Direct optimal control of flexible structures with application to adaptive optics systems

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Direct Optimal Control of Flexible Structures With

Application to Adaptive Optics Systems

by

Chellabi Abdelkader

Eng. Dip., Ecole NationaJ Polytechnique (A lgeria), 1988 MASc University Laval, Quebec, 1992

A Thesis S u b m itted in P artial Fulfillm ent of th e R equirem ents for the Degree of

Do c t o r o f Ph i l o s o p h y

in the

M echanical Engineering.

We accept this thesis as conform ing to th e required standard

____

Dr. Y. Stepanenko, Supervisor (Mechanical Engineering)

pervisor (M echanical Engineering)

r. R. Podhorodeski, D ep artm en t Member (M echanical Engineering)

Dr. P. Agaüioklis, O utside M em ber (Electrical & C o m p u te r Engineering)

______________________________________________________________

Dr. JVl. Èpste^n, E x te m ^ ^ x a m in e r (Mechanical Engineering, U niversity of Calgary)

© C h e l l a b i A b d e l k a d e r , 1998 University of V ictoria

All rights reserved. T his th esis m ay not be reproduced in whole or in p art, by photocopy or o th e r m eans, without th e perm ission of th e author.

11

Supervisors: Dr. Y. Stepanenko, and Dr. S. Dost

A b str a c t

An adaptive optics system consists mainly of a wavefront sensor to detect optical aber­ rations, a control system to reconstruct the wavefront and compute a correction, and a deformable mirror to apply the correction. In this dissertation, the problem of optimal control of an adaptive optics system is investigated. A direct optimal control approach is used in the controller design.

The direct optimal control methodology developed for discrete parameter systems is extended in this study to distributed parameter systems, where the Rayleigh-Ritz method is used for both spatial and temporal variables. The displacement field is written as the product of spatial functions (mode shapes for a vibrating structure, and Zemike modes for deformable mirror) and the generalized coordinates. These generalized coordinates and the control input functions (voltages) are written as simple series expansions in time in terms of selected functions and unknown coefficients. Substitution of these selected functions and their variations into Hamilton's law of varying action results in algebraic equations of motion (AEM) of the structure. These AEM are then considered as the algebraic state equations where the unknown expansion coefficients of the time series (assumed time-modes) for the generalized coordinates axe recognized as the states and those of the input functions are recognized as the controls.

Using the space-time assumed mode method, the usual variational optimal control prob­ lem is transformed into an equivalent algebraic problem. Optimal solutions are then ob­ tained in a closed form and the solution is a global optimum within the time period con­ sidered. The solution procedure does not lead to any Ricatti equation or alike. The direct method proved to be simple, computationally efficient, attractive from implementation point of view, and it is general and allows a deterministic modelling of many physical problems.

Applied to active vibration control of plates with piezoelectric transducers, the direct methodology exhibits results similar to those obtained through conventional methods. Ac­ tive shape control of a deformable mirror using the direct approach results in high perfor­ mance of the controller. The method allows direct control of Zemike modes, and highlights the relationship between the control inputs and Zemike modes through an algebraic con­ trollability measurement index. Robustness of the controller is shown through simulation of smooth and severe random variations of the optical aberrations.

In the same line of thought, a space-time finite element m ethod is developed and applied to stm ctural optimal control problems. Finite element method is used for both spatial and temporal discretizations. The unique feature of this method is its ability to analyse the stracture-control interaction in the same mathematical framework, which allows simultane­ ous control and stm ctural model design iterations. However, due to its high dimensionality, the space-time finite element method is computationally less efficient than its counterpart assumed mode.

I l l

Exajniners:

Dr. Y. Stepanenko, Supervisor (M echanical Engineering)

D n^-StsD ostr-Co-S^ervisor (M echanical Engineering)

)r. R. P odhorodeski, D epartmD epartm ent M em ber (Mechanical Engineering)

________________________________________________

Dr. P. A g ^ o k lis , O utside M em ber (Electrical & C om puter Engineering)"

xam iner (Mechanical Engineering, U niversity of Calgary)

