The feedback interconnection of multivariable systems: Simplifying theorems for stability

We consider the stability of the feedback interconnection of possibly unstable n-input n-output subsystems whose interconnection is described by e
1
= u
1
- y
2
, e
2
= u
2
+ y
1
and y
i
= G
i
(e
i
), i = 1,2. We give three theorems which simplify the stability tests. Theorem 1 deals with nonlinear time-varying subsystems. It gives conditions on G
2
so that the stability of u
1
↦ y
1
guarantees that of the feedback system. The other two theorems consider continuous-time linear time-invariant subsystems. It is noted that in the multivariable case, the stablity of u
i
↦ y
i
, i = 1,2 is not sufficient to guarantee the stability of the feedback system, and Theorem 2 specifies some additional requited conditions. Theorem 3 shows that if G
^{^}
2
and G
^{^}
1
(I + G
^{^}
2
G
^{^}
1
)
^-1
are in some special stable classes, so is the transfer function of the feedback system. In both theorems, corollaries specialize the results to lumped and single-input single-output cases. The paper ends by showing how these results can be translated for the discrete-time case.