On the Stability of Interconnected Systems

On the Stability of Interconnected Systems* By I. W. SANDBERG (Manuscript received March 3, 1978) Theorems are presented concerning conditions for the input-output stability of interconnected dynamical systems. Results in the area of input-output stability are often partitioned into two categories: small-gain type-results and passivity-type results. The main theorem given here does not fall into either of these categories, but is most closely related to the passivity-type results. The theorem involves a new class of interconnection operators that is a substantial generalization of the familiar set of nonnegative operators defined on a space of vectorvalued functions. I. INTRODUCTION In this paper, theorems are presented concerning the input-output stability of interconnected systems.+ Results in the area of input-output stability are often partitioned into two categories: small-gain type results and passivity-type results. The main theorem given here does not fall into either of these categories, but is most closely related to the passivity-type results. The theorem involves a new class of interconnection operators that is a substantial generalization of the familiar set of nonnegative operators defined on a space of vector-valued functions. The mathematical model considered throughout the paper is described in Section II, and results of a general nature concerning the model are given in Section III. The case in which the interconnection operator has a certain matrix representation is treated in considerable detail in Section IV.