Equivalence relations between learnability, output-dissipativity and strict positive realness

Abstract

This paper was originally motivated by aiming at explicating why a simple iterative learning control scheme for complicated robot dynamics with strong non-linearities works well in acquiring any given desired motion over a finite or infinite time duration or any periodic motion. To gain a physical insight into the problem, a class of linear dynamical systems with specified input and output of the same dimension is treated by defining two properties: output-dissipativity and learnability. It is then shown that the former implies the latter and furthermore, for a class of linear systems with single input and single output, they are equivalent to each other and each of them is also equivalent to strict positive realness of input-output transfer function. For a class of MIMO (multiple inputs and multiple outputs) systems, it is possible to prove that each of these properties is equivalent to strict positive realness of the input-output transfer function matrix if it is strictly proper or otherwise its direct term from input to output satisfies an extra condtion.