Analytical expressions for Linear Quadratic aeroelastic flight control eigenvalues

Several obstacles limiting the use of contemporary control design techniques in production vehicles are noted. These obstacles restrict one from obtaining insight into the control law augmentation of the vehicle. For example, with the Linear Quadratic (LQ) state feedback method, detail effects from cost function weight adjustment is not easily understood. Further, the role of each feedback loop is not readily apparent. Fundamental to overcoming this problem is factoring of the LQ characteristic polynomial. An approximate analytical factoring technique, previously developed for open-loop applications, is considered as a tool for this closed-loop problem. The first two terms of a Taylor series are used to capture the polynomial coefficient dependencies upon the polynomial factors. By analytically inverting the first order sensitivity matrix, corrections to preliminary approximate factors are generated. Expressions for the closed-loop factors are in terms of basic parameters such as stability and control derivatives, structural vibration damping ratio and natural frequency, and cost function weights, allowing key relationships to be uncovered between the "design knobs" and important closed-loop features. Introduction Increased reliance upon flight control systems to meet mission design requirements, while retaining exceptional flying qualities, has been the trend for several decades. * In advanced vehicle concepts, where overall flight characteristics are influenced by various factors such as rigid-body motion, structural vibration, unsteady aerodynamic flow, and propulsion system behavior, multiple feedback loops are designed to actively control, in an integrated fashion, key dynamic features in the vehicle system. Often these control loops must operate near maximum performance levels with stability margins approaching minimum requirement limits for economic or functional viability of the concept. The theoretical control community has been concentrating on contemporary design techniques for such applications for over three decades. Powerful methods such as LQR/LQG/LTR, H2 and H^ have been created to close multiple feedback loops simultaneously in a well coordinated manner/* 3 t Assistant Professor. Senior Member AIAA. * Doctoral Student. Student Member AIAA. Recent activity has concentrated on tuning existing methods to provide closed-loop robustness properties. Two obstacles limiting the use of these methods in production vehicles are 1) the lumped nature of the algorithms masking individual loop significance and 2) theoretical-mathematical sophistication linking relevant closed-loop properties to the "design knobs". These characteristics limit insight into the control law's stabilization and/or augmentation of the vehicle, and if design results suggest the need for modifications, the flight control designer is often unclear as to how to proceed, short of quasi-iterative strategies. For example, with standard Linear Quadratic (LQ) state feedback control, once the input-output suite is specified, the designer must select cost function weights which provide acceptable trades between stability and performance, in both time and frequency domain measures. Effects from adjusting a particular weight, as it passes through the Riccati equation and into a specific feedback gain or closed-loop characteristic, is not easily understood. Further, the role each feedback loop plays in stabilizing/augmenting specific plant modes is not readily apparent, as compared with a conventional, sequential loop closure strategy. The overall objective of this work is to initiate a small step towards bridging this gap between theory and application for specific flight control problems. LQ-based root locus theory provides an efficient origin for pursing this activity. It is well understood how the closed-loop characteristic polynomial is explicitly defined in terms of the plant transfer function numerator and denominator polynomials and cost weightings. Therefore, analytical factoring of the closed-loop characteristic polynomial is fundamental to fostering useful relationships between the design knobs and resulting closed-loop features.^ For single-loop systems, this relationship is portrayed graphically as the cost weight ratio is varied/ For multi-loop systems, this graphical information is quite sparse, existing only in an asymptotic sense.""" Exact symbolic factoring of polynomials is only possible for polynomials of degree four or less. Noting the symmetric nature of the LQ characteristic polynomial, this restricts exact analysis to dynamic systems of order four or less. However, the cubic and quartic formulas are not easily used for problem understanding. Therefore, attention is turned to an approximate analytical factoring technique which has been used successfully to develop relationships for Copyright © 1997 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. open-loop elastic airframe transfer function poles and zeros in terms of basic parameters such as stability and control derivatives and structural vibration parameters like damping ratio and natural frequency. 10-15 in this technique, the first two terms of a Taylor series are used to capture the polynomial coefficient dependencies upon the polynomial factors. By analytically inverting the first order sensitivity matrix, corrections to preliminary approximate factors can be generated. In principle, the technique is applicable to higher order systems. Relatively simple and accurate analytical expressions conducive for obtaining insight into the vehicle physics have been obtained by this technique. The strategy here is to explore the utilization of this factoring tool in closed-loop settings, and to generate analytical expressions, albeit approximate, between the cost function weights and closed-loop eigenvalues, for a LQ-based aeroelastic flight control law. Through these expressions, the goal of the research is to foster improved understanding of the cost weighting selection process, and to provide practical, relevant design information to the flight control engineer.*""!' If successful, the results may contribute to increased application and implementation of contemporary design techniques to advanced vehicle concepts requiring multiple, coordinated feedback loops. Flight Dynamics Modeling Within the commercial flight industry, there exists considerable interest for the development of a long range, high speed, high capacity vehicle which can serve global transportation markets while remaining a viable concept both economically and environmentally."^ The top end speeds of this vehicle, coupled with the use of composite materials for the primary structure, will lead to significant interaction between rigid-body and structural dynamics. Lowest frequency structural modes are expected to be well within one frequency decade (rad/s) of the short period dynamics. This type of vehicle is a prime candidate for, and may necessarily require, a multivariable flight control system." Therefore, an aeroelastic flight control application is chosen for the analytical expressions work presented here. To capture the multidisciplinary aspects of rigidbody and vibrational motion, modeling efforts must return to the fundamental governing principles, such as in Ref. 21. The nonlinear dynamic model generated by this process can be linearized and reduced in order, resulting in a linear model appropriate for flight control design activities. Ref. 12 contains a model of this flavor for a large, high speed, elastic vehicle. Configuration geometry consists of a low-aspect ratio swept wing, conventional aft tail, and small canard. A fifth order polynomial matrix realization is given in Eq. (1). This model involves the small perturbation longitudinal dynamics of the closely spaced, effective short period and first Table 1. Vehicle Numerical Model Definition