Structural Mode Significance using INCA Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3346 Frank H. Bauer*, NASA/Goddard Space Flight Center John P. Downing’, NASA Goddard Space Flight Center Christopher J. Thorpe( Fairchild Space Company TIROS-N (ATN) [l] weather satellite series, and the Cosmic Background Explorer (COBE) spacecraft and instruments. Structural finite element models are often too large to be used in the design and analysis of control systems. Model reduction techniques must be applied to reduce the structural model to manageable size. In the past, engineers either performed the model order reduction by hand or used distinct computer programs to retrieve the data, to perform the significance analysis and to reduce the order of the model. To expedite this process, the latest version of INCA has been expanded to include an interactive graphical structural mode significance and model order reduction capability. The Interactive Controls Analysis (INCA) program was developed at NASA’s Goddard Space Flight Center (GSFC) as a computer aided control system design tool primarily for use by engineers in the Guidance and Control Branch. Initial development of the INCA program began in 1982 with the first release of the program to members of the Guidance and Control Branch in 1983. Since then, INCA has been used extensively to design or analyze control systems for all of Goddard’s spacecraft programs. Numerous flight proven designs have been developed or validated using INCA’S analytic capabilities. These include the Earth Radiation Budget Satellite (ERBS) spacecraft, the Advanced *Head, Project Support Section, Guidance & Control Branch, Member AIAA ‘Physicist, Member AlAA #Svstern Prowammer. Member AIAA Copyright 1990 by the American Institute of Aeronautics and Astronautics, lnc. N o copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner. 285 In 1985 INCA was first delivered to COSMIC [2], NASA’s computer program dissemination source, for general distribution. Its excellent user interface, versatile graphics system, and robust controls algorithms have made INCA a popular tool for use in universities, private industry and the government. Since it is a public domain program with source code provided, user enhancements are possible. A number of the user enhancements have been incorporated in the COSMIC version of INCA; the most notable enhancement in this category is the describing function analysis capability which was developed by Dr. Bong Wie and Mr. Tobin Anthony of the University of Texas at Austin [3]. INc4 overview INCA was developed for use on VAX/VMS computers. The program was written primarily in Pascal; however the matrix and multivariable control algorithms, obtained from the SAMSAN [4] subroutine package, are written in FORTRAN. Since INCA was written to perform control system design analysis on very expensive spacecraft and instruments, emphasis was placed on doing the analysis right the first time. Thus, the program. utilizes algorithms which strive for numerical accuracy and attempt to prevent ill conditioned situations. These algorithms may be less efficient than others available, but when expensive spacecraft are at stake accurate results are of supreme importance. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3346 Figure 1 illustrates the program interface. INCA incorporates a menu driven command structure. These commands can be used to access the graphic and editor modes, query the help library, retrieve data from input files, and create and manipulate transfer functions and matrices. Plots are displayed on local graphics terminals and can be printed using standard hard copy capabilities. Using INCA the controls engineer can quickly generate system models. INCA provides the capability to analyze continuous, sampled-data and hybrid (continuous/sampled-data) systems. Systems with computational delays or transport lags can be analyzed via the modified z-transform. The resultant linear models can then be manipulated to modify the design or can be used to determine system stability, parameter robustness or control system performance. Standard filter and control law templates have also been included to expedite the system design. INCA provides a comprehensive host of standard classical controls analysis techniques including Root Locus, Frequency Response (Bode, Nichols, Nyquist) and Linear Time Response. Frequency response analyses of nonlinear systems can be performed using the describing function method [3]. Analysis results can be presented in graphical or tabular form. Figure 2 is a sample design analysis plot generated by INCA. Matrix data in various ASCII and binary file formats can be easily merged in the INCA data base. Supported matrix arithmetic operations include addition, subtraction, multiplication, inversion, transpose and exponentiation. Standard linear algebra routines are also available to the user including eigenvalue/eigenvector computations, solving the determinant of a matrix, trace computations and singular value decomposition calculations. Controls specific MIMO capabilities include transfer function t o s t a t e variable transformations, state variable to transfer function matrix computations, and a structural finite element model reduction capability. The mode significance module was designed to expedite the model reduction process from loading the finite element data through to generating reduced order flexible body plant transfer matrices. Structural dynamic models, such as that provided by the NASA Structural Analysis program, NASTRAN, [5] can be described in the following form: m + D X + K x = F (1) The present