Gain Weighted Eigenspace Assignment

This report presents the development of the Gain Weighted Eigenspace Assignment methodology. This provides a designer with a systematic methodology for trading off eigenvector placement versus gain magnitudes, while still maintaining desired closed-loop eigenvalue locations. This is accomplished by forming a cost function composed of a scalar measure of error between desired and achievable eigenvectors and a scalar measure of gain magnitude, determining analytical expressions for the gradients, and solving for the optimal solution by numerical iteration. For this development the scalar measure of gain magnitude is chosen to be a weighted sum of the squares of all the individual elements of the feedback gain matrix. An example is presented to demonstrate the method. In this example, solutions yielding achievable eigenvectors close to the desired eigenvectors are obtained with significant reductions in gain magnitude compared to a solution obtained using a previously developed eigenspace (eigenstructure) assignment method. 1.0 INTRODUCTION The Direct Eigenspace Assignment (DEA) method (Davidson 1986) is currently being used to design lateral-directional control laws for NASA's High Angle-of-Attack Research Vehicle (Davidson 1992). This method allows designers to shape the closed-loop response by choice of desired eigenvalues and eigenvectors. During this design effort DEA has been demonstrated to be a useful technique for aircraft control design. The control laws developed using DEA have demonstrated good performance, robustness, and flying qualities during piloted simulation. These control laws are scheduled for flight test at NASA Dryden Flight Research Center in 1994. During the control law design effort, one limitation of the DEA method became apparent. Using DEA the designer has no direct control over augmentation gain magnitudes. Often it is not clear how to adjust the desired eigenspace in order to reduce individual undesirable gain magnitudes. To reduce undesirable gain magnitudes the designer must rely upon a strong physical insight into the dynamics or is forced to iterate on the design. This report presents the development of an eigenspace (eigenstructure) assignment method that overcomes this limitation. This method, referred to as Gain Weighted Eigenspace Assignment (GWEA), allows a designer to place eigenvalues at desired locations and trade-off the achievement of desired eigenvectors versus feedback gain magnitudes. This report is organized into four sections. Background information on how eigenvalues and eigenvectors influence a system's dynamic response and a review of the Direct Eigenspace Assignment methodology is presented in the following section. The development of the Gain Weighted Eigenspace Assignment methodology is presented in section 3. Concluding remarks are given in the final section.