IV

T a b le o f C o n te n ts

A b str a ct ii

Table o f C on ten ts iv

List o f T ab les vii

List o f F igu res viii

A ckn ow led gem ents x

D e d ic a tio n xi

1 In tro d u ctio n 1

1.1 System D e s c rip tio n ... 1

1.2 Problem D e s c r i p t i o n ... 4

1.3 L iterature Survey and Proposed A p p r o a c h ... 6

1.4 Thesis C o n trib u tio n ... 11

1.5 Thesis O rg a n isa tio n ... 12

2 A d a p tiv e O ptics S y stem s 14 2.1 In tro d u c tio n ... 14

2.2 P hase C o n j u g a t i o n ... 15

2.3 R epresentation of th e W a v e fro n t... 17

2.4 Wavefront Sensing ... 20

2.4.1 H artm ann-Shack W avefront S e n s o r ... 21

2.4.2 Curvature S e n s o r ... 21

2.5 Wavefront C o rre c tio n ... 25

2.5.1 Modal T ilt C o r r e c t i o n ... 25

2.5.2 Modal Higher-order C o r r e c t i o n ... 26

2.5.3 M ultichannel C o r r e c t i o n ... 26

T A B L E O F C O N T E N T S v

2.6.1 Phase from wavefront s l o p e s ... 30

2.6.2 Modes from wavefront s lo p e s ... 32

2.6.3 Modes from Curvature M e a su re m e n ts... 35

2.6.4 Modes from wavefront p h a a e ... 37

2.7 Control S y s t e m ... 37

2.7.1 ZonaJ control from continuous p h a s e ... 38

2.7.2 M odal control from continuous p h c is e ... 39

2.7.3 ZonaJ control from m odal p h a s e ... 39

2.7.4 Zonal control from wavefront s lo p e s ... 40

2.7.5 Modal control from wavefront c u r v a t u r e ... 41

2.8 Concluding R e m a r k s ... 42

2.9 Conclusion of C h ap ter 2 ... 44

3 S y s te m D yn am ics: A ssu m ed M o d e s 47 3.1 System R e p r e s e n ta tio n ... 47

3.2 Electrom echanical C onstitutive R e l a t i o n ... 48

3.3 Variables D e f in itio n s ... 51

3.3.1 Generalized Mechanical C oordinates ... 51

3.3.2 Generalized Electrical C o o r d i n a t e ... 54

3.3.3 Generalized Applied Forces ... 54

3.4 S tatem ent of H am ilton’s Law ... 55

3.5 T h e Variations ... 56

3.5.1 Generalized C o o r d in a te s ... 57

3.5.2 Kinetic E n e r g y ... 57

3.5.3 P otential E n e r g y ... 58

3.5.4 Work done by External F o r c e s ... 59

3.5.5 B oundary T e r m s ... 60

3.6 Algebraic E quations of Motion ( A E M ) ... 60

3.6.1 A ctuator E q u a tio n s ... 61

3.6.2 Sensor E q u a ti o n s ... 62

3.7 Generalized Coordinates C ontinuity E q u a tio n s ... 63

3.7.1 H am ilton’s Law and Initial Value P ro b lem s... 66

3.8 C o n tro lla b ility ... 67

3.8.1 C o n tro llab ility ... 67

3.8.2 T rajectory C o n tr o lla b ility ... 70

4 D irect O ptim al C on trol 72 4.1 Algebraic Perform ance M e a s u r e ... 73

4.2 Form ulation of C o n s tra in ts ... 75

T A B L E O F C O N T E N T S vi

4.4 O ptim al S o l u t i o n ... 77

4.4.1 O p tim al Solution via Direct S u b s titu tio n ... 77

4.4.2 O p tim al Solution via Lagrange M u ltip lie r s ... 78

4.4.3 O p tim al Gains and Physical C o n tr o ls ... 79

4.5 Closed Loop S y s t e m ... 80

4.6 Active V ibration Control of a P l a t e ... 82

4.7 Concluding R e m a r k s ... 91

5 D irect O p tim al S h a p e C ontrol 96 5.1 Direct O p tim al Tracking C o n t r o l ... 96

5.2 Prescribed T r a je c to r ie s ... 97

5.3 O bjective F u n c t i o n ... 98

5.4 Direct O p tim al Tracking Control S t a t e m e n t ... 99

5.5 O ptim al S o l u t i o n ... 100

5.5.1 O p tim al Solution via Direct S u b s titu tio n ... 100

5.5.2 O p tim al Solution via Lagrange M u ltip lie r s ... 102

5.5.3 O p tim al Gains and Physical C o n tr o ls ... 103

5.5.4 Concluding R e m a r k s ... 104

5.6 Tracking C ontrol of a Deformable M ir r o r ... 105

5.7 Concluding R e m a r k s ... 116

6 Sp ace-T im e F in ite E lem en t Formulation 118 6.1 Variational E q u a tio n s ... 119

6.2 Finite Elem ent D is c r e tiz a tio n ... 119

6.3 Equations of M o t i o n ... 121

6.3.1 A ctu ato r E q u a tio n s ... 123

6.3.2 Sensor E q u a t i o n s ... 124

6.4 Two Nodes T im e E l e m e n t ... 125

6.5 O ptim al C ontrol F o r m u l a t i o n ... 126

6.6 Active v ibration control of a p l a t e ... 128

6.7 O ptim al Tracking C o n t r o l ... 129

6.8 Concluding R e m a r k s ... 132

7 C losing C o m m en ts 133

vu

L ist o f T a b les

2.1 First 10 Zemike P o ly n o m ia ls... 18

4.1 M aterial P r o p e r t i e s ... 84

6.1 O ptim al Regulator Control Problem with free End S t a t e ... 128

vm

L ist o f F ig u r e s

1.1 A daptive O ptics S y s t e m ... 2 2.1 Block diagram of an AO s y s t e m ... 15 2.2 Principle of phase c o n ju g a tio n ... 16 2.3 Principle of H artm ann-S hack Wavefront Sensor: a) Plane Wave b)

D isturbed W a v e ... 22 2.4 Principle of th e C u rv atu re S e n s o r ... 23 2.5 Segm ented m ultichannel m i r r o r ... 27 2.6 Discrete A ctuators m irrors: a) discrete position actuators; b) discrete

force actu ato r; c) bending m om ent a c t u a t o r s ... 28 2.7 A m em brane m i r r o r ... 29 2.8 A bim orph m i r r o r ... 46 2.9 Configuration of zonal wavefront reconstruction a) Hudgin; b) Fried;

c) S o u th w e ll... 46 3.1 Electroelastic C ontinuum S y s t e m ... 48 4.1 G eom etry of th e square p la te w ith p ie z o e lec tric s... 85 4.2 Controlled and U ncontrolled responses at point (5,5)cm of th e plate,

and the control voltages of the four piezoelectric patches for th e con­ trolled response; Case 1 ... 88 4.3 Shape of th e p late a t different tim e f r a m e s ... 89 4.4 Controlled and U ncontrolled Displacements a t th e m iddle of th e plate.

and the control voltages of th e four actuators; Case I I ... 93 4.5 Controlled an d U ncontrolled Displacements a t th e m iddle of th e plate,

and the control voltages o f th e four actuators; Case I I I ... 94 4.6 Controlled D isplacem ent a t th e center of th e p late, and th e control

voltages; Case I V ... 95 5.1 Bim orph m irror w ith sixe e le c tro d e s ... 106

L IS T O F F IG U R E S ix

5.2 Tracking an d response a t th e center of the m irror for two different trajectories ... 109 5.3 Control voltages corresponding to trajectory ^ 1 ... 110 5.4 Control voltages for the random ly generated t r a j e c t o r y ... I l l 5.5 Surface shape of th e m irror for 10 modes and 6 e le c tro d e s... 114 5.6 Surface shape of th e m irror for 10 modes and 17 e le c tr o d e s ... 115 6.1 Displacem ent a t th e center of th e plate, and the control voltages . . . 130

X

A ck n o w led g em en t s

First and form ost, I would like to express my sincere appreciation to the patient guidance of ray supervisor Dr. Yury Stepanenko. I thank him for his encouragem ent, advice and m oral support I received during the course of th is research, an d for his help in the p rep aratio n of this dissertation. Financial assistance received from Professor Yury Stepanenko through IRIS-Precran is also gratefully acknowleged.

I would like also to thank My Co-Supervisor Dr. Sadik Dost, for his guidance, encouragem ent, assistance in the preparation of this work, an d for creating a pleasant atm osphere for th e success of this research.

My thanks are extended to Dr. Behrouz Tabarrok who was of g reat help from day one of my Ph.D . program till this point, for his assistance, and for reviewing some m aterial of th e thesis. Dr. Ronald Podhorodeski, and Dr. P an ajo tis Agathoklis deserve special thanks for their encouragem ent, support and assistance.

1 also would like to thank Dr. Marcelo Epstein for accepting to be my external examiner.

Lastly, I would like to thank all my friends who m ade my stay in V ictoria enjoyable among th em A m irouche Cherfi and Belkacem Chergui.

Finally, I th a n k my wife Naim a for her patience and su pport during the critical times of this work.

XI

C h a p te r 1

I n tr o d u c tio n

1.1

S y s te m D e s c r ip tio n

As suggested by th e title of this thesis, the focus of this investigation is th e adaptive optics control system . P articularly, the problem of optim al control of a deformable m irror is studied.

In general term s, a d a p tiv e optics deals with the control o f light in a real time

closed-loop fashion and refers to optical components whose characteristics axe con­

trolled during actual o p eratio n in order to improve th e quality o f an optical signal [5, 6, 27, 44, 72].

A typical adaptive optics system consists of three m ain com ponents: deformable m irrors, wavefront sensors, and a control system. Figure 1.1 illu strates th e design of a typical adaptive optics system . An optical wavefront is collected by a telescope and is reflected off a deform able m irror. T he reflected wavefront is observed by a wavefront sensor. T he wavefront sensor m easures an array of local phase g ra d ie n ts/ curvatures, which are processed in a wavefront reconstructor to estim ate th e p h ase /c u rv a tu re of

C H A P T E R L IN T R O D U C T IO N Star UgbK Tilt Mirror Telescope Control System Actuator Drivers W avefront Sensor Actuator Driver

Figure 1.1: A daptive Optics System

the incom ing wavefront. These estim ates are then used by the control system driving the deform able m irror to compensate for th e incoming wavefront distortions.

