Alain C. Carrier

Second-order algorithm for optimal model order reduction

A second-order algorithm is given for optimal model order reduction of continuous single-input/single-output systems. Given a transfer function of order n, it finds the transfer function of a model of order m<n that minimizes the integral-square difference between the impulse responses of the two systems. It is shown that this is exactly equivalent to two other problems: 1) minimizing the integral-square difference between the frequency responses of the two systems and 2) minimizing the mean-square difference between the statistical steady-state responses of the two systems to a white-noise input. The algorithm is based on the known explicit solution of the governing Lyapunov equation, which yields analytical expressions for the first and second derivatives of the performance index with respect to the residues and poles of the reduced-order model. These derivatives are then used in a Newton-Raphson algorithm. The algorithm is applied to find exact solutions for four examples, one of which has an unstable mode. The examples corroborate previous research indicating that Moores truncated balanced realization is not always close to the optimal reduced order model. Matching step response instead of impulse response requires only a slight change in the algorithm.

MODAL CHARACTERIZATION OF THE ASCIE SEGMENTED OPTICS TEST BED: NEW ALGORITHMS AND EXPERIMENTAL RESULTS

New frequency response measurement procedures, on-line modal tuning techniques, and off-line modal identification algorithms are developed and applied to the modal identification of the Advanced Structures/Controls Integrated Experiment (ASCIE), a generic segmented optics telescope test-bed representative of future complex space structures. The frequency response measurement procedure uses all the actuators simultaneously to excite the structure and all the sensors to measure the structural response so that all the transfer functions are measured simultaneously. Structural responses to sinusoidal excitations are measured and analyzed to calculate spectral responses. The spectral responses in turn are analyzed as the spectral data become available and, which is new, the results are used to maintain high quality measurements. Data acquisition, processing, and checking procedures are fully automated. As the acquisition of the frequency response progresses, an on-line algorithm keeps track of the actuator force distribution that maximizes the structural response to automatically tune to a structural mode when approaching a resonant frequency. This tuning is insensitive to delays, ill-conditioning, and nonproportional damping. Experimental results show that it is useful for modal surveys even in high modal density regions. For thorough modeling, a constructive procedure is proposed to identify the dynamics of a complex system from its frequency response with the minimization of a least-squares cost function as a desirable objective. This procedure relies on off-line modal separation algorithms to extract modal information and on least-squares parameter subset optimization to combine the modal results and globally fit the modal parameters to the measured data. The modal separation algorithms resolved modal density of 5 modes/Hz in the ASCIE experiment. They promise to be useful in many challenging applications.

Modal characterization of the ASCIE segmented optics test bed: new algorithms and experimental results

New frequency response measurement procedures, on-line modal tuning techniques, and off- line modal identification algorithms are developed and applied to the modal identification of the Advanced Structures/Controls Integrated Experiment (ASCIE), a generic segmented optics telescope test-bed representative of future complex space structures. The frequency response measurement procedure simultaneously uses all the actuators to excite the structure and all the sensors to measure the structural response so that all the transfer functions are measured simultaneously. Structural responses to sinusoidal excitations are measured and analyzed to calculate spectral responses. The spectral responses in turn are analyzed as the spectral data become available and, which is new, the results are used to maintain high quality measurements. Data acquisition, processing, and checking procedures are fully automated. As the acquisition of the frequency response progresses, an on-line algorithm keeps track of the actuator force distribution that maximizes the structural response to automatically tune to a structural mode when approaching a resonant frequency. This tuning is insensitive to delays, ill-conditioning, and nonproportional damping. Experimental results show that it is useful for modal surveys even in high modal density regions. For thorough modeling, a constructive procedure is proposed to identify the dynamics of a complex system from its frequency response with the minimization of a least-squares cost function as a desirable objective. This procedure relies on off-line modal separation algorithms to extract modal information and on least-squares parameter subset optimization to combine the modal results and globally fit the modal parameters to the measured data. The modal separation algorithms resolved modal density of 5 modes/Hz in the ASCIE experiment. They promise to be useful in many challenging applications.

