Singular perturbation analysis of system order reduction via system balancing

In this paper we first present several alternative techniques for system order reduction based on system balancing by employing the method of singular perturbations. The techniques obtained have the same robustness accuracy evaluated with respect to the H_∞ norm of the reduced-order system as the two techniques existing in the control literature known as the direct truncation and balancing residualization methods. We propose how to perform order-reduction of a balanced system using theory of singular perturbations that can produce very good accuracy at high frequencies particularly for systems that have lightly damped highly oscillatory modes. The result obtained for such kind of systems requires that the modes with higher Hankel singular values should be neglected and those with smaller Hankel singular values should be retained, which is counter-intuitive result in the spirit of H _∞ theory, but an intuitive result in the spirit of theory of singular perturbations. This result raises an important question about validity of the system robustness estimate based on the H _∞ norm for linear systems with lightly damped highly oscillatory modes