A Design-Oriented Mode Acceleration Method for Lyapunov's Equation Based Random Gust Stresses in Aeroservoelastic Optimization

Abstract

The present paper describes a method for obtaining accurate design-oriented stress and stress sensitivity information from reduced order Linear Time Invariant (LTI) state space models of integrated aeroservoelastic systems, using Lyapunov’s Equation for calculating covariance matrices of the displacement and stress responses. A complete formulation of the reduced-order stress gust response problem for aeroservoelastic design synthesis, tailored toward integration with control system design techniques based on modern control, is presented. It includes an adaptation of the Mode Acceleration method, reduced-order analytic sensitivities of stress covariances, and efficient approximations to be used in a Nonlinear Programming / Approximation Concepts (NLP/AC) approach to design optimization. A realistic aeroservoelastic model of a typical passenger airplane is used as a test case, and the paper includes results of convergence studies for the assessment of order reduction effects on the accuracy of the integrated structure / aerodynamic / control models. NOTATION [ ] ) s ( A Laplace transformed full order influence coefficient matrix (associated with full order structural model) u u n n × [ ] φ i A Roger aerodynamic matrices (real) for fitting [ ][ ] φ ) s ( A (Eq. 8) q u n n × [ ] L A defined in Eqs. 53, 54. [ ][ ][ ][ ] D , C , B , A state space model matrices (with appropriate subscripts to designate sensors, actuators, control laws, gust filter, and overall system) g i s i c , c aerodynamic lag terms for structural and gust forces, respectively  Copyright © 2002 by Frode Engelsen and Eli Livne. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. [ ] C viscous damping matrix (structural) [ ] D aerodynamic matrix in a Minimum-State rational approximation form (Eq. 9) [ ] E aerodynamic matrix in a Minimum-State rational approximation form (Eq. 9) { } L F defined in Eqs. 53, 54 [ ] I identity matrix [ ] K stiffness matrix [ ] M mass matrix A n order of the state space model of the actuators c n number of active control surfaces CO n order of state space model of the MIMO control law block g n order of the state space model of the gust filter Ls n number of lag terms in a minimum state unsteady aerodynamic force rational approximation (structural and control surface motions) Lg n number of lag terms in a minimum state unsteady aerodynamic gust force rational approximation q n number of reduced order structural degrees of freedom u n full order number of structural degrees of freedom NLs – number of aerodynamic lag terms { } q vector of reduced order generalized coordinates [ ] Q generalized aerodynamic matrix s – Laplace variable ref S wing reference area [ ] S matrix used to calculate stresses from full order displacements { }T xy yy xx s s s vector of local stresses in a planestress thin structural element D q dynamic pressure [ ] Q intensity matrix of the white noise driving the gust filter. { } u full order displacement vector ∞ U flight speed 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con 22-25 April 2002, Denver, Colorado AIAA 2002-1477 Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. 2 American Institute of Aeronautics and Astronautics [ ] 2 V Eqs. 51,52 [ ] 2 V Eqs. 39, 40 w vertical speed of atmospheric gusts { } x state vector for complete aeroservoelastic system { } i x state vector for sub-system i. [ ] X covariance matrix for complete aeroservoelastic system with structural degrees of freedom reduced by mode displacement basis. { } meas y true structural responses at the points of measurement { } δ vector of commands to the actuators [ ] φ matrix containing (column by column) reduced order modal basis q u n n × [ ][ ][ ] 2 1 0 , , Φ Φ Φ matrices defining (Eq. 17) the true responses on the structure at the points of response measurement Subscripts and superscripts: A actuator g – gust G – gust filter c – control degrees of freedom cc – control-control partition of system matrices CO – control law cs – control-structural coupling in system matrices s – structural degrees of freedom sc – structural-control coupling in system matrices SE sensor sg – structural-gust partition of system matrices ss – structural-structural partition of system matrices T transpose