The applications and a general solution of a fundamental matrix equation pair

Equation TA − FT = LC (F is stable) is necessary and sufficient for the output of a feedback compensator (F, L, KZ, Ky) to converge to a state feedback (SF) signal Kx(t) for a constant K, where (A, B, C, 0) is the open loop system and TB is the compensator gain to the open loop system input. Thus equation TB = 0 is the defining condition for this feedback compensator to be an output feedback compensator. Equation TB = 0 is also the necessary and sufficient condition to fully realize the critical loop transfer function and robust properties of SF control if K is systematically designed. Furthermore, because B is compatible to the open loop system gain to its unknown inputs and its input failure signals, TB = 0 is also necessary for unknown input observers and failure detection and isolation systems. Finally, this equation pair is the key condition of a really systematic and explicit design algorithm for eigestructure assignment by static output feedback control. This paper presents a general and exact solution which is uniquely direct, simple, and decoupled, to this matrix equation pair. An approximate solution which is general and simple, and which can be simply added to the exact solution to increase the row dimension of this solution, is also presented.