic excitations with the

output noises uncorrelated to the input signal yields the

following computations:

(4 • 10)

( 4 • 11)

( 4 • 12 )

In this case, Equation 4.12 shows the necessary computations

in terms of spectral density estimates.

4.6 Digital Results

In this section, the results of the digital simulation

identification process are presented for each of the

excitation signals introduced. Here, only the magnitudes of

the frequency response functions are computed and displayed.

However, this lays the groundwork for a more extensive

investigation in the hybrid experiment.

63

The first results presented in Figure 4.8 are due to

unit digital pulse excitation. The flexible modes are

identified clearly at the expected frequencies in accordance

with Figure 4.3. The noticeable spreading of the 0.1 Hz.

peak occurs because the spectral resolution at this

frequency is less than desirable. Since the digital

experiment is limited to 256 data samples per record at a

sampling period of 0.2 seconds, the record time window is

51.2 seconds. This implies a spectral resolution of about

0.02 Hz. which is a significant amount at the 0.1 Hz.

frequency. The corresponding analysis for the hybrid

experiment shows that the resolution will be increased

fourfold and will not be nearly so objectionable.

Next, the results for an increased magnitude pulse are

shown in Figure 4.9. Since no noise was added to the

measurements for the digital simulation, this response is

essentially the same as the response for the unit digital

pulse. Later in the hybrid experiment, the advantages of

this signal in the presence of noise will be demonstrated.

The response function for the increased duration pulse

is displayed in Figure 4.10. As anticipated, satisfactory

results are achieved at low frequencies; but at frequencies

approaching the half-sample frequency, the spectral nulls of

the excitation introduce aberrations into the identified

30

20

10

o

-10

-20

-30

-40

-50

-60

-70

-80

64

l-

•--...

i

-- ~

~l-

I-

I~

~------..If

\~. ~I- ~ r--~•I- \l-

•I-

- \l-

I-

l-

I-

~..~

'-

I- \~

~\I-

~

.. \...~

I- '",.. [\.... )\.. ~....

" ~.... ......,/

\....l-

I-

I- \'- ~~..

• 1

FREQUENCY IN HZ

1

Figure~ Frequency response function obtained usingunit digital pulse.

30

20

10

o

-10

-20

-30

-40

-50

-60

-70

-80

65

l-

I-

l-

I-f

I- ~

\I-

~

/I-

t--~II

\, ~"""" ~ ~....... I---..

- \I-~

l-

I- \l-

I-

l-

I-

~l-

•I-

- \I-

~\l-

I-

I- 1\I-

~

I- \""""

r\, }\l-

I- ~I- i"\~

I-1---""";

\l-

I-

l-

I- '\I- ~I--

• 1

FREQUENCY IN HZ

1

Figure~ Frequency response function obtained usingincreased magnitude pulse.

:sa

20

10

-10

-20

-30

-40

-50

-60

-70

-80

66

I-

~

l-

I-~

- I- \l-

I....

~f

\~ ~~ ....-. :...--- ~

- \---- \I-..-I-

1\---~ \...

\-..- \- \-~

.. \-J

.. \ t.. -,~ \ V

\J..--- \J

- V~..• 1

FREQUENCY IN HZ

1

Figure 4.10. Frequency response function obtained usingincreased duration pulse.

67

frequency response function. If modes need to be identified

at these frequencies, e.g. the 1.5 Hz. mode, a different

excitation signal is indicated.

For the binomial triangle excitation results presented

in Figure 4.11, the low frequency fidelity is again

apparent. Also, the peak at 1.5 Hz. may be attributed to

the response function since the excitation spectrum does not

have any nulls below 2.5 Hz. However, this mode is on the

borderline for confident identification.

In Figure 4.12 for the bipolar triangle excitation, the

low frequency results are erroneous, but the high frequency

mode is clearly defined. Again, this suggests using these

shaped pulses in tandem for broad-spectrum identification

applications. This is another point that is developed in

the hybrid experiment.

Finally, Figure 4.13 shows the results of the digital

experiment for a single record stochastic identification.

Due to the small number of samples in the record and not

averaging several results, the response function is not

reliable for the identification of either mode. However,

there is considerable evidence of the 1.5 Hz. mode. A more

realistic application of stochastic identification is

provided in the following chapter.

30

29

10

e

-19

! -20

-30

-40

-50

-70

68

~

~

lIIIO..II

I- 1• \I- jI-

~f

\- ,---- ",..

.",..;

- .......

~ \ r-:I---lIIIO \I- J~

~

~

\ j"""-~

.. '\

I- '\~

~

I-

~ ~l-

I-

~ \~ i\..I- " JI- r-,

~

.... '"'J..

...-

• 1

FREQUENCY IN HZ

1

Figure 4.11. Frequency response function obtained usingbinomial triangle shaped pulse.

30

20

10

e

-10

! -20

-30

-40

-60

-70

70

l-

t-l-

II- r\~ v \=/ \v

~ ~lI-

~

I-

~ 1\ ~ \ J\I-

~ ~~~

l- V'~

I-

~ ~-l- Vy f~

I-

~~

I- ~I-

~~

~l-I

I- IJ ~I Jl-

I- ~... , ~ ~il I ~i- ~r ~~l-

I-

~

I-

"'""l-

I-

• 1

FREQUENCY IN HZ

1

Figure 4.13. Frequency response function obtained usingunity white noise.

60

49

20

o

-20

-40

-60

-80

-100

-120

69

--- -~'\

I- \..~..

~ \.. \~ ,..\....

...

.- \~

~..\

0- ~~

-..~

\~I-

-,\-..

l-

I-..

• 1

FREQUENCY IN HZ

1

Figure 4.12. Frequency response function obtained usingbipolar triangle shaped pulse.

71

CHAPTER 5.

EXPERIMENT IMPLEMENTATION

5.1 Hybrid Experiment

The hybrid experiment hardware configuration is shown in

Figure 5.1. The continuous system dynamics are modeled on

the EAI Analog Computer Model TR-20. The digital controller

is a Hewlett-Packard (HP) HP-3852A Data Acquisition/Control

unit (DACU). The D/A converter is an HP-44727A Digital to

Analog Voltage Converter, and the A/D converter is an

HP-44702A High Speed Voltmeter. Both the converters are

option modules installed in the HP-3852A DACU. Finally, the

digital computer is an HP Series 9000 Model 300 computer

which is linked to the DACU via the HPIB parallel bus

(IEEE-488) .

The hybrid experiment system identification is accom

plished in two steps. First, the real-time simulation is

executed to elicit time responses due to the desired

excitation signal. During this part of the process, the

DACU drives the system and logs the necessary data, while

the digital computer generates a video display "strip chart"

of the first twenty seconds of the simulation. After the

required number of time records are collected and stored on

diskette, the second part of the identification process,

Digital COMputer

II

IIII

x(kT) I HPIBI

I

I

I

I

81(t),I,

,.(kT) 1'(t) 83(t)

Digital Controller "'- D/A .. Analog' COMPuterp:-

(HP-44?2?A)

(HP-3852A) (EA I TR-20)

81(kT) ,

83(kT)

AID

(HP-44?02A)

Figure 5.1. Hybrid experiment hardware configuration.

72

73

frequency analysis and response function computation, may be

executed. The hybrid experiment incorporates all the input

signals that the digital simulation accommodated. In

addition, averaging of results for several records is

possible in this implementation. Also, there is sufficient

noise inherent in the sampled-data system to demonstrate the

fidelity differences in response function estimates. The

function of each of the major hardware components and its

associated program(s) are summarized next.

5.2 Analog Computer

The oisk-3 and Disk-l velocities' transfer functions are

programmed on the TR-20 analog computer as mentioned in

Chapter Four. An investigation of the analog computer

realization was conducted with the digital simulation to

find the range for each differential equation variable. A

pseudo-random gaussian sequence of zero mean and unity

variance was used as the excitation signal. Based on the

results of this examination, a magnitude-scaled program was

developed for the analog computer. The program is

represented in functional block form in Figure 5.2 using

integrators, summers/inverters, and potentiometers [8]. The

variables shown in this figure are normalized "x" variables

corresponding to the SUbscript variable. Also, the

nUmbering of the various components in this figure

X'T

74

SUJ"'IMer/Inverter

LEGENDn : ta9' nUMberg : gaIn

oPotentioMeter Integ-rator

Figure~ Analog computer simulation functional blockdiagram.

75

corresponds to the actual hardware tag numbering. The

potentiometers provide magnitude-scaling of the differential

equations' variables, feedback gains, and scaling for linear

combinations of modal components. The resultant values

determined for the potentiometers are recorded in Table 5.1.

Analog Computer Potentiometer settings

Pot. Setting Pot. Setting Pot. Setting

1 none 7 0.398 13 1.000

2 0.100 8 0.001 14 1.000

3 1.000 9 1.000 15 0.200

4 0.079 10 0.224 16 0.200

5 0.888 11 0.500 17 0.002

6 0.200 12 0.500 18 0.007

Table 5.1. Analog computer potentiometer settings.

5.3 Data Acquisition/Control unit

The DACU is loaded with a data acquisition/control

subroutine and the specific excitation sequence by the

digital computer. During the execution of the subroutine,

the DACU performs the following functions:

76

* Reads and stores the system outputs via the A/D.

* Computes the closed-loop error signal and sends it

to the system via the D/A.

* Sends the first twenty seconds of data to the

digital computer via the HPIB for strip chart

display.

since the DACU program is downloaded from the digital

computer, it is part of the SIMULATION program in the hybrid

experiment software listed in Appendix B.

5.4 Digital Computer

There are three programs listed in Appendix B that are

used by the digital computer. The first is SIMULATION,

which is an interactive program to drive the time data

acquisition part of the identification process. The second

is SIUDI (System Identification Using Deterministic Inputs),

which does the frequency domain analysis, frequency response

function computations, and averaging for deterministic

excitations. The third is SIUSI (System Identification

Using Stochastic Inputs), which performs the corresponding

functions for stochastic excitation analysis.

The SIMULATION program performs the following major

functions:

77

* Prompts user to specify the excitation signal and

number of data records to add to the storage

archive.

* Downloads subroutine and excitation signal to DACU.

* Displays strip chart of time data.

* Retrieves time data records from DACU and stores

them in the archive.

The hardware and software outlined above were used to

evaluate the effectiveness of the signals discussed in

Chapter Three for identifying this hybrid system. The

results of this investigation are considered next.

5.5 Hybrid Results

The results of the hybrid experiment presented in this

section are divided into two parts. First, the signal set

used in the digital experiment is evaluated in this case

with the addition of phase characteristics. Then, a group

of signals is selected whose characteristics are desirable

for this specific system. An extensive identification

process is conducted with these signals which includes

averaging multiple results. Throughout this section,

graphical analysis is again employed to judge the relative

effectiveness of the various' methods considered.

78

First, Figure 5.3 displays the unit digital pulse's

characteristics, and Figure 5.4 presents the identified

frequency response function. Both the magnitude and phase

functions are very noisy and contain little information.

The 1.5 Hz. mode's magnitude peak is discernible, but on the

whole this response function is very unsatisfactory.

Next, the increased magnitude pulse data is given in

Figures 5.5 and 5.6 respectively. There is remarkable

improvement in the results. The low frequency magnitude and

phase are close to the ideal values, and the high frequency

magnitude peak is clearly identifiable. The slight shift in

frequency of the low frequency mode from the target value of

0.1 Hz. is most likely due to slight inaccuracies inherent

in setting the analog computer potentiometers.

The increased duration pulse results are provided in

Figures 5.7 and 5.8. The low frequency fidelity compares

favorably with that achieved in the increased magnitude

case, but the spectral nulls of the input interfere with

high frequency identification.

Now, the binomial triangle shaped pulse excitation is

considered in Figures 5.9 and 5.10. This signal performs

better than the unit digital pulse at low frequencies, but

it has no merit at high frequencies. The increased

magnitude and increased duration pulses show superior

Unit Digital Pulse (txt)Exc;tation Slgnal Spectrum

79

!························~················_·· ..t···········..·..·..··....·....··1··..···..··· ..··········~· ..·..·· ......·..····_·! ..···..·..···_··....·········..·j················..··--f··..···········..···....1····

: i j + ,;, t ;

. ; ! : : . } :·······..·······..·.. ···1'··· ..···..··········· ..1..· ·..··..·..···..·..··· 1..····..················~..·····..·····..····..· ·~ ···..·····..·····..···· I· .. ·· ..·..·· ..··········i···..· ; .

r + r · f ~ ~ ;·······..···············1'···..···············..·1···········.. ···..······..·..···]'······················1·..·····.. ··············~·· ..·····..··················..··i.. ·················....·1..··.... ·· ..············1'···

+ r f I ~ ! r j························r·············..···..···1·--··········..········_·..·1'····..···_··..····..··1· ·..· ··..······r..··..·..···..·····..······ 1' ·..·..-1"··· ..·····..··· ··..1..··

• • + • .. • ~:::: r::':I ! ~ J ~ I . .·....···..··..·..··..···r·········_·········..r ..······....········......l···..·····....···_··_~· .. ···..···..·..······.. ~..·····..····..·..··..···········r..········..····......r ....·..·········..rt r t • ; • ~ :

-100"------......---------------------'t£IS)

40

20

'300-0

c -20.-0

-0 -40:J+J.-CC"J -60..,I:

-80

Frequency in Hertz ..-Unit Digital Pulse ( Lx L)

Excitation Signal Spectrum180

135

90

(J)11) 45~s,0')ll) e(J

c.--45

d)V)

~

~ -90a..

-135

; i

..........._._ L..._ .t.. _._ _.: _.._ : J_ _..: .1.. Lttl : 1 ~ I :

···· ············ ..·..i..····..·_··· ..·..·..·· ·..·..· ··..· j i..·· ···· .. ···.· ..·.·· ..····.i··.··· ..····· .. ··· ..·..· j .: : : ~ : . ~ :

• + I + ~ • :. : ; : : : : ~...... ·,········· .. ··_···~······· ..················1·········· _~ ~ ~ _ ~ _~ _ ~._.

+ • j. + t. . ~ ~ ~ ~

.... ~.;.

t t t t t t ~ t························r······ ·············~······· · · · · · · ·__··········_··1························4!'························f····_·__·····················~····· ~ ~ .

····· ..· ··········;····..····..·..··..·····f············ ! .;. ···i··..· ····· ..·.._·········~ ..···..·····..···..·····~··· ; .

.- t .;. ~ ~

. : .. . . . . :··· ·.. ··.. ··········;·· ··.. ·····..····..·t············ j ; ·i· ..······· ..····..··..···· ..·· .. ·j························~ j ••..

t t :.; : f t ~ ~~

("\JI"'\J-~-180 '------------------------------...1lJ'")~I~

Frequency In Hertz

Figure~ Spectral characteristics of unit digitalpulse.

Disk-3IJn; t

VelocityDi g; ta 1

Frequency Response tunc.Pulse (txt) Excitation

80

(,\J

i+

~ ~ . . : . i :............. " : ••••••••••••• II" + ~ ..........•.......•.....~•..•.••.••••..•.•....... ~ ..•••.....•..._....•.•........••~..•..•.•......._ ~ ~...•

; . i . : :: 1 1:

··············.. ········~············ ..····.. ··..t..··.. ···..·····················:·········..···..······..r·········· !..t r

..............._ + "1 -·················'j'····..··_-····-f-·--···..··..···1 __ - +·'"j 1 I j

r . r t ...................................................; _ _ ····..·..··..·..·..···,··..· ·.····..···..········.··i····.·· __ .

~ I ~ ~ ~ ~ ~ f. : : ; i : :

r t r r • r····..··..········..····i·..·..··· ·· ···..j""..··· ····· i..···..·················!"··..··..··..·····..··r··..·..··············..··..··..I····..·..·..·· ····t·..··..········..·..··..: .

. . . t

-100 '----.......----------------------........-----........,\[)

~

40

20

aco-0

c -20

w'"C --to:J+>

C0") -6e'"z

-80

Frequency in Hertz

Disk-3IJn i t

Velocity Frequecny Response Func.Ili q t t a l Pu l s e CLx l ) Ex c t t at t o n

.: ......,:

..;.. ...

ti : :c:_.1~. ii:: :i: ii::.. ~;~~ H~~~

: ~;~ i~ ~ ~ ~ rg~~i~!~i ~1~5~iii ii g 1 ~ i :;;iiiii~f i~i~ai~

..· ·..··..·..···..··i..· ····..··..~.1--..::··..·_··..··•·•·· ~ + _ ~.._ ····++~··~·..·~·+·~~·f~~~+·H~~~i·~~• : ..t ~~ : ~~~.;~ : ~ ;:~ii·~~

... ~. .... .... ~ :'1 ' i:: ":f l:-18e I----......-.---""----------------------~

I.J")

~

180

1"iC'oJ.J

90

v.tt1)

~511)

~

mOJ

0I~

c

-451)(/J

It1J: -90a.

-135

Frequency In Hertz

Figure 5.4. Frequency response function using unit digitalpulse.

Magnitude 10 Pulse (10xl)Excitation Signal Spectrum

81

I:'\J

........................! '! ··········.._···············~························1········_·· ..-········i···· ···························~··············,···_····t························1····

; ~ : : : : : :

w::;:;; ~.'::!:.' ~ ~ ~ ~. . r r I . . r························l..···········..·········i ···..······..· ···· ;..··..···..··..···..····r···..······..·..········:·· · ····· ..·········.. ··~········ ..··· ·· ·I· ~ .

r r + r : • t 1···········..·..······-r···················....r··..··········..·····......·····r······....··..·..·..···T···············..·······i..·······..················· ..·..1'..·······......······..7···....·······..····....1'..·

~ 1 l ! ~ ~ i ~

r r 1 t r · r r··..·····.._·..····..··r-···..·········..·····:··············_·_········_)·········_······_··:..·..··_···········_·f········..······················r·······..········..·_·T· _ _ : .

r f r r r · t ;..·········· ········r····..······..··· ··r·..·..· ···..· - _: 1' ·..·..······ 1'..··..·..·······..· 1··· ········ · 1·_·

• r r r r t • •-10e "-----------------------------'

U'1~0'.)

