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CONTROL OF THE NASA LANGLEY 16 FOOT TRANSONIC TUNNEL WITH THE SELFORGANIZING FEATURE MAP By MARK A MOTTER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIRE:MENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1998
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To the memory ofmy parents
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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. Loe VuQuoc. 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. Horne. Finally, I would like to thank my wife, Lisa, for her unwavering support, and Benjamin, who makes every day new. 111
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TABLE OF CONTENTS p a ge A C"KN' 0 WLEDG ME.NT S .... .. ...... ..... .. . .......................................................................... m LIST OF TABLES .............. ... .. . .. . .... . .. .... ....... ... .... .......... .. . .. . .... ........ ... ...... .... .. ... VI LIST OF FIGURES ... .. .. .. . ... ..... ........................... .. .......... .. . . .. .... ........ .. ..... . .. . ....... Vll ABSTRACT .................................... .. .... .. .. .... ........ .. ........ ...... .......... .. .... .. .... .. .. .. .............. .. ...................... .. .... .. .... ...... ...... .. ......................... ........... .. ........................ ... .. .. x 1 .IN'TRODUCTION . .. .... .. .... .. . .. . ... ... ..... ....... . . .. . . .. . .. .. ... .... ... .. . .. . ..... . ............. 1 Motivation . ............. . . ..... .... .. . .. .... .... . .. ......... .... . ....... .. .... ...... ...... ................ ....... 1 Evolution of the Research ..... ... ... .. . .. . .. . .. ... . ... . ...... . . .... .. ........ . .......................... 2 Background . ... ...... .. .... . .. .... ....... . .. ...... ... ...... .. . . .. ...... . .. ..................... .. . .. . ...... .. 5 Control Chillenges .. .. ....... .... .... . .. .. .. .. ................ .. . .. ............ .. ... .. . . .. ...... ... ......... 10 Experimental F ramework ..... . .. ....... ...... . .. . .... ...... .... .. . . . ......... . .... .... .. ...... . .. . .. .. . 18 Overview of the Dissertation .. . ..... ..... . ... . .. .... ... .. .. .. .. .... .. ................................ ...... 19 2 REVIEW OF LITERATUR.E ........................ ............................. ... . ............ . .. . .. .... .. 2 1 Introduction .. .. .... .. .. ............ .. . .. . .. ......... . .. .. . .. . ....... ... . ..... . .................. . .... .. . .. . 21 Self Organizing Feature Map .... ....... . .. .... .... .. . ... .. ... .. ..... .. ........ . . .. .... . .. . .. .... ... 2 1 Practical Aspect s for the Application of the SOFM Algorithm .. . .. ......... . ........ 2 5 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 Application s of Neural Networks for Control ....... .. ... . .. .... . . . .. ... ......... .. .... 41 3 MODELLING THE TUNNEL DYNAMICS .. .. .... . .. ......... .. .. . .... ......... ... .. .. . ....... 43 Introduction ............ .. ... .. .. .. . ..... ... ................. ......... .. . .. . .. . ......... ...... .. . .............. 43 Review of Local Dynamic Modeling with SOFM ................. ... . .. ................ ....... ... 45 Modifications for SO FM ba se d Predictive Control ... ....... .. ....... ... . . ......... .... .... ..... 48 Partitioning the Control Input Space ...... . .... . .... .. . . ....... .... .. ... ... .. .. . .. ... .. ... ............ 51 Clustering the Ma c h Number Re s ponse s . ................... .............. ............................... 53 Convergence of the Input Neural Fields ...... .... .............. .. ................ ................. .. .. 65 SOFM Selection for Local Model Identification ......... . .... .. . .... .... ........................... 69 Prediction of Tunnel Response Using Local Model s ... ......... .. .. ............ .. . . .............. 70 IV
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4 PREDICTIVE CONTROLLER . . . . . . .. . ....... .... .... ......... .... . .. . . .. .. ... . .. . ............ 73 Introduction .............................. ... ... . .. . . ... .................... .. ... .. . .. . .. . .... ......... ...... 73 Model Predictiv e Control Background . ... . . . . .... . ...................... .... ...... .. ................. 7 4 SOFMbased Predictive Controller . ..... . ..... ... ...... .. .. ... .. . . .. ...... . .. ...... .... .... .. . . .. 76 Operating Point Changes . . .. .. . .. .... .. ... .. . . ....... .. .. ......... . ................................ 79 Regulating About an Operating Point .. ....... . . ...... .... . .. ......... ......... . .. . . ........... 80 5 EXPER.IMENT AL RES UL TS .... .. .. ... .. . .... ............ .... .. . .. . . ... . . .... ...................... 81 Experimental Setup . . . .. . .. ............ ......... .... . ..... ...... .... ................. ...... ... . .... . .. .... 82 Mach Number Measurements ....... ......... .............. .. . . . .. .... .............. ...... .. .... . .... . 83 Experimental R esu lts of Controlling the Ma ch Number ........... . .. ............................. 85 Comparison of PMMSC to Existing Controller and Expert Operator .. . . .. . .. .......... 92 Experimental R es ults of Mod e ling the T11nnel Dynami cs .. ... ................... .. . .. . ....... 97 6 CONCLUSIONS AND FUTURE RESEARCH .. ... .. ................ .... ... . ....... . .............. 116 Conclu s ions . . .............. ... . . .... .. . .... .. .... .. . .... .. . .... .. .. . ........... . . .. .. . .... .. ........ 116 Future Research ............................................................. . .. ... .. . ......... .... . .. ...... . .. ... 1 18 APPENDIX ST ABILITY CONSIDERATIONS .. ... ......... .. . .. . . ................................ 119 LIST OF REFERENCES ............... . .. .. ... .. ............ . .... .. ............................... .. .. ..... 126 BI OGRAPID CAL S "KETCH ... ..... .. ....... ... .... .... . .. . .. . . .. .. ... .. .. . ...... ... . .. . . .. . .. . .. . .. .. 130 V
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LIST OFT ABLES Table p a ge 1 Variation of Ma c h number ratelimited increase while ramping up . . .. ... .. .. ......... 12 2 Changes in control input effectiveness for blocked conditions . . .. .. ... . ... .... . .. ... .. 18 3. Prototype Control vectors . .. .... .. . .. .. .. . .. ..... ... . .. ...... .. . . .. . .. . .. . .. .... .... .... .. .. 52 4. Training exemplar s for each input cla ss . .. . ....... . .. . .. .. .. . ... ..... .... .... .... .. . .. . .. 54 5 Difference between interval m e ans of adjacent input neural field s .... .... .... ........ ... 66 6 Interval mean s of SOFM input fields . ....... . .. ... .. . .. . . ... . . .. . ... .. .. . ... .. ..... ........ 68 7 Euclidean norm of SOFM input neural fields . .. . .. . .. ..... .. ....... .. ........ .... .. . ....... . 69 8 Candidate Control s equences and associated parameter s ... .. .... . . .. .. .... . . . ... ...... 7 9 9 Standard and maximum deviati o n of Mach number during calibration .. .. ......... . .. . 84 10 Statistics of time historie s 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 point s . ... . ......... .. .. ... ... .. .. . .. 96 13 Distribution among input cla ss e s for Figure s 31 and 45 ..... ....... . .. ...... .. . .. .. .. . 101 14. Multistep prediction errors for all input cla ss es ... ......... . . .... . ....... .. ... .. .... .. . .. 102 Vl
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LIST OF FIGURES Figure p a ge 1 Aerial View of the 16Foot Tran so nic Tunnel ................. . .. . ... .. .... ........... ............... 6 2. Test Section with model in place ...... .. ........ .. . ....... . .. . . . ... . . .. . ... .. . .. . ....... .. . ..... 6 3. Arrangement of Langley 16Foot Transonic Tunnel . .. . .. . . .. ......... ......... .. .............. 7 4. Inside View of the 16Foot Tunnel down s tream of the sec ond set of turning vanes .. 8 5 Mach Number and Tunnel Drive Control Input s during a typical s ub s oni c run ......... 9 6. Tunnel conditions during a typical run with s teady ramping to the desired te st Mach number, M=0 .95. The Mach number is to be held to within 0.003 of the de sire d value while varying the angle of attack . . .. . .. . .... .. ... .... .. . . ....... . ....... ... ... .............. 13 7. Out of tolerance Mach number extends the test duration during the last 3.5 minut es ofthetest@M 0 .9 5 ..... ..... . .... . .. .. . .... ....... . .... .. ......... . .. . . .. .. ... . ... . . .............. 15 8. Regulating the Mach numb e r at M = 0.8 with large di st urbances from th e model Angle of Attack .... ........ ... .. .... .. . .. ... ..... . . .. .. . .... ....... . ... .. . .... .. . ............ . .. . .. .... 17 9. Experimental Framework with PMMSC .. .... .. . .. .... ........... . .. . .... .... . .................... 2 0 10 SOFM with a onedimensional array of neuron s ........... ....... ................................. 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. Di s tribution of training inputs among 20 converged SOFM clusters ............ .. .... .. 28 15. Structure of th e multiple model control with s witching .... ... .... ............................... 39 16. The SO FM ba se d Modeling Architecture for Tim e Series .. ... .... ...... . .. . . ..... ......... 4 7 17 50point prototype co ntrol input s ... .. .... .... ................. . .. . .... ...... . .. .... ........ .. .. ..... 51 Vil
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18. A sing le tap of the Mach number preprocessor ........................ . . .. .. . .. .. ............... 55 19 Mach number responses and corresponding SOFM for input class_O . .. .. .... .... .. 56 20. Mach number response s and corresponding SOFM for input class_l .... .... .... .... .. 57 21. Mach number responses and corresponding SOFM for input _c lass _2 .... . .. ........... 58 22. Mach number responses and corresponding SOFM for input_class _3 ... .. . .. .... ...... 59 23. Mach number responses and corresponding SOFM for input_class 4 ................... 60 24. Mach number responses and corresponding SOFM for input _c lass 5 .... .. . .... .. . . 61 25. Mach number responses and corresponding SOFM for input_class 6 .. . .. ... .. .. .... 62 26. Mach number responses and corresponding SOFM for input _c la ss_ 7 .... .. . .. ......... 63 27. Mach number responses and corresponding SOFM for input _c lass 8 .............. ... .. 64 28. Selection of SOFM by input_class . .. .... .... . .. .... ........................ .... . .... ....... . .. .. . 70 29 Candidate Control Sequences ......................... . ......... ................. . .... . ....... . .. .... ...... 72 30 Experimental Setup ...... .... ....... .. . . .. ...... .. ... .. ....... . .. ...... .. . ......... .. ...... ..... ... .. .. 82 31. Mach number controlled by PMMSC during a three hour test .... . .. ...... ............ ... 86 32. Variations of angleofattack and angleofsideslip during te s t ... .... ....... .... ....... .. .. 87 33. Winning node s 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 3 8. Comparison for controlling to s everal different set points ......... .. . ....... ...... .. ...... .. 95 39. Comparison of Control Densities during set point changes . .. ......... ...... ........ .. .. ... 96 40. SOFM O winning nodes .. . .. . ..... ... .. . .. ................ .. ............................ ................ .. 98 41 SOFM_l and SOFM _2 winning nodes ... . .. . .... ................ .. ............. .... . .. . .... .... ..... 98 42. SOFM _3 and SOFM_ 4 winning nodes ...... .. . .. ... ....... . ....... ........ ....... ...... . .... .. .... 99 Vlll
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43 SOFM 5 and SOFM 6 winning node s . ... .. .... .... . .. ... .. . .. . ..... ........... ..... . . .. .. . 99 44. SOFM 7 and SOFM 8 winning node s .. ... .... ... ..... .. . .. . .. . . .. .. . .... .... .... . .. . .. .. 99 45 Ramping up with PMMSC control .. .... .... .. ........ .. .......... . .. ... . .. ....... .... . .. . ... 1 00 46. Prediction s, re s ponses and prediction error for input cl as s O .. .. . .............. ......... 1 0 3 47 Predictions re s ponse s and prediction e1Tor for input _c las s_ l .. . .... . ... . . . ........ . 1 04 48. Predictions, re s pon s e s, and prediction error for input_class 2 .. ......... . .. . .. .... .. 1 05 49. Prediction s, re s ponses, and prediction error for input cla ss_ 3 .. . .. . . .. ....... . .. ... 1 06 50. Predictions, r es ponse s and pr e diction error for input clas s_ 4 .. .... .. .. .. .. ... .. . .. . . 1 07 51. Prediction s re s ponses, and prediction error for input cla ss_ 5 .. . .. . .. . .... .... ....... 108 52. Prediction s r es ponse s, and prediction error for input cla s s 6 ........ .... . .. .... . .. .. 1 09 53. Predictions re s ponses and prediction error for input clas s_ 7 ....... .. . . ... ... . .. . .... 1 10 54. Predictions, re s pon s es, and prediction error for input class 8 . . ......... .. ... .. .. . ..... 111 55. SOFM Prediction s of Mach number in set point regulation .... .... .... . .. .... . .. ...... .. 1 13 56 SOFM Prediction s of Mach number in set point regulation .... .. . ....... . ...... .. .. . .. 114 5 7 SOFM Predictions of Mach number in ramping ......... .. .. . . .... ..... ... .... .. ...... ..... 1 15 58. A s imple nonlinear sy s tem with f e edback .. .. . ......... . .. . .. .. . ...... .. . . ..... . .. . .. ... . 1 2 0 lX
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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 16FOOT TRANSONIC TUNNEL WITH THE SELFORGANIZING FEATURE MAP Chairman: Dr. Jose C. Principe By MARK A. MOTTER May 1998 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 X
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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. Xl
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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 fmite 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 1
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2 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 l ocal 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 costeffective 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 [N arendra 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 16Foot Transonic Tunnel at the NASA Langley Research Center in Hampton, Virginia The NASA Langley 16Foot 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
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3 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 timedependent, inputoutput data. After suitably extracting the underlying dynamical model from the tunnel inputoutput 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 fmally applied as the control input to the wind tunnel. The frrst major task was to find a suitable neural architecture for modeling the wind tunnel dynamics based solely on inputoutput 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 steadystate wind tunnel response at various operating points when driven by similar control inputs. A refmement 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
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4 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 bangzerobang ( + 1, 0, and 1 ) permissible values of the control signal. If the control input sequence is considered to be a pcomponent vector with each element having a value of+ 1, 1, or zero, then there are 3 P 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 selforganizing 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
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5 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, singlereturn, continuousflow, 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 number s 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 10Foot 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 i s 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. ..
