No abstract

This paper addresses the issue of robustness of linear uncertain systems. In addition to the conventional notion of robustness based on quantitative information, in this paper, a new and novel perspective of qualitative robustness is introduced. The qualitative robustness measure is inspired by ecological principles and is based on the nature of interactions and interconnections of the system. Thus, using the proposed framework, the robustness of engineering systems can be assessed both from quantitative as well as qualitative information. This type of analysis from both viewpoints sheds considerable insight on the desirable nominal system in engineering applications. Using these concepts it is shown that a specific quantitative set of matrices labeled ‘Target Sign Stable Matrices’ are the best nominal matrices. These concepts are then extended to closed loop control systems and problem of control design and for ease in design a new set of matrices ‘Target Pseudosymmetric Matrices’ are introduced which enhance the class of desirable closed loop system matrices. Examples are included to illustrate these concepts.

No abstract

This paper revisits the issue of robust stability analysis of linear interval parameter matrices, which used to be a highly active research topic in the eighties and nineties. The reason for this revived interest in this topic is that the recent research by the authors on Qualitative Stability, a topic of interest in the field of population/community dynamics in ecology is shown to shed considerable insight with possible new results in the robust stability of matrix families. Thus in this paper, we expand on the two notions of robustness introduced recently by the authors, namely 'Qualitative Robustness' and 'Quantitative Robustness' and investigate their interdependence. Specifically, it is shown that for a class of matrix families with specified 'Qualitative Robustness' indices, it is sufficient to check the stability of only 'vertex' matrices (i.e. an extreme point solution) to guarantee the robust stability of the entire interval matrix family. This is indeed deemed important and significant because with this result, we can easily identify for which 'interval matrix families' we need to resort to more sophisticated stability check algorithms, and for which families we can get away with a ' vertex matrix check' (i.e. an 'extreme point solution'). It turns out that this class of 'qualitative stable' matrices that admit 'vertex solution' for its 'quantitative robustness' is quite large. Thus the results of this paper offer new insight into the nature of interactions and interconnections in a matrix family on its robust stability. Encouraged by the results of this paper, continued research is underway in using this interdependence of 'qualitative robustness' and 'quantitative robustness' in the design of robust controllers for engineering systems.

In this paper, we briefly review some fundamental qualitative features of ecological sign stability and transform these principles of ecology to a set of mathematical results in matrix theory with quantitative information, which is usually encountered in engineering sciences. This type of cross fertilization of ideas of life sciences and engineering sciences is deemed to be highly beneficial to both fields. In particular, we show in this paper what effect the signs of elements of a matrix have on the matrix properties such as eigenvalues and condition number. Similarly, it is also shown that under some assumptions on the magnitudes of the elements, predator-prey phenomenon in ecology renders some special properties like ‘normality’ to matrices. It is also shown that these predator-prey models have better robustness properties when compared to other matrices. The results presented in this paper can assist in the use of ecological system principles to build highly robust engineering systems.