Stability trapezoid and stability-margin analysis for the second-order recursive digital filter

• We propose a stability trapezoid for guaranteeing a specified stability-margin.
• We analyze the relation between stability margin and stability-margin parameter.
• We use a demonstrative example to verify the stability margin.
• We have verified the consistence between stability margin and simulation results.

To design a variable recursive digital filter whose stability is always guaranteed, it is necessary to ensure that the stability conditions are always satisfied in the tuning process. Furthermore, it is also necessary to keep a certain margin for the stability (stability margin) in such a way that some unpredictable environmental changes and coefficient-value deviations will not cause instability. To add a stability margin to the stability of the second-order (2nd-order) recursive filter, this paper first introduces a stability trapezoid by trimming the well-known stability triangle of the 2nd-order recursive digital filter. Then, we quantitatively analyze the upper bound for the stability margin by using a stability-margin parameter. This theoretical stability-margin analysis is fundamental to the design of a variable recursive filter with an expected stability margin. Finally, we utilize a demonstrative example to verify the consistence between the theoretical upper bound of the stability margin and the computer simulation results.