Digital filter realizations absent of self-sustained oscillations

As shown in [9], for a state-space realization (A,B,C,d) of a digital filter there are no self-sustained oscillations (i.e., limit cycles) if there exists some diagonal matrix D ≫ 0 such that D-A^T DA ≥ 0. Based on this result, we show that the optimal roundoff realizations are free of limit cycles. For any given realizations, a method is proposed to check the existence of such a D. A novel class of robust state-space realizations is derived and characterized, which are free of limit cycles and yield a minimal error propagation gain. The optimal realization problem for this class of realizations is formulated and solved analytically. Two examples are presented to test the efficiency of the proposed method and to demonstrate the behavior of the obtained optimal realization.