Computation of system gramians for linear time-varying systems

摘要

This paper considers the problem of reachability and observability gramian computation for linear time-varying systems. Both gramians are obtained by using system trajectories. The reachability gramian is computed from the state impulse response and the observability gramian is obtained from the output response. The relation with the linear time invariant case is discussed and three different approaches for gramian computation are presented: the dual system approach, the empirical gramian approach and integral operator approach. The general theory of linear time-varying systems is shortly presented and the role of the transient matrix for solving the linear state equation is shown. An algorithm for computation of the transient matrix is proposed, which is based on the integration of the state equation by the method of Runge Kutta. The role of the adjoint system is also discussed and the two basic impulse response characteristics, namely the normal and the adjoint state impulse responses are presented.