Effects of displacement and rate saturation on the control of statically unstable aircraft

Abstract

Methodologies are presented for the analysis and design of stability augmentation control laws for aircraft in which hard displacement and rate limiting are significant. Candidate control laws are derived using the linearquadratic (LQ) regulator. Analytical and computational estimates of the stability limits imposed by control saturation are presented using state trajectories, as well as describing functions and eigenvalue computation. Analysis also includes an investigation of the interaction of the state-space saturation and stability boundaries for various choices of LQ weighting matrices. For minimum energy control, the saturation and stability boundaries are shown to be parallel. In this case, there is a direct relation between the solution to the matrix Riccati equation and the aircraft's open-loop dynamics. V IRTUALLY all aircraft control system actuation mechanisms possess hard limits on deflection and rate of deflection, and accounting for these limits is particularly important for aircraft designed with relaxed or negative static stability. Relaxed static stability generally leads to reduced weight and trim drag, both of which enhance the aircraft's performance.1 Relaxed static stability also may lead to increased control activity.2 For such aircraft, consideration must be given to the effects of control saturation on the initiation and, more importantly, the arrestment of dynamic motions, if departure from controlled flight is to be prevented. It may be possible to provide hardware actuation limits large enough so that normal disturbances and command inputs do not force the controls and to their mechanical, hydraulic, or electrical stops; however, there is no guarantee that commanded control deflections or rates will not exceed any established limit. Therefore, it is important to define the region within which stability and satisfactory response can be assured. This paper explores the effects of deflection and rate saturation on the stability of a statically unstable aircraft. This paper also reports on the development of methodologies for the analysis and design of stability augmentation control laws, with emphasis on response to initial conditions. (Command response is presented in Ref. 3 is the subject of a separate paper.11) Candidate stability augmentation system (SAS) control laws are derived using optimal control theory, with the linear-quadratic (LQ) regulator taken as the fundamental solution. Analytical and computational estimates of the stability limits imposed by control saturation are presented.