Bilinear System Control

Bilinear system is the simplest nonlinear systems and most approximates to linear system. Many actual field models (such as economy, biology, zoology, chemical industry, population, etc.) can be concluded or approximated to a bilinear system. In recent several decades, the control problems for bilinear system has been widely studied by nationwide researchers, but there are few results in resolving singular bilinear system control problems. From the theory of singular system, bilinear system is a special form of singular bilinear system model, for this reason, studying for singular bilinear system not only has application values, but also has theory significance.This thesis give some discusses only on the stability control problem of singular bilinear systems. The main content may be summarized as follows:(i) The globally asymptotically stable problem of a class of discrete-time singular bilinear systems is studied. For the systems, we give a simple feedback control law, such that the resulting closed-loop systems is globally asymptotically stability. The assumptions in the literature are weakened, which states that the obtained result has more widely applicative range.(ii) The quadratically stable problem for a class of discrete-time singular bilinear systems with uncertain parameters under exogenous inputs is studied. Based on passivity theory and by means of linear matrix inequalities, a sufficient condition is given for uncertain discrete-time singular bilinear systems to be generalized quadratically stable and strictly passive. From which, a sufficient condition is given for the closed-loop systems under state feedback to be generalized quadratically stable and strictly passive.(iii) The zero solution asymptotically stable problem for a class of time-delay singular bilinear systems under exogenous inputs is studied. By means of linear matrix inequalities, a sufficient condition is given for time-delay singular bilinear systems to be zero solution asymptotic stable and passive. From which, The state feedback control law, which insures that the resulting closed-loop systems is both zero solution asymptotically stable and passive is designed. Moreover, the structure of such a state feedback control law is presented rely on the matrix inequalities.