Table Of Content

NASA Conference Publication 10065 Part 1 4th NASA Workshop | on Computational Control of Flexible Aerospace Systems Compiled by c_— -— , Lawrence W. Taylor, Jr. Langley Research Center Hampton, Virginia Proceedings of a workshop sponsored by the National Aeronautics and Space Administration and held at the Kingsmill Resort Williamsburg, Virginia July 11, 1999 oA MARCH 1991 NASA Nat Ona! Aer Nauti © ang | Space Admunistratior Langley Research Center Hampton, Virgina 23665-5225 Foreword ) The practice of modeling and controlling flexible aerospace systems grows in importance as the performance needed of active control systems increases. As the size of spacecraft increases and the j demands of control systems become more exacting, the accuracy required of the models used for analysis also increases. The increased complexity, the increased model accuracy, and the demands for more precise and higher control system performance result in an increased burden on the part of the analyst. Although this burden is somewhat alleviated by advances in software, there remains the pressure for assuring system stability and performance under conditions of plant uncertainty. Although robust considerations are included in many synthesis techniques, the price in terms of reduced system performance is often prohibitive. Because similar difficulties and concerns are encountered for different applications, it is valuable to enhance the exchange of information with regard to aircraft, spacecraft and robotic applications. This is the fourth workshop in a series which has emphasized the computational aspects of controlling flexible aerospace systems. It is hoped that the reports contained in this proceedings will be useful to practicioners of modeling and controlling flexible systems. Lawrence W. Taylor, Jr. NASA Langley Research Center TABLE of CONTENTS - Part 1 Structures /Control Theory “Spillover, Nonlinearity and Flexible Structures" - Robert W. Bass and Dean Zes "Stabilization of Large Space Structures by Linear Reluctance Actuators” - Saroj K. Biswas and Henry M. Sendaula 1S “Optimal Control of Systems with Capacity-Related Noises” - Mifang Ruan and Ajit K. Choudhury "Querying Databases of Trajectories of Differential Equations II: Index Functions” - Robert Grossman 35 Computational Issues "A Fast Algorithm for Control and Estimation Using a Polynomial State-Space Structure” - James R. Shults, Thomas Brubaker and Gordon K. F. Lee 41 “Supercomputer Optimizations for Stochastic Optimal Control Applications” - Siu-Leung Chung, Floyd B. Hanson and Huihuang Xu = 557 "A Fast, Reliable Algorithm for Computing Frequency Responses of State Space Models” - Matt Wette 71 “Coupled Riccati Equations for Complex Plane Constraint” - Kristin M. Strong and John R. Sesak 79 “Optimal Controllers for Finite Wordlength Implementation” - K. Liu and Robert Skelton 91 ili BLANK NOT FILMED Table of Contents - Continued Page Multi-Flex Body Simulation "System Dynamic Simulation of Precision Segmented Reflecior’ - Choon-Foo Shih and Michael C. Lou 133 "Flexible Body Dynamic Stability for High Performance Aircraft” - E. A. Goforth, H. M. Youssef, C. V. Apelian and S. C. Shroeder 145 "A Recursive Approach to the Equations of Motion for the Maneuvering and Control of Flexible Multi-Body Systems” - Moon K. Kwak and Leonard Meirovitch 157 "Serial and Parallel Computation of Kane's Equations for Multibody Dynamics” - Amir Fijany 181 | "A Generic Multi-Flex-Body Dynamics, Controls Simulation Tool for Space Station” | - Ken W. London, John F. L. Lee, Ramen P. Singh and Buddy Schubele 211 | Control-SInttergrautecd tOputirmizeatsio n "An Integrated Control/Structure Design Method Using Multi- Objective Optimization” - Sandeep Gupta and Suresh M. Joshi 231 “Combined Structures-Controls Optimization of Lattice Trusses” - A. V. Balakrishnan 253 “Control and Dynamics of a Flexible Spacecraft During Stationkeeping Maneuvers" - D. Liu, J. Yocum and