The generalized state-space description of positive realness and bounded realness

Positive realness and bounded realness are two important issues in system theory. Some related applications about these notions include analysis and synthesis of passive networks, optimal designs in control and estimation, stability investigations in linear/nonlinear closed-loops; etc.. In this paper, we propose the equations of constant matrices, which are stabilizable and detectable realizations of impulse-free generalized state-space systems, to describe the positive real and bounded real properties. The established generalized positive real lemma and generalized bounded real lemma are necessary and sufficient. They are the extensions of results by B.D.O.Anderson nd S.Vongpanitlerd (1973), in which the state-space realizations are considered. The impulse-free generalized state-space systems contain both finite and nondynamic infinite modes. The state-space systems contain finite modes only. To express those algebraic restrictions among state variables is easier in the generalized state-space