On solutions of bounded-real LMI for singularly bounded-real systems

In this paper, we provide an algorithm to compute the solutions of the LMI arising from the bounded-real lemma for a special class of bounded-real systems that do not admit algebraic Riccati equations due to singularity condition in the feedthrough term. We call such systems singularly bounded-real systems. We show that unlike strictly bounded-real systems, the solutions of the bounded-real LMI for singularly bounded-real systems can be computed by a suitable arrangement of the controllability and observability matrices of the system. This is intrinsically linked to the Markov parameters and relative degree of the system. Further, we also show that the same algorithm can be used to compute the solutions of the bounded-real LMI corresponding to allpass systems, as well.