Negative imaginary functions for infinite dimensional systems

This paper studies the operator-valued irrational complex symmetric negative imaginary (CSNI) systems. The complex symmetric negative imaginary function is defined using the complex symmetric operator. In this case, a matrix-valued symmetric negative imaginary transfer function $G(s)$: $\mathbb{C}\rightarrow \mathbb{C}^{m\times m}$ is identical to an operator-valued CSNI transfer function defined on a finite dimensional Hilbert space. A necessary and sufficient condition for operator-valued CSNI transfer functions is given by using the frequency-domain properties. A specific illustration of an operator-valued CSNI function is shown.