Control of the NASA Langley 16-foot transonic tunnel with the self-organizing feature map

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Control of the NASA Langley 16-foot transonic tunnel with the self-organizing feature map
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Self-organizing systems   ( lcsh )
Transonic wind tunnels   ( lcsh )
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Thesis:
Thesis (Ph.D.)--University of Florida, 1998.
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Includes bibliographical references (leaves 126-129).
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by Mark A. Motter.
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Typescript.
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Vita.

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CONTROL OF THE NASA LANGLEY 16-FOOT TRANSONIC TUNNEL
WITH THE SELF-ORGANIZING FEATURE MAP












By

MARK A. MOTTER


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA
































To the memory of my parents















ACKNOWLEDGMENTS


I would like to thank Dr. Jose C. Principe, my advisor and supervisory committee

chairman, for his encouragement, guidance, patience, and insight during the course of this

research and the writing of this dissertation. I am also grateful for the time and patience

of my committee members, Dr. Gijs Bosman, Dr. Thomas E. Bullock, Dr. John G. Harris,

and Dr. Loc Vu-Quoc.

I would also like to acknowledge the support of this research by NASA

Langley Research Center, and encouragement from both my former branch head,

Kenneth L. Jacobs, and the current branch head, Carl E. Home.

Finally, I would like to thank my wife, Lisa, for her unwavering support, and

Benjamin, who makes every day new.














TABLE OF CONTENTS


page


ACKNOW LEDGM ENTS............................................................. ............................iii

L IST O F T A B L E S ......................................................................... ............................. vi

LIST OF FIGURES....................... ...............................................vii

ABSTRACT.... ................ ............................................................................... x

1 INTRODUCTION .................................................................................................... 1
M otivation.................. .......................................................................... 1
Evolution of the Research .................... ... ....................................... 2
B background ..................................... ............................................... ................... 5
C control C hallenges.......................................................................... ...................... 10
Experimental Framework............................................................................ 18
Overview of the Dissertation.......................................................................... 19

2 REVIEW OF LITERATURE................................................................................... 21
Introduction ................................................................... ........................ 21
Self-Organizing Feature M ap.................................................................................. 21
Practical Aspects for the Application of the SOFM Algorithm....................... 25
Magnification Factor....................... .. ......................... 28
Applications of the SOFM .................................................................................... 29
A Brief Review of Adaptive Control ........................................... ......................... 32
Linear Adaptive Control.......................................................... ........................ 34
Control Using Multiple Models and Switching ........................ ........... ............. 37
Other Applications of Neural Networks for Control......................... ............. 41

3 MODELLING THE TUNNEL DYNAMICS ...................... ..................... ... 43
Introduction ................................... .................................................. ................ 43
Review of Local Dynamic Modeling with SOFM............................... ......... .. 45
Modifications for SOFM-based Predictive Control........................ ......................... 48
Partitioning the Control Input Space............................................... ...................... 51
Clustering the Mach Number Responses ....................................... ....................... 53
Convergence of the Input Neural Fields.......................................... ...................... 65
SOFM Selection for Local Model Identification...................................................... 69
Prediction of Tunnel Response Using Local Models.............................................. 70








4 PREDICTIVE CONTROLLER .............................................. 73
Introduction ............................................................................. .............................. 73
Model Predictive Control Background............................................... 74
SOFM-based Predictive Controller............................................... 76
Operating Point Changes........................................................ 79
Regulating About an Operating Point ................................ ..................... 80

5 EXPERIM ENTAL RESULTS ................................................. ......................... 81
Experim mental Setup ................................................................. ........................ 82
Mach Number Measurements ....................................... .... .............................. 83
Experimental Results of Controlling the Mach Number........................................... 85
Comparison of PMMSC to Existing Controller and Expert Operator...................... 92
Experimental Results of Modeling the Tunnel Dynamics ........................................ 97

6 CONCLUSIONS AND FUTURE RESEARCH.................................... ........... 116
Conclusions ................................................................ ....................... 116
Future Research.......................................... .................. .............. 118

APPENDIX STABILITY CONSIDERATIONS ...................... ...................... 119

LIST OF REFERENCES ............... ........................................................... 126

BIOGRAPHICAL SKETCH............................................... .. 130














LIST OF TABLES


Table page

1. Variation of Mach number rate-limited increase while ramping up .................... 12

2. Changes in control input effectiveness for blocked conditions.............................. 18

3. Prototype Control vectors...................................................... 52

4. Training exemplars for each input class............................. ....................... 54

5. Difference between interval means of adjacent input neural fields ....................... 66

6. Interval means of SOFM input fields ........................................ ..................68

7. Euclidean norm of SOFM input neural fields ........................ ......... ............ 69

8. Candidate Control sequences and associated parameters.................................... 79

9. Standard and maximum deviation of Mach number during calibration................. 84

10. Statistics of time histories of steady state Mach number measurements................ 85

11. Comparison of existing automatic control, expert operator, and PMMSC control.. 93

12. Comparison for controlling to several different set points..................................... 96

13. Distribution among input_classes for Figures 31 and 45..................................... 101

14. Multi-step prediction errors for all inputclasses.............................................. 102














LIST OF FIGURES


Figure page

1. Aerial View of the 16-Foot Transonic Tunnel...................... ........ .............. 6

2. Test Section with model in place ................... .. .... ........... ............ 6

3. Arrangement of Langley 16-Foot Transonic Tunnel..................................... ............. 7

4. Inside View of the 16-Foot Tunnel downstream of the second set of turning vanes.. 8

5. Mach Number and Tunnel Drive Control Inputs during a typical subsonic run......... 9

6. Tunnel conditions during a typical run with steady ramping to the desired test Mach
number, M=0.95. The Mach number is to be held to within 0.003 of the desired
value while varying the angle of attack............................. .. ....................... 13

7. Out of tolerance Mach number extends the test duration during the last 3.5 minutes
of the test @ M =0.95. ........................... ....... ..... ................................. 15

8. Regulating the Mach number at M = 0.8 with large disturbances from the model
A ngle of A attack ....................................................................... .... ......................... 17

9. Experimental Framework with PMMSC.............................. ....................... 20

10. SOFM with a one-dimensional array of neurons ................................... ......... 23

11. Input exemplars for training the example SOFM................................................ 26

12. Learning rate and Neighborhood function during training.................................... 27

13. SOFM during various points in the training.................................................. 27

14. Distribution of training inputs among 20 converged SOFM clusters ....................... 28

15. Structure of the multiple model control with switching..................................... 39

16. The SOFM-based Modeling Architecture for Time Series..................................... 47

17. 50-point prototype control inputs................................................. ....................... 51








18. A single tap of the Mach number preprocessor............................... ............... 55

19. Mach number responses and corresponding SOFM for inputclass_0....................56

20. Mach number responses and corresponding SOFM for inputclass_l .................... 57

21. Mach number responses and corresponding SOFM for inputclass_2.................... 58

22. Mach number responses and corresponding SOFM for inputclass_3.................... 59

23. Mach number responses and corresponding SOFM for inputclass_4....... ........ 60

24. Mach number responses and corresponding SOFM for input_class_5 ................ 61

25. Mach number responses and corresponding SOFM for inputclass 6.................... 62

26. Mach number responses and corresponding SOFM for input_class_7.................. 63

27. Mach number responses and corresponding SOFM for input_class_8................ 64

28. Selection of SOFM by input_class................................................................... 70

29. Candidate Control Sequences..................................... ............. ........................ 72

30. Experim ental Setup ............................................................................................ 82

31. Mach number controlled by PMMSC* during a three hour test.......................... 86

32. Variations of angle-of-attack and angle-of-sideslip during test........................... 87

33. Winning nodes for SOFM_5 and SOFM_6 during test ......................................... 88

34. Fan RPM and Tunnel temperature during test ................................. ........... 88

35. A 15 minute interval of the test.................................................. ........................ 90

36. Comparison of PMMSC to existing control and expert operator...................... 91

37. Comparison of Control Densities............................................... ........................ 94

38. Comparison for controlling to several different set points..................................... 95

39. Comparison of Control Densities during set point changes.................................... 96

40. SOFM _0 winning nodes..................................................................................... 98

41. SOFM_1 and SOFM_2 winning nodes....................................... ....................... 98

42. SOFM_3 and SOFM_4 winning nodes................. .......................... 99








43. SOFM_5 and SOFM_6 winning nodes................................ ..................... 99

44. SOFM_7 and SOFM_8 winning nodes............................... ...................... 99

45. Ramping up with PMMSC control .............................................. 100

46. Predictions, responses, and prediction error for inputclass_0............................ 103

47. Predictions, responses, and prediction error for inputclass_l .............................. 104

48. Predictions, responses, and prediction error for input_class_2............................. 105

49. Predictions, responses, and prediction error for inputclass_3............................ 106

50. Predictions, responses, and prediction error for input_class_4............................ 107

51. Predictions, responses, and prediction error for input_class_5............................ 108

52. Predictions, responses, and prediction error for input_class_6............................ 109

53. Predictions, responses, and prediction error for input_class_7............................ 110

54. Predictions, responses, and prediction error for inputclass_8............................ 111

55. SOFM Predictions of Mach number in set point regulation ................................. 113

56. SOFM Predictions of Mach number in set point regulation .............................. 114

57. SOFM Predictions of Mach number in ramping....................... ...................... 115

58. A simple nonlinear system with feedback......................... ........... ........... 120

















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

CONTROL OF THE NASA LANGLEY 16-FOOT TRANSONIC TUNNEL
WITH THE SELF-ORGANIZING FEATURE MAP

By

MARK A. MOTTER

May 1998


Chairman: Dr. Jose C. Principe
Major Department: Electrical and Computer Engineering


A predictive, multiple model control strategy is developed based on an ensemble

of local linear models of the nonlinear system dynamics for a transonic wind tunnel. The

local linear models are estimated directly from the weights of a self organizing feature

map (SOFM). Local linear modeling of nonlinear autonomous systems with the SOFM is

extended to a control framework where the modeled system is nonautonomous, driven by

an exogenous input. This extension to a control framework is based on the consideration

of a finite number of subregions in the control space.

Multiple self organizing feature maps collectively model the global response of

the wind tunnel to a finite set of representative prototype controls. These prototype

controls partition the control space and incorporate experiential knowledge gained from








decades of operation. Each SOFM models the combination of the tunnel with one of the

representative controls, over the entire range of operation. The SOFM based linear

models are used to predict the tunnel response to a larger family of control sequences

which are clustered on the representative prototypes. The control sequence which

corresponds to the prediction that best satisfies the requirements on the system output is

applied as the external driving signal.

Each SOFM provides a codebook representation of the tunnel dynamics

corresponding to a prototype control. Different dynamic regimes are organized into

topological neighborhoods where the adjacent entries in the codebook represent the

minimization of a similarity metric which is the essence of the self organizing feature of

the map. Thus, the SOFM is additionally employed to identify the local dynamical

regime, and consequently implements a switching scheme than selects the best available

model for the applied control.

Experimental results of controlling the wind tunnel, with the proposed method,

during operational runs where strict research requirements on the control of the Mach

number were met, are presented. Comparison to similar runs under the same conditions

with the tunnel controlled by either the existing controller or an expert operator indicate

the superiority of the method.













CHAPTER 1
INTRODUCTION

Motivation


The initial motivation for this research was to extend neural network based

methods that had proven successful in modeling autonomous nonlinear dynamical

systems [Principe and Kuo, 1994; Principe, Hsu, and Kuo, 1994; Principe, Kuo, and

Celebi, 1994] to the modeling of nonautonomous dynamical systems. The temporal state

evolution of an autonomous system is functionally dependent only on the system state,

but nonautonomous systems allow for an explicit dependence on an independent variable,

usually taken to be time [Jackson, 1989] or some function of time, in addition to the

system state. For this study, this independent variable is taken to be an external, or

exogenous driving signal, referred to as the control input. For an autonomous system, it

is reasonable to assume that the future behavior, or output, of the system can be predicted

over some finite interval from a finite number of observations of past outputs [Takens,

1980]. In contrast, predictions of the behavior of a nonautonomous system require

consideration of not only the past outputs in response to past inputs, but the future input

to the system as well.

It was also desired to develop a global representation of the underlying

nonautonomous dynamic system, that is, a model, or a collection of models that fit all of

the state space. This is in contrast to a local representation which is valid only in a








restricted region of the state space. The desired global representation may be achieved by

a single model if the underlying system is simple, but most complex, nonlinear dynamical

systems can only be represented in a localized region of the state space by a single model.

