## Evaluation of Control Laws and Actuator Locations for Control Systems Applicable to Deformable Astronomical Telescope MirrorsNational Aeronautics and Space Administration, 1973 - Orbiting astronomical observatories - 60 pages Some of the major difficulties associated with large orbiting astronomical telescopes are the cost of manufacturing the primary mirror to precise tolerances and the maintaining of diffraction-limited tolerances while in orbit. One successfully demonstrated approach for minimizing these problem areas is the technique of actively deforming the primary mirror by applying discrete forces to the rear of the mirror. A modal control technique, as applied to active optics, has previously been developed and analyzed. The modal control technique represents the plant to be controlled in terms of its eigenvalues and eigenfunctions which are estimated via numerical approximation techniques. The report includes an extension of previous work using the modal control technique and also describes an optimal feedback controller. The equations for both control laws are developed in state-space differential form and include such considerations as stability, controllability, and observability. These equations are general and allow the incorporation of various mode-analyzer designs; two design approaches are presented. The report also includes a technique for placing actuator and sensor locations at points on the mirror based upon the flexibility matrix of the uncontrolled or unobserved modes of the structure. The locations selected by this technique are used in the computer runs which are described. The results are based upon three different initial error distributions, two mode-analyzer designs, and both the modal and optimal control laws. |

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### Contents

SUMMARY | 1 |

MODAL CONTROL TECHNIQUE | 7 |

DEVELOPMENT OF OPTIMAL CONTROLLER | 14 |

3 other sections not shown

### Common terms and phrases

20-actuator active optics actuator and sensor actuator forces analyzer astronomical telescopes coefficient matrices complex conjugate computer results computer runs conjugate transpose contour maps control channels control matrix control system described eigenvalues eigenvectors equation 22 error curve error distribution number error number evaluated flexibility matrix force-transformation compensation matrix function of number grid point 88 H nn identity matrix invert matrix Langley Research Center location of actuators matrix see eq matrix Snn mirror displacements mirror surface modal and optimal modal control law modal control loop modal control technique modal-displacement coefficients modal-displacement vector mode shapes mode-analyzer designs NASA NASTRAN number of actuators number of modes number of sensors numerical analysis observed optimal control law paper Percent of rms perfect figure-error sensor performance index potential energy primary mirror rms error left sensor points state-space differential steady-state error Substituting equation tions total number tt nn uncontrolled modes