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 e1= u1- y2, e2= u2+ y1and yi= Gi(ei), i = 1,2. We give three theorems which simplify the stability tests. Theorem 1 deals with nonlinear time-varying subsystems. It gives conditions on G2so that the stability of u1↦ y1guarantees 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 ui↦ yi, 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^{^}2and G^{^}1(I + G^{^}2G^{^}1)^-1are 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.