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<inf>1</inf>= u<inf>1</inf>- y<inf>2</inf>, e<inf>2</inf>= u<inf>2</inf>+ y<inf>1</inf>and y<inf>i</inf>= G<inf>i</inf>(e<inf>i</inf>), 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<inf>2</inf>so that the stability of u<inf>1</inf>↦ y<inf>1</inf>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<inf>i</inf>↦ y<inf>i</inf>, 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<sup>^</sup><inf>2</inf>and G<sup>^</sup><inf>1</inf>(I + G<sup>^</sup><inf>2</inf>G<sup>^</sup><inf>1</inf>)<sup>-1</sup>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.