Generic Solvability of the Decoupling Problem
E. Fabian, Walter Murray Wonham · SIAM Journal on Control · 1974
For the linear multivariable system $\dot x = Ax + Bu$, $z_i = D_i x(i \in {\bf k})$, decoupled (non-interacting) control is shown to be achievable for “almost all” sets of real matrices $\{ A(n \times n),B(n \times m);D_i (q_i \times n),i \in {\bf k}\} $, having the dimensions shown, if and only if \[\sum\limits_{i = 1}^k {q_i \leqq \min (n,m - 1 + \mathop {\min }\limits_{1 \leqq i \leqq k} q_i ).} \] With $(n,m,q_1 , \cdots ,q_k) $ subject to the inequality, the exceptional matrix sets for which decoupling is impossible belong to a proper algebraic variety in $\mathbb{R}^N $, $N = n^2 + nm + (q_1 + \cdots + q_k )n$.