A daptive optics includes correction of both am p litu d e/in ten sity and phase of a light beam . A m plitude/intensity variations (also called scintillation) contribute to image quality degradation much less th a n phase variations and are therefore generally ignored in the planning and evaluation of adaptive optics system s [17].

W avefronts represent surfaces of con stant phase for the electrom agnetic field.

Wavefront for plane and spherical waves are considered as a reference. Any deviation from a reference sphere or plane results in an aberrated wavefront. This deviation occurs when th e index of refraction of the propagating m edium , the atm osphere, changes due to density and tem perature changes. As a result, th e wavefront of light traveling through the m edium is distorted. An adaptive optics system seeks to adjust the shape of th e deform able mirror so as to cancel out this deviation exactly; this is called phase conjugation.

C H A P T E R I. IN T R O D U C T IO N 3

Deform able m irrors axe typically continuous surface m irrors deformed by actuators to have peaks and valleys. These actuators are usually piezoelectric actuators which are attach ed to the b o tto m of th e mirror at preselected points. One form of frequently used actu ato rs is the stacked piezoelectric actuator. W hen a voltage is applied, th e piezoelectric m aterial e ith e r expands or shrinks (depending on the polarity of th e voltage) and shapes the surface of the deformable m irror. Instead of stacked actuators,

bimorph m irrors use two thin plates of oppositely polarized piezoelectric m aterials

[31, 50, 41, 55]. The plates are bonded together and controlled by a set of electrodes deposited on th e back side of one plate.

In ad ap tiv e optics, th e re axe mainly two types of wavefront sensors, namely wave- front slope sensors which m easure the two-dimensionaJ sp atial gradients of the phase at a discrete num ber of points, and curvature sensors which measure local curva­ tures of th e phase at discrete locations. The inform ation ab out phase, phase gradient and/or curvature is used in the control system to reconstruct the wavefront. There are various types of com m on wavefront sensors which differ in th e manner by which the gradients are m easured [34, 49, 45, 73].

A daptive optics finds its m ajor application in astronom y. There axe m any tele­ scopes in the world which use adaptive optics, am ong th em th e Keck telescope in Hawaii, the B onnette system a t the Dominion observatory in V ictoria (Canada), and the European Southern O bservatory in Germany and Chile. O ther application fields of adaptive optics include communications, biom edicine and laser welding. Atmo­ spheric distortions affect ground-based, free space laser com m unications. Medical devices (e.g., endoscopes) image tissue through bodily fluids th a t cause optical aber­ rations and degrade im age quality. These aberrations also affect precise delivery of laser power in endoscopic and retinal surgery. Likewise, heat induced aberrations affect th e delivery of laser power in welding and cu ttin g applications.

C H A P T E R I. I N T R O D U C T I O N 4

1.2

P r o b le m D e s c r ip tio n

Adaptive com pensation in large optical telescopes, or in airb o rn e imaging or missile system s, requires deform able m irrors capable of correcting large optical path errors at high speed. T his is due to the requirem ent on the control bandw idth to be lOkHz or more [26].

Controlling a deform able flexible m irror is part of stru c tu ra l control. A critical problem in stru c tu ra l control is the interaction between th e a ctiv e control system and th e structural dynam ics. T h e overlap of the control bandw idth w ith modaJ frequency spectrum is a m ajo r issue in the active control of flexible stru c tu re s. T he classical m ethod for avoiding control instability relies on having a w ide separation between the lowest-frequency resonance and the highest frequency for th e control closed loop response. The effectiveness of the control system depends on th e degree of interaction between the control system and th e stru ctu ral resonances, which, in tu rn , depends on the details of th e control law used.

An equally critical problem in adaptive optics control design, is th e random na­ tu re of the tu rb u le n t atm osphere which prohibits a d eterm inistic expression relating turbulent effect to optical image quality. The quality of th e received image depends on such factors as w avelength, refractive-index stru ctu re c o n sta n t, zenith a n g le to the source, wind velocity, and wind velocity distribution over a ltitu d e . T h e combination of these effects results in a random ly distorted wavefront. P a st research [19,63, 32, 68] has always considered a stochastic description of the wavefront distortions. A possi­ ble representation for describing atm ospheric phase distortion is by a set of Zernike polynomials. T h e Zernike function space provides an orthogonal basis set of func­ tions corresponding to aberrations commonly studied in optics [63]. T h e total phase

C H A P T E R I. IN T R O D U C T IO N 5

distortion (f>{r,0,t) is described by a linear com bination of the Zem ike functions:

N

<i>{r,e,t) = Y . M t ) Z i { r , e ) 1 = 0

where Z i{r,6 ) are th e Zemike basis functions (which will be explained in th e next chapter) spanning th e space within the a p e rtu re as functions of radial and an g u lar coordinates r and 0, and Ai{t) are random tim e-varying coefficients. However, control design using determ inistic -as opposed to stochastic- control laws, requires th a t th e phase d isto rtio n present in the wavefront is to b e described determ inistically.

For high speed applications several types of m irrors have been proposed, such as th in -p late m irrors on piezoelectric stacks, and bim orph mirrors. T h e bim orph m irro r is especially appealing since it provides large-am plitude continuous deform ations at high speed w ith low voltages [31].

T h e p o te n tia l of using bim orph m irrors for high speed adaptive com pensation has been convincingly dem onstrated in some experim ents [31] where th e device per­ forms to frequencies in excess of 10 kHz. N evertheless, there are m any issues which still need to be resolved before appropriate control design procedures are developed. Specifically, given a piezoelectric m aterial, electrom echanical coupling, th e s tru c tu re (m irror) a n d th e control domain, a designer should be able to obtain an o p tim u m procedure for controlling the shape of the given m irro r’s surface. To m eet th is de­ sign goal, one m ust be able to obtain -determ inisticaily- a measure of th e wavefront ab erratio n s and correlate them to the variables used in th e controller design.

To resolve some of these issues, the research in this thesis is directed tow ards th e following m a jo r objectives:

I. T h e first objective is to develop models for th e active structure (i.e., deform able m irro r) consisting of a thin-plate stru c tu re and piezoceramic actuators. This

C H A P T E R 1. IN T R O D U C T IO N 6

also includes a determ inistic representation of the wavefront ab erratio n s (i.e. m irror's surface).

2. The second objective is to develop an optim ai control methodology for dis­ trib u ted param eter system s based on th e developed m odels. This also includes optim al regulator problem for active vibration control of a tip /tilt m irror, and an optim al tracking problem control for active shape control of a deform able mirror.

