Optimal Angle Reduction - A Behavioral Approach to Linear System Approximation

We investigate the problem of optimal state reduction under minimization of the angle between system behaviors. The angle is defined in a worst-case sense, as the largest angle that can occur between a system trajectory and its optimal approximation in the reduced order model. This problem is analysed for linear time-invariant finite dimensional systems, in a behavioral l2-setting, without reference to input/output decompositions and stability considerations. The notion of a weakest past-future link is introduced and it is shown how this concept is applied for the purpose of model reduction. A method that reduces the state dimension by one is presented and shown to be optimal. Specific algorithms are provided for the numerical implementation of the approximation method.