Eigenvalues are Sign Pattern Blind making them Unsuitable as Measures of Real State Variable Convergence for Linear Time Invariant State Space (LTISS) systems
Rama Krishna Yedavalli · IFAC-PapersOnLine · 2024
In this paper (with its IP protected contents) we revisit the current literature's Transformation Compliant (TC) methods such as the Routh-Hurwitz Criterion, Cayley Hamilton Theorem and the Lyapunov Matrix Equation method (which are all equivalent to each other) and demonstrate that eigenvalues, which are used as the sole measures of stability (real state variable convergence in Linear Time Invariant State Space (LTISS) systems) are highly unsuitable for assessing the All Real State Variable Convergence, i.e. Convex Stability, of LTISS systems since they are sign pattern blind. For this purpose, we borrow the Qualitative Sign Stability (QLSS) concepts being used in the fields of ecology and economics and by juxtaposing the TC methods with QLSS philosophy, prove that eigenvalue based Hurwitz stability criterion is erroneous and leads to misleading conclusions about the actual ARSVC (Convex stability) property of any LTISS system. An illustrative counterexample to the Cayley-Hamilton Theorem's statement is presented to prove this important observation. Further research results on the TA approach are explained and documented in the RES developed TA Approach Control Systems Design Toolbox in the website of RES.