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6 Figure 1 Aerial View of the 16Foot Transonic Tunnel Figure 2. Test Section with model in place
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I TS 814 2 Turning vanes (third set) TS 747 2 ,Air exchange TS 500.3 TS 434 1 1 Turn i ng O : : vanes rive : l ~ Pressure drop screen ~ Return passage (second set) motor no. 2 I Airflow ___ tllL 1 4 < __ __ I t+rr l I I ~ _ .. ~ a:=.. . 1 I I , \ I 58 0 .,I I i I I I I I I I tc~ :: : :.=:~ >j Plenum valve ~ S ..,_ L.....,' lcd b.l urge 7, 1 Entrance cone '\>control Exhaust Motor 1 : : : : : I ..1.l 1 Qu1escent :=== ~ :::=:!:= ~ 1 I A"rfl I 1 chamber 1 ', 1 ow Catcher ,..,... ..... : ', \ .. 1 ""c"T' ""11 I , \ 1 <::t"r00n ti I,~ ~ I , Turning vanes (fourth set ) _, l ..c:.......,...... ".,c 7~ ~4++i +~ Plenum Y I Turning Drive TS 899.2 TS 0 0 vanes motor TS 134 0 I TS 179 0 (first set) no 1 Q TS285 9 TS 40.0TS 70 0 TS 107.0 TS 154 0 Antiturbulence screen \ ) Section through TS 134.0 (looking downstream) TS 349.1 F i g u re 3. Arr a n geme nt of Lan g l ey 16 Fo ot Tran so ni c T unn e l Dim e n s i o n s ar e in f ee t. ..J 85.0
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Figure 4 vanes 8 Inside View of the 16Foot Tunnel downstream of the second set of turning
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9 The test s ection Mach number generally referred to as the Mach number i s computed f rom a calibrated ratio of two measured quantitie s the airstream s tagnation pressure ~ ragnation and the plenum static pressure, ~ tatic The s e two mea s ured quantit ie s are used to calculate the plenum Mach number. A tabulated windtunnel calibration provides the correlation between the test s ection airstream Mach number and the plenum Mach number Th e relationship between the two measured pre s sure s and the plenum Mach number M, i s [John 1984 ; Mercer et al. 1984]: r 2 p stagnation r1 r 1 ~ tag n ation 7 1+ M 2 M= 5 1 p static 2 p static y air = l 4 ( 1 ) A large volume of test data relating the tunnel fan drive s ystem control input ( + 1, 0, 1 ), and the Mach number, i s available for nominal operating condition s over mo s t of the operating range. Data from a typical sub s onic run i s shown in Figure 5. 1 0 9 0 8 0.7 0 6 ii m 0.5 ,0.4 0.3 0 2 0 1 5 Mach number during a typical subsonic test ' ' 10 15 20 25 Time (min) . . 30 35 11~ 0.8 0 6 0 4 m 0.2 i 0 la Ir 0.2 0.4 0.6 0.8 1 0 5 Control Input during a typical subsonic test 10 15 20 25 30 Time (min) Figure 5. Mach Number and Tunnel Drive Control Input s during a typical s ub s on i c run 35
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10 Control Challenges The problem of controlling the Mach number at the 16Foot Transonic Tunnel presents the following challenges to any control scheme, including a human operatorin theloop: 1. Both the linear and nonlinear characteristics of the tunnel dynamics vary significantly over the operational range of the tunnel The ratelimited 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 openloop 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 bangzerobang ( + 1 raise 0 to maintain speed, 1 lower ) 3. The effectiveness of the contro l 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
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11 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 s ection Mach number computed from pressure measurements i s 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 drive s the Fan RPM up for approximately five minutes until the desired Mach number is attained Table 1 shows the variation of the ratelimited 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
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12 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 value s 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 st ill 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. t (seconds) M(Mach) 11.M (11.M I ~t) 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 .5 8 240 0.8285 0.0721 2.40 270 0.9076 0.0791 2.63 282 0 9421 0 0345 2.87 Table 1. Variation of Mach number ratelimited increase while ramping up
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13 Ramping t o and regulati n g O Ma ch = 0.95 1.......Angle of Attacl( 2 .5 ..~, 0.9 2 0.8 1 .5 0.7 1 0.5 0.6 i .s:: 0 Cl Q) 0.5 0 0.5 0.4 1 0.3 1 .5 2 0.1,...__~~ 2.5....._ _____ __.__ __ ___ ..__ _____ __. 0 5 10 1 5 0 5 1 0 15 Time (min) Time ( m in) Fan RP M 350.,,, 1.6,.. :=;;~, Pitch Actuator 300 250 ~200 a: 150 100 1 0 8 0 6 0.4 lo 0.2 l o a a: 0.2 0 4 0 6 .. .. .. .. 0 8 .. 1 0 5 10 Time (min) Con1rol Comm ends from Operator . 5 10 Time (min) ., Q) 1 0 .5 0 Ql 0 0.5 1 1 5 ''..__ _____ __. 15 0 5 10 15 . . I II~ 15 Time (min) Tunnel Temp e rature 100 ,..~ 160 14 0 130 u. g120 "O 110 100 90 5 1 0 15 Time (min) Figure 6. Tunnel conditions during a typical run with steady ramping to the desired te st 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.
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14 The relationship of Mach number to temperature is embedded in the definition of Mach number [John 1984] : V V M== a ,.jyRT ( 2 ) where Vis velocity of the air, a is the speed of sound, y and R are constant s for air. For a constant air velocity, the variation of Mach number with temperature is: dM dT with T in degrees Kelvin or Rankine. At the conditions for this test 1 dM dT M = 0 95 2 T = l45F 1 2 M T M T 0.000786 o p which corresponds to a 0.003 decrease in Mach number for a 3 8 degree F increa s e in temperature ( 3 ) Figure 7 illustrates in greater detail the last 3.5 minutes of the test. The te s t 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 drive s the Mach number out of tolerance, which is compensated for by the operator with a longer duration corrective input During thi s interval, the effect of the moving the model is relatively small compared to the effect o f the rising temperature but the two can act in combination as illustrated in this example
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15 Angle of Attack Set point regulation O M=0 .95 2.2r.,,,~~,,, 0.954 2 ..... ...... .... .. . ............................. . .. .. . 0 952 .,:; i 0.95 0.948 0.946 1 8 1 2 . .. 0.944 ''''''''''3' 0 8 0 0 0 5 1 1 .5 2 2 5 3.5 0.5 1 1 5 2 2 5 3 3 5 341.6 341.4 341 2 341 340 8 0.. a: 340 6 340 4 340.2 340 .. .. .. .. 339.8 0 1 0 8 0.6 0 4 0 4 0.6 0 8 1 ... .. .. .. .. 0 0 5 0 5 Time (min) Fan RP M . 1 1 5 2 2.5 Time (min) Control Commands from Ope r ator ' 1 1 .5 2 2.5 Tima (min) Time (min) Pitch A cllJator 0 6.,.,., ,,.,,,,,, 0 7>0 8 0 9 '" I 1 .. "' G) 0 1 1 .. . 1 2 . 1.3'" '. 3 o I I 1.4 ''L'L''' 3.5 0 0 5 1 1.5 2 2 5 3 3.5 Time (min) Tunnel Temperature 151~.,..,., . 150 149 u.. J148 . 147 . 146 . 3 3.5 145'''''''''' 0 0 5 1 1 5 2 2 5 3 3 5 Time (min) Figure 7. Out of tolerance Mach number extends the test duration during the last 3.5 minute s of the test @ M=0.95.
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16 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 nonadaptive automatic controller tuned to the nominal, unblocked dynamic s 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 twentyfive 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 liM. The control effort applied to compensate for the disturbance I, u, which is the samplebysample 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 liM I I.u 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.
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1 7 Regulating at M = 0.8 with angle of attack disturbance 0.82,,.;, 25 .c lll 0.815 0.81 0 805 0 795 0.79 0.785 0 78 . . .. . ........ . .. . .. .. . ... . .. 20 15 ii 10 .. 0, 5 0 5 o.n5._ __ __ __ _._ ______ ~._~ 10 0 5 10 15 0 Ir .. al !!! ., 330 325 320 315 0 1 0.8 0.6 0.4 0.2 0 a: 0.2 0.4 0.6 0.8 1 0 Time (min} Fan RP M . 5 10 Time (mi n ) Control Commands from Operator s 10 Time (m i n} I . l 25 20 r 15 10 5 r 0 r 5 r 1 0 15 0 1 43 142 141 140 u. 139 0, ., "0 138 137 136 135 16 134 0 Angle ot Attack . I 5 1 0 Time (min) Pitch Actuator I 5 10 Time (min) Tunnel Temperature 6 10 Time (min) Figure 8 Regulatin g the Mach number at M = 0.8 with larg e di s turban c e s from th e model Angl e of Att ac k \ . 15 . . 15 15
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18 AOA M1 LU Mf/I:u ( 10 4 ) 05 0.005 1 50 515 0 020 27 7.41 1520 0 017 39 4.36 2025 0 010 29 3 45 Table 2 Changes in control input effectivene s s 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 switche s between multiple, SOFM based models which, collectively describe the global inputoutput behavior of the tunnel. The tunnel re s ponse 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 fredictive Multiple Model ~witching ontroller, is shown in Figure 9. It i s composed of the following major function s : 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 pa s t n + I Mach number measurements, M(k ) M(k 1 ) .. M(kn ) and chooses a winning processing element (PE)
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19 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 l) steps of the p stepsahead 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 pstep 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 MsP 11 where Mp = M(k + l + 1), M(k + l + 2), ... M(k + p) and Msp is a (pl) length constant vector of M sP 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
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20 responses during operational research runs. Chapter 6 wi l l summarize the result s and indicate directions for future resear c h. co ntr o l u(k M ac h numb e r M (k) J ~<> <:. .,. ._., ~ V , . .; ,. Candidate controls Figure 9 Experimental Framework with PMMSC
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CHAPTER2 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 fallow. SelfOrganizing Feature Map The selforganizing 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 21
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22 The SOFM belongs to the class of artificial neural networks that use competitive or unsupervised learning. In contrast to supervised learning, the SOFM inputoutput behavior is not learned from a set of training examples which specify the desired output y E R'n, for a given input x E R", 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 defmes 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 winnertakeall (WT A) 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.
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23 Let the input be a vector x E R n : (4) With each neuron j there corresponds a vector of synaptic weights w E R n : ( 5 ) The winner is identified by the index i(x) that corresponds to the neuron whose synapti c weights are the best match to the input x: i(x)= arg min xw j j=l,2, ... ,N ( 6 ) J where 11 I 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. i(x)= arg min xw j j=l,2, ... ,N J Figure 10 SOFM with a onedimensional array of neurons
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24 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 1 < x ) identifies the neurons around the winner that will be updated as well. The width or radius of N i< x > i s usually varied over the training phase [Kohonen, 1990]. To achieve good global ordering, N i( x ) is made very wide at the beginning of the training, on the order of the onehalf the map, and then shrinks monotonically as the training progresses. The rationale for this [Kohonen, 1990] is that the wide initial N i< x > corresponding to a coarse spatial resolution in the learning process, fust induces a rough global ordering over the weight vectors. Then, as the N i( x ) 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 : if j Ni < x > ( 7)
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25 with a(k) a scalar learning rate parameter, 0< a(k) < 1, similar ,to the gain used in 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 j (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 i( x > 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 fineadjustment 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
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26 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 adju s ted during 10,000 pre s entations of the training s et with the learning rate, a ( n) and neighborhood function N i( x > varied as shown in Figure 1 2. 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 provide s a smooth organization of the weights in the neural field, in contrast with the input pattern s for training. The distribution of the training inputs among the converged SOFM clusters is shown in Figure 14 .... 0 02 0 015 0 0 1 5. 0 005 C: 0 0.005 0.01 200 Inputs f or tralng the SOF M 20 1 0 Exemplar# 0 0 n Figure 11. Input exemplars for training the example SOFM 50 4 0 30
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27 Leaming rate Nolghbomood Function 1 ' . 20 ' 0 9t18 o.e~ 16 0 1 .. 14 0.6c 12 i 10 ~0 .5 "' i5 0 4 8 0.3t6 0 2 .. 4 0.1 c 2. 0 ' ' 0 ' ' . ' 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 n n Figure 12 Learnin g rate and Neighborhood fun c tion durin g trainin g SOFM after 100 training cycles SOFM attar 500 training cycles X 10 "" X 10 a 6 12 5 10 4 8 3 j ., 2 6 4 1 2 0 0 1 20 2 20 50 40 50 30 ~10 20 20 Cluster# 0 O 10 n Clustar # 0 0 n SOFM attar 1000 training cycias Converged SOFM x 10 ..., X 10 3 12 12 10 8 8 6 I 4 .g, 6 I 4 2 2 0 0 2 2 20 20 50 50 20 Cluster# 0 O 10 n Cluster# 0 0 n Figure 13. SOFM du r ing various po i nt s in the trainin g
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28 Distribution of t r aining e n semble inpu t s t o SOF M ou t pu t s 14 1 2 '" N = 1 55 10 '" r. ,rr ,,,. 4 2 '" r. ' I 5 10 15 20 25 Clus t er# Figure 14. Distribution of training inputs among 20 converged SOFM clu s ter s Magnification Factor The input di s tribution of the ve ctor s x or the multidimen s ional probabili ty d e n si ty function ( pdf) of x p ( x ), i s repre s ented by the total N neuron s in the output layer of th e SOFM. The input v ec tors x are drawn from an ndimen s ional input space X. The pd f of x, integrat e d over all of X mu s t equal unity : 00 f p ( x ) dx = 1 (8) oo The corre s ponding density of neuron s in the output layer of the SOFM i s r e f e rred to a s the magnifi c ati o n fa c t o r m ( x ) defin e d a s the numb e r of neuron s in a s mall volume dx o f the input s pace X The integral of th e magnifi ca tion f ac tor o v er the entire input s pa ce mu s t equal the total n umber of neur o n s N : 00 f m ( x ) dx = N (9) oo
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29 For the SOFM to match the input density exactly, the magnification factor must be directly proportional to the input pdf: m(x) oc p(x) ( 10) Si and Lin [ 1997], have recently shown, for multidimensional input, the fl converged SOFM weights have a magnification factor proportional to p(x) 11 + 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 n proportional to p(x) 11 + 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 instrt1mentation, for both monitoring and control 2) Medical applications, for diagnostic methods, prostheses, and modeling
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30 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 realworld measurements, process control, robotics, chemistry, physics, elecuoniccircuit 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 controlspecific 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 lookup table, where the input pattern
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31 identified by the ''winner'', specifies an SOFM location associated with specified values of control parameters, which were learned adaptively. The two dimensional coordinates, x 1 and x 2 of a target point in the image planes of two cameras were combined into a fourdimensional, stereoscopic input vector x and used as the input to the SOFM. A threedimensional 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 0 = [ 0 1 0 2 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: ( 11) Here b e is the configuration vector corresponding to the location m e A c is the 3x4 Jacobian matrix, m e is from the weights of the SOFM winner, and (11) gives the first two terms of the Taylor series expansion of 0(x) around m e Linearization is carried out around m e and is valid in the whole Voronoi set of x values around m e Ritter et al., developed a learning scheme where the control parameters A c b e 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, b e to regions of the input vectors, x, having a higher density of lattice points where the control must be more accurate.
