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A modified positive-real type stability condition

Lightly damped structures with colocated (position) sensors and (force) actuators can typically be modeled by a (possibly infinite) sum of second order transfer functions. Since the relative degree of these systems is more than unity, they are not directly amenable to standard positive real analysis and since these systems are highly resonant, application of the small gain theorem would be usually very conservative. However, a condition for the internal stability of a positive feedback interconnection of two such systems has been available in the theory of positive position control. This result has important implications on stability robustness due to spill-over dynamics. In this paper, we propose a new stability analysis result which generalizes the positive position control stability result to the feedback interconnection of systems satisfying a certain modified positive-real type condition. Broadly speaking, the result states that a necessary and sufficient condition for the internal stability of the feedback interconnection of such systems is that the DC loop gain is less than unity.