This naturally leads to the use of multiple local models to represent the global

characteristics of a system with some method employed to smoothly patch together the

local models [Principe and Wang, 1995] in a system identification context, or to switch

between models [Narendra, Balakrishnan, and Ciliz, 1995] in a control context.

Another prime motivation in the research was to develop models that would be

amenable for control of the underlying system, as opposed to models developed solely for

system identification. It was desired to have a system model that would provide a

computationally cost-effective means of determining the input signal to be applied to the

system in order to achieve a desired state. In this context, an approximate model that is

linear in the control input is more desirous than an exact model which has a nonlinear

dependence on the control [Narendra and Mukhopadhyay, 1997].

Finally, the combined modeling and control scheme was to be implemented in

software and experimental tests conducted using the actual dynamical system under

study.

Evolution of the Research


The dynamical system considered for this study is the 16-Foot Transonic Tunnel

at the NASA Langley Research Center in Hampton, Virginia. The NASA Langley

16-Foot Transonic Tunnel, simply referred to as the tunnel in the sequel, is driven by a

simple control input which provides the function of setting the desired output, which is








the Mach number, while compensating for any external disturbances. The task of

modeling and controlling the Mach number with an artificial neural network system was

undertaken with the vision to capture the underlying dynamics of an nonautonomous

system from observations of time-dependent, input-output data. After suitably extracting

the underlying dynamical model from the tunnel input-output data, predictions of the

response to future control inputs are based on this model. A control input sequence

which minimizes the error between the desired response and the predicted response, over

a reasonable time horizon, is then selected from a set of candidate input sequences. This

input sequence is finally applied as the control input to the wind tunnel.

The first major task was to find a suitable neural architecture for modeling the

wind tunnel dynamics based solely on input-output data. Our initial studies investigated

the use of several dynamic neural networks to identify the dynamics of the wind tunnel

response to control inputs, at one particular operating point [Principe and Motter, 1994].

The most promising architecture from this study was investigated further, using a single

global dynamic neural network for system identification over a wide range of operating

points [Motter and Principe, 1994]. This model was reasonably successful in predicting

the steady-state wind tunnel response at various operating points when driven by similar

control inputs. A refinement of this model came when the wind tunnel responses were

first clustered using a competitive neural network [Motter and Principe, 1995]. A

competitive neural network was used to cluster the tunnel responses at several operating

points to similar control inputs, thereby extracting pertinent features of the response. The

clustering of the wind tunnel dynamic responses provided a basis for developing a set of

predictors that collectively captured the dynamics of the wind tunnel response for a single








class of similar control inputs. At this point, it became clear that a significant

improvement in the prediction accuracy could be realized from an ensemble of local

models, each derived from a clustering of the tunnel dynamic responses.

The control input space was partitioned manually, based on experience and the

bang-zero-bang (+1, 0, and -1) permissible values of the control signal. If the control

input sequence is considered to be a p-component vector with each element having a

value of +1, -1, or zero, then there are 3 possible control sequences to be considered.

The idea was to partition the control input space by manually constructing representative

prototype vectors for the control sequence. The goal of this partitioning was to provide a

set of control inputs capable of driving the tunnel from one operating point to another,

regulating about a given operating point, rejecting disturbances, while eliminating control

sequences known to be experimentally of no practical interest. Limiting the number of

candidate controls to be evaluated by the predictive controller was a major consideration

in partitioning the control input space. Initially this partitioning was done with five

control input prototypes, but later, in the implementation of the experiment, the

partitioning was extended to nine control input prototype vectors, to provide the desired

control accuracy.

For each these control input classes, the tunnel Mach number responses were

clustered using Kohonen's self-organizing feature map (SOFM) [Kohonen; 1990, 1995].

The SOFM is a competitive neural architecture that imposes a topographic ordering of the

output neural field corresponding to features of the input patterns, which are in this case,

the Mach number responses. For prediction purposes, the SOFM's advantage is that the

topographic ordering imposes a similarity measure over the input neural field. This








similarity can be exploited in the construction of local linear models from the input neural

field corresponding to the winning output. The construction of local linear models

facilitated the evaluation of the wind tunnel response to a larger set of candidate controls

than could have been realized with multiple dynamic models.


Background


The 16-Foot Transonic Tunnel at the NASA Langley Research Center, Hampton,

Virginia, is a closed circuit, single-return, continuous-flow, atmospheric tunnel with a

Mach number capability from 0.20 to 1.30. When the tunnel began operation in

November 1941, it had a circular test section that was 16 feet in diameter and maximum

Mach number of 0.71 [Peddrew, 1981]. Numerous upgrades to both the test section and

drive system have expanded the test envelope of this facility. Currently, Mach numbers

up to 1.05 are achieved using the tunnel main drive fans only. Mach numbers from 1.05

to 1.3 require the combination of test section plenum suction with the tunnel fans. The

tunnel fans, 34 feet in diameter, are driven from 60 to 372 rpm by a 50 MW electric drive

system. An air removal system using a 30 MW compressor and 10-Foot diameter

butterfly valve provides test section plenum suction. At Mach numbers above 1.275, the

10-Foot valve is fully open and increases in Mach number are obtained from increased

power to the tunnel main drive fans. Figure 1 is an aerial view of the tunnel. Figure 2 is

a view of the tunnel test section with a model inserted. Figure 3 shows the arrangement

of the major components of the tunnel. Figure 4 shows a view from the inside of the

tunnel near the second set of turning vanes.





























Figure 1. Aerial View of the 16-Foot Transonic Tunnel


Figure 2. Test Section with model in place





7









I-l



c.) 0 0E

CD "0
1 .
(a a T







k I5 ^


o o
Ta (E
oA?

0 0







CD 0









-*--






-- g
a. c c rag
I> cs
* =II
.r_ v N






























































Figure 4. Inside View of the 16-Foot Tunnel downstream of the second set of turning
vanes








The test section Mach number, generally referred to as the Mach number, is

computed from a calibrated ratio of two measured quantities, the airstream stagnation

pressure, Ps,,,, and the plenum static pressure, Pc. These two measured quantities

are used to calculate the plenum Mach number. A tabulated wind-tunnel calibration

provides the correlation between the test section airstream Mach number and the plenum

Mach number. The relationship between the two measured pressures and the plenum

Mach number, M, is [John, 1984; Mercer et al., 1984]:



p,.= 1+ M-; M= -1 ; ,, ,=1.4. (1)





A large volume of test data relating the tunnel fan drive system control input (+1,

0, -1), and the Mach number, is available for nominal operating conditions over most of

the operating range. Data from a typical subsonic run is shown in Figure 5.

Ma .atl l*lber dga lubo C ont Icnp dunp g aI. subMa Uior


Figure 5. Mach Number and Tunnel Drive Control Inputs during a typical subsonic run








Control Challenges


The problem of controlling the Mach number at the 16-Foot Transonic Tunnel

presents the following challenges to any control scheme, including a human operator-in-

the-loop:

1. Both the linear and nonlinear characteristics of the tunnel dynamics vary

significantly over the operational range of the tunnel. The rate-limited slewing

of the tunnel Mach number varies by 50% over the subsonic range, as shown

in Table 1. Linearized models identified at individual subsonic operating

points contain a set of complex poles with damping ratios ranging from

0.4 0.7, and natural frequencies between 1/3 to 1/8 Hz. On the positive side,

the open-loop plant is stable, so the control problem is concerned mainly with

regulation about the desired set point

2. The control input to the tunnel fan drive system is bang-zero-bang

(+1 raise, 0 to maintain speed, -1 lower)

3. The effectiveness of the control input varies by a factor of five over the

nominal dynamic range

4. The effectiveness of the control input varies due to degradation of the drive

system components, replacement of components, and routine maintenance

5. There is transport lag (pure delay) that varies from 0.3 to 3 seconds over the

operational range

6. The Mach number varies with the temperature of the air for a fixed fan RPM








7. The dynamics can change dramatically and abruptly at any given operating

point from a particular combination of model attitude and Mach number. This

abnormal condition is referred to as a "blocked" tunnel condition

8. The effectiveness of the control input can abruptly change by an order of

magnitude for blocked conditions

9. The test section Mach number computed from pressure measurements is noisy

and nonstationary

10. The Mach number is to be controlled to within +/- 0.003 of set point

11. Research data is taken with the tunnel in an equilibrium condition, i.e. all

Mach number transients have decayed to a minimum, with zero control input

to the drive system

12. Power consumption is significant: 20 MW @ Mach 0.7, 80 MW @ Mach 1.3,

so the potential for reduction in operating costs is high.

Figure 6 shows a typical operating scenario, with the tunnel under control of an

expert human operator. The tunnel is being ramped up from a cold startup condition to a

subsonic Mach number of 0.95. A steady raise command from the operator drives the

Fan RPM up for approximately five minutes until the desired Mach number is attained.

Table 1 shows the variation of the rate-limited increase in Mach number. Once the Mach

number is within the 0.003 tolerance, the attitude of the aircraft model under test is

stepped through the desired range. For this particular test, the angle of attack was varied

directly with the pitch actuator. The tunnel operator is required to make frequent

corrective inputs to regulate the Mach number to within the 0.003 tolerance, primarily








due to the rising temperature of the air in the tunnel. The tunnel Mach number is required

to be within 0.003 of the set point at each of the angle of attack test values.

This operating scenario highlights the effect of unsteady temperature of the air on

the stability of the tunnel Mach number. The tunnel is initially at 85 degrees Fahrenheit,

and the temperature at the end of this test sequence is just over 150 degrees and still

rising. The rate of temperature rise while ramping to M = 0.95 exceeds 10 degrees per

minute. The rate of temperature rise decreases rapidly after the initial ramp, but still

exceeds one degree per minute at the end of this interval.


Variation of Mach number rate-limited increase while ramping up


t (seconds) M (Mach) AM (AM / At) *10-3

0 0.1119 -

30 0.2226 0.1107 3.69

60 0.3333 0.1107 3.69

90 0.4251 0.0918 3.06

120 0.5134 0.0883 2.94

150 0.5971 0.0837 2.79

180 0.6789 0.0818 2.73

210 0.7564 0.0775 2.58

240 0.8285 0.0721 2.40

270 0.9076 0.0791 2.63

282 0.9421 0.0345 2.87


Table 1.











ampiWn andrguta0ch 0.95






0.7






0.4

03 0




0 5 10 15
Thm (min)


Tin (min)


02





-2,sL

0 S 10
Tin.. (nrn)


Trm- (m)


Figure 6. Tunnel conditions during a typical run with steady ramping to the desired test

Mach number, M=0.95. The Mach number is to be held to within 0.003 of the desired

value while varying the angle of attack.







The relationship of Mach number to temperature is embedded in the definition of

Mach number [John, 1984]:

V V
M= -- -
a (2)

where V is velocity of the air, a is the speed of sound, y and R are constants for air. For a

constant air velocity, the variation of Mach number with temperature is:

dM 1 (M
3=2[T1) (3)
dT 2 (M)

with T in degrees Kelvin or Rankine.

At the conditions for this test

dM I (M) -0.000786
T 145,F 2 T ) F

which corresponds to a 0.003 decrease in Mach number for a 3.8 degree F increase in

temperature.

Figure 7 illustrates in greater detail the last 3.5 minutes of the test. The test point

taken at an angle of attack of one degree takes more than two minutes to acquire. Four

corrective inputs applied over a period of more than a minute are required to regulate the

Mach number to just barely within the tolerance required for this test point. The next

increase in the angle of attack drives the Mach number out of tolerance, which is

compensated for by the operator with a longer duration corrective input. During this

interval, the effect of the moving the model is relatively small compared to the effect of

the rising temperature, but the two can act in combination as illustrated in this example.











Sal poln negulaon a M=o


Pitc Acl, i
PkleA~njal







-07










-1.31
-1 4



0 06 1 S15 2 2.S 3 35
Tkn. (mal)


Figure 7. Out of tolerance Mach number extends the test duration during the last 3.5

minutes of the test @ M=0.95.


Ang of Allck








Figure 8, from a different test, illustrates the effect of large changes in angle of

attack disturbing the Mach number under relatively steady temperature conditions. The

large change in angle of attack from 5 to 15 degrees in the middle of the test produces a

Mach number disturbance of approximately 0.02, or seven times the required tolerance.

Here the variation in temperature accounts for only six percent of the total disturbance.

