Balancing and model reduction for second-order form linear systems

Model reduction of second-order form linear systems is considered where a second-order form reduced model is desired, The focus is on reduction methods that employ or mimic Moore's balance and truncate (1981). First, we examine second-order form model reduction by conversion to first-order form and obtain a complete solution for this problem. Then, new Gramians and input/output (I/O) invariants for second-order systems are motivated and defined. Based on these, direct second-order balancing methods are developed. This leads naturally to direct second-order form analogs for the well-known first-order form balance and truncate model reduction method. Explicit algorithms are given throughout the paper.