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 analyzed 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. The concepts and results are explicitly translated to an input–output setting, and related to balancing, Hankel norm reduction and normalized doubly coprime factorizations.