MATH319 Slides 65 A determinant formula for realization by SISO 67 Realising MIMO

66 MIMO as a feedback system, without differentiators

Differentiators are sometimes regarded as bad components to have in a linear system since they can introduce noise. Integrators are preferable, since they depend upon the long term history of system. So it is advantageous to produce linear systems without differentiators.

Proposition

The MIMO system $(A,B,C,D)$ can be realised as a feedback system involving taps, matrix amplifiers, summing junctions and integrators, but no differentiators. Proof. We write

{{d}\over{dt}}X=AX+BU

in the form

X=A\int X+B\int U

and realise this as a feedback with $\int$ and the matrix amplifiers $A$ and $B$ .