1.3

L itera tu re S u rv ey a n d P r o p o se d A p p ro a ch

Since the adaptive optics system consists of a m ulti-actuator deformable m irro r and multiple wavefront sensor grad ien t/cu rv atu re m easurem ents, it is a m ulti-dim ensional control system . T he required speed of this system is d ictated by the frequency con­ tent of the incoming optical wavefront [38, 39]. As the signal bandw idth increases, the control system will be required to respond faster. C u rrently control bandw idths of a few hundred Hertz are in use for most applications [33, 48]. The word band­ width is often cited in adaptive optics literature, without regard to the m ultivariable nature of the system . Some articles assume th a t the overall system operates at one bandwidth, thus analyzing th e adaptive optics system as a single-input single-output system [38, 39, 74], which results in lower performance when applied to m u lti-in p u t m ulti-output systems. In some other papers it is recognized th a t there are m ulti­ ple loops and therefore m ultiple “bandw idths” but w ithout considering the possible coupling between the different loops [4, 22, 64, 36, 51]. H uang [48] considered the m ultivariable n atu re of the system and proposed m ultivariable H<x> control design, but this was com putationally non-efficient due th e high dim ensionality of th e controller.

C H A P T E R 1. IN T R O D U C T IO N 1

sensors with a bimorph m irror to remove th e interm ediate stage of reconstructing the wavefront. This is done by connecting th e o u tp u t of each curvature detector to the corresponding input of th e bim orph which results in a very fast tim e response of the system . This claim was contested by Shwartz [69] who showed th at indeed an interm ediate stage is necessary to correct for th e mismatching term s between th e bimorph m irror surface and th e wavefront. Furtherm ore, the m ultivariable problem for the bim orph mirror coupled with a curvature sensor remain unresolved to d a te , and the existing controller was designed based only on the quasi-static model of th e mirror.

Control of a deformable m irror is strongly related to control of flexible structures for which a well established theory and algorithm s exist in the literatu re. Meirovitch [59] has w ritten what is already a classical book on control of flexible structures. He develops models of flexible stru ctu res and gives frequency and tim e dom ain m ethods of systems analysis and synthesis. Porter [65] introduces modal control which can be considered as a predecessor to balanced control. Junkins and Kim [54] give an up-to-date introduction to control of flexible structure. Also Junkins [53] edited a monograph th a t consists of up-to date contributions to the dynam ics, identification and control of flexible structures. Lin [58] gives a good and wide-ranging review of methods used in advanced system analysis and synthesis; H2 and controllers, robust design , nonlinear system s fuzzy controllers, and control using neural network. The monograph by Joshi [52] presents his own developments in the area of control of flexible space structures, supported by num erous applications; it concentrates on robust dissipative controllers a n d LQG controllers. However, apphcation of these conventional control approaches to adaptive optics was not possible w ithout resort to stochastic m ethods in describing th e random variation of the atm ospheric aberrations.

C H A P T E R L IN T R O D U C T I O N 8

as an a ltern a tiv e to th e stochastic problem of th e wavefront representation. T h e foremost featu re of th e proposed methodology is th a t th e conventionad differential sta te space form ulation is replaced by an equivalent algebraic representation. T h e source of in spiration here has been the work of Adiguzel [1] who developed the direct optim al approach for discrete systems. His developm ent is based on th e analyses of response problem s of mechanics introduced for th e first tim e by Bailey [8. 9] an d consists in essence of utilizing H am ilton’s Law of V arying Action, in short HLVA.

To help u n d erstan d th e premise of the direct m ethodology via HLVA, a brief sum m ary would be ap p ro p riate at this point. HLVA was set forth by W illiam Rowan H am ilton in 1834-1835 in his classical papers on a general m ethodology in dynam ics [42, 43]. It m anifests a natu ral law of mechanical system s, m athem atically expressing m inim ization of an energy functional of the system . As in extrem ization of any functional, one can th en bring in m athem atical tools of calculus of variations to a tte m p t a solution. W hen this is done w ithout deducing o r resorting to any differential equation of m otion, th e approach is referred to as direct method. W hen th e governing laws are expressed by differential equations, which can also be obtained from HLVA, the approach is sim ply indirect [Ij.

C entral to generating the solution directly to th e response problem of mechanics espoused by Bailey, is th e concept of expressing the displacem ent field during a tim e interval of m otion in th e form of tru n cated simple power series in tim e w ith constant expansion coefficients which are treated as the unknowns of th e response problem . In essence, this is th e m ethod of Ritz [12], [11], [1]. A pplication of this concept to the tim e varying coefficients in Zemike expansion represents an alternative to th e stochastic problem .

Following B ailey ’s papers in mid and late 1970’s, a tre n d has em erged and o th ers have obtained d irect solutions to dynam ics problems eith er based on HLVA or ref­

C H A P T E R L IN T R O D U C T I O N 9

erencing to it. From th e latest articles, to mention a few, axe those of Borri et al. [21], Ben-Tal and Bax-Yoseph [18], Bax-Yospeh et al. [15], A tilgan and Hodges [3], and Hodges and Hou [47] w ith vaxious em phases, who successfully applied th e direct approach and provided num erical results th a t could get as close to known exact so­ lutions as desired. T he rem arkable feature of this approach is not only th e capability to solve these problems (solution to some of them are available by o th e r m eans), but also the sim plicity of th e procedure w ith which the results are o b ta in e d considering their accuracy.

Since the direct m eth o d in response axialysis proved successful, addressing the control (inverse response) problem directly via HLVA becomes an ap p ealin g technique to investigate. In paxticulax, its dem o n strated simplicity, generality and accuracy provides th e m otivation to search for possibilities of devising a direct control approach to mechanical systems. To this end , a direct open-loop control m ethodology using HLVA was dem onstrated by Oz and Raifie [62]. Adiguzel [1] successfully developed and dem onstrated through a few exam ples, a direct optim al control m ethodology for discrete systems. It was concluded in [1] th a t: Many issues still rem ain to be addressed

. . . Perhaps more practically and readily, an immediate extension would be the study o f distributed parameter system s using the direct control concept o f this study. An on-line implem entation task can be undertaken. A comparative stu d y can be pursued by applying the proposed direct method and others. And maxiy o th e r questions need

to be studied as to how such a direct (control) approach would fit in to th e problem of controllability, observability, optim ality, and feedback control, e tc . All of these aspects of the direct control problem are investigated in this study.

A nother sim ilar b ut somehow different approach, is th e space-tim e fin ite element

optimal control approach. Nagurka and W ang [60], and Yen and N agurka [77] proposed

C H A P T E R 1. IN T R O D U C T IO N 10

of discretization of th e equations of m otion ajid the perform ance index by expanding the states in term s of a finite num ber of prespecified basis functions and undeterm ined param eters. This form ulation leads to a constrained q u ad ratic program m ing prob­ lem th a t is solved analytically. An altern ativ e approach for solving o p tim al control problems by discretization in time was studied by Hodges and Bless [46]. In their approach, th e optim al control problem was form ulated via temporal finite elem ents resulting in a two point boundary value problem whose solution yields a discrete­ time control law. This was applied only to discrete system s. Following Ben-Tal’s [18] approach, the optim al control problem is solved while performing sim ultaneous tem poral and spatial discretization. Therefore, this m ethod can be viewed as a gen­ eralization and extension of the above approaches. Ben-Tal et. ai [18] use dynam ic program m ing m ethod to solve for the control law, while in this study th e control law is sim ply derived using the conventional m ethods of discrete optim al control theory.

