An overview of the applications and solutions 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 (1) the defining condition for this feedback compensator to be an output feedback compensator. Equation TB=0 is also the necessary and sufficient condition to (2) 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 (3) unknown input observers and (4) failure detection and isolation systems. Finally, this equation pair (TA−FT=LC, TB=0) is the key condition of a really systematic and explicit design algorithm for (5) eigenstructure assignment by static output feedback control. This paper reviews the existing solutions of this equation pair, and points out that a general and exact solution is uniquely direct, simple, and decoupled. This paper also points out that these unique features also enable two decisive advantages: (1) the systematic compensator order adjustment and (2) a simple and approximate solution which is general to all systems (A,B,C,0) and which can be simply added to the exact solution whether it exists or not.