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32 For our application, the SOFM discretizes an ndimensional space composed of output sequences of the system, y (k) y(k1 ) . y(k(n1 )), which are considered to be the responses of the system to a prototype control input u(k1) u(k2), ... u(km). Thus, the prototype input is the control parameter associated with all the nodes in the lattice, which is here, onedimensional 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: ( 12) where M c is the winner, A c is the Jacobian, derived directly from the SOFM, u c 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 timeinvariant 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
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33 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, timevarying 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 1equired 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 online 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 timeinvariant 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 online
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34 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 S c~ 2 n 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 timeinvariant 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 inputsingle output (SISO) linear timeinvariant system with unknown parameters, described by the state equations: x(k + 1) =Ax(k) +bu(k) y(k) = cx(k) (13)
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35 correspond s to the case where some or all of the parameters of the matrix A and vectors b and c are unknown. Alternate! y if the system is de s cribed by the n th order difference equation : n 1 n1 y(k + 1) = I,a ; y(ki) + Lf3 j u(kj) ( 14 ) i=O j=O where u(k) and y (k) represent the input and output respectively at time k, the parameters a ; and {3 j are as s umed 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 P (z), of the plant described by equation ( 14) i s : f.? n1 f.? n 2 f.? JJ oZ +M z + .. + /J nl W P (z ) = n n 1 Z a oz . a n I ( 15 ) The order of the sy s tem i s n and if {3 0 ::t O, then the relative degree i s one If, however, f3o = /3 = A = ... = {3 d 2 = 0 and f3 d i ::t O, then the relative degree i s d and the input u(k) affect s the output at time instants greater than or equal to k+d. It is best to frr s t consider the case when the relative degree is one then extend the results to the ca s e when the relative degree i s 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 in s tant k has to be chosen so that ( 16 )
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36 In model reference adaptive control (MRAC), y (k) is generally chosen as the output of a reference model. The simp lest 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 nonadaptive problem, if the plant is described by equation (14) and the parameters a and f3 are known, the control law can be chosen as : 1 u(k)= f3o n 1 n 1 L,a ;y(ki)L,{3 j u(kj)+y*(k+l) i=O }= I (17) 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+ 1, and that the output of the plant converges to the desired output in one instant. For the adaptive case where the parameter s a and f3 are assumed constant but unknown, the indirect approach can be employed and requires the estimation of the A parameters a and f3. If a; ( k) and {3 j ( k) represent the estimates of a and f3 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 continuoustime and discretetime 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:
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37 1) An upper bound on the order n is known 2) The relative degree of the plant is known 3) The sign of {3 0 as well as an upper bound on the absolute value of /3 0 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 f3 j (k) re s ult in a similar control law: 1 u ( k) = ,.. f3o 11 l n 1 La ;y( ki)LP j u(kj ) + y*( k + 1) i=O j = I where the outputy(k ) follow s y*(k) asymptotically. Control Using Multiple Models and Switching (1 8 ) 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
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38 The need for multiple models in the control of dynamic systems is further elaborated by [Narendra, 1996] as: 1) Many physical systems can be repre sented 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 infotmation 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 perf onnance. A proper combination of the two may result in a stable system with better performance. The architecture of the Narendra's multiple model switchi ng controller is shown in Figure 15. I 1 / 2 , / n are N predictive models of the plant which have been obtained by observing the system over a long period of time. C 1 C 2 , C 11 are the corresponding controllers, designed offline and stored in memory. If the plant output is y(k) and the output of model I j is y j (k), the output error is defined as e j = y j (k)y(k). Based on some performance index J ( e j ) evaluated for j = l, 2, ... N the model to be used at any instant is chosen. If l ; ( k) = min j J ( e i ( k)) the model I ; 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
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39 p r event arbitrarily fast switching between models. In a more recent paper [N arendra and Balakrishnan, 1997], stability results for an allfixed 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 nonzero waiting time between switches from one model to another then the overall system is stab le, given that each fixed model is stabilized by its corresponding controller. Un Mode) I n Model / 2 Model/ 1 u Contro ll er C 1 Controller C 2 Controller C n Plant ,.. Yn + I: en ,.. e 2 Y 2 I: + ,.. Yi + I: el y Desired Output y .. Figure 15. Structure of the multiple model control with s witching Control Error An even more recent paper [N arendra and Mukhopadhyay, 1997] introduces two classes of approximate nonlinear inputoutput models which reduce the computational complexity of designing a controller based on the fact that the approximate model s are
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40 linear in the control input. This was essentially the approach taken in this experiment, where the converged SOFM provides multiple, approximate models of the inputoutput 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 nonlinear dynamical system using state equations: L : x(k+l)=f[x(k),u(k)] y(k)=h[x(k)] where {u(k)},{x(k)}, and {y(k)} are discretetime sequences with (1 9) x(k)E9tn' u(k)E9t,y(k)E9t, f :9t n x9t 9t n ,h:9tn 9t,andf ,hEC 00 The origin is assumed to be an equilibrium state of ( 4 ), hence f (0,0) = 0 The linearization of L L of L is described by the linear state equations: L L : x(k + 1) = Ax(k) + bu(k) y(k)= cx(k) (2 0) where the (n x n) mat1ix A and the (n x 1) and (1 x n) vectors b and care defined by df(x,u) Jx 0 0 =A a f (x,u) du 0 0 =b dh(x) Jx 0 =c Given this parameterization, the general state of knowledge about the system L can fall into one of the following categories: 1) f and hare known, and the state x(k) is accessible 2) f and hare unknown, and the state x(k) is accessible (21)
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41 3) f and hare 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 inputoutput 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 3050 % 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 policyiteration optimization scheme to provide suggested process setpoints for recovery from longterm drift in equipment used in the plasma etch process. The combined optimization scheme suggested ''reasonable low cost solutions'' for what were considered outofcontrol 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 perceptron 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 neuralbased control to the targeted process.
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42 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 neuralnet automation of windtunnel 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 3060% overall performance improvement over the conventional controller normally used. Their results were based on a pstep 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 closedloop 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 net s are wellsuited to process control since they can be used to build predictive models from multivariate sensor data generated by process monitors.''
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CHAPTER3 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 timedependent, inputoutput data. The motivation for this approach came from previous work by Principe and Wang [1995], using the selforganizing 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 predeterrnined 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 43
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44 the control space, C, S x C c 9\ 11 x 9\ to a tangent space Tc 9\ n If a closedform solution for (22) exists, that is : : S x C S then for a given initial condition x(O) and u(t) specified for all t, (x(O), u(t)) represents a statespace 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 statedependent, or statefeedback, 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 nonlinear dynamical system using state equations was stated in Chapter 2, repeated here for convenience: :t: x(k+l)=f[x(k),u(k)] y(k)=h[x(k)] where { u(k)}, { x( k)}, and { y( k)} are discretetime sequences with (25) x(k)E9\ 11 u(k)E9\,y(k)E9\, f :9\n x9\ 9\ 11 ,h:9\n 9',andf ,hEC 00 Here/is a
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45 map from the space of system states and input to the space of system states 9t n x9t 9t n and his a map from the space of system states to the output 9t n 9t 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 Uc where : I (2 6 ) is the ith candidate control sequence, and : (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 observation s : I) 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.
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46 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 wa s 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 winnertakesall neuron was identified by the competition. Each neuron in the neural field layer corresponded to a ~ specific processor F; :[a i ,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.
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Approximation of Dynamics Neural Field A Competitive Selection i (x) 47 ~ F(x) Input x Figure 16. The SOFMbased Modeling Architecture for Time Series Reconstruction of the state space from the training signal. Following the approach by [Takens, 1980], a sequence of d + I dimensional state vectors [x(n) x(n +r)] was created from the given training time series, where x(n)=[x(n(d 1)r), x(n(d 2)r), ... ,x(n)] and 'l' is the appropriate time delay where d ~d A and d A is the dimension of the underlying dynamical process. Mapping the state space in the neural field. This step was accomplished via the Kohonen learning process. With each vectorscalar pair [x(n) x(n +r)] presented as the input to the network, the Kohonen algorithm adaptively discretizes the continuous input space X c R d+t into a set of disjoint cells A to construct the mapping : 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
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48 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 i EA its local linear predictor in terrr1s of [a ir ,b;] is estimated based on a i c A, which is a set of L i neurons in the neighborhood of u ; including u i itself. Each of them has a corresponding weight vector [wr ; ,w,. (d+l)]r ERd+l where w r =[w 1 (l),w (2), ... ,w ( d)]. The 1 i 1 1 i j 1 1 local prediction model [a; r ,bi] is fitted in the leastsquare sense to the set of weights in a : I (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) = F'; (x(n)) = a 1 r x(n) + b ; (29) where i is the winnertakeall neuron identified by competition in (6 ). Modifications for SOFMbased Predictive Control From (25), consider the output of the nonlinear system L : y(k) = h[x(k)]=\J' 1 [x(k)] y(k + 1) = h[f (x(k ), u(k ))] = \J' 2 [x(k ), u(k )] y(k+2) = h[f( f(x(k),u(k)), u(k+l)]=\J' 3 [x(k),u(k),u(k+l)] (30) y(k+n) = h O fn[.,.]=\J'n+i[x(k),u(k),u(k+l), ... ,u(k+nl)]
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49 where f n is an ntimes iterated composition of f Denoting the sequence y(k+ 1), ... y(k + n) by :i,: (k) and the sequence u(k), u(k+ 1), . u(k + n l) as U n (k) (30) can be expressed as : ( 31 ) For SOFMbased predictive control, the thesis is that a set of feature maps can, collectively, be a global representation of these ntimes 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: ( 32) where U P U c I is the LI norm of the difference between the prototype control, U P and the candidate control, U c and V P is the Jacobian with respect to the control, extracted from the converged SOFM weights Ideally, perhaps, there would be an individual SOFM, i for each candidate control, Uc =[u c, (k) u c 1 (k + 1) ... u c 1 (k + pl) ] and predictions of the tunnel
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50 response, MP 1 =[ y P ( k + 1) y P t (k + 2) .. y P t (k + p) ] would be made using the SOFM winners: (33) This would not have explored the concept of being able to extract a model that wa s locally linear in the control from the SOFM and would have required exce ss ive 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. Fir s t, the control input space was manually partitioned by the construction of s ignificant 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 respon s e to a set of control input prototypes provides a s ufficiently complete modeling infrastructure given the de s ired objective of predictively controlling the tunnel. There is no need to provide an infrastructure capable of modeling the response to all possible 3 P control s equences of length p, becau se it is a ss umed that the control input s applied to the tunnel, at least in th e PMMSC mode of operation, will come from the known set of candidate controls, which
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51 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 50point control prototypes. The 10point prototypes were composed of ei t her all+ 1' s or all 1 's. .... (l) ;= 0 ' ,..... I a> en CtS cc ,..... + 1 0.5 ..... 0 ..... 0.5 ..... 1~ 0 50 sample protoype control vectors 2 4 6 10 20 8 0 Prototype n Figure 17. 50point prototype control inputs 40 50 30
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52 For convenience, the prototypes were assigned labels such as input_class_O, input_class_l, input_class_2, etc Input_class O, input_class_l, and input_class 2 are 50point control sequences consisting of all zeroes, all + 1 's and all 1 's, respectively. Table 3 lists the features of the prototype control vectors. input_class n Composition Control function input_class_O 50 Fifty zeroes Steadystate input class_l 50 Fifty+ 1 s (Raise) Ramp up input_class 2 50 Fifty 1 's (Lower) Ramp down input_class_3 50 Ten + 1 's, forty zeroes End of Ramp up input_class_ 4 50 Ten 1 's, forty zeroes End of Ramp down input class_5 50 69 zeroes, 14 + 1 's, forty zeroes Positive correction input class_6 50 69 zeroes, 14 1 's, forty zeroes Negative correction input_class_ 7 10 Ten +1 's Positive transition input_ class_ 8 10 Ten 1 '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
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53 practical interest, particularly when considering the desire to minimize control activity while regulating about an operating point. An alternating sequence of+ 1 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 accompli s hed with input _c lasse s_ l, 2, 3, and 4. Input cla sses_ 5 and 6 would be used for regulating about a given operating point as well as rejecting distwbance s Input _c lass O provide s the control input for the ideal s teadystate condition with no disturbance, requiring no control action over a 50point sample interval. Input _c lasses 7 and 8 provide a tran s ition from the zeroinput class to the ramping input s of input _c la ss es_l and 2 and provide identification of the tunnel response over a shorter, more recent interval of time Clu s tering the Mach Number Responses For each of the control input classes, ensembles of Mach number re s pon ses resulting from the application of each control prototype were extracted from the wind tunnel te s t data. Next, each ensemble of responses wa s 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 s ample intervals Collectively, the SOFM (s) were trained u s ing data extracted from more than 20 hours of actual wind tunnel response data Table 4 li s t s th e number of exemplars for each class.