To address the optim al control problem s directly, th e spatial part of displacem ent fields are prescribed and curve fitted by known shape functions which represent the Zernike polynomials in an adaptive optics system or th e m ode shape in a vibrating structure. The tem poral part is expanded in power series up to desired accuracy. This concept is extended to the input fields, which are th e ultim ate unknowns in a control problem. The spatial expansion is prescribed according to the geom etrical distribution of the actuators, for the tem p o ral part, the expansion is in sim ple power series w ith unknown coefficients, and th e series are tru n c ate d to m atch th e order of the accelerations. T he expansion coefficients of displacem ent and control field in their respective time-series, together, c o n stitu te the control unknowns of th e problem . Direct application of HLVA to these yields a set of algebraic equations for th e control problem. Thus w ithout any resort to differential equations, the control problem can also be studied directly via such algebraic equations. Through th e proposed

C H A P T E R I. IN T R O D U C T IO N 11

direct methodology, the conventional variational optim al problem involving integral functionals, is transform ed to an equivalent algebraic optim al problem, from which, solutions are obtained in closed form with a straightforw ard procedure. T he space­ tim e finite elem ent -another direct methodology- based o p tim al control m ethodology is also developed and analysed through some examples.

1.4

T h e sis C o n trib u tio n

The contribution of this thesis can be summ erized as follows:

I. An optim al control design for high speed adaptive optics com pensation has been developed. The design methodology is applicable to a wide sp ectru m of problems, w hether the problem considered is determ inistic or stochastic. It is simple, com putationally efficient and possesses a great flexibilty for hardw are im plem entations.

II. As a consequence of this study, a new technique for m odelling stochastic prob­ lems by determ inistic models is developed. Application of this technique to conventional stochastic control problems could simplify th e controller design for such problems.

III. Beside the direct optim al control methodology, the space-tim e optim al control approach newly developed in the course of this thesis, offers another a ltern a ­ tive for stru ctu ral control for both stochastic and determ inistic problems. T he unique feature of this m ethod is its ability to analyse th e structure-control in­ teraction in th e same m athem atical framework, allowing a b e tte r understanding of this interaction, which results in b etter control designs.

C H A P T E R 1. IN T R O D U C T IO N 12

1.5

T h e sis O rg a n isa tio n

C hapter 2 provides a description in a m ore detailed fashion ab o u t ad ap tiv e optics system s, in p articu lar the concept of atm ospheric aberrations and th e ir polynomial representation. A daptive optics using phase conjugation is described along w ith the com ponents which constitute an adaptive optics system. W avefront sensing m ethods, and the corresponding correction m ethods associated w ith each sensing system are introduced. A ttention is finally drawn to adaptive system s using bim orph mirrors and curvature sensor which co n stitu te the m otivating application of th is study. The chapter closes w ith some rem arks and conclusions.

C hapter 3 is dedicated solely to the descriptions and definitions for th e flexible stru ctu re in term s of the assum ed m ode technique. T he p relim in ary formulations of system dynam ics via HLVA and the resulting algebraic s ta te equations (based on assumed mode expansion) are derived. E quations of tim e co n tin u ity required in a tim e m arching process are also included, followed by discussions on controllability from the perspective of direct control approach. This discussion is undertaken to shed light on th e controllable and uncontrollable modes of a deform able structure.

In chapter 4, th e optim al regulator control problem is stu d ied . T h e formulation of an algebraic optim al control problem for a classical linear regulator is introduced after the introduction of the assum ed modes into the control perform ance measure. Then a straightforw ard solution of the o p tim al problem is given an d an illustrative exam ple dem onstrating the m ethod is included. Form ulation an d applications of the direct optim al control are then extended to tracking problem s in C h ap te r 5, where an optim al controller is designed for a flexible deform able m irror. A sim ulation of th e deformable m irror (bim orph m irror) w ith an d without th e controller is included at the end of th e chapter.

C H A P T E R 1. I N T R O D U C T I O N 13

C hapter 6 introduces th e space-tim e finite elem ent formulation applied to stru c­ tures with d iscrete a n d /o r continuous piezoelectric transducers. T he op tim al control is reconsidered here. An o ptim al controller is designed using space and tim e discretiza­ tion, and th e resulting controller is com pared w ith th e direct optim al controller in overall perform ance via sim ulation of the deform able mirror introduced previously. Some concluding rem arks close this chapter.

This study is concluded in chapter 7 w ith an overall summary and assessm ent of the findings and directions for future research.

14

C h a p te r 2

A d a p tiv e O p tic s S y s te m s

In this chapter, we will present a general overview of an ad ap tiv e optics system as found in the literatu re. We will describe and explain in a m ore detailed fashion, the m ajor functions and com ponents th a t constitute an adaptive optics system . We first define a block diagram representation of the closed loop adaptive optics system , then using the diagram , we will explain the function of each com ponent in th e diagram. Since the control system is the focus of this study, reconstruction and control of the wavefront are given m ore atten tio n .

2.1

In tr o d u c tio n

Figure 2.1 represents a block diagram of the closed loop adaptive optics system . The atmospheric ab erratio n s axe represented as additive disturbances by th e vector d. Note that all th e com ponents of d can have different and perhaps independent tem ­ poral variations th a t m u st be com pensated for by th e adaptive optics control system. The wavefront is reflected from a deformable m irror, and th e resulting wavefront

C H A P T E R 2. A D A P T IV E O P T IC S S Y S T E M S 15

D/A Amplifier Circuit

Wavefiont Sensor

Deformable Mirror Control System

Wavefiont Reconstruction

Figure 2.1: Block diagram of an AO system

error vector e is introduced into th e sensor. T h e wavefront sensor measures local subaperture slopes in the x and y directions a n d /o r wavefront curvature which are represented by th e vector s. T he wavefront reconstructor com bines these measure­ ments and produce an estim ate of th e wavefront error ê. T h e wavefront estim ates are passed to th e control system , which selects control signals for each actuator (for discrete a c tu a to r mirror) or electrode (for a bim orph m irror) to reduce the estim ated error ê. T he o u tp u t of the control system V is amplified by an amplifier circuit, which produces th e mirror drive voltages Va- T h e voltages drive th e piezoelectric actuators which deflect the surface of the deform able m irror. W hen th e closed loop system is o perating properly, th e shape of the deform able m irro r will m atch the shape of the incom ing disturbance in steady state, producing zero wavefront error.

2.2

P h a s e C o n ju g a tio n

The principle of phase conjugation is the core of adaptive optics. It can be analyzed by a num ber of ways. T he m ethod can be shown by observing F igure 2.2. T he wavefront of a beam entering from the left (a) is distorted by a piece of glass because the index of refraction is higher than I. T he wavefront is retarded as it goes through the gleiss

C H A P T E R 2. A D A P T IV E O P T IC S S Y S T E M S 16

W avefront

A berrator

M irror

D eform able M irror

C H A P T E R 2. A D A P T IV E O P T IC S S Y S T E M S 17

(b). After reflectioa from th e m irror (c), the wavefront has th e sam e shape but it is propagating in the opposite direction. As it traverses th e glass again, it receives the same retardation as before. T h e exiting wavefront (e) is greatly distorted, since it passed through the a b e rra to r twice.