The expert operator's response is quite effective in compensating for this disturbance,

whereas a non-adaptive automatic controller tuned to the nominal, unblocked dynamics

would be unacceptably slow in compensating for this type of disturbance. The

effectiveness of the control input decreases abruptly as the model is moved from an angle

of attack of five degrees to an angle of attack of fifteen, twenty and twenty-five degrees,

respectively. Table 2 lists the changes of control input effectiveness from the nominal

condition at five degrees as the model angle of attack is increased. For each large step

change in the angle of attack (AOA), the corresponding disturbance is AM. The control

effort applied to compensate for the disturbance I u, which is the sample-by-sample sum

of the control inputs required to return the Mach number to within tolerance. The

effectiveness of the control input is evaluated for each of these cases as AM/ Eu, simply

the ratio of the change in Mach number over the cumulative control effort required to

regulate the Mach number. This value is seen to vary by more than an order of magnitude

over the test conditions listed in the table. This is a prime example of the kind of

variation that motivates the need for multiple models to represent rapidly varying

conditions of the plant to be controlled.































































Figure 8. Regulating the Mach number atM = 0.8 with large disturbances from the
model Angle of Attack


Pm (.hn)








AOA AM u AM/I u

0-5 0.005 1 50

5-15 0.020 27 7.41

15-20 0.017 39 4.36

20-25 0.010 29 3.45


Table 2. Changes in control input effectiveness for blocked conditions




Experimental Framework


The experimental framework that evolved was essentially a predictive control

scheme that used multiple models of the plant with switching. The controller switches

between multiple, SOFM-based models which, collectively, describe the global

input-output behavior of the tunnel. The tunnel response to a set of candidate controls is

predicted p steps ahead, using the currently selected model. The overall system, which

will be referred to in the sequel as the PMMSC, for Predictive Multiple Model Switching

Controller, is shown in Figure 9. It is composed of the following major functions:

1. The recent control input, u(k 1),u(k 2),...,u(k m), is clustered on a set of

prototype control inputs which will choose one of the Kohonen self-

organizing feature maps (SOFM)

2. The selected SOFM identifies the local dynamics of the tunnel based on the

past n + 1 Mach number measurements, M(k), M(k -1)... M(k -n), and

chooses a winning processing element (PE)








3. A linear predictor associated with each PE predicts the Mach number response

p steps into the future for each of the candidate controls

4. The predicted effectiveness of the candidate control inputs is evaluated over

the last (p 1) steps of the p steps-ahead predictions

5. The control input that provides the best response with respect to the Mach

number set point is chosen as the next control, u(k).

The local model associated with the winning PE captures the dynamical regime of

the wind tunnel. The controller still must decide what is the most appropriate control

input to meet the set point specification. The controller sends candidate input sequences

for p-step ahead prediction to the predictor of the winning node. The controller evaluates

the relative effectiveness of the candidate control inputs in reducing the error between the

predicted Mach number sequence, Mp, and desired Mach number, Msp. This is

accomplished by a suitable metric, the Euclidean norm over the error, I~MP -Msp1

where Mp = M(k+I+l),M(k+ +2),... M(k+p) and Ms, is a (p-l) length


constant vector of Msp. Finally, the control input that provides the smallest error is sent

to the wind tunnel fan control.

Overview of the Dissertation


The dissertation is composed of six chapters. Chapter 2 will survey the literature.

Chapter 3 will focus on the modeling of the tunnel dynamics. Chapter 4 explains the

development of the predictive controller from the local linear models. Chapter 5

describes the experimental setup and results from controlling and modeling the tunnel






responses during operational research runs. Chapter 6 will summarize the results and
indicate directions for future research.


Mach number, M (k)




A_


U(k + 1),...,u{k + p)
mmmmmmmmm


Figure 9. Experimental Framework with PMMSC


W-11,


i^ ^L1

VMIW \ __ "<















CHAPTER 2
REVIEW OF LITERATURE

Introduction


Two major ideas from the existing literature embodied in our system are the Self

Organizing Feature Map (SOFM), credited to Kohonen [Kohonen, 1995], and control

using multiple models and switching, credited to Narendra [Narendra, Li, and Cabrera,

1994]. In Narendra's multiple model control scheme, an external switching scheme is

used to select the model to be used at any given instant of time. In the experiment

described in this dissertation, the SOFM is used as the modeling infrastructure, with

selection of the model done by the activity of the output neural field or winner. A

description of both of these topics, as well as a brief review of adaptive control, SOFM

applications to control, and more general review of applications of neural networks to

control follow.

Self-Organizing Feature Map


The self-organizing feature map (SOFM) was adopted as the neural architecture

for the experiment. The SOFM was chosen based on its ability to transform an incoming

signal of arbitrary dimension into a lower dimensional, discrete, topologically ordered

map, one dimensional in this case. The spatial location of the neurons, arranged in a one

dimensional lattice, or linear array, corresponds to intrinsic features of the input signal.








The SOFM belongs to the class of artificial neural networks that use competitive

or unsupervised learning. In contrast to supervised learning, the SOFM input-output

behavior is not learned from a set of training examples which specify the desired output

y E R', for a given input x E Rn, where the parameters of the network are adjusted by

the backpropogation algorithm [Rumelhart, Hinton, and Williams, 1986; Werbos, 1990].

In feedback networks [Hopfield, 1982], the other major category of artificial neural

networks, the input defines an initial state of activity of a feedback system which settles

to a final asymptotic state that represents the response to the given input. In the SOFM,

however, neurons compete to respond to the input signal, with the result that only one

output neuron is fired or activated. The output neuron activated in response to a

particular input is called the winner, while all the other neurons are inhibited,

representing a winner-take-all (WTA) structure. During the training phase, the SOFM

becomes topologically ordered by adapting the weights not only of the winner, but those

of the neighboring neurons as well. This is inspired by lateral inhibitory feedback in

biological neurons [Willshaw and von der Malsburg, 1976], but implemented in the

SOFM by a computational shortcut, referred to as the neighborhood function. Not only

do the individual neurons in the SOFM become specifically tuned to input patterns by

means of this emulation of lateral feedback among neighboring units, but the locations of

responses become ordered along the coordinates of the map, corresponding to intrinsic

features of the input.








Let the input be a vector x e R" :

x= [x,x2.... ,x, T. (4)

With each neuron j there corresponds a vector of synaptic weights we R":

W, =[wj w,2 ...., w'. (5)

The winner is identified by the index i(x) that corresponds to the neuron whose synaptic

weights are the best match to the input x:

i(x)= arg min x-w ,j, j=1,2,...,N (6)


where | denotes the Euclidean norm. Thus, the response of the network can be

considered to be the index of the winning neuron, representing its location, or,

equivalently, the synaptic weight vector that is closest to the input vector in a Euclidean

sense [Haykin, 1994]. In this experiment, the latter interpretation of the network response

is more appropriate.

xeR"


W,

i(x)=arg min -x-w j=1,2,...,N
/(x)=argmin x-w^ j=l,2w...,N
j N


Figure 10. SOFM with a one-dimensional array of neurons







For the formation of an ordered map, it is crucial that the weights of the winner

are not updated independently from the weights of the other neurons, as is the case of

other competitive learning or vector quantization schemes. In the SOFM, the adaptation

or updating of the weight vectors is done over a topologically related subset, resulting in

weight vectors that are ordered along the output dimension of the network. At each

learning step, the network is presented a sample x, drawn from the input distribution. The

winner is determined as specified in (6), and a neighborhood set N,,,, identifies the

neurons around the winner that will be updated as well. The width or radius of N ,,) is

usually varied over the training phase [Kohonen, 1990]. To achieve good global

ordering, N,(,, is made very wide at the beginning of the training, on the order of the

one-half the map, and then shrinks monotonically as the training progresses. The

rationale for this [Kohonen, 1990] is that the wide initial N,(x), corresponding to a coarse

spatial resolution in the learning process, first induces a rough global ordering over the

weight vectors. Then, as the N,(,) narrows, the spatial resolution of the map improves

without destroying the acquired global order. Thus the use of the neighborhood function

emulates the formation of a localized response in biological neurons by initially applying

a strong positive lateral feedback corresponding to an ordering phase, followed by

negative lateral feedback which corresponds to a convergence phase.

The updating of the weight vectors in discrete time proceeds as :


w( (k) + a(k)[x(k)-w(k)] if jeN( (7)
w,(k) if jeN,&)








with a(k)a scalar learning rate parameter, 0
stochastic approximation processes [Robbins and Monroe, 1951], and should decrease

over the training interval.



Practical Aspects for the Application of the SOFM Algorithm

1. The initial weight vectors w, (0) are set to random values.

2. Samples x are drawn from the input distribution and presented to the network.

3. The best matching neuron is determined by (6).

4. The weight vectors of all the neurons are updated by (7).

5. Steps 2 through 4 are repeated until no noticeable changes are observed.

The "rules of thumb" are that for approximately the first 1000 steps, a(n) should

be close to unity, then decrease monotonically. The actual rule for the decrease is not

critical. The ordering of the map takes place during this period. The neighborhood

function N,,, should be fairly wide initially, perhaps on the order of half the map, and

decrease linearly to one unit during this ordering phase. After the first thousand steps, a

much longer convergence or fine-adjustment phase of the training proceeds with the

learning rate a(n) slowly decreased to a value near 0.01. During this phase the

neighborhood function may contain the nearest neighbors of the winner, with the final

stages of the convergence phase updating only the winner. A rule of thumb for the

number of steps to achieve convergence is at least 500 times the number of network units.

The following figures illustrate an example of training an SOFM used in the

experiment. The inputs to the SOFM are a 50 sample window of the Mach number








response. There were 155 exemplars in the training set, shown in Figure 11. The SOFM

consisted of 20 neurons, arranged in a linear array, similar to Figure 10, shown earlier.

The SOFM weights were adjusted during 10,000 presentations of the training set, with

the learning rate, a(n), and neighborhood function N,,i varied as shown in Figure 12.

The SOFM is shown at 100, 500, and 1000 training cycles, with the converged SOFM,

after 10,000 training cycles, in Figure 13. The converged SOFM provides a smooth

organization of the weights in the neural field, in contrast with the input patterns for

training. The distribution of the training inputs among the converged SOFM clusters is

shown in Figure 14.

Inputs for trading te SOFM


Exemplar #


Figure 11. Input exemplars for training the example SOFM











_Figure 12. Leaing rate and Neighborhood function during training

09 iB



o0 14



Sos ji

04 a 1







0 10 00 00 4000 5000 60 7000 9000 5n 0, o 0oo 1 2 30 4o 5 70e 0 BO 00 10000




Figure 12. Learning rate and Neighborhood function during training




so0 tfu 70o 005I so5e 0 401101r I so .g


SOFM er 100s0 mtrin 0


Figure 13. SOFM during various points in the training








Distribution of training ensemble inputs to SOFM outputs


Figure 14. Distribution of training inputs among 20 converged SOFM clusters


Magnification Factor

The input distribution of the vectors x, or the multidimensional probability density

function (pdf) of x, p(x), is represented by the total N neurons in the output layer of the

SOFM. The input vectors x are drawn from an n-dimensional input space X. The pdf of

x, integrated over all of X, must equal unity:


jp(x)dx =


The corresponding density of neurons in the output layer of the SOFM is referred to as

the magnification factor, m(x), defined as the number of neurons in a small volume dx of

the input space X. The integral of the magnification factor over the entire input space,

must equal the total number of neurons N:


Jm(x)dx = N (9)


N -155












5 1 1 15 20 25
Cluster #








For the SOFM to match the input density exactly, the magnification factor must

be directly proportional to the input pdf:

m(x) p(x) (10)

Si and Lin [1997], have recently shown, for multidimensional input, the


converged SOFM weights have a magnification factor proportional to p(x)+2.

Kohonen [1995] makes the point that in most practical applications that the input data

vectors have high dimensionality, on the order of dozens to hundreds, and compares the

result to classical vector quantization (VQ), where the asymptotic point density is


proportional to p(x)'+2 as well. For this experiment, the input dimension n, is n = 50, so

it was expected that the input distribution would be well matched by the locations of the

output neurons of the SOFM. From a control viewpoint, this has the beneficial effect of

providing a higher density of neurons in regions of the input space where the statistical

frequency of input features is correspondingly higher, with fewer neurons assigned to

regions of the input space with features of lower statistical frequency.




Applications of the SOFM


Three major practical application areas suggested by Kohonen [Kohonen, 1995]

where the SOFM could be used effectively are:

1) Industrial and other instrumentation, for both monitoring and control

2) Medical applications, for diagnostic methods, prostheses, and modeling








3) Telecommunications, for allocation of resources to networks, transmission

channel equalization, and adaptive equalization.

