Robust Stability Analysis of Structured Convex Combinations of Matrices with Aerospace Applications
Rama Krishna Yedavalli · AIAA Guidance, Navigation, and Control Conference and Exhibit · 2006
This paper presents a practical, computationally aware ‘vertex solution’ to the problem of checking the stability of families of matrices described by convex combinations of Hurwitz stable ‘vertex’ matrices. The convex combinations resulting from the interval parameter matrix family are labeled as ‘structured’ convex combinations to distinguish them from the so called ‘unstructured’ convex combinations which are convex combinations of ‘user specified’ Hurwitz stable vertex matrices. In this paper, we first present a motivation for solving this problem using an application in aerospace flight vehicle dynamics. Then, for the first time in the literature, in this paper, explicit expressions for the convex combination coecients in terms of the interval parameters are derived. These expressions help to clarify and explain the misconceptions that currently exist in the research community about the nature of the convex combination coecients induced by the interval parameters and shed significant insight into the ‘correct’ scenario for this case. A previously presented ‘vertex algorithm’ by the author for this tough problem was derived under the misunderstood mapping of the parameter space to the matrix element space that currently exists in the literature (in the absence of the explicit expressions derived in this paper). Based on the correct mapping presented in this paper, a computationally aware vertex solution is oered which accounts for the ‘discrepancy’ in the results for some ‘ill-conditioned’ problems because the eigenvalues which are supposed to be calculated in the noncompact matrix element space are calculated using the hitherto held assumption of a compact matrix element space. This ‘discrepancy’ arises for some ill-conditioned problems because in the noncompact matrix element space, the theoretically present coupling between real eigenvalues and complex (real part) eigenvalues, is destroyed or weakened. In this paper, this ‘practical, computationally aware’ algorithm is presented in its ‘final’ form. Several examples are given which clearly demonstrate eectiveness of the new algorithm, even for ill-conditioned problems. It is concluded that with this new insight provided by the correct mapping presented in this paper, in a future paper it is possible to present a more elegant vertex algorithm that takes into account the noncompactness of the new mapping.