Development and demonstration of a precision latch for deployable optical systems

The Lockheed Martin/Advanced Technology Center (LM/ATC) developed a lightweight, compact, high-load capable and yet high precision latch for use on deployable optical systems such as the Next Generation Space Telescope (NGST). The design allows precise self-centering and control of the stiffness at the latch interface. It also incorporates unique capabilities to evaluate the effects of gravity loads, latch preload level, creep, and very low vibration loads on the dynamics and microdynamics of the deployed instrument. The stiffness, nonlinearity and hysteresis characteristics of the latch and its catch flexure assembly were thoroughly tested in 6 axes down to the nanometer level at room temperature using the LM/ATC Compliance Measurement Device. The latch is stiff enough to hold an NGST-size mirror segment cantilevered against gravity allowing only small gravity sag when the primary mirror is horizontal, thus enabling end-to-end performance verification in 1-G in that orientation. The latch hysteresis is less than 1.0 nm/N under mechanical loads less than 25 N, which meets the NGST stability requirements with significant margin (20 nm at the tip of the petal in space environment). Several of these latches were integrated and demonstrated at the petal assembly level on a Single Petal Test-bed and the experimental results obtained on that test-bed are consistent with the component level results described in this report. We experimentally demonstrated that the latch engagement performance is not affected by exposure to cryogenic temperatures down to 20K, as required for use of the device on cryogenic infrared optical instruments such as NGST. A structural model of the latch was developed using Finite Element Analysis. Good correlation was obtained between the linear components of the analytical and of the experimental results: the model can therefore reliably be used in future NGST or other mission design efforts. This paper includes a brief description of the LM/ATC latch hardware and its principle of operation as well as the results of the modeling and the experimental characterization work performed on that hardware in the NGST Phase I formulation.

Development and Microdynamics Characterization of a Deployable Petal Assembly at Full Scale

As part of its risk mitigation efforts related to large, future space-based deployable optics such as NGST, Lockheed Martin developed, implemented, and evolved a full-scale, lightweight, deployable petal structure and associated deployment mechanisms for cryogenic and microdynamic stability testing. The test-bed features a single petal assembly for an 8-meter diameter telescope, including a flight-like mirror support structure and full-size hinges and latches. The work completed on this test-bed include: 1) Characterization of the dynamics and microdynamics response of the full-scale petal and its hinge/latch interface to low-level vibration sources down to 0.1 nanometer, 2) Evaluation of petal deployment repeatability, 3) Evaluation of the performance of simple passive damping strategies for petal vibration control at cryogenic temperatures. In all respects, including microdynamics, deployment repeatability and stability, the hardware demonstrated performance well in excess of the NGST requirements. In this paper, we summarize the development and the results of the performance testing completed during the NGST Phase I formulation, including testing of hysteresis and deployment repeatability at room temperature.

Experiments in modeling and control of the ASCIE segmented reflector

This paper reports on experiments in modeling and control of the Advanced Structures/Controls Integrated Experiment.

Damping Characteristics of Composite Petal Structure for an 8-m Diameter Telescope at Cryogenic Temperature

Concerns have been raised in the engineering community about the potentially extremely low levels of structural damping in structures at cryogenic temperatures. Experiments conducted on material coupons have shown that material damping at those temperatures can be orders of magnitude lower than that at room temperature. Whether structural damping in built-up structures at those temperatures can be that low is unknown, but if it was, the telescope resonances could exacerbate microdynamics originating from the structure itself and residual vibrations propagating from the instrument module to the telescope. Since the effect of those vibrations might not be compensated for optically, the observatory might not meet its wavefront and jitter error budgets. The structural damping characteristics of built-up structures in the micrometer to nanometer regime and at cryogenic temperatures are to a large extent unknown. Characterization on structures traceable to future flight designs is therefore necessary to develop an understanding of these characteristics, as well as devise means to mitigate those effects. To address those concerns and to reduce the technical risks in these areas, Lockheed Martin tested the dynamics characteristics of its Single Petal Testbed (SPT) flight-like petal structure at full-scale, from room temperature down to -175C (98K). The SPT was designed by the Lockheed Martin Advanced Technology Center and fabricated by Programmed Composites Inc. Significant changes in dynamics characteristics with temperature were observed, but primarily in mode shapes as opposed to modal frequencies and modal dampings. The modal damping remained fairly constant throughout the temperature range and, to the extent changes could be detected, the trends were more towards an increase than a decrease in damping at 98K, which was highly unexpected. A detailed analysis of these results extracted from dynamics tests conducted during the cool down portion of the last thermal cycle is presented in this report. The levels of damping observed in the built-up petal structure are 10 to 20 times higher than those measured by Marie Levine at JPL on all-composite coupons of the petal panels provided by Lockheed Martin.