A survey of the diverse applications of the SOFM [Kohonen, 1995] highlights the

following areas: machine vision and image analysis, optical character and script reading,

speech analysis and recognition, acoustic and musical studies, signal processing and radar

measurements, telecommunications, industrial and other real-world measurements,

process control, robotics, chemistry, physics, electronic-circuit design, medical

applications without image processing, data processing, linguistic and AI problems,

mathematical problems, and neurophysiological research. The reported applications in

process control were of interest, but, for the most part, the research focused on monitoring

the process state rather than effecting some control action. Some general problems

addressed in this area are: identification of process state [Kasslin, Kangas, and Torkkola,

1992], process error detection [Alander et al., 1991], and diagnosis of machine vibrations

[Wu et al., 1991]. Some specific examples of industrial applications are: monitoring

paper machine quality [Lampinen and Taipale, 1994], flow regime identification [Cai,

Toral, and Qiu, 1993], grading of beer quality [Cai, 1994], and estimation of torque in

switched reluctance motors [Garside et al., 1992]. In a more recent application to process

control, [Matthews and Warwick, 1995] the SOFM was used for separating fault types

and monitoring the process state. In [Warwick, 1996] the SOFM is proposed again as a

classifier for fault indications as opposed to a system identification tool.

One of the most control-specific applications of the SOFM reported in the

literature is the visuomotor control of a robot arm by [Ritter, Martinetz, and Schulten,

1992]. In this application, the SOFM is used as a look-up table, where the input pattern,








identified by the "winner", specifies an SOFM location associated with specified values

of control parameters, which were learned adaptively.

The two dimensional coordinates, x, and x,, of a target point in the image planes

of two cameras were combined into a four-dimensional, stereoscopic input vector x and

used as the input to the SOFM. A three-dimensional SOFM was used to form the spatial

representation of the target point. The three joint angles, one about the vertical axis for

motion in the horizontal plane, and two for motion in the vertical plane, comprise a

configuration vector 8= [01, 0 3 ]. The basic goal of their approach was to find the

transformation 0(x) that would bring the tip of the robot arm to the target point, where

the cameras can get the observation x. The configuration vector is determined by a

linearization about the origin determined by the "winner" location c:

0 =Ac(x-m,)+b,. (11)

Here b, is the configuration vector corresponding to the location m,, A, is the

3x4 Jacobian matrix, mr is from the weights of the SOFM winner, and (11) gives the

first two terms of the Taylor series expansion of O(x) around mc. Linearization is

carried out around m, and is valid in the whole Voronoi set of x values around mn.

Ritter et al., developed a learning scheme where the control parameters A,, b, were

updated simultaneously with the formulation of the SOFM. The importance of the SOFM

in their problem was the discretization of the input space, in particular, the allocation of

the configuration vectors, be, to regions of the input vectors, x, having a higher density of

lattice points where the control must be more accurate.








For our application, the SOFM discretizes an n-dimensional space composed of

output sequences of the system, y(k), y(k-1), ... y(k-(n-1)), which are considered to be the

responses of the system to a prototype control input u(k-1), u(k-2), ... u(k-m). Thus, the

prototype input is the control parameter associated with all the nodes in the lattice, which

is here, one-dimensional corresponding to the single control input to the system, u. In our

application, the linearization is done around the "winner" to predict responses to

candidate controls:

M, = A(u -uc)+ M, (12)

where M, is the winner, Ac, is the Jacobian, derived directly from the SOFM, u, is the

control prototype associated with the SOFM and u, is one of the candidate control

sequences. In our application we replace the slow adjustment of control parameters by an

external scheme, as in Ritter's application, with the ability to switch, at discrete intervals,

among the discrete local linear models associated with each node in the SOFM. This

highlights the difference between a slowly adaptive control scheme, and our application,

which is designed to switch rapidly to accommodate abrupt changes in the system

characteristics.



A Brief Review of Adaptive Control


The adaptive identification and control of dynamical systems has been

extensively developed for linear time-invariant systems with unknown parameters over

the past three decades. The development of adaptive control for linear systems is a

logical consequence of the diversity of mathematical tools available for the analysis of the








properties of linear systems. The choice of parameterization of the plant model and the

controller in such problems were naturally based on results from linear systems theory. In

the 1980's, the theory of adaptive control focused on the design of stable adaptive control

laws which are robust in the presence of unmodeled disturbances, time-varying

parameters and unmodeled dynamics [Narendra and Annaswamy, 1989]. A good

understanding exists for the design of stable adaptive controllers for linear systems with

unknown parameters.

Two major approaches to the adaptive control of linear systems, direct and

indirect, have developed over the past twenty years. The direct approach seeks to

minimize some performance criteria, usually based on the error between the output of the

system and some desired output, by direct adjustment of the controller parameters. The

indirect approach attempts to explicitly identify the dynamics of the system to be

controlled, and then modifies the parameters of the controller based on this identification.

Both of these methods traditionally used a single, linear, parameterized model of the

system being controlled, or plant. One of the major drawbacks of both these approaches,

is the time required for adaptation of the controller parameters in the direct case, or the

identification of the parameters of the plant in the indirect case, to achieve the desired

control. This is particularly troublesome when the method is to be applied on-line to

control processes whose dynamic behavior is known to change abruptly.

As a result of the shortcomings mentioned above, a more recent approach to the

adaptive control of an uncertain linear time-invariant system (LTI), is the use of multiple

models with switching [Narendra and Balakrishnan, 1997]. Although this was not the

first time that the individual concepts of multiple models, with switching and on-line








tuning of some models, had been proposed, this framework proposed to improve the

transient response of adaptive systems in a stable fashion [Narendra and Balakrishnan,

1994]. The recent results present the problem in the context of model reference adaptive

control (MRAC) [Narendra and Annaswamy, 1989] of a LTI system, and the principle

results are the proofs of stability for various assumptions on the coverage of the space

Sc %~2n of the plant parameters by either the initial parameter values of a set of adaptive

models or the parameters of a set of fixed models, and various combinations of both fixed

and adaptive models. The multiple model and switching framework is quite general and

applies to both linear and nonlinear systems, but stability results are only currently

available for the linear time-invariant plants.

The development of nonlinear adaptive control has for the most part, paralleled

the linear case, usually with even more restrictive assumptions about plant than the linear

case. The usual approach is to perform a linearization of the plant model around some

point in the state space, determine the localized characteristics of the linearized system,

and the region in the state space where the linearization is valid.

Linear Adaptive Control

A single input-single output (SISO) linear time-invariant system with unknown

parameters, described by the state equations:

x(k + 1) =Ax(k) +bu(k)
y(k) = cx(k) (13)








corresponds to the case where some or all of the parameters of the matrix A and

vectors b and c are unknown. Alternately, if the system is described by the n'* order

difference equation:

n-1 nM-
y(k + 1) = a,y(k-i) + fu(k j) (14)
i=0 j=o



where u(k) and y(k) represent the input and output respectively at time k, the

parameters a, and /, are assumed to be unknown. The objective then is to determine the

control input u(k) so the output y(k) behaves in some desired fashion.

The transfer function, W, (z), of the plant described by equation (14) is:


AoZ-+Azn-2 + ... + ,-
W, (z) = z (15)




The order of the system is n and if f, # 0, then the relative degree is one. If,

however, A= = A = A =...= id-2 = 0 and f_, # 0, then the relative degree is d and the

input u(k) affects the output at time instants greater than or equal to k+d. It is best to first

consider the case when the relative degree is one, then extend the results to the case when

the relative degree is greater than one.

A bounded signal y (k) is specified as the desired output of the plant and the

input u(*) is to be determined. Alternately, u(k) at instant k has to be chosen so that


lim y(k)-y'(k) =0.
k-~-










In model reference adaptive control (MRAC), y'(k) is generally chosen as the

output of a reference model. The simplest reference model that can be satisfied by (16)

above is z-d where d is the relative degree of the plant. For the case where the relative

degree is one, the reference input r(k) to the reference model is y'(k + 1) and is assumed

to be known at time k.

For the non-adaptive problem, if the plant is described by equation (14) and the

parameters a and f are known, the control law can be chosen as :


u(k)- a y(k -i)- Pu(k- j)+y*(k+l) (17)
Ai) -8= = ju(k IJ7



and then the output y(k) = y(k). The control input is merely a linear combination of n

past values of the input and output as well as the desired signal at instant k+l, and that the

output of the plant converges to the desired output in one instant.

For the adaptive case where the parameters a and P are assumed constant but

unknown, the indirect approach can be employed and requires the estimation of the

parameters a and p. If d, (k) and /3 (k) represent the estimates of a and f respectively,

these can be used to compute the control input. However, it is no longer obvious that the

overall system will be stable and that the condition (16) will be satisfied. This problem

was resolved for both continuous-time and discrete-time systems in 1980 [Narendra, Lin,

and Valavani, 1980; Morse, 1980]. However the stability of the overall system in the

discrete case requires the following assumptions about the plant transfer function:







1) An upper bound on the order n is known

2) The relative degree of the plant is known

3) The sign of fA as well as an upper bound on the absolute value of AF are

known

4) The zeroes of the plant transfer function are within the unit circle (minimum

phase condition).

Given these assumptions, stable adaptive laws for the adjustment of the estimates

a, (k) and f, (k) result in a similar control law:


u(k)=-[-- d,y(k-i)- u(k-j)+y*(k+l1) (18)




where the output y(k) follows y'(k) asymptotically.



Control Using Multiple Models and Switching


The multiple model structure with switching has been proposed by [Narendra et

al.; 1994, 1995] when the overall system is required to operate in multiple environments.

Sudden changes in parameter values, failures of sensors or subsystems, and external

disturbances taken to be the output of an unforced stable dynamical system, can be

considered as different environments a control system may be required to cope with. In

these cases, the need to use multiple models arises naturally, since a different

mathematical model may be needed to represent the behavior of the plant in each of the

environments.








The need for multiple models in the control of dynamic systems is further

elaborated by [Narendra, 1996] as:

1) Many physical systems can be represented by interpolating between local

models. Gain scheduling is the control paradigm based on this concept

2) Multiple models may be needed to detect different changes in the plant and

initiate the appropriate control action

3) In some cases, all the information concerning the plant, such as the order or

the relative degree, may not be available to compute the input. Multiple

models may be needed to obtain the appropriate information

4) The advantages of individual models may be combined in a multiple model

controller. One model may assure stability, while another heuristically

designed may provide better performance. A proper combination of the two

may result in a stable system with better performance.

The architecture of the Narendra's multiple model switching controller is shown in Figure

15. 1,12,...., I are N predictive models of the plant which have been obtained by

observing the system over a long period of time. C,, C2,..., C are the corresponding

controllers, designed off-line and stored in memory. If the plant output is y(k) and the

output of model Ij is ,j (k), the output error is defined as ej = j, (k)- y(k). Based on

some performance index J(ej), evaluated for j= 1,2,... N, the model to be used at any

instant is chosen. If J,(k)=mini J(ej(k)), the model Ii and the corresponding

controller C, are chosen at instant k. This corresponds to the switching part of the

scheme. The implementation of the switching scheme employs some hysteresis to








prevent arbitrarily fast switching between models. In a more recent paper [Narendra and

Balakrishnan, 1997], stability results for an all-fixed models controller was established

for linear systems under some mild assumptions. In particular, it is shown that if there is

at least one model that is sufficiently close to the actual plant and there is a non-zero

waiting time between switches from one model to another, then the overall system is

stable, given that each fixed model is stabilized by its corresponding controller.




Y2 ee
Model / +_




Un Model 1,

u y e,

u2 Control Error

Desired Output y"


Controller C2


Controller C,
-- I Controller C, [




Figure 15. Structure of the multiple model control with switching


An even more recent paper [Narendra and Mukhopadhyay, 1997] introduces two

classes of approximate non-linear input-output models which reduce the computational

complexity of designing a controller based on the fact that the approximate models are








linear in the control input. This was essentially the approach taken in this experiment,

where the converged SOFM provides multiple, approximate models of the input-output

behavior of the plant for a given class of input. These approximate models were then

used as the basis for linear predictions of the response to a set of control candidates to

determine the control input that minimized the error between the predicted output and the

desired output.

The development of these models begins by considering the representation of an

arbitrary, discrete non-linear dynamical system using state equations:

x: x(k+l)=f[x(k),u(k)]
y(k)=h[x(k)] (19)

where {u(k)},{x(k)), and {y(k)} are discrete-time sequences with

x(k)ER'", u(k)e9, y(k)E9) f:9V" x9 -+ 9" ,h:9t" 91,and f,hE~C The origin

is assumed to be an equilibrium state of (4), hence f(0,0)=0 The linearization of L of

I is described by the linear state equations:

1, : x(k+l)=Ax(k)+ bu(k)
y(k)= cx(k) (20)

where the (n x n) matrix A and the (n x 1) and (1 x n) vectors b and c are defined by

df(x,u) A df(x,u)l dh(x)
=A [=b d=c (21)

Given this parameterization, the general state of knowledge about the system

I can fall into one of the following categories:

1) fand h are known, and the state x(k) is accessible

2) fand h are unknown, and the state x(k) is accessible








3) fand h are unknown, and only the input u(k) and the output y(k) are accessible.