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54 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 _c lasses 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 input _c lass_O, 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 input _c lasse s, 5 & 6, represent prototype po s itive and negative correction s which provide disturbance rejection and regulation about a set point, with the desired features of the control se quence, i e minimum control effort and minimum number of switchings or transitions from one s tate 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 exemp lar s for each input class
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55 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 samp le intervals as the application of the control prototype, M=MM(tn); M = M ( t), M(t1), ... M(tn); (3 4 ) (3 5 ) and n is either 50 or 10. Thus, M represents the output of an ntap delay line, where the value at the nth tap is su btracted 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 de component yielding the change in Mach number over the past n samples. Both the training sample s and the online Mach number respon ses were preprocessed in this fashion. M(t) 1 z 1 z ' 1 z Figure 18. A single tap of the Mach number preprocessor M (tn)
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Q) 0 5 100 4 3 2 .c 1 CG 0 a> 0 1 2 3 20 X10 3 5 80 60 Exemplar 15 10 Cluster# 56 Mach responses tor input c/ass o 40 30 40 20 0 0 10 n SOFM weights for input class o 20 30 40 0 0 10 n Figure 19. Mach number responses and corresponding SOFM for input_class_O 50 50
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.c (.)
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0 01 0 .c 0.01 0 co :!E 0.02 .s Q) o 0.03 0.04 0.05 0 0 0 01 0.02 .c g E 0.03 co .... Q) o 0.04 0.05 ...... 0 06 o20 40 5 58 Mach responses for input class 2 10 20 60 30 80 40 100 50 n Exemplar SOFM weights for input class 2 10 10 15 30 40 20 50 n Cluster# Figure 21. Mach number responses and corresponding SOFM for input_class_2 0 0
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.c 0 (tJ (tJ Q) 0 025 0.02 0 015 0.01 0 0.005 0 0.005 0 025 0.02 "fi 0.015 (tJ !!! cu 0.01 0 0 .00 5 0 20 40 ..., 15 59 Mach responses for in put _class_3 Ir \ .,, Ir .,.. ,,, rv ., "' .,.. ,,,, r I/ rr \ '~ r ~ J, _Yr .,, /'I ,r I .l .. ~:#l / 'lliN I I I ~ r ,}~ Dr .. \,, y ., V ,., V :,, r JI r '
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o oos 0 .c o oos 0 cu 0 01 Jg CD a 0 015 0.02 0. 025 0 005 0 0 .c o oos 0 cu 0. 01 Jg CD C0 015 0.02 0 025_.....,. 0 ..,,,. ~; ... .., 1N+HITT!~ lJ 'I 1 [ l 10 .. 20 Exemplar 5 10 15 Cluster# 60 Mach responses for input_class_ 4 ,..,. I ,\ A;"rti / ,..'>... I ~'~U.,~HJ"Ji ~~,,,,...__ !/ ,'\ ,,, \ "' "'~ ~... it: '\.'>,,I" > UJ 111' ,, r~ ., ... .J TT Iii, ,.,._._., uk )I ... '" ) '\ V .le')( ,,. ... }.., ., \~ \ ~, \ I) ~s \I{',, '' r;~'/1, , 20 30 30 40 40 so n 10 20 40 20 so n Figure 23. Mach number responses and corresponding SOFM for input_class 4 0 10 0
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.c 0 n, :E n, Cl) a ..c 0 02 0.015 0 01 0 005 0 0.005 0 01 200 10 8 6 X 10 3 4 E 2 Cl) a 0 2 4 20 61 Mach responses for input class 5 50 100 40 30 20 10 Exemplar 0 0 n SOFM weights for input class 5 10 40 so 30 20 10 Cluster# 0 0 n Figure 24. Mach number responses and corresponding SOFM for input_class_5
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.c 0 co 5 0 Q) 0 5 .c 0 co 5 0 X 10 3 5 co ::: Q) 0 10 5 62 Mach responses for input class 6 80 100 50 Exemplar 40 SOFM weights for input class 6 10 15 40 20 50 Cluster# 0 30 n 0 10 20 30 n Figure 25. Mach number responses and corresponding SOFM for input_class_6
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.c 15 10 Cd Q) 0 0 5 100 0 014 0 012 0 01 g 0 008 E 0.006 Q) 0 0 004 0 002 X 10 3 0.... 20 60 40 Exemplar 10 Cluster# 63 Mach responses for input class 7 8 6 4 2 0 0 n SOFM weights for input_c la ss_7 8 6 5 4 2 0 0 n Fig11re 26. Mach number responses and corresponding SOFM for input_class 7 10 10
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.c 0 ro ro Q) 0 2 0 2 4 6 8 10 12 0 0 0 002 0.004 .c 0 ro 0 006 0 008 (l) 0 0.01 0.012 0.014........._ 0 20 40 5 64 Mach responses for input class 8 2 4 60 80 8 100 10 n Exemplar SOFM weights for input _c lass 8 2 10 4 6 15 8 20 10 Cluster# n Figure 27. Mach number responses and corresponding SOFM for input_class 8 0 0
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65 Convergence of the Input Neural Fields The number of nodes for the SOFMs, each representing a cl u ster of the Mach number responses, was adjusted during t he 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 thi s 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 samplesahead 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: 50 LM : (i,j) ( 36 ) i=2 l 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_O was even less than the above classes, and the mean separations for the ramping input classes SOFMs were deemed sufficient for the relatively coarser control required to move from one set point to another.
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66 1 i= lO 1 50 50 (10 3 ) L M:(i,j+l) L M;(i,j) c10 3 ) L < ) 30 10 i=I .... i =2l i=21 k=O k=l k=2 k=3 1 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 0.447 2 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 .43 8 2.220 0 3767 0.4339 0 .2 112 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 .2 699 8 0.0129 0.3392 0.5479 0 8365 0.2759 0.3067 0.0531 0 1421 0.1933 9 0 0 1 66 0.8414 0.1769 0.3579 0.5014 0.4656 0.4 1 98 0.089 0.041 6 10 0 .1 377 0.8744 0 3810 0 1971 0.5801 0.2794 0.4 1 87 0.0292 0.1017 11 0.3549 0.8427 0.8745 0.4618 0.4409 0 0 1 69 0. 1 808 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. 1 219 1.009 0 5736 0 6931 0.96 1 6 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 0.4 1 75 1 130 0 7018 0.4792 0.6299 0 7110 0 771 1 0.2827 0.3107 17 0.3741 2 039 1.055 0.3583 0.8048 0 5912 1.417 0 .3 475 0.2851 1 8 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 .7 670 0.7153 0.6259 0.6221 0 6809 0 2003 0 2550 T a b le 5. D iffe r e n ce betwee n interval mea n s of a d jace n t i n p u t n e ur al fie l ds
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67 ,... 1 50 ,... ( 10 3 ) LM ; ( i ,j) 1 10 lO ~ M;(i,j) 3 0 i 21 11 1000 k=O k=l k=2 k=3 J 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 1 0.48 0.24 5 2 0.4755 2.698 3.117 6 0.5549 28.30 30.25 1 1.87 1 2.67 0 62 1 9 0.9095 2.910 3.331 7 0.3210 29.24 31.03 12.47 13.82 0.8897 1 1 30 3.010 3.536 8 0.2645 29.85 31 82 1 3.54 14.26 1.104 1 124 3.403 3.806 9 0.2519 30 19 32.37 1 4.37 1 4.54 1 4 11 1.1 77 3.545 4.000 10 0.2349 31.03 32.54 14.73 15.04 1 .876 1 .597 3.456 4.041 1 1 0.0973 3 1 .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 1 6.64 2 250 2 4 1 4 4.002 4.698 14 0.6830 35.12 35.04 1 6.64 17 .59 2.444 2.998 4.236 4.859 15 0.7 1 69 36.01 35.60 1 7.25 1 8.33 2.887 3.631 4.382 4.906 1 6 0.8387 36.72 36 78 17.70 1 8.79 3.584 4.149 4.520 5.111 1 7 1 .256 37.85 37.50 1 8.18 1 9.42 4 295 4.9 1 9 4.803 5.421 1 8 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 1 9.76 20 9 1 6.441 8.806 5.625 6.309 20 2 703 40.74 42.39 21 15 2 1 .35 9.176 11.06 6. 1 50 6.939 T a bl e 6. Int erval m ea n s o f SO F M i n put fi e ld s
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68 c 1 o3) I M ; (i, j) I k=O k=l k=2 k=3 J 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. 1 3 30 09 6.997 4.947 5.456 5.658 3 8.921 141.6 158.5 39.4 1 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 7 1 3 6.448 12 5 1 14.13 9 2.933 176.0 188.6 8 5. 83 88 76 8.406 6 576 13.17 14.82 1 0 2.632 180.9 189.5 88.40 91.33 10.55 8.931 1 3.21 15.38 11 1.902 185.9 191.9 90.99 94.44 1 2.09 1 1.24 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 1 4.24 14.92 17.24 14 4.774 204.4 204.2 100.3 106. 1 14.36 17.37 15.58 17.88 1 5 5.547 210 0 209.8 1 04.7 1 11.2 1 6.69 20 30 16.41 18.44 16 6.890 213.9 214.6 1 07.5 115.1 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 1 8 10 63 231.9 225.3 113.3 1 23.4 27.28 36.12 18.92 21.10 19 1 3.57 241.6 236.9 1 1 9.0 1 26.7 35.86 49.03 20.78 23 18 20 1 7.47 236.9 247. 1 126 3 1 28.2 50.95 6 1 .49 22.89 25.40 T abl e 7 E u c lid e an norm of S O FM in p u t n e ur a l f i e ld s
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69 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 weight s. Table 7 enumerates the Euclidean norm for all the converged neural fields as well. With the exception of SOFM O, the norms steadily increase (or decrease) along the output field of the map. SOFM O 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 norrr1 between the control input history U = u( t 1 ), u( t 2 ), .. u( t m ) and the se t of prototype control vectors U i; i= l,n : input cl as s i = m 1 n U U . { I (37) If more than one prototype control vector matches identically, i.e. U U i = 0 for more than one i both SOFM (s) are excited with the appropriate length M. This can occur for SOFM l (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 l ( 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 s ingle point in the control space.
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70 The SOFM metric for the winner is the minimum Euclidean norm between M and the SOFM prototype vectors, Mi for the SOFM selected by input_class : mach class i = minjlM Mi j. i= l : 2 0 ( 38) ( t 1 ) ( t 2 ) SOFM O I I u u 1 I z input clas s_ i ( t m ) __ s0FM 1 (t) (t1) M ___ soFM s 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: ( 39) where w* is the prototype response vector, or weights, of the winning node.
