L_2Optimal Model Reduction for State Space Symmetrc

This paper deals with the problem of finding a lower-order linear continuous time-invariant stable state space symmetric system to approximate a given higher-order linear continuous time-invariant stable state space symmetric system so that the L_2 norm of the model mismatch is minimized. It is still an open problem whether this problem has a solution or not. This paper attempts to find a lower-order state space symmetric model whose model reduction cost differs from the optimal model reduction cost by a prescribed precision. It is shown that this task can be tackled by solving a smooth constrained optimization problem which is guaranteed to admit a global minimum. A gradient flow algorithm is proposed to solve the latter problem. A numerical example is presented to demonstrate the effectiveness of this approach.