Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  here extend the ll known Positive Real Lemma (also known as the Kalman-Yakubovich-Popov Lemma) to complex matrix-valued general- ized positive rational function, when non-minimal realizations are considered.  then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive func- tions. As a by-product,  partition all state space realizations into subsets: Each is identified with a set of matrices satisfying the same Lyapunov inclusion and thus form a convex invertible cone, cic in short. Moreover, this approach enables us to characterize systems which may be brought to be generalized positive through static output feedback. The formulation through Lyapunov inclusions suggests the introduction of an equivalence class of rational functions of various dimensions associated with the same system matrix.
['Daniel Alpay', 'Izchak Lewkowicz']
I am writing a research paper on a idea THE POSITIVE REAL LEMMA AND CONSTRUCTION OF ALL REALIZATIONS OF GENERALIZED POSITIVE RATIONAL FUNCTIONS
[{"from": "human", "value": "I am writing a research paper on a idea THE POSITIVE REAL LEMMA AND CONSTRUCTION OF ALL REALIZATIONS OF GENERALIZED POSITIVE RATIONAL FUNCTIONS"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  here extend the ll known Positive Real Lemma (also known as the Kalman-Yakubovich-Popov Lemma) to complex matrix-valued general- ized positive rational function, when non-minimal realizations are considered.  then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive func- tions. As a by-product,  partition all state space realizations into subsets: Each is identified with a set of matrices satisfying the same Lyapunov inclusion and thus form a convex invertible cone, cic in short. Moreover, this approach enables us to characterize systems which may be brought to be generalized positive through static output feedback. The formulation through Lyapunov inclusions suggests the introduction of an equivalence class of rational functions of various dimensions associated with the same system matrix."}]