If we want to achieve a plane wavefront after a beam passes through the glass twice, there may be a way to alte r th e surface of th e m irror in such a way as to invert the wavefront so th a t th e second passage leaves no residual distortion. Looking at figure 2.2, we can see th a t a bum p in the m irror at just the right place and at just

the right amount caji cause th e leading edge of the wavefront to be reversed. W hen

this wavefront (d) passes through the aberrator again, th e final wavefront (e) is once again plane. This is called conjugation of the phase which comes from the fact th a t the correction is proportional ajid inverse in sign to the am o u n t of aberration in the wavefront. Note th a t if th e aberration is dynam ic -in reality it is-, we must place the phase conjugate on the beam at the right time [72].

2.3

R e p r e s e n ta tio n o f th e W a v efro n t

For imaging applications, atm ospheric turbulence (which is referred in the literature to the changes of air density due to tem perature variations) along th e propagation p ath causes continuous sp atial and tem poral variations in th e index of refraction. Such variations in index of refraction result in spatial an d tem p o ral m odulation of intensity and phase of th e optical image.

The random n atu re of th e tu rbulent atm osphere prohibits a determ inistic expres­ sion relating tu rbulent effects to optical image quality. T h e q uality of the received image depends on such factors as wavelength, refractive-index stru ctu re constant, zenith angle to the source, wind velocity, and wind velocity d istribution over alti­

C H A P T E R 2. A D A P T IV E O P T IC S S Y S T E M S 18

tude. The com bination of these effects results in a random ly distorted wavefront. Thus it becomes necessary to have means of m athem atically describing the phase distortion present in the wavefront.

A number of m athem atical constructs are used to describe th e phase of a beam . These include power series representation and a set polynomials called Zemike Poly­ nomials.

T he power series representation is, unfortunately, not an orthonorm al set over a circle. Many such series exist. T he set of polynomials th a t is orthonorm al over a circle introduced by Zernike has some very useful properties. T h e series called Zemike

series, is composed of sums of power series term s w ith appropriate normalizing factors.

A detailed description of th e Zemike series is given by Bora an d Wolf [20], and an analysis of Zernike Polynomials and atm ospheric turbulence including their Fourier transforms was done by Noll [61].

n m Expression {p. 9) Expression (x, y) Description

0 0 1 1 P iston

1 1 2/9 cos 9 2x y T ilt

1 1 2/9 sin 9 •2y X T ilt

2 0 v/3[2/>^ — 1] v ^ [2x^ + 2t/2 - 1] Defocus

Zs 2 2 cos 29 \ /6[x^ — y^] A stigm atism

Ze 2 2 y/6p^ sin 29 \/62xy Astigm atism

Zr 3 1 \/8[(3p^ — 2p) cos 9\ \/8[3x(x^ + y^) — 2x] Com a Zs 3 1 \ /8[(3/9'^ - 2/9) sin 9] \/8[3y(x^ + y^) - 2y] Com a Zg 3 3 \ /8[/9^ cos 30] v/^[x(x2 - 3y2)] Trefoil Zio 3 3 \ /8[/»^ sin 30] \/8[y(3x^ - y^)] Trefoil

Table 2.1: First 10 Zem ike Polynomials

T he general Zem ike series contains all ab erratio n term s, including piston ^ and tilt. T he analytic expressions of the first 10 Zem ike term s are given in table 2.1. In

C H A P T E R 2. A D A P T IV E O P T IC S S Y S T E M S 19

polar coordinates, these polynomials are defined by [61, 75]

Zeven j = ^2(71 + l)72^(/9) COS ; m / 0

Zodd j = ^2(71 + s i n ; ttz ^ 0

Zj = yj2{n + l ) i2^ (p ) ; m = Q (2.1)

Where /» = ^ , r is a radial distajice, R is th e a p ertu re radius , 9 is the azim uth angle.

n is the degree of radial m ode, m is the éizirauthal frequency, and

_________ ( - 1 ) '( 7 Z - S ) ! _________ 25 (2.2) è o 5![(fi + m ) / 2 - s ] ! [ ( 7 7 - 7 7 i ) / 2 - s j !

Therefore, a wavefront W can be described by a linear com bination of th e Zemike functions as:

N

W { r , 9 , t ) = Ao{t ) + Y i Ak i t ) Z k { r , 9 ) (2 . 3)

t=l

where Zk { r , 9 ) are the Zem ike basis functions spanning the space w ithin th e aper­ ture as functions of radial and angular coordinates r and 6, and Ak{ t ) are random time-varying coeflScients which have to be determ ined using inform ation ab o u t the wavefront obtained through phase m easurem ents.

M athem atically, an infinite num ber of Zem ike functions are required to character­ ize the wavefront completely. However, approxim ately 92% of th e root m ean square (rms) phase inform ation is contained in th e first 14 Zernike modes excluding th e pis­ ton (zeroth mode) [63]. T h e piston m ode represents the average phase w ithin the aperture. It is nondistortive and unobservable by the wavefront sensor. As such the piston is removed from th e sum m ation in Eq. 2.3.

An im p o rtan t property of the Zernike functions is th a t they form an orthogonal basis set of functions th a t satisfy the relation [63]

C H A P T E R 2. A D A P T IV E O P TIC S S Y S T E M S 20

W here 6{j is th e Kroneker symbol:

6ij = < ^ (2.5)

0, i^ j

From Eqs. (2.3), (2.4) it can be shown th a t th e rm s value of phase across a n ap ertu re is simply th e root sum square of th e Zernike coefficients:

a = \ / ( c f + c ^ + c § + . . . + c ^ ) (2.6)

T he Zernike coefficients for an arb itra ry value of p hase are obtained as:

4 _ f d0 f ^ r , e ) W ( r . 0 , t ) Z i i r , 9 ) r d r

' f d 0 f ^ { r , d ) r d r ’

W here $ ( r , 0) is th e ap ertu re weighting function defined in this study as: I, r < R

0(r,g ) = “ (2.8)

0, r > R

2.4

W a v efro n t S en sin g

The wavefront inform ation that is derived from m easured data, will be used by the control system for phase correction. T he actual reconstruction of the phase from this d a ta is discussed in subsequent sections. Two basic types of wavefront inform ation are used. W hen th e wavefront is expressed in te rm s of O ptical P a th Differences (OPD) over a sm all spatial area, or zone, th e w avefront is said to be zonal. W hen the wavefront is expressed in term s of coefficients of th e m odes of a polynom ial expansion (such as Zernike Polynom ials) over th e entire pupil, it is said to be modal.

The wavefront sensors (W PS) in ad ap tiv e o p tics do not m easure d irectly the wavefront but its first a n d /o r second sp atial derivatives. H artm ann-Shack sensors are the m ost com m only used for th e m easurem ent of th e first spatial derivative (or tilt), and cu rv atu re sensors are used to m easure th e second spatial derivative which is the curvature of th e wavefront.

