Semistability Analysis Of Continuous Linear Systems

The semistability of normal and singular continuous linear systems is discussed in this paper. Semistability is the relevant notion of stability for a system that possesses a continuum of equilibria. It is the property whereby every trajectory that starts in the neighborhood of a Lyapunov stable equilibrium point converges to a (possibly different) Lyapunov stable equilibrium, which is not determined com-pletely by the system dynamics, but depends on the initial state as well. Hence, a central notion of semistability theory is that of convergence in which every solution converges to a limit point that may depend upon the initial condition.For the normal continuous linear systems. Firstly, we give the definition of semistability and some basic concepts, such as semiobservable, weakly κ-semiobserv-able, and then the sufficient and necessary conditions of semistability are addressed using the Lyapunov equation. Secondly, a sufficient condition of semistability for linear continuous systems is given using the Lyapunov equation and a rank condition. Finally, a simulation example is presented to verify the feasibility of the proposed method.For the singular continuous linear systems. Firstly, the definition of semista-bility for singular continuous linear systems is given based on the the equivalent conditions of semistability for the normal continuous linear systems using the finite eigenvalues of matrix pencil. Similarly, some sufficient conditions of semistability for singular linear continuous systems are addressed using the generalized Lyapunov equations and a rank condition. Finally, a simulation example is presented to verify the feasibility of the proposed method as well.