Absolute response error bounds for floating point digital filters in state space representation

A deterministic study of the zero input asymptotic behavior of second order state space digital filters with floating point arithmetic is carried out. Using a consecutive bound reduction method, asymptotic bounds on the state response are derived. It is shown, that by using a relatively small number of mantissa bits, for many systems, limit cycles occur only in underflow conditions. The results in this paper confirm previous knowledge obtained from direct form filters, that limit cycles in any linearly stable system implemented in floating point arithmetic can be made arbitrarily small by choosing a sufficiently large mantissa and exponent register length. The analysis also shows, that the quantization and reformatting scheme have little effect on the derived error bounds