The third case is the one of interest here, where system identification and control have to

be carried out using only input-output data.



Other Applications of Neural Networks for Control


Three recently reported neural network applications for control appeared in the

July 1997 edition of the IEEE Transactions on Neural Networks. The first paper,

"Reliable Roll Force Prediction in Cold Mill Using Multiple Neural Networks" [Cho,

Cho, and Yoon, 1997] reported the use of multilayer perceptrons to predict the roll force

and a corrective coefficient used to improve prediction accuracy by 30-50 % compared to

an existing mathematical model used in the cold rolling mill process for steel. The

second paper, "Dynamic Neural Control for a Plasma Etch Process" [Card, Snidermann,

and Klimasauskas, 1997] described the use of a cascade (feedforward) neural network

and a policy-iteration optimization scheme to provide suggested process setpoints for

recovery from long-term drift in equipment used in the plasma etch process. The

combined optimization scheme suggested "reasonable low cost solutions" for what were

considered out-of-control situations. The third paper, "Neural Intelligent Control for a

Steel Plant" [Bloch et al., 1997] suggests incorporating the skill of the human operators in

neural models, at various levels of control. A feedforward multilayer perception is

developed as a model of the annealing furnace, from which a static inverse model is

derived. None of the three papers had any experimental results from actually employing

the neural-based control to the targeted process.








The most specific reference citing the use of neural networks for wind tunnel

control was [Buggele and Decker, 1994] where neural networks where used to interpret

shadowgraph images, a type of flow visualization, in order to tune parameters in existing

controllers. They concluded that their exercise was too complicated to demonstrate

neural-net automation of wind-tunnel operations. Another reference citing the use of

predictive control of Mach number at the National Aerospace Laboratory in Amsterdam,

The Netherlands, [Soeterboek et al., 1991] demonstrated a 30-60% overall performance

improvement over the conventional controller normally used. Their results were based on

a p-step ahead prediction scheme, using a single operating point model (Mach 0.8), scaled

to accommodate small variations in operating point (Mach 0.7 to 0.9).

In [Cooper et al., 1992], a vector quantizing neural classifier is used to identify

process error due to both step and oscillating disturbances and adapt a single gain

parameter in a simulated continuous stirred tank reactor (CSTR). Their approach

demonstrated the ability of such a classifier to distinguish between the resulting error

transients associated with these disturbances and adapt the gain of the closed-loop system

to reduce the effect of the disturbances.

An overview of manufacturing applications of neural networks [May, 1994],

reports positive results of researchers at DuPont Electronics and AT&T Bell Laboratories

in plasma etch modeling for semiconductor manufacturing. Arc welding, machining

operations, color printing, and linear accelerator beam positioning are given as examples

of successful process control applications of neural network based control. "Neural nets

are well-suited to process control since they can be used to build predictive models from

multivariate sensor data generated by process monitors."














CHAPTER 3
MODELLING THE TUNNEL DYNAMICS

Introduction


In the opening chapter, it was stated that the task of controlling the Mach number

in the tunnel was undertaken with a vision to capture the underlying dynamics of a

nonautonomous system from observations of time-dependent, input-output data. The

motivation for this approach came from previous work by Principe and Wang [1995],

using the self-organizing feature map as the infrastructure for local dynamic modeling of

chaotic time series. Their work focused on modeling autonomous systems, that is

systems where the state trajectory evolves without an external, or exogenous input signal

driving the trajectory from one region to another in the state space. That work is adapted

here to provide localized predictions of the system response, p steps ahead, to a

predetermined set of input or control sequences which will drive the system toward the

desired region of operation.

The assumption is that the state of the underlying nonautonomous system can be

described as a differential equation of the form:

dx(t)
dt f(x(t),u(t)) (22)

where x(t) are the system states, u(t), the control signal, is an exogenous input to the

system, and f is the vector field that maps a Cartesian product of the state space, S, and








the control space, C, Sx C c W" x S, to a tangent space Tc 9". If a closed-form

solution for (22) exists, that is : (: S x C S, then for a given initial condition x(0) and

u(t) specified for all t, ((x(0),u(t)), represents a state-space trajectory of the system, or

system flow.

For an autonomous system, there is no exogenous u(t), and the evolution of the

system is assumed to be described by :

dx(t)
dt f(x(t)) (23)

Often, at this point the exogenous input u(t) is expressed as a function of the states:

u(t)=g(x(t)) (24)

whereby the nonautonomous system becomes autonomous. This is particularly useful for

considering the stability characteristics of the system under the influence of a

state-dependent, or state-feedback, signal u(t) as in (24) above. This will be elaborated

upon in the appendix to gain some insight into the stability of the overall system. The

approach in this chapter, however, will be to model the system response to a set of

candidate control sequences applied as a function of time over a specified interval.

The representation of an arbitrary, discrete non-linear dynamical system using

state equations was stated in Chapter 2, repeated here for convenience:

Z: x(k+l)=f[x(k),u(k)]
y(k)=h[x(k)] (25)

where {u(k) }, {x(k)}, and {y(k) } are discrete-time sequences with

x(k)E"n, u(k)e9,y(k)E91, f:9V"x9t n,h:9 -4 9,and f,heC" Herefis a








map from the space of system states and input to the space of system states

9t" x91 9t", and h is a map from the space of system states to the output 91" 9 .

Our goal here is to determine the system output y(k), over p steps into the future,

in response to the application of a set of candidate control sequences U, where :

Uc, =[u,, (k) u (k + 1) ... u (k + p-) ] (26)

is the ith candidate control sequence, and:

Mp =[y, (k +1) y (k + 2) ... y (k + p)] (27)

is the predicted response from the ith candidate control sequence.




Review of Local Dynamic Modeling with SOFM


As stated earlier, the previous work by Principe and Wang [1995] provided the

starting point for the modeling architecture. Their objective was to construct a neural

architecture capable of capturing the underlying dynamics of a chaotic time series. They

employed the SOFM as the modeling infrastructure based on the following observations:

1) The SOFM is a localized representation of a signal constructed through

competitive learning

2) The converged neural field bears a stronger global resemblance to the input

space than other competitive learning, due to the neighborhood function

3) The positioning of each neuron is more strictly constrained by the overall

statistical distribution of the signal, which helps to smooth out the irregular

spacing of local data samples in the state space.








Their basic idea was to embed the given input space into a compact neural field

through the Kohonen SOFM algorithm. Then a simple model estimation process was

performed to construct the linearized local models for each response region. The global

description of the dynamics was composed of all these local models pieced together. The

whole process was composed of two separate procedures: the embedding process of the

input space into the neural field followed by the local model estimation.

Their architecture was composed of three layers: input layer x, neural field layer

A, and the layer of local linear models F(x) as shown in Figure 16. The time series was

embedded in a state space to create a state vector x. The function i (x) was realized by

the SOFM. That is to say that the input was fully connected to the nodes of the second

layer through a set of weight vectors w,, where the winner-takes-all neuron was

identified by the competition. Each neuron in the neural field layer corresponded to a

specific processor F :[a ,b, ], which represented the linear approximation of the local

dynamics.

In this architecture, the SOFM performed two major functions: the positioning of

the local models in the state space, and the identification of the matched local model for

the current input state x. The first function is accomplished during the training phase of

the SOFM, while the second is accomplished during the modeling phase. The

construction of the overall architecture was composed of three consecutive steps:

reconstruction of the state space, mapping the state space in the neural field, and

estimation of local linear predictors.








Approximation F(x)
of Dynamics






Neural Field 0000000
A 0000000



Competitive
Selection i'(x)
Input x
Figure 16. The SOFM-based Modeling Architecture for Time Series


Reconstruction of the state space from the training signal. Following the

approach by [Takens, 1980], a sequence of d+ 1 dimensional state vectors

[x(n) x(n + )] was created from the given training time series, where

x(n)=[x(n-(d 1)), x(n-(d 2)r),...,x(n)] and r is the appropriate time delay

where d >d, and dA is the dimension of the underlying dynamical process.

Mapping the state space in the neuralfield. This step was accomplished via the

Kohonen learning process. With each vector-scalar pair [x(n) x(n + ) ] presented as the

input to the network, the Kohonen algorithm adaptively discretizes the continuous input

space XcRd into a set of disjoint cells A to construct the mapping D:X A. This

process continues until the learning rate decreases close to zero and the neighborhood

function covers one unit. After learning, the neural field representation A of the input








space X via the constructed mapping relationship ( is formed in terms of disjoint units

topologically organized in the output space.

Estimation of the locally linear predictors. For each neuron u, eA, its local

linear predictor in terms of [a, ,b, ] is estimated based on a, cA, which is a set of L,

neurons in the neighborhood of u, including u, itself. Each of them has a corresponding

weight vector [wi w, (d + )]T eRd where w r =[w (1), w (2),...,w, (d)]. The

local prediction model [a, ,b, ] is fitted in the least-square sense to the set of weights in

a,:

w, (d + 1) = b + a'wT, (28)

After the above construction procedure, a modeling network is obtained with a

global functional map composed of a set of local linear equations

x(n +1) = (x(n))= a, x(n) + b, (29)

where i is the winner-take-all neuron identified by competition in (6).



Modifications for SOFM-based Predictive Control


From (25), consider the output of the nonlinear system I :

y(k) = h[x(k)]=- ,[x(k)]
y(k + 1) = h[f(x(k),u(k))]E_,2[x(k),u(k)]
y(k + 2) = h[f( f(x(k),u(k)), u(k + 1)] =s3[x(k),u(k),u(k + 1)] (30)


y(k+n) = h o f [.,.]= +, [x(k),u(k),u(k +1)...,u(k+n -1)]








where f "is an n-times iterated composition of f Denoting the sequence

y(k+l),..., y(k + n) by Y,(k) and the sequence u(k), u(k+1),..., u(k + n 1) as U, (k),

(30) can be expressed as :

T[x(k),U,(k)]= Y (k) (31)

For SOFM-based predictive control, the thesis is that a set of feature maps can,

collectively, be a global representation of these n-times iterated compositions off, where

an SOFM winner represents the localized response of the system to a prototype control

sequence, belonging to a larger set of control sequences, the candidate controls.

Thus, the embedded state space is mapped into a neural field corresponding to a prototype

control.

The second major point in the thesis is that predictors that are locally linear in the

control can be constructed from the SOFM winners. The construction of the locally

linear predictors associated with the SOFM winners is essentially a linearization around

the weights of the winner:


M, = <,[x(k),U,]+ VO,[x(k),|U, -U,.] (32)

where U, -U, is the LI norm of the difference between the prototype control,

U, and the candidate control, U,, and VO is the Jacobian with respect to the control,

extracted from the converged SOFM weights.

Ideally, perhaps, there would be an individual SOFM, c,, for each candidate

control, U, =[uc, (k) u, (k +1) ... u, (k + p -1) ], and predictions of the tunnel








response, M, =[y, (k + 1) y, (k + 2) ... y, (k + p)] would be made using the SOFM

winners:

M, =Q,[x(k),U,] (33)

This would not have explored the concept of being able to extract a model that

was locally linear in the control from the SOFM and would have required excessive

amounts of training data that was not available, i.e. an ensemble of responses for each

candidate control over the entire operational range.

Thus the approach to modeling the tunnel dynamics evolved into a procedure

consisting of two major components. First, the control input space was manually

partitioned by the construction of significant prototype control vectors assumed to be

capable of producing the general features of the desired wind tunnel response. Second,

for each such partition of the control input space, a SOFM was constructed from an

ensemble of tunnel dynamic responses, i.e. the resulting Mach number response, covering

the operating range. Each ensemble of Mach number responses was extracted from over

20 hours of actual wind tunnel data, covering the entire operational range. Collectively,

the SOFM(s) form an atlas of the global wind tunnel response due to the prototype

control inputs.

The assumption here is that having an atlas for the system response to a set of

control input prototypes provides a sufficiently complete modeling infrastructure, given

the desired objective of predictively controlling the tunnel. There is no need to provide

an infrastructure capable of modeling the response to all possible 3 control sequences of

length, because it is assumed that the control inputs applied to the tunnel, at least in the

PMMSC mode of operation, will come from the known set of candidate controls, which








are either the control prototypes themselves, or close enough to the prototypes, by design,

so as to predict the tunnel response by local models constructed from the response

embedded in the input neural field of the corresponding SOFM.




