A Necessary and Sufficient Condition for the Existence of Static Output Feedback Stabilization Gain via Non-Lyapunov, Transformation Allergic (TA) Approach for the Null Plant Matrix (NPM) Case

In this paper, with its IP protected contents, we present a complete solution to the problem of Static Output Feedback (SOF) Stabilization, which is touted as one of the unsolved problems in the stability theory of Linear Time Invariant State Space (LTISS) systems. We use a Non-Lyapunov philosophy, specifically labeled as the Transformation Allergic (TA) Approach (or philosophy). In this proposed TA Approach, the definitions of
$A^{0}$
and the State Transition Matrix (STM) are quite different from the traditional Lyapunov/Routh-Hurwitz/Cayley Hamilton Theorem based Transformation Compliant (TC) methods. The complete solution to the SOF Stabilization problem is made possible using the concept of Convex Stability recently proposed in the sign pattern based TA Approach of this author, which in turn works with the new definitions for the
$A^{0}$
matrix and the State Transition Matrix (STM) of any LTISS system, unencumbered by the popular eigenvalue and frequency domain based control theory of the current literature. In this paper, we give necessary and sufficient conditions for the Null Plant Matrix (NPM) LTISS system only, relegating the case of Non-NPM case (needing a different set of necessary and sufficient conditions), to a separate set of papers.