D. S. Kang 291 "Transform Methods for Precision Continuum and Control Models of Flexible Space Structures” - Victor D. Lapi, James D. Turner and Hon M. Chun 331 "Structural Representation for Analysis of a Controlled Structure” - Paul A. Blelloch 341 iv Table of Contents - Continued Page “"PDEMOD - Software for Controls-Structures Optimization" - Lawrence W. Taylor, Jr. and David Zimmerman 359 "Maneuver Simulations of Flexible Spacecraft by Solving Two-Point Boundary Value Problems” - Peter M. Bainum and Feiyue Li 393 “Control Effort Associated with Model Reference Adaptive Control for Vibration Damping” - Scott Messer and Raphael Haftka 419 “Component Mode Damping Assignment Techniques” - Allan Y. Lee 443 Table of Contents - Part 2* Aircraft Active Control Applications “An Overview of the Active Flexible Wing Program” - Stanley R. Cole, Boyd Perry Ill and Gerald Miller 459 “Aeroelastic Modelling of the Active Flexible Wing Wind-Tunnel Model” - Walter A. Silva, Jennifer Heeg, Robert M. Bennett 497 “Design and Test of Three Active Flutter Suppression Controllers” - David M. Christhilf, William M. 4dams, Martin R. Waszak, S. Srinathkumar and Vivek Mul:hopadhyay 535 “Roll Plus Maneuver Load Alleviation Control System Designs for the Active Flexible Wing Wind-Tunnel Model” - Douglas B. Moore, Gerald D. Miller, Martin J. Klep] 561 "Development, Simulation Validation and Wind-Tunnel Testing of a Digital Controller System for Flutter Suppression” - Sherwood T. Hoadley, Carey S. Buttrill, Sandra M. McGraw and Jacob A. Houck 583 * Published under separate cover Table of Contents - Continued Page t and Testing of a Controller Performance Evaluation Methodology for Multi-Input/Multi-Output Digital Control Systems” - Anthony Potozky, Carol Wieseman, Sherwood Tiffany Hoadley and Vivek Mukhopadhyay 615 Active Control and Passive Damping “Active Versus Passive Damping in Large Flexible Structures” - Gary L. Slater and Mark D. McLaren 655 "Vibration Suppression and Slewing Control of a Flexible Structure” - Daniel J. Inman, Ephrahim Garci and Brett Pokines 663 “Candidate Proof Mass Actuator Control Laws for the Vibration Suppression of a Frame” - Jeffrey W. Umland, Daniel J. Inman 673 “Simulator Evaluation of System Identification with On-Line Control Law Update for the Controls and Astrophysics Experiment in Space” - Raymond C. Montgomery, Dave Gosh, Michael A. Scott and Dirk Warnaar 691 “Dynamics Modeling and Adaptive Control of Flexible Manipulators” - J. Z. Sasiadek 727 “Active and Passive Vibration Suppression for Space Structures" - David C. Hyland 743 “Real Time Digital Control and Controlled Structures Experiments” - Michael J. Rossi and Gareth J. Knowles 781 SIydsetenmtsi fication/Modeling "Finite Element Modelling of Truss Structures with Frequency- Dependent Material Damping” - George A. Lesieutre 795 "An Experimental Study of Nonlinear Dynamic System Identification” - Greselda |. Stry and D. Joseph Mook 813 "Time Domain Modal Identification/Estimation of the Mini-MAST Testbed" - Michael J. Roemer and D. Joseph Mook 825 vi Table of Contents - Continued Page “An Overview of the Essential Differences and Similarities of System identification Techniques” - Raman K. Mehra 845 “Likelihood Estimation for Distributed Parameter Models for the NASA Mini-MAST Truss” - Ji Yao Shen, Jen Kuang Huang and Lawrence W. Taylor, Jr. 881 “Spatial Operator Approach to Flexible Multibody System Dynamics and Control” - G. Rodriguez 907 "A Model for the Three-Dimensional Spacecraft Control Laboratory Experiment” - Yogendra Kakad 921 vil N91-22308 SPILLOVER, NONLINEARITY, & FLEXIBLE STRUCTURES , Robert W. Bass Dean Zes Ror -well International Science Center, McDonnell: Douglas Helicopter Co. PO. Box 1085, ThousanOadks , CA 91358 5000 McDowell Road, MesAaZ ,852 05 ABSTRACT Many systems whose evolution in time is governed by Partial Differential Equations (PDEs) are linearized around a known equilibrium before Computer Aided Control Engineering (CACE) is considered. In this case there are infinitely many | independent vibrational modes, and it is intuitively evident on physical grounds that | infinitely many actuators would be needed in order to control al! modes. A more precise, general formulation of this gra:ve d ifficulty (the “spillover” ifs # which is the infinitesimal genera transition operators Tit) for non-negative times {; and let 8 denote a bounded linear operator acting on another separable Hilbert space U (the control space) with range in #. Now consider the control problem dx/dt = Ax + Bu, with x(0) given. Then according to Balakrishnan this system is not exactly controllable if B is compact. A possible route to circumvention of this difficulty lies in leaving the PDE in its original nonlinear form, and adding the essentially finite-dimensional control action Bu prior to linearization. In many cases it can be shown that the nonlinearity couples the system's modes in such a manner that only a finite-dimensional subset of the modes is functionally independent, with the remaining higher-order modes nonlinearly dependent upon them. Herce control of ali modes can be achieved by controlling only finitely many modes. One possibly applicable technique is the Liapunov-Schmidt rigorous reduction of singular infinite-dimensional implicit function problems to finite-dimensional implicit function problems. Such a procedure was employed by Leon Lichtenstein in the 1930's to prove the existence of a solution of the Navier-Stokes equations for a sufficiently small time-interval 0 =f < «. Omitting details of Banach-space rigor, the formalities of this approach are as follows. Let £ be a Fredholm operator with pseudo-inverse K; then there exist idempotent projection operators F = 3 - K¥, Q = 9 + ZK whose ranges are finite-dimensional and such that a NASC for (*) Ze ~- Fix) = O is that (e*#) x = Pe + KFix) & OFix) = 0. Thus one may set x = u + v where v satisfies the auxiliary equation v = K#iu + v) and u the bifurcation equations |Verzweigungsgicichungen| QFiu + v) = 0. Typically one may solve the auxiliary equation by (contraction) iterations to find v = Siu) where now v is infinite-dimensional but u is finite-dimensional and then insert the result into the (finitely many) bifurcation equations to define a finite-dimensional vector function fiu) = QOFiu + Giu)) such that (+) is equivalent to Siu) = 0. In summary, a NASC for («) is {eoe) x = u + Glu), Lu =O, PRlu) =O, Pueseu, slu) = O, As an illustration the auxiliary equation and bifurcation equations for the problem of deflection of an in tension la 0) EXTENSIBLE beam (a, > O) is considered, including viscous damping la, > 0) and Balakrishnan-Taylor damping la, » 0). Here : ZimeAdel u+¢a‘u *“a-‘u + [a ta fugrar say [fue )-ax] }-u : tt > 2 > oe 0 i e * 4 . * «tt Da vtz2o,0s x 6s L. As the dimension N of the bifurcation equations increases, the result approaches an N-dimensional truncated cigenexpansion (provided that the initial defiections and their initial spatial and temporal rates of change are not too large). _<' Preface The basic idea behind the present paper is simply: SUGGET SIO N Don’t linearize a PDE until after its reduction to a finite-dimensional ODE. This idea can be implemented by means of the following analytical procedure: LIAPUNDY-SCHFIDT BIFURCATION EQUATIONS: A rigorous reduction of a singular infiniie-dimensional implicit equation to the problem of an equivalent, merely finite-dimensional implicit equation. This suggestion is presented as a possible technique for circumvention of the famous “Sp.liover Problem.” Introduction if the problem of control of a flexible structure is linearized before one considertsh e control aspects, then frequently it leads to an abstract problem in functional analysis of the type of the following system of ordinary differential equations: . dx . 