40

20

0CQ"'0

c -20.-Il)

"0 -49:3+>.-c0')

-60,~

~

-S0

Frequency in Hertz

Magnitude 10 Pulse (10xl)Excitation Signal Spectrum

t • • t + • + •......·· ·..···· ·].. ···· · ·f··.. ·· .. ·· .. ·· ..····· ·····..1..·· ·..·· ··..·+····· ··_ .._···.. i · ····..•..·l..···· .. ········_·· t·········· ·.. ·····.. ~· ..·

• t • t • • • ~

........................I. J 1... : _ : L J L..! f ~ t • t • •

····..· ·········..··~·······..·········..·..·t······························ ..~··..··· ··..······..·t ·········..· ! ! ·..······· .. ·..·..·r ····· .. ·· .. ······· ..i.. ··I ' · ~ - I :

.................__ ~ _ ~ ···..····.._····i· ..···..····· ·····f..····· ..···········..··1..· ·..··..·.. ······ ·..·..~············ .. ·········•·· ~ ..

r : · · · r I ·.' ~.- ~ _ ": , ·T········_··············I········_············· ·.. ·1··· "' ~ ~ .

J ! · t f I I i

• ..:

..

I • • •······..····· ..·..·t..···· .. ················~ ....·······.. ······.. ······.... ··~· ..·....·· .. ··· .. ·······~ .. ·· .... ····· .. ······· ..r..·• f •

........................i ~ ··..i·

180

135

90

V)

11) 45l1.)s,0')l1)

0~

c.-

-45/l)(J)

10,- -90a:

-135

-I$)

-180~-------------------------_....-u;I'S)IS)

Frequencv in Hertz ~

~-----------------------------------Figure 5.5. Spectral characteristics of increased magni

tude pulse.

Disk-3 Velocity Frequency Response Func.Magn;tude 10 Pul~e (lax!) Exc;tat;on

82

,.. . .: , : '~A: ' : '20 ,......,.:] 1,....····...·..·r·.._..·..··..·..··.........:..·....,..·..··..r..··....·..·..·..'···r'a

('\J

t t t t t-60 ·_..·_ ·i..· ·· ·~ · · l + +-_ j ..

i : iii ;

t t t t t t .-80 1· ··· t · · l ·..· ·· t..· ·..·..· !..· ·..· ·..·:..· r +.

t t t' ,-10e"'-----------------------------"'"

L(')

~

F~equency In Hertz

D1Sk-3 Velocity Frequency Response Fune.M-a~~.+"~. 11A C •• l .... t"1C\v1't C-v,..;+-a+;o""\""• • ....~ •• 'I v ....... -." .. '-' I " .... ,.. • I ... ,-, "-' • v ... ·." • .. ••

: ;

::Il ; ': j l. f180 r----~---~----~---~---~----__:_---~---~

135

45

I:"\Ju;

-180~ -.l

u-,''::l~

Frequency In Her~z

Figure 5.6. Frequency response function using increasedmagnitude pulse.

Duration 10 Pulse (lx10)Excitation Signal Spectrum

83

40,---~-------.....--~-------~--~--~

-100"----------------------------~Ui

~

c

. . . .

20 r------j':---~;----~j--~:~~.·..~··..~.··;,:,::~.:.-··t:······==.:::·······..·· ..·,····.···················r·····················,.. ··. ~ . . . \' ; ~

o ······················· ·····················r·················..···········....··_···············r····················r··························· ·y~:~:l-·· ",r

-20 : 1.. ·············..···········,·······..···············t···· r-.............................. ...

1 t • t r •-40 ·····················..r:..·········r···························..r······················r·····················T··········· ··..···..·····..···..r·············· . . ! .

.~ t! r t t r f t~ -60 ··· ·_·····..·······f······_·····_-·..·..·r..·····..······ ~ + _ ! · ·..··.._·········· ..·f········· ·······..-f········..··..·..···..··f····!: .!: .j i .i : .J :

!.:. • ~:: I:~ . i i ! . j .-80 ···········..··· ····!..·· ····..···..·····f······..·· ·· ·· ··· ..·f·..··..··..·..···..·····+······..··········..··..f··········· ··_..·· ··..··~······· ..····.. ····..·..+ ··· _ ~ .

~ 1 : ~ : ' ! j+ r t + • ! ! +

Frequency in HertzDur at ion 10 Pu 1 s e (1 x 10 )

Excitation Signal Spectrum180---~---~----------~------------

135

«>(f)

I'd.ca.

I + • I" I +····· ..·..···..·.. ··....!·..·....····· ....·......t··· ..···.. ·····....·..·..··..···:··············..··....··t·..·........·....·..··..i··..·......·..·..·..··· ..·..··..·i· ..·....····..······....7··....·....········..·..;....

: I I ~ · I : r90 ·················r·················:······················..···T············r···············r······················r······t··········--r·

45 .. ···········..·..··..··!·..··············· t· ····..··.._ f _ r _ ~ ~ + ~: r~.,~ + .. + .;:

7 ~ ~ t ~ '. :o ::::·-:::::::::::::::::::+:,::::::::,,:::::,,::,·::::.t::: : f" j _.... .. _. ..ft·..···..· ~... ..; ~ :~ • ~ ".. ' .. f.. + :. :

-45 ··..·..· ·..···j·_· · ·t..·..·..·..······ ·······..··j······· ·· ·..··t· .:z:-··::··.. •·..!··· ..·..····· :, ~ ~ .f ~ ! ~ '~

t • ~ t r I: ;-90 _..···· ..·· ..· ···! ···· ···· ..T..··..··..······ ···· ~ ·..·t··..·..·.. ··· .. ·········:·..··..·.. ~··~·· ······ ·I · , ..• ·· ·.:4 ..

!. ~.: : .

......................+ ~ ·············..i·· ················1·····················1··············· :::1 tt t-

-180lSi - ~"'J

CS\ ~ ~I~

-135

Frequency In Hert~

Figure 5.7. Spectral characteristics of increased durationpulse.

Disk-3 Velocity Frequency Response Fune.Duration 10 Pulse (lx10) Excitation

84

........................! ~ ! + ~ ! + ! .\ : ;. . ~ :

~ r

; ..:

...::I:1;:C:~:::::::::::::I::::::::::::::::::::::::::::::I:::::::::::::::::::::-:::::::::::U'::::::::-::::1 .. ; \ ~

: : '. :

........................: u u..' u u'»K: u u.f + ~ + t·v. ~~IJ J1 ~ i i ~ Jyf'\J.. ····.. ·~······· ..·..t ..····..·..·· t..· ·· ·-r···..··· · f.. ·········..·..·····..·t·..·..·········..·..·· ·· ·t..··..··

i

t : t .... ···.. ·· ..·..·····..···t·..···..····· ..·..··· .. ~·· ······ ·..· j _ + ~ j + j .

-100 '---- ----.1

L(')

i

40

20

aCO"'0

c -20

tl.'1:1 -40:J~

COJ -60Ittz

-80

Frequency in Hertz

Disk-3 Velocity Frequency Response Func.Duration 10 Pulse lx10) Excitation

180

135

90

VlOJ 45l1)

~

01tD aU

c

-45IJj(/.t

m.£ -90c,

-135

(\J-180 '-------------------------------------....

lr')~~

Frequency in Hertz

Figure~ Frequency response function using increasedduration pulse.

Shaped Pulse (lx10)Excitatton Slgnal Spectrum

85

N

i ~:.;:.~. ir · :.......................; + : ~ ; ········..········+·····················r············ +..

··· ····..····..··..r ·····..·····..·1 ··· · ·..· j···· ····..····..··..r·..··..··..····· ···i-··..· ·..·····..·· ···..r·..····_·..·.. ..: .: : : i : : . :It_ _ ~_ + _....•.........•~..............•...._.•.~....•.•......•.•.•......:...••._.........•...••..........:..It" •••••••••••••••••••*!r......... . _~ .. ,.

~ i 1 ~ ! ! ~ .r r 1 r t ! t !

.._ + _ _..+ ! t- ! _ ! +....... . ! .I 1 1 1 i : ~ .

J r r r t : r !

···..·····..············1'..·_···· ····..··1·····..····..·_·····..··..···r·····..· ·········..r..·..···············..··r·········..········..·······_·1'······..·· ···..·····r..···· - j .

+ + + ! ~". + • +! 1 1 1 ; 1 . :..··_···_--····r..........··......·..T...._..·····......···..·..·T....·......·..····....T·....·_..····....·..r·······..··..···..·..·..···..T··..····..·..·........f·..··..···....·......"(..r r t t • ! r h

-10e'---------------------------~i

'40

20

aa.1"'0

c -20.-Il)

-c -40::J~.-cm -69toL

-80

frequency in Hertz

Shaped Pulse (lx10)Excitation Signal Spectrum

180

135

90

f/JIJJ 454)s,0')(1)

0-0

c'- -45ll)(/)

l'O..c -90a...

-135

· • t : t : ,.. •·_·········· ·····..1·····················_·t..················..···..·······j·············.. ·········t······_···_ ·······:··..········..·_····_· ···~···! .\···· ·····~··· ..········..··t·\···l..··

• t • t t t t : +.;, .;, .;, ,;,"" ~ :t~ : ~':. .

...........·············!·····················..·t···.._ ~ ·· ..·················t····.. ···· j ·•· .. ·i···~········ ······\· ..·t·· ·t.. ··~i'···

• I + . t I·' .1;....····..·····..·· : ···..···· ·_·T···..··..··..·..·· ·· ~ ! ······..·· rr..·..··..·····..l ..··..····..·..·7·····t ·

~ t ~ : ::.

Ui

Frequency in Hertz

Figure~ Spectral characteristics of binomial triangleshaped pulse.

Dlsk-3 Velocity Frequency Response Func.Shaped Pulse (lx10) Excitation

86

t·....·......·....····J.·..

U"')

........................~ + ··························r-····..···········-i··············..········,···························..····t···· .1...

::::::::::::::::::::::-::.:::::::::::::':::::::r::::::::::::::::::::::::::.I::::::::::::::::::::::;::::::~::::·::::::::::::~:::I:::::::::::::::]no ..~ ~ ~ f ; l

t + t r i

···..·······........··..t........·....··..·..····t...... ·· ..····..··..······_··..t···....·..···..·..·..···f·..··..··..····....·....t....·....·..· ._.... ..j i i !

t . t . . . ' .•............- -·············_···········4····················_··.···· _ _.......................•- ·..· ···.•..-...· t••••

j i j ~ ~ 1 ; ;: : i :: :.. ...... ..I . i J It.: r...................·····1··_····· f' •••••.! ! _ r: ! _ -~ I •••• - • ..··r···_·····..··· !.. fl

~ ~ ~ ~ ~

f f' t\ i 1

-100 '--__IIlIIi-- ~

i

40

20

aCO'"'0

c -20

Q)

'"'0 -49~

+l'

c0)

-60t'C11:

-80

Frequency in HertzD1Sk-3 Velocity Frequency Response Fune.

Shaped Pulse (lx18) Excitation

: .t~ ... ~ iL n~~ d

•.... ::. .: :::; ::: ii :: ;:; :i:: :: '31

./~ ..! • . • -If .' ;tm~~ ~ln il ~....·..··::····-'· r · · ··~ ·· ..·..· ·r · ··· ..·~ r· ··· ·:!r.. ·..·r·!nnrn r·'f: . ::i+ ~ ;f ~ ~ :

-180 ......------- -..l

\Jj~l~

180

135

99

fI)

IV 45(1)~

UJt[I

0"0

c

-45d)

Chf~

.c -90a..

-135

Frequency In Hertz

Figure 5.10. Frequency response function using binomialtriangle shaped pulse.

87

results; however, it should be noted that their spectral

magnitudes are arbitrarily higher in this case. In the

second group of signals considered in this section, the

relative strengths of the excitation signals are chosen more

equitably.

In the complementary case of the bipolar triangle pulse,

the results are shown in Figures 5.11 and 5.12. Again, at

this excitation level the results show no merit.

The final result from the first group is the stochastic

excitation response function of Figure 5.13. Here the

magnitUde peaks of the two modes are noticeable, but the low

frequency peak is badly spread and the high frequencies are

quite noisy. The phase characteristic is also quite noisy,

but it tracks the expected phase somewhat at low

frequencies.

For the second part of the experiment, the signals were

tailored for use under the constraints of the given system.

Recall that the analog computer realization was magni

tude-scaled based on the results of the digital simulation

excited with zero mean, unity variance gaussian noise.

Therefore, the digital simulation was used to scale the

magnitudes of the excitation signals so the linearity

constraints of the analog computer would be observed.

Furthermore, due to modes of interest near the half-sampling

Shaped Pulse (-lx10)Excitation Signal Spectrum

88

40,...---~--~---~--~--~---~~-----~

20

('\J

-20 ···..· ·····1" ·..··..·..·····+..····..· · ·· ·1..····· ···..···· r ·..·..· ·..·\"· · · ···.. ·····.'1" _ · ·..· 1..

• t r · · r . t-40 ~ _ ~ _.~ ~ ! !.................. ._.! ! .

~ ! ~ ~ ~ ~ I f ~

.~ t t r r r Tit r~ ~: ···..r..· ·r _·r..·_· ·..·..·r..·..·_r..-- · _·· ·~) -1"' 1"'.

..............· r· · r ··· ···· f· ···..·_· ·r · _·..·r ·..·..·..~ · r..·..·..··.-100 '-------------------"-----------~

iFrequency in Hertz

Shaped Pulse (-lx10)Exc;tation Signal Spectrum

• t.. :.: t. ~ t. • *. . . . ! :. ;. ,::...... ·· ~..· ··..·..·· r..·..·..·..·..· T..·..·..··..·..·..·· ..I..· · · ·..r ·· ······ ·..··• ·~· ..··· ~ : ~ .• t t ! • \ ~..............._ ; ~ ·..·..··..····i..·..···..······ ··t·..·····..·····..··· ..··l·· ~ ..:~ ;j .• t t t + .: ~ •

180~--o:___--~---~--~--~------~--~; i+_:.; _:.; ! t ·.i.·· · 1.. ; ~ ~

135 ..·· · ·( · ·..·· t..L\ l _·t · · · 1" · ·..·..1" · ·;..·:1 · ··..[1..

90 1 I··t ····..'. .r :;..: ; .I. ; !..\ :.[.,::..';T •..• ·..··t· ·.......... t .1. :

+/ ...... ~... . t4S ··.. ··.... ·············~ ..··..···....··..····_·ft..······.. ·_··..·_··.. ·_··~·· ..···..····..·..··_··f..···::·':··-:..·..-:--····f·..·_···..·..·..·..·····....····1......····..······~· ....~·····~ ..···.. ···....·1!····

t 1 + t "!- +' : ~o1---- :-"..::::·::·::·..·l · ·..· ·..·..· ;· ··T ·..·..· 1" ,·,:~ · T ·T T ·~ ·..·..·t

-45

-90

(''\,J("\JIi")~

. ; ;:·..··..· ·T ..·..· r · ·· ·T · · T ·..· · T · · ·r ....+· ..! · ·;:::~·..F

-135

-180'-------------------------------~lJ')IS)l~l

Frequency in Hertz

Figure 5.11. Spectral characteristics of bipolar triangleshaped pulse.

Disk-3 Velocity Frequency Response Func.Shaped Pulse (-lx10) Excitatlon

89

40r---~--~---~--~--~---~--~--~

".J

20 , · · i.... ....\ t ; ·· r· ··1'..

: :o . ....· ·T ·..·..··· · r..· \~ ····· T ···..·..· ·..··..·..·..1 · ·· .r..·..· · 1..·

........................i .l : 1 :...; t) 1 i 1

· . t . r . .·•·..···················1"·····················..1································1························1··········..··········..I..···_·..·····················..r· · ! .

. r . r : . . .·..······· ······..1'·····..····· ·······1..·..·..·· ·_········· ·r..·..····..··········..l..···..·····..······..··1'·· 1' ···.. ···..··r..······..············..r···• ••iii ;

• = • ; ; . ~ •...................._ ", _ _ - ········t····..·········· -•............., -•....: i 1 j 1 1 j ~

+ t . r t •-100 "--__"'--__-..-__.......i ----.__---i.__---..J

!

CQ'"C

c -20

~

"'0 -49:J~

CC') -69IU

1:

-80

Frequency in HertzDisk-3 Velocity Frequency Response Func.

Shaped Pulse (-lx10) Excitation180

135

99

"~ 45~

'-0')

o0-0

c

-45Q)(/J

~

.c -90a..

-135

,....J-1813 ~------------------------------------~1.0

~

Frequency 1n Hertz

Figure 5.12. Frequency response function using bipolartriangle shaped pulse.

DiSk-3 Velocity rrequency Response ~unc.

Unity White Noise Excitatton90

40r---~--~---~--~~--~---~--~--~.....

a

: ~

-20 _ ~ ~ ········_········..··1···..···· ·········_+············ ;. . ) _+ ~ ..~ :! j : 1 :

. . . . r t-40 ······ ·T ·· ··· r ···..· ·r ·..· · r ·..· 1".._..·· ..

.. .. . .. ..

! i 1 ! I .-£0 ..·..·..·..· ·..··T·..·_· ·-1"·..·..·_·..· · 1' ·.._·..··..·..·r..··· 1-··..···..·..·· · ·····1..··· .

r r I 1 r . .-80 ! '!' ! _.~ _ ~ _ _ ~ ~...................... ...

1 i 1 j i . j+ t + t t

NIt,)~

-100--------------------------------~If')cgIS)

Frequency in Hertz

Dlsk-3 Velocity Frequency Response Func.Unity ~hlte Noise Excitation

180

135

90

f/Jo

~5Q)

'-rsJQJ

B~

c

-4511)V)

Itti: -90a.

-135

t ;·. : . . ~ ..::: .:::;~ !:If!::

: ~

.......................,t. .J J t J. _ .L. ~.~.fJ.J ~: . . ~; : . ~ ~ j \f j

~1 : .:: :: :....................~ ~ _··,::-··.._· ··_·:· ;..; ···:·;··::;:r· · ·· ~ · ~ ·· ·ljj·lT'..:il· i · · r ·..·..l"' r · ·..r..··_· -r ·..l.rfl'···..ff!1Ii=

t t t~ ~1 ;~ j-180 '------------------------------'

t.(')

~

Frequency In Hertz

Figure 5.13. Frequency response function using unity whitenoise.