Partitioning the Control Input Space


The control input space was partitioned by the construction of prototype vectors.

Experimentally, it was found that nine prototype vectors were required to achieve the

desired control to the specified tolerance. Seven of the control prototypes were 50 sample

periods in length, with two shorter prototypes which were 10 samples long. Figure 17

shows the seven 50-point control prototypes. The 10-point prototypes were composed of

either all +l's or all -l's.

50 sample prototype control vectors





0.5





S-0.5


Prototype


Figure 17. 50-point prototype control inputs








For convenience, the prototypes were assigned labels such as input_class_0,

input_class_l, input_class_2, etc. Input_class_0, input_class_l, and input_class_2 are

50-point control sequences consisting of all zeroes, all +1's and all -1's, respectively.

Table 3 lists the features of the prototype control vectors.



inputclass n Composition Control function

input_class 0 50 Fifty zeroes Steady-state

inputclass1_ 50 Fifty +'s (Raise) Ramp up

inputclass_2 50 Fifty -l's (Lower) Ramp down

input_class_3 50 Ten +1's, forty zeroes End of Ramp up

input_class_4 50 Ten -l's, forty zeroes End of Ramp down

input_class_5 50 6-9 zeroes, 1-4 +1's, forty zeroes Positive correction

inputclass_6 50 6-9 zeroes, 1-4 -l's, forty zeroes Negative correction

input_class_7 10 Ten +1's Positive transition

input_class_8 10 Ten -l's Negative transition


Table 3. Prototype Control vectors



The idea here, as discussed in the introduction, was to partition the control input

space by manually constructing prototype vectors for the control sequence. The goal of

this partitioning was to provide a set of control inputs capable of ramping the tunnel from

one operating point to another, regulating about a given operating point, rejecting

disturbances, while eliminating control sequences known experimentally to be of no








practical interest, particularly when considering the desire to minimize control activity

while regulating about an operating point. An alternating sequence of +l's and -1's

might provide the desired regulation of the output, but would be highly undesirable in

terms of control effort. This will be elaborated upon in the following chapters.

Ramping the tunnel from one operating point to another would be accomplished

with input_classes_l, 2, 3, and 4. Input_classes_5 and 6 would be used for regulating

about a given operating point as well as rejecting disturbances. Input_class_0 provides

the control input for the ideal steady-state condition with no disturbance, requiring no

control action over a 50-point sample interval. Input_classes_7 and 8 provide a transition

from the zero-input class to the ramping inputs of input classes_l and 2, and provide

identification of the tunnel response over a shorter, more recent interval of time.




Clustering the Mach Number Responses


For each of the control input classes, ensembles of Mach number responses

resulting from the application of each control prototype were extracted from the wind

tunnel test data. Next, each ensemble of responses was clustered using a SOFM. The

SOFM imposes a topographic ordering of the output neural field corresponding to

features of the input patterns, which are in this case, the Mach number responses, taken

over the past n sample intervals. Collectively, the SOFM(s) were trained using data

extracted from more than 20 hours of actual wind tunnel response data. Table 4 lists the

number of exemplars for each class.








Input_classes_3 and _4 have the fewest number of exemplars because they only

occur at the end of the transition from one set point to another. Input_classes_l, _2, _7,

and _8 have the greatest numbers of exemplars due to the relatively long transition times

from one operating point to another, requiring steady ramping up or down. Next in

frequency of application is inputclass_0, representing the most desirable, minimum

control effort over the 50 sample interval (15 seconds) when the Mach number is within

the desired tolerance. The remaining two inputclasses, _5 & _6, represent prototype

positive and negative corrections which provide disturbance rejection and regulation

about a set point, with the desired features of the control sequence, i.e. minimum control

effort and minimum number of switching or transitions from one state to another.

Input class # exemplars

0 10,158

1 15,332

2 13,464

3 41

4 31

5 155

6 198

7 17,393

8 16,694


Table 4. Training exemplars for each input class








The following figures (19 through 27) show ensembles of Mach number responses

from the application each control prototype, and their corresponding SOFM. The Mach

number response, M, is taken over the same n sample intervals as the application of the

control prototype,

M = M M(t -n); (34)

M = M(t), M(t -),..., M(t- n); (35)

and n is either 50 or 10. Thus, M represents the output of an n-tap delay line, where the

value at the nth tap is subtracted from all the values in the delay line. The output at a

single tap is shown in Figure 18. This is essentially a bank of comb filters which

preprocesses the Mach number responses, particularly for removing the dc component,

yielding the change in Mach number over the past n samples. Both the training samples

and the on-line Mach number responses were preprocessed in this fashion.



M(t)M I" 1" I--' M(t-n)


Figure 18. A single tap of the Mach number preprocessor




































Exemplar


56


Mach responses for Inputclass_0





















40
30
20 10 20
0 0
n

SOFM weights for inputclass_0





















4 -5o


10 30
5 ~~ :20
10
Cluster# 0 0

Figure 19. Mach number responses and corresponding SOFM for inputclass_0
Figure 19. Mach number responses and corresponding SOFM for inputsclass_0






57


Mach responses for input_class_l




0.06,

0.05,

0.04,

0.03-



0.01.

0-

-0.01
100
80 --" 50


Exemplar


SOFM weights for inputclass_1


Cluster # 0 0

Figure 20. Mach number responses and corresponding SOFM for inputclas
Figure 20. Mach number responses and corresponding SOFM for input-class_1






58


Mach responses for inputclass_2


-0.01

2-0.02

Q -003-


60
6 800 -40
100 50
Exemplar

SOFM weights for inputclass 2


Cluster #


Figure 21. Mach number responses and corresponding SOFM for inputclass_2






59


Mach responses for inputclass_3




0.025

0.02

0,015

S0.01

o 0.005

0

-0.005
40

30 50


10 20
10
Exemplar 0 0


SOFM weights for lnputclass_3




0.025


0.02


1 0.015


0.01


0.005


0
20

50




Cluster# 0 0
n


Figure 22. Mach number responses and corresponding SOFM for input__class.3






60


Mach responses for input_class_4














-0.025>





0 00
0. 10

3020 40

40 50
Exemplar







0.005

01



X-001 -

S-0.015-

-0.02-

-0.025
0


Cluster #


20 50


Figure 23. Mach number responses and corresponding SOFM for input_class_4






61



Mach responses for inputclass_5





0.02

0.015

0.01

0.005

o 0

-0.005

-0.01
200

15050
100 30
50 20
10
Exemplar 00 n


SOFM weights for inputclass 5



x 10
10,

8,

6,

4


0
0-

-2-

-4


15 50
10 30
5 20
10
Cluster # 0 0



Figure 24. Mach number responses and corresponding SOFM for inputclass_5









Mach responses for inputclass_6


0


60 30
80 40
100 50
Exemplar


SOFM weights for inputclass_6


Cluster #


Figure 25. Mach number responses and corresponding SOFM for input_class_6






63


Mach responses for inputclass_7



x10
15


101





0



-5
100



00


Exemplar 0 0


SOFM weights for inputclass_7




0.014

0.012

0.01

0.008,

0.006

0.004,

0.002

0O
20
15 1
10 6 8

5 4

Cluster# 0 0
n


Figure 26. Mach number responses and corresponding SOFM for input_class_7









Mach responses for inputclass_8


Exemplar

SOFM weights for inputclass_8


Cluster #


Figure 27. Mach number responses and corresponding SOFM for inputclass_8








Convergence of the Input Neural Fields


The number of nodes for the SOFMs, each representing a cluster of the Mach

number responses, was adjusted during the training phase to achieve an average

separation between the adjacent converged neural input fields, or more simply, the

weights of the SOFM. The topographic ordering imposed by the SOFM was key in this

phase of the development. The number of nodes were adjusted so that the separation

between the adjacent input neural fields corresponded to the desired goal of controlling

the Mach number, based on 50 samples-ahead predictions, to better than the required

0.003 tolerance. Thus, the major focus was to determine the number of classes for the

SOFMs for input_class_5 & _6, which provide the basis for regulation and disturbance

rejection. Each of these SOFMs were trained with 155 and 198 exemplars. It was found

experimentally that 20 nodes or clusters provided adequate separation based on

considering the separation between the adjacent means of each neural field over the last

30 point interval:

1 [ 50
3 M k(i,j+l) M(i,j) (36)
=L 121 i=21

Nodes were added to the SOFM until the mean separation, taken over the entire

map, was well below 0.001 for input_classes_5 & _6, as listed in Table 5. The resulting

20 node SOFM structure was implemented for all the input_classes, and the resulting

separations between adjacent input neural fields were considered adequate. The mean

separation for input_class 0 was even less than the above classes, and the mean

separations for the ramping inputclasses SOFMs were deemed sufficient for the

relatively coarser control required to move from one set point to another.









