Determination of Most Desirable Nominal Closed Loop State Space System via Qualitative Ecological Principles

This paper addresses the issue of determining the most desirable ‘Nominal Closed Loop Matrix’ structure in linear state space systems, by combining the concepts of ‘Quantitative Robustness’ and ‘Qualitative Robustness’. The qualitative robustness measure is based on the nature of interactions and interconnections of the system. The quantitative robustness is based on the nature of eigenvalue/eigenvector structure of the system. 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 ‘Quantitative Ecological Stable (QES) Matrices’ have features which qualify them as the most desirable nominal closed loop system matrices. Thus in this paper, we expand on the special features of the determinant of a matrix in terms of self-regulation, interactions and interconnections and specialize these features to the class of ‘Quantitative Ecological Stable (QES)’ matrices and show that for checking its Hurwitz stability, it is sufficient to check the positivity of only the constant coefficient of the characteristic polynomial of a matrix in a higher dimensional ‘Kronecker’ space. In addition, it is shown that these matrices possess the most attractive property among any matrix class, namely that their Determinants possess convexity property. Establishment of this optimal nominal closed loop system matrix structure paves the way for designing controllers which qualify as robust controllers for linear systems with real parameter uncertainty. The proposed concepts are illustrated with many useful examples.