91

frequency, the protracted constant amplitude pulse was

omitted. These constraints lead to consideration of the.

following excitation signals:

* Increased magnitude pulse (10x1).

* Binomial triangle shaped pulse (2x10).

* Bipolar triangle shaped pulse (-2x10).

* Unity white gaussian noise N(O,l).

Additionally, the benefits from averaging results are

demonstrated by comparing the average response functions for

one, four, and sixteen records in each case.

First, the results for the increased magnitude pulse are

shown in Figures 5.14, 5.15, and 5.16. The reduction of

noise in the response function estimate due to averaging is

clearly demonstrated in both the magnitude and phase plots,

especially at high frequencies where the signal to noise

ratio is smaller.

Next, the binomial triangle excitation is considered in

Figures 5.17, 5.18, and 5.19. These results improve with

averaging at low frequencies, but the high frequency results

are not usable.

D1Sk-3 Velocity Frequency Response rune.Magnitude 10 Pulse (lext) Excitation

92

·······················f······················r···········..···············1································· >:~!:. ;····..·..·········.. ·· .. ·····l..······················~··· _L.; :....·....·· ..·..........···..················....r·..·····..·..··..·..··········t..··..··....·....···..t..··....·..·· ..···..·..·i..··..·..........·.....r ! .

··········..·· ·· t ······· ····· t..··..····· ..····..·· ···..··i ············ - ..··..·..···· ..·..· ·!..·· ~ .i i .::

r r· ··..· · T..····..· ·· t · · · ··· ·.. '1'..· ·· ·..·..r: · ·'1"· ·· i.. L +..t i' '

40

29

0

£II1)

c -20

dl'7J

-~0:J~

Cm -60ttl

1:

-80

(\J

Frequency In Hertz

Disk-3 Velocity Frequency Response Func.Magnitude 10 Pulse (lexl) Excitation

180

135

90

CI)

l1) 45~s,0)

QJ0-0

c

-45C)(/J

f(j

.c -900...

-135

(\JIf")t\JLfjC;).

('\jIS)

-180 :- ....... ~~..J

l("JIS)tSI

Frequency In Hertz

Figure 5.14. Frequency response function using singleincreased magnitude record.

Average (4) rrequency Response runctionMagnitude 10 Pulse (t0xl) Excitation

93

N

········· ··j_..·_-_·_·..·t.._·..· ··· ·..·..·l·· _·-· ·1 'l _ i ~ : .

. i

·······..············..i· ·..··_···· ····t···········..···· ·····..····t·····..·· ··..··..···t······..······..···· !......................... ··!·..······..·······..····t·····..····· !._.1 ! ! 1 1 !

: 1 t t t . '•.............._ -...- _ -- _.............•......_.•..............._ .1 [ ~ 1 j j

1 r f f r : ; ......................... !•............_ ~ _ ! ..- ~ _ _ ! _ _ ~ ······t························!····i ! 1 ill ~ 1: l : :: ~:

~ t t r r . t-100""------------------------------......

Uj

~

40

20

0a:J"'0

c -20

4)

'7J -49~....cen -S0Itj

1:

-80

~requency in HertzAverage (4) Frequency Response Function

Magnitude 10 Pulse (l0xl) Excitation

uj-~

.; ~:.

-~.:: . : :: a:HE: ~!!

. . ~ . . : ~ ~.: Hjill

· . . ...-_ ....-· 1. ~ ! mi!i~!! !! ~

. . . f~1. .: j ll~r!! 'i I=........· · + _.._..t · · +..· ·_· tT · ..\ ·..i- ··-i;~rt~·ff·TI=-

: I Y : . .1!1t 1 If ~I;:::::.1:..·-....::::.:::..::..::..r:::::::::::::::::..::·1······· ....··· ..·······T··..····_..···· ..··············\······..········t..1TT···:····rlhi· . : ' :... 116............ ··· .. ····.. ·~ .... ··..····..··..···..·r·····..·····....····.. ········-r·····..········..·..·..r·· ..·····..······..·····1·....···..·....·····..·....······r··.. ···········+·..!..r·..···..·····'1Ir:~.

~ + , t t ~ ~ ~ img ~ ~~t t ~ . .:: ;.~~. ffil

-180 ~----------------------------'IJ1s

180

135

90

"It) 454)~

mII) e-0

c::

-45~III~

s: -90a.

-135

Frequency In Hertz

Figure 5.15. Frequency response function using fourincreased magnitude records.

Average (16) Frequency Response ~unct1on

Magnitude 10 Pulse (lext) Excitation

94

(\J

················~..·_····..·· · t··· ..···_·..·..····..·_· i..·..··..··_··..·······t i i ~ _;••••

..............._1 =:= t \ ..t. ...- _ L. L.: : : : x : : : ;r t r t ~ t T t

..· · ·..·j· · · l ·..·..· ·r ·..l· ·..r~ ·..r..· + · ·r..

.. ···..· ········..··!·..······..····· ··..t···· ..····· ··..···· ···t..······· ·..·..· t·····..···..··..·· f· ···................. .. ~ t".......... ·..······f..·j; ~ i ! it t t t + t~ ~ , 1 ; .._ _.................•..........- .- _ _ - .J J ! : ! ;; : : : : :

f r t ttl........................! + ! + ! ! + ..

~ ! ~ ~ ; 1 ~

. 1 .• •-100 .......-------------------------......

i

40

29

0CO

"c -29

G)

""C -40~

+oi

C0') -60tfj

~

-80

~requency in HertzAverage (16) Frequency Response Function

Magnitude 10 Pulse (10xl) Excitation

(\j('\J(\JtS)-~

t :: ~f

T . , . . . . ~~~ j~..........................! ~ .•.........._ _-~ __ +......•........••..••... ~......•••..._......•.•.......••• ~...•.•.....•.......••...~ :+:.~:

~ ~ ~ ~ ; ~ ; ! g:1 ~

•• •• +:':;". :'::::, - , ·"IE..:; 1 j ~~~~7 T T !?iii:. : ; :: . :::=:

t ~ , 1\ + . ;+1' i ~ i

..............· · , · · ·..·f· · · · i · _ t.L · ·; · · ·j" · 'l...· _... L ~'

":'; ~:' : : ;: :iii!t-----.......---.;.---:.,:.:::..:..: :.--:::+ :.:::::.•::::.•:::.~•.••........•••...••.... ~ ··········~························~······~······f··;;

: :: :===t ... III

~

........................! ; ~ ! ! ! ········~······ ..·····iiif!·: : : ~ . .. . :::=:

t . t t "1~ ~~.t ~ ~ ~ 1TI-t8e Lo- ........ --.I

L(')

~

180

135

90

~

11) 45t1)~

enl1)

0'"0

c

-45lU"4)Itt.c -90a..

-135

Frequency in Hertz

Figure 5.16. Frequency response function using sixteenincreased magnitude records.

Disk-3 Velocity Frequency Response Func.Shaped Pulse (2xlB) Excitation

95

40,......--~--~---~--~--~----~--~--~

(\J

20 ····················..·i···············~····_·~········ · · · · · · · · · · · · · · · · · · ..····i························+····· ..··· ··········~···· ·..·..·..·..·····..·.._·i·· ..··..·· ·· + ~ ..

-40 ..·..·..·..···· ..··..·..r·····..··············..r ··········_··_.. ·····.._·j"··_··············..·.. j·..··· ······· ···r· .: : : : ;

r ~ ; . r . . .-60 ·..····················t··_·················T-················..··········r- · ·····..r· ·..·..-r _ '1" ·_..r..· ·..--r... .. ..

. . I . I J . J-80 ..· · · T· · ·_..· ~..··..·_ ·_·· ·-T..· T_·..· ·..1"··..·..··.._..· · ~ ···..··..·..···..·..r·..· ·r..+ t t t . ti ~ \ ~

-20

-100 "----------------------------------'Lf")

~

Frequency in HertzDisk-3 Velocity Frequency Response Fune.

Shaped Pulse (2x10) Excitation180

135

90

~

C) 45~s,

m(I

0'1J

c

-45«)

("111.c -90a.

-135

-180~ - (\j l(")

~ IS) IS)IS)

Frequency In Hertz

Figure 5.17. Frequency response function using singlebinomial triangle record.

Average (4) Frequency Response ~unct1on

Shaped Pulse (2x10) Excitation

96

N('\J('\JIS)

· .

· i ' j 1 \

........................1"' + ······················r· ···..············+·······················T ·····························r···················..·t .

r . r . . i·····..··..····..·······j ·······..·····..·..·1 ···················..·······r········..······· r..···..··..··· ..···· 1' .

1 • : • : , . .·······················I···············..··•··..r..···..···..·..·····..···· ..··..~·· ..··· ··· ..······T·····..· ·..···· 1' _ 1' ·..7·········__·..··..··..r

t• • : . I : . •..··..··· ·· ·····1'···········..·····..···1·· j j ·..······..···..····r·················..·····..······r..·..· ·····..··1··..···.._·_·····_·[..·

• i

-10a '---------------------------......Lni

40

20

0t:O"'0

c -20

l1)

'"7J -49~

+'

cC"J -60ffj

L

-80

Frequency in Hertz *Average (4) Frequency Response Function

Shaped Pulse C2x10) Excitatl0n180

135

90

VJt) 4SQ.)s,

mo a-0

c

-45Q)

VJtfj .:

s: -90Q..

-135

-180IJ") N If") ('"I..j

~ l::) l=s;) ~lSI

.. , 1~~

Frequency In Hertz

Figure 5.18. Frequency response function using fourbinomial triangle records.

Average (16) Frequency Response FunctionShaped Pulse (2x10) Excitat;on

97

<'J

" ".................................................! ! ~ ~ .. ·············-f······_····· ~ . ~ : ~ ~

, 1 . , . 1.......................~ _r..· ·..·······..r····.. ·· ..··· ·.. ·f ···..··· ········r..·····..·..· · ··· ..~ _.._.

· r i ! r . . 1··········..··· ·..l-··········_·_····..·j·..·..·..···..···-· ··..·..·I..····_·· ·····..·•········..········..····r·· _.._.._ _J_ _. ..~.: ···_··············-1;: .: : i ::

· r 1 f r t···•·..····..··..·..····!··_..···········..•·..·r..···········..··· · ·..··!····..···..··..··..··..·r..··· ····..·..······l..···· ··..····..····..··..t:..· ·..·.._·t··· ..·..· ·· ~ ..

t t t t-100 '--------------------------....

~

40

20

0CD-0

c -20

G)

'"0 -40:J+ol

C0) -6£1t'd

1:

-80

Frequency in HertzAverage (16) Frequency Response Function

Shaped Pulse (2x18) Excitation180r---~--~---~--~--~---~--~--~

90

(1)

Il) 45~

~

C')

o e"1J

c

-45d.)

CI.I

""...

s: -90a..

-135

-180lJ'j .-4 N In N U1 ('\Jf3) eg cs:> t'S)1St

Frequency in Hertz

Figure 5.19. Frequency response function using sixteenbinomial triangle records.

98

The bipolar triangle signal's performance is displayed

in Figures 5.20, 5.21, and 5.22. These results near the

half-sampling frequency improve with averaging, but the low

frequency regions are not meaningful. In Figure 5.22, the

high frequency mode is identifiable. The spliced results of

the complementary pulse pair are shown in Figure 5.23. In

this method, both the flexible modes are distinguishable.

Finally, the results from noise excitation are presented

in Figures 5.24, 5.25, and 5.26. Here, the low frequency

results are not of the quality achieved with deterministic

excitations. However, the high frequency fidelity improves

dramatically with averaging.

The results of the experiment show that there are some

clear distinctions in frequency response function estimate

fidelity due to the excitation signal. For low frequency

fidelity, a deterministic method using a signal of high

spectral magnitude is desirable. Also, a deterministic tack

may be indicated if averaging is not to be performed, or if

only a few records will be used in an average. On the other

hand, with a moderate number of records to include in an

average, excellent high frequency fidelity is achievable

with stochastic methods. Of course, for broad-spectrum

applications, a combination of deterministic and stochastic

Dlsk-3 Velocity rrequency Response rune.Shaped Pulse (-2x10) Excitation

99

40..----~--.....,..---~---~--~---~--~---~

NN

·· ..··············..··..I..··················..··t-······_·····_···········T"7\·····.._·_··+······················l· _ : .

T ! ~\, , r T !

-1.... ·····..r..··..---··..········..·;······..·· ~·· ..·..·..··········t················..···············!..·····················T·..·················..·j····

..................._..I I 1. 1 r.. f. . 1. .

r . · · . , ... ····..· ·.._···..·r···..····..···· ·..·r ··..~ ·_.._····· ··!..·· ········..·_·;·······..·..·· ·······r······..· !' : .

r r . . . r :····..·.._···..···--r········..·..····..····r....··..·······..····..·.._·..l·····_····..····....···I-··..··_..········..··r·····..··········....··....····r··..··..···..····..··..1..··..··..·····.....

t r f f r , · .···..·..·..·..·......···i..··..···..··....·..····r-·....··_·....·....···..······r·....··..···..·_·..···I···..·..············....r·....·..·..···..····..·····..···!·....·..·....···......··~·....·.....··..···_..r-·t! ! t

20

-100'----.-....----...--------------------~\i)

i

t11"0

c -20

4)

1J -49:J.,:-

C0') -60,'1L:

-80

Frequency in Hertz

D1Sk-3 Velocity Frequency Response Func.Shaped Pulse (-2x10) Excitation

t :

...:.•..:: .~.::': ::::.. • ! . J.., 1

1 I .! ': HUi :i;' :;..·········-······ ~.:········ .._···..·······..Jt.. ··..··..· ··-..· · ·~.·r.·.; ~.; i ·..•..........•. t····.······..·.,.·· -M ••••••• -,.

! ~ ~ L ~ ~ i H~Hi.: ~h ~: ji

! it !II ~ ...····..···_·..··....··t·....·····....·..·..·r·tr....··_........·..·..··.. ··.. ·t·r~1 ..·..··......_....·+··..·..·_·..··....·..··t· ....··.... ·I#·· ~.....

! {f1 !U

180

135

90

(I)

Q,) 4S4)

~

0)

lIJ0-0

c

-45()

iIJl1j

.c -90a.

-135

N-CS)

-180 .......---.......----..........----------....0.----------..~

Frequency in Hertz

Figure 5.20. Frequency response function using singlebipolar triangle record.

Average (4) rrequency Response FunctionShaped Pulse (-2x10) Excitation

100

40.----~--~-----.---~--~----~--"""!"---~

('\oj

..1·.. ···......··..·....···1....

.......................

·······················r······..······..·····~········· ; + : .-." ·······T.· ~ ..

\. :..·..I..··..··_···_··..·..·····..·t·····..· ·~·~·..·· ·..;..··· ..

........................j'" ··..·..·..f· ! ..

· . • r . . .·..·····..····..·..··..·1'..··..·_· ·..···1..·..·_··_··..·······..··;·· · · ·····r..···..·· ····..···..!..··..·..···..·..·· .,. - .

· . r . . . .··········..····_····r·········....·....·····f··......·..···_··......·..····r-..·..·....···....·..·1..···_·..·..··_·..···r·_..··..···....··....····..··..r·······..··....·······"7···....········ ···....·1·

· . . r , r , t·..···..·..·············r·· ·· ···..··..··~· ..···..···..·..·..·..···..·····[..·..··..········· ··1····· ·..·········.. ·1 1' ) ~ ..

t 1

" .

20

-100"'-----------------------------'U1

i

ttl1'J

c -20

G)

'"'C --49:J.+oJ

c0') -6910~

-80

rrequency in Hertz

Average (4) Frequency Response FunctionShaped Pulse (-2x10) Excitation

180

135

90

UJI) 45vs,

enQ)

0"'0

c

-45e(/)

~

..c -90a.

-135

-180Ln~~

N

Frequency in Hertz

Figure 5.21. Frequency response function using fourbipolar triangle records.

Average (IS) Frequency Response FunctionShaped Pulse (-2x10) Excitation

101

('\J('\J

. .

··················_·-1······················1·········· -r J..... + . 1

: 1 · . . ' .························r·················..··..1·_···..····..····..·····..····1..····..····..··..···..·1····..········· ·1·· ····· ·· ···..···..·····1..··..·· ··· ·_·1·..

I f I . . . .·..···..·..··..·--····r·....·········--···l·········...··....·..·..······..1'·..·__··__·······1·..········_....··....·[..·....·..···..····....··....·..r···..···...._·..···..·1..········......! + + + + . .1 1 1 1 r ; . .··..··..··--·..·-r···..···· ·..···..r·..·..·..··..··· ·.._·..··~ _·..···..r· ! 1'· ····· ..·..·..· 1·..···..····"'-'''1''

+ r t-100 "'-"---------------------------~

i

40

20

0CO-0

c -20

l)

~ -49:J~

Cm -b1)10~

-S0

rrequency in HertzAverage (16) Frequency Response Function

Shaped Pulse (-2x10) Excitation180

135

30

ent) 45G)s,

O"J11)

0-0

c

-45Q)(J)

~

s: -90a.

-135

('\J

:: 10:

('\J-100'--------------------------------.........

lJj

iFrequency in Hertz

Figure 5.22. Frequency response function using sixteenbipolar triangle records.

40

20

0ID1'J

c -20

wf.J -40::J.+oJ

cen -60~~

"-

-80

Average (16) trequency Response ~unc.

Spliced Shaped Inputs (2x10) & (-2x10)

: :·····_····-r········ ~·..··.._..·r"·"···..·....····_·T..·....·_··..·....T·"·....·........·_·r

! i ~ j ~ j ! j

·..···· · ···· ·..· ···r..·· ·· ··..· ·!···..· " +......................... . !..................... • 1"..

+ r . ; •..····,,·····..····1·· ·..·· 1'·..·_··..····..·····1-..··"·..·_.._··t..·_··· _· ·..•· ·..··· _..·..·,,..t·..·........ .. .-t-......_ .

