Model reduction of second order systems

where the matrix M ∈ RN×N is assumed to be invertible. Models of mechanical systems are often of this type since (1.1) then represents the equation of motion of the system. For such a system M = MT , C = CT and K = KT are respectively the mass, damping and stiffness matrices, f(t) ∈ RN×1 is the vector of external forces, and x(t) ∈ RN×1 is the vector of internal generalized coordinates (see [4] and [7] for more information on such models). In civil engineering or aeronautics, the size N of the model (obtained using for instance finite elements techniques [4], [7]) is often so high that many analysis and design problems can not be solved anymore within a reasonable computing time. It is then advisable to construct a reduced order model [5] that nevertheless keeps the “mechanical” structure of the system. Since (1.1) is a particular case of a linear time-invariant system, one may consider its corresponding (linearized) state-space model (see section 2) and apply the techniques of model reduction known for statespace models. In doing so, the reduced-order system is generally not of the same type anymore and the symmetry of the data is lost. Since from a physical point of view it makes sense to impose the reduced-order system to be of the same type, we propose in this paper new methods of model reduction that preserve the second order form and (if needed) its symmetry. When writing the motion equation in the Laplace domain, the characteristic polynomial matrix P (s) appears : P (s)X(s) = F (s), P (s) . = Ms + Cs+K. (1.2)