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71 A is the least square approximation of the winner's prototype response to a dsampl e delayed unit step sequence U = [0 0 ... 0, 1,1, . l], where d represents the maximum ~v' \, I "v d pd relative degree or delay from input to output and p is the total number of samp l es ahead for which the prediction is made : A =bU r ( 40 ) where bis fit in the lea s t square sense or alternately ( 41 ) and O denotes the MoorePenrose p s eudoinverse of the vector U,. By inspection of the SOFM for all input classes, d was chosen con s ervatively to be greater than any observed delay d = 20 A single constant, a c s cales A based on the ratio of the Ll norm of the candidate control vector U c and the Ll norm of the control sequence U producing the response M: a = C a = C ( 42 ) Thus, a c A provides the difference in the predicted Mach number response due to the to the distance between the control sequence, U and the i' h candidate control sequence U c For the simplest ca s e, U c = U the value of a is zero, and the Mach number response is predicted directly from the input neural field. Thi s linear model i s driven by the candidate control input s, shown in Figure 29
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a.... Q) 0 .J Tl ........ Q) U'> cu cc T0.5 + 1 0 1 \ 10 20 Candidate# ,11111 11111 IU ' ' _, ' ~ ', 'I ,.,. 30 '\ 0 ' ' 72 Candidate Control Inputs u(t+n) ' ' ' ...I ,. ....l '11''.....J, ...JI .. ,. 'I ..l _l ___: .. _:i .....: _, \ '\ 10 20 ' , _l ll ~ ' ' ' 30 Figure 29. Candidate Control Sequence s n 'I ...II ]I __J ' _, ' _, ~ ' \ 40 so Comparison s of the prediction s of the Mach number to the actual tunnel re sponses as a re s ult of the application of the candidate controls will be pre se nted in Chapter 5 Experimental Re s ul ts
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CHAPTER4 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 input s The advantage of partitioning the control input space using a set of prototype controls becomes more apparent when compared to modelbased 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 multis tep prediction by this method allows prediction for relatively long (50 samples ahead) control sequences, or contr o l hori z on 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 ofMPC is provided as a basis for comparison to SOFMbased predictive control using control prototypes. 73
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74 Model Predictive Control Background Most inputo utput model based predictive control schemes [Clarke, Mothadi and Tuft s, 1987] begin with the a s sumption of a linear model ( ARMA, or Autoregre ss ive Moving Average ) : n n1 y( k ) = L,a iy( k i ) + L,b j_ 1 u ( k j ) i = I j= I with an additional di s turbance term in moving average form : nc d ( k ) = f,(k ) + L,c ; /;, ( ki ) i=I where f, ( k ) i s an un c orrelated random s equence Combining ( 43 ) and ( 44 ) and introducing the polynomials A, B and C in the backward s hift operator q 1 : B ( l) b b 1 b nb q = 0 + l q + ... + nb q C ( I) 1 1 n c q = + c, q + . + c,. c q yield s ( 43 ) ( 44 ) ( 45 ) which is referred to in the literature as the CARMA (Controlled AutoRegres s ive and Moving Average ) model, a variation on the ARMAX ( AutoRegre s sive Moving Avera g e with exogenou s input ) model. A further refinement to the disturbance model to accommodate nonstationary di s turbances such as random step s occurring at random time s i s :
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75 d ( k) = C(q 1 )~(k ) IL\ where L\ = I q 1 the differencing operator Combining ( 45 ) and ( 46 ) yields the CARIMA (Controlled AutoRegressive Integrated Moving Average ) model used in Generalized Predictive Control ( GPC): where: At thi s point it is useful to introduce a s calar co s t function J : Ni Nu 1 = L[ y( k + i)w ( k + i ) ] 2 + L A ( j ) [u ( k + j1 ) ] 2 j = l y is the predicted respon s e from the control input sequence u N, i s the beginning of the costing horizon ; N 2 i s the end of the costing horizon ; N u is the control horizon ; A ( } ) is a controlweighting sequence. N,, N 2 N 1 , and A ( j) represent tuning knob s which can be adjusted by the control ( 46) ( 47 ) ( 48 ) de s igner to tailor the c ontrol action for the de s ir e d re s pon s e c hara c te1 istic s. Rule s of thumb provide s ome guideline s for initial selection N 1 i s u s ually picked to be greater than the largest anticipated time delay between the input u(k) and its response in the output y (k ) N 2 i s deter1nined by the longe s t settling time as s ociated with the pul s e o r s tep re s pon s e of the model N 11 = 1 is quite often cho s en for open loop s table non minimum ph as e plant s but thi s often repre s ent s a c ompromi s e betwe e n the computational burden a ss ociated with longer control horizon s.
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76 The minimization of J, given a future set point sequence w, where : w = [ w(t + I),w(t + 2), ... w(t + N)] leads to the control law : The matrix G is of dimension N x NU: G 0 0 0 ( 49) ( 50) ( 51) This requires the inversion of an NU x NU matrix at each sample time, or at least for each identified change in the g parameters, which are the coefficients of the ztransform of the plant's step response. f is a linear combination of values of u(t) and y(t) up to time t. SOFMbased Predictive Controller The function of the SOFMbased 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 50pointsahead predicted Mach number responses and the desired Mach number set point :
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77 err _norm_i = Mp, [21:50]MsP ( 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 steadystate matching. The control sequence associated with the minimum nonn 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 = p N2 L,[y P (k + i) w(k + i)] 2 i=N 1 1 /2 with N 1 = 21 N 2 = 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 u = 50 The control U P that generates y P is selected for the minimum J P ( 53) 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 fust point in the sequence. Implicitly, thi s se l ection 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
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78 minimum control effort (i.e. zero) over the control horizon, the control sequence is updated by selection of the predictionerror 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+ 1), u(t+2), ... u(t+50)] SOFM k 1 [ (50) + 1 's] 1 1 2 [ ( 11) + 1 's (39) zeros] 3 1 3 [ (10) +l's (40) zeros] 3 50 4 [ (9) +l's (41) zeros] 5 50 5 [(8)+1's (42) zeros] 5 50 6 [ (7) + 1 's (43) zeros] 5 50 7 [ +1 +1 +1 +1 +1 +1 (44)zeros] 5 50 8 [ +1 +I +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 + I ( 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
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79 18 [ 1 (49) zeros] 6 50 19 [ 1 1 (48) zeros] 6 50 20 [ 1 1 1 ( 4 7) zeros ] 6 50 21 [ 1 1 1 1 ( 46) zeros ] 6 50 22 [ 1 1 1 1 1 ( 45) zeros] 6 50 23 [ 1 1 1 1 1 1 ( 44) zeros] 6 50 24 [ (7) 1 s ( 4 3) zeros ] 6 50 25 [ (8) l's (42) zeros] 6 50 26 [(9)l's (4l)zeros] 6 50 27 [ (10) l's (40) zeros] 4 50 28 [ (11)l's (39) zeros] 4 1 29 [ (50) 1 '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 1 '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_l (ramp
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80 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 controltick 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 predictionerror minimizing control. The s e control sequences (#3 or #27 ) are applied for their entire 50point duration, allowing for a s mooth 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 consi s ting of an active or non zero s egment of 1/3 to 10 sample periods, followed by the corresponding number of zeroes during the inactive segment, i s applied a s 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 50sample Mach number response is then input to the corre s ponding SOFM and future prediction s are made from the output neural field of the SOFM winner a s described in Chapter 3. If the s equence of all zeroes is selected the control is updated at the next s ampl e period. The 50 sample Mach number response is input to SOFM O for identification o f the local dynamics by the SOFM O winner. Prediction and control sequence selection is perfonned at each s ample period until an active ( nonz ero) control s equence i s s elected to regulate the Mach number.
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CHAPTERS EXPERIMENT AL RESULTS In Chapter 3 and 4, the SO FMbased 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 3 P 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 gainscheduled 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 controlinput partitioned SOFM architecture. This will be accomplished by comparing the Mach number response predicted by the SOFMderived local model to the actual response after application of the errorminimizing control sequence determined by the predictive controller 81
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82 Experimental Setup The experimental setup consisted of a 48633 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 8086based microcomputer. The existing automatic control implemented in the tunnel micro is a highly tuned but fixed table lookup of drive motor commands based on the error at a given Mach number [Capone et a l ., 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. C coded PMMSC Mach number Control commands I Mach I number Control commands I I I I I I Existing controls I I I I I I I I I I I I I I ___ __ ____ _ __, 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
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83 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 fmer 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 8ft calibrated test section length (CSTL). Flow uniformity was parameterized by both the standard and maximum deviation of spatially local Mach number from a leastsquares straightline 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 leastsquares straightline fit to the Mach number data, corresponding to the midpoint of the test section The standard
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84 deviation is a measure of the average discrepancy along the test section length. The maximum deviation represents the worst departure from the leastsquares fit along the selected length of test section. The document reporting the results of the calibration lists 2a values, i.e. twice the positive square root of the variance. Mr 2a amax 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 steadystate 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, Mis the mean value of 200 consecutive Mach number measurements taken under steady
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85 conditions with no control input applied. The standard deviation is the sample standard deviation: l N 2 a 2 = L,(M (i)M) ;a= [ci1; N 1 i= l where 2a is used for direct comparison to the calibration results. M 2a 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 ( 54) Figure 31 shows the wind tunnel Mach number being controlled by the PMMSC fo r 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
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86 shown as dashed lines The PMMSC regulated the steady state Mach number to within the research requirement of 0.003 of the s et point during the interval shown PMMSC command s are s hown with magnitudes les s than one for control commands who s e duration was less than the 0.3 second sampling period. Control pulse s of 0.1 second are s hown with magnitude 0.33 and 0 2 second pulses are shown with magnitude 0 66 1 0 9 .c 0 8 (.) co 0.7 ... ... ... ... 0 6 0 5 0 .... 0.5 Q) ?: 0 ..J Q) 0 en co a: 0 5 I I I I I I 2 0 40 60 20 40 60 Mach number I I I I I I I I I I I I i 80 100 120 140 160 NNCPC command 80 100 120 140 160 Time ( min ) Figure 31. Mach number controlled by PMMSC during a three hour test 180 180 T he PMMSC wa s previously referred to a s NNCPC so thi s acronym appears in som e of the plot s.
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87 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. (/) Q) 10 5 C) Q) "'O 0 Angle of Attack (alpha) 5 LJL......__1. __ __._.:__ __.__ _ _,__ __ __._ __ _._ __ .....__J 0 20 40 60 80 100 120 140 160 180 10 5 ._ (/) C) 0 Q) "'O 5 t10 0 I I I ,J I I I 20 40 60 Angle of Sideslip (beta) I I I I I I I I I I 80 100 120 140 160 Time (min) F i gure 32. Variations of angleofattack and angleofsideslip during test 180 The variations are of two general types, referred to as an ''alpha sweep '' or ''beta sweep''. During an alpha sweep, the model angleofattack or ''alpha'', is stepped through some range from 4 degrees to+ 12 degrees in this test, while maintaining a constant angleofsideslip, or ''beta' During a beta sweep, the model angle o fsideslip i s 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 requirement s. The variation in model attitude produced some modest
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88 ( < 0.001/degree) di s turbance in the tunnel Mach number, particularly at anglesofattack above five degrees although the onset of this disturbance was dependent on the test Mach number Figure 33 show s the SOFM winning nodes for positive and negati ve corrections a s determined by the PMMSC during the run Figure 34 s hows the Fan RPM and tunnel temperature during the run. 20 15 10 5 0 0 20 15 10 5 0 0 J 20 40 M 20 40 Raise correct i on c l ass ' ' ' 60 80 10 0 Lower correction class ' ' 60 80 100 T ime (min) , r lr ' 1 20 140 160 180 I l 120 14 0 1 60 18 0 Figure 33. Winning node s for SOFM 5 and SOFM_6 during te s t u. 340 320 300 280 260 2 4 0 ... ... ... ,0 1 20 1 10 g,100 'O 90 ... ' I 20 40 60 ' RP M I . . ' 80 10 0 1 20 140 1 60 180 Tunnel T emp erature ' \ 80'__._. __ ...... __ __.. ___ .__. __ .t_, __._, _ _... __ ....J'L_J 0 20 40 60 80 10 0 1 20 140 160 180 Time (min) Figure 34. Fan RPM and Tunnel temperature during test