..._ ; 1.. _._ ~ + ! \ L..- +-

102

-100~-------------------------_-..~~

rrequency in Hertz

Average (16) Frequency Response Func.Spl iced Shaped Inputs (2x10) & (-2x10)

180

135

90

I/J1]) 45~

~

0')~

0"'0

c

-45Q)r/)

rU..c -90CL

-135

.. ;

.:: • :::::jt,.;

. ~ uL ~ i i:: ~m~!H L.......................j ~ " :.. + " t· · · · ·..· ·ri~lI~J,l..li!

: , ' : T. ~ ,:L! :~iggjIL rdf~

! :

.......................l _ " l " L I L L ....ll!..I:.\ii + ; :,! · i If IiIi'll.

..·..· ····..··::L:···.:·:·.·· .:s:"".+" "".:r: " L ~ ~,i.··..··..·..· ·..i l+..i-H·t-III: :;~ ;:: :;~: =c:s::=, ., : = :::: ::::::=: .: ·1 :;:: ;'i:=:~:

: : ~ : : :' :: ; ~~ ;~ ;~ i!·! g~ ;................................................. ") ~ ... .. .'j lHl ~!~,

N-180 '-------------------------------

lJ)IS)(!j

Fr~quency in Hertz

Figure 5.23. Frequency response function using sixteenspliced triangle records.

Djsk-3 Velocity Frequency Response rune.Unity White Noise Excitation

103

40

20

'3~'"0

c -20

II"'0 -40~

+-'

c0) -60~

L

-80

..·····················..··..··················T·····..···················.. ·~··.._··_············r··..·· ~ _ _ ~ _~ .

........................: ·..· ·· ·..t ····..K ·..· ·· · ·..· i · r ..

~::: .: : ; I . . : .·····..··· ..· ·.. ····~ ..· ···..·..·····1.. ·····..········· , ~ , · ···..··..·..·..· r..· · ··_..·•·..·..· · ·..· .

('\Ju')I:'\J-100 \.----------------------------....

lJ')lSI~

rrequency in Hertz

DiSk-3 Velocity Frequency Response Func.Unity White Noise Excitation

180,.---~--~---~-~--~---~------ .....

90

45

; .: j! .: :::::: ;

. . . ~ !:;. i di~ ~1h;135 ··..·······; ········ ···· t..·..·..·· ·..··· ..······;··.._ ··..·····..···t· ..···..·· ··· ..·····~ ..·········· ..· ·..·i..···i ·; ·f·t·.. ·~··j ..t~ .. ~ ~ ilffi

.)

-45 ··················· ..··~····· ....·······..·..··T·......·..·..········..·......r·····..···_..···..···!!·:··· ··..··· .i.,..··..·....··..··· ·..: ···..·······..·t~ ~f..l=!

-913 -,,·_'-,·,-·· ·--·f ·""·····..··;·:l· · · ·..··..·..·..·..·· ·..·· · ········.. ···· _·HI·_·H~: ~~~ I: ~fil

; ~ : ~i :: ;;g iii-135 ,·.. ··· ..·· 1 _..···..··..·..··t· · · ·· ··..···i.. ······.._..·..·_·..·t..·..· ·········..·t··· .. ·· ·.. ·· ·-!..·· · ···..·..··ti~··..r··..ff·ill:.

~ f r ! t t t~ ~ ff j~,r+ +? ; ~ t; ~j

('\J-180~----------------------------'

If":\c:gIS)

Frequency in Hertz

Figure 5.24. Frequency response function using singleunity white noise record.

Average (4) Frequency Response FunctionUnity White Noise Excitation

104

('"\J('\J~

. . . . , . , .···..······..·..·..·....1..······ ········..··..··t············..··..··......··....1......·.._··.. ··..-··r..·.._·········..·_..l···..·..··..······....····..····· t····..·················r·················.._··\·..·

. i _.:.: ;

: ! ~

······················r····················r···············_··········r···..·····..········r··············_·····r············~~···············T··· .. ······ C'

............._.........•..............._......•............._ - _._ - _ , _.................................. ..: : : ! ! :

I t f f ;. . . L j : : i

••••••••••••••••••_ _ _ ••••••••••• t· .. ••••.. -···,,···_·_·.····_········_·.. ·-··.············ ••••••••••••••••••••.,.•••••.••••••••••••••••••••••••••- ••••.••••.•••• t· •

j ~ . ~ 1 j j 1

. + r ~ i

-100 L....-__......._ ....... ......... ........ ........

If')

~

40

29

0CO-0

c -20

Q)

~ -40~

~

cr.n -60~

z

-80

Frequency ;n HertzAverage (4) ~r.quency Response Function

Unity White Noise Excitation

('\J-rs-

-45

-180 '--__...i--__....... --'

LI"J

~

-135

180

135

90

II)

o 4511)

\..0')l1) e""0

c

'1)(,')

1'0

5: -90 ·:::.:..._".::::::.:.:.:::..:::-::.c........

Frequency in Hertz

Figure 5.25. Frequency response function using four unitywhite noise records.

Average (IS) rrequency Response ~unction

Unity White Noise Excitation105

-~

I. i

···•···········..······j····..····_·..········i···············..··············r···.._·· ..··········r·····..·············T- ·······················..···')""··..·····..········..·r..··········_·······;····~t:\ t

:::::::::.:.::::::.I:::::::·::.:.:::~I::::::::::.:.::...:::::::I.:::::::::::..::·:I::.:~:::::.:]:::::.:::::::.:~ ..:':::::I::: ~ ~ ~ : t ~ :

, ~ • • . r 1........................! _ ··········r..···· '" ·······r· ~ -1' __ ···..····..1 ·· ··_··1 ·············

r ! t ;-lee L- ........

Lf':l

i

40

20

0CO-0

c -20

tD'7J -40:5+"

Cen -60~

L

-80

Frequency in Hertz

Average (16) Frequency Response FunctionUnity ~hite Noise Excitation

t'\J-(S)

•.! ~ ~~l:. ~

......:.::.....

f :

..................·I..· ···..·..+ .•:.:.:.:.:.:.:.·:·.J.t::::::::::::·:::::::I.~:::::·::::·::'::::':::::::':::::!.::::::::::':"'::"":~.:.i·':::::::-:··::::.-::::'-:-a-:::-::':. .-

• \~e; iae

........................ j t..·········· ·····..·· ·j······· ········..···! j ·t..··· ····+· ·..· t·j~

r ! ~ l~'f: ~jr~:::~: :-180 L..-. .......,

If')

~

180

135

90

~

to 45l1)

~

0')G)

0fJ

c

-454)CI)

tf:J.c -90a.

-135

Frequency in Hertz

Figure 5.26. Frequency response function using sixteenunity white noise records.

106

approaches may be employed. In any case, the results of

this chapter should provide some insight into the successful

selection of suitable excitation signals.

107

CHAPTER 6.

CONCLUSION

6.1 Observations

In this thesis, methods of improving the fidelity of

empirical sampled-data frequency response function estimates

have been investigated. The successful identification of a

high-fidelity system response function is the cornerstone of

the LSS design methodology, Design-To-Performance, which was

outlined in Chapter One. The analytical methods reviewed in

Chapter Two permit the identification of the frequency

response function matrix necessary for the One-Con

troller-At-A-Time design procedure discussed in the

introduction.

In order to achieve high-fidelity estimates, several

methods of enhancing the properties of excitation signals

were considered in Chapter Three. These methods include

increasing the signal's magnitude and/or duration, shaping

the signal, and using complementary signal pairs. Other

means of improving results include averaging several

responses and combining deterministic and stochastic methods

advantageously.

The system identification process and signals of

interest were explored via digital simulation in Chapter

108

Four. Finally, the practical application of the methods

discussed was demonstrated with the hybrid system experiment

and results presented in Chapter Five.

6.2 Applications

The methods of enhancing excitation signals used in

system identification discussed in this thesis were

developed from the viewpoint of LSS applications. However,

the same principles apply to any application that requires

the empirical identification of a sampled-data frequency

response function. Likely applications include very high

order systems or systems whose dynamics are not well

understood. For instance, many biomedical systems may lend

themselves to a similar analysis.

As a direct application of the results detailed in this

thesis, the identified response functions could be used to

design controllers for the hybrid system. In this manner,

the influence of the identification methods on the

closed-loop system performance could be investigated. since

this is the purpose for the identification process, it is

the true test of adequate fidelity of the identified

response function estimate. As a more ambitious project,

the analog computer simulation could be replaced with a

109

physical system, e.g. a pointing system. Such a demonstra

tion should establish the usefulness of the methods

described herein.

110

REFERENCES

1. Jerrel R. Mitchell, Sherman M. Seltzer, and H. Eugene

Worley, "Design-to-Performance", Proceedings of the 1985

IFAC Conference on Automatic Controls in Space,

Toulouse, France, June 1985.

2. Jerrel R. Mitchell and J.C. Lucas, "l-CAT: A MIMO Design

Methodology", Workshop on Structural Dynamics and

Control Interaction of Flexible structures, Sponsored by

OAST and MSFC, NASA Marshal Space Flight Center,

Alabama, April 1986.

3. Julius S. Bendat and Allan G. Piersol, Engineering

Applications of Correlation and Spectral Analysis, John

Wiley & Sons,Inc., New York, 1980.

4. Julius S. Bendat and Allan G. Piersol, Random Data

Analysis and Measurement Procedures, Second Edition,

John Wiley & Sons, Inc., New York, 1986.

5. Charles P. Plant, Schemes for Improving the Fidelity of

Empirical Sample-data Frequency-domain Models of Large

Space Structures, Masters Thesis, Mississippi State

University, Mississippi, May 1987.

6. Jerrel R. Mitchell, Advising Session, Ohio University,

Ohio, April 1988.

111

7. William P. Maggard, Adaptive Control of Flexible Systems

Using Self-tuning Digital Notch Filters, Masters Thesis,

Ohio University, Ohio, September 1987.

8. Electronics Associates, Inc., "Model TR-20 Desk Top

Analog Computer Operation Manual", Electronics Associ

ates, Inc., Long Branch, New Jersey.

9. Hewlett-Packard Co., "HP-3852A Data Acquisition and

Control unit Plug-in Accessories Configuration and

Programming Manual", Hewlett-Packard Co., Everett,

Washington, 1983.

10. Raymond G. Jacquot, Modern Digital Control Systems,

Marcel Dekker, Inc., New York, 1981.

11. Daniel Graupe, Identification of Systems, Krieger

PUblishing Co., Inc., New York, 1972.

12. Charles L. Phillips and H. Troy Nagle, Jr., Digital

Control System Analysis and Design, Prentice-Hall, Inc.,

New York, 1984.

13. L. Ljung, "Identification: Basic Problem", System .i

Control Encyclopedia: Theory, Technology, Applications,

Editor-in-Chief M.C. Singh, Pergamon Press, U.K., 1987.

112

14. K. Glover, "Identification: Frequency Domain Methods",

System ~ Control Encyclopedia: Theory, Technology,

Applications, Editor-in-Chief M.C. Singh, Pergamon

Press, U.K., 1987.

15. G.S. Hope and O.P. Malik, "Identification: Pseudo-random

Signal Method", System ~ Control Encyclopedia: Theory,

Technology, Applications, Editor-in-Chief M.C. Singh,

Pergamon Press, U.K., 1987.

16. J.C. Lucas, I-CAT: A MIMO Design Methodology for the

Control of Large Space Structures, Masters Thesis,

Mississippi State University, Mississippi, December

1986.

17. Daniel DeBra, Jerrel R. Mitchell, Sherman M. Seltzer,

and H. Eugene Worley, "Practical Dynamics & Control of

Large Space structures", AIAA Professional study Series

Book Three, Huntsville, Alabama, August 1986.

APPENDIX A

Digital Simulation Source Code

113

114

This appendix contains the source code listings for the

digital simulation of Chapter Four. The programs are exe

cutable command files for the MatrixX control system analy

sis and design software package.

The three disk system described in Chapter Four is in

corporated into MatrixX variables in the ANCMPINI program,

which initializes the digital simulation of the analog com

puter. The programs for the simulation provide three major

functions. The CALCFRSP program permits the generation of

continuous and sampled-data frequency response functions for

the system. The RTLOCUS program provides the capability to

examine the continuous and discrete root loci of the system.

Finally, the SYSTEMID program accesses several subprograms

to perform a system identification simulation.

There are five subprograms called by the SYSTEMID pro

gram. The first is GETPARAM, which gets the system identi

fication parameters for the user. The DISCRTCL module forms

the discrete closed-loop system. Then, the GENINPUT module

generates the input signal defined by the user. Finally,

the time simulation and frequency analysis are performed by

the DTRMNRSP program for deterministic analysis or by the

STOCHRSP program for stochastic analysis.

115NOTE: Comment lines in MatrixX cannot be placed in com-

pound statements. In the listings that follow, comments

precede the compound statements which they describe, e.g.

WHILE loops.

II PROGRAM116

--------~-~~~--~~~--~~-~-~~-~~~-~~~-~-~~~--~~~~--~~------~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-~~~~-~~~~~~~~~~~~~~~~~~~~~~---

II NAME: ANCMPINI

II

II AUTHOR: T T HAJVIIVIOND

II

II DATE: 10 APR 1988

II

II PURPOSE:

II This file is a MatrixX executable command file. Its purpose is

II to initialize the MatrixX main program with a state-space

II description of the system analyzed in the thesis. A modal

II realization is used paralleling that implemented on the analog

II computer. This permits sirrulation of the analog ccmputer's

II responses to various signals. In order to effectively magnitude

II scale the analog computer simulation~ the dynamic ranges of the

II states of the realization must be determined. Then the variables

I I generated by t he ana log canputer can be norma1; zed to prevent

II "their exceeding the dynamic range of the computer's amplifiers.

II Additionally, the MatrixX variables initialized by this program

I I are SAVEd in the file ANCMPSIM. VAR for use by other program

1,1 modu1es .

II The rigid body mode of the system's response is modeled by

II subsystem A. The 0.1 Hz flexible body mode ;s modeled by

I I subsystem 8, and the 1.5 Hz flexible body mode is modeled

I I by subsystem C.

II

II VARIABLES ==--==--===========================--===============--========

II AA, AS, AC

II

II BA, 88, EC

II

II CA, C8, CC

II

II DA, 08, DC

II

II NSA, NSB,

II NSC

II SA, 58, SC

II

117coefficient matrices of state vectors in the

state equations of the subsystems

coefficient matrices of input vectors in the

state equations of the subsystems

coefficient matrices of state vectors in the

OLJtput equatrions of the subsystems

coefficient matrices of input vectors in "tt1e

output equations of the subsystems

integers indicating the subsystems T orders

matrices composed of coefficient matrices "to form

the Matrix)( representatioras of the subsystems

II

II BEGIN ===============================================================

II CLEAR SCREEN AND DISPLAY PROGRAM FUNCTION -------------------------

ERASE

DISPLAY(TINITIALIZING ANALOG COMPUTER SIMULATION ... T)

SHORT E

II DEFINE ANALOG COMPUTER SIMULATION SUBSYSTEMS -----------------------

AA= [0 1 ; 0 OJ ;

SA = [0; 1/3];

CA = [1 0; 0 1] ;

DA = [0 ; 0] ;

SA = [AA SA; CA DAJ;

NSA = 2;

A8 = [0 1; -0.39478 OJ;

88 = [0; -0.33482];

118C8 = [1 0; 0 1] ;

08 = [0; 0] ;

S8 = [A8 88; CB D8J;

NS8 = 2;

AC = [0 1 ; -88.82686 OJ ;

Be= [0; 1.48807E-3];

CC = [1 0; 0 1] ;

DC = [0; OJ ;

SC = [AC Be; CC OC];

NSC = 2;

II CLEAR SUBSYSTEM COEFFICIENTS AND SAVE SUBSYSTEM DESCRIPTIONS ------

CLEAR AA A8 AC SA 88 Be CA C8 CC DA DB DC

SAVE(tANCMPSIM.VAR')

II CLEAR VARIABLE STACK ----------------------------------------------

CLEAR

II END OF ANCMPINI PROGRAM =======================--====================RETURN

II

II

PROGRAM119

-----~~-~~~~~~~-~~-~~--~~~-~~~----------~~~~~~~~~--~~~---~~~~~~~~~~~~~~~~~-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I I NAME: CALCFRSP

II

I I AUTHOR: T T HAMMOND

II


II

II PURPOSE:

II This file is a MatrixX executable command file. Its purpose is to

II generate the calculated frequency response for the continuous-time

II analog computer or the discretized analog computer system. The

II file ANCMPSIM.VAR must be placed in the current directory prior to

I I "the executian of this program.

II The response functions calculated are for Disk-3 velocity.

II

II VARIABLES

II CHOICE

II DELTAT

II DSNFRESP

II DSFRESP

II K

II KFB

II MFRESP

II NAB, NABC

// NFRESP

II NSA~ NSB,

-----------~~~--~-~~~----------~-~~-~-~~~~--~-~-~~-~----~~~~~~~~~~~~~~~~~~~~~~~~~~---~~~~~~~~~~~~~~~~~-~~~~-~~-~~~

index of menu selection

sampling period in seconds

order of discrete system

MatrixX description of discrete system

compensator gain

feedback matrix

connect function input matrix

intermediate subsystems f orders

connect function output matrix

120II NSC

II NSFRESP

II SA, S8, SC

II

II SAB, SABC

II SFRESP

II

integers indicating the subsystems' orders

order of analog system

matrices composed of coefficient matrices to form

the MatrixX representations of the subsystems

intermediate subsystem matrices

MatrixX description of analog system

II CONSTANTS ==========================================================

II

II SELECTION MENUS:

TI MEMENU = [f * * MENU * * f

, CONTINUOUS r

DISCRETE f];

II

IVOREMEJ\IU = [f * * MENU * * r

MORE

QUIT -i.

