We derive an error bound in the gap metric for positive real balanced truncation and positive real singular perturbation approximation. We prove these results by working in the context of dissipative driving-variable systems, as in behavioral and state/signal systems theory. In such a framework no prior distinction is made between inputs and outputs. Dissipativity preserving balanced truncation of dissipative driving-variable systems is addressed and a gap metric error bound is obtained. Bounded real and positive real input-state-output systems are manifestations of a dissipative driving-variable system through particular decompositions of the signal space. Under such decompositions the existing bounded real and positive real balanced truncation schemes can be seen as special cases of dissipative balanced truncation and the new positive real error bounds follow.
Guiver, C., & Opmeer, M. R. (2013). Error bounds in the gap metric for dissipative balanced approximations. Linear Algebra and its Applications, 439(12), 3659–3698. https://doi.org/10.1016/j.laa.2013.09.032