50 so1 -0
(103) M (i,j+l) ZMk(i,j) E (103) ()
21 i=21 l i0

j k=0 k=l k=2 k=3 k=4 k=5 k=6 k=7 k=8

1 0.6342 5.284 -1.352 1.648 -1.407 0.5741 -0.2934 0.6155 -0.7840

2 0.6194 3.111 -3.625 2.639 -2.557 0.6314 -0.4436 0.7627 -1.007

3 0.2973 1.686 -1.270 1.032 -1.249 0.2768 -0.4233 0.2835 -0.4472

4 0.1652 1.377 -0.8680 2.655 -1.547 0.3332 -0.3458 0.2418 -0.2451

5 0.2856 0.8588 -0.8589 1.438 -2.220 0.3767 -0.4339 0.2112 -0.1888

6 0.2339 0.9396 -0.7762 0.5972 -1.116 0.2678 -0.2206 0.1906 -0.2295

7 0.0565 0.6088 -0.7927 1.073 -0.4461 0.2143 0.0058 0.3037 -0.2699

8 0.0129 0.3392 -0.5479 0.8365 -0.2759 0.3067 -0.0531 0.1421 -0.1933

9 0.0166 0.8414 -0.1769 0.3579 -0.5014 0.4656 -0.4198 -0.089 -0.0416

10 0.1377 0.8744 -0.3810 0.1971 -0.5801 0.2794 -0.4187 0.0292 -0.1017

11 0.3549 0.8427 -0.8745 0.4618 -0.4409 0.0169 -0.1808 0.2533 -0.2718

12 0.3034 1.368 -0.6687 0.5598 -0.5765 0.0771 -0.2167 0.2644 -0.2841

13 0.1219 1.009 -0.5736 0.6931 -0.9616 0.1947 -0.5848 0.2335 -0.1611

14 0.0339 0.9442 -0.9577 0.6108 -0.7398 0.4424 -0.6323 0.1458 -0.0474

15 0.1218 0.6524 -0.7976 0.4443 -0.4548 0.6975 -0.5178 0.1386 -0.2038

16 0.4175 1.130 -0.7018 0.4792 -0.6299 0.7110 -0.7711 0.2827 -0.3107

17 0.3741 2.039 -1.055 0.3583 -0.8048 0.5912 -1.417 0.3475 -0.2851

18 0.4796 1.665 -2.076 1.221 -0.6797 1.554 -2.468 0.4746 -0.6025

19 0.5925 -0.8114 -1.757 1.390 -0.4480 2.735 -2.253 0.5246 -0.6308

m 0.2768 1.303 -1.058 0.9839 -0.9282 0.5656 -0.6363 0.2819 -0.3319

SD 0.2039 1.234 0.7670 0.7153 0.6259 0.6221 0.6809 0.2003 0.2550



Table 5. Difference between interval means of adjacent input neural fields









(103) (D ) 10o
3021


*1000

j k=0 k=l k=2 k=3 k=4 k=5 k=6 k=7 k=8

1 -2.556 15.98 -22.28 2.456 -3.719 -1.570 1.031 0.7942 -0.6336

2 -1.922 21.26 -23.63 4.105 -5.126 -0.9963 0.7372 1.409 -1.417

3 -1.303 24.38 -27.25 6.745 -7.683 -0.3649 0.2936 2.172 -2.425

4 -1.006 26.06 -28.52 7.777 -8.933 -0.0881 -0.1297 2.456 -2.872

5 -0.8406 27.44 -29.39 10.43 -10.48 0.2452 -0.4755 2.698 -3.117

6 -0.5549 28.30 -30.25 11.87 -12.67 0.6219 -0.9095 2.910 -3.331

7 -0.3210 29.24 -31.03 12.47 -13.82 0.8897 -1.130 3.010 -3.536

8 -0.2645 29.85 -31.82 13.54 -14.26 1.104 -1.124 3.403 -3.806

9 -0.2519 30.19 -32.37 14.37 -14.54 1.411 -1.177 3.545 -4.000

10 -0.2349 31.03 -32.54 14.73 -15.04 1.876 -1.597 3.456 -4.041

11 -0.0973 31.90 -32.93 14.93 -15.62 2.156 -2.016 3.485 -4.142

12 0.2576 32.74 -33.78 15.39 -16.06 2.173 -2.197 3.738 -4.414

13 0.5610 34.11 -34.47 15.95 -16.64 2.250 -2.414 4.002 -4.698

14 0.6830 35.12 -35.04 16.64 -17.59 2.444 -2.998 4.236 -4.859

15 0.7169 36.01 -35.60 17.25 -18.33 2.887 -3.631 4.382 -4.906

16 0.8387 36.72 -36.78 17.70 -18.79 3.584 -4.149 4.520 -5.111

17 1.256 37.85 -37.50 18.18 -19.42 4.295 -4.919 4.803 -5.421

18 1.631 39.89 -38.55 18.54 -20.23 4.886 -6.337 5.151 -5.706

19 2.110 41.55 -40.63 19.76 -20.91 6.441 -8.806 5.625 -6.309

20 2.703 40.74 -42.39 21.15 -21.35 9.176 -11.06 6.150 -6.939


Table 6. Interval means of SOFM input fields









(10)1 Mk(ij) 1

j k=O k=l k=2 k=3 k=4 k=5 k=6 k=7 k=8

1 16.78 92.73 129.3 14.83 22.78 9.669 8.075 3.195 2.594

2 12.90 123.3 136.9 24.13 30.09 6.997 4.947 5.456 5.658

3 8.921 141.6 158.5 39.41 44.14 2.808 2.318 8.344 9.363

4 6.794 151.9 166.3 46.01 51.61 1.048 1.322 9.629 10.92

5 5.625 160.1 171.6 62.03 61.78 3.099 3.247 10.74 12.02

6 4.136 165.2 176.3 70.51 76.33 4.611 6.642 11.26 12.96

7 2.839 170.6 180.4 74.66 82.47 5.177 7.177 11.58 13.58

8 2.623 174.0 185.1 81.11 85.71 6.713 6.448 12.51 14.13

9 2.933 176.0 188.6 85.83 88.76 8.406 6.576 13.17 14.82

10 2.632 180.9 189.5 88.40 91.33 10.55 8.931 13.21 15.38

11 1.902 185.9 191.9 90.99 94.44 12.09 11.24 13.65 16.03

12 2.070 190.6 197.3 94.72 97.32 12.26 12.45 14.40 16.64

13 3.504 198.4 201.0 96.88 100.6 13.35 14.24 14.92 17.24

14 4.774 204.4 204.2 100.3 106.1 14.36 17.37 15.58 17.88

15 5.547 210.0 209.8 104.7 111.2 16.69 20.30 16.41 18.44

16 6.890 213.9 214.6 107.5 115.1 20.09 23.23 17.22 19.27

17 8.854 220.4 219.2 110.9 119.0 23.92 28.27 17.93 20.19

18 10.63 231.9 225.3 113.3 123.4 27.28 36.12 18.92 21.10

19 13.57 241.6 236.9 119.0 126.7 35.86 49.03 20.78 23.18

20 17.47 236.9 247.1 126.3 128.2 50.95 61.49 22.89 25.40


Table 7. Euclidean norm of SOFM input neural fields







In order to quantify the topological ordering of the converged neural fields,

Table 6 lists the mean taken over each 30 or 10 sample interval of the SOFM weights.

Table 7 enumerates the Euclidean norm for all the converged neural fields as well. With

the exception of SOFM_0, the norms steadily increase (or decrease) along the output field

of the map. SOFM_0 displays increasing distance from the center of the map, outward,

corresponding to the symmetry of the interval means about the center of the map shown

in Table 6.




SOFM Selection for Local Model Identification


After the application of a candidate control, one of the nine SOFM is used to

cluster the Mach number response, M, over the past n sample intervals. The selection of

the SOFM is based on the minimum Euclidean norm between the control input history

U = u(t 1), u(t 2), ...,u(t m) and the set of prototype control vectors U; il,n :
1; i=1,n

input_ class i = mjnl U Ui (37)


If more than one prototype control vector matches identically, i.e. U Ui = 0 for more

than one i, both SOFM(s) are excited with the appropriate length M. This can occur for

SOFM_1 (or _2) and SOFM_7 (or_8), where the SOFM_7 (or 8) winner represents the

response over the 10 most recent samples, while the SOFM_I (or _2) winner represents

the response over the past 50 samples. Additionally, the regulating control classes were

clustered on a region of the control space, defined in Table 3, as opposed to a single point

in the control space.








The SOFM metric for the winner is the minimum Euclidean norm between M

and the SOFM prototype vectors, M,, for the SOFM selected by inputclass :


mach_ class_ i = l M -Mill.
i=l,2011M


Figure 28. Selection of SOFM by input_class





Prediction of Tunnel Response Using Local Models


The Mach number responses are predicted by a linear model:

M, = acA' + W';

where W* is the prototype response vector, or weights, of the winning node.








A' is the least square approximation of the winner's prototype response to a d-sample

delayed unit step sequence, U = [0,0,...,0,1,1,...1], where d represents the maximum
d p-d

relative degree or delay from input to output and p is the total number of samples ahead

for which the prediction is made:

A'=bU, (40)

where b is fit in the least square sense, or, alternately,

b=W' (U,) (41)

and 0 denotes the Moore-Penrose pseudoinverse of the vector U,.

By inspection of the SOFM for all input classes, d was chosen conservatively to

be greater than any observed delay, d = 20. A single constant, ac, scales A' based on

the ratio of the L1 norm of the candidate control vector Uc and the L1 norm of the

control sequence U, producing the response M:


ac = !U 1 u ITJ, 0
SIIll J (42)
ac = 0 'Itll, =0



Thus, acA' provides the difference in the predicted Mach number response due

to the to the distance between the control sequence, U, and the i' candidate control

sequence, U ,. For the simplest case, U =U, the value of a is zero, and the Mach

number response is predicted directly from the input neural field. This linear model is

driven by the candidate control inputs, shown in Figure 29.











Candidate Control Inputs u(t+n)


T o,



" -0.5,



-1
0
-'J


Candidate #

Figure 29. Candidate Control Sequences


Comparisons of the predictions of the Mach number to the actual tunnel responses

as a result of the application of the candidate controls will be presented in Chapter 5,

Experimental Results.















CHAPTER 4
PREDICTIVE CONTROLLER

Introduction


Given the model of the tunnel response developed in the previous chapter, the

predictive controller evaluates the relative effectiveness of the candidate control inputs.

The advantage of partitioning the control input space using a set of prototype controls

becomes more apparent when compared to model-based predictive control (MPC)

[Clarke, Mohtadi, and Tuffs, 1987]. In our method, predicted responses from a set of

candidate control inputs can be extracted either directly from the SOFM's output neural

field or from the derived local model. The controller then applies the control sequence

which minimizes the error between the desired output and the predicted output over some

finite number of steps into the future. The low computational cost of multi-step

prediction by this method allows prediction for relatively long (50 samples ahead) control

sequences, or control horizon, in the terminology of MPC, using relatively simple

computing hardware. This is in contrast to MPC, which requires the inversion of an

NU x NU matrix at each step for a control horizon NU steps into the future. A brief

background of MPC is provided as a basis for comparison to SOFM-based predictive

control using control prototypes.








Model Predictive Control Background


Most input-output model based predictive control schemes [Clarke, Mothadi, and

Tufts, 1987], begin with the assumption of a linear model (ARMA, or Autoregressive-

Moving Average):


y(k) = a,y(k -i) + bju(k j) (43)


with an additional disturbance term in moving average form:


d(k) = (k) + c,j(k -i) (44)


where 4(k)is an uncorrelated random sequence.

Combining (43) and (44) and introducing the polynomials A, B, and C in the

backward shift operator q~':

A(q-') = 1+a,q-'+...+a,q-"

B(q- ) = b +bq- +...+b,,q -

C(q ')= +c,q-'+...+cq-"

yields

A(q-' )y(k) = B(q-' )u(k 1) + C(q-1 )(k) (45)

which is referred to in the literature as the CARMA (Controlled Auto-Regressive and

Moving Average) model, a variation on the ARMAX (Auto-Regressive Moving Average

with exogenous input) model.

A further refinement to the disturbance model to accommodate non-stationary

disturbances such as random steps occurring at random times is :








d(k)= C(q-')(k)/A (46)

where A = 1- q-', the differencing operator. Combining (45) and (46) yields the

CARIMA (Controlled Auto-Regressive Integrated Moving Average) model used in

Generalized Predictive Control (GPC) :

A(q~')y(k) = B(q-')u(k 1) + C(q-' )(k) / A. (47)

At this point it is useful to introduce a scalar cost function J:

N, Nu
J = X[y(k +i) w(k +i)]2 + JX(j)[u(k + j -1)]2 (48)
I=NI j=l

where:

y is the predicted response from the control input sequence u

N, is the beginning of the costing horizon;

N2 is the end of the costing horizon;

N, is the control horizon;

A(j) is a control-weighting sequence.

N,, N2, N., and X(j) represent tuning knobs which can be adjusted by the control

designer to tailor the control action for the desired response characteristics. Rules of

thumb provide some guidelines for initial selection. N, is usually picked to be greater

than the largest anticipated time delay between the input u(k) and its response in the

output y(k). N2 is determined by the longest settling time associated with the pulse or

step response of the model. N, = 1 is quite often chosen for open-loop stable non-

minimum phase plants, but this often represents a compromise between the

computational burden associated with longer control horizons.








The minimization of J, given a future set point sequence w, where :

w = [w(t + 1),w(t + 2),...,w(t + N)] (49)

leads to the control law :

u= [GT G+A;I]- GT (w-f) (50)

The matrix G is of dimension N xNU:

go 0 -.. 0
gl go ... 0


G= go (51)


gN-I gN-2 '" N-NU


This requires the inversion of an NUx NU matrix at each sample time, or at least

for each identified change in the g parameters, which are the coefficients of the

z-transform of the plant's step response. f is a linear combination of values of u(t) and

y(t) up to time t.



SOFM-based Predictive Controller


The function of the SOFM-based predictive controller is to evaluate the relative

effectiveness of the candidate control inputs in reducing the error between the desired

Mach number and Mach number predicted by the current SOFM winners. This is done

by evaluating the Euclidean norm of the difference between the last 30 points of the 50-

points-ahead predicted Mach number responses and the desired Mach number set point:








err_ norm_ i= JIMp [21:50]- MS, (52)

for all i candidate control sequences, as specified in the prediction section. The

evaluation over the last thirty points of the prediction is to emphasize steady-state

matching. The control sequence associated with the minimum norm of all i sequences is

then applied as the control to the tunnel.

This is similar to the scalar cost function for GPC (48) :


J= +l,(k +i) w(k +i)2] (53)
i=N, I

with N, = 21 N2= 50 and yp is the predicted Mach response for the pth candidate

control sequence. Both the constraints on the permissible values of the control (+1, 0,

and -1) as well as the minimization of the control cost is embedded in the set of all p

candidate control sequences with control horizon N,= 50. The control Up that

generates y, is selected for the minimum J,.

In the set of candidates we included controls to ramp the set point up and down

for large changes in operating point, as well as the regulating control sequences for

disturbance rejection. The candidate control sequences, their associated SOFMs for

prediction, and their control update parameters are listed in Table 8. The control update

parameter for each candidate control determines whether the entire 50 sample control

sequence is applied as the control, or just the first point in the sequence. Implicitly, this

selection is done based on the error between the Mach number set point and the predicted

responses. If the selected candidate control corresponds to either the two largest control

efforts over the control horizon, or if the selected candidate control represents the







minimum control effort (i.e. zero) over the control horizon, the control sequence is

updated by selection of the prediction-error minimizing control at the next sample period.

For all other cases, the entire 50 sample selected candidate is applied. The two cases of

regulating about an operating point and operating point changes illustrate the differences.

