I am studying MIMO control systems, and I have found that in the Nyquist stability criterion for MIMO system, one of the condition for stability is that:

$det(I+L(j\omega))\neq 0$

and also that

$det(I+L(j\omega))$ has a number of encirclements of the origin equal to the number of poles in the RHP.

but what is $det(I+L(j\omega))$?

In SISO systems I know that we have to look at the encirclements of $L(j\omega)$ around $-1$.

But why in MIMO we have to look at $det(I+L(j\omega))$?

I put here the reference in which I found this: link

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Nyquist's stability theorem is actually given in terms of $1+L(j\omega)$, which is the denominator polynomial of the close-loop transfer function. For SISO system, close-loop stability is examined by looking at the encirclement of $1+L(j\omega)$ around $0$, which is simplified as looking at the encirclement of $L(j\omega)$ around $-1$. Now for MIMO systems, we cannot just use $\det(L(j\omega))$ because that will cause pole-zero cancellation, for example with the plant $\mathrm{diag}\left(\frac{s-1}{s+3},\frac{1}{s-1}\right)$.

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