multivariable capabilities incorporated in INCA perform rudimentary Multi-Input-Multi-Output (MIMO) analyses. Additional MIMO enhancements will follow later. INCA will accept real or complex matrices. INCA’s dynamic transfer function capability has been extended to include matrices. The dynamic matrices work like a spreadsheet. As an independent variable changes, the matrix is recomputed to incorporate this change. INCA’s MIMO extension accepts arrays of 1st order (vectors), 2nd order (standard mxn matrices), 3rd order (mxnxo arrays) and beyond. where M, D, and K represent the system mass, damping and stiffness matrices respectively, x represents the flexible system’s translation and rotation degrees of freedom, and F represents the external forces and torques which act on the body. Using the modal coordinate transformation: where 4 represents the mass normalized eigenvector matrix, the system finite element model of equation (1) can be expressed as: q t Cq + Xq = 4’F (3) where x is the diagonal matrix of system 286 eigenvalues (the square of the modal frequencies), C is the diagonal matrix representing the modal damping and +T is the transpose of the mass normalized eigenvector matrix defined in (2). The mode significance analysis in INCA relies on the system modeling and modal coordinate definitions described in equations 1-3. Data input to INCA includes , C, and the sensor and actuator locations defined in x. Row vectors +s and 4A,representing the respective sensor and actuator locations to be analyzed, are obtained from the eigenvector matrix of equation (2). Since each column of represents a specific structural mode, the numeric data in the i'th column of the row vectors cPs and +A represent the participation or influence of mode i to the sensor or actuator in question. The significance of each structural mode is then computed in INCA for each requested sensor/actuator pair using the above row vectors and the model reduction criteria selected. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3346 +, + where s=jiz and 8 is the transfer function obtained from the rigid body modes such that: R 8 = C. +si +Ai (7) i=l Where R represents the number of rigid body modes kept in the analysis. The modal significance criteria for the Gregory's model reduction criteria is computed for each sensor/actuator pair using the following formula: + Several model reduction techniques are available. The simplest analytic technique is the modal gain criteria [6]. For the modal gain criteria, the significance of each mode, oi is computed for a specific sensor/actuator pair as follows: Frequency weighted criteria include the Peak Amplitude criteria [7l and Gregory's model reduction criteria [7l [8]. For the peak amplitude criteria, the significance of each mode, oi is computed at the frequency for which the amplitude of each mode is a maximum. This frequency is: where wi is the flexible mode's natural frequency, and r is the structural damping. Thus, the significance for each modal frequency becomes: 287 User-defined frequency weighting criteria such as sensor dynamics frequency shaping or control bandwidth shaping can be used to augment the criteria described above. Mode significance analysis results can be displayed through the use of two and three dimensional color plots (figure 3) or by viewing the results in tabular form. These results can then be used to automatically or manually select the modes to keep in the model. Commands have been included to automatically generate the plant transfer matrices from the reduced order model. Mode Sigm- EjcMtple The GOES (Geosynchronous Operational Environmental Satellite) I/J/K/L/M satellite series, scheduled for first launch in June 1991, contains two instruments which observe at infrared and visible wavelengths: the Imager and the Sounder. See figures 4 and 5. The instrument design represents a challenging structural-control interaction problem due to the tight jitter and pointing accuracy requirements imposed to meet the needs of the National Weather Service. The instruments are functionally similar in design and both scan the Earth by positioning their mirrors using a twoaxis gimbal pointing system. See figure 6 . EastWest scans across the Earth are performed using Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3346 the elevation gimbal. North-South motion is performed by rotating the azimuth gimbal. The pointing performance specifications result in instrument bandwidth requirements on the order of 50 Hz. A detailed NASTRAN model with over 8,000 degrees of freedom was developed to emulate the instrument scanner dynamics. See figure 7. A cursory model reduction was performed by first reducing the degrees of freedom to include only actuator and sensor grid points required to perform the controls analysis. Structural modes above 3 KHz were also deleted from the model. The resultant model containing 125 modal frequencies and 8 degrees of freedom was specified in an eigenvector and eigenvalue format. All structural modes were initially specified with a 0.1% damping ratio (r = .001). This model was retrieved by INCA and used to perform the significance analysis. Once the instrument engineering model frequency response tests were completed, the damping ratio and frequency of each mode was fine tuned to provide better correlation with the test data. These adjustments are reflected in the presented mode significance analysis. Model reduction for the instrument was accomplished interactively using Gregory’s model reduction criteria. Two structural models were generated for the GOES instruments. A high fidelity, high order model was developed to design the