Candidate # [u(t+l), u(t+2), ... u(t+50)] SOFM k

1 [(50) +1's] 1 1

2 [(11)+l's (39) zeros] 3 1

3 [(10)+l's (40) zeros] 3 50

4 [(9)+1's (41) zeros] 5 50

5 [(8)+l's (42) zeros] 5 50

6 [(7)+l's (43) zeros] 5 50

7 [+1 +1 +1 +1 +1 +1 (44)zeros] 5 50

8 [+1 +1 +1 +1 +1 (45) zeros ] 5 50

9 [+1 +1 +1 +1 (46) zeros ] 5 50

10 [+1+1+1 (47) zeros] 5 50

11 [+1+1 (48) zeros] 5 50

12 [+1 (49) zeros) ] 5 50

13 [0.66 (49) zeros ] 5 50

14 [0.33 (49) zeros ] 5 50

15 [ 50 zeros] 0 1

16 [-0.33 (49) zeros] 6 50

17 [ -0.66 (49) zeros ] 6 50








18 [-1 (49) zeros] 6 50

19 [-1-1 (48) zeros] 6 50

20 [-1-1-1 (47) zeros] 6 50

21 [-1 -1-1-1 (46) zeros] 6 50

22 [-1 -1 -1 -1 (45) zeros] 6 50

23 [ -1 -1 -1-1 -1 (44) zeros] 6 50

24 [(7)-l's (43)zeros] 6 50

25 [(8) -'s (42) zeros] 6 50

26 [(9) -'s (41) zeros] 6 50

27 [(10)-l's (40) zeros] 4 50

28 [(11) -'s (39) zeros] 4 1

29 [(50)-l's] 2 1


Table 8. Candidate Control sequences and associated parameters


Operating Point Changes

The typical set point change is greater in magnitude than 0.1, which is several

times greater than the largest Mach number change associated with any of the SOFM

input fields. Set point changes of this magnitude produce the selection of either candidate

control #1 (50 +1's) or #29 (50 -i's). These control sequences are updated at each

sample interval, which means that the controller decides at each sampling instant whether

to extend the series of raise or lower commands to achieve the desired set point. If the

only selection was between the continued ramping associated with either SOFM_1 (ramp








up) or SOFM_2 (ramp down), and the next prototype control associated with SOFM_3

(end of ramp up) and SOFM_4 (end of ramp down), the transition between set points

would indeed be rather coarse. The inclusion of candidates # 2 and #28 provide a one

control-tick resolution between continued ramping and the transition to regulating about

the desired set point. Ramping continues on until candidates #3 is selected over #2 or

#27 is selected over #28 as the prediction-error minimizing control. These control

sequences (#3 or #27) are applied for their entire 50-point duration, allowing for a smooth

transition to regulation about the set point.

Regulating About an Operating Point

When actively regulating about an operating point, the entire 50 sample control

sequence selected from the set of candidates, consisting of an active or non-zero segment

of 1/3 to 10 sample periods, followed by the corresponding number of zeroes during the

inactive segment, is applied as the control input for the next 50 sample periods. Thus, the

selected candidate control is applied open-loop over the entire 50 sample control horizon,

with the next control update occurring 50 sample periods later. The resulting 50-sample

Mach number response is then input to the corresponding SOFM, and future predictions

are made from the output neural field of the SOFM winner as described in Chapter 3.

If the sequence of all zeroes is selected, the control is updated at the next sample

period. The 50 sample Mach number response is input to SOFM_0 for identification of

the local dynamics by the SOFM_0 winner. Prediction and control sequence selection is

performed at each sample period until an active (non-zero) control sequence is selected to

regulate the Mach number.













CHAPTER 5
EXPERIMENTAL RESULTS



In Chapter 3 and 4, the SOFM-based modeling of the tunnel dynamics and the

resulting predictive controller were developed. The control input space was manually

partitioned by the use of prototype control sequences and SOFM's were trained to cluster

the corresponding Mach number responses. Thus, the output neural field of each SOFM

represents a collection of local models of the tunnel dynamics for the corresponding

prototype control input. During the experiment, while actually controlling the tunnel with

the PMMSC, the control inputs were chosen from the set of candidate control sequences,

making the task of identifying the local dynamic model more straightforward than in the

more general case of all allowable 3P control sequences of length p.

Thus, the experimental results are composed of two parts. The first part is to look

at the results of controlling the Mach number during actual experimental tests conducted

in January 1996. These results will be compared to control of the tunnel with an existing

gain-scheduled automatic controller as well as control by an expert human operator. The

second part is to explicitly examine the results of modeling the tunnel dynamics with the

control-input partitioned SOFM architecture. This will be accomplished by comparing

the Mach number response predicted by the SOFM-derived local model to the actual

response after application of the error-minimizing control sequence determined by the

predictive controller.








Experimental Setup


The experimental setup consisted of a 486-33 MHz PC connected via a serial port

to the existing control computer at the wind tunnel, referred to as the "tunnel micro". The

tunnel micro is an early 1980's vintage 8086-based microcomputer. The existing

automatic control implemented in the tunnel micro is a highly tuned but fixed table

look-up of drive motor commands based on the error at a given Mach number [Capone et

al., 1995]. The tunnel micro also communicates with the wind tunnel data acquisition

system. The data acquisition system provides the Mach number measurements at a

nominal sample interval of 0.3 seconds. Figure 30 shows a block diagram of the

experimental setup.


Mach number


Control commands


I Mach
Number





Ex


\ Control
\ commands


isting controls


Figure 30. Experimental Setup


The PMMSC was implemented as a C program, compiled and run on the PC. The

output of the program, at each sample interval, is a control command which is








communicated to the tunnel micro and then applied to the drive system for the tunnel

fans. The control command can take on the values of; +1 to raise the tunnel fan RPM, -1

to lower the fan RPM, or zero to maintain fan RPM. Further, the command duration may

be specified to be either the full sample interval, 0.3 seconds, or less than the full sample

interval in 0.1 second increments, (i.e. either 0.1 or 0.2 second duration). This

subdivision of the sample interval was required to provide finer control of the tunnel Fan

RPM. Control inputs of less than 0.1 second duration are generally ineffective in

producing a change in the tunnel fan RPM. Additionally, the PC was used to record the

time histories of the tunnel state, control inputs, and PMMSC internal variables such as

the predicted response and SOFM winning nodes.



Mach Number Measurements


The Mach number is computed from the a calibrated ratio of stagnation pressure

to static pressure measured in the plenum surrounding the test section, as described earlier

in Chapter 1, equation (1).

The most recent calibration of the wind tunnel was performed in 1990 [Capone,

et. al, 1995]. This calibration used 30 static pressure measurements taken along the

nominal 8-ft calibrated test section length (CSTL). Flow uniformity was parameterized

by both the standard and maximum deviation of spatially local Mach number from a

least-squares straight-line fit. The results of this calibration are listed in Table 9. In this

table, the test section Mach number M,, is the value of a least-squares straight-line fit to

the Mach number data, corresponding to the midpoint of the test section. The standard








deviation is a measure of the average discrepancy along the test section length. The

maximum deviation represents the worst departure from the least-squares fit along the

selected length of test section. The document reporting the results of the calibration lists

2cr values, i.e. twice the positive square root of the variance.


Mr 27 aOmax

0.3015 0.000560 0.001088
0.4006 0.000754 0.001688
0.5014 0.000943 0.002158
0.6018 0.001152 0.002381
0.6544 0.001291 0.002833
0.7030 0.001388 0.003085
0.7537 0.001415 0.003350
0.7795 0.001478 0.003413
0.8000 0.001422 0.003015
0.8284 0.001481 0.003552
0.8555 0.001514 0.003632
0.8809 0.001576 0.003756
0.9038 0.001428 0.003700
0.9304 0.001584 0.003749
0.9579 0.001539 0.003684
0.9816 0.001422 0.003211

Table 9. Standard and maximum deviation of Mach number during calibration


From this table, it can be seen that both the standard and maximum deviation of

Mach number measured along the CTSL vary significantly over the subsonic range. This

spatial variation corresponds to the temporal variation of steady-state Mach number

measurements. This is illustrated by taking the standard deviation of a time series of

Mach number measurements calculated from the calibrated ratio of stagnation pressure to

plenum static pressure under steady conditions during operational tests. In Table 10, M is

the mean value of 200 consecutive Mach number measurements taken under steady








conditions with no control input applied. The standard deviation is the sample standard

deviation:


l 2 N 2
- 2 ( = o-M(i)- ;
N-l =1


where 2c is used for direct comparison to the calibration results.


M 2(

0.2979 0.000546
0.3968 0.000520
0.5999 0.000908
0.8014 0.001820
0.8518 0.001497
0.8850 0.001977
0.9003 0.001577
0.9504 0.001822
0.9819 0.002652


Table 10. Statistics of time histories of steady state Mach number measurements



Experimental Results of Controlling the Mach Number


Experimental results were obtained while controlling the wind tunnel with the

PMMSC at several subsonic Mach numbers. These tests were conducted during the

period of January 10th through January 23rd, 1996.

Figure 31 shows the wind tunnel Mach number being controlled by the PMMSC

for a period of three hours, during a normal operational tunnel run, where aerodynamic

research data was being taken. Mach number set points of 0.95, 0.9, 0.85, and 0.6 are









shown as dashed lines. The PMMSC regulated the steady-state Mach number to within

the research requirement of 0.003 of the set point during the interval shown. PMMSC

commands are shown with magnitudes less than one for control commands whose

duration was less than the 0.3 second sampling period. Control pulses of 0.1 second are

shown with magnitude 0.33 and 0.2 second pulses are shown with magnitude 0.66.

Mach number
1

0.9 -

S0.8 -
20.7-

0.6 -L

0.5 -
0 20 40 60 80 100 120 140 160 180

NNCPC command
1






C1 5


0 20 40 60 80 100 120 140 160 180
Time (min)


Figure 31. Mach number controlled by PMMSC* during a three hour test


*The PMMSC was previously referred to as NNCPC, so this acronym appears in some of
the plots.








During these tests, the aircraft model attitude was varied to achieve the desired

aerodynamic research data. Figure 32 shows typical variations of model attitude at each

Mach number.


Angle of Attack (alpha)


Angle of Sideslip (beta)


60 80 100 120
Time (min)


140 160 180


Figure 32. Variations of angle-of-attack and angle-of-sideslip during test


The variations are of two general types, referred to as an "alpha sweep" or "beta

sweep". During an alpha sweep, the model angle-of-attack or "alpha", is stepped

through some range, from -4 degrees to +12 degrees in this test, while maintaining a

constant angle-of-sideslip, or "beta". During a beta sweep, the model angle-of-sideslip is

stepped through some range, from -6 to +6 degrees for this test, while maintaining a

constant alpha. Mach number must be within 0.003 of the desired Mach number to

satisfy the research requirements. The variation in model attitude produced some modest










(< 0.001/degree) disturbance in the tunnel Mach number, particularly at angles-of-attack


above five degrees, although the onset of this disturbance was dependent on the


test Mach number. Figure 33 shows the SOFM winning nodes for positive and negative


corrections as determined by the PMMSC during the run. Figure 34 shows the Fan RPM


and tunnel temperature during the run.


Raise correction class

15





0 20 40 60 80 100 120 140 160 180

Lower correction class








0 20 40 60 80 100 120 140 160 180
Time (min)


Figure 33. Winning nodes for SOFM_5 and SOFM_6 during test



RPM
340


300 -
280
260

0 20 40 60 80 100 120 140 160 180

Tunnel Temperature

110

io


80' -'-'-- -,------_ i_
0 20 40 6o 0o 10c 120 140o I 180
Time (min)


Figure 34. Fan RPM and Tunnel temperature during test








In order to illustrate the operation of the PMMSC, a shorter interval of the run is

shown in Figure 35. Figure 35 shows the Mach number being controlled to a set point of

0.85 over a 15 minute interval. The angle of attack, alpha, is being steadily increased

during this interval, while beta is maintained at zero. Fan RPM and tunnel temperature

are steady as shown. At t = 98, the increases in alpha begin to cause the Mach number to

drop. At t=99, a short duration (0.1 sec) raise correction brings the Mach number back to

within tolerance. Further increases in alpha result in another decrease in Mach number.

Another small raise correction minimizes the error. At t=104 a longer duration corrective

pulse is applied after the Mach number response from the previous short duration pulse is

classified as much less effective than the previous corrections. This is seen where the

raise correction SOFM winner changes from 11 to 3. The corresponding raise correction

SOFM winner change is shown in Figure 35. The raise correction SOFM winner

corresponding to the Mach number response to the longer pulse is node 18.

A rather large decrease in angle-of-attack, from 12 degrees to near zero, causes the

Mach number to jump up even further. Successive lower corrective pulses bring the

Mach number back, while changing the SOFM winner of the lower corrective response,

seen in Figure 35.




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