Identification/reduction to a balanced realization via the extended impulse response Gramian

A new state space model reduction algorithm for linear, discrete time-invariant (LDTI) systems is proposed which is based on the extended impulse response Gramian first defined here for LDTI systems. The reduction algorithm employs singular value decomposition to retain states corresponding to dominant singular values of these Gramians. The proven properties of the reduced order models include convergence to a balanced realization with conditional controllability, observability and asymptotic stability. A suboptimal property of the model is also demonstrated. Finally, an error bound is obtained for this algorithm. In addition to model reduction, the proposed technique can also be applied to system identification from impulse response data of deterministic or stochastic nature.