II

I I QUERY RESPONSES:

NO = fN';

y = 'Y":

YES = fyt;

II

II BEGIN ===============--==============================================


ERASE

121DISPLAYC'FREQUENCY RESPONSE FUNCTION GENERATION ... T)

II RETRIEVE SYSTEM SIIVlILATION AND FORM DISK-3 VELOCITY OUTPUT -------

LOAD('ANCMPSIM.VAR')

[SAB, NABJ = APPEND(SA~ NSA, S8, NSB);

[SABe, NA8CJ = APPEND(SA8, NAB, SC, NSC);

MFRESP = [1 1 1]';

NFRESP = [0 1 0 1 0 1J;

KFB = [0 0 0 0 0 0; 0 0 000 0; 000 0 0 OJ;

[SFRESP, NFRESP] = CONNECT(SA8C~ NABe, KFB, MFRESP, NFRESP);

II WHILE LOOP COMMENTS ==========~=====================================

II SET UP LOOP FOR MULTIPLE RESPONSES --------------------------------

II QUERY USER FOR CONTINUOUS OR DISCRETE RESPONSE --------------------

II SPACE CONTINUOUS FREQUENCY DATA LOGARITHMICLY ---------------------

II DISPLAY CONTINUOUS FREQUENCY RESPONSE FUNCTION --------------------

II INQUIRE IF PRINTING IS DESIRED ------------------------------------

II QUERY USER FOR SAMPLING PERIOD ------------------------------------

II SPACE DISCRETE FREQUENCY DATA LOGARITHMICLY ----------------------~

II DISPLAY SAMPLED-DATA FREQUENCY RESPONSE FUNCTION ------------------

II INQUIRE IF PRINTING IS DESIRED ------------------------------------

II QUERY USER FOR MORE RESPONSES -------------------------------------

CHOICE = 0;

WHILE CHOICE <> 2; ...

CHOICE = MENU(TIMEMENU,1); ...

IF CHOICE = 1; ...

GEOMRATIO = 10**(1/250); ...

FRANGE = [O.OS*PI O*ONES(1,500)J' ; ...

FOR 1=1:500, FRANGE(I+1)=GEOMRATIO*FRANGE(I); END, ...

122H = FREQCSFRESP, NFRESP, FRANGE); ...

FHZ = FRANGE/(2*PI); ...

PLOT(FHZ, 20*LOGCA8S(H))/LOG(10), ...

'TITL/DISK 3 VELOCITY CONTINUOUS FREQUENCY RESPONSE FUNCTION/ ...

YLAB/dB/ ...

XLAB/FREQjENCY IN HZ/...

LOGX t ) ; •••

INQUIRE CHOICE 'DO YOU WANT TO PRINT THIS FUNCTION

IF CHOICE = 'Y' , ...

PLOT('PRINTER HOLD'); ...

PLOT(FHZ, 20*LOG(A8SCH))/LOG(10), ...

, .. , ...

'TITL/DISK 3 VELOCITY CONTINUOUS FREQUENCY RESPONSE FUNCTION/ ...

YLA8/dB/ ...

XLAB/FREQJENCY IN HZ/...

ERASE; ...

PLOTC'OISPLAY HOLD'); ...

END; •..

ERASE; ...

ELSE INQUIRE DELTAT TENTER SAMPLING PERIOD IN SECONDS'; ...

[DSFRESP, DNFRESPJ = DISCRETIZE(SFRESP, NFRESP, DELTAT); ...

GEOMRATIO = 10**(1/250); ...

FRANGE = [O.OS*PI*OELTAT O*ONES(1,SOO)J'; ...

FOR 1=1:500, FRANGE(I+1)=GEOMRATIO*FRANGE(I); END~ ...

H = FREQ(DSFRESP, DNFRESP, FRANGE, 'DIS'); ...

PHZ = FRANGE/(2*PI*OELTAT); ...

PLOT(FHZ, 20*LOG(A8S(H))/LOG(10), ...

123'TITL/DISK 3 VELOCITY SAMPLED-DATA FREQUENCY RESPONSE FUNCTION/ ...

YLAB/dB/ ...

XLAB/FREQUENCY IN HZ/...

LCX3X' ) ; •••

INQUIRE CHOICE '00 YOU WANT TO PRINT THIS FUNCTION

IF CHOICE = 'y t~ •••

PLOT(tPRINTER HOLD'); ...

PLOT(FHZ~ 20*LOG(A8S(H))/LOG(10)~...

, .. ,. ....

'TITL/DISK 3 VELOCITY SAMPLED-DATA FREQUENCY RESPONSE FUNCTION/ ...

YLAB/dB/ •..


ERASE; •..

PLOT('DISPLAY HOlD t) ; •••

END; ••.

ERASE; •..

END; •.•

CHOICE = MENU(MOREMENU,1); ...

END

II CLEAR VARIABLE STACK -----------------------------------------------

CLEAR

/ / END OF CALCFRSP PROGRAM ================--==========--===

RETURN

124II PROORAM

/ / NAME: RTLOCUS

II

II AUTHOR: T T HAMVIOND

II


1,1

I I PURPOSE:


II generate the r-oot; locus for ,the con"tinuous-'time analog canputer

II simulation or the discretized analog computer system. The file

II ANCMPSIM.VAR must be placed in the current directory prior to

I I the execution of this program.

II

II VARIABLES

II CHOICE

II DELTAT

II DSNRTLOC

II DSRTLOC

II EV

II GAIN

II K

II KFB

II MRTLOC

1/ NAB, NAOC

II NRTLOC

1/ NSA p NSB p

~~-~-~~-~~~~-~-~---~~~~~~~~~~~-~~~~~~~-~--~-~------~~~~~~~~~~~~~-~~--~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-~~~~~-



order of discrete system

MatrixX description of discrete system

matrix of discrete poles corresponding 'to GAIN vector

row vector of gains for desired discrete roots

compensator gain

feedback matrix

connect function input matrix

intermediate subsystems' orders

connect function output matrix

125II NSC

II NSRTLOC

II SA, S8, SC

II

II SA8, SABC

II SRTLOC

II


order of ana log system

rllc."9trices composed of coefficient matrices to form

the Matrix)( representatior1s of the subsystems

inotermediate subsystem matrices

MatrixX description of analog system

II CONSTANTS ==--=======================================================

II

II SELECTION MENUS:

TIMEMENU = ['* * MENU * *'

, CO\ITI NUOUS t

DISCRETE '];

II

IVOREMENU = [t * * MENU * * t

MORE

QUIT T];

II

II QJERY RESPOI\ISES:

N = TN';

Y = Ty';

YES = Ty';

II

II BEGIN

/1 CLEAR SCREEN AND DISPLAY PROGRAM FUNCTION -------------------------

ERASE

126DISPLAY('ROOT LOCUS GENERATION ... ')

II RETRIEVE SIMULATED SYSTEM AND FORM DISK-1 VELOCITY OUTPUT ----------

LOAD('ANCMPSIM.VAR')

[SAB~ NAB] = APPEND(SA t NSA, S8, NS8);

[SABe, NABeJ = APPEND(SAB, NAB, SC, NSC);

MRTLOC = [1 1J';

NRTLOC = [0 1 0 -1.9909 0 -3.4905E-3];

KF8 = [0 0 0 0 0 0; 0 0 000 0; 0 0 0 0 0 OJ;

[SRTLOC, NRTLOCJ = CONNECT(SABC, NABC, KF8, MRTLOC, NRTLOC);

II WHILE LOOP COMMENTS ================================================

II SET UP LOOP FOR MULTIPLE LOCI --------------------------------------

II QUERY USER FOR CONTINUOUS OR DISCRETE ROOT LOCUS -------------------

II DISPLAY CONTINUOUS ROOT LOCUS --------------------------------------

II QUERY USER FOR PRINTING CONTINUOUS ROOT LOCUS ----------------------

II INQUIRE FOR SAMPLING PERIOD AND GAIN VALUES ------------------------

II DISPLAY DISCRETE ROOT LOCUS ----------------------------------------

II QUERY USER FOR PRINTING DISCRETE ROOT LOCUS ------------------------

II QUERY USER FOR MORE LOCI -------------------------------------------

CHOICE = 0;

WHILE CHOICE <> 2; ...

CHOICE = MENU(TIMEMENU,1); ...

IF CHOICE = 1; ...

K = RLOCUS(SRTLOC, NRTLOC); ...

INQUIRE CHOICE 'DO YOU WANT TO PRINT THIS LOCUS



EV = RLOCUS(SRTLOC, NRTLOC, K); ...

, .. , ...

127ERASE; ...

PLOT(TDISPLAY HOLD'); ...

END; ...

ERASE; ...

ELSE INQUIRE DELTAT 'ENTER SAMPLING PERIOD IN SECONDS'; ...

[DSRTLOC, DNRTLOC] = DISCRETIZE(SRTLOC, NRTLOC, DELTAT); ...

INQUIRE GAIN 'ENTER VECTOR FOR GAINS OF ROOTS TO BE SHOWN'; ...

ev = DRLOCUS( DSRTLOC, DNRTLOC, GAIN); ...

GAIN, ...

INQUIRE CHOICE 'DO YOU WANT TO PRINT THIS LOCUS



EV = DRLOCUS( DSRTLOC, DNRTLOC, GAIN); ...

ERASE; ...

PLOT('DISPLAY HOLD'); ...

END; ...

ERASE; ...

END; ..•

CHOICE = MENU(MOREMENU,1); ...

END

, .. , ...

II CLEAR VARIABLE STACK -----------------------------------------------

CLEAR

II END OF RTLOCUS PROGRAM =============================================RETURI\I

128II PRCX3RAM --

II

II NAME: SYSTEMI0

II

II AUTHOR: T T H.AJVI'JtOI\ID

II

II DATE: 9 APR 1988

II

II PURPOSE:

II This file is a MatrixX executable command file. Its purpose is

I I to drive the system identification process. This is accompl ished

II by calling smaller MatrixX executable command files to perform

I I required tasks.

II Preparation to use this file must include creating the ANCMPSIM.VAR

II file in the DOS default directory. That file contains the analog

I I computer simulation MatrixX description. It is created by

II executing the ANCMPINI.EXC file.

II Files read during the execution of this file must be in the OOS

II default directory. Files written during execution will be placed

II in this same directory.

II

II VARIABLES

II ANSWER

II TYPE

II

II

~~~~~~-~~~~--~~~~~~~~~~~~~---~~~-~~-~~----~-~~~~----~~~-~~~~~~~~~~~~~~~---~~~~~-~~~~--~~~~~~~~~~~~~~~~~~~~~~~-~~~

response to question

= for deterministic input signal

= 2 for stochastic input signal

I I CO\ISTANTS .................-..-...-

129N = TN t ;

NO = TN t ;

II

y = 'yt;

YES = "( ' ;

II

II BEGIN

II CLEAR SCREEN AND DISPLAY PROGRAM INSTRUCTIONS ---------------------

ERASE

II

* * * * * * * * * *')DISP('* * * * * * * * * SYSTEM IDENTIFICATION

DISP('*

DISP('* This program simulates the sampled-data frequency

DISP( '* response function method of system identification

DISP('* applied to a highly flexible structure.

DISP('*

*' )*' )

*' )

*' )

*' )

DISP('* A variety of input signals may be specified by menu *t)

DISP('* responses. Also -the sampling period and length of -time *')

DISP(f* for each sampled record is selectable. The maximum *')

DISP(T* number of samples must be restricted to 256.

DISP('*

DISP('* The program pauses at several statements until the user *')

is ready to continue. To cant;nue~ enter <RETURN> at

DISP('* the PAUSE> prompt.

DISP('*

*' )

*' )

*' )

DISP(T* Time responses and frequency responses of pertinent *' )

DISP('* data are displayed graphically. These are plotted on *T)

130DISP('* the default plotting device (video display at power-on). *')

*' )

DISP('* * * * * * * * * T T HAMMOND 21 APR 88 * * * * * * * * *')

II

II WAIT FOR USER RESPONSE ---------------------------------------------

OISP(' ')

PAUSE

II

II WHILE LOOP COMMENTS =============================================--==

II SET UP LOOP FOR MULTIPLE IDENTIFICATIONS -------------~-------------

I I GET SYSTEM AND INPUT SIGI\IAL PARAMETERS FROM USER -------------------

II FORM DISCRETE CLOSED-LOOP SYSTEM -----------------------------------

II GENERATE INPUT SIGNAL SEQUENCE -------------------------------------

II DETERMINISTIC INPUT SIGNAL SYSTEM IDENTIFICATION -------------------

II STOCHASTIC INPUT SIGNAL SYSTEM IDENTIFICATION ----------------------

II QUERY USER FOR ANOTHER IDENTIFICATION ------------------------------

ANSWER = 'Y';

WHILE ANSWER = 'Y' , ...

EXECUTE('GETPARAM.EXC'), ...

EXECUTE('DISCRTCL.EXC')~...I

EXECUTE('GENINPUT.EXC'), ...

IF TYPE = 1, ...

EXECUTE('DTRMNRSP.EXC'), ...

ELSE~ ...

EXECUTE('STOCHRSP.EXC'), ...

EJ\lD t •••

INQJIRE Ar\ISWER 'DO YOU WANT TO DO ANOTHER RESPO\ISE . . ,. , ...

131END

II END OF SYSTEMID PROGRAM =============================================RETURN

II

II

PROGRAM132

----------~-~~~~~~~~-~~~~-~~~~------~~-~~~-~~~~~~-------~--~~~~~~~~~~--~~~~~~~~~~~~~~~~~~~--~~~~~~~~~~~~~~~~~~~~~~~~~~~

I I NAME: GETPARAM

II

I I AUTHOR: T T HAJV1IVtO\I 0

II

II DATE: 6 APR 1988

II

II PURPOSE:


II establish values for the parameters necessary to run the thesis

II simulation. Default parameters are set and the user is prompted

I I to change any parameters that re/she desi res .

II

II VARIABLES

II CHOICE

II DELTAT

II NSAMPLES

II PULSAMPLTD

II PULSDURATN

II RECORWR

II SHAPE

II

II

II TYPE

II

II

~-~-~~-~~-~~~~~~~-~~~~---~~-~~--~-~~~~------~~~~--~~-----~~~~~~~~~~~~~~~~~~~~----~~~~~~~~~~~~~~~~~~~~--~~~-~~-~~~~

index of menu se1set;on


number of samples rounded to next higher power of 2

input pulse amplitude (normalized -10 to 10)

length of inpu"t pu l se in sample periods

1ength of time s;gna1s observed in seconds

= for rectangular input pulse

= 2 for binomial triangle shaped input pulse

= 3 for bipolar triangle shaped input pulse


= 2 for stochastic input signal

133II CONSTANTS ==========================================================BLANK15 = '

II

II SELECTION MENUS:

CHANGEMENU = [' * MENU *

, ACCEPT

'CHANGE '];

II

SHAPEMENU == [t * SHAPE MENU *

RECTANGLE

BINOMIAL TRIANGLE'

BIPOLAR TRIANGLE f];

II

TYPEMENU == I.'* TYPE MENU *

DETERMINISTIC

f STOCHASTIC '];

II

II BEGIN ===============================================================

II

SHORTI

II

II SET DEFAULT PARAMETERS ---------------------------------------------

RECORDUR == 40;

DELTAT = 0.2;

NSAMPLES ::: 2**ROUND(0.5 + LOG(RECORDUR/DELTAT)/LOG(2));

TYPE::: 1;

PULSOORATN = 1;

134SHAPE = 1;

PULSAMPLTD = 1;

II

ERASE

II QUERY USER FOR RECORD DURATION ------------------------------------

OISPLAY( 'CURRENT TIME RECORD DURATIO\I IS') t RECOROOR

PAUSE

CHOI CE = MENU (CHANGEMENU r 1) ;

IF CHOICE = 2, ...

INQUIRE RECORDUR 'ENTER TIME RECORD DURATION IN SECONDS' , ...

RECOROORt • • •

NSAMPLES= 2**ROUND(O.5 + LOG(RECORDUR/DELTAT)/LOG(2)); ...

PAUSE, ...

END

II

ERASE

II QUERY USER FOR SAMPLING PERIOD ------------------------------------

DISPLAY('CURRENT SAMPLING PERIOD IS'), DELTAT

PAUSE

CHOI CE = MENU (CHANGEMENU , 1) ;

IF CHOICE = 2, ...

INQUIRE DELTAT 'ENTER SAMPLING PERIOD IN SECONDS' , ...

DELTAT, ...

NSAMPLES= 2**ROUND(O.5 + LOG(RECORDUR/DELTAT)/LOG(2)); ...

PAUSE, ...

END

II

135ERASE

II QUERY USER FOR DETERMINISTIC OR STOCHASTIC EXCITATION -------------

DISPLAY('CURRENT INPUT SIGNAL TYPE IS')

DISPLAY(TYPEMENU(TYPE + 1,:))

PAUSE

CHO I CE = MENU (CHANGEMENU r 1) ;

IF CHOICE = 2, ...

TYPE = MENU(TYPEMENU,1), ...

DISPLAY(TYPEMENU(TYPE + 1,:)), ...

PAUSE, ...

END

II

II DETERMINISTIC SIGNAL SECTION COMMENTS ==============================

II QUERY USER FOR PULSE DURATION -------------------------------------

II QUERY USER FOR PULSE SHAPE ----------------------------------------

II QUERY USER FOR PEAK PULSE AMPLITUDE -------------------------------

IF TYPE = 1, ...

ERASE, .•.

DISPLAY('CURRENT PULSE DURATION IS'), PULSDURATN, ...

PAUSE, •••

CHOICE = MENU(CHANGEMENU,1); ...

IF CHOICE = 2, ...

INQJIRE PULS{)JRATN 'EJ\ITER PULSE DURATION IN SAMPLE PERIODS T , •••

PULS()JRATN = ROJND(PULSDURATN), ...

PAUSE, ...

END, ...

ERASE, •••

136DISPLAY('CURRENT PULSE SHAPE IS'), .

DISPLAY(SHAPEMENU(SHAPE + 1,:)), .

PAUSE, •••

CHOICE = MENU(CHANGEMENU,1); ...

IF CHOICE = 2, ...

SHAPE = MENU(SHAPEMENU,1), ...

DISPLAY(SHAPEMENU(SHAPE + 1,:)), ...

PAUSE, ••.

END, .

ERASE, .

DISPLAY('CURRENT PULSE AMPLITUDE IS'), PULSAMPLTD, ...

PAUSE, •••

CHOICE = MENU (CHANGEMENU , 1); ...

IF CHOICE = 2, ...

INQUIRE PULSAMPLTD TENTER PEAK PULSE AMPLITUDE (-10 TO 10)T , ...

PULSAMPLTD, ...

PAUSE, .•.

EJ\ID, •••

END

II

II CLEAR UNNECESSARY VARIABLES ---------------------------------------

CLEAR BLANK15 CHANGEMENU TYPEMENU. SHAPEMENU CHOICE RECORDUR

II END OF GETPARAM PROGRAM ============================================

RETURJ\l

137II PROGRAM ============================================================

II

II NAME: DISCRTCL

II

II AUTHOR: T T HAMMOND

II

II DATE: 5 APR 1988

II

II PURPOSE:


II create a MatrixX description of "the sampled-data system examined

II in the tt'1esis. Tlie MatrixX description of the analog computer

II realization rrust be available in the file ANCMPSIM.VAR in the OOS

II default directory.

II This program incorporates a stabilizing feedback loop to the input

II using collocated, complementary sensing (i.e. the rotational

II velocity of disk one is the feedback to the torque input of disk

I lone).

II

II VARIABLES

II DA8, DA8C

II DELTAT

II DSA, DS8,

II DSC

II K

/1' KF8

II NAB, NAOC

----------~~~~---------~~-~~~~~~~~~~~~~~-~~~~~-~---------~~~~~~~~~~~-----~~~~~~~------~~~~~~~~~~~~~~~~~~-~~~~-~~~~~

intermediate subsystem descriptions


descriptions of digital subsystems

compensator gain

feedback matrix

intermediate subsystems' orders

138II NCL

II NSA,. NS8,

II NSC

II SA, S8, SC

II

II SCL

II

order of c Iosed-Toop system description


matrices composed of coefficient matrices to form

,the MatrixX represerttations of the subsystems

closed-loop system IVIatrixX descriptiQli

II CONSTANTS ==========================================================I I LOOP CLOSING PARAMETERS:

INGAIN = [1 1 1J';

K = -1;

KFB = K*[O 0 -1.9909 0 -3.4905E-3; .

o 0 -1.9909 0 -3.4905E-3; .

o 0 -1.9909 0 -3.4905E-3];

II

II BEGIN ==============================================================


ERASE

DISPLAy(tDISCRETIZING THE CLOSED-LOOP SYSTEM ... T)

SHORT E

II RETRIEVE SIMULATED SYSTEM AND DISCRETIZE EACH SUBSYSTEM -----------

LOAD(tANCMPSIM.VAR t)

DSA = DISCRETIZE(SA, NSA, DELTAT);

058 = DISCRETIZE(SB, NS8, DELTAT);

DSC = DISCRETIZE(SC, NSC, DELTAT);

II CLOSE DISK-1 VELOCITY FEEDBACK LOOP -------------------------------

[DAB, NABJ = APPEND(DSA, NSA, DSB, NSB);

139[DABC, NA8C] = APPEND(DAB, NAB, DSC, NSC);

[SCL, NCLJ = CONNECT(DABC, NABe, KFB, INGAIN);

II CLEAR UNNECESSARY VARIABLES --------------~------------------------

CLEAR DAB DABC DSA DSB DSC KFB NAB NABC NSA NS8 NSC SA S8 SC

II STORE CLOSED-LOOP DISCRETE SYSTEM ON DISKETTE ---------------------

SAVECtDISCRTCL.VAR t)

II END OF DISCRTCL PROGRAM =========================================--==RETURN

140II P~RAM ============================================================

II

II NAME: GENINPUT

II


II

II DATE: 7 APR 1988

II

II PURPOSE:


I I gerterate tile inpu"t vector TAU for the digita1 system simulation.

II The input signal is a rectangular pulse, a binomial triangle

II shaped pulse, or unity white noise signal as defined by the user

II in the calling program.

II

II VARIABLES

II CHOICE

II COLUMNOX

II DELTAT

II LEVEL

II NEXTPASCAL

II NOISEQ

II NSAMPLES

1/ PPS::,AL