controller and determine system stability margins. A lower order structural model was incorporated in the instrument nonlinear simulation to determine system performance. A cutoff significance of 1% was chosen for the high fidelity model and a significance of 10% was used for the lower order model. These criteria reduced the high and low order models to 17 modes and 8 modes respectively. Figure 8 lists the 17 modes selected for the high fidelity model. Figure 9 is a graphical plot of the significance results for the first 30 modes. This plot is used in INCA to interactively select structural modes or to change the significance criteria. For GOES, the mode significance analyses for the azimuth and elevation gimbal axes were 288 merged together to form a single “pool”of modes from which the plant model was generated. The plant transfer matrix was computed and stored in the INCA database for future design analysis support. Bode plots of the elevation (figure 10) and azimuth gimbal axes were generated. This plant transfer matrix was used in INCA to perform the scanner control system design analysis, structural mode compensation and linear performance studies. Futwe P h The structural mode significance and model reduction capability described in this paper is incorporated in the latest version of INCA. Enhancements planned beyond this version include extending the multivariable and mode significance capabilities. A PC and Macintosh derivative of INCA is also under development at NASA Goddard called ASTEC (Analysis and Simulation Tools foe Engineering Controls). ASTEC, is planned for release to COSMIC in 1991. The Interactive Controls Analysis (INCA) program couples a user friendly interface with excellent, well conditioned computational algorithms to provide control system engineers with tools which are simple to use, quick, and provide accurate results. The program’s algorithms are quite robust and have been spaceflight proven. The latest version of INCA includes an interactive graphical structural mode significance and model order reduction capability. As shown in the GOES instrument servo control system example, the INCA model reduction capability allows the controls designer to reduce a complex high order structural model to a manageable size. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3346 1. Bauer, F. H. and Downing, J. P., "Control System Design and Analysis using the INteractive Controls Analysis (INCA) Program", AIAA Guidance, Navigation and Control Conference Paper 87-2517, 1987. 2. Bauer, F. H. and Downing, J. P., "INteractive Controls Analysis (INCA) Version 2.0", Program Number GSC-12998, COSMIC, University of Georgia, Athens GA., 1985, updated to Version 3.13, 1989. 3, Anthony, T., Wie, B., and Carroll, S., "PulseModulated Controller Synthesis for a Flexible Spacecraft", AIAA Paper No. 893433, 1989. 4. Frisch, H. P., and Bauer, F. H., "Modem Numerical Methods for Classical Sampled Systems Analysis (SAMSAN Version 2) User's Guide", Program Number GSC- 12827, COSMIC, University of Georgia, Athens, GA., 1984. 5. "The NASTRAN Theoretical Manual", NASA SP-221-(06), COSMIC, University of Georgia, Athens, GA., January 1981. 6. McGlew, D. E., "MODESIG, a Computer Program to Determine the Significant Flexible Modes", GSFC Guidance and Control Branch Report No. 324, November 1982. 7. Class, B. F., Bauer, F. H., Strohbehn, IC,and Welch, R. V., "Space Infrared Telescope Facility/Multimission Modular Spacecraft Attitude Control System Conceptual Design", AAS Paper No. 86-031, AAS Guidance and Control Conference, 1986. 8. C. 2.Gregory, "Reduction of Large Flexible Spacecraft Models Using Internal Balancing Theory," J. Guidance and Control, Vol. 7, NO. 6, NOV-DW1984, pp. 725-732. 289 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3346 INCA Data & Command Flow INCA -1 DATA OUTPUT DATA SAVE (inpuffoutput) FILES Figure 1 m = 58d8 n 1.416 92 Hz Y= 2.869588 Hz 0dB HACN I Tk U= 164.3621 Hz -58dB I Y= 2392.361 Hz - 188d I Frequency Response Example Figure 2 290 MODE SICNIflCANCE S I RTF/MMS A Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3346 6 Sensor/Torquer Pair Structural Rode Number 3-D Mode Significance Plot Figure 3 GOES I/J/K/L/M Satellite Figure 4 291 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3346 SOUNDER SENSOR ASSEMBLY IMAGER SENSOR A GOES Imager and Sounder Instruments Figure 5 Instrument Scan Assembly Figure 6 292 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3346 Scan Assembly NASTRAN Model Figure 7 Mode ---3 4 5 6 7 18 20 63 65 66 67 68 98 113 115 118 119 59.00 56.65 96.00 111.72 174.90 621.60 536.00 1601.94 1500.00 1681.18 1681.18 1650.00 1679.97 2665.88 2710.12 2761.65 2765.09 0.0010 0.0010 0.0010 0.0010 0.0200 0.0010 0.0030 0.0010 0.0070 0.0010 0.0010 0.0010 0.0070 0.0010 0.0010 0.0010 0.0010 100.00 3.69 12.16 7.97 1.29 3.22 1.03 10.17 14.21 100.00 38.69 2.11 12.19 1.44 2.02 61.22 8.33 SELECT SELECT SELECT SELECT SELECT SELECT SELECT SELECT SELECT SELECT SELECT SELECT SELECT SELECT SELECT SELECT SELECT ---- ---------- ------ ------ ........................ Mode Significance Results Figure 8 293 A z i m u t h E l e v a c l o n 0 . 1 0 - 0 . 5 0 0 . 5 0 - 1.00 1 . 0 0 - 2.00 I C .. 1 2 2 . 0 0 - 5 . 0 0 5 . 0 0 - 1 0 . 0 0 1 0 . 0 0 - 2 0 . 0 0 2 0 . 0 0 - 5 0 . 0 0 1 3 1 4 1 5 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3346 1 6 I " 1 9 2 0 2 1 2 2 U H D . d l c a i n 2 3 G r . 9 2 4 o r y s 2 5 A m p l l t d d . 2 6 I f 2 d 2 9 3 0 3 1 3 2 2 1 1 . 1 e 5 0 1 GOES Instrument Mode Selection Plot Figure 9 1.23 -189 19-JUN-1998 -- dB -158 18 188 frequency ( Hz 1 Bode Plot of Instrument Elevation Axis Figure 10 294 1090

1/--страниц