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89 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 c l assified 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 angleofattack, from 12 degrees to near zero, causes the Mach number to jump up even further. Successive lower corrective pulses bring the Mach number back, whi l e changing the SOFM winner of the lower corrective response, seen in Figure 35
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0 856 0 854 ii 0 852 Ill ;:e 0.85 0 848 0 846 95 1 0 5 I o Ill ai a: 0 5 1 95 Mach number 100 NNCPC command . 1 100 Time (min) 105 10 5 110 90 RPM 314.r=,,, 313 .5 313 312.5 312..._ ______ _._ _______ ..__ ______ _,, 95 100 105 110 Tunnel Temperature 105.,.., 104 5 104 Cl G) '0 103 5 103 102 5 ________ .__ ______ __, _______ __ 110 95 100 105 110 Time (min) Raise correction class 15r,..~ Angle of Attaclc (a lpha ) 20 100 105 Angle of Sldeslip (bela) 15 .. 10 .. 5 0 110 95 . ' 100 105 110 Lower correction class 2.. ...~ 18,, ...... ,, 01.J\.... 16 14 6 ____.,. 100 105 Time (mi n) 10 .....__ _______ _. c._i. ______ ____, 110 95 100 105 110 Time (min) Figure 35 A 15 minute interval of the test
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91 0,456;,,., Mach !MIiter O 0.85 lilt! exlsfng contollar 0,856;,,,, 0 856~. 0.854 O.B52 OM6 0.8 ...... ........ ................. .. . ....... ., .. ..... .. Melll : 0.!497 Slandald demon: 0 001527 Out oJ illlw"101 : 46.5 NCOndl 0.854 0 852 0.841 .... 00 00 .... 4 O oo O h ,. ., ,, ,,,,,,. ,.,,.,,. .,,,. 0 ,, 0 ,.,,,0+00,00,0 0 0 ,, .. ..., h ,...,,, Meen : 0.8497 Standald drillfbn : 0 001226 Out al lolarance : 33.23 sec:ond1 0 854 ............................. .... ... ................................... _.,,....... 0 846 Mean : 0.8500 9andalU dMllk>n: 0 001358 0.844 0 5 10 15 0 6 TIIN(!m) 10 15 2 6 a 10 1 2 14 16 18 1 0.8 0 6 OA 0.2 0 0.2 0 6 0.B 1 0 L 1 nonn : 10 63 5 10 12 10 8 r 0 6 16 1 0 8 0 6 0.4 0.2 r 0 0 .! I[ 0 2 0 & 0 8 1 0 f1111(mil) l1 nomc 6 33 s 10 l1me (llin) Ar,Je al allact (8'lha) 14.., 1 0 8 J 6 4 2 2 ''.....1..' 5 10 15 0 5 10 15 Tl 1111 (mil ) Tinl(m) fme(llin) 1 o a 0 6 0.4 0.4 L1 nonn: 12.33 0 6 1 15 2 4 6 8 10 12 14 16 Tme(mil) 14r,,,,,,,.,, 10 I s r ,, 6 2 4 6 8 10 Time (min) 12 14 16 18 Figure 36 Comparison of PMMSC to existing control and expert operator 18
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92 Comparison of PMMSC to Existing Controller and Expert Operator Figure 36 compares the performance of the existing schedu led control, an expert operator, and the PMMSC under similar conditions over a nominal fifteen minute interval. Mach number, control commands, and test model angleofattack (disturbance) are shown for the existing control, an expert operator, and PMMSC control. Derived metrics to quantify the comparisons between the three cases are the time out of tolerance and the Ll norm of the control input, u. The time out of tolerance is cumulative sum of time that the measured Mach number deviates beyond the required tolerance of 0.003 : k = N time out of tolerance = l:lXLit (k) ; k = l where a=O if M(k)M s p (k) 0.003; and a= I otherwise ; t 3 (t(O),t(l), ... ,t(N)); ~t(k)=t(k)t(k 1). The Ll norm of the control commands is : k = N LI[u]= Llu(k) k = O (55) (56) For this particular model, the angleofattack begins to mildly disturb the Mach number at approximately 5 degrees. Table 11 lists the reduction in the standard deviation of the Mach number, time out of tolerance, control effort, and time required to complete the sweep through the desired range of angleofattack while maintaining Mach number
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93 steady for this comparison. For this particular test condition, the PMMSC performs slightly better than the expert operator, but with much less control effort and less time to complete the alpha sweep. Compared to the existing automatic control, the PMMSC maintains the Mach number within the desired tolerance much better with less control effort, completing the alpha sweep in less time, which is the most important figure of merit for the utilization of the facility. Existing Expert Operator PMMSC % Reduction controller Auto I manual Mean 0.8497 0.8500 0.8497 SD 0.001527 0.001358 0.001226 20 I IO Time out of 46.5 s 34.52 s 33 2 s 29 I 4 tolerance LI nonn [u] 10.6 I2.33 6 3 40 I 49 Aloha sweep 886 s 930 s 806 s 9 I 13 Table 11. Comparison of existing automatic control, expert operator, and PMMSC control An additional metric on the control, the control density ; was calculated by taking the sum of the absolute value of the control over a 50 sample sliding window: i = 4 9 ; (k) =; [u(k)]= L, ju(k i) i=O ( 57) The control density is used to compare the sparseness of the control between the PMMSC, the existing controller, and an expert operator, shown in Figu1e 37. This quantity measures the accuracy of the present control input, so in the PMMSC case it is a measure of the local linear models to predict the tunnel dynamics. The PMMSC is clearly the most sparse, but allows for increased density of the control when demanded by external disturbance, similar to the variation in control density employed by the expert
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94 operator. This is in contrast to the existing automatic control, with fixed gains for a particular operating point resulting in a narrow range of control density. Control densly for axl$tlng control 4 . . 4 . . ' 3.5 3.5 3.5 3 3 3 2.5 2.S 2.5 2 2 2 1.5 1.5 1.S 1 1 1 0 5 0.5 0.5 0 0 ' . 2 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 1 2 14 16 18 Tme(rin) Tme (min) Trne (m) Figure 37 Comparison of Control Densities Figure 38 compares the results of controlling the Mach number to several different set points over a nominal 28 minute interval. Mach number set points of 0.95, 0.9, and 0.6 are common to all three controllers. The PMMSC controls the Mach number to 0.85 versus 0.8 for the operator and existing controller. This difference is minimal and still provides a reasonable basis for comparison of the controllers. The angleofattack was varied extensively during all three runs. Again, the PMMSC maintains the Mach number within tolerance for a higher percentage of the time, with less expenditure of control effort. Table 12 lists the figures for time out of tolerance and control effort for the three runs. The PMMSC reduces the time out of tolerance on the order of 15 20 percent compared to the existing controller or an expert operator The control effort is reduced by 12 percent compared to the existing controller, and 20 percent compared to an expert operator
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1 0 95 0 9 0.85 ... .x C 0 8 1 0.75 0.7 0 65 0 6 0 0.8 0 6 0.4 ... 0.2 f J~.: 0.4 0.6 ~ 8 1 0 iO I i' .,, 0 so 5 ' I I OU1 ol l:>leran:e: 328 8 MC . 5 10 5 10 10 19 6 % 15 Tl!B ( nin) 15 Trne(irin) Arqeol A !ld 15 Time (min) ' I I I I I I I I I I I I I I I I I I I 20 25 l1 norm: 25 20 1 0 !15 0 9 0.85 1 o.a 2 > 0.75 > 0 7 0.65 0 8 30 0 1 0.8 0 6 0 4 > ... 0.2 t 0 ~ 2 30 ~ ~ 6 ~ 8 1 0 ... 8 6 4 2 0 0 30 40 5 95 . 1 ,. I I I I I p. Out of lderarca: 310.2 MC 5 10 5 10 Iii ,. 10 18 62% 15 TITIII ( nin) 15 Tine (min) Arq,eof Alladc I ... 16 fime(,m) r I I I I I I I I I I I I I 20 25 ' L 1 nonn : '466.21 20 2S ' J 20 25 1 0 95 0 9 > I 0.85 5 0 7 0 65 > 0 6 0.55 30 0 5 I 0 8 0 6 0 4 ... 0 2 f i 0 ~.2 ~ 6 ~.B 1 30 0 5 12 10 a 6 : 4 i' .,, 2 0 t,t,jepc conlnll tor sel poln1 chllngea ' I CM ol 101&1111C1 : 266 35 MC 10 10 15 9% 15 TIIT8 (nil) 15 Time (mlo) AtltjiJolAl!ack ' I 20 2S L1 nomt :174 27 25 61''''''....... 30 0 5 10 15 Tine{,m) 20 2S 30 Figure 38. Comparison for controlling to several different s et points 30
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96 Existing Expert Operator PMMSC % Redu c tion controller Auto I manual Out of 329 s 310 s 266 s 19.1 / 16.5 tolerance Ll norm [u] 424.2 466.2 374.3 11.7 / 19.7 Table 12. Compari s on for controlling to several different s et points The differences in the control density for the three cases are illustrated in Figure 39. The variation in the control density is greatest for the expert operator and lea s t for the existing controller The PMMSC falls between the two cases in terms of variation of the control density, while requiring less overall control effort to provide less time out of tolerance 50 40 35 30 26 20 15 10 s 1, 0 0 n 6 ' ,\. n "" ,t, ... ... ft n. .... .. n ri1 10 15 20 25 ftne(mii) 50 45 40 35 30 25 20 15 10 s 0 30 0 Cltrd densly for axpart operalor ' ' 50 40 36 30 25 20 15 10 I 11 ll 11.o .A ..1 n M.Jt s 10 15 20 25 s 30 o 5 Trne (nm) Figure 39. Comparison of Contt ol Densities during set point changes Cavol density tor NNCPC ' n 10 20 30 Tltne ( mil }
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97 Experimental Results of Modeling the Tunnel Dynamics Although the results of controlling the tunnel using the PMMSC imply some degree of success at modeling the tunnel dynamics, in this section we will explicitly compare the predicted Mach number responses to actual responses recorded during the experimental testing. This will provide some insight into the relation between the prediction error and the actual error observed while controlling the tunnel with the control sequence that was selected based on minimizing the predicted error. The experimental results presented in this section are based on all the predicted and actual responses for the three hour control test shown earlier in Figure 31. This test consisted of rather lengthy segments where the Mach number was controlled to within the 0.003 tolerance at M=0.95, 0.90, 0.85, and 0.6. The control inputs during this test are predominately from input_class_O (all zeroes), input_class_2 (ramp down), input_class_ 4 (end of ramp down), input_class 5 (small positive correction), input_class_6 (small negative correction), and input class_8 (negative transition). A second control test, shown in Figure 45, provides results from predicting responses to input_class_l (ramp up), input_class_3 (end of ramp up), and input_class_7 (positive transition). Table 13 lists the distribution among input classes for the two tests. Then, for each input_class, the relative frequency of the associated SOFM winners were determined and displayed in the corresponding histogram plots, Figures 40 through 44.
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2 000 1800 .... 1 60 0 140 0 '1200 '1 00 0 '800 .... 60 0 40 0 L. 2 0 0 0 0 98 H isto gr a m for winn i ng n o d e o f S OFM O I r I N =20859 5 1 0 1 5 C lu s t e r # Figure 40. SOFM O winning nodes l 20 90 Histogram for winning node of SOFM 1 His1ogram for winning node of SOF M _2 80 .. 70 .. 60 50 40 .. 30 .. 20 .. 10 5 10 N ri11 ,r 7 15 20 Cluster# 70 60 50 .. . 40 .. 30 .. 20 .. 10 0 25 0 NE200 __ r, 5 F i gure 41 SOFM l and SOFM 2 winning nod es n r. 10 15 Cluster # 25 . . 20 25
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Histogram for winning node of SOF M _3 4 3 .5 .. 3 IN 2.5 I2 I1 5 l 0.5 .. 5 10 15 Cluster II 20 99 3 2.5 2 I. .5 l I. 0.5 0 25 0 Figure 42. SOFM 3 and SOFM 4 winning node s Histogram f or winning nod e o f SOF M _S 18 . 18 16 I18 IN 155 14 14 .. ,12 .. 12 ,10 .. ,. 10 .. ,... ,... ,,B 8 B .. B 4 4 2 2 5 10 1 5 20 25 o Cluster # Figure 43. SOFM _5 and SOFM _6 winning nodes 100 90 80 70 BO so 4 0 30 20 I.. .. 10 N 28 ,... nnn 5 Histogram f or winning node of SOFM 7 ,,,... ,10 15 Cluster 25 N 3111 20 ,. 15 .. 10 I. 5 20 25 o Figure 44. SOFM 7 and SOFM_8 winning nodes ,. 5 Histogram for win n ing noda o f SO FM 4 N . . 5 10 15 20 25 Cluster II Histogram for winning node of SOF M B . ,... N 150 ,... ,,,... ,. ' 5 10 15 20 2 5 Cl uster# Histogram f or winning node of SOF'M B ,,,. 10 15 20 25 Clust er#
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0 .9 0 8 0.7 0.6 0.4 0 .3 0 .2 0 1 1 0 8 0.6 0.4 .._ 0 .2 0 U)
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101 input_class Figure 31. Figure 45. 0 20859 64 1 0 668 2 200 0 3 2 4 4 3 0 5 155 27 6 150 10 7 2 828 8 319 0 Table 13. Distribution among input_classes for Figures 31 and 45. As a measure of the error between the predicted and actual responses to the control input sequence selected by the predictive controller, an average of the multistep prediction error over the la st 30 steps of the prediction was calculated by: 1 11 =5 0 = L M(k+n)M *(k +n)j 30 11 =2 I ( 58 ) This measure is the average absolute value of the stepbystep prediction error over the last 30 prediction steps. This i s the same interval over which the predictive conuoller evaluated the re s ponses to candidate controls, thu s providing a direct mea s ure of the difference between the predicted re s pon s e and the actual respon se. The average
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102 absolute value of the error is a more intuitive choice over the Euclidean norm here, given the small absolute value of the control tolerance. The SOFM' s associated with input classes_ 7 and _8 are only used to predict the Mach number response over the next ten sample periods, so the average multistep prediction error is modified to cover only the frrst ten points of the predicted response : 1 11=20 10 =I M(k+n)M *( k+n) 10 n= l 1 ( 59) Figures 46 through 54 show ensembles of predicted and actual Mach number responses for all the input_classes. Additionally, the average multistep prediction error for each prediction is also shown. The mean value and standard deviation for taken over all predictions for each input_class, are listed in Table 14. input_class N Mean SD Control Function 0 10000 0.000947 0.000401 Steady State 1 668 0.0065 0.0041 Ramp up 2 200 0.0039 0.0030 Ramp down 3 4 0.0046 0.0018 End of ramp up 4 3 0.0031 0.0019 End of ramp down 5 155 0.0018 0.0012 Positive correction 6 150 0.0017 0.0012 Negative correction 7 828 0.0014 0.0010 Positive transition 8 319 0.0016 0.0010 Negative transition Table 14. Multistep prediction errors for all input_classes