~~~~~---------~~~~~---~~~~~~~~~~~--~~~~~~~~~~~~~~~~~~-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


column index counter


level of Pascal's triangle

next level of Pascal's triangle

noise sequence


level of Pascal's triangle

II PULSAMPLTD input pulse amplitude (normalized -10 to 10)

II PULSDURATN length of input pulse in sample periods

II SHAPE = 1 for rectangular input pulse

141

determinis"tic vector input signal ~ or

= 2 for stochastic inpu"t signal

= 2 for binomial triangle shaped input pulse

= 3 for bipolar triangle shaped input pulse

stochastic matrix of vector input noise signals


II

II

II TAU

II

II TYPE

II

II

II BEGIN ==--===--=======================================================


ERASE

DISPLAy(tGENERATING SYSTEM INPUT SIGNALS ... f)

II SIGNAL GENERATION COMMENTS =========================================

II GENERATE DETERMINISTIC SIGNAL -------------------------------------

II GENERATE RECTANGULAR SIGNAL ---------------------------------------

II GENERATE SHAPED SIGNAL --------------------------------------------

II SCALE PEAK MAGNITUDE OF DETERMINISTIC SIGNAL -----------------------

II ALTERNATE SIGNS FOR BIPOLAR TRIANGLE ------------------------------

II GENERATE PSEUDO-RANDOM UNITY WHITE NOISE SEQUENCE SIGNAL ----------

IF TYPE ;: 1~ ...

I F SHAPE = 1 ~ ...

TAU ;: [PULSAMPLTD*ONES(1 ~PULSDURATN)J T; •••

ELSE PASCAL = 1; •••

LEVEL;: 1; ...

ltt-tILE LEVEL < PULSruRATN, ...

NEXTPASCAL = [1 O*ONES(1~LEVEL-1) 1J; ...

COLUM\IOX = 2; •••

If.tiILE COLUM\lOX < LEVEL+1 ~ ...

142NEXTPASCAL(1,COLUMNDX) = PASCAL(1,COLUMNDX-1)+ ...

PASCAL(1,COLUMNDX); ...

COLUMNDX = COLUMNDX+1; ...

END, •••

PASCAL = NEXTPASCAL; ...

LEVEL = LEVEL +1; ...

END, •••

TAU = PULSAMPLTD*[PASCAL/MAX(PASCAL)Jf; ...

I F SHAPE = 3, ...

FOR COLUMI\IDX = 1: PULSDURATN , ...

TAU(COLUMNDX,1)=TAU(COLUMNDX,1)*(-1)**(COLUMNDX-1); ...

EJ\ID; •••

EJ\ID; •••

END, .••

TAU = [TAU; O*ONES(NSAMPLES-PULSDURATN, 1)J; ...

ELSE RAND(tNORMAL t) , •••

TAU = RAND(NSAMPLES, 1); ...

END

II

II CLEAR. UNNECESSARY VARIABLES ---------------------------------------

CLEAR COLUMNDX

CLEAR LEVEL NEXTPASCAL NOISEQ PASCAL PULSAMPLTD PULSDURATN SHAPE

ERASE

II END OF GENINPUT PROGRAM ============================================RETURN

II

II

PROORAM143

--~~~~~~-~~~~--~~-~~~~~--~-~~--~--~-~~~~~---------~~~~~~~~-~~~~~~~~~~~~~~~~~~~-~~~~~~~~~~~~~~~~~~~~~~~~--~~~~~~~~~~~~~~~

II NAJVIE: DTRMNRSP

II

II AUTHOR: T T HAMVIOND

II

II DATE: 9 APR 1988

II

II PURPOSE:

II This file is a Matrix)( executable command file. Its purpose is to

I I compute the time and frequency responses for the exper i menta1

II system simulation due to deterministic inputs. Also the desired

I I frequency response function is extracted from the c Iosed-Toop

II configuration.

II The system description variables and input signal must be provided

II by the calling program.

II

II VARIABLES

II DELTAT

II FHZ

II G5FEST

II K

II IVIODES

II NCL

II NOISEQ

II NSAMPLES

II NSIM

--~~~~~~~~--~~--~-~~-~~-~~~~~-~-~~~--~~-~~~~---~~~----~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


frequency in Hz

es"timated disk 3 velocity frequency response function

comper1sator gain

rncda1 responses

order of closed-lcx:>p system descriptiQr1

noise sequence


discrete simulatiQr1 index vector

144II SCL closed-loop system MatrixX description

II TAU deterministic vector input signal~ or

II stochastic matrix of vector input noise signals

II TIME time in seconds

II VEL1CL disk closed-loop velocity

II VEL1CLF disk c1osed-loop veloci ty f r-equency s pect rum

II VEL3CL disk 3 closed-loop velocity

II VEL3CLF disk 3 closed-loop velocity frequency spectrum

II

II BEGIN ==============================================================


ERASE

DISPLAY('COMPUTING SYSTEM RESPONSES ... ')

II GENERATE TIME RESPONSES --------------------------------------------

[NSIM, MODESJ = DLSIM(SCL, NCL, TAU);

TIME = NSIM*DELTAT;

CLEAR NSIM

II DISPLAY INPUT SIGNAL AND DISK-3 RIGID BODY MODES ------------------

PLOT(TIME, [TAU MODES(:,[1 2J)J, ...

'TITL/DETERMINISTIC INPUT AND DISK-3 RIGID BODY MODES/ ...

YLAB/MAGNITUDE/ ...

XLAB/TIME IN SECONDS/')

PAUSE

ERASE

II DISPLAY DISK-3 LOW FREQUENCY MODES --------------------------------


'TITL/DETERMINISTIC INPUT AND DISK-3 0.1 HZ MODES/ ...

145YLAB/MAGNITUDE/ ...

XLAB/TIME IN SECONDS/')

PAUSE

ERASE

II DISPLAY DISK-3 HIGH FREQUENCY MODES -------------------------------


fTITL/DETERMINISTIC INPUT AND DISK-3 1.5 HZ MODES/ ...

YLAB/MAGNITUDE/ ...

XLAB/TIME IN SECONDS/ r )

PAUSE

ERASE

CLEAR TIME

I/next section added to find max values for normalization==============

I/P3L=MAX(A8S(MODES(:,3)))

/IPAUSE

I/P3H=MAX(A8S(MODES(:,5)))

I/PAUSE

//V3R=MAX(A8S(MODES(:,2)))

IIPAUSE

//V3L=MAX(A8S(MODES(:,4)))

//PAUSE

//V3H=MAX(A8S(MODES(:,6)))

//PAUSE

I/A3L=MAX(A8S(-O.335*TAU-O.395*MODES(:,3)))

//PAUSE

I/A3H=MAX(A8S(O.0015*TAU-88.8269*MODES(:,5)))

//PAUSE

146Ilend first max values section

II DELETE POSITION RESPONSES FOR MEMORY CONSERVATION -----------------

MODES = [MODES(:,2) MODES(:,4) MODES(:,6)];

II FORM DISK-1 AND DISK-3 VELOCITY SIGNALS ---------------------------

VEL1CL = [MODES(:,1)-1.9909*MODES(:,2)-3.4905E-3*MODES(:,3)];

VEL3CL = [MODES(:,1)+MODES(:,2)+MODES(:,3)];

Iisecond max values section begins

1/V1CL=MAX(ABS(VEL1CL))

IIPAUSE

1/V3CL=MAX(ABS(VEL3CL))

IIPAUSE

Ilend second max values section

II CLEAR SUBSYSTEM TIME RESPONSES FOR MEMORY CONSERVATION ------------

CLEAR MODES

II PERFORM FFTs ON INPUT SIGNAL, AND DISK-1 AND DISK-3 VELOCITIES ----

TAUF = FFT(TAU, TRECT T);

CLEAR TAU

TAUF = [TAUF([2:(NSAMPLES/2)],1)J;

VEL1CLF = FFT(VEL1CL, 'RECT T) ;

CLEAR VEL1CL

VEL1CLF = [VEL1CLF([2:(NSAMPLES/2)],1)];

VEL3CLF = FFT(VEL3CL, TRECT T) ;

CLEAR VEL3CL

VEL3CLF = [VEL3CLF([2:(NSAMPLES/2)],1)];

I I COMPUTE DISK-3 VELOCITY OPEN-LOOP FREQUENCY RESPONSE FUNCTION -----

G5FEST = [VEL3CLF./CTAUF+K*VEL1CLF)];

FHZ = [1:(NSAMPLES/2)-1]f/(NSAMPLES*DELTAT);

147// DISPLAY INPUT SIGNAL SPECTRUM --------------------------------------

PLOT(FHZ~ 20*LOG(A8S(TAUF))/LOG(10), ...

fTITL/DETERMINISTIC INPUT FREQUENCY SPECTRUM! ...

YLAB/dB! ...


LOGX t)

PAUSE

ERASE

II DISPLAY DISK-1 CLOSED-LOOP VELOCITY SPECTRUM -----------------------

PLOT(FHZ 7 20*LOG(A8S(VEL1CLF))/LOG(10), ...

fTITL/DISK-1 CLOSED-LOOP VELOCITY FREQUENCY SPECTRUM/ ...

YLAB/dB/ ...


LOGX t)

PAUSE

ERASE

II DISPLAY DISK-3 CLOSED-LOOP VELOCITY SPECTRUM -----------------------

PLOT(FHZ~ 20*LOG(A8S(VEL3CLF))/LOG(10), ...

fTITL/DISK-3 CLOSED-LOOP VELOCITY FREQUENCY SPECTRUM/ ...

YLA8/d8/ ...


LOGX t)

PAUSE

ERASE

1/ DISPLAY DISK-3 OPEN-LOOP VELOCITY FREQUENCY RESPONSE FUNCTION ------

PLOT(FHZ~ 20*LOG(A8S(G5FEST))/LOG(10), ...

TTITL/DISK-3 OPEN-LOOP VELOCITY FREQUENCY RESPONSE FUNCTION/ ...

148YLAB/d8/ ...

XLAB/FREQJENCY IN HZ/ ..•

LOGX T)