C H A P T E R 2. A D A P T I V E O PTIC S S Y S T E M S 21

2.4.1

H a r tm a n n -S h a c k W a v efro n t S e n so r

The principle of H artm ann-Shack W FS is presented in F igure 2.3. A lenslet array is placed in a conjugate pupil plane in order to sample th e incoming wavefront. If the wavefront is plane, each lenslet forms an image of th e source a t its focus Figure 2.3a. If the wavefront is distorted, each lenslet receives a tilte d wavefront in the first approxim ation and forms an image out of axis in its focal plane Figure 2.3b. The measure of the image position gives directly th e angle of arrival of the wave at each lenslet.

Usually, th e Shack-H artm ann W FS requires the use of a reference plane wave generated from a reference source in the instrum ent, in o rd er to calibrate precisely the focus positions of th e lenslet array. A good feature of th is sensor is the simultaneous determ ination of the x an d y slopes by th e m easurem ent o f th e image position (x and

y coordinates)

A num ber of m ethods can be used to m easure the H artm ann-S hack images formed by the lenslet array. T h e simplest technique is to use a quad-cell detector for each subaperture [2],[72]. T h e m ain drawbacks of this technique are generally the lim ited dynamic range and spot size dependent response. A n o th er solution is the use of a CCD as a detecto r to record sim ultaneously all the im ages. T h e good feature of a CCD is the perfectly determ ined pixel positions and th e 100% fill factor. Even a CCD can be used as an array of quad-cells. B ut it allows in principle to calculate the centroid of th e spot a t th e price of a large num ber of pixels per subaperture. [2]

2.4.2

C u r v a tu r e S en sor

This type of sensor was developed by R oddier [67] to m easu re curvature instead of wavefront slope m easu iem en t as in H artm ann-Shack W FS described previously. The

C H A P T E R 2. A D A P T IV E O P T IC S S Y S T E M S 22

Focal Plane Pupil Plane

Lanslet Array Plane W ave a ) oc f D isturbed Wave b)

Figure 2.3: Principle of H artm ann-Shack Wavefront Sensor: a) Plane Wave b) Dis­ turbed Wave

C H A P T E R 2. A D A P T IV E O P T IC S S Y S T E M S Telescope 23 LackofiUumioatico Excess of illummüon Field Lens

Figure 2.4: Principle of th e C urvature Sensor

Laplacian of the wavefront together with th e wavefront tilts at the a p e rtu re edge are measured, providing d a ta to solve Poisson’s equation to reconstruct th e wavefront. C ontrary to what is claimed in the literatu re [2] this type of sensor cannot be used directly with a bim orph m irror to solve Poisson’s equation. R ather, an interm ediate stage is needed for correction [69].

The principle of this sensor is presented in Figure 2.4. T he telescope of focal length / images the source in its focal plane. T he cu rv atu re sensor consists of two image detectors placed out-of-focus. T he first d etecto r records th e irradiance distrib u tio n in plane Pi at a distance I before th e focal plane. T he second one records th e irradiance distribution in plane P2 at the sam e distance / behind th e focus. A local wavefront curvature will produce an excess of illum ination in one plane and a lack of illum ination in the other (Figure 2.4). A field lens is used for sym m etry in order to reimage th e pupil. The planes Pi and P2 can be also seen as two defoculized pupil planes. It can

C H A P T E R 2. A D A P T IV E O P T IC S S Y S T E M S 24

be shown th a t in th e geom etrical optics approxim ation, th e difference between th e two irradiance distributions is a m easure of th e local wavefront inside th e beam and th e radial first derivative a t th e edge of th e b eam [2, 67]. The geom etrical optics is approxim ated by:

f - I I

y - < r o - (2.9)

ro f

T he m easured signal S is th e norm alized difference between the illum inations Ii{r) and /^ (r) in th e planes P\ and P2 an d is given by [67]

W here r is a position vector, W = W { r ) is th e m easured wavefront surface, V^VF is the Laplacian of th e wavefront, is the wavefront slope a t the edge of th e aperture, and 6c is a linear im pulse d istrib u tio n around th e pupil edge.

For high order aberration m easurem ents, th e distance I must be laxger th an for th e case of low o rd er aberrations m easurem ents. For extended sources, th e distance / m ust also be larger th an the case of point sources. An increase of the distance / means a decrease in th e sensitivity (and an increase o f th e dynam ic) of the curvature sensor as expressed by Eq. 2.10. T h e distance / of th e curvature sensor is very sim ilar to the lenslet focal length of th e H artm ann-S hack sensor. N ote th a t when th e distance

I is decreased to th e m inimum, th e curvature sensor is only able to m easure tilts and

is reduced to quad-cell. In th e lim iting case th e cu rv atu re sensor provides four edge m easurem ents an d no more curvature.

The setup used at the U niversity of Hawaii [37] employs a variable curvature m irror placed a t th e focus of the telescope as a field lens. It allows to reim age on the detector array th e inside and outside focus b lu rred pupil images by its concave and convex deform ation. This produces a m odulation of the illum ination on th e detector

C H A P T E R 2. A D A P T IV E O P T IC S S Y S T E M S 25

array. T he cu rv atu re signals are recovered by synchronous detection. T he pixels inside th e beam give the local curvatures, the pixels on the edge of th e beam give the local wavefront slopes. The m odulation frequency is th e tem poral sam pling frequency of th e wavefront, th e deform ation am p litu d e directly determ ines th e working distance /. A good feature of this device is its capability to modify easily th e sensitivity of the sensor by changing the am p litu d e of the m irror vibrations (i.e. I). Because of the low num ber of subapertures in th e sensor, photo-counting avalanche photodiodes are used as d etecto rs, taking advantage of their q u an tu m efficiency. For curvature sensing, only two detectors are required per su baperture [2].

2.5

W avefron t C o r r e c tio n

As w ith sensing, th e re are both m odal and zonal means of correction as well. T he need to engineer b e tte r system s, with higher spatial resolution, more stroke capability, and higher operational bandw idth, has been a catalyst in developing actu a to rs, faceplate m aterials, and analytical techniques th a t apply to m any other areas of engineering.

M irrors th a t have an actively controlled reflective surface are th e m ost common devices used for wavefront correction.

2 .5 .1

M o d a l T ilt C o r r e c tio n

The sim plest form of wavefront correction is variation of the beam direction, or the tilt of th e wavefront. The am ount of energy needed to control th e tilt m ode in an optical system is d irectly related to th e stroke and bandw idth requirem ents for the steering m irror [72]

C H A P T E R 2. A D A P T I V E O PTIC S S Y S T E M S 26

2 .5 .2

M o d a l H ig h er-o rd er C o r r e c tio n

T he next m ost com m on form of correction is an active focus system. W hile peering through a telescope, an image is brought into focus by one’s knowledge th a t a ‘Tuzzy’’ image is out of focus while a “sharp” image is not. By cycling through the position of the sharpest focus, th e system can be made to a d ap t to variations in the receiver’s lens system , th e o bserver’s eye (or a cam era).

It can be shown th a t «istigmatism can be corrected by moving a single lens or m irror. If a cylindrical m irror or lens is aligned w ith th e beam and moved along the optic axis, astig m atism is changed. By having two m irrors or lenses with their cylinder axes oriented 45° to each other, astigm atism for all orientations can be corrected [72].

