Nonconservative evaluation of uniform stability margins of multivariable feedback systems

Abstract

This paper discusses concepts of stability margins of multivariable feedback systems. Independent and uniform stability margins are defined. A previous conjecture that the uniform margins may be computed by using the eigenvalue magnitudes instead of the singular values in the robust stability criteria is theorized. The nonconservatism provided by this theory in the evaluation of uniform margins is discussed, along with limitations of the uniform margins. Also presented is a method of using the uniform margins to extend the region of stability beyond what can be specified by singular values. Results are demonstrated numerically in an example of a lateral attitude control system for a drone aircraft. attempt to relax the conservatism in the evaluation of stability margins of a two-input two-output lateral attitude control system of a drone aircraft, Mukhopadhyay and Newsom5 experimented with using the magnitudes of the eigenvalues instead of the singular values in the robustness criterion. By examining the Nyquist plot of the eigenvalues of the return difference matrix of the control system in their study, they conjectured that the eigenvalue-based gain (or phase) margins are limits within which the gains (or phases) of all feedback loops vary uniformly without destabilizing the feedback system while the phase angles (or gains) remain at their nominal values. In fact, since the spectral radius (maximum of the modulii of the eigenvalues) of a matrix is the greatest lower bound of all norms of that matrix, the con- jecture of Ref. 5 is the least conservative for the evaluation of the uniform stability margins by means of norm-bounded robust stability criteria. The uniform variations of multiloop gains and phases are also interesting in that the uniformity constraints give the multiloop variations a single-variabl e nature. Hence, the regions of stability in the gain and phase spaces degenerate into line segments. At each stable nominal operating point in the gain and phase spaces one such line segment may be constructed. In this fashion, the uniform gain and phase margins facilitate a nonconservativ e but discrete represen- tation of the regions of stability in multidimensional gain and phase spaces. ! In this paper, the conjecture given in Ref. 5 is proved. Uniformity in the gain and phase variations in feedback loops may be viewed as a special structure in the perturbation of an open-loop transfer matrix. Therefore, weighted £; and 4, norms3 are used in the norm-bounded stability robustness criteria to derive the formulas from which the uniform stability margins are computed. The concept and the one- dimensional characteristics of the uniform stability margins and their use in discretizing the regions of stability in multidimensional gain and phase spaces are demonstrated.