0 y at Ax + Bu, xtO) =x. (1) Here x € ” is an element of an infinite-dimensional state space taken to be a separable Hilbert space. Also u € U is an element of the control space, taken to be another separable Hilbert space. We take A D + M” to denote a closed linear mapping of the dense linear-subspace domain D into M” = D which is the infinitesimal generator 4 a strongly continuous semi-group of transition operntors Tit) for t 2 0. Finally we require that B: U + M” be a bounded linear operator. The celebrated “Spillover Probnowl heas man ”exa ct formulabyt mieaons no f: THEOREM. (Balakrishnan, [5), p. 233.) If B is compact, then ¥ is NOT ewactiy controllable. A compact operator is one which can be approximated arbitrarily closely by an operator whose range is finite-dimensional. Therefore the practical import of the preceding theorem can be phrased as: if a linear system has infinitely many independent modes of motion, it cannot be controlied completely with a finite- dimensional actuator suite. This suggests that complete control of a flexible structure by a finite actuator suite is foredoomed to impossibility. However, there may be a way to circumvent this difficulty. Note that the preceding theorem has been proved only in the case that the dynamical system ¥ is linear. The purpose of the present paper is to demonstrate that for many flexible structure problems, such nonlinear mechanisms as Balakrishnan-Taylor damping will couple the higher order modes of motion to the lower order modes in such a way that only a finite number of the lower order modes is functionally independent. This suggests that a finite actuator suite could control such a system. However, we defer consideration of the control problem and deal here only with the free motion of an wuncontrolied, but intrinsically nonlinear system. Our purpose is to stimulate further research into this approach rather than to present a finished theory. Liapunov-Schmidt Bifurcation Theory For the reader's convenience we recall the salient features of purely formal point of view. The details of Banact space rigor cen 173-177 of Deimling [2] and other texts on nonlinear functional analysic (3), 14! Let Z denote a Fredholm operator, which may be singular, ic. KK eK, *2Z (2) Now define projection operators Ped - kz, Q=)- 2K; (3) it is readily verified that #” «= 7, Q = Q, ie. these operators are idempotent, (*) fx = FHixh We 0, QE = O, (4) and the problem (e*) x= Px + Kix), QRix) = O. (5) Now define u = Px and verify that Pu = u; then we can replace (ee) by (def inition) xeureyv, (6) (AUXILIARY EQUATION) v = KFiu * wv), (7) (BIFURCATION EQUATION) QFiu + v) = O. (8) Another name for the Bifurcation Equations is Branching Equations. Suppose that the right-hand side of (7), regarded as a function of v, has a global Lipschitz constant less than unity. Then by the well-known principle of geometric convergence of Contraction Mappings we may, for each fixed u, define a nonlinear mapping v = Siu) as vetinv, Me KHuev), v0, (k= 0,1230,5° 2. (9) Here & is the resolvent of the auxiliary equation in the sense that Ge XFiu «+ G). (10) Hence we may eliminate the auxiliary equation and replace v in the bifurcation equation by & to obtain a new finite-dimensional equation Siu) © OFiu + Giu)) = 0, (11) which is equivalent to the original infinite-dimensional implicit equation. Thus (eo) eo (ee) oe (eee) wx e# ue Siu), Zu = 0, PRlud = 0, Pueu, slu) = 0. (12) if the original functional equation was analytic, then the § final finite-dimensional equation flu) = 0 will also be analytic. if £ was non-singular, then X = £', whence P « 0, Q = 0, u = 0, the bifurcation equation does not arise, and the resolvability of the auxiliary equation is equivalent to the resolvability of the original equation: fx « Hx « «x «= £'Hx) = G60). (13) Finally, if 2 was singular, then the linear part F «= f to) . (of 70x) of fix) is necessarily also singular. Typically then the solutions of fix) = 0 will not be wnique and one studies the branching of these solutions by such methods as Newton's polygon.