Stability analysis of feedback interconnections between systems with “mixed” properties

Sufficient conditions for the finite-gain stability of positive feedback interconnected systems are given when the subsystems have a certain mixed dissipative property, such as “mixed” small gain and passivity, “mixed” small gain and negative imaginary, “mixed” passivity and negative imaginary. In addition, the converse of the integral quadratic constraint (IQC) theorem involving nonlinear systems is provided on the basis of the S-procedure lossless theorem. Furthermore, a collection of converse results for mixed dissipative theorems is derived by the converse IQC theorem and the decomposition of the multipliers. It is demonstrated that if the feedback interconnection of a linear time-invariant (LTI) system with an arbitrary system satisfying some mixed dissipative property is finite-gain stable, then the given system must have a more strict version of the same mixed dissipative property. Meanwhile, the converse IQC theorem can cover the converse theorems of small-gain, passivity and $(Q,S,R)$-dissipativity.