PAUSE

ERASE


CLEAR DELTAT FHZ G5FEST K NCL NSAMPLES SCL TAU TAUF

CLEAR TYPE VEL1CL VEL1CLF VEL3CL VEL3CLF

II END OF DTRMNRSP PROGRAM ============================================RETURJ\I

II PROGRAM149

~~~~~~~~-------~~~~~~~~~~~--~-~~~~~~~~~~~~~~~~~~~--~~~-----~~~~~~~~~~~~~~~~~~--~~~~~~~~~~~~~~--~~~~~~~~~~~~~~~~~~~~~~--

II

II NAME: STOCHRSP

II


II


II

II PURPOSE:


I I ccmout.e the time and frequency responses for the exper trnerrce 1

I I system sirrulation due to stochastic inputs . Also the desired

I I frequency response funct i on is extracted from the C 1osed-l oop

II configuration.

II The system description variables and input signal must be provided

II by the calling program.

II

II VARIABLES

II DELTAT

II FHZ

II G5FEST

II K

II MODES

II NCL

II NOISEQ

II NSAMPLES

II NSIM

-~~~~-~---~~~-----~----~~~-~-----~~~~~-----------~~-~----~~~~~~~~~~~~--~~~~~~~-~~~~~~~~---~~~~~~~~~~~~~~~-~~~~~~~-~


frequency in Hz

estimated disk 3 velocity frequency response function

compensator gain

mcxJa 1 responses

order of closed-loop system description

noise sequence


discrete simulation index vector

150II SCL

II TAU

II

II TIME

II VEL1CL

II VEL1CLF

II VEL3CL

II VEL3CLF

II

closed-loop system MatrixX description

deterministic vector input signal, or

stochastic ma"trix of vector input noise signals

time ; n seconds

disk closed-loop velocity

disk closed-loop velocity frequency spectrum

disk 3 closed-loop velocity

disk 3 closed-loop velocity frequency spectrum

II BEGIN ==============================================================


ERASE

DISPLAy(tCOMPUTING SYSTEM RESPONSES ... t)

II GENERATE TIME RESPONSES --------------------------------------------

[NSIM, MODESJ = DLSIM(SCL, NCL, TAU);

TIME = NSIM*DELTAT;

CLEAR NSIM

II DISPLAY INPUT SIGNAL AND DISK-3 RIGID BODY MODES ------------------


tTITL/STOCHASTIC INPUT AND DISK-3 RIGID BODY MODES/ ...

YLA8/MAGNITUDE/ ...

XLAB/TIME IN SECONDS/ f)

PAUSE

ERASE

II DISPLAY DISK-3 LOW FREQUENCY MODES --------------------------------


fTITL/STOCHASTIC INPUT AND DISK-3 0.1 HZ MODES/ ...

151YLAB/MAGNITUDE/ ...

XLAB/TIME IN SECONDS/ T)

PAUSE

ERASE

// DISPLAY DISK-3 HIGH FREQUENCY MODES -------------------------------

PLOT(TIME~ [TAU MODES(:,[5 6])], ...

fTITL/STOCHASTIC INPUT AND DISK-3 1.5 HZ MODES/ ...

YLAB/MAGNITUDE/ ...

XLAB/TIME IN SECONDS/f)

PAUSE

ERASE

CLEAR TIME

Iinext section added to find max values for normalization===============

I/P3L=MAX(ABS(MODES(:~3)))

//PAUSE

//P3H=MAX(A8S(MODES(:,5)))

//PAUSE

//V3R=MAX(A8S(MODES(:,2)))

I/PAUSE

//V3L=MAX(A8S(MODES(:~4)))

//PAUSE

//V3H=MAX(A8S(MODES(:,6)))

//PAUSE

//A3L=MAX(ABS(-O.335*TAU-O.395*MODES(:,3)))

//PAUSE

//A3H=MAX(A8S(O.0015*TAU-88.8269*MODES(:~5)))

//PAUSE

152Ilend first max values section

II DELETE POSITION RESPONSES FOR MEMORY CONSERVATION -----------------

MODES = [MODES(:,2) MODES(:,4) MODES(:,6)];

II FORM DISK-1 AND DISK-3 VELOCITIES ---------------------------------

VEL1CL = [MODES(:,1)-1.9909*MODES(:,2)-3.4905E-3*MODES(:,3)];

VEL3CL = [MODES(:,1)+MODES(:,2)+MODES(:,3)];

Iisecond max values section begins

I/V1CL=MAX(A8S(VEL1CL))

IIPAUSE

//V3CL=MAX(A8S(VEL3CL))

I/PAUSE

Ilend second max va 1ues section

II CLEAR TIME RESPONSES FOR MEMORY CONSERVATION ----------------------

CLEAR MODES

II FORM FEEDBACK ERROR SIGNAL ----------------------------------------

U = TAU + K*VEL1CL;

II COMPUTE AUTo-PSD OF INPUT SIGNAL -----------------------------------

[TAUF, FIN] = SPECTRUM(TAU, TAU);

CLEAR TAU

TAUF = [TAUF([(NSAMPLES/2+2):(NSAMPLES)])];

II COMPUTE AUTo-PSD OF DISK-1 VELOCITY --------------------------------

[VEL1CLF, FIN] = SPECTRUM(VEL1CL, VEL1CL);

CLEAR VEL1CL

VEL1CLF = [VEL1CLF([(NSAMPLES/2+2):(NSAMPLES)])];

II COMPUTE CROSS-PSD OF ERROR SIGNAL AND DISK-3 VELOCITY -------------

[UVEL3CLF, FIN] = SPECTRUM(VEL3CL, U);

CLEAR VEL3CL U

153UVEL3CLF = [UVEL3CLF([(NSAMPLES/2+2):(NSAMPLES)])];

II COMPUTE DISK-3 OPEN-LOOP VELOCITY FREQUENCY RESPONSE FUNCTION -----

G5FEST = [UVEL3CLF./(TAUF+K*VEL1CLF)];

FHZ = FIN([(NSAMPLES/2+2):(NSAMPLES)])/(DELTAT);

II DISPLAY INPUT SIGNAL SPECTRUM --------------------------------------

PLOT(FHZ, 20*LOG(ABS(TAUF))/LOG(10), .

'TITL/STOCHASTIC INPUT AUTO-PSD/ .

YLAB/dB/ ...


LOGX f)

PAUSE

ERASE

II DISPLAY DISK-1 CLOSED-LOOP VELOCITY SPECTRUM ----------------------

PLOT(FHZ, 20*LOG(A8S(VEL1CLF))/LOG(10), ...

'TITL/DISK-1 CLOSED-LOOP VELOCITY AUTO-PSD/ ...

YLAB/dB/ ...


LOGX t)

PAUSE

ERASE

1/ DISPLAY DISK-3 CROSS-PSD ------------------------------------------

PLOT(FHZ, 20*LOG(A8S(UVEL3CLF))/LOG(10), ...

'TITL/ERROR SIGI\IAL AND DISK-3 CLOSED-LOOP VELOCITY CROSS-PSD/ ...

YLAB/dB/ ...

XLA8/FREQUENCY IN HZ/...

LOGX T)

PAUSE

154ERASE

II DISPLAY DISK-3 OPEN-LOOP VELOCITY FREQUENCY RESPONSE FUNCTION -----

PLOT(FHZ, 20*LOG(A8S(G5FEST))/LOG(10), ...

'TITL/DISK-3 OPEN-LOOP VELOCITY FREQUENCY RESPONSE FUNCTION/ ...

YLAB/dB/ ...


LOGX')

PAUSE

ERASE


CLEAR DELTAT FHZ G5FEST K NCL NSAMPLES SCL TAU TAUF

CLEAR TYPE VEL1CL VEL1CLF VEL3CL VEL3CLF

II END OF STOCHRSP PROGRAM ============================================R8U~

APPENDIX 8

Hybrid Simulation Source Code

155

156

This appendix contains the source code listings for the

hybrid simulation of Chapter Five. The programs are written

in HP BASIC for use on a HP Model 9000 computer.

The programs listed perform three major tasks. The SIM

ULATION program drives the hybrid simulation to collect the

time data for the identification process. The SIUDI (System

Identification Using Deterministic Inputs) program performs

frequency analysis and response function computation for

deterministic analysis. The SIUSI (System Identification

Using Stochastic Inputs) program provides the corresponding

functions for stochastic analysis.

15710 PROGRAM ======================================================

20 NaMe: SIMULATION

30 Author: T T HAMMOND

40 Date: MAY 1988

50 Purpose: This prograM drives the three-dIsk saMpled-data

60 control sY5teM 5iMulation. First the HP-3852A Data Acquisi-

70 tion/Control Unit is readied by initializing the HP-44702A

80 HIgh Speed VoltMeter and HP-44727A Olgital to Analog Voltage

90 Converter, and downloading the data acquisition/control

100 routine. Next the user is queried for the Input 5Ignal flle-

1 10 naMe and the record nUMbering sequence for the archIve of tlMe

120 records. At thIS pOInt) the Data AcquisitIon/Control UnIt 15

130 cOMManded to run the siMulatIon, and this prograM concurrently

140 displays a strip chari view of the lnput signal and the

150 Measured velocIties of disk one and disk three. DurIng the

160 siMulation, the user is proMpted to operate the analog

170 COMputer and confIrM these actions. FInally, the data col-

180 lected 15 stored on diskette in the approprlate archive} and

190 the prograM recycles to begin another siMulatIon If the user

200 desires.

210

220 SUBPROGRAMS ==================================================

230 The following subprograMs froM the HP Data AcquI5ition Manager

240 OACQ/300 package are used:

280

290

300

310

330

340

Y_M.in ,Y_Ma>·< ,[ ,Buf_5ize[ ,Strlp_buf$J J)

158

350

360

37G

390

400

410

420

Arch_open[ . Ar-ch i ve name s I ,Arch.i'lesize[ ,Last_pageslze ] J

Page_store(BooknaMe$( ,Inst_addr[ ,Pagelabel$[,

Des t inat ion$]] ] )

430 VARIABLES ====================================================

440 ! J)e.l1 ,')e13 d i s k r ! , disk-3 ve l c c i t i e s In radians/sec

450

460 ! Scandata

analog COMputer norMall:lng voltage ln volts

raw d r s k r ] ,disk-3 vo l t ne t er- readings in v o l t s

470

480

I Insignal input signal sequence in radians/sec

chart abSCIssa in seconds

490

500

520

530

Last rQr'

i Record_count

saMple nUMber Index

first record nUMber to append to archive

last record nUMber to append to archIve

record nuno e.r i ride x

strip chart graph record

540

550

560

570

! Stri.p_count

! SaMpling

! InsQnl$

strip chart saMple nUMber index

=1 while s anp l i no data

=0 c t her u i s e

naMe of input signal vector

159

5812) More$

590 I Title$

600 ! Sub titleS

6i0 ! Trace_labels$

620 ! X label$

630 Y' Lab e l s

640

query response

5trip chart title

5trip chart 5ubtitle

strtp chart trace labels

strlp chart abscissa label

strIp c har t ordlnate label

650 REAL Vell(0:1023),Ue13(0:1023),NorM_volt

560 REAL Scandata(0: 1)

670 REAL Insignal(0:1023)

710 OIM Insgnl$[10J ,More$(3J

7:0 DIM Tltle$[40J ,-Sub_title$[40J ,Tr3ce_labels$[40J

730 OI~l X_label$[ 20 J ,Y_label$[ 20]

740 CONSTANTS ====================================================

750

760

770

! Nor-nve l t

I NorMve13

! NorMsgnl

disk-l velocity norMallzlng constant

disk-3 velocity norMalizing constant

input signal norMalizing constant

790 REAL Nor-mve l l .Nornve l S .Nor-n s qn I

800 NorMvell=5

160

810 Nornv e 13-=5

820 NorMsgnl=10

830 BEGIN ========================================================

840 INITIALIZE DATA ACQUISITION SUBPROGRAMS ----------------------

850 Sys_inlt

860 CONFIGURE HP-3852A HARDWARE ----------------------------------

870 OUTPUT 709;"RST"

880 OUTPUT 709; JIIJSE 200"

890 OUTPUT 709; iI CONF DC\) /I

90!~ OUTPUT 709 j II PACER 0.200 J 1()24 iI

9110 C)UTPt.JT 709; II PDELAY 0.200 It

920 OUTPUT 709. IlSWRITE 100, 2,13"

930 OUTPUT 709;IISIJJRITE 100,6,00,00 11

940 UNMASK HP-3852A SOFTWARE INTERRUPT ---------------------------

95fb OUTPUT 709; II RQS FPS 11

960 DECLARE HP-3852A VARIABLES -----------------------------------

970 OUTPUT 709; II REAL C:NTRLERR, I NS I GNAL ( 11:Q23 ) II

980 ;JUTPUT 709 ; iJ REAL 'S GNLSeAL, t) EL t 1(02 3 ) 1 l.)EL3 ( 1023 ) Ii

990 OUTPUT 709; "REAL SCANDATA( 1 ) ~ FBGAIN II

1000 OUTPUT 709;;1 I NTEGER COUNT II

1010 SET FEEDBACk GAIN --------------------------------------------

1020 OUTPUT 709; /I FBGAIN = 1 II

1030 HP-3852A DATA ACQUISITION AND CONTROL ROUTINE ----------------

1040 OUTPUT 709;"SUB CNTRLOOP"

1;J50 OUTPUT 709;!' FOR COUNT=0 TO 1023"

1060 OUTPUT 709;IlCNTRLERR=0.667*SGNLSCAL*INSIGNAL(COUNT)-0.333*FBGAIN*S

161

CANOATA(0)"

1070 OUTPUT 709; II INDEX SCANDATA 0 11

1080 OUTPUT 709; III;jAITFOR PACERII

1090 OUTPUT 709;IIMEAS DCV, 401-412'2 INTO SCANDATA"

1100 OUTPUT 709; 11 APPLY DCIJ 100, CNTRLERR"

1110 OUTPUT 709;lIlJEL1(COUNT)=SCANDATA<0)"

112i~ OUTPUT 709; II VEL3 ( COUNT) =SCANOATA ( 1 ) II

1130 OUTPUT 709; II I F COUNT 100 THE~r'

1 i 40 OUTPUT 709; II ~JREAD SCANDATA II

1150 OUTPUT 709; !lEND IFII

1160 OUTPUT 709;"NEXT COUNTlI

1170 OUTPUT 709;IlSRQII

118f~ OUTPUT 709; "SUBENO"

1190 DEFINE INTERRUPT SERVICE ROUTINE -----------------------------

1200 ON INTR 7 GOSUB Service_routine

1210 LOOP WHILE MORE SIGNALS TO BE APPLIED ------------------------

1230 t~ HI LE i10r e $ [ 1 , 1 J=1\ Y"

1240 GET uSER SUPPLIED INFORMATION --------------------------------

1250 BEEP

1260 OISP "ENTER NAME OF INPUT SIGNAL '.JECTOR, e.g. RT1Xl_\)EC 11

1270 LINPUT II 1'1Insgnl$

12B0 BEEP

1290 OISP IlENTER FIRST RECORD NUMBER";

1310 BEEP

1320 DISP "ENTER LAST RECORD NUMBER";

162

1330 INPUT II II ,Last_rec

1340 IF Insgnl$<>·IINOISE_RECIl THEN

1350 Get_rarray(Insgnl$,Insignal<*»

1360 MAT Insignal= Insignal/(NorMsgnl)

1370 END IF

1380 LOOP FOR EACH RECORD -----------------------------------------

1400 PARSE NOISE SIGNAL -------------------------------------------

1410 IF Insgnl$=ilNOISE_REC" THEN

E_UEC" )

1430 Get_rarray( "NOISE_I)EC" ,Insignal( *»

1440 MAT Insignal= Inslgnal/(NorMsgnl)

1450 END IF

1460 INITIALIZE STRIP CHART ---------------------------------------

1470 Strip_init(Strip_rec(*),3,0,20,1 ~-10,10)

150 0 Tit 1e $ =il Thr e e - 0 i 5 k S i MU 1a t ion II

15i0 Sub_title$=lISa Mp l e d- da t a Strip Char t "

1530 Y_label$="Radians per second"

1540 Trace_labels$=IIInput:Vel-t :Vel-3 11

1560 OUTPUT ~~BD;CHR$(255)&Jlf<";

163

1570 GRAPHICS ON

1580 Graph_show(Strip_rec(*))

1590 INITIALIZE HP-3852A VARIABLES --------------------------------

1600 OUTPUT 709; "0ISP OFF"

1610 OUTPUT 709;J1SCANDATA(0)=0 11

1620 OUTPUT 709;/IINDEX SCANDATA, 0 11

1630 OUTPUT 709;"INOEX INSIGNAL, 0"

1640 FOR SaMple_count=0 TO 1023

1650 OUTPUT 709;1I\)WRITE INSIf3NAL "&VAL$(Inslgnal(SaMple_count»)

1670 ENABLE INTERRUPT FOR HP-3852A FINISH -------------------------

1680 SaMpling=1

1690 ENABLE INTR 7;2

1700 PROMPT USER TO ACTIVATE ANALOG COMPUTER ----------------------

1710 BEEP

1720 BEEP

1730 DISP llACTIIJATE ANALOG COMPUTER AND PRES'S RETURN .. II,

174 0 LIN PUT II II ,N u 1$

1750 DISPLAY NUMBER OF RECORD BEING RECORDED ----------------------

1760 DISP IIRECORDING RECORD NUMBER ";Record_count

1770 MEASURE ANALOG COMPUTER NORMALIZING VOLTAGE ------------------

1780 OUTPUT 709; II MEAS DCtJ, 403 INTO S.GNLSCAL II

1790 OUTPUT 709; II ()READ SGNLSCAL"

1800 ENTER 709;NorM_volt

1810 =TART HP-3852A ROUTINE ---------------------------------------

1820 OUTPUT 709;"PTRIG"

1830 OUTPUT 709;"CALL CNTRLOOplI

164

1840 ADO DATA POINTS TO STRIP CHART FOR 100 SAMPLES ---------------

1850 FOR Strip_count=0 TO 99

1860 ENTER 709;Scandata(*)

1880 S t r i p_p 0 i n t 5 ( S t. r i p_rec ( * ) ,S t r i p_ t i ne , II 1 : 2 : 3 II .Nor n 5 gn 1* In 5 i gn

al (Strip_count ) .Normve l l *Scandata( 0) ,NorMve13*Scandata( 1 »

1890 NEXT Strip_count

1900 WAIT UNTIL HP-3852A FINISHES AND INTERRUPTS ------------------

1910 WHILE SaMpling=l

1920 END WHILE

1930 PROMPT USER TO RESET ANALOG COMPUTER -------------------------

1940 BEEP

1950 BEEP

1 1360 BEEP

1!370 0ISP II SA~1PL I NG COMPLETED: RESET ANALOf3 COr1PUTER II

1980 GET RAW UELOCITY MEASUREMENTS FROM HP-3852A ------------------

1990 OUTPUT 709; IIlJREAO UEL1 II

2000 ENTER 709;Ve11 *)

2010 OUTPUT 709;IIVREAD VEL3 u

2020 ENTER 709;Ve13(*)

2030 CONVERT UELOCITIES TO RADIANS/SEC ----------------------------

2040 ~1AT \Jell = (NorMvell )*\)e11

2050 MAT Vell= Vell/(NorM_volt)

2060 MAT Ve13= (NorMve13)*Ve13

2070 MAT Vel3= Ve13/(NorM_volt)

2080 STORE VELOCITIES IN ARCHIVE ----------------------------------

165

2090 Save_rvect (VeIl (*) ,1I0I51<_1 VEL II)

2110 Arch_open( Insgnl$[ 117J&"ARC

Il)

2140 ACTIVATE HP-3852A DISPLAY ------------------------------------

2 150 0UTPUT 709; II 0 I SP 0N II

2160 END OF LOOP FOR EACH RECORD ----------------------------------

2170 NEXT Record_count

2130 QUERY USER FOR MORE SIGNALS TO BE APPLIED --------------------

2190 BEEP

2200 OISP -oc YOU WISH TO 00 ANOTHER INPUT SI(3NAL? (Y or N)";

2210

2220

2230

2240

2270

2230

2290

2300

2310

2320

L I NPUT II II ~ More$

END OF LOOP FOR MORE SIGNALS ---------------------------------

END WHILE

STOP

HP-3852A INTERRUPT SERVICE ROUTINE ---------------------------

ServIce_routine:

Ser_poll=SPOLL 709)

OUTPUT 709;IlSTA?"

ENTER 709;Stat_word