2 .5 .3

M u ltic h a n n e l C o rrectio n

W hen it is necessary to correct for modes higher th an astigm atism , the wavefront can be divided spatially. Each part of the beam is corrected by applying the required strength of correction to th a t part. Devices th a t work in this m anner are cailed

multichannel correctors.

S egm en ted m irrors

As reported in [72], the earliest im plem entation of a m ultichannel corrector was the segmented m irror. This is a mirror made up of a num ber of closely spaced small mirrors with piston or tilt correction capabilities. See Figure 2.5. In this way the higher-order m odes of correction can be applied by determ ining the individual con­ tribution of each of the segments. Some segm ented m irrors of this type have been operated in a piston-only mode, whereby each segm ent is confined to simple up-and- down motion. T h e piston/tip/tilt has been shown also. Each segment operates with

C H A P T E R 2. A D A P r r V E O P T IC S S Y S T E M S 27

Mirror Faceplates

Actuators

Piston Only Piston plus Tilt

Figure 2.5: Segmented m ultichannel m irro r

three degrees of freedom: th e up/dow n piston mode and two orthogonal éixes of tilt. .4 further consideration is th e shape of th e segments. Square, hexagonal, and circular segments have been used.

The discontinuities (gaps) between segm ents have an im p act on overall perfor­ mance. Energy is lost through the gaps. It is im portant th a t th e area of th e gaps is below 2% [72]. Because each segment operates independently, there is no cross­ coupling or need for actu ato r preloading. T h e step response for a segment can be as low as lOO/zs

C ontinuous Faceplate M irrors

The continuous faceplate m irrors as opposed to segm ented m irrors, autom atically m aintain continuity and therefore can work w ith a reduced n um ber of actuators. Ac­ tuators are generally of the push-pull type using mostly piezoelectric or electrostrictive m aterials. T h e actuators can be discrete actu ato rs perpendicular to the surface as in Figure 2.6 o r discrete actuators on the edges th a t im part bending m om ents. T hey can aJso be continuous such as in m em brane m irror or bim orph m irror. See Figures

C H A P T E R 2. A D A P T IV E O P T IC S S Y S T E M S 28

a

Figure 2.6: D iscrete A ctuators m irrors: a) discrete position actu ato rs; b) discrete force actu ato r; c) bending moment actu ato rs

2.7, and 2.8

D iscrete A c tu a to r s D eform ab le M irrors

SADM {Stacked Actuator Deformable Mirror) correctors, with d iscrete sets of actua­ tors in the form of a stack of piezoceram ic disks su pported on a rigid base were first developed in th e la te 1970’s [30] to address the large stroke requirem ents of infrared system s. A m irro r deform ation of 0.8^m was obtained, b u t the to ta l voltage required for each stack is m ore th an 1 kV. In th e beginning of I980’s the range of displacement is increased to ± 8^ m . T h e high supply of voltage required for control, th e large hys­ teresis and th e lim itatio n on the n u m b er of actuators m ade this ty p e of mirrors not an ideal one. M in iatu rizatio n of a ctu ato rs, low operating voltages, position accuracy, and low hysteresis becom e then im p o rta n t param eters to consider. A review m ade by Eaiey [29] a n d Raybova and Zakharenkov [66] discussed o th er developments of

C H A P T E R 2. A D A P T IV E O P T IC S S Y S T E M S deformable m irrors. 29 M em brane M irrors Membrane Electrode Control Electrodes

Figure 2.7: A membrane m irror

New types of deform able m irror designs have also been developed in recent years, such as the membrane m irro r [40]. From Figure 2.7, a reflective surface is positioned between a tran sp aren t electrode an d a series of electrodes at the back of the m irror. T he deformation is caused by electro static forces. T h e m em brane is suspended in a partially evacuated environm ent to reduce damping. For large-stroke designs, dam p­ ing has been found to be needed since the surface tends to be unstable and pin the m em brane against th e electrodes. Additionally a m em brane m irror requires the use of a window which com plicates applications in the infrared.

C H A P T E R 2. A D A P T IV E O P T IC S S Y S T E M S 30

A bimorph m irror is m ade of two th in plates of piezoelectric m aterials bonded to­ gether. The two plates are oppositely poled. When a voltage is applied, th e piezoelec­ tric effects leads to an opposite vaadation of the transverse length of th e two plates, and this results in bending of th e m irror. See Figure 2.8. T he use of curvature adap­ tive optics is strongly advocated for correction of the lower order aberrations on the basis of its economy, of its sim plicity of control, and because its influence function which is global across the en tire m irror has a spatial behavior sim ilar to th a t of the Kolmogorov-based atm ospheric wavefront structure [17]

2.6

W avefront R e c o n s tr u c tio n

Wavefront sensors, described earlier m easure the condition of the wavefront and pro­ duce signals th a t represent th e wavefront. Both zonal and modal sensors may be used in a particular system. It is up to th e reconstructor to sort out the m eaning of those signals, and then it is the control system th a t must determ ine how to tre a t the signais and relay them to the ap p ro p riate correction device.

2 .6 .1 P h a s e from w a v efro n t slo p es

Retrieving the phase from slope m easurem ents is the determ ination of the phase at various points from the knowledge of the wavefront slopes at other points of the wavefront plane. This is a fundam ental problem of adaptive optics. T h e most com­ mon wavefront sensors (H artm ann-Shack sensors or shearing interferom eters) produce o u tp u t signals proportional to th e wavefront slopes.

The num ber of unknown phase points is N and the num ber of wavefront slope m easurem ents is M . T he problem of phase determ ination from wavefront slopes

C H A P T E R 2. A D A P T I V E O P TIC S S Y S T E M S 31

depends upon th e geom etry of the problem. T h e phase estim ation a t N locations is

0n- T he m easured slopes a t M locations are Sm- T h e m atrix connecting the two is

determ ined by the geom etry of the problem. For instance, for th e geom etry given by Figure 2.9a, th e m easured slopes from the wavefront sensor are represented by or The superscripts represent the slope in th e x or y direction. T he phéise is to be determ ined at th e points 0_i.o, . . . . w here the subscripts represent the coordinates of th e unknow n points. A set of equations can then b e constructed

■Sj = 0 - 1 . - 1 — 0 0 , - 1

•^2 = 0 0 , - 1 — 0 0 ,0

th a t can be w ritten in a m atrix form as

3 = B<t> (2.11)

W here the m atrix B is th e reconstruction matrix. Often, B is sim ply composed of sums and differences based on the geom etrical configuration of wavefront sensor positions and phase d eterm ination positions. M any of the elem ents are +1 or -1. In some cases, however, B can be quite complex. In wavefront sensors, th e slope m easurem ents is seldom a sim ple difference between two different points, but usually it is a spatial average over th e subaperture. In ad ap tiv e optics control system problems, one often encounters th e case where th e num ber of equations M is greater than the num ber of unknowns N . This ovevdetermined system is the basis for much research in num erical techniques, m any of which apply to ad ap tiv e optics system s.

T h e problem reduces to calculating the values of th e unknowns such th a t th e error between the m easured, or known param eters s,- an d th e actual values of s is small. An inverse for a non-square m atrix cannot be calculated directly. A n approxim ation