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0.01 0.005 0 005 0 01 0 0.01 0.005 0 005 0.01 _...__ 0 2 4 X 10 4 2 2 2 1 8 1 6 1 4 1 2 1 0 8 0 6 0 4 0 50 10 3 Predicted M ach number responses for lnpuLclass_O 200 40 4 00 600 0 Prediction # sample number k Actual M ach number responses for lnpULclass_o 40 200 30 300 20 10 500 0 Respanse # sample number k 100 Multistep prediction error for lnpu t class o 150 200 250 300 Pr e diction # 350 400 450 500 Figure 46. Prediction s, responses and prediction error for input_class_O 60 50
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104 Predicted M ach number responses for lnpuLclass_ 1 0 07 0.06 0 05 0 04 "' 0.03 0 02 0 01 0 0 50 200 0 P rediction # sample number k Actual mach number responses for lnpuLclass 1 0 06 0.05 0.0 4 .c 0 03 g 0 02 0 01 0 0.01 0 50 4 0 100 30 150 200 0 Response# sample number k M uftistep prediction e rror f or I npuLclass 1 0 01 4 ~~.7~.~,~ ,, 0.012 '0 01 0 008 0 006 '0 004~ n n 0 002"" 20 40 60 80 100 120 1 40 160 180 200 Prediction # Figure 47. Prediction s responses and prediction error for input class l
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0.01 0 0 01 0 02 "' 0 03 0.04 0.05 0 06 0 0 01 0 0 01 .c 0 02 :;I 0 03 0 04 0 05 0. 06 0 105 Predicted M ach number responses for lnpuLclass 2 40 30 20 10 Prediction # 200 O sample number k Actual Mach number responses for lnpu L class 2 40 30 20 10 Response# 200 O sample number k Multi step predict i on error for lnpuLclass...2 0 014 ,, ,.,,,,r,r,, 0 .01 2 0 01 0 008 0. 006 0 004 0 002 0 0 .__ __. 20 ~ 4 _..o _ a .._ o _ a ._ 0 __ 1 .... o o, .... 20 __ 14 .... o , ~ 6 ._ 0 __ 1 .... a o ~200 Prediction # 50 50 Figure 48 Predictions responses and prediction error for input_class _2
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0 025 0.015 1 0 01 0.005 0 50 0.035 0 03 0 025 ..c 0 02 g 0 015 0 01 0 005 0 50 106 Predicted M ach number responses for lnpuLcl ass 3 30 3.5 3 2.5 2 1 5 sample number k 0 1 Prediction # Actual M ach number responses f or ln put ctass 3 3.5 3 2 5 2 sa mple number I< 0 1 1 5 Response# 10 4 Mult is t ep prediction error f or lnpuLcl ass 3 7 5;;.X....:...:;. ___ ..,...r.., 7 6 5 6 5 5 5 2 5''''...._ ___ __._ ___ __, 1 1 5 2 2 5 3 3.6 4 P r ed iction # 4 4 Figure 49. Predictions, responses, and prediction error for input _c lass_3
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107 Predicted Ma ch number responses for in pu Lclass_ 4 0 0.005 0.01 0.015 1 0.02 ~o.02s 0 _r~;7;;;.,:=:7 ,L 10 20 30 40 3 50 Prediction # 0 0.005 0 01 ~0 015 0 02 0.025 samole number k Actual Mach number responses for lnpuLclass_ 4 0.030r~7;.,:;::,7""'.~ I 10 20 1 30 40 3 50 Response# samole number k Multi s tep prediction error for tnpuLclass_ 4 x10 4 5,5,..;....;;_,r..rrrr~., 5 4 5 4 3 5 3 2 5 2 1 5 1 L'''L___.__ __ __._ __ _.__ __ _,__ __ .,__ __ .__ ___. 1 1 2 1.4 1 6 1 8 2 2 2 2 4 2 6 2 8 3 Predi ction # Figure 50. Predictions responses and prediction error for input _c lass 4
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108 Predicted Mach number responses for lnput class S X 1 o 8 6 4 2 0 2 4 0 50 40 100 150 200 0 sample number k Prediction It A ctual Ma ch number responses for lnpuLctass_5 10 5 0 5 ........._ 0 50 40 100 30 10 200 0 Response# sample number k M ultistep pred iction error for lnput_class_S ' 7 '. 6 '. 5 '. II 3 2 t 1 ... I V 00 ' 20 40 60 80 100 120 140 1 60 Prediction # Figure 51 Predictions, responses, and prediction error for input _c lass_5
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x10 4 2 0 2 j 4 6 8 10 0 0 01 0.005 0.005 0 01 0 7 x 10 4 2 1 0 0 50 50 109 Predicted Mach number responses for lnpuLclass 6 50 40 100 150 0 sample number k Prediction # Actual Mach number responses for lnpuLclass 6 50 100 150 0 Response# sample number k Multistep prediction error for lnpuLclass 6 . . . f~ v.f I 50 100 ,so Prediction # Figure 52. Predictions, responses and prediction error for input_class 6
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15 10 0 5 10 15 10 0 5 10 3 ... 1 0 0 110 Predicted Mach number responses for lnpuLclass_7 x10 a 6 100 50 sample number k 0 0 Prediction # Actual Mach number responses for lnput class 7 x10a 100 50 sample number k 0 0 Response# Mufti step prediction error for lnput class 7 . "'" ('I ,,~ ,., 20 40 60 80 100 120 140 160 Prediction # 200 150 200 150 . 180 200 Figure 53. Predictions, responses, and prediction error for input_class_7
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111 Predicted M ach number responses for lnput_class_a 0 0.002 0.004 ..:: 0 006 0 "' :i! 0.008 0.0 1 0 0.012 0.014 0 2 4 6 8 10 200 P rediction # sample number k Ac t ual M ach number responses for lnpuLclass a 5 0 5 "' ::ii 10 0 15 200~2i:,.;:,r,.L 1 4 6 8 200 10 Response# sample number k ..:, 6 x 10 M ultlstep prediction error for lnpuLclass_8 . s~ . 32 1 f'I \J J I, \ ,. 0 . ' 0 20 4 0 60 80 100 120 140 160 180 200 Prediction # Figure 54 Predictions, respon s e s and prediction error for input cla ss_ 8
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112 Figures 55 through 57 show typical predictions, the actual Mach number response, and the corresponding control input. The examples show both ramping and regulating cases at several Mach numbers. The Mach number is shown in solid and the predicted value as a dotted line. The goal of the linear model is to predict the steady state response of the tunnel, so the plots can be effectively divided into an initial transient phase corresponding to the first twenty samples and the following steady state phase, corresponding to the last 30 samples, which is the interval which the predictive controller evaluates the candidate sequences. The examples shown are illustrative of the performance found during operation, and we observe that the local linear models are predicting well the resulting responses. Although examples of typical predictions are illustrative, consideration of the multistep prediction errors and their corresponding statistics provides more insight into the overall accuracy of the predictions and consequently the accuracy of the control provided by the PMMSC. The predictions for both the regulatory cases, input_class 5 and input_class_6, have multistep prediction errors with a mean plus one standard deviation less than the required control tolerance of 0.003. Figure 43, the histogram for these SOFM, indicates a good distribution of winners across these maps. Taken together with the steady state control tolerance achieved by the PMMSC, we can conclude that the locally linear predictors and consequently, the underlying discretization of the tunnel dynamics by the corresponding SOFM provides both sufficient resolution and coverage of state and control spaces.
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113 Predicted and actual M ach number response O M =0 9 5 0953 0 952 0 951 j 0 95 0 949 0 948 .. pradlcted . . .,....""\ . .. .. . ... .. . . ~ . . 0 9 4 7 0 5 1 0 15 o o o o .. .. 20 25 k Co nt rol nput 30 35 40 45 . 50 Predicted a nd actual M ach number response O M =0 9 0 903 .. predicted 0 902 0 901 0. 9 ? 0 899 0 898 0 897 0 1 .. o i,... o e 11 o .. I. . _ _ ..t : . 5 1 0 .. .. 15 ._ 2 0 . . . . 25 k 30 35 40 45 . 50 l o a \ .. 0 ......, _____________ ......... ___ .. _______________________ I': 0..2 j o a o O a 1:. ....... ......... ___,. ........ ~ 4lil o .. '"' o e ... o e 1 0 0 853 0 852 0 851 I 0 85 0 849 0 848 0 84 7 0 o e loo o o o .. o 1 0 .. .. 0 .. 0 Predicted and actual M ach number response O M ..0 85 . .. .. ... predict9(l . ~ ~ .. . . . . .. ..., 5 10 15 20 25 k 30 35 4 0 4 5 50 . I o .. 0 .. o.a .. o .. O e 0 I __________ .,.. __________________________________ 10 2 0 ae .. so . .. 0 .. 1o e1o a 11 0 0 10 10 2 0 aa k P r edicted and actu al M a c h number response O M 6 .. . 0 603 pred ict ed 0 602 0 601 .. . . . . . . 0 6 7~ ~ ~ ~0 599 0 598 0 597 0 1 o e .. o . .. o .. .. I 5 1 0 . . 15 . : 20 25 k Oon tf' ot npul 30 35 40 45 . 50 1 o ... 1 o : : _____________________________ ..,. ................... _..,..,..___..,." o e 11 .. 0 1 0 10 a o aa .. 3 0 o .. . Figure 55. SOFM Predictions of Mach number in set point regulation
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114 Predicted and actual M ach num b er response O M <=0 95 0 953 0 952 0 951 .. predicted . .... . J 0 95 0 949 0 948 ... . . ~ . . ... 0947 _ _ 0 o o e 1.. o L.. 5 1 0 15 20 25 k Conlrol lnpv..t 30 35 4 0 45 50 . I o ot;,.."' ~ .. .. o eL.. _, 0 ,. ,.0 ,.. k "' .. o Predicted and actual M ach number response O M a0. 85 0 853 .. pr ed icted 0 852 .... 0 85 1 i 085 2 . 0 849 0 848 0847 _ 0 o e o e o .. .. 5 10 15 20 25 k 30 35 40 4 5 50 . 0 905 0 904 0 903 .. ... Pr ed icted a nd actua l M ach number response O M %0 9 .. .. .... .. .. .. p r edicted 0 902 .c .. j 0 901 I . 0 9 .. 0 899 0 898 0 897 0 .. o ..... 5 1 0 15 20 25 k ea. ,.. ,., tnpu1 .... 30 35 40 . 45 . . 50 I o ... Ot. ,... ___________ .,. ___ .,. ___________ ,.. ___ ' ' ' o e .._ 0 I ' 0 ,o .. ao .. .. Predlated and actual Ma ch number response O M ..0 6 ... 0603 . .. predk:1ed 0 602 .. 0 601 0 6 0 599 0 598 '. . _..,:.._ '"~ . . . 0 597 __ 0 5 1 0 .. o . .... o . ... o .. ... 15 20 25 k Cont,c,t l npUI 30 35 40 45 50 . I o a 0 .....~, 0 ~ :a: I o a, oi,.. ~ 11 o a,./ .. .. 0 4 0 0 o .. 0 ,o .. 20 o ... o .. ao o o o o a .. 0 ,o "' 20 a .. Figure 56. SOFM Predictions of Mach number in s et point regulation .... .. 0 .. . 60
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115 Predicted and actual Ma ch number response O M= 0.9 50polnts ahead prediction of Ma ch number wh il e ramp i ng to M .. 0 6 0 85.....~~,,, I 0903 ~ . .. pred icted 0 902 0.901 09 . 0.899 0.898 ' .. .. 0.8 0.75 .. .. . i 0 7 0 65 ... predic t ed 0 897 0.6 0.898 0 5 10 15 20 25 30 35 40 45 50 0 660 50 100 150 200 250 k k Con t rol Inpu t Co ntrol Inpu t 7 , 1 .... 1 1. I 0.8 I 0.8 I 0 6 0.6 .. I I 0. 4 I 0.4 .. 0.2 16 0.2 .. l I i ...J 0 Oi} ll. 0.2 I a: 0.2 .. 0.4 0. 4 .. 0.6 0.6 0.8 0.8 I 1 1 . 0 5 1 0 15 20 25 30 35 4 0 4 5 50 0 50 100 160 200 250 k k Figure 57. SOFM Predictions of Mach number in ramping
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CHAPTER6 CONCLUSIONS AND FUTURE RESEARCH Conclusions Modeling and controlling systems with a wide range of dynamic characteristics is a rich problem with many possible approaches. In this research, the method of local linear modeling based on the selforganizing feature map has been extended to a control framework as an approach to this problem. The SOFM based modeling method was employed to develop a set of models which, collectively, described the system dynamic characteristics over the entire range of operation, but individually, represented the response of the system in some restricted region of both the state and control spaces of the system. The extension of the method allowed us to predict the system response to a small, but effective set of inputs, using the model which best describes the local dynamics. The input corresponding to the prediction that best satisfied the requirements at the output was then applied as the control. The overall result was the development of a controller, the PMMSC, which predicted the system response by switching to the best available model. Two problems which naturally arise from this approach are: how to guarantee that the collection of models adequately cover all the dynamic regimes of the system, and bow to select the model which best describes the local dynamical regime. Our SOFM based local linear modeling approach addresses both the problems with a computationally 116
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117 efficient method The SOFM guarantees that the repertoire of dynamics u s ed for training are represented by the collection of local models and serves to identify the local dynamic regime. In a sense the diverse plant dynamics aie captured in a compact table lookup of linear models. The combination of the locally linear models and a small s et of candidate control sequences provided a computationally efficient method for multi step predictive control This was contrasted with generalized model predictive control, particularly in an environment which requires switching between model s For this application, the fundamental control problem i s one of regulating the tunnel Mach number to within tolerance of the desired set point under non s tationary loads. There were s everal characteri s tics of this problem that made it an attractive candidate for application of our method. The first was that the openloop plant wa s stable, so the focu s wa s on improved regulation. Secondly, there was a wealth of data to train a locally linear dynamic model of the tunnel under different dynamic conditions Third, the control input is quantized to three values, which allowed for a meaningful clustering on a small set of representative control prototypes that were derived from the experiential know ledge of the tunnel. The PMMSC was implemented on inexpensive computing hardware and used to control the wind tunnel to within the strict research requirements for three s eparate runs of three hours. The performance of the PMMSC wa s compared to both the existing controller and expert human operators by several metrics The PMMSC provided improved perf orrnance with decreased control effort over both the exi s ting controller and expert humaninthe loop control.
