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The paper considers the problem of reachability and observability finite interval gramians computation and approximation for linear time-varying systems. Both gramians are obtained from system trajectories. The reachability gramian is derived from the adjoint state impulse response and the observability gramian is obtained from the zero-input output response. The application of the adjoint system is discussed and the two basic impulse response characteristics, namely the regular and the adjoint state impulse responses are presented. The relation with the linear time-invariant case is also discussed and the role of the state transition matrix for computing the gramians is shown. An algorithm for derivation of the state transition matrix is proposed, which is based on the integration of the state equation by using the method of Runge-Kutta. The gramians are approximated in terms of Legendre orthogonal series representations of system trajectories.

A Simple Method for Orthogonal Polynomial Approximation of Linear Time-Varying System Gramians