Admissible, Stabilizability, and Bicoprime Factorization

In [1], a system $G=egin{bmatrix} G_{11} & G_{12}\ G_{21} & G_{22}end{bmatrix}$ is admissible if the characteristic determinant (i.e., determinant of the denominator) of a coprime factorization of $G$ is equivalent to the characteristic determinant of a coprime factorization of $G_{22}$. It says that admissibility plays the same roles as joint stabilizability/detectability plays in the state-space theory. 

Let $G = NM^{-1}= ilde{M}^{-1} ilde{N}$ is the right and left coprime factorization of $G$, respectively. Then [2] shows that $G$ is stabilizable is equivalent to that

$left(M,~egin{bmatrix} 0 & Iend{bmatrix}N ight)$ is right-coprime and $left(M,~egin{bmatrix}egin{smallmatrix} 0 \ I end{smallmatrix}end{bmatrix} ight)$ is left-coprime,

or

$left( ilde{M},~ ilde{N}egin{bmatrix}egin{smallmatrix} 0 \ Iend{smallmatrix}end{bmatrix} ight)$ is left-coprime and $left( ilde{M},~egin{bmatrix} 0 & I end{bmatrix} ight)$ is right-coprime.

Note that if the coprimeness holds, both $left(egin{bmatrix} 0 & Iend{bmatrix}NM^{-1}egin{bmatrix}egin{smallmatrix} 0 \ Iend{smallmatrix}end{bmatrix} ight)$ and $left(egin{bmatrix} 0 & I end{bmatrix} ilde{M}^{-1} ilde{N}egin{bmatrix}egin{smallmatrix} 0 \ Iend{smallmatrix}end{bmatrix} ight)$ are actually bicoprime factorizations (right-left coprime factorizations) of $G_{22}$. From [3], the characteristic determinant of $G_{22}$ is $det M$, this means that the admissibility of [1] is equivalent to the stabilizability of [2], as the characteristic determinant of $G$ is also $det M$.

Example. Suppose $G = egin{bmatrix} G_{11} & G_{12}\ G_{21} & G_{22}end{bmatrix}$ and $G_{11} = G_{12}= G_{21} = G_{22}$, then $G$ is stabilizable.

Let $G_{22} = NM^{-1}$ be a right coprime factorization of $G_{22}$. Then it is easy to show that

egin{align*}
egin{bmatrix} I \ I end{bmatrix} N egin{bmatrix} I & I end{bmatrix} egin{bmatrix} M & M-I \ 0 & I end{bmatrix}^{-1}
end{align*}

is a right coprime factorization of $G$. The rest is to show that this factorization indeed satisfies the left and right coprimeness conditions above. Thus, $G$ is stabilizable/admissible.

Reference

[1] Carl N. Nett (1986). Algebraic aspects of linear control system stability. IEEE TAC, 31(10), 941-949.

[2] B. A. Francis (1987). A Course in $H_infty$ Control Theory. Springer Berlin Heidelberg.

[3] M. Vidyasagar (1985). Control System Synthesis: A Factorization Approach, The MIT Press.