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118 Future Research In this research, the method of SOFM based local linear modeling has been extended to a control framework. However, the resulting predictive, multiple model switching controller developed for this application does not represent a general control architecture, which points to directions for further work 1 The control space for this application was partitioned by the construction of representative prototypes This represented the incorporation of a priori knowledge about the operation of the system. A clustering of the inputs using the SOFM would enhance the generality of the method 2. The plant being controlled in this application was stable. The method should be investigated for stabilization of an unstable plant, at least in simulation studies. This would open the possibility of identifying stabilizing and destabilizing manifolds of the control space, under the first suggestion for further work 3 Incorporation of an online adaptive model should be investigated. These could use the best SOFM based model as a s tarting point to speed the up the convergence 4. The SOFM algorithm could be implemented online and used to reformulate the maps based on accumulated knowledge a s the system explores operating regions not covered in the training data
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APPENDIX ST ABILITY CONS ID ERA TIO NS An approach for considering the stability of the overall system is to analyze the stability characteristics of a simple system with bangzerobang (r, 0, +r ) input, controlled using feedback and a combination of simple nonlinear functions The system, considered in discrete time formalism, consists of an integrator, d pure delays, and a nonlinear controller implemented by : 1) a symmetric dead zone with zero output for inputs in the closed interval [, +] 2 ) a signum function f (x) = X if X > f ( X) = X if X < f ( x) = 0 if lxl with > 0 f (x) = r if x>O f ( X) = r if X < 0 f (x) = 0 if x=O with r > 0 The system is shown in block diagram format in Figure 58. 119 ( 60) (61)
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120 r=O u (k) e c_ r Dead Zone Signum Figure 58. A simple nonlinear system with feedback d z Delay z z + 1 Integrator y(k) Consider the sys tem with no delays, d = 0. The nonlinear control law yields the system equations: y(k+l) = y(k) r if y(k) > e y(k+l)=y(k)+r if y(k)<e y(k + 1) = y(k) if jy(k)j e (62) In order to prove the stabi lity of the system, a simple candidate Lyapunov function of the system is chosen: 1 V[ y(k)] = 2 y 2 (k) (63) For stability about the origin, y = 0, we require that [LaSalle 1986] : 1) V[ y] > 0 for all y y ;t; 0 2) V[y] =0 for y=O (64) 3) V 0 along the trajectory of (62) for ally where V = V[y( k + 1) ] V[ y( k)] (65) Conditions 1 ) and 2) are true by inspection for (64). Condition 3) needs to be considered in three distinct cases corresponding to (62) :
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121 Case 1: y(k) > AV=[y(k)rj2 [y(k)]2 1 2 1 2 1 2 ~V=2y (k)y(k)r+2r 2y (k) 1 2 = r y(k)r 2 1 1 Thus~V~Oif y(k) ~r, whichimpliesthate ~r 2 2 Case 2: y(k) < e ~V=_!.[y(k) + r] 2 _!.[y(k)] 2 2 2 1 2 1 2 1 2 ~V=2y (k)+y(k)r+2r 2y (k) 1 2 =r +y(k)r 2 1 1 Thus~ V ~Oif y(k) 2 r which implies that e ~ 2 r Case 3: y(k) < e ~V=_!.[y(k)] 2 _!_[y(k)] 2 2 2 ~V=O Thus ~V~Oif e ~o Thus this system, with d=O, will be stable and within the bound e of the origin if the 1 discretized control input at each instant is either + r, r, or O, and e is greater than 2 r
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122 Now we consider the cases of a single delay d = 1. The nonlinear control law yields the system equations: y(k + 2) = y(k + 1) + r if y(k) > y(k + 2) = y(k + 1) r if y(k) < y(k + 2) = y(k+l) if ly(k) 5: e (66 ) For this case the Lyapunov candidate function includes the additional state associated with the delay: 1 V[y(k)] = 2 y 2 (k) + 6V = V[y(k + 1)] V[y(k)] 1 2 1 2 = 2 y (k+2)2 y (k) and Again we evaluate the three distinct cases corresponding to (65), with the additional consideration for the added delay in the system: Case 1 : y(k)>e If y(k1) > e y(k + 2) = y(k) 2r If y(k 1) 5: y(k +2) = y(k) r If y(k1)
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Case 2: y(k)< If y(k 1) < 123 y(k + 2) = y(k) + 2I' If y(kl) ~ y(k+2) = y(k) +r If y(k 1) > y(k + 2) = y(k) Again, the lower bound on is deterrmined by: LiV = [y(k)+2r )' [y(k) J2 1 1 ~v =2 y 2 (k) +2y(k)I' +2r 2 2 y 2 (k) = 2r 2 +2y ( k)I' =2I' (I'+ y(k)) Thus~ V Oif y(k) r, which implies that r
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Case 3: y(k) If y(k1)< y(k+2) = y(k) + r If I y(k 1) y(k + 2) = y(k) If y(k 1) > y(k+2) = y(k)+r 124 6V= [y(k)+r)' [y(k)]2 1 2 1 2 1 2 ~V=2y (k)+y(k)r+2r 2y (k) 1 2 =r +y(k)r 2 1 =r( 2 r+y(k)) 1 1 Thus~ V ~Oif y(k) 2 r, which implies that 2 r Thus this system, with d=l, will be stable and within the bound of the origin if the discretized control input at each instant is either + r, r, or O, and is greater than r. By induction, for the case of d delays we get the result that: 6V = [y(k) (d + I)r)' [y(k) J2 where is for y(k) > and + is for y(k) < then 1 1 1 ~V=2y 2 (k)(d + l)y(k)r 2(d + l) 2 r 2 2y 2 (k) 1 = 2(d + l)r 2 (d + l)y(k)r =(d + l)r d+l 2 r y(k) Thus~ V ~Oif y(k)j d+l 2 r, which implies that d+l 2 r
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The general result is that if: 125 d+l 2 r (69) then the system will be stable and y( k) as k 00 i.e. the system will converge to a region around the origin bounded by For a system with long delays, this can impose a large loss of precision in the control of the output in order to guarantee stability. A strategy to circumvent this los s of precision while still providing stability is to apply sequences of control commands composed of m sample intervals of r followed by d sample intervals of zero input, for a total s equence length of (m + d). Thus control sequences are determined every (m +d ) samples, and applied openloop over the (m+d)T interval, where Tis the sample interval duration Then if at time k the controller selects a control sequence: e r, r, .. r ), o o, .. OJ ( 70) 111 s an1pl es d s a,npl es then the output y(k + m+d) is: y( k+m+d) = y( k) mr (7 1 ) Similar to the earlier result ( 69 ), the system controlled in this fashion will be stable to within of the origin if : where m = 1, 2, 3, ... m r 2 (7 2 )
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127 Haykin, Simon, (1994) Neural Networks: A Comprehensive Foundation. New York: Macmillan College. Hopfield, J.J., (1982). ''Neural Networks and Physical Systems with Emergent Collective Computational Abilitites," Proceedings of the National Academy of Sciences of the U.S.A., Volume 81, pp.30883092. Jackson, E.A., (1989). Perspectives of nonlinear dynamics. Cambridge: Cambridge University Press. John, James E.A., (1984). Gas Dynamics. Newton, MA: Allyn and Bacon, Inc. Kasslin, M., Kangas, J., Simula, 0., (1992). ''Process State Monitoring Using SelfOrganizing Maps," Artificial Neural Networks, 2, Vol. II, pp. 15311534. Kohonen, T., (1990). ''The SelfOrganizing Map," Proceedings of the IEEE, Vol. 78, No. 9. Kohonen, T., (1995). SelfOrganizing Maps, Berlin, Heidelberg, Gerrr1any: Springer Verlag. Lampinen, J., and Taipale, 0., (1994). ''Optimization and Simulation of Quality Properties in Paper Machine with Neural Networks," Proc. ICNN'94, Int. Conj. on Neural Networks, pp. 38123815. LaSalle, J P., (1986). The Stability and Control of Discrete Processes. New York: SpringerVerlag. Matthews, C.P., and Warwick, K., (1995). ''Practical Application of Self Organising Feature Maps to Process Modelling," Proceedings of Engineering Applications of Neural Networks. May, Gary S., (1994). ''Manufacturing IC's the Neural Way," IEEE Spectrum, Vol. 31, No. 9. Mercer, C.E., Berrier, B.L., Capone, F.J., Grayston, A. M., Sherman, C.D., (1984). Computations for the 16Foot Transonic Tunnel NASA Langley Research Center NASA Technical Memorandum 86319. Morse, A.S., (1980). ''Global Stability of Parameter Adaptive Systems," IEEE Transactions on Automatic Control, Vol.25, pp 433439. Motter, M., and Principe, J.C., (1994). ''A Gamma Memory Neural Network for System Identification," Proceedings of IEEE International Conference on Neural Networks, Vol. 5, pp. 32323237.
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128 ___ ., (1995) ''Classification and Prediction of Wind Tunnel Mach Number Responses using both Competitive and Gamma Neural Networks," Proceedings of 1995 World Congress on Neural Networks, Vol. 2, pp. 2529. Narendra, K S., and Annaswamy, A., (1989). Stable Adaptive Systems. Englewood Cliffs, New Jersey: Prentice Hall Narendra, K.S. and Balakrishnan, J., (1994). ''Intelligent Control using Switching and Tuning,'' Proceedings of the Eighth Yale Workshop on Adaptive and Learning Systems, June 1315. Narendra, K.S., Balakrishnan, J., (1997). ''Adaptive Control Using Multiple Models, IEEE Transactions on Automatic Control, Vol.42, No. 2, pages 171187 Narendra, K.S., Balakrishnan, J., and Ciliz, M.K., (1995). ''Adaptation and Learning Usng Multiple Models, Switching, and Tuning," IEEE Control Systems Maga z ine, Vol 15, No. 3. Narendra, K.S., Li, S., and Cabrera, J.B.D., (1994). ''Intelligent Control using Neural Networks,' Proceedings of the Eighth Yale Workshop on Adaptive and Learning Systems, June 1315. Narendra, K.S., Lin, Y.H., and Valavani, L.S., (1980) ''Stable Adaptive Controller DesignPart II: Proof of Stability," IEEE Transactions on Automatic Control, Vol. 25, pp 440448 Narendra, K S., Mukhopadhyay, S., (1997). ''Adaptive Control Using Neural Networks and Approximate Models,'' IEEE Transactions on Neural Networks, Vol.8, No.3, pages 475485. Peddrew, Kathryn H (1981). A User's Guide to the Langley 16Foot Transoni c Tunnel. NASA Technical Memorandum 83186. Principe, J., Hsu, H., and Kuo, J., (1994). ''Analysis of Short Term Neural Memory Structures for Nonlinear Prediction," NIPS, pp. 10111018. Principe, J., and Kuo, JM., (1994) ''Dynamic Modeling of Chaotic Time Series with Neural Networks," NIPS, pp. 311318. Principe, J C., Kuo, J., Celebi, S., (1994). ''An Analysis of the Gamma Memory in Dynamic Neural Networks," IEEE Transactions on Neural Networks, Vol. 5, No. 2.
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129 Principe, J.C., and Motter, M., (1994). ''System Identification with Dynamic Neural Networks,'' Proceedings of the 1994 World Congress on Neural Networks Principe, J., and Wang, L., (1995). ''Nonlinear Time Series Modeling with SelfOrganizing Feature Maps." IEEE Workshop on Neural Networks for Signal Processing pp. 1120. Robbins, H., and Monroe, S., (1951). ''A Stochastic Approximation Method," Annals of Mathematical Statistics, Vol. 22, pp.400407. Ritter, H., Martinetz, M., Scholten, K., (1992). Neural Computation and SelfOrgani zing Maps. Reading, MA: AddisonWesley. Rumelhart, D.E., Hinton, G.E., and Williams, R.J., (1986). ''Learning Internal Representations by Error BackPropagation," Parallel Distributed Processing D.E. Rumelhart and J.L. McClelland, editors. Cambridge, MA: The MIT Press. Soeterboek, Ronald A. M., Pels, A.F., Verbruggen, H.B., A. van Langen, G.C., ( 1991 ) ''A Predictive Controller for the Mach Number in a Transonic Wind Tunnel," IEEE Control Systems Magazine, Vol. 11, Number 11, pp. 6372. Si, J., and Lin, S., (1997). ''Weight Convergence and Weight Density of the MultiDimensional SOFM Algorithm," Proceedings of the 1997 American Control Conference, CD edition. Takens, F., (1980). ''Detecting Strange Attractors in Turbulence," Dynamical Systems and Turbulence, ed. by D.A. Rand and L.S. Young, Springer Lecture Notes in Mathematics, pp.365381, SpringerVerlag, New York. Warwick K ., (1996). ''System Identification using neural networks," Proceedings of the Conference on Identification in Engineering Systems, M.I.Friswell and J.E. Mottershead, editors. Swansea, UK:University Wales Swansea. Werbos, P.J., (1990). ''Back Propagation Through Time : What It Does and How To Do It," Proceedings of the IEEE 78(10), 15501560. Willshaw, D.J and C. von der Malsburg, (1976). ''How Patterned Neural Connections Can Be Set Up by SelfOrganization," Proceedings of the Royal So ciety of London, Series B, 194, 431445. Wu, J.M., Lee, J .Y., Tu, Y.C., and Liou, C.Y., (1991). ''Diagnoses for Machine Vibrations Based on SelfOrganization Neural Network," Proc. /ECON '91, Int. Conf on Industrial Electronics Control and Instrumentation, Vol. II, pp. 15061510.
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BIOGRAPHICAL SKETCH Mark A. Motter was born in Columbia Pennsylvania, on September 3, 1955 He s erved in the United States Navy from 1973 until 1979 and was honorably discharged at the rank of Electronics Technician First Cla ss. He then began his formal engineering education at Old Dominion Univer s ity in Norfolk, Virginia, receiving his BSEE magna cum laude, and MSEE in 1983 and 1985, respectively. Since 1985 he ha s been employed at NASA Langley Research Center, primarily involved in the modeling and control of wind tunnels and associated experimental equipment. He is a member of the IEEE and a registered Professional Engineer. 130
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosop ,; I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Phllosoph Gijs Bosman Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinio11 it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy Thomas E. Bullock Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinion it confor1ns to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ohn G. Harris Assistant Professor of Electrical and Computer Engineering J
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I certify that I have read this s tudy and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a di ss ertation for the degree of Doctor of Philosophy. Loe VuQuoc Associate Professor of Aerospace Engineering, Mechanics and Engineering Science This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted a s partial fulfillment of the requirements for the degree of Doctor of Philo s ophy May, 1998 Winfred M. Phillips Dean College of Engineering Karen A. Holbrook Dean, Graduate School
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