SaMpling=0

RETURN

END OF HP-3852A INTERRUPT SERVICE ROUTINE --------------------

::30 END OF SIMULATION PROGRAM :===================================

2340 END

10 PROGRAM ===================================================)=~ 6

20 NaMe: SIUDT (SysteM Identification U5ing DeterMinistic Input)

30 Author: T T HAMMOND

40 Date: MAY 1988

50 Purpose: This prograM COMputes and averages the saMpled-data

60

70

80

90

frequency re5ponse function for disk-3 velocity for deterMin

Istic excitations. First a raM disk is established for COMpU-

tational efficiency. Next the U5er is querIed for input

filenaMe and nUMber of records to average. Then the FFT of

100 the input signal is perforMed. Now FFTs for the disk-1 and

1 0 disk-3 ~elocitie5 for each record to be averaged are perf8r~ed

120 and sUMMed. Finally the average disk-3 open-loop frequency

130 response function is calculated and stored on diskette.

140

150 SUBPROGRAMS ==================================================

160 The following subprograMs froM the HP Data Acqu151tlon Manager

170 DACQ/300 package are used:

200 Arch_query(ArchivenaMe$~BooknaMe$,Pagelabel$,FieldnaMe$~

:10 Record$1Results$)

230

240 The following subprograMs appear at the end of this prograM:

250 FNMag_db(Real_part ,IMaQ_part,Nr_saMples)

270

167

280 VARIABLES ====================================================

290 I SaMple_rec$ input 5ignal filenaMe

310

320

330

I Nr_records

! All zeros

/ Frequencies

nUMber of records in average

nUMber of records to average

array of zeros

FFT frequency array

350 ! Disk_l,iMag real and iMaginary parts of FFT for velocity

370 ! Disk_3iMc.g resl and iMaginary parts ~f FFT for ~elaclt ~

390 real and IMaginary SUM5 of FFTs for velOCIty

410 real and IMaginary SUM5 of FFTs for velOCIty j

430 real and iMaginary parts of FFT for lnput

real and iMaginary parts of denOMinator

460

470

480

490

500

510

52C0

OenOM faz

NUMer fa::

denOMInator MagnItude In dB

denOMinator phase in degrees

nUMerator Magnitude In dB

nUMerator phase in degrees

frequency response function Magnitude in dB

frequency response function phase in degrees

168

540 REAL Nr_records

550

560 CONSTANTS ====================================================

570 ! Nr_saMples nUMber of saMples

580 I Nr sectors nUMber of raM disk sectors

saMple period in seconds

650 SaMple_per=.2

660

670 BEGIN ========================================================


700 SET UP RAM DISK FOR EFFICIENT EXECUTION ----------------------

710 INITIALIZE ":MEMORY,f~,14"1Nr_5ector5

720 QUERY USER FOR INPUT SIGNAL NAME AND NUMBER OF RECORDS TO ----

730 AVERAGE ------------------------------------------------------

74.0 BEEP

750 OISP ~ENTER NAME OF INPUT SIGNAL VECTOR USED FOR SYSTEM EXCITATION

760 LINPUT 1111 ,SaMple_rec$

770 BEEP

780 0ISP II ENTER NUMBER OF RECORDS TO I NCLUOE IN At)ERAGE II ;

790 INPUT 1111 ,Nr_records

169

800 GENERATE ZERO VECTOR AND FREQUENCY VECTOR --------------------

810 ALLOCATE REAL All_zero5( 1 :Nr_sBMples),Frequencie5( 1 :Nr_saMples)

820 FOR 1=1 TO Nr_saMples

840 Fr euue nc i e e t I )=( I-l )/( SaMple_per*Nr_saMples)

850 NEXT I

860 Save_rvec t ( AII_zero5 ( *' ) , II ALL_ZEROS: r1EMORY ~ fD ,14 II )

870 Save_r'v'ec t ( Frequenc i e s ( * ) , II FREQ_I')ECT: ,700 , 1 II )

880 DEALLOCATE Frequencies(*)

890 PERFORM FFT ON INPUT SIGNAL ----------------------------------

900 USE RECTANGULAR WINDOW -------------------------------------

REAL: MEMORY) 0 ~ 14: INPUT_I MAG:MEMORY, 0 ~ 14" , II NIf )

920 SET UP LOOP TO PROCESS NUMBER OF RECORDS SPECIFIED -----------

gS0 ALLOCATE REAL Oisk_1real(1:Nr_sBMples),Disk_1iMag(1:r\ir_saMple:=,)

970 MAT SUM_lreal= All zer05

990 MAT SUM_3real= AII_zero5

1000 MAT SUM_3iMag= All zer05

1010 FOR J=l TO Nr_records

1020 RECALL SAMPLED-DATA FOR OISK-l ,OISK-3 VELOCITIES FROM ARCHIVE

1030 --------------------------------------------------------------.

170

VAL$ ( J ) 1 II * II , II * II 1 '10 I SK_l UEL : r1EMORY ,0 ,14 : 0 I S~~_31')EL : MEMORY, 0 , 14 11)

1050 PERFORM FFT ON DISr~-l CLOSED-LOOP VELOCITY SIGNAL ------------


SK_1 REAL: MEMORY J 0 , 14:0I S~:_l I MAG: MEMORY, 0 , 14 11, l/ Nl/ )

1080 PERFORM FFT ON OISK-3 CLOSED-LOOP VELOCITY SIGNAL ------------


1100 \)ec t_ f f t ( II FFT II , if 0I Sr<_3VEL : MEMORY' ,0 ,14 I ALL_ZEROS: MEMORY, 0 , 1411, II 0 I

S~~_3REAL: MEMORY, 0 , 14: 0 I -S~~_3 I r1AG: MEMORY ~ 0 ,14" , II Nil)

1110 ACCUMULATE SUM OF VELOCITY FFT'S -----------------------------

1130 !3et_rarray( 1I0IS~~_1 IMAG:MEMORY,0, 14 11 ,Oisk_1 iMag("')

1140 Get_rarray( 1I0ISt<_3REAL:MEMORY,0, 14" ,Disk._3real( *))

1150 Get_rarray( IIDIS~~_3IMAG:MEMORY,0, 14 11 ,Dlsk_31Mag( *))

1160 MAT SUM_1real= SUM_lreal+Oisk_lreal

1170 MAT SUM_liMag= SUM_liMag+Disk_liMag

1230 COMPUTE OISK-3 VELOCITY OPEN-LOOP FREQUENCY RESPONSE FUNCTION

1240 DENOMINATOR --------------------------------------------------

1250 ALLOCATE REAL Input_real( 1:Nr_saMples),Input_iMag( 1:Nr_saMples)

1270 ALLOCATE REAL DenoM_~ag( 1:Nr_5aMple5),Oeno~_faz( 1 :Nr_50Mples)

1280 Get_rarray< "INPUT_REAL:MEMORY ,O,14" ,Input_real<*))

171

1300 MAT Input_real= (Nr_records)*Input_real

1310 MAT Input_iMag= (Nr_records/*Input_iMag

1320 MAT Denom- real= Input_real-SuM_lreal

1330 MAT DenoM_iMag= Input_iMag-Su~_liMag

1340 FOR I =1 TO Nr_saMples

5 ) )

1370 NE/T I

1380 DEALLOCATE Input_real(*),Input_i~ag(*)

1390 DEALLOCATE OenoM_real(*),DenoM_iMag(*)

1400 DEALLOCATE SUM_lreal(+),SuM_liMag(*)


1420 -------------------------------------------------------------

1430 ALLOCATE REAL NUMer_Mag( 1:Nr_sBMple5),NuMer_faz( 1:Nr_sBMples)

1440 ALLOCATE REAL Xferfn_Mag( 1 :Nr_scMples),Xferfn_faz( 1:Nr_sBMples


1470 NUMer_faz(! )=FNPhase_deg«(SuM_3real( I») ,(SuM_3iMBg( I»))

i480 Xferfn_Mag( I )=NuMer_Mag(I )-DenoM_Mag(I)

1490 Xferfn_faz(I )=NuMer_faz(I)-DenoM_faz(I)

1500 IF ABS(Xferfn_faz(! »))=180 THEN Xferfn_faz(I )=Xferfn fa:(! )-360*

1510 NEXT I

ON

172

1520 STORE OISK-3 OPEN-LOOP VELOCITY FREQUENCY RESPONSE FUNCTION

1530 DISKETTE -----------------------------------------------------

1540 IF Nr_record5<10 THEN

1550 Nr"'_a···.;g$="_11 &UAL$( Nr_records)

1560 ELSE

1570 Nr_avg$=VAL$(Nr_records)

1580 END IF

1610 END OF SIMULATION PROGRAM ====================================

1620 END

1630 SUBPROGRAMS ==================================================

1640 COMPUTE MAGNITUDE IN dB OF REAL AND IMAGINARY FFT RESULTS

1650 OEF FNMaQ_db(REAL Real_part ,IMaQ_part ,INTEGER Nr_sBMples)

166 If) RETURN 20* LGT( Nr _ 5 amp 1e 5 *" SQR< Rea l_p art ,. 2+ I Ma Q_P art ,\:: ) )

1670 FNENO

1580

1590 COMPUTE PHASE IN DEGREES OF REAL AND IMAGINARY FFT RESULTS ---

1720 CASE -1 ,1

1730 IF Real_part 0 THEN

1740 RETURN 180*ATN(IMag_part/Real_part )/PI

1750 ELSE

1770 END IF

1780 CASE 0

173

1790 IF Real_part<>0 THEN

1800 RETURN 180*( l-SGN(Real_part))

181r3 ELSE

1820 RETURN 90*SGN(IMag_part)

1830 END IF

1840 END SELECT

1851~ FNEND

1860

i870 END OF SUBPROGRAMS ===========================================

10

20

30

PROGRAM ====================================================~Z 4

NaMe: SIUSI (SysteM Identiflcation Using Stochastic Input)

Author: T T HAMMOND

40 Date: ~A 1\ \ !:IM r i988

50

60

70

90

Purpose: This prograM COMputes and averages the saMpled-data

frequency response function for disk-3 velocity for stochastic

excitations. First a raM disk is established for COMputa-

tional efficiency. Next the user is queried for input 5ignal

filenaMe and nUMber of records to average. Then FFTs are

100 perforMed on the input signal and disk-l and dlSk-3 velocitIes

! 10 for each record. Also tne nUMerator and denOMInator of the

120 average response function are forMed. Finally the average

130 disk-3 open-loop frequency response function 15 calculated

140 and stored on diskette.

150

160 SUBPROGRAMS ==================================================

170 The following subprograMs frOM the HP Data Acquisition Manager

1B0 OACQ/300 package ar-e u s e d :

220 Arch_query(ArchivenaMe$,BooknaMe$,Pagelabel$,FieldnaMe$,

230 Record$,Result$)

250

250 The following subcragraM5 appear at the end ~f thIS prograM:

270

175

280

290

300

310 lJARIABLES ====================================================

320

330

340

360

, Nr_records

I All zer05

j Frequencies

input signal filenaMe

nUMber of record5 in average

nUMber of record5 to average

array of zeros

FFT frequency array

330 I T +".. npu ,,_1 Mag real and iMaginary part5 of FFT for input

390 I Disk-"'real,

400 real and iMaginary parts of FFT for velocIty

410 Disk-3real,

.4.30 j Error_rea 1 1

Error_1Mag

rea 1 and i Mag i ne r y par t 5 Gf FFT f ":J r "/ e 1eel t y -..i

real and iMaginary parts or error signal

460 Error_1Mag_sq: squares of real and IMaginary part5 of error

470 DenoM_5uM_real: cUMulative denOMinator of response funct:on

481~ ! NUMer 5UM rea 1. ~

490 NUMer_5uM_IMag: cUMulative nUMerator of response function

530 Product 4 teMporary product terMS

540

550

560

570

580

! OenoM_Mag

I NUMer_Mag

I NUMer_faz

I Xferfn fa:

denOMinator Magnitude in dB

nUMerator Magnitude in dB

nUMerator phase in degrees

frequency response function Magnitude in dB

frequency response function phase in degrees

176

590

600 OIM SaMple_rec$[10J ,Nr_avg$[2J

510 REAL Nr_records

620

530 CONSTANTS ====================================================

640

650

650

670

I Nr_saMples

l'··lr_sectors

nUMber of saMples

nUMber of sectors

saMple perIod in seconds

730

740 BEGIN ========================================================


760 5Y5 i n i t

770 SET UP RAM OISK FOR EFFICIENT EXECUTION ----------------------

73(~ INITIALIZE 11:MEMORY,Z),14Il

JNr _ s e c t or s

790 QUERY USER FOR INPUT SIGNAL NAME AND NUMBER OF RECORDS TO ----

800 AVERAGE ------------------------------------------------------

177

810 BEEP

820 OISP ~ENTER NAME OF INPUT SIGNAL VECTOR USED FOR SYSTEM EXCITATION

830 LINPUT II 1I ,SaMple_rec$

840 BEEP

850 0 I SP II ENTER NUMBER OF RECORDS TO I NCLUDE IN A'.JERAGE \I ;

860 INPUT II II ;!Nr_records

870 GENERATE ZERO VECTOR AND FREQUENCY UECTOR --------------------

:2·80 ALLOCATE REAL AII_zeros ( 1 : Nr_sarflp 1es ) .Frequenc i e5 ( 1 : Nr_saMp 1es

900 All zeros < I )=fl)

9 1(2) Fr eq ue nc i e 5 ( I )=( 1-1 ) / ( Sa mp 1e_p er '*' Nr _ 5 anp 1e 5 )

930Save_rvec t ( All_zeros ( * ) , II ALL_ZEROS: MEMORY, 0 ,14 'I :>

940 Save_rvec t ( Frequenc i e s ( * ) , II FREQ__'JECT: ,700 , i \I )

950 DEALLOCATE Frequencles(*)

960 SET UP LOOP TO PROCESS NUMBER OF RECORDS SPECIFIED -----------

1370 ALLOCATE REAL Input_real(1:Nr_5aMple5),Input_iMag<1:~\Jr_sa!"lple5)

980 ALLOCATE REAL Disk_lreal< 1 :Nr_saMples),Olsk_liMag( 1 :Nr_saMple5)

990 ALLOCATE REAL Oisk_3real(1:Nr_saMples)10isk_3iMag(1:Nr_saMples}

1000 ALLOCATE REAL Error_real( 1:Nr_saMples),Error_iMag( 1 :Nr_sBMple5)

1010 ALLOCATE REAL Error_real_sq( 1:Nr_saMples)lError_iMaQ_sq( 1:Nr_sdMpl

e s )

pIes)

178

1040 ALLOCATE REAL Product_1( 1:Nr_saMples),Product_2( 1:Nr_soMples)

1050 ALLOCATE REAL Product_3( 1:Nr_saMples),Product_4( 1:Nr_sBMples)

1060 r1AT OenoM- SUM- real= All- zeros

1070 MAT NUMer- 5UJ'r) ._-real:: All - zeros

108!J MAT NUMer_5uM_iMag= All - zer05

1090 FOR J=l TO Nr- records

1100 RECALL INPUT SIGNAL FOR JTH RECORD ---------------------------

1120 PERFORM FFT ON INPUT SIGNAL ----------------------------------

1130 USE HANNING WINDOW -----------------------------------------

1140 l,}ec t._ f f t ( II FFT" , II INPUT_SGNL : i1EMOR\{ ~ 0 , i 4 I ALL_ZEROS: MEMOR'Y ,0 ,14 \I , q I

NPt.JT_REAL: MEMORY, 0 , 14: INPUT_ I 11AG : r1EMORY ,0 , 14 II , \I U II )

1150 RECALL SAMPLED-DATA FOR DIS~~-l ,OISt<-3 UELOCITIES, FROM ARCHIl..)E

1150 --------------------------------------------------------------

1180 PERFORM FFT ON OISK-l CLOSED-LOOP VELOCITY SIGNAL ------------

1 90 USE HANNING WINDOW -----------------------------------------

·S.~~_1 REAL: t1EMORY ,0 ,14: or S~~_1 Ir1AG: r1EMORY .e ,14" ,II!.J II )

1210 PERFORM FFT ON DISk-3 CLOSED-LOOP VELOCITY SIGNAL ------------

1220 USE HANNING WINDOW -----------------------------------------

1230 Vect_fft( IIFFTII,"DIS~~_3VEL:MEMORY,0,14:ALL_ZEROS:MEMORY ,0~14" ,ClDI

1240 ACCUMULATE NUMERATOR AND DENOMINATOR OF RESPONSE FUNCTION ----

1250 SUMS ---------------------------------------------------------

1260 Get_rarray( "INPUT_REAL:t1EMORY,0, 14 11 ,Input_.real( *))

1270 Get_rarr'ay( "INPUT_IMAG:MEMORY ,0,14" ,Input_iMBg( *)

12:80 Get_rarray( "0IS~<_1REAL:l'-'lEMORY,0,14" ,Oisk_1real(*)")

1290 Get_rarray( 1I0IS~~_1 IMAG:MEMORY,0, 14" ,Oisk_l iMag( *)

1301" Ge t_rarray ( "0 I Sr<_3REAL: ~1EMORY ,0 ,14 II ,0 i 5 k_3rea 1( * ) )

1310 6et_rarray ( II 0 I Sr<_3 I MAG: r1EMORY ,0 ,14 II ,0 i 5 k_3 i Mag ( * ) )

1320 MAT Error_real= Input_real-Oi5k_~real

1370 MAT OenoM_5uM_real= OenoM_5uM_real+Error_iMsg_sq

1380 MAT Product 1= Error_real. Oisk_3real

1390 MAT Product 2= Error_iMsg . Oisk_3iMag

1400 MAT Product_3= Error_real. Olsk_3iMag

1410 MAT Product 4= Error_iM8g . Oisk_3real

1430 MAT NUMer_5uM_real= NUMer 5UM real+Product 2

1460 NEXT J

179

180

1520 DEALLOCATE Product_l(*)~Product_2(*)

1530 DEALLOCATE Product_3(*),Product_4(*)


1550 ------------------------------------------------------------.--

1560 ALLOCATE REAL OenoM_Mag( 1 :Nr_saMples)


1580 OenoM_Mag(I }=FNMag_db«(OenoM_5uM_realCI )),(All_zeros(I)),(Nr_saM

p l e s : )

1590 NE)<T I

15 1(0 ALL0CATERE AL NU ne r _M a g ( 1: Nr _ 5 amp 1e 5 ) ,NU jY) e I~_ fa: ( 1: Nr _ 5 aMp 1e 5 )

1620 ALLOCATE REAL Xferfn_Mag( 1:Nr_sBMples),Xferfn_fa:( 1 :Nr_saMples)

1630 FOR 1=1 TO Nr_sBMples

1680 IF ABS(Xferfn_faz(I))=180 THEN Xferfn_faz<I )=Xferfn_faz( I )-360*

SG~j(Xferfn_faz<I))

1630 NEXT I

1700 STORE DISK-3 OPEN-LOOP VELOCITY FREQUENCY RESPONSE FUNCTION ON

1710 DISKETTE -----------------------------------------------------

1720 IF Nr_records 10 THEN

1730 Nr_avg$=ll_"&UAL$(Nr_records)

1740 ELSE

181

1750 Nr_avg$=VAL$(Nr_records)

1760 END IF

1790 END

1800 SUBPROGRAMS ==================================================

1810 COMPUTE MAGNITUDE IN dB OF REAL AND IMAGINARY FFT RESULTS

1·850

1860 COMPUTE PHASE IN DEGREES OF REAL AND IMAGINARY FFT RESULTS ---

1870 DEF FNPhase_deg(REAL Real_part 1IMag_part)

1880 SELECT SGN(Real_part*IMag_part)

: 8 gO CASE -1 11


1910

1940 Ei'·jD IF

1950 CASE 0


1970 RETURN 180*( 1-SGN(Real_part)

1980 ELS·E

2000 END IF

182

2010 END SELECT

2020 FNENO

2030

2040 HANNING WINDOW -----------------------------------------------

2060 RETURN .5-.5*COS(2*PI*EleMent_nuM/Total nUM)

2070 FNENO

2~80 END OF SUBPROGRAMS ===========================================

L§AMPLED-DATA FREQUENCY RESPONSE SYSTEM ...

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