Observability gramian for nonlinear systems

In this paper, we propose a novel approach for the construction of observability gramian for nonlinear systems. For linear systems observability gramian is obtained as a solution of matrix Lyapunov equation, for nonlinear systems we show that the observability gramian information can be obtained from the solution of Lyapunov measure equation. Lyapunov measure equation is introduced in (U. Vaidya, 2007) to provide necessary and sufficient condition for almost every where stability of an invariant set in nonlinear systems. Gramian result for the linear systems forms a special case of the proposed observability gramian approach using transfer operator. For system with output measurement we use set oriented numerical methods are used for the finite dimensional approximation of the gramian.