Model reduction based on limited-time interval impulse response Gramians

In this paper, three new Gramians are introduced namely - limited-time interval impulse response Gramians (LTIRG), generalized limited-time Gramians (GLTG) and generalized limited-time impulse response Gramians (GLTIRG). GLTG and GLTIRG are applicable to both unstable systems and also to systems which have eigenvalues of opposite polarities and equal magnitude. The concept of these Gramians is utilized to develop model reduction algorithms for linear time-invariant continuous-time single-input single-output (SISO) systems. In the cases of GLTIRG and GLTG based model reduction, the standard time-limited Gramians are generalized to be applied to unstable systems by transforming the original system into a new system which requires the solution of two Riccati equations. Two numerical examples are included to illustrate the proposed methods. The results are also compared with standard techniques.